Modular forms of weight 8 for Γ g (1, 2)

Currently there are two proposed ansätze for NSR superstring measures: the Grushevsky ansatz and the OPSMY ansatz, which for genera g ≤ 4 are known to coincide. However, neither the Grushevsky nor the OPSMY ansatz leads to a vanishing two point function in genus four, which can be constructed from the genus five expressions for the respective ansätze. This is inconsistent with the known properties of superstring amplitudes.In the present paper we show that the Grushevsky and OPSMY ansätze do not coincide in genus five. Then, by combining these ansätze, we propose a new ansatz for genus five, which now leads to a vanishing two-point function in genus four. We also show that one cannot construct an ansatz from the currently known forms in genus 6 that satisfies all known requirements for superstring measures.

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“…The OPSMY ansatz from [35] is constructed using lattice theta series, defined as follows for any lattice Λ ⊂ R n :…”

Section: For a Theta Characteristicmentioning

confidence: 99%

“…Then, the second ansatz was formulated in terms of theta series for 16-dimensional self-dual lattices by M. Oura, C. Poor,R. Salvati Manni and D. Yuen (OPSMY) in [35]. This second ansatz, however, is only defined for genera g ≤ 5.…”

Section: Introductionmentioning

confidence: 99%

NSR superstring measures in genus 5

2013

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“…This Ansatz can be satisfied through g ≤ 5 but is thought unlikely to extend further [18]. Over the hyperelliptic locus, however, the corresponding conditions are solved for all g by a family of binary invariants, see [19].…”

Section: An Applicationmentioning

confidence: 99%

The Chiral Superstring Siegel Form in Degree Two Is a Lift

Poor

Yuen

2012

Journal of the Korean Mathematical Society

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Abstract. We prove that the Siegel modular form of D'Hoker and Phong that gives the chiral superstring measure in degree two is a lift. This gives a fast algorithm for computing its Fourier coefficients. We prove a general lifting from Jacobi cusp forms of half integral index t/2 over the theta group Γ 1 (1, 2) to Siegel modular cusp forms over certain subgroups Γ para (t; 1, 2) of paramodular groups. The theta group lift given here is a modification of the Gritsenko lift.

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“…For g > 5 some issues arise due to the presence of holomorphic roots of modular forms in the definition of the chiral measure, and it is not clear whether such roots are well defined and have the correct modular properties. In [14], Oura, Poor, Salvati Manni and Yuen (OPSMY) proposed an alternative construction for the chiral measure up to g = 5, using lattice theta series rather than theta constants, as done by Grushevsky. They also proved that the solution to the constraints is unique up to g = 4.…”

Section: Introductionmentioning

confidence: 99%

“…Another simple consequence is that the proposed three-point amplitude does not vanish at genus four as requested by the non-renormalisation theorem. We conclude this section by considering the OPSMY ansatz for the superstring measure in terms of theta lattices [14].…”

Section: Introductionmentioning

confidence: 99%

Getting superstring amplitudes by degenerating Riemann surfaces

Matone

Volpato

2010

Nuclear Physics B

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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.

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Modular forms of weight 8 for Γ g (1, 2)

Cited by 15 publications

References 27 publications

NSR superstring measures in genus 5

NSR superstring measures in genus 5

The Chiral Superstring Siegel Form in Degree Two Is a Lift

Getting superstring amplitudes by degenerating Riemann surfaces

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