String Field Theory and the Fuzzy Sphere

We use boundary string field theory to study open string tachyon condensation on a three-sphere closed string background. We consider the closed string background described by $SU(2)_k$ WZW model in the limit of large $k$. We compute the exact tachyon potential and analyse the decay modes.


Introduction
Non-BPS D-brane systems and the process of open string tachyon condensation have been extensively studied in recent years [1]. In one line of research, it has been shown that by using boundary string field theory (BSFT) [2,3,4,5] the study of open string tachyon condensation simplifies considerably [6,7,8]. In this case the system has only one field, the tachyon, and one can compute exact properties such as profiles and tensions of lower dimensional branes. So far, boundary string field theory has been applied in the cases of a flat target space. The inclusion of a nonzero closed string B-field in the BSFT framework has been analysed in [9,10,11], and the inclusion of the open string gauge fields has been studied in [12,13,14].
It is clearly of interest to extend the available methods to consider curved closed string backgrounds. This is of particular interest since it can touch on the nature of the background independence of the boundary string field theory formalism.
In this letter we will make a step into this direction. We will consider as the closed string background the three-sphere S 3 . Closed strings on S 3 are described by SU(2) k WZW model. Tachyon condensation in this background has been discussed in [15]. We will be interested in the boundary string field theory description of D-brane systems wrapping 2-cycles in S 3 . Exact analysis of these systems when the level k is finite is technically difficult. However, in the limit of large k the system simplifies considerably, and allows an exact analysis. We will compute the exact tachyon effective action up to two derivatives in the tachyon field, and in particular the exact tachyon potential. We will then use the results to study the possible decay products.
The letter is organised as follows. In section 2 we briefly review D-branes in SU(2) k WZW model and the structure of their function algebra in the large k limit. In section 3 we will consider the bosonic boundary string field theory approach to D-branes wrapping S 2 ⊂ S 3 . We will perform exact computations in the large k limit and will obtain the exact tachyon potential. We will then discuss the decay products. In section 4 we will consider the supersymmetric string case. The relevant systems are D-branes and anti D-branes wrapping the same 2-cycle or different ones, and systems of non-BPS D-branes obtained from the branes-antibranes systems by projection. We will compute the exact tachyon potential in all these cases and analyse the decay products.
2 SU (2) WZW and the fuzzy sphere Consider closed strings propagating on a three-sphere S 3 with radius R S 3 = √ α ′ k, and a nonzero NSNS 3-form H field proportional to the volume form ω S 3 . The metric on S 3 is given by This system is described by an SU(2) k WZW model.
Symmetry preserving D-branes 1 on S 3 are characterised by boundary conditions on the currents where Solutions to these boundary conditions can be labelled by an index α = 0, 1, ..., k, with each D-brane having a world volume being an SU(2) conjugacy class. Geometrically these conjugacy classes are two-spheres S 2 ⊂ S 3 specified by an angle θ given by where h I = I(I+1) k+2 is the conformal dimension of V [Y I i ](x 1 ) and [:::] are the Clebsch-Gordan coefficients of SU (2). The structure constants c k,α IJK are given by the qdeformed 6J symbols of SU(2) [18], We will consider the system in the limit of large k. In this case the q-deformation parameter goes to one and the structure constants c k,α IJK become the ordinary 6J symbols. We expand around small conformal weights, such that the OPE (5) depends on the insertion points only through the order of the operators To leading order the OPE of the vertex operators then reads The OPE (8) for a brane α is isomorphic to the matrix algebra M α+1 (C) [16] where * is the ordinary matrix product (see appendix A for a detailed discussion). This simplification can be used in order to calculate the expectation values of products of vertex operators via where . Note that since for a stack of N branes on the same world volume we have to add the Chan Paton matrices λ, the effective matrix algebra is enlarged from M α+1 (C) to M N (α+1) (C).

The bosonic BSFT
In this section we will consider the boundary string field theory approach to Dbranes wrapping S 2 ⊂ S 3 in the large k limit. We will consider the bosonic case, compute the exact tachyon potential and analyse the decay modes. In general the D-branes may have extra world volume coordinates in M 6 . This can be taken into account simply by considering matrix valued functions on these coordinates. In the following we will ignore dependence on M 6 .

The tachyon action
Consider the two-dimensional action where S 0 denotes an open plus closed conformally invariant background (the bulk action), and V is a general boundary perturbation. In our case S 0 is a WZW action, while V is ∂D denotes the boundary of the disk D. Following [2,3,4,5] one constructs the partition function on the disk The proposed space-time action S(T ) is defined by where β T are the β-functions of the couplings T I i .
We will work in the large k limit described in the previous section. The β-function for (the SU(2) part of) the tachyon reads where L a , a = 1, 2, 3 are the rescaled angular momentum operators and The string partition function (13) to linear order in α ′ reads where α labels the S 2 conjugacy classes on the S 3 and tr is taken on the (α + 1) × (α + 1) matrices. The normalisation factor is chosen such that for T = 0 one gets the mass of the brane. The mass is given by the noncommutative brane tension. The usual string coupling g s is replaced by the non commutative string coupling G s = gs sin πα k along the lines of [19] times the volume of the brane. a 1 is a numerical constant that will be fixed shortly.
The boundary string field theory relates the string partition function to the space-time action (14) Let us show that by using appropriate field redefinitions the tachyon actions, to all orders in T and to second order in "derivatives" L a , can thus be recast in the form We start with an action for the tachyon of the form where a i (T ) are some polynomials in T . We can use a field redefinition where b i (T ) are some polynomials in T , which generates, to leading order in L a L a , an orbit of actions These orbits of actions thus differ by a kinetic term of the form where C is a constant, which cannot be generated by field redefinitions. Thus, the most general tachyon action is and as we have seen it can be reached by field redefinitions from the action with a constant in front the kinetic term. To fix the constant we can use the consistency of the field equations derived from S(T ) and the β-function [6] β T ∼ δS(T ) δT .

Decay modes
We have seen above that for a D-brane wrapping the α conjugacy class, the tachyon potential is given by For a single D-brane, T is a hermitian (α + 1) × (α + 1) matrix. For N D-branes we have to increase the matrix size to N(α + 1) × N(α + 1). The potential (30) is the exact (string tree level) tachyon potential in the large k limit. It has the same form as for a flat bosonic D-brane and the flat bosonic D-brane with a constant B-field in the non-commutative limit. What differs is the function algebra, which in our case is simply a matrix algebra. In the following we discuss the possible decay modes of the system. For this analysis we can neglect the kinetic term and analyse the potential.
Minima of the tachyon potential satisfy In order to analyse (31) we can diagonalise the tachyon. Solutions of (31) are matrices with zero and infinite eigenvalues. The top of the tachyon potential is when T is the zero matrix. The absolute minimum of the tachyon potential is reached when the tachyon's eigenvalues are all infinite It corresponds to reaching the closed string vacuum. Intermediate decay products correspond to tachyon configurations where not all eigenvalues are infinite Such a configuration corresponds to the decay of the α D-brane to the α−k D-brane. The number of zeros is also the number of D0-branes from which the spherical D2brane is built. In particular the trivial tachyon T = 0 describes the α D2-brane which is made of α + 1 D0-branes.
In the language of perturbations (12) the analysis of decay modes is simple. Since we work in the large k limit we associate for a given tachyon perturbation an (α+1)×(α+1) matrix, using the results in section 2 and the details in the appendix. We then diagonalise this matrix and analyse its eigenvalues as above. This provides us with the information on the decay mode associated with the perturbation and a simple picture of the endpoint of the two-dimensional RG flow. Let us illustrate this with a simple example.
We consider a D2-brane wrapping the α = 1 2-sphere. The tachyon field is a hermitian 2×2-matrix T , which we can expand in terms of the matrix representation of the α = 1 matrices {Y J j } ab given by (66). When T has two positive eigenvalues the system condenses to the vacuum. This happens, for instance, if T = iY 0 0 . To get an α = 0 brane at the endpoint of the condensation we can take a tachyon configuration of the form T = Y 1 1 + Y 1 −1 − i √ 2Y 0 0 which has eigenvalues zero and one.
We note that this simple picture is really a feature of the large k limit. For finite k, such perturbations which are typically not free, are much harder to analyse. In particular, we cannot simply look for the vanishing locus of the tachyon profile expressed via the spherical harmonics in order to analyse the decay product, as done for the free perturbations used in the flat target space case. This can be seen from the above example. The spherical harmonic iY 1 0 (θ, φ) has zeros at the poles ψ = 0, π. Thus one might suspect the α = 1 brane to decay into two separated α = 0 branes on the poles. However, the matrix iY 1 0 has nonzero eigenvalues ±1/ √ 2. It seems plausible to assume that for a given tachyon perturbation, the endpoint of the two-dimensional RG flow for finite k will not differ from the large k one. In such a case, while the finite k RG-flow is complicated to analyse, at least the end-points of the flow have a simple picture as described by the large large k limit above.

Supersymmetric BSFT
In this section we will consider the supersymmetric boundary string field theory approach to DD-branes and non-BPS branes wrapping S 2 ⊂ S 3 in the large k limit.

The tachyon action
The boundary vertex operators for N α branes and N β antibranes can be constructed from [20,21], with, where ψ a and j a form the supercurrent The tachyon fields T are complex (T = 1 2 (T 1 + iT 2 )) products of Y J j s and Chan Paton matrices of dimensions N α × N β . The superconnection structure appears in(34), as in [22,23,13,14,24].
The operators L a T act as the generators of the rotation group on the respective branes. For the tachyon field they are defined as . This is natural as the tachyons transform in the bifundamental representation of the rotation groups on the respective branes α, β. Note that the Chan Paton factors are not changed under the rotation. The normalisation is such that the OPEs read For later use we introduce M 0 and M 1 for the θ independent and the θ dependent part of M = M 0 + θM 1 . M 0 (M 1 ) is related to the picture (−1) (picture (0)) vertex operators of the sum of the tachyon and the gauge field.
The world sheet action reads where the boundary fieldsη,η have to be integrated over in the path integral. We use the notation dτ = dτ dθ, D = ∂ θ + θ∂ τ and the fermionic superfieldsη a = η a + θχ a .
The integration over the auxiliary fields χ rearranges the boundary action, such that one finds the path ordered product of M 2 0 + M 1 , For zero gauge fields A 1 , A 2 one finds From this (43) the tachyon potential, which is the leading order term in the 1/k expansion, can be read. First order corrections have two origins: corrections from the OPE (5) and contributions from the off diagonal entries in (43).
The kinetic terms are understood to be ordered in a symmetric way. The constants c 1 , c 2 will be fixed below. The BSFT action S of the super string is conjectured to be [8] (see also [25,26]) Consistency with the β function can be used to fix the coefficients c 1 = 1/4, c 2 = −α ′ /2. This is because to quadratic order in the tachyon the action has to reproduce the mass formula −α ′ m 2 = J(J + 1)/k − 1/2. The kinetic term gives The constant c 2 is fixed such that the quadratic term in the tachyon potential produces the − 1 2 in the open string mass formula. Finally, the tachyon action up to second order in 1/k reads Next we will derive the tachyon action for a N α non-BPS branes on a α sphere S 2 ⊂ S 3 . To this end one gauges the (−1) F L symmetry of the system of N α brane anti brane pairs [1]. This equates the gauge fields A α = A α ′ and selects the tachyon field to be real T = T † .
The boundary perturbation simplifies to Analogous steps as above lead to the tachyon action

Decay modes
We can distinguish three different systems: (1) Coinciding brane-antibrane system wrapping the same conjugacy class α, which we denote by D(α)D(α).
Consider the decay modes of these systems.

D(α)D(α) system
The tachyon energy of the D2D2 system wrapping the α 2-cycle is given by As in the bosonic case we are interested in the extrema of this potential, they fulfil the equations The tachyon can be diagonalised by two unitary matrices U, W with UT W † . The global minimum corresponds to tachyon perturbations with nonzero eigenvalues that become infinite at the endpoint of the perturbation Intermediate decay products of D2D2 system wrapping a 2-cycle β < α correspond to the tachyon matrices with some zero eigenvalues. They are bi-unitary transformations of T = λ diag{t 1 , .., t k , 0, .., 0} α+1 , λ → ∞, t i = 0 .
Tachyons of the form (55) correspond to a perturbation from a D2D2 system wrapping a 2-cycle α to a D2D2 system wrapping a 2-cycle β = α − k. One can also interpret the zeros of the tachyon matrix as counting the number of D0D0 constituent states of the condensed D2D2 system. As usual the zero tachyon corresponds to the top of the potential with no condensation.

D(α)D(β) system
We consider a brane wrapping a 2-cycle α and an antibrane wrapping a 2-cycle β. Without loss of generality we assume that α > β and that the branes are concentric. The energy of the system in this case reads E(α, β, T, T † ) = C α tr α e − 1 4 T T † + C β tr β e − 1 For extrema the conditions have to hold. We can think of the tachyon T and its conjugate T † as maps T : E β → E α and T † : E α → E β , where E α and E β are the vector bundles on the α and β 2-cycles corresponding to the brane and antibrane respectively. The relevant operators for the discussion are T T † and T † T . In our case, they are matrices of size (α+1)×(α+1) and (β +1)×(β +1). Their zeros determine the number of D0-branes andD0-branes constituents, respectively. Thus, the index Index(E, T ) = dim Ker T T † − dim Ker T † T , counts the net D0-brane charge.
As to the decay modes, the analysis is as before. A matrix T T † of the form T T † = λ diag{t 1 , .., t k , 0, ..., 0} α+1 , λ → ∞, t i = 0 , corresponds to a perturbation that will reduce the 2-cycle α to a 2-cycle α − k. A T † T matrix of the form corresponds to a perturbation that will reduce the 2-cycle β to a 2-cycle β − k.

Non-BPS branes
where T = T † . The analysis of the decay modes is as before. A tachyon of the form initiates a flow to a non-BPS branes wrapping the α − k 2-cycle.