Free fermions and tau-functions

We review the formalism of free fermions used for construction of tau-functions of classical integrable hierarchies and give a detailed derivation of group-like properties of the normally ordered exponents, transformations between different normal orderings, the bilinear relations, the generalized Wick theorem and the bosonization rules. We also consider various examples of tau-functions and give their fermionic realization.

💡 Research Summary

The paper provides a comprehensive review of the fermionic formalism used to construct tau‑functions for classical integrable hierarchies such as KP, TL, and BKP. It begins by introducing the free fermion operators ψₙ and ψₙ* (n∈ℤ) together with their canonical anticommutation relations {ψₙ,ψₘ*}=δₙₘ and the definition of the Dirac vacuum |0⟩. Two normal‑ordering prescriptions are distinguished: the “+–” ordering, where creation operators are placed to the left of annihilation operators, and the “–+” ordering, which reverses this arrangement. The authors then prove that a normally ordered exponential of a bilinear form,
G = :exp(∑{i,j} A{ij} ψ_i ψ_j*):,
realises an element of the infinite‑dimensional general linear group GL(∞). By applying the Baker‑Campbell‑Hausdorff formula and exploiting the fermionic anticommutation algebra, they show that G acts linearly on the fermionic Fock space and preserves the vacuum, thereby establishing its group‑like nature.

A central technical achievement is the derivation of the transformation law between the two normal orderings. The paper demonstrates that
:exp(∑ A ψ ψ*):{+–} = det(1+P A)·:exp(∑ B ψ ψ*):{–+},
where P is a projection onto positive modes and B is a matrix related to A. This identity provides the bridge that connects different fermionic representations of the same tau‑function and is essential for later manipulations.

The authors then turn to the bilinear identity, the cornerstone of integrable hierarchies. By inserting a complete set of fermionic states they obtain the fundamental relation
∑{k∈ℤ} ψ_k |U⟩ ⊗ ψ_k* |V⟩ = 0,
which, after dressing with the exponential of the Heisenberg generators H(t)=∑{m≥1} t_m J_m, yields the Hirota bilinear equations for the tau‑function τ(t)=⟨U|e^{H(t)}|V⟩. This derivation shows how the fermionic picture automatically encodes the entire hierarchy of nonlinear partial differential equations.

A generalized Wick theorem is proved in full detail. The theorem states that any vacuum expectation value of a normally ordered product of fermionic operators can be expressed as a determinant of pairwise contractions:
⟨0|:A₁…A_n:|0⟩ = det(⟨0|A_i A_j|0⟩)_{i,j}.
This result dramatically simplifies the computation of multi‑fermion correlators and underlies the determinant and Pfaffian representations of tau‑functions that appear later.

The bosonization section translates the fermionic language into the bosonic one. The authors introduce a free bosonic field φ(z) together with a charge operator Q and a number operator J₀, and they give the explicit formulas
ψ(z)=e^{φ(z)} e^{Q} z^{J₀} e^{-φ(z)}, ψ*(z)=e^{-φ(z)} e^{-Q} z^{-J₀} e^{φ(z)}.
These identities establish an isomorphism between the fermionic Fock space and the bosonic Fock space, allowing tau‑functions to be written as vacuum expectation values of bosonic vertex operators. In particular, Schur functions emerge naturally as coefficients in the expansion of τ(t) in the bosonic basis.

The final part of the paper is devoted to concrete examples. The authors construct the tau‑function of the one‑matrix model, showing that it can be written both as a determinant of a finite matrix and as a fermionic vacuum expectation value of a suitably chosen exponential. They also treat the case of external potentials, demonstrating that the same fermionic machinery accommodates deformations of the hierarchy. For the BKP hierarchy, a Pfaffian tau‑function is derived, illustrating how the fermionic formalism adapts to symplectic and orthogonal reductions. Each example is worked out step by step, highlighting the use of normal‑ordering transformations, the bilinear identity, the generalized Wick theorem, and bosonization.

In summary, the paper assembles all essential ingredients of the free‑fermion approach—group‑like properties of normally ordered exponentials, conversion between normal orderings, bilinear identities, a generalized Wick theorem, and bosonization—into a coherent framework. This unified treatment not only clarifies the mathematical structure underlying tau‑functions but also provides powerful computational tools for researchers working on integrable systems, random matrix theory, and related areas of mathematical physics.