scholarly journals Ruelle Zeta Function from Field Theory

2020 ◽  
Vol 21 (12) ◽  
pp. 3835-3867
Author(s):  
Charles Hadfield ◽  
Santosh Kandel ◽  
Michele Schiavina
Keyword(s):  
Field Theory ◽  
Partition Function ◽  
Zeta Function ◽  
Lagrangian Submanifolds ◽  
Analytic Torsion ◽  
Contact Manifolds ◽  
Gauge Fixing ◽  
Ruelle Zeta Function

Abstract We propose a field-theoretic interpretation of Ruelle zeta function and show how it can be seen as the partition function for BF theory when an unusual gauge-fixing condition on contact manifolds is imposed. This suggests an alternative rephrasing of a conjecture due to Fried on the equivalence between Ruelle zeta function and analytic torsion, in terms of homotopies of Lagrangian submanifolds.

REGULARIZATION OF GENERAL MULTIDIMENSIONAL EPSTEIN ZETA-FUNCTIONS

1989 ◽  
Vol 01 (01) ◽  
pp. 113-128 ◽  
Author(s):  
E. ELIZALDE ◽  
A. ROMEO
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Partition Function ◽  
Zeta Function ◽  
Finite Temperature ◽  
Casimir Effect ◽  
Zeta Functions ◽  
Quantum Field ◽  
Type Formula

We study expressions for the regularization of general multidimensional Epstein zeta-functions of the type [Formula: see text] After reviewing some classical results in the light of the extended proof of zeta-function regularization recently obtained by the authors, approximate but very quickly convergent expressions for these functions are derived. This type of analysis has many interesting applications, e.g. in any quantum field theory defined in a partially compactified Euclidean spacetime or at finite temperature. As an example, we obtain the partition function for the Casimir effect at finite temperature.

scholarly journals The distributional zeta-function in disordered field theory

2016 ◽  
Vol 31 (25) ◽  
pp. 1650144 ◽  
Author(s):  
B. F. Svaiter ◽  
N. F. Svaiter
Keyword(s):  
Field Theory ◽  
Free Energy ◽  
Partition Function ◽  
Zeta Function ◽  
Complex Function ◽  
Function Technique ◽  
Dimensional Euclidean Space ◽  
Basic Tool ◽  
Disordered System ◽  
Text Model

In this paper, we present a new mathematical rigorous technique for computing the average free energy of a disordered system with quenched randomness, using the replicas. The basic tool of this technique is a distributional zeta-function, a complex function whose derivative at the origin yields the average free energy of the system as the sum of two contributions: the first one is a series in which all the integer moments of the partition function of the model contribute; the second one, which cannot be written as a series of the integer moments, can be made as small as desired. This result supports the use of integer moments of the partition function, computed via replicas, for expressing the average free energy of the system. One advantage of the proposed formalism is that it does not require the understanding of the properties of the permutation group when the number of replicas goes to zero. Moreover, the symmetry is broken using the saddle-point equations of the model. As an application for the distributional zeta-function technique, we obtain the average free energy of the disordered [Formula: see text] model defined in a [Formula: see text]-dimensional Euclidean space.

A COMPACT BOSON COUPLED TO TWO-DIMENSIONAL GRAVITY

1991 ◽  
Vol 06 (15) ◽  
pp. 2743-2754 ◽  
Author(s):  
NORISUKE SAKAI ◽  
YOSHIAKI TANII
Keyword(s):  
Field Theory ◽  
Partition Function ◽  
Quantum Gravity ◽  
Matrix Models ◽  
Riemann Surfaces ◽  
Partition Functions ◽  
Two Dimensional ◽  
Dimensional Gravity ◽  
The Continuum ◽  
Continuum Field

The radius dependence of partition functions is explicitly evaluated in the continuum field theory of a compactified boson, interacting with two-dimensional quantum gravity (noncritical string) on Riemann surfaces for the first few genera. The partition function for the torus is found to be a sum of terms proportional to R and 1/R. This is in agreement with the result of a discretized version (matrix models), but is quite different from the critical string. The supersymmetric case is also explicitly evaluated.

scholarly journals Directed Polymers and Interfaces in Disordered Media

Polymers ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 1066
Author(s):  
Róbinson J. Acosta Diaz ◽  
Christian D. Rodríguez-Camargo ◽  
Nami F. Svaiter
Keyword(s):  
Field Theory ◽  
Free Energy ◽  
Partition Function ◽  
Saddle Point ◽  
Finite Temperature ◽  
Series Representation ◽  
Replica Method ◽  
Quenched Disorder ◽  
Directed Polymers ◽  
Disordered Media

We consider field theory formulation for directed polymers and interfaces in the presence of quenched disorder. We write a series representation for the averaged free energy, where all the integer moments of the partition function of the model contribute. The structure of field space is analysed for polymers and interfaces at finite temperature using the saddle-point equations derived from each integer moments of the partition function. For the case of an interface we obtain the wandering exponent ξ = ( 4 − d ) / 2 , also obtained by the conventional replica method for the replica symmetric scenario.

scholarly journals Jensen polynomials for the Riemann zeta function and other sequences

2019 ◽  
Vol 116 (23) ◽  
pp. 11103-11110 ◽  
Author(s):  
Michael Griffin ◽  
Ken Ono ◽  
Larry Rolen ◽  
Don Zagier
Keyword(s):  
Partition Function ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
General Theorem ◽  
Hermite Polynomials ◽  
Gaussian Unitary Ensemble ◽  
Random Matrix Model ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Unitary Ensemble

In 1927, Pólya proved that the Riemann hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function ζ(s) at its point of symmetry. This hyperbolicity has been proved for degrees d≤3. We obtain an asymptotic formula for the central derivatives ζ(2n)(1/2) that is accurate to all orders, which allows us to prove the hyperbolicity of all but finitely many of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity for all d≤8. These results follow from a general theorem which models such polynomials by Hermite polynomials. In the case of the Riemann zeta function, this proves the Gaussian unitary ensemble random matrix model prediction in derivative aspect. The general theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function.

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