Closed string tachyon potential and $tt^*$ equation

Recently Dabholkar and Vafa proposed that closed string tachyon potential for non-supersymmetric orbifold $\C/\Z_3$ in terms of the solution of a $tt^*$ equation. We extend this result to $\C^2/\Z_n$ for $n=3,4,5$. Interestingly, the tachyon potentials for $n=3$ and 4 are still given in terms of the solutions of Painleve III type equation that appeared in the study of $\C^1/\Z_3$ with different boundary conditions. For $\C^2/\Z_5$ case, governing equations are of generalized Toda type. The potential is monotonically decreasing function of RG flow.


Introduction
The study of localized tachyon condensation [1,2,3,4,5] has been considered with many interesting developments. 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 a geometric description as the resolution of the spacetime singularities.
Following the line of Vafa's reformulation of the problem in terms of Mirror Landau-Ginzburg theory, we worked out the detailed analysis on the fate of spectrum and the background geometry under the tachyon condensation as well as the question of what is the analogue of c-theorem with the GSO-projection in a series of papers [7,8,9]. In all these works, superconformal invariance was used very heavily so that we were working at the string theory on-shell level.
On the other hand, recently, Dabholkar and Vafa [4] proposed that the tachyon potential is given by the maximal charge. Strictly speaking, the U(1) charge in the question is defined only on the conformal point not on the off-shell. On the other hand, the decay process considered as a renormalization group flow is an off-shell process and one needs to extend the concept of charge in order to define the tachyon potential. The way to extend the charge is to consider it as a semi-index which is contributed by BPS objects only [10]. These are not just topological since it has anti-holomorphic dependence as well as holomorphic dependence on the deformation parameters. The way to calculate this quantity is to show that it satisfies tt * equations [12] and solve it if possible. In [4], the process where C/Z 3 decays to C 1 was discussed.
In this paper we extend the result of [4] to C 2 /Z n for n = 3, 4, 5. Interestingly, the tachyon potential for n = 3 and 4 is still given in terms of the solutions of Painleve III type that appeared in the study of C 1 /Z 3 with different boundary conditions. For C 2 /Z 5 the governing equation is that of generalized Toda type. The potential is monotonically decreasing function of RG flow, therefore expected so as function of real time as well. For LG model associated to C 2 /Z 5 without orbifold projection, tt * equation contains the Bullough-Dodd equation, whose solution is known [14].
In this letter, we do not attempt to give all necessary background. For the set up of tachyon condensation in terms of mirror symmetry, see [2,7]. For the application of tt * to the tachyon condensation, see [4].
In this section we study the tachyon condensation in C 2 /Z n with Landau-Ginzburg (LG) description by calculating and solving the corresponding tt * equations. First we consider the simplest of all C 2 /Z n , namely n = 3. The mirror of this is a LG whose superpotential is given by with an imposition of an orbifold constraint.
Before we consider the orbifolded LG theory, we work out the tt * equations for LG theory without orbifolding for later interests. The fundamental variables are not x, y but log x, log y [4,11] so that the chiral ring consists of from which charges of the elements can be read off to give NS-charges The R-charges in the topological strings are related to those in NS by the spectral flow For C 2 /Z 3 , we have n = 2 so we get This superpotential has symmetries x → ωx, y → ω 2 y; and x → ω 2 x, y → ωy, (2.6) where ω = e The topological metric η ij is given by residue paring with superpotential W , which in this case is calculated to be From the reality constraint (2.10) The chiral ring coefficient are defined by φ i φ j = C k ij φ k and if we denote the matrix for the multiplication by xy by C t , then (2.11) Putting all these into tt * equation In terms of Y and z = |ζ| the last equation can be written as which is already known as Painleve III equation. We can further rewrite the eq.(2.16) as the sinh-Gordon equation by introducing u by u = 2 log y. Now, since we are interested in the scaling behavior of the solutions, we look at the dependence on z = |ζ| of the eq.(2.18), where the prime denote the derivative in z. The solution to this is well known [13]. The real solutions are classified by their asymptotic behavior in z → 0 In our case, the regularity of a 11 requires The precise form of regular solution can be written in terms of a convergent expansion with θ 2n+1 ≡ θ 1 and r = −4/3.
In order to define the charge matrix, we need to look at the scaling behavior of the system.
The scaling z → λz induces d 2 zd 2 θ → λ d 2 zd 2 θ, which is equivalent to the field redefinition and coupling change such that W → λW . For the given superpotential we can identify the necessary field redefinitions and coupling change as by which W can be written as In this way changing t is equivalent to the changing the scale. The metric components in old and new basis are related by with n = 3. The off-shell 'charge' of the system was defined [12] as where τ = log λ and the metric components in use are inx,ỹ basis, namely b ij 's. From the regularity condition of the metric in x, y basis at t = 0, one can verify that the charge at the starting point t = λ = 0 is encoded correctly to give the result listed in eq.(2.5).
Orbifolded LG model: In this case, the chiral ring generated by xy: {xy, x 2 y 2 } and NS-charges are { 2 3 , 4 3 }. The metric and charges can be obtained from the previous subsection by simply discarding a 12 and a 21 .
Following [4], the tachyon potential is proposed to be where we identified ζ = λ by eqs. The topological metric and metric from sec.3.1 are given by Then, the reality condition gives a 22 = 1, a 32 = 1/a 12 , a 13 = 1/a 31 , a 23 = 1/a 21 , a 33 = 1/a 11 + |t| 4 a 11 /4, b = t 2 a 11 /4. (3.33) C t can also be calculated easily and given by which determines the value r = − 2k k+2 . Now from eq.(2.28) the components of charge matrix is given by Using a ij = |λ| (i+j)/2 b ij , a −1 ij ∼ y l for some l and y ∼ z k/(k+2) e u/2 with identification z = |λ|, Notice that (ij) is not the matrix index but the vector index. Using eq.(3.41), the value of the charge at t = 0 is which confirms that we are in the right track. Now we apply this result to our system. i.e, a 12 = a 21 . This supports the exchange symmetry a ij = a ji . With this, the eq.(3.36, 3.38) are the case of m = 1, k = 2; by setting a −1 12 = y = |t|e u/2 and ζ = t 2 /2, we get sinh-Gordon equation. l = 1 lead us to q 12 (z) = q 21 (z) = 1 4 z du(z) dz . For a 31 , by z = t 2 and y = a −1 3 we get sinh-Gordon and the solution is y = e u/2 . It is easy to see q 31 (z) = 1 4 z du(z) dz . Notice that q 31 (0) = q 13 (0) = 0. The monotonicity of the charge in t suggests that q 31 (z) = 0. In fact the exchange symmetry a 31 = a 31 and the reality condition a 31 = 1/a 31 sets a 31 = 1.

Mirror of C 2 /Z 4 :
So far, we have been considering the LG model without orbifolding action. To consider the mirror of C 2 /Z 4 with generator xy, we need to consider a ii i = 1, 2, 3. Since a 22 = 1 and a 33 is given by a 11 , we only need to consider the equation for a 11 , which is given by eq.(3.35).

C 2 /Z 5
Here again, we first analyze the general LG model and at the end we comments on the orbifolded case. The superpotential is given by The chiral ring is give by x 4 − ty = 0 and y 4 − tx = 0. We order the basis by dictionary order in charge pair (i, j): R = {xy, x 2 y, x 3 y, x 4 y, xy 2 , x 2 y 2 , x 3 y 2 , x 4 y 2 , xy 3 , x 2 y 3 , x 3 y 3 , x 4 y 3 , xy 4 , x 2 y 4 , x 3 y 4 , x 4 y 4 }. The η ij can be readily written and we avoid to writing it down. The metric g ij has 16 diagonal components which we denote by a ij = x¯iyj|x i y j , i, j = 1, 2, 3, 4, and two non-vanishing off diagonal elements b,b as before. The coupling matrix C t in this basis is given by  Then, both eq.(4.9) eq.(4.10) are reduced to which are known as Bullough-Dodd equation which is a degenerate Painleve III, which also appears in the case The Bullough-Dodd equation has been studied extensively and the properties of the asymptotically regular solutions were given in [14]. The solutions are parametrized by four complex numbers g 1 , g 2 , g 3 , and s satisfying The asymptotic forms are given by τ → ∞; e u ∼ 1 + 3 π s 2 (3τ ) − 1 4 e −2 √ 3τ , g 1 = g 2 = 0, g 3 = 1, .

(4.22)
Regularity of the metric as τ → 0 can fix s. Notice that in terms of q ij and λ(= ζ), and if we look at the |λ| dependence only, the tachyon potential can be identified as V = 2Q max = −ζ∂ ζ q 11 (ζ). (4.28) We expect that this is monotonically decreasing from the value 5/3 at t = 0 to 0 at t → ∞.
So far, the mathematical literature on the solution to the equation is not available and the qualitative behavior we suggested above is from physical intuition that in the final stage of tachyon condensation there is no nontrivial chiral primaries with charge other than 0.

Discussion
In this paper, we calculated tt * equations for C 2 /Z n with n=3,4,5. In n = 3, 4 cases, they reduce to a Painleve III equation with different boundary conditions. In n = 5 case, they reduce to a simple Toda system whose explicit solutions are not known yet. Non-orbifolded LG model associated to n = 5 case involves a Bullough-Dodd equation.
As a limitation of this paper, we mention that we considered the string theories without GSO projection only. Since GSO projection does not provide a supersymmetry immediately in the orbifold background, there is not much point on restricting ourselves to GSO projected theory.
According to the rule given in [9], xy term considered in this paper is projected out for type II, and we need to consider the deformation by higher operator, which result in highly non-trivial equations due to the algebraic complexity of reality condition. Another very interesting case is the one where the daughter theory is also an orbifold. This also results in a highly non-trivial equations even for C 1 /Z n background. We wish to report on these issues in later publications.