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JHEP09(2017)044

Published for SISSA by Springer

Received: May 31, 2017

Revised: August 8, 2017

Accepted: August 15, 2017

Published: September 12, 2017

Duality twisted reductions of Double Fi el d Theory of

Type II strings

Aybike C¸atal-

Ozer

Department of Mathematics,

Istanbul Technical University,

Maslak 34469,

Istanbul, Turkey

E-mail:

ozerayb@itu.edu.tr

Abstract: We study duality twiste d reductions of the Double Field Theory (DFT) of

the RR sector of massless Type II theory, with twists belonging to the duality group

Spin

(10, 10). We determine the action and the gauge algebra of the resulting th eor y

and determine the conditions for consistency. In doing this, we work with the DFT action

constructed by Hohm, Kwak and Zwiebach, which we rewrite in terms of the Mukai pairing:

a natural bilinear form on the space of spinors, which is manifestly Spin(n, n) invariant.

If the duality twist is introduced via the Spin

(10, 10) element S in the RR sector, then

the NS-NS sector should also be deformed via the duality twist U = ρ(S), wh er e ρ is the

double covering homomorphism between P in(n, n) and O(n, n). We show that the set of

conditions required for the consistency of the reduction of the NS-NS sector are also c r uc i al

for the consistency of the reduction of the RR sector, owing to the fact that the Lie algebras

of Spin(n, n) and SO(n, n) are isomorphic. In addition, requirement of gauge invariance

imposes an e x tr a constraint on the ﬂuxes that determine the deformations.

Keywords: Flux compactiﬁcati ons , String Duality

ArXiv ePrint: 1705.08181v2

Open Access,

 The Authors.

Article funded by SCOAP

ht tp s: //doi.org/10.1007/JHEP09(2017)044

JHEP09(2017)044

Contents

1 Introduction

2 Preliminaries on spin representations and the spin group 3

3 Democratic formulation of Type II theories and the Double Field Theory

extension

3.1 Democratic formulation 12

3.2 Double Field The ory extension 14

3.3 The DFT action of the RR sector rewrit te n with the Mukai pairing 18

4 Duality twisted reductions of DFT: Gauged Double Field Theory 21

4.1 Review of the reduction of the NS-NS sec tor 22

4.2 Reduction of the RR sector 23

4.2.1 Reduction of the Lagrangian 26

4.2.2 Reduction of the gauge algebra 27

4.2.3 Consistency of the reduced theory 28

5 Conclusions and outlook 31

1 Introduction

Double Fie l d Theory (DFT) is a ﬁeld theory deﬁned on a doubled space, where the usual

coordinates conjugate to momentum modes are supplemented with dual coordinates that

are conjugate to winding modes [

1–4]. DFT was originally constructed on a doubled torus,

with the aim of constructing a manifestly T-duality invariant theory describing the massless

excitations of closed strin g theory [

1, 2]. Later, this action was shown to be background

independent [

3], allowing for more gener al doubled spaces than the doubled torus. Obvi-

ously, the dual coordinates might not have the interpretation of being conjugate to winding

modes on such general spaces. Constr uc t i on of DFT builds on earli e r work, see [

5–13]. For

reviews of DFT, see [

14–17].

On a general doubled space of d i me nsi on 2n, the DFT action has a manifest O(n, n)

symmetry, under which the standard coordinates combined with the dual ones transform

linearly as a vector. The doubled coordinates must satisfy a set of constraints, called the

weak and the strong constraint and the th eor y is consiste nt only in those frames in which

these constraints are satisﬁed. It is an important challenge to relax these constraints, es-

pecially the strong one, as in any such frame the DFT becomes a rewriting of standard

supergravity, related to it by an O(n, n) transformation. Even in this case, DFT has the

virtue of exhibiting already in ten dimensions (part of) the hidden symmetri e s of super-

gravity, that would only appear upon dimensional reduction in i ts standard formulation.

– 1 –

JHEP09(2017)044

This virtue should not to be unde r e st i mate d, as it provides the possibility of implementing

duality twisted reductions of ten dimensional supergravity with duality twi st s belonging to

a larger symmetry group, that would normally be available only in lower dimension s.

Duality twisted reductions (or generalized Scherk-Schwarz reductions) are a general-

ization of Kaluza-Klein reductions, which introduces into the reduced theory mass terms

for various ﬁelds, a non-Abelian gauge symmetry and generates a scalar potential for the

scalar ﬁelds [

18, 19] . This is possible if the parent theory has a global symm et r y G, and the

reduction anzats for the ﬁelds in the theory is deter m i ne d according to how they transform

under G. It is natural to study duality twisted reductions of DFT, as i t comes equipped

with the large duality group O(n, n), and indeed this line of work has been pursued by

many groups so far [20–24]. In [20, 21] it was shown that the duality twist ed reductions of

DFT gives in 4 dimensi ons the electric bosonic sector of gauged N = 4 supergravity [

25].

A curious fact which was noted in these works was that the weak and the strong constraint

was never needed to be imposed on the doubled internal space. This (partial) relaxation

of the strong constraint made the twist ed reductions of DFT even more attractive. Later,

in [22], thi s was made more explicit, as they showed that the set of conditions to be sat-

isﬁed for the consistency of the twisted reduction are in one-to-one correspondence with

the constraints of gauged supergravity, consti t uti n g a weaker set of constraint s compared

to the strong constr ai nt of DFT. Following this, in [

26], it was shown that the weakening

of the strong constraint in the twisted reductions of DFT implies that even non-geometric

gaugings of half-maximal supergravity (meaning that they cannot be T-dualize d to gauged

supergravities arising from conventional compactiﬁcations of ten-dimensional supergravity)

has an uplift to DFT. Such non-geometric gaugings also arise from compactiﬁcat i ons of

string theory with non-geome t r i c ﬂux (see, for example [

27–29]) and the relation of such

compactiﬁcations with twisted compactiﬁcations of DFT was ex pl or e d in various papers,

including [

30–33]. We should also note that, the results of [22] was also obtained by [34],

by considering the duality twisted reductions of the DFT action they constructed i n terms

of a torsionful, ﬂat generalized connection, called the Weitzenb¨ock connection.

In all of the works cited above, only the reduction of the DFT of the NS-NS sector of

massless string theory was studied.

The fundamental ﬁelds in this sector are the gener-

alized metric (comprising of the Riemannian metric and the B-ﬁeld) and the generalized

dilaton. In a frame in which there is no dependence on the dual coor di nat e s, this sector

becomes the NS-NS sector of string theory. We will hereafter refer to this frame as the “su-

pergravity frame”. On the other hand, the DFT of the RR se c tor of Type II string theory

has also been constr uc t ed by Hohm, Kwak and Zwiebach [

37, 38] (an alternative formula-

tion of the RR sector, called the semi-covariant f ormulation is given in t he papers [

39, 40]).

This formulation of the DFT action has the ad d ed advantage that it alrea d y includes an extra term,

which has to be added by hand in the original formulation . This extra term is needed in order to match

the 4 dimensional half-maximal gauged supergravity with the theory t h a t results from the duality twisted

reduction of the DFT action.

An exception is the work of [

23], where th ey also include the reduction of the RR sector. However, their

methods are diﬀerent from ours, as they perform the twisted reduction in the semi-covariant formalism of

DFT [35, 36].

– 2 –

JHEP09(2017)044

Likewi se , in the supergravity frame, th i s action reduce s to the action of the democratic for-

mulation of the RR sector of Type II supergravity. The fundamental ﬁelds of this sector are

two SO(10, 10)-spinor ﬁelds, S and χ. The latter is a spinor ﬁeld which encodes the mass-

less p-form ﬁelds of Type II theory. It has to have a ﬁxed chirality, depending on whether

the theory is to describe the DFT of the massless Type II A theory or Type IIB theory. The

ﬁeld S is the spinor representative of the generalized metric, that is, under the double cov-

ering homomorphism between P in(n, n) and O(n, n), it projects to the generalized metric

of the NS-NS sector. The action of this sector has manifest Spin(10, 10) invarianc e (not

P in(n, n)) in order to preserve the ﬁxed chirality of χ. The action has to be supplemented

by a self-duality condition, which further reduces the duality group to Spin

(10, 10).

The aim of this pape r is to study the duality twisted reductions of the DFT of the RR

sector of massless Type II theory, with twists belonging to t he duality group Spin

(10, 10).

We study how the action and the gauge transformation rules reduce and determ i ne the con-

ditions f or the consistency of the reduction and the closure of the gauge algebra. We also

construct the Dirac operator associated with the Spin

(10, 10) covariant derivative that

arises in the RR sector. In ﬁnding the redu ce d theory, we ﬁnd it useful to rewrite the action

of [

37, 38] in terms of the Mukai pairing, which is a natural bilinear form on the space of

spinors [

42–44]. The advantage of this reformulation is that the Mukai pairing is m ani fe st l y

Spin(n, n) invariant. If the duality twist is introduced via the Spin

(10, 10) element S in

the RR sector, the consistency requi r e s that t he NS-NS sector should also be deformed, via

a duality twisted anzats introduced by U = ρ(S). Here, ρ is the double covering homomor-

phism between P in(n, n) and O(n, n). The fact that Lie algebras of Spin(n, n) and SO(n, n)

are isomorphic plays a crucial role in all the calculations. We show that the set of conditions

required for the consistenc y of the reduction of the NS-NS sector are also crucial for the con-

sistency of the reduction of the RR se c tor . In addit i on, t he deformed RR sector is gauge in-

variant only when the Dirac operator is nilpotent, which in turn imposes an extra constraint

on the ﬂuxes that de te r mi ne the deformations . The fact that such a constraint should arise

in the presence of RR ﬁelds has already been noted in [

20] and was veriﬁed in [23].

The plan of the paper is as follows. Section 2 is a preliminary section on spin rep-

resent ati ons and the spin group. Most of the mater i al needed in the calculations for the

reduction is reviewed in this section. In the ﬁrst part of section

3, we present a brief re v i ew

of both sectors of DFT, with a special emphasis on the RR sector. As the DFT of the

RR sector reduces to the democratic formulation of Type II theory in the supergravity

frame, we start this section by a brief review of the democratic formulation of Type II

supergravity. The rewriting of the action of [37, 38] in terms of the Mukai pairing is also

explained in this section. Section

4 is t he main section, where we study the reduction of

the action and the gauge algebra and discuss the conditions for consistency and clos ur e of

the gauge algebra. We ﬁnish wi th a discussion of our r e sul t s in section

2 Preliminaries on spin representations and the spin group

The purpose of this preliminary section is to review the material, which we will need in

the later secti ons of the paper. We closely follow [

45].

– 3 –

JHEP09(2017)044

Let V be an even dimensional (m=2n) real vector space with a symmetric non-

degenerate bilinear form (a metric) Q on it. Then the orthogonal group O(V, Q) is the

space of automorphisms of V preserving Q:

O(V, Q) = {A ∈ Aut(V ) : Q(Av, Aw) = Q(v, w), ∀v, w ∈ V } (2.1)

If we restrict this set to the aut omor phi s ms of determinant 1, then we get the subgroup

SO(V,Q). The corresponding orthogonal Lie algebras so(Q) = o(Q) are then the endomor-

phisms A : V → V such that

Q(Av, w) + Q(v, Aw) = 0 (2.2)

for al v,w in V. The standard methodology in constructing t he spin representations of the

orthogonal Lie algebra is to embed it in the Cli ﬀ ord algebra on V associated to the bilinear

form Q and use the well-known isomorphisms between the Cliﬀord algebras and the matrix

algebras.

Given the vector space V and the metric Q, one can deﬁne the Cliﬀord algebra C =

Cl(V, Q) as the universal algebra which satisﬁes the property

{v, w} ≡ v.w + w.v = 2Q(v, w) (2.3)

Here . is the product on the Cliﬀord algebra. Cl(V, Q) is an associative algebra with

unit 1 and as such it determines a Lie algebra, with bracket [ a, b] = a.b − b.a. Cli ﬀor d

algebras enjoy nice isomorphisms with various matrix algebras (the form of which depends

on V and Q) under which the Cliﬀord product becomes the matrix multiplication. If

, ··· , e

form a basis of V , then the unit element 1 and the products e

= e

. ··· .e

for I = {i

< i

< ··· < i

} form a basis for the 2

dimensional algebra Cl(V, Q). The

images of these basis elements (of V ) under the isomorphisms wit h the matrix algebras

are usually called Γ-matrices in the physics literature. The Cliﬀord algebra is a Z

graded

algebra and it splits as C = C

even

⊕ C

odd

, where C

even

is spanned by products of an even

number of eleme nts in V and C

odd

is spanned by an odd number of ele me nts of V . The

space C

even

is also a subalgebra and it has half the dimension of C, that is, it is an algebra

of dimension 2

m−1

The orthogonal Lie algebra so(Q) embeds i n the even part of the Cliﬀord algebra

as a Lie subalgebra via the map (for a proof, see [

45]) ψ ◦ ϕ

−1

: so(Q) → C

even

, where

ψ : ∧

V → Cl(V, Q),

ψ(a ∧ b) =

(a.b − b.a) = a.b − Q(a, b) (2.4)

and

ϕ : ∧

V −→ so(Q) ⊂ End(V ) (2.5)

a ∧ b 7−→ ϕ

a∧b

(2.6)

Here we i d entify the dual space V

∗

with V via the bilin ear form Q and hence ∧

V ⊂ End(V ) =

V ⊗ V

∗

∼

V ⊗ V .

– 4 –

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