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Register a new userGenus expansion of open free energy in 2d topological gravity
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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 Buryaks 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 Buryaks 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.
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We study multi-boundary correlators of Witten-Kontsevich topological gravity in two dimensions. We present a method of computing an open string like expansion, which we call the t Hooft expansion, of the $n$-boundary correlator for any $n$ up to any order by directly solving the Korteweg-De Vries equation. We first explain how to compute the t Hooft expansion of the one-boundary correlator. The algorithm is very similar to that for the genus expansion of the open free energy. We next show that the t Hooft expansion of correlators with more than one boundary can be computed algebraically from the correlators with a lower number of boundaries. We explicitly compute the t Hooft expansion of the $n$-boundary correlators for $n=1,2,3$. Our results reproduce previously obtained results for Jackiw-Teitelboim gravity and also the t Hooft expansion of the exact result of the three-boundary correlator which we calculate independently in the Airy case.
We define and study a holographic dual to the topological twist of $mathcal{N}=4$ gauge theories on Riemannian three-manifolds. The gravity duals are solutions to four-dimensional $mathcal{N}=4$ gauged supergravity, where the three-manifold arises as a conformal boundary. Following our previous work, we show that the renormalized gravitational free energy of such solutions is independent of the boundary three-metric, as required for a topological theory. We then go further, analyzing the geometry of supersymmetric bulk solutions. Remarkably, we are able to show that the gravitational free energy of any smooth four-manifold filling of any three-manifold is always zero. Aided by this analysis, we prove a similar result for topological AdS$_5$/CFT$_4$. We comment on the implications of these results for the large $N$ limits of topologically twisted gauge theories in three and four dimensions, including the ABJM theory and $mathcal{N}=4$ $SU(N)$ super-Yang-Mills, respectively.
Torsion gravity is a natural extension to Einstein gravity in the presence of the fermion matter sources. In this paper we adopt Walds covariant method of Noether charge to construct the quasi-local energy of the Einstein-Cartan-fermion system, and find that its explicit expression is formally independent of the coupling constant between torsion and axial current. This seemingly topological nature is unexpected and is reminiscent of similar nature of quantum Hall effect and topological insulator. However, the coupling dependence does enter when evaluating it on-shell, and thus the topological nature is pseudo. Based on the expression of the quasi-local energy, we evaluate it for a particular solution on the entanglement wedge and find the agreement with the holographic relative entropy obtained before. This shows that the equivalence of these two quantities in the Einstein-Cartan-fermion system. Moreover, the quasi-local energy in this case is not always positive definite so that it provides an example of swampland in torsion gravity. Based on the covariant Noether charge, we also derive the nonzero fermion effect on Komar angular momentum. The implication of our results to the tests of torsion gravity in the future gravitational wave astronomy is also discussed.
The quenched free energy, $F_Q(T){=}{-}Tlangle ln Z(T)rangle$, of various JT gravity and supergravity theories is explored, taking into account the key non-perturbative physics that is accessible using their matrix model formulations. The leading low energy physics of these systems can be modelled by the Airy and (a family of) Bessel models, which arise from scaling limits of matrix ensembles. The $F_Q(T)$s of these models are directly computed by explicit sampling of the matrix ensembles, and how their properties are connected to the statistical mechanics of the underlying discrete spectrum of the ensembles is elucidated. Some of the low temperature ($T$) features of the results confirm recent observations by Jassen and Mirbabayi. The results are then used as benchmarks for exploring an intriguing formula proposed by Okuyama for computing $F_Q(T)$ in terms of the connected correlators of its partition function, the wormholes of the gravity theory. A low $T$ truncation of the correlators helps render the formula practical, but it is shown that this is at the expense of much of its accuracy. The significance of the statistical interpretation of $F_Q(T)$ for black hole microphysics is discussed.
We quantize the two-dimensional projectable Horava-Lifshitz gravity with a bi-local as well as space-like wormhole interaction. The resulting quantum Hamiltonian coincides with the one obtained through summing over all genus in the string field theory for two-dimensional causal dynamical triangulations. This implies that our wormhole interaction can be interpreted as a splitting or joining interaction of one-dimensional strings.
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