ACT \alpha - One of the less talked about correspondences between algebraic geometry and quantum gravity is the relation of finding the volume of the moduli space, intersection theory on these moduli, random matrix theory, and JT gravity (a two-dimensional solution of gravity), at least on this website. A bit of history can be traced in this paper by Witten from 91. In recent years, we have seen more of these interplays, especially see the following papers
- https://arxiv.org/abs/math/9902051
- https://arxiv.org/abs/0812.0544
- https://link.springer.com/article/10.1007/s00222-006-0013-2
- https://arxiv.org/abs/1907.03363
- https://arxiv.org/abs/1903.11115
- https://arxiv.org/abs/1804.03275
- https://arxiv.org/abs/0705.3600
- https://arxiv.org/abs/2004.05183
- https://arxiv.org/abs/2006.13414
In this paper, Witten tries to argue for an analogy of the volume of moduli space of Riemann surfacthe es to volume of moduli space of super-Riemann surfaces. But let us first see the example of an ordinary Riemann surface. It is defined over a bundle PSL(2,\mathbb{R}). The volume of the moduli space (denoted by \mathcal{M}) of these Riemann surfaces is given by
V = \int_\mathcal{M} Pf(\omega) = \int_\mathcal{M} e^\omega
where moduli space \mathcal{M} is defined for some genus g>1 cause it admits a symplectic form called Weil-Petterson form and \omega is that form given by
\omega= \frac{1}{4\pi} \int_M Tr \delta A \wedge \delta A
where M is a Riemann surface and A is the connection on the flat PSL(2, \mathbb{R}) connection.
In the super-Riemann surfaces, the moduli spaces have a clear analogy of volume too, the moduli spaces of super-Riemann surfaces \mathfrak{M} for genus g>1 have a similar symplectic form
\hat{w} = \frac{1}{4\pi} \int_M Tr \delta A \wedge \delta A
but now the connection A is on the OSp(1|2) connections. The volume of \mathfrak{M} is defined by
\hat{V} = \int_\mathfrak{M} \sqrt{Ber\ \hat{w}}
where Ber is Berezinian, an analog of determinant here for the supersymmetric case.
Here, in the ordinary moduli space \mathcal{M}, volume V is given by the intersection theory on \mathcal{M} (the details of this fact will be discussed soon in some other post or somewhere else). The essence is that the Weil-Petterson form (a tautological class) can be computed if one knows the intersection numbers on \mathcal{M}. And for physicists, it is that random matrix theory and intersection theory are related. Also, see this work by Konstevich.
Here, the Riemann surface M is defined with geodesic boundary of length and then the intersection numbers and volumes are related to these lengths. Moreover, the two-dimensional gravity theory, namely JT gravity, is dual to a random matrix theory. This crucial result is due to Saad, Shenker, and Stanford which uses recursion relations on JT gravity on a disk. More crucially, they replace the notion of a function of energy levels of a particular Hamiltonian by the average level density of an ensemble of Hamiltonians in a random matrix theory. This was surprisingly related to the moduli space of Riemann surfaces and thus to the volume of moduli space. The result can be explained by these papers by Eynard & Orantin and Mirzakhani. This paper is helpful too.
Anyhow, the analogy for the super-Moduli space is not straightforward. And no trivial counting of intersection numbers exists. However, a hopeful topological recursion relation still exists and was computed by Stanford and Witten. Now instead of a matrix ensemble, one studies a supersymmetric matrix ensemble dual to super-JT gravity. Then mimicking the result of Eynard & Orantin and Mirzakhani, one can find the volume of super-Moduli space \mathfrak{M}.
