Degenerate operators in JT and Liouville (super)gravity

Abstract

We derive explicit expressions for a specific subclass of Jackiw-Teitelboim (JT) gravity bilocal correlators, corresponding to degenerate Virasoro representations. On the disk, these degenerate correlators are structurally simple, and they allow us to shed light on the 1/C Schwarzian bilocal perturbation series. In particular, we prove that the series is asymptotic for generic weight h ∉ −ℕ/2. Inspired by its minimal string ancestor, we propose an expression for higher genus corrections to the degenerate correlators. We discuss the extension to the $$ \mathcal{N} $$ N = 1 super JT model. On the disk, we similarly derive properties of the 1/C super-Schwarzian perturbation series, which we independently develop as well. As a byproduct, it is shown that JT supergravity saturates the chaos bound λL = 2π/β at first order in 1/C. We develop the fixed-length amplitudes of Liouville supergravity at the level of the disk partition function, the bulk one-point function and the boundary two-point functions. In particular we compute the minimal superstring fixed length boundary two-point functions, which limit to the super JT degenerate correlators. We give some comments on higher topology at the end.

Gravity factorized

“…First we have FZZT branes, the most commonly studied brane in minimal string theory [19,[69][70][71][72][73][74][75][76][77][78].…”

Gravity without averaging

“…They come in two types. The first one is the contact term contributions and has segments with different boundary condition separated by marked points [56,72,106,107]. The second type is simply the coming from the multi-trace contributions in the first line of (5.15).…”

Microstructure in matrix elements

“…Then each flavor of EOW particles corresponds with a D-brane, and interaction vertices correspond naturally with insertions of boundary operators T n ij ; these are open string Tachyons stretching between D-branes, with Chan-Paton indices i and j. Only the simplest of the chiral vertex operators T 1 ij , known as marking operators, have a known JT gravity interpretation [51,[54][55][56] when stretching between Dbranes with FZZT boundary conditions [8,[57][58][59]. 13 Marking operators then correspond with the β " 0 limit of a fixed length boundary.…”

“…The example which we choose is sufficiently complex so that the generalization to all amplitudes should 39 In stringy language the k flavors of interior modes are ordinary Chan-Paton indices, because all D-branes coincide at µ. 40 Recently an educated guess was made in [54] which perhaps deserves further study.…”

“…1 , E 2 and E 3 38. This corresponds in random matrix theory with the following observable[51,[54][55][56] Now consider a disk with thermal boundary length α 1 `β1 `α2 `β2 `α3 `β3 , which corresponds in random matrix language with Trpexpp´pα 1 `β1 `α2 `β2 `α3 `β3 qHqq. Laplace transforming some segment of thermal boundary gives a segment of FZZT boundary, this is obviously true from the matrix integral formulas.…”