Abstract
We exploit the correspondence between twisted homology and quantum group to construct an algebra explanation of the open string kinematic numerator. In this setting the representation depends on string modes, and therefore the cohomology content of the numerator, as well as the location of the punctures. We show that quantum group root system thus identified helps determine the Casimir appears in the Knizhnik-Zamolodchikov connection, which can be used to relate representations associated with different puncture locations.
Highlights
Both sets of numerators of a φ3 are replaced by kinematic numerators, the result correctly reproduces the amplitude of gravitons coupled to dilatons and B fields
We exploit the correspondence between twisted homology and quantum group to construct an algebra explanation of the open string kinematic numerator
As we should see in the following discussions that the tensor of a string Hilbert state with an SL(2, R) fixed vertex operator V (z) serves as a non-trivial two-particle representation of the quantum universal enveloping algebra (QUEA) 1 ⊗ 2 with the deformation parameter given by braiding factor q = e−iπα produced by string monodromy, while the rest of the string insertions, being integrated over C-shaped contours in the kinematic numerator, act on the two particle state as symmetry algebra generators Ei
Summary
We briefly review the notion of quantum groups. Historically quantum groups first appeared in the inverse scattering problem of integrable systems [88,89,90] and was independently discovered by Drinfeld [91] and Jimbo [92] when generalising classical Lie algebra through deformations. As we should see in the following discussions that the tensor of a string Hilbert state with an SL(2, R) fixed vertex operator V (z) ( known as an evaluation module [83]) serves as a non-trivial two-particle representation of the QUEA 1 ⊗ 2 with the deformation parameter given by braiding factor q = e−iπα produced by string monodromy, while the rest of the string insertions, being integrated over C-shaped contours in the kinematic numerator, act on the two particle state as symmetry algebra generators Ei. In this paper we do not assume particular value for the inverse string tension α , especially we rely on α being able to adjust freely to obtain field theory amplitudes from string theory ones as their point particle limit, so that generically q not being a root of unity, in which case the representation of the quantum algebra is known to be given by the Verma module Mλq = IndH⊕F Cλ: starting with a highest weight, namely highest eigenvalue, 1-dimensional complex vector vλ,. The action of antipode and counit are represented by reversing and removing the contour of a screening respectively
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