Enumeration of maps with tight boundaries and the Zhukovsky transformation

The Weil–Petersson volumes of moduli spaces of hyperbolic surfaces with geodesic boundaries are known to be given by polynomials in the boundary lengths. These polynomials satisfy Mirzakhani’s recursion formula, which fits into the general framework of topological recursion. We generalize the recursion to hyperbolic surfaces with any number of special geodesic boundaries that are required to be tight. A special boundary is tight if it has minimal length among all curves that separate it from the other special boundaries. The Weil–Petersson volume of this restricted family of hyperbolic surfaces is shown again to be polynomial in the boundary lengths. This remains true when we allow conical defects in the surface with cone angles in (0, π) in addition to geodesic boundaries. Moreover, the generating function of Weil–Petersson volumes with fixed genus and a fixed number of special boundaries is polynomial as well, and satisfies a topological recursion that generalizes Mirzakhani’s formula. This work is largely inspired by recent works by Bouttier, Guitter, and Miermont [Ann. Henri Lebesgue 5, 1035–1110 (2022)] on the enumeration of planar maps with tight boundaries. Our proof relies on the equivalence of Mirzakhani’s recursion formula to a sequence of partial differential equations (known as the Virasoro constraints) on the generating function of intersection numbers. Finally, we discuss a connection with Jackiw–Teitelboim (JT) gravity. We show that the multi-boundary correlators of JT gravity with defects are expressible in the tight Weil–Petersson volume generating functions, using a tight generalization of the JT trumpet partition function.

We consider planar maps with three boundaries, colloquially called pairs of pants. In the case of bipartite maps with controlled face degrees, a simple expression for their generating function was found by Eynard and proved bijectively by Collet and Fusy. In this paper, we obtain an even simpler formula for \emph{tight} pairs of pants, namely for maps whose boundaries have minimal length in their homotopy class. We follow a bijective approach based on the slice decomposition, which we extend by introducing new fundamental building blocks called bigeodesic triangles and diangles, and by working on the universal cover of the triply punctured sphere. We also discuss the statistics of the lengths of minimal separating loops in (non necessarily tight) pairs of pants and annuli, and their asymptotics in the large volume limit.

We prove that single G-weighted mathfrak {b}-Hurwitz numbers with internal faces are computed by refined topological recursion on a rational spectral curve, for certain rational weights G. Consequently, the mathfrak {b}-Hurwitz generating function analytically continues to a rational curve. In particular, our results cover the cases of mathfrak {b}-monotone Hurwitz numbers, and the enumeration of maps and bipartite maps (with internal faces) on non-oriented surfaces. As an application, we prove that the correlators of the Gaussian, Jacobi and Laguerre beta -ensembles are computed by refined topological recursion.

In this paper, we obtain new d-dimensional and asymptotically flat wormhole solutions by assuming a specific form of the energy density distribution. This is addressed by considering the generalization of the so-called Dymnikova model, originally studied in the context of regular black holes. In this way, we find constraints for the involved parameters, namely, the throat radius, the scale associated to the matter distribution, and the spacetime dimension, to build those wormholes. Following, we study the properties of the obtained solutions, namely, embedding diagrams as well as Weak and Null Energy Conditions (WEC and NEC). We show that the larger the dimension, the larger the flatness of the wormhole and the more pronounced the violation of these energy conditions. We also show that the corresponding fluid behaves as phantom-like for d ≥ 4 in the neighborhood of the wormhole throat. In addition, we specialize the employed model for d = 4 spacetime, associating it with the gravitational analog of the Schwinger effect in a vacuum and correcting the model by introducing a minimal length via Generalized Uncertainty Principle (GUP). Thus, we obtain a novel traversable and asymptotically flat wormhole solution by considering that the minimal length is very tiny. The associated embedding diagram shows us that the presence of this fundamental quantity increases the slope of the wormhole towards its throat compared with the case without it. That correction also attenuates the WEC (and NEC) violations nearby the throat, with the fluid ceasing to be a phantom-type at the Planck scale, unlike the case without the minimal length.

A direct relation between the enumeration of ordinary maps and that of fully simple maps first appeared in the work of the first and last authors. The relation is via monotone Hurwitz numbers and was originally proved using Weingarten calculus for matrix integrals. The goal of this paper is to present two independent proofs that are purely combinatorial and generalise in various directions, such as to the setting of stuffed maps and hypermaps. The main motivation to understand the relation between ordinary and fully simple maps is the fact that it could shed light on fundamental, yet still not well-understood, problems in free probability and topological recursion.

Conformal boundary loop models

We derive an efficient way to obtain generating functions of bipartite maps of arbitrary genus and boundary length using a spectral curve as initial data for the framework of topological recursion. Based on an earlier result of Chapuy and Fang counting these maps and having a structural proximity to topological recursion, we deduce the corresponding spectral curve which has a strong relation to the spectral curve giving rise to generating functions of ordinary maps. In contrast to ordinary maps, the spectral curve is an irregular one in the sense of Do and Norbury. It generalises the irregular curve for the enumeration of Grothendieck’s dessins d’enfant.

The purpose of this paper is to compare the minimal marked length spectrum and the Laplace spectrum on functions and on forms for Riemannian two-step nilmanifolds. A Riemannian nilmanifold is a closed manifold of the form ( \G, g), where G is a simply connected nilpotent Lie group, is a cocompact (i.e., \G compact) discrete subgroup of G, and g arises from a left invariant metric on G. Examples of nilmanifolds include flat tori and Heisenberg manifolds. The Laplace spectrum of a closed Riemannian manifold (M, g) is the set of eigenvalues of the Laplace–Beltrami operator , counted with multiplicity. The Laplace–Beltrami operator may be extended to act on smooth p-forms by = dδ + δd, where δ is the metric adjoint of d. Two manifolds have the same marked length spectrum if there exists an isomorphism between the fundamental groups such that corresponding free homotopy classes of loops can be represented by smoothly closed geodesics of the same length. Two manifolds have the same minimal marked length spectrum (resp., maximal marked length spectrum) if there exists an isomorphism between the fundamental groups such that the smallest (resp., longest) closed loops in corresponding free homotopy classes have the same length. The main theorem of this paper is the following (see Theorem 2.5).

In this paper, a Naveir-Stokes equation is solved by using the Adomian's decomposition method (ADM) , modifled Adomian's decom- position method (MADM), variational iteration method (VIM), modifled variational iteration method (MVIM), modifled homotopy perturbation method (MHPM) and homotopy analysis method (HAM). The approxi- mate solution of this equation is calculated in the form of series which its components are computed by applying a recursive relation. The exis- tence and uniqueness of the solution and the convergence of the proposed methods are proved. A numerical example is studied to demonstrate the accuracy of the presented methods.

Recursion Theory as a Branch of Model Theory

For a given spectral curve, the theory of topological recursion generates two different families \omega_{g,n}ωg,n and \omega_{g,n}^\veeωg,n∨ of multi-differentials, which are for algebraic spectral curves related via the universal x-yx−y duality formula. We propose a formalism to extend the validity of the x-yx−y duality formula of topological recursion from algebraic curves to spectral curves with exponential variables of the form e^x=F(e^y)ex=F(ey) or e^x=F(y)e^{a y}ex=F(y)eay with FF rational and aa some complex number, which was in principle already observed in [Commun. Number Theory Phys. 13, 763 (2019); J. Lond. Math. Soc. 109, e12946 (2024)]. From topological recursion perspective the family \omega_{g,n}^\veeωg,n∨ would be trivial for these curves. However, we propose changing the n=1n=1 sector of \omega_{g,n}^\veeωg,n∨ via a version of the Faddeev’s quantum dilogarithm which will lead to the correct two families \omega_{g,n}ωg,n and \omega_{g,n}^\veeωg,n∨ related by the same x-yx−y duality formula as for algebraic curves. As a consequence, the x-yx−y symplectic transformation formula extends further to important examples governed by topological recursion including, for instance, Gromov-Witten invariants of \mathbb{C}^3ℂ3 (or, equivalently, triple Hodge integrals), orbifold Hurwitz numbers, and stationary Gromov-Witten invariants of \mathbb{P}^1ℙ1. The proposed formalism is related to the issue topological recursion encounters for specific choices of framings for the topological vertex curve.

Kostka, Littlewood-Richardson, Kronecker, and plethysm coefficients are fundamental quantities in algebraic combinatorics, yet many natural questions about them stay unanswered for more than 80 years. Kronecker and plethysm coefficients lack “nice formulas”, a notion that can be formalized using computational complexity theory. Beyond formulas and combinatorial interpretations, we can attempt to understand their asymptotic behavior in various regimes, and inequalities they could satisfy. Understanding these quantities has also applications outside of combinatorics. On the one hand, the asymptotics of structure constants is closely related to understanding the [limit] behavior of vertex and tiling models in statistical mechanics. More recently, these structure constants have been involved in establishing computational complexity lower bounds and separation of complexity classes like VP vs VNP, the algebraic analogs of P vs NP in arithmetic complexity theory. Here we discuss the outstanding problems related to asymptotics, positivity, and complexity of structure constants focusing mostly on the Kronecker coefficients of the symmetric group and, less so, on the plethysm coefficients. This expository paper is based on the talk presented at the Open Problems in Algebraic Combinatorics coneference in May 2022.

We prove, in a purely combinatorial way, the spectral curve topological recursion for the problem of enumeration of bi-colored maps, which are dual objects to dessins d'enfant. Furthermore, we give a proof of the quantum spectral curve equation for this problem. Then we consider the generalized case of four-colored maps and outline the idea of the proof of the corresponding spectral curve topological recursion.

We extend the theory of topological recursion by considering Airy ideals (also known as Airy structures) whose partition functions are highest weight vectors of particular $$\mathcal {W}$$ W -algebra representations. Such highest weight vectors arise as partition functions of Airy ideals only under certain conditions on the representations. In the spectral curve formulation of topological recursion, we show that this generalization amounts to adding specific terms to the correlators $$ \omega _{g,1}$$ ω g , 1 , which leads to a “shifted topological recursion” formula. We then prove that the wave-functions constructed from this shifted version of topological recursion are WKB solutions of families of quantizations of the spectral curve with $$ \hslash $$ ℏ -dependent terms. In the reverse direction, starting from an $$\hslash $$ ℏ -connection, we find that it is of topological type if the exact same conditions that we found for the Airy ideals are satisfied. When this happens, the resulting shifted loop equations can be solved by the shifted topological recursion obtained earlier.

A classic problem in circuit theory first studied by German physicist Kirchhoff more than 170 years ago is the computation of resistances in resistor networks. Nowadays, resistor network has been an important model in the fields of natural science and engineering technology, but it is very difficult to calculate the equivalent resistance between two arbitrary nodes in an arbitrary resistor network. In 2004, Wu F Y formulated a Laplacian matrix method and derived expressions for the two-point resistance in arbitrary finite and infinite lattices in terms of the eigenvalues and eigenvectors of the Laplacian matrix, and the resistance results obtained by Laplacian matrix method is composed of double sums. The weakness of the Laplacian matrix approach is that it depends on the two matrices along two orthogonal directions. In 2011, Tan Z Z created the recursion-transform (RT) method, which can resolve the resistor network with arbitrary boundary. Using the RT method to compute the equivalent resistance relies on just one matrix along one direction, and the resistance is expressed by single summation. In the present paper, we investigate the equivalent resistance and complex impedance of an arbitrary mn cylindrical network by the RT method. Firstly, based on the network analysis, a recursion relation between the current distributions on three successive vertical lines is established through a matrix equation. In order to obtain the eigenvalues and eigenvectors of the matrix, and the general solution of the matrix equation, we then perform a diagonalizing transformation on the driving matrix.Secondly, we derive a recursion relation between the current distributions on the boundary, and construct some particular solutions of the matrix equation. Finally by using the matrix equation of inverse transformation, we obtain the analytical solution of the branch current, and gain the equivalent resistance formula along the axis of the arbitrary mn cylindrical network, which consists of the characteristic root and expressed by only single summation. As applications, several new formulae of equivalent resistances in the semi-infinite and infinite cases are given. These formulae are compared with those in other literature, meanwhile an interesting new identity of trigonometric function is discovered. At the end of the article, the equivalent impedance of the mn cylindrical RLC network is also treated, where the equivalent impedance formula is also given.