The KLT relations in string theory

3.1 The KLT relations in string theory

The KLT relations between open and closed string theory amplitudes can be motivated by the observation that any closed string vertex operator for the emission of a closed string state (such as a graviton) is a product of open string vertex operators (see e.g. Ref. [70

]),

This product structure is then reflected in the amplitudes. Indeed, the celebrated Koba–Nielsen form of string amplitudes [89 ], which may be obtained by evaluating correlations of the vertex operators, factorize at the level of the integrands before world sheet integrations are performed. Amazingly, Kawai, Lewellen, and Tye were able to demonstrate a much stronger factorization: Complete closed string amplitudes factorize into products of open string amplitudes, even after integration over the world sheet variables. (A description of string theory scattering amplitudes and the history of their construction may be found in standard books on string theory [70

, 110 , 111 ].)

As a simple example of the factorization property of string theory amplitudes, the four-point partial amplitude of open superstring theory for scattering any of the massless modes is given by

where $α′$ is the open string Regge slope proportional to the inverse string tension, $g$ is the gauge theory coupling, and $K$ is a gauge invariant kinematic coefficient depending on the momenta $k1,...,k4$ . Explicit forms of $K$ may be found in Ref. [70

]. (The metric is taken here to have signature $(+, − ,− ,− )$ .) In this and subsequent expressions, $2 s = (k1 + k2)$ , $2 t = (k1 + k2)$ , and $u = (k1 + k3)2$ . The indices can be either vector, spinor or group theory indices and the $ξA$ can be vector polarizations, spinors, or group theory matrices, depending on the particle type. These amplitudes are the open string partial amplitudes before they are dressed with Chan–Paton [106

] group theory factors and summed over non-cyclic permutations to form complete amplitudes.(Any group theory indices in Eq. (6

) are associated with string world sheet charges arising from possible compactifications.) For the case of a vector, $ξA$ is the usual polarization vector. Similarly, the four-point amplitudes corresponding to a heterotic closed superstring [76 , 77 ] are

( ) closed 2 α′πt Γ (α′s∕4)Γ (α ′t∕4) Γ (α ′t∕4)Γ (α ′u∕4 ) M 4 = κ sin ----- × -------′-------′-----× --------′------′----- ′ 4′ ′ Γ (1′ + α s∕4 + α t∕4) Γ (1 + α t∕4 + α u∕4) × ξAA ξBB ξCC ξDD KABCD (k1∕2, k2∕2,k3∕2,k4∕2 ) ×KA ′B′C′D ′(k1∕2, k2∕2,k3∕2,k4∕2 ), (7 )

where $α′$ is the open string Regge slope or equivalently twice the close string one. Up to prefactors, the replacements $ξAξA′ → ξAA ′$ and substituting $ki → ki∕2$ , the closed string amplitude (7

) is a product of the open string partial amplitudes (6

). For the case of external gravitons the $AA ′ ξ$ are ordinary graviton polarization tensors. For further reading, Chapter 7 of Superstring Theory by Green, Schwarz, and Witten [70 ] provides an especially enlightening discussion of the four-point amplitudes in various string constructions.

As demonstrated by KLT, the property that closed string tree amplitudes can be expressed in terms of products of open string tree amplitudes is completely general for any string states and for any number of external legs. In general, it holds also for each of the huge number of possible string compactifications [102 , 103 , 49 , 50 , 87 , 3 ].

An essential part of the factorization of the amplitudes is that any closed-string state is a direct product of two open-string states. This property directly follows from the factorization of the closed-string vertex operators (5 ) into products of open-string vertex operators. In general for every closed-string state there is a Fock space decomposition

In the low energy limit this implies that states in a gravity field theory obey a similar factorization,

For example, in four dimensions each of the two physical helicity states of the graviton are given by the direct product of two vector boson states of identical helicity. The cases where the vectors have opposite helicity correspond to the antisymmetric tensor and dilaton. Similarly, a spin 3/2 gravitino state, for example, is a direct product of a spin 1 vector and spin 1/2 fermion. Note that decompositions of this type are not especially profound for free field theory and amount to little more than decomposing higher spin states as direct products of lower spin ones. What is profound is that the factorization holds for the full non-linear theory of gravity.