Topological contributions in two-dimensional Yang-Mills theory: from group averages to integration over algebras A. Bassetto, S. Nicoli and F. Vian Dipartimento di Fisica “G.Galilei”, Via Marzolo 8, 35131 Padova, Italy INFN, Sezione di Padova, Italy 1 0 0 2 Abstract n a J We show that keeping only the topologically trivial contribution to the aver- 9 1 age of a class function on U(N) amounts to integrating over its algebra. The v 2 goal is reached firstby decompactifying an expansion over the instanton basis 5 0 1 and then directly, by means of a geometrical procedure. 0 1 0 / h t - p e h : v DFPD 00/TH 55 i X PACS numbers: 02.20.Qs, 11.15.-q, 11.15.Tk r a Keywords: Yang-Mills in two dimensions, invariant integration over groups and related algebras. 1 I. INTRODUCTION Yang-Mills theory in two dimensions without dynamical fermions (YM ) shares important 2 features with topological theories. When considered on the plane and quantized in the light- conegauge(LCG)itlooksindeedtrivial,wereitnotfortheverysingularnatureofcorrelators at large distances. When infrared singularities are regularized via compactification, namely by putting the theory on a (partially or totally) compact euclidean manifold, dynamics gets hidden in peculiar topological properties. These features emerge quite naturally if we consider the vacuum average of a regular non self-intersecting Wilson loop. Thanks to the invariance of YM under area-preserving 2 diffeomorphisms, this average turns out to be insensitive to the shape of the contour. Ac- tually one can prove that it can be obtained via invariant integration over the group man- ifold (in the following we shall limit ourselves to U(N), the generalization to SU(N) being straightforward [1]) of a class function (the heat kernel), the outcome exhibiting a pure area dependence [2]. By class functiononU(N) we meananexpression which enjoys the property P(U) = P(hUh†), U ∈ U(N), ∀h in the fundamental representation of U(N). This implies that P is a symmetric function of the eigenvalues of U. The goal of deriving Wilson loops by means of integration over the group can be reached either through a geometrical construction, generalizing techniques which are drawn from the lattice [3–5] to the continuum, or by resumming a perturbative series when the theory is quantized on the light front [6]. Performing a modular inversion [7–10], one can eventually interpret the result as an infinite sum over non-analytic contributions (instantons), tightly related to the group curvature. If the theory is again quantized in LCG, but at equal times, and the same Wilson loop is computed by resumming the corresponding perturbative series, a quite different behaviour ensues [11]. In particular, in the large-N limit, confinement is lost. On the other hand, in Ref. [12], it has been shown that perturbation theory, in the equal-time formulation, can only account for the zero-instanton sector (a truly perturbative result). 2 A further step has been performed in Ref. [13], where the zero-instanton contribution to the average of the class function describing a Wilson loop in two different representations, is obtained by trading integration over the group manifold with integration over its algebra. The purpose ofthis note isto elucidate further the geometrical meaning of thisprocedure aswellastoextendittothevacuumaverageofamoregeneralsetofclass-invariantfunctions. II. FROM THE GROUP MANIFOLD TO THE GROUP ALGEBRA As is well known, the partition functions on compact surfaces of genus zero, have a twofold expression, either as an expansion in the characters of the irreducible representations of the gauge group [4], or as a series of terms behaving exponentially with respect to the inverse coupling constant squared 1/g2 [1,4,14,8,9,15]. For example, the partitionfunction ofYM on a cylinder of area Awith fixed holonomies 2 at the boundaries g , g reads 1 2 g2A K(A; g , g ) = χ† (g )χ (g ) exp − C (R) (1) 2 1 R 2 R 1 4 2 R (cid:20) (cid:21) X and g2A (g2A)−N +∞ K(A;g2,g1) = N exp N(N2 −1) 2 (−1)(N−1) klk 48 J(θ)J(φ) (cid:20) (cid:21) {liX}=−∞ P N 1 × ǫ(σ)exp − (θ −φ −2πl )2 . (2) g2A i σ(i) i " # σ∈Π(1,...,N) i=1 X X C (R) is the quadratic Casimir invariant of the irreducible representation R , θ and φ are 2 i i the invariant angles corresponding to g and g , respectively, N is a normalization factor 1 2 and J(θ) ≡ J(θ ,...,θ ) = 2sin θi−θj . In the equation above, σ ∈ Π denotes 1 N i<j 2 (cid:16) (cid:17) a permutation and ǫ(σ) its sigQnature. Both the expressions (1), (2) are solutions of the heat-kernel equation 4 ∂ K(A;g ,g ) = △ K(A;g ,g ), (3) g2∂A 2 1 θ 2 1 3 △ being the laplacian over the group manifold, with the boundary condition (which fixes θ the normalization constant N) lim K(A;g ,g ) = δ(g −g ), (4) 2 1 2 1 g2A→0 δ(g −g ) being the class-invariant δ-distribution. 2 1 Equations (1), (2) are linked by what is called a modular inversion [7–10]. Actually they represent two different, unitarily equivalent, harmonic analyses of the class function K. In turn the kernel K is the basic quantity for writing the partition function on the sphere of area A Z(A) = K(A;11,11) as well as, more generally, the expectation value of a non self-intersecting Wilson loop in the R representation, enclosing an area A and winding n times [4,5,16] 1 Z(A)W (A ,A) = DU K(A ;11,U) K(A−A ;U,11)Tr(Un). (5) n 1 1 1 R Z Since our goal will be to single out the zero-instanton contribution to any class function in L2(U), as a warm-up we begin by considering the simple case of the Wilson loop in Eq. (5). It is immediately realized that Eq. (2) provides the natural starting point, being a series of exponentials of 1 . However the limit g → 11 is to be performed carefully; the g2 2 result is K(A;11,g1) = N˜(g2A)−N22 exp g2AN(N2 −1) (6) 48 (cid:20) (cid:21) × +∞ i<j[θi −θj +2π(li −lj)] exp − 1 N (θ +2πl )2 . i i J(θ+2πl) g2A {liX}=−∞ P " Xi=1 # Hereafter Eq. (5), in the case of the fundamental representation, becomes 2π ∆(θ +2πl) ∆(θ +2πm) W (A ,A)Z(A) = dθ ...dθ |∆(eiθ)|2 (7) n 1 1 N J(θ +2πl) J(θ +2πm) {m} {l} Z0 X X N N N 1 1 × exp − (θ +2πl )2 − (θ +2πm )2 eiθkn, i i i i g2(A−A ) g2A " 1 1 # i=1 i=1 k=1 X X X 4 where ∆(a) = (a − a ) is the Vandermonde determinant and |∆(eiθ)|2dθ ...dθ is i<j i j 1 N the group invariQant measure. Notice that, in Eq. (7), the product J(θ + 2πm)J(θ + 2πl) simplifies with |∆(eiθ)|2 leaving the phase factor exp[iπ(N −1) (l −m )], and that an i i i area dependent normalization factor in W was dropped since itPcan be absorbed in Z by n suitably rescaling the integration variables. In the sequel it will be understood that the expression of Z(A) is recovered by performing the average of the identity. It is now easy to get rid of the sums over l by enlarging the range of integration over θ ; exploiting the i i symmetry over the angles, one has +∞ WnZ = eiπ(N−1) imi dθ1...dθN ∆(θ)∆(θ +2πm) (8) {m} P Z−∞ X N N 1 1 × exp − θ2 − (θ +2πm )2 eiθ1n. g2(A−A ) i g2A i i " 1 1 # i=1 i=1 X X After performing suitable shifts and rescalings over the integration variables, we obtain n2g2A A 1 2 W Z = exp − exp iπ(N −1) m (9) n i 4A " # (cid:20) (cid:21) {m} i X X 4π2 N A +∞ 1 N × exp − m2 −2πi 2nm dθ ...dθ exp − θ2 g2A i A 1 1 N g2A i " i=1 # Z−∞ " i=1 # X X A ing2 A A 2 2 2 × ∆ θ −2π m + A A , θ −2π m ,...,θ −2π m 1 1 1 2 2 2 N N A 2 A A r 1 r 1 r 1 ! p A ing2 A A 1 1 1 × ∆ θ +2π m + A A , θ +2π m ,...,θ +2π m 1 1 1 2 2 2 N N A 2 A A r 2 r 2 r 2 ! p where A = A−A . Using now the the identity 2 1 +∞ − 1 z2 dz1 ... dzN e g2A i ∆(zi +ai/h) ∆(zi +bih) (10) Z−∞ P +∞ − 1 z2 = dz1 ... dzN e g2A i ∆(zi +ai) ∆(zi +bi) Z−∞ P with a ,b and h complex quantities, which can be proven for instance by expanding the i i Vandermonde determinants in terms of Hermite polynomials, Eq. (9) becomes +∞ A A WnZ = eiπ(N−1) imi dθ1...dθN ∆(θ1 −2πm1 −ing2 2− 1, ... , θN −2πmN) 4 {m} P Z−∞ X 5 N A A 1 nθ × ∆(θ +2πm +ing2 2− 1,...,θ +2πm ) exp − θ2 +i 1 1 1 4 N N g2A i 2 ! i=1 X n2g2(A −A )2 4π2 N A × exp 2 1 exp − m2 −2πi 2nm . (11) 16A g2A i A 1 " # (cid:20) (cid:21) i=1 X Eq. (11) coincides with the result of Ref. [12], derived from the representation in terms of group characters (our Eq. (1)) via a Poisson transformation. Nevertheless we found it instructive to derive it from the representation (2) since the same procedure can be easily transferred to more general situations. We can now single out the zero-instanton contribution from the above formula by retain- ing only the term with m = 0, ∀i. i In Ref. [13] it is shown how this zero-instanton contribution can be written as an integral overthegroupalgebraforaWilsonloopinthefundamentalandintheadjointrepresentation, namely 1 A (A−A ) W0(A ,A)Z0(A) = DF exp − Tr F2 Tr exp igF 1 1 (12) n 1 2 2A Z (cid:18) (cid:19) " r # = DF e−12Tr(F2) χfund(eigEF) Z and 2 A (A−A ) W0(A ,A)Z0(A) = DF Tr exp igF 1 1 −1 (13) n 1  2A  (cid:12) " r #(cid:12) Z (cid:12) (cid:12) (cid:12) (cid:12) × exp −1 T(cid:12)(cid:12)r F2 = DF e−21Tr(F2) χ(cid:12)(cid:12)adj(eigEF), 2 (cid:18) (cid:19) Z respectively, with E = A1(A−A1), and F a Hermitean N ×N matrix. 2A q We want to extend this property to a generic class function S(U) ≡ P(eiθj), P being a L2-summable symmetric function of the eigenvalues, which can be expanded in the group characters P = b χ (eiθj). (14) R R R X 6 The set of group characters {χ } represents an orthogonal basis in the space of class func- R tions. Alternatively, recalling that each character is a symmetric polynomial in e±iθ, one can write P = p S , (15) {q} {q} {q} X where S{q} = σ∈Π(1,...,N) eiθσ(1) q1... eiθσ(N) qN and q1,...qN are integers. Generalizing Eq. (7) we getP (cid:0) (cid:1) (cid:0) (cid:1) 2π SZ = (−1)(N−1) i(mi−li) dθ1...dθN ∆(θ+2πl)∆(θ+2πm) (16) {m} {l} P Z0 X X N N 1 1 × exp − (θ +2πl )2 − (θ +2πm )2 P(eiθj). i i i i g2A g2A " 1 2 # i=1 i=1 X X We can now repeat the previous procedure to obtain +∞ 1 N SZ = p{q} (−1)(N−1) imi dθ1...dθN exp −g2A θi2 (17) {q} σ∈Π(1,...,N){m} P Z−∞ " i=1 # X X X X × ∆ θ −2πm −iαq ,..., θ −2πm −iαq 1 1 σ(1) N N σ(N) × ∆(cid:0)θ1 +2πm1 +iαqσ(1),..., θN +2πmN +iαqσ(N)(cid:1) e2i kθkqσ(k) P (cid:0) g2(A −A )2 N 4π2 N (cid:1)A × exp 1 2 q2 exp − m2 −2πi 2 q m 16A i g2A i A σ(k) k " # " # i=1 i=1 k X X X with α = g2(A−2A ). By retaining only the zero-instanton contribution (m = 0,∀i), we 4 1 i find +∞ 1 N S0Z0 = p{q} dθ1...dθN ∆(θ)2 exp −2 θi2 eigE kθkqσ(k) (18) {q} σ∈Π(1,...,N)Z−∞ " i=1 # P X X X +∞ 1 N = dθ ...dθ ∆(θ)2 exp − θ2 p (eigEθ1)qσ(1)...(eigEθN)qσ(N) 1 N 2 i {q} Z−∞ " i=1 # {q} σ∈Π(1,...,N) X X X +∞ = DFe−21Tr[F2]P(eigEF). Z−∞ To get such a result, we have used the relation [13] +∞ 2g2 2g2 dθ He (θ −iαq ) He (θ +iαq ) (19) k rk−1 k σ(k) A sk−1 k σ(k) A Z−∞ r r 7 ×exp −21θk2 + 2i g22A qσ(k)θk = exp −g2(A126−AA1)2 qσ2(k) (A−A1)rk−2skA1sk−2rk r ! (cid:20) (cid:21) +∞ 1 A (A−A ) × dθ exp − θ2 +ig 1 1 q θ He (θ )He (θ ) k 2 k 2A σ(k) k rk−1 k sk−1 k Z−∞ r ! and taken into account that r = s . k k k k ThusfarwehaveshownthPatretainPingonlythezero-instantoncontributiontotheaverage of a class function amounts to integrate over the group algebra. One may wonder how this happens. We see from Eq. (17) that the zero-instanton contribution dominates in the limit g2A → 0. Intuitively, in this limit both heat-kernel solutions become the class-invariant δ-distribution, and integration over the group manifold is turned into integration over its tangent space, namely its algebra. III. A GEOMETRICAL APPROACH This issue can be made mathematically more precise according to the following argument. The exact heat-kernel solution lives on the topologically non-trivial U(N) manifold. Thus, neglecting instantons amounts to map the manifold U(N) into a topologically trivial one. Of course this mapping cannot be a global diffeomorphism; nevertheless we may require it to be local, in order to preserve the original differential structure. We can consider the map U → −ilogU, which is a multivalued immersion of the group into its algebra, infer the image ∆ of the laplacian operator ∆ and seek for solutions of the heat-kernel equation X θ 4 ∂ ∆ H(A;θ) = H(A;θ) (20) X g2∂A in the algebra manifold, obeying lim H(A;θ) = δ(θ). (21) g2A→0 The group algebra manifold is RN2 with canonical coordinates (ω ,...,ω ). Consider 1 N2 the exponential map 8 (ω ,...,ω ) 7−→ eiωaTa. (22) 1 N2 This is a locally invertible differentiable map, and therefore defines a local diffeomor- phism. The metric on the group manifold is given by [1,17] N N ds2 = Tr(U−1dU (U−1dU)†) = dθ2 + |eiθj −eiθk|2|(V†dV) |2, (23) k jk k=1 j6=k=1 X X where V is a unitary matrix such that U = Vdiag(eiθk)V†, whereas on the tangent space it reads N N ds2 = Tr(dF dF) = dθ2 + |θ −θ |2|(V†dV) |2. (24) k j k jk k=1 j6=k=1 X X The laplacian operator acting on class functions over the group N 2 1 ∂ 1 △ = J(θ)+ N(N2 −1) (25) θ J(θ) ∂θ 12 k=1(cid:18) k(cid:19) X becomes 1 N ∂2 △ = ∆(θ ). (26) X ∆(θ ) ∂θ2 k k k=1 k X Then the solution of the heat-kernel equation on the algebra manifold is N H(A;θ) = (πg2A)−N22 exp −g21A θk2 (27) ! k=1 X and consequently, generalizing Eq. (5), we check that S0Z0 = DF H(A ;θ)H(A ;θ)P(eiF) (28) 1 2 Z = DFe−g21A1 TrF2e−g21A2 TrF2P(eiF) Z = DFe−2g21E2 TrF2P(eiF) Z is equivalent to Eq. (18), as expected. 9 IV. CONCLUSIONS In this letter we showed that the topologically trivial contribution to the average over U(N) of a class function belonging to L2(U) corresponds to its average over the group algebra and thereby to a matrix model. This was first derived generalizing the harmonic analysis of the Wilson loop average in the fundamental and in the adjoint representations performed in Ref. [13]. We started from an expansion in terms of instantons and retained only the trivial sector. This procedure makes clear that instantons are indeed related to the group curvature. Then we presented a direct approach relying on purely geometrical considerations: we mapped the heat equation from the group manifold to its tangent space and exploited the basic heat-kernel sewing property. In our procedure, when singling out the trivial topological sector, the entire group U(N) underwent decompactification. It might be worth examining the consequences on the in- stanton patterns ensuing from partial decompactifications, namely understanding to what extent different instanton sectors turn out to be intertwined. ACKNOWLEDGEMENTS Discussions with L. Griguolo are gratefully acknowledged. This work was carried out in the framework of the NATO project “QCD Vacuum Structure and Early Universe” granted under the reference PST.CLG 974745. 10