Reparametrization modes in 2d CFT and the effective theory of stress tensor exchanges
PPrepared for submission to JHEP
Reparametrization modes in 2d CFT andthe effective theory of stress tensor exchanges
Kevin Nguyen
Department of Mathematics, King’s College London, London, United Kingdom
E-mail: [email protected]
Abstract:
We study the origin of the recently proposed effective theory of stress tensorexchanges based on reparametrization modes, that has been used to efficiently computeVirasoro identity blocks at large central charge. We first provide a derivation of the non-linear Alekseev–Shatashvili action governing these reparametrization modes, and arguethat it should be interpreted as the generating functional of stress tensor correlations onmanifolds related to the plane by conformal transformations. In addition, we demonstratethat the rules previously prescribed with the reparametrization formalism for computingVirasoro identity blocks naturally emerge when evaluating Feynman diagrams associatedwith stress tensor exchanges between pairs of external primary operators. We make a fewcomments on the connection of these results to gravitational theories and holography. a r X i v : . [ h e p - t h ] F e b ontents An effective action used to describe stress tensor exchanges in conformal field theories(CFT) has been recently constructed by Haehl, Reeves and Rozali [1, 2]. In two dimensions,it reads W = − c π (cid:90) d x (cid:15) i ( δ ij (cid:3) − ∂ i ∂ j ) (cid:3) (cid:15) j + O ( (cid:15) ) , (1.1)where (cid:15) i ( x ) is known as the reparametrization mode , and δ ij is the flat euclidean backgroundmetric. The reparametrization mode is interpreted as the generator of the infinitesimaltransformation δg ij = ∂ i (cid:15) j + ∂ j (cid:15) i − δ ij ∂ k (cid:15) k , (1.2)resulting from a change of coordinate followed by a Weyl rescaling of the backgroundmetric. When (cid:15) i ( x ) is a conformal Killing vector field, it is the symmetry parameter ofa conformal transformation and the effective action (1.1) correspondingly vanishes. Bycontrast, generic configurations (cid:15) i ( x ) do not generate spacetime symmetries and acquirea nonzero action due to the Weyl anomaly. Hence one can think of (cid:15) i ( x ) as a pseudo-Goldstone mode resulting from broken Weyl invariance. The effective action (1.1) admitsa nonlinear extension known as the Alekseev–Shatashvili action [3]. Quite remarkably,Cotler and Jensen showed that this nonlinear extension describes pure gravity in threedimensions with anti-de Sitter (AdS) asymptotics [4], whose symmetries are well-knownto be that of a two-dimensional CFT [5]. Through this specific example, they thereforeprovided a completion of (1.1). The first goal of the present work will be to explain theorigin and role of Alekseev–Shatashvili action from a CFT perspective.The primary interest of the reparametrization mode formalism is that it provides anefficient way to compute Virasoro identity blocks, i.e., to evaluate the contributions to cor-relation functions coming from stress tensor exchanges between pairs of external primary– 1 –elds. To achieve this, one first introduces a set of bilocal vertex operators that are closelyrelated to reparametrized primary two-point functions. Within the reparametrization for-malism, these bilocal operators are used as effective couplings between a pair of identicalprimary operators and the reparametrization mode itself. One can use them together withthe dynamics dictated by the effective action (1.1) to compute Virasoro identity blocks.Agreement with previously known results at large central charge [6–9] was found in [4],thereby demonstrating the utility of the reparametrization mode formalism. Finally, apartial justification of the above procedure was provided through the identification of thereparametrization mode with the shadow operator of the stress tensor [2].The reparametrization mode formalism builds on a series of works aiming at a uni-versal description of out-of-time-order correlators (OTOCs) in maximally chaotic quantumsystems [1, 4, 10–13]. The exponential Lyapunov behavior displayed by OTOCs in such sys-tems being universally controlled by their temperature [14], it was therefore natural to lookfor a universal effective description in terms of reparametrization modes. Agreement wasindeed found between this effective description and more conventional CFT methods [1].Although this is not the primary focus of the present paper, the strong connection betweenmaximal chaos and the AdS/CFT correspondence is worth mentioning as it appears thatholographic CFTs are maximally chaotic [15–22].The aim of the present work is to clarify the origin of the reparametrization modeformalism. In particular, we provide a derivation of the nonlinear version of (1.1) fromfirst principles, starting from the Polyakov action as the universal generating functional forstress tensor correlations in any 2d CFT [23]. We review basic properties of the Polyakovaction in section 2, and describe in section 3 how it reduces to the Alekseev–Shatashviliaction when the background metric, considered as a source for the stress tensor, is generatedfrom the flat metric by a finite version of (1.2). We show that this nonlinear extension ofthe effective action (1.1) should still be viewed as a generating functional for (holomorphic)stress tensor correlations on manifolds related to the plane by conformal transformations,with the derivative of the reparametrization mode acting as the corresponding source . Insection 4, we revisit the computation of Virasoro identity blocks in the reparametrizationformalism. In particular, we show that the prescriptions given in [2, 4, 25] naturally emergewhen evaluating Feynman diagrams involving stress tensor exchanges between externalpairs of identical primary operators. This provides a new justification for these otherwisemysterious prescriptions, independent from the earlier one based on the shadow operatorformalism and originally presented in [2]. We end with a discussion of the results, andpoint towards possible further developments and applications of the formalism describedhere. We also comment on connections to theories of gravity and holography. Conventions.
We work in euclidean signature. We use the shorthand notations T ≡− πT zz for the holomorphic component of the stress tensor, δ ( z ) ≡ δ (2) ( z, ¯ z ) for the deltadistribution normalized as (cid:82) d z δ ( z ) = 1, and z ij ≡ z i − z j for relative distances. To avoid A similar use of the reparametrization mode as a source for the stress tensor has been made in thecontext of T ¯ T deformations, however without reference to a nonlinear action for the reparametrizationmode [24]. – 2 –lutter, we sometimes suppress the functional dependence on coordinate labels, and writeexpressions such as ¯ ∂(cid:15) ≡ ∂ ¯ z (cid:15) ( z , ¯ z ). We make repetitive use of the magic distributionalidentity ∂ ¯ z (cid:18) z (cid:19) = 2 πδ ( z ) , (1.3)which will be our main computational weapon. The starting point for the construction of the effective action (1.1) in [2] was the generatingfunctional for the connected stress tensor two-point function on the complex plane, W [ δg ] = 18 (cid:90) d x d y δg ij ( x ) δg mn ( y ) (cid:104) T ij ( x ) T mn ( y ) (cid:105) plane + O ( δg ) , (2.1)where the metric perturbation δg ij is a source for the stress tensor T ij . Upon insertion of(1.2), it was shown to reduce to the effective action (1.1) in terms of the reparametriza-tion mode (cid:15) i [2]. It is reasonable to expect that its nonlinear extension can be obtainedby considering the generating functional of all connected stress tensor correlators. Thisgenerating functional appears to be universal in two dimensions and has been derived inclosed form long ago by Polyakov [23], which we briefly review in this section.Stress tensor correlation functions on the plane are fully constrained by conformalsymmetry, the only dependence on a given theory occurring through the central charge c .Up to its value, the generating functional W [ g ij ] of all connected stress tensor correlatorsis therefore universal. Polyakov’s starting point for its construction is the anomalous traceof the stress tensor expectation value on a space with arbitrary background metric g ij andcurvature R , c π R = g ij (cid:104) T ij (cid:105) = − √ g g ij δWδg ij , (2.2)where the last equality follows from the very definition of the generating functional W [ g ij ].Integrating this equation, Polyakov obtained [23] W [ g ij ] = − c π (cid:90) d x d y (cid:112) g ( x ) (cid:112) g ( y ) R ( x ) G ( x, y ) R ( y ) (2.3a)= − c π (cid:90) d x (cid:112) g ( x ) R ( x ) 1 (cid:3) R ( x ) , (2.3b)where G ( x, y ) is the Green function solution to (cid:3) G ( x, y ) = δ (2) ( x − y ) (cid:112) g ( x ) . (2.4)The Polyakov action (2.3) is manifestly nonlocal in the background metric g ij that acts asa source for the stress tensor. It can be put in an alternative form through the introductionof an auxiliary variable φ solving (cid:3) φ = R, (2.5)– 3 –uch that the generating functional coincides with the action of a Liouville theory, W [ g ij ] = − c π (cid:90) d x (cid:112) g ( x ) (cid:18)
12 ( ∂φ ) + φR (cid:19) . (2.6)We stress that the Liouville field φ is not an independent variable, but rather a nonlocalfunctional of the metric through (2.5). The stress tensor expectation value may be com-puted from (2.6) by functional differentiation, and is found to coincide with the classicalLiouville stress tensor (cid:104) T ij (cid:105) = − √ g δWδg ij = c π (cid:20) ∂ i φ ∂ j φ − ∇ i ∇ j φ + g ij (cid:18) (cid:3) φ −
14 ( ∂φ ) (cid:19)(cid:21) = T φij . (2.7)Consistently, one recovers the trace anomaly which we started from, g ij (cid:104) T ij (cid:105) = c π R. (2.8)It is worth mentioning that covariance of (2.5) under a Weyl rescaling g ij (cid:55)→ e ω g ij , (2.9)implies that φ must transform by a shift φ (cid:55)→ φ − ω . It is therefore natural to interpret φ as the pseudo-Goldstone mode associated to broken Weyl symmetry. The expectationvalue (2.8), or equivalently the configuration φ determined through (2.5), labels one of thebroken vacua. Due to explicit breaking of Weyl symmetry by the central charge c , thispseudo-Goldstone mode acquires a nonzero action (2.6).Functional differentiation of the generating functional W [ g ij ] yields connected stresstensor correlators on a background with fixed metric g , (cid:104) T ij ( x ) ...T mn ( x n ) (cid:105) g = ( − n (cid:112) g ( x ) ... (cid:112) g ( x n ) δ n Wδg ij ( x ) ...δg mn ( x n ) (cid:12)(cid:12)(cid:12) g = g + ... , (2.10)where the dots refer to contact terms resulting from functional differentiation of the metricdeterminant of the type δδg ij ( x k ) (cid:32) (cid:112) g ( x l ) (cid:33) . (2.11)For instance, we can compute the stress tensor two-point function on the plane equippedwith flat metric euclidean metric. A straightforward computation yields (cid:104) T ij ( x ) T mn ( y ) (cid:105) plane = − c π (cid:0) δ ij (cid:3) x − ∇ xi ∇ xj (cid:1) ( δ mn (cid:3) y − ∇ ym ∇ yn ) ln µ | x − y | , (2.12)where µ is an arbitrary energy scale introduced such that the argument of the logarithmis dimensionless. With complex coordinates ds = dz d ¯ z, (2.13)one recovers in particular the standard expression (cid:104) T ( z, ¯ z ) T ( w, ¯ w ) (cid:105) = c z − w ) . (2.14)– 4 – eyl non-invariance. The Polyakov action (2.3) arises from the breaking of Weyl in-variance by quantum effects. It should therefore be expected that it transforms nontriviallyunder Weyl rescalings. Using R [ e ω g ij ] = e − ω ( R g − (cid:3) g ω ) , det e ω g = e ω det g, (2.15)one indeed finds that (2.3) transforms in a nontrivial way, W [ e ω g ij ] = W [ g ij ] + c π (cid:90) d x (cid:112) g ( x ) (cid:18)
12 ( ∂ω ) + ωR g (cid:19) . (2.16)As one could have anticipated from the previous discussion, the term spoiling Weyl invari-ance is precisely of the form of a Liouville action for the conformal factor ω . Conformal invariance.
A conformal field theory is invariant under conformal transfor-mations. These are generated by conformal Killing vector fields whose infinitesimal actionwas briefly discussed around (1.2). On the complex plane equipped with the flat metric(2.13), a holomorphic conformal transformation is generated by a coordinate transforma-tion z (cid:55)→ Π( z ) , (2.17)followed by a Weyl rescaling (2.9) with conformal factor ω = − ln ∂ z Π( z ) . (2.18)In this way, the background metric is indeed left invariant, dz d ¯ z (cid:55)→ d Π d ¯ z = ∂ z Π( z ) dz d ¯ z (cid:55)→ e ω d Π d ¯ z = dz d ¯ z . (2.19)The same holds for anti-holomorphic conformal transformations generated by ¯ z (cid:55)→ ¯Π(¯ z ).Due to the Weyl anomaly, one could have feared that conformal transformations are notsymmetries of the Polyakov action, and therefore not true symmetries of the quantumtheory. This is however not the case, and it can be explicitly checked that the second term in(2.16) vanishes provided that Π( z ) reduces to a PSL(2, C ) global conformal transformationat infinity, lim z →∞ Π( z ) = az + bcz + d , ad − bc = 1 . (2.20)It is instructive to compute the energy in the family of vacua related by the aboveconformal transformations. By convention, the vacuum energy on the complex plane isnormalized to zero and the corresponding value of the Liouville field therefore vanishes.Performing a conformal transformation induces a Weyl rescaling with conformal factor(2.18). As mentioned below (2.9), the Liouville field shifts to the new value φ = − ω = ln ∂ z Π( z ) , (2.21)such that, on the manifold obtained by conformal transformation from the complex plane,the vacuum energy (2.7) reduces to (cid:104) T ( z, ¯ z ) (cid:105) Π − (plane) = c (cid:18) ∂ z Π ∂ z Π − (cid:18) ∂ z Π ∂ z Π (cid:19)(cid:19) . (2.22)We have recovered the well-known expression in terms of the Schwarzian derivative of Π.– 5 – The nonlinear action governing reparametrization modes
Having reviewed the Polyakov action and its basic properties, we are now ready to de-rive the nonlinear version of the effective action (1.1) governing reparametrization modes.Much in the same way that the infinitesimal reparametrization mode (cid:15) i ( x ) is defined asparametrizing the infinitesimal metric variation (1.2), the finite reparametrization modeΠ( z, ¯ z ) is defined as parametrizing a change of coordinate z (cid:55)→ Π( z, ¯ z ) , (3.1)followed by a Weyl rescaling (2.9) with parameter ω = − ln ∂ z Π( z, ¯ z ) . (3.2)Of course, when Π is a holomorphic function, it is a symmetry parameter with vanishingaction. For generic configurations however, it induces a nontrivial transformation of thebackground metric and generates a nonzero curvature. With the help of (2.16), we canevaluate the Polyakov action associated with this curved metric, resulting in W [Π( z, ¯ z )] = c π (cid:90) d z ∂ z Π ∂ ¯ z ∂ z Π( ∂ z Π) . (3.3)As anticipated in [4] from the study of three-dimensional gravity with AdS asymptotics,this nonlinear extension of (1.1) is a complex version of the Alekseev–Shatashvili action,which was originally understood as the action of a particle on the vacuum coadjoint orbitof the Virasoro group [3]. Here, we derived it from first principles without appealing togravity or the AdS/CFT correspondence, simply starting from the Polyakov action. Ofimportance for the reparametrization mode formalism to be discussed in section 4, up toboundary terms the action (3.3) is invariant under PSL(2, C ) transformations [3],Π (cid:55)→ a (¯ z )Π + b (¯ z ) c (¯ z )Π + d (¯ z ) , ad − bc = 1 . (3.4)In the remainder of this section, we illustrate how the action (3.3), still viewed as agenerating functional, can be used to compute stress tensor correlations on the plane or onthe cylinder. Correlations on the plane.
We recall that the reparametrization mode Π generatesa coordinate transformation followed by a Weyl rescaling. If we consider an infinitesimalversion of such a transformation around the identity,Π( z, ¯ z ) = z + (cid:15) ( z, ¯ z ) , (3.5)it induces the metric variation δg zz = − ∂(cid:15) , δg ¯ z ¯ z = δg z ¯ z = 0 . (3.6)We should therefore expect that the action (3.3), when expanded in powers of (cid:15) , allowsto compute correlations of T zz on the plane. Of course, T ¯ z ¯ z correlations can be computed– 6 –rom W (cid:2) ¯Π (cid:3) = ( W [Π]) ∗ obtained by complex conjugation. For instance, to cubic order in (cid:15) the action reduces to W = − c π (cid:90) d z (cid:0) ∂ (cid:15) ¯ ∂(cid:15) − ∂ (cid:15) ∂ (cid:15) ¯ ∂(cid:15) (cid:1) + O ( (cid:15) ) . (3.7)The quadratic piece coincides with the action (1.1) when evaluated in flat complex coor-dinates. Treating (3.7) as the generating functional with sources (3.6), one recovers thecorrect expressions for two- and three-point functions on the plane, (cid:104) T ( z , ¯ z ) T ( z , ¯ z ) (cid:105) = ( − π ) δ Wδ ¯ ∂(cid:15) δ ¯ ∂(cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 = c z , (3.8) (cid:104) T ( z , ¯ z ) T ( z , ¯ z ) T ( z , ¯ z ) (cid:105) = ( − π ) δ Wδ ¯ ∂(cid:15) δ ¯ ∂(cid:15) δ ¯ ∂(cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 = cz z z . (3.9)Thus, the Alekseev–Shatashvili action (3.3) is the generating functional for correlationfunctions of the holomorphic stress tensor component. Correlations on the cylinder.
The Alekseev–Shatashvili action can be used to com-pute stress tensor correlation functions on manifolds related to the plane by a conformaltransformation. We illustrate this for the cylinder, covered by the real coordinates τ ∈ R , σ ∈ [0 , β ) , (3.10)As is well-known, one can map the plane to the cylinder by a conformal transformationassociated with the change of coordinateΠ( z ) = e − i πβ z , z = σ + iτ . (3.11)Said differently, Π( z ) is now the coordinate covering the plane while z is the coordinatecovering the cylinder. In order to compute stress tensor correlations on the cylinder fromthe Alekseev–Shatashvili action (3.3), we need to consider infinitesimal reparametrizationmodes on top of the finite conformal mapping (3.11). This is conveniently achieved bywriting Π( z, ¯ z ) = e − i πβ f ( z, ¯ z ) , f ( z, ¯ z ) = z + (cid:15) ( z, ¯ z ) , (3.12)where (cid:15) is periodic and asymptotes to a constant at infinity, (cid:15) ( σ + β, τ ) = (cid:15) ( σ, τ ) , lim τ →±∞ (cid:15) ( σ, τ ) = cst . (3.13)Plugging (3.12) into the nonlinear action (3.3) and making use of the conditions (3.13) todiscard total derivative terms, we obtain W [ f ( z, ¯ z )] = c π (cid:90) d z (cid:32) − (cid:18) πβ (cid:19) ∂ z f ∂ ¯ z f + ∂ z f ∂ ¯ z ∂ z f ( ∂ z f ) (cid:33) . (3.14) One simple way to proceed is to use the magic formula (1.3) in order to express (3.7) as a nonlocalfunctional of the field ¯ ∂(cid:15) alone. For the quadratic part of the generating functional for instance, we have (cid:90) d z ∂ (cid:15) ¯ ∂(cid:15) = 12 π (cid:90) d z d z ¯ ∂ (cid:18) z (cid:19) ∂ (cid:15) ¯ ∂(cid:15) = 12 π (cid:90) d z d z ∂ (cid:18) z (cid:19) ¯ ∂(cid:15) ¯ ∂(cid:15) . – 7 –nterestingly, this alternative form of the Alekseev–Shatashvili action coincides with theaction of a particle on the first exceptional coadjoint orbit of the Virasoro group [3]. Notethat it naturally inherits the PSL(2, C ) invariance described in (3.4) through the identifi-cation Π = e − i πβ f . As for the plane, we expand this action in powers of the infinitesimalreparametrization mode (cid:15) (3.12) that is appropriate to the cylinder. To quadratic order,we get W = − c π (cid:18) πβ (cid:19) (cid:90) d z (cid:32) ¯ ∂(cid:15) + ∂(cid:15) ¯ ∂(cid:15) + (cid:18) β π (cid:19) ∂ (cid:15) ¯ ∂(cid:15) (cid:33) + O ( (cid:15) ) . (3.15)In particular, we recover the quadratic action for reparametrizations of the thermal cylinderconstructed in [2]. In addition, we also find a linear term which turns out to account forthe Casimir energy of the cylinder, (cid:104) T ( z, ¯ z ) (cid:105) = − π δWδ ¯ ∂(cid:15) ( z, ¯ z ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 = c (cid:18) πβ (cid:19) . (3.16)We can similarly recover the stress tensor two-point function, (cid:104) T ( z , ¯ z ) T ( z , ¯ z ) (cid:105) = (2 π ) δ Wδ ¯ ∂(cid:15) δ ¯ ∂(cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 = − c (cid:34)(cid:18) πβ (cid:19) ∂ + ∂ (cid:35) (cid:18) z (cid:19) (3.17a)= c (cid:34) z + (cid:18) πβ (cid:19) z (cid:35) = (cid:18) πβ (cid:19) c πβ z , (3.17b)where the last equality holds up to non-singular terms that are irrelevant.In summary, in this section we have derived the nonlinear extension of the generatingfunctional (1.1), starting from the Polyakov action. We have illustrated how it can be usedto compute correlations of the holomorphic stress tensor component on manifolds relatedto the complex plane by a conformal transformation. We now come to the description of stress tensor exchanges in the reparametrization modeformalism, arguably its main interest from a computational perspective. As usual, theholomorphic and anti-holomorphic dependencies of correlation functions factorize. Forsimplicity, we will only discuss their holomorphic part, but an analogous reasoning obvi-ously applies to their anti-holomorphic counterpart as well.We start by reviewing the prescriptions for computing Virasoro identity blocks withinthe reparametrization formalism, following [2, 4, 25]. Ultimately, our goal will be to derive these rules from Feynman diagrams describing stress tensor exchanges between externalprimary operators. As a preliminary step to the reparametrization formalism, one considersprimary two-point functions on a manifold related to the complex plane by a conformaltransformation (2.17) with symmetry parameter Π( z ), (cid:104)O h (1) O h (2) (cid:105) Π − (plane) = (cid:18) ∂ z Π( z ) ∂ z Π( z )(Π( z ) − Π( z )) (cid:19) h . (4.1) I thank Jordan Cotler and Jakob Salzer for discussions on this point. – 8 – bilocal vertex operator B h is then introduced by promoting the symmetry parameterΠ( z ) to an arbitrary reparametrization mode Π( z, ¯ z ), B h (1 , ≡ (cid:18) ∂ z Π( z , ¯ z ) ∂ z Π( z , ¯ z )(Π( z , ¯ z ) − Π( z , ¯ z )) (cid:19) h . (4.2)For the purpose of computing stress tensor exchanges on the plane, we again expand Π( z, ¯ z )around the identity. As was pointed out in [25], an infinitesimal reparametrization mode (cid:15) exponentiates into a finite mode through Π = e (cid:15)∂ z = z + (cid:15) + (cid:15)∂(cid:15) + ... such that theexpansion of the bilocal vertex (4.2) takes the form B h (1 ,
2) = 1( z ) h (cid:88) n ≥ B ( n ) h (1 , , (4.3)where the first few terms are given by B (0) h (1 ,
2) = 1 , (4.4a) B (1) h (1 ,
2) = b (1) h (1 , , (4.4b) B (2) h (1 ,
2) = 12! (cid:16) b (1) h (1 , (cid:17) + b (2) h (1 , , (4.4c)... B ( n ) h (1 ,
2) = 1 n ! (cid:16) b (1) h (1 , (cid:17) n + lower orders in h , (4.4d)with b (1) h (1 ,
2) = h (cid:18) ∂(cid:15) + ∂(cid:15) − (cid:15) − (cid:15) z (cid:19) , (4.5a) b (2) h (1 ,
2) = h (cid:18) (cid:15) ∂ (cid:15) + (cid:15) ∂ (cid:15) − (cid:15) ∂(cid:15) − (cid:15) ∂(cid:15) z + ( (cid:15) − (cid:15) ) z (cid:19) . (4.5b)More details and higher order terms of this expansion can be found in [25].Remarkably, the bilocal vertex operator B h can be used to straightforwardly computethe contribution of the Virasoro identity block V to four-point functions involving pairsof identical operators, (cid:104) V (1) V (2) W (3) W (4) (cid:105) = 1( z ) h V ( z ) h W (cid:88) O C V V O C W W O V h O ( u ) , (4.6)where u = z z z z and we again only displayed the holomorphic part of the correlator.In (4.6), the four-point function is expressed as a sum over Virasoro conformal blocks V h O ( u ), where the sum runs over all primary operators O and where C V V O and C W W O are fusion coefficients that characterize any particular CFT. Note that conformal blocksare purely kinematical objects that only depend on the conformal dimensions h V , h W , h O of the various operators involved. A prescription to compute the contribution from theVirasoro identity block V based on the reparametrization mode formalism has been putforward in [1, 2, 4] and further developed in [25]. It involves the following ingredients: It has been recently argued that the bilocal operator (4.2) can be formally identified with the Virasoroidentity OPE block, but still requires proper renormalization [26]. – 9 –
A re-interpreation of the reparametrization mode (cid:15) ( z, ¯ z ) as a dynamical field insteadof as a source for the stress tensor, together with a re-interpretation of the action (3.7)as that governing its dynamics. Accordingly, the reparametrization mode propagatoris found to be (cid:104) (cid:15) ( z , ¯ z ) (cid:15) ( z , ¯ z ) (cid:105) = 6 c z ln µ | z | , (4.7)with µ an arbitrary energy scale that cannot be determined from the theory buteventually drops out from the four-point functions of interest. • A gauging of the PSL(2, C ) symmetry (3.4), resulting in the physical gauge-invariantand purely holomorphic propagator G (cid:15) ( z , z ) ≡ (cid:104) (cid:15) ( z , ¯ z ) (cid:15) ( z , ¯ z ) (cid:105) phys = 6 c z ln µz . (4.8)This physical propagator can be alternatively obtained by a monodromy projectionof (4.7) as described in [2]. • The identification of the Virasoro identity block as a connected correlation functionof bilocal vertex operators, C V V T C W W T ( z ) h V ( z ) h W V ( u ) ≡ (cid:104)B h V (1 , B h W (3 , (cid:105) c , (4.9)where the vertices are viewed as functionals of the dynamical reparametrization field (cid:15) with physical propagator (4.8). The fact that this universal formula describes thefusion coefficients C V V T , C
W W T together with the Virasoro identity block V shouldnot come as a surprise, since the coupling between primary operators and the stresstensor is universally dictated by conformal symmetry.Following the above set of prescribed rules, the Virasoro identity block contribution tothe normalized four-point function F ≡ (cid:104) V (1) V (2) W (3) W (4) (cid:105)(cid:104) V (1) V (2) (cid:105)(cid:104) W (3) W (4) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) V , (4.10)can be computed order by order in a 1 /c expansion. Indeed, each propagator G (cid:15) comeswith a factor of 1 /c such that the reparametrization formalism naturally organizes as aperturbative expansion at large central charge. In addition, terms in the expansion of thebilocal vertex operator (4.3) that contribute to a given order in 1 /c are easy to identify.At zeroth order, we simply have F (cid:12)(cid:12) O (1) = (cid:104)B (0) h V (1 , B (0) h W (3 , (cid:105) = 1 . (4.11)At subleading order, we have [4] F (cid:12)(cid:12) O (1 /c ) = (cid:104)B (1) h V (1 , B (1) h W (3 , (cid:105) = (cid:104) b (1) h V (1 , b (1) h W (3 , (cid:105) (4.12a)= 2 h V h W c u F (2 , , u ) , (4.12b)– 10 –hich is recognized as the global identity block contribution [27, 28]. We will also discussthe terms appearing at order O (1 /c ) without explicitly evaluating them, F (cid:12)(cid:12) O (1 /c ) = (cid:104)B (2) h V (1 , B (2) h W (3 ,
4) + B (3) h V (1 , B (1) h W (3 ,
4) + B (1) h V (1 , B (3) h W (3 , (cid:105) c (4.13)+ (cid:104)B (2) h V (1 , B (1) h W (3 ,
4) + B (1) h V (1 , B (2) h W (3 , (cid:105) c . Upon replacement of the bilocal vertices by their expressions in terms of reparametrizationmodes, the first line contains terms involving two propagators (cid:104) (cid:15)(cid:15) (cid:105) while the second linecontains terms involving a single three-point function (cid:104) (cid:15)(cid:15)(cid:15) (cid:105) . Except from the first one,all terms suffer from ultraviolet (UV) divergences since they contain (cid:15) correlators eval-uated at coincident points. To make sense of these, one would need to supplement thereparametrization formalism with a regularization procedure. We will not try to remedythis here, and we will restrict our attention to the first regular term instead. We make afew comments regarding divergences and their regularization in the discussion section.As shown in [4], the above prescription successfully reproduces known results at largecentral charge c in the ‘light-light’ limit h V , h W = O ( √ c ) [6] and in the ‘heavy-light’limit h V = O (1) , h W = O ( c ) [7–9]. Furthermore, an appropriate modification of thereparametrization mode formalism to Lorentzian signature similarly led to a successfuldescription of the maximal Lyapunov growth displayed by out-of-time-order (OTOC) cor-relators at large central charge [1, 2, 4].In spite of these successes, the origin of the above set of rules seems rather mysterious atfirst sight. A convincing justification was nonetheless provided by Haehl, Reeves and Rozaliby showing that these rules are those of the shadow operator formalism upon identificationof (cid:15) ( z, ¯ z ) with the shadow of the stress tensor T ( z, ¯ z ) [2]. However, their argument onlyapplied to the first nontrivial term in the bilocal vertex expansion (4.3), i.e., to B (1) h (1 , (cid:15) ( z, ¯ z ) as a dynamical field,which we find somewhat awkward given its meaning of source for the stress tensor whenfirst introduced. The alternative derivation which we propose is straightforward and sim-ply consists in computing contributions to the normalized four-point function (4.10) from(position-space) Feynman diagrams involving stress tensor exchanges between the two pairsof identical operators. These diagrams are shown in Figure 1. As will be shown below, wefind perfect agreement with the reparametrization formalism. We believe that the alter-native method developed below conceptually clarifies the results of the reparametrizationmode formalism, and provides compelling evidence of its validity.The atomic ingredients that we need are the stress tensor propagator (3.8) together The three-point function (cid:104) (cid:15)(cid:15)(cid:15) (cid:105) , whose explicit expression may be found in [25], scales like 1 /c . – 11 – (1) V (2) W (3) W (4) T (a) V (1) V (2) W (3) W (4) TT (b) V (1) V (2) W (3) W (4) TTTT (c)
Figure 1 : Feynman diagrams corresponding to stress tensor exchanges between two pairsof identical operators. The grey circles indicate that (cid:104) V (1) V (2) ˆ T ( w ) ... ˆ T ( w n ) (cid:105) are exact rather than free vertices. (a) Single exchange diagram A . (b) Double exchange diagram A . (c) Multiple exchange diagram A n .with the partially amputated (2 + n )-point correlation functions (cid:104) V (1) V (2) ˆ T ( w ) ... ˆ T ( w n ) (cid:105) = (cid:18) − πc (cid:19) n n (cid:89) i =1 ∂ ¯ w i ( ∂ w i ) − (cid:104) V (1) V (2) T ( w ) ...T ( w n ) (cid:105) , (4.14)where the two matter insertions on the left-hand side are unamputated while the n stresstensor insertions are amputated. We denote amputated stress tensor insertions with a hat.The above equality can be derived by first writing a correlator with unamputated i -th legas the convolution of its amputated counterpart with the stress tensor propagator, (cid:104) ... T ( w i ) ... (cid:105) = (cid:90) d y (cid:104) ... ˆ T ( y ) ... (cid:105)(cid:104) T ( y ) T ( w i ) (cid:105) . (4.15)Equation (4.14) is obtained after invoking the magic identity (1.3) in order to rewrite thestress tensor propagator as (cid:104) T ( z ) T ( w ) (cid:105) = c z − w ) = − c ∂ w (cid:18) z − w (cid:19) = − πc ∂ w ( ∂ ¯ w ) − δ ( z − w ) . (4.16)The partially amputated correlation functions (4.14) will be used as vertices in evaluatingthe Feynman diagrams of interest. Because they are exact rather than free vertices, weindicate them with large grey circles in Figure 1.– 12 –efore turning to their evaluation, let us comment on the overall power of 1 /c as-sociated with a Feynman diagram involving n stress tensor exchanges. Such a diagramcontains two vertices (4.14) and n stress tensor propagators (4.16), so that it has an overallfactor of (1 /c ) n . In the reparametrization mode formalism, this factor would be associatedto n reparametrization propagators (4.8). Of course, we will discover that this is not acoincidence. Single exchange.
We first evaluate the Feynman diagram of Figure 1a containing asingle stress tensor exchange. Patching together the vertices (4.14) and the stress tensorpropagator, we have A = (cid:90) d w d w (cid:104) V (1) V (2) ˆ T ( w ) (cid:105)(cid:104) T ( w ) T ( w ) (cid:105)(cid:104) ˆ T ( w ) W (3) W (4) (cid:105) (4.17)= (cid:18) πc (cid:19) (cid:90) d w d w (cid:104) V (1) V (2) ¯ ∂T ( w ) (cid:105) ∂ − w ∂ − w (cid:104) T ( w ) T ( w ) (cid:105)(cid:104) ¯ ∂T ( w ) W (3) W (4) (cid:105) . Remarkably, we observe that the kernel of the second line coincides with the ‘physicalreparametrization propagator’ (4.8), ∂ − w ∂ − w (cid:104) T ( w ) T ( w ) (cid:105) = c w ln µw = (cid:16) c (cid:17) G (cid:15) ( w , w ) . (4.18)We stress that we never had to consider any kind of coupling to a dynamical reparametriza-tion mode in order to witness the appearance of this propagator. In this approach, G (cid:15) is an‘emergent’ quantity derived from the stress tensor propagator. To simplify (4.17) further,we use the conformal Ward identity (cid:104) ¯ ∂T ( w ) V (1) V (2) (cid:105) = − π (cid:88) i =1 , (cid:104) h V ∂ w δ (2) ( w − z i ) − δ (2) ( w − z i ) ∂ z i (cid:105) (cid:104) V (1) V (2) (cid:105) (4.19a)= − πh V (cid:104) V (1) V (2) (cid:105) (cid:20)(cid:18) ∂ w + 2 z (cid:19) δ ( w − z ) + ( z ↔ z ) (cid:21) . (4.19b)Plugging (4.19) into (4.17) and integrating by parts, we find F (cid:12)(cid:12) O (1 /c ) = A (cid:104) V V (cid:105)(cid:104)
W W (cid:105) = (cid:90) d w d w D h V w (1 , D h W w (3 , G (cid:15) ( w , w ) , (4.20)where we defined the differential operator D hw (1 , ≡ h (cid:20) δ ( w − z ) (cid:18) ∂ w − z (cid:19) + ( z ↔ z ) (cid:21) . (4.21)Of course, the delta distribution in (4.21) allows to trivially perform the integrals in (4.20).Doing so and comparing with the bilocal vertex operators in (4.5), the formula (4.12)obtained from the reparametrization formalism emerges before our eyes, F (cid:12)(cid:12) O (1 /c ) = (cid:104) b (1) h V (1 , b (1) h W (3 , (cid:105) . (4.22)A successful derivation of the reparametrization mode prescription is thus provided byevaluating the Feynman diagram containing a single stress tensor exchange.– 13 – ouble exchange. To further test the correspondence with the reparametrization for-malism uncovered at order O (1 /c ), we evaluate the contribution coming from two stresstensor exchanges. The corresponding Feynman diagram is shown in Figure 1b. Dividingby the appropriate symmetry factor of 2 associated with the interchange of internal lines,and applying the same line of reasoning as above, we find A = 12(2 π ) (cid:90) (cid:89) i =1 d w i (cid:104) V (1) V (2) ¯ ∂T ( w ) ¯ ∂T ( w ) (cid:105) G (cid:15) ( w , w ) (4.23) × G (cid:15) ( w , w ) (cid:104) ¯ ∂T ( w ) ¯ ∂T ( w ) W (3) W (4) (cid:105) . The four-point function (cid:104) ¯ ∂T ¯ ∂T V V (cid:105) is derived in appendix A from the conformal Wardidentity. Plugging its expression in (4.23), a tedious but straightforward computationyields F (cid:12)(cid:12) O (1 /c ) ⊃ A (cid:104) V V (cid:105)(cid:104)
W W (cid:105) = 12 (cid:90) (cid:89) i =1 d w i D h V w (1 , D h W w (3 , G (cid:15) ( w , w ) D h V w (1 , D h W w (3 , G (cid:15) ( w , w ) (4.24a)+ (cid:90) (cid:89) i =1 d w i D h V { w ,w } (1 , D h W w (3 , G (cid:15) ( w , w ) D h W w (3 , G (cid:15) ( w , w ) (4.24b)+ (cid:90) (cid:89) i =1 d w i D h W { w ,w } (3 , D h V w (1 , G (cid:15) ( w , w ) D h V w (1 , G (cid:15) ( w , w ) (4.24c)+ 2 (cid:90) (cid:89) i =1 d w i D h V { w ,w } (1 , D h W { w ,w } (3 , G (cid:15) ( w , w ) G (cid:15) ( w , w ) , (4.24d)where D hw was given in (4.21) and we have introduced a second differential operator, D hw ,w (1 , ≡ h (cid:20) δ ( w − z ) δ ( w − z ) (cid:18) ∂ w − ∂ w z + 1 z (cid:19) (4.25) − δ ( w − z ) δ ( w − z ) z + ( z ↔ z ) (cid:21) . As before, the integrals in (4.24) localize due to the delta distributions. Like at order O (1 /c ), we want to make the comparison with the reparametrization formalism. Moreprecisely, because (4.24) involves two propagators G (cid:15) whose legs are connected to both pairs of operators, it should be compared to (cid:104)B (2) h V (1 , B (2) h W (3 , (cid:105) c = 12 (cid:16) (cid:104) b (1) h V (1 , b (1) h W (3 , (cid:105) (cid:17) + 12 (cid:104) (cid:16) b (1) h W (3 , (cid:17) b (2) h V (1 , (cid:105) c (4.26)+ 12 (cid:104) (cid:16) b (1) h V (1 , (cid:17) b (2) h W (3 , (cid:105) c + (cid:104) b (2) h V (1 , b (2) h W (3 , (cid:105) c . Looking again at the definitions of the bilocal vertices in (4.5), a careful comparison showsthat the four different terms in (4.26) exactly coincide with the four terms in (4.24). Theagreement occurs term by term such that (4.24) and (4.26) are just two ways the write thesame things. Once again, the reparametrization formalism has effectively emerged whenevaluating Feynman diagrams describing stress tensor exchanges.– 14 – xponentiation in the light-light limit.
One of the successes of the reparametrizationformalism was to correctly reproduce the leading term of the Virasoro identity block in thelight-light limit h = O ( √ c ) [4], F = exp (cid:18) h V h W c u F (2 , , u ) (cid:19) + O (1 / √ c ) . (4.27)Hence, the Virasoro identity block contains a contribution which is the exponentiated globalidentity block (4.12). This result was first derived in [6].Since it is an important result, it is worth deriving it within the alternative formalismproposed here. For this, we consider the contributions resulting from an arbitrary number n of stress tensor exchanges. The corresponding Feynman diagram is shown in Figure 1c. Weuse the expression for the vertex (cid:104) ¯ ∂T ( w ) ... ¯ ∂T ( w n ) V (1) V (2) (cid:105) given in (A.5) which holds inthe light-light limit h = O ( √ c ), and integrate it against n stress tensor propagators. Takinginto account the symmetry factor of n ! associated with interchanges of internal lines, andafter a straightforward computation similar to that for single and double exchanges, wefind F = (cid:88) n n ! (cid:16) (cid:104) b (1) h V (1 , b (1) h W (3 , (cid:105) (cid:17) n + O (1 / √ c ) . (4.28)Upon insertion of (4.12), we indeed recover the expected exponential (4.27). We have discussed several aspects of the reparametrization mode formalism. After re-viewing some of the basic properties of the Polyakov action in section 2, we provided afirst principle derivation of the Alekseev–Shatashvili action as a nonlinear extension of theeffective action (1.1) governing the reparametrization modes. We have further argued insection 3 that the correct interpretation of the reparametrization mode is that of a sourcefor the holomorphic component of the stress tensor, and that the Alekseev–Shatashvili ac-tion is the generating functional for its connected correlation functions on manifolds relatedto the complex plane by conformal transformations. We then turned to the computation ofVirasoro identity blocks within the reparametrization mode formalism in section 4 wherewe showed that the otherwise mysterious prescriptions of that formalism naturally emergewhen evaluating Feynman diagrams associated with stress tensor exchanges between pairsof identical primary operators. Several interesting open problems deserve further investi-gation, which will help bring the program initiated here to further completion.
Comparison with other formalisms.
Although the approach proposed here and basedon the evaluation of Feynman diagrams will look familiar to anyone having studied pertur-bative quantum field theory, it is quite unconventional from the common perspective on2d CFTs. In fact, we are not aware of any other similar use of Feynman diagrams madein this context. It would therefore be very interesting to connect it to more conventionaltechniques used to compute Virasoro blocks in 2d CFTs [6, 8, 29, 30]. In particular, theformalism developed in [26, 30] based on gravitational Wilson lines and the AdS/CFTcorrespondence seems very close in spirit to our approach. Indeed, it was shown that– 15 –he expectation value of the gravitational Wilson line coincides with the reparametrizedtwo-point function (4.1), while a natural interpretation in terms of Feynman diagrams andstress tensor exchanges also emerged in that picture.
UV divergences and their regularization.
As mentioned in section 4, some termsarising from the central formula (4.9) for computing Virasoro identity blocks within thereparametrization formalism suffer from ultraviolet divergences. A general regularizationprocedure of some sort is needed, which has not been provided so far . A similar issue po-tentially arises when evaluating Feynman diagrams. At order O (1 /c ) for instance, one canconsider the diagrams of Figure 2a-2b in addition to that of Figure 1b. The appearance ofthe stress tensor running in loops implies that these also suffer from ultraviolet divergences.One should in fact identify the diagrams displayed in Figure 2a and Figure 2b with theterms (cid:104)B (3) h V (1 , B (1) h W (3 , (cid:105) c and (cid:104)B (2) h V (1 , B (1) h W (3 , (cid:105) c in (4.12), respectively. Since bothapproaches require regularization, a natural strategy would consist in applying one of thetextbook regularization procedures to the evaluation of Feynman diagrams and deduce thecorresponding rules within the reparametrization formalism. But one could also argue thatthese diagrams should be discarded altogether on the basis that they seem to correct thevertices appearing in the diagram at order O (1 /c ) displayed in Figure 1a. Since these ver-tices are already exact as previously emphasized, they may not need to be renormalized.We illustrate this in Figure 2c. This reasoning seems in agreement with the regulariza-tion procedure of the gravitational Wilson line formalism [30], where one only keeps termscorresponding to stress tensors propagating between both pairs of primary operators. Wehope to come back this issue in the future. Gravitational theories and holography.
We end this discussion by mentioning therelevance of reparametrization modes to gravitational theories and holography, which hasbeen our initial motivation to perform the present study. Especially clear is their rolewithin the AdS /CFT correspondence, where it was shown that the gravitational on-shell action coincides with the Liouville version (2.6) of the Polyakov generating functional[31, 32]. This makes perfect sense since the AdS/CFT dictionary precisely identifies thebulk on-shell action with the generating functional of a dual CFT [33, 34]. Hamiltonianreductions of three-dimensional gravity with AdS asymptotics were also shown to yieldeither Liouville theory [35] or the Alekseev–Shatashvili action [4] . Here, we gave a unifiedview of the different forms taken by the generating functional of stress tensor correlationsfrom the perspective of 2d CFTs. The Alekseev–Shatashvili action also appeared fromHamiltonian reductions of three-dimensional gravity with de Sitter asymptotics [36] andof the superrotation sector of four-dimensional gravity with flat asymptotics [37]. We be-lieve that these constitute important hints to the holographic nature of these gravitationaltheories away from the well-understood and heavily studied AdS/CFT correspondence. In the heavy-light limit, background subtraction has been successfully applied [4]. Away from this limit,there is however no obvious reference background to subtract from. The authors of [4] further argued that the Alekseev–Shatashivili action may be successfully quantized,leading to a quantum theory of boundary gravitons. – 16 – (1) V (2) W (3) W (4) TT (a) V (1) V (2) W (3) W (4) TT T (b)
VV TT + VV TT T ⊂ VV T (c)
Figure 2 : (a)-(b) Additional Feynman diagrams that potentially contribute to Virasoroidentity blocks at order O (1 /c ). (c) The effect of these diagrams is to renormalize theleft vertex of the diagram displayed in Figure 1a and occurring at order O (1 /c ). Since thelatter is already exact, one might want to simply discard them. Acknowledgments
I thank Jordan Cotler and Jakob Salzer for useful comments on a draft of this paper,and Jakob Salzer for collaboration on related topics. I also thank Felix Haehl and PeterWest for interesting discussions. This work is supported by a grant from the Science andTechnology Facilities Council (STFC).
A Conformal Ward identities
We recall the conformal Ward identity for n stress tensor insertions [38] (cid:104) T ( w ) ...T ( w n ) O h (1) O h (2) (cid:105) (A.1)= n (cid:88) i =2 (cid:18) w i + ∂ w i w i (cid:19) + (cid:88) j =1 , (cid:18) h ( w − z j ) + ∂ z j w − z j (cid:19) (cid:104) T ( w ) ...T ( w n ) O h (1) O h (2) (cid:105) + n (cid:88) i =2 c/ w i (cid:104) T ( w ) ...T ( w i − ) T ( w i +1 ) ...T ( w n ) O h (1) O h (2) (cid:105) . – 17 –n the case of two stress tensor insertions, it yields (cid:104) ¯ ∂T ( w ) ¯ ∂T ( w ) O h (1) O h (2) (cid:105) (A.2)= (2 π ) (cid:104)O h (1) O h (2) (cid:105) (cid:104) C hw (1 , C hw (1 ,
2) + C h { w ,w } (1 , (cid:105) , with C hw (1 ,
2) = − h (cid:20)(cid:18) ∂ w + 2 z (cid:19) δ ( w − z ) + ( z ↔ z ) (cid:21) , (A.3) C hw ,w (1 ,
2) = − C w (1 , ∂ w δ ( w − w ) + δ ( w − w ) ∂ w C w (1 ,
2) (A.4)+ h (cid:2) δ ( w − z ) (cid:18) ∂ w + 2 ∂ w z + 2 z (cid:19) δ ( w − z ) − δ ( w − z ) δ ( w − z ) z + ( z ↔ z ) (cid:3) . Note that the symmetrizer { w , w } = [( w , w ) + ( w , w )] ensures that (A.2) is sym-metrical under w ↔ w as it should. The careful reader already sees the structure ofthe reparametrization formalism appearing at this stage. Indeed, integration of C hw (1 , (cid:15) propagators yields the bilocal vertex operator b (1) h (1 , b (2) h (1 , h = O ( √ c ), repetitive use of theconformal Ward identity (A.1) yields (cid:104) ¯ ∂T ( w ) ... ¯ ∂T ( w n ) O h (1) O h (2) (cid:105) (A.5)= (2 π ) n (cid:104)O h (1) O h (2) (cid:105) (cid:104) C hw (1 , ...C hw n (1 ,
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