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

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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→∞ 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 hyperkählerfour-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 hyperkählermoduli 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). These differential forms are necessarily harmonic and square-integrable, i.e., the spacetime BPS states would correspond to L2-harmonic forms which span the L2-cohomology of (some natural compactification of) MG (M). An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A–B Groups

Note that MG (M) consists of components labeled by (a, ρ0, ρ∞); that is, one can write M M (M) = Mρ0,a (M), (9) G G,ρ∞ a,ρ0,ρ∞

where a is the instanton number, and ρ0,∞ are conjugacy classes of the homomorphism Zk → G that one can pick at 0, ∞ of 4 M = R /Zk . Note that a is not really independent of ρ0 and ρ∞, as we shall now explain. 1 By reducing along Sn to type IIA string theory, one can have (half-BPS) D0-branes within the M × Rt worldvolume of the D4-branes. Since D0-branes correspond to static particle-like BPS conﬁgurations on M × Rt which thus present themselves as instantons on M, counting them would yield the formula a = kn0(i − j) + b˜(λ,¯ λ¯) − b(¯µ, µ¯). (10) An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A–B Groups

In summary, the generic Hilbert space HBPS of spacetime BPS states in the M-theory compactification (1) is given by M λ,µ M ∗ λ 4 HBPS = HBPS = IH U(MG,µ(R /Zk )), (11) λ,µ λ,µ ∗ λ 4 where IH U(MG,µ(R /Zk )) is the intersection cohomology (which can be identified with the L2-cohomology) of the Uhlenbeck λ 4 compactification U(MG,µ(R /Zk )) of the component λ 4 4 MG,µ(R /Zk ) of the highly singular moduli space MG (R /Zk ) labeled by the triples λ = (k, λ,¯ i) and µ = (k, µ,¯ j) which can be interpreted as dominant coweights of the corresponding affine Kac-Moody group Gaff of level k. Notice that because we cannot have a negative number of D0-branes, i.e., a ≥ 0, via (10) and the condition i ≥ j, we have λ ≥ µ. (12) An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A–B Groups

Let us now turn our attention to the other M-theory compactification (2) with k coincident M5-branes. One can proceed as before to ascertain the spacetime BPS states by computing the ground states of the M5-brane quantum 1 R→0 worldvolume theory over Rt × Sn × TNN . This would allow us to derive a 4d-4d relation that is a McKay-type correspondence of the intersection cohomology of the moduli space. However, since we would like to derive a geometric Langlands duality for surfaces which is a 4d-2d relation, we shall seek a different description of these states. Specifically, we shall seek a 2d QFT description of these states. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A–B Groups

To this end, recall that the low-energy limit of the worldvolume theory is a 6d N = (2, 0) Ak−1 superconformal field theory of massless tensor multiplets. Hence, where the ground states are concerned, one can regard the worldvolume theory to be conformally-invariant. Since it is conformally-invariant, one can rescale the worldvolume to bring the region near infinity to a finite distance close to the origin without altering the theory. Thus, one can, for the purpose of computing ground states, simply analyze the physics near infinity. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A–B Groups

1 R→0 Near infinity where the radius of the S -fiber of TNN tends to zero, we have a reduction to the following type IIA configuration:

5 1 3 IIA : R × Sn × Rt × R . (13) | {z } 1 I-brane on Sn × Rt = ND6 ∩ kD4 Here, we have a stack of N coincident D6-branes whose 5 1 worldvolume is given by R × Sn × Rt , and a stack of k coincident 1 3 D4-branes whose worldvolume is given by Sn × Rt × R . 1 The two stacks intersect along Sn × Rt to form a D4-D6 I-brane system, where there is 2d N = (8, 0) supersymmetry inherited from the ambient spacetime supersymmetries, which ought to also be associated with the eight supersymmetries of the original M5-branes worldvolume theory in (2) which underlies the I-brane. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A–B Groups

In other words, our sought-after M5-branes worldvolume ground states ought to be given by massless excitations of the 2d I-brane 1 theory along Sn × Rt . These massless excitations in question are furnished by the massless 4-6 strings that live along the I-brane. Indeed, 4-6 open strings which stretch between the D4- and D6-branes descend from open M2-branes whose topology is a disc 1 with an SR boundary that ends on the M5-branes, whence the interval ﬁlling the disc and thus, the tension of these open M2-branes, goes to zero as the 4-6 strings approach the I-brane and become massless. That is, the massless 4-6 strings which live along the I-brane descend from tensionless self-dual closed strings 1 of topology SR that live in the M5-branes worldvolume, and in their R → 0 low-energy limit, their spectrum would give the M5-branes worldvolume ground states that we seek. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A–B Groups

The massless modes are well-known to be chiral fermions on the 2d I-brane [15, 16]. If we have k D4-branes and N D6-branes, the kN complex chiral fermions

ψ (z), ψ† (z), i = 1,..., k, a = 1,..., N, (14) i,¯a ¯i,a

will transform in the bifundamental representations (k, N¯ ) and (k¯, N) of U(k) × U(N). Their action is given (modulo an overall coupling constant) by Z 2 † ¯ I = d z ψ ∂A+A0 ψ, (15)

where A and A0 are the restrictions to the I-brane worldsheet 1 Sn × Rt of the U(k) and U(N) gauge ﬁelds associated with the D4-branes and D6-branes, respectively. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A–B Groups

Now, a system of kN complex free fermions has central charge kN (n) and gives a direct realization of ub(kN)1 , the integrable module (n) over the Zn-twisted aﬃne Lie algebra u(kN)aﬀ,1 of level 1, and this has a conformal embedding

(n) (n) (n) (n) u(1)aff,kN ⊗ su(k)aff,N ⊗ su(N)aff,k ⊂ u(kN)aff,1. (16)

In other words, the total Fock space F ⊗kN of the kN complex free fermions can be expressed as

⊗kN F = WZW (n) ⊗ WZW (n) ⊗ WZW (n) , (17) bu(1)kN sub (k)N sub (N)k

where WZW (n) , WZW (n) and WZW (n) are the spectra bu(1)kN sub (k)N sub (N)k of states furnished by the relevant chiral WZW models. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A–B Groups

Note that F ⊗kN is the Fock space of the kN complex free fermions which have not yet been coupled to A and A0. In our case, only the U(k) gauge field associated with the D4-branes is dynamical; the U(N) gauge field associated with the D6-branes should not be dynamical as the geometry of the R→0 underlying TNN is fixed in our description. Therefore, the free fermions will, in our case, couple dynamically to the gauge group U(k) = U(1) × SU(k). Schematically, this means that we are dealing with the following partially gauged CFT

(n) (n) (n) u(kN)aff,1/[u(1)aff,kN ⊗ su(k)aff,N ]. (18)

(n) (n) In particular, the u(1)aﬀ,kN and su(k)aﬀ,N chiral WZW models will be replaced by the corresponding topological G/G models. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A–B Groups

Thus, the eﬀective overall partition function of the I-brane theory will be expressed solely in terms of the chiral characters of (n) sub (N)k (as the topological G/G models only contribute constant complex factors). In other words, the sought-after spectrum of spacetime BPS states in the M-theory compactiﬁcation (2) would, for n = 1, be realized by M ˜ WZWsu(N)k = WZW λ . (19) b sub (N)k,µ˜ λ,˜ µ˜

where the corresponding dominant aﬃne weights are such that λ˜ > µ˜.

Since su(N)aff is isomorphic to its Langlands dual counterpart ∨ ˜ su(N)aff, λ andµ ˜ are also dominant weights of the Langlands dual ∨ affine Kac-Moody group SU(N)aff of level k whence we can identify them with λ and µ of (11), respectively. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A–B Groups

λ As WZW λ is furnished by su(N) , and since sub (N)k,µ b k,µ ∨ λ su(N)aff ' su(N)aff whence sub (N)k,µ is isomorphic to the L λ ∨ submodule sub (N)k,µ over su(N)aff, the principle that the spectra of spacetime BPS states in the physically dual M-theory compactifications (1) and (2) ought to be equivalent will mean, from (11) and (19), that

∗ λ 4 L λ IH U(MSU(N),µ(R /Zk )) = sub (N)k,µ (20)

Note that this, together with (10), coincide with [2, Conjecture 4.14(3)] for simply-connected G = SU(N)! This completes our purely physical M-theoretic derivation of a geometric Langlands duality for surfaces for the SU(N) = AN−1 groups. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A–B Groups

Let us now restrict ourselves to even N, and consider n = 2 whence (11) can be written as M M M Heﬀ = IH∗U(Mλ ( 4/ )), (21) BPS SO(N+1),µν R Zk λ ν=0,1 µν while (19) gets replaced by M M M WZW (2) = WZW ˜ . (22) su(N) su(N)(2),λ b k b k,µ˜ν λ˜ ν=0,1 µ˜ν

Here, the overhead bar means that we project onto Z2-invariant states; ν = 0 or 1 indicates that the sector is untwisted or twisted, respectively. (2) ∨ ˜ Since su(N)aff is isomorphic to so(N + 1)aff, it would mean that λ andµ ˜ν are also dominant weights λ and µν of the Langlands dual ∨ affine Kac-Moody group SO(N + 1)aff of level k. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A–B Groups

As WZW (2),λ is furnished by (the -invariant projection of) su(N) Z2 b k,µν su(N)(2),λ, and since su(N)(2) ' so(N + 1)∨ whence (the b k,µν aff aff -invariant projection of) su(N)(2),λ is isomorphic to the Z2 b k,µν submodule Lso(N + 1)λ over so(N + 1)∨ , the principle that the b k,µν aff spectra of spacetime BPS states in the physically dual M-theory compactifications (1) and (2) ought to be equivalent will mean, from (21) and (22), that

IH∗U(Mλ ( 4/ )) = Lso(N + 1)λ (23) SO(N+1),µν R Zk b k,µν for ν = 0 and 1. Thus, together with (10), we have arrived at a G = SO(N + 1) generalization of [2, Conjecture 4.14(3)]! This completes our purely physical M-theoretic derivation of a geometric Langlands duality for surfaces for the SO(N + 1) = BN/2 groups. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the C–D–G Groups

Likewise, the principle that the spectra of such spacetime BPS states in the physically dual M-theory compactifications (3) and (4) ought to be equivalent, will lead us to a geometric Langlands duality for surfaces for the C–D–G groups. Via the same arguments as before in the A–B case, the Hilbert space of spacetime BPS states in the M-theory compactification (3) is, for n = 1, 2, 3 (with N = 4), given by M ∗ λ 4 HBPS = IH U(MSO(2N),µ(R /Zk )), (24) λ,µ M M M Heff = IH∗U(Mλ ( 4/ )), (25) BPS USp(2N−2),µν R Zk λ ν=0,1 µν 2 M M M Heff = IH∗U(Mλ ( 4/ )), (26) BPS G2,µν R Zk λ ν=0 µν

where λ > µ, µν. Instanton number a is again given by (10). An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the C–D–G Groups

Let us now turn our attention to the other M-theory compactification (4) with k coincident M5-branes. 1 R→0 Near infinity, the SR circle fiber of SNN has radius R → 0, and we have a reduction to the following type IIA configuration:

5 1 3 IIA : R × Sn × Rt × R /I3 . (27) | {z } 1 − I-brane on Sn × Rt = ND6/O6 ∩ kD4 Here, we have a stack of N coincident D6-branes on top of an − 5 1 O6 -plane whose worldvolume is given by R × Sn × Rt , and a stack of k coincident D4-branes whose worldvolume is given by 1 3 3 Sn × Rt × R /I3 (where I3 acts as ~r → −~r in R ). 1 − These two stacks intersect along Sn × Rt to form a D4-D6/O6 I-brane system. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the C–D–G Groups

As before, the I-brane theory is a theory of massless free chiral fermions couple to gauge ﬁelds associated with the gauge groups that appear along the D4-D6/O6− system. Via T-duality, one can understand that there ought to be, in the presence of the O6−-plane, an SO(α) and SO(2N) gauge group on the k D4- and N D6-branes, respectively, where α depends on k. To determine what α is, note that the total central charge of the real chiral fermions should not change as we move the stack of coincident D4- and D6-branes away from the O6−-plane [17], whence we eﬀectively have the U(k) × U(N) theory described by (14)–(15). Thus, α must be such that the total central charge of the real chiral fermions is kN. As a single real chiral fermion will contribute 1/2 to the central charge, we ought to have a total of 2kN real chiral fermions. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the C–D–G Groups

Since the 2kN real chiral fermions are furnished by the massless modes of the 4-6 open strings, they necessarily transform in the bifundamental representation of SO(α) × SO(2N); this would mean that α = k. In short, the 2kN real chiral fermions ought to be given by

ψi,a(z), where i = 1,..., k, and a = 1,..., 2N, (28) which transform in the bifundamental representation (k, 2N) of SO(k) × SO(2N). Their action is given (modulo an overall coupling constant) by Z 2 ¯ I = d z ψ∂A+A0 ψ, (29)

where A and A0 are the restrictions to the I-brane worldsheet 1 Sn × Rt of the SO(k) and SO(2N) gauge ﬁelds associated with the k D4- and N D6-branes. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the C–D–G Groups

The system of 2kN real free chiral fermions of central charge kN (n) gives a direct realization of sob (2kN)1 , the integrable module over (n) the Zn-twisted aﬃne Lie algebra so(2kN)aﬀ,1 of level 1, and this has a conformal embedding:

(n) (n) (n) so(k)aff,2N ⊗ so(2N)aff,k ⊂ so(2kN)aff,1, (30)

In other words, the total Fock space F ⊗2kN of the 2kN real free fermions can be expressed as

⊗2kN F = WZW (n) ⊗ WZW (n) , (31) sob (k)2N sob (2N)k

where WZW (n) and WZW (n) are the spectra of states sob (k)2N sob (2N)k furnished by the relevant chiral WZW models. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the C–D–G Groups

Note that F ⊗2kN is the Fock space of the 2kN real free fermions which have not yet been coupled to A and A0 In our case, only the SO(k) gauge field associated with the D4-branes is dynamical; the SO(2N) gauge field associated with the D6-branes/O6−-plane should not be dynamical as the R→0 geometry of SNN is fixed in our description. Therefore, the free fermions will, in this case, couple dynamically to the gauge group SO(k) only. Schematically, this means that we are dealing with the following partially gauged CFT

(n) (n) so(2kN)aﬀ,1/so(k)aﬀ,2N . (32)

(n) In particular, the so(k)aﬀ,2N chiral WZW model will be replaced by the corresponding topological G/G model. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the C–D–G Groups

Thus, the eﬀective overall partition function of the I-brane theory will be expressed solely in terms of the chiral characters of (n) sob (2N)k ( as the topological G/G models only contribute constant complex factors). In other words, the sought-after spectrum of spacetime BPS states in the M-theory compactiﬁcation (4) would, for n = 1, 2, 3 (with N = 4), be realized by M ˜ WZWso(2N)k = WZW λ , (33) b sob (2N)k,µ˜ λ,˜ µ˜ M M M WZW (2) = WZW ˜ , (34) so(2N) so(2N)(2),λ b k b k,µ˜ν λ˜ ν=0,1 µ˜ν M M M WZW (3) = WZW ˜ , (35) so(8) so(8)(3),λ b k b k,µ˜ν λ˜ ν=0,1,2 µ˜ν

where λ˜ > µ,˜ µ˜ν. An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the C–D–G Groups

(2) (3) ∨ Since so(2N)aff, so(2N)aff , so(8)aff are isomorphic to so(2N)aff, ∨ ∨ ˜ usp(2N − 2)aff, g2 aff, it will mean that λ,µ ˜ andµ ˜ν are also dominant weights of the Langlands dual affine Kac-Moody group ∨ ∨ ∨ SO(2N)aff , USp(2N − 2)aff , G2aff , whence we can identify them with λ, µ and µν of (24), (25), (26), respectively.

As WZW λ , WZW (2),λ , WZW (3),λ are furnished by so(2N) so(2N) so(8) b k,µ b k,µν b k,µν so(2N)λ , so(2N)(2),λ, so(8)(3),λ, and since b k,µ b k,µν b k,µν ∨ (2) ∨ so(2N)aff ' so(2N)aff, so(2N)aff ' usp(2N − 2)aff, so(8)(3) ' g∨ whence so(2N)λ , so(2N)(2),λ, so(8)(3),λ are aff 2 aff b k,µ b k,µν b k,µν isomorphic to the submodules Lso(2N)λ , Lusp(2N − 2)λ , b k,µ d k,µν (Lg )λ over so(2N)∨ , usp(2N − 2)∨ , g∨ , respectively, b2 k,µν aff aff 2 aff An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the C–D–G Groups

the principle that the spectra of spacetime BPS states in the physically dual M-theory compactiﬁcations (3) and (4) ought to be equivalent will mean, from (24), (25), (26) and (33), (34), (35), that ∗ λ 4 L λ IH U(MSO(2N),µ(R /Zk )) = sob (2N)k,µ (36)

IH∗U(Mλ ( 4/ )) = Lusp(2N − 2)λ (37) USp(2N−2),µν R Zk d k,µν

IH∗U(Mλ ( 4/ )) = (Lg )λ (38) G2,µν R Zk b2 k,µν Note that (36), (37), (38) is [2, Conjecture 4.14(3)] for G = SO(2N), USp(2n − 2), G2! This completes our purely physical M-theoretic derivation of a geometric Langlands duality for surfaces for the SO(2N) = DN , USp(2N − 2) = CN−1 and G2 groups. An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

Now, let k = 1 in (1), (2), (3) and (4).

In (1) and (3), turn on Omega-deformation with real parameters 1 and 2 along the z = x2 + ix3 and w = x4 + ix5 planes, respectively, via a ﬂuxbrane as described in [18, 19]:

0 1 2 3 4 5 6 7 8 9 10 N M5’s/OM5 − − − − − − ﬂuxbrane × ⊗ 1 2 3 × × ◦ (39) 1 Here, the ‘×’s denote the ﬂuxbrane directions; ‘⊗’ denotes the Sn circle direction; and ◦ denotes the “eleventh circle”. In addition, there is also a rotation along the u-plane with rotation parameter 3 = 1 + 2, and it is tantamount to a topological twist (that involves an R-symmetry) which helps preserve some supersymmetry that would otherwise be completely broken by the (1, 2) rotations along the (z, w) planes. An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

Repeating in the presence of this fluxbrane, the chain of arguments that gave us the dual compactifications (1)-(4), we can express the dual configurations (2) and (4) in the presence of the now dual fluxbrane, as

0 1 2 3 4 5 6 7 8 9 10 1 M5 − − − − − − dual ﬂuxbrane × ⊗ 1, 2 3 × × ◦ (40) Here, “1, 2” along the x2-x3-x4-x5 directions means that there are two simultaneous rotations along the x4-x5 plane with rotation parameters 1 and 2. An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

In short, we can write (1) and (3) in the presence of the ﬂuxbrane denoted in (39) as

4 1 5 4 1 5 R |1,2 × Sn × Rt ×R |3; x6,7 , R |1,2 × Sn × Rt ×R |3; x6,7 . | {z } | {z } N M5-branes N M5-branes + OM5-plane (41) Also, we can write (2) and (4) in the presence of the dual ﬂuxbrane denoted in (40) as

5 3 1 R→0 5 3 1 R→0 R |x4,5 ×Rt × Sn × TNN |3; x6,7 , R |x4,5 ×Rt × Sn × SNN |3; x6,7 . | {z } | {z } 1 M5-branes 1 M5-branes (42) An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

Now, recall that the sought-after spacetime BPS states in (41) are furnished by a topological sigma-model with worldsheet 1 Σ = Sn × Rt , so we are free to deform Σ into a short cylinder 1 Σn,t = Sn × It , where It is an interval whose length is much 1 smaller than β, the radius of Sn. Since the far past and far future are now brought to ﬁnite distances whence the eleven-dimensional ﬁelds no longer decay to zero at the beginning and end of time, one would need to specify nontrivial boundary conditions at the ten-dimensional ends of It . For our purpose of deriving the AGT correspondence, we choose a common half-BPS boundary condition that preserves only one-half of the sixteen worldvolume supersymmetries, i.e., we insert a pair of M9-branes [20]. An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

In other words, in place of (41), we have

4 5 4 5 R |1,2 × Σn,t ×R |3; x6,7 , R |1,2 × Σn,t ×R |3; x6,7 , | {z } | {z } N M5-branes N M5-branes + OM5-plane (43) where the M9-branes intersect the M5-branes/OM5-plane along 1 4 5 Sn × R |1,2 and span R |3; x6,7 . Then, Omega-deformation in the worldvolume theory can be understood as follows. At low-energy distances much larger than 1 It , as one traverses around the Sn circle, the (z, w) planes in 4 R |1,2 would be rotated by angles (1, 2) together with an SU(2)R -symmetry rotation of the resulting G gauge theory along 4 R |1,2 . An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

The partition function of spacetime half-BPS states in (43) (given by a trace that is tantamount to gluing the two ends of 1 1 Σn,t = Sn × It into a two-torus Sn × St ) would be given by the following 5d (since St β) worldvolume expression (c.f. [21, eqns. (29) and (43)]) X ~ ZBPS(1, 2,~a, β) = TrHm exp β(1J1 + 2J2 + ~a · T ), (44) m ~ where T = (T1 ..., Trank G ) are the generators of the Cartan subgroup of G; ~a = (a1,..., arank G ) are the corresponding purely 4 imaginary Coulomb moduli of the G gauge theory on R |1,2 ; J1,2 are the rotation generators of the (z, w) planes, corrected with an appropriate amount of the SU(2)R -symmetry to commute with the two surviving worldvolume supercharges; and Hm is the space of holomorphic functions on the moduli space MG,m of G-instantons 4 on R with instanton number m. An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

According to the duality of the six-dimensional compactiﬁcations (41) and (42), the dual of (43) would be given by

5 R→0 5 R→0 R |3; x4,5 ×Σn,t × TNN |3; x6,7 , R |3; x4,5 ×Σn,t × SNN |3; x6,7 , | {z } | {z } 1 M5-branes 1 M5-branes (45) where we have a pair of M9-branes whose worldvolumes at the tips of It span the ten directions transverse to it, and as one traverses 1 around the Sn circle, among other things, the x4-x5 plane in 4 R |3; x4,5 would be rotated by an angle of 1 + 2 = 3 together with an SU(2)R -symmetry rotation of the gauge theory along 4 R |3; x4,5 . An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

We shall now derive, purely physically, a pure AGT correspondence for the A–B groups. From (43) and (45), we have the following physically dual M-theory compactiﬁcations

4 5 5 R→0 R |1,2 × Σn,t ×R |3; x6,7 ⇐⇒ R |3; x4,5 × C × TNN |3; x6,7 , | {z } | {z } N M5-branes 1 M5-branes (46) 1 where C is a priori the same as Σn,t = Sn × It , and we have an M9-brane at each tip of It . Let us ﬁrst ascertain the spectrum of spacetime BPS states on the LHS of (46) that deﬁne ZBPS(1, 2,~a, β) in (44).

Note that (44) means that as one traverses a closed loop in Σn,t , there would be a g-automorphism of MG , where g ∈ U(1) × U(1) × T , and T ⊂ G is the Cartan subgroup. An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

Consequently, the spacetime BPS states of interest would, in the presence of Omega-deformation, be captured by the topological sector of a non-dynamically g-gauged version of our sigma-model on instanton moduli space. Hence, via the same argument as that employed to derive a geometric Langlands duality for surfaces, we can express the Hilbert space of spacetime BPS states on the LHS of (46) as

Ω M Ω M ∗ HBPS = HBPS,m = IHU(1)2×T U(MG,m), (47) m m

∗ where IHU(1)2×T U(MG,m) is the (Zn-invariant in the sense of (21) when n = 2) U(1)2 × T -equivariant intersection cohomology of the Uhlenbeck compactiﬁcation U(MG,m) of the (singular) moduli 4 space MG,m of G-instantons on R with instanton number m. An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

Let us next ascertain the corresponding spectrum of spacetime BPS states on the RHS of (46). Again, via the same argument as that employed to derive a geometric Langlands duality for surfaces, we ﬁnd that the spacetime BPS states would be furnished by the I-brane theory in the following type IIA conﬁguration:

5 3 IIA : R |3;x4,5 × C × R |3;x6,7 . (48) | {z } I-brane on C = ND6 ∩ 1D4 Here, we have a stack of N coincident D6-branes whose 5 worldvolume is given by R |3;x4,5 × C, and a single D4-brane whose 3 worldvolume is given by C × R |3;x6,7 . An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

Let us for a moment turn off Omega-deformation in (48), i.e., let 3 = 1 + 2 = 0. Then, from our earlier arguments, we learn that the spacetime BPS states would be furnished by a chiral WZW level 1 ∨ model at level 1 on C, WZW ∨ , where g is the Langlands gaff aff dual of the affine G-algebra gaff . Now turn Omega-deformation back on. As indicated in (48), as one traverses around a closed loop in C, the x4-x5 plane in 4 5 R |3; x4,5 ⊂ R |3; x4,5 would be rotated by an angle of 3 together with an SU(2)R -symmetry rotation of the supersymmetric SU(N) 4 gauge theory along R |3; x4,5 . As discussed below (46), we find that there is a g0-automorphism of MSU(N),m as we traverse around a closed loop in C, where 4 MSU(N),m is the moduli space of SU(N)-instantons on R with instanton number m; g0 = U(1) × T 0, where T 0 ⊂ SU(N) is a Cartan subgroup. An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

In fact, since MSU(N),m is also the space of self-dual connections 4 of an SU(N)-bundle on R , and since these self-dual connections correspond to differential one-forms valued in the Lie algebra su(N), this also means that there is a g0-automorphism of the space of elements of su(N) and thus SU(N), as we traverse a closed loop in C. Since a G WZW model on Σ is a sigma-model on Σ with target the G-manifold, it would then mean that there is a g0-automorphism of level 1 the target space of WZW ∨ as we traverse a closed loop in C. gaff In short, in the presence of Omega-deformation, we would have to level 1 0 non-dynamically gauge WZW ∨ by U(1) × T . gaff An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

That being said, notice also from (48) that Omega-deformation is also being turned on along the D4-brane. As before, because the U(1) gauge field on the D4-brane (unlike the SU(N) gauge field on the D6-branes) is dynamical, one had to reduce away in the I-brane system the U(1) WZW model associated with the D4-brane. As such, the Omega-deformation along the D4-brane would act to 0 reduce the U(1) × T Omega-deformation factor by R = U(1) × T, where U(1) ⊂ R is associated with the 3-rotation of the x6-x7 3 plane in R |3; x6,7 , and T ⊂ R is the Cartan of the gauge group on the D4-brane, i.e., T = U(1). In short, we would in fact have to non-dynamically gauge level 1 0 0 WZW ∨ not by U(1) × T but by T ⊂ T . gaff An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

0 As SU(N)/T ' SL(N, C)/B+, where B+ is a Borel subgroup, it 0 would mean that SU(N)/T' (SL(N, C)/B+) × (T /T ). Also, T 0/T is never bigger than the Cartan subgroup C ⊂ B+ = C × N+, where N+ is the subgroup of strictly upper triangular matrices which are nilpotent and traceless whose Lie algebra is n+. Altogether, this means that our gauged WZW model which corresponds to the coset model SU(N)/T , can also be studied as an S-gauged SL(N, C) WZW model which corresponds to the coset model SL(N, C)/S, where N+ ⊆ S ⊂ B+. Physical consistency of our system implies that S is necessarily a connected subgroup of G, i.e., S = N+. Therefore, what we ought to ultimately consider is an N+-gauged SL(N, C) WZW model. An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

Schematically, our N+-gauged SL(N, C) WZW model can be expressed as the partially gauged chiral CFT

(n) (n) sl(N)aﬀ,1/n+aﬀ,p (49) on C, where the level p would necessarily depend on the 0 0 Omega-deformation parameters 1 = β1 and 2 = β2.(p, being a purely real number, should not depend on the purely imaginary parameter ~a0 = β~a).

Notice that the level p of the affine N+-algebra deviates from the allowed value of 1; physical consistency of our system then implies that there ought to be a corresponding shift in its central charge arising from a curvature along C to “absorb” this. In other words, Omega-deformation ought to deform the a priori flat C = Σn,t into a curved Riemann surface with the same topology, i.e., the effective geometry of C is S2/{0, ∞}. An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

∨ The partially gauged chiral CFT in (49) realizes W(gaff ) – a Zn-twisted version of the affine W-algebra obtained from sl(N)aff via a quantum Drinfeld-Sokolov reduction. Therefore, the Hilbert space of spacetime BPS states on the RHS of (46) can be expressed as

Ω0 ∨ HBPS = Wc(gaff ), (50) ∨ ∨ where Wc(gaff ) is s Verma module over W(gaff ). In summary, the physical duality of the compactifications in (46) will mean that (47) is equivalent to (50), i.e.,

M ∗ ∨ IHU(1)2×T U(MG,m) = Wc(gaﬀ ) (51) m Thus, we have a 4d-2d duality relation in the sense of (20) and (23). An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

From the fact that (i) the level p must depend on the 0 Omega-deformation parameters 1,2; (ii) the compactiﬁcations in 0 0 (46) are symmetric under the exchange 1 ↔ 2; (iii) we have a 00 0 0 0 0 −1 geometrical g = exp[(1 + 2)J3] = exp[(λ1 + λ2)λ J3] automorphism associated with the Omega-deformation in (48), whilst the central charge is independent of the x4-x5 plane rotation generator J3; we can deduce that the central charge and level of ∨ W(gaﬀ ) are

2 3 (1 + 2) cA,1,2 = (N − 1) + (N − N) (52) 12

0 −2 p k = −N − b and b = 1/2 (53) An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

∨ Wc(gaff ) is generated by the application of creation operators (si ) Wm<0 on its Zn-twisted highest weight state |∆i, where m ∈ Z/n. (2) From the additional fact that (i) W0 generates translations along 1 4 the Sn fiber in C, (ii) there is a rotation of an R space and the 1 gauge field over it as we go around the Sn; (iii) (51) would mean (2) that the symmetries of the β-independent W0 ought to be compatible with the symmetries of ZBPS(1, 2,~a, β) in (44); we can deduce that (2) (2) W0 |∆i = ∆ |∆i (54) where (N3 − N) ( + )2 γ~a2 ∆(2) = 1 2 − (55) 24 12 12 for some real constant γ. An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

P Writing (44) as ZBPS = m ZBPS,m, the Nekrasov instanton partition function will be

∨ X 2mhg 0 Zinst(Λ, 1, 2,~a) = Λ ZBPS,m(1, 2,~a, β → 0), (56) m where Λ can be interpreted as the inverse of the observed scale of 4 0 the R |1,2 space on the LHS of (46), and ZBPS,m is just a rescaled version of ZBPS,m. Note that equivariant localization [22] implies that ∗ IHU(1)2×T U(MSU(N),m) must be endowed with an orthogonal 0 basis {|~pmi}. Thus, since ZBPS,m is a weighted count of the states Ω ∗ in HBPS,m = IHU(1)2×T U(MG,m), it would mean from (56) that

Zinst(Λ, 1, 2,~a) = hΨ|Ψi, (57)

∨ L mhg L ∗ where |Ψi = m Λ |Ψmi ∈ m IHU(1)2×T U(MG,m). An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

In turn, the duality (51) and the consequential observation that |Ψi is a sum over 2d states of all energy levels m, mean that

|Ψi = |q, ∆i, (58)

∨ where |q, ∆i ∈ Wc(gaﬀ ) is a coherent state, and from (57),

Zinst(Λ, 1, 2,~a) = hq, ∆|q, ∆i (59)

Since the LHS of (59) is defined in the β → 0 limit of the LHS of (46), |q, ∆i and hq, ∆| ought to be a state and its dual associated with the puncture at z = 0, ∞ on C, respectively (as these are the 1 points where the Sn fiber has zero radius). This is depicted in fig. 1. An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

Moreover, since we have N D6-branes and 1 D4-brane wrapping C (see (48)), we eﬀectively have an N × 1 = N-fold cover ΣSW of C. This is also depicted in ﬁg. 1.

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Figure 1: C and its N-fold cover ΣSW with the coherent states hq, ∆| and |q, ∆i at z = 0 and ∞

Incidentally, ΣSW is also the Seiberg-Witten curve which underlies Zinst(Λ, 1, 2,~a)! An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

For n = 1, i.e., G = SU(N), ΣSW can be described in terms of the algebraic relation N N−2 ΣSW : λ + φ2(z)λ + ··· + φN (z) = 0, (60)

where λ = ydz/z (for some complex variable y); the φs (z)’s are (s, 0)-holomorphic diﬀerentials on C given by dz j ΛN dz N φ (z) = u and φ (z) = z + u + , j j z N N z z (61) where j = 2, 3,..., N − 1. This is consistent with our results that we have, on C, the following (si , 0)-holomorphic diﬀerentials (s ) ! s X W i dz i W (si )(z) = l , where s = 2, 3,..., N. zl z i l∈Z (62) An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

4 In fact, the U(1) R-symmetry of the 4d theory along R |1,2 on the 1 LHS of (46) can be identiﬁed with the rotational symmetry of Sn; the duality relation (46) then means that the corresponding U(1) R-charge of the φs (z) operators ought to match the conformal dimension of the W (s)(z) operators on C, which is indeed the case.

Thus, we can naturally identify, up to some constant factor, φs (z) with W (s)(z), and by recalling the facts used to derive (52)-(55), we can deduce that

(s) Wl≥2 |q, ∆i = 0, for s = 2, 3,..., N (63)

u W (s)|q, ∆i = s |q, ∆i, for s = 2, 3,..., N (64) 0 s/2 (12)

ΛN W (N)|q, ∆i = q|q, ∆i, q = (65) 1 N/2 (12) An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

When n = 2 (with even N) whence we have G = SO(N + 1), instead of (61), we now have

dz s ΛN dz N φ = u , φ˜ = z1/2 + , (66) s s z N z1/2 z

where the φ˜s (z)’s are also (s, 0)-holomorphic diﬀerentials on C. This is again consistent with our results that for n = 2 (with even N), we have, on C, the following (si , 0)-holomorphic diﬀerentials

 (si )  (si ) ! si W˜ si (s ) X Wl dz (s ) X l+1/2 dz W i = , W˜ i =   . zl z zl+1/2 z l∈Z l∈Z (67) An M-Theoretic Derivation of the Pure AGT Correspondence for A–B Groups

Thus, we can naturally identify, up to some constant factor, φs (z) (s) (s) with W (z) and φ˜s (z) with W˜ (z), whence we can again deduce that

(s) Wl≥1 |q, ∆i = 0, for s = 2, 3,..., N (68)

˜ (s) Wl≥3/2 |q, ∆i = 0, for s = 2, 3,..., N (69)

ΛN W˜ (N)|q, ∆i = q|q, ∆i, q = (70) 1/2 N/2 (12)

In arriving at the above boxed relations, we have just furnished a fundamental physical derivation of the pure AGT correspondence for the AN−1 and BN/2 groups! An M-Theoretic Derivation of the Pure AGT Correspondence for C–D–G Groups

We shall now derive, purely physically, a pure AGT correspondence for the C–D–G groups. From (43) and (45) that we have the following physically dual M-theory compactiﬁcations 4 5 5 R→0 R |1,2 × Σn,t ×R |3; x6,7 ⇐⇒ R |3; x4,5 × C × SNN |3; x6,7 , | {z } | {z } N M5-branes + OM5-plane 1 M5-branes (71) 1 where C is a priori the same as Σn,t = Sn × It , and we have an M9-brane at each tip of It . As before, the Hilbert space of spacetime BPS states on the LHS of (71) is given by Ω M Ω M ∗ HBPS = HBPS,m = IHU(1)2×T U(MG,m), (72) m m ∗ where IHU(1)2×T U(MG,m) is the Zn-invariant (in the sense of (25) and (26) when n = 2 and 3) cohomology. An M-Theoretic Derivation of the Pure AGT Correspondence for C–D–G Groups

Next, to ascertain the corresponding spectrum of spacetime BPS states on the RHS of (71), notice that instead of (48), we now have 5 3 IIA : R |3;x4,5 × C × R /I3|3;x6,7 . (73) | {z } I-brane on C = ND6/O6− ∩ 1D4 Here, we have a stack of N coincident D6-branes on top of an − 5 O6 -plane whose worldvolume is given by R |3;x4,5 × C, and a 3 single D4-brane whose worldvolume is given by C × R /I3|3;x6,7 3 (where I3 acts as ~r → −~r in R ). Repeating the arguments that led us to (49), we ﬁnd that instead of (49), we now have

(n) (n) so(2N)aff,1/n+aff,p, (74) and the effective geometry of C would again be S2/{0, ∞} due to Omega-deformation. An M-Theoretic Derivation of the Pure AGT Correspondence for C–D–G Groups

∨ The partially gauged chiral CFT in (74) realizes W(gaff ) – a Zn-twisted version of the affine W-algebra W(so\(2N)) obtained from so(2N)aff via a quantum Drinfeld-Sokolov reduction. Ω0 Therefore, the Hilbert space HBPS of spacetime BPS states on the RHS of (71) can be expressed as Ω0 ∨ HBPS = Wc(gaff ), (75) ∨ ∨ where Wc(gaff ) is s Verma module over W(gaff ). In summary, the physical duality of the compactifications in (71) will mean that (72) is equivalent to (75), i.e.,

M ∗ ∨ IHU(1)2×T U(MG,m) = Wc(gaﬀ ) (76) m Thus, we have a 4d-2d duality relation in the sense of (36), (37) and (38). An M-Theoretic Derivation of the Pure AGT Correspondence for C–D–G Groups

In repeating the arguments which led us to (52)–(59), we have the ∨ central charge and level of W(gaﬀ ) as

2 2 (1 + 2) cD,1,2 = N + (2N − 2)(2N − N) (77) 12

0 −2 p k = −2N + 2 − b and b = 1/2 (78) Also, (2) (2) W0 |∆i = ∆ |∆i (79) where

(2N − 2)(2N2 − N) ( + )2 γ0~a2 ∆(2) = 1 2 − (80) 24 12 12

for some real constant γ0. An M-Theoretic Derivation of the Pure AGT Correspondence for C–D–G Groups

Last but not least, the Nekrasov partition function

Zinst(Λ, 1, 2,~a) = hq, ∆|q, ∆i (81)

∨ where |q, ∆i ∈ Wc(gaff ) is a coherent state. The state |q, ∆i and its dual hq, ∆| ought to be associated with the puncture at z = 0, ∞ on C, respectively. This is depicted in fig. 2. Note also that if we only have N D6-branes and 1 D4-brane wrapping C in (73), we would just have an N × 1 = N-fold cover of C (as explained earlier). In the presence of the O6−-plane however, there will be a mirror image of this configuration on the “opposite side” whence this cover is doubled, i.e., in (73), we effectively have a 2(N × 1) = 2N-fold cover ΣSW of C. This is also depicted in fig. 2. An M-Theoretic Derivation of the Pure AGT Correspondence for C–D–G Groups

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Figure 2: C and its 2N-fold cover ΣSW with the states hq, ∆| and |q, ∆i at z = 0 and ∞

Incidentally, ΣSW is also the Seiberg-Witten curve which underlies Zinst(Λ, 1, 2,~a)! An M-Theoretic Derivation of the Pure AGT Correspondence for C–D–G Groups

For n = 1, i.e., G = SO(2N), ΣSW can be described in terms of the algebraic relation 2N 2N−2 2 2 ΣSW : λ + φ2(z)λ + ··· + φ2N−2(z)λ + φN (z) = 0, (82)

where λ = ydz/z (for some complex variable y); the φs (z)’s are (s, 0)-holomorphic diﬀerentials on C given by dz j Λ2N−2 dz 2N−2 φ (z) = u , φ (z) = z + u + , j j z 2N−2 2N−2 z z (83) where j = 2, 4,..., 2N − 4, N. This is consistent with our results that for G = SO(2N), we have, on C, the following (si , 0)-holomorphic diﬀerentials (s ) ! s X W i dz i W (si )(z) = l , where s = 2, 4,..., 2N − 2, N. zl z i l∈Z (84) An M-Theoretic Derivation of the Pure AGT Correspondence for C–D–G Groups

Thus, we can naturally identify, up to some constant factor, φs (z) with W (s)(z), whence we can deduce that

(s) Wl≥2 |q, ∆i = 0, for s = 2, 4,..., 2N − 2, N (85) u W (s)|q, ∆i = s |q, ∆i, for s = 2, 4,..., 2N − 2, N 0 s/2 (12) (86) 2N−2 (2N−2) Λ W1 |q, ∆i = q|q, ∆i, q = N−1 (87) (12) When n = 2 whence G = USp(2N − 2), we have

(s) Wl≥1 |q, ∆i = 0, for s = 2, 4,..., 2N − 2, N (88)

˜ (s) Wl≥3/2 |q, ∆i = 0, for s = 2, 4,..., 2N − 2, N (89) An M-Theoretic Derivation of the Pure AGT Correspondence for C–D–G Groups

2N−2 ˜ (2N−2) Λ W1/2 |q, ∆i = q|q, ∆i, q = N−1 (90) (12)

When n = 3 (with N = 4) whence G = G2, we have

(s) Wl≥1 |q, ∆i = 0, for s = 2, 4, 6 (91)

˜ (s) Wl≥2/3 |q, ∆i = 0, for s = 2, 4, 6 (92)

6 ˜ (6) Λ W1/3|q, ∆i = q|q, ∆i, q = 3 (93) (12)

In arriving at the above boxed relations, we have just furnished a fundamental physical derivation of the pure AGT correspondence for the DN , CN−1 and G2 groups! An M-Theoretic Derivation of the AGT Correspondence with Matter

Let us now extend our derivation of the pure AGT correspondence to include matter. For concreteness, we shall restrict ourselves to the A-type superconformal quiver gauge theories described by Gaiotto in [23]. To this end, ﬁrst note that our derivation of the pure AGT correspondence is depicted in ﬁg. 3.

$$#% &'(!)

# " " ! ! & & "# " ! ! " # " %' +* %' (*), (*),

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 derivation of the pure AGT correspondence. An M-Theoretic Derivation of the AGT Correspondence with Matter

Here, l and ~a are the instanton number and Coulomb moduli of 4 9 the underlying 4d gauge theory along R |1,2 ⊂ X |i ; Vq,∆ and ∗ Vq,∆ is a vertex operator and its dual with higher order poles that represent the coherent state |q, ∆i and its dual hq, ∆| in (59) of the CFT on C; and the two points on C where the vertex operators are located are also where the two instantonic M9-branes which are dual to the two original M9-branes, sit. This coincides with the fact that in Gaiotto’s construction, there 4 are defects realized by M-branes which span all of R |1,2 that sit ∗ at the points where Vq,∆ and Vq,∆ are inserted. In turn, this means that we can deﬁne Gaiotto’s theories – and therefore their corresponding Nekrasov instanton partitions Zinst(G, 1, 2,~a, m) – in the same way he did using building blocks represented by Riemann surfaces with holes and punctures. An M-Theoretic Derivation of the AGT Correspondence with Matter

In particular, the building blocks of our derivation of the AGT correspondence with matter is as shown in ﬁg. 4 below.

!!"# #$%"&

# #

! ! !

$ "% $ "% $ "% ( " ( &'&'%! &'&'%! &'&'%!

!!"# #$%"&

)#* ($

! ! $ ( " $ "% "%

&'&'%! &'&'%!

Figure 4: Building blocks with “minimal” M9-branes for our derivation of the AGT correspondence with matter An M-Theoretic Derivation of the AGT Correspondence with Matter

Here, an M9-brane is “minimal” in the sense that the D8-brane it reduces to as β → 0 [24] supports a minimal number of D0-branes within the stack of N D4-branes it intersects, whence the correspondence between l and the conformal weight of CFT states on C derived hitherto, means that the vertex operators in ﬁg. 4 are all primary operators.

V~ai ,~ai+1 is associated with a M9-brane domain wall that transforms the theory with ~ai to the adjacent one with ~ai+1. As the worldvolume gauge theory in ﬁg. 4b is asymptotically-free, length ∼ 1/g 2. As the CFT on C with W-algebra symmetry can be thought of as a conformal Toda ﬁeld theory with background charge Q, with an appropriate metric on C, one can localize Q to the poles [25]. An M-Theoretic Derivation of the AGT Correspondence with Matter

Consider a conformal necklace quiver of n SU(N), N > 2.

! "!#

"! " ##$$!

"! " ##$

! "!# ! ! !

"! " ##!

"! " ## "

! "!#

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" %!& %$

! !

" %!& %!

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Figure 5: The necklace quiver diagram and the various steps that lead us to the overall Riemann surface Σ on which our 2d CFT lives. An M-Theoretic Derivation of the AGT Correspondence with Matter

! &$&' '()#*

" $$$ #! $$! " #$$! " #! $ ) *# "!#" " #" " !## $$% + " + # #! $ " #$ "!#! "!# " # "#! $$% #!% , ! ! " ) *# $ ++# % " " #!% %# % %# - " !!!"#$ " % (#$ !!!"#$ % (#$ %# % (#% %# ) +% *#% !!!"#$ !!! ) *# , "#$ $ , -..

Figure 6: The eﬀective 4d-2d correspondence In the case of a necklace quiver of n SU(N) gauge groups, X Z neck(q, , ,~a, m) = ql1 ··· qln Z neck ( , ,~a, m, β → 0) inst 1 2 1 n BPS,l1...ln 1 2 l1,l2...,ln (94) Here, l is the instanton number of the SU(N) , and Z neck is i i BPS,l1...ln the partition function of BPS states on LHS of ﬁg. 6. An M-Theoretic Derivation of the AGT Correspondence with Matter

Z neck can be viewed as a sum over BPS states that propagate BPS,l1...ln around the diagram which undergo transformations of the kind eﬀected by the operators

Φ~ai ,mi ,~ai+1 : H~ai → H~ai+1 , (95)

where

M ∗ H~ap = IHU(1)2×T U(MSU(N),l ) ⊗ C(1, 2,~ap) (96) l In other words, one can also write

neck l1 Z (q, 1, 2,~a, m) = TrH (q1 ... qn) Φ ··· Φ , inst ~a1 ~a1,m1,~a2 ~an,mn,~a1 (97) where l1 is an instanton number operator whose eigenvalue is the instanton number l1 associated with the BPS states. An M-Theoretic Derivation of the AGT Correspondence with Matter

However, the duality relation (51), (52), and the map (95), also mean that

Φ~ai ,mi ,~ai+1 : Vj(~ai ) → Vj(~ai+1), (98)

where Vj(~ap) is the Verma module over W(su(N)aﬀ ) of highest weight j(~ap) and central charge

( + )2 c = (N − 1) + (N3 − N) 1 2 (99) 12

Consequently, Φ can also be interpreted as a primary vertex operator V acting on V, hence the correspondence between Φ~ ~ and V~ in ﬁg. 6. ai−1,mi−1,ai ji An M-Theoretic Derivation of the AGT Correspondence with Matter

Therefore,

neck neck D E Zinst = Z (q, i , m) · V~ (1)V~ (q1) ... V~ (q1q2 ... qn−1) j1 j2 jn T2 (100) neck The independence of the factor Z on ~a is because the ~ap’s have already been “contracted” in the correlation function (see the RHS of (97)).

Via the fact that (i) the ap’s and the ms ’s have the same

dimension, (ii) Vjs ought to depend on ms−1, (iii) Q vanishes where V is inserted, one can conclude that ~js

−im~ s−1 ~js = √ for s = 1, 2,..., n (101) 12

where m~ 0 = m~ n, and the N − 1 component vector m~ k depends on mk . An M-Theoretic Derivation of the AGT Correspondence with Matter

As we have N D6-branes and 1 D4-brane wrapping Ceff (which one can see by “gluing” the configuration in (48) according to our description above), i.e., we effectively have an N × 1 = N-fold cover ΣSW of Ceff .

$ !"#

% ($" &! ($! "') ($! )#! % % ! &( "') ) ! # & ! % '! &!! % &! % " * +,,

Figure 7: Ceﬀ and its N-fold cover ΣSW

Incidentally, ΣSW is also the Seiberg-Witten curve which underlies neck Zinst (q, 1, 2,~a, m)! An M-Theoretic Derivation of the AGT Correspondence with Matter

Like in the pure AGT case, we have a correspondence between the φk (z)’s (which define ΣSW such that near the puncture z = zs , (2) −2 2 (k) φ2(z) ∼ us (z − zs ) dz ) and the W (z)’s (which define W(su(N)aff )) on Ceff , whence we can deduce that

~2 ~ js js · i~ρ (1 + 2) (2) − √ = us , where s = 1, 2,..., n (102) 2 12

from which we can ascertain the explicit form of the mass vectors m~ s in (101). Thus, in arriving at the above boxed relations, we have just derived the AGT correspondence for a conformal necklace quiver of n SU(N) gauge groups! A “Fully-Ramiﬁed” Pure AGT Correspondence for G and Quantum Aﬃne Toda Systems

In the derivation of a “ramified” pure AGT correspondence, we introduce fluxbranes in (5)-(8), and repeat our analysis in the presence of 4d worldvolume defects. In particular, for a defect which corresponds to the “fully-ramified” case, in place of (51) and (76), we have

M ∗ ∨ IH U(M 0 ) = g , (103) U(1)2×T G,T ,a baﬀ,kAB a0

M ∗ ∨ IH U(M 0 ) = g , (104) U(1)2×T G,T ,a baff,kCDG a0 0 where a is the “ramified” instanton number, MG,T ,a0 is the moduli space of T -“ramified” instantons, and the levels

2 2 kAB = −N − , kCDG = −2N + 2 − (105) 1 1 A “Fully-Ramiﬁed” Pure AGT Correspondence for G and Quantum Aﬃne Toda Systems

Thus, in the Nekrasov-Shatashvili limit 2 = 0, i.e., kAB = −N, kCDG = −2N + 2, the “fully-ramiﬁed” Nekrasov instanton partition function is now

Zinst(G, 1, 0,~a, T ) = h1, ∆n|1, ∆ni, (106)

∨ where |1, ∆ni ∈ bgaﬀ,crit, an integrable module at the critical level.

Note that on ΣSW in ﬁg. 1, 2, there are (twisted) operators

(s ) a ...a S i (z) =: d 1 si (k)(J ... J )(z), s = 1,..., rank(g), a1 asi i (107) ∨ where the (twisted) currents Jai generate gaff,crit, with d being a g-invariant tensor. (s ) The S i ’s are Casimir operators which act as order-si differential operators on correlation functions that at critical level, just serve to multiply them by a c-number. A “Fully-Ramified” Pure AGT Correspondence for G and Quantum Affine Toda Systems

Notice that the RHS of (106) can be interpreted as a two-point correlation function of coherent state operators.

Hence, Zinst(G, 1, 0,~a, T ) ought to be a simultaneous eigenfunction of rank(g) commuting differential operators D1, D2,... Drank(g) derived from the Casimirs of the Langlands dual of an affine G-algebra with spectral curve ΣSW . Also, the RHS of (106) is a Whittaker function associated with ∨ gaff,crit, whence according to [26, §2],

(l) (l) DToda · Zinst(G, 1, 0,~a, T ) = EToda Zinst(G, 1, 0,~a, T ) (108)

(l) where the DToda’s are quantum Toda Hamiltonians and their (l) complex eigenvalues EToda which define a quantum affine Toda system with spectral curve ΣSW . A “Fully-Ramified” AGT Correspondence with Matter and Hitchin Systems

Now note that a pure G theory can also be interpreted as the m → ∞, q = e2πiτ → 0 limit of a G theory with an adjoint hypermultiplet of mass m and complexiﬁed gauge coupling τ. In this limit, the (twisted) elliptic Calogero-Moser system associated with the G theory with matter reduce to the quantum Toda system deﬁned by (108) [27]. For G = SU(N), the untwisted elliptic Calogero-Moser system is known to be equivalent to the Hitchin system on C [28]. These three points and (108) therefore imply that

(l) (l) DH · Zinst(G, q, 1, 0,~a, m, T ) = EH Zinst(G, q, 1, 0,~a, m, T ) (109) Thus, Zinst(SU(N), q, 1, 0,~a, m, T ) is also a simultaneous eigenfunction of the quantum Hitchin Hamiltonians for SU(N). This conﬁrms the conjecture by Alday-Tachikawa in [29]. A “Fully-Ramiﬁed” AGT Correspondence with Matter and the Geometric Langlands Correspondence for Curves

Now in this case, where n = 1 on the LHS of ﬁg. 6, the 6d worldvolume of the N M5-branes that underlie Zinst, is

part of M9-plane z 2 }| 2 { 2 R |2=0 × R |1 × Tmarked,β→0, (110) | {z } 4d defect

2 where Tmarked,β→0 is a degenerate torus with a marked point over which the (spatial part of the) M9-plane sits. 4 Because the twisting of the 6d theory along the R subspace is 2 trivial, we can replace the above R ’s with cigars:

part of M9-plane z }| { 2 D × DR,1 × Tmarked,β→0 . (111) | {z } 4d defect A “Fully-Ramiﬁed” AGT Correspondence with Matter and the Geometric Langlands Correspondence for Curves

Macroscopically at low-energies whence the curvature of the cigar 1 ˜1 tips is not observable, D × DR,1 is eﬀectively a trivial S × S ﬁbration of R+ × I. Therefore, where the minimal energy limit of the M5-brane worldvolume theory is concerned, we can simply take (111) to be

part of M9-plane z }| { 1 ˜1 2 S × R+ × S × I × Tmarked,β→0 . (112) | {z } 4d defect

According to [30, 31], we have an A-model on Σ = R+ × I with target space MH , the “tamely-ramified” Hitchin fibration 2 associated with SU(N) and Tpunc with A-branes at ends of I. A “Fully-Ramified” AGT Correspondence with Matter and the Geometric Langlands Correspondence for Curves

The distinguished A-brane at the tip of DR, is a space-ﬁlling canonical coisotropic brane Bcc . Hence, on one end of I, we have the brane Bcc .

Since the brane BL at the other end cannot be Bcc (as (Bcc , Bcc )-strings represent differential operators and not states, which are what we are seeking), it must be middle-dimensional wrapping either base or fiber of MH . As our correspondence means that the c-number acting S(k)’s can be identified with the commuting Hamiltonians Hk underlying MH [32], we have Hk − hk = 0, where hk ∈ C. This condition defines the fiber F of MH [33]. k ∗ Thus, as Hk ∼ Tr Φ , where Φ of the “fully-ramified” N = 2 SU(N) theory on S1 × S˜1 × Σ survives as a sigma-model scalar field on Σ after compactifying on S1 × S˜1, one can deduce that BL = F. Hence, on the other end of I, we have the brane F. A “Fully-Ramified” AGT Correspondence with Matter and the Geometric Langlands Correspondence for Curves

From the above four points, one can conclude that the minimal energy spectrum of the M5-brane worldvolume theory which is captured by Zinst(SU(N), q, 1, 0,~a, m, T ), is furnished by the space of (Bcc , F) strings.

In turn, since the space of (Bcc , F) strings in MH also furnishes a D-module [33], we find that Zinst(SU(N), q, 1, 0,~a, m, T ) is also a D-module in the “tamely-ramified” geometric Langlands 2 correspondence for SU(N) on Tpunc. One could also generalize the analysis to arbitrary n, where the only change is MH = MH,1 → MH,n. neck In summary, Zinst (SU(N), q, 1, 0,~a, m, T ) is also a D-module in the “tamely-ramified” geometric Langlands correspondence for 2 SU(N) on Tpunc,n, where n ≥ 1! Generalizations and Future Work

One can generalize our setup to (i) derive, in a similar manner, a 6d and 5d AGT correspondence; (ii) demonstrate their connection to relativistic and elliptized integrable systems; and (iii) obtain various mathematically novel and interesting relations involving the double loop algebra of SU(N), elliptic Macdonald operators, equivariant elliptic genus of instanton moduli space, and more. See my paper [arXiv:1309.4775] JHEP12(2013)031. In our recent work [arXiv:1512.07629], we used a QFT approach to study the M5-brane worldvolume theory, whence we were able to derive a completely novel 4d-2d duality involving CDO’s on instanton moduli space and integrable representations of toroidal Lie algebras. If we could derive this using our purely M-theoretic setup, we would be able to obtain the corresponding generalizations presented in this talk, where this would be very interesting and signiﬁcant. THANK YOU FOR YOUR TIME AND ATTENTION.

HERE’S WISHING ALL A FABULOUS 2016 AHEAD! [1] H. Nakajima, “Instantons on ALE Spaces, Quiver Varieties, and Kac-Moody Algebras”, Duke Math. 76 (1994) 365-416. [2] A. Braverman and M. Finkelberg, “Pursuing the Double Aﬃne Grassmannian I: Transversal Slices via Instantons on Ak−1 Singularities”, Duke Math. 152, Number 2 (2010), 175-206, [arXiv:math/0711.2083]. [3] L.F. Alday, D. Gaiotto, Y. Tachikawa, “Liouville Correlation Functions from Four-dimensional Gauge Theories”, Lett. Math. Phys. 91: 167-197, 2010, [arXiv:0906.3219].

[4] N. Wyllard, “AN−1 conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories”, JHEP 11 (2009) 002, [arXiv:0907.2189]. [5] O. Schiffmann and E. Vasserot, “Cherednik algebras, W algebras and the equivariant cohomology of the moduli space 2 of instantons on A ”, [arXiv:1202.2756]. [6] D. Maulik and A. Okounkov, “Quantum Groups and Quantum Cohomology”, [arXiv:1211.1287]. [7] R. Dijkgraaf, L. Hollands, P. Sulkowski, C. Vafa, “Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions”, JHEP 02 (2008) 106, [arXiv:0709.4446]. [8] E. Witten, “Duality from Six-Dimensions I, II, III”, lectures delivered at the IAS in Feb 08. Notes for the lectures taken by D. Ben-Zvi can be found at: [http://www.math.utexas.edu/users/benzvi/GRASP/lectures/IASterm.html]. [9] O. Chacaltana, J. Distler, Y. Tachikawa, “Nilpotent orbits and codimension-two defects of 6d N=(2,0) theories”, [ arXiv:1203.2930]. [10] H. Kanno and Y. Tachikawa, “Instanton counting with a surface operator and the chain-saw quiver”, JHEP 06 (2011) 119, [arXiv:1105.0357]. [11] R. Dijkgraaf, E. Verlinde, H. Verlinde, “BPS Quantisation of the Five-Brane”, Nucl. Phys. B486: 89-113, 1997, [arXiv:hep-th/9604055] [12] R. Dijkgraaf, E. Verlinde, H. Verlinde, “BPS Spectrum of the Five-Brane and Black Hole Entropy”, Nucl. Phys. B486 (1997) 77-88, [arXiv:hep-th/9603126]. [13] R. Dijkgraaf, “The Mathematics of Fivebranes”. International Congress of Mathematicians (ICM 98), Berlin, Germany, Doc. Math. J. DMV, 1999, [arXiv:hep-th/9810157] [14] Y. Tachikawa, “On S-duality of 5d super Yang-Mills on S1”, [arXiv:1110.0531]. [15] C. Bachas, M. Green, A. Schwimmer, “(8, 0) Quantum mechanics and symmetry enhancement in type II superstrings”, JHEP 01 (1998) 006, [arXiv:hep-th/9712086]; L. Hung, “Comments on I1-branes”, JHEP 05 (2007) 076, [arXiv:hep-th/0612207]. [16] M. Green, J. Harvey, G. Moore, “I-brane Inflow and Anomalous Couplings on D-branes”, Class. Quant. Grav. 14 (1997) 47-52, [arXiv:hep-th/9605033]. [17] N. Itzhaki, D. Kutasov, N. Seiberg, “I-brane Dynamics”, JHEP 01 (2006) 119, [arXiv:hep-th/0508025]. [18] S. Reffert, “General Omega Deformations from Closed String Backgrounds”, [arXiv:1108.0644]. [19] S. Hellerman, D. Orlando, S. Reffert, “The Omega Deformation From String and M-Theory”, [arXiv:1204.4192]. [20] E.A. Bergshoeff, G.W. Gibbons, P.K. Townsend, “Open M5-branes”, Phys. Rev. Lett. 97: 231-601, 2006, [arXiv:hep-th/0607193]. [21] N. Nekrasov, “Lectures on nonperturbative aspects of supersymmetric gauge theories”, Class. Quantum Grav. 22 (2005), S77-S105. [22] M.F. Atiyah and R. Bott, “The moment map and equivariant cohomology”, Topology 23 (1984) 1-28. [23] D. Gaiotto, “N = 2 Dualities”, [arXiv:0904.2715]. [24] E. Bergshoeff, E. Eyras, R. Halbersma, C.M. Hull, Y. Lozano, J.P. van der Schaar, “Spacetime-Filling Branes and Strings with Sixteen Supercharges”, Nucl. Phys. B564 (2000) 29-59. [25] S.V. Ketov, “Conformal Field Theory”, World Scientific Press, Singapore, (1997). [26] P. Etingof, “Whittaker functions on quantum groups and q-deformed Toda operators”, [arXiv:math/9901053]. [27] E. D’Hoker, D.H. Phong, “Seiberg-Witten Theory and Calogero-Moser Systems”, [arXiv:hep-th/9906027]. [28] R. Donagi, “Seiberg-Witten Integrable Systems”, [arXiv:alg-geom/9705010]. [29] L.F. Alday, Y. Tachikawa, “Affine SL(2) conformal blocks from 4d gauge theories”, Lett. Math. Phys. 94: 87-114, 2010, [arXiv:1005.4469]. [30] D. Nanopoulos and D. Xie, “Hitchin Equation, Singularity, and N=2 Superconformal Field Theories”, JHEP 03 (2010) 043, [arXiv:0911.1990]. [31] N. Nekrasov and E. Witten, “The Omega Deformation, Branes, Integrability, and Liouville Theory”, [arXiv:1002.0888]. [32] D. Gaiotto, G.W. Moore, A. Neitzke, “Wall-crossing, Hitchin Systems, and the WKB Approximation”, [arXiv:0907.3987]. [33] S. Gukov and E. Witten, “Gauge Theory, Ramification, And The Geometric Langlands Program”, Current Developments in Mathematics Volume 2006 (2008), 35-180. [arXiv:hep-th/0612073].