Getting superstring amplitudes by degenerating Riemann surfaces

Abstract

We explicitly show how the chiral superstring amplitudes can be obtained through factorisation of the higher genus chiral measure induced by suitable degenerations of Riemann surfaces. This powerful tool also allows to derive, at any genera, consistency relations involving the amplitudes and the measure. A key point concerns the choice of the local coordinate at the node on degenerate Riemann surfaces that greatly simplifies the computations. As a first application, starting from recent ansätze for the chiral measure up to genus five, we compute the chiral two-point function for massless Neveu-Schwarz states at genus two, three and four. For genus higher than three, these computations include some new corrections to the conjectural formulae appeared so far in the literature. After GSO projection, the two-point function vanishes at genus two and three, as expected from space-time supersymmetry arguments, but not at genus four. This suggests that the ansatz for the superstring measure should be corrected for genus higher than four.

NSR superstring measures in genus 5

“…, by the method used in [27]. We show that this, together with the condition of vanishing genus 5 cosmological constant leads to our main formula (6.25): a unique ansatz built from the known modular forms in genus 5.…”

“…, 5. However, it was shown by M. Matone and R. Volpato in [27] that the genus 4 two-point function obtained by degeneration from the OPSMY ansatz in genus 5 does not vanish, contrary to requirement c). The results of the present paper imply that the same problem occurs with the Grushevsky ansatz as well.…”

“…The first term in the q-expansion is easy to obtain, and we will obtain the q-linear term as in [27] by writing Now the first terms of the series in q will be:…”

“…Because the norm 2 vectors are the same as those from D 8 , we can use the fact that 28) which is mentioned and used in [27]. We then obtain…”

“…The second statement holds assuming that the Matone-Volpato method [27] is the correct way to obtain the genus 4 two-point function from the genus 5 zeropoint function. Then, we look at the situation in genus 6.…”