V. A. Fateev, A. V. Litvinov
In these notes we consider relation between conformal blocks and the Nekrasov partition function of certain $\mathcal{N}=2$ SYM theories proposed recently by Alday, Gaiotto and Tachikawa. We concentrate on $\mathcal{N}=2^{*}$ theory, which is the simplest example of AGT relation.
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The AGT conjecture claims an equivalence of conformal blocks in 2d CFT and sums of Nekrasov functions (instantonic sums in 4d SUSY gauge theory). The conformal blocks can be presented as Dotsenko-Fateev beta-ensembles, hence, the AGT conjecture implies the equality between Dotsenko-Fateev beta-ensembles and the Nekrasov functions. In this paper, we prove it in a particular case of beta=1 (which corresponds to c = 1 at the conformal side and to epsilon_1 + epsilon_2 = 0 at the gauge theory side) in a very direct way. The central role is played by representation of the Nekrasov functions through…
The recent AGT suggestion to use the set of Nekrasov functions as a basis for a linear decomposition of generic conformal blocks works very well not only in the case of Virasoro symmetry, but also for conformal theories with extended chiral algebra. This is rather natural, because Nekrasov functions are introduced as expansion basis for generalized hypergeometric integrals, very similar to those which arise in expansion of Dotsenko-Fateev integrals in powers of alpha-parameters. Thus, the AGT conjecture is closely related to the old belief that conformal theory can be effectively described in …
We consider the AGT relation, expressing conformal blocks for the Virasoro and W-algebras in terms of Nekrasov's special functions, in the simplest case of the 4-point functions for the first non-trivial W_3 algebra. The standard set of Nekrasov functions is sufficient only if additional null-vector restriction is imposed on a half of the external $W$-primaries and this is just the case when the conformal blocks are fully dictated by W-symmetry and do not depend on a particular model. Explicit checks confirm that the AGT relation survives in this restricted case, as expected.
Recently Alday, Gaiotto and Tachikawa have proposed relation between 2- and 4-dimensional conformal field theories. The relation implies that the Nekrasov partition functions of N=2 superconformal gauge theories are equal to conformal blocks associated with the conformal algebra. Likewise, a counterpart in pure super Yang-Mills theory exists in conformal field theory. We propose a simple relation between the Shapovalov matrix of the W_3-algebra and the Nekrasov partition function of N=2 SU(3) Yang-Mills theory.
We propose an extension of the Alday-Gaiotto-Tachikawa-Wyllard conjecture to 5d SU(N) gauge theories. A Nekrasov partition function then coincides with the scalar product of the corresponding Gaiotto-Whittaker vectors of the q-deformed W_N algebra.
Using recursive relations satisfied by Nekrasov partition functions and by irregular conformal blocks we prove the AGT correspondence in the case of N=2 superconformal SU(2) quiver gauge theories with N_f = 0,1,2 antifundamental hypermultiplets
Given a 4d N=2 SUSY gauge theory, one can construct the Seiberg-Witten prepotentional, which involves a sum over instantons. Integrals over instanton moduli spaces require regularisation. For UV-finite theories the AGT conjecture favours particular, Nekrasov's way of regularization. It implies that Nekrasov's partition function equals conformal blocks in 2d theories with W_{N_c} chiral algebra. For $N_c=2$ and one adjoint multiplet it coincides with a torus 1-point Virasoro conformal block. We check the AGT relation between conformal dimension and adjoint multiplet's mass in this case and inve…
We extend the proof from arXiv:1012.3137, which interprets the AGT relation as the Hubbard-Stratonovich duality relation to the case of 5d gauge theories. This involves an additional q-deformation. Not surprisingly, the extension turns out to be trivial: it is enough to substitute all relevant numbers by q-numbers in all the formulas, Dotsenko-Fateev integrals by the Jackson sums and the Jack polynomials by the MacDonald ones. The problem with extra poles in individual Nekrasov functions continues to exist, therefore, such a proof works only for β= 1, i.e. for q=t in MacDonald's notation. For …
In the limit of large central charge $c$ the 4-point Virasoro conformal block becomes a hypergeometric function. It is represented by a sum of chiral Nekrasov functions, which can also be explicitly evaluated. In this way the known proof of the AGT relation is extended from special to generic set of external states, but in the special limit of c=\infty.
We use AGT correspondence between N=2 SUSY Yang-Mills theory on ${\mathbb R}^4/{\mathbb Z}_2$ and two-dimensional CFT model with the algebra $ {\cal H} \oplus \hat{sl}(2)_2 \oplus \text{NSR}$ to obtain the explicit expressions for 4-point NSR conformal blocks including Ramond fields in terms of Nekrasov partition functions and correlation functions of $\hat{sl}(2)_2$ WZW model.
The five dimensional AGT correspondence implies the connection between the q-deformed Virasoro block and the 5d Nekrasov partition function. In this paper, we determine a q-deformation of the four-point block in the Coulomb gas representation from the 5d Nekrasov function, and obtain an expression of the q-deformed vertex operator. If we use only one kind of the q-vertex operators, one of the insertion points of them must be modified in order to hold the 2d/5d correspondence.
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.
The AGT conjecture relates \mathcal{N}=2 4d SUSY gauge theories to 2d CFTs. Matrix model techniques can be used to investigate both sides of this relation. The large N limit refers here to the size of Young tableaux in the expression of the gauge theory partition function. It corresponds to the vanishing of Omega-background equivariant deformation parameters, and should not be confused with the t'Hooft expansion at large number of colors. In this paper, a saddle point approach is employed to study the Nekrasov-Shatashvili limit of the gauge theory, leading to define beta-deformed, or quantized…
Once famous and a little mysterious, AGT relations between Nekrasov functions and conformal blocks are now understood as the Hubbard-Stratanovich duality in the Dijkgraaf-Vafa (DV) phase of a peculiar Dotsenko-Fateev multi-logarithmic matrix model. However, it largely remains a collection of somewhat technical tricks, lacking a clear and generalizable conceptual interpretation. Our new claim is that the Nekrasov functions emerge in matrix models as a straightforward implication of superintegrability, factorization of peculiar matrix model averages. Recently, we demonstrated that, in the Gaussi…
A matrix model approach to proof of the AGT relation is briefly reviewed. It starts from the substitution of conformal blocks by the Dotsenko-Fateev beta-ensemble averages and Nekrasov functions by a double deformation of the exponentiated Seiberg-Witten prepotential in beta \neq 1 and g_s \neq 0 directions. Establishing the equality of these two quantities is a typical matrix model problem, and it presumably can be ascertained by investigation of integrability properties and developing an associated Harer-Zagier technique for evaluation of the exact resolvent.
The actual definition of the Nekrasov functions participating in the AGT relations implies a peculiar choice of contours in the LMNS and Dotsenko-Fateev integrals. Once made explicit and applied to the original triply-deformed (6-dimensional) version of these integrals, this approach reduces the AGT relations to symmetry in q_{1,2,3}, which is just an elementary identity for an appropriate choice of the integration contour (which is, however, a little non-traditional). We illustrate this idea with the simplest example of N=(1,1) U(1) SYM in six dimensions, however, all other cases can be evide…
The AGT conjecture identifying conformal blocks with the Nekrasov functions is investigated for the spherical conformal blocks with more than 4 external legs. The diagram technique which arises in conformal block calculation involves propagators and vertices. We evaluated vertices with two Virasoro algebra descendants and explicitly checked the AGT relation up to the third order of the expansion for the 5-point and 6-point conformal blocks on sphere confirming all the predictions of arXiv:0906.3219 relevant in this situation. We propose that U(1)-factor can be extracted from the matrix element…
We discuss an analog of the AGT relation in five dimensions. We define a q-deformation of the beta-ensemble which satisfies q-W constraint. We also show a relation with the Nekrasov partition function of 5D SU(N) gauge theory with N_f=2N.
This note deals with the five-dimensional pure SU(2) AGT conjecture proposed by Awata and Yamada. 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.
We give some evidences of the AGT-W relation between SU(3) quiver gauge theories and A_2 Toda theory. In particular, we derive the explicit form of 5-point correlation functions in the lower orders and confirm the agreement with Nekrasov's partition function for SU(3)xSU(3) quiver gauge theory. The algorithm to derive the correlation functions can be applied to general n-point function in A_2 Toda theory which will be useful to establish the relation for more generic quivers. Partial analysis is also given for SU(3)xSU(2) case and we comment on some technical issues which need clarification be…
A complete construction of the AFLT states is proposed. With this construction and for all the cases we have checked, the AGT conjecture on the equivalence of Nekrasov Instanton Counting (NIC) to the $Vir\oplus u(1)$ conformal block has been verified to be true.
An extension of the AGT relation from two to three dimensions begins from connecting the theory on domain wall between some two S-dual SYM models with the 3d Chern-Simons theory. The simplest kind of such a relation would presumably connect traces of the modular kernels in 2d conformal theory with knot invariants. Indeed, the both quantities are very similar, especially if represented as integrals of the products of quantum dilogarithm functions. However, there are also various differences, especially in the "conservation laws" for integration variables, which hold for the monodromy traces, bu…
We find generalized Jack polynomials for the group $SU(3)$ and verify that their Selberg averages for several first levels are given by Nekrasov functions. To compute the averages we derive recurrence relations for the $sl_3$ Selberg integrals.
Conformal blocks and their AGT relations to LMNS integrals and Nekrasov functions are best described by "conformal" (or Dotsenko-Fateev) matrix models, but in non-Gaussian Dijkgraaf-Vafa phases, where different eigenvalues are integrated along different contours. In such matrix models, the determinant representations and integrability are restored only after a peculiar Fourier transform in the numbers of integrations. From the point of view of conformal blocks, this is Fourier transform w.r.t. the intermediate dimensions and this explains why such quantities are expressed through tau-functions…
Recently, an intriguing family of the one-point toric conformal blocks AGT related to the $\mathcal{N}=2^*\,\, SU(2)$ Nekrasov functions was discovered by M. Beccaria and G. Macorini. Members of the family are distinguished by having only finite amount of poles as functions of the intermediate dimension/v.e.v. in gauge theory. Another remarkable property is that these conformal blocks/Nekrasov functions can be found in closed form to all orders in the coupling expansion. In the present paper we use Zamolodchikov's recurrence equation to systematically account for these exceptional conformal bl…
It is shown that the deformed Seiberg-Witten curve equation after Fourier transform is mapped into a differential equation for the AGT dual 2d CFT cnformal block containing an extra completely degenerate field. We carefully match parameters in two sides of duality thus providing not only a simple independent prove of the AGT correspondence in Nekrasov-Shatashvili limit, but also an extension of AGT to the case when a secondary field is included in the CFT conformal block. Implications of our results in the study of monodromy problems for a large class of $n$'th order Fuchsian differential equa…
We compute the light asymptotic limit of $A_{n-1}$ Toda conformal blocks by using the AGT correspondence. We show that for certain class of CFT blocks the corresponding Nekrasov partition functions in this limit are simplified drastically being represented as a sum of a restricted class of Young diagrams. In the particular case of $A_{2}$ Toda we also compute the corresponding conformal blocks using conventional CFT techniques finding a perfect agreement with the results obtained from the Nekrasov partition functions.
We study five dimensional AGT correspondence by means of the q-deformed beta-ensemble technique. We provide a special basis of states in the q-deformed CFT Hilbert space consisting of generalized Macdonald polynomials, derive the loop equations for the beta-ensemble and obtain the factorization formulas for the corresponding matrix elements. We prove the spectral duality for Nekrasov functions and discuss its meaning for conformal blocks. We also clarify the relation between topological strings and q-Liouville vertex operators.
Operator product expansion (OPE) of two operators in two-dimensional conformal field theory includes a sum over Virasoro descendants of other operator with universal coefficients, dictated exclusively by properties of the Virasoro algebra and independent of choice of the particular conformal model. In the free field model, these coefficients arise only with a special "conservation" relation imposed on the three dimensions of the operators involved in OPE. We demonstrate that the coefficients for the three unconstrained dimensions arise in the free field formalism when additional Dotsenko-Fatee…
The Seiberg-Witten prepotentials for N=2 SUSY gauge theories with N_f<2N_c fundamental multiplets are obtained from conformal N_f=2N_c theory by decoupling 2N_c-N_f multiplets of heavy matter. This procedure can be lifted to the level of Nekrasov functions with arbitrary background parameters epsilon_1 and epsilon_2. The AGT relations imply that similar limit exists for conformal blocks (or, for generic N_c>2, for the blocks in conformal theories with W_{N_c} chiral algebra). We consider the limit of the four-point function explicitly in the Virasoro case of N_c=2, by bringing the dimensions o…
We prove the connection between the Nekrasov partition function of N=2 super-symmetric U(2) gauge theory with adjoint matter and conformal blocks for the Virasoro algebra, as predicted by the Alday-Gaiotto-Tachikawa relations. Mathematically, this is achieved by relating the Carlsson-Okounkov Ext vector bundle on the moduli space of rank 2 sheaves with Liouville vertex operators.Our approach is geometric in nature, and uses a new method for intersection-theoretic computations of the Ext operator.
In these notes we consider integrable structure of the conformal field theory with the algebra of symmetries $\mathcal{A}=W_{n}\otimes H$, where $W_{n}$ is $W-$algebra and $H$ is Heisenberg algebra. We found the system of commuting Integrals of Motion with relatively simple properties. In particular, this system has very simple spectrum and the matrix elements of special primary operators between its eigenstates have nice factorized form coinciding exactly with the contribution of the bifundamental multiplet to the Nekrasov partition function for U(n) gauge theories.
An intriguing coincidence between the partition function of super Yang-Mills theory and correlation functions of 2d Toda system has been heavily studied recently. While the partition function of gauge theory was explored by Nekrasov, the correlation functions of Toda equation have not been completely understood. In this paper, we study the latter in the form of Dotsenko-Fateev integral and reduce it in the form of Selberg integral of several Jack polynomials. We conjecture a formula for such Selberg average which satisfies some consistency conditions and show that it reproduces the SU(N) versi…
AGT conjecture connects Nekrasov instanton partition function of 4D quiver gauge theory with 2D Liouville conformal blocks. We re-investigate this connection using the central extension of spherical Hecke algebra in q-coordinate representation, q being the instanton expansion parameter. Based on AFLT basis together with interwiners we construct gauge conformal state and demonstrate its equivalence to the Liouville conformal state, with careful attention to the proper scaling behavior of the state. Using the colliding limit of regular states, we obtain the formal expression of irregular conform…
This note announces results on the relations between the approach of Beilinson and Drinfeld to the geometric Langlands correspondence based on conformal field theory, the approach of Kapustin and Witten based on $N=4$ SYM, and the AGT-correspondence. The geometric Langlands correspondence is described as the Nekrasov-Shatashvili limit of a generalisation of the AGT-correspondence in the presence of surface operators. Following the approaches of Kapustin - Witten and Nekrasov - Witten we interpret some aspects of the resulting picture using an effective description in terms of two-dimensional s…
We present a summary of current knowledge about the AGT relations for conformal blocks with additional insertion of the simplest degenerate operator, and a special choice of the corresponding intermediate dimension, when the conformal blocks satisfy hypergeometric-type differential equations in position of the degenerate operator. A special attention is devoted to representation of conformal block through the beta-ensemble resolvents and to its asymptotics in the limit of large dimensions (both external and intermediate) taken asymmetrically in terms of the deformation epsilon-parameters. The …
We study the representation theory of the Ding-Iohara algebra $\calU$ to find $q$-analogues of the Alday-Gaiotto-Tachikawa (AGT) relations. We introduce the endomorphism $T(u,v)$ of the Ding-Iohara algebra, having two parameters $u$ and $v$. We define the vertex operator $Φ(w)$ by specifying the permutation relations with the Ding-Iohara generators $x^\pm(z)$ and $ψ^\pm(z)$ in terms of $T(u,v)$. For the level one representation, all the matrix elements of the vertex operators with respect to the Macdonald polynomials are factorized and written in terms of the Nekrasov factors for the $K$-theor…
We describe relationships between integrable systems with N degrees of freedom arising from the AGT conjecture. Namely, we prove the equivalence (spectral duality) between the N-cite Heisenberg spin chain and a reduced gl(N) Gaudin model both at classical and quantum level. The former one appears on the gauge theory side of the AGT relation in the Nekrasov-Shatashvili (and further the Seiberg-Witten) limit while the latter one is natural on the CFT side. At the classical level, the duality transformation relates the Seiberg-Witten differentials and spectral curves via a bispectral involution. …
AGT correspondence gives an explicit expressions for the conformal blocks of $d=2$ conformal field theory. Recently an explanation of this representation inside the CFT framework was given through the assumption about the existence of the special orthogonal basis in the module of algebra $\mathcal{A}=Vir\otimes\mathcal{H}$. The basis vectors are the eigenvectors of the infinite set of commuting integrals of motion. It was also proven that some of these vectors take form of Jack polynomials. In this note we conjecture and verify by explicit computations that in the case of the Virasoro central …
AGT relations imply that the four-point conformal block admits a decomposition into a sum over pairs of Young diagrams of essentially rational Nekrasov functions - this is immediately seen when conformal block is represented in the form of a matrix model. However, the q-deformation of the same block has a deeper decomposition - into a sum over a quadruple of Young diagrams of a product of four topological vertices. We analyze the interplay between these two decompositions, their properties and their generalization to multi-point conformal blocks. In the latter case we explain how Dotsenko-Fate…
We study an analog of the AGT relation in five dimensions. We conjecture that the instanton partition function of 5D N=1 pure SU(2) gauge theory coincides with the inner product of the Gaiotto-like state in the deformed Virasoro algebra. In four dimensional case, a relation between the Gaiotto construction and the theory of Braverman and Etingof is also discussed.
Original proofs of the AGT relations with the help of the Hubbard-Stratanovich duality of the modified Dotsenko-Fateev matrix model did not work for beta different from one, because Nekrasov functions were not properly reproduced by Selberg-Kadell integrals of Jack polynomials. We demonstrate that if the generalized Jack polynomials, depending on the N-ples of Young diagrams from the very beginning, are used instead of the N-linear combinations of ordinary Jacks, this resolves the problem. Such polynomials naturally arise as special elements in the equivariant cohomologies of the GL(N)-instant…
Let ${\mathcal B}^{\, p, \, p^{\prime}, \, {\mathcal H}}_{N, n}$ be a conformal block, with $n$ consecutive channels $χ_ı$, $ı= 1, \cdots, n$, in the conformal field theory $\mathcal{M}^{\, p, \, p^{\prime}}_N \! \times \! \mathcal{M}^{\mathcal{H}}$, where $\mathcal{M}^{\, p, \, p^{\prime}}_N$ is a $\mathcal{W}_N$ minimal model, generated by chiral fields of spin $1, \cdots, N$, and labeled by two co-prime integers $p$ and $p^{\prime}$, $1 < p < p^{\prime}$, while $\mathcal{M}^{\mathcal{H}}$ is a free boson conformal field theory. $\mathcal{B}^{\, p, \, p^{\prime}, \mathcal{H}}_{N, n}$ is the …
Recently exact formulas were provided for partition function of conformal (multi-Penner) beta-ensemble in the Dijkgraaf-Vafa phase, which, if interpreted as Dotsenko-Fateev correlator of screenings and analytically continued in the number of screening insertions, represents generic Virasoro conformal blocks. Actually these formulas describe the lowest terms of the q_a-expansion, where q_a parameterize the shape of the Penner potential, and are exact in the filling numbers N_a. At the same time, the older theory of CIV-DV prepotential, straightforwardly extended to arbitrary beta and to non-pol…
We investigate the duality between the Nekrasov function and the quantized Seiberg-Witten prepotential, first guessed in [1] and further elaborated in [2] and [3]. We concentrate on providing more thorough checks than the ones presented in [3] and do not discuss the motivation and historical context of this duality. The check of the conjecture up to $o (\hbar^6, \ln (Λ))$ is done by hands for arbitrary $N$ (explicit formulas are presented). Moreover, details of the calculation that are essential for the computerization of the check are worked out. This allows us to test the conjecture up to $\…
We give a concise summary of the impressive recent development unifying a number of different fundamental subjects. The quiver Nekrasov functions (generalized hypergeometric series) form a full basis for all conformal blocks of the Virasoro algebra and are sufficient to provide the same for some (special) conformal blocks of W-algebras. They can be described in terms of Seiberg-Witten theory, with the SW differential given by the 1-point resolvent in the DV phase of the quiver (discrete or conformal) matrix model (β-ensemble), dS = ydz + O(ε^2) = \sum_p ε^{2p} ρ_β^{(p|1)}(z), where εand βare r…
We study N=2 super-conformal field theories in four dimensions that correspond to mass-deformed linear quivers with n gauge groups and (bi-)fundamental matter. We describe them using Seiberg-Witten curves obtained from an M-theory construction and via the AGT correspondence. We take particular care in obtaining the detailed relation between the parameters appearing in these descriptions and the physical quantities of the quiver gauge theories. This precise map allows us to efficiently reconstruct the non-perturbative prepotential that encodes the effective IR properties of these theories. We g…
Proof of the AGT Conjecture at Generic $β$
该研究通过证明广义Jack多项式Selberg平均的全阶因子化公式，结合广义Cauchy恒等式，完成一般β下带四个基本超多重态的四点SU(2)AGT对应关系的全阶证明，将双分划项与Nekrasov不动点贡献对应。
The Nekrasov conjecture predicts a relation between the partition function for N=2 supersymmetric Yang-Mills theory and the Seiberg-Witten prepotential. For instantons on R^4, the conjecture was proved, independently and using different methods, by Nekrasov-Okounkov, Nakajima-Yoshioka, and Braverman-Etingof. We prove a generalized version of the conjecture for instantons on noncompact toric surfaces.
On the convergence of Nekrasov functions
本文研究四维 N=2 规范理论中 Nekrasov 配分函数的收敛性，证明共形理论有有限收敛半径、渐近自由理论有无限收敛半径，并推广到 W_N 代数共形块和 Painlevé τ-函数。
We study refined B-model via the beta ensemble of matrix models. Especially, for four dimensional N=2 SU(2) supersymmetric gauge theories with N_f=0,1 and 2 fundamental flavors, we discuss the correspondence between deformed disk amplitudes on each Seiberg-Witten curve and the Nekrasov-Shatashvili limit of the corresponding irregular one point block of a degenerate operator via the AGT correspondence. We also discuss the relation between deformed annulus amplitudes and the irregular two point block of the degenerate operator, and check a desired agreement for N_f=0 and 1 cases.
We consider the AGT correspondence in the context of the conformal field theory $M^{\, p, p^{\prime}}$ $\otimes$ $M^{H}$, where $M^{\, p, p^{\prime}}$ is the minimal model based on the Virasoro algebra $V^{\, p, p^{\prime}}$ labeled by two co-prime integers $\{p, p^{\prime}\}$, $1 < p < p^{\prime}$, and $M^{H}$ is the free boson theory based on the Heisenberg algebra $H$. Using Nekrasov's instanton partition functions without modification to compute conformal blocks in $M^{\, p, p^{\prime}}$ $\otimes$ $M^{H}$ leads to ill-defined or incorrect expressions. Let $B^{\, p, p^{\prime}, H}_n$ be a c…
5D AGT conjecture for circular quivers
本文通过自由场形式将环形量子图的5维AGT猜想与顶点算子积分表示等价，验证了泛化和退化情况下的一致性。
In this thesis, we obtain the formula for the Kac determinant of the algebra arising from the level $N$ representation of the Ding-Iohara-Miki algebra. This formula can be proved by decomposing the level $N$ representation into the deformed $W$-algebra part and the $U(1)$ boson part, and using the screening currents of the deformed $W$-algebra. It is also discovered that singular vectors obtained by its screening currents correspond to the generalized Macdonald functions. Moreover, we investigate the $q \rightarrow 0$ limit of five-dimensional AGT correspondence. In this limit, the simplest 5D…
In this note, using Nekrasov's gauge origami framework, we study two different versions of the the BPS/CFT correspondence - first, the standard AGT duality and, second, the quiver W algebra construction which has been developed recently by Kimura and Pestun. The gauge origami enables us to work with both dualities simultaneously and find exact matchings between the parameters. In our main example of an A-type quiver gauge theory, we show that the corresponding quiver qW-algebra and its representations are closely related to a large-n limit of spherical gl(n) double affine Hecke algebra whose m…
It was recently conjectured that the AGT correspondence between the $U(r)$-instanton counting on $\mathbb R^4/\mathbb Z_p$ and the two-dimensional field theories with the conformal symmetry algebra $\mathcal A(r,p)$ can be considered as a root of unity limit of its K-theoretic analogue. From this point of view, the algebra $\mathcal A(r,p)$ and a special basis in its representation are limits of the Ding-Iohara algebra and the Macdonald polynomials respectively. In this paper we confirm this conjecture for the special case $r=1$. We uncover the implicit $\mathcal A(1,p)$ symmetry in this limit…
We give some evidences which imply that W(1+infinity) algebra describes the symmetry behind AGT(-W) conjecture: a correspondence between the partition function of N=2 supersymmetric quiver gauge theories and the correlators of Liouville (Toda) field theory.
In their recent paper \cite{Alday:2009aq} Alday, Gaiotto and Tachikawa proposed a relation between $\mathcal{N}=2$ four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories. As part of their conjecture they gave an explicit combinatorial formula for the expansion of the conformal blocks inspired from the exact form of instanton part of the Nekrasov partition function. In this paper we study the origin of such an expansion from a CFT point of view. We consider the algebra $\mathcal{A}=\text{\sf Vir}\otimes\mathcal{H}$ which is the tensor product of mutually com…
As anticipated in [1], elaborated in [2-4], and explicitly formulated in [5], the Dotsenko-Fateev integral discriminant coincides with conformal blocks, thus providing an elegant approach to the AGT conjecture, without any reference to an auxiliary subject of Nekrasov functions. Internal dimensions of conformal blocks in this identification are associated with the choice of contours: parameters of the DV phase of the corresponding matrix models. In this paper we provide further evidence in support of this identity for the 6-parametric family of the 4-point spherical conformal blocks, up to lev…
Highest-weight vectors and three-point functions in GKO coset decomposition
本文重新审视GKO余构造，推导了最高权向量公式、范数及顶点算符矩阵元，导出的共形块关系经AGT对应等价于带表面缺陷的Nekrasov配分函数blowup关系，可用于证明Painlevé tau-函数的Kyiv公式。
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