Holomorphic anomaly and matrix models

TL;DR

This paper demonstrates that non-holomorphic amplitudes in matrix models satisfy holomorphic anomaly equations, supporting the Dijkgraaf--Vafa conjecture linking matrix models to topological string theory on Calabi--Yau threefolds.

Contribution

It establishes the holomorphic anomaly equations for matrix models' amplitudes, including open string sectors, providing evidence for the Dijkgraaf--Vafa conjecture at all genera.

Findings

Non-holomorphic amplitudes are modular invariant and depend on the spectral curve geometry.

Holomorphic anomaly equations are satisfied by these amplitudes.

Results support the matrix model and topological string theory correspondence.

Abstract

The genus g free energies of matrix models can be promoted to modular invariant, non-holomorphic amplitudes which only depend on the geometry of the classical spectral curve. We show that these non-holomorphic amplitudes satisfy the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vafa. We derive as well holomorphic anomaly equations for the open string sector. These results provide evidence at all genera for the Dijkgraaf--Vafa conjecture relating matrix models to type B topological strings on certain local Calabi--Yau threefolds.