The Magnus representation and homology cobordism groups of homology cylinders

A homology cylinder over a compact manifold is a homology cobordism between two copies of the manifold together with a boundary parametrization. We study abelian quotients of the homology cobordism group of homology cylinders. For homology cylinders over general surfaces, it was shown by Cha, Friedl and Kim that their homology cobordism groups have infinitely generated abelian quotient groups by using Reidemeister torsion invariants. In this paper, we first investigate their abelian quotients again by using another invariant called the Magnus representation. After that, we apply the machinery obtained from the Magnus representation to higher dimensional cases and show that the homology cobordism groups of homology cylinders over a certain series of manifolds regarded as a generalization of surfaces have big abelian quotients. In the proof, a homological localization, called the acyclic closure, of a free group and its automorphism group play important roles and our result also provides some information on these groups from a group-theoretical point of view.Comment: 21 pages, 2 figures, results on the algebraic closure of a free group are added, final version, to appear in Journal of Mathematical Sciences, the University of Toky

The Bloch-Okounkov correlation functions of classical type

Bloch and Okounkov introduced an n-point correlation function on the infinite wedge space and found an elegant closed formula in terms of theta functions. This function has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, etc, and it can also be interpreted as correlation functions on integrable gl_\infty-modules of level one. Such gl_\infty-correlation functions at higher levels were then calculated by Cheng and Wang. In this paper, generalizing the type A results, we formulate and determine the n-point correlation functions in the sense of Bloch-Okounkov on integrable modules over classical Lie subalgebras of gl_\infty of type B,C,D at arbitrary levels. As byproducts, we obtain new q-dimension formulas for integrable modules of type B,C,D and some fermionic type q-identities.Comment: v2, very minor changes, Latex, 41 pages, to appear in Commun. Math. Phy

Multiplicative slices, relativistic Toda and shifted quantum affine algebras

We introduce the shifted quantum affine algebras. They map homomorphically into the quantized $K$-theoretic Coulomb branches of $3d\ {\mathcal N}=4$ SUSY quiver gauge theories. In type $A$, they are endowed with a coproduct, and they act on the equivariant $K$-theory of parabolic Laumon spaces. In type $A_1$, they are closely related to the open relativistic quantum Toda lattice of type $A$.Comment: 125 pages. v2: references updated; in section 11 the third local Lax matrix is introduced. v3: references updated. v4=v5: 131 pages, minor corrections, table of contents added, Conjecture 10.25 is now replaced by Theorem 10.25 (whose proof is based on the shuffle approach and is presented in a new Appendix). v6: Final version as published, references updated, footnote 4 adde

A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case

This article is concerned with the derivation and the mathematical study of a new mean-field model for the description of interacting electrons in crystals with local defects. We work with a reduced Hartree-Fock model, obtained from the usual Hartree-Fock model by neglecting the exchange term. First, we recall the definition of the self-consistent Fermi sea of the perfect crystal, which is obtained as a minimizer of some periodic problem, as was shown by Catto, Le Bris and Lions. We also prove some of its properties which were not mentioned before. Then, we define and study in details a nonlinear model for the electrons of the crystal in the presence of a defect. We use formal analogies between the Fermi sea of a perturbed crystal and the Dirac sea in Quantum Electrodynamics in the presence of an external electrostatic field. The latter was recently studied by Hainzl, Lewin, S\'er\'e and Solovej, based on ideas from Chaix and Iracane. This enables us to define the ground state of the self-consistent Fermi sea in the presence of a defect. We end the paper by proving that our model is in fact the thermodynamic limit of the so-called supercell model, widely used in numerical simulations.Comment: Final version, to appear in Comm. Math. Phy