A Note on Perturbative and Nonperturbative Instabilities of Twisted Circles

Physics Letters B 545 (2002) 379–383 www.elsevier.com/locate/npe

A note on perturbative and nonperturbative instabilities of twisted circles

Michael Gutperle a,b

a Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA b Department of Physics, Stanford University, Stanford, CA 94305, USA 1 Received 9 August 2002; accepted 28 August 2002 Editor: M. Cveticˇ

Abstract In this short Letter we compare the endpoint of tachyon condensation of twisted circles with the endpoint of nonperturbative brane nucleation in the Kaluza–Klein Melvin spacetime.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction Letter we will compare what happens to the twisted circle under perturbative tachyon condensation [3] to Twisted circles are constructed using an identifi- what happens after nucleation of a spherical brane cation where a rotation in a plane (by a rational an- in a Melvin background [17]. Although the regimes gle) is combined with a shift along an orthogonal real where the analysis of the instabilities is valid are very line. This is an interesting construction because on the different we find that the end result of the tachyon one hand, the freely acting orbifold is smoothing out condensation and nucleation of branes is the same: the conical singularity which would occur by an iden- the twisted circle untwists itself and the radius of the tification by rotation without a shift [1]. This there- compact circle increases. This might be considered as fore provides a nice arena to study localized tachyon some evidence that these two seemingly very different condensation (see also [1–3]). On the other hand a processes are actually related, a conjecture made in Kaluza–Klein reduction produces a Melvin fluxbrane [11,13]. spacetime [4] which has gotten a lot of attention recently [5–16]. Methods of semi-classical quantum gravity can be 2. The nonperturbative instability used to analyze nonperturbative instabilities [5,6,11] corresponding to the nucleation of KK-branes. In this Starting with the d-dimensional flat space metric

E-mail addresses: [email protected], 2 =− 2 + 2 +···+ 2 + 2 ds dt dx1 dxd−4 dr [email protected] (M. Gutperle). 1 Address after August 1st, 2002. + r2 dϕ2 + R2 dy2, (2.1)

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02599-6 380 M. Gutperle / Physics Letters B 545 (2002) 379–383 where y is periodic with period 2π, one reduces along The absence of a conical singularity at r = r+ then the orbits of the Killing vector ∂y + q∂ϕ, which means determines the radius R of the Kaluza–Klein di- that a translation y → y + 2π is accompanied by a rection xd . The second quantity characterizing the rotation ϕ → ϕ + 2πγ,whereγ = qR.Itisuseful black hole solution is the (analytically continued) an- to introduce a new single-valued angular variable ϕ˜ = gular momentum Ω.Intermsofm and k,these ϕ − qRy which has standard periodicity. In the new are coordinates the metric becomes 6−d d−5 2mr+ kr+ 2 2 2 2 2 = = ds =−dt + dx +···+dx − + dr R 2 ,Ω . 1 d 4 (d − 3)r+ − (d − 5)k2 m + r2 dϕ˜ + qRdy 2 + R2 dy2. (2.2) (2.7) Using the standard formulae for Kaluza–Klein reduc- Note that the physical range of R, Ω is restricted by tion, |ΩR| 1. Since (2.5) is asymptotically flat one can embed the black hole in a Melvin fluxbrane by twist- √ 4 φ 2 2 = d−2 + µ 2 dsd e dy 2Aµ dx ing − 4√ φ + e (d−3) d−2 ds2 . (2.3) sgn(Ω) d−1 q = Ω − . (2.8) Rescaling brings the metric into the following canoni- R cal form Under the twisted identification the twist angle is given 1 by ds2 = 1 + b2r2 d−3 d−1 = = − 2 2 2 γ qR ΩR sgn(Ω), (2.9) × −dt + dx +···+dx − 1 d 4 r2 hence Ω and γ are really periodic variables, which are + dr2 + dϕ˜2 , 1 + b2r2 identified modulo 2/R and 2, respectively. We are interested in the Minkowskian evolution √ 4 φ e d−2 = R2 1 + b2r2 , of the spacetime after the nucleation of a brane. To br2 q achieve this one analytically continues one of the = = − Aϕ˜ d−2 ,b 1 . (2.4) angular variables of the d 4 sphere results into the 2R d−3 (1 + b2r2) R d−3 time coordinate of the Minkowskian solution after In [5] it was shown that the gravitational instanton nucleation.3 The Lorentzian metric post-nucleation is mediating the creation of KK-branes in a Melvin then given by background is given by the Euclidean Myers–Perry 1 Σ [18] black hole, ds2 = Λ d−3 dr2 + Σdθ2 2 − 2 − 5−d 2 r k mr 2 m 2 2mk sin θ ds = 1 − dx − dxd dϕ + 2 2 − 2 + 2 2 rd−5Σ d rd−5Σ r cos θ dt cosh tdΩd−5 Σ R2 + dr2 + Σdθ2 + 2 2 − 2 − 5−d 2 2 − 2 − 5−d sin θ r k mr dϕ . r k mr Λ 2 θ m + sin 2 − 2 − 2 2 2 (2.10) r k Σ − k sin θ dϕ Σ rd 5 Where Λ is given by 2 2 + r cos θdΩd−4, (2.5) 2 2 2 where Σ = r −k cos θ. Under analytic continuation 2 As explained in [5] there is a second choice of twist corre- the horizon of the Minkowskian black hole becomes sponding to supersymmetry breaking boundary conditions on the an Euclidean ‘bolt’, with radius r+ defined by compactification circle, however, we will not discuss this case here. 2 − 2 − m = 3 Recently these spacetimes have been proposed as good labora- r+ k d−5 0. (2.6) r+ tories for studying time-dependence in string theory [19]. M. Gutperle / Physics Letters B 545 (2002) 379–383 381 m 2mk sin2 θ Λ = R2 1 − − q and hence the new angular momentum and twist angle rd−5Σ rd−5Σ are given by 2 sin θ σ(Ω) + q2 r2 − k2 Σ = − Ω 2Ω , Σ R − − mr5 d k2 sin2 θ . (2.11) γ = qR = ΩR − σ(Ω). (2.17)

As explained in [5] this metric for our choice q describes a spherical D6-brane expanding in a flux 3. Relation between perturbative and 7-brane background. It is natural to ask what the nonperturbative instabilities ‘leftover’ spacetime after the D6-brane has moved to infinity looks like. This question was addressed in [17] In [3] a twisted circle in ten-dimensional type II and we refer the reader there for more details. The string theory was discussed. The endpoints of tachyon metric (2.10) has an acceleration horizon and only condensation for twisted circles was analyzed (fol- covers the region of spacetime inside it. To continue lowing [20,21]) using a N = 2 gauged linear sigma past it, it is useful to make some coordinate changes. model [22–24]. The fields of the GLSM consist of a Firstly define z = r cosθ and r˜ = f(r)sin θ,where U(1) gauge field, two chiral fields Φ−n, Φm of charge (−n, m) andan‘axion’P which transforms by imag- 1 df r = . (2.12) inary shifts under U(1) gauge transformations. The 2 − 2 − 5−d f dr r k mr gauged linear sigma model has the following action Secondly, define Rindler√ like coordinates X, T in 1 2 4 mV −nV terms of z, t by z = X2 − T 2 and t = arctanh(T/X) . S = d σd θ Φme Φm + Φ−ne Φ−n 2π The exact form of the metric in the new coordinates k 1 is very complicated, but we are only interested in + P + P+ V 2 − |Σ|2 . T →∞limit where one can analyze the leading part 4 2e2 of the solution, dropping sub-leading terms of order (3.1) 1/T. Integrating out the gauge fields, the vacuum manifold Now, in order to get a static metric in terms of is given by the solutions to the D-term condition the new coordinates, the old radial coordinate has to (modulo the U(1) gauge transformations which are behave as responsible for the twisted identifications). In the low- energy limit a nonlinear sigma model is defined by the r˜ 1/ch 1 rˆ 2 = + + = + + massless fluctuations about the vacuum manifold. r r r 2 , (2.13) T 4ch T 2 2 m|φ1| − n|φ−n| + kp1 = 0. (3.2) d−4 where we have defined c = Rr+ /(2µ).Itwas h = + shown in [17] that as T →∞, the metric can again be The role of the complex axion P p1 ip2 is brought into the canonical form (2.4) with parameters twofold. The imaginary part p2 is used to construct √ the circle whereas the real part p1 is an auxiliary − 1 φˆ − d 2 direction. In [3] it was proposed that motion along the b = qΩ d−3 ,e0 = Ω 2 . (2.14) auxiliary direction is equivalent to RG-flow and this Where here and in the following the parameters correspondence was used to determine the endpoint characterizing the ‘leftover’ spacetime are primed. of the tachyon condensation. In the following we will compare the endpoints of perturbative tachyon 1 q = q, R = . (2.15) condensation to the nonperturbative brane nucleation |Ω| in several cases. From (2.8) it follows that (a) The gauged linear sigma model with charges (−n, 1),wheren is an odd integer, interpolates be- σ(Ω ) σ(Ω) q = Ω − = Ω − , (2.16) tween a twisted circle with twist γ =−1 + 1/n and R R 382 M. Gutperle / Physics Letters B 545 (2002) 379–383

radius R for p1 →∞and an untwisted circle γ = 0 systems which do not behave this way. However, the and radius R = nR for p1 →−∞. For the Melvin underlying theory has a spin structure which breaks s background this twist can be realized by choosing supersymmetry to start with. For example, consider re- peating the analysis for 1 but with n even. From (3.4) 1 1 R, Ω = ,γ=−1 + . (3.3) it follows that after the nucleation on ends up with nR n R = nR and γ =−1. From (3.2) its easy to see this Using formulas (2.15) and (2.17) for the radius and from the gauged linear sigma model too. For charge twist after the nucleated brane has accelerated away to (−n, 1) with n even, in the IR one has a twist −1 infinity one finds (this depends on the extra −1inthetwistwehaveto have for proper a proper GSO projection, see [2] for a R = nR, discussion). This twist corresponds to supersymmetry 2 1 2 − n Ω = − = , breaking boundary conditions on the circle [25]. The nR R nR nonperturbative instability is associated with Witten’s γ = 3 − n. (3.4) bubble of nothing [26]. Note, however, that for a twist =− Since n is odd, γ is even and by periodicity equivalent γ 1 the bounce is given by a Euclidean Schwarz- to γ = 0. Hence the end point of the nucleation is schild black hole and the analysis of Section 2 will not an untwisted circle of n times the original radius. work since the leftover spacetime is not of the form of Note that the action of the instanton diverges for a Kaluza–Klein Melvin solution. = the untwisted circle γ 0andthereisnofurther (d) If the twist angle γ is irrational, the gauged lin- nonperturbative instability. This agrees with the fact ear sigma model analysis cannot be applied. From the that the end point of the evolution is supersymmetric nonperturbative bounce one finds that the nucleation type II theory compactified on a circle. does not stop after a finite amount of steps and the (b) In the gauged linear sigma model with charges radius increases monotonically and one ends up with (−n, n − 2), with n odd, it follows from (3.2) that (supersymmetric) theory at infinite circle. It is tempt- one flows from twisted circle with twist γ =−2/n ing to speculate that this would also be the case for and radius R in the UV to a twisted circle with twist the perturbative tachyon condensation on the twisted γ =−2/(n − 2) and radius R = nR/(n − 2).Note circles. that such a flow corresponds to turning the lightest tachyon which does not completely untwist the circle. For the fluxbrane one chooses 4. Discussion n − 2 2 R, Ω = ,γ=− , (3.5) nR n In this Letter we have pointed out that for twisted circles the endpoint of tachyon condensation (using the spacetime after the brane has accelerated away has RG-flow which is believed to give the same result nR n − 4 2 as on shell time evolution) and nonperturbative brane R = ,Ω= ,γ=− . n − 2 nR n − 2 nucleation (where the nucleated brane accelerates off (3.6) to infinity) are very similar. In particular in both cases the twist becomes smaller and the radius of the With exactly parallels the result for the perturbative circle grows, even the quantitative features of the two tachyon condensation. Note that the resulting twisted endpoints agree. This might suggest that the twisted circle theory contains itself tachyons or is nonpertur- circle really wants to untwist itself, whether it takes batively unstable. Using the analysis above it is easy a perturbative of a nonperturbative mechanism to do to see that after (n − 3)/2 further bounces one ends up so. In this Letter we have considered the classical with (3.4), i.e., the stable end point is supersymmetric (tree level) perturbative tachyon and its condensation type II theory on a circle of n times the original radius. and the nonperturbative semi-classical instabilities. (c) The two examples discussed above decay to- However, we have not studied perturbative quantum ward a supersymmetric end state. There are different instabilities coming from tadpoles at higher loops. M. Gutperle / Physics Letters B 545 (2002) 379–383 383

The importance of those effects (in particular in [8] J.G. Russo, A.A. Tseytlin, hep-th/0110107. comparison with the nonperturbative effects) is an [9] J.G. Russo, A.A. Tseytlin, Nucl. Phys. B 611 (2001) 93, important open question (see [27] and [28] for related hep-th/0104238. [10] J.G. Russo, A.A. Tseytlin, Nucl. Phys. B 461 (1996) 131, discussion of this issue). hep-th/9508068. [11] M.S. Costa, M. Gutperle, JHEP 0103 (2001) 027, hep- th/0012072. Acknowledgements [12] P.M. Safﬁn, Phys. Rev. D 64 (2001) 024014, gr-qc/0104014. [13] M. Gutperle, A. Strominger, JHEP 0106 (2001) 035, hep- This work is supported in part by DOE grant th/0104136. [14] M.S. Costa, C.A. Herdeiro, L. Cornalba, hep-th/0105023. DE-FG02-91ER40655. I wish to thank M. Headrick [15] T. Suyama, hep-th/0110077. and S. Minwalla for useful conversation. I gratefully [16] Y. Michishita, P. Yi, Phys. Rev. D 65 (2002) 086006, acknowledge the hospitality of the Stanford ITP while hep-th/0111199. this work was ﬁnalized. [17] R. Emparan, M. Gutperle, JHEP 0112 (2001) 023, hep- th/0111177. [18] R.C. Myers, M.J. Perry, Ann. Phys. 172 (1986) 304. [19] O. Aharony, M. Fabinger, G.T. Horowitz, E. Silverstein, References hep-th/0204158. [20] C. Vafa, hep-th/0111051. [1] A. Adams, J. Polchinski, E. Silverstein, hep-th/0108075. [21] A. Dabholkar, C. Vafa, JHEP 0202 (2002) 008, hep- [2] J.A. Harvey, D. Kutasov, E.J. Martinec, G. Moore, hep- th/0111155. th/0111154. [22] E. Witten, Nucl. Phys. B 403 (1993) 159, hep-th/9301042. [3] J.R. David, M. Gutperle, M. Headrick, S. Minwalla, JHEP [23] K. Hori, C. Vafa, hep-th/0002222. 0202 (2002) 041, hep-th/0111212. [24] K. Hori, A. Kapustin, JHEP 0108 (2001) 045, hep-th/0104202. [4] M.A. Melvin, Phys. Lett. 8 (1964) 65. [25] R. Rohm, Nucl. Phys. B 237 (1984) 553. [5] F. Dowker, J.P. Gauntlett, G.W. Gibbons, G.T. Horowitz, Phys. [26] E. Witten, Nucl. Phys. B 195 (1982) 481. Rev. D 53 (1996) 7115, hep-th/9512154. [27] M. Fabinger, P. Horava, Nucl. Phys. B 580 (2000) 243, [6] F. Dowker, J.P. Gauntlett, G.W. Gibbons, G.T. Horowitz, Phys. hep-th/0002073. Rev. D 52 (1995) 6929, hep-th/9507143. [28] S.P. De Alwis, A.T. Flournoy, hep-th/0201185. [7] G.W. Gibbons, D.L. Wiltshire, Nucl. Phys. B 287 (1987) 717, hep-th/0109093.