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# Advances in Theoretical and Mathematical Physics

## Volume 20 (2016)

### Number 2

### Twistorial topological strings and a $\mathrm{tt}^*$ geometry for $\mathcal{N} = 2$ theories in $4d$

Pages: 193 – 312

DOI: https://dx.doi.org/10.4310/ATMP.2016.v20.n2.a1

#### Authors

#### Abstract

We define twistorial topological strings by considering $\mathrm{tt}^*$ geometry of the 4d $\mathcal{N} = 2$ supersymmetric theories on the Nekrasov–Shatashvili $\frac{1}{2} \Omega$ background, which leads to quantization of the associated hyperKähler geometries. We show that in one limit it reduces to the refined topological string amplitude. In another limit it is a solution to a quantum Riemann–Hilbert problem involving quantum Kontsevich–Soibelman operators. In a further limit it encodes the hyperKähler integrable systems studied by GMN. In the context of AGT conjecture, this perspective leads to a twistorial extension of Toda. The 2d index of the $\frac{1}{2} \Omega$ theory leads to the recently introduced index for $\mathcal{N} = 2$ theories in 4d. The twistorial topological string can alternatively be viewed, using the work of Nekrasov–Witten, as studying the vacuum geometry of 4d $\mathcal{N} = 2$ supersymmetric theories on $T^2 \times I$ where $I$ is an interval with specific boundary conditions at the two ends.

Published 16 August 2016