4.3 Non-geometric fluxes
5.1.2 Conformal field theory with H-flux
Instead of the exactly solvable WZW-model used in [74] we consider a simpler example of flat space with constant H-flux (see also section 4.3.3). Let us again recall the metric and flux to set our conventions
ds2 =
N
X
a=1
dXa2
, H = 2
α02 abcdXa∧dXb∧dXc. (5.1.2) Note that this ansatz does not solve the string equations of motion. Looking for example at the renormalization group equation for the graviton,
0 =α0Rab−α0
4 HacdHbcd+ 2α0∇a∇bΦ +O(α02), (5.1.3) we see that the ansatz (5.1.2) is only a solution up to first order in theH-flux. It turns out that this is true also for the other equations of motion and therefore we expect to get a proper conformal field theory description for this background up to first order in the flux. In the following we are going to formulate this theory in order to implement T-duality by a simple reflection of the right moving part of the closed string. Starting out by the geometric model (5.1.2), we can then compute correlation functions of coordinate fields and vertex operators in T-dual situations and try to extract information on their target space interpretation.
1For a phase-space realization of this type of algebra, see [40].
5.1 Conformal field theory with H-flux and T-duality 59 Sigma model and classical solution
Let us start by the closed string sigma model S = 1
2πα0 Z
Σ
d2z gab+Bab
∂Xa∂Xb , (5.1.4)
and take an the following background fields to implement the ansatz (5.1.2):
gab = δab , Bab = 1
3HabcXc. (5.1.5)
The classical equations of motion for the fields Xa are given by
∂∂X¯ a= 1
2Habc∂Xb∂X¯ c, (5.1.6) where we raise and lower indices with the metric gab =δab. Thus, at zeroth order in the H-flux, the B-field vanishes and we get the free theory of a closed string on the sphere. Its solution can be split into left- and right-moving parts
Xa0(z, z) = XaL(z) +XaR(z). (5.1.7) This is not possible at linear order in the flux, for which a solution to the equations of motion is readily calculated to be
Xa1(z, z) = Xa0(z, z) + 12HabcXbL(z)XcR(z). (5.1.8) This is already a hint that for a conformal field theory also at linear order in the flux, we have to redefine the coordinate fields in an appropriate way. We will see this in the definition of currents in the following subsections.
Perturbation theory
Let us shortly recall our way to compute correlation functions. The sigma model action can be split into a free and an interaction part if we use the ansatz (5.1.5):
S = 1 2πα0
Z
Σ
d2z δab∂Xa∂X¯ b+1
3HabcXa∂Xb∂X¯ c
=: S0+S1 , (5.1.9) where the first part S0 determines the (free) propagator, which we recall to be
Xa(z1, z1)Xb(z2, z2)
0 =−α0
2 log|z1−z2|2δab , (5.1.10)
and the second part S1 is treated as a perturbation for small flux, as it is for example the case in the large volume or dilute flux limit. Correlation functions can then be computed in the standard way by path integration:
hO1. . .ONi= 1 Z
Z
[dX]O1. . .ONe−S[X], (5.1.11) where we denote the vacuum functional by Z = R
[dX]e−S[X]. Expanding the expression (5.1.11) into a power series in the perturbation and denoting the ex-pectation value with respect to the free action by h. . .i0 gives, up to first order in the flux,
hO1. . .Oni=hO1. . .Oni0−
hO1. . .OnS1i0− hO1. . .Oni0hS1i0 + 1
2
hO1. . .OnS12i0− hO1. . .Oni0hS12i0
−
hO1. . .OnS1i0− hO1. . .Oni0hS1i0
hS1i0+. . .
=hO1. . .Oni0− hO1. . .OnS1i0+O(H2),
(5.1.12)
where in the second step we used the fact thathS1i0 = 0 because S1 is a product of an odd number of fields. We will use this formula in the following to compute two- and three-point correlation functions of various fields.
Currents and correlation functions
The standard currents of the free theory Ja =i∂Xa and ¯Ja=i∂X¯ a are not holo-morphic and anti-holoholo-morphic objects any more. Even at linear order in the flux, the classical equations of motion (5.1.6) show, that theH-flux mixes holomorphic and anti-holomorphic parts.
To get proper holomorphic and anti-holomorphic currents we have to add ad-ditional terms toJa,J¯a that compensate the mixed terms in (5.1.8) at least up to linear order in the flux. We therefore propose the following currents:
Ja(z,z) =¯ i∂Xa(z,z)¯ − 2iHabc∂Xb(z,z)¯ Xc(z,z)¯ , J¯a(z,z) =¯ i∂X¯ a(z,z)¯ − 2iHabcXb(z,z) ¯¯ ∂Xc(z,z)¯ .
(5.1.13) By using (5.1.8), it is easy to see that they are holomorphic and anti-holomorphic fields up to linear order in the flux. We therefore introduce the notation Ja(z) and ¯Ja(¯z) meaning (anti-)holomorphicity up to linear order, i.e.
∂J¯ a(z) = 0 +O(H2), ∂J¯(¯z) = 0 +O(H2). (5.1.14)
5.1 Conformal field theory with H-flux and T-duality 61 Let us now compute two-point functions of the currentsJ and ¯J. The interaction Lagrangian S1 in (5.1.9) has an odd number of fields and therefore, up to linear order in the flux, only the first term in (5.1.12) is relevant:
Ja(z1)Jb(z2)
=
i∂Xa(z1, z1)i∂Xb(z2, z2)
0 = α0 2
1
(z1−z2)2 δab , Ja(z1)Jb(z2)
=
i∂Xa(z1, z1)i∂Xb(z2, z2)
0 = α0 2
1
(z1−z2)2 δab , Ja(z1)Jb(z2)
=
i∂Xa(z1, z1)i∂Xb(z2, z2)
0 = 0 .
(5.1.15)
We observe that up to linear order in the flux, the two-point functions of the new currents are the same as for the free theory. Let us therefore move on to the three-point functions. Now, the interaction Lagrangian in (5.1.12) contributes to the result. As an example, for three holomorphic currents, we have to compute:
Ja(z1)Jb(z2)Jc(z3)
= i
∂Xa(z1,z¯1)∂Xb(z2,z¯2)∂Xc(z3,z¯3)S1
0
= i Hpqr 6πα0
Z
Σ
d2z
∂Xa(z1,z¯1)∂Xb(z2,z¯2)∂Xc(z3,z¯3)×
× Xp(z, z)∂Xq(z, z)∂Xr(z, z)
0 .
(5.1.16)
The last expression can now be evaluated by using Wick’s theorem and the free propagator (5.1.10). The computation is straightforward and we only want to men-tion the mixed holomorphic and anti-holomorphic derivatives of the propagator, as it involves the two-dimensional delta function:
∂z1∂z2log|z1−z2|2 =−2π δ(2)(z1−z2). (5.1.17) In contrast to the free theory, there are now non-vanishing three-point functions of purely holomorphic and purely holomorphic currents. Taking the anti-symmetry of the H-flux into account, the result of applying Wick’s theorem is given by:
Ja(z1)Jb(z2)Jc(z3)
=−iα02
8 Habc 1 z12z23z13 , Ja(z1)Jb(z2)Jc(z3)
= +i α02
8 Habc 1 z12z23z13 ,
(5.1.18)
and all the mixed holomorphic and anti-holomorphic currents vanish. We use the standard notationzij =zi−zj. Note again that for the non-vanishing of the above three-point correlators already at first order in the flux, the interaction term S1 was crucial.
Basic three-coordinate correlator
We take the holomorphicity and anti-holomorphicity of the currentsJaand ¯Ja re-spectively as a motivation to introduce new coordinatesXa, which are the integrals of the currents, i.e. we define them by the following relation:
Ja(z) = i∂Xa(z, z), Ja(z) = i∂Xa(z, z). (5.1.19) As we will see later, the currentsJa are the proper conformal fields in our theory and therefore we propose the coordinates defined by (5.1.19) to be the right ob-jects to get information about the target space geometry. Whereas the two-point function does not change in comparison to the uncorrected fieldsXaat linear order in the flux, the tree-point function can be determined by integrating e.g. the first correlator in (5.1.18). To state the result, we use the Rogers dilogarithm function of a complex variable L(z), defined by:
L(z) = Li2(z) + 1
2log(z) log(1−z), (5.1.20) where Li2(z) is the standard dilogarithm function2. The three-point correlator of the fields Xa is now given by:
Xa(z1, z1)Xb(z2, z2)Xc(z3, z3)
= α02 12 Habc
L
z12
z13
+L z23
z21
+L z13
z23
−c.c.
+F . (5.1.21) Here, we included the integration constants in a single functionF. It is determined by the condition ∂i∂j∂kF = 0, where i ∈ {z1,z¯1}, j ∈ {z2,z¯2}, k ∈ {z3,z¯3}. This is similar to the propagator for standard coordinate fields Xa(z,z). It is only¯ determined up to integration constants:
Xa(z1, z1)Xb(z2, z2)
0 =−α0 2
log|z1−z2|2+f(z1, z1) +f(z2, z2) δab.
(5.1.22) But in the two-point case, one can show that physical amplitudes do not depend on the integration constants and therefore they can be set to zero. At the moment a similar statement for the three-point function is not possible but we still set F = 0 in the following.
2We refer the reader to the appendix for a short introduction and properties of the Rogers-and stRogers-andard dilogarithm functions.
5.1 Conformal field theory with H-flux and T-duality 63 To further simplify the result (5.1.21), we introduce the flux parameter θabc =
(α0)2
12 Habcand use the following function which is composed of Rogers dilogarithms with characteristic arguments:
L(z) =L(z) +L
1− 1 z
+L
1 1−z
. (5.1.23)
Taking this into account, we can rewrite the three-point correlator in the following compact form:
Xa(z1, z1)Xb(z2, z2)Xc(z3, z3)
=θabch L zz12
13
− L zz12
13
i
. (5.1.24) This remarkable result will be the main source of a deformed product on the target space as we will see later by computing scattering amplitudes of tachyon vertex operators. But before moving on in this direction, we will first show how a conformal field theory can be constructed up to linear order in the H-flux and how T-duality is realized.
Conformal field theory linear in H
To first order in the flux, we were able to define holomorphic and anti-holomorphic currents Ja(z) and ¯Ja(¯z). The goal of this section is to give the main arguments that, up to linear order in H, it is possible to construct a conformal field theory.
We will analyse the operator product expansions of the currents and then define an energy momentum operator for which the currents are primary fields of dimension one, which is known as the Sugawara construction. Finally, in the next subsection, we will introduce tachyon vertex operators.
The operator product expansion3 (OPE) of two currents Ja(z1), Jb(z2) and their anti-holomorphic counterparts can be derived by computing correlation func-tions with other fields. The singular part of the OPE can be fixed by looking at the two- and three-point functions (5.1.15) and (5.1.18) of the currents:
Ja(z1)Jb(z2) = α0 2
δab
(z1 −z2)2 − α0 4
i Habc
z1−z2 Jc(z2) + reg. , Ja(z1)Jb(z2) = α0
2
δab
(z1−z2)2 +α0 4
i Habc
z1 −z2 Jc(z2) + reg. .
(5.1.25)
We denoted the regular part by “reg.” and the OPE of a holomorphic and anti-holomorphic current is purely regular. As a next step, let us construct the energy
3For standard techniques in conformal field theory we refer the reader to the literature, e.g.
[75, 76, 77].
momentum tensor in a way that the Ja are the right currents of the theory:
T(z) = 1
α0 δab :JaJb: (z), T(z) = 1
α0 δab :JaJb: (z). (5.1.26) As we are working only up to first order in the flux, we have to check whether all the axioms of an energy momentum tensor are really obeyed up to this order. At first, the OPEs for two energy momentum tensors have the right form, as we can check using the anti-symmetry of H:
T(z1)T(z2) = c/2
(z1−z2)4 + 2T(z2)
(z1−z2)2 +∂T(z2)
z1−z2 + reg. , T(z1)T(z2) = c/2
(z1−z2)4 + 2T(z2)
(z1−z2)2 +∂T(z2)
z1−z2 + reg. ,
(5.1.27)
whereas the OPE of a holomorphic and an anti-holomorphic energy-momentum tensor is purely regular. This result has the canonical form and thus, as in the standard case, we get two copies of the Virasoro algebra with the same central charge as for the free theory (for the case of the three-dimensional background we have c = 3). For the second step, we have to check whether the currents are conformal primary of dimension one. Calculating the OPE with the energy-momentum tensor (5.1.26), we get
T(z1)Ja(z2) = Ja(z2)
(z1−z2)2 +∂Ja(z2)
z1−z2 + reg. , T(z1)Ja(z2) = reg. .
(5.1.28)
Similar OPEs can be computed for the anti-holomorphic parts. To put it in a nutshell, up to linear order in the flux, we are able to define an energy-momentum operator (5.1.26) with respect to which the currentsJa(z) and ¯Ja(¯z) are conformal primary fields of dimension (1,0) and (0,1), respectively. Furthermore, their OPEs (5.1.25) have the form of a non-Abelian current algebra with structure constants Habc. Up to linear order in the flux, we therefore made the first steps to construct a conformal field theory framework. In the following we want to denote this theory by CFTH.