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

Authors

Sergio Cecotti (International School of Advanced Studies (SISSA), Trieste, Italy)

Andrew Neitzke (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)

Cumrun Vafa (Jefferson Physical Laboratory, Harvard University, Cambridge, Massachusetts, U.S.A.)

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.

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