Story of Maryam’s mathematical magic


Gravity is a difficult concept to understand as a quantum theory. Therefore physicists (and their mathematician friends) study quantum gravity in 2 dimensions where equations are more tractable. Just like any good quantum theory, the path integral for quantum gravity is defined by Z(\beta) = \frac{1}{\text{vol}}\int \mathcal{D}\phi\mathcal{D}g e^{-I} where “vol” is the volume of the diffeomorphism group implying that the integration should be done on \phi and g upto diffeomorphism and I is the action of a model knowns as Jackiw-Titelboim (JT) theory. Once the partition function is known, a quantum theory can be solved.

The partition function for the 2 dimensional quantum gravity (after imposing appropriate boundary conditions on two manifold U with the topology of a disc) turns out to be Z_{U}(\beta) = \int_{0}^{\infty}dE\rho_0(E)e^{-\beta E} where \rho_0=\frac{e^S}{4\pi^2}\sinh^2(2\pi\sqrt{E}). S is the renormalized coefficient for Einstein-Hilbert term (ignore it if you don’t know what it means).

Holographic duality

From AdS/CFT correspondence we know that the 2 dimensional quantum gravity is dual to 1 dimensional simple quantum system on the boundary of U. This implies that the partition function that we computed earlier form JT model should be Z_{U}(\beta) = \text{Tr}_{\text{H}}e^{-\beta \hat{H}} where \hat{H} is the Hamiltonian operator of the associated 1 dimensional quantum system. In other words, Z_{U}(\beta)=\int_0^\infty dE\sum \delta(E - E_i)e^{-\beta E}. This clearly is not equal to the partition function computed in the previous paragraph and that is problematic. The dual theories should have same partition function

Random Matrix theory

The idea here is to consider an ensemble of random matrices (with certain properties) and define the ensemble average of the partition function \langle Z(\beta)\rangle = \langle \text{Tr}_{\text{H}}e^{-\beta \hat{H}} \rangle. In the matrix literature it is common to work with the inverse Laplace transform of the partition function given by R(E) = -\int_0^\infty d\beta Z(\beta)e^{\beta E} known as the resolvent. So the resolvent correlation function has the “genus” expansion as follows \langle R(E)\rangle = \sum_{g=0}^{g=\infty}\frac{R_{g,n}(E)}{L^{2g+n-2}} (it is asymptotic series expansion) where L is the dimension of the matrices of the ensemble. The terms R_{g, n} can be computed systematically from what are known as “loop equations”, in terms of \rho_0(E) which is the “average density of the eigenvalues of random matrix of the ensemble”. The loop equations are streamlined into simple recursion relations which yield the correlation function upto all orders (in 1/L^2)  once \rho_0(E) is specified.

2 dimensional gravity again

In the JT model, there is a prescription to compute the quantity \langle Z(\beta) \rangle which involves integrating over connected 2d geometries with specified boundary length and value of dilation field at the boundary. The partition function turns out to be \langle Z(\beta)\rangle = e^{S_0}Z^{\text{disc}}_{\text{sch}} + \sum_{g=1}^{\infty} e^{(1-2g)S_0}bdbV_{g, 1}(b)Z^{\text{trumpet}}_{\text{sch}}(\beta, b) where b is the length of a geodesic and V_{g, 1}(b) is the Weil-Petersson volume of moduli space of Riemann surface with genus g.

In the phenomenal paper Maryam Mirzakhani showed that the Weil-Petersson volume follow certain recursion  relation which was shown to be equivalent to the recursion relation of matrix ensemble, we saw earlier, if we use the leading (genus 0) contribution of the JT path integral as \rho_0(E) thus resolving the problem described earlier.

There is analogous story for supersymmetric quantum gravity but I am in no mood to write about it. If you are interested read this paper by Witten and Stanford.

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