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Register a new usert Hooft expansion of multi-boundary correlators in 2D 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.
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We continue the systematic study of the thermal partition function of Jackiw-Teitelboim (JT) gravity started in [arXiv:1911.01659]. We generalize our analysis to the case of multi-boundary correlators with the help of the boundary creation operator. We clarify how the Korteweg-de Vries constraints arise in the presence of multiple boundaries, deriving differential equations obeyed by the correlators. The differential equations allow us to compute the genus expansion of the correlators up to any order without ambiguity. We also formulate a systematic method of calculating the WKB expansion of the Baker-Akhiezer function and the t Hooft expansion of the multi-boundary correlators. This new formalism is much more efficient than our previous method based on the topological recursion. We further investigate the low temperature expansion of the two-boundary correlator. We formulate a method of computing it up to any order and also find a universal form of the two-boundary correlator in terms of the error function. Using this result we are able to write down the analytic form of the spectral form factor in JT gravity and show how the ramp and plateau behavior comes about. We also study the Hartle-Hawking state in the free boson/fermion representation of the tau-function and discuss how it should be related to the multi-boundary correlators.
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.
At classical level, dynamical derivation of the properties and conservation laws for topologically non-trivial systems from Noether theorem versus the derivation of the systems properties on topological grounds are considered as distinct. We do celebrate any agreements in results derived from these two distinct approaches: i.e. the dynamical versus the topological approach. Here we consider the Corrigan-Olive-Fairlie-Nuyts solution based on which we study the stability of the t Hooft- Polyakov outer field, known as its Higgs vacuum, and derive its stability, dynamically, from the equations of motion rather than from the familiar topological approach. Then we use our derived result of the preservation of the Higgs vacuum asymptotically to derive the stability of the t Hooft-Polyakov monopole, even if inner core is perturbed, where we base that on observing that the magnetic charge must be conserved if the Higgs vacuum is preserved asymptotically. We also, alternatively, note stability of t Hooft-Polyakov monopole and the conservation of its magnetic charge by again using the result of the Higgs vacuum asymptotic preservation to use Eq.(5) to show that no non-Abelian radiation allowed out of the core as long as the Higgs vacuum is preserved and restored, by the equations of motion, if perturbed. We start by deriving the asymptotic equations of motion that are valid for the monopoles field outside its core; next we derive certain constraints from the asymptotic equations of motion of the Corrigan-Olive-Fairlie-Nuyts solution to the t Hooft-Polyakov monopole using the Lagrangian formalism of singular theories, in particular that of Gitman and Tyutin. The derived constraints will show clearly the stability of the monopoles Higgs vacuum its restoration by the equations of motion of the Higgs vacuum, if disturbed.
We study the perturbative series associated to bi-local correlators in Jackiw-Teitelboim (JT) gravity, for positive weight $lambda$ of the matter CFT operators. Starting from the known exact expression, derived by CFT and gauge theoretical methods, we reproduce the Schwarzian semiclassical expansion beyond leading order. The computation is done for arbitrary temperature and finite boundary distances, in the case of disk and trumpet topologies. A formula presenting the perturbative result (for $lambda in mathbb{N}/2$) at any given order in terms of generalized Apostol-Bernoulli polynomials is also obtained. The limit of zero temperature is then considered, obtaining a compact expression that allows to discuss the asymptotic behaviour of the perturbative series. Finally we highlight the possibility to express the exact result as particular combinations of Mordell integrals.
We study higher-form global symmetries and a higher-group structure of a low-energy limit of $(3+1)$-dimensional axion electrodynamics in a gapped phase described by a topological action. We argue that the higher-form symmetries should have a semi-strict 4-group (3-crossed module) structure by consistency conditions of couplings of the topological action to background gauge fields for the higher-form symmetries. We find possible t Hooft anomalies for the 4-group global symmetry, and discuss physical consequences.
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