Green-Schwarz superstring on doubled-yet-gauged spacetime

We construct a world-sheet action for Green-Schwarz superstring in terms of doubled-yet-gauged spacetime coordinates. For an arbitrarily curved NS-NS background, the action possesses $\mathbf{O}(10,10)$ T-duality, $\mathbf{Spin}(1,9)\times\mathbf{Spin}(9,1)$ Lorentz symmetry, coordinate gauge symmetry, spacetime doubled-yet-gauged diffeomorphisms, world-sheet diffeomorphisms and Weyl symmetry. Further, restricted to flat backgrounds, it enjoys maximal spacetime supersymmetry and kappa-symmetry. After the auxiliary coordinate gauge symmetry potential being integrated out, our action can consistently reduce to the original undoubled Green-Schwarz action. Thanks to the twofold spin groups, the action is unique: it is specific choices of the NS-NS backgrounds that distinguish IIA or IIB, as well as lead to non-Riemannian or non-relativistic superstring a la Gomis-Ooguri which might deserve the nomenclature, type IIC.

coordinates,x µ , to the conventional ones, x µ , to form doubled (D+D)-dimensional coordinates, x M = (x µ , x ν ) , M = 1, 2, 3, · · · , D+D . (1.1) On the doubled coordinate space, T-duality becomes a run-of-the-mill O(D, D) rotation. 1 However, despite of the doubling, the physical dimension of the spacetime should be undoubled: the doubled coordinates must describe D-dimensional physics. One governing geometric principle, proposed in [13] and pursued in this work, is the notion of doubled-yet-gauged coordinate system: the doubled coordinate space is gauged by an equivalence relation, called coordinate gauge symmetry, such that it is a gauge orbit that represents a single physical point. Hereafter, Φ s , Φ t and Φ u denote arbitrary fields and their arbitrary derivative descendants which must belong to the theory employing the doubledyet- In Double Field Theory, the equivalence relation, (1.2), is realized by requiring that all the fields in the theory are invariant under the coordinate gauge symmetry shift, This invariance is then equivalent, i.e. necessary [13] and sufficient [14], to the 'section condition' [10], 2 1 Yet, we stress that the doubled coordinates are not restricted to the description of strings but equally applicable to point-like particle dynamics, see e.g. [12]. 2 The equivalence basically follows from the power series expansion of (1.4). It is worth while to note that the former (strong) constraint in (1.5) implies the latter (weak) one, since ∂A∂ B Φs ∂B∂ C Φs = 0 means that ∂A∂ B Φs is a nilpotent matrix and hence is traceless. On the other hand, replacing Φu by the product, ΦsΦt, the latter gives the former. which are the differential constraints required for the consistency of DFT. 3 Upon the section condition, the generalized Lie derivatives given by [10,18] (c.f. [19,20]), (1.6) are closed under commutations: That is to say, the generalized Lie derivative generates the diffeomorphisms on the doubled-yet-gauged coordinate system (see [13,[21][22][23][24][25][26] for finite transformations). Then, in a parallel manner to Riemannian geometry, by taking the whole massless NS-NS sector as the geometric fields, the relevant torsion-free diffeomorphism connection (i.e. "Christoffel symbols"), covariant derivatives, a two-indexed curvature (i.e. "Ricci curvature") and a scalar curvature have been constructed [27] (c.f. [28]). 4 By now, the formalism has been well developed, such that D = 10 maximally supersymmetric DFT has been constructed to the full order in fermions [30], and the Standard Model itself has been 'double-field-theorized' to covariantly couple to the massless NS-NS sector of the gravitational DFT [31] (c.f. [32][33][34][35][36] for related earlier works). In particular, the maximally supersymmetric DFT not only contains and unifies type IIA and IIB supergravities but can also feature 'non-Riemannian' geometry, as we review below.
The massless NS-NS sector enters (bosonic) DFT in the form of a symmetric O(D, D) element, called "generalized metric", along with a scalar density, e −2d , having the weight of unity. Combined with the O(D, D) invariant metric, the generalized metric can produce a pair of orthogonal and complete symmetric projectors, The unification of IIA and IIB is due to the facts that i) the local Lorentz spin group in DFT is twofold, Spin(1, D−1) × Spin(D−1, 1) (basically one for P M N and the other forP M N ), ii) the maximally supersymmetric DFT is chiral with respect to both spin groups, Spin (1,9) and Spin(9, 1), iii) hence, the theory is unique: it admits IIA, IIB and non-Riemannian backgrounds as different types of solutions. In this sense, the last type might deserve the nomenclature, type IIC.
On the other hand, in doubled sigma models where the doubled coordinates are dynamical, the coordinate gauge symmetry (1.2) calls for the relevant gauge connection rather explicitly [14], As in any gauge theory, the gauge potential, A M , should meet precisely the same property as the gauge generator which is, in the present case, ∆ M in (1.4). Hence, similarly to the section condition (1.5), the coordinate gauge symmetry potential satisfies Respecting these constraints, the coordinate gauge symmetry is realized as Further, while dX M is not a diffeomorphism covariant vector, DX M is so: i) world-sheet action for a string [14], ii) world-line action for a point-like particle [12], The former result (1.14) was essentially a re-derivation of the doubled string action proposed by Hull [8], with the coordinate gauge symmetry interpretation added. Especially upon Riemannian backgrounds, the Euler-Lagrangian equation of the coordinate gauge symmetry potential, A iM , implies the self-duality (i.e. chirality) over the entire doubled spacetime, c.f. (3.20), 16) and the Euler-Lagrangian equation of X M gets simplified to give the stringy geodesic equation, where Γ LM N is the stringy Christoffel connection obtained in [27], and is worth while to note that the world-sheet topological term in (1.14) transforms to total derivatives under the coordinate gauge symmetry (1.12) as well as under the diffeomorphisms (1.13) [14], The kinetic terms in (1.14) and ( In the above doubled sigma models, the gauge potentials are all auxiliary. After they are integrated out, the doubled sigma models consistently reduce to the conventional undoubled string and particle actions. It is the purpose of the present paper to supersymmetrize the above doubled string action (1.14), or equivalently to formulate the renowned Green-Schwarz superstring action [39] on the doubled-yet-gauged spacetime, as the complementary world-sheet counterpart to the maximally supersymmetric DFT [30].
Since we do not include spin connections, the Spin(1, 9) × Spin(9, 1) Lorentz symmetry is going to be global rather than local. Nevertheless, the global twofold spin structure ensures to unify IIA and IIB superstrings: different choices of the NS-NS backgrounds give rise to IIA or IIB, as well as non-Riemannian IIC superstrings. Once again, after the auxiliary coordinate gauge symmetry potential being integrated out, our action reduces consistently to the Green-Schwarz type IIA/B superstring action if the background is Riemannian. Alternatively, upon a non-Riemannian background, our action leads to the supersymmetric extension of the Gomis-Ooguri non-relativistic string [38].
For further inspiring precursors, we refer readers to [8,40] for the world-sheet supersymmetries, [41] for the construction of chiral affine (super-)Lie algebras, [42] for the T-duality supergroup, OSp(D, D|2s), as well as [43] for a doubled Hamiltonian sigma model and [44,45] for the Born reciprocity. We also refer the work by Bandos [46,47] on the construction of a PST superstring action in doubled superspace.

Green-Schwarz superstring in terms of doubled-yet-gauged coordinates
In this section, firstly we present our main result, i.e. 'the construction of the Green-Schwarz superstring action on the doubled-yet-gauged spacetime', and then provide the relevant explanations, such as the conventions, the field contents, the target-spacetime supersymmetry and the kappa-symmetry. The reductions to the undoubled type IIA, IIB and non-relativistic IIC superstrings will be discussed in the next section.

Main result
We propose the Green-Schwarz superstring action on the doubled-yet-gauged spacetime, with the Lagrangian, Here, equipped with the map from the string world-sheet to the doubled-yet-gauged target-spacetime, and a pair of Majorana-Weyl spinors, θ α for Spin (1,9) and θ ′ᾱ for Spin(9, 1), we set For an arbitrarily curved NS-NS background, the action possesses the manifest O(10, 10) T-duality, the Spin(1, 9) × Spin(9, 1) global Lorentz symmetry, the coordinate gauge symmetry, the target-spacetime doubled-yet-gauged diffeomorphisms over any Killing direction, world-sheet diffeomorphisms and Weyl symmetry.
Moreover, when the background is flat, the action is invariant under 16+16 global target-spacetime supersymmetry, as well as 16+16 local fermionic kappa-symmetry, (2.6) In the above, we set a pair of world-sheet projection matrices, Further, Π kM projected means the projection of Π kM to the coordinate gauge symmetry value, such that Concretely, without loss of generality up to O(10, 10) rotations, if we choose the section as and thus, Surely, ε, ε ′ are constant Majorana-Weyl Spin (1,9), Spin(9, 1) spinors having the same chiralities as θ, θ ′ respectively, while κ i , κ ′ j are local Majorana-Weyl spinors with the opposite chiralities. As stressed by Hull [8], the string tension on a doubled space should be halved, i.e. (4πα ′ ) −1 . Further explanations are in order in the following subsections.

Conventions and field contents
Our conventions, especially for the indices, are identical to [30,36] and summarized in Table 1.
The NS-NS background of the action is given by the DFT-vielbeins satisfying four defining properties: That is to say, they are normalized, orthogonal and complete. They correspond to the "square-roots" of the projectors (1.9), as while the generalized metric is given by the difference,
It is worth while to note that, using the properties of the coordinate gauge symmetry potential (1.11), we may rewrite the world-sheet topological term as

Target-spacetime supersymmetry and Wess-Zumino term
For flat NS-NS backgrounds where the DFT-vielbeins are all constant, Π M i is target-spacetime supersymmetry invariant, under (2.5), 23) and the Lagrangian transforms to total derivatives, implying the invariance of the action, (2.24) In the above, the second equality follows essentially from the Fierz identity (2.14) which enables us to write In fact, extending the two-dimensional world-sheet to a fictitious three-dimensional space and using identities due to (2.14) like we may straightforwardly compute the 'exterior derivative' of the topological term, where we set the field strength of the coordinate gauge symmetry potential, The resulting 'three-form' on the right hand side of the equality in (2.27) then corresponds to the Wess-Zumino term [48] for Green-Schwarz superstring [49] now on doubled-yet-gauged spacetime. As desired, it is manifestly invariant under the global target-spacetime supersymmetry (2.5).

Fermionic kappa-symmetry
For the systematic derivation of the kappa-symmetry, we start with generic variations of the spinors, δθ, δθ ′ , and the auxiliary fields, δA iM , δh ij , while we set, with the opposite sign compared to the target-spacetime supersymmetry (2.5), It follows straightforwardly upon flat backgrounds, 29) and the kinetic term transforms as (2.30) On the other hand, the Fierz identity (2.25) implies for arbitrary δθ and δθ ′ , which in turn enable us to organize the variation of the world-sheet topological term as (2.32) Combining (2.30) and (2.32), with (2.7), (2.8), we obtain where the world-sheet projection matrices, h ij ± (2.7), naturally appear. They satisfy There are four terms on the right hand side of the equality in (2.33). The last term is total derivative and hence harmless. The first term is quadratic in Π iM and needs to be canceled by other two terms (i.e. second and third). For this, the variations of the fermions need to be linear in Π M i , such as which fix the kappa-symmetry transformations of the fermions, δ κ θ, δ κ θ ′ , completely as (2.6). Consequently the second line determines the variation of the world-sheet metric (2.6), up to Weyl transformations, which we rewrite here, For consistency with The vanishing of the right hand side of the above equality then should fix the kappa-symmetry transformation of the coordinate gauge symmetry potential. Yet, since the potential is constrained to satisfy A M i ∂ M = 0 and A M i A jM = 0 (1.11), it does not take the naive form one might be tempted to put: Instead, we must "double" this and project Π lM to the coordinate gauge symmetry value, Without loss of generality up to O(10, 10) rotations, if we choose the section by 47) and thus, the kappa-symmetry transformation of the coordinate gauge symmetry potential (2.43) reads explicitly, After all, under the kappa-symmetry, the Lagrangian transforms to the total derivative, parametrization even locally at all [14,37] (c.f. [51]).
Hereafter, for concreteness, yet without loss of generality, we fix the section as (2.10):
where we put, like (2.45), and we set without the coordinate gauge symmetry potential, The on-shell value of the coordinate gauge symmetry potential is, from the last line of (3.7) which is a 'perfect square' of the potential, Therefore, after the auxiliary potential being integrated out, our action reduces to where the standard string tension, (2πα ′ ) −1 , is restored. The last term, as total derivative, is the topological term introduced in [52] and [8].
In this way, setting B µν = 0, up to the world-sheet topological term and constant rescaling of the fermions, θ,θ → 4 √ 2 θ, 4 √ 2θ ′ , the reduced action (3.11) can be identified as the original undoubled Green-Schwarz superstring action.
Self-duality over the entire doubled-yet-gauged spacetime.
This gives, contracting with B λµ , and further separately, contracting with g λµ Adding (3.17) and (3.18), we obtain Then, as in the case with the bosonic string action of [14], (3.16) and (3.19) imply that the full set of the self-duality relations hold over the entire doubled-yet-gauged spacetime coordinate directions -although the coordinate gauge symmetry is a constrained field -when the NS-NS background is Riemannian, In the generic cases, i.e. not necessarily Riemannian, the equation of motion of the coordinate gauge symmetry potential gives a priori only the half of the self-duality relations, from (2.33), Then the above result (3.20) tells us that when the NS-NS background admits Riemannian interpretation, the other half of the self-duality relations is automatically satisfied, Π iν = 0. It is useful to note that, contracting with the DFT-vielbeins, the self-duality (3.20) decomposes into

Type IIC : non-Riemannian and non-relativistic backgrounds
While the non-Riemannian NS-NS background was first noted in [14] and subsequently shown in [37] to lead to the Gomis-Ooguri non-relativistic bosonic string [38], until now there is no systematic classification of it. Decomposing the DFT-vielbeins in terms of 10 × 10 square matrices, such as V M p = (V µ p , V νp ) and V Mp = (V µp , V νp ), the defining relations of them (2.16), especially the last one, imply This shows that V µ p is invertible if and only ifV µp is so.
In this subsection, we focus on the non-Riemannian background for the Gomis  The corresponding DFT-vielbeins are essentially, As 4 × 2 matrices, these represent the genuinely non-Riemannian 'hatted' part of the full DFT-vielbeins.
The master Lagrangian (2.2) reduces, upon the non-Riemannian NS-NS background, to where we set a ten-dimensional target-spacetime constant metric, g µν := 1 2ημν , δ µ ′ ν ′ . (3.28) The resulting superstring action is then, 29) and is subject to the chirality condition for the hatted untilde directions, xμ = (t, x 1 ) : This is the action for the Green-Schwarz superstring on the non-Riemannian background which supersymmetrizes the Gomis-Ooguri non-relativistic string.
Restricted to flat backgrounds of constant DFT-vielbeins, the action is further invariant under maximal spacetime global supersymmetry and also under local fermionic kappa-symmetry. After the auxiliary coordinate gauge symmetry potential being integrated out, the action can consistently reduce to the undoubled original Green-Schwarz action upon a Riemannian background. Thanks to the twofold spin groups, the action is unique: the two fermions, θ α and θ ′ᾱ , are Majora-Weyl spinors for Spin (1,9) and Spin(9, 1) respectively. It is then specific choices of the NS-NS backgrounds that distinguish Riemannian IIA, IIB and non-Riemannian IIC. Upon the Riemmanian IIA/IIB backgrounds, the Euler-Lagrangian equation of the coordinate gauge symmetry potential implies the self-duality over the entire doubled-yet-gauged spacetime.
Investigating the supersymmetry, the Killing spinor equations of the maximally supersymmetric DFT [30] should appear naturally. The computations of the one-loop beta function and the partition function are worth while to perform: we expect to derive the equations of motion of the maximally supersymmetric DFT [30]. Related to this, we refer readers to earlier works [53][54][55] on bosonic doubled sigma models, along with [37] for the matching of the fluctuation spectrum between DFT and the bosonic world-sheet action (1.14) around the non-Riemannian background for the Gomis-Ooguri string. Promoting the global Spin(1, 9) × Spin(9, 1) Lorentz symmetry to the local symmetry seems desirable. We leave quantization as for future work.