O(D, D) gauge fields in the T-dual string Lagrangian

We present the string Lagrangian with manifest T-duality. Not only zero-modes but also all string modes are doubled. The gravitational field is an O(D, D) gauge field. We give a Lagrangian version of the section condition for the gauge invariance which compensates the O(D, D) transformation from the gravitational field and the GL(2D) coordinate transformation. We also show the gauge invariance of the line element of the manifest T-duality space and the O(D, D) condition on the background. Different sections describe dual spaces.


JHEP02(2019)010
coordinates or half of the worldsheet chiral currents. Instead of solving the selfduality condition gauging the antiselfdual mode allows to preserve both target space doubling and the worldsheet covariance [18]. In this paper we have applied this formulation to the doubled coordinate space. These covariance are preferable for the quantum computation. The obtained new T-theory Lagrangian with the O(D, D) gravitational fields E M A is The Lagrangian includes both the selfdual current J + A and the antiselfdual current J − A .
The current has the worldsheet index and the target space index The worldsheet index is m = (τ, σ) and the worldsheet tangent-space index is a = ±. The 2D-dimensional target space index is M and the target tangent-space index is A = (Ā, A) with left/right indicesĀ, A. The worldsheet zweibein is e a m with e = det e a m . The nonchiral treatment [18] has some advantages in the worldsheet covariance comparing to the chiral treatment: the Lagrangian has the manifest 2-dimensional Lorentz symmetry and the canonical σ derivative coincides with the chain rule σ derivative without the selfduality condition.
Under the general coordinate transformation X M → X ′M = X M − Λ M with the infinitesimal parameter Λ M (X), the gauge invariance of the action requires the Lagrangian version of the section condition  T-duality is a duality between long and short distances in D-dimensional spacetime, so the distance of the T-theory spacetime is double-faced. Invariance under the general coordinate transformation is the guiding principle to determine the line element of the manifest T-duality space (T-space). We propose the line element of the T-space and the orthogonal condition Different sections of the metric (1.6) give dual space metrics. The organization of the paper is the following: in the next section the T-theory Lagrangian is proposed. The Lagrangian includes both the selfdual and the antiselfdual currents. Selfduality is chirality, where the selfdual current is the chiral left moving current. Contrast to the previous Hamiltonian formulation which contains only selfdual currents [11], the new T-theory Lagrangian is non-chiral. The worldsheet covariance is manifest where the Weyl-Lorentz gauge parameters are used for the worldsheet reparametrization invariances [19,chapter XI.A.3., Exercise XIA3.1]. In section 3 the gauge invariances of the T-theory Hamiltonian and the Lagrangian under the spacetime coordinate transformation are shown. The spacetime coordinate invariance of the Hamiltonian requires two types of the conditions; the Lagrangian version of the section condition and the weaker form of the selfduality condition. The spacetime coordinate invariance of the Lagrangian requires the new condition. In section 4 constraint algebras of the Virasoro constraints and selfduality constraints are presented. Subsidiary conditions are clarified. The Hamiltonian is non-chiral including both the covariant derivative and the symmetry generator. Non-chiral description makes the worldsheet covariance manifest. The comparison with the chiral description in our previous work [11] is also explained. In section 5 the line element of the T-space and the orthogonal condition on the background are derived from the Virasoro operators. Although the expression of the line element was used to examine solutions [20], the gauge invariance was not examined yet. The O(D, D) gauge invariance of the line element was tried to be realized by integrating out a compensating field [21]. The selfduality condition eliminates ∂ σ X by ∂ τ X resulting the zero-mode limit (particle limit) of the Virasoro operators. Their gauge invariance requires the new condition. The covariant and contravariant vectors are transformed with O(D, D) matrices. This is different from the bilinear of the GL(2D) matrices proposed in [22] which is inconsistent under the multiple transformations and non-associative. The finite O(D, D) transformations are not included there. The correspondence between the Buscher's T-duality transformation and the sectioning in the T-space is also shown by taking an AdS space as a simple example.

Lagrangian
The Hamiltonian form of the Lagrangian for a bosonic string in a flat space is written by where vectors are contracted by the usual D-dimensional Minkowski metric. We use the simple notation . X = ∂ τ X and X ′ = ∂ σ X only for obvious situations. It reduces to or more explicitly To derive the manifestly covariant Lagrangian for T-theory from the Hamiltonian, it's useful to note the orthogonality of the background field E A M with respect to the metric η, Indices are raised and lowered with η M N and η M N . The double space metric field G M N includes the B field. The doubled (left/right) indices A = (Ā, A) can be covariantly divided with respect to the tangent-space symmetry, with follows directly from that without. (The background can be treated effectively as constants for purposes of varying just the P 's.) The Hamiltonian in the flat doubled space can thus be written as The left/right D-dimensional vectors are contracted by the left/right D-dimensional Minkowski metrics asV 2 =VM ηMNVN and V 2 = V M η M N V N . The g's play the same role as before for both theĀ (left) and A (right) pieces of X and P denoted by (X,P ) (left) and (X, P ) (right). The λ's impose selfduality by killing the antiselfdual pieces, which carry opposite signs forĀ and A.
The Hamiltonian in (2.7) is rewritten as (2.9) In curved space the vielbein is included as

and the
Hamiltonian becomes (2.10)

JHEP02(2019)010
The derivation of the Lagrangian is then a double copy of the usual string case, with the appropriate substitutions for the g's. This suggests using independent zweibeins for left and right The final result for the T-theory Lagrangian is then where the currents with background arē Another interesting gauge for half these Weyl/Lorentz invariances is (2.14) Then in terms of the "usual" currents that use only a single zweibein the above T-theory Lagrangian becomes which, directly in the Lagrangian formalism, shows theλ's as selfduality multipliers. Thesê λ's are related to the previous by as seen by Weyl/Lorentz rescaling all ∂ τ coefficients to 1.

JHEP02(2019)010 3 Spacetime coordinate invariance
We show the spacetime coordinate invariances of the T-theory Hamiltonian and Lagrangian requiring new conditions. The transformation of the spacetime background vielbein under X M → X ′M = X M − Λ M was already given as the new Lie derivative in the Hamiltonian formalism [4,5]: The only difference from the usual Lie derivative comes from the last term which follows E A M to be orthogonal.
At first we examine the spacetime coordinate invariance of the T-theory Hamiltonian (2.10). The general coordinate transformation rule is obtained by taking commutator The infinitesimal transformation case of M N M is given by (3.1) as On the other hand the covariant derivative is transformed Next we examine the spacetime coordinate invariance of the T-theory Lagrangian. In Lagrangian formulation J A m should be also gauge invariant The transformation matrices are given by On the other hand the usual chain rule of the derivative gives These require the following condition This condition is the worldsheet covariant version of the last conditions in (3.4). with ∂ σ δ(2−1) = ∂ ∂σ 2 δ(σ 2 −σ 1 ). There is a normalization ambiguity of the currents in (2.8). The commutator of derivative currents gives to the particle derivative as

Virasoro and selfduality constraints
The selfduality condition . This condition is equal to vanishing of the antiselfdual current. The antiselfdual current is nothing but the symmetry generator. The selfduality condition is modes [18]. This term is added in the Hamiltonian (2.7), (2.9) and (2.10) where Lagrange multipliers may be functions. The Hamiltonian in curved space (2.9) is also written by where H τ,σ = 0 are the Virasoro constraints and h τ,σ = 0 are the selfduality constraints generating shift of the antiselfdual mode The Virasoro algebra is given by where the σ derivatives in the canonical formalism and the usual chain rule coincide (4.6) No subsidiary condition on fields is required. The bilinears of the selfduality constraints satisfy the following algebras  Commutators of the Virasoro operators and the selfduality constraints are given by

JHEP02(2019)010
with and The closure of the algebras requires the section conditions and the weaker form of the selfduality condition Let us compare the chiral description in [11]. The Virasoro constraints by the selfdual current only are given by (4.13) They satisfy the Virasoro algebra where the σ derivative is given by the commutator with H σ in the canonical formalism On the other hand the usual chain rule gives  In the chiral formulation the dimensional reduction constraint ⊲ M − ⊲ M = 0, which is first class, give the following Hamiltonian which leads to the worldsheet covariant string action as shown in [11]. Naively the dimensional reduction constraint supplies the antiselfdual currents.

JHEP02(2019)010 5 Stringy geometry
The currents in the Lagrangian formulation are related to the currents in the Hamiltonian formulation by the following derivative relations If the selfdual condition ⊲ = 0 is imposed then X M are chiral scalars, but it is not the case in this paper.
Taking variation with respect to the Lagrange multipliers of the Lagrangian (2.16) gives Virasoro constraints T ++ = 0 = T −− and selfduality constraintsh = 0 The T-theory space is defined by the covariant derivatives ⊲ A in Hamiltonian formulation, while the Lagrangian is described by the selfdual currentsJ +Ā and J − A . Imposing the selfduality constraintsJ −Ā ≡ 0 ≡ J + A leads to the particle (zero-mode) limit of the selfduality currentsJ The gauge invariant operators which do not vanish in the zero-mode limit are the Virasoro operators. The zero-mode limit of the Virasoro operators in the gauge g + = g − = 1 leads to the line element of the manifest T-duality space and the orthogonal condition on backgrounds

JHEP02(2019)010
Let us examine the zero-mode limit of the curved space covariant derivative (3.9). While the vielbein is transformed with the O(D, D) matrix as (3.5) the tangent vector is transformed with the GL(2D) matrix as In order to be gauge invariant the section condition is required It is also mentioned that the doubled coordinate X M does not satisfy the section conditions. The zero-mode limit of the curved space one form (3.10) should be also gauge invariant The Lagrangian version of the section condition as the zero-mode limit of (3.13) is required Concrete examples are shown as follows: • Under the infinitesimal coordinate transformation by using with the constraint the curved space one form is gauge invariant the curved space covariant derivative (5.8) and the curved space one form (5.12) are gauge invariant. The line element is invariant and the orthogonal condition in (5.15) is also invariant from ΞηΞ T = η.
• The finite coordinate transformation with only x µ dependence is examined. For convenience the O(D, D) invariant metric η M N is off-diagonal The coordinate transformation is given by a GL(2D) matrix By using the section condition (5.9) It is also denoted that the contravariant vector V M (X) is transformed by the new Lie derivative (3.1) and the covariant vector is its inversely transformed. The relation to the Buscher's transformation [23,24] is also mentioned by taking AdS space as a simple example. The Lagrangian for a string in the D-dimensional AdS space is given by

Conclusions
The string action in curved space with manifest T-duality is presented. The gravitational field is O(D, D) gauge field and the gauge invariance of the Lagrangian is shown. Its gauge invariance requires the Lagrangian version of the section condition and the weaker form of the selfduality condition. Finally the line element and the orthogonal condition in the manifest T-duality space are presented and their gauge invariances are also shown. This formulation will be useful for quantum computation to explore T-duality symmetric situations.