Fermionic construction of partition function for multi-matrix models and multi-component TL hierarchy

We use $p$-component fermions $(p=2,3,…)$ to present $(2p-2)N$-fold integrals as a fermionic expectation value. This yields fermionic representation for various $(2p-2)$-matrix models. Links with the $p$-component KP hierarchy and also with the $p$-component TL hierarchy are discussed. We show that the set of all (but two) flows of $p$-component TL changes standard matrix models to new ones.

💡 Research Summary

The paper develops a unified fermionic framework for a broad class of multi‑matrix models by employing p‑component fermions (p ≥ 2). The authors start by introducing a set of creation and annihilation operators ψ_i^{(a)} and ψ_i^{(a)†} (a = 1,…,p; i ∈ ℤ) that satisfy the canonical anticommutation relations and generate an infinite‑dimensional Clifford algebra acting on a Fock space with vacuum |0⟩. Each of the (2p‑2) matrix variables X₁,…,X_{2p‑2} is represented as a bilinear combination of two fermionic components; this mapping translates matrix multiplication and trace operations into normal‑ordered products of fermionic operators.

With this correspondence the (2p‑2)N‑fold integral that defines the partition function of a (2p‑2)‑matrix model can be rewritten as a fermionic vacuum expectation value
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