Alday-Gaiotto-Tachikawa Research Articles

Recurrence relation for instanton partition function in SU(N) gauge theory

We derive a residue formula and as a consequence a recurrence relation for the instanton partition function in mathcal{N} = 2 supersymmetric theory on ℂ2 with SU(N) gauge group.The particular cases of SU(2) and SU(3) gauge groups were considered in the literature before. The recurrence relation with SU(2) gauge group is long well known and was found as the Alday-Gaiotto-Tachikawa (AGT) counterpart of the Zamolodchikov relation for the Virasoro conformal blocks. In the SU(3) case a residue formula for the term with the minimal number of instantons was found and basing on it a recurrence relation was conjectured.We give a complete proof of the residue formula in all instanton orders in presence of any number of matter hypermultiplets in the adjoint and fundamental representations. The recurrence relation however describes only theories with not too much matter hypermultiplets so that the behaviour at infinity is moderate. The guideline of the proof is an algebro-geometric interpretation of the mathcal{N} = 2 supersymmetric gauge theory partition function in terms of the framed torsion-free sheaves. Lead by this interpretation we formulate a refined version of the residue formula and prove it by direct algebraic manipulations.

Seiberg–Witten theory and monstrous moonshine

Abstract We study the relation between the instanton expansion of the Seiberg–Witten (SW) prepotential for D = 4, ${\cal N}=2$SU(2) SUSY gauge theory for Nf = 0 and 1 and the monstrous moonshine. By utilizing a newly developed simple method to obtain the SW prepotential, it is shown that the coefficients of the expansion of q = e2πiτ in terms of $A^2=\frac{\Lambda ^2}{16 a^2}$ (Nf = 0) or $\frac{\Lambda ^2}{32a^2}$ (Nf = 1) are all integer-coefficient polynomials of the moonshine coefficients of the modular j-function. A relationship between the Alday–Gaiotto–Tachikawa (AGT) c = 25 Liouville conformal field theory (CFT) and the c = 24 vertex operator algebra CFT of the moonshine module is also suggested.

Affine super Yangians and rectangular W-superalgebras

Motivated by the Alday-Gaiotto-Tachikawa (AGT) conjecture, we construct a homomorphism from the affine super Yangian Yε1,ε2(sl̂(m|n)) to the universal enveloping algebra of the rectangular W-superalgebra Wk(gl(ml|nl),(l(m|n))) for all m ≠ n, m, n ≥ 2 or m ≥ 3, n = 0. Furthermore, we show that the image of this homomorphism is dense, provided that k + (m − n)(l − 1) ≠ 0.

W$W$‐algebras associated to surfaces

We define an integral form of the deformed W $W$ -algebra of type gl r ${\mathfrak {gl}}_r$ , and construct its action on the K $K$ -theory groups of moduli spaces of rank r $r$ stable sheaves on a smooth projective surface S $S$ , under certain assumptions. Our construction generalizes the action studied by Nakajima, Grojnowski and Baranovsky in cohomology, although the appearance of deformed W $W$ -algebras by generators and relations is a new feature. Physically, this action encodes the Alday–Gaiotto–Tachikawa correspondence for 5-dimensional supersymmetric gauge theory on S × $S \times$ circle.

Generalized Gibbs Ensemble of 2D CFTs with U(1) charge from the AGT correspondence

The Generalized Gibbs Ensemble (GGE) is relevant to understand the thermalization of quantum systems with an infinite set of conserved charges. In this work, we analyze the GGE partition function of 2D Conformal Field Theories (CFTs) with a U(1) charge and quantum Benjamin-Ono2 (qBO2) hierarchy charges. We use the Alday-Gaiotto-Tachikawa (AGT) correspondence to express the thermal trace in terms of the Alba-Fateev-Litvinov-Tarnopolskiy (AFLT) basis of descendants, which diagonalizes all charges. We analyze the GGE partition function in the thermodynamic semiclassical limit, including the first order quantum correction. We find that the equality between GGE averages and primary eigenvalues of the qBO2 charges is attainable in the strict large c limit and potentially violated at the subleading 1/c order. We also obtain the finite c partition function when only the first non-trivial charge is turned on, expressed in terms of partial theta functions. Our results should be relevant to the eigenstate thermalization hypothesis for charged CFTs, Warped CFTs and effective field theory descriptions of condensed matter systems.

Toward AGT for Parabolic Sheaves

Abstract We construct explicit elements $W_{ij}^k$ in (a completion of) the shifted quantum toroidal algebra of type $A$ and show that these elements act by 0 on the $K$-theory of moduli spaces of parabolic sheaves. We expect that the quotient of the shifted quantum toroidal algebra by the ideal generated by the elements $W_{ij}^k$ will be related to $q$-deformed $W$-algebras of type $A$ for arbitrary nilpotent, which would imply a $q$-deformed version of the Alday-Gaiotto-Tachikawa (AGT) correspondence between gauge theory with surface operators and conformal field theory.

Generalized Macdonald Functions on Fock Tensor Spaces and Duality Formula for Changing Preferred Direction

An explicit formula is obtained for the generalized Macdonald functions on the $N$-fold Fock tensor spaces, calculating a certain matrix element of a composition of several screened vertex operators. As an application, we prove the factorization property of the arbitrary matrix elements of the multi-valent intertwining operator (or refined topological vertex operator) associated with the Ding--Iohara--Miki algebra (DIM algebra) with respect to the generalized Macdonald functions, which was conjectured by Awata, Feigin, Hoshino, Kanai, Yanagida and one of the authors. Our proof is based on the combinatorial and analytic properties of the asymptotic eigenfunctions of the ordinary Macdonald operator of $A$-type, and the Euler transformation formula for Kajihara and Noumi's multiple basic hypergeometric series. That factorization formula provides us with a reasonable algebraic description of the 5D (K-theoretic) Alday-Gaiotto-Tachikawa (AGT) correspondence, and the interpretation of the invariance under the preferred direction from the point of view of the $SL(2,\mathbb{Z})$ duality of the DIM algebra.

The affine Yangian of [formula omitted] revisited

The affine Yangian of gl1 has recently appeared simultaneously in the work of Maulik–Okounkov [11] and Schiffmann–Vasserot [20] in connection with the Alday–Gaiotto–Tachikawa conjecture. While the presentation from [11] is purely geometric, the algebraic presentation in [20] is quite involved. In this article, we provide a simple loop realization of this algebra which can be viewed as an “additivization” of the quantum toroidal algebra of gl1 in the same way as the Yangian Yh(g) is an “additivization” of the quantum loop algebra Uq(Lg) for a simple Lie algebra g. We also explain the similarity between the representation theories of the affine Yangian and the quantum toroidal algebras of gl1 by generalizing the main result of [10] to the current settings.

S-duality as Fourier transform for arbitrary ϵ1, ϵ2

The Alday–Gaiotto–Tachikawa relations reduce S-duality to the modular transformations of conformal blocks. It was recently conjectured that, for the four-point conformal block, the modular transform up to the non-perturbative contributions can be written in the form of the ordinary Fourier transform when β ≡ −ϵ1/ϵ2 = 1. Here I extend this conjecture to general values of ϵ1, ϵ2. Namely, I argue that, for a properly normalized four-point conformal block the S-duality is perturbatively given by the Fourier transform for arbitrary values of the deformation parameters ϵ1, ϵ2. The conjecture is based on explicit perturbative computations in the first few orders of the string coupling constant g2 ≡ −ϵ1ϵ2 and hypermultiplet masses.

On the integrability of four dimensional $ \mathcal{N}=2 $ gauge theories in the omega background

We continue to investigate the relationship between the infrared physics of N=2 supersymmetric gauge theories in four dimensions and various integrable models such as Gaudin, Calogero-Moser and quantum spin chains. We prove interesting dualities among some of these integrable systems by performing different, albeit equivalent, quantizations of the Seiberg-Witten curve of the four dimensional theory. We also discuss conformal field theories related to N=2 4d gauge theories by the Alday-Gaiotto-Tachikawa (AGT) duality and the role of conformal blocks of those CFTs in the integrable systems. As a consequence, the equivalence of conformal blocks of rank two Toda and Novikov-Wess-Zumino-Witten (WZNW) theories on the torus with punctures is found.

Three-dimensional extensions of the Alday-Gaiotto-Tachikawa relation

An extension of the two-dimensional (2d) Alday-Gaiotto-Tachikawa (AGT) relation to three dimensions starts from relating the theory on the domain wall between some two S-dual supersymmetric Yang-Mills (SYM) models to the 3d Chern-Simons (CS) theory. The simplest case of such a relation would presumably connect traces of the modular kernels in 2d conformal theory with knot invariants. Indeed, the two quantities are very similar, especially if represented as integrals of quantum dilogarithms. But there are also various differences, especially in the “conservation laws” for the integration variables holding for the monodromy traces but not for the knot invariants. We also consider another possibility: interpreting knot invariants as solutions of the Baxter equations for the relativistic Toda system. This implies another AGT-like relation: between the 3d CS theory and the Nekrasov-Shatashvili limit of the 5d SYM theory.

Five-dimensional SU(2) AGT conjecture and recursive formula of deformed Gaiotto state

This paper deals with the five-dimensional pure SU(2) Alday-Gaiotto-Tachikawa (AGT) conjecture proposed by Awata and Yamada [J. High. Energy Phys. 1001, 125 (2010)]. We give a conjecture on a recursive formula for the inner product of the deformed Gaiotto state. We also show that the K-theoretic pure SU(2) Nekrasov partition function satisfies the same recursion relation. Therefore the five-dimensional AGT conjecture is reduced to our conjectural recursive formula.

Calculations in conformal theory needed for verifying the Alday—Gaiotto—Tachikawa hypothesis

An explicit check of the AGT relation between the W_N-symmetry controlled conformal blocks and U(N) Nekrasov functions requires knowledge of the Shapovalov matrix and various triple correlators for W-algebra descendants. We collect simplest expressions of this type for N=3 and for the two lowest descendant levels, together with the detailed derivations, which can be now computerized and used in more general studies of conformal blocks and AGT relations at higher levels.

On AGT description of $ \mathcal{N} = 2 $ SCFT with N f = 4

We consider Alday-Gaiotto-Tachikawa (AGT) realization of the Nekrasov partition function of $ \mathcal{N} = 2 $ SCFT. We focus our attention on the SU(2) theory with N f = 4 flavor symmetry, whose partition function, according to AGT, is given by the Liouville four-point function on the sphere. The gauge theory with N f = 4 is known to exhibit SO(8) symmetry. We explain how the Weyl symmetry transformations of SO(8) flavor symmetry are realized in the Liouville theory picture. This is associated to functional properties of the Liouville four-point function that are a priori unexpected. In turn, this can be thought of as a non-trivial consistency check of AGT conjecture. We also make some comments on elementary surface operators and WZW theory.

Recursion relations in CFT and N=2 SYM theory

Based on prototypical example of Al. Zamolodchikov's recursion relations for the four point conformal block and using recently proposed Alday-Gaiotto-Tachikawa (AGT) conjecture, recursion relations are derived for the generalized prepotential of = 2 SYM with f = 0,1,2,3,4 (anti) fundamental or an adjoint hypermultiplets. In all cases the large expectation value limit is derived explicitly. A precise relationship between generic 1-point conformal block on torus and specific 4-point conformal block on sphere is established. In view of AGT conjecture this translates into a relation between partition functions with an adjoint and 4 fundamental hypermultiplets.