arXiv subjects

Jonas Matuzas

Publications and source records attributed to Jonas Matuzas.

8 recordsLinked to original sources

arXiv2026

Shifted poles and chamber cancellation for classical Witten zeta functions

We determine two infinite families of poles on the positive real axis for classical single-variable Witten zeta functions. In type $A_r$, for $r \geq 5$, the point $q_r^A = 2(r-4)/(r^2+r-4)$ is a simple pole except in ranks $12$ and $20$, where it is a double pole. In type $D_r$, for $r \geq 4$, the point $q_r^D = (r-3)/(r(r-1)-1)$ is a simple pole except at $D_8$, where it is double. In root-product normalization, we express the simple residues and the leading coefficients of these three double poles in terms of gamma, trigonometric, and Riemann zeta values. For $r \geq 4$, the functions of types $B_r$ and $C_r$ are holomorphic at $q_r^{BC} = (r-3)/(r^2-1)$ except possibly in ranks $7$ and $11$, where any pole is simple. These pole and holomorphy statements arise from quadratic normal Taylor coefficients. For every fixed higher even normal degree, we also determine exactly when the associated continued $B/C$ chamber sum is nonzero; this auxiliary result does not by itself classify poles of the full Witten function. The proofs combine exhaustive support classifications, explicit integration-by-parts identities, and finite chamber relations. No numerical nonvanishing estimate is used.

math.RT↗

arXiv2026

Second-pole wall periods for Witten zeta functions in the classical families

Let Phi be an irreducible reduced crystallographic root system of rank r, and let N be the number of positive coroots. A previous theorem (arXiv:2608.16363, Theorem 3.3) identifies the first pole below 2/h as q_2 = (r-1)/(N-1) and expresses its residue as a sum of periods attached to the simple walls of the dominant chamber. We evaluate that wall-period sum for all four classical families. The identity q_2 (N-1) = r-1 removes the radial variable from every wall integral. The resulting projective integrals reduce to mixed Dotsenko-Fateev chambers in type A_r, to those chambers together with a Selberg endpoint in types B_r and C_r, and to a single chamber family in type D_r. The chamber recurrences give explicit sine weights in types A, B, and C. In type D, a terminating basic-hypergeometric sum at a root of unity reduces to a finite cyclotomic product. We also show that the corresponding exceptional wall restrictions are not reflection arrangements, so this particular reduction does not extend directly to the exceptional types.

math.RT↗

arXiv2026

The second pole of Witten zeta functions and exact evaluations in types F4 and D5

Let Phi be an irreducible reduced crystallographic root system of rank r at least 2, let N = |Phi^+| be the number of positive roots, and let h be its Coxeter number. For the normalized Witten zeta function xi_Phi, we determine the first distinct pole below the leading pole 2/h. It is located at q_2(Phi) = (r-1)/(N-1) = 2(r-1)/(rh-2), is simple, and receives contributions precisely from the codimension-one faces of the dominant chamber. Its residue is zeta_R(q_2)/(N-1) times the sum of the corresponding wall periods, where zeta_R denotes the Riemann zeta function. These periods are finite and positive, so the residue is strictly negative. We also prove a Stokes relation for projective hyperplane-arrangement periods. Let A be an essential central real arrangement in R^n with weights lambda_H strictly between 0 and 1. Suppose that the sum of lambda_H over all H in A equals n, and that for every nonzero proper intersection flat X the sum of lambda_H over those H containing X is strictly less than the codimension of X. Then the vector of positive chamber periods lies in the kernel of the Varchenko matrix with weights exp(pi i lambda_H). Applying this relation, we evaluate the relevant wall periods in types F4 and D5. A two-orbit reduction and Dixon's 3F2(1) summation give a gamma-product evaluation in type F4, while a four-orbit reduction and Selberg's integral give a gamma-product evaluation of the complete wall sum in type D5. Consequently, we obtain exact formulas for the normalized and ordinary Witten zeta residues at 3/23 and 4/19.

math.RT↗

arXiv2026

Generic polar divisors and flag residues for root-system zeta functions

Let $\Phi$ be an irreducible crystallographic root system, and let $Z_\Phi(\mathbf{s})$ denote the untwisted Komori-Matsumoto-Tsumura zeta function with one exponent for each positive coroot. For a nonempty set $S$ of simple nodes, let $H_{S,\ell}$ be the hyperplane on which the exponents of the roots meeting $S$ sum to $|S|-\ell$. We prove that every proper-support hyperplane $H_{S,\ell}$ is a genuine polar divisor at a generic point, whereas exact homogeneity leaves only the unshifted full-support divisor. The residue on $H_{S,\ell}$ is expressed as a finite Taylor-jet sum of reduced projective periods and polynomially weighted complementary root-system zeta functions. On the maximal support wonderful model, boundary terms are indexed by strict decorated flags. We derive recursive flag residues, component-mass gamma factors, and an incidence-complete formula for the Laurent coefficients on any transverse affine slice. In particular, the pole order is determined by the first nonzero aggregate coefficient, not by the largest order of an individual flag. The general formulas recover the classical $A_2$ and $A_3$ singular data, Zhao's Euler-Zagier residues, and the rank-two $C_2$ and $G_2$ residue functions. For $B_3$ and $C_3$ we derive the carrier geometry and the lower-rank factorizations of the positive residues, identify the three-term cancellation at $s=1/8$, and show that negative half-integers are the only possible locations of double poles. The known $B_3$ double coefficient at $-1/2$ is recovered in the flag normalization.

math.RT↗

arXiv2026

A universal leading-residue formula for Witten zeta functions

Let $\Phi$ be an irreducible crystallographic root system of rank $r$, with Coxeter number $h$, Weyl group $W$, Cartan matrix $C_\Phi$, and invariant degrees $2=d_1\leq\cdots\leq d_r=h$. We prove that Au's normalized Witten zeta function has a simple pole at $2/h$ and evaluate its residue as $\frac{2(2\pi)^{r/2}\sqrt{\det C_\Phi}}{h|W|}\cdot\frac{\prod_{i<r}\Gamma(1-d_i/h)}{\Gamma(1-1/h)^r}$. The central step evaluates the critical chamber integral in gamma values from the boundary pole of the Macdonald-Mehta-Opdam identity; two preparatory sections pass from the dominant-weight lattice to that convergent integral. This proves Au's conjecture on algebraic multiples of products of gamma values at rational arguments, including his $A_4$ prediction.

math.RT↗

arXiv2026

A strict gap above the Brito--Chacon--Naveira bound for the Sasaki volume on higher odd spheres

For a smooth unit vector field $V$ on the round sphere $S^{2m+1}$, its Sasaki volume is the volume of its graph in the unit tangent bundle. Brito, Chac\'on and Naveira proved that this volume is at least $c_m\operatorname{vol}(S^{2m+1})$, where $c_m=4^m/\binom{2m}{m}$. We prove that, for every $m\ge2$, the infimum over smooth unit fields is strictly larger than this value. The proof constructs a global closed comass-one form on $T^1S^{2m+1}$ and classifies all of its equality planes. A hypothetical sequence approaching the bound has a calibrated integral-current limit. Its positive horizontal-Jacobian part is a multiplicity-one graph, whereas its projection-degenerate residual is confined to a base-rank-one equality plane. Coordinate test forms show that this residual contributes no singular part to the distributional derivative of the graph section. The section is therefore Sobolev and satisfies a scalar concircular equation; a weak rigidity theorem identifies it with a radial distance-gradient field. The radial graph has the two pole fibers as boundary, while a local boundary--mass identity forces the residual to vanish, contradicting the cycle condition. We also prove a sharp current-level flat-boundary inequality for fillings of the pole fibers. Consequently, every sequence of smooth unit fields converging in measure to a radial field has lower limiting volume at least twice the Brito--Chac\'on--Naveira bound. The strict gap obtained here is qualitative; no explicit gap constant is claimed.

math.DG↗

arXiv2026

A Non-Reciprocal Elliptic Spectral Solution of the Right-Angle Penetrable Wedge Transmission Problem

We study the two-dimensional time-harmonic scalar transmission problem for an impedance-matched penetrable right-angle wedge: the exterior medium has wavenumber k_0 and the interior sector |theta| < pi/4 has wavenumber k_1 = nu*k_0 with nu > 1, with continuity of the total field and its normal derivative across each face. A Sommerfeld-Malyuzhinets reduction leads to a 2x2 matrix Riemann-Hilbert (RH) problem on the Snell surface Sigma_nu. For the right angle the surface has genus one, and we give an explicit theta-function uniformization and a closed-form Mittag-Leffler construction of the full family of elliptic RH solutions with finite forcing (prescribed poles and residues), subject to a single residue-sum constraint encoding the Meixner edge condition. We then consider the additional forcing data required to model plane-wave incidence. Numerical reciprocity tests show that the minimal one-point plane-wave prescription does not yield a physically closed solution: the natural sheet-swap pairing u# = u + omega_2 produces nontrivial scattered densities but violates the far-field reciprocity benchmark, whereas the Hardy pairing u# = omega_1 - u enforces reciprocity but collapses the scattered field to zero. The paper therefore provides an exact elliptic spectral framework and a reproducible reciprocity diagnostic, while identifying what must be added (multi-point forcing and/or modified pairing incorporating both wedge faces) to obtain a reciprocity-consistent plane-wave solution.

math-ph↗

arXiv2026

A Non-Reciprocal Elliptic Spectral Solution of the Right-Angle Penetrable Wedge Transmission Problem for $\nu=\sqrt{2}$

We consider the two-dimensional time-harmonic transmission problem for an impedance-matched (\rho = 1) right-angle penetrable wedge at refractive index ratio \nu = \sqrt{2}, in the integrable lemniscatic configuration (\theta_w ,\nu,\rho) = (\pi/4,\sqrt{2},1). Starting from Sommerfeld spectral representations, the transmission conditions on the two wedge faces yield a closed spectral functional system for the Sommerfeld transforms Q(\zeta) and S(\zeta). In this special configuration the associated Snell surface is the lemniscatic curve Y^2 = 2(t^4 + 1), uniformized by square-lattice Weierstrass functions with invariants (g_2,g_3) = (4,0). We construct an explicit meromorphic expression for a scattered transform Q_{scat} as a finite Weierstrass--\zeta sum plus an explicitly constructed pole-free elliptic remainder, with all pole coefficients computed algebraically from the forcing pole set. A birational (injective) uniformization is used to avoid label collisions on the torus and to make the scattered-allocation pole exclusion well posed. The resulting closed form solves the derived spectral functional system and satisfies the local regularity constraints imposed at the physical basepoint. However, numerical reciprocity tests on the far-field coefficient extracted from Q_{scat} indicate that the construction is generally non-reciprocal; accordingly we do not claim that the resulting diffraction coefficient coincides with the reciprocal physical transmission scattering solution. The result remains restricted to this integrable lemniscatic case; the general penetrable wedge remains challenging (see [10--12] and, in a related high-frequency penetrable-corner setting, [13]).

math-ph↗