Virasoro constraints and flavor-topology duality in QCD

Abstract

We derive Virasoro constraints for the zero momentum part of the QCD-like partition functions in the sector of topological charge . The constraints depend on the topological charge only through the combination N f ϩ␤/2 where the value of the Dyson index ␤ is determined by the reality type of the fermions. This duality between flavor and topology is inherited by the small-mass expansion of the partition function and all spectral sum rules of inverse powers of the eigenvalues of the Dirac operator. For the special case ␤ϭ2 but arbitrary topological charge the Virasoro constraints are solved uniquely by a generalized Kontsevich model with the potential V(X)ϭ1/X. DOI: 10.1103/PhysRevD.64.054002 PACS number͑s͒: 12.39.Fe, 11.30.Rd, 12.38.Lg I. FLAVOR-TOPOLOGY DUALITYThrough the work of 't Hooft we know that the lowenergy limit of QCD is dominated by light flavors and topology ͓1͔. We expect that the same will be the case for the low-lying eigenvalues of the Dirac operator. Indeed, for massless flavors, the fermion determinant results in the repulsion of eigenvalues away from ϭ0. It is perhaps less known that the presence of exactly zero eigenvalues has the same effect. The reason is the repulsion of eigenvalues which occurs in all interacting systems and has probably best been understood in the context of random matrix theory where the eigenvalues obey the Wigner repulsion law ͓2͔.In QCD, the fluctuations of the low-lying eigenvalues of the Dirac operator are described by chiral random matrix theory ͑chRMT͒ ͓3-5͔. This is a random matrix theory with the global symmetries of the QCD partition function. It is characterized by the Dyson index ͓6͔ ␤ which is equal to the number of independent variables per matrix element. For QCD with fundamental fermions we have ␤ϭ1 for N c ϭ2 and ␤ϭ2 for N c Ͼ2. For QCD with adjoint fermions and N c у2 the Dirac matrix can be represented in terms of selfdual quaternions with ␤ϭ4. The main ingredient of the chRMT partition function is the integration measure which includes the Vandermonde determinant. In terms of Dirac eigenvalues i k it is given byTherefore, the presence of N f massless flavors, with the fermion determinant given by ͟ k k 2N f , has the same effect on eigenvalue correlations as ϭ2N f /␤ zero eigenvalues. More precisely, the joint eigenvalue distribution only depends on the combination 2N f ϩ␤ ͓4,7͔. Based on the conjecture ͓3,4͔ that the zero-momentum part of the QCD partition function, Z N f , (m 1 , . . . ,m N f ), is a chiral random matrix theory, it was suggested ͓8͔ that its mass dependence obeys the duality relation ͑for ␤ϭ2͒This relation, which is now known as flavor topologyduality, is a trivial consequence of the flavor dependence of the chRMT joint eigenvalue distribution. The mass dependence of the chRMT partition function can be reduced to a unitary matrix integral which is known from the zero momentum limit of chiral perturbation theory ͓9,10͔. Starting from this representation of the low energy limit of the QCD partition function, also known as the finite...

QCD3 and the replica method

“…In section 4.1 it is shown that results for the microscopic spectral density and microscopic spectral correlation functions obtained in this way are in exact agreement with previous results form Random Matrix Theory [2]. An exact calculation of the resolvent of the QCD 3 partition function from its finite volume partition function is thus possible by using the recent results for the QCD 4 finite volume partition function [31] (see section 4.2). In sections 5 and 6 we perform explicit calculations in QCD 3 based on the replica method.…”

“…For non-integer values of ν we define the QCD 4 partition function according to the analytical continuation in ν of the Bessel functions [31], 9) and not by analytical continuation in ν of the unitary matrix integral (2.12). The integral (2.12) satisfies an important relation known as flavor-topology duality [38].…”

“…Although the replica method has been challenged [25], this method has received a considerable boost in the context of condensed matter physics during the past year [26,27,28,29]. Recently, the replica method has also been applied to the field theory corresponding to the low-energy limit of QCD 4 [30,31,32]. The advantage of the fermionic replica method is that we are familiar with the pattern of spontaneous chiral symmetry breaking, and it is straightforward to define the corresponding low-energy field theory.…”

Dirac spectrum of the Wilson-Dirac operator for QCD with two colors

“…At vanishing lattice spacing a the matrix D W has ν zero modes and n pairs of imaginary eigenvalues ±iy. This ensemble is known as the chiral Gaussian orthogonal ensemble [42,43,61]. For small lattice spacing |a| 1 we can apply first order perturbation theory.…”

“…which is expressed in terms of the partition function at a = 0, see [42,46]. This expression has some interesting relations to the microscopic partition function of the chiral Gaussian orthogonal random matrix ensemble (chGOE) which is up to a factor the same as the partition function Z N f ν (M + iσ, a = 0),…”