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

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Two Dimensional Supersymmetric Models and Some of Their Thermodynamic Properties from the Context of Sdlcq 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 By 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 ther- modynamic 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, re- spectively. ................................. 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
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