Free fermions and tau-functions

Abstract

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.

67 pages, 3 figures

References in corpus (7)

• Generation of Matrix Models by W-operators
• Matrix Models for Random Partitions
• Melting Crystal, Quantum Torus and Toda Hierarchy
• Quantum hydrodynamics and nonlinear differential equations for degenerate Fermi gas
• Coupled Modified KP Hierarchy and Its Dispersionless Limit
• Hypergeometric tau functions $\tau (\mathbf{t},T,{\mathbf{t}}^{\ast })$ as $\infty$-soliton tau function in T variables
• Canonical and grand canonical partition functions of Dyson gases as tau-functions of integrable hierarchies and their fermionic realization

Cited by in corpus (47)

• Fredholm determinants, full counting statistics and Loschmidt echo for domain wall profiles in one-dimensional free fermionic chains
• Toda hierarchies and their applications
• Open intersection numbers, Kontsevich-Penner model and cut-and-join operators
• The free boundary Schur process and applications I
• Instanton R-matrix and W-symmetry
• Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra
• The Cauchy two-matrix model, C-Toda lattice and CKP hierarchy
• Microscopic approach to a class of 1D quantum critical models
• Open intersection numbers, matrix models and MKP hierarchy
• $\mathcal{N}=2^{\ast }$ gauge theory, free fermions on the torus and Painlevé VI
• Bilinear equations in Darboux transformations by Boson-Fermion correspondence
• Fermionic approach to weighted Hurwitz numbers and topological recursion
• Free fermions, W-algebras and isomonodromic deformations
• On the Dynamics of Free-Fermionic Tau-Functions at Finite Temperature
• Microstates of a $2d$ Black Hole in string theory
• Grothendieck polynomials and the Boson-Fermion correspondence
• BKP hierarchy and Pfaffian point process
• $b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral
• ABJM Matrix Model and 2D Toda Lattice Hierarchy
• W-representations of the fermionic matrix and Aristotelian tensor models
• Free-fermions and skew stable Grothendieck polynomials
• Finite temperature spin diffusion in the Hubbard model in the strong coupling limit
• Combinatorial models for moduli spaces of open Riemann surfaces
• KP integrability of triple Hodge integrals. II. Generalized Kontsevich matrix model
• Hurwitz numbers and matrix integrals labeled with chord diagrams
• From $r$-Spin Intersection Numbers to Hodge Integrals
• Correlations for symplectic and orthogonal Schur measures
• Matrix model for the total descendant potential of a simple singularity of type $D$
• Free fermions and Schur expansions of multi-Schur functions
• Matrix integrals and Hurwitz numbers
• Nonlinear Luttinger liquid: Exact result for the Green function in terms of the fourth Painlevé transcendent
• Discrete Painleve equation, Miwa variables, and string equation in 5d matrix models
• Fermionic limit of the Calogero-Sutherland system
• Cubic Hodge integrals and integrable hierarchies of Volterra type
• Virasoro constraints revisited
• Integrals of tau functions
• $Q$-Boson model and relations with integrable hierarchies
• Quantum KdV hierarchy and shifted symmetric functions
• Lattice Models, Hamiltonian Operators, and Symmetric Functions
• Classifying fermionic states via many-body correlation measures
• Matrix model for the stationary sector of Gromov-Witten theory of ${\mathbf{P}}^1$
• Comments on Slavnov products, Temperley-Lieb open spin chains, and KP tau functions
• On affine coordinates of the tau-function for open intersection numbers
• $\tau$-Functions, Birkhoff Factorizations and Difference Equations
• Intertwining operator and integrable hierarchies from topological strings
• 2D Toda $\tau$ Functions, Weighted Hurwitz Numbers and the Cayley Graph: Determinant Representation and Recursion Formula
• Free fermionic probability theory and K-theoretic Schubert calculus