Virasoro constraints and flavor-topology duality in QCD

Virasoro constraints and flavor-topology duality in QCD

D. Dalmazi*

UNESP, Guaratingueta´, S.P., 12.500.000, Brazil J. J. M. Verbaarschot†

Department of Physics and Astronomy, SUNY, Stony Brook, New York 11794

共Received 8 January 2001; published 31 July 2001兲

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 Nf

⫹␤␯/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␤⫽2 but arbitrary topological charge the Virasoro constraints are solved uniquely by a generalized Kontsevich model with the potentialV(X)⫽1/X.

DOI: 10.1103/PhysRevD.64.054002 PACS number共s兲: 12.39.Fe, 11.30.Rd, 12.38.Lg

I. FLAVOR-TOPOLOGY DUALITY

Through the work of ’t Hooft we know that the low-energy limit of QCD is dominated by light flavors and topol-ogy 关1兴. 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 re-pulsion of eigenvalues away from ␭⫽0. 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 关2兴.

In QCD, the fluctuations of the low-lying eigenvalues of the Dirac operator are described by chiral random matrix theory共chRMT兲 关3–5兴. This is a random matrix theory with the global symmetries of the QCD partition function. It is characterized by the Dyson index关6兴␤ which is equal to the number of independent variables per matrix element. For QCD with fundamental fermions we have ␤⫽1 for Nc⫽2

and ␤⫽2 for Nc⬎2. For QCD with adjoint fermions and

N_c⭓2 the Dirac matrix can be represented in terms of

self-dual quaternions with ␤⫽4. 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 by 兿k⬍l兩␭k 2_⫺␭ l 2_兩␤␭ k ␤␯⫹␤⫺1_.

Therefore, the presence of Nf massless flavors, with the

fer-mion determinant given by 兿k␭k

2N_f

, has the same effect on eigenvalue correlations as␯⫽2Nf/␤zero eigenvalues. More

precisely, the joint eigenvalue distribution only depends on the combination 2Nf⫹␤␯ 关4,7兴. Based on the conjecture

关3,4兴 that the zero-momentum part of the QCD partition function, Z_N

f,␯(m1, . . . ,mNf), is a chiral random matrix

theory, it was suggested 关8兴 that its mass dependence obeys the duality relation共for␤⫽2兲

Z_N

f,␯共m1, . . . ,mNf兲

⬃

兿

f

m_f␯ZN_f⫹␯,0共m1, . . . ,mN_f,0, . . . ,0兲. 共1兲

This relation, which is now known as flavor topology-duality, 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 关9,10兴. Starting from this representation of the low energy limit of the QCD partition function, also known as the finite volume partition function, flavor-topology duality was first proved for Nf⫽2 in 关8兴. However, its generalization to

arbi-trary Nfand␯ has only been achieved for␤⫽2 关11,12兴. The

relation共1兲 has been particularly useful for establishing rela-tions between correlation funcrela-tions of Dirac eigenvalues and finite volume partition functions关12–16兴.

For␤⫽2 the unitary matrix integral, which represents the low-energy limit of the QCD partition function, is also known as the one-link integral of two-dimensional QCD共see for example 关17兴兲 or the Brezin-Gross-Witten model 关18兴. For zero topological charge it can also be represented as a generalized Kontsevich model 关19兴 with potential V(X) ⫽1/X. This model has been discussed extensively in the con-text of topological gravity共for reviews see 关23–25兴兲. In this case the one-link integral has been analyzed 关19,20兴 by means of Virasoro constraints which are based on its invari-ance properties. The main issue we wish to address in this article is whether flavor-topology duality of the one-link in-tegral can be understood without relying on its chRMT rep-resentation. We will do this by deriving Virasoro constraints for arbitrary topological charge. For␤⫽2 we find a nonper-turbative solution of these constraints in the form of a gen-eralized Kontsevich model.

In Sec. II we will derive the Virasoro constraints for arbi-trary␯ and three different values of␤. We observe that they satisfy the flavor-topology duality relations. A recursive so-lution of these relations is presented in Sec. III. It also pro-vides us with an efficient derivation of sum rules for the *Email address: dalmazi@feg.unesp.br

inverse Dirac eigenvalues to high order. In the second part of this section we discuss the uniqueness of the nonperturbative solution of the Virasoro constraints for ␤⫽2. Concluding remarks are made in Sec. IV.

II. VIRASORO CONSTRAINTS

In the simplest case, ␤⫽2 and vanishing topological charge ␯⫽0, the small-mass Virasoro constraints were first found in关19兴 after an identification of the appropriate unitary integral and the corresponding generalized Kontsevich model 共GKM兲. Here we work entirely in the context of uni-tary integrals and derive a simple form of the Virasoro con-straints valid for arbitrary topological charge and␤. They are obtained by expanding the partition function in powers of the masses.

A.␤Ä2

The low energy limit of QCD in the phase of broken chiral symmetry is a gas of weakly interacting Goldstone bosons. Its partition function is determined uniquely by the invariance properties of the Goldstone fields. For quark masses m and space-time volume V in the range

1

⌳ ⰆV1/4Ⰶ 1

m_␲ 共2兲

共where m␲⫽冑2m⌺/F2 is the mass of the Goldstone modes

with F the pion decay constant and⌺ the chiral condensate兲 the partition function for the Goldstone modes factorizes 关9,10兴 into a zero-momentum part, also known as the finite volume partition function, and a nonzero momentum part. For fundamental fermions and a gauge group with N_c⭓3, the finite volume partition function in the sector of topologi-cal charge␯ is given by the following integral over the uni-tary group关9,10兴

Z_␯␤⫽2共M,M†_兲⫽

冕

U苸U(N_f)

dU共det U兲␯e(1/2) Tr (MU†⫹M†U), 共3兲 where M⫽MV⌺. Here and below we always take ␯⭓0. The quantity M stands for the original unscaled quark mass matrix. The partition function is normalized such that

Z_␯␤⫽2(M,M†)→det␯(M) for M→0.

Especially for␯⫽0, the integral 共3兲 has been studied ex-tensively in the literature. In the context of lattice QCD it is known as the one-link integral关18,21,19兴 共see also 关17兴 for a review兲. Of particular interest is the fact that for Nf→⬁ the

partition function Z_␯␤⫽2 undergoes a phase transition 关18兴 from a small mass phase (M→0) to a large mass phase (M→⬁). The large mass expansion is asymptotic and its coefficients can be calculated, even for finite Nf, by

expand-ing about a flavor-symmetric saddle point. For Re(M)→⬁ such expansion can be carried out in a efficient way by means of the共large-mass兲 Virasoro constraints found in 关22兴. For Im(M)→⬁, at fixed value of Re(M), other flavor-nonsymmetric saddle points contribute to the partition func-tion as well, and it is not known whether the expansion can

be carried out by means of Virasoro constraints 共for an ex-plicit calculation including nonsymmetric saddle points see 关26兴兲. On the other hand, the small-mass expansion has a non-zero radius of convergence, and it can be determined by different methods for any finite N_f. For example, one can expand the exponential in Eq. 共3兲 and calculate the corre-sponding unitary integrals systematically using a character expansion 关27,28兴. Another efficient technique is again the use of the 共small-mass兲 Virasoro constraints found in 关19,20,12兴. By generalizing such constraints to arbitrary to-pological charge ␯ we will show that they naturally lead to flavor-topology duality.

First of all, using the unitary invariance of the measure in Eq. 共3兲 we can deduce the covariance properties of the par-tition function under a redefinition of the mass matrix M

→V⫺1_MU,

共det M兲⫺␯Z_␯␤⫽2共M,M†_兲 ⫽ 共det V⫺1_MU兲⫺␯_Z

␯

␤⫽2_共V⫺1_MU,U⫺1_M†_{V兲. 共4兲} This equation implies that (detM)⫺␯Z_␯␤⫽2(M,M†) is a symmetric function of the eigenvalues of the Hermitian ma-trix Lab⬅(MM†)ab. Following关19兴 we can introduce the

infinite set of variables

tk⫽ 1 kTr

冉

MM† 4

冊

k , k⭓1, 共5兲

which are explicitly symmetric with respect to permutations of the eigenvalues and write

共det M兲⫺␯Z_␯␤⫽2共M,M†_兲⫽G

␯共tk兲, 共6兲

where G_␯(tk) has the Taylor expansion

G_␯共t_k兲⫽1⫹a₁t₁⫹a₂t₂⫹a₁₁t₁2⫹•••. 共7兲

A simple consistency check on Eq. 共6兲 is that unitary inte-grals are only nonvanishing if the powers of U and U† are equal. Following 关19兴 we can find the coefficients of the Taylor expansion共7兲 from the differential equation

⳵2_Z ␯ ␤⫽2 ⳵Mba⳵Mac † ⫽ ␦cb 4 Z␯ ␤⫽2_{, 共8兲}

which follows directly from Eq. 共3兲. As a consequence,

G_␯(tk) satisfies the equation

冋

⳵2 ⳵Mba⳵Mac † ⫹␯Mab⫺1 ⳵ ⳵Mac †

册

G␯共tk兲⫽ ␦cb 4 G␯共tk兲. 共9兲 Using the chain rule

⳵ ⳵Mac † ⫽_k

兺

⭓1

冋

共MM†_兲k⫺1_M 4k

册

_ca⳵k, ⳵ ⳵Mba ⫽

兺

k⭓1

冋

M†_共MM†_兲k⫺1 4k

册

ab ⳵k, 共10兲

with⳵k⫽⳵/⳵tk we immediately obtain, from Eq.共9兲,

兺

s⫽1 ⬁

冉

MM† 4

冊

_cb s⫺1 关Ls␤⫽2⫺␦s,1兴G␯共tk兲⫽0, 共11兲 where Ls␤⫽2⫽

兺

k⫽1 s⫺1 ⳵k⳵s⫺k⫹

兺

k⭓1 kt_k⳵_s⫹k⫹共N_f⫹␯兲⳵_s, s⭓1, 共12兲 and our convention throughout this paper is that terms like the first one in the expression for Ls vanish for s⫽1. The

operatorsLs obey a sub-algebra of the Virasoro algebra

关Lr,Ls兴⫽共r⫺s兲Lr⫹s 共13兲

without central charge and r,s⭓1.

At fixed value Nf the tk are not independent, and the

coefficients of the expansion 共7兲 are not determined uniquely.1In Eq.共11兲 the matrix elements of MM†and the

t_kare not independent so that we cannot conclude from Eq. 共11兲 that the coefficients of (MM†₎s _{vanish. Indeed, the}

corresponding equations for Nf⫽1 and ␯⫽0 are

inconsis-tent. However, in order to fix the coefficients in Eq. 共7兲 uniquely, we may supplement Eq.共11兲 with additional equa-tions. We do that by requiring that all coefficients of (MM†)s vanish,

关Ls␤⫽2⫺␦s,1兴G␯共tk兲⫽0, s⭓1. 共14兲

This procedure is justified provided that the all equations are consistent. Indeed, for Nf→⬁ the matrix elements and the tk

are independent so that Eq.共14兲 must be valid. By inserting the Taylor expansion共7兲 into Eq. 共14兲 we obtain an inhomo-geneous set of linear equations for the coefficients which depend on the parameter Nf⫹␯. Inconsistencies can only

arise for isolated values of N_f⫹␯ for which the homoge-neous part of the equations becomes linearly dependent. This is indeed what happens in the example Nf⫽1 and␯⫽0.

Because of the commutation relations共13兲 the constraints for s⭓3 follow from the constraints for s⫽1 and s⫽2. The coefficients of the Taylor expansion共7兲 can be found recur-sively by solving the constraints for s⫽1 and s⫽2. This will be carried out in Sec. III after deriving the Virasoro con-straints for␤⫽4 and␤⫽1.

B.␤Ä4

Similarly, we now derive the Virasoro constraints for ad-joint fermions (␤⫽4). For fermions in the adjoint represen-tation, the zero-momentum Goldstone modes belong to the coset space SU(Nf)/SO(Nf). These Goldstone fields are

conveniently parametrized by UUTwith U苸SU(Nf). In the

sector of topological charge␯, we thus find the finite volume partition function关10,29兴,

Z_␯¯␤⫽4共MS,MS

†_兲

⫽

冕

U_苸U(Nf)

dU共det U兲2␯¯e(1/2)Tr„MS(UUt)†⫹MS †

UUt…_,

共15兲 where¯␯⫽Nc␯ andMS⫽M⌺V. In this case the mass matrix

MS is an arbitrary symmetric complex matrix. From MS

→Vt_M

SV we obtain the transformation law

共det MS兲⫺␯¯Z_␯¯ ␤⫽4_共M S,MS †_兲 ⫽共det Vt_M SV兲⫺␯¯Z_␯¯ ␤⫽4_„Vt_M SV,V⫺1MS †_共Vt_兲⫺1_…. 共16兲 This implies that (detMS)⫺␯¯Z_␯¯

␤⫽4

(MS,MS

†

) is a symmet-ric function of the eigenvalues of MSMS

† . We thus have that 共det MS兲⫺␯¯Z_␯¯␤⫽4共MS,MS †_兲⫽G ␯¯ ␤⫽4_共t k S_兲⫽1⫹a 1 S_t 1 S_⫹a 2 S_t 2 S ⫹a11 S_共t 1 S_兲2_{⫹•••, 共17兲}

where in analogy with Eq.共5兲 we have defined

tk S_⫽1 kTr

冉

MSMS † 4

冊

k , 共18兲

and G_␯¯␤⫽4(t_kS) will be determined by the differential equation ⳵2_Z ␯¯ ␤⫽4 ⳵MS ba⳵MS ac † ⫽␦bcZ␯¯ ␤⫽4_{. 共19兲}

After substituting the expansion共17兲 we obtain a differential equation for G_␯¯␤⫽4(t_kS) which can be written in the form of the Virasoro constraints,

关Ls␤⫽4⫺2␦s,1兴G_␯¯␤⫽4共tk S_{兲⫽0, s⭓1, 共20兲} where Ls␤⫽4⫽2

兺

k⫽1 s_⫺1 ⳵k S_⳵ s⫺k S _⫹

兺

k⭓1 kt_kS⳵_sS_⫹k⫹共Nf⫹2¯␯⫹s兲⳵s S_, 共21兲 and ⳵_kS⫽⳵/⳵tS_k. One can easily check thatL_s␤⫽4 andL_s␤⫽2 satisfy the same algebra.

C.␤Ä1

For QCD with fundamental fermions and Nc⫽2 the

La-grangian can be written in terms of fermion multiplets of length 2Nf containing Nf quarks and Nf anti-quarks. The

chiral symmetry is thus enhanced to SU(2N_f). Since the fermion condensate is anti-symmetric in this enlarged flavor space, the coset space of the Goldstone modes is given by

SU(2Nf)/S p(2Nf). This coset manifold is parametrized by

UIUt where U苸SU(2Nf) and I is the 2Nf⫻2Nf

antisym-metric unit matrix

1_{Of course, they are determined uniquely up to the order that the}

I⫽

冉

0 1

⫺1 0

冊

. 共22兲

The partition function in the sector of fixed topological charge ␯ is then given by integrating over the group mani-fold U(2N_f) instead of SU(2N_f),

Z_␯␤⫽1共M˜ _A,M˜ A †_兲 ⫽

冕

U_苸U(2Nf) dU共det U兲␯e(1/4)Tr„M˜A † UIUt⫹M˜_A(UIUt)†…_, 共23兲 where the mass matrix M˜ A⫽MV⌺ is an arbitrary

anti-symmetric complex matrix. In addition to the usual mass-term given by

冉

0 M

⫺M 0

冊

共24兲

it contains di-quark source terms in its diagonal blocks. Be-low, in the calculation of the mass dependence of the parti-tion funcparti-tion, the di-quark source terms will be put equal to zero. The covariance properties ofZ_␯␤⫽1 are given by

共det M˜ A兲⫺␯/2Z_␯␤⫽1共M˜ A,M˜ A †_兲 ⫽共det Vt_M˜ AV兲⫺␯/2Z_␯␤⫽1共VtM˜ AV,V⫺1M˜ A †_Vt⫺1_兲. 共25兲 We thus have that (detM˜ A)⫺␯/2Z␤⫽1␯ (M˜ A,M˜ A

†

) is a sym-metric function of the eigenvalues of M˜ AM˜ A

† . This results in the expansion 共det M˜ _A兲⫺␯/2_Z ␯ ␤⫽1_共M˜ A,M˜ A †_兲 ⫽G_␯␤⫽1共tk A_兲 ⫽1⫹a1 A t₁A⫹a₂At₂A⫹a₁₁A共t₁A兲2⫹•••, 共26兲 where t_kA⫽ 1 2kTr

冉

M˜ _AM˜ A † 4

冊

k . 共27兲

The factor 12 in the above definition of tk A

takes into account that M˜ is a 2Nf⫻2Nf matrix for ␤⫽1. Substituting Eq.

共26兲 in the differential equation ⳵2_Z ␯ ␤⫽1 ⳵M˜ _{A ba⳵M}˜ A ac † ⫽ ␦bc 4 Z␯ ␤⫽1_{, 共28兲}

we deduce the differential equations for G_␯␤⫽1(t_kA) in the form of Virasoro constraints:

冉

Ls␤⫽1⫺ ␦s,1 2

冊

G␯ ␤⫽1_共t k A_{兲⫽0, 共29兲} with Ls␤⫽1⫽ 1 2 k

兺

_⫽1 s⫺1 ⳵k A ⳵s⫺k A _⫹

兺

k_⭓1 kt_kA⳵_sA_⫹k⫹

冉

Nf⫹ ␯ 2⫺ s 2

冊

⳵s A , 共30兲

and ⳵_kA⫽⳵/⳵t_kA. The L_s␤⫽1 satisfy the same algebra as the two other values of␤ discussed in previous sections.

III. SOLVING THE CONSTRAINTS

In the first part of this section we discuss the recursive solution of the Virasoro constraints. Flavor-topology duality and nonperturbative solutions are discussed in the second part of this section.

A. Recursive solution of Virasoro constraints

Remarkably, in all three cases,␤⫽1, ␤⫽2 and␤⫽4, the constraints共14兲 , 共20兲 and 共29兲 can be written in the unified form:

冉

Ls␤⫺

1

␥ ␦s,1

冊

G共␣,␥兲⫽0, 共31兲

where关see Eqs. 共12兲, 共21兲 and 共30兲兴

Ls␤⫽ 1 ␥ k

兺

⫽1 s⫺1 ⳵k⳵s⫺k⫹

兺

k⭓1 ktk⳵s⫹k ⫹1_␥关␣⫹␥共2⫺s兲⫹s⫺1兴⳵s, 共32兲

and, motivated by chRMT results of 关31兴, we have intro-duced the notation

␣⫽共Nf⫺2兲2/␤⫹␯⫹1,

␥⫽2/␤. 共33兲

The tk are defined as in Eqs. 共5兲, 共18兲 and 共27兲 for ␤

⫽2,4,1, respectively. From Eq. 共31兲 we conclude that in all three cases the Virasoro constraints only depend on N_fand␯ through the combination Nf⫹␤␯/2 (␯→¯␯ for ␤⫽4).

Proceeding further, we can recursively solve Eq.共31兲 by sub-stituting the Taylor expansion of the form共7兲 for G(␣,␥) in Eq.共31兲. By treating the t_kas independent variables, we find the following relations between the expansion coefficients:

␥

兺

k_⫽1 ⬁ k共nk⫹1⫹1兲an₁•••n_k⫺1n_k⫹1⫹1n_k⫹2••• ⫹共␣⫹␥兲共n1⫹1兲an₁⫹1n₂•••⫽an₁n₂•••, 共34兲 for s⫽1 and 共n1⫹2兲共n1⫹1兲an₁⫹2n₂•••⫹␥

兺

k⫽1 ⬁ k共nk⫹2⫹1兲 ⫻an₁•••n_k⫺1n_k⫹1n_k⫹2⫹1•••⫹共␣⫹1兲共n2⫹1兲 ⫻an₁n₂⫹1n₃•••⫽0, 共35兲

for s⫽2. 共The subscript n1n2. . . of the coefficients

an₁n₂. . . is a shorthand for the partition 1n12n2. . . .兲 All higher order Virasoro constraints are satisfied trivially through the Virasoro algebra. If we denote the level of the coefficients by n⬅兺kknk and the total number of partitions

level n⫹2 is equal to p(n⫹2) whereas the total number of inhomogeneous equation 共for s⫽1) is equal p(n⫹1) and the total number of homogenous equations 共for s⫽2) is equal to p(n). The total number of partitions satisfies the recursion relation 关30兴 p共n⫹2兲⫽p共n⫹1兲⫹p共n兲⫺p共n⫺3兲⫺p共n⫺5兲⫹••• ⫺共⫺1兲k_p

冉

_n⫹2⫺3k 2_⫺k 2

冊

⫺共⫺1兲 k_p

冉

_n⫹2 ⫺3k 2_⫹k 2

冊

⫹•••. 共36兲

Since the number of partitions is a monotonic function of n, we have that

p共n⫹2兲⭐p共n⫹1兲⫹p共n兲, 共37兲

and the number of equations is always larger than or equal to the number of coefficients. In Table I we give the total num-ber of coefficients and the total numnum-ber of equations up to level n⫽10.

From normalization condition a0⫽1 we conclude that

a₁⫽1/(␣⫹␥). At level n⫽2 we obtain one equation each from Eqs. 共34兲 and 共35兲, respectively,

共␣⫹1兲a2⫹2a11⫽0,

共38兲 ␥a2⫹2共␣⫹␥兲a11⫽a1⫽1/共␣⫹␥兲,

which are solved by

a11⫽ 共1⫹␣兲 2␣共␣⫹␥兲共␣⫹␥⫹1兲, 共39兲 a2⫽ ⫺1 ␣共␣⫹␥兲共␣⫹␥⫹1兲. Up to order (MM†)4 we find Z_␯,N␤ _f共M,M†_{兲⫽共det M兲}␯

冋

_1⫹Tr共MM †_兲 4共␣⫹␥兲 ⫹ 共␣⫹1兲共TrMM†_兲2 32␣共␣⫹␥兲共␣⫹␥⫹1兲 ⫺ Tr共MM†兲2 32␣共␣⫹␥兲共␣⫹␥⫹1兲 ⫹ 2Tr共MM †_兲3 433␣共␣2⫺␥2兲共␣⫹␥⫹1兲共␣⫹␥⫹2兲⫺ 共␣⫺␥⫹2兲Tr共MM†_兲Tr共MM†_兲2 432␣共␣2⫺␥2兲共␣⫹␥⫹1兲共␣⫹␥⫹2兲 ⫹ 关共␣⫺␥兲共␣⫹3兲⫹2兴关Tr共MM †_兲兴3 436␣共␣2⫺␥2兲共␣⫹␥⫹1兲共␣⫹␥⫹2兲 ⫹ 共␥⫺5␣⫺6兲Tr共MM †_兲4 45␣共␣⫹1兲共␣2⫺␥2兲共␣⫹␥⫹1兲共␣⫹␥⫹2兲共␣⫹␥⫹3兲共␣⫺2␥兲 ⫺ 关2␣ 2_{⫺4␣共␥⫺2兲⫺7␥⫹6兴Tr共MM}†_兲Tr共MM†_兲3 443␣共␣⫹1兲共␣2⫺␥2兲共␣⫹␥⫹1兲共␣⫹␥⫹2兲共␣⫹␥⫹3兲共␣⫺2␥兲 ⫹ 关␣ 2_{⫹␣共5⫺3␥兲⫹2␥}2_{⫺␥⫹6兴关Tr共MM}†_兲2_兴2 452␣共␣⫹1兲共␣2⫺␥2兲共␣⫹␥⫹1兲共␣⫹␥⫹2兲共␣⫹␥⫹3兲共␣⫺2␥兲 ⫹关3␣ 2_{共␥⫺2兲⫺␣}3_⫺6⫺4␥2_{⫹13␥⫹␣共⫺11⫹14␥⫺2␥}2_{兲兴共TrMM}†_兲2_Tr共MM†_兲2 45␣共␣⫹1兲共␣2⫺␥2兲共␣⫹␥⫹1兲共␣⫹␥⫹2兲共␣⫹␥⫹3兲共␣⫺2␥兲 ⫹关6⫹␣ 4_⫹␣3_{共7⫺3␥兲⫺25␥⫹18␥}2_⫹␣2_{共17⫺21␥⫹2␥}2_{兲⫹␣共17⫺43␥⫹14␥}2_{兲兴共TrMM}†_兲4 45_{6␣共␣⫹1兲共␣}2_⫺␥2_{兲共␣⫹␥⫹1兲共␣⫹␥⫹2兲共␣⫹␥⫹3兲共␣⫺2␥兲} ⫹O„共MM†_兲6_…

册

_{. 共40兲}

TABLE I. The total number of unknown coefficients 关P(n)兴 and the total number of equations关P(n⫺1)⫹P(n⫺2)兴 at level n.

n P(n) P(n⫺1)⫹P(n⫺2) n P(n) P(n⫺1)⫹P(n⫺2) 1 1 1 6 11 12 2 2 2 7 15 18 3 3 3 8 22 26 4 5 5 9 30 37 5 7 8 10 42 52

For␤⫽1 the mass matrix M˜ has been expressed in terms of the standard N_f⫻N_f mass matrix that occurs in the QCD partition function 关see Eq. 共24兲兴. For ␤⫽2 and ␤⫽4 the matrix M is a complex Nf⫻Nf matrix as well, but for ␤

⫽4 it is symmetric. We have checked that all coefficients in Eq. 共40兲 agree with the corresponding expansion for ␤⫽2 obtained in 关20兴. When the masses are degenerate (MM†)ab⫽␮2␦ab exact expressions for Z␯,N␤ _f are known

for all three values of the Dyson index. For instance, in the special case Nf⫽2 and ␤⫽1 the finite volume partition

function is given by关29兴 Z_␯,2␤⫽1 ⫽ 2

兺

k_⫽0 ⬁

冉

␮2 4

冊

(␯⫹k) _{关2共␯⫹k兲兴!} k!共␯⫹k兲!共␯⫹k⫹2兲!共2␯⫹k兲!, 共41兲 which is a convergent series forZ_␯,2␤⫽1. The reader can check that Eq. 共40兲 only differs from Eq. 共41兲 by the overall nor-malization factor 21⫺2␯/(␯!(␯⫹2)!). We have also made similar checks for␤⫽4. The poles that appear in the expan-sion 共40兲 at finite Nf, the so called de Wit–’t Hooft poles

关32兴, cancel after rewriting the expansion in powers of prod-ucts of the eigenvalues ofM†M. They can be regulated by choosing a noninteger value for the topological charge 关20,33兴.

The expansion in terms of traces of the quark mass matrix can be directly compared with

具

兿_aN_⫽1f _det(D⫹m

a)

典

Yang-Mills from which we derive spectral sum-rules for inverse powers of the eigenvalues of the Dirac operator关10兴. With␨krelated

to the eigenvalues ⫾i␭k of the Dirac operator through ␨k

⫽V⌺␭k, the first six sum rules for ma⫽0 are given by

冓

兺

n 1 ␨n 2

冔

⫽ 1 4共␣⫹␥兲,

冓

兺

n 1 ␨n 4

冔

⫽ 1 16␣共␣⫹␥兲共␣⫹␥⫹1兲,

冓

冉

兺

n 1 ␨n 2

冊

2

冔

⫽_{16␣共␣⫹}␣⫹1_{␥兲共␣⫹␥⫹1兲},

冓

兺

n 1 ␨n 6

冔

⫽ 1 32␣共␣2⫺␥2兲共␣⫹␥⫹1兲共␣⫹␥⫹2兲,

冓

兺

n 1 ␨n 2

兺

_m 1 ␨m 4

冔

⫽ ␣⫺␥⫹2 64␣共␣2⫺␥2兲共␣⫹␥⫹1兲共␣⫹␥⫹2兲,

冓

冉

兺

n 1 ␨n 2

冊

3

冔

⫽ 共␣⫺␥兲共␣⫹3兲⫹2 64␣共␣2⫺␥2兲共␣⫹␥⫹1兲共␣⫹␥⫹2兲. 共42兲 The sum is over the positive eigenvalues only. It is clear that all sum rules will depend on the topological charge through the combination Nf⫹␤␯/2. The first formula in Eq.共42兲 had

already been obtained in 关29兴 from the low-energy QCD

fi-nite volume partition functions. For ␤⫽2 our sum rules agree with the ones derived in 关20兴 obtained by means of Virasoro constraints and flavor-topology duality. The remain-ing sum rules have never been derived before from the finite volume partition functions for ␤⫽1,4, but all except the fourth one have been obtained from chRMT by means of Selberg integrals for arbitrary␤ 关31兴.

B. Flavor-topology duality

So far for the sum rules, we now return to the flavor-topology duality relations which is our main result. To this end we construct a (N_f⫹␤␯/2)⫻(N_f⫹␤␯/2) matrix M¯ from the Nf⫻Nf original mass-matrix M by adding zeros.

We thus have that Tr(MM†)k⫽Tr(M¯ M¯ †)k. The flavor-topology duality relation then follows from the observation that all coefficients in the small-mass expansion共40兲 depend on the number of flavors explicitly through the combination

Nf⫹␤␯/2 and can be written as

Z_␯,N

f

␤ _共M,M†_{兲⫽共det M兲}␯_Z 0,N_f⫹␤␯/2

␤ _{共M¯ ,M¯} †_{兲, 共43兲} where ␯→¯ for␯ ␤⫽4. For the cases where ␤␯/2 is not an integer, the flavor-topology duality should be understood as an analytical continuation2 in␯. Strictly speaking, we have only shown the duality relation 共43兲 for the recursion rela-tions for the coefficients of the small-mass expansion of the partition functions. To complete our proof we have to show that the Virasoro constraints uniquely determine all coeffi-cients and that the expansion is a convergent series. The convergence of the small mass expansion follows immedi-ately from the compactness of the unitary group. Indeed, for ␤⫽2, the small-mass expansion can be resummed to the ␶-function of a KP hierarchy关19,16兴. This result can also be obtained from an exact reconstruction ofZ_␯␤⫽2关28兴 from the character expansion关27兴 which is an alternative to the small-mass expansion derived here. The uniqueness of the solution of the Virasoro constraints is a much more difficult question 关34兴. We have checked explicitly to the order (MM†₎8_that this is indeed the case for all values of the parameters.

For␤⫽2, however, the uniqueness follows from the ex-plicit solution关35兴 of the equation

⳵2 ⳵Mba⳵Mab † Z␯ ␤⫽2_共M,M†_兲⫽Nf 4Z␯ ␤⫽2_共M,M†_兲, 共44兲 obtained from the trace of Eq. 共8兲 or the Schwinger-Dyson equation关21兴

2_{The analytical continuation in␯ is subtle. The correct}

continua-tion is obtained by 关26,12兴 expressing the finite volume partition function in terms of modified Bessel functions and make an ana-lytical continuation in the index of the Bessel functions.

Mcd † _M db ⳵2 ⳵Mba⳵Mac † Z␯ ␤⫽2_共M,M†_兲 ⫽Mba † _M ab Nf 4Z␯ ␤⫽2_共M,M†_{兲, 共45兲} obtained by contracting Eq. 共8兲 with M and M†. In both cases, the invariance of the partition function can be used to reduce the equations to a separable differential equation in the eigenvalues of M†M. Because we are looking for a solution that is symmetric in the eigenvalues satisfying the boundary condition G_␯(0)⫽1, it is determined uniquely by the solution of the single particle equation which happens to be the Bessel equation.

Let us discuss this in more detail for Eq. 共44兲. The first step is to write the partition function as

Z_␯␤⫽2共M,M†_兲⫽

冉

detM detM†

冊

␯/2

Z˜_{␯共⌳兲, 共46兲}

whereZ˜_␯(⌳) is a function of the eigenvalues x_k ofM†_M only 关⌳⫽diag(x1, . . . xN_f)兴. It satisfies the differential

equation

冋

⳵2 ⳵Mba⳵Mab † ⫺

兺

k ␯2 4x_k2

册

Z ˜_{␯共⌳兲⫽}Nf 4 Z˜␯共⌳兲. 共47兲 This equation can be further simplified by introducing the reduced partition function

z_␯共⌳兲⫽⌬共x_k2兲

兿

k

冑

xkZ˜␯共⌳兲, 共48兲

where the Vandermonde determinant is defined by ⌬(x_k2) ⫽兿k_⬍l(xk

2_⫺x

l

2_{). We thus have that z}

␯(⌳) is a completely

antisymmetric function of the eigenvalues with boundary condition such that Z˜_␯(⌳)⬃兿kxk␯ for xk→0. The reduced

partition function satisfies the separable differential equation

兺

k

冋

⳵k

2_⫺4␯ 2_⫺1

4x_k2

册

z␯共⌳兲⫽Nfz␯共⌳兲. 共49兲 The solution can thus be expressed as a Slater-determinant

z_␯共⌳兲⫽detkl关␾k共xl兲兴, 共50兲

where the ␾k are the regular solutions of the single particle

equation up to terms that vanish in the determinant. This requires further discussion. The solution of the single particle equation D_B␾_k共x兲⬅

冋

⳵_x2⫺4␯ 2_⫺1 4x2

册

␾k共x兲⫽␻k 2_␾ k共x兲, 共51兲

is the modified Bessel function

␾k共x兲⫽冑xI␯共␻kx兲. 共52兲

In order to cancel the Vandermonde determinant for xk→0

we necessarily have that all ␻_k2 are equal and thus ␻k⫽1.

Because of the antisymmetry of the columns in the Slater determinant 共50兲 we can allow for single particle solutions that satisfy the equations

DB␾1共x兲⫽␾1共x兲,

DB␾2共x兲⫽␾2共x兲⫹␮21␾1共x兲,

DB␾3共x兲⫽␾3共x兲⫹␮31␾1共x兲⫹␮32␾2共x兲,etc. 共53兲 By redefining ␾3,␾4, . . . we obtain the equations 共for k ⭓2)

DB␾k共x兲⫽␾k共x兲⫹␮k k⫺1␾k⫺1共x兲, 共54兲

where the ␮_{k k⫺1}␾_k⫺1(x) cancel in the expression for the Slater determinant. The solution of these equations follows immediately from the recursion relation

DB关xk⫹1/2I␯⫹k共x兲兴⫽xk⫹1/2I␯⫹k共x兲⫹2kxk⫺1/2I␯⫹k⫺1共x兲.

共55兲 We thus find that

␾k⫽xk⫹1/2I␯⫹k共x兲 共56兲

and ␮_{k k⫺1}⫽2k. Since the Bessel equation has only one regular solution, we conclude that this solution is unique. The Virasoro constraints together with the boundary condi-tions determine the partition function uniquely and is given by a Slater-determinant of modified Bessel functions 共50兲 which agrees with the known result for the one-link intergral 关21,19,10,35兴.

Actually the solution of the Virasoro constraints, although not its uniqueness can be obtained in a simpler way by iden-tifying them with constraints which are satisfied by the gen-eralized Kontsevich model 共GKM兲. By changing variables from the matrix elements ofM and M† to the matrix ele-ments of the Hermitian matrix L⬅(MM†_{/4) we can} re-write Eq. 共9兲 as

冋

⳵ ⳵Lbe Lde ⳵ ⳵Ldc ⫹␯_⳵L⳵ bc

册

G_␯共L兲⫽␦bcG␯共L兲. 共57兲

This equation admits a solution in the form of a GKM with potential V(X)⫽1/X where X is a N_f⫻N_f normal matrix. More explicitly we have3

G_␯共L兲⫽

冕

dX

共det X兲N_f⫺␯e

Tr[LX_{⫹1/X], 共58兲}

which is a simple modification of the ␯⫽0 case studied in 关19兴. In order to verify that Eq. 共58兲 is a solution of Eq. 共57兲

3_{The contour of integration is such that the integral converges for}

all values of L, and that an integral over a total derivative of X vanishes.

we have to insert a total derivative in the integral over X and multiply it by a second order differential operator as follows:

⳵ ⳵Lbe ⳵ ⳵Ldc

冕

dX ⳵ ⳵Xed

冉

1 共det X兲N_f⫺␯e Tr[LX⫹1/X]

冊

_⫽0. 共59兲 Concluding, for␤⫽2 the Virasoro constraints naturally lead to a simple and rigorous proof of flavor-topology duality共43兲 which is an alternative to the proof presented in关11,12兴. For ␤⫽1 and␤⫽4 a direct solution of the Virasoro constraints is not known, although we believe that further progress can be achieved at least in the case of equal quark masses where exact formulas for the partition function are known关29兴. For non-degenerate masses the obstacle is the lack of a generali-zation of the Itzykson-Zuber formula for integrals over sym-metric and anti-symsym-metric unitary matrices. Therefore our proof of flavor-topology duality for ␤⫽1 and ␤⫽4 is only valid under the assumption of the uniqueness of the solution of the Virasoro constraints.

IV. CONCLUSIONS

In a parameter range where the main contribution to the QCD partition function comes from the zero-momentum component of the Goldstone modes, the partition functions reduce to zero-dimensional integrals over the group mani-folds SU(N_f), SU(N_f)/O(N_f) and SU(2N_f)/S p(2N_f), for fermions in the fundamental representation and Nc⭓3 (␤

⫽2), Nc⫽2 (␤⫽1), and adjoint fermions with Nc⭓2 (␤

⫽4), respectively. In the sector of fixed topological charge␯ we have shown that these partition functions satisfy differ-ential equations in the form of Virasoro constraints which determine the small-mass expansion for arbitrary values of the topological charge. We have calculated the coefficients of the small mass expansion recursively up to the order m8 and have obtained new spectral sum rules for inverse powers of the eigenvalues of the Dirac operator for ␤⫽1, ␤⫽2 and ␤⫽4. For all three values of␤the constraints depend on the topological charge through the combination Nf⫹␤␯/2 which

demonstrates flavor-topology duality for the corresponding small-mass (m→0) expansions. For the special case of ␤

⫽2 the uniqueness of the solution of the Virasoro constraints and the convergence of the small mass expansion implies flavor-topology duality for the full 共finite m) partition func-tion. In this case, we have found a non-perturbative solution of the Virasoro constraints in the form of a generalized Kont-sevich model for arbitrary topological charge ␯ which complements the␯⫽0 results of 关19兴. We have not been able to find an analogous solution for␤⫽1 and␤⫽4. Based on numerical evidence we believe that for these two cases the solution of the Virasoro constraints is unique as well, but a rigorous proof could not be given. The existence of exact expressions for partition functions with degenerate masses (MM†₎

ab⫽␮2␦ab indicates that further progress is

pos-sible.

It is worth commenting that, even for␤⫽2, although or-der by oror-der the character expansion is equivalent to the perturbative solution of the small-mass Virasoro constraints, it does not provide us with a natural proof of flavor-topology duality共see 关28兴兲. It should also be pointed out that the GKM are known to satisfy both small-mass and large-mass Vira-soro constraints. Thus, in view of our results, we also expect to find large-mass constraints for ␯⫽0 at least for ␤⫽2. Indeed we have explicitly obtained such constraints entirely within the context of unitary integrals and they depend qua-dratically on the topological charge␯, contrary to the linear dependence of the small-mass constraints. The generalization of the large-mass constraints to␤⫽1 and␤⫽4 seems to be within reach. Due to the absence of an explicit Nf

depen-dence of the expansion coefficients they are the most rel-evant ones from the point of view of both the replica limit 关20,26,12兴 and the classification of universality classes 关36兴. In particular, for applications in 2D gravity they correspond to the continuum Virasoro constraints which appear after the double scaling limit.

ACKNOWLEDGMENTS

This work was partially supported by the U.S. DOE grant DE-FG-88ER40388. The work of D.D. is supported by FAPESP共Brazilian Agency兲. Poul Damgaard, Kim Splittorff and Dominique Toublan are acknowledged for useful discus-sions.

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