Segmented strings coupled to a B-field

In this paper we study segmented strings in AdS$_3$ coupled to a background two-form whose field strength is proportional to the volume form. By changing the coupling, the theory interpolates between the Nambu-Goto string and the $SL(2)$ Wess-Zumino-Witten model. In terms of the kink momentum vectors, the action is independent of the coupling and the classical theory reduces to a single discrete-time Toda-type theory. The WZW model is a singular point in coupling space where the map into Toda variables degenerates.

In this paper we study segmented strings in AdS3 coupled to a background two-form whose field strength is proportional to the volume form. By changing the coupling, the theory interpolates between the Nambu-Goto string and the SL(2, R) Wess-Zumino-Witten model. In terms of the kink momentum vectors, the action is independent of the coupling and the classical theory reduces to a single discrete-time Toda-type theory. The WZW model is a singular point in coupling space where the map into Toda variables degenerates.

I. INTRODUCTION
Segmented strings in flat space are piecewise linear classical string solutions: at a given time their shape is a union of straight string segments. Kinks between the segments move with the speed of light and their worldlines form a lattice on the worldsheet. This idea can be generalized to AdS 3 (or more generally to (A)dS n ) where the embedding is built from AdS 2 patches [1,2]. The construction provides an exact discretization of the non-linear string equations of motion (as opposed to approximate discretizations used for instance in [3]). For recent developments, the reader is referred to [4][5][6][7].
In [8], the area of segmented strings has been computed using cross-ratios constructed from the kink momentum vectors. The cross-ratios were expressed in terms of purely left-handed (or right-handed) Toda variables. In this way, classical Nambu-Goto string theory in AdS 3 could be reduced to an integrable time-discretized relativistic Toda-type lattice.
In [5], Gubser pointed out that the segmented string evolution equations simplify if they are derived from the SL(2, R) Wess-Zumino-Witten action. In this theory, strings couple to the three-form field strength which supports the AdS 3 geometry. (For the quantum theory see [9].) If the three-form field strength comes from the Ramond-Ramond sector, then the WZW theory describes the motion of D1-branes, whereas the Nambu-Goto action is appropriate for fundamental strings (F1).
In this paper we generalize [8] by considering F1-D1 bound states. Our starting point is the action [6] (1) where τ 1 is the tension. In order to simplify the formulas, we set the prefactor to one (τ 1 = −2). Y M are coordinates on AdS 3 , h and G are the worldsheet and background metrics, respectively. B is the background two-form with field strength proportional to the volume form of AdS 3 . Finally, κ is its coupling to the worldsheet. * Electronic address: dvegh@cmsa.fas.harvard.edu On causal grounds [6], its value is restricted to κ ∈ [−1, 1]. We can without loss of generality restrict κ to be non-negative.
The equations of motion derived from (1) are exactly solved by segmented strings. These can be built by gluing diamond-shaped worldsheet patches. Each patch borders four others along null kink lines.
Let us consider the patch in FIG. 1. The four vertices in the R 2,2 embedding space of AdS 3 are labeled by V ij . We have V 2 ij = −1. The boundary of the worldsheet patch consists of four null kink lines. Let us define the following kink momentum vectors These vectors satisfy The latter equation can be interpreted as "momentum conservation" during the scattering of two massless scalar particles with initial and final momenta p 1,2 and p 3,4 , respectively.
Points on the surface are given by the interpolation ansatz [6] which solves the equations of motion where From the interpolation ansatz, we have This equality constitutes a discrete evolution equation for segmented strings. The set of points in AdS 3 null separated from both V 10 and V 01 is a one-dimensional locus, conveniently parametrized by V 11 (κ). Note that at κ = 0, the interpolation ansatz gives an AdS 2 patch embedded into AdS 3 .
In the next section, we evaluate the action for a single patch. The action can be written in terms of Mandelstam variables corresponding to kink momentum vectors. Section III computes the Euler character for the string. Section IV discusses the singular κ = 1 case where the map into Toda variables degenerates. We finish with a short discussion of the results.

II. THE ACTION OF SEGMENTED STRINGS
The value of the action evaluated on the patch is analogous to a scattering amplitude in four dimensional Minkowski spacetime. Instead of the Lorentz symmetry, however, the action is invariant under the SO(2, 2) isometry group of AdS 3 and R 2,2 . The only independent invariants are the Mandelstam variables s = (p 1 + p 2 ) 2 and u = (p 1 − p 4 ) 2 . The patch action then takes the form where L is the AdS 3 radius (henceforth set to one) and F (x) is a dimensionless function. In the following, we will determine this function by evaluating the action for patches that have been moved into a certain position.

A. Fixing the patch
Using the SO(2, 2) symmetry we can rotate/boost any patch such that V 00 , V 10 , V 01 are parametrized by two numbers c andc as follows These points satisfy V 2 ij = −1. It is easy to check that the corresponding difference vectors from (2) indeed satisfy p 2 i = 0. The interpolation ansatz in (3) gives the embedding of the patch into R 2,2 The fourth vertex is computed from V 11 = X(1, 1). The Mandelstam variables are found to be for the argument of F (x). Note that κ = 1 is a special point, since at this value the ratio is independent of cc. This is precisely the Wess-Zumino-Witten theory.

B. Evaluating the action
Let us now evaluate action (1) for our worldsheet patch. The metric and a canonical B-field are The field strength three-form is Since H and the background metric are both SO(2, 2) invariant, we expect that the patch action will also be invariant and therefore it can be expressed in terms of Mandelstam variables. The κ = 0 case is straightforward and the results were presented in [8]. At κ = 0, however, S(B 0 ) depends separately on c andc and thus it cannot be expressed in terms of Mandelstam variables which are functions of cc. This is due to the fact that the bulk action alone is not gaugeinvariant if the worldsheet has boundaries. In order to preserve gauge invariance, point particles with opposite charges must be attached to the string endpoints. These particles are charged under a one-form gauge field A. The minimal coupling is described by where ∂Σ is the worldsheet boundary. The variation of the bulk worldsheet action under a gauge transformation Λ can be canceled if we let A transform according to Our strategy will be the following. For a given B, we choose A such that S + S A is SO(2, 2) invariant (and thus a function of s, t, u). Clearly, A does not affect closed string motion because those worldsheets have no boundaries. Using the gauge transformation above, we set A = 0. As a result, S A vanishes and the entire patch action will come from S. When the dust settles, all we have done was a gauge transformation on B (and we can forget about S A ).
Consider the following B-field It is gauge-equivalent to B 0 and -as we will see -gives an SO(2, 2) invariant result. Let us now evaluate (1) on the patch given by the interpolation ansatz (5) using the expression above for the B-field. The purely geometrical part of the integrand gives The second term (proportional to the B-field) gives Integrating I 1 + I 2 over the patch gives the value of the action. We get Combining these results with (6) results in the covariant formula Note that the expression is independent of κ.

C. Toda variables
The Mandelstam variables may be expressed in terms of Toda variables as in [8]. In order to do this, the lightlike kink momenta p are written as products of helicity spinors. We define We will call λ left-handed spinors andλ right-handed spinors. In terms of these two-component variables, the patch action can be written as There is a similar formula in terms of right-handed spinors. The spinor modulus drops out of the action. Thus, by defining the angles α i via The α angles are the global Toda variables. Let us further define the left-handed Poincaré Toda variables by For a kink momentum vector p ∈ R 2,2 , the left-handed and right-handed Poincaré Toda variables are simply given by In terms of these fields the patch action becomes The total action is then the sum of all patch contributions where the i, j indices indicate the position of the kink edge in the two-dimensional lattice (see FIG. 3). The previously used one-index a k are

D. Equation of motion
The equation of motion computed from (12) is [ This equation is independent of κ. It has been obtained in [10] as the equation of motion of a time discretization of a relativistic Toda-type lattice.
The validity of the above equation of motion can be checked directly. Let us consider two adjacent patches as in FIG. 2 and fix four vertices V 10 , V 11 , V 01 , V 02 in R 2,2 such that neighboring vertices are lightlike separated. The interpolation ansatz (3) then gives 1 V 00 and V 12 . The left-handed Toda variables a ij can be computed from the kink momentum vectors. We claim that these variables satisfy the above equation of motion.
The calculation is straightforward using a computer algebra program, but the formulas are too large to present here. Therefore we only sketch the proof. One might start with the following "forward null triple" supplemented by another vertex V 10 which satisfies Altogether there are six real constants parametrizing the vertices. Note that V 01 is fixed, but apart from this the ansatz is generic. In principle, one could start with a completely generic null triple ansatz, but then the formulas become prohibitively complicated. Instead, we fix V 01 , and we perform a global SO(2, 2) = SL(2) L × SL(2) R transformation on the kink momentum vectors before the final step when the Toda variables are computed. Note that left-handed variables are invariant under SL(2) R and thus it is enough to consider SL(2) L rotations. The resulting Toda variables do satisfy the equation of motion.

III. EULER CHARACTER
In this section, for pedagogical reasons, we study the Euler character Let us consider a worldsheet with torus topology 2 . Such worldsheets are known to give χ = 0. In the following, we will check this result for segmented strings. Using the interpolation ansatz (3) and σ ± = 1 2 (τ ± σ), the induced metric is diagonal with components The Ricci scalar computed from g is constant away from kink collision vertices Let us decompose the Euler character as an integral away from the vertices plus integrals at the vertices Here the sum is over kink vertices and V i labels an infinitesimally small area around the i th vertex. Let us momentarily set κ = 0. From the appendix of where φ i is defined such that cos φ i is the scalar product of the normal vectors of the two space-like separated patches around the i th collision point. Denote the kink collision point by X M ∈ R 2,2 . Then, two kink momentum vectors emanating from this point span an AdS 2 patch with normal vector [8] Here α and β are the left-handed global Toda variables corresponding to the two kink momentum vectors. Using this formula, φ i can be expressed and we arrive at where a The result (15) holds even for κ = 0. Recall that in [8], the integrated Ricci scalar was computed in flat background space and then the result was expressed in terms of AdS 3 quantities (i.e. R 2,2 normal vectors). The curvature of the target space did not matter since the collision of kinks was instantaneous. The Christoffel symbols can therefore be neglected in a limit where we zoom in on the collision point. Similary, we can argue that the threeform field strength can also be neglected in this limit and thus (15) should be independent of κ.
By performing the sum over kink collisions we get i Vi where now the indices of a ij label positions in the kink lattice, see FIG. 3. In the RHS, using the results of the previous section, we recognize twice the total action (12). Plugging (16) into (14) and using (13) we get where A is the worldsheet area. From (7) and (8) the integrand in the bulk action is which, after integration, yields S total = (1 − κ 2 )A. Plugging this result back into (17) finally gives χ = 0.

IV. DEGENERATE WZW LIMIT
Classical solutions for the SL(2) Wess-Zumino-Witten model are given by How can such a (classically) trivial theory be mapped into the non-trivial Toda-type theory? The answer is that at κ = 1 the map is not surjective: segmented strings are mapped into a smaller subspace of the Toda phase space 3 . The fact that this point in coupling space is singular can already be seen from (6) that gives u/s = −1 which is independent of the c andc patch parameters.
FIG . 3 shows the trivial subspace of the left-handed Toda phase space. The a ij variables sitting on the black dots depend only on i + j and they are independent of i − j. This means that the black dots grouped together (blue shading) have the same values. This is clearly a lower dimensional subspace. We have not included a separate (mirror) figure, but the right-handed variablesã ij are similarly degenerate: the ones sitting on white dots depend only on i − j (and not on i + j).
Let us sketch the proof of degeneracy discussed above. The calculation is similar to the one in section II D. Consider the following forward null triple The interpolation ansatz then gives the fourth vertex, We now perform a global R ∈ SO(2, 2) = SL(2) L × SL(2) R transformation on the kink momentum vectors which transforms V 00 into a generic position. Left-handed variables are invariant under SL(2) R and thus it is enough to consider w ∈ SL(2) L rotations. The left-handed Toda variables computed from the difference vectors are where a(p) denotes the Poincaré Toda variable corresponding to a kink momentum vector p, see eqn. (10).
The other two variables a 2 and a 3 depend on κ. They are easy to compute, but the formulas are too large to present here. They satisfy There are degeneracies in the right-handed variables which can be proven in a similar fashion.
We finish this section with the following observation. Let us exchange black and white dots in the lattice. This duality exchanges patches and kink collision vertices and changes the string embedding. After the transformation, left-handed variables sitting on white dots along the same kink line will be equal. This configuration correspond to trivial left-moving kinks (edges with black dots), since they do not cause a time delay when they cross a rightmoving kink. Thus, this string embedding only contains right-moving shockwaves and in the κ = 0 case it is equivalent to Mikhailov's construction [11].

V. DISCUSSION
In this paper, we have computed the action of segmented strings in AdS 3 . The worldsheet is coupled to a background two-form whose field strength is proportional to the volume form. We have used the interpolation ansatz of [6] to parametrize elementary patches. Segmented string solutions are obtained by gluing the patches along null boundaries (kink lines).
The null kink momentum vectors in the embedding space R 2,2 can be decomposed using helicity spinors. Then, the action can be expressed in terms of crossratios of the spinor angles. We have called both these angles and their tangents "Toda variables". Time evolution of segmented strings can be described by the evolution equation of a discrete-time Toda-type lattice. This equation was presented in section II D.
Interestingly, the final form of the action does not depend on the two-form coupling κ. Thus, the theory in terms of Toda variables treats the classical Nambu-Goto theory (κ = 0) and the SL(2) Wess-Zumino-Witten model (κ = 1) on the same footing. However, the latter theory is a special one, because the map from the segmented string into Toda variables degenerates as κ → 1. This is seen in FIG. 3: the variables grouped together will become equal at κ = 1. By performing a duality that exchanges black and white dots in the lattice, WZW solutions can be mapped to string solutions with purely left-or purely right-moving kinks.
The results generalize those in [8] which can be obtained by setting κ = 0. We have not discussed the reconstruction of string embeddings from solutions of the Toda-like lattice. The procedure should be analogous to the κ = 0 case.
An interesting question is how these ideas generalize to other spacetimes (e.g. dS n or AdS n ) and what kind of Toda-like theories one would get by a similar reduction. We leave this for future work.