Green-Schwarz Mechanism for String Dualities

We determine the complete spacetime action to first order in α for the massless fields of bosonic string theory compactified on a d-dimensional torus. A fully systematic procedure is developed that brings the action into a minimal form in which all fields apart from the metric enter only with first-order derivatives. T-duality implies that this action must have a global O(d, d,R) symmetry, and we show that this requires a Green-Schwarz type mechanism for α-deformed O(d, d,R) transformations. In terms of a frame formalism with GL(d) × GL(d) gauge symmetry this amounts to a modification of the three-form curvature by a ChernSimons term for composite gauge fields.


Introduction
String theory is a particularly promising candidate for a consistent theory of quantum gravity, but it is fair to say that its underlying principles remain elusive. An important clue is that, even classically, string theory modifies general relativity in two significant ways. First, string theory features novel dualities which imply that theories defined on seemingly different backgrounds are actually equivalent. Second, the classical field equations receive an infinite number of higherderivative corrections governed by the dimensionful (inverse) string tension α ′ . Whatever the fundamental formulation of string or M-theory may be, it seems clear that it would have to accommodate these two features as core principles. In this Letter we report on results for a fully systematic procedure to determine the duality invariant spacetime action for massless string fields to higher order in α ′ and point out a curious interplay between string dualities and α ′ corrections. We discuss only the main results; the technical details will be presented in Ref. [1].
The simplest duality shared by all closed string theories is T-duality. It states that string backgrounds containing a d-dimensional torus are mapped under the group O(d, d, Z) to physically equivalent backgrounds. This duality includes, for a single circle, the inversion of the radius R → α ′ /R. The T-duality property of closed string theory implies that the spacetime action for the massless string fields on such backgrounds features a global O(d, d, R) symmetry [2]. To lowest order in α ′ this symmetry was first displayed in the cosmological setting (reduction to one dimension) in Ref. [3] and later generalized to arbitrary d in Ref. [4]. Subsequent seminal work by Meissner revealed an O(d, d, R) invariance to first order in α ′ in the cosmological setting, with the group action being undeformed thanks to a judicious choice of field variables [5]. In Ref. [6] the case of a single circle was investigated, while Ref. [7] includes a general torus but truncates to 'internal' field degrees of freedom.
More recently, α ′ corrections have been investigated in the extended framework of double field theory [8][9][10], which is the duality-covariant formulation of spacetime actions before compactification. Notably, two unexpected new features appear: 1) the gauge transformations need to be α ′ -corrected, and 2) there is no background independent formulation in terms of the familiar O(d, d, R) matrix encoding metric and B-field [11][12][13][14][15][16]. Rather, the general formulation of double field theory to first order in α ′ employs a frame formalism with α ′ corrected tangent space transformations. While in the dimensionally reduced theories determined so far there is a choice of field basis for which the O(d, d, R) action is undeformed at order α ′ , it has remained an open question whether α ′ -deformations of O(d, d, R) may be required in general dimensionally reduced theories. The reduction of α ′ -deformed double field theory has been investigated in Ref. [17], however, without extracting the consequences for the realization of O(d, d, R) in the dimensional reduction of conventional (non-extended) theories.
In this Letter we give the complete action to first order in α ′ for arbitrary d and show that in general the O(d, d, R) transformations are α ′ -deformed according to a novel Green-Schwarz type mechanism. The original Green-Schwarz mechanism, which triggered the first superstring revolution, is needed to show that gravitational and gauge anomalies can be cancelled, for gauge groups SO(32) or E 8 × E 8 , by an α ′ -deformation of the gauge transformations of the (singlet) B-field, thereby modifying the classical (tree-level) theory [18]. Similarly, we show here that O(d, d, R) invariance of the spacetime action reduced on a d-dimensional torus requires a non-trivial transformation for the singlet B-field already classically, with a Chern-Simons type modification of the 3-form curvature. This suggests that Green-Schwarz type mechanisms may be much more ubiquitous than expected.

Two-derivative action
We consider the two-derivative effective action for the bosonic string in D + d dimensions, with metricĝμν , antisymmetric Kalb-Ramond fieldBμν and the dilatonφ where indicesμ run over the (D + d) dimensional space, andĤ 2 =ĤμνρĤμνρ with the field strengthĤμνρ = 3 ∂ [μBνρ] . For compactification on the d-dimensional torus, we use the index split and drop the field dependance on the internal coordinates y m . For the metricĝμν, we consider the standard Kaluza-Klein ansatẑ in terms of the D-dimensional metric g µν , Kaluza-Klein vector fields A (1)m µ and the internal metric G mn . Similarly, for the 2-formBμν, we use the ansatz (2.4) in terms of D-dimensional scalars B mn , vector fields A (2) µ m , and a 2-form B µν . After dimensional reduction, the action (2.1) may be cast into the manifestly O(d, d, R) invariant form [4] and M is the associated abelian field-strength. Finally, the D-dimensional

Four-derivative action
At first order in α ′ , the correction to the action (2.1) is given by [19] up to field redefinitions. Here,Rμνρσ is the Riemann tensor, and we definedĤ 2 µν ≡ĤμρσĤνρσ. In the following, we compactify this action on the d-dimensional torus and seek to bring it into a manifestly O(d, d, R) invariant form. In the presence of higher order corrections this construction appears far less straightforward than in the two-derivative case due to the many potential ambiguities from field redefinitions and partial integrations.
Let us outline our systematics, which generalize those in Refs. [20,21] to arbitrary dimensions. We first determine a basis of independent O(d, d, R) invariant four-derivative terms. In D dimensions, every four-derivative term carrying the leading two-derivative contribution from the field equations descending from the two-derivative action (2.5) can be eliminated by a field redefinition in order α ′ . Moreover, different four-derivative terms in the Lagrangian can be related by integration by parts. Upon a systematic count, dividing out this freedom, we determine the independent four-derivative terms that can be built from  7). Details of this rather lengthy computation will appear in Ref. [1]. As a result, we find that after compactification the action (2.10) can be brought into the form where I 1 takes the manifestly O(d, d, R) invariant form Tr (∇ µ S∇ ν S) Tr (∇ µ S∇ ν S) with Ω µνρ given by (2.14) This 3-form has a remarkable structure owing its existence to the non-vanishing cohomology    [5,20]. Once truncated to scalars only (i.e. setting A µ M = 0 = B µν , g µν = η µν ), the action (2.12) reduces to two terms which provide an equivalent compact rewriting of the action given in Ref. [7]. However, the need for the Green-Schwarz type mechanism (2.19) is not visible in any of these limits, as the mechanism mixes the Kalb-Ramond field B µν and the scalars B mn . It is instructive to inspect the above Green-Schwarz deformation in view of the Z 2 invariance of bosonic string theory that sendsB → −B.

Frame formulation
We have seen that the rigid O(d, d, R) transformations need to be α ′ -deformed via a Green-Schwarz type mechanism. We now show that the theory can be reformulated, by means of a frame formalism, so that instead the gauge group of local frame transformations gets α ′deformed. This formulation then follows the standard Green-Schwarz mechanism more closely, albeit with composite gauge fields.
We introduce a frame field E ≡ (E M A ) with inverse E −1 ≡ (E A M ) in terms of which the matrix (2.6) is given by where flat indices are split as A = (a,ā), and we assume κ AB to be block-diagonal with components κ ab and κāb. Since we make no further a priori assumption on κ there is a local GL(d) × GL(d) frame invariance, acting as Gauge fixing κ AB = δ AB the above reduces to the familiar frame formalism with local SO(d) × SO(d) invariance but for our present application an alternative gauge fixing is more convenient: using matrix notation, we set and Defining the Maurer-Cartan forms one finds that the P µ transform as GL(d) × GL(d) tensors while the Q µ transform as GL(d) × GL(d) connections: , and analogously for the barred version. For the gauge choice (3.4) the explicit form of these connections reads, in matrix notation, (3.7) We can now consider the Chern-Simons 3-form built from these connections: which transforms under Eq. (3.6) as The barred formulas are analogous. Working out the Chern-Simons-form for Eq. (3.7) one finds precisely Eq. (2.14), up to a global factor 3. We thus define the three-form curvature with Chern-Simons modification as The matrixZ has the effect of exchanging the two GL(d) factors or, equivalently, to exchange the role of unbarred and barred indices. Indeed, a quick computation shows that under Eq. (3.12) the Maurer-Cartan forms (3.5) transform as P µ ↔P µ and Q µ ↔Q µ , in agreement with the explicit form (3.7). Thus, thanks to the relative sign in Eq. (3.10) the total Chern-Simons form is Z 2 odd, which together with B µν → −B µν implies Z 2 invariance.

Conclusions
We have shown that Green-Schwarz type mechanisms may be necessary even in classical string theory, which includes α ′ corrections, in order to realize its global duality symmetries. While the original Green-Schwarz mechanism of Ref. [18] does modify the classical theory, here the need for an α ′ -deformation of gauge transformations only follows by requiring cancellation of quantum anomalies. The observation that already invariance of the purely classical theory requires Green-Schwarz type mechanisms suggests that these may play a much more general role, perhaps in order to revisit U-duality in the presence of higher order corrections.