The fluctuation determinant, the preexponential factor for the instanton transition, has been computed several years ago in the Abelian Higgs model, using a method based on integrating the Euclidean Green' function. A more elegant method for computing functional determinants, using the Gel'fand-Yaglom theorem, has been applied recently to a variety of systems. This method runs into difficulties if the background field has nontrivial topology, as is the case for the instanton in the Abelian Higgs model. A shift in thre effective centrifugal barriers makes the s-wave contribution infinite, an infinity that is compensated by the summation over the other partial waves. This requires some modifications of the Gel'fand-Yaglom method which are the main subject of this work. We present here both, the Green' s function and the Gel'fand-Yaglom method and compare the numerical results in detail.
The discrete Gel'fand--Yaglom theorem was studied several years ago. In the present paper, we generalize the discrete Gel'fand--Yaglom method to obtain the determinants of mass matrices which appear current works in particle physics, such as dimensional deconstruction and clockwork theory. Using the results, we show the expressions for vacuum energies in such various models.
I present a partly pedagogic discussion of the Gel'fand-Yaglom formula for the functional determinant of a one-dimensional, second order difference operator, in the simplest settings. The formula is a textbook one in discrete Sturm-Liouville theory and orthogonal polynomials. A two by two matrix approach is developed and applied to Robin boundary conditions. Euler-Rayleigh sums of eigenvalues are computed. A delta potential is introduced as a simple, non-trivial example and extended, in an appendix, to the general case. The continuum limit is considered in a non--rigorous way and a rough compa…
The bulk reconstruction formula for a Euclidean anti-de Sitter space is directly related to the inverse of the Gel'fand-Graev-Radon transform. Correlation functions of a conformal scalar field theory in the boundary are thereby related to correlation functions of a self-interacting scalar field theory in the bulk at different loop orders.
We apply the procedure of Magri and Weinstein to produce an infinity of compatible Poisson structures on a bihamiltonian manifold, to the case of the KdV phase space. The higher Gel'fand-Dikii structures thus obtained contain non local terms, which we express with the help of the r.h.s. of the KdV hierarchy. We also give a generating function for all these Poisson structues, in terms of the Baker-Akhiezer functions. Finally we describe the symplectic leaves of these Poisson structures.
The Gelfand-Yaglom formula relates functional determinants of the one-dimensional second order differential operators to the solutions of the corresponding initial value problem. In this work we generalise the Gelfand-Yaglom method by considering discrete and continuum partial second order differential operators in higher dimensions. To illustrate our main result we apply the generalised formula to the two-dimensional massive and massless discrete Laplace operators and calculate asymptotic expressions for their determinants.
The string theory calculation of the 1/2-BPS circular Wilson loop of N=4 SYM in the planar limit at next to leading order at strong coupling is revisited in the ratio of its semiclassical string partition function and the one dual to a latitude Wilson loop with trivial expectation value. After applying a conformal transformation from the disk to the cylinder, this problem can be approached by means of the Gel'fand-Yaglom formalism. Using results from the literature and the exclusion of zero modes from a modified Gel'fand-Yaglom formula, we obtain matching with the known field theory result. As…
Bulk reconstruction formulas similar to HKLL are obtained for de Sitter and anti-de Sitter spaces as the inverse Gel'fand Graev Radon transform. While these generalize our previous result on the Euclidean anti-de Sitter space, their validity in here is restricted only to odd dimensions in both instances. The exact Wightman function for the de Sitter space is then derived analytically. The GGR transform fixes the coefficient of the Wightman function. For the anti-de Sitter space it is shown that a reconstruction formula exists for the case of time-like boundary as well. The restriction on the d…
The principal series of unitary representations of the Lorentz group has been considered in the helicity basis. Decompositions of tensor products of the spinspaces are studied in the framework of projective representations of the symmetric group. Higher-spin Gel'fand-Yaglom equations are defined in the helicity basis over an arbitrary representation space. Applications of decomposable and indecomposable Gel'fand-Yaglom equations to particle physics are discussed.
In this paper we find an explicit formula for the most general vector evolution of curves on $RP^{n-1}$ invariant under the projective action of $SL(n,R)$. When this formula is applied to the projectivization of solution curves of scalar Lax operators with periodic coefficients, one obtains a corresponding evolution in the space of such operators. We conjecture that this evolution is identical to the second KdV Hamiltonian evolution under appropriate conditions. These conditions give a Hamiltonian interpretation of general vector differential invariants for the projective action of $SL(n,R)$, …
We introduce and study action of quantum groups on skew polynomial rings and related rings of quotients. This leads to a ``q-deformation'' of the Gel'fand-Kirillov conjecture which we partially prove. We propose a construction of automorphisms of certain non-commutaive rings of quotients coming from complex powers of quantum group generators; this is applied to explicit calculation of singular vectors in Verma modules over $U_{q}(\gtsl_{n+1})$. We finally give a definition of a $q-$connection with coefficients in a ring of skew polynomials and study the structure of quantum group modules twist…
It is shown that the Gel'fand-Tsetlin realization of irreducible representations of the $A_n$ algebra is directly connected with a linear exactly integrable system in the n-dimensional space. General solution for this system is explicitly given.
A new, so called odd Gel'fand-Zetlin basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra gl(n|n). The related Gel'fand-Zetlin patterns are based upon the decomposition according to a particular chain of subalgebras of gl(n|n). This chain contains only genuine Lie superalgebras of type gl(k|l) with k and l nonzero (apart from the final element of the chain which is gl(1|0)=gl(1)). Explicit expressions for a set of generators of the algebra on this Gel'fand-Zetlin basis are determined. The results are extended to an explicit construction of a class of…
The problem of construction of irreducible representations of quantum $A^q_n$ algebras is solved at the level of explicit integration of the linear (inhomogeneous) system in finite differences in the n-dimensional space. The general solution of this system is given explicitly and particular ones, which correspond to the irreducible representations are selected.
The space of physical states in relativistic scattering theory is constructed, using a rigorous version of the Dirac formalism, where the Hilbert space structure is extended to a Gel'fand triple. This extension enables the construction of ``a complete set of states'', the basic concept of the original Dirac formalism, also in the cases of unbounded operators and continuous spectra. We construct explicitly the Gel'fand triple and a complete set of ``plane waves'' -- momentum eigenstates -- using the group of space-time symmetries. This construction is used (in a separate article) to prove a gen…
This talk reports work done in collaboration with Jin Hur, Choonkyu Lee and Hyunsoo Min concerning the computation of the precise mass dependence of the fermion determinant for quarks in the presence of an instanton background. The result interpolates smoothly between the previously known chiral and heavy quark limits of extreme small and large mass. The computational method makes use of the fact that the single instanton background has radial symmetry, so that the computation can be reduced to a sum over partial waves of logarithms of radial determinants, each of which can be computed numeric…
An explicit construction of all finite-dimensional irreducible representations of the Lie superalgebra gl(1|n) in a Gel'fand-Zetlin basis is given. Particular attention is paid to the so-called star type I representations (``unitary representations''), and to a simple class of representations V(p), with p any positive integer. Then, the notion of Wigner Quantum Oscillators (WQOs) is recalled. In these quantum oscillator models, the unitary representations of gl(1|DN) are physical state spaces of the N-particle D-dimensional oscillator. So far, physical properties of gl(1|DN) WQOs were describe…
We compute the one-loop effective action of string configurations embedded in $AdS_4\times\mathbb{CP}^3$ which are dual to $\frac{1}{6}$-BPS latitude Wilson Loops in the ABJM theory. To avoid ambiguities in the string path integral we subtract the $\frac{1}{2}$-BPS case. The one-loop determinants are computed by Fourier-decomposing the two dimensional operators and then using the Gel'fand-Yaglom method. We comment on various aspects related to the regularization procedure, showing the cancellation of a hierarchy of divergences. After taking into account an IR anomaly from a change in topology,…
We analyze GKZ(Gel'fand, Kapranov and Zelevinski) hypergeometric systems and apply them to study the quantum cohomology rings of Calabi-Yau manifolds. We will relate properties of the local solutions near the large radius limit to the intersection rings of a toric variety and of a Calabi-Yau hypersurface. (Talk presented at "Frontiers in Quantum Field Theory", Osaka, Japan, Dec.1995)
We derive simple new expressions, in various dimensions, for the functional determinant of a radially separable partial differential operator, thereby generalizing the one-dimensional result of Gel'fand and Yaglom to higher dimensions. We use the zeta function formalism, and the results agree with what one would obtain using the angular momentum cutoff method based on radial WKB. The final expression is numerically equal to an alternative expression derived in a Feynman diagrammatic approach, but is considerably simpler.
The character of the principal series of representations of SL(n,R) is evaluated by using Gel'fand and Naimark's definition of character. This representation is realized in the space of functions defined on the right coset space of SL(n,R) with respect to the subgroup of real triangular matrices. This form of the representations considerably simplifies the problem of determination of the integral kernel of the group ring which is fundamental in the Gel'fand-Naimark theory of character. An important feature of the principal series of representations is that the `elliptic' elements of SL(n,R) do…
After an introduction to some basic issues in non-commutative geometry (Gel'fand duality, spectral triples), we present a "panoramic view" of the status of our current research program on the use of categorical methods in the setting of A.Connes' non-commutative geometry: morphisms/categories of spectral triples, categorification of Gel'fand duality. We conclude with a summary of the expected applications of "categorical non-commutative geometry" to structural questions in relativistic quantum physics: (hyper)covariance, quantum space-time, (algebraic) quantum gravity.
