Nekrasov and Argyres–Douglas theories in spherical Hecke algebra representation

Abstract

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 conformal states corresponding to Argyres-Douglas theory, which involves summation of functions over Young diagrams.1 the simultaneous eigenvector of two generators L 1 and L n in [8]. In addition, Virasoro irregular state of higher rank n (simultaneous eigenstate of L k with n ≤ k ≤ 2n) is suggested in [9], while some of the coefficients for the representation are not fixed.Colliding limit, a limiting procedure to obtain the irregular state from the regular state is clarified in [10]. The decoupling limit in [7] and also in the matrix model [11] is a special case of the colliding limit. The colliding limit turns out to be a very efficient tool to investigate the irregular state to find the correct representation of the irregular state of rank greater than 1. Indeed, the coefficients undetermined in [9] are fixed by irregular matrix model in [12] which obeys consistency conditions of Virasoro generators of lower positive modes L k with 0 ≤ k < n. The irregular matrix model analysis is extended to W-symmetry in [13].The irregular matrix model analysis, however, provides indirect information because the partition function of the matrix model is equivalent to the inner-product of two states. Direct process to find the irregular state is more desirable. For this goal, we resort to the representation of spherical double degenerate affine Hecke algebra (spherical DDAHA or SH for short). DDAHA is generated by z i and D i = z i ∇ i + j<i σ ij (i = 1, · · · , N ) where ∇ i is the Dunkl operator and σ ij is the transposition of z i to z j . Spherical DDAHA (SH) is restricted to the symmetric part of product of z i 's and D i 's. SH also allows central extension, which is considered in this text and is still denoted as SH instead of SH c for simplicity. More details refer to [14]. The algebraic elements and their commutation relations are given in section 2.In this paper, we elaborate and generalize the procedure presented in [15] to irregular Virasoro state of arbitrary rank m using SH algebra based on AFLT orthonormal basis [16] and interwiners [17,18] . For this purpose we construct the regular conformal state |T m in q-coordinates (q being the instanton expansion parameter), which is the counter-part of Liouville conformal state |R m used in [10]. The equivalence relation is manifest after the proper scaling behavior is compensated. After this, one can find the irregular state |I m of rank m using the colliding limit. This paper is organi...

Argyres-Douglas theories, S-duality and AGT correspondence

“…With this modified charge lattice, the minimal electric charge is now half the charge of fundamental quark, 27 which is consistent with our T-transformation (6.2).…”

“…26 The "magnetic charge" here is simply associated with the SU(2) vector multiplet without mixing with those arising from the (A1, D4) sectors. 27 This can be seen from the discussions around eq. (2.64) of [8], where deg X and rank X are respectively the (modified) electric and magnetic charges naturally acted on by PSL(2, Z).…”

“…While the partition functions of conformally gauged AD theories have not been evaluated, there exists a series of non-conformally gauged AD theories whose partition functions were evaluated via a generalization [17,18] of the AGT correspondence [19,20] (See [21][22][23][24][25][26][27][28][29][30] for recent developments on this generalization). In particular, for the theory described by the quiver in figure 2, the partition function was evaluated as the inner product of so-called "irregular states" of Virasoro algebra.…”

Argyres-Douglas Theories, S-duality and AGT Correspondence

“…The curve correctly reduces to the curve of the IR theory (4.17) (for n = 2) in the limit M, q → ∞ with (5.39) kept finite. 24 Note here that, by standard arguments, the dynamical scale Λ of the mass-deformed theory and the gauge coupling e…”

“…While the partition functions of conformally gauged AD theories have not been evaluated, there exists a series of non-conformally gauged AD theories whose partition functions were evaluated via a generalization [14,15] of the AGT correspondence [16,17] (See [18][19][20][21][22][23][24][25][26][27] (A 1 , D 2n )…”