Abstract
We show that a recently conjectured expression for the superstring three-point amplitude, in the framework of the Cacciatori, Dalla Piazza, van Geemen–Grushevsky ansatz for the chiral measure, fails to vanish at three-loop, in contrast with expectations from non-renormalization theorems. Based on analogous two-loop computations, we discuss the possibility of a non-trivial correction to the amplitude and propose a natural candidate for such a contribution. Thanks to a new remarkable identity, it is reasonable to expect that the corrected three-point amplitude vanishes at three-loop, recovering the agreement with non-renormalization theorems.
Highlights
In the last years there has been a considerable progress in the conceptual understanding and in derivation of explicit formulas for multiloop superstring amplitudes
Analogous procedures led them to prove the non-renormalization of the cosmological constant and of the n-point functions, n ≤ 3, up to 2-loops, as expected by space-time supersymmetry arguments [7]
The 4-point amplitude has been computed and checked against the constraints coming from S-duality [8]
Summary
We provide the basic background on theta functions and Riemann surfaces necessary for the subsequent derivations. Given a Riemann surface C with marking, one can canonically associate to each theta bundle Lδ on characteristic C such that. Let δ be a non-singular theta characteristic (that is, such that at least one among θ[δ](z) and its first partial derivatives does not vanish at z = 0) and, for an arbitrary y ∈ C, set fδ,y(x) := θ[δ](x − y). It can be proved that such a divisor class [(fδ,y) − y], and Lδ, is independent of y ∈ C, so that for each marked Riemann surface we have a correspondence δ → Lδ. For each fixed b ∈ C, Sδ(a, b) is the unique meromorphic section of Lδ with a single pole of residue −1 at b and holomorphic elsewhere Such a characterization implies that, for a fixed spin bundle L, S(a, b; L) is independent of the marking. A, b, x ∈ C, the Abelian 1-differential of the second kind with single poles on a and b with residue +1 and −1, respectively, holomorphic on C \ {a, b}, and with vanishing α-periods
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