Two Dimensional Supersymmetric Models and Some of Their Thermodynamic Properties from the Context of Sdlcq

Home , Bound state, Canonical quantization, Super QCD

TWO DIMENSIONAL SUPERSYMMETRIC MODELS AND SOME OF THEIR THERMODYNAMIC PROPERTIES FROM THE CONTEXT OF SDLCQ

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

Yiannis Proestos, B.Sc., M.Sc.

*****

The Ohio State University

2007

Dissertation Committee: Approved by

Stephen S. Pinsky, Adviser Stanley L. Durkin Adviser Gregory Kilcup Graduate Program in Robert J. Perry Physics © Copyright by

Yiannis Proestos

2007 ABSTRACT

Supersymmetric Discrete Light Cone Quantization is utilized to derive the full mass spectrum of two dimensional supersymmetric theories. The thermal properties of such models are studied by constructing the density of states. We consider pure super Yang–Mills theory without and with fundamentals in large-Nc approximation.

For the latter we include a Chern-Simons term to give mass to the adjoint partons.

In both of the theories we ﬁnd that the density of states grows exponentially and the

theories have a Hagedorn temperature TH . For the pure SYM we ﬁnd that TH at inﬁ-

2 g Nc nite resolution is slightly less than one in units of π . In this temperature range, q we ﬁnd that the thermodynamics is dominated by the massless states. For the system with fundamental matter we consider the thermal properties of the mesonic part of the spectrum. We ﬁnd that the meson-like sector dominates the thermodynamics.

We show that in this case TH grows with the gauge coupling. The temperature and

coupling dependence of the free energy for temperatures below TH are calculated. As expected, the free energy for weak coupling and low temperature grows quadratically with the temperature. The ratio of free energies at strong coupling compared to weak

coupling, rs w, for low temperatures grows quadratically with T. Our data indicate − that rs w tends to zero in the continuum at low temperatures. Finally a super-QCD − model realized from weakly coupled D3-D7 branes is proposed as a future project in

SDLCQ.

ii To my sweet wife and life superpartner Lefkoula from my heart!

iii ACKNOWLEDGMENTS

The ﬁrst person that I am indebted to and wish to express my deepest gratitude is my advisor, Stephen Pinsky, who cordially welcomed me to his research group and introduced me to the beautiful ﬁelds of supersymmetry and SDLCQ. His constant guidance, enthusiasm and experience in explaining complex things in a simple manner undoubtedly helped me to accomplish this work. Moreover, something which I will never forget is that with his always positive attitude was helping me overcome stressful situations related to my research. In addition, I would like to give many thanks to my collaborators, John Hiller, Nathan Salwen and Uwe Trittmann, without whom the results of my research would have not been made possible. A special thank you goes to Uwe for his insightful comments and support during our multi-hour discussions.

In particular, I frankly thank my cohort Moto Harada for the fruitful and always friendly conversations. Finally, I would like to convey my wholehearted and sincere thankfulness to my family. Especially to my wife Lefki who always supported and boosted me with her immense love, patience and understanding during my journey at The Ohio State University. And foremost to my parents, Haralambos and Aglaia, who despite their suﬀerings, always taught me that dignity and humility is above any degree and made everything they could possibly do in order for me to have a good education. The support from both my sweet wife and my beloved parents was the single critical factor for me being in the position I am today.

iv VITA

July 19, 1975 ...... Born - Lemesos, CYPRUS

1999 ...... B.Sc. Physics, University of Cyprus

2001 ...... M.Sc. Physics, University of Cyprus

2001-present ...... Graduate Teaching Associate, The Ohio State University.

PUBLICATIONS

Research Publications

J. R. Hiller, S. Pinsky, Y. Proestos, N. Salwen, U. Trittmann, Spectrum and thermodynamic properties of the = (1, 1) SYM coupled to fundamental matter and N 1+1 Chern-Simons form, Phys. Rev. D70(4) (2007), [arXiv:hep-th/0702071].

J. R. Hiller, S. Pinsky, Y. Proestos, N. Salwen, = (1, 1) super Yang-Mills the- N ory in 1+1 dimensions at ﬁnite temperature, Phys. Rev. D70 (2004) 065012, [arXiv:hep-th/0407076].

A. Bode, H. Panagopoulos, Y. Proestos, (a) improved QCD: The Three loop beta O function, and the critical hopping parameter, Nucl. Phys. Proc. Suppl. 106 (2002) 832, [arXiv:hep-lat/0110225].

H. Panagopoulos, Y. Proestos, The Critical hopping parameter in (a) improved O lattice QCD, Phys. Rev. D65 (2002) 014511, [arXiv:hep-lat/0108021].

v FIELDS OF STUDY

Major Field: Physics

Specialization: Elementary Particles and High Energy Theory

vi TABLE OF CONTENTS

Page

Abstract...... ii

Dedication...... iii

Acknowledgments...... iv

Vita ...... v

ListofTables ...... x

ListofFigures ...... xii

Chapters:

1. Introduction...... 1

2. Introductory remarks on Super Quantum Mechanics ...... 8

2.1 FormulationofSUSYQM ...... 10 2.1.1 A first glimpse of superspace in quantum mechanics . . . . . 10 2.1.2 =2 SUSYQM: Action, Supercharges and hamiltonian . . 16 N 2.1.3 PropertiesofSUSYQM ...... 18 2.1.4 SUSYQM in action: An example ...... 23 2.2 Supersymmetry breaking and non-zero temperature effects..... 26 2.2.1 Witten’s index and SUSYQM ...... 26 2.2.2 β regularization of the Witten Index ...... 30 − 2.2.3 SUSYQM and finite temperature effects ...... 33 2.3 Discussion...... 39

vii 3. Abriefpreviewof(S)DLCQ ...... 40

3.1 LightConeparametrization ...... 40 3.2 Supersymmetric Discrete Light Cone Quantization: A few details . 43 3.2.1 QuantizationrulesontheLightCone...... 44 3.3 Formulation of the supersymmetric bound state problem ...... 46 3.3.1 Matrixstructureofsupercharges ...... 48 3.3.2 An example: pure super Yang-Mills theory meets SDLCQ . 51 3.4 Statistical Mechanics on the Light Cone ...... 54 3.5 Discussion...... 56

4. SYM-ThermodynamicpropertiesI ...... 59

4.1 The pure super Yang–Mills model ...... 62 4.2 Densityofstates ...... 65 4.3 Numerical results for the density of states ...... 66 4.4 Finite temperature in 1+1 dimensions ...... 70 4.5 Discussion...... 73

5. SYM-ThermodynamicpropertiesII ...... 77

5.1 SYM with fundamental matter and Chern-Simons 3-form...... 80 5.1.1 Formulation of the theory and its supercharges ...... 80 5.1.2 SDLCQ eigenvalue problem ...... 84 5.2 Mesonandglueballspectra ...... 87 5.2.1 Limitingcases ...... 87 5.2.2 Comparison of meson and glueball spectra ...... 91 5.3 Density of states and Hagedorn temperatures revisited ...... 94 5.3.1 A method of estimating the density of states ...... 95 5.3.2 Fitstothespectrum ...... 98 5.3.3 Hagedorntemperature ...... 104 5.4 Finite temperature results in 1+1 dimensions ...... 106 5.4.1 Thefreeenergy...... 106 5.4.2 An analytic result: The free energy for the free theory . . . 110 5.4.3 Numerical results for nonzero coupling ...... 112 5.4.4 Temperature dependence of the free energy ...... 112 5.4.5 Coupling dependence of the free energy ...... 115 5.5 Discussion...... 123

viii 6. A possible future direction for SDLCQ calculations ...... 127

6.1 =2 Super QCD in d =3+1 from D3-D7 branes ...... 130 N 6.2 Supercurrentandsupercharges ...... 136 6.3 Quantization rules and momentum space expansions ...... 139 6.4 A comment regarding the expansion of Q± ...... 142 6.5 Discussion...... 145

Appendices:

A. UsefulresultsrelatedtoSUSYQM ...... 147

A.1 Component ﬁeld variations under supersymmetry transformations . 147 A.2 Thesuperchagres...... 148 A.3 Deriving the hamiltonian ...... 151 A.4 Witten index: Discrete spectrum cases ...... 151 A.5 Witten index: Continuous distribution of states ...... 152 A.6 Various derivations regarding the ﬁnite temperature model . . . . . 154

B. Gamma matrices and Spinors in arbitrary spacetimes ...... 158

B.1 Generalremarks ...... 158 B.2 Equivalencerelations...... 162 B.3 Minimal or irreducible spinors ...... 168 B.4 Extended superalgebra from dimensional compactiﬁcation ..... 172 B.5 Four dimensional Majorana and Weyl representations in light cone coordinates ...... 176

C. Conventions and derivations for Chapters 4 and 5 ...... 179

C.1 Constraintequationforthegaugeboson ...... 179 C.2 The pure super Yang-Mills conserved current ...... 183 C.3 Free gas thermodynamics in D spacetime dimensions ...... 187

Bibliography ...... 190

ix LIST OF TABLES

Table Page

5.1 The free energy as a function of the temperature in the meson and the glueballsectorsinthefreetheory...... 112

5.2 Free energy as a function of temperature T at K = 14 in the -even T sector for weak coupling, g = 0.1: ˜ is obtained by summing Fspect. over the eigenvalues in the interval M 2 [0.00001, 9.10283]κ2; ˜ ∈ Ffit is obtained by the DoS method described in Section 5.3.1. ˜< and Ffit ˜ are the contributions to the latter states below the mass gap (i.e., F0 M 2 < 0.00113κ2) and of a single supersymmetric massless state, respectively...... 115

5.3 Results for free energy as a function of temperature T at K = 16 and strong coupling, g =4.0. ˜ corresponds to the overall free energy F including both symmetry sectors. The third column shows the overall contribution of the -even sector, the fourth the contribution from T the single nearly massless state, M 2 0.0362κ2, and the last column ≈ is the contribution that a supersymmetric massless state would make, ifitwerepresent...... 116

5.4 Data for strong to weak coupling ratio rs w for K = 13, 14, 15, 16 at − various temperatures. We show rs w(K = 16) as a function of T in − Fig. 5.19(b)...... 123

x B.1 The values for and η are shown for arbitrary spacetimes. Particu- larly for Minkowski spacetimes (d = s +1) η = +1, whenever the Majorana condition is allowed. We have used superscripts + and − in the Majorana condition column to emphasize the sign of η one has

to use so that ∗ = ½ holds. The spinor condition column refers B B to what type of irreducible spinor is allowed: W -Weyl, M-Majorana, MW -Majorana-Weyl, in a certain Minkowski spacetime. Note that the superscripts s and c in the Weyl constraint column (W ) refer to self- and complex-chiral spinors, respectively. The number of real degrees of freedom of the minimal spinor is shown in the last column. For completeness, although non-irreducible spinor representations, the dimensions where symplectic structures can be deﬁned are also provided. Except from the last column every row has a periodicity 8, i.e., d mod8...... 173

B.2 Some useful results for gamma matrices and spinor ﬁelds in the Ma- jorana representation that have been used in Chapter 6...... 176

B.3 The Weyl representation of gamma matrices along with some useful results for the light cone coordinates are shown in this table. Spinor conventions and deﬁnitions are also shown...... 177

xi LIST OF FIGURES

Figure Page

1.1 The three forms of quantization as introduced by Dirac are depicted in a two dimensional spacetime (x0 ct, x1). The dashed (blue) line ≡ corresponds to the light-cone boundary, while the (red) arrows indicate the direction of time-evolution. The leftmost diagram corresponds to the instant form, the central one to the light cone quantization (with + coordinates (x , x−)) and the rightmost to the point form scheme. Notice that in the case of light front parametrization the time evolution surface (line in this case) coincides with x+ = 0 and in the case of the point form the quantization takes place on a hyperbolic curve (x0)2 (x1)2 = a2 > 0, x0 > 0. For a more elaborate and complete − discussion on this matter consider review article [1]...... 4

2.1 Thermal ground state: (a) The thermal ground state of the oscillator H along with the thermal ground state of the whole system for small values H of T and (b) the same plot but for higher range of temperatures. For the interaction term we have used λ = 1.3 and g = 0.01. The fact that − supersymmetry is broken at ﬁnite temperatures is manifest in these diagrams. 38

4.1 The distribution functions (a) N(M 2,K) for K = 12 and (b) f(M 2, K, 100) 2 2 for K = 12, 14, and 17, as functions of M in units of g Nc/π. In (a) we also show the error-function ﬁt a erf(b(x c)) + d, and in (b), a − polynomialﬁt...... 67

4.2 The normalized density of states (a) for M 2 100 and (b) extrapo- ≤ lated to all masses. The data points are a convenient way of displaying the continuous functions calculated from the ﬁts to the CDF...... 69

4.3 The Hagedorn temperature TH , as obtained from exponential fits, α exp M α, to the universal fits to N(M 2,K), as shown in (a) TH − for K= 17 , and (b) a linear plot against 1/K for fits in the range 11 K 17...... 70 ≤ ≤ xii 4.4 The free energy (a) and the heat capacity (b) as functions of temperature for each resolution. Both functions are normalized to the number of massless states, N = 2(K 1). The data points are a convenient 0 − way to display the continuous functions calculated from fits to the CDF. 73

5.1 Mass spectra for (a) mesonic bound states and (b) glueball bound states as a function of the inverse resolution for 3 K 23 when ≤ ≤ κ =1 and g =0 (freetheory)...... 90

5.2 Mass spectra for (a) the meson ( -even, -even) sector and (b) the P T glueball ( -even, -even) sector as a function of the inverse resolution P T for 3 K 16 when κ =0 and g =1...... 91 ≤ ≤ 5.3 Mass spectra for (a) the meson ( -even) as a function of the inverse T resolution for 3 K 15 sector and (b) the glueball ( -even) sector ≤ ≤ T as a function of the inverse resolution for 3 K 9. In both cases ≤ ≤ κ =1 and g =1. We note also a small splitting in the masses due to the presence of CS term which breaks explicitly the symmetry. . . 92 P

5.4 CDF of the free (g = 0) mesonic -odd ( −) sector at K = 13. T T Crosses (boxes) refer to numerical (analytical) calculation of the bound- statemasses...... 96

5.5 CDF of the -even ( +) sector at K = 14 and g = 0.1. Shown are T T data (dots) and a ﬁt to the data: (a) all states in units of 105 states; (b) range of masses just above the mass-gap in units of 102 states; (c) statesbelowthemassgap...... 99

5.6 Same as Fig. 5.5 but for the density of states, in units of (a) 103 states, (b) 102 states,and(c)1state...... 100

5.7 CDF of the -odd ( −) sector at K = 16 and g = 0.5. Shown are T T data (dots) and a fit to the data: (a) all states in units of 106 states with point of inflection; (b) range of masses just above the mass-gap in units of 102 states; (c) states below the mass gap. The relatively poor fit near M 2 = 10 in (b) does not have a significant effect on the results...... 101

5.8 Same as Fig. 5.7, but for the DoS, in units of (a) 104 states, (b) and (c)10states...... 101

xiii 5.9 CDF of the -even sector at K = 16 and g = 4.0. Shown are data T (dots) and a ﬁt to the data: (a) all states in units of 106 states; (b) range of masses just above the mass-gap in units of 102 states; (c) statesbelowthemassgap...... 103

5.10 Same as Fig. 5.9, but for the DoS, in units of (a) 103 states, (b) and (c)1state...... 103

5.11 (a) Logarithm of the CDF versus M. The approximately linear part of this logarithmic CDF is ﬁt to f(M)= αM + 1 . (b) Extrapolated TH Hagedorn temperature for coupling g =0 with T ∞(0) 0.52κ. . . . 106 H ≈ 5.12 Hagedorn temperature (a) plotted versus 1/K at couplings g = 0.1 (crosses), 0.5 (boxes), 1.0 (triangles), 4.0 (diamonds) in units where κ =1, and (b) extrapolated in K as function of g with a ﬁt to T2(g)= 0.52+0.39g +0.054g2 (dashed line). In (a), the dots at 1/K = 0 are the continuum values. For clarity we have included only four representative values of g in(a)...... 107

5.13 The free energy ˜ for the free theory (g =0, κ =1) as (a) compared F between the analytic and numerical results for K = 16 and (b) a function of 1/K. In (a) the temperature ranges from 0.015 to 0.5, in units where κ = 1, by steps of ∆T = 0.015κ. The dashed line represents an exact match, the solid line the actual relation between ˜ and ˜ . In (b) the temperatures are T =0.1κ (crosses), 0.3κ Fspect. Ffit (boxes), 0.5κ (triangles)...... 111

5.14 Free energy ˜ at weak coupling (g =0.1) as a function of T at K = 12 F (crosses), 14 (boxes), 16 (triangles) for (a) all temperatures T < TH (dashed vertical line) and (b) low temperatures with a quadratic ﬁt F (T )= α T 2, with α 0.338...... 114 w w ≈ 5.15 Same as Fig. 5.14, but for strong coupling (g = 4.0). In (b) the low 4 temperature behavior is described by a quartic ﬁt F (T )= αsT , with α 0.321...... 117 s ≈

xiv 5.16 The free energy as a function of the coupling g at temperature T = 0.1κ (with κ =1) and for resolutions K = 12 (crosses), 13 (boxes), 14 (triangles) with two diﬀerent vertical scales. In (b) we see along with the data the contribution to the free energy that would be made by a pair of exactly massless superpartners. For fairly large values of g and at this temperature, the overall free energy is small compared to the contribution of a single pair of massless states. This is expected at this coupling region because the masses are very large and are suppressed by the modiﬁed Bessel function, K1(x)...... 118

5.17 Same as Fig. 5.16, but for T = 0.3κ. From (b) it is clear that for values of g > 1.0 and at this temperature, the main contribution to the free energy again comes from the nearly massless state in the - T even sector and it resembles the result from an exactly massless state (dashedlines)...... 119

5.18 Same as Fig. 5.16, but for T = 0.5κ. From (b) it is clear that more states contribute to the free energy at higher temperature, and at relatively large values of g...... 120

5.19 (a) Comparison of the free energies ˜(g = 4, κ = 1; K = 13) and 2 F F 0(g = 1, κ = 0; K = 13) 4 ˜ at various temperatures between ≡ F∗ 0.015κ and 0.5κ with steps of ∆T = 0.015κ, for κ = 1. The dashed line represents a perfect match, the solid line the best linear ﬁt ˜(g = F 4, κ =1)=0.97F 0(g =1, κ = 0). (b) Strong/weak coupling ratio rs w − as a function of the temperature at K = 16...... 121

6.1 A schematic diagram showing the D3-D7 brane system. The Nc D3 branes are coincident at the origin, while the brane probe is set at the distance L from the origin; their separation distance. The bound state meson spectrum is dictated by the ﬂuctuations of (3, 7) and (7, 3) bi- fundamental strings. The (3, 3) strings describe the pure adjoint sector ofthefourdimensionalgaugetheory...... 130

6.2 The ﬁrst picture (a) illustrates the structure of state Ψ which cor- | 0i responds to two disjoint strings of fundamental partons. The initial fundamental state has broken into two pieces; the form is not conserved. In picture (b) the structure of Ψ is depicted. This one is | 2i clearly of the same form as the initial state Ψ ...... 145 | 1i

xv CHAPTER 1

INTRODUCTION Αφού δεν είπες τίποτα κύριε ποιητή, γιατί ενόχλησες τις λέξεις; –Κ. Μόντης

The abstract notion of supersymmetry was implicitly realized in the mid sixties- early seventies, after the introduction of super Lie (or graded) algebras [2–5]. That is, people were trying to ﬁnd ways to evade the restrictions posed by the Coleman-

Mandula [6] theorem on symmetry generators and their nature. In particular, they were looking for models that one could extend Poincaré algebra so that it can admit anti-commuting charges obeying fermi statistics, consistent with the symmetries allowed by the S-matrix. At about the same time the idea of supersymmetry/graded algebras, found an application to the so called dual resonance model theory of scattering [7,8], the precursor of string theory. However, the most notable breakthrough came when Wess and Zumino [9–11] managed to build a lagrangian that was invariant under supergauge transformations. Soon after, supersymmetry, the symmetry between fermions and bosons, as is commonly known, started evolving rapidly.

Although supersymmetry as a mathematical concept is undoubtedly elegant, there must be some convincing arguments that such a theory can be introduced to describe physical phenomena. Primarily, the studies of supersymmetric ﬁeld theories gave a new direction towards a uniﬁed theory of nature since it could serve as a

1 principal model for grand uniﬁcation as well as a low energy limit of string theories.

More speciﬁcally, let us name a couple of the original motivating reasons why supersymmetry has been considered as a legitimate candidate for a theory which can help to solve the puzzle of nature. Apart from succeeding in unifying the fundamental building blocks of matter and building blocks of force, from the phenomenological standpoint it provides a work-around to the hierarchy problem. As it is known, there is a disturbingly huge energy gap between the Planck scale,

1 ~c5 2 E = =1.2 1019 GeV, Planck G × Newton where gravitational effects are as important as the quantum effects, and the characteristic Standard Model energy ( 300 GeV). The heart of the hierarchy problem ∼ reflects the fact that scalar fields (e.g., Higgs boson) seem not to be protected from acquiring large bare masses by any of the known symmetries of the standard model.

To this end it is diﬃcult to decide why their masses are not of the order of 1016 GeV.

However, if these scalars appear in supermultiplets with fermions in chiral representation of some gauge symmetry, then supersymmetry would require zero bare masses for both the fermions and the scalars. Then all the masses in the regime of the standard model would be related to the energy scale where supersymmetry is broken.

Moreover, a scenario where the electroweak and strong forces of nature are uniﬁed, is realized by requiring that supersymmetry exists. Yet another reason which makes supersymmetry more attractive is that radiative corrections become less important in supersymmetric models due to cancellations between fermionic and bosonic loops and consequently quantities that are small at the classical level will remain so if higher orders of perturbation theory are taken into account. In addition to the above

2 reasons which make supersymmetry attractive and accessible, local supersymmetry is required by string theory in order to describe gravitational eﬀects.

Meanwhile, over the last couple of decades a great deal of development has oc- cured, which improve signiﬁcantly our understanding of strongly coupled supersymmetric gauge theories. To mention a couple of those developments we note Seiberg and Witten’s seminal work [12,13] on electric-magnetic duality for a special class of supersymmetric theories and Maldacena’s [14] groundbreaking work, on AdS/CFT correspondence where in the large-N limit a (UV-ﬁnite) = 4 super (conformal) c N Yang-Mills theory is equivalent (or dual) to a type-IIB closed string theory. There- fore the nonperturbative aspects of supersymmetric models and their implications are of paramount importance and need to be explored, because this way we will gain valuable information about the underlying mechanisms of such theories.

As the ﬁeld of supersymmetry has progressed, the need for more detailed quantitative results has increased. However, the tools available for further and more detailed non-perturbative calculations are limited. Lattice calculations, for example, provide a powerful platform for studying gauge theories in the strong coupling regime. However, the success of lattice gauge theory is rather slow in the context of supersymmetry not to mention even in the context of QCD. An alternative tool to study some of the non-perturbative aspects of gauge ﬁeld theories and also to obtain more quantitative results is the method of Discrete Light Cone Quantization

(DLCQ) [1].

Dirac suggested in 1949 [1, 15, 16] that in order to study the quantization of relativistic ﬁeld theories one has only three available parametrizations:

◆ instant form – canonical (or equal-time) quantization

3 Figure 1.1: The three forms of quantization as introduced by Dirac are depicted in a two dimensional spacetime (x0 ct, x1). The dashed (blue) line corresponds to the ≡ light-cone boundary, while the (red) arrows indicate the direction of time-evolution. The leftmost diagram corresponds to the instant form, the central one to the light + cone quantization (with coordinates (x , x−)) and the rightmost to the point form scheme. Notice that in the case of light front parametrization the time evolution surface (line in this case) coincides with x+ =0 and in the case of the point form the quantization takes place on a hyperbolic curve (x0)2 (x1)2 = a2 > 0, x0 > 0. For a − more elaborate and complete discussion on this matter consider review article [1].

◆ light front form – light cone quantization

◆ point form, that are not covariantly connected, that is they cannot be mapped onto each other by a Lorentz transformation. In particular, in the instant form the time–like direction is the one that deﬁnes naturally the canonical time, while in light front form the canonical time is deﬁned by the light-like direction, i.e., the light cone (see diagrams in Fig. 1.1 above, Section 3 and also reference [1] for more details). Although the the- oretical framework of light cone quantization was laid down by Dirac, the technique of DLCQ was introduced, with success, much later [17] in an attempt to construct numerically the bound state spectrum and the wavefunctions of (1+1)-dimensional

4 field theories. In this dissertation we set out to discuss non-perturbative properties of supersymmetric gauge field theories in (1+1) spacetime dimensions using a modified version of DLCQ suitable for such models. This method is called Supersymmetric

Discrete Light Cone Quantization (SDLCQ) [18–20]. It was noted by the authors of [19] that the square of the supercharge could be used instead of the light cone hamiltonian. We know from superalgebra that indeed the light cone hamiltonian is proportional to the square of the supercharge. Although one can study ﬁeld theories with supersymmetry in the framework of DLCQ [21–23], we choose to work with SDLCQ mainly because this formulation has the advantage of preserving supersymmetry upon discretizing momentum. Moreover, this approach provides faster numerical convergence, thus it reduces computational time.

The dissertation we present here is mainly divided into two parts. In the ﬁrst part, Chapter 2, we review some of the rudiments of supersymmetric theories. Our primary example for doing so, is from the viewpoint of supersymmetric quantum mechanics. In the second part we study properties of dimensionally reduced super

Yang-Mills (SYM) theories in the context of SDLCQ.

In particular, in Chapter 3 we introduce the basics of light cone quantization and also provide a review on the technique of SDLCQ. We use this elegant method to study the bound state spectrum at large-Nc, of d =2+1 super Yang-Mills models that are dimensionally reduced to d =1+1 dimensions. The mass spectrum that is numerically calculated is used to study finite temperature effects of pure SYM models and SYM theory with matter in the fundamental representation. Namely, Chapter 4 is a first attempt to discuss thermodynamic properties of a pure SYM theory in 1+1 dimensions. The spectrum consists of a supersymmetric ideal gas of free bosons and

5 free fermions, for the large-Nc approximation assumed in our SDLCQ calculations.

The calculation of thermal quantities, such as the free energy, requires us to consider the full spectrum of the theory. This is a computationally demanding and non-trivial task. However, using advanced numerical techniques we achieve full diagonalization for values of harmonic resolution up to K = 17. The property of the spectrum is characterized by an exponential behavior, suggesting that a Hagedorn temperature exists, which we calculate numerically from our data. Moreover, we discuss various results regarding finite temperature effects below the Hagedorn temperature. To do so we propose a systematic approach: The discrete spectrum, and more specifically the cumulative distribution, can be approximated by a continuous fit from which we can calculate the density of states of the invariant mass spectrum. Therefore instead of summing over thousands of states we just integrate over a continuous density function.

Our study of thermodynamic properties of SYM theories continues in Chapter 5.

Namely we consider a SYM model that includes matter ﬁelds in the fundamental representation of the gauge group SU(Nc) as well as a Chern-Simons (CS) form. The latter is used to generate mass for the adjoint ﬁelds. The meson-like spectrum of this theory is calculated in the context of SDLCQ. As we mentioned before, calculating statistical mechanics requires the full spectrum, which is quite challenging in this particular model, simply because the number of states are far more than those we encountered in the pure SYM sector of the theory. To this end we introduce a new numerical approximation which emulates the full spectrum of the theory, avoiding this way the need for full diagonalization. This numerical method is explained in detail. The thermodynamic behavior of the theory is studied using a similar approach

6 applied in Chapter 4. Namely, we construct the density of states from the cumulative distribution of SDLCQ spectrum. Again we realize a Hagedorn temperature for this theory and we explore thermodynamics below this temperature and for various values of the gauge coupling in the theory, because here the gauge coupling is not an overall multiplicative constant as it was in the pure SYM sectror.

In Chapter 6 we brieﬂy discuss a possible candidate model that can be studied numerically in the context of SDLCQ. Earlier successes of SDLCQ [24–27] indicate that the method has the potential to be used as a direct means to solve numerically challenging physics problems drawn from the context of string theory. In fact, the model we explore in the last chapter derives from the lagrangian of a weakly coupled

D3-D7 brane system [28]. Then this four dimensional =2 theory is dimensionally N reduced to two dimensions giving rise to a = (4, 4) D1-D5 model. This particular N lagrangian resembles a supersymmetric QCD theory with a rich ﬁeld content. More speciﬁcally, we present one of the two supercurrents for the D3-D7 theory and we provide the corresponding supercharges from the two dimensional point of view (i.e.,

D1-D5 model). The study of this particular model, is rather ambitious, both computationally and theoretically. Nonetheless, it can be used to test numerically, to some extent, the validity of AdS/CFT correspondence and compared against string theory calculations.

Finally, we included for completeness a few appendices in order to provide some further details and elucidate, whenever possible, results we introduce and discuss in the main part of this dissertation.

7 CHAPTER 2

INTRODUCTORY REMARKS ON SUPER QUANTUM MECHANICS

Η ποίηση είναι το καταφύγιο που φθονούμε. –Κ. Καρυωτάκης

The idea of supersymmetry1 was first introduced in quantum field theories in an attempt to help our efforts in understanding the unification of the fundamental forces of nature. Physically speaking, supersymmetry dictates that fields corresponding to different spin statistics can coexist in the same irreducible multiplet, namely the supersymmetric partner of a boson is a fermion and vice versa. Mathematically speaking, it is the set of transformations, that extend the Poincaré group in accord with the S-matrix symmetries, and their primary function is to interchange bosonic and fermionic states that belong in the same multiplet (see e.g., references [30,31]).

However, since the invention of supersymmetry there has not been observed a (mass) degeneracy between fermionic and bosonic degrees of freedom. Nonetheless, it is a general belief that supersymmetry is a part of nature’s rich tapestry, however it must be spontaneously broken. If supersymmetry is indeed a theory of nature, then according to estimates the lightest superpartners are expected to be observed at energies on the order of 100GeV 1TeV, or so. To this end the long awaited − 1In fact, as an idea it was introduced in mathematics in 1882 by Darboux [29].

8 experiment at LHC, CERN, is expected to shed some light, hopefully in the upcoming year.

The research area that the ﬁeld of supersymmetry covers is vast. Here we will just introduce supersymmetry using the language of quantum mechanics. Namely, the main scope of this chapter is to provide an overview to the idea of Supersym- metric Quantum Mechanics (SUSYQM) [32–34], which can be thought of as the non-relativistic limit of supersymmetric quantum ﬁeld theory. On the other hand we will try to unveil, as much as possible, the beautiful and elegant concepts of supersymmetry as they apply in quantum mechanics. An additional objective of this pedagogical introduction is to help motivate the subsequent discussion related to supersymmetry as this is studied in the framework of DLCQ.

This review consists of two main parts. The first one deals with the formulation and the preliminaries of one-dimensional SUSYQM. The emphasis is given mainly to the superfield lagrangian formalism as well as the hamiltonian approach. In the former the realization of supersymmetry is manifest, while in the latter the properties of the supersymmetric system can be studied more clearly. At the same time the second part is focused on some example applications such as the Witten Index, which is introduced as a tool for testing supersymmetry breaking. Furthermore, finite temperature effects are discussed, addressing the question of supersymmetry breaking at non-zero temperatures. For completeness, in Appendix A we have included some of the mathematical details that were left out of the main part of this chapter.

9 2.1 Formulation of SUSYQM

The concept of Supersymmetry in quantum mechanics made its first appearance in the literature almost three decades ago. Nicolai [32] applied the idea of supersymmetry in spin systems and thus incorporated it into non-relativistic quantum mechanics. This work was done after the first successful attempt by Wess and Zumino [9,10] to construct a field theory that is invariant under supersymmetry transformations2.

However, the blooming of SUSYQM came after Witten [33,34] introduced and used

SUSYQM as a laboratory for investigating supersymmetry breaking. Two decades after this seminal work, SUSYQM found its way into many branches of physics rang- ing from condensed matter to high energy theory3. In what follows, however, we will focus mainly on Witten’s quantum mechanical model. In the next few paragraphs we will lay down the necessary mathematical background.

2.1.1 A ﬁrst glimpse of superspace in quantum mechanics

The notion of superspace with all its ingredients (superﬁelds, grassmann numbers etc.) is very elegant. Superspace can be thought of as a generalization of Minkowski space, where Poincaré transformations act. Therefore it can be viewed as the space where super-Poincaré transformations act [30, 37, 38]. Simply stated, it is a formulation that enables us to write down the action of a theory in a compact and, in a sense super-covariant form, that is both supersymmetric and gauge invariant.

One dimensional quantum mechanics can be regarded as equivalent to ﬁeld theory in D (t + s) = (1+0) dimensions, i.e., when the only variable of this space is ≡ 2We refer the reader to the reference [35] for an excellent discussion of the subject of supersymmetry in the context of ﬁeld theory. 3For an extensive list of references see [36].

10 the time. Now the superspace is spanned by the time plus two anti-commuting grassmann variables [39] i.e., t (t, θ, θ¯), where in the jargon of modern physics it is → 4 called the =2 superspace (consider e.g., [33,39,40]). Note also that θ∗ θ¯ is by N ≡ deﬁnition the conjugate of θ, thus the combination θθ¯ is regarded as real. Consider now grassmann variables θi and θ¯i, where i =1, 2, 3,.... These variables are deﬁned such that they obey the following rules:

θ , θ¯ = θ, θ¯ = θ, θ = θ,¯ θ¯ = [θ, t] = [θ,¯ t]=0. (2.1) { i j} { } { } { }

Moreover we need to give deﬁnitions for diﬀerentiation and integration over such anti-commuting objects. Some of the basic ones are listed below:

∂ 1=0, ∂ θ = δ , ∂ (θ θ )= δ θ θ δ , (2.2) θ θi j ij θi j k ij k − i jk

dθi = dθ¯i =0, dθiθj = dθ¯iθ¯j = δij, (2.3) Z Z Z Z ¯ ¯ ¯ ¯ ¯ ¯ dθidθj = dθidθjθk = dθidθjθk =0, dθidθj(θkθl)= δjkδil. (2.4) Z Z Z Z The transition to superspace also requires another ingredient, the notion of a super-

ﬁeld φ(t, θ, θ¯). We may now expand the scalar function φ(t, θ, θ¯) about θ = θ¯ =0,

∂φ ∂φ 1 ∂2φ ∂2φ φ(t, θ, θ¯) = φ(t)+ θ + θ¯ + θθ¯ ∂θ ∂θ¯ 2 ∂θ∂θ¯ − ∂θ∂θ¯ x(t)+ θψ(t)+ ψ¯(t)θ¯ + θθF¯ (t), (2.5) ≡ where it is by construction real5 and with (x(t), A(t)) being real bosonic ﬁelds while

(ψ(t), ψ¯(t)) are fermionic ones. Note also that it is the most general superﬁeld we can have in terms of (θ, θ¯) since other terms in the expansion, such as θ(θ¯θ¯)=0,

4In fact if we were introducing one grassmann variable, then the resulting theory would be trivial corresponding to the case of a free particle; =1. N 5 Throughout this chapter the (complex) conjugation is deﬁned as: (ξ¯)∗ = ξ.¯

11 will vanish due to θ2 = 0. This is a direct consequence of their anti-commuting character. But the crux of this “device” is its form, i.e., the balance between bosonic and fermionic degrees of freedom, the fundamental principle of supersymmetry, is manifestly realized.

Another tool we need is the invariant (or covariant) derivative, which by deﬁnition acts on the superspace,

D = ∂ iθ∂¯ , D¯ = ∂¯ + iθ∂ . (2.6) θ θ − t θ − θ t

Now the action of Dθ and its conjugate on the superﬁeld (2.5) will produce the following terms,

D φ(t, θ, θ¯) = ∂ θψ(t) θ¯ψ¯(t) θθF¯ (t) iθ∂¯ x(t)+ θψ(t) θ¯ψ¯(t) θθF¯ (t) θ θ − − − t − − = ψ + ∂ (θθ¯)F + i θθ¯ψ˙ θ¯x˙ = ψ + θF¯ iθ¯x˙ + iθθ¯ψ,˙ (2.7) θ − − ˙ (D φ)∗ D φ = ψ¯ + θF + iθx˙ iθθ¯ψ.¯ (2.8) θ ≡ θ −

After having developed some basic mathematical tools lets try to determine the supersymmetry transformations (see for instance [35, 41]). A ﬁnite element of the super-translation algebra can be written as

¯ ¯ (t, θ, θ¯) = ei[θQ+Qθ+tP0], (2.9) T where (Q, Q¯) are the supersymmetry generators responsible for producing “super- ¯ translations” on the grassmann coordinates θ, θ and P0 denotes the hamiltonian H responsible for the time development of the quantum mechanical system. A standard trick to help us determine the (inﬁnitesimal) form of the changes on the superspace variables (t, θ, θ¯) is to consider the composition law of group elements. Left multiply

12 Eq. (2.9) with (t, ξ, ξ¯). Their product will result in a new translation, which will T have the following form

¯ ¯ ¯ ¯ i[ξQ+Qξ+t2H] i[θQ+Qθ+t1H] (t0, θ0, θ¯0) = e e T i (θ+ξ)Q+Q¯(θ¯+ξ¯)+(t +t )H+ i [ξQ,Q¯θ¯]+[Q¯ξ,θQ¯ ]+[Qξ,Qθ ]+[ξ¯Q,¯θ¯Q¯] = e [ 1 2 2 ( )]

i (θ+ξ)Q+Q¯(θ¯+ξ¯)+(t1+t2+i(ξθ¯ θξ¯))H = e [ − ]

t + t + i(ξθ¯ θξ¯), θ + ξ, θ¯ + ξ¯ (2.10) ≡ T 1 2 − In the above derivation, two things are being employed. The ﬁrst one is the Baker-

Campbell-Hausdorﬀ formula (BCH):

A B A+B+ 1 [A,B]+... e e = e 2 , AB = BA, (2.11) ∀ 6 while the second but most important is the supersymmetry algebra [30] that is obeyed by the supercharges

Q, Q = Q,¯ Q¯ =0, Q, Q¯ =2H. (2.12) { } { } { }

This is the heart of supersymmetry, i.e., the extension to the usual Poincaré transformations that one meets in ordinary supersymmetric ﬁeld theory. Note also that the energy is now part of the symmetry algebra. It also worth pointing out that it is the combination of two odd-valued elements6 of grassmann algebra, which results in an even element; the hamiltonian, H in our case. Their character is essentially fermionic. Furthermore, an immediate consequence of Eq. (2.12) is that

[Q, H] = [Q,¯ H]=0. (2.13)

6These are deﬁned such that they obey anti-commutation relations where together with the even members (the elements which obey commutation relations) form a Z2 graded algebra. For a nice discussion see reference [40].

13 This should be expected since they are generators of a symmetry in the system, described by H; they have to commute with the hamiltonian. We will discuss the above statements in a qualitative fashion in Section 2.1.3. Now from the above algebra and the anti-commutation rules in Eq. (2.1) we can simplify the result of

Eq. (2.10). For instance, left and right multiply the anti-commutator Q, Q =0 by { } the anti-commuting parameters ξ and θ, respectively to get:

ξQQθ + ξQQθ =0 (ξQ)( θQ) θξQQ =0 (ξQ)( θQ)+(θQ)(ξQ)=0, ⇒ − − ⇒ − therefore,

[ξQ, θQ]=0. (2.14)

In a similar fashion we evaluate the rest of the commutators that appear in Eq. (2.10).

The remarkable observation here is that without the anti-commuting relations between the supercharges the group multiplication law would be violated and the algebra wouldn’t be closed. It should also be pointed out that terms of the form [A, [A, B]] and higher in the BCH-formula vanish due to the anti-commuting nature of grassmann parameters7. To finish this discussion, we observe that the infinitesimal translation on superspace can be read off Eq. (2.10). In particular, the super-coordinate transformations are:

t t + i(ξθ¯ θξ¯), θ θ + ξ, θ¯ θ¯ + ξ¯ . (2.15) → − → → We would like now to obtain a diﬀerential representation for the supercharges (Q, Q¯).

Any superﬁeld operator is canonically transformed under the action of the generator

7We have terms like [A, [A, B]] ξ2(. . .)=0. ∼

14 (t0, θ0, θ¯0) = exp i(Qξ + ξ¯Q¯) as follows: T 1 φ(t0, θ0, θ¯0)= (ξ, ξ¯)φ(t, θ, θ¯) − (ξ, ξ¯) φ iξθ¯ iθξ,¯ θ + ξ, θ¯ + ξ¯ , (2.16) T T ≡ − where the argument of the ﬁeld on the right-hand side is a consequence of Eq. (2.10).

Then consider the Taylor expansion of the right hand side of the above up to ﬁrst order terms8

φ(t0, θ0, θ¯0) 1+(iξθ¯ iθξ¯)∂ + ξ∂ ∂¯ξ¯ φ(t, θ, θ¯) ' − t θ − θ 1 + iξ(θ∂¯ i∂ )+i( θ∂ + i∂¯)ξ¯ φ(t, θ, θ¯). (2.17) ' t − θ − t θ Expanding up to ﬁrst order in ξ, ξ,¯ the left hand side of Eq. (2.16), we get:

φ(t0, θ0, θ¯0) = (1+i(ξQ + Q¯ξ¯)) φ (1 i(ξQ + Q¯ξ¯)) − = 1 + iξ[Q, ] + i[Q,¯ ]ξ¯ φ(t, θ, θ¯), (2.18) · · where the notation [Q, ] is understood as an operator acting on a function of space- · time variables. Comparing Eq. (2.17) and Eq. (2.18) we read oﬀ the diﬀerential representation of the supercharges, namely

ˆ Qˆ = i∂ + θ∂¯ , Q¯ = i∂¯ θ∂ , (2.19) − θ t θ − t where we easily prove that Q, Q¯ = 2i∂ 2H, as we expected! Moreover here { } t ≡ is a test whether or not the covariant derivatives are invariant under the super- translations Eq. (2.15). Indeed, application of the chain rule yields 0 ∂ ∂θ ∂ ∂t ∂ ∂θ¯7 ∂ ∂ ∂ 0 0 0 ¯ ∂θ = + + ¯ = iξ , ≡ ∂θ ∂θ ∂θ0 ∂θ ∂t0 ∂θ ∂θ0 ∂θ0 − ∂t0 0 0 ∂ ∂t ∂ ∂θ7 ∂ ∂θ¯7 ∂ ∂ 0 0 0 ∂t = + + ¯ = . ≡ ∂t ∂t ∂t0 ∂t ∂θ0 ∂t ∂θ0 ∂t0 8We ignore the pure time translation since it is associated with H, as we know.

15 Substituting these relations along with θ¯ = θ¯0 ξ¯ into Eq. (2.6) we obtain −

¨¨ ¨¨ D =(∂ iξ∂¯ ) i(θ¯0 ξ¯)∂ = ∂ iθ¯0∂ ¨iξ∂¯ + ¨iξ∂¯ = ∂ iθ¯0∂ , (2.20) θ θ0 − t0 − − t0 θ0 − t0 − t0 t0 θ0 − t0 which is clearly covariant. We have now equipped with the appropriate formulation that will allow us to construct the SUSYQM action that is manifestly invariant under supersymmetry-transformations as we will see in the next section.

2.1.2 =2 SUSYQM: Action, Supercharges and hamiltonian N An ansatz for the =2 SUSYQM action is the following N 1 S[φ]= dtdθ¯dθ D φD φ + w(φ) , (2.21) 2 θ θ Z where w(φ) is the superpotential, a function only depending on the superﬁeld φ(t, θ, θ¯).

Thus the super-covariance is manifest in the proposed action. Let us now expand the kinetic and superpotential terms in order to perform the integrations over the grassmann variables and then realize the form of the Lagrangian in terms of the component

“ﬁelds”. We ﬁrst expand the covariant derivatives. We note from the beginning that we only need terms proportional to θθ¯ having in mind the relations (2.3)-(2.4). The kinetic term gives

D φD φ = θθ¯ F 2 +x ˙ 2 + i(ψ¯ψ˙ ψψ¯˙ ) + θ(. . .)+ .... (2.22) θ θ − Then expand the superpotential about θ = θ¯ = 0 so w(φ) becomes a pure function of x(t) i.e.,

dw(x) 1 d2w(x) w(x(t)+ y) = w(x)+ y + y2 dx 2 dx2 1 = θθ¯ Fw0(x)+ w00(x)[ψ, ψ¯] , (2.23) 2

16 where y = θψ(t)+ ψ¯θ¯(t)+ θθF¯ (t). Therefore, after integrating out the odd-elements, the super-action becomes

2 1 2 i ˙ 1 F S [x, ψ, ψ¯] = dt x˙ + (ψ¯ψ˙ ψψ¯ )+ w00(x)[ψ, ψ¯]+ + Fw0(x) SUSYQM 2 2 − 2 2 Z 1 2 i ˙ 1 2 1 = dt x˙ + (ψ¯ψ˙ ψψ¯ ) (W (x)) + W 0(x)[ψ, ψ¯] ,(2.24) 2 2 − − 2 2 Z where the redundant ﬁeld F (t) was eliminated using its equation of motion (F (t)=

w0(x) W (x)). Notice the ﬁrst two terms in the action, they are just kinetic − ≡ − terms, that describe the bosonic and fermionic degrees of freedom of the particle

(state).

Before we set out to ﬁnd the hamiltonian lets calculate the supercharges in terms of the component ﬁelds. To do so we need the variation of the latter due to supersymmetry transformations (for the details of the derivation see Appendix A.1) d δx = ξψ + ψ¯ξ,¯ δψ =(F ix ˙)ξ,¯ δψ¯ =(F + ix ˙)ξ, δF = i (ψξ + ψ¯ξ¯). (2.25) − dt

A crucial observation from the above formulae is that under supersymmetry transformation (e.g., Eq. (2.15)) the “ﬁelds” of this one-dimensional model are interchanged.

For instance, the component x(t) is transformed into a fermionic degree of freedom

ψ(t) and the fermionic onto a bosonic. This is essentially one of the cornerstones of supersymmetry, i.e., the balance between fermions and bosons.

We know a priori, see Appendix A.1, that the supersymmetry generators in terms of the superﬁeld components can be expressed as follows Q¯ = ψ¯(x ˙ + iW (x))=iψ¯(∂ + W (x)), Q = ψ(x ˙ iW (x))=iψ(∂ W (x)), (2.26) x − x − where πˆx =x ˙ = i∂x. Finally, it is straightforward to determine and canonically quantize the hamiltonian9 corresponding to the action Eq. (2.21). Leaving the details

9 For practical purposes the physical constants such as ~,m, ω,kB are set to one trough out these notes.

17 in Appendix A.3, the hamiltonian is:

1 2 1 2 1 Hˆ = x˙ + (W (x)) W 0(x)[ψ, ψ¯], (2.27) 2 2 − 2 subject to the following canonical quantization rules:

[ˆx, πˆ ]= i, [ˆx, xˆ]=[ˆπ , πˆ ]=0; (ˆπ xˆ˙ = i∂ ) x − x x x ≡ x ψ,ˆ ψ¯ˆ =1, ψ,ˆ ψˆ = ψ,¯ˆ ψ¯ˆ =0, [ˆx, ψˆ]=0, etc. (2.28) { } { } { }

The above construction is essentially the celebrated one dimensional SUSYQM model proposed by Witten [33]. The properties of this model will be discussed in the next paragraph.

2.1.3 Properties of SUSYQM

The properties of SUSYQM are better realized in the hamiltonian formulation.

We have seen previously that SUSYQM is built and motivated by the fact that there exists a set of a fermion-like generators that obey the following anti-commutation relation

Q , Q =2δ H, (i, j = 1,..., ) (2.29) { i j} ij N which is a special case of the more general supersymmetry algebra, Q , Q¯ = { α β} 2(σµ) P , in D = 1+3 spacetime dimensions. Here stands for the number αβ µ N of supersymmetries. So the supercharges Q are responsible for the so called N i extended supersymmetry and this explains the notion =2 introduced earlier N − N in Section 2.1.1, where indeed we had two such symmetry generators. As a remark note that in the above deﬁnition Eq. (2.29) the supercharges are Hermitian in contrast to the supercharges deﬁned earlier (see e.g., equations (2.12), (2.27)and(2.28)) where

18 they were regarded as Hermitian conjugates of each other10. As a ﬁrst observation

2 we note that from the deﬁning anti-commutation rule is implied that H = Qi for all i. This further suggests that

[H,ˆ Qˆ ] = [Qˆ2, Qˆ ]= Qˆ (2δ Hˆ Qˆ Qˆ ) (2δ Hˆ Qˆ Qˆ )Qˆ =2δ Qˆ Hˆ j i j i ji − j i − ji − i j i ji i QˆQˆ Qˆ 2δ Hˆ Qˆ + QˆQˆ Qˆ =2Qˆ Qˆ2 2Qˆ Qˆ2 =0 − i j i − ji i i j i j j − j j [H,ˆ Qˆ ]=0. (2.30) ⇒ j

So by construction, the supercharges leave the system invariant or state it in an equivalent way, they are constants of motion (assuming they have no explicit time dependence.) For the rest of our discussion let us restrict ourselves to the = 2 N model of Witten [33]. To this end it is more convenient to use the non-Hermitian supercharges which obey the relations (2.12) and (2.13). A consequence of Eq. (2.12) is that the hamiltonian (operator) is semi-positive deﬁnite, i.e., it supports only non- negative eigenvalues. To see this we evaluate the expectation value of H in some arbitrary state ... , | i

... Q, Q¯ ... = ... QQ¯ ... + ... QQ¯ ... = Q ... 2 + Q¯ ... 2 0, (2.31) h |{ }| i h | | i h | | i k | ik k | ik ≥ therefore it follows that E = H 0, so the spectrum of such a theory is strictly h i ≥ positive including a zero-energy state. According to the last statement the vacuum state can be either zero or attain some positive value. Concentrating now only on the ground state of the system, it is inferred from the above result that supersymmetry will be spontaneously broken if Eg > 0, otherwise the symmetry is respected. A ﬁrst check whether supersymmetry is (spontaneously) broken or not can be made using

10 1 Recall that we may always introduce Q = (Q +iQ ) and its Hermitian conjugate Q† Q¯ = √2 1 2 ≡ 1 (Q iQ ) whenever = even. √2 1 − 2 N 19 the supersymmetry generators. Indeed, if supersymmetry is not violated then it must be annihilated by the supercharges, otherwise if neither of the generators annihilate the vacuum state, then the symmetry is violated by the ground state. It is readily seen from Eq. (2.31) that for H vac =0 we have that | i Q vac = Q¯ vac =0, (2.32) | i | i something expected as long as the vacuum respects the symmetry. Alternatively we may state that broken supersymmetry signals that the ground state is degenerate. We verify this reasoning by considering 0 to be our ground state with some eigenvalue | i

= 0. Let us also assume that Q 0 = 00 with = 0 and that 0 = 00 . Using 0 6 | i | i 6 | i 6 | i Eq. (2.30) we have

0 = [H, Q] 0 = HQ 0 QH 0 = H 00 00 H 00 = 00 (2.33) | i | i − | i | i − 0 | i ⇒ | i 0| i which implies that = 0, and thus the two states are degenerate. The last statement also leads to the conclusion that the supersymmetric spectrum is energy degenerate.

The system is still supersymmetric in the sense that there is a two-fold degeneracy among the positive energy states ( > 0.) On the other hand the ground state for an unbroken supersymmetric theory has to be singlet.11 We would also like to point out here a crucial difference between supersymmetric theories and conventional field theories. In the latter an infinite zero-point energy may be removed for the reason being that measurement of physical quantities considers only energy differences, i.e., experimental measurements are insensitive to the zero-point energy. However, in supersymmetric theories the zero value of the ground state is important and it cannot

11This is related to the fact that the irreducible reps. of the algebra (2.29)-( 2.30) can be either zero energy singlets (i.e., non-degenerate) or positive energy multiplets that contain an equal number of fermionic and bosonic states with the same energy. In particular the number of constituents in ( 2)/2 ( 3)/2 these multiplets grows like NB = NF = 2 N− for even and 2 N− for odd, for every 3. N N N ≥ 20 be lifted since this would violate automatically the supersymmetry algebra. To sum up, the lesson from this discussion is that whenever exists (at least) a zero-energy state supersymmetry is not broken while at the same time if there is no such a state then supersymmetry is spontaneously broken. We will say more on the supersymmetry breaking when we discuss Witten’s index.

Let us now come back to the strictly positive states that exhibit a two-fold degeneracy (or boson-fermion degeneracy). One may observe that due to the nilpotent character of supercharges, i.e., Q2 = Q¯2 = 0, the positive energy states can be split into two sectors12 . The spectrum degeneracy may also be inferred from the fact that these charges commute with the hamiltonian. Consider now the supercharges from

Eq. (2.25). These can be expressed via the following matrix representation

0 1 0 A Q = (x ˙ iW (x)) , (2.34a) 0 0 ⊗ − ≡ 0 0 A ψ | {z } 0 0 0 0 Q¯ = | {z } (x ˙ + iW (x)) . (2.34b) 1 0 ⊗ ≡ A† 0 ψ¯ | {z } With the aim of the above representation and using the supersymmetry relation given by Eq. (2.12), the hamiltonian (2.27) of our = 2 model can be written as N follows

ˆ 1 2 2 1 1 1 AA† H = px +(W (x)) W 0(x) σ3 = 2 ⊗ − 2 ⊗ 2 0 A†A H 0 = − H H+, (2.35) 0 H+ ≡ − ⊕ 12It is worth noting here that we use the word sector in order to distinguish from the ﬁeld theory case were supersymmetry relates fermions and bosons, which are particles. In SUSYQM the symmetry relates diﬀerent Hilbert spaces, however the analogy is evident.

21 ¯ where we remind that x˙ = px and also [ψ, ψ]= σ3. In addition we deﬁne

1 2 2 1 H = px + W (x) W 0(x). (2.36) ± 2 ± 2 The derivation of the last equation is based on the fact that [i∂x, W (x)] = iW 0(x).

Now in this diagonal representation the splitting of the whole system into the two distinct sectors is manifest. These two spaces “communicate” with each other by the means of supersymmetry generators. In this notation the state vector of the system is naturally a two-component Pauli spinor

ψ (x) 0 Ψ = Ψ + Ψ+ = − + , (2.37) | i | −i | i 0 ψ+(x) with ψ+(x) and ψ (x) being eigenstates of H+ and H respectively. The mapping − − between “bosonic” (ψ+(x)) and “fermionic” (ψ (x)) states follows naturally. In par- − ticular, utilizing Eq. (2.34) we see that

A ψ+(x) Q Ψ = Q Ψ+ = Ψ H , (2.38) | i | i 0 ∝| −i ∈ − and consequently Q turns a state that originally belongs in the H+ sector into a state which belongs in the H sector. Similarly the other charge, Q¯, will do the opposite − job. Now are these interchanged states degenerate? The answer is partially answered previously but nevertheless let us emphasize it once more since the degeneracy of

(positive energy) states in supersymmetry is a central issue. Assuming that the

“rotated” state, Ψ has eigenvalue > 0, then because of [H, Q] Ψ =0, it follows | +i | +i that H Ψ = Ψ . In other words, the newly obtained state Ψ has the same | −i | −i | −i energy as the original state Ψ . Thus the role of supercharges as “intertwining” | +i operators is clear.

Let us further continue our exploration of SUSYQM properties. Assume ﬁrst that

g g T there exists a zero-energy state of the form Ψg =(ψ (x), ψ+(x)) and since there | i − 22 is such a state then condition (2.32) must hold true, that is

d x g g g ( dx0 W (x0)) Aˆ† ψ (x)= 0 + W (x) ψ (x)=0 ψ (x)=N e − , (2.39a) − ⇒ dx − ⇒ − − R d x Aˆ ψg (x)= 0 W (x) ψg (x)=0 ψg (x)=N e(+ dx0 W (x0)). (2.39b) + ⇒ dx − + ⇒ + + R Then according to the postulates of quantum mechanics we require that the would be ground state is square integrable (normalizable). Normalizability is related to the form of the superpotential W (x) and it is worth emphasizing that all the properties of such a supersymmetric system are ultimately related to the behavior of the superpotential. Speciﬁcally, assuming the asymptotic behavior of W (x) we distinguish three possible cases:

x g 2 g i. if lim dx0 W (x0) + then dx ψ (x) < but ψ (x) is non-normalizable, x + →±∞{ }→ ∞ | − | ∞ R x R g 2 g ii. if lim dx0 W (x0) then dx ψ (x) < but ψ (x) is non-normalizable, x + →±∞{ }→−∞ | | ∞ − R x Rx iii. if lim dx0 W (x0) = lim dx0 W (x0) ; no normalized state can exist; no x x + →−∞{ } − → ∞{ } zero-energyR state. R

The first two cases are satisfied for W (x) x2n+1, an odd polynomial and the ∼ third one is satisfied for even valued superpotential. We also conclude that the ground state is not degenerate, either it belongs to the H+ or H sector; it is manifestly − singlet in this context. This will be basically the key for attacking the example that is discussed next.

2.1.4 SUSYQM in action: An example

In this section we consider a rather simple but nevertheless illustrative example that can be solved using the ideas we have just presented above. We focus in the case of one dimensional potentials. For instance, assume that we are given the

23 inﬁnite potential well and that we are asked to ﬁnd (if any) its “super-partner”. From the previous discussion on Witten’s model we recall formula (2.36) for the partner hamiltonians,

1 d2 H = + V (x), (2.40) ± −2 dx2 ±

1 2 where we deﬁne V = (W W 0(x)), known in the literature as the supersymmetric ± 2 ± partner potentials [39,42]. Assume now that we know the ground state of one of the two partner hamiltonians and let this be the zero-energy state of the supersymmetric system that we are about to set up. Say that we know the zero-energy state, Φg of

H+, then it follows from Schrödinger’s equation that H+Φg(x)=0. As a consequence of the last result we have A Φg(x)=0 (see Eq. (2.35)) from which we extract the superpotential

d Φ0 (x) W (x) Φ (x)=0 W (x)= g . (2.41) dx − g ⇒ Φ (x) g We stress at this point that the validity of the above result is based on the fact that the (chosen) ground state energy is exactly zero, otherwise if it is not such, we have to shift it to zero in order to satisfy A Φg(x)=0; see inside reference [42,43]. Consider now the particular choice of the inﬁnite well of width π. Recall that the ground state of the inﬁnite well is: Φ (x) = 2/π sin(x) with 0 x π with corresponding g ≤ ≤ 13 1 p eigenvalue given by Eg = 2 . Further, we assume that this state is the ground state of the supersymmetric system under construction. However, according to the above remark, we have to adjust this energy because is not exactly zero. Namely, we want

to have H Φ = (HIW E )Φ = 0. After this modiﬁcation the state and energy + g − g g 13 ~2π2 Upon restoring the units the energy should be Eg = 2mL2 , where m is the mass of the particle and L is the width of the potential.

24 spectrum of the inﬁnite well become

n2 1 Φ(+)(x)= 2/π sin(nx), E(+) = − , 0 x π, (2.42) n n 2 ≤ ≤ p where n =1, 2 .... Using Eq. (2.41) the superpotential is found to be: W (x) = cot(x), 0 x π. (2.43) ≤ ≤

From this we can easily derive the the super-partner potentials V , ± 1 1 2 V+ = (adjusted inﬁnite well); V = csc (x) 1 . (2.44) −2 − 2 − The next step is to outline the procedure of obtaining the wave functions corresponding to V . Supersymmetry implies that the spectra of the two potentials will possess − an 1 1 pairing for states with E > 0. From Schrödinger’s equation for the sector − where we have the wave functions, namely in the case at hand H+, we have

(+) 1 (+) (+) H Φ = A†AΦ E , (n = 1). (2.45) + n 2 n ≡ n 6

The expected degeneracy of spectrum due to supersymmetry implies that

(+) 1 (+) (+) (+) (+) ( ) H (A Φn )= AA†(A Φn )= En (A Φn ) En Φn−+1, (2.46) − 2 ≡ where we realize that the role of the operator A is indeed to map the wave functions of H+ into the other sector. It is readily seen, that the (normalized) wave functions and energies of H can be obtained according to the following pattern: −

( ) 1 (+) Φm− (x)= A Φm+1(x), (2.47) (+) 2Em+1 q ( ) (+) where for the energies we have Em− = Em+1 (m = 1, 2 . . .). Note that the case

(+) Em=0 = 0 is understood to be the unpaired ground state of this simple supersymmetric model. For example the lowest state corresponding to H is easily found to − 25 ( ) 2 2 ( ) (+) be: Φ − (x)= 2i sin (x) with energy E − = E =3/2. It is remarkable that, 1 − 3π 1 2 q although the two superpartner potentials are quite diﬀerent, their spectrum is the same apart from the ground state. One may now consider H to be the hamiltonian − with the lowest state (after an adjustment) and set up another pair of superpartners by applying the same procedure as outlined above. So supersymmetry allows us to built up a group of exactly solvable potentials just by starting from a potential that its solution is known. In other words, the subsequent members of the hierarchy are supersymmetric partners. For good reviews on various techniques and more applications [36,39,42,43].

2.2 Supersymmetry breaking and non-zero temperature effects

We have already hinted out about the spontaneous supersymmetry breaking in

Section 2.1.3. Of course, in nature supersymmetry (if exists) must be (spontaneously) broken since it has not been found yet any boson-fermion degeneracy. However, we are hoping to observe such effects at some (high) energy scale. In fact, SUSYQM introduced by Witten [33, 34] as a laboratory for investigating the phenomenon of supersymmetry breaking, i.e., it served as a first tool for testing whether or not the vacuum energy is zero. Specifically in the context of SUSYQM one has to search for a (unique) normalizable ground state (2.39).

2.2.1 Witten’s index and SUSYQM

An important order parameter for determining the breakdown of supersymmetry was introduced by Witten [34], subsequently known as the Witten index. Recall that a necessary and suﬃcient condition for good supersymmetry is that the supercharges

26 must annihilate the vacuum state. On the other hand, we know that for the positive energy states we have boson-fermion degeneracy. In this case we can only have a supersymmetry invariant state if it is a linear combination between boson and fermion states, i.e., s ( bos + fer ). Remembering the role of supercharges in | i ∝ | i | i interchanging a fermionic into a bosonic state and vice-versa we expect the above state to be invariant under a supersymmetry transformation. Indeed this is the case

E>0 E>0 and the number of bosons (must) equals the number of fermions say Nb = Nf ; exact pairing. However, the zero modes are singlets under supersymmetry variations and thus they are invariant themselves separately. Namely, in this case the number

E=0 E=0 of bosons need not be equal to the number of fermions, i.e., Nb and Nf may have diﬀerent values. To that end Witten [34] suggested the possibility of looking for some topological invariants that may exist in the theory. Physically speaking these are quantities in the theory that are insensitive under “small” deformations of the superpotential caused by variations of the volume, mass or coupling constants.

These changes will cause the energy of the states to change, but as long as the supersymmetry algebra is not aﬀected then no new states will be added (or removed) to (from) the system. We will rather have a rearrangement of the boson-fermion pairs.

For instance, a boson-fermion pair with E > 0 may now be forced to acquire E =0

E=0 E=0 so immediately both Nb and Nf increase by the same amount. In the meantime some zero-energy states may gain some energy but again due to supersymmetry they

E=0 E=0 have to appear in a doublet, so both Nb and Nf decrease the same. Therefore, in both situations any (reasonable) variations of the superpotential will not affect the difference between the bosonic zero-modes and the fermionic zero-modes. Thus the quantity (N E=0 N E=0) is qualified as a topological invariant. One first observation b − f

27 that can be made is that if this quantity is non-vanishing then it is guaranteed that supersymmetry is not spontaneously broken. However, in the case where this quantity vanishes then we can not arrive to a deﬁnite conclusion. In such a case supersymmetry can be either broken or not. We may distinguish two possibilities,

E=0 E=0 (i) if both Nb and Nf are zero then supersymmetry is certainly (spontaneously)

E=0 E=0 broken and (ii) if Nb and Nf are equal but non-zero then supersymmetry is a good symmetry. At this point, it should be emphasized that for the case of a purely

= 2 quantum mechanical system the only possible values for the N or N are N b f zero or one and so can be the value of the index14. In other words either there is a zero energy state or not; in the context of SUSYQM the role of particle is replaced by the state.

Now in order to make the counting of zero-energy modes plausible, a usual trick one uses is to consider the theory in a finite volume. In such case the spectrum of the theory becomes discrete and in principle countable due to the imposed energy cut-off. We note here that if supersymmetry is not broken at infinite volume then it is also not broken at any finite volume, since the zero-energy is simply independent of the volume; it is just zero. Moreover, in putting the theory on a finite volume requires some special choice of boundary conditions on the fields, having in mind that they should not break explicitly the super-Poincaré algebra. This can easily be fixed by imposing periodic boundary conditions on both type of fields, i.e., we enforce boundary conditions that respect translational invariance.

14Considering the case of SUSYQM, at the tree level we may have more that one would be ground states. For example, in a potential of the form W x3 we have 3 such states but 2 will be lifted due to tunnel eﬀect (instanton calculation [44].) ∼

28 Let us now give a more appropriate definition of the Witten index. We first need to introduce the operator ( 1)Fˆ , which is defined such that it has eigenvalues − 1 in order to make distinction between fermions and bosons (or between H and ± −

H+ sectors). Fˆ is the fermion number operator, where in the context of SUSYQM is deﬁned by: Fˆ = 1 (1 σ ) such that its eigenvalues are 0 (for bosons) or 1 (for 2 − 3 fermions). The claim now is that (N E=0 N E=0) can be thought of as the trace of the b − f operator ( 1)Fˆ , where the trace extends over all the spectrum of the theory15; bound − and continuum states. Usually one considers the following standard representation for the index

∆= Tr(1 2Fˆ). (2.48) −

However, in principle we have to regard the trace over an inﬁnite number of states and thus the trace is ill-deﬁned. In order to circumvent this issue we need to regulate it. The following possible regularization was suggested by Witten

Fˆ βHˆ ∆(β)= Tr ( 1) e− , (2.49) − where β is the regulating parameter. In the language of SUSYQM ( 1)Fˆ is well − ¯ represented by the [ψ, ψ]= σ3, Pauli matrix. Therefore Eq. (2.49) becomes ∆(β)=

βH βH+ Tr(e− − e− ). At least, in the case of the infinite well this definition yields − immediately ∆(β) = ∆(0) = 1, which is in agreement with our findings (see Ap- pendix A.4 for more examples.) However, it is not always true that the index will be independent of β as we will see in an example later on. Actually, it is “temperature” independent only when the spectrum of the theory is discrete and only then the definition (N E=0 N E=0) is consistent. At this point, it is worth mentioning that some b − f 15Properly speaking it is defined as: Tr( 1)Fˆ = n ( 1)Fˆ n . − n h | − | i P 29 issues may raise some concerns regarding the validity of the above arguments, in the context of field theory . For instance one might ask whether or not ultraviolet diver- gencies have an effect on the index Tr( 1)Fˆ. In this case only high energy states are − affected by the ultraviolet cut-off and it is expected that they will have no impact on

Tr( 1)Fˆ because it deals only with low-lying energy states. This is true provided that − supersymmetry still holds in the inﬁnite volume cut-oﬀ. Another issue which is important here is the case where the deformations in the superpotential are not “small”.

In particular, if the variation in the parameters (of the superpotential) is such that the asymptotic behavior of the superpotential is altered, then new states may enter causing a discontinuity in the index. To that end, consider W (x)= λ x(x2 β2) and − suppose that we add a quadratic term in W (x). This will not aﬀect the number of zeroes of W (x), since in the large-x limit W (x) x3, so it is an allowed deformation. ∼ On the other hand if we add a quartic term the asymptotic behavior will now be

W (x) x4 and this is not allowed. As a rule of thumb, the acceptable deforma- ∼ tions for keeping the index unaﬀected, are those that do not introduce terms in the hamiltonian, which might be of higher order (in the large-x limit) than those already existing in the theory.

2.2.2 β regularization of the Witten Index − Here we consider the case where the Witten index can be “temperature”16 dependent. It has been found by R. Akhouri and A. Comtet (consider references [45, 46] for more technical details) that the Witten index is regulator dependent when the spectrum of the theory, apart from discrete states contains also continuum ones. In

16Here the regulating parameter β can be thought of as the inverse temperature without loss of generality.

30 particular, the above authors showed that when the superpotential is of the form

W (x )= W , then the index shows an “anomalous” behavior. The regular- → ±∞ ± o ized Witten index can be put in the following general form

E=0 E=0 ∞ βE ∆(β)=(Nb Nf )+ dEe− (ρ+(E) ρ (E)), (2.50) − − − ZEc where the second term is the contribution of the continuum spectrum of states, while

ρ (E) represents the continuum distribution density of states. According to [45] the ± Witten index can be calculated using the following relation (see also Appendix A.5 for a derivation)

d∆(β) 1 d βE(k) + = dx e− 2E(k)ψ (x)ψ−(x) (2.51) dβ 2 dx k k Z k=0 X6 p Let us now consider a particular example where the Witten index exhibits this anomalous behavior. Namely, for the superpotential W (x) = tanh x the corresponding superpartner hamiltonians are

2 2 1 d 1 2 1 d 1 H+, H + (1 2sech x), + , { −}≡{−2 dx2 2 − −2 dx2 2} and see for instance Eq. (2.40). The respective eigenfunctions17, obtained from

H ψ (x)= E(k)ψ (x), (2.52) ± ± ± after imposing periodic boundary conditions (ψ±(L) = ψ±( L)) are found [45] to n n − be:

1 + 1 ψ1−(x)= cos(k1x); ψ1 (x)= (k1 sin(k1x) + tanh x cos(k1x)), (2.53) √π πE(k1)

1 + 1 ψ2−(x)= sin(k2x); ψ2 (x)= p ( k2 cos(k2x) +tanh x sin(k2x)), (2.54) √π πE(k2) −

17 2 Note here that H+ has one bound state,p which happens to be the zero-energy state ( sech ) of this supersymmetric system. However, formula (2.51) precludes this state from the sum.∝

31 1 2 with energy constraints (from equation (2.52)) E(ki)= 2 (1+ ki ), i =1, 2. Substitu- tion of Eq. (2.54) into Eq. (2.51) and taking the boundaries at inﬁnity (L ) the →∞ regulated Witten index is indeed “temperature” dependent (for detailed derivation see Appendix A.5): β d∆(β) e− 2 = (2.55) dβ √2πβ

So the above result in the limit where β agrees with the fact that there is → ∞ a supersymmetric vacuum in this model; ∆(β) 1. It is generally true [45] that → the Witten index is cut-oﬀ dependent when the super hamiltonian possesses a continuum spectrum of states in addition to the discrete ones. The reason for being

β dependent is mainly due to the diﬀerent contributions of the scattering states − of the two partner hamiltonians. Therefore the mismatch in the two (continuum) spectra is responsible for this “anomaly”. The result that we just presented seems contradicting with our earlier discussion. However that discussion was mainly concerned with models that appear to have discrete spectra; case where the index is calculated reliably. Thus as a concluding remark we point out that even with an appropriate regularization the index is β dependent for cases where the spectrum − contains a continuous distribution of states. Moreover, this peculiar behavior may be related to some kind of anomalies in supersymmetry. However, the Witten index, is still a powerful measure for determining whether or not supersymmetry is broken since by deﬁnition is a global invariant of the theory and in principle can be calculated in a reliable manner by considering suitable limits. We will also consider

Witten index in the next section.

32 2.2.3 SUSYQM and ﬁnite temperature eﬀects

We will now investigate how supersymmetry is incorporated and behaves when

finite temperature effects take place. We will analyze this from the point of view of SUSYQM. More specifically one would like to find out what happens in a supersymmetric system that is immersed in a heat bath. Is supersymmetry conserved or broken when finite temperature effects take place or will any additional interactions restore or break supersymmetry? We address such questions in the context of

SUSYQM by the means of an example (see Refs. [46–49].) In particular we will ex- amine an interacting theory by utilizing the Thermoﬁeld dynamics formalism [47–50] where we calculate the thermal vacuum energy. The non-vanishing of the vacuum energy signals that supersymmetry is not violated.

The model we wish to discuss is described by the following hamiltonian [47].

p2 1 1 1 = x + x2 [ψ, ψ¯] + (λ + g x2)2 +2gx(λ + g x2) g x[ψ, ψ¯] (2.56) H 2 2 − 2 2 −

H Hint The above| construction{z corresponds} | to the SUSYQM{z hamiltonian (2.27) for} a superpotential (x) = x +(λ + gx2). Without any loss of information we, may think W 2 of it as the simple harmonic oscillator plus corrections of order (x ). Namely, Hint O is assumed to be a small perturbation to the (supersymmetric) harmonic oscillator.

First we should point out that the constant λ that appears in the hamiltonian, controls in a way the supersymmetry breaking; its sign is an indicator whether or not supersymmetry is broken. Namely, when λ> 0 one expects spontaneous breakdown of supersymmetry and when λ< 0, supersymmetry is good. In addition g is the coupling for the Yukawa-like terms as well as the quartic interaction terms. Intuitively, one would expect that supersymmetry would be broken at ﬁnite temperature, ﬁrst

33 because bosons and fermions obey diﬀerent statistics and thus their thermal contributions will not necessarily cancel but rather added together, and second at ﬁnite temperature Lorentz invariance is lost because of the existence of a preferred frame, e.g., heat bath at rest. These observations are in contrast with the expectations one has for the internal symmetries, where if they are broken at T =0 then they are restored above some high temperatures. We will set out to investigate these issues but

ﬁrst lets digress for a while in order to introduce the notion of Thermoﬁeld dynamics.

Without going into the technical details, we brieﬂy mention that the advantage of this formalism is that enables us to study the structure of the vacuum at ﬁnite temperature in a more direct way in terms of the harmonic oscillator (creation/annihilation) operator language. For instance, statistical ensemble averages can be expressed in terms of a temperature dependent -themal vacuum.- In other words any ensemble average of an observable may be formally obtained by its vacuum expectation value in the thermal vacuum,i.e.,

ˆ βH ˆ Tr e− ˆ β = O βH = Ω(β) Ω(β) . (2.57) hOi Tr e− h | O| i

One arrives at this form by introducing a fictitious (tilde) system with the same properties as the original one. Namely, one doubles the degrees of freedom of the system and an arbitrary state may be described by18 N , N ; N˜ , N˜ with the vacuum de- | b f b f i fined as Ω ω ω˜ . Then the thermal vacuum can be obtained by operating | i≡| i ⊗ | i on Ω with a (unitary) Bogoliupov transformation, . In particular, the thermal | i G vacuum is defined to be:

i (β) θ(˜aa a a˜ ) θ¯(˜bb b ˜b ) Ω e− G Ω = e− − † † − − † † Ω (2.58) | iβ ≡ | i | i 18 Here Nb = a†a and Nf = b†b denote bosonic and fermionic number operators, respectively. This is actually the familiar case of the bosonic/fermionic oscillator.

34 with tan θ = tanh θ¯ = exp( β/2). Now any observable will be “evolved” from zero − temperature to an arbitrary one

i i ˆ = e− G ˆ e G, (2.59) Oβ O like any observable does, under the action of a canonical transformation. For the case at hand, we use the Bogoliupov transformation in order to obtain expressions for the thermal creation/annihilation operators in terms of the zero temperature ones. After some algebraic manipulations (see Appendix A.6 for details) we obtain

a†(β)= a† cosh θ¯ a˜ sinh θ,¯ a(β)= a cosh θ¯ a˜† sinh θ,¯ (2.60a) − −

a˜†(β)=˜a† cosh θ¯ a sinh θ¯ (2.60b) − and

b†(β)= b† cos θ ˜b sin θ, b(β)= b cos θ ˜b† sin θ, (2.60c) − −

˜b†(β)= ˜b† cos θ + b sin θ; (2.60d)

Moreover, the operators a˜†, a˜ (˜b†, ˜b) obey the same (anti-)commutation relations as the ordinary ones and also the thermal vacuum Ω remains normalized assuming | iβ that the non-thermal was normalized, too. These are consequences of the Bogoliupov transformation, which by definition is canonical and therefore the operator algebra remains unaltered after such a transformation. To be more specific: [a, a†]=[ã, a˜†]=

[˜a(β), ˜a†(β)]=1= b, b† = ˜b, ˜b† = ˜b(β), ˜b†(β) and also the bosonic (fermionic) { } { } { } operators of the non-tilde system are taken to commute (anti-commute) with the like ones of the tilde system. Furthermore, we remind that: a ω = b ω = 0 and | i | i

35 therefore the temperature dependent annihilation operators would also destroy the thermal vacuum, a(β) Ω = b(β) Ω =0. | iβ | iβ In the speciﬁc example, the supercharges corresponding to the unperturbed hamiltonian can be expressed in terms of the (super harmonic oscillator) ladder operators,

1 1 i.e., Q = √2a†b and Q† = √2b†a with a† = (p ix), a = (p + ix), b = ψ and √2 − √2 b† = ψ.¯ Using the above deﬁnitions our (full) hamiltonian can be written in terms of creation/annihilation operators as follows:19

1 g 2 2 g 2 =(a†a + b†b)+ (λ (a† a) ) + i√2(a† a)(λ (a† a) ) H 2 − 2 − − − 2 − ig (a† a)[b, b†]. (2.61) − 2√2 −

In order to calculate the thermal vacuum energy, E = Ω ˆ Ω , we have to express g h | H| iβ the non-thermal operators in terms of the thermal ones. We can do this quite easily, just by inverting relations (2.60); for instance combining the ﬁrst and the last one of the top row we get: a† = a†(β) cosh θ¯ +˜a(β) sinh θ.¯ Similarly by combining the

ﬁrst and the last relation of the second row we get: b† = b†(β) cos θ + ˜b(β) sin θ.

The rest can be obtained just by conjugation. Then the procedure is straightforward bearing in mind that by construction Ω Ω =1 and also that the thermal vacuum h | iβ is destroyed by the annihilation operators. Finally after some algebra, which is outlined in Appendix A.6, we arrive at the following ﬁnal result for the thermal vacuum energy:

Eg = E + Eint β β β 2 2 β 2 e− e− 1 g 1 + e− g 1 + e− = + + λ + + . (2.62) 1 e β 1 + e β 2 2 1 e β 2 1 e β − − − ( − − − − ) 19 One can easily check that Q, Q¯ = 2(a†a + b†b)=2H, but [Q, ] = 0 since Q does not represent the whole system. Also{ note} that in writing the hamiltonian inH terms6 of ladder operators 1 1 we used p = (√2)− (a + a†), x = i(√2)− (a† a). − 36 The above result is alarming from the point of view that supersymmetry is broken, at finite temperature, not only due to the introduced interaction terms, but is also spontaneously broken before the perturbation was turned on; see Fig. 2.1. Namely, the first term of the above equation is non-vanishing at T = 0. Moreover the su- 6 persymmetry breaking controlling parameter λ has no effect in this case since the contribution from the interaction terms appears to be strictly positive, thus eliminating any hope for cancellations among the various terms in Eg. A further analysis in the limit of high temperatures shows that the leading term is

3g2 Eg = + (1/β), (2.63) β 0 2 → 2β O where it is readily observed that is independent of the supersymmetry breaking parameter λ. So even if it was unbroken at T =0, at very high temperatures is always broken. In other words, supersymmetry is not good at high temperatures and it is not restored either. Therefore as we have speculated from the beginning, supersymmetry is broken at ﬁnite temperatures, at least this is the outcome of the SUSYQM model we have just examined. To this end one would expect Witten index to produce the correct results in the two extreme limits, i.e., at low and high temperatures. A simple calculation (for λ = g = 0) using the (non-regularized) form of the Witten index, which is deﬁned in Eq. (2.48), yields

β 2 1 e− ∆(β)= (1 2Fˆ) = (1 2b†b) =1 2 sin θ = − , (2.64) h − iβ h − iβ − 1 + e β − where we made use of the result obtained in the derivation of E and also Fˆ N . g ≡ f It is worth noting that this result becomes fractional for arbitrary values of β and it is clearly not appropriate for an index, which by deﬁnition assumes only integer values; this has to do with the regularization one uses (see Section 2.2.2) Nevertheless,

37 E, Eg E, Eg

20 2 E E

E 15 E 1.5 g g

10 1

0.5 5

T T 0.2 0.4 0.6 0.8 1 1.2 1.4 5 10 15 20 (a) (b)

Figure 2.1: Thermal ground state: (a) The thermal ground state of the oscillator H along with the thermal ground state of the whole system for small values of T and (b) the same H plot but for higher range of temperatures. For the interaction term we have used λ = 1.3 − and g = 0.01. The fact that supersymmetry is broken at ﬁnite temperatures is manifest in these diagrams.

for T it gives ∆(β)=0 and for T 0 gives ∆(β)= 1. Both of these → ∞ → results are in agreement with the above calculation, that is for high temperatures supersymmetry is (spontaneously) broken and for zero temperature supersymmetry is good. Therefore we are lead to the following conclusion regarding the behavior of supersymmetry at ﬁnite temperature: supersymmetry is spontaneously broken even in the unperturbed case and so is also the result when a small interaction is turned on.

The interaction term does not cancel the thermal contributions which already exist in the unperturbed theory, but it rather adds energy to the system. Of course, one could go further and check whether or not the supercharges annihilate the thermal vacuum. For the case at hand we do not expect the supercharges, that generate , H to annihilate the thermal vacuum, since supersymmetry is spontaneously broken.

38 2.3 Discussion

We have explored some of the basic ideas and foundations of SUSYQM. In particular, we have seen how SUSYQM is formulated in terms of superfield language as well as in terms of the hamiltonian formalism, which stems out of the former naturally. We have also worked out an example from quantum mechanics and saw how the ideas of supersymmetry can be applied to solve quantum mechanical systems in a neat way. Specifically, one can start from a solvable potential and by utilizing supersymmetry can find the solution to a series of hamiltonians. Furthermore, we introduced the notion of Witten index, which by definition counts the difference between the number of fermionic and bosonic zero-energy states. Witten noticed that such quantity is insensitive in (reasonable) deformations of the superpotential and therefore can be used reliably to test supersymmetry breaking. Moreover, we discussed its calculation when a theory contains scattering states. Finally, we studied

SUSYQM at ﬁnite temperature by the means of an example. The outcome from this problem revealed that supersymmetry is (spontaneously) broken at ﬁnite temperature because the thermal vacuum energy has non-vanishing value. This result is in accordance with the Witten index at the high temperature limit.

Although for the rest of this dissertation we are dealing with supersymmetric gauge ﬁeld theories, we hope that this introduction from the super quantum mechanics perspective will help motivate further the discussion regarding the numerical technique (SDLCQ) we employ in order to solve complicated SYM models. For more information related to the principles and technical details of supersymmetric ﬁeld theories we refer the reader to a few of the very many excellent treatments on the subject [35,40,41,51–57].

39 CHAPTER 3

A BRIEF PREVIEW OF (S)DLCQ

Η ποίηση έχει τις ρίζες τις στην ανθρώπινη ανάσα. –Γ. Σεφέρης

In this chapter we introduce light cone coordinates, the basics of DLCQ [1,17,58] and its generalization to supersymmetric systems conﬁned in two dimensions [19,20].

We choose to work with two dimensional systems obviously because they are much simpler than their higher dimensional counterparts, more manageable computationally and also can be used as a testing laboratory for string theory models in the future. More speciﬁcally, we discuss: quantization rules on light front, the formulation of the bound state problem that is of our interest in this work, an example where the technique is applied to pure two dimensional SYM theory and ﬁnally we discuss the implications of performing thermal calculations on the light front.

3.1 Light Cone parametrization

Unlike non-relativistic dynamics, where the time development of a system is de- termined solely by the hamiltonian (instant form), in the relativistic scenario one can prove [59] that the covariantly inequivalent forms of relativistic dynamics are ﬁve.

40 This number is exactly specified by the five subgroups of the Poincaré group20. Note that although these three parametrizations are not connected by a Lorentz transformation, they are however unitary equivalent as required by quantum field theory.

Before that was known, Dirac [15] found the three non-trivial distinct parametrizations of spacetime that guarantee the relativistic invariance of a quantum dynamical system. One of these choices, as mentioned earlier is the light cone (or light front) quantization. One conventional choice [60] of coordinates, that can be used as a possible parametrization of the light cone coordinate system21 is the following

x0 x+ 1 1 0 x0 x0 + x1 1 1 1 1 0 1 x x− = 1 1 0 x = x x . (3.1)     √     √   x2 7−→ x2 2 0− 0 1 x2 2 x−2           In other words, this deﬁnition amounts to a rotation of the plane x0x1 by φ = π/4 − followed by a reﬂection of the coordinate x1, see also Fig. 1.1. Then the corresponding spacetime metric becomes

01 0 ηµν ηˆµν = 10 0 . (3.2)   7−→ 0 0 1 −   Let us now focus our attention in the two dimensional plane22 formed by the light

+ cone (longitudinal) coordinates x and x−. Namely, we choose the light cone time to be the ﬁrst one. Next, following the rule given by Eq. (3.1) we can deﬁne the light

20A subgroup of this kind is formally known by the name stability group (or little group) of the Poincaré group and consists of all the kinematical generators of the group; the rest are the dynamical ones. Dirac used to call the dynamical generators of Poincaré group Hamiltonians, for obvious reasons. Note that the largest stability group of the Poincaré group is realized in the light cone formulation. Namely in d =3+1 dimensions the stability group consists of 7 generators. 21Here and for the rest of our discussion, unless otherwise stated, we consider Minkowski (2 + 1)- dimensional spacetime. Recall also that the instant form’s, metric is ηµν = diag(1, 1, 1). In addition we point that we work in units where ~ = c =1. − − 22Alternatively one can ﬁnd a Lorentz frame where the transverse component x2 is zero.

41 cone energy-momentum vector p0 p+ 1 1 0 p0 p0 + p1 1 1 1 1 0 1 p p− = 1 1 0 p = p p . (3.3)     √     √   p2 7−→ p2 2 0− 0 1 p2 2 p−2           Having deﬁned spacetime, energy-momentum vectors and the metric tensor in light

µ front let us calculate the energy-momentum dispersion relation p pµ,

µ 2 0 2 1 2 + p p M =(p ) (p ) =2p p−, (3.4) µ ≡ − where M 2 is the invariant mass, and Casimir of Poincaré group. This relation deserves some attention. First note that the following is true

p0 = M 2 +(p1)2 p1, (3.5) ≥ p 0 1 and as a result p p 0, therefore p± 0. This result is not familiar to the ± ≥ ≥ equal-time formulation and it deserves some more explanation. This is according to Dirac [15] the spectral property and it has tremendous impact on the light cone vacuum. Since the vacuum has zero energy, the vacuum in the light come regime creates only massless particles, as opposed to the equal time formulation where the vacuum can create massive particles such that the sum of their energies adds up to zero. In other words, the light cone vacuum (with P + = 0) is an eigenstate of the full hamiltonian, which admits no radiative corrections.

The second observation has to do with the choice of energy among the momentum

+ components (p−,p ) on the light cone. Naturally we choose p− as the light cone energy, since it is the conjugate variable to the light cone time x+. This choice is

µ 0 0 1 1 + + + also supported by the fact that: p x = p x p x = p−x + p x−. Thus p is the µ − longitudinal light cone momentum. We can now introduce light cone derivatives. In

∂ ∂ particular, the spacelike derivative will be ∂ and the timelike one ∂+ + . − ≡ ∂x− ≡ ∂x 42 3.2 Supersymmetric Discrete Light Cone Quantization: A few details

We are interested in constructing numerically the bound state spectrum of two dimensional SYM theories. To this end we employ the method of Discretized Light

Cone Quantization where SYM is realized as a matrix model. This method resorts in a hamiltonian approach where the calculations are done in momentum representation and ﬁeld quantization is assumed on equal light cone times. DLCQ refers to

23 compactification of the longitudinal light cone spatial direction x− on a lightlike circle (x− x− +2L) of radius L/π. This is of course, equivalent to choosing the ∼ light cone box where x− is restricted in the finite interval [0, 2L) (or [ L, L), etc.). − Since we consider two dimensional SYM theories [19], in order not to violate supersymmetry the scalar Φ (e.g., gauge boson Aµ(x)) and spinor Ψ superpartners must obey the same boundary conditions. For the present work we impose periodic boundary conditions (b.c) on the fields

+ + + + Φ(x , x−)=Φ(x , x− +2L), Ψ(x , x−)=Ψ(x , x− +2L) (3.6)

As a consequence the single particle (light cone longitudinal) momentum becomes discrete24 and is given by π k+ = n, (3.7) n L with25 n =1, 2, 3,....

23 By compactifying the x− direction we violate Lorentz invariance, hence causality is violated too. The latter is restored of course after we extrapolate the results we obtain, to the continuum limit, L . However, P ± are still preserved so the states in the discrete theory are characterized → ∞ by additional quantum numbers corresponding to P ±.

+ − 24The plane wave expansion and the periodic b.c essentially tell us that: eik x ei2nπ = + − + eik x ei2Lk k+L = nπ. ⇒ 25Although the zero mode problem (n = 0) is of critical importance in studying topics such as the vacuum condensate or spontaneous symmetry breaking, we do not consider the study of this

43 3.2.1 Quantization rules on the Light Cone

Let us now impose quantization rules as these appear in the light cone prescription. As usual, following Dirac [61, 62], we write the canonical equal light cone time x+ = y+ commutation/anti-commutation relations for the bosonic Φˆ and the fermionic Ψˆ ﬁeld operators respectively

ˆ i 1 [Φij (x−), πˆΦ,kl (y−)] = δ(x− y−) δikδjl δijδkl , (3.8) 2 − − Nc ˆ i 1 Ψij(x−), πˆΨ,kl (y−) = δ(x− y−) δikδjl δijδkl , (3.9) { } 2 − − Nc ∂ where, π = L . We have also assumed that the ﬁeld operators are transforming Φ δ(∂+Φ)

26 under the adjoint representation of SU(Nc) color gauge group . Of course, the 1/Nc factor drops oﬀ once we consider the large-Nc limit; the case in our calculations.

Hereafter we drop factors proportional to 1/Nc.

The next step is to expand the ﬁelds in terms creation/annihilation operators in momentum representation. In light cone quantization, for a lightlike hypersurface with x+ =0, the ﬁelds have the following expansions [19,22],

+ 1 ∞ dk + + + ik x− + ik x− Φˆ ij(x−, 0) = aij(k )e− + a† (k )e , (3.10) + ji √2π 0 √2k Z 1 ∞ + + ˆ + + ik x− + ik x− Ψij(x−, 0) = dk bij(k )e− + bji† (k )e . (3.11) 2√π 0 Z The canonical (anti-)commutation relations (3.9) are then satisﬁed by

+ + + + + + [a (k ), a† (k˜ )] = b (k ), b† (k˜ ) = δ(k k˜ )δ δ . (3.12) ij lk { ij lk } − il jk problem in this thesis since our primary concern is the calculation of the bound states. With regards to our theory the elimination of the longitudinal spacelike component of the gauge ﬁeld via its constraint equation introduces an infrared diverging factor 1/∂ (as k+ 0). We regularize this term by simply dropping the zero mode. For more discussion concerning− → this issue see [19,20]. 26 Here and throughout this document the normalization of the SU(Nc) generators is taken to be 2 Nc 1 C Tr(TaTb)= Cδab with corresponding completeness relation − (Ta)ij (Ta)kl = Cδikδjl δij δkl, a=1 Nc with C =1. − P 44 Recall at this point that the operators we deﬁned above, which represent adjoint

ﬁelds, are complex valued objects and hermiticity of the SU(Nc) generators implies ˆ ˆ ˆ ˆ that Φij† = Φji; same for the fermion operator. Therefore, the reality of Φij† and Ψij† imply that hermitian conjugation on creation/annihilation operators is just equivalent to complex conjugation.

Now in order to employ DLCQ we replace the integral with a sum over the modes n =1, 2, 3,... from Eq. (3.7)

∞ π ∞ dk+ , (3.13) → L 0 n=1 Z X where it is understood that the values of k+ are given by Eq. (3.7) and that δ-

+ + L functions δ(k k˜ ) are replaced by Kronecker deltas, δ ˜. Therefore the discrete − π kk forms of Eq. (3.10) and Eq. (3.11) are, respectively

∞ 1 1 iπnx /L iπnx /L Φˆ = A (n)e− − + A† (n)e − , (3.14) ij √ √n ij ji 4π n=1 X ∞ 1 iπnx /L iπnx /L Ψˆ = B (n)e− − + B† (n)e − , (3.15) ij √ ij ji 4L n=1 X with π π A (n)= a (k+), B (n)= b (k+). (3.16) ij L ij ij L ij r r Then the discrete version of formula (3.12) will become

[A (n), A† (n0)] = B (n), B† (n0) = δ δ δ , (3.17) ij lk { ij lk } nn0 il jk where it is understood that we have used the discrete representation of delta function.

Finally one last thing we want to mention is the gauge ﬁxing. In particular, for the SYM model that we consider for our calculations we simply set In light cone formalism we set the timelike component of the gauge boson Φ A+ equal to ij → ij 45 zero. For a nice account on this matter consider the corresponding Appendix of reference [1].

3.3 Formulation of the supersymmetric bound state problem

µ One’s wish is to be able to solve analytically the inﬁnite matrix problem, P Pµ =

M 2, in order to obtain the bound state spectrum of a ﬁeld theory. Let us formulate the problem in terms of two dimensional theories. In the context of light cone quantization the mass eigenvalue problem reads as follows

+ 2 2P P − Φ = M Φ . (3.18) | i | i

Although there are approximate methods, from numerical analysis, to attack special inﬁnite matrices, unfortunately this is a “no-go” problem for the case for quantum

ﬁeld theories. To this end, we restrict ourselves in studying this problem by utilizing

(S)DLCQ technique, which reduces the problem of inﬁnite matrix diagonalization to a ﬁnite matrix eigenvalue problem.

In particular, for the two dimensional supersymmetric models [19,22,63,64] that we focus our attention in this work, we note that the = (1, 1) superalgebra relates N + light cone (total) momentum P and hamiltonian P − through the following relations

Q±, Q± =2√2P ±, (3.19) { }

+ Q , Q− =0. (3.20) { }

Concerning supersymmetric theories, it is in general a diﬃcult task to diagonalize directly the hamiltonian P −. However, superalgebra dictates that

2 (Q−) P − = , (3.21) √2 46 2 and as it turns out diagonalizing (Q−) is the easiest way to resolve the spectrum of a supersymmetric theory. This is in fact the crux of supersymmetric discretized light cone quantization. So the problem now reduces to the calculation of the numerical entries of matrix Q−. We emphasize once more that SDLCQ is more convenient in our case than traditional DLCQ because it preserves supersymmetry and it is numerically faster, a necessity when we are dealing with very large matrices.

Since we decided that the light cone hamiltonian is going to be expressed in terms of the supercharge Q− let us choose a convenient Fock basis. By the fact that

+ [P −,P ]=0 ( of Poincaré algebra) then a state Φ is a simultaneous eigenvector ⊂ | i for both operators. In addition we know that the total light cone momentum is a kinematic charge, i.e., it does not involve any interactions, thus it can be easily diagonalized as opposed to the hamiltonian P −. Recall that

∞ ∞ π π P + = k+ = n K, (3.22) ni L i ≡ L i=1 i=1 X X where K = 1, 2, 3,..., is the harmonic resolution. Therefore it is convenient to deﬁne Fock states that carry a deﬁnite value of momentum P + (or K). That is,

P + Φ = π K Φ , is already diagonal in this basis. Moreover, by considering the | i L | i large-Nc limit of SU(Nc) gauge (color) group the physical states, bosonic or fermionic, are strictly color singlets. In other words, they are represented by a single trace over the color indices [19,22]

1 m Φ ; K = Tr ˆ†(n ) ˆ†(n ) 0 ; K = n , (3.23) | m i m/2 O 1 ··· O m | i i Nc √s i=1 h i X where ˆ†(n ) can be either A†(n ) or B†(n ) with unit of momentum n . Note O m m m m that 0 is the vacuum of the full theory and it is annihilated by the oscillators A or | i ij m/2 Bij and also the normalization factors Nc ( Nc is the number of colors) and √s (s

47 is the symmetry factor), are chosen such that the states are orthonormal [19,22]. An important consequence from using SU(Nc) gauge group is that the minimum number of partons or oscillators allowed in Φ is two. On the other hand the maximum | mi number of partons allowed in a state is equal to the harmonic resolution K. This last fact is a consequence of integer composition theory. It is clear that as far as the problem considers a ﬁnite value of harmonic resolution the Fock space of states is ﬁnite dimensional and we numerically solve for the eigenvalues. To sum up, after combining equations (3.18), (3.21) and (3.22), our task is to compute the eigenvalues of the following equation

√2π 2 2 K(Q−) Φ = M Φ , (3.24) L | i | i for a particular value of the harmonic resolution K. Of course, K corresponds →∞ to the continuum, thus the spectrum of the full (physical) field theory will eventually be realized in this limit. Nonetheless, the eigen-solutions from the finite matrix problem, it is assumed that can describe quite well the low lying spectrum of the quantum field theory. What one typically does, since it is almost “forbidden”, due to computational constraints to reach resolution K , is to study the spectrum as → ∞ a function of 1/K and then extrapolate the results to the continuum.

3.3.1 Matrix structure of supercharges

We will now try to construct the generic form of the matrix problem that we have to solve. The formulation of the matrix structure in inspired by the discussion in Chapter 2, especially Section 2.1.3. Recall at this point that the supersymmetry generators, by deﬁnition change the type of particle, that is Q± acting on a bosonic state will transform it into a fermionic and vice versa. From this fact we infer that

48 the matrix form of the supercharges is given by the following equations

i; b i; f | i | i i; b | i 0 M † (3.25) Q± = ± , i; f  M 0  | i ±   where the i fermionic and i bosonic blocks are denoted by i; f and i; b , respectively. | i | i To conﬁrm the above matrix structure of the supercharge, which stems from the action of Q± on boson or fermion states, let us consider Q± acting on the following column of (supersymmetric) states

0 M † i; b M † i; f ± | i = ±| i , (3.26)  M 0 · i; f   M i; b  ± | i ±| i      where the bosonic states occupy the ﬁrst half of the rows. From this form we verify that the matrix M (M † ) interchange bosons (fermions) into fermions (bosons), so ± ± we generically write

M † i; f = i; b (3.27) ±| i | i M i; b = i; f , (3.28) ±| i | i thus the choice of Eq. (3.25) is justiﬁed.

As we stated earlier, the quantity of main interest in SDLCQ is the Q− supercharge where the diagonalization of its squared will yield the supersymmetric bound state spectrum. Using the particular matrix form of Q− from Eq. (3.25), the bound state matrix problem (3.24) reduces to the following block diagonal form,

√2πK M † M 0 M 2 = − − , (3.29) L  0 M M †  − −   which according to our convention the bosonic spectrum is given by the block M † M − − and the fermionic by M M † . At this point we should emphasize the resemblance of − − 49 this relation, Eq. (3.29), and the hamiltonian (2.35) we explored in the context of quantum mechanics in Section 2.1.3. Due to supersymmetry it suffices to diagonalize either M † M , or ˜ M M † in order to realize the spectrum. More precisely, M ≡ − − M ≡ − − assuming that the hermitian positive definite matrices , ˜ are diagonalizable, M M then there exists a unitary transformation U 0 = (3.30) V 0 V 2 1 2 such that (Q−) − (Q−) , yields D ≡ V V 1 U − U 0 0 M = D , (3.31)  1    0 V − ˜ V 0 M D where is again positive definite, as it should be! Now let us discuss a few noteworthy D comments about the properties of the matrices M . From superalgebra (3.19) and ± particularly from the anticommutator that is satisfied by Q+, we deduce that

˜ ˜ M+M+† = ½, (3.32)

1 K is unitary, with M˜ 2 4 π M . Farther, by using Eq. (3.20) we obtain + ≡ L + q ˜ ˜ M † = M+† M M+† , (3.33) − − − or equivalently ˜ ˜ = M+† M M+† M . (3.34) M − − − We are about to ﬁnd the transformation that relates the unitary matrices U and

V, namely the bosonic and fermionic subspaces. Let us substitute either Eq. (3.33), or (3.34) into (3.31)

1 1 ˜ ˜ U − U = U − M+† M M+† M U M − − − 1 ˜ ˜ = U − M+† M M † M+U − − 1 =(M˜ U)− ˜ (M˜ U), (3.35) + M + 50 which suggests that the matrix M˜+ is the transformation that maps bosonic eigenstates onto fermionic ones,

M˜ U = V, M˜ i; b0 = i; f 0 . (3.36) + +| i | i

Therefore, the supercharge Q+ is responsible for relating bosonic/fermionic states with the same masses.

Finally before we continue, to provide an example, let us summarize the characteristic property of SYM theory spectrum stemming from the superalgebra. Namely,

2 if the state Φ corresponding to invariant mass M is an eigenstate of P −, then Φ | i | i + 2 is also eigenstate of operators Q± and Q Q− with eigenvalue M as well. This is indeed a consequence of the fact that

+ [P −, Q±] = [P −, Q Q−]=0. (3.37)

3.3.2 An example: pure super Yang-Mills theory meets SDLCQ

We have already mentioned that in SDLCQ approximation supersymmetry is manifest, thus the bound state spectrum that we obtain is exactly supersymmetric with equal number of bosonic and fermionic color singlet states. To help motivate the matrix problem let us consider an example for harmonic resolution K =3, a case that can be solved analytically. Simple combinatorics (i.e., composition of integers) and the various restrictions on the bosonic/fermionic operators (i.e., fermion statistics) as well as the trace cyclic property imply that the Fock basis is eight dimensional; four states are fermionic and four states are bosonic. Namely, the four fermionic

51 color singlet states (see Eq. (3.23)) i; f are {| i} 1 Tr 1; f = 3/2 A†(1)A†(1)B†(1) 0 , (3.38a) | i Nc | i 1 Tr 2; f = 3/2 B†(1)B†(1)B†(1) 0 , (3.38b) | i Nc √3 | i 1 3; f = Tr A†(2)B†(1) 0 , (3.38c) | i Nc | i 1 Tr 4; f = 1/2 B†(2)A†(1) 0 , (3.38d) | i Nc | i and the four states for the bosonic color singlet states i; b {| i} 1 Tr 1; b = 3/2 A†(1)A†(1)A†(1) 0 , (3.39a) | i Nc √3 | i 1 Tr 2; b = 3/2 A†(1)B†(1)B†(1) 0 , (3.39b) | i Nc | i 1 3; b = Tr A†(2)A†(1) 0 , (3.39c) | i Nc | i 1 Tr 4; b = 1/2 B†(2)B†(1) 0 . (3.39d) | i Nc | i Having found the basis states we can now calculate the matrix form of the supercharge Q−. For the sake of simplicity consider a term that appears in the pure SYM

27 supercharge (see Eq. (4.6)), Q−,

q˜− Tr B†(1)B†(1)B(2) Q−. (3.40) ≡ ⊂ To obtain matrix elements, in general, we act with q˜ on the basis states. Take for instance q˜− 4 | ib

q˜− 4 = B† B† B B† B† 0 | ib 1,ij 1,jk 2,ki 2,mn 1,nm| i

= B† B† B† 0 1,ij 1,jk 1,ki| i = 2 , (3.41) | if 27 1 g√L 2 The dimensions L− in this term is set by an overall factor i 21/4π . Here we simply ignore Q−+ this factor. For a detailed derivation of the supercharge, Q = Q− , consider the Appendix C.2 and [19, 20]. 52 where we have dropped state normalization factors and also in the last step we have used the anti-commutator from equation (3.17). Notice the striking fact, that our supercharge operator q˜ when acted upon a bosonic state it transformed it into a fermionic one. Thus, the essence of super-transformation in our example is explicitly satisfied! Moreover, we point out the fact that Q− changes the fermion number as well as the total parton number in a given trace (state), from even to odd and vice versa. Following the same procedure one can construct M (M † ,) the matrix − − elements of which are obtained by calculating i; f Q− j; b ( j; b Q− i; f ). For the h | | i h | | i particular example at resolution K =3, one obtains [19] the following expression for the 4 4 matrix M × − 0 0 00 9  0 0 0  −2 g√NcL M = i  3  . (3.42) − − 1/4 √   2 π 3  0 3 0 0   − r2   3   √   0 3 0 0   2    Diagonalization of M † M (or M M † ) yields the supersymmetric mass spectrum in − − − − units of g Nc/π. Namely, we find two massless states and two massive states with p √ 9 masses M = 18 g Nc/π and M = 2 g Nc/π. Moreover, using the map defined by equation (3.36)p we can find the supersymmetricp eigenstates that correspond to the above masses. As an example consider the following term (see equation (4.5) and [19])

+ + q˜ Tr B†(1)A(1) Q (3.43) ≡ ⊂ acting on the massless state 1; b . Then it turns out that the fermionic state with | i zero mass is just 1; f . One can proceed similarly and ﬁnd the rest of the pairs. | i

53 3.4 Statistical Mechanics on the Light Cone

Undoubtedly light cone quantization has some advantages concerning practical calculations in high energy physics. Apart from the fact that the light cone ground state coincides with the ground state of the full theory there is also another important fact concerning boost invariance [1]. This elegant feature is a characteristic of light cone quantization, which immediately makes it a natural laboratory for the study of systems where boost invariance is of major concern, such as the ﬁreballs created from heavy ion collisions experiments. Therefore the study of thermodynamic functions and properties can in principle be studied on light cone in a straightforward manner. Especially when one is able of calculating the invariant mass spectrum with the help of DLCQ technique. Since our primary concern in this dissertation is the study of supersymmetric theories at ﬁnite temperatures, we would like to discuss the formulation of statistical mechanics in the light cone framework.

In the conventional, equal-time, realization of statistical ﬁeld theory one chooses a system where the heat bath is at rest and writes the density matrix describing, say a canonical ensemble, as follows28

βP 0 ρ(β) = e− , (3.44) where P 0 is the hamiltonian of the system under consideration. Then one may be tempted to write the density matrix in light cone parametrization in analogy to the above deﬁnition of instant form

βLC P − ρ(βLC ) = e− . (3.45)

28Note that β =1/T in our units.

54 However, the last equation leads to singular results [65–69] and must be modiﬁed properly. In fact we know [1, 15] that the instant form and light front cannot be mapped onto each other by a Lorentz transformation, thus we cannot set P + = 0; no heat bath at rest on light cone!

One way to express the density matrix in light cone coordinates is the following

+ βLC P −+µP ρ(βLC ) = e− , (3.46) where P + is the total longitudinal light cone momentum of the thermal ensemble and it is one of the conserved (Poincaré group) charges on the light cone29. Thus one may introduce it through a chemical potential µ [63, 64]. This way since the partition function, which is a trace over the density matrix, is a Lorentz scalar we have

β P +µP + βP 0 = Tr[e− LC − ] Tr[e− ]. (3.47) Z ≡

0 + Hence by using the fact that √2P = P + P −, we obtain

β βLC = , µ =1. √2 An immediate interpretation of the chemical potential in this case is the following.

For µ =1 this corresponds to a rotation of the quantization axis from light cone to instant form where the heat bath is at rest. Moreover for µ = 1 the quantization 6 takes place in a boosted frame.

An equivalent approach is discussed in [70]. In this consideration we recall the quantum Liouville theorem regarding the equilibrium state of a system, which is expressed by

∂+ρ(β)=i[ρ(β),P −]. (3.48)

29Note that P + is a kinematic generator of the Poincaré group, thus not aﬀected by interactions in the theory.

55 Thermal equilibrium is reached when the commutator on the right-hand side of the above equation is zero. So the vanishing of the commutator implies that ρ is a function, f( ), of operators that can only commute with P −. Moreover, we know O from statistical mechanics in thermal equilibrium that f( ) is a linear combination O of the conserved charges. As a matter of fact in our case this requirement is fulﬁlled

+ by P − and P together. In particular, in light cone coordinates way may write,

ν + + β(ˆvν P ) β(ˆv P +ˆv P ) ρ(β) = e− = e− − − , (3.49)

where vˆν is the unit velocity vector. Thus a comparison with Eq. (3.47) reveals that

ν + 1 vˆ =(v , v−)= (1, 1). √2

Therefore, heat bath at rest (vˆν = (1, 0)) in equal-time quantization corresponds to vˆν = 1 (1, 1) √2 in light cone quantization. However, in this particular study we use (S)DLCQ as a mean which we can calculate the invariant mass spectrum of the theory in hand. The use of Eq. (3.44) is legitimate because the role of (S)DLCQ is ﬁnished by the time we have the (invariant) bound state spectrum. This same route was used by the authors of [71], where they studied the two dimensional QED. In our calculations of thermal properties regarding

SYM theories we follow this approach.

3.5 Discussion

We have concisely reviewed various aspects of light cone quantization that are important for this dissertation. In particular, we have introduced the basic facts regarding the formulation of SDLCQ, which is a supersymmetric generalization of

DLCQ. The diﬀerence between the two approaches lies in the fact that SDLCQ

56 approach uses the square of the supercharge, while in the traditional DLCQ approximation the light cone hamiltonian is being employed. In the former the formulation preserves supersymmetry, while the latter breaks it explicitly.

The main objective of (S)DLCQ is to solve the bound state eigenvalue problem, that is to extract numerically the wavefunctions and mass spectrum of a quantum theory starting from ﬁrst principles. Realization of the full mass spectrum will be essential for the thermodynamic calculations we set out to explore in the following chapters. Of course the use of SDLCQ is not limited to these certain quantities that are of our interest in this work. At this point we name a few other physical quantities that SDLCQ can calculate at almost no expense, such as structure functions, correlation functions and parton number distributions [27,72,73]

An important advantage of using (S)DLCQ is that because the light come momenta are positive definite the total momentum of the partons in a state is fixed by the harmonic resolution. This is in fact the deep reason why we are able to construct a finite matrix eigenvalue problem, which can in principle be solved numerically. An- other important advantage that is being used directly in our calculations, but has not been mentioned above, is that in light cone formulation the non-dynamical degrees of freedom are removed from the theory through their constraint equations of motion, thus one avoids problems of having ghost fields. This is of particular significance when dealing with supersymmetric theories.

We restrict our discussion to two dimensional SYM models, which are the products of higher dimensional theories dimensionally reduced to 1+1 spacetime dimensions. Moreover, the models we set to explore with SDLCQ are taken in the large-Nc limit. Generalization of SDLCQ approximation to higher dimensions is feasible e.g.,

57 by discretizing dimensions transverse to light cone axes [73]. Of course by discretizing transverse momenta on a spatial circle or box is totally diﬀerent from the discrete light cone momentum, namely the transverse momentum modes can take negative values.

Finally the light cone formulation of statistical mechanics is discussed. Although we do not calculate thermodynamical functions and properties of supersymmetric spectra directly through the light cone regime, this overview might be useful for future practical purposes.

58 CHAPTER 4

SYM-THERMODYNAMIC PROPERTIES I

Η ποίησις είναι ανάπτυξις στίλβοντος ποδηλάτου. –Α. Εμπειρίκος

The free energy of = 4 super Yang–Mills (SYM) theory at a large number N of colors Nc is larger at strong coupling by a factor 4/3 compared to weak coupling [74, 75]. The weak-coupling result is calculable in perturbation theory, while the strong-coupling result can be derived from black-hole thermodynamics. It would be interesting to be able to directly solve this theory at all couplings and see the transition between the weak-coupling and strong-coupling limits [75–77]. In the light of this finding, one can ask if other SYM theories exhibit a similar behavior. An analytic calculation of both the strong and the weak coupling limit of a field theory is generally not possible, although there have been a number of proposals for methods to obtain solutions of finite-temperature supersymmetric quantum field theories [78]. We will instead resort in a numerical method based on Supersymmetric Discrete Light-Cone

Quantization (SDLCQ) [19, 20], which preserves the supersymmetry exactly as we already mentioned in Chapter 3. Currently this is the only method available for numerically solving strongly coupled super Yang–Mills theories. Conventional lattice methods have diﬃculty with supersymmetric theories because of the asymmetric way

59 that fermions and bosons are treated (i.e., fermion doubling), and progress [79–81] in supersymmetric lattice gauge theory has been relatively slow.

Given that SDLCQ makes use of light-cone coordinates, with x+ =(x0 + x3)/√2

0 3 the time variable and p− = (p p )/√2 the energy, we must take some care in − deﬁning thermodynamic quantities if we want to work directly in the light cone

β p regime. The seemingly natural choice [82] of e− LC − as the partition function has been shown by Alves and Das [65, 66, 83] to lead to singular results for well known

β p quantities that are ﬁnite in the equal-time approach. They argue that using e− LC − for the partition function implies that the physical system is in contact with a heat bath that has been boosted to the light-cone frame and that this is not equivalent to the physics of a system in contact with a heat bath at rest.

A more direct way to see this is that, since the light-cone momentum p+ =

(p0 + p3)/√2 is conserved, the partition function must include it in the form = Z β (p +µp+) e− LC − , with µ its chemical potential. The interpretation of the chemical potential is that of a rotation of the quantization axis. Thus µ = 1 corresponds to quantization in an equal-time frame, where the heat bath is at rest and the inverse temperature β = √2β , and µ =1 corresponds to quantization in a boosted frame LC 6 where the heat bath is not at rest. Thus µ corresponds to a continuous rotation of the axis of quantization, and µ = 0 would correspond to rotation all the way to the light-cone frame. We emphasize that a rotation from an equal-time frame to the light-cone frame is not a Lorentz transformation. It is known that such a transformation can give rise to singular results for physical quantities. This appears to be consistent with the results found in [83]. A number of related issues have

60 been extensively discussed by Weldon [67,68,84]. The method has also been applied successfully to the Nambu–Jona-Lasinio model [85].

However, we can escape these diﬃculties if we compute the equal-time partition

βp0 function = Tr(e− ), as proposed much earlier by Elser and Kalloniatis [71]. The Z computation may, of course, still use light-cone coordinates. Elser and Kalloniatis did this with ordinary DLCQ [1,17] as the numerical approximation to (1+1)-dimensional quantum electrodynamics. Here, as we already mentioned we will be employing a similar approach using SDLCQ to calculate the spectrum of = (1, 1) super Yang– N Mills theory in 1+1 dimensions [18, 86]. Though this calculation is done in 1+1 dimensions, it is known that SDLCQ can be extended in a straightforward manner to higher dimensions [72,73,87].

We have overviewed the SDLCQ numerical method in the previous chapter, which has been also discussed in a number of other places, hence we will not present a detailed discussion of the method here; for a nice review, consider [20]. We just remind, those familiar with DLCQ [1, 17], it suffices to say that SDLCQ is similar, since both have discrete momenta and cutoffs in momentum space, x− [ L, L]. ∈ − In 1+1 dimensions the discretization is specified by a single integer K = (L/π)P +, the resolution, such that longitudinal momentum fractions are integer multiples of

1/K. However, the main advantage of SDLCQ over DLCQ, is that it is formulated in such a way that the theory keeps supersymmetry intact. Exact supersymmetry brings a number of very important numerical advantages to the method; in particular, theories with enough supersymmetry are ﬁnite. We have also seen greatly improved numerical convergence in this approach.

61 In Section 4.1 we briefly review super Yang–Mills theory in 1+1 dimensions. The discussion in Section 4.2 describes the method we use to extract the density of states from the numerical spectrum. The calculation of the density of states is presented in Section 4.3. We fit the data to smooth analytical functions, and we find that the theory has a Hagedorn temperature TH [88], which we calculate. In Section 4.4, we use the analytic fit to the density of states to calculate the free energy, the energy, and the specific heat, up to the Hagedorn temperature. Section 4.5 contains a discussion of our results and the prospects for future work using these methods.

4.1 The pure super Yang–Mills model

We will start by providing a brief review of = (1, 1) supersymmetric Yang–Mills N in 1+1 dimensions. The lagrangian of this theory is

1 = Tr F F µν + iΨ¯ γ DµΨ , (4.1) L −4 µν µ where for more details regarding super Yang-Mills in several spacetime dimensions the reader is referred to review [89]. The two components of the real spinor Ψ =

1/4 ψ 2− χ are in the adjoint representation, and we will work in the large-Nc limit. The field strength and the covariant derivative are F = ∂ A ∂ A + ig[A , A ] µν µ ν − ν µ µ ν and Dµ = ∂µ + ig[Aµ, ]. The most straightforward way to formulate the theory in 1+1 dimensions is to start with the theory in 2+1 dimensions and then simply dimensionally reduce to 1+1 dimensions by setting φ = A and ∂ 0 for all fields. 2 2 → In the light-cone gauge, A+ =0, we find that the supercharges are reduced to

+ 1/4 Q =2 dx− Tr (φ∂ ψ ψ∂ φ) , (4.2) − − − Z 3/4 1 1 Q− =2 g dx−Tr i[φ, ∂ φ] ψ +2ψψ ψ . (4.3) − ∂ ∂ Z − − 62 The mode expansions in two dimensions are

+ 1 ∞ dk + + + ik x− + ik x− φij(0, x−) = aij(k )e− + a† (k )e , + ji √2π 0 √2k Z h i 1 ∞ + + + + ik x− + ik x− ψij(0, x−) = dk bij(k )e− + bji† (k )e . (4.4) 2√π 0 Z h i Then as a result of using the above expansions and the conventions introduced in the previous chapter, the normal ordered discretized versions of supersymmetry generators can be found in a straightforward manner. That is for Eq. (4.2) we obtain,

∞ + + 1/4 π Q =(Q )† =2 i √n B† (n)A (n) A† (n)B (n) , (4.5) L ij ij − ij ij r n=1 X h i whilst for the hamiltonian generating on, i.e., Eq. (4.3), we have

1/4 2− ig√L ∞ n1 +2n2 Q− =(Q−)† = A† (n )B† (n )A (n + n ) − π im 2 mj 1 ij 1 2 n ,n =1 2n1 n2(n1 + n2) 1X2 B† (n )A† (n )A p(n + n ) A† (n + n )A (n )B (n ) − im 1 mj 2 ij 1 2 − ij 1 2 im 2 mj 1

+ Aij† (n1 + n2)Bim(n1)Amj(n2) n1 1 ∞ − n1 2n2 + − B† (n )A (n )A (n n ) ij 1 im 2 mj 1 − 2 n =3 n =1 2n1 n2(n1 n2) X1 X2 − p A† (n )A† (n n )B (n ) − im 2 mj 1 − 2 ij 1 ∞ 1 1 + B† (n )B† (n )B (n + n )+ B† (n + n )B (n )B (n ) − n n im 2 mj 1 ij 1 2 ij 1 2 im 2 mj 1 n ,n =1 1 2 1X2 n1 1 ∞ − 1 + B† (n )B (n )B (n n ) n ij 1 im 2 mj 1 − 2 n =2 n =1 1 X1 X2

+ B† (n )B† (n n )B (n ) . (4.6) im 2 mj 1 − 2 ij 1 ! To obtain the spectrum, we solve the mass eigenvalue problem

+ + 2 2 2P P − ϕ = √2P (Q−) ϕ = M ϕ . (4.7) | i | i | i

63 This theory has two discrete symmetries, besides supersymmetry, that we use to reduce the size of the hamiltonian matrix we have to calculate. S-symmetry, which is associated with the orientation of the large-Nc string of partons in a state [21], gives a sign when the color indices are permuted

S : A (n) A (n), B (n) B (n). (4.8) ij → − ji ij → − ji

P -symmetry is what remains of parity in the x2 direction after dimensional reduction

P : A (n) A (n), B (k) B (k). (4.9) ij → − ij ij → ij

Thus, all of our states can be labeled by the P and S sector in which they appear.

+ 2 We convert the mass eigenvalue problem 2P P − M = M M to a matrix eigen- | i | i value problem by introducing a discrete basis where P + is diagonal. We will always

2 2 state M in units of g Nc/π. In SDLCQ the discrete basis is introduced by ﬁrst discretizing the supercharge Q− and then constructing P − from the square of the su-

2 percharge: P − =(Q−) /√2. To discretize the theory, we impose periodic boundary conditions on the boson and fermion fields alike and obtain an expansion of the fields with discrete momentum modes. We define the discrete longitudinal momenta k+ as fractions nP +/K of the total longitudinal momentum P +, where K is an integer that determines the resolution of the discretization and is known in DLCQ as the harmonic resolution [17]. Because light-cone longitudinal momenta are always positive, K and each n are positive integers; the number of constituents is then bounded by K. The continuum limit is recovered by taking the limit K . →∞ In constructing the discrete approximation, we drop the longitudinal zero-momentum mode. For some discussion of dynamical and constrained zero modes, see the review [1] and previous work [90]. Inclusion of these modes would be ideal, but the

64 techniques required to include them in a numerical calculation have proved to be dif-

ﬁcult to develop, particularly because of nonlinearities. For DLCQ calculations that can be compared with exact solutions, the exclusion of zero modes does not aﬀect the massive spectrum [1]. In scalar theories it has been known for some time that constrained zero modes can give rise to dynamical symmetry breaking [1], and work continues on the role of zero modes and near zero modes in these theories [91–96].

4.2 Density of states

The thermodynamic functions will be written as sums over the spectrum M { n} of the theory. The most convenient way to calculate such sums is to represent each sum as an integral over a density of states ρ(M 2),

∞ ρ(M 2)dM 2. (4.10) → n=1 X Z From our numerical solutions we can approximate the density of states by a continuous function. The remaining integrals in the thermodynamic functions are then done by standard numerical integration techniques, which are fast and convenient.

We can look at the density for a series of increasing resolutions K in the SDLCQ approximation and thereby discuss the convergence of the density in the limit K → . The maximum mass that we can reach in the SDLCQ approximation increases ∞ as we increase the resolution. We report results for 11 K 17. ≤ ≤ Convenient functions to extract from the spectral data [97] are the cumulative distribution function (CDF) N(M 2,K) and the normalized cumulative distribution

2 2 function (NCDF) f(M ,K,Mr ). The CDF is the number of massive states at or below M 2 at resolution K, and the NCDF is this number divided by the total number

65 2 of massive states below an arbitrary normalization point Mr , again at resolution K:

2 2 2 N(M ,K) f(M ,K,Mr )= 2 . (4.11) N(Mr ,K)

The function f turns out to be very smooth and can be ﬁt by a single smooth analytic form. By deﬁnition, the density of states is given by

dN (M 2,K) ρ(M 2,K)= . (4.12) dM 2

It is also convenient to deﬁne a normalized density of states [97]

df (M 2,K,M 2) ρ˜(M 2,K,M 2)= r . (4.13) r dM 2

It is well known that if the density of states grows exponentially with the mass of the state,

ρ(M 2) exp(M/T ), (4.14) ∼ H the theory will have a Hagedorn temperature, TH [88] that regulates the exponential

30 growth of the spectrum . Above the temperature TH , the thermodynamic integrals diverge.

4.3 Numerical results for the density of states

The numerical results presented in this section are the ﬁrst from our new code, which was rewritten to run on clusters. Most of these results were produced on our six-processor development cluster. While this cluster was suﬃcient for this problem, we expect to be able to handle larger problems by moving to larger clusters. In

30Note that if ones wants to check for possible phase transitions due to the exponential growth in the spectrum then it is suggestive to use the more general relation

ρ(M 2) M αexp(M/T ). ∼ H

66 fact, it is now so easy to generate the hamiltonian up to resolution K = 17, that we only used one node in our cluster for that purpose. What made this calculation challenging numerically was that we needed to extract a large number of eigenvalues.

For the largest values of the resolution, this was done with a specially tuned Lanczos diagonalization code based on the techniques of Cullum and Willoughby [98,99]. We should note that, prior to the development of the new matrix-generation code, the highest resolution results presented for this theory were for resolution K = 10 [18,86].

Here we will present only resolutions K 11. All our earlier results are now trivially ≥ reproduced.

(a) (b)

Figure 4.1: The distribution functions (a) N(M 2,K) for K = 12 and (b) 2 2 2 f(M , K, 100) for K = 12, 14, and 17, as functions of M in units of g Nc/π. In (a) we also show the error-function ﬁt a erf(b(x c)) + d, and in (b), a polynomial ﬁt. −

In the SDLCQ approximation, the portion of the spectrum that we can see at any

finite resolution will naturally be cut off. As we approach the cutoff, the approximation limits the number of states that are available and distorts the density of states.

In Fig. 4.1a, we present the data for the CDF at resolution K = 12, where we can

67 diagonalize the entire hamiltonian. We clearly see evidence of this distortion. By the midpoint, the cutoff is already diminishing the number of states available in the approximation. Interestingly, we can find a fit to this data with a simple universal function of the form a erf(b(x c)) + d. Clearly the fit shown is excellent; the fit is − so good that one cannot separately see the data and the fitted curve on this scale.

At low masses, there is a mass gap, which has been discussed elsewhere [20]. The mass gap closes linearly with 1/K. For very small values of M 2, the decrease of the mass gap with increasing resolution adversely aﬀects the quality of the universal

fit, and it is convenient in that region to improve the quality of the fit by using a polynomial function of the form p(K) f(x, K, 100) = α xδK +pΘ(x x (K)). (4.15) p − min p=0 X In Fig. 4.1b we show the fit to the NCDF for some representative values of the resolution K. We have only shown the data at resolutions K = 12, 14, and 17 to keep the figure uncluttered. One can see how the endpoints tend to lower values as the mass gap closes with increasing K.

At larger values of K it is difficult to completely diagonalize the entire hamiltonian. We have limited ourselves to states with M 2 100. However, once we know the ≤ universal form of the function that fits the CDF, we can fit just the region M 2 100 ≤ and extrapolate to all masses. At large M 2, the CDF approaches the total number of bound states. The total number of states in the SDLCQ approximation at any resolution, and, in any symmetry sector in the large-Nc approximation, is exactly calculable; see for instance Section 5.2.1. We use this asymptotic value of the CDF, in addition to the behavior for M 2 100, in making the fits. We have done this at ≤ all resolutions up to K = 17. In Fig. 4.2a we show the normalized density of states

68 (a) (b)

Figure 4.2: The normalized density of states (a) for M 2 100 and (b) extrapolated ≤ to all masses. The data points are a convenient way of displaying the continuous functions calculated from the ﬁts to the CDF.

calculated from the NCDF, for M 2 100. In Fig. 4.2b we show the normalized ≤ density of states for 11 K 17, extrapolated to the full range of masses. ≤ ≤ Inspecting these curves, we see that on the up-slope part of the density of states, where we believe our numerical approximation is a valid approximation to the actual density of states, the shape appears exponential. As we go to larger and larger values of the resolution K, the size of this region grows. This suggests that the density of states ultimately becomes simply an exponential, and, therefore, this theory has a

Hagedorn temperature. To ﬁnd the Hagedorn temperature, we ﬁt the NCDF in this up-slope region with a function of the form

M f(M 2, K, 100) = α exp α. (4.16) T − H

In Fig. 4.3b we plot TH against 1/K. This yields a good linear fit, which we extrapolate to infinite resolution. We find that the Hagedorn temperature at infinite

2 resolution is slightly less than one in units of g Nc/π. This value serves as a limiting temperature for the region of validity in the calculationp of thermodynamic quantities.

69 (a) (b)

Figure 4.3: The Hagedorn temperature TH , as obtained from exponential fits, α exp M α, to the universal fits to N(M 2,K), as shown in (a) for K = 17, TH − and (b) a linear plot against 1/K for fits in the range 11 K 17. ≤ ≤

We study thermodynamics below this temperature limit and borrowing terminology from QCD, we thus consider thermodynamics of the “hadronic gas” phase.

4.4 Finite temperature in 1+1 dimensions

In the large-Nc approximation, the numerical solution of a theory is a set of non- interacting bound states. Therefore, the thermodynamics of such supersymmetric theories is simply the thermodynamics of a free gas of a large number of species of degenerate bosons and fermions. In principle, one could go beyond the calculation of the standard set of the thermodynamic functions and calculate a variety of matrix elements. These calculations would require the wave functions of the bound states, which can be calculated as part of the SDLCQ calculation. We will, however, not exploit this detailed information here. We will focus on the calculation of standard thermodynamic quantities that can be obtained from the density of states. The light

70 cone plays no role beyond the calculation of the density; the thermodynamics is that of a system at rest.

Let us now brieﬂy review the thermodynamics of free bosons and fermions; for a more general derivation refer to Appendix C.3. We assume that our system has constant volume V and is in contact with a heat bath of constant temperature T .

31 The free energy in units with kB =1 is

(T,V )= T ln . (4.17) F − Z

The contribution of a single bosonic oscillator to the free energy FB is

∞ dp3 p0/T F =2V T ln 1 e− , (4.18) B 2π − Z0 2 2 where p0 = M + p3 and the factor of 2 compensates for integrating over only p positive values of p3. It is convenient to change variables from p3 to p0:

p0 dp3 = dp0. (4.19) p2 M 2 0 − The limits of integration are changedp from 0 p < to M p < . We may ≤ 3 ∞ ≤ 0 ∞ also use the following representation for the logarithm that appears in the integrand

∞ jp0/T p0/T e− ln 1 e− = , (4.20) − − j j=1 X

p0/T since p is positive and 0 e− < 1. Finally, we obtain an expression for the total 0 ≤ bosonic free energy just by summing over the energy spectrum

jp0/T V T ∞ ∞ ∞ p0 e− B = dp0 . (4.21) F − π 2 2 j n=1 j=1 Mn p0 Mn X X Z − 31 p Note that in the case of ﬁnite-Nc where we have a supersymmetric interacting gas the formulae for thermodynamics that we present here and the next chapter do not apply and thus have to be modiﬁed appropriately.

71 The calculation of the fermionic contribution to the free energy proceeds analogously, and we ﬁnd the identical result with the exception that there is a factor of ( 1)j − inside the summation. We can separate out the massless states from these expressions and calculate their contribution explicitly. We know that for resolution K there are

(K 1) massless bosons and (K 1) massless fermions. Thus the contribution to − − the free energy from massless states is

(K 1)π (K 1)π 0 = − V T 2, 0 = − V T 2. (4.22) FB − 6 FF − 12

After doing the integral over p0, we ﬁnd for the total free energy

Mn (T,V ) (K 1)π 2 ∞ ∞ K1 (2l + 1) F = − M T . (4.23) V T 2 − 4 − T π n (2l + 1) n=1 l=0 X X The even terms, where j = 2l in the original sum, cancel between the fermion and boson contributions. We have also factored out the temperature dependence of the massless contribution and the volume dependence.

We can now rewrite the free energy in terms of the density of states. The sums involving the Bessel function are cut off at a few terms; generally l 2 will be cut ≤ sufficient. We find

lcut M (T,V ) (K 1)π 2 ∞ 2 2 K1 (2l + 1) T F 2 = − dM ρ(M )M . (4.24) V T − 4 − πT M 2 (2l + 1) min l=0 Z X The free energy may now be used to calculate all the thermodynamic functions.

The internal energy and heat capacity are given by,

∂ln ∂ (T,V )= T 2 Z , (T,V )= E . (4.25) E ∂T CV ∂T V V See Appendix (C.3) for the exact formulae corresponding to the above quantities.

It is straightforward, given the density of states, to calculate the thermodynamic functions. Fig. 4.4a shows the free energy, and Fig. 4.4b shows the heat capacity. We

72 (a) (b)

Figure 4.4: The free energy (a) and the heat capacity (b) as functions of temperature for each resolution. Both functions are normalized to the number of massless states, N = 2(K 1). The data points are a convenient way to display the continuous 0 − functions calculated from ﬁts to the CDF.

2 expect the free energy to diverge as Nc and therefore must normalize our results to extract a ﬁnite number. In most of the region below the Hagedorn temperature the thermodynamic functions are totally dominated by the massless states. It therefore seems appropriate to normalize the thermodynamic functions to the total number of massless bound states, which is a function of the resolution and is 2(K 1). Alter- − natively, we could normalize by the number of states in any region. It is conceivable that at very high resolutions, where the mass gap is signiﬁcantly less than one, that the massive states may make an important contribution to the thermodynamics. In that case we would not choose to normalize by the massless states.

4.5 Discussion

The large-N SDLCQ solution of = (1, 1) super Yang–Mills theory in 1+1 c N dimensions gives a set of non-interacting bound states. From this set of bound states

73 it is in principle possible to calculate the thermodynamics of this theory. Central to this calculation is the calculation of the density of states. At resolutions K = 12 and below, where we can completely diagonalize the hamiltonian, we find that the entire cumulative distribution function can be fit with a single erf -function. From the cumulative distribution function, it is straightforward to calculate the density of states. For K larger than 12, it is difficult to calculate the entire spectrum; therefore, our calculations are confined to a fixed range of masses, M 2 100 g2N /π. Using ≤ c the known form of the distribution, we only need to fit a section of the cumulative distribution function to get a very good fit to the entire distribution. We know analytically the total number of bound states at any resolution, and this information can also be used in conjunction with a fit to a section of the distribution to produce the fit to the entire distribution.

The density of states that are found by this procedure grow sharply at small masses, then level off and decrease at larger masses. The peak of the density of states grows as we increase the resolution. Our understanding of this behavior is that the cutoff of the theory is forcing the density of states to level off, turn over, and then decrease. The true behavior of the density of states is reflected in the region of the density of states that is rapidly increasing, because it is this region that is increasing in size with the resolution K.

It appears that the cumulative distribution function, and, therefore, the density of states, are growing exponentially with the mass. To confirm this and find the asymptotic values of this growth, we fit the cumulative distribution function with an exponential at each K. We extrapolate these results to infinite resolution to find the asymptotic behavior of the density of states. The coefficient of the exponential

74 growth is the reciprocal of the Hagedorn temperature. We ﬁnd that this temperature

2 is about 0.854 g Nc/π/kB. The thermodynamic functions calculated from this data p are expected to produce valid results up to a temperature that is around TH.

It is now straightforward to calculate a standard set of thermodynamic functions from this density of states. The best estimate of the thermodynamics is obtained by using the exponential fits to the density of states. What we see is that, for resolutions up to K = 17, all of the massive bound states are well above the Hagedorn temperature. The thermodynamics below T is therefore controlled by the K 1 H − massless boson bound states and K 1 massless fermion bound states. − We can speculate on what will happen as the resolution goes to infinity. We have seen that the mass gap closes linearly with 1/K. So, for a resolution of order 100, there will be massive bound states below the Hagedorn temperature. This, of course, assumes that the estimate of the Hagedorn temperature is not changed by the higher resolution calculations. We found, however, that the actual number of massive bound states in a fixed mass range may grow slowly. For resolutions 11 to 17 we are able to find excellent fits with both exponential and linear growth as a function of the resolution K for masses with M 2 100. If the number of massive states grows only ≤ linearly with K, the contribution to the thermodynamic functions below TH might become significant but not dominant.

These calculations indicate that = (1, 1) super Yang–Mills theory in 1+1 N 2 dimensions has a Hagedorn temperature of about one in units of g Nc/π/kB. p

75 More generally, we found that SDLCQ can be used to ﬁnd interesting properties of

ﬁnite-temperature supersymmetric ﬁeld theories. The extension of this method to theories with more supersymmetry and in higher dimensions appears to be straightforward but may be computationally challenging.

76 CHAPTER 5

SYM-THERMODYNAMIC PROPERTIES II Η ποίηση δεν είναι ο τρόπος να μιλήσουμε αλλά ο καλύτερος τοίχος να κρύψουμε το πρόσωπό μας. –Μ. Αναγνωστάκης

As in the previous chapter we have used the approach which is based on Supersym- metric Discrete Light-Cone Quantization [19, 20], which preserves supersymmetry exactly, so we will be doing in this chapter as well. Currently, this is the only method available for numerically solving strongly coupled SYM theories. Despite some recent eﬀorts [79], conventional lattice methods have diﬃculties with supersymmetric gauge theories mainly because of the fermion doubling and the loss of translational invariance, facts that are responsible for violation of supersymmetry. However, results from the lattice community are welcomed and necessary.

Previously, in Chapter 4, we presented some thermodynamic properties of pure glue = (1, 1) SYM theory in 1+1 dimensions [63]. Here we extend the calculations N to include a sector with fundamental partons, which is our main focus in this chapter.

In the large-Nc limit the bound states in this sector of the theory are chains in color space with a fundamental parton at each end. The links in the chain are adjoint partons. Bound states of this type will be called mesonic because they have two fundamental partons, whereas solutions with only adjoint partons will be called

77 glueballs. We have also extended the calculation to include a Chern–Simons (CS) term, which gives mass to the adjoint partons.

The mesons and glueballs constitute two sectors of the same theory; both contribute to the thermodynamics. In the large-Nc limit the sectors decouple. Due to the cyclic redundancy of single-trace glueball states, there are many more meson states, which are in turn likely to dominate the thermodynamic properties of the system.

In previous work on the glueball sector [63] we found that the system possesses a

Hagedorn temperature.

Recall that SDLCQ makes use of light-cone coordinates, with x+ =(x0 + x3)/√2

0 3 the time variable and p− = (p p )/√2 the energy. One must also be careful − in deﬁning thermodynamic quantities on the light-cone. As we noted in an earlier section (Sec. 3.4) and also in the previous chapter this can be realized in a more direct way by noting that, since the light-cone momentum

(p0 + p3) p+ = √2 is conserved, the partition function must include the conserved quantity and is of the form

β (p +µp+) = Tr(e− LC − ), Z where µ is the chemical potential corresponding to the conserved quantity. The simply amounts to a rotation of the quantization axis. Thus µ = 1 corresponds to quantization in an equal-time frame, where the heat bath is at rest and the inverse temperature is β = √2β , and µ =1 corresponds to quantization in a boosted frame LC 6 where the heat bath is not at rest. Therefore µ corresponds to a continuous rotation of the axis of quantization, and µ =0 would correspond to rotation all the way to the

78 light-cone frame. Here we do not consider statistical mechanics on light cone, we will just employ SDLCQ to calculate the spectrum of = (1, 1) super Yang–Mills theory N in 1+1 dimensions; see for example [86] as well as Chapters 3 and 4. As is customary, we will assume that the single-trace bound states of our large-Nc approximation are single-particle states, that is exactly what we did in the previous chapter.

The calculation of thermodynamic quantities requires summing over the spectrum of available states, which we represent by a density of states (DoS). We will use a new numerical approach to estimate the density of states. The new approach is more eﬃcient than the method used in previous work [63], because it allows us to extract the density of states without fully diagonalizing the hamiltonian, a computationally challenging task. This innovation will enable us to pursue calculations at higher values of the resolution K.

We ﬁnd that the two-dimensional SYM theory with fundamentals and a Chern–

Simons term exhibits a Hagedorn temperature TH [88], and calculate TH for several values of the resolution K and Yang–Mills coupling g. Extrapolating to the continuum limit, we obtain TH as a function of the coupling. The Hagedorn temperature is used as an upper limit for the temperatures we can use to calculate the thermodynamic properties of the system.

In Section 5.1 we provide a review of the formulation of super Yang–Mills theory with fundamental matter and a Chern–Simons term in 1+1 dimensions. In Sec- tion 5.2 we discuss some of the properties of the SDLCQ spectra in some limiting cases and provide comparisons between glueball and mesonic sectors of the theory.

The discussion in Section 5.3 presents the methods for estimating the density of

79 states and the Hagedorn temperature. In Section 5.4, we summarize our formulation of the thermodynamics and the formulae we use to calculate the free energy.

We then present the numerical results for the free energy, which we obtained using the DoS approximation to the spectrum, at various values of SYM coupling g up to the Hagedorn temperature. Finally, in Section 5.5 we conclude by summarizing our results and the prospects for future work using these methods.

5.1 SYM with fundamental matter and Chern-Simons 3-form

5.1.1 Formulation of the theory and its supercharges

We start by considering = 1 supersymmetric gauge theory in 2+1 dimen- N sions coupled to fundamental matter and a Chern–Simons three-form. The action is denoted by

S2+1 = SYM + Sf.matter + SCS, (5.1) with

1 i S = dx3 Tr F F µν + ΛΓ¯ µD Λ , (5.2a) YM −4 µν 2 µ Z 3 µ µ S = dx D ξ†D ξ + iΨD¯ Γ Ψ g ΨΛ¯ ξ + ξ†ΛΨ¯ , (5.2b) f.matter µ µ − Z κ 2i S = dx3 Tr µνλ A ∂ A + gA A A +2ΛΛ¯ . (5.2c) CS 2 µ ν λ 3 µ ν λ Z

The SYM part of the action describes a system of gauge bosons Aµ and their superpartners, the Majorana fermions Λ. Both ﬁelds are (N N ) matrices trans- c × c forming under the adjoint representation of SU(Nc); hereafter, unless indicated otherwise, we treat these ﬁelds as matrices, and thus we suppress the color indices

(i, j, k). Additionally, we have two complex ﬁelds, a scalar ξ, and a Dirac fermion Ψ,

80 all transforming according to the fundamental representation of the gauge group. In matrix notation the covariant derivatives and the gauge ﬁeld strength are deﬁned as follows32:

DµΛ= ∂µΛ + ig[Aµ, Λ], Dµξ = ∂µξ + igAµξ, DµΨ= ∂µΨ + igAµΨ, (5.3a)

D ξ† = ∂ ξ† igξ†A , D Ψ† = ∂ Ψ† igΨ†A , F = ∂ A + ig[A , A ]. (5.3b) µ µ − µ µ µ − µ µν [µ ν] µ µ

The action (5.1) is invariant under supersymmetry transformations parameterized by a constant two-component Majorana spinor ε (ε ,ε )T;ε ¯ εTΓ0: ≡ 1 2 ≡ i 1 1 δA = ε¯Γ Λ, δΛ= F Γµνε, δΛ=¯ ε¯ΓµνF , (5.4a) µ 2 µ 4 µν −4 µν i i δξ = ε¯Ψ, δξ† = Ψ¯ ε, (5.4b) 2 −2 1 µ 1 µ δΨ= Γ εD ξ, δΨ=¯ D ξ†ε¯Γ , (5.4c) −2 µ −2 µ where Γµν , the spinor generator of the Lorentz group, is written as

µν 1 µ ν µνλ +2 Γ = [Γ , Γ ] = i Γ ; (− = 1). (5.5) 2 λ

Using standard Noether techniques (see e.g., Appendix (C.2)), we construct the spinor supercurrent corresponding to the above supersymmetric ﬁeld variations

µ i αβ µ i µ i µν ε¯ = ε¯Γ Γ Tr (ΛF )+ D ξ†ε¯Ψ+ ξ†ε¯Γ D Ψ S 4 αβ 2 2 ν i i Ψ¯ εDµξ + D ΨΓ¯ µνεξ. (5.6) − 2 2 ν

For the remainder of the chapter we assume that the ﬁelds are independent of the

2 space-like dimension x , i.e. ∂2(...)=0, thereby dimensionally reducing the theory to

32 Note that the notation for the gauge covariant derivatives is in accord with the SU(Nc) matrix notation. Recall that under a ﬁnite SU(Nc) transformation Uc the fundamentals transform as 1 ξ Ucξ, while the anti-fundamentals as ξ† ξ†Uc− . As for the adjoint ﬁelds we have, as usual, → 1 µν µν → A U A U + i(∂ U )U − , F UF U −1. µ → c µ c µ c c → 81 two dimensions. Thus the = 1 supersymmetry in 2+1 dimensions is naturally N expressed in terms of = (1, 1) supersymmetry in 1+1 dimensions. N We will implement light-cone quantization, which means that initial conditions as well as canonical (anti-) commutation relations will be imposed on the light-like surface x+ = const. In particular, we construct the supercharge by integrating the supercurrent (5.6) over the light-like surface

i αβ + i i +ν εQ¯ = dx− ε¯Γ Γ Tr (ΛFαβ)+ D ξ† ε¯Ψ+ ξ†ε¯Γ DνΨ 4 2 − 2 Z i + i +ν ε¯Ψ†D ξ ε¯Γ D Ψ† ξ . (5.7) − 2 − 2 ν Note that because we have taken the ﬁelds to be independent of x2, the integration over this coordinate resulted in a constant factor, which rescaled our original ﬁelds.

By choosing the following imaginary (Majorana) representation for the Dirac matrices in three dimensions:

0 1 2 Γ = σ2, Γ = iσ1, Γ = iσ3, (5.8)

T the Majorana spinor ﬁeld Λ is manifestly real, i.e. Λ† = Λ . At this point it is convenient to introduce the component form for the spinors:

T T + T Λ= λ, λ˜ , Ψ= ψ, ψ˜ , Q = Q , Q− . (5.9) In terms of this decomposition, the superalgebra is realized explicitly in its = (1, 1) N form, namely

+ + + + Q , Q =2√2P , Q−, Q− =2√2P −, Q , Q− =0, (5.10) { } { } { }

+ where Q (Q−) are left (right) Majorana–Weyl spinors, each characterizing the small- est spinor representation in 1+1 dimensions (for further discussion on this issue consider e.g., Appendix B.4).

82 To readily eliminate the nondynamical ﬁelds, we impose the light-cone gauge,

A+ =0. In this case the supercharges can be read oﬀ (5.7) and are given by

+ 1 2 i i i i Q = dx− Tr(λ∂ A )+ ∂ ξ†ψ ψ†∂ ξ ξ†∂ ψ + ∂ ψ†ξ , (5.11a) 2 − 2 − − 2 − − 2 − 2 − Z 1 i Q− = dx− Tr(λ∂ A−) iξ†D2ψ + iD2ψ†ξ ∂ (ψ˜†ξ ξ†ψ˜) . (5.11b) √2 − − − √2 − − Z

Notice that the right-movers (ψ˜) appear in the supercharge Q− only in the total derivative term. This is a consequence of the light-cone formulation, which singles out the non-dynamical fermion degrees of freedom, leaving in the expression only the physical spinor ﬁelds (λ and ψ). Among the equations of motion that follow from the action (5.1), in the light-cone gauge, three serve as constraints rather than as dynamical equations. Namely, for λ˜ and ψ˜, respectively, we have

ig 2 ∂ λ˜ = [A ,λ] + iξψ† iψξ† iκλ , (5.12) − −√2 − − ig g ∂ ψ˜ = A2ψ + λξ. (5.13) − −√2 √2

While for A− we obtain

2 ∂ A− = gJ, (5.14) − with

2 2 1 κ 2 J = i[A , ∂ A ]+ λ,λ i(∂ ξ)ξ† + iξ(∂ ξ†)+ √2ψψ† + ∂ A . (5.15) − − √2{ } − − − g −

Note that the ﬁeld A− has to be eliminated from the supercharges, in favor of the physical degrees of freedom. This can be done by inverting (5.14).

The only contribution from the Chern–Simons term enters into the supercharges via equation (5.15), because δ = 0 under the supersymmetry transformations LCS 83 (5.4). The inclusion of a Chern–Simons term in our theory is important, since it eﬀectively generates mass for the adjoint partons proportional to the coupling κ.

5.1.2 SDLCQ eigenvalue problem

The bound-state spectrum is obtained by solving the following mass eigenvalue equation:

+ + 2 2P P − ϕ = √2P (Q−) ϕ | i | i + 2 2 = √2P g(Q˜− + Q˜− ) + iκQ˜− ϕ M ϕ , (5.16) SYM f.matter CS | i ≡ | i where the various pieces of Q−, after dropping the surface terms and eliminating the non-dynamical ﬁelds using the constraint (5.14), may be expressed as follows:

˜ i 2 2 i 1 QSYM− = dx− [A , ∂ A ]+ λ,λ λ, (5.17a) √2 − √2{ } ∂ Z − 1 1 ˜ √ Qf−.matter = dx− i(∂ ξ)ξ† iξ(∂ ξ†) 2ψψ† λ √2 − − − − ∂ Z − 2 2 + ξ†A ψ + ψ†A ξ , (5.17b) ˜ i 2 1 QCS− = dx−(∂ A ) λ, (5.17c) √2 − ∂ Z − where a trace over color space is understood.

The strategy for solving equation (5.16) is to cast it as a (ﬁnite) matrix eigenvalue problem. This is achieved by employing a discrete basis where the longitudinal light- cone momentum P + is diagonal. The discrete basis is introduced by ﬁrst discretizing

33 the supercharge Q− and then constructing P − from the square of the supercharge:

2 P − = (Q−) /√2. The two dimensional theory is compactiﬁed on a light-like circle

33 Note the relative phase between QSY− M and QCS− . QSY− M is deﬁned as Hermitian and QCS− is deﬁned to be anti-Hermitian such that Q− remains Hermitian.

84 ( L x− < L), and periodic boundary conditions are imposed on all dynamical − ≤ degrees of freedom. This leads to the following ﬁeld mode expansions:

∞ 2 1 1 inπx /L inπx /L A (0, x−)= a (n)e− − + a† (n)e − , (5.18) ij √ √n ij ji 4π n=1 X 1 ∞ inπx−/L inπx−/L λij(0, x−)= 1 bij(n)e− + bji† (n)e , (5.19) 4 √ 2 2L n=1 X ∞ 1 1 inπx /L inπx /L ξ (0, x−)= c (n)e− − +˜c†(n)e − , (5.20) i √ √n i i 4π n=1 X 1 ∞ inπx−/L ˜ inπx−/L ψi(0, x−)= 1 di(n)e− + di†(n)e . (5.21) 4 √ 2 2L n=1 X In the above expressions34 we introduced the discrete longitudinal momenta k+ k ≡ as fractions nP +/K = nπ/L; (n = 1, 2, 3,...) of the total longitudinal momentum

P +, where K is the integer that determines the resolution of the discretization. The color indices were made explicit as well. Because light-cone longitudinal momenta are always positive, K and each n are positive integers. The number of constituents is thus bounded by K. The continuum limit is reached by letting K . →∞ The time direction in the light-cone formalism is taken to be the x+ direction.

Thus the (anti-)commutation relations between ﬁelds and their conjugate momenta are assumed on the surface x+ = 0. Quantization is achieved by imposing the following relations:

2 2 1 Aij(0, x−), ∂ Akl(0,y−) = i δilδjk δijδkl δ(x− y−), (5.22) − − N − c 1 λ (0, x−),λ (0,y−) = √2 δ δ δ δ δ(x− y−), (5.23) ij kl il jk − N ij kl − c ξi(0, x−), ∂ ξj(0,y−) = iδijδ(x− y−), (5.24) − − ψ (0, x−), ψ (0,y−) = √2δ δ(x− y−). (5.25) i j ij − 34We remind the reader that the inclusion of zero modes is beyond the scope of the present work.

85 The above (anti-) commutators can also be expressed, with the help of equations

(5.18)-(5.21), in terms of creation-annihilation operators

1 aij(k), akl† (k0) = bij(k), bkl† (k0) = δikδjl δijδkl δ(k k0), (5.26) − Nc − h i n o c (k),c†(k0) = c˜ (k), c˜†(k0) = d (k),d†(k0) = d˜ (k), d˜†(k0) =δ δ(k k0). (5.27) i j i j i j i j ij − h i h i n o n o The expansion of the supercharge Q− in terms of creation and annihilation operators is a straightforward exercise. For instance, the decomposition of QCS− in terms of Fourier modes gives the following normal ordered expression:

iκ√L ∞ 1 Q− = a† (n)b (n)+ b† (n)a (n) . (5.28) CS −25/4√π √n ij ij ij ij n=1 X

Similarly, for the supercharge, Qf−.matter, that controls the behavior of the fundamental matter ﬁelds and also generates the hamiltonian, we obtain

ig√L ∞ (n2 + n3) Q− = c˜†(n )˜c (n )b (n ) f.matter − 25/4π 2n n n i 3 j 2 ji 1 n ,n ,n =1( 1√ 2 3 1 X2 3 c˜†(n )b† (n )˜c (n )+ b† (n )c†(n )c (n ) c†(n )b (n )c (n ) − i 2 ij 1 j 3 ji 1 i 2 j 3 − i 3 ij 1 j 2 1 + d˜†(n2)b† (n1)d˜j(n3)+ d˜†(n3)d˜j(n2)bji(n1)+ b† (n1)d†(n2)di(n3) √2n i ij i ij j 1 i + d†(n3)bij(n1)dj(n2) + c†(n3)aij(n2)dj(n1) i 2√n n i 2 3 ˜ ˜ + aij† (n2)dj†(n1)ci(n3)+ dj†(n1)aji† (n2)˜ci(n3)+˜ci†(n3)dj(n1)aji(n2) i + aji† (n2)ci†(n1)dj(n3)+ dj†(n3)aji(n2)ci(n1) 2√n1n2 ˜ ˜ + dj†(n3)˜ci(n1)aij(n2)+˜ci†(n1)aij† (n2)dj(n3) δn3,n1+n2 . (5.29) ) Although this calculations can be done in a straightforward fashion following the procedure outlined in Chapter 3, fortunately our computer code carries out these expansions automatically.

86 Apart from supersymmetry, the theory we set out to explore possesses another symmetry,35 which may be used to reduce the size of the hamiltonian matrix we need

to produce and diagonalize. Namely, we have a Z symmetry that is associated 2 T with the orientation of the large-Nc string of partons in a state [21,100]. It gives a sign when the gauge group indices are permuted

a (n) T a (n), b (n) T b (n). (5.30) ij → − ji ij → − ji

In this chapter we will discuss numerical results obtained in the large-Nc limit, i.e. terms of order 1/Nc in the above expressions are dropped. Note that corrections on the order of 1/Nc are expected to lead to interesting eﬀects [101]; however, they are beyond the scope of this work.

5.2 Meson and glueball spectra

5.2.1 Limiting cases

We ﬁrst investigate the strong-coupling (g κ) and weak-coupling (g 0) limits → of the theory. In the strong-coupling limit, we previously found [102] that there are approximate BPS (aBPS) glueball bound states with masses (squared)

M 2 = n2κ2, n =2, 3,..., (K 1). aBPS −

In the meson sector under investigation in the present chapter, nearly all the masses grow with g. However, we see evidence for a state that remains near zero mass as g . →∞ The free theory can be solved analytically, and the results are shown in Fig. 5.1.

Its free-meson spectrum, Fig. 5.1(a), has K 1 massless states in each symmetry − 35Note that when κ =0, parity ( ) is also conserved. P 87 sector. Each such state is made out of two fundamental partons. All states in the corresponding glueball sector, Fig. 5.1(b), are massive. The mass scale for all states is set by the CS coupling κ. To obtain the spectrum, we consider sets of free partons that form mesonic color-singlet multi-parton combinations. There are many other combinations of such free partons that belong to the non-singlet sector of the Hilbert space, which we can omit in the large-N limit; for g = 0 the only viable states are c 6 color-singlets. In other words, the free color-singlet combinations are expected to become bound states as soon as the coupling is turned on. Mesonic multi-parton color-singlet states are of the form

f †(m )f˜†(m ) 0 , f †(m ) † (n )f˜†(m ) 0 , f †(m ) † (n ) † (n )f˜†(m ) 0 ,..., i 1 i 2 | i i 1 Oij 1 j 2 | i i 1 Oik 1 Okl 2 l 2 | i where the operators f †, f˜† create fundamental partons, while † creates adjoint par- O tons; fermions or bosons. In our theory we have four types of fundamental partons and two types of adjoint partons. In the large-Nc limit we can have the following four types of mesonic multi-parton color-singlet states:

c†(adj.) c˜† 0 ,c†(adj.) d˜† 0 ,d†(adj.) d˜† 0 ,d†(adj.) c˜† 0 , i ij j| i i ij j| i i ij j| i i ij j| i where (adj)ij can be any string of adjoint partons. Only the adjoint partons contribute to the mass of a state and do so proportional to the CS coupling κ. Note also that the ﬁrst pair of fundamental partons above forms a massless combination; there are 4(K 1) of these at each value of the resolution. − To construct the spectrum and ﬁnd the degeneracies of each mass state, we utilize combinatorics and the DLCQ multi-particle formula

jmax 2 2 1 Mjmax (K)= κ K . (5.31) nl Xl=1 88 Here 1 j (K 2) is the number of adjoint partons. Speciﬁcally, we calculate ≤ max ≤ − K K 1 the compositions Cj = j −1 (i.e. ordered partitions) of the integer K into j = −

(2 + jmax) parts, where the factor two counts the two massless fundamental partons in a particular state. For example, at K = 5 the compositions that give massive

5 5 5 5 multi-particle states are: C3 ,C4 ,C5 . The ﬁrst composition in the list, C3 , yields

2 2 2 5 2 (modulo parton type) three states with M1 = 5κ , two states with M1 = 2 κ , and

2 5 2 one state with M1 = 3 κ . Thus, taking into account the several types of partons

5 3 that can form C3 , the total number of states we get for this case is 2 (3+2+1) = 48.

The total number of states – including the massless ones – as a function of K is thus

K k K N(K)= 2 Ck . (5.32) Xk=2 The dimension N(K) of the Hilbert space of states grows exponentially with K, e.g.,

N(16) = 28, 697, 812.

As an example, consider a four parton state, consisting of two fundamental and two adjoint partons, with mass

n + m M 2 = K, n, m =1, 2, 3,.... 4 nm

The two adjoint partons have mass unity (κ2 = 1), while the two fundamentals are

2 35 massless. Thus at K =7, we have four-parton states starting at M = 6 . The glueball spectrum is evaluated in a similar fashion. However, the free glueball multiparticle color-singlet states form closed loops made out of adjoint fermion

(bij† ) and boson (aij† ) partons, with a mass easily obtained by (5.31). Obviously, there are no massless states in the glueball sector. The cyclic symmetry of the color trace reduces the total number of states that are available, so the free mesons will dominate the free energy. Due to supersymmetry it suﬃces to count only the fermions.

89 20. 20.

15. 15. 2 10. 2 10. M M

5. 5.

0. 0.

0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25 0.3 1K 1K (a) (b)

Figure 5.1: Mass spectra for (a) mesonic bound states and (b) glueball bound states as a function of the inverse resolution for 3 K 23 when κ = 1 and g = 0 (free ≤ ≤ theory).

Using the combinatorics above, we arrive at the number of fermionic states with j

partons [103],

∞ (2q + 1) K j N (K; j)= C˜ ; . (5.33) f j f 2q +1 2q +1 q=0 X

The function C˜f is deﬁned recursively as

∞ j 1 K K j C˜ (K; j)=2 − C C˜ ; . f j − f 2q +1 2q +1 q=1 X ˜ K j Note that Cf 2q+1 ; 2q+1 is zero if none of its arguments is an integer. The total number of states at a speciﬁc K is found by summing over the number of partons j.

For example, at K =5 and j =3, we have a total of 16 states. Eight of these states

2 2 2 35 2 have mass squared M = 10κ and the other eight have M = 3 κ .

90 30.

8. 25.

6. 20. 2 2 M M 15. 4.

10. 2. 5.

0. 0.

0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25 0.3 1K 1K (a) (b)

Figure 5.2: Mass spectra for (a) the meson ( -even, -even) sector and (b) the P T glueball ( -even, -even) sector as a function of the inverse resolution for 3 K 16 P T ≤ ≤ when κ =0 and g =1.

5.2.2 Comparison of meson and glueball spectra

The generic meson and glueball spectra for nonzero g are shown in Figs. 5.2 and

5.3. In the glueball sector with ﬁnite coupling but vanishing CS coupling, there

are 2(K 1) massless BPS states. The number of partons in these states grows − with the resolution K, and there is a mass gap between these massless states and

the lowest massive states that decreases with increasing resolution. When the CS

coupling is not vanishing (g = 0, κ = 0), the BPS massless glueball bound states 6 6 become approximate BPS states [102], with bound-state masses nearly independent

of the gauge coupling. The masses of the remaining states in this sector grow rapidly

with the coupling.

91 14. 60.

12. 50.

10. 40. 8. 2 2

M M 30. 6.

20. 4.

2. 10.

0. 0.

0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25 0.3 1K 1K (a) (b)

Figure 5.3: Mass spectra for (a) the meson ( -even) as a function of the inverse T resolution for 3 K 15 sector and (b) the glueball ( -even) sector as a function ≤ ≤ T of the inverse resolution for 3 K 9. In both cases κ =1 and g =1. We note also ≤ ≤ a small splitting in the masses due to the presence of CS term which breaks explicitly the symmetry. P

92 In earlier work [63] we studied the thermodynamics of this sector with vanishing

CS coupling. Here, we are considering the mesonic sector of this SYM theory. From a previous work [104,105] we know that for non-zero coupling there is a mass gap in the low-mass sector. The low-mass sector consists of the states that become massless bound states of two fundamental partons in the limit that the coupling goes to zero.

This mass gap decreases as the resolution increases. Of the K 1 massless states in − each symmetry sector at vanishing coupling only one remains at ﬁnite coupling.

In the large-Nc limit, the mesonic and glueball sectors decouple. The thermodynamics of the theory is generated by the partition function which is the product of the partition functions of the two sectors, and the free energy is the sum of the two free energies. The glueball bound states are closed loops in color space; their cyclic symmetry greatly reduces the number of Fock basis states. Therefore, the number of glueball states relative to the meson states at a particular K is very small. One would thus expect the mesonic bound states to dominate the thermodynamics.

There are several ways that the glueball bound states may aﬀect the thermodynamics of the full theory. At very low temperature, the thermodynamics will be dominated by the very low mass states. At small coupling there are many more light mesonic states than glueballs. At strong coupling there are many approximate BPS glueball bound states, while only one of the mesonic states remains massless. Thus at strong coupling and at temperature high enough to be inﬂuenced by the approximate BPS states, the thermodynamics will eventually be dominated by the glueball sector.

93 5.3 Density of states and Hagedorn temperatures revisited

We calculate thermodynamic quantities from the partition function, which we express as a sum of Boltzmann factors weighted by the density of states (DoS),

ρ(M 2,K). The discrete spectrum is estimated numerically, and a ﬁt to the data is used to calculate the DoS. The discrete spectrum is used to calculate the cumulative distribution function (CDF), N(M 2,K), which is the number of states with mass squared below M 2 at resolution K. The DoS is related to the CDF by

dN (M 2,K) ρ(M 2,K) ρ (M 2)= , (5.34) ≡ K dM 2 with dimensions of L2.

A comment regarding the units of the invariant mass squared eigenvalues M 2 is in order. From (5.16) it is inferred that the hamiltonian is of the form

2 2 2 2 g g P − = g A + κgB + κ C = κ A + B + C , (5.35) n n n κ2 n κ n n where g stands for g Nc/π. Thus it is a function of a dimensionless ratio, g/κ, and the dimensionful36 parameterp κ. The latter sets the mass scale. Here we simply ﬁx the value of κ to unity, while we numerically investigate the spectra for several values of g. So the quantities we calculate are expressed in units where κ =1.

2 2 It is interesting that for g large and κ =1 we have Mi = g Ai, so the eigenvalues scale with g2. Therefore, we may determine the strong coupling properties of the

36Choosing the CS coupling to set the mass scale is quite natural for the problem at hand, since among others we investigate the case where g =0 and κ = 1, where the mass squared eigenvalues are proportional to κ2, see (5.31). In general we have the freedom of choosing the parameters such that they suit the problem in hand. For instance, we may perform the following general reparametrization M 2 = α2 β2A + (1 β)βB + (1 β)2C , n n − n − n 1 κ κ − where α(g,κ)= g 1+ g with α [κ, ) and β(g,κ)= 1+ g with β [0, 1]. Hence the relevant overall parameter is the ratio between∈ ∞ the couplings, whereas the overall∈ multiplicative factor sets the mass scale.

94 theory from the solution of the g = 1, κ = 0 theory, a numerically much simpler problem. For some associated numerical results, see Section 5.4.3.

5.3.1 A method of estimating the density of states

The DoS can be estimated by diagonalizing P −, computing the CDF from the spectrum, and differentiating a smooth fit to the CDF. This is what was done in previous work [63]. The size of the matrix representation of P − increases with K and with the number of fields. Eventually, the computational cost becomes too high.

To ameliorate the situation, we adapted a Lanczos-based algorithm to estimate the

CDF directly.

We start by writing the density of states as

ρ(M 2)= d δ(M 2 M 2), (5.36) n − n n X where dn is the degeneracy of the mass eigenvalue Mn. The CDF is just

M 2 N(mass2 M 2)= dM¯ 2ρ(M¯ 2). (5.37) ≤ Z

iP x+ The density can be written in the form of a trace over e− − as follows:

2 + + + 2 1 2 + 1 ∞ iM x /2P iP −x + ρ(M )= d δ(M /2P P −)= e d e− n dx 2P + n − n 4πP + n n n X Z−∞ X 1 ∞ iM 2x+/2P + iP x+ + = e Tr e− − dx . (5.38) 4πP + Z−∞ To approximate the trace, we use an average over a random sample of vectors [106,

107]. Deﬁne a local density for a single vector s as | i

2 1 ∞ iM 2x+/2P + iP x+ + ρ (M )= e s e− − s dx , (5.39) s 4πP + h | | i Z−∞

95 2.5 ´ 105 numerical analytical 2. ´ 105

1.5 ´ 105 N

1. ´ 105

5. ´ 104

0. 0. 20. 40. 60. 80. 100. 120. 140. M 2

Figure 5.4: CDF of the free (g = 0) mesonic -odd ( −) sector at K = 13. Crosses T T (boxes) refer to numerical (analytical) calculation of the bound-state masses.

so that the average can be written

1 S ρ(M 2) ρ (M 2). (5.40) ' S s s=1 X The sample eigenstates s can be chosen as random phase vectors [108], meaning | i that the coeﬃcient of each Fock state in the basis is a random number of modulus one.

2 D iM 2x+/2P + iP x+ + ρ (M )= e u e− − u dx , (5.41) s 4πP + h 1| | 1i Z 96 iP x+ and u e− − u can be approximated by the (1, 1) element of the exponentiation h 1| | 1i of the Lanczos tridiagonalization of P −. Let Ps− be this tridiagonal matrix, and solve the eigenvalue problem 2 s Msn s P −~c = ~c . (5.42) s n 2P + n

A diagonal matrix Λ is related to Ps− by the usual similarity transformation Ps− =

2 1 s Msn UΛU − , where Uij = (cj)i and Λij = δij 2P + . This means that the (1, 1) element is given by

+ 2 + + iPs−x s 2 iMsnx /2P e− = (cn)1 e− . (5.43) 11 | | n X The local density is

2 D iM 2x+/2P + s 2 iM 2 x+/2P + + ρ (M ) e (c ) e− sn dx s ' 4πP + | n 1| n Z X D (cs ) 22πδ(M 2/2P + M 2 /2P +) (5.44) ' 4πP + | n 1| − sn n X w δ(M 2 M 2 ), ' sn − sn n X where w D (cs ) 2 is the weight of each Lanczos eigenvalue. Note that only the sn ≡ | n 1| extreme Lanczos eigenvalues are good approximations to eigenvalues of the original

P −; however, the other Lanczos eigenvalues and eigenvectors provide a smeared representation of the full spectrum.

The contribution to the cumulative distribution function is

M¯ 2 N (M 2) dM¯ 2ρ(M¯ 2) w θ(M 2 M 2 ). (5.45) s ≡ ' sn − sn n Z X The full CDF is then approximated by the average

1 N(M 2) N (M 2). (5.46) ' S s s X

97 In forming the full CDF, one has to decide how to combine theta functions. This

2 is done by using the ﬁrst sample run as a template for values M1n at which to evaluate

N. The contributions of the other samples to N at these values are estimated by

2 linear interpolation in cases where the Lanczos eigenvalues Msn are not the same as those in the ﬁrst set. Also, in cases where duplicate eigenvalues are generated by the

Lanczos iterations, only one is included in the template and the associated weights are added together.

The convergence of the approximation is dependent on the number of Lanczos iterations per sample, as well as the number S of samples. Test runs indicate that the recommended value [106,107] of 20 samples is suﬃcient. The number of Lanczos iterations is kept at 1000 per sample; using only 100 leaves errors on the order of

1-2%.

A check for the validity of this approach is the comparison between the CDFs for the free theory, where the analytic solution is available. Figure 5.4 clearly shows that the numerical technique introduced here gives a CDF almost identical to the one obtained by the analytical calculation. The very few points that appear to be extraneous have no impact on the ﬁtting algorithm we use to calculate the ﬁts to the

CDFs.

5.3.2 Fits to the spectrum

In an earlier work related to thermodynamics [63], we split the spectrum into low and high-mass regions, separated by the mass gap. The bound-state spectrum for this problem has similar characteristics. For instance, for small values of the

98 coupling, namely g . 1, the mass gap separates the K 1 nearly massless color- − singlet states evolving from the massless states of the free theory from the rest of

the spectrum. For those values of the coupling these states have M 2 . 1. Thus our

2 2 2 density of states, ρK (M ), is zero in the mass gap (M1 , M2 ), and the CDF has the

following generic form:

2 2 2 2 N1(M ,K), Mmin M M1 2 2 ≤ 2 ≤ 2 N(M ,K)=  const., M1 < M < M2 (5.47) 2 2 2 2  N2(M ,K), M M M . 1 ≤ ≤ max 

14.

17.5 7. 12. 6. 15. 10. 5. 12.5 8.

N 4. 10. 6. 3. 7.5 4. 2. 5. 2. 1. 2.5

2.5 5. 7.5 10. 12.5 15. 2.5 5. 7.5 10. 12.5 15. 0.2 0.4 0.6 0.8 1. 1.2 1.4 M 2 @´10D M 2 M 2 @´10-3D (a) (b) (c)

Figure 5.5: CDF of the -even ( +) sector at K = 14 and g =0.1. Shown are data T T (dots) and a ﬁt to the data: (a) all states in units of 105 states; (b) range of masses just above the mass-gap in units of 102 states; (c) states below the mass gap.

99 2.5 10. 3.

8. 2.5 2.

2. 6. 1.5 Ρ 1.5 4. 1. 1.

2. 0.5 0.5

2.5 5. 7.5 10. 12.5 15. 2.5 5. 7.5 10. 12.5 15. 0.2 0.4 0.6 0.8 1. 1.2 1.4 2 M @´10D M 2 M 2 @´10-3D (a) (b) (c)

Figure 5.6: Same as Fig. 5.5 but for the density of states, in units of (a) 103 states, (b) 102 states, and (c) 1 state.

We ﬁt the low-lying mass spectrum, Ml, of the CDF using the following function

p(K) 2 2p N1(Ml ,K)= αp Ml , (5.48) p=0 X

while the logarithm of the CDF for higher masses, Mh, is ﬁt to the following function

p(K) ln[N (M 2,K)]=(x + a )γ exp[ b M 2δ] α M 2p. (5.49) 2 h h 1 − 1 h p h p=0 X The other parameters in our ﬁt functions are computed using standard non-linear ﬁt

algorithms. Typical results are shown in Figs. 5.5-5.10.

100 7. 14.

6. 4. 12.

5. 10. 3. 4. 8. N

3. 2. 6.

2. 4. 1. 1. 2.

5. 10. 15. 20. 25. 30. 2.5 5. 7.5 10. 12.5 15. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 M 2 @´10D M 2 M 2 (a) (b) (c)

Figure 5.7: CDF of the -odd ( −) sector at K = 16 and g = 0.5. Shown are T T data (dots) and a fit to the data: (a) all states in units of 106 states with point of inflection; (b) range of masses just above the mass-gap in units of 102 states; (c) states below the mass gap. The relatively poor fit near M 2 = 10 in (b) does not have a significant effect on the results.

6. 8. 7. 5. 6. 6. 4. 5.

4. Ρ 3. 4. 3. 2. 2. 2. 1. 1.

5. 10. 15. 20. 25. 30. 2.5 5. 7.5 10. 12.5 15. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 2 M @´10D M 2 M 2 (a) (b) (c)

Figure 5.8: Same as Fig. 5.7, but for the DoS, in units of (a) 104 states, (b) and (c) 10 states.

101 The spectrum exhibits some structure at relatively small M 2, as we can see from

Figs. 5.5(b) and (c). These states dominate the thermodynamics in the range of temperatures at which we perform the free-energy calculations. Noteworthy is also the point of inflection in the CDF plot and the peak in the DoS plot. The data beyond this point show the effect of the cutoff imposed by the resolution K.

From the plots of the CDF and the DoS, and in particular the ﬁgures that depict the DoS for g & 0.5, one may predict the result for the free energy . The main F contribution to comes from the one nearly massless state37 of the spectrum. This F state exists only in the -even sector, and consequently this sector dominates the T thermodynamics at low temperatures. Therefore, we do not expect the ﬁt (e.g., see

Fig. 5.10(c)) to give us an accurate result for the free energy, because it essentially leaves out the contribution from the nearly massless state. We will return to this point when we discuss the numerical results for the free energy in Section 5.4.3.

37This very light state, compared to the rest of the spectrum, can be considered instead as an exactly massless states, when it comes to calculations of thermodynamics. This is a legitimate approximation as long as K is large enough. Speciﬁcally, from studying the spectrum behavior this very light state tends to become massless as K . → ∞

102 7. 1.2 10.

6. 1. 8. 5. 0.8 4. 6.

N 0.6 3. 4. 0.4 2. 2. 1. 0.2

1. 2. 3. 4. 5. 6. 7. 25. 50. 75. 100. 125. 150. 175. 10. 20. 30. 40. 50. 2 3 M @´10 D M 2 M 2 (a) (b) (c)

Figure 5.9: CDF of the -even sector at K = 16 and g =4.0. Shown are data (dots) T and a ﬁt to the data: (a) all states in units of 106 states; (b) range of masses just above the mass-gap in units of 102 states; (c) states below the mass gap.

0.4 1.5 2. 1.25 0.3 1.5 1. Ρ 0.75 0.2 1.

0.5 0.5 0.1 0.25

1. 2. 3. 4. 5. 6. 7. 25. 50. 75. 100. 125. 150. 175. 10. 20. 30. 40. 50. 2 3 M @´10 D M 2 M 2 (a) (b) (c)

Figure 5.10: Same as Fig. 5.9, but for the DoS, in units of (a) 103 states, (b) and (c) 1 state.

103 5.3.3 Hagedorn temperature

We ﬁnd that the physical spectrum in the theory grows approximately exponentially with the mass of the state

ρ (M 2) exp(M/T ), (5.50) H ∼ H and therefore has a Hagedorn temperature TH . The partition function has the following general form

M M¯ exp[M 1 1 ] dMρ¯ (M¯ ) exp − T T TH − T . (5.51) Z ∝ H T ∝ H T T Z − H The partition function obviously diverges as T T , and T sets the region of Z → H H validity for the calculation of thermodynamic properties. Thus TH serves as an upper limit for the temperatures we can use to calculate the thermodynamic functions.

We will calculate TH by fitting the CDF with an exponential function, and then determining its exponent as a function of the resolution and coupling. At fixed coupling we extrapolate to infinite resolution and obtain the continuum Hagedorn temperature as a function of the coupling. We fit the CDF in the region lying above the mass gap and below the point of inflection. The number of states below the mass gap is closely related to the number of free massless states and therefore not a factor in the Hagedorn domain. The number of states above the point of inflection are significantly reduced because of the cutoff imposed by the finite resolution and are therefore not useful in determining the Hagedorn temperature.

In Fig. 5.11(a) we show a typical fit to the CDF in this region of M 2. The data is fit well with an exponential. The particular figure deals with the -odd sector of T the spectrum at coupling g = 0.1 and K = 13. A similar behavior occurs for the

-even sector and other values of K. It is clear from our data that the spectral CDF T 104 and the DoS exhibit a Hagedorn behavior. We plot the logarithm of the CDF versus the bound-state mass obtained from our numerical calculations. Then we estimate the range of the mass values M where the plot is approximately linear, and ﬁt this

1 region to a linear fit of the form αM + TH− . The non-linear part of the distribution, for high values of M, is cut off because of the finite resolution K. The extrapolated result for the -even sector is very similar to the result of the -odd sector, so we T T take the average of the two values, for each K.

In Fig. 5.11(b) we show the Hagedorn temperature as a function of the inverse resolution for the massive, free theory (g = 0, κ = 1). The data appear to be following a straight line and have been extrapolated to the continuum, where we found T (g = 0) 0.52 in units where κ = 1. We show the Hagedorn temperature H ≈ in Fig. 5.12 for several cases where (g =0, κ = 1) and for resolution K [11, 16]. As 6 ∈ a check we note that for the cases g =0 and g =0.1 one expects the corresponding extrapolated Hagedorn temperatures to be comparable.

For the variety of values of coupling g that are considered here, the upper bound for temperatures is set by the Hagedorn temperature of the free theory, T ∞(0) H ≈ 0.52κ and thus we calculate the thermodynamic properties of the theory below this limit. From Fig. 5.12 we glean that TH∞(g) grows with the coupling. For larger values of g we may therefore access a signiﬁcantly larger region in T. However, we will leave the discussion of these cases for future work.

105 15. 1. TH1KL f HML 12.5 0.9

10.

D 0.8 N H @

7.5 T Log

0.7 5.

2.5 0.6

0. 2. 4. 6. 8. 10. 12. 14. 0. 0.02 0.04 0.06 0.08 0.1 M 1K (a) (b)

Figure 5.11: (a) Logarithm of the CDF versus M. The approximately linear part of this logarithmic CDF is ﬁt to f(M) = αM + 1 . (b) Extrapolated Hagedorn TH temperature for coupling g =0 with T ∞(0) 0.52κ. H ≈

5.4 Finite temperature results in 1+1 dimensions

5.4.1 The free energy

We now introduce the basic formulation necessary for our ﬁnite temperature calculations. Note that our approach here deviates slightly from our earlier work [63], mainly in the way the free energy and the mass squared are normalized. We consider a system with constant volume which is in contact with a heat bath of constant

38 temperature. The free energy, in units where kB =1, is given by

(T,V )= T ln . (5.52) F − Z 38Although our focus is the free energy, , it is straightforward to use our numerical techniques F 2 ∂ln to calculate other thermodynamic functions, such as the internal energy (T, V )= T ∂TZ V,the ∂ E entropy S = ( )/T , and the heat capacity (T, V )= E , cf. [63]. E−F CV ∂T V 106 4. 3.5 T2HgL

3. 3. 2.5

2. H H

2. T T

1.5

1. 1.

0.5

0. 0. 0.02 0.04 0.06 0.08 0.1 1. 2. 3. 4. 5. 1K g (a) (b)

Figure 5.12: Hagedorn temperature (a) plotted versus 1/K at couplings g = 0.1 (crosses), 0.5 (boxes), 1.0 (triangles), 4.0 (diamonds) in units where κ = 1, and (b) 2 extrapolated in K as function of g with a ﬁt to T2(g)=0.52+0.39g+0.054g (dashed line). In (a), the dots at 1/K = 0 are the continuum values. For clarity we have included only four representative values of g in (a).

107 For the large-Nc system at hand, the thermodynamics is described by a canonical ensemble of non-interacting glueball and meson-like states. Furthermore, the bound states of the theory constitute a supersymmetric two-dimensional ideal gas. The canonical free energy for such a free gas in D space-time dimensions is given by

D 1 ∞ d − p 1 2 2 √p +Mn = T ln 1 e− T (5.53) Fb (2π)D 1 − n=1 − X Z D 1 ∞ d − p 1 2 2 √p +Mn = T ln 1 + e− T (5.54) Ff − (2π)D 1 n=1 Z − X for bosons and fermions, respectively, where M 2 in the expression for ( ) is the n Fb Ff invariant bosonic (fermionic) mass spectrum. The integral is performed by expanding the logarithm and using the integral representation of the modiﬁed Bessel function

39 of the second kind Kν(x), to ﬁnd

D/2 ∞ ∞ MnT qMn b,M = 2VD 1 KD/2 , (5.55) F − − 2πq T n=1 q=1 X X D/ 2 ∞ ∞ q+1 MnT qMn f,M = 2VD 1 ( 1) KD/2 . (5.56) F − − − 2πq T n=1 q=1 X X Here VD 1 is the volume in D 1 space dimensions. The total free energy is obtained − − by adding these two expressions. Because the spectrum is supersymmetric, the sums over masses Mn traverse the same spectrum, and the total free energy takes the form

D/2 ∞ ∞ MnT (2q + 1)Mn tot,M = 4VD 1 KD/2 . (5.57) F − − 2π(2q + 1) T n=1 q=0 X X For our calculations we use a rescaled form of Eq. (5.57), with D =2

1 ∞ ∞ M T (2q + 1)M ˜ Ftot,M = n K n F ≡ −4(K 1)L 2(K 1) (2q + 1)π 1 T n=1 q=0 − − X X 1 ∞ ∞ d M T (2q + 1)M = k k K k . (5.58) 2(K 1) (2q + 1)π 1 T q=0 − Xk=1 X 39Consider Appendix C.3 for a detailed derivation.

108 In the last line we introduced the factor dk which counts degeneracies of mass eigenvalues. This equation is most eﬃcient in the present calculation, because it expresses the free energy solely as a function of the numerically evaluated bound-state masses

1 M . We have chosen to normalize by ( 4(K 1)L)− , since 2n 4(K 1) is the k − − 0 ≡ − total number of massless states of the free, massive theory (with g = 0 and κ = 1).

In practice, we can truncate the sum over Bessel functions at q = 10 due to fast convergence. Obviously, the sum over states is ﬁnite at any ﬁnite K.

The contribution of the massless states to the free energy in D = 2 dimensions can be calculated analytically to be

nbL ∞ po 2 π b,0 = dpo = n0LT (5.59) F − π e po/T 1 − 6 Z0 − nf L ∞ po 2 π f,0 = dpo = n0LT (5.60) po/T F − π 0 e +1 − 12 Z π ˜ = n T 2 . (5.61) F0 0 16(K 1) − for bosons, fermions, and the contribution to the rescaled total, respectively. Thus one may separate this contribution from the rest of Eq. (5.58).

Finally, the sum over the states is replaced by an integral over the density of states, and Eq. (5.58) becomes

2 1 M ∞ MT¯ (2q + 1)M¯ ˜ = dM¯ 2ρ(M¯ 2) K , (5.62) F 2(K 1) (2q + 1)π 1 T q=0 − Z X where ρ(M 2) = ρ (M 2) = ρ (M 2) for supersymmetric systems. The symmetry b f T splits the bosonic and fermionic sectors into halves. Thus when calculating the free energy, or other thermodynamic properties, we can write

˜ =( ˜b + ˜f ) + +( ˜b + ˜f ) = ˜tot + + ˜tot . (5.63) F F F T F F T − F T F T −

109 5.4.2 An analytic result: The free energy for the free theory

Let us start by exploring the free, massive theory (g = 0, κ = 1,) which can be solved analytically. We will compare the contributions of the meson and glueball sectors to the free energy. In particular, using the free-meson sector we can check the validity of our approach to replace the sums over discrete spectra with a density of states function and check how good our numerical results are compared to an analytic calculation.

First, we compare the free energy obtained by the means of the analytic method outlined above, ˜ , to the free energy extracted from the numerical approach, Fspect. ˜ , using the DoS. The graph presented in Fig. 5.13(a) compares analytic and Ffit numerical results at K = 16 for temperatures 0.015 T 0.5, in units where κ =1. ≤ ≤ We deduce from the plot that the agreement is within 1%, which is a typical result.

Cutoﬀ dependence is very mild: ˜ / ˜ (K =13)=1.012, while ˜ / ˜ (K = Fspect. Ffit Fspect. Ffit 16) = 1.015. In Fig. 5.13(b) we show the free energy of the free, massive theory as a function of the inverse resolution at diﬀerent temperatures. It seems that the free energy ˜ converges for low and intermediate temperatures, while it diverges for F temperatures close to the Hagedorn transition in the continuum limit, as expected.

We can extract the contributions to the free energy of diﬀerent parts of the spectrum by using Eq. (5.58). It is interesting to compare the contributions of the two non-interacting sectors of the theory. At temperature T = 0.5κ and K = 5,

2 2 the three-parton meson states contribute 1.9 10− κ to the free energy, while the × 4 2 corresponding K = 5, three-parton glueball state contributes only 3.9 10− κ . × Results at diﬀerent temperatures are listed in Table 5.1. The free energy ˜ Fmesons associated with the free meson sector dominates the corresponding ˜ . This is Fglueball

110 0.3 0.2 T=0.1 T=0.3 0.25 T=0.5

0.15 L

. 0.2 spect ; F

T 0.15 H 0.1 F

0.1

0.05 0.05

0. 0. 0. 0.05 0.1 0.15 0.2 0.1 0.15 0.2 0.25 0.3 F HT; fitL 1K (a) (b)

Figure 5.13: The free energy ˜ for the free theory (g = 0, κ = 1) as (a) compared F between the analytic and numerical results for K = 16 and (b) a function of 1/K. In (a) the temperature ranges from 0.015 to 0.5, in units where κ = 1, by steps of ∆T = 0.015κ. The dashed line represents an exact match, the solid line the actual relation between ˜ and ˜ . In (b) the temperatures are T = 0.1κ (crosses), Fspect. Ffit 0.3κ (boxes), 0.5κ (triangles).

111 a consequence of the fact that the mesonic spectrum has 4(K 1) massless states − that contribute π T 2 to ˜ and that the meson sector possesses many more states than 8 F the glueball sector, especially low-mass states which are important in the present calculation.

T/κ 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55

˜ /κ2 0.009 0.016 0.026 0.040 0.059 0.089 0.133 0.199 Fmeson ˜ /κ2 0.000 0.000 0.001 0.004 0.011 0.026 0.053 0.101 Fglueball

Table 5.1: The free energy as a function of the temperature in the meson and the glueball sectors in the free theory.

5.4.3 Numerical results for nonzero coupling

We now discuss numerical results for the free energy at ﬁnite values of g. These results were obtained by applying the DoS method, i.e. replacing the sum over states by an integral over the bound-state masses times a density of states computed from a ﬁt to the numerically obtained CDF. First, we discuss the temperature dependence and then the coupling dependence of the free energy ˜, for temperatures that lie F below the zero-coupling Hagedorn temperature T (g = 0) 0.52κ. H ≈ 5.4.4 Temperature dependence of the free energy

The DoS method works also for the interacting theory (g > 0). Using either the discrete spectrum approach (sum over the states; (5.58)), or the DoS ﬁt to the spectrum yields good agreement, at least for weak to medium couplings. The

112 disagreement between the two approaches is typically below 1%, as seen in Table 5.2.

This table also presents contributions to the free energy from about a thousand states up to M 2 =9.10283κ2, which belong to the -even sector at resolution K = 14 and T weak coupling g = 0.1. We also consider, in Table 5.3, the free energy at resolution

K = 16 for large coupling g =4.0.

It is veriﬁed from these tables that, at low temperatures, the major contribution to the free energy comes from the low-lying states, i.e. the states below the mass gap.

In the case of Table 5.3, we have ten such states. As we increase the temperature, more states from above the mass gap will contribute signiﬁcantly. The results are shown in Fig. 5.14 for small coupling, g = 0.1, and in Fig. 5.15 for large coupling, g = 4.0. For K = 16 we ﬁnd that, at a coupling value of g = 0.1, the free energy is quadratic in T. This is expected since for low values of temperature the quasi-massless modes dominate and their contribution should be similar to the massless states of the free theory. The fourth column of Table 5.2, which considers contributions from states below the mass gap (e.g., M 2 [0.0001, 0.00113]κ2), shows clearly that these ∈ states dominate.

At large coupling (g = 4.0) we obtain free energies that are about a hundred times smaller than the available weak coupling free energies. This is less than the free energy ˜ that would be contributed from a single massless state! However, F0 it can be justiﬁed from the spectrum at these temperatures. Recall Figs. 5.9 and

5.10, especially those plots depicting the spectrum below the mass gap. For the example shown in Table 5.3, the dominant symmetry sector is -even, the sector T which includes the lightest state. We denote the contribution of this lightest state by

. This single state accounts for almost all the contribution to the free energy for F1

113 0.3

K=12 FHTL 0.4 0.25 K=14 K=16 T H0.1L»0.515 0.2 H 0.3 D 2 -

0.15 10 ´ F @ 0.2 0.1 F

0.05 0.1

0. 0. 0. 0.1 0.2 0.3 0.4 0.5 0.02 0.04 0.06 0.08 0.1 0.12 T T (a) (b)

Figure 5.14: Free energy ˜ at weak coupling (g =0.1) as a function of T at K = 12 F (crosses), 14 (boxes), 16 (triangles) for (a) all temperatures T < TH (dashed vertical line) and (b) low temperatures with a quadratic ﬁt F (T )= α T 2, with α 0.338. w w ≈

0 T 0.5κ. The latter result is “comparable” to the contribution ˜ , that would ≤ ≤ F0 be made by a single, exactly massless mode, suggesting that this very light state may

be approximated with a massless state. To this end, we point out that this ( +) T lone state, for large g, although much smaller than the rest of the spectrum, it will

only become massless for very large resolution K and this is the reason that at least

for small temperatures ˜ and ˜ do not seem to match. More discussion on this F1 F0 issue follows in the next section. Finally, at this coupling and K = 16, we show in

Fig. 5.15(b) the behavior of the free energy at low T, which appears to be quartic.

114 ˜ ˜ ˜< ˜ T spect. fit fit 0 F 2 2 F 2 2 F 2 2 F2 2 [κ] [10− κ ] [10− κ ] [10− κ ] [10− κ ]

00 0 0 0 0.025 0.00658 0.00654 0.00654 0.0009 0.05 0.03459 0.03448 0.03448 0.0038 0.075 0.08522 0.08499 0.08499 0.0085 0.1 0.15852 0.15810 0.15809 0.0151 ...... 0.225 0.89782 0.89487 0.86245 0.0765 0.25 1.14656 1.14269 1.07109 0.0944 0.275 1.44823 1.44337 1.30232 0.1142 0.3 1.81661 1.81078 1.55613 0.1359 ...... 0.425 5.33793 5.33113 3.16396 0.2728 0.45 6.54702 6.54176 3.55327 0.3059 0.475 7.98870 7.98597 3.96515 0.3408 0.5 9.69247 9.69351 4.39962 0.3776

Table 5.2: Free energy as a function of temperature T at K = 14 in the -even sector T for weak coupling, g =0.1: ˜ is obtained by summing over the eigenvalues in the Fspect. interval M 2 [0.00001, 9.10283]κ2; ˜ is obtained by the DoS method described ∈ Ffit in Section 5.3.1. ˜< and ˜ are the contributions to the latter states below the Ffit F0 mass gap (i.e., M 2 < 0.00113κ2) and of a single supersymmetric massless state, respectively.

5.4.5 Coupling dependence of the free energy

The behavior of the free energy as a function of the coupling is summarized in

Figs. 5.16-5.18. For relatively low temperatures (T 0.1) and for values of g on ≈ the order of one and above (see Fig. 5.16), the DoS ﬁt misses the most important contribution, which is expected from the single lightest state in the -even sector. T For instance, for the resolution K = 16 at coupling g =4.0, the lightest bound state

115 T ˜ ˜ + ˜1 ˜0 F2 2 FT2 2 F2 2 F2 2 [κ] [10− κ ] [10− κ ] [10− κ ] [10− κ ]

00 0 0 0 0.025 0 0 0 0.0008 0.05 0.0002 0.0002 0.0002 0.0033 0.075 0.0011 0.0011 0.0011 0.0074 0.1 0.0033 0.0033 0.0033 0.0131 ...... 0.225 0.0384 0.0383 0.0383 0.0663 0.25 0.0505 0.0503 0.0503 0.0818 0.275 0.0644 0.0641 0.0639 0.0990 0.3 0.0803 0.0796 0.0793 0.1178 ...... 0.425 0.1959 0.1867 0.1813 0.2364 0.45 0.2283 0.2149 0.2068 0.2651 0.475 0.2647 0.2458 0.2340 0.2953 0.5 0.3056 0.2795 0.2628 0.3272

Table 5.3: Results for free energy as a function of temperature T at K = 16 and strong coupling, g = 4.0. ˜ corresponds to the overall free energy including both F symmetry sectors. The third column shows the overall contribution of the -even T sector, the fourth the contribution from the single nearly massless state, M 2 ≈ 0.0362κ2, and the last column is the contribution that a supersymmetric massless state would make, if it were present.

has M 2 =0.0362κ2 and the next available state is at M 2 =4.86κ2. Although the ﬁt in Fig. 5.9(c) seems to capture quite well the behavior of the states below the mass gap, states which are expected to dominate the thermodynamics at low temperatures,

6 2 it yields ˜(T =0.1) 10− κ . This is not what we expect from the CDF data. The F ≈ free energy should be close to the contribution of a pair of massless supersymmetric

π 2 partners, 16(K 1) T . A way to improve the calculation of the free energy is to use − the discrete spectrum and sum over the states instead of approximating this part of

116 0.5 K=12 0.4 K=14 FHTL K=16 0.4 H L 0.3 TH 0.1 »0.515 D D 4 2 -

- 0.3 10 10 ´ ´ @ @ 0.2 F F 0.2

0.1 0.1

0. 0. 0.1 0.2 0.3 0.4 0.5 0.02 0.04 0.06 0.08 0.1 0.12 T T (a) (b)

Figure 5.15: Same as Fig. 5.14, but for strong coupling (g = 4.0). In (b) the low temperature behavior is described by a quartic ﬁt F (T )= α T 4, with α 0.321. s s ≈

the spectrum with a ﬁt function. By extracting the states’ degeneracies from the

5 2 CDF data and by utilizing Eq. (5.58), we obtain ˜(T =0.1) 3.33 10− κ , which F ≈ × matches the expectations much better. In fact, the value of ˜ is the contribution of F one supersymmetric -even state. The -odd sector does not contribute signiﬁcantly T T to ˜, since its lightest state (M 2 =3.651κ2) is heavily suppressed due to the Bessel F

factor K1(M/T ) at T =0.1.

Although failing here, generally (at relatively weak couplings and small temper-

atures) the ﬁt does a good job, mainly because the states below the mass gap are

very light compared to those for large g, and therefore not suppressed by the Bessel

function, K1(M/T ), of Eqs. (5.58) and (5.62). Therefore, for large values of the cou-

pling, we see that as the temperature is gradually being increased, the contribution

117 40. 2. 35. K=12

K=13 30. 1.5 K=14 25. D D 4 4 - - 10 10 ´ 20. ´ @ @ 1. F F 15.

10. 0.5

1. 2. 3. 4. 1.5 2. 2.5 3. 3.5 4. g g (a) (b)

Figure 5.16: The free energy as a function of the coupling g at temperature T =0.1κ (with κ =1) and for resolutions K = 12 (crosses), 13 (boxes), 14 (triangles) with two diﬀerent vertical scales. In (b) we see along with the data the contribution to the free energy that would be made by a pair of exactly massless superpartners. For fairly large values of g and at this temperature, the overall free energy is small compared to the contribution of a single pair of massless states. This is expected at this coupling region because the masses are very large and are suppressed by the modiﬁed Bessel function, K1(x).

to the free energy becomes similar to the one coming from an exact massless mode;

this contribution is included as dotted lines in Figs. 5.16—5.18. These results are

also in accord with the results presented in Tables 5.2 and 5.3.

As a check, we have compared results for the massive, strongly coupled theory (g

large, κ =1) and the massless theory (g =1, κ =0), where g is the only scale factor.

We expect the strongly coupled theory to have masses M related to the masses M ∗ of the massless theory by M 2 = g2M 2. At g =4.0, we ﬁnd M 2 4.052M 2. The free ∗ ≈ ∗

118 4. K=12 K=13 0.5

3. K=14 0.4 D D 2 2 - - 10 10 ´ ´ 0.3 @ 2. @ F F

0.2

1. 0.1

1. 2. 3. 4. 1.5 2. 2.5 3. 3.5 4. g g (a) (b)

Figure 5.17: Same as Fig. 5.16, but for T =0.3κ. From (b) it is clear that for values of g > 1.0 and at this temperature, the main contribution to the free energy again comes from the nearly massless state in the -even sector and it resembles the result T from an exactly massless state (dashed lines).

119 2.5 K=12 0.25 K=13

2. K=14 0.2 D D 1 1 - - 10 1.5 10 0.15 ´ ´ @ @ F F 1. 0.1

0.5 0.05

1. 2. 3. 4. 1.5 2. 2.5 3. 3.5 4. g g (a) (b)

Figure 5.18: Same as Fig. 5.16, but for T = 0.5κ. From (b) it is clear that more states contribute to the free energy at higher temperature, and at relatively large values of g.

energies are related by

˜(T, M 2; K)= g2 ˜ (T/g,M 2; K). F F∗ ∗

We also calculated the free energy with the DoS method described earlier, and we

found that it matches quite well the free energy of the theory with g =4.0 and κ =1.

This is shown in Fig. 5.19(a). Therefore, by solving a numerically less challenging

problem, i.e. the model with no CS term, we were able to determine the strong

coupling behavior of the theory with a CS term.

Having established that g =4.0 is a relatively strong coupling, and by knowing the

exact, weak-coupling (g = 0) free energy, let us calculate the strong/weak coupling

free-energy ratio rs w at K = 16, the highest available resolution in our calculations. −

120 5. 175.

4. 150. L

13 125. D = 4

K 3. - 10 4; = ´ 100. @ g , 1 w - = 2. s Κ r

H 75. F

50. 1.

25.

0. 0. 1. 2. 3. 4. 5. 0.1 0.2 0.3 0.4 0.5 F’ HΚ =0, g=1; K=13L T (a) (b)

Figure 5.19: (a) Comparison of the free energies ˜(g = 4, κ = 1; K = 13) and 2 F F 0(g = 1, κ = 0; K = 13) 4 ˜ at various temperatures between 0.015κ and 0.5κ ≡ F∗ with steps of ∆T =0.015κ, for κ =1. The dashed line represents a perfect match, the solid line the best linear ﬁt ˜(g =4, κ =1)=0.97F 0(g =1, κ = 0). (b) Strong/weak F coupling ratio rs w as a function of the temperature at K = 16. −

121 3 At low temperature, T = 0.1κ, we get rs w(K = 16) 8.47 10− , and at T = − ≈ × 3 0.5κ we obtain rs w(K = 16) 9.46 10− . Results for several temperatures are − ≈ × summarized in Fig. 5.19(b). A quartic ﬁt to the strong-coupling data reveals that, at resolution K = 16, the ratio is

˜ s 2 rs w = F 0.95T . − ˜ ≈ Fw This is consistent with the fact that at low temperatures and weak coupling, g, the massless states dominate and make a contribution proportional to T 2 to ˜ . Fw Our CDF data suggest that in the continuum limit, however, the lightest (nearly massless) state of the strongly coupled theory will become exactly massless, and also yield a contribution proportional to T 2 at low temperatures. On the other hand, we know that in the weak coupling (g 0) theory for ﬁnite K there are exactly ∼

2(K 1) massless pairs of bound states and the ratio rs w, for large K, becomes − − 1 rs w(K) = (2(K 1))− . Thus, we conclude that in the continuum limit − −

K rs w →∞ 0, − −→ and the discrepancy between the strongly and weakly coupled theories becomes max- imal. However, we cannot exclude the possibility that we may have more than one pair of massless states in the strongly coupled sector of the theory for large values of

K, namely a number of massless states proportional to K. Although from Fig. 5.2(a), which refers to the strongly coupled system (dual theory), one may try to argue in favor of the latter statement that the mass of states at small K seem to follow a trend towards the massless limit for relatively large K. However, this is not a deﬁnitive result, at least from our data, because the highest resolution we have is only up to K = 16. Therefore, as far as thermodynamics is concerned in this chapter, we

122 T rs w(K = 13) rs w(K = 14) rs w(K = 15) rs w(K = 16) − 4 − 4 − 4 − 4 [κ] [10− ] [10− ] [10− ] [10− ]

0.025 0.12 0.19 0.29 0.50 0.05 10.44 11.32 12.89 16.56 0.075 42.95 41.24 42.71 49.97 0.1 84.74 76.62 75.77 84.72 0.15 161.82 138.05 130.57 139.74 0.2 217.32 180.08 166.48 174.25 0.3 262.02 208.95 186.77 188.81 0.4 235.75 180.94 154.97 149.70 0.45 207.06 155.59 129.71 121.84 0.5 174.62 128.31 103.76 94.57

Table 5.4: Data for strong to weak coupling ratio rs w for K = 13, 14, 15, 16 at − various temperatures. We show rs w(K = 16) as a function of T in Fig. 5.19(b). −

will just assume that we only have one massless pair in the continuum limit of the strongly interacting sector. Further results for rs w for several values of T and K − are presented in Fig. 5.19(b) and Table 5.4. It seems that rs w decreases with K − starting at medium temperatures.

5.5 Discussion

We have studied the thermodynamics of = (1, 1) super Yang–Mills theory in N 1+1 dimensions with fundamentals and a Chern–Simons term that gives mass to the adjoint partons. We used SDLCQ to solve the theory in the large-Nc approximation.

The theory has two classes of bound states: glueballs, which form a closed strings in color space, and meson-like states, which form open strings. In the large-Nc approximation, these two sectors do not interact with each other, and make independent

123 contributions to the thermodynamics. We previously calculated the contribution of the glueball sector but without a CS term. We found that the meson-like sector dominates the glueball sector for combinatorial reasons, and, therefore, the results presented here represent the full thermodynamics of the theory. Adding a CS term to the theory introduces an additional parameter, and thus allows us to inquire about the coupling dependence of the theory.

We have been able to take the calculation up to resolution K = 16, which eﬀec- tively means that we are diagonalizing matrices that are of order 7 106 by 7 106 × × in our approximation of the continuum ﬁeld theory. We introduced a new Lanczos method, which is particularly valuable in our calculation of the Hagedorn temperature.

It is interesting that the spectrum for this theory has a mass gap, which we have discussed extensively. The states below the mass gap dominate the low temperature behavior of the theory while the states above the mass gap and below the point of inﬂection of the CDF determine the Hagedorn temperature. In fact, the very low temperature behavior is dominated by a few massless or nearly massless states in the theory.

The determination of the Hagedorn temperature from the states beyond the mass gap requires a detailed understanding of the SDLCQ spectrum and careful ﬁtting techniques. We have checked our numerical methods by comparing the solutions of the free, massive theory obtained numerically to those extracted analytically.

We extrapolated the Hagedorn temperature at ﬁxed coupling to the continuum limit. The process is repeated at various values of the coupling to determine the coupling dependence of the Hagedorn temperature. We ﬁnd that it increases with

124 1 the coupling from a value of about TH = 2 κ at g = 0 to a value of nearly 3.0κ at a coupling of g =4.0.

We calculate the free energy of the theory as a function of both the temperature and the coupling. As the coupling vanishes, the bound state spectrum can be obtained analytically; the analytic results agree with our SDLCQ calculations. The theory has 4(K 1) massless fermionic bound states and an equal number of bosonic − bound states. At low temperature and near-zero coupling, the free energy is simply given by the contribution of these massless states, which can be calculated analytically. As the temperature increases, the free energy grows quadratically and starts to diverge as the temperature approaches the Hagedorn temperature. As we discussed above, this point of divergence increases with the coupling.

At strong coupling and very low temperature, the nearly massless bound states dominate the free energy. We ﬁnd one such fermionic and one such bosonic state.

These states have very small masses at the highest resolution. Their masses appear to decrease with increasing resolution, suggesting that they will become massless in the continuum limit. Since the free energy at low temperatures is proportional to the number of nearly massless states, the free energy at strong coupling is independent of the resolution and therefore has this ﬁxed value in the continuum limit. On the other hand, at weak coupling the number of light states grows with the resolution and diverges in the continuum limit, as does the free energy. We therefore ﬁnd that at low temperatures the ratio of the free energies at strong and weak coupling goes to zero as we approach the continuum.

In summary, the relevance of the present work discussed in this chapter is three- fold. We veriﬁed that supersymmetric Yang-Mills theories can have very diﬀerent

125 thermodynamic properties in the weakly- and strongly-coupled regimes, as expected.

We made progress on the way towards studying more “physical” models, i.e. models that share certain features with (supersymmetric) QCD. Thirdly, we improved our methods to evaluate thermodynamic properties, in the sense that their calculation in higher dimensions is very similar to the one presented here, provided that we work in the large-Nc approximation. The obvious caveat is the condition that the mass spectrum be computable numerically within our SDLCQ framework. While issues of renormalization creeping in at higher dimensions are a road-block for most generic theories, one can circumvent the problem by adding more supersymmetry. Looking ahead, a study of SYM with a Chern-Simons term in three dimensions seems feasible within our framework [111], while we will be hard pressed to overcome convergence problems in theories with too many degrees of freedom, such as extended supersymmetric theories or theories in even higher dimensions. Apart from tackling these more realistic theories, there is merit in studying the thermal properties of two-dimensional models, like super-QCD with extended supersymmetry. Interest in this theory stems from the fact that it is related to the theory of a D1-D5 brane system. The theory’s mesonic spectrum calculated with SDLCQ could be compared against the spectrum obtained from string theory calculations [112]. The latter will be the subject of the following chapter.

126 CHAPTER 6

A POSSIBLE FUTURE DIRECTION FOR SDLCQ CALCULATIONS Η ποίηση είναι η αιτία που φθείρει το κάθε τι από το μη είναι στο είναι. –Πλάτωνας

SDLCQ is a powerful and elegant tool, which has already been used previously by Pinsky et al. [27, 113, 114] for some particular supersymmetric models in order to test numerically the Maldacena or AdS/CFT correspondence [14,115], that is the conjecture that strongly coupled gauge theory is dual to a weakly coupled string theory. Already, using the proposed duality, a lot of work has been done regarding the dynamics of supersymmetric gauge theories at strong and weak coupling regimes.

These work has provided some valuable information about the behavior of such theories. Yet the study and calculation of the full massive bound state spectrum at any gauge coupling has to be explored as well. Following the previous work done with the help of SDLCQ we propose the study of a potential candidate model that can serve as a laboratory for understanding gauge and gravity duality, especially because we will have a direct access to nonperturbative numerical results.

The theory that we want to describe is a supersymmetric Yang-Mills, QCD-like model, with ﬁelds in the fundamental representation of the SU(Nc) color gauge

127 group40. Before we proceed, let us describe briefly the background of this theory by using “brane” language. As we will see below the super QCD-like models arise naturally from intersecting D-brane configurations and give us some hope when we study possible extensions to the Standard Model or when we try to explain cosmo- logical problems. D-branes are useful tools in deriving connections between string theory and supersymmetric gauge field theories [116]. In particular the AdS/CFT correspondence [14,115] conjectures the duality between a string theory and a confor- mally supersymmetric theory on special spacetime backgrounds, a.k.a Anti-de Sitter spacetime. Namely, the correspondence has been very successful in providing insight to the strongly coupled gauge field theories. One particular example, which we are interested in, that can be used to establish a quantitative measure for gauge/string duality is the following D3-D7 brane configuration living in a nearly flat41 ten dimensional spacetime background, depicted schematically below. Specifically the cross symbol represents the directions where the brane extends, while the dot represents the directions where the brane is pointlike.

xµ 0123456789 D3 (6.1) ××××· · · ··· D7 ××××××××· · 40I acknowledge at this point that part of this work is a product of our research group’s commu- nication with R. Myers, R. Thomson and D. Mateos of the Perimeter Institute. I would also like to thank my collaborator M. Harada for useful discussions regarding this work. 41 In fact this is guaranteed in the large-Nc limit [14] while at the same time keeping the number of brane probe ﬂavors low ( Nc) as well as the string coupling gs small. This case scenario (i.e., weak string coupling or probe limit) is suitable for studying strongly coupled supersymmetric gauge theories, for it avoids gravitational eﬀects. For nice discussions related to ’t Hooft’s limit and the D-branes consider references [115,117–119].

128 Our brane system assumes Nc coincident D3 branes and one intersecting D7 brane probe with the D7 brane situated parallel to the D3 brane’s worldvolume coordinates, (x0, x1, x2, x3). Now the brane excitations are described by open (oriented) strings with the constraint that their end-points have to end on the branes and cannot be free. For instance, we can have an oriented string originating from D3 brane and ending on D7. Such state is denoted by (3, 7). In the problem under consideration we can have the following four types of oriented strings: (3, 3), (3, 7), (7, 7), and (7, 3).

One is interested in the decoupling limit of the brane system, which is the case where the four dimensional pure SYM living on the D3 brane couples to fundamental matter (e.g., dynamic quarks) with mass proportional to the separation distance between

D3 and D7 branes. In other words, one wants to ﬁnd the supersymmetric lagrangian and eventually the supercharges describing the (3, 3), (3, 7), (7, 3) strings. From the supergravity side of the correspondence one can calculate the meson spectrum by studying the string excitations on the D7 brane [112,118,120,121]. In particular the

D3-D7 system can be reduced to a D1-D5 after compactifying two spatial coordinates (T -duality). The system on the ﬁeld theory side will be described by a two dimensional SYM lagrangian with matter ﬁelds in the fundamental representation.

Therefore it is tempting to utilize SDLCQ in order to study the problem from the gauge ﬁeld theory point of view, that is one can, in principle, compute the meson bound state spectrum and then compare it with the result obtained from the string theory calculation. According to the gauge/string duality the latter is expected to be the same for both theories. As a remark we note that in the large-Nc limit the

Yang-Mills coupling g and the string coupling gs for a Dp brane are related by the

129 Figure 6.1: A schematic diagram showing the D3-D7 brane system. The Nc D3 branes are coincident at the origin, while the brane probe is set at the distance L from the origin; their separation distance. The bound state meson spectrum is dictated by the ﬂuctuations of (3, 7) and (7, 3) bi-fundamental strings. The (3, 3) strings describe the pure adjoint sector of the four dimensional gauge theory.

following condition

2 p 2 p 3 g = (2π) − gsls − , (6.2)

2 where ls is the fundamental string length. Thus for p =3, we have g =2πgs.

6.1 =2 Super QCD in d =3+1 from D3-D7 branes N

On a D3 brane there exists a four dimensional = 4 super Yang-Mills theory N which is broken down to =2 due to the existence of the D7 brane, that is half of N the original sixteen supercharges remain intact after the D7 brane probe is embedded in the x8x9 plane. The general =2, d =3+1 lagrangian describing the SU(N ) N c supersymmetric theory existing on the D3-D7 system is given, in terms of superﬁeld

130 language by

1 1 2 α α˙ 4 2Vs 2Vs = Tr d θ W W + W¯ W¯ +2 d θ Φ†e− Φ e L g2 4 α α˙ i i s=(3,3),(7,7) " Z Z X √2i 2 + d θ ijk ΦiΦjΦk Φi†Φj†Φk† 3 − # Z s 4 2V 2V 2V 2V + d θ (Q†e− (3,3) Qe (7,7) + Qe˜ (3,3) Q˜†e− (7,7) ) Z 2 + d θ √2i(Q˜Φ Q Q†Φ† Q˜†), (6.3) 1,s − 1,s s=(3X,3),(7,7)Z where the summation over s is necessary because we have two gauge theories one living on D3 brane and the other on D7 and with the trace being taken over the color gauge group indices [28,51–53]. The superfields hold all the information about the component fields. For the fields transforming in the adjoint representation of

the color group SU(Nc), the chiral Φi(y, θ) (i = 1, 2, 3) and real vector VWZ (y, θ) superﬁelds are deﬁned as follows42

Φi(y, θ)= φi(y)+ √2θψi(y)+ θθFi(y), (6.4a) 1 V (x, θ)= θσµθA¯ (x) + iθ2θ¯λ¯(x) iθ¯2θλ(x)+ θ2θ¯2D(x), (6.4b) WZ µ − 2

µ µ µ with y = x + iθσ θ¯. The chiral superﬁeld (Φ = Φ†) contains the complex scalar i 6 i

φi, a Weyl (two-component) spinor ψi and a complex (auxiliary) scalar ﬁeld Fi, while

µ the vector superﬁeld (VWZ = VWZ† ) contains the gauge boson A , a Weyl spinor λ and a real (auxiliary) scalar ﬁeld D. Note also that the vector multiplet is taken,

42For completeness we also note here the deﬁnition of the non-abelian gauge superﬁeld strength

1 2 2V 2V W = D¯ e W Z D e− W Z , α 8 α with super-covariant derivatives chosen as follows:

D = ∂ + iσµ θ¯α˙ ∂ , D = ∂¯ iσµ θα∂ . α α αα˙ µ α α˙ − αα˙ µ The two component Weyl notation is understood with spinor index α =1, 2.

131 as usual, in the Wess-Zumino gauge; for nice accounts on the subject consider the following references [35, 40, 51–57]. This way gauge invariance and supersymmetry are manifest in our lagrangian. The = 2 hypermultiplet, corresponding to the N (3, 7), (7, 3) strings, is summarized by the following complex chiral superﬁelds

Q(y, θ)= q(y)+ √2θχ(y)+ θθf(y), (6.5a)

Q˜(y, θ)=˜q(y)+ √2θχ˜(y)+ θθf˜(y), (6.5b) transforming under the fundamental and anti-fundamental representations of the gauge group SU(Nc). In particular, q(y) (q˜(y)) is a complex scalar, χ(y) (χ˜(y)) is a Weyl spinor and f(y) (f˜(y)) is a complex scalar auxiliary ﬁeld; all transforming under the fundamental (anti-fundamental) representation of SU(Nc).

Assuming now the situation where D7 brane decouples (see Fig. 6.1) from the coincident D3 branes Eq. (6.3), becomes

1 1 2 α α˙ 4 2Vs 2Vs = Tr d θ W W + W¯ W¯ +2 d θ Φ†e− Φ e L g2 4 α α˙ i i " Z Z √2i 2 + d θ ijk ΦiΦjΦk Φi†Φj†Φk† 3 − # Z 4 2V 2V + d θ (Q†e− (3,3) Q + Qe˜ (3,3) Q˜†) Z 2 + d θ i(√2Q˜Φ Q + LQQ˜ √2Q†Φ† Q˜† LQ†Q˜†). (6.6) 1,(3,3) − 1,(3,3) − Z

The separation eﬀect between the Nc coincident D3 branes and the D7 probe, which takes place in the large-Nc limit, is in fact the mechanism which gives mass to the bi-fundamental hypermultiplets (3, 7) and (7, 3), respectively. In other words, a vev is introduced for Φ i.e., Φ = L and the associated “quark” mass is 1,(7,7) h 1,(7,7) i √2 L 2 , where α0 ls is the string tension [115–117]. As a consequence of the decou- 2πα0 ≡ pling, terms involving (7, 7) ﬁelds have been dropped, meaning that the gauge theory

132 that is left is purely four dimensional. To this end we simply drop the summation over the index s and also set V(7,7) = 0. We can alternatively phrase the above in a diﬀerent way. For instance, working in ’t Hooft’s limit, that is the large-Nc approximation, while keeping the coupling, λ g2N , for the four dimensional gauge 3 ≡ c theory described by the D3 branes, ﬁxed, guarantees that at the end we will have

Nc coincident copies of D3 branes decoupled from the D7 brane probe. At the same

time the corresponding ’t Hooft coupling for the D7 brane, in terms of λ3 is given

λ3 4 by λ7 = (2π√α0) . This vanishes for α0 0, which can be viewed as the condition Nc → that deﬁnes the decoupling limit between the intersecting branes. We stress that this whole procedure is consistent with supersymmetry, hence the resulting lagrangian is still supersymmetric, describing a QCD-like theory.

Then one rewrites Eq. (6.6) in terms of the component fields with the help of equations (6.4a)-(6.5b), a straightforward but tedious task. Next by eliminating auxiliary fields, Fi and D through their equations of motion and recombining two component Weyl spinor fields into Majorana or Dirac spinors, then our lagrangian

Eq. (6.6) takes the following Dirac covariant form

1 i i = Tr(F F µν ) Tr(Ψ¯ ΓµD Ψ ) Tr(Ψ¯ ΓµD Ψ ) iΨ¯ ΓµD Ψ L −4g2 µν − 2g2 λ µ λ − 2g2 ψi µ ψi − χ µ χ 1 µ µ µ + Tr[(D φ )(D φ )†]+(D q)†(D q)+(D˜ q˜)(D˜ q˜)† g2 i µ i µ µ

1 2 1 Tr([φ†,φ ] )+ Tr([φ†,φ† ][φ ,φ ]) − 2g2 i i g2 j k j k

√2 1 Γ5 1 Γ5 1+Γ5 1+Γ5 i Tr Ψ¯ − Ψ Ψ¯ − Ψ φ† Ψ¯ Ψ Ψ¯ Ψ φ − g2 λ 2 ψi − ψi 2 λ i− ψi 2 λ − λ 2 ψi i " # √2 1 Γ5 1+Γ5 i Tr(φ Ψ¯ − Ψ Ψ¯ Ψ φ†) − g2 ijk i ψj 2 ψk − ψk 2 ψj i 1 Γ5 1+Γ5 1+Γ5 i√2q†Ψ¯ − Ψ + i√2Ψ¯ Ψ q i√2˜qΨ¯ Ψ − λ 2 χ χ 2 λ − λ 2 χ

133 1 Γ5 + i√2Ψ¯ − Ψ q˜† √2Lq†(φ† + φ )q √2Lq˜(φ + φ†)˜q† χ 2 λ − 1 1 − 1 1 1 Γ5 1+Γ5 1+Γ5 i√2˜qΨ¯ − Ψ + i√2Ψ¯ Ψ q˜† + i√2q†Ψ¯ Ψ − ψ1 2 χ χ 2 ψ1 ψ1 2 χ 1 Γ5 1 Γ5 1+Γ5 i√2Ψ¯ − Ψ q i√2Ψ¯ φ − Ψ + i√2Ψ¯ φ† Ψ − χ 2 ψ1 − χ 1 2 χ χ 1 2 χ

2q†φ† φ q 2˜qφ φ†q˜† + q†[φ†,φ ]q q˜[φ†,φ ]˜q† +2 (˜qφ†φ† q q†φ φ q˜†) − 1 1 − 1 1 i i − i i 1jk j k − j k 2 1 g (q†qq†q +˜qq˜†q˜q˜†) q†q˜†qq˜ +2˜qq˜†q†q − 2 − 2h 2 1 Γ5 1+Γi 5 L q†q L q˜q˜† iLΨ¯ − Ψ + iLΨ¯ Ψ , (6.7) − − − χ 2 χ χ 2 χ where the gamma matrices assumed here, are in the Majorana representation43 and note the minus sign in front of the kinetic fermion terms, which differs from the convention we adopted in the previous chapter. See also Appendix B.5 for various conventions and definitions regarding the gamma matrices and the spinors. Thus the Majorana spinor fields, which transform under the adjoint representation of the color group have the following conventional decomposition,

1 θλ 1 θψ Ψ =2 4 , Ψ =2 4 i . (6.8) λ ϑ ψi ϑ λ ψi

Similarly the Dirac spinor Ψχ, which transforms in the fundamental of SU(Nc) is deﬁned as

1 θχ Ψ =2 4 . (6.9) χ ϑ χ Also with regards to the spinor ﬁelds we note that the ones transforming in the ¯ adjoint representation (e.g., Ψλ, Ψψi ) are Majorana spinors, while Ψχ (Ψχ) which transforms under the Nc (N¯c) representation of the gauge group is a Dirac spinor. In

43Notice that although the superfields are written in terms of two component Weyl spinors we can always perform a similarity transformation that changes the Weyl representation into a Majorana representation; see Appendix B. The latter is particularly useful for our calculations mainly because the Majorana fields are manifestly real. For an alternative approach using four-component superfield language see [55,56].

134 addition, the notation as well as the deﬁnition of gauge covariant derivatives and the

ﬁeld strength have been modiﬁed slightly compared to equation (5.3) from Chapter 5.

In this example we adopt the following conventions

F F D F = ∂ + i[A , ],D q = ∂ q + iA q, D˜ q˜ = ∂ q i˜qA , (6.10) µ adj. µ adj. µ adj. µ µ µ µ µ − µ

Fµν = ∂[µAν] + i[Aµ, Aµ]. (6.11)

As usual, the following task is to derive the equations of motion for the ﬁeld content in Eq. 6.7. In particular in light cone consideration one is interested in constraint equations that can be used to eliminate non-physical degrees of freedom. We will just present the result for the gauge boson Aµ(x)

1 1 ∂ F µν = i(A F νµ + F µνA )+ Ψ¯ ΓνΨ g2 µ ab g2 µ µ λ λ ν ν ν + Ψ¯ Γ Ψ i[φ , (D φ )†] + i[D φ ,φ†] ψi ψi − i i i i ab ν ν ν ν ν iq ( q)† + i(D q) q† i˜q†(D q˜) + i(D q˜)† q˜ Ψ¯ Γ Ψ (6.12) − a D b a b − a b a b − χ,b χ,a where the gauge group indices, a, b, e, f, on the ﬁelds have been made explicit. The derivation of the above equation falls along the lines of the example provided in

Appendix C.1. Since we set out to consider SDLCQ we employ the light cone gauge

A+ =0 with ν =+. Expressing everything in terms of light cone coordinates we ﬁnd that A+ab obeys

2 ab ∂ A+ab J , (6.13) − ≡ with

+ T T Jab Jab = i[XI , ∂ XI ]ab + 2(θλ)ac(θλ)cb + 2(θψ )ac(θψi )cb ≡ − i

+ i[φi, ∂ φi†]ab i[∂ φi,φi†]ab − − −

135 2 + g 2θχθχ† + iq∂ q† i∂ qq† +i˜q†∂ q˜ i∂ q˜†q˜ , (6.14) − − − − − − ab where the spinors appear in two component form after we compactiﬁed the x1 and x3 directions. Then the ﬁelds are assumed not to depend on those coordinates. Namely,

µ XI (I = 1, 3) are the SO(2) vector bosons, remnants of the gauge boson A . This

+ expression for the longitudinal current Jab is familiar in the light cone gauge. The only exception is that this formula has a rich ﬁeld content compared to our previously mentioned models. As usual in the SDLCQ consideration of the constraint

Gauss’ law Eq. (6.13) plays an important role in obtaining the ﬁnal expression for

√ the supercharge that is proportional to P −. Namely, Aab− is the only non-dynamical ﬁeld component that has to be removed from the supercharge expression as we will see below. This is done relatively easy ignoring the zero mode problem. The left

moving fermion components (i.e., ϑλ, ϑψi , ϑχ) are non-dynamical as well, fortunately, they do not appear in the ﬁnal expressions of the supercharges so we do not have to worry about them. We emphasize here that unlike the equal-time approach to quantization, light cone formulation has the inherent property of allowing us to remove the unphysical degrees of freedom hence no ghost ﬁelds appear and most importantly in our case, supersymmetry is manifest.

6.2 Supercurrent and supercharges

In SDLCQ approximation the square of the supercharge is used to resolve the bound state problem of the (supersymmetric) theory in hand, instead of the hamiltonian P −. The procedure of deriving the supercurrent in a general four dimensional supercurrent is analogous to the procedure we describe in Appendix C.2 for the three dimensional pure SYM case. In addition, we follow a detailed recipe, concerning the

136 derivation of the =2 supercurrent, provided in the review [122], which is speciﬁc N for the problem we investigate here. After employing the prescription from [122] we arrive to the following expressions for the supercurrent,

2 ρ i µν ρ ρ g S = Γ Γ Tr[F Ψ ]+Γ Γ Tr[Ψ [φ†,φ ]]

adj 2 µν λ 5 λ i i ½ µ ρ ½ Γ5 µ ρ +Γ5 + √2Γ Γ − Tr[Ψ D φ†]+ √2Γ Γ Tr[Ψ D φ ]

2 ψi µ i 2 ψi µ i ½ ρ ½ +Γ5 ρ Γ5 +2 Γ Tr[Ψ φ†φ† ] 2 Γ − Tr[φ φ Ψ ]

ijk 2 ψi j k − ijk 2 k j ψi ½ 2 ρ ½ +Γ5 2 ρ Γ5 2g Γ q†Ψ q˜† +2g Γ − q˜Ψ q, (6.15) − 2 ψ1 2 ψ1

ρ ρ ρ S = Γ Γ q†Ψ q +Γ Γ q˜Ψ q˜†

f.matter − 5 λ 5 λ ½ µ ρ ½ Γ5 µ ρ +Γ5 c + √2Γ Γ − (D˜ q†)Ψ + √2Γ Γ Ψ (D q)

2 µ χ 2 χ µ ½ ρ ½ +Γ5 c L c ρ Γ5 L 2Γ (Ψ φ†q˜† + Ψ q˜†)+2Γ − (˜qφ1Ψχ + q˜Ψχ)

− 2 χ 1 √2 χ 2 √2 ½ µ ρ ½ Γ5 c µ ρ +Γ5 + √2Γ Γ − Ψ (D q˜†)+ √2Γ Γ (D˜ q˜)Ψ

2 χ µ 2 µ χ ½ ρ ½ +Γ5 L ρ Γ5 c L c 2Γ (q†φ†Ψχ + q†Ψχ)+2Γ − (Ψ φ1q + Ψ q). (6.16) − 2 1 √2 2 χ √2 χ

where the first describes the supercurrent for the adjoint fields VWZ and Φi, and the second which corresponds to the fundamental matter fields Q and Q.˜ In light cone coordinates the supercharge is obtained from integrating the light cone time Q ρ ρ ρ component (ρ =+) of S (S + S ) over the spatial coordinate x− ≡ mat adj

+ + + s1 Qα = dx−S = dx− + . (α =1, 2) (6.17) Q s ≡ Q− Z Z 2 α In particular, we ﬁnd in the light-cone gauge, A = A+ = 0 and assuming that the − ﬁelds do not depend on x1, x3, that the upper component (+) of the supercharge is

1/4 + 1 − − Tr 2 Qα = dx 2 2(∂ XI )βI θλ (g − Z h 137 ½ √ ½ √ + 2( σ2)θψi ∂ φi† + 2( + σ2)θψi ∂ φi − − −

i ½ + √2(½ σ2)(∂ q†)θχ + √2( + σ2)θχ∗ ∂ q

− − − ½ + √2(½ σ2)θχ∗ ∂ q˜† + √2( + σ2)(∂ q˜)θχ , (6.18) − − − )α while for the lower one ( ), the one we mainly interested in, we obtain −

1/4 1 + 1 1 Tr √ √ † 2− Qα− = dx− 2 i 2J σ2θλ [XI ,XJ ]βJ βI σ2θλ + 2θλ[φi ,φi] (g ∂ − √2 Z h − + i(σ iσ )θ [X ,φ†]+i(σ + iσ )θ [X ,φ†] 1 − 3 ψi 1 i 3 1 ψi 3 i + i(σ + iσ )θ [X ,φ ]+i(σ iσ )θ [X ,φ ]

1 3 ψi 1 i 3 − 1 ψi 3 i ½ + √2 (σ + ½)θ φ†φ† √2 (σ )φ φ θ ijk 2 ψi j k − ijk 2 − k j ψi i 1 √2(σ + ½)q†θ q˜† + √2(σ )˜qθ q − 2 ψ1 2 − ψ1

√2q†θ q + √2˜qθ q˜† − λ λ

i(σ iσ )q†X θ i(σ + iσ )q†X θ − 1 − 3 1 χ − 3 1 3 χ

+ i(σ + iσ )θ∗ X q + i(σ iσ )θ∗ X q 1 3 χ 1 3 − 1 χ 3

L L ½ √2(σ2 + ½)(θ∗ φ† q˜† + θ∗ q˜†)+ √2(σ2 )(˜qφ1θχ + qθ˜ χ) − χ 1 √2 χ − √2

+ i(σ iσ )θ∗ X q˜† + i(σ + iσ )θ∗ X q˜† 1 − 3 χ 1 3 1 χ 3 i(σ + iσ )˜qX θ i(σ iσ )˜qX θ − 1 3 1 χ − 3 − 1 3 χ

L L ½ √2(σ2 + ½)(q†φ†θχ + q†θχ)+ √2(σ2 )(θ∗ φ1q+ θ∗ q) (6.19) − 1 √2 − χ √2 χ )α where J + is deﬁned by (6.14), β σ3, β σ1, and 1 [X ,X ]β β σ θ = 1 ≡ 3 ≡ − − √2 I J J I 2 λ i√2[X ,X ]θ . The SO(2) vector boson indices I,J =1, 3 are being understood as − 1 3 λ i 2 the labels of the compact dimensions. We have also used the fact that 2 [β1, β3]σ = ½. A comment follows concerning the supercharges of the dimensionally reduced

T version of D3-D7, i.e., D3-D7 1,3 D1-D5, follows. Here we only considered one set → of available supercharges, that is only one of the two = (2, 2) sectors that our model N 138 possesses44. More speciﬁcally, the theory is invariant under another set of = (2, 2) N supercharges 0 that are obtained by making the following replacements [55] in the Q expression for Q

θ θ , θ θ , (6.20a) λ → ψ1 ψ1 → − λ

φ φ† , φ φ† , (6.20b) 2 → 3 3 → − 2

q q˜†, q˜† q, (6.20c) → → − which is essentially the crux of =2 supersymmetry realized in the four-dimensional N theory. For some discussion concerning extended supersymmetry from dimensional reduction consider [123] and Appendix B.4. The SDLCQ bound state spectrum of a similar super Yang-Mills theory was considered by the authors of [25,124]. Note that the choice of which component of Qα− is to be used for extracting the SDLCQ spectrum may raise some concerns. In fact, as was found earlier by the authors of [25,124] the ambiguity is lifted since we can always ﬁnd a unitary transformation such that the two possible hamiltonians (P1−,P2−) are equivalent, thus resulting in the same spectrum.

6.3 Quantization rules and momentum space expansions

SDLCQ program requires that ﬁelds are promoted into quantum ﬁeld operators and expanded in momentum representation as we already discussed in Chapter 3.

For reference we note the quantization rules imposed on ﬁelds in the light cone gauge A+ =0 and at equal light cone times x+ = y+. First we derive the conjugate

44 The = (2, 2) realization of superalgebra is now expressed by Q±,Q± = 2√2δ P ± and N { α β } αβ Q±,Q± = 0, where we have assumed that momentum modes in the compact directions vanish, { α β } P 1 = P 3 =0.

139 ∂ momenta (e.g., πf = L ) from our lagrangian Eq. (6.7). Take as an example the ∂(∂+f) scalar Majorana spinor ﬁeld Ψλ,

∂ i π = Tr[Ψ¯ Γ+∂ Ψ ] Ψλ L 2 λ + λ ∂(∂+Ψλ) −g i 0 σ2 0 0 = θT ϑT −g2 λ λ σ2 0 σ2 0 i = θT 0 . (6.21) −g2 λ Note that the transposition symbol on the two-component real object θλ does not aﬀect the color indices and also notice that the non-propagating component ϑλ is not present as we argued earlier. Proceeding along the same lines we can get rest of the conjugate momenta, regarding the physical degrees of freedom. For reference they are listed below

1 i T i T πX = ∂ XI , πθ = θλ, πθ = θψ , (6.22a) I g2 − λ −g2 ψi −g2 i 1 1 † πφ = ∂ φi , πφ = ∂ φi, πθχ = iθχ† , (6.22b) i g2 − i† g2 − −

πq = ∂ q†, πq = ∂ q, (6.22c) − † −

πq˜ = ∂ q˜†, πq˜ = ∂ q.˜ (6.22d) − † −

Therefore by assuming the large-Nc limit and according to Chapter 3, the canonical equal light come time (anti-) commutation rules on the various quantum ﬁelds are postulated as follows

2 I J ig [Xab(x−), ∂ Xcd(y−)] = δIJ δ(x− y−)δadδbc, (6.23a) − 2 − 2 α β g (θ ) (x−), (θ ) (y−) = δ δ(x− y−)δ δ , (6.23b) { λ ab λ cd } − 2 αβ − ad bc 2 α β g (θ ) (x−), (θ ) (y−) = δ δ δ(x− y−)δ δ , (6.23c) { ψi ab ψj cd } − 2 ij αβ − ad bc 2 i j i j ig [φab(x−), ∂ φcd†(y−)]=[φab† (x−), ∂ φcd(y−)] = δijδ(x− y−)δadδbc, (6.23d) − − 2 − 140 α β 1 (θ ) (x−), (θ† ) (y−) = δ δ(x− y−)δ , (6.23e) { χ a χ b } −2 αβ − ab i [∂ q˜b†(y−), q˜a(x−)]=[∂ qb(y−), qa†(x−)] = δ(x− y−)δab, (6.23f) − − −2 − i [˜qa†(x−), ∂ q˜b(y−)]=[qa(x−), ∂ qb†(y−)] = δ(x− y−)δab, (6.23g) − − 2 − where I,J = 1, 3 are the indices of compact directions, a,b,c,d are color indices,

α, β = 1, 2 are spinor indices, and i, j = 1, 2, 3 refer to the diﬀerent types of ﬁelds,

ψi,φi. The expansions of the physical ﬁelds in terms of creation/annihilation operators in momentum space at a surface where x+ =0, are given below

+ I g ∞ dk I + ik+x I + ik+x X (x−)= [a (k )e− − + a †(k )e − ], (6.24a) ab + ab ba √2π 0 √2k Z + g ∞ dk + + α α + ik x− α + ik x− (θλ) = [b (k )e− + b †(k )e ], (6.24b) ab √2π √2 ab ba Z0 + g ∞ dk + + α α + ik x− α + ik x− (θψ ) = [(vi) (k )e− +(vi) †(k )e ], (6.24c) i ab √2π √2 ab ba Z0 + i g ∞ dk i + ik+x i + ik+x φ (x−)= [c (k )e− − +˜c † (k )e − ], (6.24d) ab + ab ba √2π 0 √2k Z + 1 ∞ dk + + α α + ik x− α + ik x− (θχ) = [d (k )e− + d˜ †(k )e ], (6.24e) a √2π √2 a a Z0 + 1 ∞ dk + + + ik x− ˜ + ik x− qa = [ha(k )e− + h† (k )e ], (6.24f) + a √2π 0 √2k Z + 1 ∞ dk + + + ik x− + ik x− q˜† = [ma(k )e− +m ˜ † (k )e ], (6.24g) a + a √2π 0 √2k Z where in the last expansion instead of using qã we have used qã†, which transforms as a column under the gauge group action; i.e., fundamental representation. Re- member also how the hermitian conjugation affects the adjoint fields, for instance

I I (Xab)† = Xba. Moreover, this deﬁnition allows the color traces to swap order i.e.,

I J K K J I (XabXbcXca)† = XacXcbXba. As a consequence of this convention, the hermitian

141 conjugation on the quantum creation/annihilation operators acts like complex conjugation so no swap of the indices on these operators takes place. Finally using

Eq. (6.23) and Eq. (6.24) one can check the canonical (anti-)commutation relations in terms of the oscillators, that is for the real ﬁelds we get

I + J + + + [a (k ), a †(p )] = δ δ δ δ(k p ), (6.25a) ab cd IJ ac bd − α + β + + + b (k ), b †(p ) = δ δ δ δ(k p ), (6.25b) { ab cd } αβ ac bd − α + β + + + (v ) (k ), (v †) (p ) = δ δ δ δ δ(k p ), (6.25c) { i ab j cd } αβ ij ac bd − while for the complex ones we have

i + j + i + j + + + [c (k ),c †(p )] = [˜c (k ), c˜ †(p )] = δ δ δ δ(k p ), (6.25d) ab cd ab cd ij ac bd − α + β + α + β + + + d (k ),d †(p ) = d˜ (k ), d˜ †(p ) = δ δ δ(k p ), (6.25e) { a b } { a b } αβ ab − + + + + + + [h (k ), h†(p )]=[h˜ (k ), h˜†(p )] = δ δ(k p ), (6.25f) a b a b ab − + + + + + + [m (k ), m†(p )]=[˜m (k ), m˜ †(p )] = δ δ(k p ), (6.25g) a b a b ab − where all the other possible (anti-)commutators vanish identically.

The next step, which is required by light cone quantization is to expand the supercharges in momentum space representation using the above relations and of course the discrete formulation will replace integrals by sums and introduce discrete light cone momenta according to Sections 3.2 and 3.2.1. This is however a relatively tedious and time consuming task, but fortunately with the help of a computer program one can carry out expansions of this kind.

6.4 A comment regarding the expansion of Q±

An important simpliﬁcation to the expression of Q± in momentum representation may occur if some of their terms can be ignored in the large-Nc approximation. It

142 suﬃces to deal with a particular example in order to demonstrate that for large-Nc, operators that contribute to the ﬁnal result, are those which conserve the form of the state upon they act. Consider a typical mesonic state Ψ . Such a state is a trace of | i a string of creation operators acting on Fock vacuum. As we have seen in an earlier chapter the main characteristic of the mesonic state is that its endpoints consist of

(unlike) fundamental particles. Assume the following example

Ψ = n d†a† h† 0 , (6.26) | 1i 1 i ij j| i where the index on the wavefunction indicates the number of adjoint partons in the state, n = 1/N is the normalization constant for Ψ and also the momentum of 1 c | 1i the state has been suppressed. We would like to check explicitly how an operator that includes fundamental partons, say,

ˆ = d† a† d + h† d†a , ( ˆ Q±) (6.27) O k kl l k l lk O ⊂ acts on the above state. It is straightforward, after using the commutator algebra,

Eq. 6.25, to obtain the answer to this question, i.e.,

ˆ 1 Ψ1 = dk† akl† dl + hk† dl†alk di†aij† hj† 0 O| i Nc | i 1 = dk† akl† dldi†aij† hj† + hk† dl†alkdi†aij† hj† 0 Nc | i 1 = dk† akl† aij† hj†δli + hk† dl†dk† hl†δilδjk 0 Nc | i use , = 0 { } 1 1 |{z} = dk† akl† alj† hj† 0 hk† dk† hl†dl† 0 Nc | i − Nc | i 1 1 Ψ2 Ψ0 = α Ψ2 + β Ψ0 . (6.28) ≡ Nc | i−| i Nc | i | i One point which is worth mentioning here is that comparing Ψ1 and Ψ2 we realize e e | i | i that the ﬁrst term of the operator ˆ acts only on the inside of the state, leaving the O 143 fundamental partons intact. In other words, it conserves the structure of the state we begun with. Such operators we call form conserving operators. On the other hand the second term of ˆ produces a state which can be thought of as two disjoint strings with O fundamentals at the endpoints without any adjoint ﬁelds (adjoints) inside. This type of operator does not conserve the form of the state we begun with. In the following lines we will prove that operators of this type can be dropped a-priori since they lead to states that can be dropped in the large-Nc limit.

45 Let us now check the normalization of the Fock states in terms of Nc, assuming that they are orthonormal and that 0 0 =1. Starting with Ψ we have h | i | 0i

2 Ψ Ψ =1= n 0 h a a d d† a† a† h† 0 h 2| 2i 2h | n mn rm r k kl lj j| i 1 2 ¨¨* 2 = n ¨0 0 δrkδmlδnjδrkδmlδnj = n δrkδmlδnj 2 h | i 2 2 2 3 = n2δrrδmmδnn = n2Nc . (6.30)

After normalization, the two states will look like (see Fig. 6.2)

1 Ψ0 = hk† dk† hl†dl† 0 , (6.31) | i Nc | i

45Note that when we invoke inner products here, we make use the fact that δ Nc δ = N . ii ≡ i=1 ii c 46 Here we define Ψp Ψp †, through hermitian conjugation. However, noticeP that the color h | ≡ | i indices are not flipped. For example, Ψ = n∗ 0 h a a d . Indeed, the opposite operation should h 2| 2h | j lj kl k give us Ψ = n (h a a d 0 )† = n d† a† a† h† 0 , which is the ket-state we had initially. This is | 2i 2 j lj kl kh | 2 k kl lj j| i consistent with the field’s definition of hermitian conjugation (A (x))† A (x), we defined earlier. ij ≡ ji

144 Figure 6.2: The ﬁrst picture (a) illustrates the structure of state Ψ which corre- | 0i sponds to two disjoint strings of fundamental partons. The initial fundamental state has broken into two pieces; the form is not conserved. In picture (b) the structure of Ψ is depicted. This one is clearly of the same form as the initial state Ψ . | 2i | 1i

1 Ψ2 = d† a† a† h† 0 (6.32) 3 k kl lj j | i Nc | i and thus, p

3 1 2 1 ˆ Ψ1 = Nc Ψ2 Nc Ψ0 = Nc Ψ2 Ψ0 . (6.33) O| i Nc | i − | i | i − √Nc | i p Since we assume the large-N limit, we see that the state Ψ dominates, by a c | 2i factor √N , over the state Ψ . Hence we drop the latter state ending up with c | 0i

Nc 1 ˆ Ψ d† a† d Ψ = N Ψ . (6.34) O| 1i ≈ k kl l| 1i c| 2i p

Therefore we have shown that in the large-Nc approximation the operators that do not conserve the form of the (Fock) state we started with, can be ignored from the expressions of Q±. This is also an indirect proof that in the large-Nc approximation the glueball and meson sectors are completely decoupled; starting with a meson

(gluon) state we end up with a meson (gluon) state upon acting with Q±.

6.5 Discussion

Inspired by the recent progress and contributions that have been made possible by calculations in the context of AdS/CFT conjecture and in particular the

145 gauge/gravity duality using brane dynamics, we considered here a supersymmetric

QCD model stemming directly from a weakly interacting D3-D7 brane system. This wealthy, in content, theory might worth the eﬀort to be studied using the SDLCQ formulation. In principle, the formulation of the problem in the large-Nc limit and by ignoring the infamous zero mode, is straightforward (e.g., see Chapter 3). In addition, this model, resembles in structure to the theory that we discussed earlier in Chapter 5. The ultimate goal of solving the eigenvalue problem (3.18) for such a theory is to compare the results with the existing results obtained from the gravity sector of the theory [112]. This is one important area where numerical results from two dimensional models can be used directly in string theory calculations.

However, tackling the speciﬁc model we introduced here, using SDLCQ is unfortunately not a trivial task mainly due to software and hardware limitations, one if not the only limitation of (S)DLCQ approach. Already the size of the supercharge, even in terms of the ﬁelds not to mention the momentum space expansion, is an indicating factor to this end. On the other hand, one might want to explore the possibility of possible (discrete) symmetries in the supercharge structure, which eventually may help reduce the size of the matrices that need to be diagonalized.

Nonetheless, SDLCQ program has matured enough and established well itself as one of the most powerful tools for numerical non-perturbative calculations by producing a lot of data over the past decade e.g., [26,64,87,124–126]. Naturally, the next major step forward for SDLCQ is to delve deeper into more realistic theories, especially in higher than two dimensions and for larger harmonic resolution, although two dimensional models are always there to serve as good testing laboratories.

146 APPENDIX A

USEFUL RESULTS RELATED TO SUSYQM

The main purpose of this appendix is to provide as much as possible some detailed calculations related to the discussion of supersymmetric quantum mechanics discussed in Chapter 2.

A.1 Component ﬁeld variations under supersymmetry transformations

The effect of supersymmetry transformations on the components of the superfield is discussed in this appendix. Then we find the (conserved) supercharges. The small changes of the constituent fields (x(t), A(t), ψ(t), ψ¯), under supersymmetry, can be found as follows. First consider the variation of the superfield (2.5) in terms of the component fields, i.e.,

δφ[x, ψ, ψ¯] i[ξQ + Q¯ξ,¯ φ] = ([iξQ, x] + ([iQ¯ξ,¯ x]) + θ([iξQ, ψ] + [iQ¯ξ,¯ ψ]) + . . . ≡ = δx + θ δψ + δψ¯ θ¯ + θθ¯ δF (A.1)

On the other hand, by making use of Eq. (2.19) and Eq. (2.5) into δφ = iξ[Q, φ]+ i[Q,φ¯ ]ξ,¯ we obtain

δφ = iξ[Q, φ].1 + i[Q,φ¯ ].ξ¯

147 = iξ( i∂ + θ∂¯ )(x(t)+ θψ(t)+ ψ¯(t)θ¯ + θθF¯ (t)) − θ t

+ i(i∂¯ θ∂ )(x(t)+ θψ(t)+ ψ¯(t)θ¯ + θθF¯ (t))ξ¯ θ − t ˙ = iξ( i∂ (θψ + θθF¯ )+ θ¯x˙ θθ¯ψ˙) + i(i∂¯( θ¯ψ¯ + θθF¯ ) θx˙ + θθ¯ψ¯)ξ¯ − θ − θ − − = iξ( iψ iθF¯ + θ¯x˙ θθ¯ψ˙)+i( iψ¯ iθF θx˙ + θθ¯ψ¯˙)ξ¯ − − − − − − =(ξψ + ψ¯ξ¯) + θ(F ix ˙)ξ¯+(F + ix ˙)ξθ¯+ θθ¯(iψξ˙ + iψ¯˙ξ¯), (A.2) − δx δψ δψ¯ δF where by comparing| {z } (A.1|) and{z (A.2} )| we{z realize} the component| {z } ﬁeld variations due to supersymmetry transformation,

d δx = ξψ + ψ¯ξ,¯ δF = i (ψξ + ψ¯ξ¯) (A.3a) dt δψ =(F ix ˙)ξ,¯ δψ¯ =(F + ix ˙)ξ. (A.3b) −

Note that the reality condition that we have adopted is satisﬁed as it should be, i.e., the variation of the ﬁeld is also real.

A.2 The superchagres

δL Here we calculate the supercharges using Noether’s technique; on = δqi. That Q δq˙i is (for on-shell variations) the charge for our supersymmetric model is

i δL δL δL δL δx + δψ + δψ¯ + δF − 2Q ≡ δx˙ δψ˙ δψ¯˙ δF˙ ψ¯ ψ =x ˙ (ξψ + ψ¯ξ¯)+( i )(F ix ˙) ξ¯+( i ) ξ (F + ix ˙) − 2 − − 2 x˙ F x˙ F = ξψ(x ˙ + i )+(˙x i )ψ¯ξ¯ − 2 2 − 2 − 2 ξψ ψ¯ξ¯ = (x ˙ iF )+(˙x+iF ) 2 − 2 = iξ ψ (x ˙ + iF ) + i ψ¯ (x ˙ iF ) ξ,¯ (A.4) ⇒ Q − Q Q¯ | {z } | {z } 148 ¯ where we have used δL = i ψ and δL = i ψ . In other words, the action is invariant, δψ˙ − 2 δψ¯˙ − 2 i.e., it can be put in the following form

d δS = dt ξ(t)Q + Q¯ξ¯(t) . (A.5) dt Z Noether charge– Q Therefore the supercharges Q and Q¯ are| given{z by: }

Qˆ = ψ(x ˙ iW (x)), Q¯ˆ = ψ¯(x ˙ + iW (x)) (A.6) − where F = W (x). The above choice of supercharges reproduce our hamiltonian (2.27) − (for x˙ = +i∂ ) i.e., the algebra Q, Q¯ closes on H. x { } To be more precise, we have to mention that the exact conserved charge may be obtained using the “canonical” Noether procedure. Namely, one has to check the variation of the Lagrangian both on-shell and oﬀ-shell. For the oﬀ-shell variation, where no use of the Euler-Lagrange equation has been made, the variation results in

∂L ∂L dQ δL = δφ + δφ˙ = off . (A.7) off i ∂φ i ˙ dt i i ∂φi X If we now make use of the Euler-Lagrange equation the above equation becomes

d ∂L dQ δL = δφ = off . (A.8) on dt i ˙ dt i ∂φi ! X The exact conserved charge is that quantity, which leaves the action unchanged, i.e.,

d δS =0= dt (Q Q ), (A.9) dt on − off Z namely = Q Q . For our SUSYQM model let us choose without loss of Q on − off generality, w0(x)= x, just for convenience. The Lagrangian now reads as

2 1 2 i ˙ 1 F L = x˙ + (ψ¯ψ˙ ψψ¯ )+ w00(x)[ψ, ψ¯]+ + Fw0(x). (A.10) 2 2 − 2 2

149 For simplicity we calculate Q¯ξ¯ and so we will need to recall from Eq. (A.3) the variations which include ξ¯, that is δx, δx,˙ δψ, δψ˙ and δF . We have

∂L ∂L ∂L ∂L ∂L δLoff (ξ¯) = δx + δx˙ + δψ + δψ˙ + δF ∂x ∂x˙ ∂ψ ∂ψ˙ ∂F i i = F ψ¯ξ¯+x ˙ψ¯˙ξ¯+(F ix ˙)ξ¯(ψ¯ + ψ¯˙)+(F˙ i¨x)ξ¯( ψ¯) + iψ¯˙ξ¯(F + x) − 2 − −2 F ψ¯˙ξ¯ x˙ψ¯˙ξ¯ F˙ ψ¯ξ¯ x¨ψ¯ξ¯ =x ˙ψ¯˙ξ¯+ F ψ¯ξ¯ F ψ¯ξ¯ i + ix ˙ψ¯ξ¯ + i + + iF ψ¯˙ξ¯+ ixψ¯˙ξ¯ − − 2 − 2 2 2 x˙ψ¯˙ξ¯ x¨ψ¯ξ¯ i = + + i(x ˙ψ¯ξ¯+ xψ¯˙ξ¯)+ (F˙ ψ¯ξ¯+ F ψ¯˙ξ¯) 2 2 2 d x˙ψ¯ξ¯ F ψ¯ξ¯ d(Q¯ξ¯) = + ixψ¯ξ¯+ i . (A.11) dt 2 2 ≡ dt Similarly for the on-shell variations we have:

∂L ∂L δLon(ξ¯) = δx + δψ ∂x˙ ∂ψ˙ i F ψ¯ξ¯ x˙ψ¯ξ¯ =x ˙ψ¯ξ¯+(F ix ˙)ξ¯( ψ¯) =x ˙ψ¯ξ¯+ i + − −2 2 2 3x ˙ψ¯ξ¯ F ψ¯ξ¯ = + i . (A.12) 2 2

Therefore, the conserved supercharge Q¯ is:

3x ˙ψ¯ξ¯ F ψ¯ ξ¯ x˙ψ¯ξ¯ F ψ¯ ξ¯ Q¯ξ¯ = + i ixψ¯ξ¯ i 2 2 − 2 − − 2 = ψ¯(x ˙ ix)ξ,¯ (A.13) − with Q¯ = ψ¯(x ˙ iW (x)). We set x w(x)= W (x). The other charge can be found − → similarly. This should be Q = ψ(x ˙ + iW (x)). First note that these supercharges produce Hˆ for the standard convention π i∂ . Second, it must be emphasized, x → − x once more, that the ordering of the derivatives is crucial since we are dealing with grassmann variables. Finally, the ﬁnal physical result is the same except from the fact that our convention for π +i∂ is not the commonly used one. x → x

150 A.3 Deriving the hamiltonian

The hamiltonian for Witten’s one-dimensional model can be found using H ≡ ˙ xπ˙ + ψπ˙ + ψπ¯ ¯ L. The fermionic type conjugate momenta must be calculated x ψ ψ − with caution since they obey anti-commuting rules. In particular we have:

δL ∂ i i πψ = = ( ψ¯ψ˙)= ψ,¯ (A.14) δψ˙ ∂ψ˙ 2 −2

i ˙ similarly for the other two we obtain π =x ˙ and π ¯ = ψ.¯ Then plugging these x ψ − 2 results into the above formula for the hamiltonian we get Eq. (2.27). Alternatively one could utilize the supersymmetry algebra, Q, Q¯ = 2H, and verify the hamiltonian { } just by employing formulae (2.28)and (2.1.3).

A.4 Witten index: Discrete spectrum cases

Let us ﬁrst give an example where the Witten index can be calculated for simple cases in the context of SUSYQM and is in agreement with our results in Section 2.1.3.

Direct substitution of our =2 hamiltonian (2.35) in Eq. (2.49) will yield N

∞ 2 2 dpdx β[p /2+W /2 σ3W (x)/2] ∆(β)= Tr σ e− − 0 . (A.15) 3 2π Z−∞

Using now the identity exp( θσ )= ½ cosh(θ/2) + σ sinh(θ/2), we have − 3 3

∞ dpdx β[p2/2+W 2/2] W 0 W 0 ∆(β)= Tr σ e− ½ cosh(β )+ σ sinh(β ) . (A.16) 3 2π 2 3 2 Z−∞ 2 We may now evaluate the traces explicitly recalling that Trσ3 = 0 and Trσ3 = 2 we

ﬁnally have

dpdx β[p2/2+W 2/2] W 0 ∆(β)= e− sinh β . (A.17) π 2 Z Now expanding the hyperbolic sine about β =0 we get

dpdx β[p2/2+W 2/2] W 0 β ∞ βp2/2 ∞ βW 2/2 ∆(0) e− β = dpe− dx W 0 e− ' π 2 2π Z Z−∞ Z−∞ 151 β ∞ βW 2/2 = dx W 0 e− . (A.18) 2π r Z−∞ Now in the special case where W (x) is an even function of x (e.g., W (x) x2m) ∝ it turns out that W 0(x) is odd and therefore the integral over x vanishes, giving

∆(0) = 0. This is expected in SUSYQM since in such case there is no normalizable state, i.e., supersymmetry is spontaneously broken. On the other hand if W (x) is an

2m+1 odd function of x (e.g., W (x) x ), then W 0(x) is even. For this case we get, ∝

β ∞ βW 2/2 ∆(0) = dx W 0 e− 2π r Z−∞ β ∞ βW 2/2 β 2π = dW e− = =1, (A.19) 2π 2π β r Z−∞ r r where in the second equality we have performed a change in variables dW = W 0 dx.

This result suggests that supersymmetry is exact, satisfying our expectations.

A.5 Witten index: Continuous distribution of states

Using “heat kernels”47 the Witten index can be regulated as follows:

∆(β)= dx[K+(x, x; β) K (x, x; β)], (A.20) − − Z

βHˆ where the heat kernels K (x, y; β) y e− ± x can be expanded in terms of the ± ≡ h | | i wave functions that make up the spectrum of the model under investigation, i.e., they may be represented as follows

βE(k) K (x, y; β)= e− ψ∗ (y,k)ψ (x, k) ± ± ± Xk βE(n) βE = e− ψ∗ (y, n)ψ (x, n)+ dEe− ψ∗ (y, E)ψ (x, E). (A.21) ± ± ± ± n X Z 47For their mathematical details consult [45].

152 In the second equality above the two terms stand for the discrete and continuum part of the spectrum respectively. Next we consider the derivative of the diﬀerence between the two kernels, with respect to β. By deﬁnition

d I(β)= (K+(x, y; β) K (x, y; β)), −dβ − − and after using the above relations for the heat kernels we may explicitly write

βE(k) I(β)= e− E(k) ψ+∗ (y,k)ψ+(x, k) ψ∗ (y,k)ψ (x, k) − − − Xk 1 βE(k) 1 1 = e− (E(k)ψ∗ )ψ + ψ∗ (E(k)ψ ) 2 2 + + 2 + + Xk d d ( i)(i) W (y) ψ∗ (y) W (x) ψ (x) − − dy − + dx − + 2 1 βE(k) 1 d 2 = e− +(W + W 0) ψ+∗ ψ+ x=y 2 2 −dx2 k X 2 1 d 2 + +(W + W 0) ψ ψ∗ 2 −dx2 + + 2 (ψ∗ )0(ψ )0 + W (ψ∗ )0ψ + W ψ0 ψ∗ W ψ∗ ψ − + + + + + + − + + 1 βE(k) = e− (ψ∗ )00ψ + ψ∗ ψ00 2W 0ψ∗ ψ −4 + + + + − + + Xk 2W (ψ∗ )0ψ 2W ψ∗ (ψ )0 + 2(ψ∗ )0(ψ )0 − + + − + + + + 1 βE(k) d = e− ((ψ∗ )0 W ψ∗ )ψ + ((ψ∗ )0 W ψ∗ )ψ −4 dx + − + + + − + + k X √2E(k)ψ √2E(k)ψ − −

1 d βE(k)| {z } | {z } = e− 2E(k)ψ (x, k)ψ+(x, k) . (A.22) −2 dx − k X p Therefore the β regulated Witten index takes following form: −

d∆(β) 1 ∞ d βE(k) = dx e− 2E(k)ψ (x, k)ψ+(x, k) . (A.23) dβ 2 dx − Z−∞ k=0 X6 p In the above derivation we have explicitly used A ψ+(x) = i(∂x W (x))ψ (x) = − −

√2Eψ+(x) and also the eigenvalue equation: H ψ (x, k) = E(k)ψ (x, k); see for ± ± ± 153 instance formulae (2.38) and (2.36). Notice also that the above formula is a general result independent of the form of W (x) and also if there exists a zero-energy state in the system, this state does not contribute; see Eq. (2.50). This is true remembering that this state is not paired like the excited states are. For the speciﬁc example that we discussed in Section 2.2.2 we replaced, in the limit L , the sum over → ∞ momentum modes with an integral over the momenta48; L dk. Speciﬁcally, we have to evaluate the following “messy” expression R

β d∆(β) 1 ∞ (k2+1) 2 = dk e− 2 √k +1 ψ (x, k)ψ+(x, k) dβ 2 − Z0 β 2 (( 1 ∞ (k +1) ((( 2 = dk e− 2 k(cos(((kx) sin(kx) + tanh x cos (kx) 2π 0 Z x= (( ∞ ((( 2 k(sin(((kx) cos(kx) + tanh x sin (kx) − x= −∞ β β e− 2 ∞ β k2 e− 2 = (tanh( ) tanh( )) dk e− 2 = , (A.24) 2π ∞ − −∞ √2πβ Z0 where we have made explicit use of the wave-functions that given in Section 2.2.2 and also the integration over x was performed trivially at the beginning.

A.6 Various derivations regarding the ﬁnite temperature model

Before we proceed to describe the procedure of calculating the thermal vacuum expectation values let us ﬁrst show how we derive expressions for the thermal creation/annihilation operators using the Bogoliupov transformation. Let us deﬁne

G θ¯(˜aa a†a˜†) and make use of BCH-formula (2.11). In particular, ≡ −

G G 1 1 a(β) = e− a e = a [G, a]+ [G, [G, a]] [G, [G, [G, a]]] + ..., (A.25) − 2 − 3!

48The normalization of the integral is essentially included in the wave functions (2.54).

154 where [G, a] is: [G, a]=θ¯ [ãa, a] [a†a˜†, a] = θ¯a˜†[a†, a] = θ¯a˜†, and similarly [G, a˜†] − − = 1 − will give: [G, ã†]=θ¯ [ãa, a˜†] [a†a˜†, a˜†] = θa.¯ Using the above identities repeatedly − | {z } in Eq. (A.25) we have

1 2 1 3 1 4 a(β) = a θ¯a˜† + θ¯ a θ¯ a˜† + θ¯ a . . . − 2 − 3! 4! − θ¯2 θ¯4 θ¯3 = a 1+ + + . . . a˜† θ¯ + + . . . 2 4! − 3! = a cosh θ¯ a˜† sinh θ.¯ (A.26) −

One can proceed in analogous fashion to extract the rest of the relations that appear in Eq. (2.60).

Now let us outline the procedure for calculating the thermal vacuum expectation value of (2.61). A standard technique that is used in similar situations, in the H context of algebraic method, is to ﬁrst normal order the expressions and then make use of the fact that the vacuum is destroyed by the annihilation operators e.g.,

Ω a†a Ω = 0 and so on. By doing so we manage to isolate the surviving terms h | | iβ that are proportional to the identity. In our case we ﬁrst normal order the non- thermal operators, where we collect only those terms that have even number of elements and are of the form a†a, a†a†aa, . . . , i.e., we keep terms that contain equal number of creation and annihilation operators. On the other hand terms of the type a†aa, a†a†a†a, . . . will give vanishing expectation values, and thus we discard them right from the beginning, before expressing them in terms of the thermal operators.

To elucidate this procedure let us consider the harmonic oscillator part of Eq. (2.61)

E = Ω (a†a + b†b) Ω 1 h | | iβ

= Ω (a†(β) cosh θ¯ +˜a(β) sinh θ¯)(a(β) cosh θ¯ +˜a†(β) sinh θ¯) Ω h | | iβ

+ Ω (b†(β) cos θ + Ω ˜b(β) sin θ)(b(β) cos θ + ˜b†(β) sin θ) Ω h | h | | iβ 155 2 = cosh θ¯ Ω a†(β)a(β) Ω + cosh θ¯sinh θ¯ Ω a†(β)ã†(β) Ω h | | iβ h | | iβ =0 =0 2 + sinh θ¯cosh θ¯ Ω a˜(β)a(β) Ω + sinh θ¯ Ω 1+ã†(β)ã(β) Ω | h |{z }| iβ h || {z | }iβ

=0 =a ˜(β)˜a†(β) 2 2 + cos θ Ω b†(β)b(β) Ω + sin θ Ω (1 b†(β)b(β)) Ω + ...... h | | |{ziβ } h | −| {z| iβ } vanishing terms =0 = ˜b(β)˜b†(β) | {z } 2 | 2{z β } 1 | 1 {z } = sinh θ¯ + sin θ = e− + , (A.27) 1 e β 1 + e β − − −

β/2 1/2 θ¯ 1 e− where in the last step we have used: e± = 1±e β/2 . After performing the same ∓ − procedure, using a computer code, the rest of the interaction terms are found to be:

2 E = λ λg(a† a)(a† a) 2 h − − − iβ ¨* 0 0 1 2 ¨ ¨¨* = λ + λg λg (a¨†.a†) +2λg a†.a λg ¨(a.a) 2h − ¨ − iβ 2 1 2 2 λ = λ + λg + 2λg(ã[β]†.a˜[β] + 1) sinh θ¯ + . . . = + λg cosh 2θ¯ 2 h iβ 2 2 β λ λg 1 + e− = + , (A.28a) 2 2 1 e β − − g2 E = 3+12a†.a +6a†.a†.a.a 6a.a + a†.a†.a†.a† 6a†.a† 4a†.a†.a†.a 3 8 − − − 3g2 g2 4a†.a.a.a + a.a.a.a = + 12a†.a +6a†.a†.a.a − β 8 8 β 2 2 3g 3g 4 = + 2(2 + 4ã[β]†.a˜[β] +ã[β]†.a˜[β]†.a˜[β].a˜[β]) sinh θ¯ 8 8 2 2 + 4(ã[β]†.a˜[β] + 1) sinh θ¯ 4 cosh θ¯ a[β]†.a[β]+ . . . − β 2 2 3g 3g 2 = 1 + 4 sinh2 θ¯ + 4 sinh4 θ¯ = 1 + 2 sinh2 θ¯ 8 8 2 2 β 2 3g 3g 1 + e− = cosh 2θ¯ = , (A.28b) 8 8 1 e β − − i g E4 = (λ ) (a† a) 0, (A.28c) √2 − 2 h − iβ → ig 3 E5 = (a† a) 0, (A.28d) −√2h − iβ → ig E6 = (a† a) b b† 0, (A.28e) √2h − iβ →

156 where the last three terms do not contribute since they consist of odd numbered strings

of operators, a behavior that is inherited also to the thermal ones. Finally, collecting

the terms and properly arranging them, the interaction hamiltonian contributes the

following:

1 λg 1 + e β g2 1 + e β 2 2g2 1 + e β 2 E = λ2 +2 − + − + − int 2 2 1 e β 4 1 e β 4 1 e β ( − − − − − − ) β 2 2 β 2 1 g 1 + e− g 1 + e− = λ + + . (A.29) 2 2 1 e β 2 1 e β ( − − − − )

157 APPENDIX B

GAMMA MATRICES AND SPINORS IN ARBITRARY SPACETIMES

We provide a review on Cliﬀord algebra and spinors in arbitrary dimensions.

Especially irreducible spinor representations are discussed, that are particularly useful in analyzing supersymmetric gauge theories. In addition we discuss the case of extended superalgebras obtained from theories that have undergone a dimensional reduction. Moreover, we summarize our conventions for spinors and gamma matrices employed in Chapter 6.

B.1 General remarks

When one deals with simple or extended supersymmetries, in arbitrary spacetime dimensions it is convenient to use spinor supercharges with the least number of independent components in a particular spacetime dimension. It is then useful to invoke some of the properties of Dirac’s gamma matrices as well as some of the spinor properties such as charge conjugation and reality conditions. In this appendix we wish to provide a self contained review of Cliﬀord algebras and spinors in arbitrary dimensions for arbitrary spacetime signatures. Detailed original work can be found in [123,127,128].

158 Let us consider t time-like and s space-like dimensions, respectively such that

D = t + s. Cliﬀord’s algebra is deﬁned as

µ ν ν µ µν + =2η ½, (B.1) G G G G where ηµν is the ﬂat metric with

diag(+,..., +, ,..., ). − − t s

Recall that the Lie algebra of the group| {zSO(}t,| s) is{z divided} into two categories, namely the class which includes the vector (or tensor) representations and the other which includes the fundamental spinor representations. Here we are interested about the latter ones. In order to study the spinor representations of SO(t, s) it is convenient to introduce Dirac’s gamma matrices Γµ (µ = 0,...,D 1) which are by defini- − tion irreducible representations of Clifford’s algebra. From the Clifford algebra it is implied that these matrices should have eigenvalues 1 and that are also traceless, ± TrΓµ = 0, then as a consequence the matrices should be of even dimension. First lets consider even D cases. Gamma matrices can be used to define a complete or- thogonal basis for the vector space of even dimensional (complex) matrices. Clifford algebra dictates that this can be achieved by defining anti-symmetric (fundamental) products of gamma matrices

Γ(n) =Γµ1µ2...µn Γ[µ1 Γµ2...Γµn] (B.2) ≡ where n =0, 1,...,D. Alternatively, we may explicitly write49

N µ µ ν µ ν λ µ1 µn (D) Γ =(½, Γ , Γ Γ , Γ Γ Γ ,..., Γ Γ ,..., Γ ), (B.3) { } { } { } { ··· } 49For example in D =3+1 we have ΓµΓν = Γ0Γ1, Γ0Γ2, Γ0Γ3, Γ1Γ2, Γ1Γ3, Γ2Γ3 . { } { }

159 by keeping µ<ν<λ<.... It is also understood that the notation introduced just above means that Γµ1,...µn = 0 if µ = µ and Γµ1,...µn = Γµ1 Γµn if µ = µ . By i j ··· i 6 j D direct counting we see that there are n independent matrices for each value of n. Thus the total number of such independent even dimensional matrices is 2D, therefore the collective index ranges between 1 N 2D. From this fact and recalling that ≤ ≤ for any general matrix of dimension m one cannot have more that m2 independent matrices (i.e., m2 2D), we see that the minimum dimension of the Dirac matrices ≤ has to be m =2D/2. Of particular interest is, Γ(D) the last element of ΓN , which can be written as

s t − 0 D 1 Γ¯ ( 1) 4 Γ . . . Γ − , (B.4) ≡ −

µ 2 such that Γ¯, Γ =0 and Γ¯ = ½. The former one follows immediately from Eq. (B.1) { } and the last one from

s t 2 − 0 D 1 0 D 1 Γ¯ =( 1) 2 Γ . . . Γ − Γ . . . Γ − − s t D(D 1) −2 2− s =( 1) ( 1) ( 1) ½ − − − s t s+t −2 2 s =( 1) ( 1)− ( 1) ½ − − −

s t 2k ½ =( 1) − ½ =( 1) − −

= ½.

Now as far as the odd case D +1 is concerned we note that the above matrix Γ¯ anticommutes with the Γµ’s of D and thus the ΓD in D+1 dimensions is proportional to Γ¯, that is one may deﬁne ¯ D Int[ s t ] Γ, s t = 1mod4 Γ = i −2 Γ=¯ ± − (B.5) iΓ¯, s t = 3mod4 (± − Moreover, we also observe that in odd dimensions the Cliﬀord algebra is degenerate in the sense that the product of all gamma matrices is trivial resulting in a scalar,

160 i.e.,

s t 0 D 1 D −4 Γ . . . Γ − Γ = ( 1) ½, ± − this fact implies that the vector space is spanned by 2D linearly independent matrices of dimension 2D/2. In other words, an odd dimensional Cliﬀord algebra D+1 contains a representation of a subalgebra of even dimension D and thus it has minimum dimension 2D/2. It is now obvious that such a basis is formed by considering Γ(n)

D antisymmetric products with n = 0, 1,..., 2 . Therefore in either even or odd d- dimensions we have a Cliﬀord algebra with 2Int[d] elements where each element has dimension 2Int[d/2].

Dirac matrices obeying algebra (B.1) in arbitrary dimensions can be constructed iteratively by the means of matrix direct products over the Pauli’s sigma σi (i =

1, 2, 3) matrices, bearing in mind some of their properties such as σi† = σi, [σi, σj]=

2 ½ i σ , σ , σ =2δ ½ , σ = σ . Without loss of generality consider timelike ijk k { i j} ij 2 i 2 ≡ 0 signature (t, 0) and even dimension D =2j. For instance we may choose the following representation50 which is in accord with Cliﬀord algebra,

Γ0 = σ σ σ , 1 ⊗ 0 ⊗···⊗ 0 Γ1 = σ σ σ , 2 ⊗ 0 ⊗···⊗ 0 Γ2 = σ σ σ σ , 3 ⊗ 1 ⊗ 0 ⊗···⊗ 0 Γ3 = σ σ σ σ , 3 ⊗ 2 ⊗ 0 ⊗···⊗ 0 . . (B.6)

50Note that the change between Euclidean and Minkowski signatures can be achieved through µ µµ µ ΓM = √η ΓE; a generalized Wick rotation. For example when we start with a Euclidean signature and then we introduce s spacetimes directions we just multiply the last s matrices with i in order to achieve the Minkowski spacetime signature.

161 Γ2m = σ σ σ σ σ , 3 ⊗···⊗ 3 ⊗ 1 ⊗ 0 ⊗···⊗ 0 m Γ2m+1 = σ σ σ σ σ , | 3 ⊗···⊗{z }3 ⊗ 2 ⊗ 0 ⊗···⊗ 0 m . . | {z }

2j 2 Γ − = σ σ σ , 3 ⊗···⊗ 3 ⊗ 1 2j 1 Γ − = σ σ σ , 3 ⊗···⊗ 3 ⊗ 2 and

Γ=¯ σ σ . 3 ⊗···⊗ 3 j The above matrices are hermitian and| in particular{z } when we introduce a Minkowski spacetime we will have, due to the i factor required for the spacelike directions that the timelike component will be hermitian while the spacelike ones antihermitian.

Note that this is not a unique a representation of gamma matrices. We will see below that one can build others, consistent with Cliﬀord algebra, that are equivalent up to similarity transformations.

B.2 Equivalence relations

The properties of gamma matrices are crucially related to the properties of spinors and therefore any restrictions on gamma matrices will enforce constraints on spinors in arbitrary dimensions. So lets discuss some of the implications regarding these matrices. Let us consider ﬁrst equivalence relations between irreducible representations, i.e.,

µ µ 1 Γ0 = SΓ S− , (B.7)

162 where S is a unique (up to a phase) non-singular (unitary) similarity transformation between the two representations. In particular, the following theorems are useful in the discussion of equivalence relations [129].

Theorem 1 For an even dimension D all the 2D/2-dimensional irreducible representations of Cliﬀord algebra ΓN and Γ0N are equivalent to one another and thus { } { } Eq. (B.7) holds true.

N N Proof Let Γ , Γ0 and be n n, n0 n0 and 2n0 2n matrices respectively such U × × × that

1 S = Γ Γ0− . (B.8) N U N N X Then based on the fact that the linearly independent elements of Cliﬀord algebra form a group with multiplication law ΓN ΓM = λNM ΓL(M,N), consider

1 1 Γ0 S = Γ0 Γ0 Γ− = λ Γ0 Γ− M M N U N MN LU N XN XN 1 1 1 = Γ0 λ Γ− = Γ0 Γ− Γ Γ− Γ (B.9) LU MN N LU L N N M N N,L X X ½ 1 = Γ0 Γ− Γ = SΓ . LU L M M | {z } XL Thus the two classes are equivalent so the proof is complete. Note that we have

51 implicitly invoked Schur’s lemma where it assures that n = n0.

Theorem 2 For an odd dimension D +1 all the 2D/2-dimensional irreducible representations of Cliﬀord algebra fall into two inequivalent classes ΓN and ΓN . { } {− } 51 Schur’s lemma states that if D1(g)A = AD2(g) for every group element g in where D1 and D2 are inequivalent irrep. representations, then A =0. Moreover if D(g)A = AD(gG) for every element

g in where D is of ﬁnite dimensional irreducible representation, then A = α½, where α is (real) constant.G Therefore when A commutes with all the elements of then it has to be proportional to the identity. For more details see inside reference [130] G

163 That is, elements of one class cannot be related to the elements of the other class by the means of Eq. (B.7).

Proof We have mentioned earlier that in odd spacetime dimensions one cannot deﬁne a Γ¯ matrix because it is trivial in the sense that it does not interwine Γµ to

Γµ, in other words Γ¯, Γµ = 0. Therefore Γ must belong to another class and − { } 6 − µ thus there is no non-singular matrix S that can connect it to Γµ.

Note that provided ΓN is a representation of Cliﬀord algebra Eq. (B.1) then the following have equal right to be so as well,

T Γ Γ† , Γ∗ , Γ . N ≡ {± N ± N ± N }

According to the above discussion we may deﬁne unitary transformations , (Dirac e A conjugate) (time reversal) and (charge conjugation) such that they connect Γ B C N to ΓN according to the following relations

µ t µ 1 Γ † = ( 1) Γ − , (B.10) e − − A A

µT 1 µ Γ = η − Γ , (B.11) − C C µ t 1 µ Γ ∗ = η( 1) − Γ , (B.12) − B B where η is a sign function ( 1) and obviously these newly introduced matrices satisfy ± Cliﬀord algebra. Note that for even dimensions both signs are acceptable while for odd only one is allowed. The reason being is that in odd, d = D +1, dimensions (D even) we know that ΓD is proportional to Γ¯ and we require that ΓD transforms under

(or equivalently ) like the rest of the gamma matrices according to Eq. (B.11) C B that is,

s t T T T ¯ −4 D 1 1 Γ =( 1) ( η) − ΓD 1 − Γ0 − − C − C···C C 164 : 1 D(D 1) s t D D − − 1 = (η( 1) ) ( 1) 2 ( 1) 4 − Γ0 ΓD 1 − − − C ··· − C D(D 1) D − 1 1 =( 1) 2 − Γ¯ =( 1) 2 − Γ¯ , (B.13) − C C − C C and therefore from comparing this result to Eq. (B.11) we ﬁnd that in odd dimensions,

D η = ( 1) 2 . (B.14) − −

The introduction of interwiner is motivated from the hermiticity property of A gamma matrices which is related to the deﬁnition of the Dirac adjoint spinor, which will be introduced later on in our discussion. First notice that for timelike directions

µ µ µ µ t, Γ † =Γ =Γ , while for spacelike directions s, Γ † = Γ =Γ . Then we deﬁne (t) µ (s) − µ

0 1 t 1 Γ Γ Γ − , (B.15) A ≡ ··· along with some of its properties

t 1 t 2 0 † =Γ − Γ − Γ A ··· (t 1)+(t 2)+ +2+1 0 1 t 1 =( 1) − − ··· Γ Γ Γ − − ··· t(t 1) − 0 1 t 1 =( 1) 2 Γ Γ Γ − , − ··· t(t 1) t(t 1) − (t 1)T 0T − T ∗ =( 1) 2 Γ − Γ =( 1) 2 , (B.16) A − ··· − A where we recall that the above deﬁnition is consistent with the regular Minkowski

0 µ T µT 1T spacetime convention =Γ . In addition, from Eq. (B.11) we have Γ = η Γ − , A − C C where 1 µT 72 1 T µT 1T Γ = η ( − )Γ ( − ), (B.17) C C C C

1 T where Schur’s lemma, implies − ½, or C C ∝

T = ; † = ∗. (B.18) C − C C − C 165 Notice also that from combining Eq. (B.10) and Eq. (B.11) one can check that

µ T t 1T µT T t 1T 1 µ T (Γ †) = ( 1) − Γ = η( 1) ( − − )Γ ( ), − − A A − A C CA

T, (B.19) B≡CA in accord with Eq. (B.12). We further want to express ∗ in terms of , η and t. B B To do so ﬁrst note that

T t t 1 t 1 0 = η ( 1) − Γ − Γ A − C ··· C t t 1 1 = η ( 1) − − − C A C t t(t+1) 1 1 = η ( 1) 2 − − , (B.20) − C A C so we obtain

t t(t+1) ∗ = η ( 1) 2 ∗ † B B − C A AC t t(t+1) = η ( 1) 2 † † − − C A AC t t(t+1) = η ( 1) 2 . (B.21) − −

1 µ1 µn Finally we observe that − Γ Γ (µ = µ ) deﬁnes a quantity which is either C ··· i 6 j symmetric or antisymmetric. In particular, this expression will enable us to sys- tematically determine the signs of η and in arbitrary spacetime dimension. For instance,

1 (n) T 1 µ1 µn T µnT µ1T 1 ( − Γ ) ( − Γ Γ ) = Γ Γ − C ≡ C ··· − ··· C

n n 1 µn µ1 = η ( 1) − Γ Γ − − C ··· n(n 1) n n − 1 µ1 µn = η ( 1) ( 1) 2 − Γ Γ − − − C ··· n(n+1) n 1 µ1 µn = η ( 1) 2 − Γ Γ , (B.22) − − C ···

166 , where we have made use of Eqs. (B.18), (B.11) as well as Cliﬀord algebra; the main tool in this whole discussion. Observe that the the above expression is periodic in n with period 4, thus we may invoke the following sums [131] involving binomial terms with period 4,

d d d 2 d/2 1 dπ n =0mod4: Σ (d)= + + . . . =2 − +2 − cos , 0 0 4 4 d d d 2 d/2 1 dπ n =1mod4: Σ (d)= + + . . . =2 − +2 − sin , 1 1 5 4 d d d 2 d/2 1 dπ n =2mod4: Σ (d)= + + . . . =2 − 2 − cos , 2 2 6 − 4 d d d 2 d/2 1 dπ n =3mod4: Σ (d)= + + . . . =2 − 2 − sin . 3 3 7 − 4 It is pointed out that adding all of the above series we ﬁnd 2d, as expected, and also notice that the periodicity in terms of d is just d mod8, which is the periodicity of

Cliﬀord algebra.

With the help of the above series and in particular combinations among them,

1 (n) T we can count the number of symmetric and antisymmetric matrices ( − Γ ) . On C the other hand, we know that there must be N+ symmetric and N anti-symmetric − matrices, respectively,

Int[d/2] 1 Int[d/2] N =2 − 2 1 . ± ± Thus by comparing the results from the two approaches we can ﬁx the signs of η and for some arbitrary spacetime dimension d. For example, when d = 2 we ﬁnd

3 symmetric and 1 antisymmetric matrices. Speciﬁcally, combinations of (Σ1(2) +

Σ (2)) = 3, and (Σ (2) + Σ (2)) = 1 will yield = η = 1, while combinations 0 3 2 −

(Σ1(2) + Σ2(2)) = 3, and (Σ3(2) + Σ0(2)) = 1, will give = η = 1. Similarly for d = 3 we also ﬁnd 3 symmetric and 1 antisymmetric matrices, where in this case

167 Σ1(2) = Σ2(2) = 3 and Σ0(2) = Σ3(2) = 1 imply that the unique value of η and is just 1.

B.3 Minimal or irreducible spinors

We will start by introducing some deﬁnitions that are needed in order to discuss the possibility of irreducible spinors in arbitrary dimensions d. A general Dirac spinor

Int[ D ] (or fundamental spinor) Ψ has 2 2 complex components and it is in general in a reducible representation of Lorentz algebra. Any conditions imposed on the Dirac spinor, which reduce its components must be consistent with the Lorentz transformation properties of spinor ﬁelds. Perhaps the most notable example of a spinor object apart from the generic spinor Ψ is the Dirac’s conjugate or adjoint spinor. For general spacetime dimensions the adjoint spinor is deﬁned by

Ψ¯ Ψ† . (B.23) ≡ A

Furthermore another spinor that one can construct and is consistent with the Lorentz transformations is the charge conjugated spinor denoted by

T Ψ Ψ∗ = Ψ¯ , (B.24) c ≡ B C where in the last equality we have used Eq. (B.19).

As for the irreducible spinors we have two general classes, namely the spinors which satisfy the chirality (or Weyl) condition and the spinors which obey the reality

(or Majorana) condition. The ﬁrst condition is inspired by introducing right and left moving spinors as follows 1 Γ¯ Ψ ± Ψ Ψ. (B.25) ± ≡ 2 ≡ P±

168 In particular, the Weyl condition requires that the spinor has deﬁnite handedness,

Ψ=Ψ , meaning that Ψ is an eigenstate of chirality matrix Γ¯, that is ±

ΓΨ¯ = Ψ (B.26) ± and such objects are called Weyl spinors and they exist only in even dimensions where the chirality matrix Γ¯ is non-trivial. One can easily observe that under an inﬁnites- imal Lorentz transformation, δ Ψ = 1 ω ΓµνΨ, δ (ΓΨ)¯ = δ Ψ i.e., the operation L 4 µν L ± L is covariant because [Γµν , Γ]¯ = 0. As a consequence of the last statement the chiral projections Γµν = Γµν are representations of Lorentz algebra, thus Weyl spinors ± P± are transforming under chiral representations. Now there is a distinction between the chiralities of Weyl spinors under charge conjugation. Consider, for example charge conjugation of constraint Eq. (B.26),

s t s t − 1 − (ΓΨ)¯ = Γ¯∗Ψ∗ = (( 1) 2 − Γ¯ )Ψ∗ =( 1) 2 Γ¯ )Ψ∗ c B B − B B − B s t ΓΨ¯ = ( 1) −2 Ψ . (B.27) ⇒ c ± − c

We deduce that

s t = 0mod4 ΓΨ¯ = Ψ , (B.28) − ⇒ c ± c s t = 2mod4 ΓΨ¯ = Ψ , (B.29) − ⇒ c ∓ c and as a result, charge conjugation, will not change (self-conjugate) the chirality of spinors in the ﬁrst case Eq. (B.28), while in the second case Eq. (B.29) the chiralities are interchanged (complex-conjugate).

52 Meanwhile, the reality condition is deﬁned by requiring Ψc =Ψ,

T Ψ= Ψ∗ = Ψ¯ , (B.30) B C 52 We could have deﬁned more generally this condition by introducing a matrix 0. However B Lorentz transformation would then require that 0 = α , where α =1. B B | | 169 which is again consistent with Lorentz transformation. Namely, it can be readily checked that δ ( Ψ∗) = (δ Ψ)∗. Comparing both sides by making a direct use of L B B L µν 1 µν δ Ψ, we arrive at the fact that Γ ∗ = − Γ , thus the reality condition is preserved L B B by the Lorentz transformation δLΨ. Then the reality requirement implies that, for

1 consistency, Ψ = (Ψ∗)∗, that is Ψ∗ = ( ∗)− Ψ∗; it is inferred that ∗ = ½. In B B B B other words equation (B.21) has to satisfy53.

t t(t+1) η ( 1) 2 =1. (B.31) − −

Note that the period on the left hand side of the above equation is t mod4, consistent with d mod8, we found earlier. From the procedure outlined in the previous section one can find the signs of (s, t) and η(s, t) for an arbitrary spacetime d and from the above equation can fix the value of t. By having the information for both d and t, then the spatial dimensions are just s = d 2t. Therefore we can find for − which spacetimes the reality condition is allowed. Spinors satisfying this condition are labeled as Majorana spinors. As an example consider s = 1 and t = 1, which yields η = 1; the case for Minkowski spacetimes. On the other hand for d = 2 we can either have = η = 1, indicating that in two dimensions Majorana spinors ± can exist. Often times in the literature [128, 132] regarding even spacetimes, there is a distinction between Majorana and pseudo-Majorana spinors, the first one is identified when η = 1, while the second when η = +1, which reflects the two − 53

There is also a possibility of having spinorial objects in accord with ∗ = ½ i.e., (Ψc)c = Ψ. In such a scenario we can not reduce the Dirac spinor. Instead we canB B introduce− 2n Majorana− spinors Ψi (i = 1, 2,..., 2n) and trade oﬀ the complex components of the fundamental spinor Ψ.

This amounts to introducing a constraint on Ψ such that Ψi = MijΨj∗, where MM ∗ = ½ has a symplectic structure. Symplectic Majorana spinors can be realizedB for example in d =− (5 + 1) spacetime dimensions. In general they exist for s t = 4mod8. −

170 choices for charge conjugation matrices . For more details and a summary of the C results consider Table B.1.

Noteworthy is also the fact that ∗ = ½ and that , are unitary, we ﬁnd that B B A C the matrix is symmetric (( )T = ), a useful result when one manipulates AC AC AC spinors. For instance with the help of this result as well as using Eqs. (B.18),(B.19),

we can deﬁne a the Dirac adjoint for Ψc,

T T 1 T 1 T 1 Ψ¯ Ψ† =Ψ † =Ψ ∗ − =Ψ ∗ − ( − ) c ≡ c A B A A C A A C − CA C

T T 1 = Ψ ∗ − − A A C

T 1 = Ψ − . (B.32) − C

Therefore it turns out that we can alternatively impose Majorana condition such that ¯ ¯ Ψc = Ψ i.e., the charge conjugate equals the Dirac adjoint. Both of the conditions we deﬁned above reduce the (independent) components of Dirac spinors in half. That

Int[ d ] is, a Dirac spinor with 2 2 complex components, after imposing either one of the

d 1 Int[ 2 ] constraints will have 2 2 complex components. Alternatively, one can introduce

1 a reduction coeﬃcient [123] r, where r = 2 corresponds to both reality and chirality conditions, while r = 1 counts for the Dirac spinor itself. The case where both

1 constraints can coexist amounts to r = 4 and it is discussed next. In the meantime, one can check for which even spacetimes the chirality and reality conditions can be imposed simultaneously. The combination of constraints (B.26) and (B.30) yields

Γ(¯ Ψ∗)= Ψ∗ B ±B 1 ( − Γ¯ )Ψ∗ = Ψ∗ ⇒ B B ± s t − ( 1) 2 Γ¯∗Ψ∗ = Ψ∗, (B.33) ⇒ − ± 171 in other words the above quantity is real for dimensions

s t = 0mod4, (B.34) − in which Majorana-Weyl spinors54 exist, see also Table B.1. Majorana-Weyl spinors play an important role in supersymmetric as well as ten dimensional string theory.

To sum up this section, we have discussed the possible reductions that Dirac spinors can have in arbitrary spacetimes, which is extremely handy when one studies supersymmetric theories and/or supergravities. The results are summarized in Table B.1.

B.4 Extended superalgebra from dimensional compactiﬁca- tion

Let us consider the scenario where an imaginary representation (or Majorana representation) for gamma matrices exists in Minkowski spacetimes [123, 128]. It indeed exists under certain situations. These representations are particularly useful for discussing the supersymmetric models in this dissertation. Such representation exists if and only if

s +1=2, 3, 4mod8. (B.35)

The special characteristic of such imaginary representation is that = ½ and = B C Γ0, which guarantee that the Majorana spinor is manifestly real, i.e., −

Ψc =Ψ∗ =Ψ. (B.36)

Moreover note that in this special case the values of and η must be both +1, see Table B.1. As an example we recall here the Majorana representation in two

0 1 0 1 dimensions: Γ = σ2, Γ = iσ1, Γ=Γ¯ Γ = σ3.

54In fact the Weyl spinors are self-conjugate in this case, in accord with Eq. (B.28). Also notice that the other possibility for s t = 2mod4, corresponds to symplectic Majorana-Weyl objects. − 172 spinor condition for t =1 min. d.o.f d mod8 η ∗ Int[ d ] B B W M MW 2r(2 2 )

1 + ! 1 − −

+ s

! ! 2 − − ! 1 + + +

3 + + + ! 2 + + + + c !

4 ! 4 + − − 5 + symplectic 8 − − + s

6 ! symplectic symplectic 8 −+ − − − 7 + symplectic 16 − − + c 8 − − ! 16

+ !− − −

Table B.1: The values for and η are shown for arbitrary spacetimes. Particularly for Minkowski spacetimes (d = s +1) η = +1, whenever the Majorana condition is allowed. We have used superscripts + and in the Majorana condition column to − emphasize the sign of η one has to use so that ∗ = ½ holds. The spinor condition B B column refers to what type of irreducible spinor is allowed: W -Weyl, M-Majorana, MW -Majorana-Weyl, in a certain Minkowski spacetime. Note that the superscripts s and c in the Weyl constraint column (W ) refer to self- and complex-chiral spinors, respectively. The number of real degrees of freedom of the minimal spinor is shown in the last column. For completeness, although non-irreducible spinor representations, the dimensions where symplectic structures can be deﬁned are also provided. Except from the last column every row has a periodicity 8, i.e., d mod8.

173 Consider now the generic form of simple ( =1) superalgebra N

Q , Q¯ = 2(Γm) P , (B.37) { A B} AB m for spacetimes of the form d = 2+I, where I corresponds to some (spatial) dimensions that we want to make compact. Notice that the Dirac indices on spinor charges and the gamma matrices have been made explicit and that we have used Latin characters for the spacetime index. Let us now compactify the I dimensions and also assume that momenta in the compact space vanish. As we have seen earlier, we can decompose consistent with Cliﬀord algebra, the gamma matrices in terms of the non-compact and compact dimensions. We follow the same procedure for the spinors as well. Namely, our spinor space becomes a Kronecker product with dimensions:

d I 2 2 =2 2 2 . We decompose gamma matrices and spinors as follows ×

µ µ Γ (Γ ) ½, (B.38) ∼ αβ ⊗ Q Qi qi, (B.39) A ∼ α ⊗ b i X where µ = 0, 1, is the non-compact spacetime index, α = 1, 2, is the spinor index referring to the non-compact two dimensional spacetime as well. The index i runs from 1 to n, where n is the number of supercharges inherited in the non-compact spacetime after the dimensional reduction. This number n is expected to be pro-

I portional to 2 2 . Ignoring for the time being the compact directions and imposing

i Majorana condition on Qα, the extended superalgebra can be casted as follows:

Qi , Q¯j =2δ (Γµ) P . (B.40) { α β} ij αβ µ

Let us now center our attention to d =2+1 and d =3+1 Minkowski spacetimes, where we assume that an imaginary representation for gamma matrices is being used.

174 In the ﬁrst case the reduction is trivial since the spinor dimensionality in two and three dimensions coincide. In this case we just have

Q1, Q¯1 Q1, Q¯2 Q , Q¯ { } { } = 2(Γµ) P , (B.41) α β ¯ ¯ αβ µ { } ≡ Q2, Q1 Q2, Q2 ! { } { } which written explicitly in terms of components yields

√ P0 P1 Q1, Q2 Q1, Q1 0 2 2 √− { } { } = 2 P0+P1 Q2, Q2 Q2, Q1 ! 2√2 0  { } { } √2  0 P +  2√2 . (B.42) ≡ P − 0 !

+ Thus the = (1, 1) supersymmetry, with (MW ) charges Q Q and Q Q−, N 1 ≡ 2 ≡ becomes manifest in Eq. (B.42). In the above equation we have used the fact that

Q¯ = QTΓ0 = (iQ , iQ ), for Q is a Majorana charge. 2 − 1 α Consider now the four dimensional simple superalgebra reduced down to two dimensions. Furthermore we take the spinor supercharge in d = 4 to be a Majorana spinor, which has four real components. In particular, by counting the real degrees of freedom in both initial and the reduced spacetimes we ﬁnd that they match if n =2

(see also Table B.1). Hence the extended superalgebra can now be described by two independent (two dimensional) Majorana spinors Q1 and Q2 . This is the = (2, 2) α α N supersymmetry algebra in two dimensions. The procedure of realizing the superalgebra in the reduced spacetime is analogous to the example in the previous case.

The result we have can now be generalized, meaning that the number of independent

Int[ d ] Majorana supercharges in the reduced sense will be given by n =2 2 r, where r is the reduction coeﬃcient reﬂecting the condition imposed on the original supercharge

QA. As a general remark, we emphasize that this formula for n is subject to the

175 condition assumed on the spinor in the compact spacetime and also the dimension of the non-compact spacetime.

B.5 Four dimensional Majorana and Weyl representations in light cone coordinates

Below we list some useful results about the Majorana representation of gamma matrices being used in the results of Chapter 6. The deﬁnitions for the spinor ﬁelds used are also shown for convenience.

A. Majorana Representation in d =3+1(ηµν = diag(1, 1, 1, 1), µ = 0, 2, 1, 3 µ =+, , 1, 3) − − − ↔ −

0 1 0 1 1 0 = = = σ3 = 1 1 0 2 1− 0 3 0 1 − 2 3 2 0 0 σ 1 σ 0 2 0 σ Γ = i(1 2)= Γ = i(½ 3)=i Γ = i(2 2)=i − ⊗ σ2 0 ⊗ 0 σ3 ⊗ σ2 0 1 2 3 σ 0 0 1 2 3 σ 0 + Γ0+Γ2 0 0 Γ = i(½ 1)= i Γ5 iΓ Γ Γ Γ = Γ = = √2 − ⊗ − 0 σ1 ≡ 0 σ2 √2 σ2 0 − 2 Γ0 Γ2 0 σ + + + 0 0 Γ− = − = √2 Γ Γ = Γ−Γ− = 0 Γ Γ5 = √2

√2 0 0 ½ 0

½ 0 0 0 0 0 Γ Γ+ = 2 Γ3Γ+ = √2 Γ1Γ+ = √2 − 0 0 σ3 0 σ1 0 2

5 ½ 5 5

½ ½ + ½ Γ σ 0 3 + Γ 1 0 0 1 + Γ 1 0 0 Γ Γ ± = ± Γ Γ ± = Γ Γ ± = − 2 0 0 2 √2 σ3 iσ1 0 2 √2 σ1 iσ3 0 ∓ ± 5 3 + ½ Γ 1 0 0 + + 0 0 1 + σ 0 Γ ± = 2 Γ −Γ = √2 2 Γ −Γ = 2i 2 √2 σ ½ 0 σ 0 0 0 ± 1 3 + σ 0 13 + 0 0 Γ −Γ = 2i − Γ Γ =i√2

0 0 ½ 0

B. Dirac (Ψχ) and Majorana (ΨM ) spinor conventions for Chapter 6

1 1 1 θλ θψi T 0 T 2 T 2 Ψλ 2 4 =Ψ∗ Ψψ 2 4 =Ψ∗ ; i = 1, 2, 3 Ψ¯ λ Ψ Γ = 2 4 ϑ σ θ σ ≡ ϑ λ i ≡ ϑ ψi ≡ λ λ λ λ ψi 0 0 T c T c C = Γ ; A = Γ = A Ψ CA Ψ∗ Ψ =Ψ∗ =Ψλ,ψ − − ≡ λ,ψi λ,ψi i

1 1 1 θχ ¯ 0 2 2 c θχ∗ Ψχ 2 4 =Ψχ∗ Ψχ Ψχ† Γ = 2 4 ϑχ† σ θχ† σ Ψχ 2 4 Ψχ∗ ≡ ϑχ 6 ≡ ≡ ϑχ∗ ≡

Table B.2: Some useful results for gamma matrices and spinor ﬁelds in the Majorana representation that have been used in Chapter 6.

176 Furthermore, we give also the corresponding relations regarding gamma matrices in Weyl representation mainly for future reference. In particular, the Majorana representation shown above and the Weyl representation shown next, are related by

µ µ 1 a unitary transformation, Γ = Mγ − , where M is given by U UM U

1 1 σ2 1 + σ2 M = − . U 2 1 + σ2 1 + σ2 −

A. Weyl Representation in d =3+1(ηµν = diag(1, 1, 1, 1), µ = 0, 2, 1, 3 µ =+, , 1, 3) − − − ↔ −

µ 0 σ µ i µ i ½ γ = σ (½, σ ) σ¯ ( , σ ) σ¯µ 0 ≡ ≡ − 0 3 + 0 3 5 0 1 2 3 ½ 0 + γ +γ 0 σ γ γ 0 σ− γ iγ γ γ γ = − γ = = √2 + γ− = − = √2 ≡ 0 ½ √2 σ¯ 0 √2 σ¯− 0 σ0 σ3 σ¯0 σ¯3 + + σ ± σ¯ ± ;σ ¯ = σ γ γ = γ γ = 0 ± ≡ √2 ± ≡ √2 ± ∓ − − σ+ 0 iσ2 + σ1 0 σ2 iσ1 0 γ γ+ = √2 γ1γ+ = 1 γ2γ+ = 1 − − 0 σ √2 0 iσ2 σ1 √2 0 σ2 iσ1 − − − −

5 5 5

½ ½ + ½+γ 0 0 + γ σ− 0 1 + +γ 1 0 0 γ γ = √2 γ γ − = √2 γ γ = − 2 0 σ+ − 2 0 0 2 √2 0 iσ2 σ1 −

5 2 1 5 5 2 1

½ ½ 1 + ½ γ 1 iσ + σ 0 2 + +γ 1 0 0 2 + γ 1 σ iσ 0 γ γ − = γ γ = γ γ − = − 2 √2 0 0 2 √2 0 σ2 iσ1 2 √2 0 0 − −

5 + 5 ½ + ½+γ 1 0 σ + γ 1 0 0 γ = γ − = 2 √2 0 0 2 √2 σ 0 −

B. Dirac (ΨD), Majorana (ΨM ) and Weyl spinor conventions

χα c 0 α αβ α˙ β˙ 2 2 Ψ =Ψ =Ψ Ψ¯ Ψ† γ = χ χ¯ = iσ ; = iσ M ≡ χ¯α˙ 6 ∗ M ≡ M α˙ ≡ αβ α˙ β˙ ≡− α αβ β˙ α˙ χ χ ; χ χ (χα) = (χα) χ¯ (χψ) = ψ¯ χ¯ = ψ¯χ¯ =χ ¯ψ¯ ≡ β α˙ ≡ α˙ β˙ † ∗ ≡ α˙ † α˙ χ χ˜ Ψ α ; Ψ¯ χ˜α χ¯ C =iγ0γ2; A = γ0 Ψc CATΨ α D ≡ χ˜¯α˙ D ≡ α˙ D ≡ D∗ ≡ χ¯α˙

Table B.3: The Weyl representation of gamma matrices along with some useful results for the light cone coordinates are shown in this table. Spinor conventions and deﬁnitions are also shown.

177 Note that Weyl representation is not particularly useful for the purposes of our calculations since we want our expressions to be in a manifest Lorentz covariant form (a.k.a. Dirac covariant form).

178 APPENDIX C

CONVENTIONS AND DERIVATIONS FOR CHAPTERS 4 AND 5

In this appendix we provide for reference some technical derivations pertaining

Chapters 4, 5 and 6. The gauge boson constraint equation in the light cone gauge is derived for the theory that includes fundamental ﬁelds. Also the derivation of the pure SYM supercurrent is explained. Furthermore the thermodynamic formulae that we use in our numerical calculations are discussed in detail.

C.1 Constraint equation for the gauge boson

The equation of motion for the gauge boson Aµ(x) is derived. From Euler-

Lagrange equations ∂ ∂ ∂ L L =0. (C.1) ρ ∂(∂ A ) − ∂(A ) ρ σ ef σ ef We just need to consider the kinetic terms of Eq. (5.2), that is

= + + + + . . . L L1 L2 L3 L4 1 µν i µ µ µ = Tr F F + Tr ΛΓ¯ D Λ + iΨD¯ Γ Ψ+(D ξ)†D ξ + ..., (C.2) −4 µν 2 µ µ µ

179 where we shall be using explicitly the gauge indices in what follows. Let us begin with : L1 ∂ 2 ∂ L1 = (F ) (F µν) ∂(∂ A ) −4 ∂(∂ A ) µν ab ba ρ σ ef ρ σ ef 1 ∂ = (∂ A ) (∂ A ) + ig[A , A ] (F µν) −2 ∂(∂ A ) µ ν ab − ν µ ab µ ν ab ba ρ σ ef 1 = (δµδνδaδb δνδµδaδb )(F µν) −2 ρ σ e f − ρ σ e f ba = (F ρσ) =(F σρ) , (C.3) − fe fe

∂ 2 ∂ L1 = (F ) (F µν) ∂(A ) −4 ∂(A ) µν ab ba σ ef σ ef ig ∂ = (A ) (A ) (A ) (A ) (F µν ) − 2 ∂(A ) µ ac ν cb − ν ac µ cb ba σ ef ig = (A ) (F σν) +(A ) (F µσ) (A ) (F µσ) (A ) (F σν ) − 2 ν fb be µ ae fa − µ fb be − ν ae fa ig = 2(A ) (F σν) (F σν) (A ) − 2 ν fb be − fb ν be = ig[A , F σν] = ig[A , F νσ] (C.4) − ν fe ν fe

We next attack the adjoint fermionic term, where in addition to the color indices, a, b, c, f, e, we make also explicit the spinor indices, i, j, k, of the Majorana ﬁeld as well,

∂ i ∂ L2 = ΛT Γ0Γµ (∂ Λ) + ig[A , Λ] ∂(A ) 2 ∂(A ) ab µ ba µ ba σ ef σ ef i ∂ = Λk Γ0 Γµ (∂ Λj) + ig[A , Λj] 2 ∂(A ) ab ki ij µ ba µ ba σ ef g k 0 µ ∂ j j = ΛabΓkiΓij (Aµ)bcΛca Λba(Aµ)ca −2 ∂(Aσ)ef − g g = Λk Γ0 Γσ Λj + Λk Γ0 Γσ Λj −2 ae ki ij fa 2 fb ki ij be

g j 0 σ T k g σ =( )( ) Λ (Γ Γ ) Λ + (ΛΓ¯ Λ) ; (extra ( ) sign due to fermion interchange) − − 2 fa kj ae 2 fe − g σ g j 0 σ k = (ΛΓ¯ Λ) + Λ Γ Γ Λ ; ((Γ0Γσ)T = Γ0Γσ) 2 fe 2 fa ji ik ae 180 g g = (ΛΓ¯ σΛ) + (ΛΓ¯ σΛ) 2 fe 2 fe σ = g(ΛΓ¯ Λ)fe. (C.5)

Moreover, the fundamental complex fermionic term yields55 L3

∂ 3 σ σ L = gΨ¯ eΓ Ψf (= gΨ¯ f Γ Ψe). (C.6) ∂(Aσ)ef − 6

Finally we calculate the last term , L4

∂ 4 µν ∂ L = g (D ξ)†D ξ ∂(A ) ∂(A ) µ ν σ ef σ ef µν ∂ = g (∂ ξ† igξ†(A ) )(∂ ξ + ig(A ) ξ ) ∂(A ) µ a − b µ ba ν a ν ac c σ ef µν ∂ 2 = g ig∂ ξ† (A ) ξ igξ†(A ) ∂ ξ + g ξ†(A ) (A ) ξ ∂(A ) µ a ν ac c − b µ ba ν a b µ ba ν ac c σ ef µσ σν 2 σν 2 µσ = igg ∂ ξ† ξ ig g ξ† ∂ ξ + g g ξ†(A ) ξ + g g ξ†(A ) ξ µ e f − e ν f e ν fc c b µ be f σ σ 2 σ σ = igξ ∂ ξ† ig ∂ ξ ξ† + g (A ) ξ ξ† + ξ ξ†(A ) (C.7) f e − f e fc c e f b be Therefore after making use of Eqs. (C.3)-(C.7), the equation of motion for the gauge boson reads as follows

(∂ F σρ) = ig[A , F νσ] + g(ΛΓ¯ σΛ) g(ΨΓ¯ σΨ) ρ fe ν fe fe − ef σ σ 2 σ σ + igξ ∂ ξ† ig ∂ ξ ξ† + g (A ) ξ ξ† + ξ ξ†(A ) (C.8) f e − f e fc c e f b be Finally, since we are studying this model in the light cone gauge, A+ = A 0, we − ≡ choose σ = +. Recall that in our convention for light cone coordinates we identify

µ =0, 1, 2 µ =+, , 2. With these in hand Eq. (C.8) becomes ↔ −

(∂ F +ρ) = ig[A , F ν+] + g(ΛΓ¯ +Λ) g(ΨΓ¯ +Ψ) ρ fe ν fe fe − ef 55 σ σ Note that Ψ¯ eΓ Ψf = Ψ¯ f Γ Ψe; this only occurs for real spinors like the ones in Eq. (C.5). In − 6 σ σ fact this can be rewritten as Ψ¯ Γ Ψ =+Ψ¯ Γ∗ Ψ∗. − e f f e

181 + + + igξ ∂ ξ† ig ∂ ξ ξ†. (C.9) f e − f e

By assuming further that the third direction, x2, is compactiﬁed on a light–like circle, we may drop derivatives with respect to this coordinate, thus Eq. (C.9) yields

+ 2+ T 0 + 0 + (∂ F −)fe = ig[A2, F ]fe + g(Λ Γ Γ Λ)fe g(Ψ†Γ Γ Ψ)ef − − + + + igξ ∂ ξ† ig ∂ ξ ξ†, (C.10) f e − f e

+ where by expanding F − in terms of the gauge boson components we arrive at:

0 0 : 0 + +> +> 2 + 0 + 2 +> 2 ∂ (∂ A− ∂−A + ig[A , A−]) = ig[A2, ∂ A ∂ A ig[A , A ]]fe − − fe − − √2 0 λ + g λ λ˜ 0 0 λ˜ fe √2 0 ψ g ψ ψ˜ − † † 0 0 ψ˜ ef + ig (ξ∂ ξ†)fe ig (∂ ξξ†)fe. (C.11) − − −

2 After sorting the terms in the above expression and using the facts that A2 = g22A =

2 A and ψ†, ψ =0, we arrive at − { e f }

2 2 2 √2 (∂ A−)fe = g i[A , ∂ A ]+ λ, λ + √2ψψ† + iξ∂ ξ† i ∂ ξξ† (C.12) − − 2 { } − − − fe Therefore we conclude that the longitudinal current, J + J, is given by ≡

2 2 √2 J = i[A , ∂ A ]+ λ, λ + √2ψψ† + iξ∂ ξ† i ∂ ξξ†, (C.13) − 2 { } − − − which completes the derivation. As a ﬁnal remark note that the inclusion of a Chern-

Simons term in our theory (see Eq. (5.2)) will just modify the above expression by introducing a term proportional to κ.

182 C.2 The pure super Yang-Mills conserved current

We derive the supercurrent for the ﬁelds in the adjoint representation, i.e., the pure super Yang-Mills theory using the supersymmetric ﬁeld variations (5.4). Let us

ﬁrst attack the oﬀ-shell variation, δ , of the lagrangian (5.2a). Namely, Loff 1 δ = (F µν ) δ(F ) , (C.14) L1 −2 ab µν ab and after making use of the facts that δ(D Λ)=D δΛ and δF =D δA D δA , µ µ µν µ ν − ν µ we arrive at

1 1 δ = (F µν ) (D δA ) (F νµ) (D δA ) L1 −2 ab µ ν ba − 2 ab ν µ ba = (F µν ) (D δA ) = (F µν ) D (¯Γ Λ) − ab µ ν ba − ab µ ν ba i = Tr (F µν¯Γ D Λ) (C.15) −2 ν µ

For the adjoint fermionic part we have a few more steps to consider. That is,

i i δ = δΛ¯ Γµ(D Λ) + Λ¯ ΓµD δΛ + igΛ¯ Γµ[δA , Λ] . (C.16) L2 2 ab µ ba 2 ab µ ba ab µ ba

Let us now work on the first and second terms of δ , since the last term, as we will L2 prove at the end vanishes. In fact, after substituting the expression for δΛ¯, we find that the first term becomes

➀ i δ = Tr (F ¯Γµν ΓρD Λ) L2 −8 µν ρ i i = Tr (¯Γµν ΓρΛD F ) D [Tr (F ¯Γµν ΓρΛ)] (C.17) 8 ρ µν − 8 ρ µν

Meanwhile the second term, after being integrated by parts, gives

➁ ➀ i δ = δ + D [Tr (F ¯ΓµνΓρΛ)] L2 L2 8 ρ µν i = Tr (¯Γµν ΓρΛD F ) . (C.18) 8 ρ µν 183 The objective now is to cast the first term in (reffirsttermDL), which also appears in (reffirsttermDL), as a total derivative. To this end we recall that

µ ν µν µνλ Γ Γ = η ½ + i Γλ, (C.19) which is the result after summing (B.1) and (5.5). In particular, by invoking the last identity we have

iµνλΓ Γ = iµνλ (η + i Γσ) = iµνρ µνλ Γσ λ ρ λρ λρσ − λρσ = iµνρ δµδν δνδµ Γσ − ρ σ − ρ σ = iµνρ δµΓν + δνΓµ. (C.20) − ρ ρ

Putting this result in δ ➁, it will become L2

➁ 1 i δ = Tr (¯Λ DρF µν ) Tr (¯ΓνΛDµF ) L2 −4 ρµν − 2 µν 0 ≡ i µν i ρν = Tr|(F ¯Γ{zν DµΛ) } Dρ [Tr (F ¯ΓνΛ)] , (C.21) 2 − 2 where the ﬁrst term in the ﬁrst equality vanishes due to the Bianchi identity

DµFνρ +DρFµν +DνFρµ =0.

Therefore, the oﬀ shell variation of Eq. (5.2a), after gathering terms from equations (C.15), (C.17) and Eq. (C.21), yields a surface term

➀ ➁ δ = δ + δ + δ Loff L1 L2 L2 i i = ∂ Tr (F ρν¯Γ Λ) Tr (F ¯Γµν ΓρΛ) (C.22) ρ −2 ν − 8 µν where we have used the fact that Dρ[Tr(. . .)] = ∂ρ(. . .). In other words, the oﬀ-shell supercurrent Kρ is just

i i Kρ = Tr (F ρν¯Γ Λ) Tr (F ¯Γµν ΓρΛ) . (C.23) −2 ν − 8 µν 184 Before we consider the on-shell variation let us prove that the third term in

Eq. (C.16) vanishes identically. We ﬁrst express the ﬁelds in terms of the gauge

a a a a a b ab group generators, Aµ = AµT , Λµ =ΛµT with TrT T = δ . Thus we have

¯ µ ¯ α µ b c a b c Tr ΛΓ [Λ, δAµ] = Λ Γ Λ δAµTr T [T , T ] =if bcdT d 1 = (Λ¯ αΓµΛb)(¯Γ Λ|c){z } −2 µ f abc = (Λ¯ αΓµΛb)(Λ¯ cΓ ) (C.24) 2 µ where the transposition identity, Λ¯ cΓ = ¯Γ Λc, for Majorana spinors have been µ − µ used. Notice that the quantity ΛbΛ¯ c, is a 2 2 matrix in spinor space, which can be × written with the help of Fierz identities56 as follows

b c 1 c b 1 c µ b Λ Λ¯ = (Λ¯ Λ )½ Γ (Λ¯ Γ Λ ), (C.25) −2 − 2 µ where the terms in parentheses are just the expansion coeﬃcients, thus can be moved around freely. We then substitute the above identity into Eq. (C.24), which reduces to

f abc f abc f abc (Λ¯ αΓµΛb)(Λ¯ cΓ ) (Λ¯ cΛb)(Λ¯ aΓµΓ ) (Λ¯ cΓρΛb)(Λ¯ aΓ Γ Γµ) 2 µ ≡ − 4 µ − 4 µ ρ f abc = (Λ¯ bΛc)(Λ¯ aΓµΓ ) − 4 µ f abc + (Λ¯ bΓρΛc)(Λ¯ aΓ Γ Γµ), (C.26) 4 µ ρ

56 Int[ d ] Int[ d ] In general dimensions d, spinor expressions of the form ΨΛ¯ are 2 2 2 2 , which can be expanded in terms of a complete Cliﬀord algebra basis Γ(n); see Appendix×B. In particular, from reference [133] we ﬁnd that the following expansion is suitable for our purposes

Int[d] n(n−1) ¯ 1 ( 1) 2 ¯ (n) ΨΛ= d − (ΛΓ(n)Ψ)Γ , − Int[ 2 ] n! 2 n=0 X d 1 where n [0, d] and n [0, − ] for d even and d odd, respectively. ∈ ∈ 2

185 where in the last equality we have used Λ¯ aΛb = Λ¯ bΛa and Λ¯ aΓµΛb = Λ¯ bΓµΛa, both − valid for Majorana spinors. Expression (C.26) can be further simpliﬁed by recalling the following identities, true for gamma matrices,

µ d = 3 µ d = 3 ½ Γ Γ = d½ = 3 , Γ Γ Γ = (2 d)Γ = Γ , (C.27) µ µ ρ − ρ − ρ and by interchanging labels c a, as well. After making these simpliﬁcations, ↔ equation (C.26) becomes

f abc(Λ¯ αΓµΛb)(Λ¯ cΓ ) f abc(Λ¯ aΛb)(Λ¯ c). (C.28) µ ≡

The above equation vanishes identically since the L.H.S is symmetric under interchange a b, while the R.H.S is anti-symmetric under this exchange. Therefore the ↔ proof has been completed. The proof can alternatively be carried out by explicitly making use of matrices, recalling the deﬁnitions (5.9) for the Majorana fermions Λ and the conventions for the Majorana representation in d =3. However using the gen-

µ eral method we applied here, one may as well show that the term, Tr ΛΓ¯ [Λ, δAµ] , vanishes also in spacetimes other than d =2+1.

Let us now start the on-shell variation of the lagrangian (5.2a), that is we want to ﬁnd the Noether current,

∂ J ρ = L δX , (C.29) ∂(∂ X ) i i ρ i X associated with the supersymmetry variations (5.4). In particular, we ﬁnd that

ρ ∂ ∂ J = L δ(Aσ)ef + L δΛef ∂(∂ρ(Aσ)ef ) ∂(∂ρΛef ) i i = (F νρ) (¯Γ Λ) + (ΛΓ¯ ρΓµν ) (F ) 2 fe ν ef 8 ef µν fe i i = Tr(F ρν¯Γ Λ) + Tr(F ΛΓ¯ ρΓµν ), (C.30) −2 ν 8 µν 186 where the second (Majorana bilinear covariant) term, in the last expression above, can be transposed as follows

ρ µν T T 0 0 µν T ρ T 0 T (ΛΓ¯ Γ ) = (Γ Γ )Γ Γ Γ Λ ; (( ) sign due to fermion exchange) − −

0 µν T 0 ρ = ¯Γ Γ Γ Γ Λ ; ((Γ0Γρ)T = Γ0Γρ) − µν ρ =¯Γ Γ Λ ; (Γµ T = Γ0ΓµΓ0, = Γ0, see Eq. (B.11)) (C.31) − C −

Therefore the Noether supercurrent is given by

i i J ρ = Tr(F ρν¯Γ Λ) + Tr(F ¯Γµν ΓρΛ). (C.32) −2 ν 8 µν

Eventually, the pure SYM supercurrent ¯ ρ, which is the difference between the J ρ S and Kρ, is given by i ¯ ρ = Tr(F ¯ΓµνΓρΛ). (C.33) S 4 µν The final task is to find the expression for the two component Majorana super-

+ + charge, dx− , in the light cone gauge A =0. Moreover, the contributions Q ≡ S coming fromR the fundamental field terms in the action (5.2) can be found by following an analogous procedure to the one we described above. The only thing which deserves some caution is when we differentiate and transposing Dirac spinors, where the spinor itself and its adjoint are treated independently as opposed to the Majorana spinor fields we encountered here.

C.3 Free gas thermodynamics in D spacetime dimensions

Let us now derive the free energy formula that is being used in our numerical calculations, which is appropriate (only) for a free (non-interacting) gas, in our case a supersymmetric one. We employ spherical polar coordinates in D spacetime di-

2 mensions. It is convenient in the case at hand since Mn is just a numerical constant

187 and p2 is just the polar vector in this space. Without loss of generality we consider only the bosonic case,

q p2+M 2 ∞ ∞ T √ n b D 1 ∞ D 2 e− F = T S − dpp − − V (2π)D 1 q − n=1 0 q=1 X Z X q √p2+M 2 D 1 ∞ ∞ ∞ e− T n D 2 2 2 = T S − dpp − ; ( p + M = p0 pdp = p0dp0) D 1 n ⇒ (2π) − q n=1 q=1 Z0 p X X q p0 ∞ ∞ D 3 T D 1 ∞ 2 2 − e− = T S − dp p (p M ) 2 ; ( p0 = Mnt dp0 = Mndt) (2π)D 1 0 0 0 − n q ⇒ − n=1 q=1 Mn X X Z D 1 ∞ ∞ M − ∞ D 3 q D−3 q D 1 n 2 − Mnt 2 M t = T S − dt t(t 1) 2 e− T ; (dv = t(t 1) 2 ,u=e− T n ) (2π)D 1 q − − − n=1 q=1 1 X X Z D 1 ∞ ∞ ∞ D 1 q D 1 Mn − dt 2 Mnt q = T S − (t 1) 2 − 2 e− T M (2π)D 1 q D 1 − T n − n=1 q=1 1 X X Z − ∞ ∞ ∞ D 1 q D 1 D 2 Mnt = S − M dt (t 1) 2 − 2 e− T (2π)D 1(D 1) n − − − n=1 q=1 Z1 X X K (qM β) ∝ D/2 n D+1 ∞ ∞ D/2 D 1 D| 2 Γ( 2 {z) KD/2(qMnβ}) = S − M (2π)D 1(D 1) n √π(qM β)D/2 − n=1 q=1 n − X X ∞ ∞ = (D) K (qM β) C D/2 n n=1 q=1 X X D/2 ∞ ∞ M =2 n K (qM β), (C.34) 2πβq D/2 n n=1 q=1 X X where, β =1/T, and we have also used the fact that the surface area of the (D 1)- − unit sphere is given by (D 1)/2 2 π − D 1 = , − D 1 S Γ( 2− ) and that

D/2 D 1 D D+1 D/2 22 π 2− M Γ( ) M (D) n 2 =2 n . D 1 D 1 D/2 D 1 D/2 D/2 C ≡ 2 π √πMn (D 1)Γ( − ) β q 2πβq − − − 2 The interchange of sum over q and the integral over p can be justiﬁed by direct term-by-term integration. The integrals are ﬁnite, at least in the range of values of

188 our interest. For completeness, we consider an example

∞ 1 ∞ q 2 ν Mnt = dt (t 1) − 2 e− T Iν=1 − Z1 q=1 ν=1 X ∞ 1 1 ∞ 1 2 2 Mnt 2 Mnt = dt (t 1) 2 e− T + dt (t 1) 2 e− T + . . . − − Z1 Z1 1 1 K1(1Mnβ)+ K1(2Mnβ)+ K1(3Mnβ)+ . . . ∝ 2Mnβ 3Mnβ ∞ 1 = K (qM β), qM β 1 n q=1 n X where term-by-term integration is explicitly justiﬁed. This can be further proved for general ν. Note also that the sum over the masses is omitted because it does not aﬀect the above procedure.

For future reference let us also provide the expressions regarding the (normalized) internal energy and entropy of the supersymmetric ideal gas we discuss in this work.

Let m˜ = (2q + 1)Mn and n˜ = Mn/(2q + 1), then we have:

D 1 nT˜ 2 m˜ m˜ m˜ ˜= m˜ K D 2 + K D+2 +T (D 2)K D , (C.35) E −2T 2π 2− T 2 T − 2 T n,q X D 1 nT˜ 2 m˜ ˜= m˜ K D+2 . (C.36) S −T 2 2π 2 T n,q X Particularly, for the speciﬁc heat, in D =2, we obtain

M 2 m˜ m˜ ˜ = m˜ 3K + K . (C.37) CV −8πT 2 1 T 3 T n,q X

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