This paper contains a proof of a recent conjecture of Artemev that connected the resonance transformations for the ( 2 , 2 p + 1 ) minimal string to the x - y swap in the theory of topological recursion.
We recall a conformal method, suggested in a previous work, for testing the stability of Einstein metrics. We then give a direct application of this method when the manifold is conformal to a Riemannian (multi-)product. Finally, we apply the procedure to the Riemannian Schwarzschild–Tangherlini metric, which turns out to be unstable in dimensions n∈{4,⋯ ,9} .
Let X be a smooth projective curve over a field k. Let 𝒢 be a parahoric group scheme on X as in [32]. The aim of this article is to establish the basic themes of the theory of conformal blocks in the parahoric setting. Using Hecke correspondences, we reduce questions on parahoric stacks to those on the reductive (or hyperspecial parahoric) stack via the Iwahori stack. Following this reduction approach, we set-up relationships between the cohomology of line bundles on various moduli stacks of torsors. These relationships give a proof of [32, Conjecture 3.7] in characteristic zero, the principle of propagation of vacua and a direct proof of the independence of central charge on base points. Projective flatness is recovered as a corollary of Faltings’ construction of the Hitchin connection. Using Teleman’s basic results ([38]), we deduce the analogous result that cohomology of line bundles on the stack of principal G-bundles vanishes in all degrees except possibly one over ℂ . Results on twisted vacua [22] are obtained as immediate consequences.
We present the first explicit derivation of a matrix Painlevé IV equation arising from semi-classical matrix orthogonal polynomials of Laguerre-type. Starting from the discrete nonlinear relations for the recurrence coefficients of the associated matrix orthogonal polynomials, established in [9] and identified there as a discrete Painlevé IV equation, we show that the recurrence coefficients satisfy a nonlinear matrix differential-difference equation in the deformation parameter t. These continuous equations provide a natural matrix analogue of the classical Painlevé IV equation, reducing to the scalar case when all coefficients commute. We conclude with a new concrete asymmetric low-dimensional example illustrating the main result.
The theory of Lie bialgebras and the classical Yang–Baxter equation plays a major role in the study of 1+1d integrable systems; many families of integrable systems can be recovered from a Lax pair which is constructed from a Lie bialgebra. A categorified, higher homotopy notion of Lie algebras has been studied, which gave rise to the notion of (strict) Lie 2-bialgebras (Lie algebra crossed-modules) and the 2-graded Yang–Baxter equations. In this paper, we use these differential-graded structures to generalize the construction of a Lax pair and introduce an appropriate notion of higher-dimensional integrability. Within this framework, we introduce a higher derived version of the affine Kac–Moody algebra, which underpins the 2-graded Lax integrability that we have developed here as a zero 2-curvature condition. As an explicit demonstration, we will consider a 3d field theory and show that it (i) is 2-graded Lax integrable and (ii) hosts symmetries governed by a Kac–Moody 2-algebra.
We construct an asymptotically flat T-fold representative of a four-dimensional dyonic black-hole charge orbit in doubled type-IIB theory. Starting from the F1-P-NS5-KKM toroidal seed, an integral parabolic T-duality monodromy is imposed on an active doubled three-torus. The non-geometric data enter only through this global patching: the Reissner–Nordstrom-type term is sourced by conserved four-dimensional electric and magnetic field strengths in the doubled Kaluza–Klein/winding local system, not by an internal algebraic Q -flux alone. Exact duality preserves the local equations, the BPS index and the O(6,6) -invariant of the NS–NS/ N=4 charge sublattice governing the entropy. The minimal Einstein–Maxwell dilaton slice contains a running scalar branch and an equal-charge Reissner–Nordstrom limit; the extremal near-horizon region is locally AdS_2× S^2 , with the compact factor globally T-duality patched.
We construct three-dimensional non-semisimple topological field theories from the unrolled quantum group of the Lie superalgebra $$\mathfrak {osp}(1 \vert 2)$$ osp ( 1 | 2 ) . More precisely, the quantum group depends on a root of unity $$q=e^{\frac{2 \pi \sqrt{-1}}{r}}$$ q = e 2 π - 1 r , where r is a positive integer greater than 2, and the construction applies when r is not congruent to 4 modulo 8. The algebraic result which underlies the construction is the existence of a relative modular structure on the non-finite, non-semisimple category of weight modules for the quantum group. We prove a Verlinde formula which allows for the computation of dimensions and Euler characteristics of topological field theory state spaces of unmarked surfaces. When r is congruent to $$\pm 1$$ ± 1 or $$\pm 2$$ ± 2 modulo 8, we relate the resulting 3-manifold invariants with physicists’ $$\widehat{Z}$$ Z ^ -invariants associated to $$\mathfrak {osp}(1 \vert 2)$$ osp ( 1 | 2 ) . Finally, we establish a relation between $$\widehat{Z}$$ Z ^ -invariants associated to $$\mathfrak {sl}(2)$$ sl ( 2 ) and $$\mathfrak {osp}(1 \vert 2)$$ osp ( 1 | 2 ) which was conjectured in the physics literature.
In this work, we study the initial-boundary value (IBV) problems for the sine-Gordon (sG) equation in the light-cone coordinates u_xt=sin u in the quarter-planes x> 0, t > 0 and x < 0, t > 0 , assuming a suitable decay as x→ +∞ or as x→ -∞ . Employing the Riemann–Hilbert (RH) problem framework, we demonstrate that these two IBV problems differ significantly with respect to the boundary data required for well-posedness. Specifically, the solution of the “right problem” ( x≥ 0 ) is uniquely determined by the initial data u(x, 0), x≥ 0 alone, whereas for the “left problem” ( x≤ 0 ), the boundary data u(0, t) have to be prescribed in addition to the initial data in order to obtain a well-posed problem. The latter problem is solved using the unified transform method (also known as the Fokas method).
In this work, we investigate the initial-boundary value problem (IBVP) of the Cahn–Hilliard-type model on the half-line, model related to heat-mass transfer phenomena and solid fluid dynamics. Recently, Chatziafratis et al. (Math. Models Methods Appl. Sci. 35 (2025) 1133-1197) employed the Fokas method to solve the IBVP for Cahn–Hilliard-type model on the half-line and derived properties such as regularity near the boundary of the domain. Inspired by this work, we rigorously prove the well-posedness in the sense of Hadamard with initial condition u_0(x) ∈ H^s(0,∞ ) , for regularity exponent 1/2
This paper contains a proof of a recent conjecture of Artemev that connected the resonance transformations for the (2,2p+1) minimal string to the x-y swap in the theory of topological recursion.
In this paper, using extension theory and cohomological approach we introduce the notion of the obstruction class for an inner post-Lie algebra being induced by a Rota–Baxter operator, and show that an inner post-Lie algebra is induced by a Rota–Baxter operator if and only if the obstruction class is trivial. Similarly, we introduce the notion of the obstruction class for an inner post-group being induced by a Rota–Baxter operator, and prove a parallel result. Finally, we give some applications of inner post-Lie algebras and inner post-groups.
In this paper, we present a detailed mathematical description of the error correction process for Kitaev’s model for finite Abelian groups. The number of errors Kitaev’s model can correct depends on the lattice and its topology. Although there is a theoretical maximum number of errors that can be corrected, we prove that correcting this number of errors, in general, is an NP-complete problem. Consequently, we introduce a polynomial-time correction algorithm that corrects a number of errors below the theoretical maximum.
This work is motivated by recent developments in celestial holography. In [15], the authors interpreted QCD collinear singularities in terms of operator product expansions in a two-dimensional CFT. We reformulate the algebraic structures arising in their work using the formalism of nonlinear Lie conformal algebras developed in [31].
Motivated by the work of Kontsevich-Soibelman on the comparison of isomorphisms conjecture for closed algebraic 1-forms, we establish a Riemann-Hilbert correspondence of Deligne-Malgrange type. As an application, we prove a variant of the comparison of isomorphisms theorem for a simple class of algebraic 1-forms on complex curves.
We consider several examples of nonautonomous systems of difference equations coming from semi-classical orthogonal polynomials via recurrence coefficients and ladder operators, with respect to various generalisations of Laguerre and Meixner weights. We identify these as discrete Painlevé equations and establish their types in the Sakai classification scheme in terms of the associated rational surfaces. In particular, we find examples which come from different weights and share a common surface type D_5^(1) but are inequivalent in two ways. First, their dynamics are generated by non-conjugate elements of W(A_3^(1)). Second, some of the examples have associated surfaces being non-generic in the sense of having nodal curves. The symmetries of these examples form subgroups of the generic symmetry group, which we compute. In particular, we find (W(A_1^(1))× W(A_1^(1)))⋊ℤ/2ℤ. These examples give further weight to the argument that any correspondence between different weights and the Sakai classification should make use of the refined version of the discrete Painlevé equivalence problem, which takes into account not just surface type, but also the group elements generating the dynamics as well as parameter constraints, e.g. those corresponding to nodal curves.
In models of phase coexistence, the precise form of the double-well potential is of central importance, yet it cannot be derived from first principles. In this paper, we investigate an inverse problem: starting from a prescribed transition layer with power-type decay at infinity, we reconstruct the structural properties of the associated double-well potential. We focus on the case of long-range interactions, where the dependence of the potential on the layer and its derivatives is particularly delicate. Our analysis establishes a correspondence between the decay rate of the transition layer and the regularity of the potential, revealing the existence of specific patterns and the possible emergence of degeneracies.
The main result of this paper is an explicit description of the stratification of the phase space of Calogero–Moser–Sutherland (CMS) integrable systems corresponding to Lie groups SU(n). The phase space decomposes into symplectic strata of dimensions 2s, where s = 0, 1, … , n - 1 . On each stratum of the positive dimension, we construct natural action-angle coordinates and compute the symplectic form explicitly, showing that every stratum is symplectomorphic to ℝ_> 0^s ×𝕋^s . The zero-dimensional stratum corresponds to the equilibrium point of the multi-time CMS dynamics.
