Check of AGT relation for conformal blocks on sphere

Abstract

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 elements of the free field vertex operators. We studied the n−point case, and found out that our results confirm the AGT conjecture up to the third order expansions.

Generalized matrix models and AGT correspondence at all genera

“…The relation between the parameters t k and the gauge couplings [63] (See also [18]) is the same as the one defined in (3.9) with t 0 = 0. It follows from (3.11) that…”

“…This proposal has been a subject of intensive investigations and refinements from different viewpoints. Evidence for this conjecture as well as complete proofs for some cases can be found in [6][7][8][9][10][11][12][13][14][15][16][17][18]. Extensions to higher rank gauge groups and Toda field theories were introduced and discussed in [19][20][21][22][23][24][25].…”

Notes on 3-point functions of A N−1 Toda theory and AGT-W relation for SU(N) quiver

“…Now the correspondence of parameters is clearly understood, then the only unclear point is so-called 'U (1) factor' in AGT-W relation [2]. At this moment, there seems to be no consensus among researchers with regard to the way of determining this factor, while some researchers propose that this factor is nothing but the free string amplitude [6].…”

“…The most basic way is to calculate the inverse Shapovalov matrix of each level as it was done in [6][7][8], but it must be not a realistic way if we want to calculate in an arbitrary high level. One choice is the calculation by Dotsenko-Fateev method, which has been recently discussed also in the context of AGT-W relation [11][12][13].…”

“…Up to now, the proof by direct calculations has been done in the following cases: For SU (2) linear quivers, the correspondence has been checked for SU (2) n quiver with n = 1, 2, 3 up to instanton level 3 [6]. For SU (3) linear quivers, it has been checked for SU (3) n quiver with n = 1, 2 and SU (3) × SU (2) quiver up to instanton level 3 [5,7,8].…”