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Tree-Level Superstring Amplitudes: The Neveu-Schwarz Sector

Sergio Luigi Cacciatori, Samuel Grushevsky, Alexander A. Voronov

Abstract

We present a complete computation of superstring scattering amplitudes at tree level, for the case of Neveu-Schwarz insertions. Mathematically, this is to say that we determine explicitly the superstring measure on the moduli space $\mathcal{M}_{0,n,0}$ of super Riemann surfaces of genus zero with $n \ge 3$ Neveu-Schwarz punctures. While, of course, an expression for the measure was previously known, we do this from first principles, using the canonically defined super Mumford isomorphism. We thus determine the scattering amplitudes, explicitly in the global coordinates on $\mathcal{M}_{0,n,0}$, without the need for picture changing operators or ghosts, and are also able to determine canonically the value of the coupling constant. Our computation should be viewed as a step towards performing similar analysis on $\mathcal{M}_{0,0,n}$, to derive explicit tree-level scattering amplitudes with Ramond insertions.

Tree-Level Superstring Amplitudes: The Neveu-Schwarz Sector

Abstract

We present a complete computation of superstring scattering amplitudes at tree level, for the case of Neveu-Schwarz insertions. Mathematically, this is to say that we determine explicitly the superstring measure on the moduli space of super Riemann surfaces of genus zero with Neveu-Schwarz punctures. While, of course, an expression for the measure was previously known, we do this from first principles, using the canonically defined super Mumford isomorphism. We thus determine the scattering amplitudes, explicitly in the global coordinates on , without the need for picture changing operators or ghosts, and are also able to determine canonically the value of the coupling constant. Our computation should be viewed as a step towards performing similar analysis on , to derive explicit tree-level scattering amplitudes with Ramond insertions.
Paper Structure (17 sections, 7 theorems, 146 equations)

This paper contains 17 sections, 7 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Basic conventions and constructions with SRSs and their moduli
  3. The super Mumford form
  4. Bases of cohomology
  5. A formula for the super Mumford form
  6. The super Kodaira-Spencer map over $\mathcal{M}_{0,n,0}$
  7. The construction
  8. Proof of the main theorem
  9. The Čech cohomology computation
  10. The dual map
  11. The Berezinian of the dual map
  12. Vertex-operator insertions
  13. Tree-level scattering amplitudes in the physics literature
  14. Conclusion
  15. The super Mumford form over $\mathcal{M}_{0,n,0}$
  16. ...and 2 more sections

Key Result

Theorem 1

The choice of conformal vertex-operator insertions VO gives for any $X\in\mathcal{M}_{0,n,0}$ a section of $\mathop{\mathrm{Ber}}\nolimits H^0(X, \newline \omega|_{\mathop{\mathrm{NS}}\nolimits})$. Then the canonical isomorphism NS-Mumford1 gives a holomorphic section of the line bundle $\omega_{\ma where and $z_j = z^0_j$, $j=1,2,3$, and $\zeta_a = \zeta^0_a$, $a=1,2$, are fixed finite values $(

Theorems & Definitions (13)

  • Theorem 1
  • Remark 1
  • Lemma 2: Folklore
  • Lemma 3: Folklore
  • proof
  • Remark 2
  • Proposition 4
  • proof
  • Lemma 5
  • Proposition 6
  • ...and 3 more