arXiv:hep-th/0612296v2 24 Jan 2007 nsgamdlR o,“eta hre action charge” “central flow, RG model sigma On 1 loa eee nttt,[email protected]. Moscow. Institute, Lebedev at Also h eta hrea h xdpit,i gemn ihteg the with agreement in points, fixed c-theorem. the at charge central the oallo res(..alodr in Perelm orders the all generalize (i.e. conditi to orders how loop a discuss all with We to dilaton fixed. the is (math. volume to essential Perelman space respect is by with which proved extremised functional action was “entropy” an monotonicity th constructing the when by flow order (one-loop) Ricci sp o leading the the To monotonicity on the indefinite. metric model the However, is sigma since couplings obvious 2d orders. not is in loop metric flow all target-space RG to the one that gradient imply constructions) action aoocio’ -hoe yeagmn adas tigt also (and argument type c-theorem Zamolodchikov’s lcetLbrtr,Ipra olg,Lno W A,U.K 2AZ, SW7 London College, Imperial Laboratory, Blackett n eemnsentropy Perelman’s and ..Tseytlin A.A. α Abstract ′ .Terslig“nrp”i qa ominus to equal is “entropy” resulting The ). 1 uk ytemetric-dilaton the ly c fmetric-dilaton of ace Gflwi simply is flow RG e nta h target- the that on nrlcamo the of claim eneral nsconstruction an’s h o fthe of flow the f Imperial-TP-AT-6-7 er effective heory dg/0211159) hudb a be should . 1 Introduction

2d sigma models containing infinite number of couplings parametrized by target-space metric tensor Gµν(x) have many interesting connections to various problems in physics and mathematics. In particular, they play an important role in , describing string propagation in curved space. In order to define the quantum stress tensor of the sigma model, i.e. its renormalization on a curved 2d space, one is led to the introduction an extra scalar coupling function φ(x) or the dilaton. The corresponding metric and dilaton RG “beta-functions” βi are then proportional to string effective equations of motion (for reviews see, e.g., [1, 2]). Assuming that the c-theorem claim [3] should apply to the sigma model (with compact euclidean target space) one should the existence of a (local, covariant) functional of the metric such that (i) its gradient is proportional to βi (with certain diffeomorphism terms added), (ii) it decreases along the RG flow and (iii) it is equal to the central charge at fixed points. While the gradient property of the flow is relatively easy to establish, its monotonicity is much less obvious. In particular, the direct generalization of Zamoldchikov’s proof of the c-theorem [4] leads to the “central charge” action which vanishes at any fixed point, instead of decreasing along the flow. One may expect that the statement of the c-theorem should apply provided one considers only the Gµν metric flow (the flow of the dilaton plays, in a sense, a secondary role). However, the technical details of the Zamolodchikov-type construction [3, 4] of the “central charge” functional do not directly apply if one ignores the dependence on the dilaton. This suggests that one needs an alternative way of constructing the corresponding RG entropy. The idea of such construction was suggested by Perelman [5] (see also a review and generalizations in [6]) on the example of the Ricci flow which is the 1-loop approximation to the full sigma model RG flow. He, in turn, was inspired by the structure of the leading terms in the metric-dilaton effective action which first appeared in the string-theory context [7, 8]. Below we shall present a generalization of the Perelman’s construction to all orders in sigma model loop expansion, i.e. suggest how to prove the c-theorem for the sigma model. We shall first review (in sect.2) the basic facts about the structure of sigma model “beta-functions” and the associated “central charge” action. In sect.3 we shall define a modification of such action which is equal to minus the Perelman’s “λ-entropy” and interpret the latter as a Lagrange multiplier for the fixed volume condition. We shall argue that this entropy should grow along the metric RG flow and is equal to minus the central charge at the fixed points.

2 2 Review of sigma model results: Weyl coefficients and “central charge” action

Let us start with a review of some known facts about renormalization of sigma model on curved 2d space. One reason to consider quantum sigma model on a curved space is to be able to define its stress tensor and its correlators which enter the standard proof of the c-theorem. Another is that to be able to define global quantities that may provide one with a “c-function” one needs to consider a 2-space of a topology of a sphere (to regularize, in particular, the IR divergences). The same problem also naturally arises in the context of the Polyakov’s approach to critical string theory when one considers propagation of a string in D-dimensional curved target space with metric Gµν [9, 7, 2]. 1 2 ab µ ν The corresponding action I = 4πα′ d z√g g ∂ax ∂ax Gµν(x) is classically invariant under Weyl rescalings of the 2d metricR gab, implying the decoupling of the conformal factor of this metric. Consistency of the critical string theory (where conformal factor of the 2d metric is not a dynamical field) then translates into the requirement of the cancellation of the Weyl anomaly. The renormalizability of the above sigma model on curved 2d background requires introduction of one extra “hidden” scalar (dilaton) coupling which has the same as the metric term [7]

1 2 ab µ ν ′ (2) I = ′ d z√g [g ∂ax ∂bx Gµν(x) + α R φ(x) ] , (2.1) 4πα Z (2) where R is the curvature of gab. The two couplings run with renormalization scale according to the corresponding beta-functions2 dϕi = βi , ϕi =(G ,φ) . (2.2) dt − µν The operator form of the Weyl anomaly relation for the trace of the 2d stress tensor is then ′ a µ a ν ¯G ′ (2) ¯φ 2πα Ta = [∂ax ∂ x βµν (x)] + α R [β (x)] , (2.3) where the β¯i are the Weyl-anomaly coefficients that differ from βi by certain diffeo- morfism terms [13, 22]

β¯G = βG + M + M , β¯φ = βφ + M µ∂ φ , (2.4) µν µν ∇µ ν ∇ν µ µ ′ Mµ = α ∂µφ + Wµ(G) . (2.5)

Wµ is a specific covariant vector constructed out of curvature and its covariant deriva- tives only (it is determined from the matrix that governs mixing under renormalization of dimension 2 operators).

2In our present notation (opposite to that of [1]) the RG evolution toward the IR corresponds to t = ln µ + (µ is a momentum renormalization scale). − → ∞

3 In dimensional regularization with minimal subtraction one finds to 2-loop order [10, 11, 7, 12, 13]3 1 βG = α′R + α′2R R λρσ + O(α′3R3) (2.6) µν µν 2 µλρσ ν 1 1 βφ = γ(G)φ + ω(G)= c α′ 2φ + α′2R Rλµνρ + O(α′3R3) , (2.7) − 0 − 2 ∇ 16 λµνρ where in critical bosonic string c = 1 (D 26) (D being the total number of coor- 0 6 − dinates xµ and 26 stands for the measure or contribution [15]). Here ω is a scalar function of− the curvature and its covariant derivatives. γ(G) is a differential operator (scalar anomalous dimension) which in the minimal subtraction scheme has the following general form [11, 14, 13]

∞ µν µ1...µn γ = Ω2 (µ ν) + Ωn (µ1 ... µn) , (2.8) ∇ ∇ n ∇ ∇ X=3 1 Ωµν = α′Gµν + p α′3Rµ Rναβγ + ... , (2.9) 2 2 1 αβγ µνρ ′4 µν αβγρ µνρλ ′4 µαβν ρ λ Ω3 = q1α DαRβ γ R + ... , Ω4 = s1α R R αβ + ... . (2.10)

In the minimal subtraction scheme Mµ gets a non-zero contribution only starting with 3 loops ′3 λνγρ Wµ = t1α ∂µ(RλνγρR )+ ... . (2.11) 3 1 The 3-loop coefficients here are p1 = 16 , t1 = 32 [16] and the 4-loop coefficients q1,s1, ... were found in [17, 18]. ¯G ¯φ In general, the Weyl anomaly coefficients βµν and β satisfy D differential identities which can be derived from the condition of non-renormalisation of the trace of the energy-momentum tensor of the sigma model [23, 13]

∂ β¯φ (δλ ρφ + V λρ)β¯G =0 , (2.12) µ − µ∇ µ λρ λρ λρ ¯G where the differential operator Vµ depends only on Gµν . To lowest order Vµ βλρ = 1 ν ¯G 1 λρ ¯G ′2 2 (βµν 2 Gµν G βλρ)+ O(α ). Eq. (2.12) implies that once the metric conformal ∇ − ¯G ¯φ ¯φ invariance equation is imposed, βµν = 0, then β =const, and thus β = 0 gives just one algebraic equation. 1 The existence of the indentity (2.12) implying D conditions between 2 D(D+1)+1 ¯G ¯φ ¯G ¯φ functions βµν and β would have a natural explanation if β and β could be obtained by variation from a covariant action functional S(G,φ) [3, 15, 12] δS δS = κ β¯j , β¯i = κij , (2.13) δϕi ij δϕj

3The 3-loop α′3 and 4-loop α′4 corrections to beta-functions were computed, respectively, in [24] and [17].

4 where κij is a non-degenerate covariant operator. Indeed, the diffeomorphism invari- ance of S implies the identity

δS 1 δS ν µ φ =0 (2.14) ∇ δGµν − 2 δφ ∇

which would then relate β¯G and β¯φ. Indeed, such action functional is easy to find to leading order in α′

D √ −2φ ′ 1 µ ′2 S = d x G e c0 α ( 4 R + ∂µφ∂ φ)+ O(α ) . (2.15) Z  −  Then

β¯G = α′(R +2 φ)+ O(α′2) , (2.16) µν µν ∇µ∇ν 1 β¯φ = c α′( 2φ + µφ φ)+ O(α′2) (2.17) 0 − 2∇ ∇ ∇µ follow from this action if κ in (2.13) is

ij 1 4GµλGνρ Gµν ′ κ = − 1 + O(α ) , (2.18) √G e 2φ Gλρ (D 2) ! 4 −

1 µλ νρ 1 µν λρ 1 µν − (G G G G ) G ′ κ = √G e 2φ 4 2 2 + O(α ) . (2.19) ij 1 G−λρ 2 2 − ! The Lagrangian in (2.15) can be written as (up to a total derivative)

β˜φ β¯φ 1 Gµνβ¯G = c 1 α′(R +4 2φ 4∂ φ∂µφ)+ O(α′2) . (2.20) ≡ − 4 µν 0 − 4 ∇ − µ This combination [13] may be interpreted as a “generalized central charge” function: it appears as the leading term in the expectation value of the trace of the stress tensor a ˜φ (2) (2π < Ta >= β R + ...) and is equal to the central charge at the conformal point where β¯G = 0 (then β˜φ = β¯φ =const). Ref. [4] put forward an argument (based on the idea of the proof of the c-theorem in [3]) that the “central charge” action4

D √ −2φ ˜φ D √ −2φ φ 1 µν G S = d x G e β (G,φ)= d x G e (β 4 G βµν) (2.21) Z Z − should have its equations of motion equivalent to β¯G = 0, β¯φ = 0 to all orders in α′ (provided one chooses an appropriate scheme, i.e. modulo a local redefinition of Gµν

4In the second equality below we used the reparametrization invariance of the action and the fact that β¯i differ from βi by diffeomorphism terms. Note also that when acting on a diffeomorphism- i δ i δ invariant functional one has β¯ i = β i ; this relation will be used below. · δϕ · δϕ 5 and φ). This was indeed confirmed by the explicit sigma model computations up to and including the 4-loop (α′4) order [16, 17, 18]. This action has a remarkable structure. In particular, it can be rewritten as 1 δ δ S = (βφ + βG ) dDx√G e−2φ . (2.22) −2 · δφ · δGµν Z

Then assuming that Gµν and φ depend on the renormalization point and using (2.2) one finds [4] 1 dV S = , V dDx√G e−2φ , (2.23) 2 dt ≡ Z i.e. that the “central charge” action evaluated on the RG running couplings is simply the RG “time” derivative of the generalized volume. More generally, based on detailed study of renormalization of the sigma model ref.[19, 20] have constructed an action that reproduces the Weyl anomaly coefficients, i.e. satisfies (2.13), to any order in α′ in an arbitrary (covariant) renormalization scheme. This action hs the following structure

D −2φ i i j S1 = d x√G e (Jiβ¯ + ρijβ¯ β¯ ) , (2.24) Z where J = 1+..., J µν = 1 Gµν +..., etc. are functions of ϕ =(φ,G ) determined in φ G − 4 i µν terms of the renormalization group quantities. The matrix ρij is, in principle, arbitrary (eq.(2.13) is satisfied for any ρij) but for a specific choice of it one can show [19] that (2.24) becomes (cf. (2.22))

1 φ δ G δ D −2φ S1 = (β + β ) d x√G e Jφ(G) , (2.25) −2 · δφ · δGµν Z where in the minimal subtraction scheme

J =1 1 α′2R Rµνλρ + O(α′3) . (2.26) φ − 4 µνλρ 1 As was pointed out in sect. 6 of [1], if one further redefines the dilaton by 2 ln Jφ the action (2.25) reduces to the simpler-looking action (2.21) or (2.22).5 Thus,− there should always exist a scheme in which the gradient of (2.21) reproduces β¯i to all α′ orders.6 Refs. [19] and [18] found explicitly the renormalization scheme in which the action (2.24) of [19] reduces to the action (2.21) of [4] at orders α′3 and α′4 respectively.

Let us note that the representation (2.23) is closely related to the interpretation of the action whose extrema are equivalent to the vanishing of the sigma model Weyl anomaly

5 i δ ′i δ One needs to note that under a coupling redefinition β i = β ′i . · δϕ · δϕ 6A possible drawback of this general argument is that the corresponding scheme choice is somewhat implicit. In the dimensional regularization with minimal subtraction the two actions begin to differ starting with α′3 order, i.e. to relate them at 3- and higher loop orders one needs a certain field redefinition.

6 coefficients as the string low-energy effective action for the and dilaton modes. The effective action can be reconstructed from the string S-matrix. Its relation to the sigma model is explained by the observation [7] that the generating functional for the string scattering amplitudes can be interpreted as a partition function Z = [dx]eiI on the 2-sphere with the string action being the sigma model with couplingsR which are the string space-time fields. The realization that a renormalisation of the sigma model corresponds to a subtraction of massless poles in the string scattering amplitudes and that a subtraction of the Mobius volume infinities can be done by differentiating over the logarithm of the 2d cutoff led to the expression for the tree-level closed string effective action S in terms of the “RG time” derivative of the renormalised sigma model partition function Z [21]: ∂Z ∂Z S = = βi . (2.27) ∂t · ∂ϕi dZ ∂Z i ∂Z Here we used the RG equation for Z: dt = ∂t β ∂ϕi = 0. Finally, there exists a scheme choice in which the renormalized value− of Z· is simply proportional to the generalized volume V in (2.23) [21, 1]. This explains the equivalence between the sigma model RG -motivated “central charge” action (2.21),(2.23) and the string partition function -motivated effective action (2.27).

3 Monotonicity of RG flow for Gµν coupling

The existence of the action (2.21),(2.25) whose derivative is proportional (2.13) to the Weyl-anomaly coefficients implies, in particular, that the RG flow of the metric Gµν and the dilaton φ couplings of the sigma model (2.1) (with a specific choice (2.4),(2.5) of the diffeomorphism terms) is a “gradient flow”. This flow is not, however, monotonic. dS Indeed, the “central charge” action (2.21) vanishes at the fixed points; also, dt = i j κijβ¯ β¯ does not have a definite sign since the “metric” κij in (2.13),(2.18) is not sign-definite.− At the same time, one may expect (in view of the Zamolodchikov’s theorem [3] for unitary 2d theories with finite number of couplings)7 that if one restricts attention just to the RG flow of the metric Gµν (and considers the case of compact euclidean signature space) there should exists an action functional (G) whose gradient is proportional to G S β and which decreases along the Gµν flow toward the IR. A natural guess is that (G) should be closely related to the functional S(G,φ), e.g., it could be found by solvingS for the dilaton, i.e. by extremising S(G,φ) in φ. The

7The proof of the Zamolodchikov’s theorem does not directly apply to the case of sigma models. If one tries to repeat the construction of [3] of the “central charge function” (whose gradient is the beta function) based on correlators of stress tensor [4] one needs to introduce the running dilaton coupling and then the metric on the space of couplings is indefinite (and also the value of the central charge function at the fixed point is zero).

7 “secondary” role of the dilaton coupling is suggested by the existence of the identity (2.12) that expresses its beta-function in terms of the metric beta-function. The idea of finding (G) by eliminating φ from S(G,φ) does not, however, work directly. At the leadingS order in α′ the combination β˜φ in (2.20) has a remarkable property that its variation over φ with the measure factor in (2.21) is zero. The variation of (2.21) over φ gives simply δS = 2√G e−2φβ˜φ =0 , (3.1) δφ − i.e. β˜φ = 0, and then the action S vanishes even before imposing β¯G = 0. The same property of β˜φ is true at least to order α′4 and should be true in general in an appropriate scheme.8 To get a non-trivial functional (G) Perelman [5] suggested to minimize S(G,φ) in φ while restricting φ to satisfy theS unit volume condition:9

V = dDx√G e−2φ =1 . (3.2) Z Imposing this condition may be in a sense interpreted as extremizing S over the con- ˜φ stant part of β or the central charge parameter c0 in (2.20). Indeed, adding the constraint (3.2) to the action (2.21) with the Lagrange multiplier λ we get the follow- ing functional

Sˆ = dDx√G e−2φ β˜φ + λ ( dDx√G e−2φ 1) , (3.3) Z Z − i.e. Sˆ = S(c c + λ) λ. 0 → 0 − Let us mention in passing another relation between actions with unit volume con- dition and without it. Starting with S(G,φ) one may formally split the dilaton into ˜ D −2φ˜ constant and non-constant parts as follows: φ(x)= φ0 + φ(x), d x√G e =1, so that V dDx√G e−2φ = e−2φ0 . Then S(G,φ)= e−2φ0 S(G, φ˜) = VS(G, φ˜) and Sˆ in ≡ R (3.3) in whichR the dilaton is constrained by the volume condition can be written as (cf. (2.23)) D √ −2φ ˜φ ˆ ˜ d x G e β (G,φ) 1 d S(G,φ)= S(G, φ)= − = ln V . (3.4) R dDx√G e 2φ 2 dt We shall not use this representation here.R Extremising Sˆ with respect to φ we then get (assuming that (3.1) is true to all orders in α′) β˜φ + λ =0 , (3.5)

8The relation (3.1) is valid, in particular, if there is a scheme in which the dependence of β˜φ on φ in (2.20) is not modified by α′ corrections to all orders; this is actually true to α′3 order but may seem to be in conflict with the α′-dependence of the operator γ in (2.7),(2.8). But the corresponding terms can be further redefined away (or integrated by parts) at the level of the action. 9For an earlier closely related suggestion in specific D = 2 case see [26].

8 so that after solving for φ, i.e. imposing (3.5), we get

Sˆ = β˜φ = λ . (3.6) − Thus λ has an interpretation of minus the effective central charge. To leading order in α′ the action Sˆ (3.3) and eq.(3.5) can be written as follows (see (2.20)) ˆ D √ ′ 2 1 ′2 S = λ + d x G Φ λ + c0 α ( + 4 R) Φ+ O(α ) , (3.7) − Z  − −∇  2 1 ′ ′−1 ( + 4 R)+ O(α ) Φ= α (c0 + λ)Φ , (3.8)  −∇  where Φ e−φ , dDx√G Φ2 =1 . (3.9) ≡ Z −φ The existence of a solution of this equation with Φ e > 0 requires that c0 + λ is 10 2 1 ≡ the minimal eigenvalue of the operator + 4 R which always exists on a compact space [5] (see also [26]). The corresponding−∇ eigenfunction will have no zeros and can be chosen positive which is what is required for the identification of it with e−φ (or the φ inverse of the effective “string ” gs = e ). Thus extremizing Sˆ in φ translates (for c0 = 0) into choosing λ as a minimal eigenvalue of the above Laplacian. To leading order in α′ when the RG flow defined by (2.17),(2.16) is simply the Ricci flow the Perelman’s definition of the functional whose gradient is β¯G (i.e. βG with an appropriate diffeomorphism term) and which grows monotonically with the RG flow toward IR (t ) is simply the minimal eigenvalue λ of (3.8) (for c = 0). Thus, in →∞ 0 view of (3.6), (G) Sˆ(G,φ(G)) = λ(G) . (3.10) S ≡ − Let us extend this definition to all orders in α′. First, the variation of (G) over G S µν is the same as the Gµν variation of Sˆ(G,φ) or S(G,φ) with φ independent of Gµν : the variation over φ vanishes as a consequence of (3.5). Then (2.13) implies that δS is δGµν ¯G proportional to βµν . In addition, we may ignore the variation of √G since its coefficient vanishes on the equation for φ. Then

¯G δ 4GµρGνσ ′2 βµν = κµν,ρσ S , κµν,ρσ = − + O(α ) . (3.11) δGρσ √G e 2φ Thus is a gradient function for the metric RG flow. S To study the monotonicity property of we note that S d G δ ¯G δ ¯G µν,ρσ ¯G = βµν S = βµν S = βµν κ βρσ . (3.12) dtS − · δGµν − · δGµν − · ·

10Let us note in passing that the conformal scalar operator in D is 2 + D−2 R so −∇ 4(D−1) that 2 + 1 R is conformal in the limit D . −∇ 4 → ∞ 9 µν,ρσ Here κ is the inverse of κµν,ρσ in (3.11): on the equation of motion for φ one need not worry about the contribution of the variation of √G term in the action and thus there is no extra term proportional to 1 G G (cf. (2.19)) − 2 µν ρσ µν,ρσ 1 √ −2φ µρ νσ ′2 κ = 4 G e G G + O(α ) . (3.13) The positivity of κµν,ρσ at leading order in α′ implies that monotonically decreases toward the IR (t ) as required of an effective centralS charge [3], while λ grows like an entropy [5].→11 ∞The positivity of κµν,ρσ is obvious in perturbation theory in α′, i.e. in sigma model loop expansion.12 It may be possible to prove it rigorously to all orders using the general properties of renormalization of the sigma model on a curved background as discussed in [19, 20]. Since the dependence on the dilaton of the beta-functions in (2.6),(2.7) is simple (linear) the same simplicity should apply to the effective action. Starting with the action in the special scheme (2.21) we shall assume that to all orders in α′ it can be put into the form similar to the leading-order action (3.7) (here we set c0 = 0)

Sˆ = λ + dDx√G Φ(λ α′∆)Φ , Φ= e−φ , (3.14) − Z − where ∆= 2 + U(G) , (3.15) −∇ 1 1 ′ µνλρ 1 ′2 µναβ λρ 4 καµβ λ ν ′3 U = 4 R + 16 α RµνλρR + 16 α (RµνλρR Rαβ 3 Rκλµν R R α β)+ O(α ). − (3.16) For R3 terms we used the result of [25, 16]. We have assumed that there should exist a scheme choice in which all higher-derivative terms which may be present in the anomalous dimension operator γ in (2.7) can be integrated by parts in the action (2.21) so that ∆ remains a canonical second-derivative scalar Laplacian as it was at the leading order in α′ in (3.7). This is indeed what happens to order α′4 as was explicitly verified in [16, 17]. The potential function U(G) is a smooth generalization of the leading-order term 1 4 R. Then for compact euclidean-signature space the operator ∆ is again positive and its spectrum should be bounded from below. Then the eigen-function Φ corresponding to its lowest eigenvalue λ/α′ can again be chosen positive, i.e. there should exist a non-singular solution for φ.13 Combined with the positivity of the metric in (3.13), λ will then provide the generalization of the Perelman’s entropy to all orders in α′, implying the irreversibility of the exact RG flow of the sigma model.

11 To leading order in α′ the relation between Perelman’s entropy and the central charge was already pointed out in [6]. The same is true also in the presence of the Bµν coupling [6]. Using the monotonicity of λ one is then able to prove the absence of periodic RG trajectories [5, 6]. 12To become negative κµν,ρσ should go through zero but that would require α′R 1, invalidating perturbation theory expansion. ∼ 13We are grateful to S. Cherkis for a clarifying discussion of this point.

10 As we have seen above in (3.6), λ has also a meaning of minus the effective central charge β˜φ, which, at the fixed point β¯G = 0, is equal to the usual central charge. This is in agreement with the general claim of the c-theorem. The construction of the sigma model “c-function” a la ref.[3] (i.e. in terms of 2-point functions of stress-tensor components) did lead [4] to β˜φ, but as we explained above following Perelman’s idea, to show that the RG flow of Gµν is monotonic one is also to solve for the dilaton and restrict its constant part by the volume condition (3.2). This then confirms the validity ′4 of the c-theorem for the Gµν RG flow of the 2d sigma model (at least to order α ). To make this proof of the c-theorem rigorous (i.e. to extend it beyond α′4 order) one is to justify our main assumption that the exact action (2.21) can be put into the form (3.14). This may be possible to achieve using the identities like (2.12) following from the renormalization properties of composite operators of the sigma model [19, 20].14

Acknowledgments

We are grateful to G. Huisken and T. Oliynyk for the invitation to the Workshop on geometric and renormalization group flows at Max Planck Institute in Potsdam in November 2006 and for creating a stimulating atmosphere during the workshop. We thank S. Cherkis and S. Shatashvili for useful remarks and discussions. We also acknowledge the support of PPARC, EU-RTN network MRTN-CT-2004-005104 and INTAS 03-51-6346 grants, and the RS Wolfson award.

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