Comments on the Fate of unstable orbifolds

We study the localized tachyon condensation in their mirror Landau-Ginzburg picture. We completely determine the decay mode of an unstable orbifold $C^r/Z_n$, $r=1,2,3$ under the condensation of a tachyon with definite R-charge and mass by extending the Vafa's work hep-th/0111105. Here, we give a simple method that works uniformly for all $C^r/Z_n$. For $C^2/Z_n$, where method of toric geometry works, we give a proof of equivalence of our method with toric one. For $C^r/Z_n$ cases, the orbifolds decay into sum of $r$ far separated orbifolds.


Introduction
The study of open string tachyon condensation [1] has led to many interesting consequences including classification of the D-brane charge by K-theory. While the closed string tachyon condensation involve the change of the background spacetime and much more difficult, if we consider the case where tachyons can be localized at the singularity, one may expect the maximal analogy with the open string case. Along this direction, the study of localized tachyon condensation was considered in [2] using the brane probe and renormalization group flow and by many others [3,4,5,6,7,8]. The basic picture is that tachyon condensation induces cascade of decays of the orbifolds to less singular ones until the spacetime supersymmetry is restored.
Therefore the localized tachyon condensation has geometric description as the resolution of the spacetime singularities.
Soon after, Vafa [3] considered the problem in the Landau-Ginzburg (LG) formulation using the Mirror symmetry and confirmed the result of [2]. In [4], the same problem is studied by using the RG flow as deformation of chiral ring and in term of toric geometry. In [3], Vafa showed that, as a consequence of the tachyon condensation, the final point of the process is sum of two orbifold theories which are far from each other but smoothly connected: one located at north and the other at the south poles of blown up P 2 singularity of the orbifold in the limit where the radius of the sphere is infinite. Schematically, we can represent this transition by C 2 /Z n(k 1 ,k 2 ) → C 2 /Z p 1 ( * , * ) ⊕ C 2 /Z p 2 ( * , * ) , (1.1) with yet unknown generators for the daughter theories.
The purpose of this paper is to determine the the decay mode of unstable orbifolds by working out the generators of orbifold action in daughter theories for C r /Z n r = 1, 2, 3. For C 1 /Z n , the transition modes are described in earlier works [2,3,4]. For C 2 /Z n(k 1 ,k 2 ) case, some examples are worked out in [4] using toric geometry and prescription in terms of continued fraction is given. In principle, it can be worked out once numbers are given explicitly. However, that method does not work for C 3 /Z n . Here, we give a simple method that works easily and uniformly for all C r /Z n . For C 2 /Z n , we give a proof of equivalence of our method with toric one. To do this we will need to know how the spectrums of chiral primaries are transformed under the condensation of a specific tachyon.
We begin by a summary of Vafa's work [3] on localized tachyon condensation. The orbifold C r /Z n is defined by the Z n action given by equivalence relation We call (k 1 , · · · , k r ) as the generator of the Z n action. The orbifold can be imbedded into the gauged linear sigma model(GLSM) [9]. The vacuum manifold of the latter is described by the D-term constraints Its t → −∞ limit corresponds to the orbifold and the t → ∞ limit is the O(−n) bundle over the weighted projected space W P k 1 ,...,kr . X 0 direction corresponds to the non-compact fiber of this bundle and t plays role of size of the W P k 1 ,...,kr .
By dualizing this GLSM, we get a LG model with a superpotential[10] Introducing the variable u i := e −Y i /n , the D-term constraint is expressed as e −Y 0 = e t/n i u k i . The periodicity of Y i imposes the identification : u i ∼ e 2πi/n u i which necessitate modding out each u i by Z n . The result is usually described by which describe the mirror Landau-Ginzburg model of the linear sigma model. As a t → −∞ limit, mirror of the orbifold is Since it is not ordinary Landau-Ginzburg theory but 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 potential is always n − 1, regardless of r.
We list some properties of orbifolded LG theory for later use.
The true variable of the theory are Y i not u i related by u i = e −Y i /n . As a consequence, monomial basis of the chiral ring is given by and u p 1 1 u p 2 2 has weight (p 1 , p 2 ) and charge (p 1 /n, p 2 /n).

Fate of the spectrum
For C 2 /Z n(k 1 ,k 2 ) case, if one consider the condensation of tachyon in the l-th twisted sector that corresponds to chiral ring element u p 1 1 u p 2 2 , with p 1 = n{lk 1 /n} and p 2 = n{lk 2 /n}, the theory is given by the super potential Consider u 2 ∼ 0 and u n 2 ∼ e t/n u p 1 1 u p 2 2 region, which should be described by [W ∼ u n 1 + e t/n u p 1 1 u p 2 2 ]//Z n . and v 2 = e t/np 2 u p 1 /p 2 1 u 2 . The single valuedness of v i induces the Z n but single valuedness of u n 1 and u p 1 1 u p 2 2 implies that v 1 , v 2 are orbifolded by Z p 2 . By substitution, we can express u q 1 1 u q 2 2 in terms of v 1 , v 2 : is linear map acting on the integrally normalized weight space and can be described by a matrix It is working near u 2 ∼ 0. It maps (n, 0) → (p 2 , 0) and (p 1 , p 2 ) → (0, p 2 ), or equivalently, One should notice that Q 1 , Q 2 are not integers in general. However, when both p and q are weight vectors of elements of orbifold chiral ring, generated by (k 1 , k 2 ), they are integers. This is because if p = (n{lk 1 /n}, n{lk 2 /n}), q = (n{jk 1 /n}, n{jk 1 /n}), s := p × q/n, then for any integers n, k, l, j. For Especially interesting case will be q = k = (1, k 2 ), in which case, we have s = [lk 2 /n] = (lk 2 − p 2 )/n. Geometrically, s is proportional to the area spanned by two vectors p and q. Therefore it is zero if p and q are parallel.
The R-charges are determined by the marginality condition. In the original theory, u i has R-charge 1/n since u n i has R-charge 1. We express this as R[u n i ] = 1. Therefore R[u p 1 1 u p 2 2 ] = (p 1 + p 2 )/n. charge space is defined by the weight space scaled by 1/n. So we use the same figure 1 to describe it. The diagonal in charge space is the line connecting A(1, 0) and B(0, 1).
Any operator whose R charge is on this diagonal corresponds to the marginal operator. The points below the diagonal correspond to the relevant operators and tachyonic and those above it correspond to the irrelevant operators. When a tachyon, P, is fully condensed, the marginal line is changed from diagonal line AB to line AP or BP. AP gives down-theory and BP gives the up-theory. ∆ + is the cone spanned by OB and OP , and similarly ∆ − is the cone spanned by OA and OP . It is defined as a two dimensional torus with size n. u n 1 and u n 2 is located at A(n,0) and B(0,n) respectively. Under the condensation of tachyon P, the parallelogram OBDP is mapped to the up-theory and OP EA is mapped to the down-theory. Translation parallel to OP is mapped to horizontal in up theory and vertical in down theory.
Let P be the point (p 1 /n, p 2 /n) in charge space that corresponds to a chiral primary that is undergoing condensation, and Q be any charge point (q 1 /n, q 2 /n) and A, B now corresponds to (1, 0) and (0, 1). One can work out the action of T − p from other point of view. If P represent the chiral primary of l-th twisted sector, (p 1 /n, p 2 /n) can be determined by its action on P and (1,0). OnceT − p is decided, we get T − p from the relation, The result of course agrees with the one given by eq. (3.5). Under this mapping, the lower triangle △P OA in figure 1 in charge space is mapped to the entire △BOA, which defines one of theory in the final stage of the tachyon condensation. We call it down-theory. 1 Similarly, by considering u 1 ∼ 0 region, we get the mappingT + p that maps the upper triangle △BOP to △BOA. By the relation T + p = (p 1 /n)T + p we can obtain the mapping in weight space: Notice that T + p leaves all the vertical lines in weight space fixed while T − p leaves horizontal lines invariant. Now we ask: given an operator with q = (q 1 , q 2 ), should we map with T + p or T − p ?
The answer is that we should use the map that gives smaller R-charge. The difference of the R-charge after the mapping is given by where ∆ + is the cone spanned by OB and OP , and similarly ∆ − is the cone spanned by OA and OP . Notice that we are condensing relevant operator p so that p 1 + p 2 < n. The line BP is mapped to the marginal line of a final theory, the up-theory, and the line AP is mapped to that of down-theory. Therefore the emerging picture is following: The parallelogram OBDP 1 Conversely, if we require thatT − p maps △P OA to △BOA,T − p is completely determined. The mapping T − in the integrally normalized weight space is induced by T − = (p 2 /n)T − . The normalization is dictated from the condition that T maps from integer vectors to integer vectors. Finally T − p (n, 0) = (p 2 , 0) and T − p (p 1 , p 2 ) = (0, p 2 ) so that the identification u n spanned by OB and OP is mapped to the up-theory whose weight space size is p 1 . Similarly, the parallelogram OP EA spanned by OP and OA is mapped to the down-theory whose weight space size is p 2 . See figure 2. From eq. (3.6), it is easy to see that chiral ring elements of Mother theory are mapped to chiral ring elements of the daughter theories, under the condensation of a chiral ring element. Any operator q ′ outside these two parallelograms can be parallel translated to inside one of above two parallelograms by the vector OP a few times if necessary. In daughter theories, if q ′ ∈ ∆ + , then T + p q ′ can be translated horizontally by p 1 a few times to a point in the up-theory. Similarly, if q ′ ∈ ∆ − , then T − p q ′ can be translated vertically by p 1 a few times to a point in the up-theory.

Fate of unstable orbifolds
We now can answer to our main question: what are the generators of final theories? We noticed that there are two theories in the final stage. These two theories are described by the difference of the marginal lines in the weight space: extension of BP or that of AP . We call the former as the up-theory, describing u 1 ∼ 0 region, and the latter as down-theory, describing the u 2 ∼ 0 region. In terms of the charge space, up-theory is obtained by mappingT + p : △BOP → △BOA and down-theory is obtained by mappingT − p : △BOP → △BOA. The up-theory is a orbifold C 2 /Z p 1 and the down theory is another orbifold C 2 /Z p 2 . Let k = (k 1 , k 2 ) be the generator of the original theory. Then the generator of the up-theory is given by T + p (k) = (k 1 , p × k/n) and that of the T − p (k) = (−p × k/n, k 2 ). Since (k 1 , k 2 ) ∼ (−k 1 , −k 2 ) as a generator, one can also use T − p (−k) = (p × k/n, −k 2 ) instead of T − p (k). Therefore we can describe the process of condensation of tachyon with charge p = (p 1 , p 2 ) as follows: (4.1) To simplify the notation, we use n(k 1 , k 2 ) for C 2 /Z k 1 ,k 2 and s = p × k/n. Then, Especially interesting cases are those when one of k i is 1.  because we can find k such that for any given l, lk 1 = j mod n and lk 2 = jk mod n for some j.
Sometimes we meet situation where s = 0, where we need more care. For example, if we condensate the generator (1, k) itself, eq.(4.3) predict that n(1, k) → 1(1, 0) ⊕ k(0, k). For the first element 1 (1,0), it is correct since the upper triangle does not contain any tachyon operator. However, for the second element, this can not be true since we have non-trivial 2 So far we proved this fact in the conformal filed theory level before GSO projection.

Equivalence of LG and Toric method in C 2 /Z n
Here we show the equivalence of our description of tachyon decay in mirror LG model with that in toric geometry [11] for the case of C 2 /Z n . We will show that the transition in LG picture with s = p ∧ (1, k)/n, s ′ = p ∧ (k −1 , 1)/n has corresponding description in toric picture  where with integer c, d satisfying cn − dk = 1. 4 Notice that it is assumed that k, n is relatively prime.
The data of weight diagram of LG model can be related to that of toric geometry by a linear map U : LG → T oric and its inverse U −1 : The weight (p 1 , p 2 ) of the condensing tachyon is related to the corresponding toric data n ′ (k ′ ) by which gives p 1 , p 2 : from which s can calculated in terms of toric data: Now, since p 1 (1, s) is trivially equal to n ′ (k ′ ), we only need to show the equivalence of p 2 (−s ′ , 1) with n ′′ (k ′′ ). The question is whether k ′′ ≡ −s ′ mod p 2 or equivalently, (cn ′ − dk ′ ) ≡ (p 1 − k −1 p 2 )/n mod p 2 (4.17) is true or not. Multiplying both sides by k, (cn ′ − dk ′ )k ≡ (kp 1 − k −1 kp 2 )/nmod p 2 . Using cn − dk = 1, s = (kp 1 − p 2 )/n and k −1 k = 1 + an, left hand side is equal to k ′ and right hand side is s − ap 2 . From s = k ′ , we now have proved eq.(4.17). Now −kk ′′ = ks ′ mod p 2 implies k ′′ ≡ −s ′ mod p 2 , provided k and p 2 are relatively prime to each other, completing the proof of our desired result.
Remark: It is interesting to observe that for a general chiral ring element q = (j, n{jk/n}), Uq = (j, k × q/n) =T + k (q/n) = (j, −[jk/n]), which means formally, U coincide with tachyon condensation mapping for generator condensation. This fact directly generalizes to the general (k 1 , k 2 ).

C 3 /Z n
We now describe what happens in C 3 /Z n case. Our method is especially useful in the present case since it applies in this case without any difficulty while toric method does not work here [7]. When a tachyon with weight vector (p 1 , p 2 , p 3 )/n, the mirror LG is described by the superpotential [W = u n 1 + u n 2 + u n 3 + e In this paper, we determined the the decay mode of unstable orbifolds by working out the generators of orbifold action in daughter theories for C r /Z n r = 1, 2, 3. We gave a simple method that works easily and uniformly for all C r /Z n . For C 2 /Z n , we give a proof of equivalence of our method with toric one. Our method trivially reproduced all of known cases worked out by brane probe [2] or toric method [4]. For C 3 /Z n cases, the unstable orbifolds decay into sum of three orbifolds.
Our discussion uses N=2 worldsheet SUSY essentially. It would be very interesting if we can get the the same result without using it.