Description

The boson-fermion correspondence can be stated as a construction of Schur functions by using vertex operators. The goal of this project is to provide a way to construct vertex operators and give their action on different vector spaces. As a result, we would would be able to show by computer the boson-fermion correspondence.

This is a meta ticket for the 2025 Google summer of code project The Boson-Fermion Correspondence and Vertex Operators to keep track of relevant issues and pull requests.

Mentor: @tscrim

Features implemented

• An infinite version of the DifferentialWeylAlgebra class
• Base class for vertex operators and a general framework for defining specific vertex operator classes.
• Creation and annihilation vertex operators
  • Computing the action of their modes (Clifford algebra action on Bosonic Fock space)
  • Lazy implementation of the full action
  • Computing matrix coefficients for these operators
• Products of vertex operators
  • Action of products of their modes
  • Lazy implementation of the full action
  • Matrix Coefficients (both lazy and approximated)

Future directions

• Implement other families of vertex operators, for instance those used in the construction of symplectic and orthogonal Schur functions (seen here)
• Incorporate existing code into Parent/Element framework to more fully implement underlying algebraic structures of vertex operators.

Relevant Issues/PRs

• Add an infinite weyl algebra class #40263
• Implement vertex operators #40545