Open-string integrals with multiple unintegrated punctures at genus one

Kaderli, André; Rodriguez, Carlos

doi:10.1007/JHEP10(2022)159

Open-string integrals with multiple unintegrated punctures at genus one

André Kaderli
(
)
,
Carlos Rodriguez
(
- Uppsala U.
)

Mar 17, 2022

83 pages

Published in:

JHEP 10 (2022) 159

Published: Oct 25, 2022

e-Print:

2203.09649 [hep-th]

DOI:

10.1007/JHEP10(2022)159

Report number:

UUITP-14/22

View in:

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Abstract: (Springer)

We study integrals appearing in intermediate steps of one-loop open-string amplitudes, with multiple unintegrated punctures on the A-cycle of a torus. We construct a vector of such integrals which closes after taking a total differential with respect to the N unintegrated punctures and the modular parameter τ. These integrals are found to satisfy the elliptic Knizhnik-Zamolodchikov-Bernard (KZB) equations, and can be written as a power series in α′ — the string length squared- in terms of elliptic multiple polylogarithms (eMPLs). In the N-puncture case, the KZB equation reveals a representation of B

_{1,N}

, the braid group of N strands on a torus, acting on its solutions. We write the simplest of these braid group elements — the braiding one puncture around another — and obtain generating functions of analytic continuations of eMPLs. The KZB equations in the so-called universal case is written in terms of the genus-one Drinfeld-Kohno algebra

\mathfrak{t} _{1,N}

⋊

\mathfrak{d}

, a graded algebra. Our construction determines matrix representations of various dimensions for several generators of this algebra which respect its grading up to commuting terms.

Note:

44+39 pages and ancillary file

Differential and Algebraic Geometry
Scattering Amplitudes
string: scattering amplitude
tree approximation
string: open
torus
modular
braid group

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