The T-dual symmetries of a bosonic string

We investigate whether the symmetry transformations of a bosonic string are connected by T-duality. We start with a standard closed string theory. We continue with a modified open string theory, modified to preserve the symmetry transformations possessed by the closed string theory. Because the string theory is conformally invariant world sheet field theory, in order to find the transformations which preserve the physics, one has to demand the isomorphism between the conformal field theories corresponding to the initial and the transformed field configurations. We find the symmetry transformations corresponding to the similarity transformation of the energy-momentum tensor, and find that their generators are T-dual. Particularly, we find that the general coordinate and local gauge transformations are T-dual, so we conclude that T-duality in addition to the well known exchanges, transforms symmetries of the initial and its T-dual theory into each other.


Introduction
One of the most important notions in theoretical physics is a symmetry. What is a symmetry of the string theory is not yet clear mainly because the theory itself is not yet formulated in a background independent way, which would enlighten its deeper principles. However, it is believed that the symmetry does exist, and that it will lead to finding the physically indistinguishable solutions to the string equations of motion and choosing the correct vacuum [1,2].
A string theory revealed a T-duality, a symmetry which is a consequence of the strings extended nature. T-duality connects seemingly different string theories by exchanging, for example the characteristics of the strings momentum and winding [3,4,5,6]. So, it should exchange the symmetries of string theories as well. If that holds, one can say that symmetries of string theories always appear in pairs.
In this paper, we will investigate symmetry of the space-time in which the bosonic string moves, using a world-sheet formulation [1]. The formalism differs from the usual, where the symmetry is a transformation of the space-time fields which leave the classical action invariant. This concept of a symmetry does not apply now, because only the world-sheet values of the space-time fields appear in the string action. Still, a symmetry should be a change in the space-time fields which does not change the physics. So, suppose one considers a string theory with some chosen space-time field configuration, and a string theory with a modified space-time configuration. How one determines if these two descriptions are physically equivalent? The string theory is conformally invariant world sheet field theory. The physics is determined by the conformal field theory, corresponding to the field configuration in question. The transformation on the fields will be a symmetry, if the corresponding conformal field theories are isomorphic [1,2].
If one is given a conformal field theory, one will obtain a physically identical conformal field theory by performing a similarity transformation on the operators of the initial conformal field theoryÔ → e −iΓÔ e iΓ .
This transformation does not change the algebraic properties, so the new theory will be physically the same as the initial theory. However, the transformation will in general make changes to the world-sheet energy-momentum tensor. If these changes can be interpreted as the changes in the space-time fields, then the latter are the symmetry transformations of the target space. This idea was introduced in [1,7], where this automorphism of the operator algebra was seen as an analogue of the change of variables in a partition function. The problem of finding symmetries was reduced to the problem of finding the operator generating the symmetry transformation. The first investigations were treating the string massless fields, but the method was then generalized to treat conformal deformations of conformal field theory [8,9].
The classical analogue of the similarity transformation is a transformation of an variable of interest by a Poisson bracket between a generator and a variable. In this paper, we will apply these transformations to the string sigma model energy-momentum tensor. We will start by considering a small modification of the space-time background Π ±µν → Π ±µν +δΠ ±µν . It will induce the transformation of the energy-momentum tensor. We will demand that the new energy-momentum tensor satisfies the Virasoro algebra as well. This means that the transformed theory is physically equivalent to the initial theory, or that these field transformations are the symmetry of space-time theory. In this way we will find a transformation of space-time fields corresponding to a similarity transformation and the generator of this symmetry.
We will consider both closed and the open string theory. For the open string we will consider a modified action which in comparison with the closed string action has an additional surface term which enables the invariance of the complete action to the general coordinate transformations and the gauge transformations, which are the symmetries of the closed string. For the open string theory, the boundary conditions can be satisfied by choosing either the Neumann or the Dirichlet boundary condition for every coordinate direction. If the choice is made, the modified action surface term is given in terms of the corresponding Neumann and the Dirichlet gauge fields. The closed string symmetries remain the symmetries of the open string theory taking the appropriate transformation of these gauge field.
If one includes T-duality into consideration, one can conclude that the general coordinate transformations and the local gauge transformations are not independent. Comparing their generators, using the T-dual coordinate transformation laws, one concludes that they are T-dual. Therefore, the symmetries are T-dual and the complete generator of symmetries is self-dual.

The bosonic string essentials
The quantization of the bosonic string theory, describing the string moving in a background consisting of a space-time metric G µν , a Kalb-Ramond field B µν and a dilaton field Φ, lead to a conclusion that in order to have a conformal invariance on the quantum level the energy-momentum tensor componentsT ± (ϕ), with ϕ = (G µν , B µν , Φ), have to obey the Virasoro algebras [2,10,11] T ± (ϕ(σ)),T ± (ϕ(σ)) = ih T ± (ϕ(σ)) +T ± (ϕ(σ)) δ ′ (σ −σ), T ± (ϕ(σ)),T ∓ (ϕ(σ)) = 0. (2.1) From these conditions follow the space-time equations of motion which space-time fields G µν , B µν , Φ have to obey. In order to obtain the symmetries of the space-time equations of motion, one does not need to find their explicit form. It is sufficient to consider the transformations which do not change the above relations.

The conformal gauge
The action which was quantized [12], for a constant dilaton field reads where the integration goes over two-dimensional world-sheet Σ with coordinates ξ α (ξ 0 = τ, ξ 1 = σ). g αβ is intrinsic world-sheet metric and x µ (ξ), µ = 0, 1, ..., D − 1 are the coordinates of the D-dimensional space-time and κ = 1 2πα ′ . Taking a conformal gauge g αβ = e 2F η αβ , the action becomes given in terms of the light-cone coordinates and therefore the canonical Hamiltonian for the theory (2.3) equals where the currents j ±µ are given by One can rewrite the Hamiltonian in terms of the energy-momentum tensor components as

The gauge invariant approach
Let us in this subsection consider a string theory without a gauge fixing. If one takes the following parametrization of the world-sheet metric tensor g αβ [13] g αβ = e 2Fĝ αβ = where Π ±µν is defined by (2.4) and the partial derivative is given bŷ Varying the action over x µ , one obtains the equations of motion with∇ ± being the covariant derivative [13], defined bŷ where x n is a scalar, vector or tensor and n is the sum of its world-sheet indices, taking 1 for plus and -1 for minus. The generalized connection is defined by One can extractẋ µ from the last equation, to obtaiṅ Using the currents (2.7), the coordinate derivatives over world-sheet parameters becomė The canonical Hamiltonian density H c = π µẋ µ − L is with T ± defined in (2.8). For h ± = ∓1, one returns to the conformal gauge.

The symmetries of space-time
In this section, we will search for the symmetries of the space-time in which the closed and the open strings propagate, as well as for their generators. We will investigate the change of the world-sheet energy-momentum tensor caused by the change in the spacetime fields. We will demand the transformed energy-momentum tensor T ± + δT ± still obeys the classical analogue of the Virasoro algebra (2.1). The energy-momentum tensor components T ± satisfy two independent copies of Virasoro algebra. To find a symmetry of the equations of motion, one should conclude what kind of transformation of fields ϕ → ϕ + δϕ, and consequently of the energy-momentum tensor conserves the Virasoro algebra. One need not know the explicit form of the space-time equations of motion to find its symmetry transformations. In order to have a conserved Virasoro algebra, one should find transformations for which the following conditions are fulfilled It is known [14] that a similarity transformation applied toT ± ensures the physical equivalence of the corresponding theories, makes the change inT ± , which corresponds to a change in the space-time fields, without changing the physics. This kind of change in the space-time fields is therefore a symmetry transformation. The similarity transformation implies that the change ofT ± is just One can confirm that the last relation solves the conditions for the Virasoro algebra conservation (3.2).
In the subsequent sections, we will be interested in finding the change in the spacetime fields, which transform T ± in a way which preserves the classical version of the Virasoro algebra. We will search for a generator Γ Λ (where Λ is some parameter) such that its Poisson bracket with energy-momentum components T ± produces the variation δT ± = {Γ, T ± }, equal to the change of energy-momentum tensor caused by the variation of fields ϕ → ϕ + δϕ. If such a generator exists, then the previous variation is a symmetry transformation of the space-time.

T-duality of the closed string symmetry generators
Our goal in this and the subsequent sections is to find the generators of the general symmetry transformations corresponding to the similarity transformation. So, let us suppose the background fields undergo a small change in value Π ±µν → Π ±µν + δΠ ±µν . Let us find the generators of the symmetries Γ, for this transformation of the background fields. The currents change by and therefore Let us determine the algebra of the currents (2.7). Using the standard Poisson brackets between the coordinates and the momenta one obtains where the generalized connection is defined by (2.15). Consequently, the Poisson brackets between T ± , defined by (2.8), and currents are Finally, we obtain the Virasoro algebra in agreement with the condition (2.1). We will suppose the generator of the symmetries in the following form (3.10) Using (3.8), one obtains the Poisson brackets between T ± and the generators If one defines the generalized covariant derivatives by one rewrites (3.11) as We know that T ± transforms as (3.5), therefore we search for a generator G = G + + G − such that which implies one obtains Using the currents (2.7) and the gauge parameter (3.16), we rewrite the generator G as G = dσ 2ξ µ π µ + 2κ(2ξ µ B µν + Λ µ G µν )x ′ν .

(3.18)
To simplify the last expression, one can define another gauge parameter so that G = 2 dσ ξ µ π µ + Λ µ κx ′µ . (3.20) In terms of the new parameter the Kalb-Ramond field transforms as 21) and the generator (3.20) is rewritten as Therefore, the closed string described by (2.11) is invariant under the general coordinate transformations with D µ ξ ν = ∂ µ ξ ν − Γ ρ µν ξ ρ , and the local gauge transformations where we omit tilde from Λ µ . If one keeps in mind T-duality, the present form of the generator offers interesting conclusions. T-duality connects physically equivalent string sigma models, and the connection between T-dual string backgrounds and their variables is simplest in the constant background case. In that case, the well known T-duality relation [15,16] π µ ∼ = κx ′µ stands. So, T-duality interchanges the sigma derivative of the coordinates with the momenta. The consequence of the above relation to the generator of symmetry (3.22), is that its constituents turn out to be T-dual as well which makes the complete generator G self-dual. Because of this, the local gauge transformations and the general coordinate transformations are T-dual also. The same relation however does not hold in more complicated backgrounds, when T-duality is performed along the nonisometry directions or in backgrounds without the global shift symmetry. These backgrounds where discussed in [17,18,19] where the generalized T-dualization procedure, applicable along the arbitrary space-time direction was presented and elaborated.

The open string and its symmetries
The open string described by the same action as the closed string is not invariant to the above symmetries. The change in the action caused by the general coordinate transformations is for the equation of motion (2.13) and the change by the local gauge transformations is (4. 2) The first expression can be rewritten as The boundary conditions of the open string are given in terms of this variable In Ref. [20] the way to gain invariance to the transformation (3.24) was offered, and in Refs. [21,22] the open string action invariant under both (3.23) and (3.24) was presented, which in comparison with the standard action has an additional surface term For each of the coordinates one can fulfill the boundary conditions (4.4), by choosing either the Neumann or the Dirichlet boundary condition. If we mark the coordinates with the Neumann condition by x a , a = 0, 1, · · · , p and the coordinates with the Dirichlet condition by x i , i = p + 1, · · · , D − 1, the surface term (4.5) reduces to where A N a and A D i are (p + 1)-and (D − p − 1)-dimensional vector gauge fields, first living on the Dp-brane and the second orthogonal to Dp-brane. The Neumann vector field is as usual coupled to the coordinate time parameter derivative and the Dirichlet vector field is coupled to variable γ (0) µ related to the boundary condition, depending on both world-sheet parameter derivatives of the coordinates.

Field strengths
It is well known that in the bosonic string action, the surface term can be rewritten in the form of the Kalb-Ramond term. If all the boundary conditions are Neumann, then the action on the boundary can be rewritten as For the arbitrary choice of the boundary conditions the action on the boundary is given by (4.7). Let us restrict our investigation to the following metric In that case, (4.7) can be rewritten using (4.4) as with In terms of the field strengths, the surface term becomes 14) i .
Comparing the boundary actions (4.14) with the action (2.11), we conclude that the addition of the surface term has changed the background fields by

The symmetry generators for the open string
The metric and the Kalb-Ramond field are changed in comparison with the closed string case to G µν → G µν = G µν + F where is a covariant derivative corresponding to the metric G µν . By B ρ µ ν we marked a field strength of the Kalb-Ramond field. The field itself is changed, however the field strength of the additional part is zero and therefore B µνρ = B µνρ .
The generator of the transformation is with Using the explicit form of the transformed metric, we obtain the connection If one choses only Neumann boundary conditions, the metric remains the same as F (s) µν = 0, so that only Kalb-Ramond field changes, but not its field strength.

Conclusion
We considered the general coordinate and the local gauge transformations of the bosonic string, and showed that they are T-dual to each other. We started with the bosonic string theory in a conformal gauge and in a gauge invariant form. One of the purposes of the latter was to find the separation of the Hamiltonian into two energy-momentum tensors satisfying two copies of the Virasoro algebras. These tensors are represented as products of currents, which are used for defining the generators of symmetries that were investigated. The generators were defined as the integrals over the spatial world-sheet parameter of the weighted currents.
Given a form of the generator, we investigated how it effects the variables, whose transformation is defined by a Poisson bracket between the generator of symmetry and the corresponding variable. We where interested in finding the explicit form of the change in the energy-momentum tensor caused by such a transformation. This form of transformation is used as a classical analogue of the quantum transformation, known to preserve the Virasoro algebra between energy-momentum tensor components. So, both initial and the transformed energy-momentum tensor describe the same physics. Because of that, we were interested in finding the small variations of background fields such that the change in energy-momentum tensor they cause is exactly the considered transformation. In fact, if there exists the generator such that the described equality is possible, then the obtained transformations of space-time fields are the symmetry of the theory.
We considered the standard closed string theory, and a modified open string theory which in comparison to the standard open string theory has an additional surface term. This term was chosen in such a way to cancel the obstacle for the closed string symmetries to be the open string symmetries as well. Introduction of the Neumann vector fields on the boundary is equivalent to the change of Kalb-Ramond field. In the case considered here however, the vector fields are not coupled only to the time derivative of the coordinates but also to the functions which define the boundary conditions on the string endpoints. Therefore, both metric and Kalb-Ramond field are changed by the surface term.
We found the transformations of the background fields, which transform the energymomentum tensor in such a way that the Virasoro algebra is preserved. It turned out, that the generators of the symmetries (3.20) can be separated in a way that one part of generator, linear in x ′µ represents the generator of the well known local gauge transformation of the Kalb-Ramond field and the other part, proportional to π µ , generates the general coordinate transformations. Since these quantities are T-dual κx ′µ ∼ = π µ , we conclude that these symmetries are T-dual. So, we showed that T-duality has an additional feature, that it interchanges the symmetries of a theory. The generator of global coordinate transformations and local gauge transformations of initial theory is T-dual to the generator of local gauge transformations and the general coordinate transformations