DOCSLIB.ORG

PHYS 652: String Theory II Homework 2

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
PHYS 652: String Theory II Homework 2 PHYS 652: String Theory II Homework 2 Due 2/27/2017 (Assigned 2/15/2017) 1 Heterotic Strings Last semester, you constructed the closed bosonic string by combining left- and right-moving copies of the open bosonic string subject to the level-matching condition. In this class we repeated that construction to obtain the type-II closed superstrings from two copies of the open type-I superstring. In the RNS formalism we employed a GSO projection to remove the tachyon and computed the massless spectrum in lightcone gauge. In the GS formalism spacetime supersymmetry extends the lightcone gauge to superspace and there is no tachyon. Despite the differences between the two formulations, the massless spectra agree. The heterotic strings are closed superstrings obtained by combining the bosonic string for the left-movers with the type-I superstring for the right-movers, again subject to level-matching. Split the 26 coordinates of the critical bosonic string into 10 X's and 16 Y 's and treat the X's as the left-moving side of the spacetime coordinate with the right-moving side identified with the 10 X's of the critical type-I superstring. Using level-matching, construct the massless spectrum in both the RNS and GS formalisms and interpret it in terms of gauge and gravity multiplets. At this point, it is not possible to determine the gauge group of this theory but you can deduce the rank. If, in addition, you were told that the gauge algebra is semi-simple and its dimension is 496, what are the possibilities for the heterotic strings? Why is this string called \heterotic"? 2 Fermionization and Picture Fixing reparameterization-invariance of the bosonic string to conformal gauge requires the in- clusion of the bc ghost system. This introduces a degeneracy of the ground state since the 0-mode of the c ghost forms an abelian super-algebra fc0; c0g = 0 (2.1) of which, the ground state should be a representation. In this simple case, the representation is two-dimensional j"#i with j"i = c0 j#i : (2.2) 1 Vertex operators U constructed from the former are called \unintegrated". A vertex operator of the form Z d2σI (2.3) is called \integrated". The two are related by I = b(U) where b(U) means the simple pole in the OPE of b with U. Use this to show that the integrated vertex operator is BRST-invariant if the unintegrated vertex operator is. Recall that both integrated and unintegrated operators were used to define the bosonic string scattering matrix. Worldsheet supersymmetry of the RNS superstring requires us to include the βγ ghost system. The NS ground state satisfies βr≥1=2 j0iNS = 0 and γr≥1=2 j0iNS = 0: (2.4) The operator-state correspondence related this state to an operator traditionally denoted δ(γ). Show that this implies that the operator products γ(z)δ(γ(0)) = O(z) and β(z)δ(γ(0)) = O(z−1): (2.5) Friedan-Martinec-Shenker ghosts allow us to write an explicit expression for δ(γ). Consider the odd ghost η(z), its conjugate ξ(z) and a holomorphic scalar φ(z): 1 η(z)ξ(0) ∼ ; φ(z)φ(0) ∼ − log(z): (2.6) z Show that the combinations β $ e−φ@ξ and γ $ eφη (2.7) satisfy the same operator products as the βγ system. Construct the ghost number current JFMS and stress-energy tensor TFMS to have the correct OPEs as those in the βγ representation. Show that δ(γ) $ e−φ (2.8) satisfies (2.5). What is the conformal weight of this operator? Define the \picture-changing operators" Z = fQ; ξg and Y = ic@ξe−2φ; (2.9) and show that if U is annihilated by Q then so is ZU. Also show that if ZU = fQ; Λg for some Λ then U = fQ; Y Λg. Conclude that if U is in the cohomology of Q then so are ZnU and Y nU for any n = 1; 2;::: . This is the picture-changing ambiguity of the RNS superstring vertex operators that needs to be addressed in the construction of superstring scattering amplitudes. 2 3 BRST vs. Cohomology Let g be a Lie algebra which we assume to be real for definiteness. Picking a basis ta for the Lie c algebra with a = 1;:::; dim(g) defines the structure constants fab of the Lie algebra in this basis. a Introduce the ghosts c as anti-commuting vectors in this basis for the Lie algebra and let ba denote the conjugate ghost momenta. Anti-symmetric tensors of the Lie algebra define the analog of differential forms. Replacing the basis of 1-forms of the Lie algebra with ghosts, we can take a form of degree p ap a1 ! $ c : : : c !a1:::ap (3.1) to be identified with a monomial of ghost-number p. Consider the collection of all such forms. This is a vector space graded by ghost number a N = c ba (3.2) with a p-form corresponding to ghost-number p.(I.e. an element of this vector space is a formal sum of p-forms of degrees p = 0; 1;:::; dim(g).) It is also a (graded abelian) algebra since products of sums of elements of the form (3.1) are still sums of elements of the form (3.1). Show that the Chevalley-Eilenberg differential a 1 a b c Q = c ta − 2 c c fab bc (3.3) acts on this algebra, squaring to 0. Note that Q has ghost-number 1. Define the cohomology H∗(g; R) = ker(Q)=im(Q); (3.4) ∗ Ldim(g) p and note that is also a graded algebra. This again splits by ghost number H = p=0 H . What does H0(g; R) the cohomology at ghost number 0 compute? Now take g to be the supersymmetry algebra. This is almost a graded (!) abelian algebra except that there is the relation c fQa;Qbg = fab Pc; (3.5) where Qa are the fermionic generators and P are the usual translation generators. Introduce the appropriate ghost system and construct the Chevalley-Eilenberg differential. It should depend on Clifford/Dirac/Pauli matrices. Under what assumption on the latter does this differential square to 0? This construction can also be carried out for affine Lie (super-)algebras. The Chevalley-Eilenberg differential for the Virarsoro algebra is worked out in GSW volume 1 where it is shown that can- cellation of the anomaly is equivalent to the condition Q2 = 0. 4 Pure Spinor Superparticle The Brink-Schwarz action for the ten-dimensional superparticle is Z µ µν S = dτ (PµΠ + ePµη Pν) : (4.1) 3 .µ . Here Πµ = x − ΘγµΘ is the supersymmetric velocity and e(τ) is an ein-bein for the world-line. This is a first-order formalism in which the momentum Pµ is an independent variable. (The square is taken with the target space Minkowski metric.) It is manifestly super-Poincar´einvariant in the target space (assuming e and P are invariant under global supersymmetry) and reparameterization invariant on the worldline. Show that it is invariant under the κ symmetry . µ µ a µ a a δX = Θγ δΘ ; δΘ = (γ κ) Pµ ; δPµ = 0 ; δe = Θ κa: (4.2) a Compute the momentum pa canonically conjugate to Θ to conclude that there is a constraint on the phase space variables of the form µ da = pa − (γ Θ)Pµ: (4.3) Show that they satisfy the supersymmetry algebra µ fda; dbg = −2i(γ )abPµ; (4.4) where the bracket is the graded Poisson bracket. Since the Hamiltonian P 2 is also a constraint (it generates reparameterizations), this implies the spinorial constraints (4.3) are not all first class. Introduce a bosonic spinor \ghost" λa, define the Berkovits differential a Q = fλ da; · g (4.5) acting on the phase space through the Poisson bracket. Find the condition on λ that makes Q2 = 0 (cf. problem3). \Wave functions" of ghost number 1 are of the form a U = λ Aa(X; Θ): (4.6) By definition, they are in the cohomology of the Berkovits differential when they are closed QU = 0 but not exact, that is, when U cannot be written as Qf for any scalar superfield f(X; Θ). Recall that exact parts get shifted by the gauge transformation δU = Qα for a gauge parameter superfield α(X; Θ). Expand (0) a (1) a b (2) α(X; Θ) = α (X) + θ αa (X) + θ θ αab (X) + :::; (4.7) and use this to show that there is a \Wess-Zumino gauge" in which1 µ µνλ b Aa = (γ θ)aAµ(X) + (θγ θ)(γµνλ)abχ (X) + :::: (4.8) Not all of the components of the gauge parameter superfield α are used up to get to this gauge. (0) In particular, the lowest component α(X) = α (X) = α(X; Θ)jΘ=0 survives. Show that in this gauge, the \photon" component has the correct gauge transformation δAµ(X) = @µα(X): (4.9) 1Recall that this means that certain components in the Θ expansion of U and Qα are of the same form. This means that those components of α can be used to remove the analogous components of A. 4.
Load more

Recommended publications
  • Off-Shell Interactions for Closed-String Tachyons
    Preprint typeset in JHEP style - PAPER VERSION hep-th/0403238 KIAS-P04017 SLAC-PUB-10384 SU-ITP-04-11 TIFR-04-04 Off-Shell Interactions for Closed-String Tachyons Atish Dabholkarb,c,d, Ashik Iqubald and Joris Raeymaekersa aSchool of Physics, Korea Institute for Advanced Study, 207-43, Cheongryangri-Dong, Dongdaemun-Gu, Seoul 130-722, Korea bStanford Linear Accelerator Center, Stanford University, Stanford, CA 94025, USA cInstitute for Theoretical Physics, Department of Physics, Stanford University, Stanford, CA 94305, USA dDepartment of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India E-mail:[email protected], [email protected], [email protected] Abstract: Off-shell interactions for localized closed-string tachyons in C/ZN super- string backgrounds are analyzed and a conjecture for the effective height of the tachyon potential is elaborated. At large N, some of the relevant tachyons are nearly massless and their interactions can be deduced from the S-matrix. The cubic interactions be- tween these tachyons and the massless fields are computed in a closed form using orbifold CFT techniques. The cubic interaction between nearly-massless tachyons with different charges is shown to vanish and thus condensation of one tachyon does not source the others. It is shown that to leading order in N, the quartic contact in- teraction vanishes and the massless exchanges completely account for the four point scattering amplitude. This indicates that it is necessary to go beyond quartic inter- actions or to include other fields to test the conjecture for the height of the tachyon potential. Keywords: closed-string tachyons, orbifolds.
    [Show full text]
  • UV Behavior of Half-Maximal Supergravity Theories
    UV behavior of half-maximal supergravity theories. Piotr Tourkine, Quantum Gravity in Paris 2013, LPT Orsay In collaboration with Pierre Vanhove, based on 1202.3692, 1208.1255 Understand the pertubative structure of supergravity theories. ● Supergravities are theories of gravity with local supersymmetry. ● Those theories naturally arise in the low energy limit of superstring theory. ● String theory is then a UV completion for those and thus provides a good framework to study their UV behavior. → Maximal and half-maximal supergravities. Maximal supergravity ● Maximally extended supergravity: – Low energy limit of type IIA/B theory, – 32 real supercharges, unique (ungauged) – N=8 in d=4 ● Long standing problem to determine if maximal supergravity can be a consistent theory of quantum gravity in d=4. ● Current consensus on the subject : it is not UV finite, the first divergence could occur at the 7-loop order. ● Impressive progresses made during last 5 years in the field of scattering amplitudes computations. [Bern, Carrasco, Dixon, Dunbar, Johansson, Kosower, Perelstein, Rozowsky etc.] Half-maximal supergravity ● Half-maximal supergravity: – Heterotic string, but also type II strings on orbifolds – 16 real supercharges, – N=4 in d=4 ● Richer structure, and still a lot of SUSY so explicit computations are still possible. ● There are UV divergences in d=4, [Fischler 1979] at one loop for external matter states ● UV divergence in gravity amplitudes ? As we will see the divergence is expected to arise at the four loop order. String models that give half-maximal supergravity “(4,0)” susy “(0,4)” susy ● Type IIA/B string = Superstring ⊗ Superstring Torus compactification : preserves full (4,4) supersymmetry.
    [Show full text]
  • Pitp Lectures
    MIFPA-10-34 PiTP Lectures Katrin Becker1 Department of Physics, Texas A&M University, College Station, TX 77843, USA [email protected] Contents 1 Introduction 2 2 String duality 3 2.1 T-duality and closed bosonic strings .................... 3 2.2 T-duality and open strings ......................... 4 2.3 Buscher rules ................................ 5 3 Low-energy effective actions 5 3.1 Type II theories ............................... 5 3.1.1 Massless bosons ........................... 6 3.1.2 Charges of D-branes ........................ 7 3.1.3 T-duality for type II theories .................... 7 3.1.4 Low-energy effective actions .................... 8 3.2 M-theory ................................... 8 3.2.1 2-derivative action ......................... 8 3.2.2 8-derivative action ......................... 9 3.3 Type IIB and F-theory ........................... 9 3.4 Type I .................................... 13 3.5 SO(32) heterotic string ........................... 13 4 Compactification and moduli 14 4.1 The torus .................................. 14 4.2 Calabi-Yau 3-folds ............................. 16 5 M-theory compactified on Calabi-Yau 4-folds 17 5.1 The supersymmetric flux background ................... 18 5.2 The warp factor ............................... 18 5.3 SUSY breaking solutions .......................... 19 1 These are two lectures dealing with supersymmetry (SUSY) for branes and strings. These lectures are mainly based on ref. [1] which the reader should consult for original references and additional discussions. 1 Introduction To make contact between superstring theory and the real world we have to understand the vacua of the theory. Of particular interest for vacuum construction are, on the one hand, D-branes. These are hyper-planes on which open strings can end. On the world-volume of coincident D-branes, non-abelian gauge fields can exist.
    [Show full text]
  • H ( Ff^L°9 . INTERNATIONAL CENTRE for \ | THEORETICAL PHYSICS
    IC/87/420 H ( ff^l°9 . INTERNATIONAL CENTRE FOR I 'V. ': \| THEORETICAL PHYSICS HARMONIC SUPERSTRING AND COVARIANT QUANTIZATION OF THE GREEN-SCHWARZ SUPERSTRING E. Nissimov S. Pacheva and INTERNATIONAL ATOMIC ENERGY S. Solomon AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION T T IC/87/420 ABSTRACT International Atomic Energy Agency Based on our recent work, we describe a new generalized model of space- and United Nations Educational Scientific and Cultural Organization time supersymmetric strings - the harmonic superstring. It has the same phys- ical content as the original Green-Schwarz (GS) superstring but it is defined on INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS an enlarged superspace: the D=10 harmonic superspace and incorporates addi- tional fennionic string coordinates. The main feature of the harmonic superstring model is that it contains covariant and irreducible first-class constraints only. This allows for a straight- HARMONIC SUPERSTRING AND COVARIANT QUANTIZATION forward manifestly super-Poincaie'covariant Batalin-Fradkin-Vilkoviski (BFV) OF THE GREEN-SCHWARZ SUPERSTRING * - Becchi-Rouet-Stora-Tyutin (BRST) quantization. The BRST charge has a re- markably simple form and the corresponding BFV hamiltonian exhibits manifest Parisi-Sourlas O5p(l,l|2) symmetry. E. Nissinsov ** In a particular Lorentz-covariant gauge we find the covariant vertex opera- International Centre for Theoretical Physics, Trieste, Italy, tors for the emission of the massless states which closely resemble the covariant "classical" expressions previously guessed by Green and Schwarz. S. Pacheva Institute of Nuclear Research and Nuclear Energy, Boul. Lenin 72, 1784 Sofia, Bulgaria and S. Solomon Physics Department, Weizmann Institute, Hehovot 76100, Israel. MIRAMARE - TRIESTE December 1987 •' To appear in "Perspectives In String Theories" (Proceedings of the Copenhagen Workshop, Denmark, October 1987).
    [Show full text]
  • The Quantized Fermionic String
    Class 2: The quantized fermionic string Canonical quantization Light-cone quantization Spectrum of the fermionic string, GSO projection . The super-Virasoro constraints, which allow to eliminate the ghosts, are implemented and analyzed in essentially the same way as in the bosonic string. One new feature is the existence of two sectors: bosonic and fermionic, which have to be studied separately. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring. Summary The fermionic string is quantized analogously to the bosonic string, although now we'll find the critical dimension is 10 . One new feature is the existence of two sectors: bosonic and fermionic, which have to be studied separately. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring. Summary The fermionic string is quantized analogously to the bosonic string, although now we'll find the critical dimension is 10 The super-Virasoro constraints, which allow to eliminate the ghosts, are implemented and analyzed in essentially the same way as in the bosonic string. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring.
    [Show full text]
  • JHEP01(2008)065 Larity Trans- January 17, 2008 January 22, 2008 January 28, 2008 (5) Pure Spinor U / 10) ) Ghosts, Together with Spin-Half RNS Fermions Alism
    Published by Institute of Physics Publishing for SISSA Received: January 17, 2008 Accepted: January 22, 2008 Published: January 28, 2008 Explaining the pure spinor formalism for the JHEP01(2008)065 superstring Nathan Berkovits Instituto de F´ısica Te´orica, State University of S˜ao Paulo, Rua Pamplona 145, 01405-900, S˜ao Paulo, SP, Brasil E-mail: [email protected] Abstract: After adding a pair of non-minimal fields and performing a similarity trans- formation, the BRST operator in the pure spinor formalism is expressed as a conventional- looking BRST operator involving the Virasoro constraint and (b, c) ghosts, together with 12 fermionic constraints. This BRST operator can be obtained by gauge-fixing the Green- Schwarz superstring where the 8 first-class and 8 second-class Green-Schwarz constraints are combined into 12 first-class constraints. Alternatively, the pure spinor BRST opera- tor can be obtained from the RNS formalism by twisting the ten spin-half RNS fermions into five spin-one and five spin-zero fermions, and using the SO(10)/ U(5) pure spinor variables to parameterize the different ways of twisting. GSO( ) vertex operators in the − pure spinor formalism are constructed using spin fields and picture-changing operators in a manner analogous to Ramond vertex operators in the RNS formalism. Keywords: Superstrings and Heterotic Strings, Topological Strings. Contents 1. Introduction 1 2. Conventional-looking pure spinor BRST operator 4 2.1 Brief review of pure spinor formalism 4 2.2 Non-minimal fields and similarity transformation 6 JHEP01(2008)065 3. GSO(−) states in the pure spinor formalism 8 3.1 GSO(+) vertex operators 8 3.2 GSO( ) vertex operators 8 − 3.3 Picture-changing operators 9 3.4 Scattering amplitudes 10 4.
    [Show full text]
  • Introduction to Superstring Theory
    Introduction to Superstring theory Carmen Nunez Instituto de Astronomia y Física del Espacio C.C. 67 - Sue. 28, 1428 Buenos Aires, Argentina and Physics Department, University of Buenos Aires [email protected] Abstract 1 Overview The bosonic string theory, despite all its beautiful features, has a number of short­ comings. The most obvious of these are the absence of fermions and the presence of tachyons in spacetime. The tachyon is not an actual physical inconsistency; it indicates at least that the calculations are being performed in an unstable vacuum state. More­ over, tachyon exchange contributes infrarred divergences in loop diagrams and these divergences make it hard to isolate the ultraviolet behaviour of the "unified quan­ tum theory" the bosonic string theory gives rise to and determine whether it is really satisfactory. Historically, the solution to the tachyon problem appeared with the solution to the other problem, the absence of fermions. The addition of a new ingredient, supersym- metry on the world-sheet, improves substantially the general picture. In 1977 Gliozzi, Scherk and Olive showed that it was possible to get a model with no tachyons and with equal masses and multiplicities for bosons and fermions. In 1980, Green and Schwarz proved that this model had spacetime supersymmetry. In the completely consisten- t tachyon free form of the superstring theory it was then possible to show that the one-loop diagrams were completely finite and free of ultraviolet divergences. While most workers on the subject believe that the finiteness will also hold to all orders of perturbation theory, complete and universally accepted proofs have not appeared so far.
    [Show full text]
  • Superstrings ETH Proseminar in Theoretical Physics
    Neveu{Schwarz{Ramond formulation Modular invariance Green{Schwarz formulation Superstrings ETH Proseminar in Theoretical Physics ImRe Majer May 13th, 2013 ImRe Majer | Superstrings 1/42 Neveu{Schwarz{Ramond formulation Modular invariance Green{Schwarz formulation Outline 1 Neveu{Schwarz{Ramond formulation Building up the spectrum GSO projection 2 Modular invariance Spin structures Partition function 3 Green{Schwarz formulation Supersymmetry Building up the spectrum ImRe Majer | Superstrings 2/42 Neveu{Schwarz{Ramond formulation Modular invariance Green{Schwarz formulation Outline 1 Neveu{Schwarz{Ramond formulation Building up the spectrum GSO projection 2 Modular invariance Spin structures Partition function 3 Green{Schwarz formulation Supersymmetry Building up the spectrum ImRe Majer | Superstrings 3/42 Neveu{Schwarz{Ramond formulation Modular invariance Green{Schwarz formulation NSR action Light cone gauge fixing The Neveu{Schwarz{Ramond action in light cone gauge 1 Z S l:c: = − d 2σ @ X i @αX i − i ¯i ρα@ i NSR 2π α α transverse coordinates i = 1;:::; 8 i is a worldsheet Majorana 2-spinor i is a spacetime vector i SO(8) rotational symmetry =) is an 8v representation of SO(8) ImRe Majer | Superstrings 4/42 Neveu{Schwarz{Ramond formulation Modular invariance Green{Schwarz formulation Building up the spectrum Tools for generating the spectrum Open string or the right-moving part of a closed string. Creation and annihilation operators: i i i X : α−n and αn n 2 Z i i i d−m and dm m 2 Z for R-sector : i i 1 b−r and br r 2 Z + 2 for NS-sector The mass2 operator: X X 1 α0m2 = αi αi + md i d i − for NS-sector −n n −m m 2 n>0 m>0 | {z } | {z } N(α) N(d) 0 2 X i i X i i α m = α−nαn + rb−r br for R-sector n>0 r>0 | {z } | {z } N(α) N(b) ImRe Majer | Superstrings 5/42 Neveu{Schwarz{Ramond formulation Modular invariance Green{Schwarz formulation Building up the spectrum Ground states NSR formulation: All creation/annihilation operators are spacetime vectors.
    [Show full text]
  • Aspects of Brane World-Volume Dynamics in String Theory
    Aspects of Brane World-Volume Dynamics in String Theory Gianni Tallarita y A thesis submitted for the degree of Doctor of Philosophy y Centre for Research in String Theory, School of Physics Queen Mary University of London Mile End Road, London E1 4NS, UK [email protected] To my Parents Abstract This thesis investigates the non-abelian dynamics of D-Brane systems in String Theory, specifically focussing on the fate of the open string Tachyon. Starting from the action of two coincident non-BPS D9-branes, we investigate kink configura- tions of the U(2) matrix tachyon field, considering both symmetrised (Str) and conventional (Tr) prescriptions for the trace over gauge indices of the non-BPS action. Non-abelian tachyon condensation in the theory with Tr prescription, and the resulting fluctuations about the kink profile, are shown to give rise to a theory of two coincident BPS D8-branes. Next we investigate magnetic monopole solutions of the non-abelian Dirac- Born-Infeld (DBI) action describing two coincident non-BPS D9-branes in flat space. These monopole configurations are singular in the first instance and require regularization. We discuss a suitable non-abelian ansatz which describes a point- like magnetic monopole and show it solves the equations of motion to leading order in the regularization parameter. Fluctuations are studied and shown to describe a codimension three BPS D6-brane, a formula is derived for its tension. Finally, we investigate the dynamics of a pair of coincident D5 branes in the background of k NS5 branes. We extend Kutasov's original proposal to the non- abelian case of multiple D-Branes and find that the duality still holds provided one promotes the radial direction to a matrix valued field associated with a non- abelian geometric tachyon and a particular parametrization for the transverse scalar fields is chosen.
    [Show full text]
  • String Creation, D-Branes and Effective Field Theory
    Preprint typeset in JHEP style - HYPER VERSION DAMTP-2008-12 String Creation, D-branes and Effective Field Theory Ling-Yan Hung Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge CB3 0WA, UK E-mail: [email protected] Abstract: This paper addresses several unsettled issues associated with string creation in systems of orthogonal Dp-D(8 p) branes. The interaction between the branes can be − understood either from the closed string or open string picture. In the closed string picture it has been noted that the DBI action fails to capture an extra RR exchange between the branes. We demonstrate how this problem persists upon lifting to M-theory. These D- brane systems are analysed in the closed string picture by using gauge-fixed boundary states in a non-standard lightcone gauge, in which RR exchange can be analysed precisely. The missing piece in the DBI action also manifests itself in the open string picture as a mismatch between the Coleman-Weinberg potential obtained from the effective field theory and the corresponding open string calculation. We show that this difference can be reconciled by taking into account the superghosts in the (0+1) effective theory of the chiral fermion, that arises from gauge fixing the spontaneously broken world-line local supersymmetries. arXiv:0802.3603v1 [hep-th] 25 Feb 2008 Keywords: String creation, light Cone Gauge. Contents 1. Introduction 1 2. Dp brane probes in D(8 p) backgrounds 4 − 2.1 Example in M-theory 5 3. Boundary states and extra duality relations 8 4. Closed string generalisation of Arvis gauge 10 4.1 The bosonic case 10 4.1.1 Spectrum 12 4.2 Superstring generalisation 13 4.2.1 RNS formalism 13 4.2.2 GS formalism 14 4.2.3 Spectrum 16 4.3 Boundary States 16 4.4 Tree-level interaction between boundary states 18 5.
    [Show full text]
  • Superstrings
    Eidgenossische¨ Technische Hochschule Zurich¨ Swiss Federal Institute of Technology Zurich Report for Proseminar in Theoretical Physics Superstrings Author: Supervisor: Imre Majer Cristian Vergu Abstract During the course of the previous chapters of the proseminar, we introduced and quantized both the bosonic and fermionic strings and determined that the theory only containing bosonic strings requires D = 26 dimensions, whereas if it contains fermionic strings as well, the critical dimension is D = 10. We have also seen that the latter is formulated such that it incorporates manifest worldsheet supersymmetry (Neveu{ Schwarz{Ramond formulation). By na¨ıvely building up the spectrum of states using the tools previously introduced, we will see that this theory in itself is inconsistent and contains several unwanted features. To fix these, a procedure called the GSO projec- tion is needed. We will first define and use it to obtain a spacetime supersymmetric spectrum which is free from the mentioned inconsistencies. Then, we will justify the seemingly arbitrary GSO conditions by requiring that the theory be modular invariant. In the final part of this report, we take a completely different approach, and formulate another theory with manifest spacetime supersymmetry (Green{Schwarz fomulation). Spacetime supersymmetry is an elegant property, and so, many constraints and restric- tions are imposed in both formulations merely for the sake of a supersymmetric result. Without rigorous proof, we will show that with these constraints, the two formulations are actually equivalent. 1 Contents 1 Neveu{Schwarz{Ramond Formulation 3 1.1 Quick Revision . .3 1.2 The Na¨ıve Spectrum . .4 1.2.1 The NS-Sector Ground State .
    [Show full text]
  • Week 15 1 GSO Projection
    Week 15 Reading material • Polchinski, chapter 10, chapter 12 • Becker, Becker, Schwartz, Chapter 4 • Green, Schwartz, Witten, chapter 4,5 1 GSO projection We have seen so far that it is possible to introduce spacetime fermions by using the R-NS formulation of the superstring: we introduce free fermions on the worldsheet. We have also seen that after we bosonize the fermions, we can describe the corresponding (left-moing) vertex operators by X Hj X j V ∼ Ξ ∼ exp(i ± ) exp(−φ/2) = exp(i H ) exp(−φ/2) (1) α 2 2 j j And we can notate the α indices by the combinations of µ pm that appear in the exponential. We also know that ∼ exp(±iHj) for some j. There are a total of 25 choices of signs, giving that a Dirac spinor in ten dimensions has 32 components. If we consider a spacetime fermion, the left mover piece will be a spinor, and the right mover piece should be an integer spin object. The reason for this is that the total spin should be half-integer. If we consider a massless particle, then the right mover piece will be roughly (@X¯ + k ) exp(ikX), and apart from the exponential piece, the conformal field has OPE's with similar conformal operators that go like integer powers of (¯z − w¯)−1. If we consider V+++++ and V−−−−−, we find that their OPE is given by 1 V (z)V (w) ∼ exp(−φ) (2) +++++ −−−−− (z − w)3=2 Remember that the V have conformal dimension one, and that exp(−φ) has conformal dimension equal to one half.
    [Show full text]