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Recently, I've mostly focussed on trying to understand four-dimensional N=2 field theories in the 1/2 Omega-background, and their relation to the Hitchin integrable system. In the resulting picture we may associate to such a theory a generalized partition function that depends on a choice of Coulomb vacuum as well as a phase. This partition function may be computed as a generating function of opers and formulated through the exact WKB analysis as a Borel sum. It encodes as its jumps the 4d BPS particles and transforms as a section of a distinguished "classical Chern-Simons" line bundle. I'm very interested in extending this picture to extended objects as well as to five-dimensional N=1 field theories dual to topological string theories. This is closely related to works of for instance Gaiotto, Moore, Neitzke, Marino, Grassi, Nekrasov, Shatashvili, Alim, Bridgeland, Teschner, Pioline etc.
Below is a selection of a few recent publications with some additional explanations:
A geometric recipe for twisted superpotentials
Lotte Hollands, Philipp Ruter and Richard J. Szabo
JHEP 12 (2021) 164, arXiv:2109.14699, 102 pages
In this paper we study some aspects of 4d N=2 theories of class S and their relation to the Hitchin integrable system. After introducing the 1/2 Omega background, spectral networks and abelianization, we explain how the partition function of the N=2 theory in the 1/2 Omega background may be formulated in terms of the exact WKB analysis and computed as a generating function of opers.
Here is an online talk given in 2021 at the Western Hemisphere Colloquium on Geometry and Physics about this paper.
Quantum curves, resurgence and exact WKB
Murad Alim, Lotte Hollands and Ivan Tulli
arXiv:2203.08249, 94 pages
In this paper we consider the 5d N=1 theory corresponding to the resolved conifold geometry in the 1/2 Omega background. We show that the Borel sum of its free energy encodes the 5d BPS particles of the N=1 theory in its jumps and defines a section of a the "classical Chern-Simons" line bundle. We furthermore formulate a new spectral problem corresponding to the resolved conifold geometry.
Here is an online talk given at the QFT for Mathematicians workshop 2022 explaining some ideas in this paper.
Exact WKB and abelianization for the T3 equation
Lotte Hollands and Andrew Neitzke
Commun. Math. Phys. 380 (2020) 1, 131-186, arXiv:1906.04271, 68 pages
This is the paper in which we explain how our abelianization method gives a novel perspective on the exact WKB method and new insights into spectral problems. We reconsider the well-known Mathieu equation from this geometric perspective and apply the technique to the T3 equation, an ordinary differential equation of degree 3 on the 3-punctured sphere, to define and study a very interesting new spectral problem.
This paper is the main subject of this talk at String-Math 2021.
All of my other preprints can be found here.
Below you find my own master thesis and PhD thesis:
My PhD thesis on topological strings and quantum curves. This thesis is based on four papers written during my PhD on topics in topological string theory and string compactifications, with a central role for Riemann surfaces. One of the main ideas is a series of dualities connecting topological string theory on a non-compact Calabi-Yau threefold modeled on a Riemann surface to a system of free fermions on a quantization of this Riemann surface.
My master thesis on topological string theory and its mathematical formulation, called Gromov-Witten theory. It is a review on various concepts such as moduli spaces, equivariant localization, topological field theory and sigma models, and discusses some simple examples of Gromov-Witten invariants and topological string correlators.
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