Exact null tachyons from renormalization group flows The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Adams, Allan , Albion Lawrence, and Ian Swanson. “Exact null tachyons from renormalization group flows.” Physical Review D 80.10 (2009): 106005. © 2009 The American Physical Society As Published http://dx.doi.org/10.1103/PhysRevD.80.106005 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/52507 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 80, 106005 (2009) Exact null tachyons from renormalization group flows Allan Adams,1 Albion Lawrence,2 and Ian Swanson1 1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2Theory Group, Martin Fisher School of Physics, Brandeis University, MS057, 415 South Street, Waltham, Massachusetts 02454, USA (Received 11 August 2009; published 20 November 2009) We construct exact two-dimensional conformal field theories, corresponding to closed string tachyon and metric profiles invariant under shifts in a null coordinate, which can be constructed from any two- dimensional renormalization group flow. These solutions satisfy first order equations of motion in the conjugate null coordinate. The direction along which the tachyon varies is identified precisely with the world sheet scale, and the tachyon equations of motion are the renormalization group flow equations. DOI: 10.1103/PhysRevD.80.106005 PACS numbers: 11.25.Àw, 11.10.Hi I. INTRODUCTION dilaton to push this strong coupling region as far into the past or future as we wish. It is an old idea that renormalization group (RG) flows in two-dimensional quantum field theories can be lifted to time-dependent solutions of string theory. When the RG II. WORLD SHEET DESCRIPTION OF NULL flow describes the evolution of couplings to relevant op- TACHYONS erators, the string theory background corresponds to a In constructing our string theory background, we begin nontrivial tachyon profile. This identification is precisely with the tensor product of a two-dimensional target space true in the limit of large dilaton slope [1–4]. Away from and a conformal field theory C. Assume this conformal this limit, the map between RG flows and real tachyon theory has a set of local primary operators Oa. Let us write profiles is modified: in particular, spacetime equations of the two-dimensional target space with the metric motion are second order in time and space, while the RG 2 þ À flow equations are first order in scale [4,5].1 One may ds ¼2dX dX : (2.1) construct the full spacetime profile in a derivative expan- We now couple these theories by:R sion for slowly varying tachyons or in conformal perturba- 2 a þ (i) Deforming C by couplings d u ðX ÞOa depend- tion theory for small tachyon expectation values [2,4]. ing only on Xþ. In this work we pursue a modification of these argu- þ À (ii) Turning on a dilaton of the form ÈðX ;X Þ¼ ments for closed string tachyons in a background with a XÀ þ È~ ðXþÞ. The coefficient is arbitrary, and null shift symmetry. It was pointed out in [6–8] that for can be shifted by rescaling XÀ ! XÀ, Xþ ! sigma models with such a symmetry, the equations of Xþ=; has mass dimension 1 if XÆ has length motion would be first order in the null directions and dimension 1. look more like renormalization group flows. A wide class As in [10–13,16–18], the dilaton is linear in XÀ. The Xþ of exact conformal field theories (CFTs) of this kind, with a dependence of the tachyon will be nonlinear, as the slope timelike linear dilaton, has been worked out in [9–15] (see must shift between Xþ ¼1to make up for the change in also [16–19] for related studies). Because of the null shift the central charge associated with the sector C [4,16]. symmetry and the relatively simple tachyon profile, the The full world sheet action is2 beta functions receive no corrections beyond one-loop in 0 1 Z pffiffiffi (much as the beta functions for the plane wave back- 2 þ À Spert ¼ SCFT þ d z g½2g @ X @ X grounds of [20] are one-loop exact). We generalize this 4 0 work to describe null tachyon profiles with a null isometry þ 0ðXÀ þ È~ ðXþÞÞRð2Þ þ uaðXþÞO : (2.2) and a timelike dilaton given any renormalization group a flow. In these flows, the direction along which the tachyon Note that Oa will generally include the identity operator. varies is mapped precisely to the world sheet scale by a Here SCFT is the action for C. However, our discussion will Lagrange multiplier constraint, and the profile satisfies first order equations equivalent to the RG equations. 2In Refs. [10–13,16–18], there is also a term ð@XþÞ2 in the Note that these backgrounds will be exact in 0, but not action corresponding to a metric Gþþ which is induced at large þ necessarily in gs. In particular, we expect the dilaton to run X . This does not appear in our calculation. We believe this þ þ 2 to strong coupling in the past or future of these solutions. amounts in part to a choice of scheme. A term GþþðX Þð@X Þ À À þ As usual, however, we can adjust the constant mode of the can be removed by a field redefinition X ! X þ fðX Þ, where @þf ¼ Gþþ, without otherwise changing the form of the action. We would like to thank A. Frey for asking about 1That is, ½R; G Þ 0. this point. 1550-7998=2009=80(10)=106005(5) 106005-1 Ó 2009 The American Physical Society ALLAN ADAMS, ALBION LAWRENCE, AND IAN SWANSON PHYSICAL REVIEW D 80, 106005 (2009) work just as well if C has no Lagrangian description. In this where latter case, the presence of SCFT merely denotes that for Z pffiffiffi 1 þ À C ~ 2 ~ þ ð2Þ a þ fixed X , X R, the theory is the conformal field theory S ¼ d g ÈðX ÞR þ u ðX ÞOa : (3.2) 2 a þ 4 perturbed by d u ðX ÞOa. The tachyon and metric couplings in (2.2) have a sym- In conformal gauge, we have À metry under shifts of X . This appears to be broken by the pffiffiffi pffiffiffi ^ ^2 term in the dilaton linear in XÀ. However, note that a shift gR ¼ g^ðR À 2@ Þ; (3.3) in XÀ simply adds a total derivative (the Euler character of where R^ is the two-dimensional curvature for the fiducial the world sheet) to the action. Thus, this action respects an ^ ^2 overall shift symmetry in XÀ, which will be reflected in the metric gp,ffiffiffi and @ pisffiffiffi the associated Laplacian. In two dimen- 2 ¼ ^ ^2 world sheet beta functions. Note, however, that the string sions, g@ g@ . We can therefore write the delta genus expansion will not respect this shift symmetry, so function in (3.1)as one must treat the strong coupling region with caution. 1 pffiffiffi ðXþ þ 0 þ QÞ; (3.4) We will assume that we know, in advance, complete detð g@2Þ information about SCFT perturbed by arbitrary couplings which might depend on the world sheet coordinates. In wherepQffiffiffi depends only on the fiducial metric. Note that particular, this means that we assume knowledge of the full while g@2 is Weyl invariant, the determinant requires a Rset of beta functions q for C perturbed by the terms regulator and thus contributes to the Weyl anomaly. 2 aO d u a, where u are constant (c-number) couplings. The main lesson of this section is that for the action at This includes the beta functions for the identity and for the hand, Xþ is identified precisely with world sheet scale. dilaton. The dilaton beta function is proportional to the This is the physical basis of the observation below that, for Zamolodchikov c-function for the perturbed CFT, as we good string backgrounds, uaðXþÞ will satisfy the first order will discuss below. RG equations, with Xþ functioning as the renormalization The beta functions are also dependent on the contribu- group scale. tion of the degreesR of freedom of the perturbed theory þ 1 2 að þÞO SCFT 4 0 d zu X a to the beta function for the þ 2 IV. BETA FUNCTIONS operator ð@X Þ (e.g., for the spacetime metric Gþþ). ¼ 0 Even so, as we will argue below, this beta function is We wish to find the conditions under which ðÞ Z : determined completely by the above information. Z Finally, we will consider (2.2) fixed to conformal gauge: þ ~ Z ¼ DX DYDbDcð½ðxÞ ððxÞSÞ namely, the world sheet metric is ðxÞ 2 1 ~ g ¼ e g^ : (2.3)  ðXþ þ 0 À QÞÞ eÀSÀSFP det@2 The partition function is given by ðqÞ þ ð ÞZ ¼ 0: (4.1) Z x Z ¼ dZ; ðqÞ Here ðxÞ denotes the part of the variation induced by Z (2.4) quantum effects3: þ À Z ¼ DXþDXÀDYDbDc eÀSpertðX ;X ;YÞSFP : ð Þ 1 pffiffiffi 1 q Z ¼ q;Ƚu; È~ ; gRð2Þ þ ðxÞ 8 8 0 Here b, c are the conformal ghosts, and SFP is their action. pffiffiffi 1 pffiffiffi C q ~ þ 2 q;a Y stands for the degrees of freedom on . (Again, this is for  þþ½u; È; gð@X Þ þ ½u gOa: ease of exposition: the central arguments of this work do 2 not require that C have a Lagrangian description.) For a (4.2) good string background, Z must be independent of .