Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence

Abstract

Abstract:We propose an asymptotic expansion formula for matrix integrals, including oscillatory terms (derivatives of theta-functions) to all orders. This formula is heuristically derived from the analogy between matrix integrals, and formal matrix models (combinatorics of discrete surfaces), after summing over filling fractions. The whole oscillatory series can also be resummed into a single theta function. We also remark that the coefficients of the theta derivatives, are the same as those which appear in holomorphic anomaly equations in string theory, i.e. they are related to degeneracies of Riemann surfaces. Moreover, the expansion presented here, happens to be independent of the choice of a background filling fraction.

Liouville quantum gravity and KPZ

“…These models have been very thoroughly studied in the physics literature, in part because of connections to string theory and conformal field theory [Pol81a, Pol81b, Pol87a, Pol89, Sei90, GM93, Dav94, Dav95, AJW95, AW95, DFGZJ95, Kle95, KH96, ADJ97, Eyn01, Dup06], and to random matrix theory and geometrical models; see, e.g., the references [BIPZ78, ADF85, KKM85, Dav85, BKKM86a, BKKM86b, Kaz86, DK88a, DK90, GK89, Kos89a, Kos89b, DDSW90, MSS91, KK92, EZ92, JM92, Kor92a, Kor92b, ABC93, Dur94, ADJ94, Dau95, EK95, KH95, BDKS95, AAMT96, Dup98, Dup99a, Dup99b, Dup99c, EB99, KZJ99, Kos00, Dup00, DFGG00, DB02, Dup04, Kos07, Kos09]. More recently, a purely combinatorial approach to discretized quantum gravity has been successful [Sch98, BFSS01, FSS04, BDFG02, BS03, AS03, BDFG03a, BDFG03b, DFG05, BDFG07, Mie09, LG07, MM07, Ber07, Ber08a, Ber08b, Ber08c, BG08a, MW08, Mie08, BG08b, LG08, BG09, LM09, BB09], as well as the so-called topological expansion involving higher-genus random surfaces [CMS09,Cha09,Cha10,EO07,EO08,Eyn09].…”

A unified approach to holomorphic anomaly equations and quantum spectral curves

“…Let us now mention some applications of our formalism. First of all, for the choice κ = (τ − τ ) −1 , we recover the holomorphic anomaly for matrix models [15,17], or more generally, when the propagator κ IJ takes the form of ∆ IJ in (4), we recover the construction of holomorphic anomaly equation in Gromov-Witten theory as discussed in [19]. For details, see Section 5.…”

“…As mentioned in the Introduction, the formal Gaussian integral (57) appeared in earlier work on holomorphic anomaly equations [15,17,19], where the propagator κ was chosen to be the Zamolodchikov metric which is non-holomorphic. Later in this paper we will see that the propagator κ admits different choices, which give us different theories in mathematical physics.…”

Complex saddle points in the Gross-Witten-Wadia matrix model

“…Among the key realizations were (i) that the set of saddle points of a matrix model allows us to describe the nonperturbative corrections to the free energy in terms of large-N instantons, i.e., migration of eigenvalues among different pieces of the eigenvalue density support, and (ii) the role played by complex saddles even in cases where the original partition functions were sums over real configurations [2,3,[7][8][9][10][11].…”