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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. SUSY gauge theory + Circa 2009, Awata-Yamada [3] | formulated 5d pure W-algebras + QGL AGT correspondence for SU(2) in terms of q-deformed Higher GL W2-algebra. Conclusion 6d/5d/4d AGT Correspondence in Physics and Mathematics Higher AGT Correspon- dences, W-algebras, and Higher Circa 2011, Awata et al. [4] | mathematically Quantum Geometric conjectured 5d AGT correspondence for conformal SU(N) Langlands Duality from linear quiver theory. M-Theory Meng-Chwan Circa 2011, Keller et al. [5] | proposed and checked Tan (partially) the 4d AGT correspondence for N = 2 pure Introduction arbitrary G theory. Review of 4d AGT Circa 2012, Schiffmann-Vasserot, Maulik-Okounkov [6, 7] 5d/6d AGT | the equivariant cohomology of the moduli space of W-algebras + SU(N)-instantons is related to the integrable Higher QGL SUSY gauge representations of an affine WN -algebra (as a theory + W-algebras + mathematical proof of 4d AGT for pure SU(N)). QGL Higher GL Conclusion 6d/5d/4d AGT Correspondence in Physics and Mathematics Higher AGT Correspon- dences, W-algebras, Circa 2013, Tan [8] | M-theoretic derivation of the 4d and Higher Quantum AGT correspondence for arbitrary compact Lie groups, and Geometric its generalizations and connections to quantum integrable Langlands Duality from systems. M-Theory Meng-Chwan Circa 2013, Tan [9] | M-theoretic derivation of the 5d Tan and 6d AGT correspondence for SU(N), and their Introduction generalizations and connections to quantum integrable Review of 4d AGT systems. 5d/6d AGT Circa 2014, Braverman-Finkelberg-Nakajima [10] | the W-algebras + Higher QGL equivariant cohomology of the moduli space of SUSY gauge G-instantons is related to the integrable representations of theory + L W-algebras + a W( gaff )-algebra (as a mathematical proof of 4d AGT QGL for simply-laced G with Lie algebra g). Higher GL Conclusion 6d/5d/4d AGT Correspondence in Physics and Mathematics Higher AGT Correspon- dences, W-algebras, and Higher Quantum Circa 2015, Iqbal-Kozcaz-Yau [11] | string-theoretic Geometric Langlands derivation of the 6d AGT correspondence for SU(2), where Duality from M-Theory 2d CFT has elliptic W2-symmetry. Meng-Chwan Tan Circa 2015, Nieri [12] | field-theoretic derivation of the Introduction 6d AGT correspondence for SU(2), where 2d CFT has Review of 4d AGT elliptic W2-symmetry. 5d/6d AGT W-algebras + Circa 2016, Tan [13] | M-theoretic derivation of the 5d Higher QGL SUSY gauge and 6d AGT correspondence for arbitrary compact Lie theory + groups, and more. W-algebras + QGL Higher GL Conclusion Motivations for Our Work Higher AGT 1). The recent work of Kimura-Pestun in [14] which furnishes a Correspon- dences, gauge-theoretic realization of the q-deformed affine W-algebras W-algebras, and Higher constructed by Frenkel-Reshetikhin in [17], strongly suggests Quantum Geometric that we should be able to realize, in a unified manner through Langlands Duality from our M-theoretic framework in [8, 9], a quantum geometric M-Theory Langlands duality and its higher analogs as defined by Meng-Chwan Tan Feigin-Frenkel-Reshetikhin in [15, 16], and more. Introduction 2). The connection between the gauge-theoretic realization of Review of 4d AGT the geometric Langlands correspondence by Kapustin-Witten 5d/6d AGT in [18, 19] and its original algebraic CFT formulation by W-algebras + Beilinson-Drinfeld in [20], is hitherto still missing. The fact Higher QGL that we can relate 4d supersymmetric gauge theory to ordinary SUSY gauge theory + affine W-algebras which obey a geometric Langlands duality, W-algebras + QGL suggests that the sought-after connection may actually reside Higher GL within our M-theoretic framework in [8, 9]. Conclusion Summary of Talk Higher AGT Correspon- dences, W-algebras, and Higher In this talk based on [13], we will present an M-theoretic Quantum Geometric derivation of a 5d and 6d AGT correspondence for arbitrary Langlands Duality from compact Lie groups, from which we can obtain identities of M-Theory various W-algebras which underlie a quantum geometric Meng-Chwan Tan Langlands duality and its higher analogs, whence we will be able to Introduction Review of 4d AGT (i) elucidate the sought-after connection between the 4d 5d/6d AGT gauge-theoretic realization of the geometric Langlands W-algebras + correspondence by Kapustin-Witten [18, 19] and its algebraic Higher QGL 2d CFT formulation by Beilinson-Drinfeld [20], SUSY gauge theory + W-algebras + QGL Higher GL Conclusion Summary of Talk Higher AGT Correspon- dences, W-algebras, and Higher Quantum (ii) explain what the higher 5d and 6d analogs of the geometric Geometric Langlands Langlands correspondence for simply-laced Lie (Kac-Moody) Duality from groups G (G), ought to involve, M-Theory b Meng-Chwan Tan and Introduction Review of 4d (iii) demonstrate Nekrasov-Pestun-Shatashvili's recent result AGT in [21], which relates the moduli space of 5d/6d 5d/6d AGT supersymmetric G (Gb)-quiver SU(Ki ) gauge theories to the W-algebras + Higher QGL representation theory of quantum/elliptic affine (toroidal) SUSY gauge G-algebras. theory + W-algebras + QGL Higher GL Conclusion Lightning Review: An M-Theoretic Derivation of the 4d Pure AGT Correspondence for Compact Groups Higher AGT Via a chain of string dualities in a background of fluxbranes as Correspon- dences, introduced in [22, 23], we have the dual M-theory W-algebras, and Higher compactifications Quantum Geometric 4 5 5 R!0 Langlands R j1,2 × Σn;t ×R j3; x6;7 () R j3; x4;5 ×C × TNN j3; x6;7 ; Duality from | {z } | {z } M-Theory N M5-branes 1 M5-branes Meng-Chwan (1) Tan where n = 1 or 2 for G = SU(N) or SO(N + 1) (N even), and Introduction 4j × Σ × 5j () 5j ×C × SNR!0j ; Review of 4d R 1,2 n;t R 3; x6;7 R 3; x4;5 N 3; x6;7 AGT | {z } | {z } N M5-branes + OM5-plane 1 M5-branes 5d/6d AGT (2) W-algebras + Higher QGL where n = 1, 2 or 3 for G = SO(2N), USp(2N − 2) or G2 SUSY gauge (with N = 4). theory + W-algebras + QGL Here, 3 = 1 + 2, the surface C has the same topology as 1 Higher GL Σn;t = Sn × It , and we have an M9-brane at each tip of It . 1 Conclusion The radius of Sn is given by β, which is much larger than It . Lightning Review: An M-Theoretic Derivation of the 4d Pure AGT Correspondence for Compact Groups Higher AGT The relevant spacetime (quarter) BPS states on the LHS of (1) Correspon- dences, and (2) are captured by a gauged sigma model on instanton W-algebras, and Higher moduli space, and are spanned by Quantum Geometric M Langlands ∗ IH 2 U(M ); (3) Duality from U(1) ×T G;m M-Theory m Meng-Chwan Tan while those on the RHS of (1) and (2) are captured by a gauged chiral WZW model on the I-brane C in the equivalent Introduction IIA frame, and are spanned by Review of 4d AGT L W( gaff ): (4) 5d/6d AGT c W-algebras + The physical duality of the compactifications in (1) and (2) will Higher QGL mean that (3) is equivalent to (4), i.e. SUSY gauge theory + W-algebras + M ∗ L QGL IHU(1)2×T U(MG;m) = Wc( gaff ) (5) Higher GL m Conclusion Lightning Review: An M-Theoretic Derivation of the 4d Pure AGT Correspondence for Compact Groups Higher AGT Correspon- dences, The 4d Nekrasov instanton partition function is given by W-algebras, and Higher Quantum _ X 2mhg Geometric Zinst(Λ; 1; 2;~a) = Λ ZBPS;m(1; 2;~a; β ! 0); (6) Langlands Duality from m M-Theory Meng-Chwan where Λ can be interpreted as the inverse of the observed scale Tan 4 of the R j1,2 space on the LHS of (1), and ZBPS;m is a 5d Introduction worldvolume index. Review of 4d AGT Thus, since ZBPS;m is a weighted count of the states in 5d/6d AGT Ω ∗ HBPS;m = IHU(1)2×T U(MG;m), it would mean from (6) that W-algebras + Higher QGL Z (Λ; ; ;~a) = hΨjΨi; (7) SUSY gauge inst 1 2 theory + W-algebras + _ L mhg L ∗ QGL where jΨi = m Λ jΨmi 2 m IHU(1)2×T U(MG;m).
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  • Instanton Counting in Class Sk

    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