Abstract
The Hilbert space of a quantum system with internal global symmetry G decomposes into sectors labelled by irreducible representations of G. If the system is chaotic, the energies in each sector should separately resemble ordinary random matrix theory. We show that such “sector-wise” random matrix ensembles arise as the boundary dual of two- dimensional gravity with a G gauge field in the bulk. Within each sector, the eigenvalue density is enhanced by a nontrivial factor of the dimension of the representation, and the ground state energy is determined by the quadratic Casimir. We study the consequences of ’t Hooft anomalies in the matrix ensembles, which are incorporated by adding specific topological terms to the gauge theory action. The effect is to introduce projective representations into the decomposition of the Hilbert space. Finally, we consider ensembles with G symmetry and time reversal symmetry, and analyze a simple case of a mixed anomaly between time reversal and an internal ℤ2 symmetry.
Highlights
We study the consequences of ’t Hooft anomalies in the matrix ensembles, which are incorporated by adding specific topological terms to the gauge theory action
This equation is precisely consistent with (4.1) and the answer for the BF theory in (3.11). This establishes that if the seed 2d gravity theory was dual to a random matrix ensemble, after coupling to BF theory, the result will be dual to a random matrix ensemble with G symmetry
We studied two-dimensional gauge theory coupled to gravity as a dual description of certain matrix ensembles with global symmetry G
Summary
In the original application of random matrix theory to physics, one views the random matrix as a model for a quantum Hamiltonian H, and one defines an ensemble that is “as random as possible” given the symmetries of H. The leading approximation to the total density of eigenvalues has an extra factor of L: ρt0otal(x) = Lρ0(x) Given these two pieces of data (the symmetry class and ρ0(x)), a universal recursion relation that follows from the loop equations determines all of the Zg(β1 . These are integrals in which the interval of support of the leading distribution of eigenvalues is the half-line x ≥ x0, and we relax the requirement of normalizability of ρ0(x) They can be obtained as limits of ordinary matrix integrals, where we take L → ∞ and adjust the potential V (H) in such a way that pointwise ρt0otal(x) → eS0 ρ0(x),. Note that both rescalings can be accomplished by shifting S0 → S0 + log λ To state this more precisely, it is helpful to define a thermal partition function restricted to a given representation sector. In the two sections, we will show how the structure in (2.13) arises in 2d gravity once we incorporate a bulk version of G symmetry
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