DOCSLIB.ORG

From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems Meng-Chwan Tan National University of Singapore January 1, 2016 Presentation Outline Introduction Dual 6d M-Theory Compactifications A Geometric Langlands Duality for Surfaces for the A{B Groups A Geometric Langlands Duality for Surfaces for the C{D{G Groups The Pure AGT Correspondence for the A{B Groups The Pure AGT Correspondence for the C{D{G Groups The AGT Correspondence with Matter The \Fully-Ramified” AGT Correspondence and Quantum Integrable Systems Generalizations & Future Work 4d-2d Dualities in Mathematics and Physics Circa 1994, Nakajima [1] | the middle-dimensional cohomology of the moduli space of U(N)-instantons on a resolved ALE space of Ak−1-type can be related to the integrable representations of an affine SU(k)-algebra of level N Circa 2007, Braverman-Finkelberg [2] | the intersection 4 cohomology of the moduli space of G-instantons on R =Zk is conjectured to be related to the integrable representations of the Langlands dual of an affine G-algebra. This conjecture was henceforth known as a geometric Langlands duality for surfaces, since it involves G-bundles over a complex surface (as opposed to a complex curve). 4d-2d Dualities in Mathematics and Physics Circa 2009, Alday-Gaiotto-Tachikawa [3] | the Nekrasov instanton partition function of a 4d N = 2 conformal SU(2) quiver theory is equivalent to a conformal block of a 2d CFT with W2-symmetry that is Liouville theory. This was henceforth known as the celebrated AGT correspondence. Circa 2009, Wyllard [4] | the AGT correspondence is partially checked to hold for a 4d N = 2 conformal SU(N) quiver theory whereby the corresponding 2d CFT is an AN−1 conformal Toda field theory which has WN -symmetry. Circa 2012, Schiffmann-Vasserot, Maulik-Okounkov [5, 6] | the equivariant cohomology of the moduli space of SU(N)-instantons is related to the integrable representations of an affine WN -algebra (as a mathematical proof of AGT for pure SU(N)). Main Objective The main objective of our talk is to present in a pedagogical manner, a fundamental M-theoretic derivation of all the above 4d-2d (conjectured) relations, their generalizations, and the connection to quantum integrable systems. Contents of talk based on my paper arXiv:1301.1977 (JHEP07(2013)171), of the same title. Approach In 2007, Dijkgraaf-Hollands-Sulkowski-Vafa gave a direct physical derivation [7] of Nakajima's result; the relevant generating functions were partition functions of BPS states in two different but dual frames in string/M-theory which could then be equated to each other. Witten, in a series of lectures delivered at the IAS in 2008 [8], argued that a geometric Langlands duality for surfaces can be understood as an invariance of the BPS spectrum of the mysterious 6d N = (2; 0) SCFT under different compactifications down to 5d. We will combine the insights from the above two works, and show that our results can be derived from the principle that the spacetime BPS spectra of string-dual six-dimensional M-theory compactifications with M5-branes, OM5-planes, fluxbranes and 4d worldvolume defects, ought to be equivalent. Dual Compactifications of M-Theory with M5-Branes By using, in a chain of dualities, (i) M-IIA theory duality, (ii) T-duality, (iii) the relation between N55-branes and Taub-NUT 4 R!1 space, (iv) the fact that R =Zk = TNk , (v) the relation between Taub-NUT space in M-theory and D6-branes in IIA theory, and (vi) IIB S-duality, we find that the following six-dimensional M-theory compactifications 4 1 5 M-theory : R =Zk × Sn × Rt ×R (1) | {z } N M5-branes and 5 1 R!0 M-theory : R × Rt × Sn × TNN ; (2) | {z } k M5-branes are physically dual. Dual Compactifications of M-Theory with M5-Branes and OM5-Plane By adding an OM5-plane and using a similar chain of dualities, one can conclude that the following six-dimensional M-theory compactifications 4 1 5 M-theory : R =Zk × Sn × Rt ×R (3) | {z } N M5-branes/OM5-plane and 5 1 R!0 M-theory : R × Rt × Sn × SNN ; (4) | {z } k M5-branes R are physically dual. Here, SNN is Sen's four-manifold which one 3 can roughly regard as TNN with a Z2-identification of its R base and S1-fiber (of asymptotic radius R). Dual Compactifications of M-Theory with M5-Branes and 4d Worldvolume Defect By adding a 4d worldvloume defect of the type studied in [9] that is characterized by a partition of N, and using a similar chain of dualities whilst noting the equivalent geometrical background of the defect [10], one can conclude that the following six-dimensional M-theory compactifications 5 1 4 M-theory : R × Rt × Sn × R =Zk (5) | {z } N M5-branes with a 4d defect and R!0 1 5 M-theory : TNN × Sn × Rt ×R ; (6) | {z } k M5-branes with a 4d defect 1 are physically dual. Here, the 4d worldvolume defect wraps Rt × Sn 4 1 and (i) the z-plane in R =Zk ' Cz =Zk × Cw =Zk , (ii) the S -fiber R!0 3 R!0 of TNN and a single direction along the R base of TNN . Dual Compactifications of M-Theory with M5-Branes, OM5-Plane and 4d Worldvolume Defect By further adding an OM5-plane, one can likewise conclude that the following six-dimensional M-theory compactifications 5 1 4 M-theory : R × Rt × Sn × R =Zk (7) | {z } N M5 + OM5 + 4d defect and R!0 1 5 M-theory : SNN × Sn × Rt ×R ; (8) | {z } k M5 + 4d defect 1 are physically dual. Here, the 4d worldvolume defect wraps Rt × Sn 4 1 and (i) the z-plane in R =Zk ' Cz =Zk × Cw =Zk , (ii) the S -fiber R!0 3 R!0 of SNN , and a single direction along the R base of SNN . An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A{B Groups 4 R!0 Notice that because R =Zk and TNN are hyperk¨ahler four-manifolds which break half of the thirty-two supersymmetries in M-theory, the resulting six-dimensional spacetime theories along 5 Rt × R in (1) and (2), respectively, will both have 6d N = (1; 1) supersymmetry. As usual, there are spacetime BPS states which are annihilated by a subset of the sixteen supersymmetry generators of the 6d N = (1; 1) supersymmetry algebra; in particular, a generic (half) BPS state in six dimensions would be annihilated by eight supercharges. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A{B Groups Since the supersymmetries of the worldvolume theory of the stack of M5-branes are furnished by the ambient spacetime supersymmetries which are unbroken across the brane-spacetime barrier { in this instance, only half of the sixteen spacetime supersymmetries are unbroken across the brane-spacetime barrier because the M5-branes are half-BPS objects { a generic spacetime BPS state would correspond to a worldvolume ground state that is annihilated by all eight worldvolume supercharges. For example, in a six-dimensional compactification of M-theory with an M5-brane wrapping K3 × S1, where K3 is a hyperk¨aher four-manifold, the generic spacetime BPS states which span the massless representations of the 6d N = (1; 1) spacetime supersymmetry algebra correspond to the ground states of the worldvolume theory of the M5-brane [11]. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A{B Groups The principle that the spectra of such spacetime BPS states in the physically dual M-theory compactifications (1) and (2) ought to be equivalent, will lead us to a geometric Langlands duality for surfaces for the A{B groups. To understand this claim, we would first need to describe the quantum worldvolume theory of the stack of M5-branes whose ground states correspond to these spacetime BPS states. The quantum worldvolume theory of l coincident M5-branes is described by tensionless self-dual strings which live in the six-dimensional worldvolume itself [12]. In the low-energy point-particle limit, the theory of these strings reduces to a non-gravitational 6d N = (2; 0) Al−1 superconformal field theory of l − 1 massless tensor multiplets. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A{B Groups Alternatively, one can also describe (using DLCQ) the quantum worldvolume theory via a sigma-model on instanton moduli space [12, 13]; in particular, if the worldvolume is given by 1 M × Sn × Rt , where M is a generic hyperk¨ahlerfour-manifold, one can compute the spectrum of ground states of the quantum worldvolume theory (that are annihilated by all of its supercharges), as the spectrum of physical observables in the topological sector of 1 a two-dimensional N = (4; 4) sigma-model on Sn × Rt with target the hyperk¨ahlermoduli space MG (M) of G-instantons on M. In the M-theory compactification (1) where l = N, we have G = SU(N) if n = 1, and G = SO(N + 1) if n = 2 and N is even [14]. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A{B Groups Since the spectrum of physical observables in the topological sector 1 of the N = (4; 4) sigma-model on Sn × Rt are annihilated by all of its eight supercharges, it would mean that the ground states of the quantum worldvolume theory and hence the spacetime BPS states, would correspond to differential forms on the target space MG (M).
Load more

Recommended publications
  • 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]
  • 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]
  • 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]
  • 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]
  • 6 9 16 20 23 11 13 16 22 ] and 10 , 9 And, Involve Gravitational Odels Respectively
    UvA-DARE (Digital Academic Repository) Non-perturbative topological strings and conformal blocks Cheng, M.C.N.; Dijkgraaf, R.; Vafa, C. DOI 10.1007/JHEP09(2011)022 Publication date 2011 Document Version Final published version Published in The Journal of High Energy Physics Link to publication Citation for published version (APA): Cheng, M. C. N., Dijkgraaf, R., & Vafa, C. (2011). Non-perturbative topological strings and conformal blocks. The Journal of High Energy Physics, 2011(9), 022. [22]. https://doi.org/10.1007/JHEP09(2011)022 General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:02 Oct 2021 Published for SISSA by Springer Received: July 4, 2011 Accepted: August 22, 2011 Published: September 5, 2011 Non-perturbative topological strings and conformal blocks JHEP09(2011)022 Miranda C.N.
    [Show full text]
  • AGT on the S-Duality Wall
    AGT on the S-duality Wall Kazuo Hosomichi 4th Taiwan String Workshop 16th Dec. 2011 @ NCTU M5-brane A fundamental DOF in M-theory, (5+1)-dim object. When many of them are put together, they support a mysterious theory --- (2,0)-theory --- on the worldvolume. Studying various compactifications of (2,0) theory * gives new insights to low-dim SUSY gauge theories * helps us understand M5-branes better 1. AGT relation A correspondence between 4D N=2 gauge theories and 2D CFTs Gaiotto's gauge theories ( Σ (τ) { ⋯ }) CFT N , g,n , Y 1, ,Y n Families of 4D N = 2 SCFTs describing N M5-branes wrapped on punctured Riemann surfaces. Σ (τ) g ,n : genus g surfaces with n punctures τ : 3g-3+n complex structure moduli ⋯ Y 1, Y 2, ,Y n : Young diagram with N boxes, height less than N N＝２ case : SU(2) generalized quivers Pants decomposition gives a Lagrangian. Pants curve ( 3g-3+n of them ) SU (2) SU (2) Pair of pants = 4 hypermultiplets SU (2) Pants curve Re(τ) Tubular region = SU(2) gauge multiplet τ : moduli = gauge coupling Im(τ) Moore-Seiberg graph m1 a2 a6 a10 a4 a a1 8 a5 a7 a9 a3 m2 m3 m4 ： ai Coulomb branch parameter ： mi matter mass To a graph with parameters, one can associate (ϵ ϵ τ) Nekrasov partition function ZNek 1 , 2 ,m,a , ϵ ϵ 1 , 2 : parameters of Ω-deformation Basic examples 1. “ Nf = 4 ” theory m1 m4 SU(2) SQCD a with 4 fund. hypers m 4-punctured sphere 2 m3 2.
    [Show full text]
  • Hep-Th] 23 Oct 2018 Ersvparameters Nekrasov Atto Ucin Unu Pcrlcurve
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by MPG.PuRe MACDONALD TOPOLOGICAL VERTICES AND BRANE CONDENSATES OMAR FODA 1 AND MASAHIDE MANABE 2 Abstract. We show, in a number of simple examples, that Macdonald-type qt-deformations of topological string partition functions are equivalent to topological string partition functions that are without qt-deformations but with brane condensates, and that these brane condensates lead to geometric transitions. 1. Introduction We recall the topological vertex, its refinements and deformations, and ask what the physical inter- pretation of a specific Macdonald-type deformation is. 1.1. A hierarchy of topological vertices. 1.1.1. Abbreviations. To simplify the presentation, we use 1. string, string partition function, vertex, etc. for topological string, topological string partition function, topological vertex, etc., which should cause no confusion, as we only consider the latter, and use topological only for emphasis when that is needed, 2. qt-string partition function, qt-quantum curve, etc. for qt-deformed string partition function, qt-deformed quantum curve, etc. 3. refined as in refined partition functions, etc., when discussing objects that are refined in the sense of [9, 10, 35]; otherwise, no refinement should be inferred, and unrefined is used only for emphasis when that is needed, 4. the qt-version of ··· for the version of an object that is deformed in the sense of [58, 18], and 5. a brane condensate, or simply a condensate is a set of infinitely-many brane insertions. 1.1.2. The original vertex as a normalized 1-parameter generating function of plane partitions with fixed asymptotic boundaries.
    [Show full text]
  • Localization Techniques in Quantum Field Theories Theory Vasily Pestun and Maxim Zabzine
    Journal of Physics A: Mathematical and Theoretical PREFACE • OPEN ACCESS Related content - Introduction to localization in quantum field Localization techniques in quantum field theories theory Vasily Pestun and Maxim Zabzine To cite this article: Vasily Pestun et al 2017 J. Phys. A: Math. Theor. 50 440301 - Localization on three-dimensional manifolds Brian Willett - Supersymmetric localization in two dimensions View the article online for updates and enhancements. Francesco Benini and Bruno Le Floch Recent citations - Correlators between Wilson loop and chiral operators in N = 2 $$ \mathcal{N}=2 $$ conformal gauge theories M. Billò et al - Mass-deformed ABJM and black holes in AdS4 Nikolay Bobev et al - Review of localization for 5d supersymmetric gauge theories Jian Qiu and Maxim Zabzine This content was downloaded from IP address 130.238.171.85 on 03/04/2018 at 15:22 IOP Journal of Physics A: Mathematical and Theoretical J. Phys. A: Math. Theor. Journal of Physics A: Mathematical and Theoretical J. Phys. A: Math. Theor. 50 (2017) 440301 (7pp) https://doi.org/10.1088/1751-8121/aa63c1 50 Preface 2017 © 2017 IOP Publishing Ltd Localization techniques in quantum field theories JPHAC5 440301 Abstract This is the foreword to the special issue on localization techniques in quantum field theory. The summary of individual chapters is given and their interrelation V Pestun et al is discussed. Localization techniques in quantum field theories 1. Summary Printed in the UK This is the summary of the special issue ‘Localization techniques in quantum field theories’ which contains 17 individual reviews*. The focus of the collection is on the localization tech- nique and its applications in supersymmetric gauge theories.
    [Show full text]
  • Introduccion a La Teoria De Cuerdas
    Curso del Posgrado en Ciencias Físicas 2021-1 CORRESPONDENCIA HOLOGRÁFICA Alberto Güijosa, [email protected] Clases: Martes y Jueves 16:00-18:00 (empezando 16:05 en punto), Sala Virtual en Zoom La correspondencia holográfica es una muy sorprendente equivalencia entre teorías cuánticas de campos (sin gravedad) y teorías de gravedad cuántica (prominentemente, teorías de cuerdas). Además de lo revolucionario que resulta desde el punto de vista conceptual el mero hecho de su existencia, esta correspondencia se ha establecido ya como una herramienta muy útil para entender el comportamiento de algunas teorías de campos en el régimen de acoplamiento fuerte, y ofrece una perspectiva novedosa sobre los aspectos problemáticos de la gravedad cuántica. Se trata de un tema extremadamente amplio (el artículo original de Maldacena, hep-th/9711200, ha recibido ya más de 16,000 citas, ¡que equivalen a casi 3 por día hábil durante 23 años!), así que solo intentaremos dar un panorama de algunas de las ideas principales. Temario Preliminar 0. Motivación y Antecedentes: ¿Qué es la correspondencia norma-gravedad? Teorías de Campo. Acoplamiento Débil y Fuerte. QED. QCD. Primas de QCD, N grande y MSYM. Gravedad Cuántica. Holografía. 1. Deducción de la Correspondencia: Breve repaso de cuerdas, branas negras y D- branas. Deducción de AdS5/CFT4 a partir de D3-branas. 2. Evidencias y Diccionario: Espacio anti-de Sitter. Mapeo de coordenadas y simetrías. Conexión UV/IR. Operadores locales. Teoría de campo conforme. Funciones de correlación. Modos normalizables y no normalizables. BMN y estados cuánticos con números grandes. Quarks. Lazos de Wilson y potencial quark-antiquark.
    [Show full text]