Remark About T-duality of Dp-Branes

This note is devoted to the analysis of T-duality of Dp-brane when we perform T-duality along directions that are transverse to world-volume of Dp-brane.


Introduction and Summary
It is well known that T-duality is central property of string theory, for review, see for example [1]. Generally, if we consider string sigma model in the background with metric G M N and NSNS two form B M N together with dilation φ and this background possesses an isometry along d−directions we find that it is equivalent to string sigma model in T-dual background with dual fieldsG M N ,B M N andφ that are related to the original fields by famous Buscher's rules [2,3] for generalization to more directions, see for example [4,5].
It is well known that string theories also contain another higher dimensional objects that transform non-trivially under T-duality. In this note we focus on Dpbranes [6,7], for more recent review, see [10]. Originally Dp-brane was defined with the open string description where the string embedding coordinates obey p + 1 Neumann boundary conditions and 9−(p+1)−Dirrichlet ones at the boundary of the string world-sheet [6] 2 . It was also shown by Polchinski that Dp-brane transforms into D(p+1)-brane when T-duality is performed along direction transverse to worldvolume of Dp-brane and Dp-brane transforms to D(p-1)-brane in case when Tduality is performed along direction that Dp-brane wraps. In other words Dp-brane transforms with very specific way under T-duality transformations.
On the other hand it is remarkable that many aspects of Dp-brane dynamics can be described by its low energy effective action which is famous Dirac-Born-Infeld action [6]. Then one can ask the question whether this description of Dpbrane dynamics could give correct description of T-duality transformation of Dpbrane. This situation is relatively straightforward when we perform T-duality along directions which Dp-brane wraps. This property is known as covariance of Dpbrane action under T-duality transformations as was previously studied in [8,9]. We generalize this approach to the T-duality along more longitudinal directions in the next section.
It is important to stress that in order to show full covariance of Dp-brane action with respect to T-duality transformation we should also study how Dp-brane effective action changes when we perform T-duality along transverse directions to its world-volume. The goal of this paper is to perform such an analysis. Our approach is based on previous works that consider description of N Dp-branes on the circle [12], for review see [11]. It was shown there that such a configuration should be described by infinite number of Dp-branes on R which is covering space of S 1 when we impose appropriate quotient conditions [11]. Since this description was performed in the context of Matrix theory [13] the low energy effective action describing dynamics of N− Dp-branes was Super Yang-Mills theory (SYM) defined on p + 1 dimensional world-volume. Then it was shown in [12] that this theory transforms under T-duality into (SYM) defined on p + 2 dimensional world-volume in T-dual background.
The goal of this paper is to generalize this analysis to the case of full DBI action for Dp-brane in the general background when we study T-duality along transverse directions. It is well known that such a T-duality can be defined when the target space-time fields do not depend on these coordinates explicitly. Since, following previous works, we should consider generalization of DBI action that describes infinite number of Dp-branes in covering space. Such an action is non-abelian generalization of DBI action that was introduced in [14]. Then we follow very nice analysis performed in [15]. Explicitly, we introduce quotient conditions and solve them in the same way as in [15]. We show that non-abelian action for infinite number of Dp-branes transforms to the action for D(p+d)-brane where d− is number of T-dual directions in the T-dual background where T-dual background fields are related to the original one by generalized Buscher's rules [4,5].
Let us outline our results. We study how Dp-brane transforms under T-duality we consider T-duality either along longitudinal or transverse directions to Dp-brane' world-volume. We show that in the first case it transforms do D(p-d)-brane while in the second one it transforms to D(p+d)-brane when all background fields transform according to generalized Buscher's rules. This fact nicely shows covariance of Dpbrane under T-duality transformations.
This paper is organized as follows. In the next section (2) we introduce Dp-brane action and study T-duality along longitudinal directions. Then in section (3) we consider T-duality along directions transverse to Dp-brane world-volume.

Longitudinal T-duality
In this section we review T-duality transformation of Dp-brane when we perform T-duality along d− longitudinal directions. Explicitly, let us consider DBI action in the general background with the metric G M N , B M N and dilaton φ. This action has the form where where we also defined pull back of G M N and B M N defined as where ξ α , α = 0, 1, . . . , p label world-volume directions of Dp-brane and where x M , M = 0, 1, . . . , 9 parametrize embedding of DBI action in the target space-time. Now we would like to perform T-duality along last d−directions when we presume that there are directions which Dp-brane wraps. The fact that these directions are longitudinal mean that Dp-brane world-volume coordinates coincide with the target space ones. Explicitly we have Then we presume that all world-volume fields do not depend on ξα only wherê where Eαβ = E µν ∂αx µ ∂βx ν , µ, ν = 0, 1, . . . , 9 − d. Then performing standard manipulation with determinant we obtain and whereẼ mn is inverse to E mn in the sense thatẼ mn E nk = δ k m . As the next step Then we can write that can be interpreted as an embedding of D(p-d)-brane in T-dual background with the background fieldsẼ Explicitly, the D(p-d)-brane action in T-dual background has the form wherẽ and where we defined tension for D(p-d)-brane in the form Note that the transformation rules for T-dual fields given in (9) coincide with the results derived previously in [4,5] and which are now derived independently using covariance of Dp-brane under T-duality transformations. However in order to see consistency of T-duality covariance of Dp-branes we should also consider opposite situation when we consider Dp-brane in general background and perform T-duality along directions that are transverse to the worldvolume of Dp-brane.

Transverse T-duality
In this section we consider opposite situation when we study Dp-brane in the background that has isometry along d−directions in the transverse space to Dp-brane world-volume. The best way how to describe such a Dp-brane in to consider infinite number of Dp-branes on the covering space of torus T d which is R d and impose appropriate quotient conditions. Further, we should also consider appropriate action for N Dp-branes which is famous Mayer's non-abelian action [14] where i, j, k, l, m, n, r, s, t, . . . = p + 1, . . . , 9 label directions transverse to the worldvolume of N Dp-branes. Note that the location of N− Dp-branes in the transverse space is determined by N × N Hermitean matrices Φ m , m = p + 1, . . . , 9 and all background fields depend on them so as for example E αβ (Φ) and so on. We use convention where Φ m are Hermitean matrices and field strength F αβ is defined as where A α is N × N Hermitian matrix corresponding to non-abelian gauge field. Finally, P [E αβ ] is a pull-back of the background E M N (Φ) defined as where D α Φ m is covariant derivative Note that non-abelian action for N Dp-branes is implicitly defined in the static gauge where world-volume coordinates ξ α coincide with the target space ones x α . Finally Str means symmetrized trace and in order to describe infinite number of Dp-branes we should divide the action (13) by the infinite order the quotient group Z d . Further, Q i j is defined as and (Q −1 ) j k is its inverse in the sense where E rs is matrix inverse to E mr defined as In order to implement T-duality along d transverse directions we follow analysis performed in [15] which we generalize to the case of non-linear non-abelian action (13). Let us presume that the background fields do not depend on x A coordinates, where A = p+1, . . . , p+d and that these coordinates are periodic with period √ 2πλ. This is natural if we recognize that all geometrical properties of the background are encoded in the the field E M N . Then we consider an infinite number of Dp-branes on compact space with coordinates x A when we impose following quotient conditions Let us presume that solution of the quotient equation corresponds to operators U A that commute In order to solve (21) it is natural to introduce an auxiliary Hilbert space on which Φ A and U B act. The simplest way is to introduce Hilbert space of auxiliary functions living on d−dimensional torus taking value in C d . Then we take U A as generators of the functions on d−dimensional torus where σ A are coordinates on the covering space of torus. Then Φ A has to be equal to since then Using these results we can now proceed to write corresponding action in T-dual background. As the first step we perform following manipulation with the determinant in the action (13) where First of all we observe that To proceed further we use the fact that D β Φ A acting on arbitrary function f (σ) defined on the space labelled by σ A is equal to and hence we can identify D α Φ A with λF A α . Using this identification we obtain and finally where we used (24) so that Now we return to the first determinant in (26) and rewrite it to the form Now we explicitly calculate components of the matrix D as To proceed further we observe that and hence we can identify expression in the bracket with the matrix inverseẼ AB to E AB so thatẼ AB E BC = δ A C . Further, let us consider following expression and multiply it with E j ′ k ′ . Then, after some calculations, we get so that we can identify expression in the bracket with matrixẼ i ′ j ′ Using these results we obtain following useful expressions Using (40) and (41) we get and In the same way we obtain and also Finally we consider following combinations of determinants that appear under square root in the action (13) det Collecting these terms together we obtain final form of D(p+d)-brane action in T-dual background in the form where we also used the relation between trace over infinite dimensional matrices and integration over coordinates σ and where the matrix D has following components