Deformations of JT gravity via topological gravity and applications

We study the duality between JT gravity and the double-scaled matrix model including their respective deformations. For these deformed theories we relate the thermal partition function to the generating function of topological gravity correlators that are determined as solutions to the KdV hierarchy. We specialise to those deformations of JT gravity coupled to a gas of defects, which conforms with known results in the literature. We express the (asymptotic) thermal partition functions in a low temperature limit, in which non-perturbative corrections are suppressed and the thermal partition function becomes exact. In this limit we demonstrate that there is a Hawking-Page phase transition between connected and disconnected surfaces for this instance of JT gravity with a transition temperature affected by the presence of defects. Furthermore, the calculated spectral form factors show the qualitative behaviour expected for a Hawking-Page phase transition. The considered deformations cause the ramp to be shifted along the real time axis. Finally, we comment on recent results related to conical Weil-Petersson volumes and the analytic continuation to two-dimensional de Sitter space.

2d TQFTs and baby universes

In this work, we extend the 2d topological gravity model of [1] to have as its bulk action any open/closed TQFT obeying Atiyah’s axioms. The holographic duals of these topological gravity models are ensembles of 1d topological theories with random dimension. Specifically, we find that the TQFT Hilbert space splits into sectors, between which correlators of boundary observables factorize, and that the corresponding sectors of the boundary theory have dimensions independently chosen from different Poisson distributions. As a special case, we study in detail the gravity model built from the bulk action of 2d Dijkgraaf-Witten theory, with or without end-of-the-world branes, and for arbitrary finite group G. The dual of this Dijkgraaf-Witten gravity model can be interpreted as a 1d topological theory whose Hilbert space is a random representation of G and whose aforementioned sectors are labeled by the irreducible representations of G.These holographic interpretations of our gravity models require projecting out negative-norm states from the baby universe Hilbert space, which in [1] was achieved by the (only seemingly) ad hoc solution of adding a nonlocal boundary term to the bulk action. In order to place their solution in the completely local framework of a TQFT with defects, we couple the boundaries of the gravity model to an auxiliary 2d TQFT in a non-gravitational (i.e. fixed topology) region. In this framework, the difficulty of negative-norm states can be remedied in a local way by the introduction of a defect line between the gravitational and non-gravitational regions. The gravity model is then holographically dual to an ensemble of boundary conditions in an open/closed TQFT without gravity.

FZZT branes in JT gravity and topological gravity

We study Fateev-Zamolodchikov-Zamolodchikov-Teschner (FZZT) branes in Witten-Kontsevich topological gravity, which includes Jackiw-Teitelboim (JT) gravity as a special case. Adding FZZT branes to topological gravity corresponds to inserting determinant operators in the dual matrix integral and amounts to a certain shift of the infinitely many couplings of topological gravity. We clarify the perturbative interpretation of adding FZZT branes in the genus expansion of topological gravity in terms of a simple boundary factor and the generalized Weil-Petersson volumes. As a concrete illustration we study JT gravity in the presence of FZZT branes and discuss its relation to the deformations of the dilaton potential that give rise to conical defects. We then construct a non-perturbative formulation of FZZT branes and derive a closed expression for the general correlation function of multiple FZZT branes and multiple macroscopic loops. As an application we study the FZZT-macroscopic loop correlators in the Airy case. We observe numerically a void in the eigenvalue density due to the eigenvalue repulsion induced by FZZT-branes and also the oscillatory behavior of the spectral form factor which is expected from the picture of eigenbranes.

Genus expansion of open free energy in 2d topological gravity

We study open topological gravity in two dimensions, or, the intersection theory on the moduli space of open Riemann surfaces initiated by Pandharipande, Solomon and Tessler. The open free energy, the generating function for the open intersection numbers, obeys the open KdV equations and Buryak’s differential equation and is related by a formal Fourier transformation to the Baker-Akhiezer wave function of the KdV hierarchy. Using these properties we study the genus expansion of the free energy in detail. We construct explicitly the genus zero part of the free energy. We then formulate a method of computing higher genus corrections by solving Buryak’s equation and obtain them up to high order. This method is much more efficient than our previous approach based on the saddle point calculation. Along the way we show that the higher genus corrections are polynomials in variables that are expressed in terms of genus zero quantities only, generalizing the constitutive relation of closed topological gravity.

Topological Gravity of Chern-Simons-Antoniadis-Savvidy in 2+1 Dimensions

Pada artikel ini akan ditunjukan analisa dari perluasan gauge invariant exact dan metric independent untuk menkontruksi lower-rank field-strength tensor. Hasil ini akan digunakan untuk mengkontruski ulang Chern-Simons-Antoniadis-Savvidy formasi (2n+1) pada dimensi genap dengan menggunakan pendekatan diferensial geometri. Selanjutnya akan dianalisa bentuk topological gravitasi 2-dimensi yang merupakan perluasan dari teorema Chern-Weil yang telah dikembangkan oleh Izurieta-Munoz-Salgado. Hasil dari penelitian ini memperlihatkan bahwa aksi Lagrangian yang sama seperti pada topological gravitasi Chern-Simons forms pada dimensi (2n+1) invariant terhadap Poincare group SO(D−1,1) SO(D−1,2). This article determine and analyess of the extended gauge invariant exact and metric independent to construct the lower-rank field-strength tensor. These results used to construct Chern-Simons-Antoniadis-Savvidy (2n+1)-forms even dimensions using a differential geometry approach. This result analyzed 2-dimensional topological gravity forms that extended Chern-Weil theorem which has been developed by Izurieta-Munoz-Salgado. These results show similary topological gravity Lagrangian action of Chern-Simons forms (2n+1)-dimension invariant under Poincare group SO(D−1,1) SO(D−1,2).Keywords: Gauge theory, field-strength tensor, Chern-Weill theorem, Chern-Simons-Antoniadis-Savvidy forms, topological gravity

Feigin–Frenkel Image of Witten–Kontsevich Points

The Witten–Kontsevich KdV tau function of topological gravity has a generalization to an arbitrary Drinfeld–Sokolov hierarchy associated to a simple complex Lie algebra. Using the Feigin–Frenkel isomorphism we describe the affine opers describing such generalized Witten–Kontsevich functions in terms of Segal–Sugawara operators associated to the Langlands dual Lie algebra. In the case where the Lie algebra is simply laced there is a second role these Segal–Sugawara operators play: their action, in the basic representation of the affine algebra associated to the Lie algebra, singles out the Witten–Kontsevich tau function within the phase space. We show that these two Langlands dual roles of Segal–Sugawara operators correspond to a duality between the first and last operator for a complete set of Segal–Sugawara operators.