Non-Perturbative Nekrasov Partition Function from String Theory

TL;DR

The paper addresses how to obtain the full Omega-deformed instanton partition function from string theory. It extends a Type I setup on K3×T^2 with D5/D9 branes by including a self-dual background to realize the complete Omega-deformation in the ADHM action. Through disc amplitudes and an auxiliary-field linearisation, the authors reproduce the refined, non-perturbative ADHM action and derive the Nekrasov partition function non-perturbatively. The results support a string-worldsheet description of the refined topological string and suggest broader connections to refinement mechanisms in string theory and related backgrounds.

Abstract

We calculate gauge instanton corrections to a class of higher derivative string effective couplings introduced in [1]. We work in Type I string theory compactified on K3xT2 and realise gauge instantons in terms of D5-branes wrapping the internal space. In the field theory limit we reproduce the deformed ADHM action on a general Ω-background from which one can compute the non-perturbative gauge theory partition function using localisation. This is a non-perturbative extension of [1] and provides further evidence for our proposal of a string theory realisation of the Ω-background.

Figures (4)

• Figure 1: Three-point disc diagrams with graviphoton bulk-insertion. Diagram (a) involves two boundary insertions from the 5-5 sector, whereas diagram (b) two insertions from the 5-9 sector. While the whole boundary of diagram (a) lies on the D5-branes, diagram (b) lies partly on the D9- and partly on the D5-branes. Notice that the latter mixed boundary conditions appear only in the space-time directions.
• Figure 2: Diagrammatic representation of the factorisation limit relating the correlator of physical operators with the one using an auxiliary field.
• Figure 3: Four-point disc diagram with bulk-insertion of the $\bar{S}'$ field strength tensor and three boundary insertions stemming from the 5-5 sector of the string setup.
• Figure 4: Factorisation channels of the disc. The diagram at the top illustrates the four-point function of physical vertices with an insertion of a PCO. The diagrams at the bottom depict various choices for the position of the latter. In case (1), the result takes the form of a contact term, leading to an effective auxiliary field vertex operator insertion on the boundary of the disc. Similarly, case (2) can be re-expressed as a sum of two contact terms, while in case (3) no such interpretation is possible.