Instanton counting, Macdonald function and the moduli space ofD-branes

Abstract

We argue the connection of Nekrasov's partition function in the Ω background and the moduli space of D-branes, suggested by the idea of geometric engineering and Gopakumar-Vafa invariants. In the instanton expansion of N = 2 SU (2) YangMills theory the Nakrasov's partition function with equivariant parameters ǫ 1 , ǫ 2 of toric action on C 2 factorizes correctly as the character of SU (2) L × SU (2) R spin representation. We show that up to two instantons the spin contents are consistent with the Lefschetz action on the moduli space of D2-branes on (local) F 0 . We also present an attempt at constructing a refined topological vertex in terms of the Macdonald function. The refined topological vertex with two parameters of T 2 action allows us to obtain the generating functions of equivariant χ y and elliptic genera of the Hilbert scheme of n points on C 2 by the method of topological vertex.

Mass-deformed T N as a linear quiver

“…For example, the web diagram corresponding to the 5d T N theory in figure 1 specifies a blow up of C 3 /Z N × Z N [7]. In this formulation, 5d BPS states come from M2-branes wrapping various two-cycles inside the toric Calabi-Yau threefold [21], and their index can be computed by the (refined) topological vertex [22][23][24][25], which can often be regarded as the 5d Nekrasov partition function of the corresponding 5d gauge theory [26][27][28][29]. This is not the end of the story, however.…”

“…More precisely, our scaling limit is 24) and then take → 0. The scaling of x is determined so that the Seiberg-Witten differential λ = xdz is fixed.…”

Topological vertex for Higgsed 5d T N theories

“…Since the T N theory does not admit a Lagrangian description, 2 it is difficult to perform the localisation computation, initiated in [16][17][18], to obtain its partition function. However, the refined topological vertex [19,20] provides a powerful tool to compute the partition function of the T N theory because its web diagram is dual to a toric Calabi-Yau threefold. After the removal of some extra factors independent of the Coulomb branch moduli it was possible to compute the partition function of the T N theory [21,22].…”

“…be computed by the powerful technique of the refined topological vertex [19,20] after eliminating what we call decoupled factor which is a contribution associated to strings between parallel external legs [21,22,25]. Technical details of the refined topological vertex as well as the decoupled factor are summarised in appendix A.…”

“…Although it is difficult to compute the higher genus topological string amplitudes directly, the M-theory interpretation of the topological string amplitude enables us to write down the all genus answer [56,57], Moreover, if the Calabi-Yau threefold is toric, the refined topological vertex [19,20] can compute the refined version of the all genus topological string partition function up to constant map contributions in a diagrammatic way. We call such a partition function the refined topological string partition function.…”