Determinants of Laplacians in real line bundles over hyperbolic manifolds connected with quantum geometry of membranes

The possibility is discussed of generalizing the Polyakov approach to strings on membranes and the connection of such a generalization with Thurston's classification of three-dimensional geometries. The important ingredients for computing a membrane path integral are the determinants of scalar Laplacians acting in real line bundles over three-dimensional closed manifolds. In the closed bosonic membrane case, such determinants are evaluated for a class of closed 3-manifolds of the H ³/ form with a discrete subgroup of isometries of the three-dimensional Lobachevsky space H ³ and they are expressed in terms of the Selberg zeta function. Some further possible implications of the results obtained are also discussed.     

On perturbation theory for regularized determinants of differential operators

A perturbation theory for determinants of differential operators regularized through the -function technique is presented. The application of this approach to the study of chiral changes in the fermionic path-integral variables is discussed.     

Determinants of Airy Operators and Applications to Random Matrices

The purpose of this paper is to describe asymptotic formulas for determinants of certain operators that are analogues of Wiener–Hopf operators. The determinant formulas yield information about the distribution functions for certain random variables that arise in random matrix theory when one rescales at the edge of the spectrum.     

Determinants of Laplacians on surfaces of finite volume

Determinants of the Laplace and other elliptic operators on compact manifolds have been an object of study for many years (see [MP, RS, Vor]). Up until now, however, the theory of determinants has not been extended to non-compact situations, since these typically involve a mixture of discrete and continuous spectra. Recent advances in this theory, which are partially motivated by developments in mathematical physics, have led to a connection, in the compact Riemann surface case, between determinants of Laplacians on spinors and the Selberg zeta function of the underlying surface (see [DP, Kie, Sar, Vor]).Our purpose in this paper is to introduce a notion of determinants on non-compact (finite volume) Riemann surfaces. These will be associated to the Laplacian shifted by a parameters(1–s), and will be defined in terms of a Dirichlet series (w, s) which is a sum that represents the discrete as well as the continuous spectrum. It will be seen to be regular atw=0, and our main theorem (see Sect. 1) will express exp as the Selberg zeta function of the surface times the appropriate -factor.A Sloan Fellow and partially supported by NSF grant DMS-8701865     

Selberg supertrace formula for super Riemann surfaces,analytic properties of Selberg super zeta-functions and multiloop contributions for the fermionic string

In this paper a complete derivation of the Selberg supertrace formula for super Riemann surfaces and a discussion of the analytic properties of the Selberg super zeta-functions is presented. The Selberg supertrace formula is based on Laplace-Dirac operators _m of weightm on super Riemann surfaces. The trace formula for allmZ is derived and it is shown that one must discriminate between even and oddm. Particularly the term in the trace formula proportional to the identity transformation is sensitive to this discrimination. The analytic properties of the two Selberg super zeta-functions are discussed in detail, first with, and the second without consideration of the spin structure. We find for the Selberg super zeta-functions similarities as well as differences in comparison to the ordinary Selberg zeta-function. Also functional equations for the two Selberg super zeta-functions are derived. The results are applied to discuss the spectrum of the Laplace-Dirac operators and to ccalculate their determinants. For the spectrum it is found that the nontrivial Eigenvalues are the same for _m and ₀ up to a constant depending onm, which is analogous to the bosonic case. The analytic properties of the determinants can be deduced from the analytic properties of the Selberg super zeta-functions, and it is shown that they are well-defined. Special cases (m=0,2) for the determinants are important in the Polyakov approach for the fermionic string. With these results it is deduced that the fermionic string integrand of the Polyakov functional integral is well-defined.     

Fredholm determinants,differential equations and matrix models

Orthogonal polynomial random matrix models ofN×N hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form ((x)(y)–(x)(y))/x–y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is the union of intervals . The emphasis is on the determinants thought of as functions of the end-pointsa _k.We show that these Fredholm determinants with kernels of the general form described above are expressible in terms of solutions of systems of PDE's as long as and satisfy a certain type of differentiation formula. The (, ) pairs for the sine, Airy, and Bessel kernels satisfy such relations, as do the pairs which arise in the finiteN Hermite, Laguerre and Jacobi ensembles and in matrix models of 2D quantum gravity. Therefore we shall be able to write down the systems of PDE's for these ensembles as special cases of the general system.An analysis of these equations will lead to explicit representations in terms of Painlevé transcendents for the distribution functions of the largest and smallest eigenvalues in the finiteN Hermite and Laguerre ensembles, and for the distribution functions of the largest and smallest singular values of rectangular matrices (of arbitrary dimensions) whose entries are independent identically distributed complex Gaussian variables.There is also an exponential variant of the kernel in which the denominator is replaced bye ^bx–e^by, whereb is an arbitrary complex number. We shall find an analogous system of differential equations in this setting. Ifb=i then we can interpret our operator as acting on (a subset of) the unit circle in the complex plane. As an application of this we shall write down a system of PDE's for Dyson's circular ensemble ofN×N unitary matrices, and then an ODE ifJ is an arc of the circle.     

Canonical fermion determinants in lattice QCD – Numerical evaluation and properties

We analyze canonical fermion determinants, i.e., fermion determinants projected to a fixed quark number q  . The canonical determinants are computed using a dimensional reduction formula and are studied for pure SU(3)$\mathrm{SU}(3)$ gauge configurations in a wide range of temperatures. It is demonstrated that the center sectors of the Polyakov loop very strongly manifest themselves in the behavior of the canonical determinants in the deconfined phase, and we discuss physical implications of this finding. Furthermore the distribution of the quark sectors is studied as a function of the temperature.     

Fermionic fields on ℤ N -curves

The line bundles of degreeg–1 on _N -curves corresponding to 1/N nonsingular characteristics are considered. The determinants of Dirac operators defined on these line bundles are evaluated in terms of branch points. The generalization of Thomae's formula for _N -curves is derived.     

A ζ-function method for weyl fermionic determinants

We present an extension of the -function method adapted to handle the regularization of Dirac operator determinants when Weyl fermions are present. The method we propose makes use of an auxiliary operator which takes into account regularization ambiguities in anomalous gauge theories. As an application, we consider a two-dimensional model where these ambiguities allow for the definition of a consistent quantum theory.     

On the quotient of the regularized determinant of two elliptic operators

We study the quotient of the regularized determinants of two elliptic operators having the same principal symbol. We prove that, under general conditions, a method recently proposed by Tamura coincides with the -function approach.Work supported by CONICET and CIC, Argentina     

Canonical Path Integral Quantization of Einstein's Gravitational Field

The connection between the canonical and the path integral formulations of Einstein's gravitational field is discussed using the Hamilton Jacobi method. Unlike conventional methods, its shown that our path integral method leads to obtain the measure of integration with no -functions, no need to fix any gauge and so no ambiguous determinants will appear.     

The transverse correlation function of anisotropicX—Y-chains: Exact results atT=∞

In this contribution we present exact results for the transverse dynamical correlation function of anisotropic spin 1/2X—Y-chains in the presence of magnetic fields at temperatureT=. Our results are obtained by an application of a new theory of block-Toeplitz determinants also developed in this paper.     

Block Toeplitz Determinants,Constrained KP and Gelfand-Dickey Hierarchies

We propose a method for computing any Gelfand-Dickey τ function defined on the Segal-Wilson Grassmannian manifold as the limit of block Toeplitz determinants associated to a certain class of symbols . Also truncated block Toeplitz determinants associated to the same symbols are shown to be τ functions for rational reductions of KP. Connection with Riemann-Hilbert problems is investigated both from the point of view of integrable systems and block Toeplitz operator theory. Examples of applications to algebro-geometric solutions are given.      

Determinants of Dirac operators and thirring model partition functions on Riemann surfaces with boundaries

The partition function of the Thirring model on a Riemann surface with boundaries is calculated using the method of Freedman and Pilch by introducing an auxiliary vector potential in the path integral of fermion representation. The Hodge decomposition on manifolds with boundaries is used to integrate over the harmonic forms. The result agrees with the bosonized calculation. The determinants of Dirac operators with mixed Neveu-Schwarz and Ramond boundary conditions are expressed in terms of the Riemann -functions of the doubled surface.     

Selberg trace formula for bordered Riemann surfaces: Hyperbolic,elliptic and parabolic conjugacy classes,and determinants of Maass-Laplacians

The Selberg trace formula for automorphic forms of weightm, on bordered Riemann surfaces is developed. The trace formula is formulated for arbitrary Fuchsian groups of the first kind with reflection symmetry which include hyperbolic, elliptic and parabolic conjugacy classes. In the case of compact bordered Riemann surfaces we can explicitly evaluate determinants of Maass-Laplacians for both Dirichlet and Neumann boundary-conditions, respectively. Some implications for the open bosonic string theory are mentioned.Address from August 1993: II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany     

One more class of scalar laplacian determinants connected with the quantum geometry of p-branes

The determinants of Laplacians acting in real line bundles over manifolds of the form

Oscillator representations of the 2D-conformal algebra

We display irreducible representations of the Virasoro algebra (group of diffeomorphisms of the circle) for any value of the central chargec (central extension defined by a cocycle) and of the highest weight, where the Ka determinants do not vanish. The construction is done in terms of a simple bosonic free field. The unitarity of the representation is discussed, and it is realized with non-trivial hermiticity properties of the free field if<(c-1)/24. In the particular case of the central charge (c=1/2) corresponding to the Ising model, the three unitary irreducible representations (=0, 1/16, 1/2) are realized in terms of the anticommuting oscillators of the free fields of the Neveu-Schwarz-Ramond model.     

Explicit Fermionic Tree Expansions

We express connected Fermionic Green's functions in terms of two totally explicit tree formulas. The simplest and most symmetric formula, the Brydges–Kennedy formula is compatible with Gram's inequality. The second one, the rooted formula of Abdesselam and Rivasseau, respects even better the antisymmetric structure of determinants, and allows the direct comparison of rows and columns which correspond to the mathematical implementation in Grassmann integrals of the Pauli principle. To illustrate the power of these formulas, we give a three lines proof that the radius of convergence of the Gross–Neveu theory with cutoff is independent of the number of colors, using either one or the other of these formulas.     

Dynamical Determinants via Dynamical Conjugacies for Postcritically Finite Polynomials

We give an analogue of Levin–Sodin–Yuditskii's study of the dynamical Ruelle determinants of hyperbolic rational maps in the case of subhyperbolic quadratic polynomials. Our main tool is to reduce to an expanding situation. We do so by applying a dynamical change of coordinates on the domains of a Markov partition constructed from the landing ray at the postcritical repelling orbit. We express the dynamical determinants as the product of an (entire) determinant with an explicit expression involving the postcritical repelling orbit, thus explaining the poles in d (z).     

Bethe Ansatz and the Spectral Theory of Affine Lie Algebra-Valued Connections I. The simply-laced Case

We study the ODE/IM correspondence for ODE associated to \({\widehat{\mathfrak{g}}}\)-valued connections, for a simply-laced Lie algebra \({\mathfrak{g}}\). We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called \({\Psi}\)-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.