Higher AGT Correspondences, W-Algebras, and Higher Quantum

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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 |1,2 × Σn,t ×R |3; x6,7 ⇐⇒ R |3; x4,5 ×C × TNN |3; 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 4| × Σ × 5| ⇐⇒ 5| ×C × SNR→0| , 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 |1,2 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) = hΨ|Ψi, (7) SUSY gauge inst 1 2 theory + W-algebras + ∨ L mhg L ∗ QGL where |Ψi = m Λ |Ψmi ∈ m IHU(1)2×T U(MG,m). Higher GL

Conclusion Lightning Review: An M-Theoretic Derivation of the 4d Pure AGT Correspondence for Compact Groups

Higher AGT In turn, the duality (5) and the consequential observation that Correspon- dences, |Ψi is a sum over 2d states of all energy levels m, mean that W-algebras, and Higher Quantum Geometric |Ψi = |q, ∆i, (8) Langlands Duality from L M-Theory where |q, ∆i ∈ Wc( gaﬀ ) is a coherent state, and from (7), Meng-Chwan Tan Zinst(Λ, 1, 2,~a) = hq, ∆|q, ∆i (9) Introduction

Review of 4d AGT Since the LHS of (9) is defined in the β → 0 limit of the LHS 5d/6d AGT of (1), |q, ∆i and hq, ∆| ought to be a state and its dual W-algebras + associated with the puncture at z = 0, ∞ on C, respectively (as Higher QGL these are the points where the S1 fiber has zero radius). This is SUSY gauge n theory + depicted in fig. 1 and 2. W-algebras + QGL Incidentally, ΣSW in fig. 1 and 2 can also be interpreted as the Higher GL Seiberg-Witten curve which underlies Z (Λ, , ,~a)! Conclusion inst 1 2 Lightning Review: An M-Theoretic Derivation of the 4d Pure AGT Correspondence for Compact Groups

# Higher AGT ! Correspon- !" dences, W-algebras, and Higher Quantum $"' $"! Geometric #%&$%& #%%&$& Langlands Duality from M-Theory

Meng-Chwan Tan Figure 1:Σ SW as an N-fold cover of C

Introduction & '$% Review of 4d AGT

5d/6d AGT

W-algebras + #%& #%! Higher QGL !!""#$ !#!"" $

SUSY gauge theory + W-algebras + QGL & '$% Higher GL

Conclusion Figure 2:Σ SW as a 2N-fold cover of C Lightning Review: An M-Theoretic Derivation of the 4d AGT Correspondence with Matter

Higher AGT Let us now extend our derivation of the pure AGT Correspon- dences, correspondence to include matter. W-algebras, and Higher Quantum For illustration, we shall restrict ourselves to the A-type Geometric Langlands superconformal quiver gauge theories described by Gaiotto Duality from M-Theory in [24]. Meng-Chwan To this end, ﬁrst note that our derivation of the pure 4d AGT Tan correspondence is depicted in ﬁg. 3. Introduction

Review of 4d $$#% &'(!) AGT

# 5d/6d AGT " " ! ! & & "# " ! ! " # " %' +* %' W-algebras + (*), (*), Higher QGL

SUSY gauge theory + W-algebras + QGL Figure 3: A pair of M9-branes in the original compactiﬁcation in the limit β → 0 and the corresponding CFT on C in the dual compactiﬁcation in our Higher GL derivation of the 4d pure AGT correspondence. Conclusion Lightning Review: An M-Theoretic Derivation of the 4d AGT Correspondence with Matter

Higher AGT This suggests that we can use the following building blocks in Correspon- dences, ﬁg. 4 for our derivation of the 4d AGT correspondence with W-algebras, and Higher matter. Quantum Geometric Langlands !!"# #$%"& Duality from M-Theory # # ! ! !

$ "% $ "% $ "% ( " ( Meng-Chwan &'&'%! &'&'%! &'&'%! Tan

Introduction

Review of 4d ! " AGT !"# #$% &

$ 5d/6d AGT )#* ( ! ! $ ( " $ " "% W-algebras + % &'&'%! &'&'%! Higher QGL

SUSY gauge theory + W-algebras + QGL Figure 4: Building blocks with “minimal” M9-branes for our derivation of Higher GL the AGT correspondence with matter Conclusion Lightning Review: An M-Theoretic Derivation of the 4d AGT Correspondence with Matter

Higher AGT Consider a conformal necklace quiver of n SU(N), N > 2. Correspon-

dences, ! "!# W-algebras, "! " ##$ ! and Higher $ Quantum "! " ##$ Geometric ! "!# ! Langlands ! !

Duality from "! " ##! M-Theory "! " ## " Meng-Chwan ! "!# Tan

Introduction

" %!& %$$!

Review of 4d " %!& %$ AGT

5d/6d AGT ! ! W-algebras +

Higher QGL " %!& %!

" %!& %" SUSY gauge theory + W-algebras + QGL Figure 5: The necklace quiver diagram and the various steps that lead us Higher GL to the overall Riemann surface Σ on which our 2d CFT lives. Conclusion Lightning Review: An M-Theoretic Derivation of the 4d AGT Correspondence with Matter

Higher AGT Correspon- dences, W-algebras, and Higher Quantum Geometric ! &$&' '()#* Langlands

Duality from " $$$ #! $$! " #$$! " #! $ ) *# "!#" " #" " !## $$% + M-Theory " + # #! $ " #$ "!#! "!# " # "#! $$% #!% , ! ! " ) *# $ ++# Meng-Chwan % " " #!% %# % %# - Tan " !!!"#$ " % (#$ !!!"#$ % (#$ %# % (#% %# ) +% *#% !!!"#$ Introduction !!! ) *# , "#$ $ , -.. Review of 4d AGT

5d/6d AGT

W-algebras + Higher QGL Figure 6: The eﬀective 4d-2d correspondence SUSY gauge theory + W-algebras + QGL

Higher GL

Conclusion Lightning Review: An M-Theoretic Derivation of the 4d AGT Correspondence with Matter

Higher AGT Correspon- dences, In the case of a necklace quiver of n SU(N) gauge groups, W-algebras, and Higher Quantum neck D E Z ∼ V (1)V (q1) ... V (q1q2 ... qn−1) (10) Geometric inst ~j1 ~j2 ~jn 2 Langlands T Duality from M-Theory where V~ (z) is a primary vertex operator of the Verma module ji Meng-Chwan L Tan Wc( su(N)aﬀ ) with highest weight

Introduction −im~ s−1 Review of 4d ~js = √ for s = 1, 2,..., n (11) AGT 12 5d/6d AGT

W-algebras + and conformal dimension Higher QGL

SUSY gauge ~2 ~ (2) js js · i~ρ (1 + 2) theory + us = − √ , where s = 1, 2,..., n (12) W-algebras + 2 QGL 1 2

Higher GL

Conclusion An M-Theoretic Derivation of a 5d Pure AGT Correspondence for A–B Groups

Higher AGT Correspon- dences, W-algebras, and Higher A pure U(1) theory can also be interpreted as the m → ∞, Quantum 0 Geometric e2πiτ → 0 limit of a U(1) theory with an adjoint Langlands Duality from hypermultiplet matter of mass m and complexified gauge M-Theory coupling τ 0, where me2πiτ 0 = Λ remains fixed. This means from Meng-Chwan Tan fig. 5 (with n = 1) that the 5d Nekrasov instanton partition

Introduction function for pure U(1) can be expressed as

Review of 4d AGT pure, 5d Zinst, U(1)(1, 2, β, Λ) = h∅|Φm→∞(1)|∅iS2 , (13) 5d/6d AGT

W-algebras + Higher QGL where Φm→∞(1) is the 5d analog of the 4d primary vertex

SUSY gauge operator V in ﬁg. 6 in the m → ∞ limit. ~j1 theory + W-algebras + QGL

Higher GL

Conclusion An M-Theoretic Derivation of a 5d Pure AGT Correspondence for A–B Groups

Higher AGT In the 5d case where β 0, states on C are no longer localized Correspon- 9 dences, to a point but are projected onto a circle of radius β. This W-algebras, and Higher results in the contribution of higher excited states which were Quantum Geometric decoupled in the 2d CFT of chiral fermions when the states Langlands Duality from were deﬁned at a point. Consequently, we can compute that M-Theory pure, 5d Meng-Chwan Zinst, U(1)(1, 2, β, Λ) = hGU(1)|GU(1)i (14) Tan with Introduction ! X 1 (βΛ)n Review of 4d |GU(1)i = exp − a−n |∅i, (15) AGT n 1 − tn n>0 5d/6d AGT where the deformed Heisenberg algebra W-algebras + Higher QGL 1 − t|p| SUSY gauge [ap, an] = p δp+n,0, ap>0|∅i = 0 (16) theory + 1 − q|p| W-algebras + QGL and √ √ Higher GL t = e−iβ 12 , q = e−iβ(1+2+ 12). (17) Conclusion An M-Theoretic Derivation of a 5d Pure AGT Correspondence for A–B Groups

Higher AGT Correspon- According to ﬁg. 1, the 2d CFT in the SU(N) case is just an dences, W-algebras, N-tensor product of the 2d CFT in the U(1) case. In other and Higher Quantum words, we have Geometric Langlands Duality from pure, 5d M-Theory Zinst, SU(N)(1, 2,~a, β, Λ) = hGSU(N)|GSU(N)i (18) Meng-Chwan Tan ni − P 1 (βΛ) a n n >0 n 1−tni −ni n Introduction |GSU(N)i = (⊗i=1e i i ) · (⊗i=1|∅ii ) (19) Review of 4d AGT where

Conclusion An M-Theoretic Derivation of a 5d Pure AGT Correspondence for A–B Groups

Higher AGT Correspon- dences, W-algebras, and Higher Note that (19)–(21) means that |GSU(N)i is a coherent state Quantum Geometric state in a level N module of a Ding-Iohara algebra [25], which, Langlands Duality from according to [26], means that M-Theory

Meng-Chwan Tan q L |GSU(N)i ∈ Wd( su(N)aﬀ) (22)

Introduction q L Review of 4d is a coherent state in the Verma module of W ( su(N)aff), the AGT q-deformed affine W-algebra associated with Lsu(N) . 5d/6d AGT aff

W-algebras + Higher QGL The relations (18) and (22) deﬁne a 5d pure AGT SUSY gauge correspondence for the A groups. theory + N−1 W-algebras + QGL

Higher GL

Conclusion An M-Theoretic Derivation of a 5d Pure AGT Correspondence for A–B Groups

Higher AGT Therefore, according to [7, 27, 28], with regard to the 2d CFT’s Correspon- dences, on the RHS of (9) and (18), we have the following diagram W-algebras, and Higher Quantum Geometric Langlands Duality from Yb(gl(1)aff,1) ⊗ · · · ⊗ Yb(gl(1)aff,1) M-Theory | {z } Wc(su(N)aff,k ) Meng-Chwan N times Tan

Introduction β → 0 β 9 0 β → 0 β 9 0

Review of 4d AGT U (Lgl(1) ) ⊗ · · · ⊗ U (Lgl(1) ) cq aff,1 cq aff,1 q 5d/6d AGT | {z } Wd(su(N)aff,k ) N times W-algebras + Higher QGL (23) SUSY gauge theory + W-algebras + where Yb(gl(1)aff,1) and Ucq(Lgl(1)aff,1) are level one modules of QGL the Yangian and quantum toroidal algebras, respectively, and Higher GL the level k(N, ). Conclusion 1,2 An M-Theoretic Derivation of a 5d Pure AGT Correspondence for A–B Groups

Higher AGT Let = 0, i.e. turn off Omega-deformation on 2d side. This Correspon- 3 dences, ungauges the chiral WZW model on C. Then, conformal W-algebras, and Higher invariance, and the remarks above (14), mean that we have the Quantum Geometric following diagram Langlands Duality from M-Theory gl[(1) ⊗ · · · ⊗ gl[(1) aff,1 aff,1 su\(N) Meng-Chwan | {z } aff,1 Tan N times

Introduction β → 0 β 9 0 β → 0 β 9 0 Review of 4d AGT 5d/6d AGT L\gl(1) ⊗ · · · ⊗ L\gl(1) ) aff,1 aff,1 L\su(N) W-algebras + | {z } aff,1 Higher QGL N times (24) SUSY gauge From diagrams (23) and (24), turning on Omega-deformation theory + W-algebras + on the 2d side effects (i) gl(1) → Y(gl(1) ) and QGL aff,1 aff,1

Higher GL su(N)aff,1 → W(su(N)aff,k ); (ii) Lgl(1)aff,1 → Uq(Lgl(1)aff,1) q Conclusion and Lsu(N)aff,1 → W (su(N)aff,k ). An M-Theoretic Derivation of a 5d Pure AGT Correspondence for A–B Groups

Higher AGT Correspon- dences, Now, with 3 = 0 still, let n = 2 in (1), i.e. G = SO(N + 1). W-algebras, and Higher Then, we have, on the 2d side, the following diagram Quantum Geometric Langlands Duality from M-Theory (2) (2) Meng-Chwan gl[(1) ⊗ · · · ⊗ gl[(1) (2) Tan aff,1 aff,1 su\(N) | {z } aff,1 Introduction N times

Review of 4d AGT β → 0 β 9 0 β → 0 β 9 0 5d/6d AGT

W-algebras + (2) (2) Higher QGL (2) L\gl(1)aff,1 ⊗ · · · ⊗ L\gl(1)aff,1) SUSY gauge L\su(N) theory + | {z } aff,1 W-algebras + N times QGL (25)

Higher GL

Conclusion An M-Theoretic Derivation of a 5d Pure AGT Correspondence for A–B Groups

Higher AGT Correspon- dences, Now, turn on Omega-deformation on 2d side, i.e. 3 6= 0. W-algebras, and Higher According to the remarks below (24), we have Quantum Geometric Langlands Duality from (2) (2) M-Theory Yb(gl(1) ) ⊗ · · · ⊗ Yb(gl(1) ) (2) aff,1 aff,1 Wc(su(N) ) Meng-Chwan | {z } aff,k Tan N times

Introduction

Review of 4d β → 0 β 9 0 β → 0 β 9 0 AGT

5d/6d AGT (2) (2) Ucq(Lgl(1) ) ⊗ · · · ⊗ Ucq(Lgl(1) ) (2) W-algebras + aff,1 aff,1 Wq(su(N) ) Higher QGL | {z } d aff,k SUSY gauge N times theory + W-algebras + (26) QGL

Higher GL

Conclusion An M-Theoretic Derivation of a 5d Pure AGT Correspondence for A–B Groups

Higher AGT Correspon- dences, Comparing the bottom right-hand corner of (26) with the W-algebras, and Higher bottom right-hand corner of (23) for the A groups, and bearing Quantum Geometric (2) ∼ L Langlands in mind the isomorphism su(N)aﬀ = so(N + 1)aﬀ, it would Duality from M-Theory mean that we ought to have

Meng-Chwan Tan pure, 5d Zinst, SO(N+1)(1, 2,~a, β, Λ) = hGSO(N+1)|GSO(N+1)i (27) Introduction

Review of 4d AGT where the coherent state

5d/6d AGT q L W-algebras + |GSO(N+1)i ∈ Wd( so(N + 1)aﬀ) (28) Higher QGL

SUSY gauge theory + The relations (27) and (28) deﬁne a 5d pure AGT W-algebras + QGL correspondence for the BN/2 groups. Higher GL

Conclusion An M-Theoretic Derivation of a 5d Pure AGT Correspondence for

C–D–G2 Groups

Higher AGT Now, with = 0, let n = 1, 2 or 3 in (2), i.e. G = SO(2N), Correspon- 3 dences, USp(2N − 2) or G (with N = 4). This ungauges the chiral W-algebras, 2 and Higher WZW model on C. Then, conformal invariance, and the Quantum Geometric remarks above (14), mean that we have, on the 2d side, the Langlands Duality from following diagram M-Theory

Meng-Chwan Tan (n) (n) (n) so[(2)aff,1 ⊗ · · · ⊗ so[(2)aff,1 Introduction so\(2N) | {z } aff,1 Review of 4d N times AGT

5d/6d AGT β → 0 β 9 0 β → 0 β 9 0 W-algebras + Higher QGL

SUSY gauge (n) (n) theory + L\so(2) ⊗ · · · ⊗ L\so(2) ) (n) W-algebras + aff,1 aff,1 Lso\(2N) QGL | {z } aff,1 N times Higher GL (29)

Conclusion An M-Theoretic Derivation of a 5d Pure AGT Correspondence for

C–D–G2 Groups

Higher AGT Correspon- dences, W-algebras, Now, turn on Omega-deformation on 2d side, i.e. 3 6= 0. and Higher Quantum According to the remarks below (24), we have Geometric Langlands Duality from M-Theory (n) (n) Yb(so(2) ) ⊗ · · · ⊗ Yb(so(2) ) (n) aff,1 aff,1 W(so(2N) ) Meng-Chwan | {z } c aff,k0 Tan N times

Introduction

Review of 4d β → 0 β 9 0 β → 0 β 9 0 AGT 5d/6d AGT U (Lso(2)(n) ) ⊗ · · · ⊗ U (Lso(2)(n) ) cq aff,1 cq aff,1 q (n) W-algebras + Wd(so(2N) 0 ) Higher QGL | {z } aff,k N times SUSY gauge theory + W-algebras + (30) QGL

Higher GL

Conclusion An M-Theoretic Derivation of a 5d Pure AGT Correspondence for

C–D–G2 Groups

Higher AGT Correspon- dences, Comparing the bottom right-hand corner of (30) with the W-algebras, and Higher bottom right-hand corner of (23) for the A groups, and bearing Quantum Geometric (n) ∼ L Langlands in mind the isomorphism so(2N)aﬀ = gaﬀ, it would mean that Duality from we ought to have M-Theory

Meng-Chwan Tan pure, 5d Zinst, G (1, 2,~a, β, Λ) = hGG |GG i (31) Introduction Review of 4d where the coherent state AGT

5d/6d AGT |G i ∈ Wq(Lg ) (32) W-algebras + G d aﬀ Higher QGL

SUSY gauge theory + The relations (31) and (32) deﬁne a 5d pure AGT W-algebras + QGL correspondence for the CN−1, DN and G2 groups. Higher GL

Conclusion An M-Theoretic Derivation of a 5d Pure AGT Correspondence for E6,7,8

and F4 Groups

Higher AGT Correspon- dences, W-algebras, and Higher By starting with M-theory on K3 with G = E6,7,8 and F4 Quantum Geometric singularity and its string-dual type IIB on the same K3 (in the Langlands Duality from presence of ﬂuxbranes), one can, from the principle that the M-Theory relevant BPS states in both frames ought to be equivalent, Meng-Chwan Tan obtain, in the limit 1 = h = −2, the relation

Introduction ∗ G L IH (M 4 ) = bg , (33) U(1)h×U(1)−h×T R aﬀ,1 Review of 4d AGT 5d/6d AGT Then, repeating the arguments that took us from (7) to (9), W-algebras + Higher QGL we have pure, 4d SUSY gauge Zinst, G (h,~a, Λ) = hcohh|cohhi. (34) theory + W-algebras + QGL

Higher GL

Conclusion An M-Theoretic Derivation of a 5d Pure AGT Correspondence for E6,7,8

and F4 Groups

Higher AGT Correspon- In turn, according to the remarks above (14), we ﬁnd that dences, W-algebras, and Higher pure, 5d Quantum Z (h,~a, Λ) = hcir |cir i (35) Geometric inst, G h h Langlands Duality from M-Theory where Meng-Chwan L Tan |cirhi ∈ Ldgaﬀ,1 (36)

Introduction L and L gaff,1 is a Langlands dual toroidal Lie algebra given by Review of 4d L AGT the loop algebra of gaff,1. 5d/6d AGT Together, (35) and (36) define a 5d pure AGT correspondence W-algebras + Higher QGL for the E6,7,8 and F4 groups in the topological string limit. SUSY gauge They are consistent with (25) and (29). theory + W-algebras + QGL The analysis for 3 6= 0 is more intricate via this approach. Left Higher GL for future work. Conclusion An M-Theoretic Derivation of a 6d AGT Correspondence for A Groups

Higher AGT Consider the non-anomalous case of a conformal linear quiver Correspon- dences, SU(N) theory in 6d. As explained in [9, §5.1], we have W-algebras, and Higher Quantum lin, 6d w v Geometric Z (q , , , m~ , β, R ) = h Φ˜ (z ) Φ˜ (z ) i 2 (37) Langlands inst, SU(N) 1 1 2 6 v 1 u 2 T Duality from M-Theory 1 1 2 1 1 Meng-Chwan where β and R6 are the radii of S and St in T = S × St , Tan β R6, and the 6d vertex operators Φ(˜ z) have a projection 2 Introduction onto two transverse circles Cβ and CR6 in T of radius β and Review of 4d R6, respectively, which intersect at the point z. Here, w, v, u AGT are related to the matter masses. 5d/6d AGT W-algebras + In the same way that we arrived at (18) and (19), we have Higher QGL SUSY gauge ˜ c ˜ ˜ ˜ ˜ ˜ ˜ theory + Φd : Fd1 ⊗ Fd2 ⊗ · · · ⊗ FdN −→ Fc1 ⊗ Fc2 ⊗ · · · ⊗ FcN , (38) W-algebras + QGL where F˜c,d is a module over the elliptic Ding-Iohara Higher GL algebra [29] deﬁned by Conclusion An M-Theoretic Derivation of a 6d AGT Correspondence for A Groups

Higher AGT Correspon- dences, |m| W-algebras, |m| 1 − t ˜ and Higher [ãm, ãn] = m(1 − v ) δm+n,0, ãm>0|∅i = 0 (39) Quantum 1 − q|m| Geometric Langlands Duality from |m| |m| M-Theory m(1 − v ) 1 − t ˜ [b˜m, bñ] = δm+n,0, b˜m>0|∅i = 0 Meng-Chwan (tq−1v)|m| 1 − q|m| Tan (40) Introduction where [ãm, bñ] = 0, and Review of 4d AGT √ √ 1 −iβ 12 −iβ(1+2+ 12) − R 5d/6d AGT t = e , q = e , v = e 6 (41)

W-algebras + Higher QGL In other words,

SUSY gauge ˜ c q,v L q,v L theory + Φd : W[ ( su(N)aff ) → W[ ( su(N)aff ) (42) W-algebras + QGL q,v q,v L Higher GL where W[ is a Verma module over W ( su(N)aff ), an L Conclusion elliptic affine W( su(N)aff )-algebra. An M-Theoretic Realization of Affine W-algebras and a Quantum Geometric Langlands Duality

Higher AGT Correspon- To derive W-algebra identities which underlie a Langlands dences, duality, let us specialize our discussion of the 4d AGT W-algebras, and Higher correspondence to the N = 4 or massless N = 2∗ case, so that Quantum Geometric we can utilize S-duality. From fig. 6 (and its straighforward Langlands Duality from generalization to include an OM5-plane), we have the dual M-Theory compactifications Meng-Chwan Tan R4| × T2 ×R5| ⇐⇒ R5| × T2 × TNR→0| , Introduction 1,2 σ 3; x6,7 3; x4,5 σ N 3; x6,7 | {z } | {z } Review of 4d N M5-branes 1 M5-branes AGT (43) 5d/6d AGT and W-algebras + Higher QGL 4 2 5 5 2 R→0 SUSY gauge R |1,2 × Tσ ×R |3; x6,7 ⇐⇒ R |3; x4,5 ×Tσ × SNN |3; x6,7 , theory + | {z } | {z } W-algebras + N M5-branes + OM5-plane 1 M5-branes QGL (44) Higher GL where T2 = S1 × S1. Conclusion σ t n An M-Theoretic Realization of Affine W-algebras and a Quantum Geometric Langlands Duality

Higher AGT Correspon- dences, W-algebras, and Higher Recall from earlier that Quantum Geometric M ∗ L L L 2 Langlands IHU(1)2×T U(MG,m) = W\aﬀ,Lκ( g), κ + hg = − . Duality from 1 M-Theory m

Meng-Chwan (45) Tan Let n = 1. From the symmetry of 1 ↔ 2 in (43) and (44), L ∼ Introduction and gaﬀ = gaﬀ for simply-laced case, we have, from the RHS Review of 4d of (45), AGT

5d/6d AGT L ∨ ∨ L L ∨ −1 W-algebras + Waﬀ,k (g) = Waﬀ,Lk ( g), where r (k + h ) = ( k + h ) Higher QGL (46) SUSY gauge ∨ theory + r = n is the lacing number, and g = su(N) or so(2N). W-algebras + QGL

Higher GL

Conclusion An M-Theoretic Realization of Aﬃne W-algebras and a Quantum Geometric Langlands Duality

Higher AGT Correspon- dences, W-algebras, ∨ and Higher Let n = 2 or 3. Eﬀect a modular transformation τ → −1/r τ Quantum 2 Geometric of Tσ in (43) and (44) which eﬀects an S-duality in the 4d Langlands Duality from gauge theory along the directions ortohgonal to it. As the LHS M-Theory 2 of (45) is derived from a topological sigma model on Tσ that is Meng-Chwan Tan hence invariant under this transformation, it would mean from (45) that Introduction

Review of 4d L ∨ L L −1 AGT Waﬀ,k (g) = Waﬀ,Lk ( g), where r (k + h) = ( k + h) ; 5d/6d AGT (47) W-algebras + h = h(g) and Lh = h(Lg) are Coxeter numbers; and Higher QGL SUSY gauge g = Lso(2M + 1), Lusp(2M) or Lg . theory + 2 W-algebras + QGL

Higher GL

Conclusion An M-Theoretic Realization of Aﬃne W-algebras and a Quantum Geometric Langlands Duality

Higher AGT Correspon- dences, W-algebras, and Higher g Quantum In order to obtain an identity for g = , i.e. the Langlands dual Geometric of (47), one must exchange the roots and coroots of the Lie Langlands Duality from algebra underlying (47). This also means that h must be M-Theory replaced by its dual h∨. In other words, from (47), one also has Meng-Chwan Tan L ∨ ∨ L L ∨ −1 Introduction Waﬀ,k (g) = Waﬀ,Lk ( g), where r (k + h ) = ( k + h ) Review of 4d AGT (48)

5d/6d AGT and g = so(2M + 1), usp(2M) or g2.

W-algebras + Higher QGL Clearly, (46) and (48), deﬁne a quantum geometric Langlands

SUSY gauge duality for G as ﬁrst formulated by Feigin-Frenkel [15]. theory + W-algebras + QGL

Higher GL

Conclusion An M-Theoretic Realization of q-deformed Aﬃne W-algebras and a Quantum q-Geometric Langlands Duality

Higher AGT Correspon- dences, W-algebras, From the relations (20) and (21), it would mean that we can and Higher Quantum write the algebra on the RHS of (22) as a two-parameter Geometric Langlands algebra Duality from q,t M-Theory Waff,k (su(N)). (49) Meng-Chwan Tan Note that as so(2)aff,1 in diagram (30) is also a Heisenberg Introduction algebra like gl(1)aff,1, it would mean that Uq(Lso(2)aff,1) Review of 4d therein is also a Ding-Iohara algebra at level 1 (with an extra AGT reality condition) that can be defined by the relations (20) and 5d/6d AGT

W-algebras + (21). Hence, we also have a two-parameter algebra Higher QGL q,t SUSY gauge W (so(2N)). (50) theory + aﬀ,k W-algebras + QGL

Higher GL

Conclusion An M-Theoretic Realization of q-deformed Aﬃne W-algebras and a Quantum q-Geometric Langlands Duality

Higher AGT Note that the change (1, 2) → (−2, −1) is a symmetry of Correspon- −iβ(1+2) dences, our physical setup, and if we let p = q/t = e , then, W-algebras, −1 and Higher the change p → p which implies q ↔ t, is also a symmetry Quantum of our physical setup. Then, the last two paragraphs together Geometric Langlands with k + h∨ = − / mean that Duality from 2 1 M-Theory q,t t,q ∨ Meng-Chwan L ∨ ∨ L L −1 W (g) = W L ( g), where r (k + h ) = ( k + h ) Tan aff,k aff, k (51) Introduction and g = su(N) or so(2N). Review of 4d AGT Identity (51) is just Frenkel-Reshetikhin’s result in [16, §4.1] 5d/6d AGT which defines a quantum q-geometric Langlands duality for the W-algebras + Higher QGL simply-laced groups! SUSY gauge The nonsimply-laced case requires a modular transformation of theory + W-algebras + T2 which effects the swop S1 ↔ S1, where in 5d, S1 is a QGL σ n t n

Higher GL preferred circle as states are projected onto it. So, (51) doesn’t

Conclusion hold, consistent with Frenkel-Reshetikhin’s result. An M-Theoretic Realization of Elliptic Aﬃne W-algebras and a Quantum q, v-Geometric Langlands Duality

Higher AGT Correspon- Similarly, from (39) and (40), we can express Wq,v (su(N) ) dences, aﬀ,k W-algebras, on the RHS of (42) as a three-parameter algebra and Higher Quantum Geometric q,t,v Langlands Waﬀ,k (su(N)). (52) Duality from M-Theory Meng-Chwan Repeating our arguments, we have Tan

Introduction q,t,v t,q,v L ∨ ∨ L L ∨ −1 Waff,k (g) = W L ( g), where r (k + h ) = ( k + h ) Review of 4d aff, k AGT (53) 5d/6d AGT and g = su(N) or so(2N). W-algebras + Higher QGL Clearly, identity (53) defines a quantum q-geometric Langlands SUSY gauge duality for the simply-laced groups! theory + W-algebras + QGL The nonsimply-laced case should reduce to that for the 5d one, Higher GL but since the latter does not exist, neither will the former. Conclusion Summary: M-Theoretic Realization of W-algebras and Higher Geometric Langlands Duality

Higher AGT Correspon- dences, In summary, by considering various limits, we have W-algebras, and Higher Quantum Geometric 2 → 0 Langlands L L Duality from Waff,k (g) = Waff,Lk ( g) Z(U(ˆg)crit) = Wcl( g) M-Theory 2 9 0 Meng-Chwan β → 0 β 9 0 β → 0 β 0 Tan 9 2 → 0 Wq,t (g) = Wt,q (Lg) q L Introduction aff,k aff,Lk Z(Uq(gˆ)crit) = Wcl( g) Review of 4d 2 9 0 AGT R6 → 0 R6 9 0 R6 → 0 R6 9 0 5d/6d AGT 2 → 0 q,t,v t,q,v L q,v L W-algebras + Waff,k (g) = Waff,Lk ( g) Z(Uq,v (gˆ)crit) = W ( g) Higher QGL cl 2 9 0 SUSY gauge theory + (54) W-algebras + QGL where g is arbitrary while g is simply-laced. Higher GL

Conclusion A Quantum Geometric Langlands Correspondence as an S-duality and a Quantum W-algebra Duality

Higher AGT Correspon- From the fact that in the low energy sector of the worldvolume dences, theories in (43) and (44) that is relevant to us, the W-algebras, and Higher worldvolume theory is topological along R4, we have Quantum Geometric Langlands Duality from DR,1 × DR,2 × Σ1 and DR,1 × DR,2 × Σ1 , (55) M-Theory | {z } | {z } Meng-Chwan N M5-branes N M5-branes + OM5-plane Tan where Σ = S1 × S1 is a Riemann surface of genus one with Introduction 1 t n

Review of 4d zero punctures. AGT Macroscopically at low energies, the curvature of the cigar tips 5d/6d AGT is not observable. Therefore, we can simply take (55) to be W-algebras + Higher QGL 2 2 SUSY gauge T1,2 × I1 × I2 × Σ1 and T1,2 × I1 × I2 × Σ1 , (56) theory + | {z } | {z } W-algebras + N M5-branes N M5-branes + OM5-plane QGL Higher GL 2 1 1 where T , = S × S is a torus of rotated circles. Conclusion 1 2 1 2 A Quantum Geometric Langlands Correspondence as an S-duality and a Quantum W-algebra Duality

Higher AGT Clearly, the relevant BPS states are captured by the remaining Correspon- dences, uncompactified 2d theory on I × I which we can regard as a W-algebras, 1 2 and Higher sigma model which descended from the N = 4, G theory over Quantum Geometric I1 × I2 × Σ , so Langlands 1 Duality from σ Σ1 L L L 2 M-Theory HI ×I (X )B = W\aff,Lk ( g) , k + h = − . (57) 1 2 G Σ1 Meng-Chwan 1 Tan Now consider Introduction ˜ 2 ˜ ˜ 2 ˜ Review of 4d T1,2 × I1 × I2 × Σ1, and T1,2 × I1 × I2 × Σ1 , (58) AGT | {z } | {z } N M5-branes N M5-branes + OM5-plane 5d/6d AGT 2 2 W-algebras + where T˜ and Σ˜ 1 are T and Σ1 with the one-cycles Higher QGL 1,2 1,2 swopped. SUSY gauge theory + W-algebras + So, in place of (57), we have QGL L σ Σ1 ∨ 1 Higher GL H (XL )L = W\aff,k (g) , r (k + h) = − . (59) I1×I2 G B Σ1 Conclusion 2 A Quantum Geometric Langlands Correspondence as an S-duality and a Quantum W-algebra Duality

Higher AGT Since (56) and (58) are equivalent from the viewpoint of the Correspon- dences, worldvolume theory, we have W-algebras, and Higher Quantum Geometric Langlands Duality from String Duality M-Theory Hσ (X Σ1 ) L\ g I1×I2 G B Waﬀ,κ( )Σ Meng-Chwan 1 Tan

Introduction 1 ∨ L L −1 S-duality τ → − r ∨τ r (κ + h) = ( κ + h) W-duality Review of 4d AGT

5d/6d AGT L Σ String Duality W-algebras + σ 1 L L H (XL )LB W\L ( g)Σ Higher QGL I1×I2 G aff, κ 1 (60) SUSY gauge theory + L W-algebras + where Waff,κ(g) is the “Langlands dual” of Waff,κ(g), an QGL affine W-algebra of level κ labeled by the Lie algebra g, and Higher GL κ + h = − / . Conclusion 2 1 A Quantum Geometric Langlands Correspondence as an S-duality and a Quantum W-algebra Duality

Higher AGT 4d/2d CFT Correspon- dences, W-algebras, and Higher Quantum Geometric Langlands

Duality from 4d/2d CFT M-Theory

Meng-Chwan Tan

Introduction (a)

Review of 4d 4d/2d AGT CFT

5d/6d AGT

W-algebras + (b) Higher QGL 4d/2d SUSY gauge theory + W-algebras + (c) QGL 4d/2d Higher GL

Conclusion (d)

Figure 7: Building blocks relevant to our discussion on the connection between the gauge-theoretic realization and algebraic CFT formulation of the geometric Langlands correspondence. (a). Dual M-theory compactifications which realize the AGT correspondence for N = 2∗ SU(N) theory and N = 2 necklace quiver theory with two SU(N) gauge groups, where β denotes the size of Σ. (b). Dual M-theory compactifications which realize the AGT correspondence for N = 2 pure SU(N) theory. (c). Gluing together g copies of the (a)-compactifications g,0 using the (b)-compactifications. (d). Finally, a single compactification on Σt,n without M9-planes on the 4d side, corresponding to a genus g surface with no punctures on the 2d side. A Quantum Geometric Langlands Correspondence as an S-duality and a Quantum W-algebra Duality

Higher AGT So, we can eﬀectively replace Σ1 with Σg in (56), (58), and Correspon- dences, thus in (57), (59), whence we can do the same in (60), and W-algebras, and Higher σ Σg A mod H (X ) = H (M (G, Σ )) = D ∨ (Bun (Σ )) Quantum I1×I2 G B I1×I2 H g Bd.c., Bα LΨ−h G g Geometric (61) Langlands Duality from where MH (G , Σg ) and BunG (Σg ) are the moduli space of G M-Theory Hitchin equations and G -bundles on Σg [18, 19], so in place Meng-Chwan C Tan of (60), we have String Duality Introduction mod L D Ψ−h∨ (BunG (Σg )) W\aﬀ,κ(g) Review of 4d L Σg AGT

5d/6d AGT L 1 L L 1 W-algebras + S-duality Ψ = − r ∨Ψ ( κ + h) = r ∨(κ+h) FF-duality Higher QGL

SUSY gauge theory + String Duality W-algebras + mod L QGL D L L ∨ (BunLG (Σg )) L \ L Ψ− h Waﬀ,Lκ( g)Σg Higher GL

Conclusion (62) A Geometric Langlands Correspondence as an S-duality and a Classical W-algebra Duality

Higher AGT Correspon- dences, Let 1 = 0 whence we would also have κ = ∞ and Ψ = 0. W-algebras, and Higher Then, (62) becomes Quantum Geometric Langlands Duality from String Duality M-Theory mod ﬂat Dcrit (BunG (Σg )) MLG (Σg ) Meng-Chwan C Tan

Introduction Review of 4d S-duality KW realization BD formulation FF-duality AGT

5d/6d AGT

W-algebras + Higher QGL String Duality ﬂat mod DL (Σg ) M (BunG (Σg )) SUSY gauge GC crit theory + W-algebras + QGL (63)

Higher GL

Conclusion A Geometric Langlands Correspondence as an S-duality and a Classical W-algebra Duality

Higher AGT Correspon- dences, Adding boundary M2-branes which realize line operators in the W-algebras, and Higher gauge theory and performing the chain of dualities would Quantum Geometric replace (43) and (44) with Langlands Duality from M-Theory ˚4 ˚1 1 ˚5 5 1 ˚1 R→0 R |0,2 × Sn × St ×R |2; x6,7 ⇐⇒ R |2; x4,5 ×St × Sn × TNN |2; x6,7 , Meng-Chwan | {z } | {z } Tan N M5 + M2 on ◦ 1 M5-branes + M0 on ◦ (64) Introduction and Review of 4d AGT 4 1 1 5 5 1 1 R→0 5d/6d AGT ˚ ˚ ˚ ˚ R |0,2 × Sn × St ×R |2; x6,7 ⇐⇒ R |2; x4,5 ×St × Sn × SNN |2; x6,7 , W-algebras + | {z } | {z } Higher QGL N M5 + OM5 + M2 on ◦ 1 M5 + M0 on ◦

SUSY gauge (65) theory + W-algebras + Here, the M0-brane will become a D0-brane when we reduce QGL M-theory on a circle to type IIA string theory [31]. Higher GL

Conclusion A Geometric Langlands Correspondence as an S-duality and a Classical W-algebra Duality

Higher AGT As such, in place of (56) and (58), we have Correspon- 1 1 dences, M2 on R+×Sn,g M2 on R+×Sn,g W-algebras, and Higher 2 z }| { 2 z }| { Quantum T0,2 × I × R+ × Σg and T0,2 × I × R+ × Σg (66) Geometric Langlands | {z } | {z } Duality from N M5-branes N M5-branes + OM5-plane M-Theory and Meng-Chwan 1 1 M2 on R ×S˜ M2 on R+×S˜ Tan + n,g n,g 2 z }| g{,0 2 z }| g{,0 Introduction ˜ ˜ ˜ ˜ T0,2 × I × R+ × Σt,n and T0,2 × I × R+ × Σt,n (67) Review of 4d | {z } | {z } AGT N M5-branes N M5-branes + OM5-plane 5d/6d AGT 1 1 Here, Sn,g is a disjoint union of a g number of Sn one-cycles of W-algebras + Higher QGL Σg . SUSY gauge loop theory + Similarly, Σ1 on the RHS of (57), (59) will now be Σg –Σg W-algebras + QGL with a loop operator that is a disjoint union of g number of 1 Higher GL loop operators around its g number of Sn one-cycles, each Conclusion corresponding to a worldoop of a D0-brane. A Geometric Langlands Correspondence as an S-duality and a Classical W-algebra Duality

Higher AGT String Duality Correspon- mod Mﬂat (Σ ) dences, D (BunG (Σg ))B L g L crit ‘t-Hooft GC Wc( g)“Wilson” W-algebras, and Higher Quantum Geometric Langlands Duality from S-duality KW realization BD formulation FF-duality M-Theory

Meng-Chwan Tan ﬂat String Duality mod D (Σ )L M (Bun (Σ )) Introduction L g BWilson crit G g W(g) GC c “’t-Hooft” Review of 4d AGT (68) 5d/6d AGT There is a correspondence in the actions of 4d line operators W-algebras + Higher QGL and 2d loop operators: SUSY gauge theory + W-algebras + B‘t-Hooft ⇐⇒ Wc(g)“’t-Hooft” (69) QGL

Higher GL L L BWilson ⇐⇒ Wc( g)“Wilson” (70) Conclusion A Geometric Langlands Correspondence as an S-duality and a Classical W-algebra Duality

L Higher AGT B‘t-Hooft: m0 → m0 + ξ( R), where magnetic ﬂux m0 and Correspon- L dences, ξ( R) are characteristic classes that classify the topology of W-algebras, 2 and Higher G-bundles over Σg and S , respectively. Thus, the ’t Hooft line Quantum mod Geometric operator acts by mapping each object in Dcrit (BunG (Σg )) Langlands L Duality from labeled by m0, to another labeled by m0 + ξ( R). M-Theory Meng-Chwan On the other hand, Wc(g) is a monodromy operator Tan “’t-Hooft” which acts on the chiral partition functions of the module Introduction mod Mcrit (BunG (Σ1)) as (c.f. [32, §3.2]) Review of 4d AGT X Zg(a) → λa,p Zg(pk ). (71) 5d/6d AGT pk W-algebras + Higher QGL where pk = a + bhk , where hk are coweights of a SUSY gauge representation R of G; and the λa,p’s and b are constants. theory + W-algebras + Therefore, W(g) maps each state in QGL c “’t-Hooft” mod Higher GL Mcrit (BunG (Σg )) labeled by a, to another labeled by a + h, L L Conclusion where h is a weight of a representation R of G. A Geometric Langlands Correspondence as an S-duality and a Classical W-algebra Duality

Higher AGT L Correspon- BWilson : e0 → e0 + θLR , where electric flux e0 and θLR are dences, L W-algebras, characters of the center of (the universal cover of) G. and Higher Quantum Because the e0-labeled zerobranes are points whence the shift Geometric Langlands e0 → e0 + θLR which twists them is trivial, the Wilson line Duality from flat M-Theory operator acts by mapping each object in DL (Σg ) to itself. GC Meng-Chwan Tan L On the other hand, Wc( g)“Wilson” is a monodromy operator Introduction which acts on the chiral partition functions of the module Review of 4d flat AGT ML (Σ1) as (c.f. [32, Appendix D]) GC 5d/6d AGT ∨ ∨ W-algebras + ZLg(a ) → λa∨ ZLg(a ), (72) Higher QGL SUSY gauge ∨ L theory + where the highest coweight vector a of g labels a submodule, W-algebras + L QGL and λa∨ is a constant. Thererfore, Wc( g)“Wilson” maps each flat Higher GL state in ML (Σg ) to itself. GC Conclusion A q-Geometric Langlands Correspondence for Simply-Laced Lie Groups

In the 5d case where β 0, in place of (66), we have Higher AGT 9 Correspon- 2 S1 2 S1 dences, T × R+ × I × Σ and T × R+ × I × Σ (73) W-algebras, 0,2 g 0,2 g and Higher | {z } | {z } Quantum N M5-branes N M5-branes + OM5-plane Geometric S1 Langlands where Σg is the compactiﬁed Riemann surface Σg (where Duality from 1 M-Theory g > 1) with an S loop of radius β over every point.

Meng-Chwan Then, Tan A S1 q L HI×R (MH (G, Σg )) β β = Wd( g)Σg , (74) Introduction or + Bc.c., Bα cl

Review of 4d AGT mod S1 S1 C (M (G, Σg )) = ML (Σg )flat, (75) 5d/6d AGT O~ H.S. G W-algebras + (O~ is a noncommutative algebra of holomorphic functions), so Higher QGL 1 SUSY gauge O (MS (G, Σ ))-module ⇐⇒ circle-valued flat LG-bundle on Σ theory + ~ H.S. g g W-algebras + QGL (76) Clearly, this defines a q-geometric Langlands correspondence Higher GL

Conclusion for simply-laced G! Quantization of Circle-Valued G Hitchin Systems and Transfer Matrices of a G-type XXZ Spin Chain

Higher AGT From diagram (54), one can also express (74) as Correspon- dences, A S1 W-algebras, H (M (G, Σg )) β β = Zb(Uq(gˆ)crit)Σ . (77) and Higher I×R+ H Bc.c., Bα g Quantum Geometric Then, if Txxz(G, Σg ) is the polynomial algebra of commuting Langlands Duality from transfer matrices of a G-type XXZ spin chain with Uq(gˆ) M-Theory symmetry on Σg , where i = 1,..., rank(g), Meng-Chwan Tan S1 O (MH.S.(G, Σg )) ⇐⇒ Txxz(G, Σg ) (78) Introduction ~ Review of 4d which relates the quantization of a circle-valued G Hitchin AGT system on Σ to the transfer matrices of a G-type XXZ spin 5d/6d AGT g

W-algebras + chain on Σg ! Higher QGL This also means that SUSY gauge theory + S1 aﬀ W-algebras + x ∈ MH.S.(G, Σg ) ⇐⇒ χq(Vi ) = Ti (z), Vi ∈ Rep [Uq (g)Σg ] QGL (79) Higher GL and T (z) is a polynomial whose degree depends on V . Conclusion i i A q-Geometric Langlands Correspondence for Simply-Laced Kac-Moody Groups

Higher AGT Note that nonsingular G-monopoles on a flat three space M Correspon- b 3 dences, can also be regarded as well-behaved G-instantons on Sˆ1 × M W-algebras, 3 1 and Higher in [33], while nonsingular G-monopoles on M3 = S × Σ Quantum 1 Geometric correspond to S -valued G Hitchin equations on Σ. Since Langlands Duality from principal bundles on a flat space with Kac-Moody structure M-Theory group are also well-defined [33], a consistent Gb version of (76) Meng-Chwan Tan would be

Introduction 1 O (MS (G, Σ))-module ⇐⇒ circle-valued flat LcG-bundle on Σ Review of 4d ~ H.S. b AGT (80) 5d/6d AGT or equivalently, W-algebras + Higher QGL Sˆ1×S1 L SUSY gauge O (MH.S. (G, Σ))-module ⇐⇒ circle-valued flat cG-bundle on Σ theory + ~ W-algebras + (81) QGL where Σ = R × S1. This defines a G version of the q-geometric Higher GL b

Conclusion Langlands correspondence for simply-laced G. Quantization of Elliptic-Valued G Hitchin Systems and Transfer Matrices of a Gb-type XXZ Spin Chain

Higher AGT In light of the fact that a -bundle can be obtained from a Correspon- Gb dences, -bundle by replacing the underlying Lie algebra g of the latter W-algebras, G and Higher bundle with its Kac-Moody generalization gˆ, from (54), it Quantum Geometric would mean that in place of (78), we now have, Langlands Duality from M-Theory Sˆ1×S1 O~(MH.S. (G, Σ)) ⇐⇒ Txxz(Gb, Σ) (82) Meng-Chwan Tan which relates the quantization of an elliptic-valued G Hitchin Introduction system on Σ to the transfer matrices of a Gb-type XXZ spin Review of 4d chain on Σ! AGT This also means that 5d/6d AGT W-algebras + Sˆ1×S1 ˆ ˆ ˆ aff Higher QGL x ∈ MH.S. (G, Σ) ⇐⇒ χq(Vi ) = Ti (z), Vi ∈ Rep [Uq (gˆ)Σ] SUSY gauge (83) theory + W-algebras + where i = 0,..., rank(g), Tî (z) is a polynomial whose degree QGL depends on Vˆ , and Uaff (gˆ) is the quantum toroidal algebra of Higher GL i q

Conclusion g. A Realization and Generalization of Nekrasov-Pestun-Shatashvili’s

Results for 5d, N = 1 G (Gb)-Quiver SU(Ki ) Gauge Theories Consider instead of the theories in figure (7), an SU(N) theory Higher AGT Correspon- with N = 2N fundamental matter, then, in place of (79), dences, f W-algebras, and Higher G,Cx,y1,y2 aff Quantum u ∈ M 1 ⇐⇒ χq(Vi ) = Ti (z), Vi ∈ Rep [Uq (g){Cx}z ,z ] Geometric S -mono,k 1 2 Langlands 1 Duality from where Cx = R × S , i ∈ IΓ, the G Dynkin vertices. (84) M-Theory Note that (83) also means that Meng-Chwan Tan G,Cx,k aff u ∈ M ⇐⇒ χq(Vî ) = Tî (z), Vî ∈ Rep [U (gˆ)C ] Introduction Sˆ1×S1-inst q x Review of 4d 1 ˆ (85) AGT where Cx = R × S , i ∈ IΓ, the affine-G Dynkin vertices. 1 5d/6d AGT Can argue via momentum around Sn (counted by D0-branes) W-algebras + ↔ 2d CFT energy level correspondence that degree of Ti (Tî ) Higher QGL is K (aK ). SUSY gauge i i theory + W-algebras + (84)/(85) are Nekrasov-Pestun-Shatashvili’s main result in [21, QGL §1.3] which relates the moduli space of the 5d G/Gb-quiver Higher GL gauge theory to the representation theory of Uaff (g)/Uaff (gˆ)! Conclusion q q A q, v-Geometric Langlands Correspondence for Simply-Laced Lie Groups

Higher AGT Correspon- In our derivation of the 6d AGT W-algebra identity in diagram dences, 1 1 W-algebras, (54), the 2d CFT is defined on a torus S × St with two and Higher 1 Quantum punctures at positions z1,2 [9, §5.1]. i.e. Σ1,2. Here, S Geometric Langlands corresponds to the decompactified fifth circle of radius β 9 0, Duality from 1 M-Theory while St corresponds to the sixth circle formed by gluing the Meng-Chwan ends of an interval It of radius R6 much smaller than β. So, we Tan effectively have a single decompactification of circles, like in Introduction the 5d case, although the 2d CFT states continue to be Review of 4d projected onto two circles of radius β and R , whence in place AGT 6

5d/6d AGT of (76), we have

W-algebras + Higher QGL S1 L O (MH.S.(G, Σ1,2))-mod ⇐⇒ elliptic-valued ﬂat G-bundle on Σ1,2 SUSY gauge ~ theory + (86) W-algebras + QGL Clearly, this deﬁnes a q, v-geometric Langlands correspondence Higher GL for simply-laced G! Conclusion Quantization of Circle-Valued G Hitchin Systems and Transfer Matrices of a G-type XYZ Spin Chain

Higher AGT Correspon- Consequently, from diagram (54), if Txyz(G, Σ1,2) is the dences, W-algebras, polynomial algebra of commuting transfer matrices of a G-type and Higher Quantum XYZ spin chain with Uq,v (gˆ) symmetry on Σ1,2, where Geometric Langlands i = 1,..., rank(g), in place of (78), we now have Duality from M-Theory S1 Meng-Chwan O (MH.S.(G, Σ1,2)) ⇐⇒ Txyz(G, Σ1,2) (87) Tan ~

Introduction which relates the quantization of a circle-valued G Hitchin Review of 4d AGT system on Σ1,2 to the transfer matrices of a G-type XYZ spin

5d/6d AGT chain on Σ1,2!

W-algebras + This also means that Higher QGL SUSY gauge S1 ell theory + x ∈ MH.S.(G, Σ1,2) ⇐⇒ χq,v (Vi ) = Ti (z), Vi ∈ Rep [Uq,v (g)Σ1,2 ] W-algebras + QGL (88) Higher GL and Ti (z) is a polynomial whose degree depends on Vi . Conclusion A q, v-Geometric Langlands Correspondence for Simply-Laced Kac-Moody Groups

Higher AGT Correspon- dences, W-algebras, Note that with regard to our arguments leading up to (86), one and Higher Quantum could also consider unpunctured Σ1 instead of Σ1,2 (i.e. Geometric consider the massless limit of the underlying linear quiver Langlands Duality from theory). Consequently, in place of (80), we have M-Theory

Meng-Chwan 1 Tan S L O~(MH.S.(Gb, Σ1))-mod ⇐⇒ elliptic-valued ﬂat cG-bundle on Σ1 Introduction (89)

Review of 4d or equivalently, AGT

5d/6d AGT ˆ1 1 O (MS ×S (G, Σ ))-mod ⇐⇒ elliptic-valued ﬂat LG-bundle on Σ W-algebras + ~ H.S. 1 c Higher QGL (90) SUSY gauge theory + This deﬁnes a Gb version of the q-geometric Langlands W-algebras + QGL correspondence for simply-laced G.

Higher GL

Conclusion Quantization of Elliptic-Valued G Hitchin Systems and Transfer Matrices of a Gb-type XYZ Spin Chain

Higher AGT Correspon- dences, Via the same arguments which led us to (82), we have W-algebras, and Higher Quantum Geometric Sˆ1×S1 Langlands O~(MH.S. (G, Σ1)) ⇐⇒ Txyz(Gb, Σ1) (91) Duality from M-Theory

Meng-Chwan which relates the quantization of an elliptic-valued G Hitchin Tan system on Σ1 to the transfer matrices of a Gb-type XYZ spin Introduction chain on Σ1! Review of 4d This also means that AGT

5d/6d AGT ˆ1 1 x ∈ MS ×S (G, Σ ) ⇐⇒ χ (Vˆ ) = Tˆ (z), Vˆ ∈ Rep [Uell (gˆ) ] W-algebras + H.S. 1 q,v i i i q,v Σ1 Higher QGL (92) SUSY gauge ˆ theory + where i = 0,..., rank(g), Ti (z) is a polynomial whose degree W-algebras + ˆ ell QGL depends on Vi , and Uq,v (gˆ) is the elliptic toroidal algebra of g. Higher GL

Conclusion A Realization of Nekrasov-Pestun-Shatashvili’s Results for 6d, N = 1 G

(Gb)-Quiver SU(Ki ) Gauge Theories

Higher AGT Note that (88) also means that Correspon- dences, W-algebras, G,Cx,y1,y2 ell u ∈ M 1 ⇐⇒ χq,v (Vi ) = Ti (z), Vi ∈ Rep [U (g) ] and Higher S -mono,k q,v {Cx}z1,z2 Quantum Geometric (93) Langlands where C = S1 × S1 and i ∈ I . Duality from x t Γ M-Theory Note that (92) also means that Meng-Chwan Tan G,Cx,k ˆ ˆ ˆ ell u ∈ Mˆ1 1 ⇐⇒ χq,v (Vi ) = Ti (z), Vi ∈ Rep [Uq,v (gˆ)Cx ] Introduction S ×S -inst (94) Review of 4d 1 1 ˆ AGT where Cx = S × St and i ∈ IΓ. 5d/6d AGT 1 Can again argue via momentum around Sn (counted by W-algebras + Higher QGL D0-branes) ↔ 2d CFT energy level correspondence that degree ˆ SUSY gauge of Ti (Ti ) is Ki (aKi ). theory + W-algebras + (93)/(94) are Nekrasov-Pestun-Shatashvili’s main result in [21, QGL

Higher GL §1.3] which relates the moduli space of the 6d G/Gb-quiver ell ell Conclusion gauge theory to the representation theory of Uq (g)/Uq (gˆ)! Conclusion

Higher AGT We furnished purely physical derivations of higher AGT Correspon- dences, correspondences, W-algebra identities, and higher W-algebras, and Higher geometric Langlands correspondences, all within our Quantum Geometric M-theoretic framework. Langlands Duality from We elucidated the connection between the gauge-theoretic M-Theory realization of the geometric Langlands correspondence by Meng-Chwan Tan Kapustin-Witten and its original algebraic CFT formulation by Beilinson-Drinfeld, also within our Introduction M-theoretic framework. Review of 4d AGT Clearly, M-theory is a very rich and powerful framework 5d/6d AGT capable of providing an overarching realization and W-algebras + Higher QGL generalization of cutting-edge mathematics and SUSY gauge mathematical physics. theory + W-algebras + QGL At the same time, such corroborations with exact results

Higher GL in pure mathematics also serve as “empirical” validation of

Conclusion string dualities and M-theory, with the former as the “lab”. Higher AGT Correspon- dences, W-algebras, and Higher Quantum Geometric Langlands Duality from M-Theory Meng-Chwan THANK YOU FOR YOUR TIME AND ATTENTION! Tan

Introduction

Review of 4d AGT

5d/6d AGT

W-algebras + Higher QGL

SUSY gauge theory + W-algebras + QGL

Higher GL

Conclusion Higher AGT [1] L.F. Alday, D. Gaiotto, Y. Tachikawa, “Liouville Correspon- dences, Correlation Functions from Four-dimensional Gauge W-algebras, and Higher Theories”, Lett. Math. Phys. 91: 167-197, 2010, Quantum Geometric [arXiv:0906.3219]. Langlands Duality from M-Theory [2] N. Wyllard, “AN−1 conformal Toda ﬁeld theory correlation Meng-Chwan functions from conformal N=2 SU(N) quiver gauge Tan theories”, JHEP 11 (2009) 002, [arXiv:0907.2189]. Introduction

Review of 4d [3] H. Awata, Y. Yamada, “Five-dimensional AGT Conjecture AGT and the Deformed Virasoro Algebra”, JHEP 5d/6d AGT 1001:125,2010, [arXiv: 0910.4431]. W-algebras + Higher QGL [4] H. Awata, B. Feigin, A. Hoshino, M. Kanai, J. Shiraishi, SUSY gauge theory + S. Yanagida, “Notes on Ding-Iohara algebra and AGT W-algebras + QGL conjecture”, Proceeding of RIMS Conference 2010

Higher GL

Conclusion Higher AGT “Diversity of the Theory of Integrable Systems” (ed. Correspon- dences, Masahiro Kanai), [arXiv:1106.4088]. W-algebras, and Higher [5] C.A. Keller, N. Mekareeya, J. Song, Y. Tachikawa, “The Quantum Geometric ABCDEFG of Instantons and W-algebras”, Langlands Duality from [arXiv:1111.5624]. M-Theory Meng-Chwan [6] O. Schiﬀmann and E. Vasserot, “Cherednik algebras, W Tan algebras and the equivariant cohomology of the moduli Introduction 2 space of instantons on A ”, [arXiv:1202.2756]. Review of 4d AGT [7] D. Maulik and A. Okounkov, “Quantum Groups and 5d/6d AGT Quantum Cohomology”, [arXiv:1211.1287]. W-algebras + Higher QGL [8] M.-C. Tan, “M-Theoretic Derivations of 4d-2d Dualities: SUSY gauge theory + From a Geometric Langlands Duality for Surfaces, to the W-algebras + QGL AGT Correspondence, to Integrable Systems”, JHEP 07 Higher GL (2013) 171, [arXiv:1301.1977]. Conclusion Higher AGT [9] M.-C. Tan, “An M-Theoretic Derivation of a 5d and 6d Correspon- dences, AGT Correspondence, and Relativistic and Elliptized W-algebras, and Higher Integrable Systems”, JHEP 12 (2013) 31, Quantum Geometric [arXiv:1309.4775]. Langlands Duality from M-Theory [10] A. Braverman, M. Finkelberg, H. Nakajima, “Instanton Meng-Chwan moduli spaces and W -algebras”, [arXiv:1406.2381]. Tan

Introduction [11] A. Iqbal, C. Kozcaz, S.-T. Yau, “Elliptic Virasoro

Review of 4d Conformal Blocks”, [arXiv:1511.00458]. AGT

5d/6d AGT [12] F. Nieri, “An elliptic Virasoro symmetry in 6d”,

W-algebras + [arXiv:1511.00574]. Higher QGL

SUSY gauge [13] M.-C. Tan, “Higher AGT Correspondences, W-algebras, theory + W-algebras + and Higher Quantum Geometric Langlands Duality from QGL M-Theory”, [arXiv:1607.08330]. Higher GL

Conclusion Higher AGT [14] T. Kimura, V. Pestun, “Quiver W-algebras”, [ Correspon- dences, arXiv:1512.08533]. W-algebras, and Higher Quantum [15] B. Feigin, E. Frenkel, “Aﬃne Kac-Moody algebras at the Geometric Langlands critical level and Gelfand-Dikii algebras”, Int. J. Mod. Duality from M-Theory Phys. A 7, Suppl. 1A, 197-215 (1992). Meng-Chwan Tan [16] E. Frenkel, N. Reshetikhin, “Deformations of W -algebras

Introduction associated to simple Lie algebras”, [ arXiv:q-alg/9708006].

Review of 4d AGT [17] E. Frenkel, N. Reshetikhin, “Quantum Aﬃne Algebras and

5d/6d AGT Deformations of the Virasoro and W-algebras”, Comm.

W-algebras + Math. Phys. 178 (1996) 237-264, [arXiv:q-alg/9505025]. Higher QGL

SUSY gauge [18] A. Kapustin and E. Witten, “Electric-magnetic duality and theory + W-algebras + the geometric Langlands program”, Comm. Numb. Th. QGL Phys. 1 (2007) 1-236, [arXiv: hep-th/0604151]. Higher GL

Conclusion Higher AGT [19] A. Kapustin, “A note on quantum geometric Langlands Correspon- dences, duality, gauge theory, and quantization of the moduli W-algebras, and Higher space of ﬂat connections”, [arXiv:0811.3264]. Quantum Geometric Langlands [20] A. Beilinson and V. Drinfeld, “Quantization of Hitchin Duality from M-Theory integrable system and Hecke eigensheaves”, preprint Meng-Chwan (ca. 1995), Tan http://www.math.uchicago.edu/arinkin/langlands/. Introduction

Review of 4d [21] N. Nekrasov, V. Pestun, S. Shatashvili, “Quantum AGT geometry and quiver gauge theories”, [arXiv:1312.6689]. 5d/6d AGT

W-algebras + [22] S. Reffert, “General Omega Deformations from Closed Higher QGL String Backgrounds”, [arXiv:1108.0644]. SUSY gauge theory + W-algebras + [23] S. Hellerman, D. Orlando, S. Reffert, “The Omega QGL Deformation From String and M-Theory”, Higher GL [arXiv:1204.4192]. Conclusion Higher AGT [24] D. Gaiotto, “N = 2 Dualities”, [arXiv:0904.2715]. Correspon- dences, W-algebras, [25] J. Ding, K. Iohara, “Generalization and Deformation of and Higher Quantum Drinfeld quantum affine algebras”, [arXiv:q-alg/9608002]. Geometric Langlands Duality from [26] B. Feigin, A. Hoshino, J. Shibahara, J. Shiraishi, S. M-Theory Yanagida, “Kernel function and quantum algebras”, Meng-Chwan Tan [arXiv:1002.2485].

Introduction [27] B. L. Feigin and A. I. Tsymbaliuk, “Equivariant K-theory Review of 4d AGT of Hilbert schemes via shuﬄe algebra”, Kyoto J. Math. 51

5d/6d AGT (2011), no. 4, 831-854.

W-algebras + Higher QGL [28] N. Guay and X. Ma, “From quantum loop algebras to

SUSY gauge Yangians”, J. Lond. Math. Soc. (2) 86 no. 3, (2012) theory + W-algebras + 683-700. QGL

Higher GL

Conclusion Higher AGT [29] Y. Saito, “Elliptic Ding-Iohara Algebra and the Free Field Correspon- dences, Realization of the Elliptic Macdonald Operator”, W-algebras, and Higher [arXiv:1301.4912]. Quantum Geometric Langlands [30] E. Frenkel, “Lectures on the Langlands Program and Duality from M-Theory Conformal Field Theory”, [arXiv:hep-th/0512172]. Meng-Chwan Tan [31] E. Bergshoeﬀ, P. Townsend, “Super D-branes”, Nucl.

Introduction Phys. B490 (1997) 145-162. [hep-th/9611173].

Review of 4d AGT [32] J. Gomis, B. Le Floch, “’t Hooft Operators in Gauge

5d/6d AGT Theory from Toda CFT”, JHEP 11 (2011) 114,

W-algebras + [arXiv:1008.4139]. Higher QGL

SUSY gauge [33] H. Garland and M. K. Murray, “Kac-Moody monopoles theory + W-algebras + and periodic instantons”, Comm. Math. Phys. Volume QGL 120, Number 2 (1988), 335-351. Higher GL

Conclusion