Exact Null Tachyons from Renormalization Group Flows

Exact null tachyons from renormalization group flows

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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.w, 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 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 (ii) Turning on a dilaton of the form ðX ;X Þ¼ ments for closed string tachyons in a background with a X þ ~ ðXþÞ. The coefficient is arbitrary, and null shift symmetry. It was pointed out in [6–8] that for can be shifted by rescaling X ! X, Xþ ! sigma models with such a symmetry, the equations of Xþ=; has mass dimension 1 if X 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 X. 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 Spert ¼ SCFT þ d z g½2g @ X @ X grounds of [20] are one-loop exact). We generalize this 40 work to describe null tachyon profiles with a null isometry þ 0ðX þ ~ 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 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 C ~ 2 ~ þ ð2Þ a þ fixed X , X R, the theory is the conformal field theory S ¼ d g ðX 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 metry under shifts of X . This appears to be broken by the pffiffiffi pffiffiffi ^ ^2 term in the dilaton linear in X. However, note that a shift gR ¼ g^ðR 2@ Þ; (3.3) in X 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 X, 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 40 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 QÞÞ eSSFP 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 Z ¼ DXþDXDYDbDc eSpertðX ;X ;YÞSFP : ð Þ 1 pffiffiffi 1 q Z ¼ q;½u; ~ ; gRð2Þ þ ðxÞ 8 80 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; ; 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 . The quantum contribution q; to the dilaton beta function comes from the determinant in (3.4), and from the pertur- III. INTEGRATING OUT X bations to C. It can thus be written as In the action (2.2), X appears as a Lagrange multiplier. c½u24 q; ¼ By analytically continuing the theory to Lorentzian signa- 6 ; (4.3) ture to do this integral, we find that ½ Z where c u is the Zamolodchikov c-function for the theory C aO Z ¼ DXþ þ DbDcDY ; perturbed by the operators u a; the factor of 24 comes from the Fadeev-Popov ghosts and from the determinant 0 pffiffiffi pffiffiffi ~ ð g@2Xþ gRÞeSSFP ; (3.1) 2 3We are using the conventions found in [4,21].

106005-2 EXACT NULL TACHYONS FROM RENORMALIZATION ... PHYSICAL REVIEW D 80, 106005 (2009) factor in (3.4) and (4.1), that arose from integrating out X. are believed to be gradients of c, but to date there is no q The beta function þþ comes from the contribution of this proof of that beyond leading order in conformal perturba- perturbed CFT to the beta function for the operator ð@XþÞ2. tion theory [22].4 Finally, q;a are the contributions to the beta functions for We can exchange the derivative of the delta function the couplings ua, coming from this perturbed CFT. These in (4.1) for an Xþ derivative and integrate by parts:

Z 1 ~ þ þ 0 SSFP DX DYDbDc ð ÞðX þ QÞ e x det@2 Z 0 þ þ 0 1 ~ ¼ þ ð þ Þ S SFP DX DYDbDcX X Q det 2 e @ Z 1 1 þ þ 0 ~ 0 ~_ þ pffiffiffi þ ¼ DX DYDbDcðX þ QÞ e S SFP ðX ðÞÞ gR þ u_ aðX ÞO ðÞ ; (4.4) det@2 4 a where the dots denote derivatives with respect to Xþ. Next, One is to note that there are no XþXþ correlators by the the classical variation of S~ with respect to is following argument. The delta function in (3.1) equates Xþ 1 pffiffiffi to the scale factor, , plus a constant. But when the beta ~ 2 ~ ð ÞS ¼ g@ functions vanish, the integrand Z is completely indepen- x 2 þ þ 0 dent of , and so the two-point function hX ðÞX ð Þi is 1 pffiffiffi € _ 0 ¼ gð~ ðXþÞð@XþÞ2 þ ~ @2XþÞ; (4.5) independent of , (the factor Q can be absorbed into ). 2 While the beta functions are computed away from the conformal point, this means that any divergences from where we have used (3.3) in the first line. This expression þ will multiply the delta function in (4.1). As applied to contractions of X must vanish at the conformal point, Eq. (4.1), the second term in (4.5) can therefore be replaced and do not yield independent terms in the beta function by equations. Another argument is as follows. It would appear that the 0 ~_ pffiffiffi path integral (3.1), together with (3.2), allows for non- g R; (4.6) þ 4 vanishing operator product expansions (OPEs) of X , since the delta function makes the dilaton coupling equiva- i.e., by a contribution to the dilaton beta function. lent (after integrating by parts) to a standard kinetic term Collecting all of the terms in (4.1), we find that the for Xþ, variation of Z vanishes if Z _ 0 þ / 2 ~ ð þÞ2 2 u_ aðX Þþq;a ¼ 0; Skin; d @X : (4.8) 0 ~_ ~ 4 þ q; ¼ 0; (4.7) Near the RG fixed points at Xþ ¼1, such a term is also 0 ~€ þ induced from integrating out the degrees of freedom in C 4 ðX Þþþþ ¼ 0: (see for example [12]), which cancels the dilaton contri- þ The left-hand sides are the beta functions for the full theory bution. As one flows in X , the second line of (4.7) (2.2). The first line is the full beta function for O, the indicates that the additional contributions to Gþþ from C second line is the dilaton beta function, and the final line are canceled by the dilaton contribution. Again, one should is the beta function for ð@XþÞ2. be careful since the beta functions are computed away from q the conformal point. As above, however, the additional We claim that the arise entirely from the divergences þ þ in the perturbed CFT due to singular operator products of divergences from X X correlators should not give any the Oa, so that as functions of u, q are the same as for additional contributions to the beta function equations constant couplings. This is true if there are no additional describing the conformal point itself. divergences arising from contractions of Xþ. (If Xþ was We should also make sure that the arguments in this instead timelike, for example, such divergences would section and in Sec. III hold for correlation functions—that appear [4].) If the beta functions are computed perturba- is, that the beta functions which we compute hold for the tively, this should be the result of the diagrammatic argu- Callan-Symanzik equation for correlation functions of 5 ments given in [9–13]. We have two additional arguments. local operators. The essential point is that for local correlators [string scattering amplitudes are finite-dimensional integrals of local correlators; recall also that three such 4Note that this proof breaks down at higher orders [4]. Furthermore, in at least one case it is likely that the beta functions are not gradients of any function of the relevant and 5We would like to thank Joe Polchinski for reminding us of marginal couplings alone [23]. this issue.

106005-3 ALLAN ADAMS, ALBION LAWRENCE, AND IAN SWANSON PHYSICAL REVIEW D 80, 106005 (2009) operators are used to fix the SLð2; CÞ invariance of the The final result is that the spacetime evolution generated world sheet CFT], the delta function constraint (3.4) should by a null tachyon profile is completely determined by the only be modified at the point of the operator insertions. For first order equations of the associated RG flow, with the example, if the world sheet was regulated with a lattice dependence of the tachyon on Xþ identical to that of the cutoff, one would integrate over all values of X point by renormalized coupling on the RG scale ¼ Xþ=ð0Þ. point. Therefore, the delta function (3.4) is only modified at the operator insertion points. In the Callan-Symanzik equa- V. CONCLUSIONS tions, modifications of the scale transformations which occur at the points of operator insertions give contributions We have shown that any RG flow arising from the to the anomalous dimensions of operators, rather than to perturbation of a CFT by a relevant operator also defines the beta functions.6 an exact CFT describing the spacetime evolution of a null Finally, Eqs. (4.7) appear to have more equations than tachyon condensate, and that the resulting tachyon and unknowns. This is typical of the beta function equations in dilaton profiles satisfy first order equations determining string theory. In fact, the left-hand side of the second their evolution along the null direction. This generalizes equation in (4.7) is conserved as a consequence of the first the work of [9–19]. As pointed out in [12], these are stringy and third equations: the vanishing of the beta functions on a ‘‘bubbles of nothing’’ (when the coupling to the identity C 7 flat two-dimensional world sheet is sufficient to ensure that operator in flows) —analogs of [25]—in which dimen- the theory is a CFT. One may then set it to zero by adjusting sions of spacetime are destroyed by an expanding domain the linear term in ~ . This fact has been checked at one- of tachyon condensate satisfying first order equations of loop for sigma models in [22], and for tachyons in [4]. motion. It would be interesting to study further examples, Furthermore, one can compute q to leading order in Xþ such as the null tachyon generated from the RG flow of the þþ CPn 8 ¨ derivatives following the discussion in [4]. Combined with model. In this example, the Kahler class is known to the leading order result [24] flow precisely logarithmically with RG scale [26,27], and it will therefore evolve linearly in Xþ. This flow goes from a c ¼ 2n sigma model in the ultraviolet to a trivial c ¼ 0 @ c½u¼242g b½u; (4.9) a ab Landau-Ginzburg theory in the infrared [27]. We should note that there often exists a scheme in which one can show that the third line in (4.7) follows from the the tachyon beta functions can be linearized, unless the first two lines, specifically by taking the derivative of the tachyons mix nontrivially under the OPEs, and are mar- second line. More generally, one can argue that the vanish- ginally relevant or satisfy some kind of resonance condi- ing of the first two equations in (4.7) implies the vanishing tion (see [4] for a review and discussion). When the beta of the last equation, as follows. The Wess-Zumino condi- functions can be linearized, the Xþ dependence of the tion implies that the derivative of the second term with þ tachyon will be a simple exponential. Furthermore, unlike respect to X is proportional to a linear combination of the the timelike examples in [4], Xþ is identified precisely with beta functions of the theory. Indeed, it should be a linear the RG scale so that the infrared fixed point will be reached combination of all of the beta functions of the theory; if only at null infinity. In this scheme there is less to be lost. any one relevant or marginally relevant coupling is turned When there are universal higher-order terms in the beta q; on, the theory will flow, and will cease to be constant. function, the Xþ dependence will of course be more Thus, if one sets the beta function for all Gþþ to zero, the complicated. derivative of the dilaton beta function will be proportional to Gþþ. q ACKNOWLEDGMENTS It is worth noting that the beta function þþ is determined by consistency of (4.7): taking the derivative of the We would like to thank A. Frey, M. Headrick, second equation, and using the first equation together with S. Hellerman, J. Polchinski, and E. Silverstein for discus- (4.3), we find that sions and comments on previous drafts. A. A. and A. L. would like to thank the Kavli Institute for Theoretical 1 q q;a Physics at UCSB ‘‘Quantum Criticality and the AdS/CFT þþ ¼ ;@ c: (4.10) 1220 a Correspondence’’ Miniprogram and the Aspen Center for Physics ‘‘String Duals of Finite Temperature and Low- It would be interesting to prove this directly. The depen- Dimensional Systems’’ for their hospitality while this dence arises from the relationship between Xþ and the work was completed. The work of A. A. and I. S. is sup- scale factor , induced by integrating out X. ported in part by funds provided by the U.S. Department of

6Similarly, if the beta functions are constrained due to a global 7We would like to thank Simeon Hellerman for reminding us symmetry, the constraints on the beta functions remain even of this. when one computes correlation functions of local operators that 8We would like to thank Eva Silverstein for suggesting this transform nontrivially under this global symmetry. example.

106005-4 EXACT NULL TACHYONS FROM RENORMALIZATION ... PHYSICAL REVIEW D 80, 106005 (2009) Energy (DOE) under cooperative research agreement DE- No. DE-FG02-92ER40706 and by the National Science FG0205ER41360. A. L. is supported in part by DOE Grant Foundation under NSF Grant No. PHY05-51164.

[1] A. R. Cooper, L. Susskind, and L. Thorlacius, Nucl. Phys. [15] P. Horava and C. A. Keeler, Phys. Rev. D 77, 066013 B363, 132 (1991). (2008). [2] C. Schmidhuber and A. A. Tseytlin, Nucl. Phys. B426, 187 [16] S. Hellerman and X. Liu, arXiv:hep-th/0409071. (1994). [17] I. Swanson, Phys. Rev. D 78, 066020 (2008). [3] E. Silverstein, Phys. Rev. D 73, 086004 (2006). [18] A. R. Frey, J. High Energy Phys. 08 (2008) 053. [4] D. Z. Freedman, M. Headrick, and A. Lawrence, Phys. [19] O. Aharony and E. Silverstein, Phys. Rev. D 75, 046003 Rev. D 73, 066015 (2006). (2007). [5] A. Adams, J. Polchinski, and E. Silverstein, J. High [20] G. T. Horowitz and A. R. Steif, Phys. Rev. Lett. 64, 260 Energy Phys. 10 (2001) 029. (1990). [6] A. A. Tseytlin, Nucl. Phys. B390, 153 (1993). [21] J. Polchinski, String Theory: An Introduction to the [7] A. A. Tseytlin, Phys. Lett. B 288, 279 (1992). Bosonic String (Cambridge University Press, Cambridge, [8] E. J. Martinec, arXiv:hep-th/0210231. England, 1998), Vol. 1, p. 402. [9] S. Hellerman and I. Swanson, arXiv:0709.2166. [22] C. G. Callan, E. J. Martinec, M. J. Perry, and D. Friedan, [10] S. Hellerman and I. Swanson, arXiv:0710.1628. Nucl. Phys. B262, 593 (1985). [11] S. Hellerman and I. Swanson, J. High Energy Phys. 07 [23] M. Headrick, Phys. Rev. D 79, 046009 (2009). (2008) 022. [24] A. B. Zamolodchikov, Yad. Fiz. 46, 1819 (1987) [Sov. J. [12] S. Hellerman and I. Swanson, J. High Energy Phys. 09 Nucl. Phys. 46, 1090 (1987)]. (2007) 096. [25] E. Witten, Nucl. Phys. B195, 481 (1982). [13] S. Hellerman and I. Swanson, Phys. Rev. D 77, 126011 [26] L. Alvarez-Gaume and P.H. Ginsparg, Commun. Math. (2008). Phys. 102, 311 (1985). [14] P. Horava and C. A. Keeler, Phys. Rev. Lett. 100, 051601 [27] S. Cecotti and C. Vafa, Phys. Rev. Lett. 68, 903 (1992). (2008).

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