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Co-algebraic methods for String Field Theory and Quantum Field Theory

Enrico Perron Cabus

TL;DR

<3-5 sentence high-level summary>

Abstract

In this work we extend the notion of co-algebra, co-algebraic Wess-Zumino-Witten formulation of Lagrangian Field Theory and the Homotopy transfer theorem to many strings and particle systems. We discuss in detail the construction of higher dimensional co-algebras and the computational methods derived from them with a special interest regarding String Field Theory and Quantum Field Theory. As a result of this work we will be able to effortlessly extend some of the newly developed tools to study the algebraic structure, compute effective actions and compute scattering amplitudes of more complicated QFTs.

Co-algebraic methods for String Field Theory and Quantum Field Theory

TL;DR

<3-5 sentence high-level summary>

Abstract

In this work we extend the notion of co-algebra, co-algebraic Wess-Zumino-Witten formulation of Lagrangian Field Theory and the Homotopy transfer theorem to many strings and particle systems. We discuss in detail the construction of higher dimensional co-algebras and the computational methods derived from them with a special interest regarding String Field Theory and Quantum Field Theory. As a result of this work we will be able to effortlessly extend some of the newly developed tools to study the algebraic structure, compute effective actions and compute scattering amplitudes of more complicated QFTs.

Paper Structure

This paper contains 58 sections, 3 theorems, 423 equations, 1 table.

Table of Contents

  1. Introduction and summary
  2. Mathematical Preliminaries
  3. Co-algebras
  4. Co-derivations
  5. Co-homomorphisms and cyclicity
  6. Types of tensor algebras
  7. Co-derivation algebra and homotopy algebras
  8. Higher order co-derivations
  9. $IBL_\infty$ algebras
  10. Homotopy transfer theorem
  11. Applications to QFT/SFT
  12. WZW Co-algebraic formulation of QFTs
  13. EFTs from the homotopy transfer theorem
  14. Amplitudes from the homotopy transfer
  15. $\Phi^3$ theory
  16. ...and 43 more sections

Key Result

Theorem 1

Given $(P,h)$, if $P$ is a quasi isomorphisms and $h$ satisfies the side conditions then there exists a suitable extension of $(P,h)$ on the tensor algebra ${\mathcal{TH}}$ such that the $A_\infty /L_\infty$ structure on ${\mathcal{TH}}$ can be transferred to a $A_\infty /L_\infty$ structure on ${\mathcal{TH}}_P$.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3