Tachyon correlation functions

We now have all the ingredients to compute n-point tachyon correlation functions in the CFT_H-framework and analyse the effect of T-duality. In particular we want to infer the product structure of functions on the target space and compare the case of the original H-flux with its complete T-dual version, where the R-flux is present.

Three tachyon correlator

Let us start by scattering three tachyons. To have winding, let us assume the tree-dimensional target space to be compact, e.g. a three-torus. We are going to compare the following two cases: First we want to analyse the standard case of pure momentum states in the presence of H-flux, where we label the momenta by 3-vectors p_i. The corresponding tachyon vertex operator was discussed in the previous two subsections and is given by

V_i^{− def}= V_pi(z_i,z¯_i) = : exp (i p_i· X(z_i,z¯_i)) : . (5.1.43) Secondly, we are interested in pure momentum state scattering in the presence of R-flux, which corresponds to pure winding state scattering in theH-flux background,

as can be seen in table 5.1. We denote the winding by three-vectors w_i and the corresponding tachyon vertex operator is given by

V_i^{+ def}= V_wi(z_i,z¯_i) = : exp

i w_i·X˜(z_i,z¯_i)

:, (5.1.44)

where the T-dual coordinate is denoted by ˜X =X_L− X_R. To proceed, let us note the following: As we are interested in scattering momentum states in an R-flux background, we setwi →pi in the vertex operator (5.1.44). The superscripth. . .i^∓ on correlators indicates that we are in theH-flux case for the minus sign and in the R-flux case for the plus sign. The result for the three-tachyon correlation function (for details of the calculation we refer the reader to the appendix) is then given by

V₁V₂V₃∓

= δ(p₁+p₂+p₃)

|z₁₂z₁₃z₂₃|² h

1−iθ^abcp_1,ap_2,bp_3,c L ^z_z¹²

13

∓ L ^z_z¹²

13

i

. (5.1.45) As we only calculated up to first order in the flux (i.e. the parameter θ^abc), we cannot make a definite statement about the full result of the correlator. But the form of (5.1.45) suggests, that it is the beginning of a power series expansion of the exponential function. To indicate that this may be possible we introduce the notation [. . .]θ, meaning that the result is only valid up to linear order in θ^abc:

V₁V₂V₃∓

= δ(p1+p2+p3)

|z₁₂z₁₃z₂₃|² exph

−iθ^abcp_1,ap_2,bp_3,c L ^z_z¹²

13

∓ L ^z_z¹²

13

i

θ. (5.1.46) Let us now investigate these results. Clearly the delta-function indicates usual momentum conservation. We first analyse the amplitude without this constraint, i.e. the off-shell correlator. Let σ∈S₃ be a permutation of three elements. Then the tachyon correlator with permuted vertex operators is given by:

V_σ(1)V_σ(2)V_σ(3)∓

= exp h

i ¹⁺₂

ησπ²θ^abcp1,ap2,bp3,c

i

V1V2V3

∓

, (5.1.47) where η_σ = 1 for an odd permutation and vanishes for an even one. In addition, the parameterindicates the background: We have=−1 for the originalH-flux, i.e. the phase vanishes in this case, as it is expected. In the case of the 3-times T-dual background, corresponding the R-flux, = 1 and there is a non-trivial phase factor. To derive this result, we used the following properties of the functionL(z), introduced in (5.1.23) as sum of Rogers dilogarithms:

L(z) = L 1−1 z

= L 1 1−z

, (5.1.48)

L(z) +L(1−z) = 3L(1) = π²

2 . (5.1.49)

5.1 Conformal field theory with H-flux and T-duality 71 These formulas follow easily from the corresponding properties of the Rogers dilog-arithm introduced in the appendix. As an example, consider exchangingz₂ and z₃ in the R-flux case:

V₁V₃V₂+

= exph

−iθ^abcp_1,ap_3,bp_2,c L ^z_z¹³

12

+L ^z_z¹³

12

i

θ

= exph

iθ^abcp_1,ap_2,bp_3,c

π²− L ^z_z¹²

13

− L ^z_z¹²

13

i

θ

= exp

iπ²θ^abcp_1,ap_2,bp_3,c V₁V₃V₂+

.

(5.1.50)

Similar calculations hold for the other permutations. Thus, off-shell (without momentum conservation), we get a non-trivial phase by performing an odd per-mutation of the vertex operators. Note that this is similar to the open string case, where off-shell permutations of tachyon vertex operators resulted in a phase which was then seen as a sign for the Moyal-Weyl star product, as was shown in chapter 3, especially in the expressions (3.1.13). In the closed string case, the correspond-ing phase is governed by the three-index objectθ^abc (5.1.47), and we will interpret this phenomenon as a hint for a three-product.

Note, that on-shell, the phase vanishes due to momentum conservation:

p_1,ap_2,bp_3,cθ^abc= 0 for p₃ =−p₁−p₂ . (5.1.51) This was expected, since in scattering amplitudes a product of field operators is radially ordered and therefore changing the order of the operators will not affect the amplitude. Even in theR-flux background this should hold, as it is one of the defining properties of conformal field theory amplitudes.

N-tachyon correlator

We are now going to generalize the results of the last subsection to the case of scattering N tachyons. It turns out that this can be done inductively by applying the formulas presented in the appendix. Before stating the general result, we want to demonstrate the logic at the four point result. The correlation function of four tachyon vertex operators is given by

V₁V₂V₃V₄∓

=

V₁V₂V₃V₄∓ 0 × exp

−iθ^abc X

1≤i<j<k≤4

p_i,ap_j,bp_k,ch L ^z_z^ij

ik

∓ L ^z_z^ij

ik

i

θ

. (5.1.52) Although the phase becomes more complicated, it turns out that permuting the vertex operators by σ∈S₄ in theR-flux case results in similar phases as the ones encountered in the previous subsection, whereas they vanish in the case of the

H-flux. We again have to use the fundamental relations of the Rogers dilogarithm (5.1.48) to extract a phase independent of the world-sheet coordinates. Only such properties are of interest for us as they reflect true target space facts.

To illustrate this statement, let us write out explicitly the holomorphic part of the phase appearing in (5.1.52) in terms of Rogers dilogarithm functions:

−iθ^abch

p1,ap2,bp3,cL z₁₂

z₁₃

+p1,ap2,bp4,c L z₁₂

z₁₄

+p1,ap3,bp4,cL z13

z₁₄

+p2,ap3,bp4,c L z23

z₂₄ i

.

(5.1.53)

Exchanging for example V3 and V4, in the R-flux background, we get a relative phase of

iπ²θ^abc(p_1,ap_3,bp_4,c+p_2,ap_3,bp_4,c) , (5.1.54) and similar for other permutations. Note that the phase vanishes on-shell due to p4 =−p1−p2−p3 and the anti-symmetry of θ^abc so that the whole amplitude is invariant under permutations of the vertex operators.

The four tachyon amplitude played an important role in the history of string theory. Many important properties of the theory were found by analyzing for example the pole structure of the Virasoro-Shapiro amplitude. In our case it is also important to do the same steps in CFT_H. It turns out that this can be done [78] and it is intriguingly connected to the properties of the so-called extended-or Neumann-Rogers dilogarithm, which we briefly mention in the appendix for completeness. This analysis goes beyond the scope of this work and therefore we want to refer the reader to the original paper [78] for more details in this direction.

Finally we can give the general N-point tachyon amplitude. It is a straightfor-ward generalization of the previous cases:

V1V2 . . .VN

∓

=

V1V2 . . . VN

∓ 0 × exp

−iθ^abc X

1≤i<j<k≤N

pi,apj,bpk,c

h L ^z_z^ij

ik

∓ L ^z_z^ij

ik

i

θ

. (5.1.55) In addition, one can extract relative phases in the case of the R-flux by doing a permutationσ ∈S_N of the vertex operators which are similar to the cases before and will not be calculated explicitly. They do not depend on the coordinates on the world sheet and vanish on-shell. We will use these properties of the N-point correlator to speculate about the existence of an N-product on the algebra of functions on the target space in case of theR-flux background in the next section.

5.1 Conformal field theory with H-flux and T-duality 73

Im Dokument Lie algebroids, non-associative structures and non-geometric fluxes (Seite 81-85)