SciPost Phys. 16, 155 (2024) ·
published 19 June 2024

· pdf
The $1/N$ expansion of matrix models is asymptotic, and it requires nonperturbative corrections due to large $N$ instantons. Explicit expressions for large $N$ instanton amplitudes are known in the case of Hermitian matrix models with one cut, but not in the multicut case. We show that the recent exact results on topological string instanton amplitudes provide the nonperturbative contributions of large $N$ instantons in generic multicut, Hermitian matrix models. We present a detailed test in the case of the cubic matrix model by considering the asymptotics of its $1/N$ expansion, which we obtain at relatively high genus for a generic twocut background. These results can be extended to certain nonconventional matrix models which admit a topological string theory description. As an application, we determine the large $N$ instanton corrections for the free energy of ABJM theory on the threesphere, which correspond to Dbrane instanton corrections in superstring theory. We also illustrate the applications of topological string instantons in a more mathematical setting by considering orbifold GromovWitten invariants. By focusing on the example of ${\mathbb C}^3/{\mathbb Z}_3$, we show that they grow doublyfactorially with the genus and we obtain and test explicit asymptotic formulae for them.
SciPost Phys. 15, 184 (2023) ·
published 3 November 2023

· pdf
Some sigma models which admit a theta angle are integrable at both $\vartheta=0$ and $\vartheta=\pi$. This includes the wellknown $O(3)$ sigma model and two families of coset sigma models studied by Fendley. We consider the ground state energy of these models in the presence of a magnetic field, which can be computed with the Bethe Ansatz. We obtain explicit results for its nonperturbative corrections and we study the effect of the theta angle on them. We show that imaginary, exponentially small corrections due to renormalons remain unchanged, while instanton corrections change sign, as expected. We find in addition corrections due to renormalons which also change sign as we turn on the theta angle. Based on these results we present an explicit nonperturbative formula for the topological susceptibility of the $O(3)$ sigma model in the presence of a magnetic field, in the weak coupling limit.
Dr Miravitllas: "We thank the referee for their..."
in Submissions  report on Large N instantons from topological strings