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Superstring Scattering Amplitudes with the Pure Spinor Formalism

Carlos R. Mafra

TL;DR

The thesis develops and exploits the pure spinor formalism to compute superstring scattering amplitudes in a manifestly supersymmetric and covariant framework, addressing long-standing inefficiencies in RNS/GS approaches. By expressing massless four-point amplitudes at tree-level, one-loop, and two-loop in a compact pure spinor superspace form, it reveals explicit relations among Kinematic factors K_tree, K_one-loop, and K_two-loop, and demonstrates their supersymmetric consistency across bosonic and fermionic external states. It shows the equivalence of the minimal and non-minimal formalisms, including a covariant proof, and derives the anomaly-related kinematic factor K_anom with its moduli-space contributions, connecting to the ε_10 and t_8 structures. The work also provides concrete calculations of two-loop amplitudes, anomaly cancellation in type I with SO(32), and preliminary results for the massless five-point amplitude, highlighting the practical power and conceptual elegance of pure spinor techniques for higher-point and higher-loop string amplitudes.

Abstract

This thesis discusses how the pure spinor formalism can be used to efficiently compute superstring scattering amplitudes. We emphasize the pure spinor superspace form of the kinematic factors, where the simplifying features of this language have allowed an explicit relation among the massless four-point amplitudes at tree-level, one- and two-loops to be found. We show how these identities elegantly simplify the task of computing the amplitudes for all possible external state combination related by supersymmetry. In particular, the two-loop amplitudes involving fermionic states had never been computed before. By explicit calculation we show that the one- and two-loop amplitudes computed with the minimal and non-minimal formalisms are equivalent. Furthermore we compute the gauge variation of the massless six-point open string amplitude and obtain the kinematic factor related to the anomaly cancellation. We also discuss some preliminary results regarding the massless five-point amplitude at one-loop.

Superstring Scattering Amplitudes with the Pure Spinor Formalism

TL;DR

The thesis develops and exploits the pure spinor formalism to compute superstring scattering amplitudes in a manifestly supersymmetric and covariant framework, addressing long-standing inefficiencies in RNS/GS approaches. By expressing massless four-point amplitudes at tree-level, one-loop, and two-loop in a compact pure spinor superspace form, it reveals explicit relations among Kinematic factors K_tree, K_one-loop, and K_two-loop, and demonstrates their supersymmetric consistency across bosonic and fermionic external states. It shows the equivalence of the minimal and non-minimal formalisms, including a covariant proof, and derives the anomaly-related kinematic factor K_anom with its moduli-space contributions, connecting to the ε_10 and t_8 structures. The work also provides concrete calculations of two-loop amplitudes, anomaly cancellation in type I with SO(32), and preliminary results for the massless five-point amplitude, highlighting the practical power and conceptual elegance of pure spinor techniques for higher-point and higher-loop string amplitudes.

Abstract

This thesis discusses how the pure spinor formalism can be used to efficiently compute superstring scattering amplitudes. We emphasize the pure spinor superspace form of the kinematic factors, where the simplifying features of this language have allowed an explicit relation among the massless four-point amplitudes at tree-level, one- and two-loops to be found. We show how these identities elegantly simplify the task of computing the amplitudes for all possible external state combination related by supersymmetry. In particular, the two-loop amplitudes involving fermionic states had never been computed before. By explicit calculation we show that the one- and two-loop amplitudes computed with the minimal and non-minimal formalisms are equivalent. Furthermore we compute the gauge variation of the massless six-point open string amplitude and obtain the kinematic factor related to the anomaly cancellation. We also discuss some preliminary results regarding the massless five-point amplitude at one-loop.

Paper Structure

This paper contains 50 sections, 9 theorems, 632 equations.

Table of Contents

  1. Introduction
  2. The Ramond-Neveu-Schwarz formalism
  3. The Green-Schwarz formalism
  4. The Pure Spinor formalism
  5. The Pure Spinor Formalism
  6. Siegel's modification of the Green-Schwarz formalism
  7. The elements which led to the pure spinor formalism
  8. Lorentz currents for the ghosts
  9. The BRST operator
  10. The pure spinor formalism
  11. The action for the ghosts
  12. Massless vertex operators
  13. Tree-level prescription
  14. Multiloop prescription
  15. The non-minimal pure spinor formalism
  16. ...and 35 more sections

Key Result

Theorem 1

If the SO(10)-covariant OPE of the Lorentz currents $N^{mn}$ is given by then the U(5)-covariant currents $(n,n_a^b,n_{ab},n^{ab})$ satisfy the following OPE's:

Theorems & Definitions (12)

  • Definition 1: Pure Spinor
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Definition 2
  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Lemma 2
  • ...and 2 more