arXiv subjects

Leon A. Takhtajan

Publications and source records attributed to Leon A. Takhtajan.

At least 19 recordsLinked to original sources

arXiv2026

Supersymmetry and trace formulas III. Frenkel trace formula

By applying the new supersymmetric localization principle introduced in \cite{Choi:2021yuz,Choi:2023pjn}, we present two complementary approaches for the path integral derivation of the `non-chiral' trace formula for a semisimple compact Lie group $G$, which generalizes the so-called Frenkel trace formula. Corresponding physical systems for each picture are the quantum mechanical sigma model on $G$ and the gauged sigma model on $G\times G$, and the approaches closely follow the spirit of the Eskin trace formula \cite{Choi:2021yuz} and the Selberg trace formula \cite{Choi:2023pjn} respectively. These methods provide a natural conceptual bridge between two seemingly independent derivations in \cite{Choi:2021yuz} and \cite{Choi:2023pjn}.

hep-th↗

arXiv2026

Supersymmetry and trace formulas II. Selberg trace formula

By extending the new supersymmetric localization principle introduced in \cite{Choi:2021yuz}, we present a path integral derivation of the Selberg trace formula on arbitrary compact Riemann surfaces, including the case of vector-valued automorphic forms of arbitrary half-integer weight corresponding to Maass Laplacian. We also generalize the method to formulate the Selberg trace formula on generic compact locally symmetric space.

hep-th↗

arXiv2025

Worldline Localization

We show that two elementary worldline path integrals-the thermal partition function of the harmonic oscillator and the one-loop effective action of scalar QED in a constant field strength-exhibit a natural form of supersymmetric localization. The mechanism relies on hidden fermionic symmetries of the worldline BRST formulation, rather than on standard BRST structure or physical supersymmetry. These symmetries localize the target-space trajectory. For the harmonic oscillator this yields an alternative localization derivation of the Jacobi-Poisson formula. Moreover, after the trajectory is localized, the remaining proper-time integral exhibits an emergent zero-dimensional supersymmetry generated by modular invariance, allowing the modulus T itself to be localized. For scalar QED the same structure provides a controlled computation of both the real and imaginary parts of the Euler-Heisenberg effective action. In particular, the imaginary part arises from a moduli space of circular worldline instantons, offering a localization perspective on the semiclassical exactness of the Schwinger pair-production result observed by Affleck-Alvarez-Manton.

hep-th↗

arXiv2025

Supersymmetry and trace formulas I. Compact Lie groups

In the context of supersymmetric quantum mechanics we formulate new supersymmetric localization principle, with application to trace formulas for a full thermal partition function. Unlike the standard localization principle, this new principle allows to compute the supertrace of non-supersymmetric observables, and is based on the existence of fermionic zero modes. We describe corresponding new invariant supersymmetric deformations of the path integral; they differ from the standard deformations arising from the circle action and require higher derivatives terms. Consequently, we prove that the path integral localizes to periodic orbits and not not only on constant ones. We illustrate the principle by deriving bosonic trace formulas on compact Lie groups, including classical Jacobi inversion formula.

hep-th↗

arXiv2024

Local index theorem for orbifold Riemann surfaces

We derive a local index theorem in Quillen's form for families of Cauchy-Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces) that are quotients of the hyperbolic plane by the action of cofinite finitely generated Fuchsian groups. Each conical point (or a conjugacy class of primitive elliptic elements in the Fuchsian group) gives rise to an extra term in the local index theorem that is proportional to the symplectic form of a new Kähler metric on the moduli space of Riemann orbisurfaces. We find a simple formula for a local Kähler potential of the elliptic metric and show that when the order of elliptic element becomes large, the elliptic metric converges to the cuspidal one corresponding to a puncture on the orbisurface (or a conjugacy class of primitive parabolic elements). We also give a simple example of a relation between the elliptic metric and special values of Selberg's zeta function.

math.AG↗

arXiv2022

Goldman form, flat connections and stable vector bundles

We consider the moduli space $\mathscr{N}$ of stable vector bundles of degree $0$ over a compact Riemann surface and the affine bundle $\mathscr{A}\to\mathscr{N}$ of flat connections. Following the similarity between the Teichmüller spaces and the moduli of bundles, we introduce the analogue of of the quasi-Fuchsian projective connections - local holomorphic sections of $\mathscr{A}$ - that allow to pull back the Liouville symplectic form on $T^{*}\mathscr{N}$ to $\mathscr{A}$. We prove that the pullback of the Goldman form to $\mathscr{A}$ by the Riemann-Hilbert correspondence coincides with the pullback of the Liouville form. We also include a simple proof, in the spirit of Riemann bilinear relations, of the classic result - the pullback of Goldman symplectic form to $\mathscr{N}$ by the Narasimhan-Seshadri connection is the natural symplectic form on $\mathscr{N}$, introduced by Narasimhan and Atiyah & Bott.

math.AG↗

arXiv2022

Logarithmic connections, WZNW action, and moduli of parabolic bundles on the sphere

Moduli spaces of stable parabolic bundles of parabolic degree $0$ over the Riemann sphere are stratified according to the Harder--Narasimhan filtration of underlying vector bundles. Over a Zariski open subset $\mathscr{N}_{0}$ of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function $\mathscr{S}$ is defined as the regularized critical value of the non-compact Wess--Zumino--Novikov--Witten action functional. The definition of $\mathscr{S}$ depends on a suitable notion of parabolic bundle `uniformization map' following from the Mehta--Seshadri and Birkhoff--Grothendieck theorems. It is shown that $-\mathscr{S}$ is a primitive for a (1,0)-form $\vartheta$ on $\mathscr{N}_{0}$ associated with the uniformization data of each intrinsic irreducible unitary logarithmic connection. Moreover, it is proved that $-\mathscr{S}$ is a Kähler potential for $(Ω-Ω_{\mathrm{T}})|_{\mathscr{N}_{0}}$, where $Ω$ is the Narasimhan--Atiyah--Bott Kähler form in $\mathscr{N}$ and $Ω_{\mathrm{T}}$ is a certain linear combination of tautological $(1,1)$-forms associated with the marked points. These results provide an explicit relation between the cohomology class $[Ω]$ and tautological classes, which holds globally over certain open chambers of parabolic weights where $\mathscr{N}_{0} = \mathscr{N}$.

math.CV↗

arXiv2021

Trace formulas for the modified Mathieu equation

For the radial and one-dimensional Schrödinger operator $H$ with growing potential $q(x)$ we outline a method of obtaining the trace identities - an asymptotic expansion of the Fredholm determinant $\mathrm{det}_{F}(H-λI)$ as $λ\to-\infty$. As an illustrating example, we consider Schrödinger operator with the potential $q(x)=2\cosh 2x$, associated with the modified Mathieu equation.

math.SP↗

arXiv2016

Weyl type asymptotics and bounds for the eigenvalues of functional-difference operators for mirror curves

We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are $H(ζ)=U+U^{-1}+V+ζV^{-1}$ and $H_{m,n}=U+V+q^{-mn}U^{-m}V^{-n}$, where $U$ and $V$ are self-adjoint Weyl operators satisfying $UV=q^{2}VU$ with $q=e^{iπb^{2}}$, $b>0$ and $ζ>0$, $m,n\in\mathbb{N}$. We prove that $H(ζ)$ and $H_{m,n}$ are self-adjoint operators with purely discrete spectrum on $L^{2}(\mathbb{R})$. Using the coherent state transform we find the asymptotical behaviour for the Riesz mean $\sum_{j\ge 1}(λ-λ_{j})_{+}$ as $λ\to\infty$ and prove the Weyl law for the eigenvalue counting function $N(λ)$ for these operators, which imply that their inverses are of trace class.

math.SP↗

arXiv2015

Potentials and Chern forms for Weil-Petersson and Takhtajan-Zograf metrics on moduli spaces

For the TZ metric on the moduli space $\mathscr{M}_{0,n}$ of $n$-pointed rational curves, we construct a Kähler potential in terms of the Fourier coefficients of the Klein's Hauptmodul. We define the space $\mathfrak{S}_{g,n}$ as holomorphic fibration $\mathfrak{S}_{g,n}\rightarrow\mathfrak{S}_{g}$ over the Schottky space $\mathfrak{S}_{g}$ of compact Riemann surfaces of genus $g$, where the fibers are configuration spaces of $n$ points. For the tautological line bundles $\mathscr{L}_{i}$ over $\mathfrak{S}_{g,n}$ we define Hermitian metrics $h_{i}$ in terms of Fourier coefficients of a covering map $J$ of the Schottky domain. We define the regularized classical Liouville action $S$ and show that $\exp\{S/π\}$ is a Hermitian metric in the line bundle $\mathscr{L}=\otimes_{i=1}^{n}\mathscr{L}_{i}$ over $\mathfrak{S}_{g,n}$. We explicitly compute the Chern forms of these Hermitian line bundles $$c_{1}(\mathscr{L}_{i},h_{i})=\frac{4}{3}ω_{\mathrm{TZ},i},\quad c_{1}(\mathscr{L},\exp\{S/π\})=\frac{1}{π^{2}}ω_{\mathrm{WP}}.$$ We prove that a smooth real-valued function $-\mathscr{S}=-S+π\sum_{i=1}^{n}\log h_{i}$ on $\mathfrak{S}_{g,n}$, a potential for this special difference of WP and TZ metrics, coincides with the renormalized hyperbolic volume of a corresponding Schottky $3$-manifold. We extend these results to the quasi-Fuchsian groups of type $(g,n)$.

math.AG↗

arXiv2015

On Bott-Chern forms and their applications

We use Chern-Weil theory for Hermitian holomorphic vector bundles with canonical connections for explicit computation of the Chern forms of trivial bundles with special non-diagonal Hermitian metrics. We prove that every del-dellbar exact real form of the type (k,k) on an n-dimensional complex manifold X arises as a difference of the Chern character forms of trivial Hermitian vector bundles with canonical connections, and that (modulo the image of del and delbar) every real form of type (k,k), k<n, arises as a Bott-Chern form for two Hermitian metrics on some trivial vector bundle over X. The latter result is a complex manifold analogue of Proposition 2.6 in the paper arXiv: 0810.4935 by J. Simons and D. Sullivan. As an application, we obtain an explicit formula for the Bott-Chern form of a short exact sequence of holomorphic vector bundles, considered by Bott and Chern in classic 1965 paper, for the case when the first term is a line bundle. We also present a very simple explicit formula for the total Chern form of a hypersurface in the complex projective space.

math.DG↗

arXiv2014

On the spectral theory of one functional-difference operator from conformal field theory

In the paper we consider a functional-difference operator $H=U+U^{-1}+V$, where $U$ and $V$ are self-adjoint Weyl operators satisfying $UV=q^{2}VU$ with $q=e^{πiτ}$ and $τ>0$. The operator $H$ has applications in the conformal field theory and in the representation theory of quantum groups. Using modular quantum dilogarithm - a $q$-deformation of the Euler's dilogarithm - we define the scattering solution and the Jost solutions, derive an explicit formula for the resolvent of the self-adjoint operator $H$ in the Hilbert space $L^{2}(\mathbb{R})$, and prove the eigenfunction expansion theorem. The latter is a $q$-deformation of the well-known Kontorovich-Lebedev transform in the theory of special functions. We also present a formulation of the scattering theory for the operator $H$.

math.SP↗

arXiv2013

Quantum Field Theories on Algebraic Curves. I. Additive bosons

Using Serre's adelic interpretation of cohomology, we develop a `differential and integral calculus' on an algebraic curve X over an algebraically closed filed k of constants of characteristic zero, define algebraic analogs of additive multi-valued functions on X and prove corresponding generalized residue theorem. Using the representation theory of the global Heisenberg and lattice Lie algebras, we formulate quantum field theories of additive and charged bosons on an algebraic curve X. These theories are naturally connected with the algebraic de Rham theorem. We prove that an extension of global symmetries (Witten's additive Ward identities) from the k-vector space of rational functions on X to the vector space of additive multi-valued functions uniquely determines these quantum theories of additive and charged bosons.

math.AG↗

arXiv2007

The first Chern form on moduli of parabolic bundles

For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen's metric and interpret it as a local index theorem for the family of dbar-operators in associated parabolic endomorphism bundles. The formula consists of two terms: one standard (proportional to the canonical Kaehler form on the moduli space), and one nonstandard, called a cuspidal defect, that is defined by means of special values of the Eisenstein-Maass series. The cuspidal defect is explicitly expressed through curvature forms of certain natural line bundles on the moduli space related to the parabolic structure. We also compare our result with Witten's volume computation.

math.AG↗

arXiv2007

Normal matrix models, dbar-problem, and orthogonal polynomials on the complex plane

We introduce a dbar-formulation of the orthogonal polynomials on the complex plane, and hence of the related normal matrix model, which is expected to play the same role as the Riemann-Hilbert formalism in the theory of orthogonal polynomials on the line and for the related Hermitian model. We propose an analog of Deift-Kriecherbauer-McLaughlin-Venakides-Zhou asymptotic method for the analysis of the relevant dbar-problem, and indicate how familiar steps for the Hermitian model, e.g. the g-function ``undressing'', might look like in the case of the normal model. We use the particular model considered recently by P. Elbau and G. Felder as a case study.

math.CA↗

arXiv2006

Quantum Liouville theory in the background field formalism I. Compact Riemann surfaces

Using Polyakov's functional integral approach with the Liouville action functional defined in \cite{ZT2} and \cite{LTT}, we formulate quantum Liouville theory on a compact Riemann surface X of genus g > 1. For the partition function and for the correlation functions with the stress-energy tensor components $<\prod_{i=1}^{n}T(z_{i})\prod_{k=1}^{l}\bar{T}(\w_{k})X>$, we describe Feynman rules in the background field formalism by expanding corresponding functional integrals around a classical solution - the hyperbolic metric on X. Extending analysis in \cite{LT1,LT2,LT-Varenna,LT3}, we define the regularization scheme for any choice of global coordinate on X, and for Schottky and quasi-Fuchsian global coordinates we rigorously prove that one- and two-point correlation functions satisfy conformal Ward identities in all orders of the perturbation theory. Obtained results are interpreted in terms of complex geometry of the projective line bundle $\cE_{c}=λ_{H}^{c/2}$ over the moduli space $\mathfrak{M}_{g}$, where c is the central charge and $λ_{H}$ is the Hodge line bundle, and provide Friedan-Shenker \cite{FS} complex geometry approach to CFT with the first non-trivial example besides rational models.

hep-th↗

arXiv2006

Holomorphic factorization of determinants of laplacians on Riemann surfaces and a higher genus generalization of Kronecker's first limit formula

For a family of compact Riemann surfaces X_t of genus g>1 parametrized by the Schottky space S_g, we define a natural basis for the holomorphic n-differentials on X_t which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n=1. We introduce a holomorphic function F(n) on S_g which generalizes the classical product \prod(1-q^m)^2 appearing in the Dedekind eta function for n=1 and g=1. We prove a holomorphic factorization formula expressing the regularized determinant of the Laplacian as a product of |F(n)|^2, a holomorphic anomaly depending on the classical Liouville action (a Kahler potential of S_g), and the determinant of the Gram matrix of the natural basis. The factorization formula reduces to Kronecker's first limit formula when n=1 and g=1, and to Zograf's factorization formula for n=1 and g>1.

math.CV↗

arXiv2004

Weil-Petersson metric on the universal Teichmuller space I: Curvature properties and Chern forms

We prove that the universal Teichmuller space T(1) carries a new structure of a complex Hilbert manifold. We show that the connected component of the identity of T(1), the Hilbert submanifold T_{0}(1), is a topological group. We define a Weil-Petersson metric on T(1) by Hilbert space inner products on tangent spaces, compute its Riemann curvature tensor, and show that T(1) is a Kahler-Einstein manifold with negative Ricci and sectional curvatures. We introduce and compute Mumford-Miller-Morita characteristic forms for the vertical tangent bundle of the universal Teichmuller curve fibration over the universal Teichmuller space. As an application, we derive Wolpert curvature formulas for the finite-dimensional Teichmuller spaces from the formulas for the universal Teichmuller space.

math.CV↗