We investigate the irreducible smooth $\widehat{\mathfrak{sl}}_{2}$-modules recently constructed in [Adv. Math. 481 (2025), 110559, 34 pages, arXiv:2404.03855], and demonstrate that these modules admit a Wakimoto-type realization at both critical and non-critical levels. In the critical level case, we identify simple quotients of these modules with the Wakimoto modules whose irreducibility was already established by Adamović. We also generalize some Wakimoto modules constructed in [Adv. Math. 289 (2016), 438-479, arXiv:1409.5354] and identify them as generalized Whittaker modules.
Let $X$ be a compact polyhedral surface (a compact Riemann surface with flat conformal metric $\mathfrak{T}$ having conical singularities). The $\zeta$-function $\zeta_\Delta(s)$ of the Friedrichs Laplacian on $X$ is meromorphic in ${\mathbb C}$ with a single simple pole at $s=1$. We define $\operatorname{reg}\zeta_\Delta(1)$ as $\lim\limits_{s\to 1} \bigl( \zeta_\Delta(s)-\frac{ {\rm Area}(X,\mathfrak{T}) }{4\pi(s-1)}\bigr)$. We derive an explicit expression for this spectral invariant through the holomorphic invariants of the Riemann surface $X$ and the (generalized) divisor of the conical points of the metric $\mathfrak{T}$. We study the asymptotics of $\operatorname{reg}\zeta_\Delta(1)$ for the polyhedron obtained by sewing two other polyhedra along segments of small length. In addition, we calculate $\operatorname{reg}\zeta(1)$ for a family of (non-Friedrichs) self-adjoint extensions of the Laplacian on the tetrahedron with all the conical angles equal to $\pi$.
We investigate the irreducible smooth bsl2-modules recently constructed in [Adv. Math. 481 (2025), 110559, 34 pages, arXiv:2404.03855], and demonstrate that these modules admit a Wakimoto-type realization at both critical and non-critical levels. In the critical level case, we identify simple quotients of these modules with the Wakimoto modules whose irreducibility was already established by Adamovic. We also generalize some Wakimoto modules constructed in [Adv. Math. 289 (2016), 438-479, arXiv:1409.5354] and identify them as generalized Whittaker modules.
Let M-p, be a circle bundle with first Chern class p[w] over a closed 4n-dimensional integral symplectic manifold (M-,w). Equivalently, M-p, is a closed contact (4n + 1)-manifold whose Reeb orbits are all closed and have the same period. For a metric gon M-p, compati ble with the symplectic structure and the geometry of the circle fiber, we use Wodzicki-Chern Simons forms on the loop space LMp, to prove that pi(1)(M-p,g)) is infinite for |p| >> 0 We also give the first high-dimensional examples of nonvanishing Wodzicki Pontryagin forms.
We conclude our work on asymptotic expansions at the soft edge for the classical $n$-dimensional Gaussian and Laguerre ensembles, now studying the gap-probability generating functions. We show that the correction terms in the asymptotic expansion are multilinear forms of the higher-order derivatives of the leading-order term, with certain rational polynomial coefficients that are independent of the dummy generating function variable. In this way, the same multilinear structure, with the same polynomial coefficients, is inherited by the asymptotic expansion of any linearly induced quantity such as the distribution of the $k$-th largest level. Whereas the results for the unitary ensembles are presented with proof, the discussion of the orthogonal and symplectic ones is based on some hypotheses. To substantiate the hypotheses, we check the result for the $k$-th largest level in the orthogonal ensembles against simulation data for choices of $n$ and $k$ that require as many as four correction terms to achieve satisfactory accuracy.
We define a (3+1)-TQFT associated with possibly non-semisimple finite unimodular ribbon tensor categories using skein theory. This gives an explicit realization of a TQFT predicted by the cobordism hypothesis, based on recent results on dualizability. State spaces are given by admissible skein modules, and we prescribe the TQFT on handle attachments. We give some explicit algebraic conditions on the input category to define this TQFT, namely to be "chromatic non-degenerate". As a by-product, we obtain an invariant of 4-manifolds equipped with a ribbon graph in their boundary, and in the "twist non-degenerate" case, an invariant of 3-manifolds. Our construction generalizes the Crane-Yetter-Kauffman TQFTs in the semi-simple case, and the Lyubashenko (hence also Hennings and WRT) invariants of 3-manifolds. The whole construction is very elementary, and we can easily characterize the invertibility of the TQFTs, study their behavior under connected sums and provide some examples.
In this paper, we compute the 1-point correlation functions of all states for the $\mathbb{Z}_2$-orbifolds of lattice vertex operator algebras.
In this paper, we consider characterisations of the class of unitary matrix integrals $\big\langle (\det U)^q {\rm e}^{s^{1/2} \operatorname{Tr}(U + U^\dagger)} \big\rangle_{U(l)}$ in terms of a first-order matrix linear differential equation for a vector function of size $l+1$, and in terms of a scalar linear differential equation of degree ${l+1}$. It will be shown that the latter follows from the former. The matrix linear differential equation provides an efficient way to compute the power series expansion of the matrix integrals, which with $q=0$ and $q=l$ are of relevance to the enumeration of longest increasing subsequences for random permutations, and to the question of the moments of the first and second derivative of the Riemann zeta function on the critical line, respectively. This procedure is compared against that following from known characterisations involving the $\sigma$-Painlevé III$'$ second-order nonlinear differential equation. We show too that the natural $\beta$ generalisation of the unitary group integral permits characterisation by the same classes of linear differential equations.
Saïd-Kobayashi-Ørsted introduced a family of $ \mathfrak{sl}_2 $-triples of differential-difference operators $ \mathbb{H}_{k,a} $, $ \mathbb{E}^+_{k,a} $ and $ \mathbb{E}^-_{k,a} $ on $ \mathbb{R}^N \setminus \{0\} $ indexed by a Dunkl parameter $ k $ and a deformation parameter $ a \neq 0 $. In the present paper, we study the behavior as the parameter $ a $ approaches $ 0 $. In this limit, the Lie algebra $ \mathfrak{g}_{k,a} = \operatorname{span}_\mathbb{R} \bigl\{\mathbb{H}_{k,a}, \mathbb{E}^+_{k,a}, \mathbb{E}^-_{k,a}\bigr\} \cong \mathfrak{sl}(2, \mathbb{R}) $ contracts to a three-dimensional commutative Lie algebra $ \mathfrak{g}_{k,0}$, and its spectral properties change. We describe the joint spectral decomposition for $ \mathfrak{g}_{k,0}$, and discuss formulas for operator semigroups with infinitesimal generators in $ \mathfrak{g}_{k,0}$. In particular, we describe the integral kernel of $ \exp\bigl(z |x|^2 \Delta_k\bigr) $ as an infinite series, which, in some low-dimensional cases, can be expressed in a closed form using the theta function.
By introducing a dynamical version of the second fundamental form, we generalize a recent result of Filip-Fisher-Lowe to the setting of magnetic systems. Namely, we show that a real-analytic negatively s-curved magnetic system on a closed real-analytic manifold has only finitely many closed totally s-magnetic hypersurfaces, unless the magnetic 2-form is trivial and the underlying metric is hyperbolic.
We discuss some examples in which symplectic monodromy (provably or conjecturally) splits off the symplectic mapping class group, hoping to illustrate different techniques and inputs to the arguments. Along the way we formulate several open questions and conjectures. Inter alia, we also construct a compact symplectic six-manifold which contains infinitely many pairwise disjoint Lagrangian 3-spheres.
In this note, we show that the weighted mass of Baldauf and Ozuch (2022) can be derived as a natural geometric mass invariant following Michel (2011), for a certain weighted curvature map. An associated weighted centre of mass definition is also derived from this. The adjoint of the linearisation of this curvature map leads to a notion of weighted static metrics, which are natural candidates for weighted mass minimisers. This weighted curvature quantity is essentially the scalar curvature of a conformally related metric that Law, Lopez and Santiago (2025) used to considerably simplify the proof of the weighted positive mass theorem. We show an equivalence between static metrics and weighted static metrics via the conformal relationship, from which we show that a uniqueness theorem holds for weighted static manifolds with weighted minimal surface boundaries. Furthermore, we show that weighted manifolds satisfy a Riemannian Penrose inequality whose equality case holds precisely for these unique weighted static metrics.
In this note, we consider contractible loops of contactomorphisms that are positive over some non-empty closed subset of a contact manifold. Such closed subsets are called immaterial. We argue that the complement of a Reeb-invariant immaterial subset can be seen as big in contact geometric terms. This is supported by two results: one regarding symplectic homology of the filling and the other regarding recently introduced contact quasi-measures.
Given a semi-simple algebra equipped with a coproduct, the Clebsch-Gordan coefficients are the elements of the transition matrices between direct product representation and its irreducible decomposition. It is well known that the Clebsch-Gordan coefficients of the Lie algebra sl2 are given in terms of the dual Hahn polynomials. Taking the reversed point of view, we show that any finite dimensional family of polynomials belonging to the Askey scheme can be interpreted as Clebsch-Gordan coefficients of an algebra. The Hahn polynomials are thus associated to the oscillator algebra with the Krawtchouk polynomials treated through a limit. The dual Hahn polynomials and Racah polynomials are seen to be associated to sl2 with a more general copro duct than the standard one. The q-Hahn polynomials are interpreted as Clebsch-Gordan coefficients of a q-deformation of the oscillator algebra and the q-Racah polynomials are seen to be connected in this way to Uq(sl2) with a generalized copro duct.
We conjecture affine or Hall-Littlewood analogues of the dual Jacobi-Trudi identities for orthogonal and symplectic Schur functions indexed by rectangular partitions of maximal height. These conjectures are then used to derive t-analogues of many known Rogers-Ramanujan identities for the characters of standard modules of affine Lie algebras. This includes t-analogues of the classical Rogers-Ramanujan identities, (some of) the Andrews-Gordon identities and the C(1) n , A(2) 2n and D(2) n+2 GOW identities. We also prove an affine analogue of the dual Jacobi-Trudi identity for Schur functions indexed by rectangular partitions of arbitrary height.
We explore a method, initiated by Guichard in 1890, which allows to generate sequences of Voss surfaces, starting from an arbitrarily chosen pseudospherical surface and a seed solution of the Moutard equation, by means of two simple transformations. In this paper, we 1) identify the Guichard transformations with the well-known recursion operator for symmetries of the sine-Gordon equation and its inverse; 2) prove a lemma which allows us to derive the length of Guichard's sequences from the invariance properties of the initial sine-Gordon solution; 3) introduce an extended class of inverted operators, expanding the class of Voss surfaces obtainable by quadratures; 4) clarify relevant aspects of Guthrie's formalism, paving the way for the future employment of the entire division algebra of recursion operators. A number of Voss nets are presented explicitly.
One of the starting points of this work was the duality of Borcea relating standard level k representations of A(1)1 and level 2k+1 of A(2) 2 . For k = 1, the combinatorial bases in both cases yield the two Capparelli identities and we wanted to see if there is a correspondence between the bases in terms of partitions for all k is an element of N. By using the vertex operator relations in the principal picture for level 5 standard A(2) 2-modules, we reduce a spanning set of Poincare-Birkhoff-Witt-type vectors in L(5 Lambda 0) by removing the leading terms of relations and rendering a list of 34 "difference" conditions for partitions. Using computer programs, we enumerated the partitions satisfying these conditions and obtained a truncated generating series agreeing with the principally specialized character for all powers of q up to 41. Although our list of leading terms is incomplete, our results show that the corresponding combinatorial identity for LA(2)2 (5 Lambda 0) drastically differs from the one for the Borcea dual LA(1) 1 (2 Lambda 0).
It is well known that for any wave function psi(x, z) of the KP hierarchy, there is another wave function called its "adjoint" such that the path integral of their product with respect to z around any sufficiently large closed path is zero. For the wave functions in the adelic Grassmannian Grad, the bispectral involution which exchanges the role of x and z also implies the existence of an "x-adjoint wave function" psi star(x, z) so that the product of the wave function, the x-adjoint, and the Hermite weight e-x2/2 has no residue. Utilizing this, we show that the sequences of coefficient functions in the power series expansion of any KP wave function in Grad and its image under the bispectral involution at t2 = 12 are always "almost bi-orthogonal" with respect to the Hermite product. Whether the sequences have the stronger properties of being (almost) orthogonal can easily be determined in terms of KP flows and the bispectral involution. As a special case, the exceptional Hermite orthogonal polynomials can be recovered in this way. This provides both a generalization of and an explanation of the fact that the generating functions of the exceptional Hermites are certain special wave functions of the KP hierarchy. In addition, one new surprise is that the same KP wave function which generates the sequences of functions is also a generating function for the norms when evaluated at t1 = 1 and t2 = 0. The main results are proved using Calogero-Moser matrices satisfying a rank one condition. The same results also apply in the case of "spin-generalized" Calogero-Moser matrices, which produce instances of matrix orthogonality.
We establish some new bilateral double-sum Rogers-Ramanujan identities involving parameters. As applications, these identities yield several new multi-sum Rogers-Ramanujan type identities. Our proofs utilize the theory of basic hypergeometric series in conjunction with the integral method.
In this paper, we use a formula obtained by Dubrovin and Yang (2017) to study certain asymptotic behaviors of GUE (Gaussian unitary ensemble) correlators. More precisely, we obtain large genus asymptotics of enumerations of ordinary graphs and ribbon graphs with 1 face.
