Abstract
We investigate the analogy between the large N expansion in normal matrix models and the asymptotic expansion of the determinant of the Hilb map, appearing in the study of critical metrics on complex manifolds via projective embeddings. This analogy helps to understand the geometric meaning of the expansion of matrix model free energy and its relation to gravitational effective actions in two dimensions. We compute the leading terms of the free energy expansion in the pure bulk case, and make some observations on the structure of the expansion to all orders. As an application of these results, we propose an asymptotic formula for the Liouville action, restricted to the space of the Bergman metrics.
Highlights
Hand is implicit already in the earlier work by Donaldson [15] on projective embeddings
We investigate the analogy between the large N expansion in normal matrix models and the asymptotic expansion of the determinant of the Hilb map, appearing in the study of critical metrics on complex manifolds via projective embeddings
For general Kahler manifolds this expansion is related to the determinant of the Donaldson’s Hilb map, which plays an important role in Kahler geometry [15]
Summary
We consider a system of non-interacting fermions in a magnetic field on a compact Kahler manifold M of complex dimension n. While more general choices of the magnetic field configuration can be considered in this context [30], this particular choice leads to a natural generalization of the two-dimensional lowest Landau levels to higher dimensions [16] This setup corresponds to a choice of positive line bundle L, its tensor power Lk and a hermitian metric hk, The curvature Rh = −i∂∂ ̄ log hk of the hermitian metric corresponds to the magnetic field strength. We assume that ωφ belongs to the cohomology class [ω0] of some reference Kahler form ω0 with the magnetic potential hk0, meaning ωφ = ω0 + i∂∂ ̄φ,. Given the choice of the reference metric, the system (2.6) above is parameterized essentially by a single function, the Kahler potential φ.
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