3.2 General B-field: Non-associativity 29
closed B-field simplified the calculation of correlation functions and resulted in the most basic example of an associative star-product.
Assuming now general fieldsgij(X) andBij(X) in the open string sigma model (4.1.3), correlators of vertex operators can be computed perturbatively by using a background field expansion ([37, 38, 39]) in the following way:
Xi(z,z) =¯ xi0+ζi(z,z)¯ ,
gij(x0+ζ) = ηij − 13Rikjlζkζl+. . . ,
Bij(x0+ζ) = Bij(x0) +∂kBij(x0)ζk+12∂k∂lBij(x0)ζkζl+. . . ,
(3.2.1)
whereζi(z,z) is a fluctuation around the constant point¯ x0 on the target space,ηij is the flat background metric andRikjl is the Riemann tensor corresponding to the metric gij(x0). The calculation of correlation functions involving the fluctuation ζi was done e.g. in [37] and results in the following product “•” of functionsf, g:
f•g =f ? g− 121 θil∂lθjk (∂i∂jf ? ∂kg+∂kf ? ∂i∂jg) +O (∂θ)2, ∂2θ
, (3.2.2)
where ? denotes the standard Moyal-Weyl product (3.1.1) and we expanded up to combinations where only one derivative of the non-commutativity parameterθ is involved. The latter is determined by the background fields η, B introduced in (3.2.1) and takes a similar form as in the last section:
θij = − 1
η−BB 1 η+B
ij
. (3.2.3)
It was observed in addition that the•-product whose terms can be computed order by order perturbatively is not associative :
(f•g)•h−f •(g•h)
= 16θimθjnθklHmnl∂if ? ∂jg ? ∂kh+O(∂2), (3.2.4) where we introduced theH-flux as the exterior differential of the two-form which we get by inverting the non-commutativity parameter θ: H := d(θ−1). The associativity of the •-product is restored in the limitH →0.
Finally, in analogy to the non-commutativity of spacetime coordinates (3.1.8) we can see the non-associativity of spacetime by introducing a bracket with three arguments which is determined by the Jacobi-identity of the commutator [·,·]• of the •-product:
[f, g, h] := [f,[g, h]•]•+ [g,[h, f]•]•+ [h,[f, g]•]• . (3.2.5)
3.2 General B-field: Non-associativity 31 Evaluating this bracket on the spacetime coordinates xi, xj, xk leads to a very simple expression [37], determined by the measure of non-associativity H:
xi, xj, xk
= θimθjnθklHmnl . (3.2.6) To sum up, in the case of generalB-field backgrounds, it is still possible to calculate star products perturbatively by using a background field expansion. However, the resulting deformation of the classical product is non-associative. The strength of this non-associativity is measured by d(θ)−1.
If we look again at the expression (3.2.3) for the non-commutativity parameter, we observe that for strongB-fields B η, the expression reduces to θ=B−1, i.e.
the non-associativity in (3.2.6) is determined by the original NS-NS H-flux.
It is well known from closed string theory that considering T-duals to configu-rations with H-flux leads to so-called geometric and non-geometric fluxes3. Thus a natural question would be if one can see similar effects of non-associativity also in the closed string case with non-vanishing H-flux and its T-dual configurations.
We will come back to this question in later chapters.
3We will introduce some important facts about geometric and non-geometric fluxes in the next chapter.
Chapter 4
Non-geometric flux backgrounds
Compactifications of string theory on geometric manifolds serve as a very rich source of interesting mathematics and physics. Calculations of correlation func-tions in the topological string lead to far reaching insights into topics on the mathematical frontier like mirror symmetry or Gromov-Witten theory. On the physical side, they were even more inspiring: Compactification on complex three-dimensional Calabi-Yau manifolds and their orientifolds together with intersecting branes are the ingredients of constructing realistic models of particle physics and cosmology.
However, from a conformal field theory point of view, geometric compactifica-tions are only a subset of possible string theory models, maybe they are not even the generic case. For example, in theories like asymmetric orbifolds, there is no geometric interpretation of the target space.
In the case of geometric compactifications involving NS-NS three-form flux H, in the last decade it became clear that T-duality may connect such compactifi-cations to backgrounds which go beyond the framework of differential geometry [40, 41, 42, 43, 44, 22, 23, 45, 20, 46, 47, 48, 49, 50]. Starting with an approximate solution to the string equations of motion given by a torus with H-flux, T-duality in an isometric direction leads to a twisted torus. Whereas this can still be des-cribed by geometric f-flux, a second T-duality results in an object whose set of transition functions between coordinate charts has to be extended to include also the T-duality group. It is called T-fold and is characterized by the non-geometric Q-flux. Even though no isometry direction being left, one can perform a formal third T-duality and there are hints that the resulting space which carries R-flux is non-commutative or even non-associative. This is often summarized in the fol-lowing chain of dualities:
Habc Ta
←−→ fabc Tb
←−→ Qcab Tc
←−→ Rabc. (4.0.1)
In this chapter, we start by describing the action of T-duality when there are
isometric directions on the target space manifold. The Buscher rules give a des-cription how to get the dual geometry in this case and we study in more detail the duality action for constant background fields in the case of compactification on a torus. We continue by giving a brief motivation for the mathematical structure of non-geometric Q- and R-fluxes, which is followed by the most prominent exam-ple of compactification on a torus with constant H-flux and its T-dual versions.
Finally, we sketch the four-dimensional effective viewpoint. The aim for having an effective superpotential whose coefficients map bijectively to each other when going from type IIA to type IIB compactifications was one of the first motivations to introduce nongeometric fluxes.
Due to the enormous amount of activity in the field, this will be far from being complete. Important topics like the formalism of double field theory (e.g.
[51, 52, 53, 54]) will not be touched. The goal of this chapter is to concentrate on the structural aspects which are important for later sections.
4.1 Introduction: The Buscher rules 35
4.1 Introduction: The Buscher rules
T-duality is an example of a symmetry which is inherent to string theory due to its fundamental objects, which are one-dimensional instead of the pointlike character of particles. Intuitively, the statement of duality is that string theories on T-dual backgrounds are equivalent, i.e. a string cannot distinguish between T-dual geometries. This has the advantage that one can extract information about exotic geometries by studying string theory on dual backgrounds. The easiest examples are circle compactifications of radii R and R−1, which we review briefly in the following.
Take for simplicity closed bosonic string theory with target space topology R1,24×SR1, i.e. compactification on a circle with radius R. Denoting by n, w the momentum and winding numbers, respectively and byN,N¯ the oscillator numbers, the mass formula for the closed string spectrum (e.g. [55, 56]) is given by:
M2 = n2
R2 + w2R2 (α0)2 + 2
α0 N + ¯N −2
. (4.1.1)
Thus the spectrum is invariant under the T-duality transformation R → α0
R, n↔w , (4.1.2)
meaning that the two theories on circles with radiiR andα0/Rare equivalent. On the level of worldsheet sigma-models, this phenomenon can be seen by Buscher’s procedure [16, 57]. To simplify notation, we set α0 = 1. Consider the sigma model on the sphere P1 with target space metric Gab and NS-NS B-field Bab:
S= 1 2π
Z
P1
d2z (Gab+Bab)∂Xa∂X¯ b . (4.1.3) We observe that the theory is invariant under the change of target space coordinate fields δXi =vi, if vi are the components of an isometry directionv of the metric and the H-field and is a small parameter:
LvG= 0,
LvH = 0 →LvB = dω , (4.1.4)
where H = dB and ω is an arbitrary one-form. In addition, if we have a dilaton term
Sd= 1 2π
Z
d2z φR(2) , (4.1.5)
the dilaton condition vi∂iφ has to be satisfied. Having an isometry of this type, we can introduce coordinates {θ :=X0, Xa} in such a way that the fields do not depend on the isometry direction θ. Consequently, we are able to introduce an action with trivial dependence onθand an additional variable ˜θwhich first appears as a Lagrange multiplier but later turns out to be the isometry coordinate of the dual theory. Together with auxiliary fields A,A¯this action is given by:
S1 = Z
d2z
G00AA¯+ (G0a+B0a)A∂X¯ a+ (Ga0+Ba0)∂XaA¯ + (Gab+Bab)∂Xa∂X¯ b+ ˜θ ∂A¯−∂A¯
.
(4.1.6)
Integrating out the Lagrange multiplier field ˜θ results in
∂A¯−∂A¯ = 0, (4.1.7)
i.e. in our case we can take ¯A = ¯∂θ, A = ∂θ as a solution and we recover the original action (4.1.3). Integrating out instead the auxiliary fieldsA and ¯A results in
A= − 1
G00(Ga0+Ba0)∂Xa+ 1 G00∂θ ,˜ A¯= − 1
G00
(G0a+B0a) ¯∂Xa− 1 G00
∂¯θ ,˜
(4.1.8)
and we arrive at adual sigma model with similar structure if we identify the field θ˜with the dual isometry direction ˜X0:
S˜= 1 2π
Z
P1
d2z
G˜ab+ ˜Bab
∂X˜a∂¯X˜b , (4.1.9) where the new fields are given by the Buscher rules [16, 57, 58]:
G˜00 = 1
G00 , G˜0a= B0a
G00, G˜ab = Gab− Ga0G0b+Ba0B0b
G00 , (4.1.10)
B˜0a= G0a
G00, B˜ab = Bab− Ga0B0b+Ba0G0b
G00 . (4.1.11)
Noting that the metric on a circle of radius R can be given by G00 = R2, we can recover the statement concluded from the mass formula (4.1.1) as a special case.
The above arguments are completely on the classical level. Quantum mechanically, the proof of the equivalence of T-dual theories is subtle (see for example [58]). We only want to mention that the transformation laws for the fields receive corrections
4.2 Torus compactification and O(d, d;Z)-duality 37