Holomorphic anomaly and matrix models
Bertrand Eynard, Marcos Marino, Nicolas Orantin
TL;DR
The paper demonstrates that matrix-model genus-$g$ free energies can be augmented to non-holomorphic, modular-invariant amplitudes $F^{(g)}(t,\bar t)$ that depend only on the classical spectral curve and satisfy the BCOV holomorphic anomaly equations. Building on the Eynard–Orantin recursion for arbitrary spectral curves, it provides both a direct antiholomorphic-variation proof and a combinatorial κ-expansion proof to show that these amplitudes obey BCOV-type relations, and derives holomorphic anomaly equations for open-string sectors in the local CY setting. This furnishes consistent, genus-by-genus evidence for the Dijkgraaf–Vafa conjecture, linking matrix-model free energies to type B topological strings on specific local Calabi–Yau manifolds, and it suggests deep connections to Seiberg–Witten/Nekrasov gravitational couplings via the same recursive framework. The work also clarifies how modularity is maintained in the non-holomorphic completion and offers tools for exploring background independence and wavefunction-like behavior of topological-string partition functions in matrix-model contexts.
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.