Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Covariant One-Loop Amplitudes in D=11

L. Anguelova, P. A. Grassi, P. Vanhove

TL;DR

The paper extends the covariant pure spinor framework to the 11d superparticle to compute one-loop amplitudes in M-theory, including $B\wedge X_8$, $C_3\wedge X_8$, and $\mathcal{F}^4$ terms, and verifies consistency by reproducing the tree-level 11d supergravity action. It develops a covariant BRST formalism, constructs 11d vertex operators, and defines a tree-level and one-loop measure that generalize known 10d results, showing agreement with established results in the appropriate limits. It also analyzes the relation between BRST and spinorial cohomology, establishing an extended BRST construction that recovers spinorial cohomology classes and clarifies how higher-derivative deformations fit into the cohomological framework. The work demonstrates the viability of the pure spinor approach for covariant loop calculations in eleven dimensions, discusses open issues such as UV regularization and membrane extensions, and outlines paths toward incorporating the full M2-brane in this covariant setting. Overall, the results solidify covariant loop techniques for M-theory effective actions and illuminate the interplay between BRST, spinorial cohomology, and higher-derivative corrections.

Abstract

We generalize to the eleven-dimensional superparticle Berkovits' prescription for loop computations in the pure spinor approach to covariant quantization of the superstring. Using these ten- and eleven-dimensional results, we compute covariantly the following one-loop amplitudes: C\wedge X_8 in M-theory; B\wedge X_8 in type II string theory and F^4 in type I. We also verify the consistency of the formalism in eleven dimensions by recovering the correct classical action from tree-level amplitudes. As the superparticle is only a first approximation to the supermembrane, we comment on the possibility of extending this construction to the latter. Finally, we elaborate on the relationship between the present BRST language and the spinorial cohomology approach to corrections of the effective action.

Covariant One-Loop Amplitudes in D=11

TL;DR

The paper extends the covariant pure spinor framework to the 11d superparticle to compute one-loop amplitudes in M-theory, including , , and terms, and verifies consistency by reproducing the tree-level 11d supergravity action. It develops a covariant BRST formalism, constructs 11d vertex operators, and defines a tree-level and one-loop measure that generalize known 10d results, showing agreement with established results in the appropriate limits. It also analyzes the relation between BRST and spinorial cohomology, establishing an extended BRST construction that recovers spinorial cohomology classes and clarifies how higher-derivative deformations fit into the cohomological framework. The work demonstrates the viability of the pure spinor approach for covariant loop calculations in eleven dimensions, discusses open issues such as UV regularization and membrane extensions, and outlines paths toward incorporating the full M2-brane in this covariant setting. Overall, the results solidify covariant loop techniques for M-theory effective actions and illuminate the interplay between BRST, spinorial cohomology, and higher-derivative corrections.

Abstract

We generalize to the eleven-dimensional superparticle Berkovits' prescription for loop computations in the pure spinor approach to covariant quantization of the superstring. Using these ten- and eleven-dimensional results, we compute covariantly the following one-loop amplitudes: C\wedge X_8 in M-theory; B\wedge X_8 in type II string theory and F^4 in type I. We also verify the consistency of the formalism in eleven dimensions by recovering the correct classical action from tree-level amplitudes. As the superparticle is only a first approximation to the supermembrane, we comment on the possibility of extending this construction to the latter. Finally, we elaborate on the relationship between the present BRST language and the spinorial cohomology approach to corrections of the effective action.

Paper Structure

This paper contains 23 sections, 145 equations.

Table of Contents

  1. Introduction
  2. From $\kappa$-symmetry to BRST symmetry for 11d superparticle
  3. Supergravity in 11 dimensions and vertex operators
  4. Vertex Operators and $N=1, D=11$ Supergravity
  5. Coupling of the action to a background
  6. Tree-level amplitudes
  7. Path integral definition for the tree-level measure
  8. Measure of integration for the pure spinor $\lambda$
  9. Tree-level action from scattering amplitudes
  10. Definition of the one-loop amplitude
  11. Measure for the conjugate momentum $w_{A}$
  12. A different representation of the $b_B$ ghost
  13. Action of the $b_B$ ghost on vertex operators
  14. One loop amplitudes
  15. $B\wedge X_8$ in ten dimensions
  16. ...and 8 more sections