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).