M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems

M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems Meng-Chwan Tan National University of Singapore January 1, 2016

Presentation Outline Introduction Dual 6d M-Theory Compactifications A Geometric Langlands Duality for Surfaces for the A B Groups A Geometric Langlands Duality for Surfaces for the C D G Groups The Pure AGT Correspondence for the A B Groups The Pure AGT Correspondence for the C D G Groups The AGT Correspondence with Matter The Fully-Ramified AGT Correspondence and Quantum Integrable Systems Generalizations & Future Work

4d-2d Dualities in Mathematics and Physics Circa 1994, Nakajima [1] the middle-dimensional cohomology of the moduli space of U(N)-instantons on a resolved ALE space of A k 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 cohomology of the moduli space of G-instantons on R 4 /Z k 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 W 2 -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 A N 1 conformal Toda field theory which has W N -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 W N -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 space, (iv) the fact that R 4 /Z k = TNk R, (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 and M-theory : R 4 /Z k S 1 n R }{{} t R 5 (1) N M5-branes M-theory : are physically dual. R 5 R t S 1 n TNN R 0, (2) }{{} k M5-branes

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 and M-theory : R 4 /Z k S 1 n R }{{} t R 5 (3) N M5-branes/OM5-plane M-theory : R 5 R t S 1 n SNN R 0, (4) }{{} k M5-branes are physically dual. Here, SNN R is Sen s four-manifold which one can roughly regard as TN N with a Z 2 -identification of its R 3 base and S 1 -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 and M-theory : R 5 R t S 1 n R 4 /Z }{{ k } N M5-branes with a 4d defect M-theory : S 1 n R }{{} t k M5-branes with a 4d defect TNN R 0 (5) R 5, (6) are physically dual. Here, the 4d worldvolume defect wraps R t S 1 n and (i) the z-plane in R 4 /Z k C z /Z k C w /Z k, (ii) the S 1 -fiber of TNN R 0 and a single direction along the R 3 base of TNN R 0.

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 and M-theory : R 5 R t S 1 n R 4 /Z }{{ k } N M5 + OM5 + 4d defect M-theory : SNN R 0 (7) S 1 n R t R 5, (8) } {{ } k M5 + 4d defect are physically dual. Here, the 4d worldvolume defect wraps R t S 1 n and (i) the z-plane in R 4 /Z k C z /Z k C w /Z k, (ii) the S 1 -fiber of SNN R 0, and a single direction along the R 3 base of SNN R 0.

An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A B Groups Notice that because R 4 /Z k and TNN R 0 are hyperkähler four-manifolds which break half of the thirty-two supersymmetries in M-theory, the resulting six-dimensional spacetime theories along R t R 5 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 S 1, where K3 is a hyperkäher 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) A l 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 M S 1 n R t, where M is a generic hyperkähler four-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 a two-dimensional N = (4, 4) sigma-model on S 1 n R t with target the hyperkähler moduli space M G (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 of the N = (4, 4) sigma-model on S 1 n R t 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 M G (M). These differential forms are necessarily harmonic and square-integrable, i.e., the spacetime BPS states would correspond to L 2 -harmonic forms which span the L 2 -cohomology of (some natural compactification of) M G (M).

An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the A B Groups Note that M G (M) consists of components labeled by (a, ρ 0, ρ ); that is, one can write M G (M) = M ρ0,a G,ρ (M), (9) a,ρ 0,ρ where a is the instanton number, and ρ 0, are conjugacy classes of the homomorphism Z k G that one can pick at 0, of M = R 4 /Z k. Note that a is not really independent of ρ 0 and ρ, as we shall now explain. By reducing along S 1 n to type IIA string theory, one can have (half-bps) D0-branes within the M R t worldvolume of the D4-branes. Since D0-branes correspond to static particle-like BPS configurations on M R t which thus present themselves as instantons on M, counting them would yield the formula a = kn (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 H BPS of spacetime BPS states in the M-theory compactification (1) is given by H BPS = λ,µ H λ,µ BPS = λ,µ IH U(M λ G,µ (R4 /Z k )), (11) where IH U(M λ G,µ (R4 /Z k )) is the intersection cohomology (which can be identified with the L 2 -cohomology) of the Uhlenbeck compactification U(M λ G,µ (R4 /Z k )) of the component M λ G,µ (R4 /Z k ) of the highly singular moduli space M G (R 4 /Z k ) 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 G aff 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 worldvolume theory over R t S 1 n TNN R 0. 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) A k 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 Near infinity where the radius of the S 1 -fiber of TNN R 0 tends to zero, we have a reduction to the following type IIA configuration: IIA : R 5 S 1 n R t R 3 }{{}. (13) I-brane on S 1 n R t = ND6 kd4 Here, we have a stack of N coincident D6-branes whose worldvolume is given by R 5 S 1 n R t, and a stack of k coincident D4-branes whose worldvolume is given by S 1 n R t R 3. The two stacks intersect along S 1 n R t 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 theory along S 1 n R t. 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 with an S 1 R 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 of topology S 1 R 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 i,ā (z), ψ (z), i = 1,..., k, a = 1,..., N, (14) ī,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 I = d 2 z ψ A+A ψ, (15) where A and A are the restrictions to the I-brane worldsheet S 1 n R t 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 and gives a direct realization of û(kn) (n) 1, the integrable module over the Z n -twisted affine Lie algebra u(kn) (n) aff,1 of level 1, and this has a conformal embedding u(1) (n) aff,kn su(k)(n) aff,n su(n)(n) aff,k u(kn)(n) aff,1. (16) In other words, the total Fock space F kn of the kn complex free fermions can be expressed as F kn = WZW (n) bu(1) WZW (n) kn bsu(k) N WZW (n) bsu(n), (17) k where WZW (n) bu(1), WZW (n) kn bsu(k) and WZW (n) N bsu(n) are the spectra 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 A. 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 underlying TNN R 0 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 u(kn) (n) aff,1 /[u(1)(n) aff,kn su(k)(n) aff,n ]. (18) In particular, the u(1) (n) aff,kn and su(k)(n) 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 su(n) (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 WZW bsu(n)k = λ, µ WZW bsu(n) λk, µ. (19) 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 and µ 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 As WZW bsu(n) λ is furnished by su(n) λ k,µ k,µ, and since su(n) aff su(n) aff whence su(n)λ k,µ is isomorphic to the submodule L su(n) λ 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 IH U(M λ SU(N),µ (R4 /Z k )) = L su(n) λ 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) = A N 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 HBPS eff = IH U(M λ SO(N+1),µ ν (R 4 /Z k )), (21) λ ν=0,1 µ ν while (19) gets replaced by WZW bsu(n) (2) k = λ ν=0,1 µ ν WZW bsu(n) (2), λ k, µν. (22) Here, the overhead bar means that we project onto Z 2 -invariant states; ν = 0 or 1 indicates that the sector is untwisted or twisted, respectively. Since su(n) (2) aff is isomorphic to so(n + 1) aff, it would mean that λ and µ ν are also dominant weights λ and µ ν 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 bsu(n) is furnished by (the Z 2 -invariant projection of) k,µν su(n) (2),λ k,µ ν, and since su(n) (2) aff so(n + 1) aff whence (the Z 2 -invariant projection of) su(n) (2),λ k,µ ν is isomorphic to the submodule L so(n + 1) λ k,µ ν over so(n + 1) 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 (21) and (22), that IH U(M λ SO(N+1),µ ν (R 4 /Z k )) = L so(n + 1) λ k,µ ν (23) 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) = B N/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 H BPS = λ,µ IH U(M λ SO(2N),µ (R4 /Z k )), (24) HBPS eff = IH U(M λ USp(2N 2),µ ν (R 4 /Z k )), λ ν=0,1 µ ν (25) HBPS eff = 2 IH U(M λ G 2,µ ν (R 4 /Z k )), λ ν=0 µ ν (26) where λ > µ, µ ν. 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. Near infinity, the S 1 R circle fiber of SNR 0 N has radius R 0, and we have a reduction to the following type IIA configuration: IIA : R 5 S 1 n R t R 3 /I }{{} 3. (27) I-brane on S 1 n Rt = ND6/O6 kd4 Here, we have a stack of N coincident D6-branes on top of an O6 -plane whose worldvolume is given by R 5 S 1 n R t, and a stack of k coincident D4-branes whose worldvolume is given by S 1 n R t R 3 /I 3 (where I 3 acts as r r in R 3 ). These two stacks intersect along S 1 n R t 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.

where A and A are the restrictions to the I-brane worldsheet S 1 n R t 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 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 ψ i,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 I = d 2 z ψ A+A ψ, (29)

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 gives a direct realization of so(2kn) (n) 1, the integrable module over the Z n -twisted affine Lie algebra so(2kn) (n) aff,1 of level 1, and this has a conformal embedding: so(k) (n) aff,2n so(2n)(n) aff,k so(2kn)(n) aff,1, (30) In other words, the total Fock space F 2kN of the 2kN real free fermions can be expressed as F 2kN = WZW (n) so(k) b WZW 2N b (n) so(2n) k, (31) where WZW (n) so(k) b and WZW (n) 2N so(2n) b are the spectra of states 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 A 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 geometry of SNN R 0 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 so(2kn) (n) aff,1 /so(k)(n) aff,2n. (32) In particular, the so(k) (n) 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 so(2n) (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 (4) would, for n = 1, 2, 3 (with N = 4), be realized by where λ > µ, µ ν. WZW so(2n)k b WZW (2) so(2n) b = λ k WZW (3) so(8) b = λ k = λ, µ WZW so(2n) λ b, (33) k, µ ν=0,1 ν=0,1,2 WZW b µ ν µ ν WZW b (2), λ so(2n) k, µν (3), λ so(8) k, µν, (34), (35)

An M-Theoretic Derivation of a Geometric Langlands Duality for Surfaces for the C D G Groups Since so(2n) aff, so(2n) (2) aff, so(8)(3) aff are isomorphic to so(2n) aff, usp(2n 2) aff, g 2 aff, it will mean that λ, µ and µ ν are also dominant weights of the Langlands dual affine Kac-Moody group SO(2N) aff, USp(2N 2) aff, G 2 aff, whence we can identify them with λ, µ and µ ν of (24), (25), (26), respectively. As WZW so(2n) b λ, WZW k,µ b so(2n) λ k,µ, so(2n)(2),λ (2),λ so(2n) k,µν so(8) (3),λ, WZW (3),λ so(8) b k,µν are furnished by k,µ ν, k,µ ν, and since so(2n) aff so(2n) aff, so(2n)(2) aff usp(2n 2) aff, so(8) (3) aff g 2 aff whence so(2n)λ k,µ, so(2n)(2),λ k,µ ν, so(8) (3),λ k,µ ν are isomorphic to the submodules L so(2n) λ k,µ, L ûsp(2n 2) λ k,µ ν, ( L ĝ 2 ) λ k,µ ν over so(2n) aff, usp(2n 2) aff, g 2 aff, respectively,

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 IH U(M λ SO(2N),µ (R4 /Z k )) = L so(2n) λ k,µ (36) IH U(M λ USp(2N 2),µ ν (R 4 /Z k )) = L ûsp(2n 2) λ k,µ ν (37) IH U(M λ G 2,µ ν (R 4 /Z k )) = ( L ĝ 2 ) λ k,µ ν (38) Note that (36), (37), (38) is [2, Conjecture 4.14(3)] for G = SO(2N), USp(2n 2), G 2! This completes our purely physical M-theoretic derivation of a geometric Langlands duality for surfaces for the SO(2N) = D N, USp(2N 2) = C N 1 and G 2 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 ɛ 1 and ɛ 2 along the z = x 2 + ix 3 and w = x 4 + ix 5 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 ɛ 1 ɛ 2 ɛ 3 (39) Here, the s denote the fluxbrane directions; denotes the S 1 n circle direction; and denotes the eleventh circle. In addition, there is also a rotation along the u-plane with rotation parameter ɛ 3 = ɛ 1 + ɛ 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 (ɛ 1, ɛ 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 ɛ 1, ɛ 2 ɛ 3 (40) Here, ɛ 1, ɛ 2 along the x 2 -x 3 -x 4 -x 5 directions means that there are two simultaneous rotations along the x 4 -x 5 plane with rotation parameters ɛ 1 and ɛ 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 R 4 ɛ1,ɛ 2 S 1 n R t R 5 ɛ3 ; x }{{} 6,7, R 4 ɛ1,ɛ 2 S 1 n R t }{{} R 5 ɛ3 ; x 6,7. 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 R 5 ɛ 3 x 4,5 R t S 1 n TNN R 0 ɛ3 ; x 6,7, R 5 ɛ 3 x }{{} 4,5 R t S 1 n SNN R 0 ɛ3 ; x 6,7. }{{} 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 Σ = S 1 n R t, so we are free to deform Σ into a short cylinder Σ n,t = S 1 n I t, where I t is an interval whose length is much smaller than β, the radius of S 1 n. 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 I t. 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 R 4 ɛ1,ɛ 2 Σ n,t }{{} N M5-branes R 5 ɛ3 ; x 6,7, R 4 ɛ1,ɛ 2 Σ n,t }{{} R 5 ɛ3 ; x 6,7, N M5-branes + OM5-plane (43) where the M9-branes intersect the M5-branes/OM5-plane along S 1 n R 4 ɛ1,ɛ 2 and span R 5 ɛ3 ; x 6,7. Then, Omega-deformation in the worldvolume theory can be understood as follows. At low-energy distances much larger than I t, as one traverses around the S 1 n circle, the (z, w) planes in R 4 ɛ1,ɛ 2 would be rotated by angles (ɛ 1, ɛ 2 ) together with an SU(2) R -symmetry rotation of the resulting G gauge theory along R 4 ɛ1,ɛ 2.

where T = (T 1..., T rank G ) are the generators of the Cartan subgroup of G; a = (a 1,..., a rank G ) are the corresponding purely imaginary Coulomb moduli of the G gauge theory on R 4 ɛ1,ɛ 2 ; J 1,2 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 H m is the space of holomorphic functions on the moduli space M G,m of G-instantons on R 4 with instanton number m. 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 Σ n,t = S 1 n I t into a two-torus S 1 n S t ) would be given by the following 5d (since S t β) worldvolume expression (c.f. [21, eqns. (29) and (43)]) Z BPS (ɛ 1, ɛ 2, a, β) = m Tr Hm exp β(ɛ 1 J 1 + ɛ 2 J 2 + a T ), (44)

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 R 5 ɛ3 ; x 4,5 Σ n,t TNN R 0 ɛ3 ; x 6,7, R 5 ɛ3 ; x 4,5 Σ n,t SNN R 0 ɛ3 ; x 6,7, }{{}}{{} 1 M5-branes 1 M5-branes (45) where we have a pair of M9-branes whose worldvolumes at the tips of I t span the ten directions transverse to it, and as one traverses around the S 1 n circle, among other things, the x 4 -x 5 plane in R 4 ɛ3 ; x 4,5 would be rotated by an angle of ɛ 1 + ɛ 2 = ɛ 3 together with an SU(2) R -symmetry rotation of the gauge theory along R 4 ɛ3 ; x 4,5.

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 R 4 ɛ1,ɛ 2 Σ n,t R 5 ɛ3 ; x }{{} 6,7 R 5 ɛ3 ; x 4,5 C TNN R 0 ɛ3 ; x 6,7, }{{} N M5-branes 1 M5-branes (46) where C is a priori the same as Σ n,t = S 1 n I t, and we have an M9-brane at each tip of I t. Let us first ascertain the spectrum of spacetime BPS states on the LHS of (46) that define Z BPS (ɛ 1, ɛ 2, a, β) in (44). Note that (44) means that as one traverses a closed loop in Σ n,t, there would be a g-automorphism of M G, 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 H Ω BPS = m H Ω BPS,m = m IH U(1) 2 T U(M G,m), (47) where IH U(1) 2 T U(M G,m) is the (Z n -invariant in the sense of (21) when n = 2) U(1) 2 T -equivariant intersection cohomology of the Uhlenbeck compactification U(M G,m ) of the (singular) moduli space M G,m of G-instantons on R 4 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: IIA : R 5 ɛ3 ;x 4,5 C R 3 ɛ3 ;x 6,7. }{{} (48) I-brane on C = ND6 1D4 Here, we have a stack of N coincident D6-branes whose worldvolume is given by R 5 ɛ3 ;x 4,5 C, and a single D4-brane whose worldvolume is given by C R 3 ɛ3 ;x 6,7.

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 ɛ 3 = ɛ 1 + ɛ 2 = 0. Then, from our earlier arguments, we learn that the spacetime BPS states would be furnished by a chiral WZW model at level 1 on C, WZW level g 1, where g aff is the Langlands aff dual of the affine G-algebra g aff. Now turn Omega-deformation back on. As indicated in (48), as one traverses around a closed loop in C, the x 4 -x 5 plane in R 4 ɛ3 ; x 4,5 R 5 ɛ3 ; x 4,5 would be rotated by an angle of ɛ 3 together with an SU(2) R -symmetry rotation of the supersymmetric SU(N) gauge theory along R 4 ɛ3 ; x 4,5. As discussed below (46), we find that there is a g -automorphism of M SU(N),m as we traverse around a closed loop in C, where M SU(N),m is the moduli space of SU(N)-instantons on R 4 with instanton number m; g = U(1) T, where T SU(N) is a Cartan subgroup.

An M-Theoretic Derivation of the Pure AGT Correspondence for A B Groups In fact, since M SU(N),m is also the space of self-dual connections of an SU(N)-bundle on R 4, 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 g -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 g -automorphism of the target space of WZW level g 1 as we traverse a closed loop in C. aff In short, in the presence of Omega-deformation, we would have to non-dynamically gauge WZW level g 1 by U(1) T. aff

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 reduce the U(1) T Omega-deformation factor by R = U(1) T, where U(1) R is associated with the ɛ 3 -rotation of the x 6 -x 7 plane in R 3 ɛ3 ; x 6,7, 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 WZW level g 1 not by U(1) T but by T T. aff

An M-Theoretic Derivation of the Pure AGT Correspondence for A B Groups As SU(N)/T SL(N, C)/B +, where B + is a Borel subgroup, it would mean that SU(N)/T (SL(N, C)/B + ) (T /T ). Also, T /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 sl(n) (n) aff,1 /n + (n) aff,p (49) on C, where the level p would necessarily depend on the Omega-deformation parameters ɛ 1 = βɛ 1 and ɛ 2 = βɛ 2. (p, being a purely real number, should not depend on the purely imaginary parameter a = 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 = Σ n,t into a curved Riemann surface with the same topology, i.e., the effective geometry of C is S 2 /{0, }.

An M-Theoretic Derivation of the Pure AGT Correspondence for A B Groups The partially gauged chiral CFT in (49) realizes W(g aff ) a Z n -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 HBPS Ω = Ŵ(g aff ), (50) where Ŵ(g aff ) is s Verma module over W(g aff ). In summary, the physical duality of the compactifications in (46) will mean that (47) is equivalent to (50), i.e., IH U(1) 2 T U(M G,m) = Ŵ(g aff ) (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 Omega-deformation parameters ɛ 1,2 ; (ii) the compactifications in (46) are symmetric under the exchange ɛ 1 ɛ 2 ; (iii) we have a geometrical g = exp[(ɛ 1 + ɛ 2 )J 3] = exp[(λɛ 1 + λɛ 2 )λ 1 J 3 ] automorphism associated with the Omega-deformation in (48), whilst the central charge is independent of the x 4 -x 5 plane rotation generator J 3 ; we can deduce that the central charge and level of W(g aff ) are c A,ɛ1,2 = (N 1) + (N 3 N) (ɛ 1 + ɛ 2 ) 2 ɛ 1 ɛ 2 (52) k = N b 2 and b = ɛ 1 /ɛ 2 (53)

An M-Theoretic Derivation of the Pure AGT Correspondence for A B Groups Ŵ(g aff ) is generated by the application of creation operators W (s i ) m<0 on its Z n-twisted highest weight state, where m Z/n. From the additional fact that (i) W (2) 0 generates translations along the S 1 n fiber in C, (ii) there is a rotation of an R 4 space and the gauge field over it as we go around the S 1 n; (iii) (51) would mean that the symmetries of the β-independent W (2) 0 ought to be compatible with the symmetries of Z BPS (ɛ 1, ɛ 2, a, β) in (44); we can deduce that W (2) 0 = (2) (54) where for some real constant γ. (2) = (N3 N) (ɛ 1 + ɛ 2 ) 2 γ a2 (55) 24 ɛ 1 ɛ 2 ɛ 1 ɛ 2

An M-Theoretic Derivation of the Pure AGT Correspondence for A B Groups Writing (44) as Z BPS = m Z BPS,m, the Nekrasov instanton partition function will be Z inst (Λ, ɛ 1, ɛ 2, a) = m Λ 2mh g Z BPS,m(ɛ 1, ɛ 2, a, β 0), (56) where Λ can be interpreted as the inverse of the observed scale of the R 4 ɛ1,ɛ 2 space on the LHS of (46), and Z BPS,m is just a rescaled version of Z BPS,m. Note that equivariant localization [22] implies that IH U(1) 2 T U(M SU(N),m) must be endowed with an orthogonal basis { p m }. Thus, since Z BPS,m is a weighted count of the states in HBPS,m Ω = IH U(1) 2 T U(M G,m), it would mean from (56) that Z inst (Λ, ɛ 1, ɛ 2, a) = Ψ Ψ, (57) where Ψ = m Λmh g Ψ m m IH U(1) 2 T U(M G,m).

An M-Theoretic Derivation of the Pure AGT Correspondence for A B Groups In turn, the duality (51) and the consequential observation that Ψ is a sum over 2d states of all energy levels m, mean that Ψ = q,, (58) where q, Ŵ(g aff ) is a coherent state, and from (57), Z inst (Λ, ɛ 1, ɛ 2, a) = q, q, (59) Since the LHS of (59) is defined in the β 0 limit of the LHS of (46), q, and q, ought to be a state and its dual associated with the puncture at z = 0, on C, respectively (as these are the points where the S 1 n 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 Σ SW of C. This is also depicted in fig. 1. #!!" $"' #% &$%& $"! #%%&$& Figure 1: C and its N-fold cover Σ SW with the coherent states q, and q, at z = 0 and Incidentally, Σ SW is also the Seiberg-Witten curve which underlies Z inst (Λ, ɛ 1, ɛ 2, a)!

An M-Theoretic Derivation of the Pure AGT Correspondence for A B Groups For n = 1, i.e., G = SU(N), Σ SW can be described in terms of the algebraic relation Σ SW : λ N + φ 2 (z)λ N 2 + + φ N (z) = 0, (60) where λ = ydz/z (for some complex variable y); the φ s (z) s are (s, 0)-holomorphic differentials on C given by ( ) dz j ) ( ) φ j (z) = u j and φ N (z) = (z + u N + ΛN dz N, z z z (61) where j = 2, 3,..., N 1. This is consistent with our results that we have, on C, the following (s i, 0)-holomorphic differentials ( ) W (s i ) W (s i ) (dz ) si l (z) = z l, where s i = 2, 3,..., N. z l Z (62)

An M-Theoretic Derivation of the Pure AGT Correspondence for A B Groups In fact, the U(1) R-symmetry of the 4d theory along R 4 ɛ1,ɛ 2 on the LHS of (46) can be identified with the rotational symmetry of S 1 n; the duality relation (46) then means that the corresponding U(1) R-charge of the φ s (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, φ s (z) with W (s) (z), and by recalling the facts used to derive (52)-(55), we can deduce that W (s) 0 q, = W (s) l 2 q, = 0, for s = 2, 3,..., N (63) u s q,, for s = 2, 3,..., N (64) (ɛ 1 ɛ 2 ) s/2 W (N) 1 q, = q q,, q = Λ N (ɛ 1 ɛ 2 ) N/2 (65)

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 ) ( ) φ s = u s, φn = (z 1/2 + ΛN dz N z z 1/2, (66) 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 (s i, 0)-holomorphic differentials ( ) W (s i ) W (s i ) (dz ) si l = z l, W (s i ) = W (s i ) ( ) l+1/2 dz si z z l+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, φ s (z) with W (s) (z) and φ s (z) with W (s) (z), whence we can again deduce that W (s) l 1 q, = 0, for s = 2, 3,..., N (68) W (s) l 3/2 q, = 0, for s = 2, 3,..., N (69) W (N) 1/2 q, = q q,, q = Λ N (ɛ 1 ɛ 2 ) N/2 (70) In arriving at the above boxed relations, we have just furnished a fundamental physical derivation of the pure AGT correspondence for the A N 1 and B N/2 groups!

where IH U(1) 2 T U(M G,m) is the Z n -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 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 R 4 ɛ1,ɛ 2 Σ n,t }{{} N M5-branes + OM5-plane R 5 ɛ3 ; x 6,7 R 5 ɛ3 ; x 4,5 C SN R 0 N ɛ3 ; x 6,7 } {{ } 1 M5-branes (71) where C is a priori the same as Σ n,t = S 1 n I t, and we have an M9-brane at each tip of I t. As before, the Hilbert space of spacetime BPS states on the LHS of (71) is given by H Ω BPS = m H Ω BPS,m = m IH U(1) 2 T U(M G,m), (72),

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 IIA : R 5 ɛ3 ;x 4,5 C R 3 /I 3 ɛ3 ;x 6,7. }{{} (73) I-brane on C = ND6/O6 1D4 Here, we have a stack of N coincident D6-branes on top of an O6 -plane whose worldvolume is given by R 5 ɛ3 ;x 4,5 C, and a single D4-brane whose worldvolume is given by C R 3 /I 3 ɛ3 ;x 6,7 (where I 3 acts as r r in R 3 ). Repeating the arguments that led us to (49), we find that instead of (49), we now have so(2n) (n) aff,1 /n + (n) aff,p, (74) and the effective geometry of C would again be S 2 /{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(g aff ) a Z n -twisted version of the affine W-algebra W( so(2n)) obtained from so(2n) aff via a quantum Drinfeld-Sokolov reduction. Therefore, the Hilbert space H Ω RHS of (71) can be expressed as BPS of spacetime BPS states on the HBPS Ω = Ŵ(g aff ), (75) where Ŵ(g aff ) is s Verma module over W(g aff ). In summary, the physical duality of the compactifications in (71) will mean that (72) is equivalent to (75), i.e., IH U(1) 2 T U(M G,m) = Ŵ(g aff ) (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(g aff ) as c D,ɛ1,2 = N + (2N 2)(2N 2 N) (ɛ 1 + ɛ 2 ) 2 ɛ 1 ɛ 2 (77) k = 2N + 2 b 2 and b = ɛ 1 /ɛ 2 (78) Also, where W (2) 0 = (2) (79) (2) = (2N 2)(2N2 N) (ɛ 1 + ɛ 2 ) 2 γ a 2 (80) 24 ɛ 1 ɛ 2 ɛ 1 ɛ 2 for some real constant γ.

An M-Theoretic Derivation of the Pure AGT Correspondence for C D G Groups Last but not least, the Nekrasov partition function Z inst (Λ, ɛ 1, ɛ 2, a) = q, q, (81) where q, Ŵ(g aff ) is a coherent state. The state q, and its dual q, 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 Σ SW of C. This is also depicted in fig. 2.

An M-Theoretic Derivation of the Pure AGT Correspondence for C D G Groups & ' $% #%&!! ""#$ #%!!#!"" $ & ' $% Figure 2: C and its 2N-fold cover Σ SW with the states q, and q, at z = 0 and Incidentally, Σ SW is also the Seiberg-Witten curve which underlies Z inst (Λ, ɛ 1, ɛ 2, a)!

An M-Theoretic Derivation of the Pure AGT Correspondence for C D G Groups For n = 1, i.e., G = SO(2N), Σ SW can be described in terms of the algebraic relation Σ SW : λ 2N + φ 2 (z)λ 2N 2 + + φ 2N 2 (z)λ 2 + φ 2 N (z) = 0, (82) where λ = ydz/z (for some complex variable y); the φ s (z) s are (s, 0)-holomorphic differentials on C given by ( ) dz j ) ( ) φ j (z) = u j, φ 2N 2(z) = (z + u 2N 2 + Λ2N 2 dz 2N 2, z 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 (s i, 0)-holomorphic differentials ( ) W (s i ) W (s i ) (dz ) si l (z) = z l, where s i = 2, 4,..., 2N 2, N. z 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, φ s (z) with W (s) (z), whence we can deduce that W (s) l 2 q, = 0, for s = 2, 4,..., 2N 2, N (85) W (s) 0 q, = u s (ɛ 1 ɛ 2 ) s/2 q,, for s = 2, 4,..., 2N 2, N (86) W (2N 2) 1 q, = q q,, q = Λ2N 2 (ɛ 1 ɛ 2 ) N 1 (87) When n = 2 whence G = USp(2N 2), we have W (s) l 1 q, = 0, for s = 2, 4,..., 2N 2, N (88) W (s) l 3/2 q, = 0, for s = 2, 4,..., 2N 2, N (89)

An M-Theoretic Derivation of the Pure AGT Correspondence for C D G Groups W (2N 2) 1/2 q, = q q,, q = Λ2N 2 (ɛ 1 ɛ 2 ) N 1 (90) When n = 3 (with N = 4) whence G = G 2, we have W (s) l 1 q, = 0, for s = 2, 4, 6 (91) W (s) l 2/3 q, = 0, for s = 2, 4, 6 (92) W (6) Λ6 1/3 q, = q q,, q = (ɛ 1 ɛ 2 ) 3 (93) In arriving at the above boxed relations, we have just furnished a fundamental physical derivation of the pure AGT correspondence for the D N, C N 1 and G 2 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 the underlying 4d gauge theory along R 4 ɛ1,ɛ 2 X 9 ɛi ; V q, and Vq, is a vertex operator and its dual with higher order poles that represent the coherent state q, and its dual q, 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 are defects realized by M-branes which span all of R 4 ɛ1,ɛ 2 that sit at the points where V q, and Vq, are inserted. In turn, this means that we can define Gaiotto s theories and therefore their corresponding Nekrasov instanton partitions Z inst (G, ɛ 1, ɛ 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, a i+1 is associated with a M9-brane domain wall that transforms the theory with a i to the adjacent one with a i+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].

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 Consider a conformal necklace quiver of n SU(N), N > 2.! "!# "! " ## $$! "! " ## $! "!#!!! "! " ##! "! " ## "! "!# " %!& % $$! %! & % $ "!! %! & %! " %! & % " "

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, Zinst neck (q, ɛ 1, ɛ 2, a, m) = q l1 1 qn ln ZBPS,l neck 1...l n (ɛ 1, ɛ 2, a, m, β 0) l 1,l 2...,l n Here, l i is the instanton number of the SU(N) i, and Z neck the partition function of BPS states on LHS of fig. 6. BPS,l 1...l n (94) is