Comments on the Fate of Unstable Orbifolds ✩

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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 integrally 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. If P represent the chiral primary of lth twisted p2)/n. Geometrically, s is proportional to the area sector, (p1/n,p2/n) := ({lk1/n}, {lk1/n}). Near u2 ∼ spanned by two vectors p and q. Therefore it is zero 0 region, the marginality condition is changed to if p and q are parallel. [ p1 p2 ]= [ n]= R u1 u2 1,R u1 1. In terms of new variable The R-charges are determined by the marginality [ p2 ]= R vi 1. The linear transformation condition. In the original theory, ui has R-charge 1/n n [ n]= − → since ui has R-charge 1. We express this as R ui 1. Tp : (q1/n,q2/n) (Q1/p2,Q2/p2), (3.7) Therefore R[up1 up2 ]=(p + p )/n. charge space 1 2 1 2 can be determined by its action on P and (1, 0).Once is defined by the weight space scaled by 1/n.So T− is decided, we get T − from the relation, T− = we use the same Fig. 1 to describe it. The diagonal p p p n T −. The result of course agrees with the one given in charge space is the line connecting A(1, 0) and p2 p B(0, 1). Any operator whose R-charge is on this by Eq. (3.5). Under this mapping, the lower triangle diagonal corresponds to the marginal operator. The POA in Fig. 1 in charge space is mapped to the entire BOA, which defines one of theory in the final stage points below the diagonal correspond to the relevant 2 operators and tachyonic and those above it correspond of the tachyon condensation. We call it down-theory. to the irrelevant operators. When a tachyon, P , is fully 2 − condensed, the marginal line is changed from diagonal Conversely, if we require that Tp maps POA to BOA, − − line AB to line AP or BP. AP gives down-theory and−−→ Tp is completely determined. The mapping T in the integrally − − BP gives the up-theory. ∆+ is the cone spanned by OB normalized weight space is induced by T = (p2/n)T .The 218 S.-J. Sin / Physics Letters B 578 (2004) 215–222

Fig. 2. Charges of 11(1, 3) (left) and 10(1, 3) (right) in Weight space.

Similarly, by considering u1 ∼ 0 region, we get the line AP is mapped to that of down-theory. Therefore + mapping Tp that maps the upper triangle BOP to the emerging picture−−→ is following:−−→ the parallelogram + = + OBDP spanned by OB and OP is mapped to the BOA. By the relation Tp (p1/n)Tp we can obtain the mapping in weight space: up-theory whose weight space size is−−→p1. Similarly,−−→ the parallelogram OPEA spanned by OP and OA is + 10q1 q1 T q = = . mapped to the down-theory whose weight space size p − × p2/n p1/n q2 p q/n is p2. See Fig. 2. From Eq. (3.6), it is easy to see that (3.8) chiral ring elements of Mother theory are mapped to + chiral ring elements of the daughter theories, under the Notice that Tp leaves all the vertical lines in weight space ﬁxed while T − leaves horizontal lines invariant. condensation of a chiral ring element. Any operator p q outside these two parallelograms can be parallel Now we ask: given an operator with q = (q1,q2), + − translated to inside one of above two parallelograms should we map with Tp or Tp ? The answer is that we −−→ should use the map that gives smaller R-charge. The by the vector OP a few times if necessary. In daughter ∈ + difference of the R-charge after the mapping is given theories, if q ∆+,thenTp q can be translated by horizontally by p1 a few times to a point in the ∈ − up-theory. Similarly, if q ∆−,thenTp q can be δ := R T +q − R T −q translated vertically by p a few times to a point in p p 1 × ∈ the up-theory. = p q + − < 0ifq ∆+, (p1 p2 n) ∈ (3.9) np1p2 > 0ifq ∆−, −−→ −−→ + where ∆ is the cone spanned by OB −−→and OP,and−−→ 4. Fate of unstable orbifolds similarly ∆− is the cone spanned by OA and OP. 2 Notice that we are condensing relevant operator p 4.1. C /Zn so that p1 + p2

Table 1 All possible tachyon condensation process in 11(1, 3) model. We should consider only the processes given by relevant operators, namely those with n − (p1 + p2)>0, otherwise it is a process by an irrelevant operator which disappears in the infrared limit

j(p1,p2)G=[3j/11] n − (p1 + p2) Process 1 (1, 3) 0711(1, 3) → 1(1, 0) ⊕ 3(1, 1) 2 (2, 6) 0311(1, 3) → 2(1, 0) ⊕ 6(2, 1) 3 (3, 9) 0 −1 irrelevant process 4 (4, 1) 1611(1, 3) → 4(1, 1) ⊕ 1(0, 1) 5 (5, 4) 1211(1, 3) → 5(1, 1) ⊕ 4(1, 1) 6 (6, 7) 1 −2 irrelevant process 7 (7, 10) 1 −6 irrelevant process 8 (8, 2) 2111(1, 3) → 8(1, 2) ⊕ 2(0, 1) 9 (9, 5) 2 −3 irrelevant process 10 (10, 8) 2 −7 irrelevant process

2 Eqs. (4.3), (4.7) are the main formula of this section. 4.2. Equivalence of LG and toric method in C /Zn When one of s,s is 0 and the other is not, we should use the non-zero one. For example, when the Here we show the equivalence of our description of condensing operator is of the form j(k−1, 1), s = 0 tachyon decay in mirror LG model with that in toric 2 and it is better to use p2(−s,k) for the exactly same geometry [11] for the case of C /Zn. We will show reason as we use p2(−s , 1) when s = 0. When ss = 0 that the transition in LG picture 4 two are equivalent in conformal ﬁeld theory level. n(1,k) −→ p1(1,s)⊕ p2(−s , 1), (4.10) We give a few examples below. If we condensate (p1,p2) an operator with p = j(1,k), its band number G := = ∧ = ∧ −1 [j/n]+[jk/n]=0ands = 0. However, s = j(1,k)∧ with s p (1,k)/n, s p (k , 1)/n has corre- (k−1, 1) =−aj = 0 unless k = 1(or,a = 0). The sponding description in toric picture transition is described as n(k) −→ n (k ) ⊕ n (k ), (4.11) (n,−k) n(1,k) −→ j(1, 0) ⊕ jk(ja,1). (4.8) j(1,k) where = = More explicitly, for p (2, 6) in 11(1, 3), j 2, n = kn − nk and − k = cn − dk (4.12) s = 0, k = 3, k−1 = 4, 4 · 3 = 11 · 1 + 1 hence a = 1 5 and s =−2sothat with integer c,d satisfying cn − dk = 1. Notice that it is assumed that k,n is relatively prime. 11(1, 3) −→ 2(1, 0) ⊕ 6(2, 1). (4.9) The data of weight diagram of LG model can be (2,6) related to that of toric geometry by a linear map Notice that 6(2, 1) contains an operator (0, 3) so that U :LG→ Toric and its inverse U −1: this is a reducible orbifold. Even in the case we start with irreducible orbifold, we can get reducible = 10 −1 = 10 U − ,U . (4.13) orbifold as a result of tachyon condensation. This k/n 1/n kn happen if and only if there is an operator sitting on The weight (p1,p2) of the condensing tachyon is the line which connect (0, 0) and the condensing one, related to the corresponding toric data n(k) by p. We tabulated all possible tachyon condensation p n n processes for model 11(1, 3) and 10(1, 3) in Tables 1 1 = U −1 = , − − (4.14) and 2, respectively. p2 k kn nk

5 + + 4 For string theory level, two prescriptions are different if s and If (c, d) is a solution of this equation, (c k m, d n m) is − → − + s does not have the same G-parity (even or oddness). we need to use also a solution. The result is the (n , k ) (n , k n m) which is just an SL Z transformation 10which corresponds the one that has the same parity as that of k. This will be discussed 2 m 1 further in later section. to a holomorphic coordinate transformation of a toric variety. S.-J. Sin / Physics Letters B 578 (2004) 215–222 221

Table 2 All possible localized tachyon condensation in model 10(1, 3) j(p1,p2)G=[3j/10] n − (p1 + p2) Process 1 (1, 3) 0610(1, 3) → 1(1, 0) ⊕ 3(0, 1) 2 (2, 6) 0210(1, 3) → 2(1, 0) ⊕ 6(0, 1) 3 (3, 9) 0 −2 irrelevant process 4 (4, 2) 1410(1, 3) → 4(1, 1) ⊕ 2(1, 1) 5 (5, 5) 1010(1, 3) → 5(1, 1) ⊕ 5(1, 2) 6 (6, 8) 1 −4 irrelevant process 7 (7, 1) 2210(1, 3) → 7(1, 2) ⊕ 1(0, 1) 8 (8, 4) 2 −2 irrelevant process 9 (9, 7) 2 −6 irrelevant process

which gives p1,p2: with weight vector (p1,p2,p3)/n, the mirror LG is described by the superpotential p1 = n ,p2 = kn − nk , (4.15)

t p p p = n + n + n + n 1 2 3 Z 2 from which s can calculated in terms of toric data: W u1 u2 u3 e u1 u2 u3 //( n) . (4.18) = ∧ s p (1,k)/n By considering uj ∼ 0regionforj = 1, 2, 3, we get (j) = (n ,kn − nk ) ∧ (1,k)/n= k . (4.16) the tachyon maps Tp ’s [12] given by Now, since p1(1,s) is trivially equal to n (k ),we − 100 only need to show the equivalence of p2( s , 1) with (1) = − Tp p2/n p1/n 0 , n (k ). The question is whether k ≡−s modp2 or −p3/n 0 p1/n equivalently, p2/n −p1/n 0 − ≡ − −1 (2) = (cn dk ) p1 k p2 /nmodp2 (4.17) Tp 010, − 0 −p3/n p2/n is true or not. Multiplying both sides by k, (cn −1 − dk )k ≡ (kp1 − k kp2)/n modp2.Usingcn − p3/n 0 p1/n −1 (3) = − dk = 1, s = (kp1 − p2)/n and k k = 1 + an,left- Tp 0 p3/n p2/n , (4.19) hand side is equal to k and right-hand side is s − ap2. 00 1 = From s k , we now have proved Eq. (4.17). Now ± 2 which play similar role of T in C /Zn.Letk = −kk = ks modp2 implies k ≡−s modp2,pro- p (k1,k2,k3) be the generator of mother theory. Then the vided k and p2 are relatively prime to each other, com- generator of the daughter theories are given by k(j) := pleting the proof of our desired result. (j) Tp k, j = 1, 2, 3. Namely the orbifold transition rule Remark. It is interesting to observe that for a general is given by chiral ring element q = (j, n{jk/n}), Uq = (j, k × + q/n) = T (q/n) = (j, −[jk/n]), which means for- n(k1,k2,k3) −→ p1(k1,s12,s13) k (p ,p ,p ) mally, U coincide with tachyon condensation mapping 1 2 3 ⊕ for generator condensation. This fact directly general- p2(s21,k2,s23) izes to the general (k ,k ). 1 2 ⊕ p3(s31,s32,k3), (4.20)

3 4.3. C /Zn where sji = pj ki − pi kj . Notice that there exists a simple formula 3 We now describe what happens in C /Zn case. Our method is especially useful in the present case (j) = + ki kiδji sji. (4.21) since it applies in this case without any difﬁculty while toric method does not work here [7]. When a tachyon This is one of the main result of this Letter. 222 S.-J. Sin / Physics Letters B 578 (2004) 215–222

5. Conclusion References

In this Letter, we determined the decay mode of [1] A. Sen, hep-th/9904207, and references therein; unstable orbifolds by working out the generators of For review, see: A. Sen, JHEP 9806 (1998) 007, hep- r = th/9803194, and references therein. orbifold action in daughter theories for C /Zn r [2] A. Adams, J. Polchinski, E. Silverstein, JHEP 0110 (2001) 029, 1, 2, 3. We gave a simple method that works easily hep-th/0108075. r 2 and uniformly for all C /Zn.ForC /Zn,wegivea [3] C. Vafa, hep-th/0111051. proof of equivalence of our method with toric one. Our [4] J.A. Harvey, D. Kutasov, E.J. Martinec, G. Moore, hep- method trivially reproduced all of known cases worked th/0111154. 3 [5] A. Dabholkar, C. Vafa, hep-th/0111155. out by brane probe [2] or toric method [4]. For C /Zn [6] A. Dabholkar, Phys. Rev. Lett. 88 (2002) 091301, hep- cases, the unstable orbifolds decay into sum of three th/0111004. orbifolds. [7] T. Sarkar, Nucl. Phys. B 648 (2003) 497, hep-th/0206109. Our discussion uses N = 2 worldsheet SUSY [8] Y. Michishita, P. Yi, Phys. Rev. D 65 (2002) 086006, hep- essentially. It would be very interesting if we can get th/0111199; S. Nam, S.-J. Sin, hep-th/0201132; the same result without using it. S.-J. Sin, Nucl. Phys. B 637 (2002) 395, hep-th/0202097; R. Rabadan, J. Simon, JHEP 0205 (2002) 045, hep- th/0203243; A.M. Uranga, hep-th/0204079; Acknowledgements T. Suyama, JHEP 0210 (2002) 051, hep-th/0210054; J.L. Barbon, E. Rabinovici, hep-th/0211212; I would like to thank Allan Adams, Keshav Das- Y.H. He, hep-th/0301162; Gupta and Eva Silverstein for discussions. This work E.J. Martinec, G. Moore, hep-th/0212059. [9] E. Witten, Nucl. Phys. B 403 (1993) 159, hep-th/9301042. is partially supported by the Korea Research Founda- [10] K. Hori, C. Vafa, hep-th/0002222. tion Grant (KRF-2002-013-D00030) and by KOSEF [11] E.J. Martinec, hep-th/0210231. Grant 1999-2-112-003-5. [12] S. Lee, S.-J. Sin, hep-th/0308029.