DOCSLIB.ORG

Dual Infrared Limits of 6D N=(2,0) Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Dual Infrared Limits of 6D N=(2,0) Theory Physics Letters B 793 (2019) 297–302 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Dual infrared limits of 6d N = (2, 0) theory ∗ Olaf Lechtenfeld a,b, , Alexander D. Popov a a Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany b Riemann Center for Geometry and Physics, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany a r t i c l e i n f o a b s t r a c t 4 2 3 ˜ 1 Article history: Compactifying type AN−1 6d N =(2, 0) supersymmetric CFT on a product manifold M × = M × S × 1 1 ˜ 1 4 Received 19 November 2018 S × I either over S or over S leads to maximally supersymmetric 5d gauge theories on M × I or 3 2 1 ˜ 1 Accepted 25 February 2019 on M × , respectively. Choosing the radii of S and S inversely proportional to each other, these Available online 3 May 2019 5d gauge theories are dual to one another since their coupling constants e2 and e˜2 are proportional to Editor: M. Cveticˇ those radii respectively. We consider their non-Abelian but non-supersymmetric extensions, i.e. SU(N) 4 × I 3 × 2 4 ⊃ 3 = × 2 Yang–Mills theories on M and on M , where M M Rt T p with time t and a punctured 2 2-torus, and I ⊂ is an interval. In the first case, shrinking I to a point reduces to Yang–Mills theory 4 or to the Skyrme model on M , depending on the method chosen for the low-energy reduction. In the 3 second case, scaling down the metric on M and employing the adiabatic method, we derive in the 2 infrared limit a non-linear SU(N) sigma model with a baby-Skyrme-type term on , which can be reduced further to AN−1 Toda theory. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license 3 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 1. Introduction and summary them. However, we are not aware of a geometric derivation of the AGT correspondence. The famous Alday-Gaiotto-Tachikawa (AGT) 2d-4d correspon- Recently, a derivation of the AGT correspondence via reduction 4 2 dence [1] relates Liouville field theory on a punctured Rie- of 6d CFT to M and was discussed in [6–9]. A relation between 2 mann surface and SU(2) super-Yang–Mills (SYM) theory on AN−1 6d CFT compactified on a circle and 5d SU(N) SYM was 4 a four-dimensional manifold M (see e.g. [2]for a nice review employed as well as further reduction to complex Chern-Simons 2 4 and references). This correspondence was quickly extended to theory on × I, where I ⊂ M is an interval. Then a generalized 2d AN−1 Toda field theory and 4d SU(N) SYM [3]. Since then version of the correspondence between 3d Chern-Simons theory various AGT-like correspondences between theories on n-and and 2d CFT was used, with the Nahm-pole boundary conditions (6 − n)-dimensional manifolds were investigated (see e.g. [4,5]) translating to the constraints reducing WZW models to Toda theo- for reviews and references). One way to interpret these correspon- ries [7–9]. N = dences is to start from 6d (2, 0) supersymmetric conformal In this paper we consider SU(N) Yang–Mills theories on man- − n × 6 n n 4 3 2 4 3 ˜ 1 field theory (CFT) on M M . The theory on M would appear ifolds M × I and M × , where M = M × S is Lorentzian − 6 n 2 ∼ 1 as a low-energy limit when M shrinks to a point, while the the- and = S × I is a two-sphere with two punctures. The two − 6 n n 3 ˜ 1 ory on M would emerge when M is scaled down to a point [2, manifolds agree in M × I but differ in the additional circle S = 1 4,5]. To be more precise, the AGT correspondence for n 4 relates versus S . The supersymmetric extensions of Abelian gauge the- 4 2 the partition function of 6d CFT of type A − on M × with N 1 ories on these two 5d manifolds originate from type AN−1 6d partition functions of theories on M4 and 2, which are all equal 4 2 1 N = (2, 0) supersymmetric CFT on M × , compactified on S or due to the conformal invariance of the 6d theory. The standard ˜ 1 ˜ S with radii R1 = εR0 or R1 = ε˜ R0, respectively [10–12]. Choos- way of establishing such correspondences consists of calculating −1 ing ε˜ = ε the two 5d theories become dual to each other, and n 6−n partition/correlation functions on M and M and to compare 2 2 their gauge coupling constants e or e˜ (in units of the reference scale R0) will be inversely proportional to one another, which is consistent with the arguments from [13]. In order to have non- * Corresponding author. 3 2 trivial vacuum solutions we take M = Rt × T with a temporal E-mail addresses: [email protected] (O. Lechtenfeld), p 2 [email protected] (A.D. Popov). direction Rt and a one-punctured 2-torus T p . https://doi.org/10.1016/j.physletb.2019.02.051 0370-2693/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 298 O. Lechtenfeld, A.D. Popov / Physics Letters B 793 (2019) 297–302 ˜ 4 For further low-energy limits we introduce two scales, R and R, Gauge fields on M × I. The gauge potential A (connection) and which determine the sizes of the two factors in the product man- the gauge field F (curvature) both take values in the Lie algebra 4 3 2 4 ifolds M × IR and M × , respectively. In the first case, via × I R˜ R˜ R su(N). On M ˜ R we have ˜ R (R, R) = (R0, εR0) and ε 1we shrink IR to a point. Depend- A = A aˆ + A ing on the choice of reduction – translational invariance along IR aˆ dx z dz and N ≥ (2.7) or adiabatic approach [14–18,4]– one obtains ( 2 extended) F = 1 F aˆ ∧ bˆ + F aˆ ∧ 4 2 aˆbˆ dx dx azˆ dx dz . Yang–Mills theory or the Skyrme model on the manifold M . In R0 ˜ For the generators I in the fundamental N × N representation the second case, taking (R, R) = (ε˜ R0, R0) and ε˜ 1we scale i 1 2 of SU(N) we use the normalization tr(Ii I j) =− δij for i, j = down the metric on Rt and T p . The adiabatic method [15–22]then 2 2 i i produces a non-linear sigma model with a baby-Skyrme-type term 1, ..., N − 1 and write A = A Ii and F = F Ii . 2 aˆbˆ aˆbˆ zz −2 on . Finally, we briefly discuss a reduction of this 2d SU(N) For the metric tensor (2.6)we have g = η and g = R . It R0 R R 2 sigma model to Toda field theory on . follows that R0 To summarize, we propose a geometric background for estab- ˆ ˆ ˆ ˆ ˆ − ˆ 4 Fab = Fab Faz = 2Faz lishing 4d-2d AGT correspondences between field theories on M R and R R . (2.8) and 2. We argue that these correspondences depend not only on The standard Yang–Mills action functional with this metric takes the topology of M4 and 2 but also on the method employed for the form deriving the low-energy effective field theories in the infrared (us- R ing symmetries, adiabatic limit, constraints etc.). =− 4 F Faˆbˆ + 2 F Fazˆ S5 d x dz tr aˆbˆ 2 azˆ , (2.9) 8e2 R 4 2. Two dual ways from 6d to 5d Yang–Mills theory M ×I 2 where e is the gauge coupling constant. It is known [13] that e 6d manifold. We consider the purported 6d Abelian N = (2, 0) 1 is proportional to the S compactification radius R1=εR0. A full CFT on the manifold supersymmetric extension of (2.9)can be found e.g. in [4]. 4 × 2 = 3 × ˜ 1 × 1 × I 3 = × 2 M M ˜ S ˜ S R R , where M ˜ Rt T p 3 2 R R1 1 R Reduction to M × . Alternatively, let us put R1 = R and shrink (2.1) S˜ 1 by taking ˜ is a temporal cylinder over a two-torus with size R and a punc- ˜ R1 = ε˜ R0 R0 with ε˜ 1 . (2.10) ˜ 1 1 ture p. The radii of the circles S and S and the length of the interval I are indicated. The metric on this space is taken as This reduction produces the 5d super-Maxwell theory on 2 a b 3 2 4 2 5 2 3 2 2 1 ds = η dx dx + (dx ) + (dx ) + (dx ) M ˜ × = Rt × T × S × IR (2.11) 6 ab R R p R ˜ 2 2 2 2 = R −(dt) + (dα) + (dβ) (2.2) with a metric ˜ 2 2 2 2 2 2 ¯ + R (dϕ) + R (dθ) + R (dz) , 2 ˜ 2 2 2 2 a¯ b 1 1 ds˜ = R −(dt) + (dα) + (dβ) + δ¯ ¯ dx dx R˜ ab (2.12) = ¯ where a, b 0, 1, 2, and all spatial coordinates in the second line with a¯, b ∈{4, 5} . range over [−π, π], with the Greek ones being periodic.
Load more

Recommended publications
  • Quantum Groups and Algebraic Geometry in Conformal Field Theory
    QUANTUM GROUPS AND ALGEBRAIC GEOMETRY IN CONFORMAL FIELD THEORY DlU'KKERU EI.INKWIJK BV - UTRECHT QUANTUM GROUPS AND ALGEBRAIC GEOMETRY IN CONFORMAL FIELD THEORY QUANTUMGROEPEN EN ALGEBRAISCHE MEETKUNDE IN CONFORME VELDENTHEORIE (mrt em samcnrattint] in hit Stdirlands) PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE RIJKSUNIVERSITEIT TE UTRECHT. OP GEZAG VAN DE RECTOR MAGNIFICUS. TROF. DR. J.A. VAN GINKEI., INGEVOLGE HET BESLUIT VAN HET COLLEGE VAN DE- CANEN IN HET OPENBAAR TE VERDEDIGEN OP DINSDAG 19 SEPTEMBER 1989 DES NAMIDDAGS TE 2.30 UUR DOOR Theodericus Johannes Henrichs Smit GEBOREN OP 8 APRIL 1962 TE DEN HAAG PROMOTORES: PROF. DR. B. DE WIT PROF. DR. M. HAZEWINKEL "-*1 Dit proefschrift kwam tot stand met "•••; financiele hulp van de stichting voor Fundamenteel Onderzoek der Materie (F.O.M.) Aan mijn ouders Aan Saskia Contents Introduction and summary 3 1.1 Conformal invariance and the conformal bootstrap 11 1.1.1 Conformal symmetry and correlation functions 11 1.1.2 The conformal bootstrap program 23 1.2 Axiomatic conformal field theory 31 1.3 The emergence of a Hopf algebra 4G The modular geometry of string theory 56 2.1 The partition function on moduli space 06 2.2 Determinant line bundles 63 2.2.1 Complex line bundles and divisors on a Riemann surface . (i3 2.2.2 Cauchy-Riemann operators (iT 2.2.3 Metrical properties of determinants of Cauchy-Ricmann oper- ators 6!) 2.3 The Mumford form on moduli space 77 2.3.1 The Quillen metric on determinant line bundles 77 2.3.2 The Grothendieck-Riemann-Roch theorem and the Mumford
    [Show full text]
  • Some Mathematical Applications of Little String Theory
    Some mathematical applications of little string theory Mina Aganagic UC Berkeley Based on joint works with Edward Frenkel and Andrei Okounkov, to appear Andrei Okounkov, arXiv:1604.00423 [math.AG] , Nathan Haouzi, arXiv:1506.04183 [hep-th] , String theory predicts existence of a remarkable quantum field theory in six dimensions, the (2,0) superconformal field theory. The theory is labeled by a simply-laced Lie algebra, . It is remarkable, in part, because it is expected to play an important role in pure mathematics: Geometric Langlands Program Knot Categorification Program The fact that the theory has no classical limit, makes it hard to extract its predictions. AGT correspondence, after Alday, Gaiotto and Tachikawa, serves well to illustrate both the mathematical appeal of the theory and the difficulty of working with it. The AGT correspondence states that the partition function of the -type (2,0) SCFT, on a six manifold of the form is a conformal block on the Riemann surface of a vertex operator algebra which is also labeled by : -algebra The correspondence further relates defects of the 6d theory to vertex operators of the -algebra, inserted at points on . x x x x x x If one is to take the conjecture at its face value, it is hard to make progress on it. Since we do not know how to describe the (2,0) SCFT, we cannot formulate or evaluate its partition function in any generality. (Exceptions are g=A_1 , or special choices of defects.) I will argue in this talk that one can make progress by replacing the 6-dimensional conformal field theory, which is a point particle theory, by the 6-dimensional string theory which contains it, the (2,0) little string theory.
    [Show full text]
  • Remarks on A2 Toda Field Theory 1 Introduction
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Remarks on A2 Toda Field Theory S. A. Apikyan1 Department of Theoretical Physics Yerevan State University Al. Manoogian 1, Yerevan 375049, Armenia C. Efthimiou2 Department of Physics University of Central Florida Orlando, FL 32816, USA Abstract We study the Toda field theory with finite Lie algebras using an extension of the Goulian-Li technique. In this way, we show that, af- ter integrating over the zero mode in the correlation functions of the exponential fields, the resulting correlation function resembles that of a free theory. Furthermore, it is shown that for some ratios of the charges of the exponential fields the four-point correlation functions which contain a degenerate field satisfy the Riemann ordinary differ- ential equation. Using this fact and the crossing symmetry, we derive a set of functional equations for the structure constants of the A2 Toda field theory. 1 Introduction The Toda field theory (TFT) provides an extremely useful description of a large class of two-dimensional integrable quantum field theories. For this reason these models have attracted a considerable interest in recent years and many outstanding results in various directions have been established. TFTs are divided in three broad categories: finite Toda theories (FTFTs) for which the underlying Kac-Moody algebra [1, 2] is a finite Lie algebra, affine Toda theories (ATFTs) for which the underlying Kac-Moody algebra [email protected] [email protected] 1 is an affine algebra and indefinite Toda theories (ITFTs) for which the under- lying Kac-Moody algebra is an indefinite Kac-Moody algebra.
    [Show full text]
  • Higher AGT Correspondences, W-Algebras, and Higher Quantum
    Higher AGT Correspon- dences, W-algebras, and Higher Quantum Geometric Higher AGT Correspondences, W-algebras, Langlands Duality from M-Theory and Higher Quantum Geometric Langlands Meng-Chwan Duality from M-Theory Tan Introduction Review of 4d Meng-Chwan Tan AGT 5d/6d AGT National University of Singapore W-algebras + Higher QGL SUSY gauge August 3, 2016 theory + W-algebras + QGL Higher GL Conclusion Presentation Outline Higher AGT Correspon- dences, Introduction W-algebras, and Higher Quantum Lightning Review: A 4d AGT Correspondence for Compact Geometric Langlands Lie Groups Duality from M-Theory A 5d/6d AGT Correspondence for Compact Lie Groups Meng-Chwan Tan W-algebras and Higher Quantum Geometric Langlands Introduction Duality Review of 4d AGT Supersymmetric Gauge Theory, W-algebras and a 5d/6d AGT Quantum Geometric Langlands Correspondence W-algebras + Higher QGL SUSY gauge Higher Geometric Langlands Correspondences from theory + W-algebras + M-Theory QGL Higher GL Conclusion Conclusion 6d/5d/4d AGT Correspondence in Physics and Mathematics Higher AGT Correspon- Circa 2009, Alday-Gaiotto-Tachikawa [1] | showed that dences, W-algebras, the Nekrasov instanton partition function of a 4d N = 2 and Higher Quantum conformal SU(2) quiver theory is equivalent to a Geometric Langlands conformal block of a 2d CFT with W2-symmetry that is Duality from M-Theory Liouville theory. This was henceforth known as the Meng-Chwan celebrated 4d AGT correspondence. Tan Circa 2009, Wyllard [2] | the 4d AGT correspondence is Introduction Review of 4d proposed and checked (partially) to hold for a 4d N = 2 AGT conformal SU(N) quiver theory whereby the corresponding 5d/6d AGT 2d CFT is an AN−1 conformal Toda field theory which has W-algebras + Higher QGL WN -symmetry.
    [Show full text]
  • An Toda Pmatcd.Pdf
    Kent Academic Repository Full text document (pdf) Citation for published version Adamopoulou, Panagiota and Dunning, Clare (2014) Bethe Ansatz equations for the classical A^(1)_n affine Toda field theories. Journal of Physics A: Mathematical and Theoretical, 47 (20). p. 205205. ISSN 1751-8113. DOI https://doi.org/10.1088/1751-8113/47/20/205205 Link to record in KAR https://kar.kent.ac.uk/40992/ Document Version UNSPECIFIED Copyright & reuse Content in the Kent Academic Repository is made available for research purposes. Unless otherwise stated all content is protected by copyright and in the absence of an open licence (eg Creative Commons), permissions for further reuse of content should be sought from the publisher, author or other copyright holder. Versions of research The version in the Kent Academic Repository may differ from the final published version. Users are advised to check http://kar.kent.ac.uk for the status of the paper. Users should always cite the published version of record. Enquiries For any further enquiries regarding the licence status of this document, please contact: [email protected] If you believe this document infringes copyright then please contact the KAR admin team with the take-down information provided at http://kar.kent.ac.uk/contact.html (1) Bethe Ansatz equations for the classical An affine Toda field theories Panagiota Adamopoulou and Clare Dunning SMSAS, University of Kent, Canterbury, CT2 7NF, United Kingdom E-mails: [email protected], [email protected] Abstract (1) We establish a correspondence between classical An affine Toda field theories and An Bethe Ansatz systems.
    [Show full text]
  • Geometric Langlands from 4D N=2 Theories
    Geometric Langlands from 4d N = 2 theories Aswin Balasubramanian DESY & U. Hamburg September 11, 2017 IMSc Seminar, Chennai based on 1702.06499 w J. Teschner + work with Ioana Coman-Lohi, J.T Aswin Balasubramanian Geometric Langlands from 4d N = 2 theories 1 / 46 Goals for my talk Part A : Light introduction to Geometric Langlands (Kapustin-Witten, Beilinson-Drinfeld, Our Motivating Questions) Part B : Review N = 2 Class S theories (Hitchin system, AGT correspondence) Part C: Aspects of Geometric Langlands from Class S Aswin Balasubramanian Geometric Langlands from 4d N = 2 theories 2 / 46 Goals for my talk Part A : Light introduction to Geometric Langlands (Kapustin-Witten, Beilinson-Drinfeld, Our Motivating Questions) Part B : Review N = 2 Class S theories (Hitchin system, AGT correspondence) Part C: Aspects of Geometric Langlands from Class S Aswin Balasubramanian Geometric Langlands from 4d N = 2 theories 2 / 46 Goals for my talk Part A : Light introduction to Geometric Langlands (Kapustin-Witten, Beilinson-Drinfeld, Our Motivating Questions) Part B : Review N = 2 Class S theories (Hitchin system, AGT correspondence) Part C: Aspects of Geometric Langlands from Class S Aswin Balasubramanian Geometric Langlands from 4d N = 2 theories 2 / 46 Part A : Approaches to Geometric Langlands Aswin Balasubramanian Geometric Langlands from 4d N = 2 theories 3 / 46 Historical Background The Langlands program has its roots in Number Theory, specifically the classification of Automorphic forms for G(Z). These generalize modular forms of SL(2; Z) . Fourier co-efficients of Automorphic forms encode very interesting Number Theoretic information. For several reasons, it was interesting to ask if there was a geometric analog.
    [Show full text]
  • Probabilistic Conformal Blocks for Liouville Cft on the Torus
    PROBABILISTIC CONFORMAL BLOCKS FOR LIOUVILLE CFT ON THE TORUS PROMIT GHOSAL, GUILLAUME REMY, XIN SUN, AND YI SUN Abstract. Virasoro conformal blocks are a family of important functions defined as power series via the Virasoro algebra. They are a fundamental input to the conformal bootstrap program for 2D conformal field theory (CFT) and are closely related to four dimensional supersymmetric gauge theory through the Alday- Gaiotto-Tachikawa correspondence. The present work provides a probabilistic construction of the 1-point toric Virasoro conformal block for central change greater than 25. More precisely, we construct an analytic function using a probabilistic tool called Gaussian multiplicative chaos (GMC) and prove that its power series expansion coincides with the 1-point toric Virasoro conformal block. The range (25; 1) of central charges corresponds to Liouville CFT, an important CFT originating from 2D quantum gravity and bosonic string theory. Our work reveals a new integrable structure underlying GMC and opens the door to the study of non-perturbative properties of Virasoro conformal blocks such as their analytic continuation and modular symmetry. Our proof combines an analysis of GMC with tools from CFT such as Belavin-Polyakov- Zamolodchikov differential equations, operator product expansions, and Dotsenko-Fateev type integrals. 1. Introduction A conformal field theory (CFT) is a way to construct random functions on Riemannian manifolds that transform covariantly under conformal (i.e. angle preserving) mappings. Since the seminal work of Belavin- Polyakov-Zamolodchikov in [BPZ84], two dimensional (2D) CFT has grown into one of the most prominent branches of theoretical physics, with applications to 2D statistical physics and string theory, as well as far reaching consequences in mathematics; see e.g.
    [Show full text]
  • Hamiltonian Reduction and Supersymmetric Toda Models
    UM-P-95/S' QMW-PH-95/7 hep-th/yymmnnn March 1995 Hamiltonian Reduction and Supersymmetric Toda Models G. Au1 and B. Spence2 School of Physics University of Melbourne Parkville 3052 Australia Abstract New formulations of the solutions of N=I and N=2 super Toda field theory are introduced, using Hamiltonian reduction of the N=I and N=2 super WZNW Models to the super Toda Models. These pa- rameterisations are then used to present the HamiJtonian formulations of the super Toda theories on the spaces of solutions. !gauOtauon.ph. unimelb.edu.au [email protected]. Current address: Dept. Physics, Queen Mary & Westfield College, London El 4NS. VQl 2 7 Hi 0 6 1 Introduction Super Toda theories [1, 2, 3, 4, 5] are two dimensional conformal field theories and are an arena for the investigation of general features of two dimensional integrable systems such as quantum supergroups, super W-algebras, and supersymmetric integrable hierachies. The study of many of these properties requires Hamiltonian formulations of the models. The Toda theories can be understood as the Hamiltonian reductions of Wess-Zumino-Novikov-Witten (WZNW) models (see ref. [6]). As the WZNW models are integrable, one can explicitly write a Hamiltonian formulation in terms of the coordinates of the spaces of solutions of the model. In order to preserve full equivalence with the space of initial data, one needs to formulate the WZNW solutions as discussed in ref. [7] (see also ref. [8]). Using this formulation, one may derive, by Hamiltonian reduction, a Hamiltonian formulation of Toda theory [9], utilising suitable coordinates on the space of solutions.
    [Show full text]
  • The Solutions of Affine and Conformal Affine Toda Field Theories
    UM-P-93/75 ' hep-th/9402079 February 1994 The Solutions of Affine and Conformal Affine Toda Field Theories G. PAPADOPOULOS* //. Institute for Theoretical Physics Lumper Chaussee 149 22161 Hamburg Germany and B. SPENCE School of Physics University of Melbourne Parkville 3052 Australia ABSTRACT We give new formulations of the solutions of the field equations of the affine Toda and con- formal afiine Toda theories on a cylinder and two-dimensional Minkowski space-time. These solutions are parameterised in terms of initial data and the resulting covariant phase spaces are diffeomorphic to the Hamiltonian ones. We derive the fundamental Poisson brackets of the parameters of the solutions and give the general static solutions for the affine theory. • E-mail: [email protected] f E-mail: [email protected] 1. Introduction It has been known for many years that the field equations of certain two-dimensional field theories can be solved exactly. Some of these theories are the Wess-Zumino-Witten model, Liouville field theory and and the various versions of Toda field theory. The field equations of these theories are non-linear, partial, hyperbolic differential equations and various dis­ cussions of their solutions have been presented in the literature. One method for solving such integrable systems is based on the theory of Lax pairs and this has been extensively studied for Toda systems [1-3]. These authors solve the field equations on two-dimensional Minkowski space-time and describe their solutions in terms of functions that depend on the light-cone co-ordinates x* of the Minkowski space-time.
    [Show full text]
  • Pos(TASI2017)015
    The String Landscape, the Swampland, and the Missing Corner PoS(TASI2017)015 T. Daniel Brennana, Federico Cartab,∗ and Cumrun Vafayc a Department of Physics and Astronomy, Rutgers University 126 Frelinghuysen Rd., Piscataway NJ 08855, USA b Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain c Jefferson Physical Laboratory, Harvard University Cambridge, MA 02138, USA Email: [email protected], [email protected], [email protected] We give a brief overview of the string landscape and techniques used to construct string com- pactifications. We then explain how this motivates the notion of the swampland and review a number of conjectures that attempt to characterize theories in the swampland. We also compare holography in the context of superstrings with the similar, but much simpler case of topological string theory. For topological strings, there is a direct definition of topological gravity based on a sum over a “quantum gravitational foam.” In this context, holography is the statement of an identification between a gravity and gauge theory, both of which are defined independently of one another. This points to a missing corner in string dualities which suggests the search for a direct definition of quantum theory of gravity rather than relying on its strongly coupled holographic dual as an adequate substitute (Based on TASI 2017 lectures given by C. Vafa). Theoretical Advanced Study Institute in Elementary Partical Physics (TASI): “Physics at the Fundamental Frontier” University of Colorado, Boulder Boulder, CO June, 5 – 30 2017 ∗La Caixa-Severo Ochoa Scholar ySpeaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
    [Show full text]
  • Instanton Counting in Class Sk
    Prepared for submission to JHEP DESY 17-189 Instanton counting in Class Sk Thomas Bourton, Elli Pomoni DESY Theory Group, Notkestraße 85, 22607 Hamburg, Germany E-mail: [email protected], [email protected] Abstract: We compute the instanton partition functions of N = 1 SCFTs in class Sk. We obtain this result via orbifolding Dp/D(p-4) brane systems and calculating the partition function of the supersymmetric gauge theory on the worldvolume of K D(p-4) branes. Starting with D5/D1 setups probing a Z` × Zk orbifold singularity we obtain the K instanton partition 4 2 2 functions of 6d (1; 0) theories on R × T in the presence of orbifold defects on T via com- puting the 2d superconformal index of the worldvolume theory on K D1 branes wrapping the T 2. We then reduce our results to the 5d and to the 4d instanton partition func- tions. For k = 1 we check that we reproduce the known elliptic, trigonometric and rational Nekrasov partition functions. Finally, we show that the instanton partition functions of SU(N) quivers in class Sk can be obtained from the class S mother theory partition func- 2πij=k tions with SU(kN) gauge factors via imposing the ‘orbifold condition’ aA ! aAe with A = jA and A = 1;:::;N, j = 1; : : : ; k on the Coulomb moduli and the mass parameters. arXiv:1712.01288v1 [hep-th] 4 Dec 2017 Contents 1 Introduction2 2 String theory description4 2.1 Type IIB realisation4 2.2 Type IIA realisation6 2.3 A 6d uplift7 2.4 On supersymmetry of the D1/D5 system9 3 4d N = 2∗ Instantons from a 2d superconformal index computation 10
    [Show full text]
  • Jhep03(2021)203
    Published for SISSA by Springer Received: November 27, 2020 Accepted: February 17, 2021 Published: March 22, 2021 RP4 From N = 4 Super-Yang-Mills on to bosonic JHEP03(2021)203 Yang-Mills on RP2 Yifan Wang Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, U.S.A. Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, U.S.A. E-mail: [email protected] Abstract: We study the four-dimensional N = 4 super-Yang-Mills (SYM) theory on the unorientable spacetime manifold RP4. Using supersymmetric localization, we find that for a large class of local and extended SYM observables preserving a common supercharge Q, their expectation values are captured by an effective two-dimensional bosonic Yang- Mills (YM) theory on an RP2 submanifold. This paves the way for understanding N = 4 SYM on RP4 using known results of YM on RP2. As an illustration, we derive a matrix integral form of the SYM partition function on RP4 which, when decomposed into discrete holonomy sectors, contains subtle phase factors due to the nontrivial η-invariant of the Dirac operator on RP4. We also comment on potential applications of our setup for AGT correspondence, integrability and bulk-reconstruction in AdS/CFT that involve cross-cap states on the boundary. Keywords: Conformal Field Theory, Supersymmetric Gauge Theory ArXiv ePrint: 2005.07197 Open Access, c The Authors. https://doi.org/10.1007/JHEP03(2021)203 Article funded by SCOAP3. Contents 1 Introduction1 2 Localization of the N = 4 SYM on RP4 3 2.1 The SYM on RP4 from involution3 2.2 The supersymmetric action for N = 4 SYM on RP4 6 2.3 Localization to 2d YM on RP2 7 JHEP03(2021)203 3 2d YM on RP2 and a new matrix model9 3.1 Partition function 10 3.2 Discrete holonomy sectors and phase factors 13 4 Conclusion and discussion 15 1 Introduction Quantum field theories with time reversal symmetry can be formulated on unorientable spacetime manifolds.
    [Show full text]