View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Physics Letters B 578 (2004) 215–222 www.elsevier.com/locate/physletb Comments on the fate of unstable orbifolds ✩ Sang-Jin Sin a,b a Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94305, USA b Department of Physics, Hanyang University, Seoul 133-791, South Korea 1 Received 21 August 2003; accepted 9 October 2003 Editor: M. Cveticˇ Abstract We study the localized tachyon condensation in their mirror Landau–Ginzburg picture. We completely determine the decay r mode of an unstable orbifold C /Zn, r = 1, 2, 3 under the condensation of a tachyon with definite R-charge and mass by r 2 extending the Vafa’s work hep-th/0111105. Here, we give a simple method that works uniformly for all C /Zn.ForC /Zn, r where method of toric geometry works, we give a proof of equivalence of our method with toric one. For C /Zn cases, the orbifolds decay into sum of r far separated orbifolds. 2003 Elsevier B.V. Open access under CC BY license. 1. Introduction until the spacetime supersymmetry is restored. There- fore the localized tachyon condensation has geometric The study of open string tachyon condensation description as the resolution of the spacetime singular- [1] has led to many interesting consequences includ- ities. ing classification of the D-brane charge by K-theory. Soon after, Vafa [3] considered the problem in the While the closed string tachyon condensation involve Landau–Ginzburg (LG) formulation using the Mirror the change of the background spacetime and much symmetry and confirmed the result of [2]. In [4], more difficult, if we consider the case where tachyons the same problem is studied by using the RG flow can be localized at the singularity, one may expect the as deformation of chiral ring and in term of toric maximal analogy with the open string case. Along this geometry. In [3], Vafa showed that, as a consequence direction, the study of localized tachyon condensation of the tachyon condensation, the final point of the was considered in [2] using the brane probe and renor- process is sum of two orbifold theories which are far malization group flow and by many others [3–8]. The from each other but smoothly connected: one located basic picture is that tachyon condensation induces cas- at north and the other at the south poles of blown up cade of decays of the orbifolds to less singular ones P 2 singularity of the orbifold in the limit where the radius of the sphere is infinite. Schematically, we can represent this transition by ✩ Work supported partially by the Department of Energy under contract number DE-AC03-76SF005515. C2 Z → C2 Z ⊕ C2 Z / n(k1,k2) / p1(∗,∗) / p2(∗,∗), (1.1) E-mail address: [email protected] (S.-J. Sin). 1 Permanent address. with yet unknown generators for the daughter theories. 0370-2693 2003 Elsevier B.V. Open access under CC BY license. doi:10.1016/j.physletb.2003.10.031 216 S.-J. Sin / Physics Letters B 578 (2004) 215–222 − Y0 = t/n ki The purpose of this Letter is to determine the expressed as e e i u . The periodicity of 2πi/n decay mode of unstable orbifolds by working out the Yi imposes the identification: ui ∼ e ui which generators of orbifold action in daughter theories for necessitate modding out each ui by Zn. The result is r 1 C /Zn r = 1, 2, 3. For C /Zn, the transition modes usually described by are described in earlier works [2–4]. For C2/Z n(k1,k2) r case, some examples are worked out in [4] using n t/n ki r−1 W = u + e u //(Zn) , (2.4) toric geometry and prescription in terms of continued i i=1 i fraction is given. In principle, it can be worked out once numbers are given explicitly. However, that which describe the mirror Landau–Ginzburg model of 3 the linear sigma model. As a t →−∞limit, mirror of method does not work for C /Zn. Here, we give a the orbifold is simple method that works easily and uniformly for all r 2 C /Zn.ForC /Zn, we give a proof of equivalence r = n Z r−1 of our method with toric one. To do this we will W ui //( n) . (2.5) need to know how the spectrums of chiral primaries i=1 are transformed under the condensation of a specific Since it is not ordinary Landau–Ginzburg theory but tachyon. an orbifolded version, the chiral ring structure of the theory is very different from that of LG model. For example, the dimension of the local ring of the super 2. Mirror symmetry and orbifolds potential is always n − 1, regardless of r. We list some properties of orbifolded LG theory for We begin by a summary of Vafa’s work [3] on later use. Cr localized tachyon condensation. The orbifold /Zn The true variable of the theory are Yi not ui related Z −Y /n is defined by the n action given by equivalence by ui = e i . As a consequence, monomial basis of relation the chiral ring is given by ∼ k1 kr p p (X1,...,Xr ) ω X1,...,ω Xr , u 1 u 2 | (p ,p ) = n{jk /n},n{jk /n} , 1 2 1 2 1 2 = 2πi/n ω e . (2.1) j = 1,...,n− 1 , (2.6) We call (k1,...,kr ) as the generator of the Zn action. p1 p2 and u u has weight (p1,p2) and charge (p1/n, The orbifold can be embedded into the gauged linear 1 2 p2/n). sigma model (GLSM) [9]. The vacuum manifold of the latter is described by the D-term constraints 2 2 3. Fate of the spectrum −n|X0| + ki|Xi | = t. (2.2) i For C2/Z case, if one consider the conden- Its t →−∞limit corresponds to the orbifold and the n(k1,k2) sation of tachyon in the lth twisted sector that cor- t →∞limit is the O(−n) bundle over the weighted responds to chiral ring element up1 up2 , with p = projected space WP . X direction corresponds 1 2 1 k1,...,kr 0 n{lk /n} and p = n{lk /n}, the theory is given by to the non-compact fiber of this bundle and t plays role 1 2 2 the super potential of size of the WP . k1,...,kr By dualizing this GLSM, we get a LG model with = n + n + t/n p1 p2 W u1 u2 e u1 u2 //Zn. (3.1) a superpotential [10] Consider u ∼ 0andun ∼ et/nup1 up2 region, r 2 2 1 2 which should be described by W = exp(−Y ), (2.3) i i=0 ∼ n + t/n p1 p2 W u1 e u1 u2 //Zn. (3.2) ∼ + where twisted chiral fields Yi are periodic Yi Yi n/p 2 = 2 = 2πi and related to Xi by Re[Yi ]=|Xi| . Introducing By introducing the new variables v1 u1 and v2 − p /p := Yi /n t/np2 1 2 the variable ui e , the D-term constraint is e u1 u2. The single valuedness of vi induces S.-J. Sin / Physics Letters B 578 (2004) 215–222 217 n p1 p2 the Zn but single valuedness of u1 and u1 u2 implies that v1,v2 are orbifolded by Zp2 . By substitution, we q1 q2 can express u1 u2 in terms of v1,v2: q1 q2 = Q1 Q2 u1 u2 v1 v2 , (3.3) where (Q1,Q2) = (−p × q/n,q2), (3.4) × = − − with p q (p1q2 p2q1). Notice that map Tp : (q1,q2) → (Q1,Q2) is linear map acting on the inte- grally normalized weight space and can be described by a matrix − − = p2/n p1/n Tp . (3.5) 01 2 Fig. 1. Integrally normalized weight/charge space for C /Zn.It It is working near u2 ∼ 0. It maps (n, 0) → (p2, 0) can be considered as the space of power of local ring elements. It p n n → n → 2 is defined as a two-dimensional torus with size n. u1 and u2 is and (p1,p2) (0,p2), or equivalently, u1 v1 and p1 p2 → p2 located at A(n, 0) and B(0,n), respectively. Under the condensation u1 u2 v2 . of tachyon P , the parallelogram OBDP is mapped to the up-theory One should notice that Q1,Q2 are not integers and OPEA is mapped to the down-theory. Translation parallel to OP in general. However, when both p and q are weight is mapped to horizontal in up theory and vertical in down theory. vectors of elements of orbifold chiral ring, generated by (k ,k ), they are integers. This is because if p = −−→ −−→ 1 2 and OP, and similarly ∆− is the cone spanned by OA (n{lk /n},n{lk /n}), q = (n{jk /n},n{jk /n}), s := −−→ 1 2 1 1 and OP. p × q/n,then Let P be the point (p1/n,p2/n) in charge space that corresponds to a chiral primary that is undergoing s = n{lk1/n}{jk2/n}−n{lk2/n}{jk1/n}∈Z (3.6) condensation, and Q be any charge point (q1/n,q2/n) for any integers n, k, l, j.Fork1 = 1, s =−l[jk2/n]+ and A, B now corresponds to (1, 0) and (0, 1).One j[lk /n]. Especially interesting case will be q = k = − 2 can work out the action of Tp from other point of (1,k2), in which case, we have s =[lk2/n]=(lk2 − view.