tt* Geometry and a Twistorial Extension of Topological Strings

TL;DR

This paper extends tt* geometry to higher dimensions and introduces a twistorial framework that unifies topological strings, Nekrasov functions, and hyperKahler geometry, revealing new structures in physical amplitudes.

Contribution

It proposes a novel twistorial extension of topological strings and Nekrasov functions, unifying various geometric and physical frameworks in a single approach.

Findings

Amplitudes extend to twistor space with physical region at the equator.

Line operator amplitudes lead to difference equations for twistorial topological strings.

Unification of tt* geometry, topological strings, and hyperKahler geometry.

Abstract

We extend the recent study of 3d tt* geometry to 5d (4d) N=1 (N=2) supersymmetric theories. We show how the amplitudes of tt* geometry lead to an extension of topological strings and Nekrasov partition functions to twistor space, where the north/south poles of the twistor sphere lead to topological/anti-topological string amplitudes. However, the equator of the twistor sphere is the physical region for the amplitudes, where the unitarity of physical amplitudes is respected. We propose that the amplitudes for line operators studied by Gaiotto, Moore and Neitzke lead to difference equations for the twistorial topological string partition function, extending the familiar one for the monodromy of Branes in the context of the standard topological strings. This setup unifies tt* geometry, topological strings and the hyperKahler geometry formulated by GMN into a single framework.