Complex Bosonic Many-body Models: Overview of the Small Field Parabolic Flow

This paper is a contribution to a program to see symmetry breaking in a weakly interacting many Boson system on a three dimensional lattice at low temperature. It provides an overview of our analysis of the "small field" approximation to the "parabolic flow" which exhibits the formation of a "Mexican hat" potential well.Comment: 36 page

A Proof of Luttinger Theorem

A rigorous and simple perturbative proof of Luttinger's theorem is sketched for Fermi liquids in two and three dimensions. It is proved that in the finite volume, the quasi-particle density is independent of the interaction strength. The thermodynamic limit is then controlled to all orders in perturbation theory.Comment: 7 page

The frequency map for billiards inside ellipsoids

The billiard motion inside an ellipsoid Q \subset \Rset^{n+1} is completely integrable. Its phase space is a symplectic manifold of dimension $2n$, which is mostly foliated with Liouville tori of dimension $n$. The motion on each Liouville torus becomes just a parallel translation with some frequency $\omega$ that varies with the torus. Besides, any billiard trajectory inside $Q$ is tangent to $n$ caustics $Q_{\lambda_1},...,Q_{\lambda_n}$, so the caustic parameters $\lambda=(\lambda_1,...,\lambda_n)$ are integrals of the billiard map. The frequency map $\lambda \mapsto \omega$ is a key tool to understand the structure of periodic billiard trajectories. In principle, it is well-defined only for nonsingular values of the caustic parameters. We present four conjectures, fully supported by numerical experiments. The last one gives rise to some lower bounds on the periods. These bounds only depend on the type of the caustics. We describe the geometric meaning, domain, and range of $\omega$. The map $\omega$ can be continuously extended to singular values of the caustic parameters, although it becomes "exponentially sharp" at some of them. Finally, we study triaxial ellipsoids of \Rset^3. We compute numerically the bifurcation curves in the parameter space on which the Liouville tori with a fixed frequency disappear. We determine which ellipsoids have more periodic trajectories. We check that the previous lower bounds on the periods are optimal, by displaying periodic trajectories with periods four, five, and six whose caustics have the right types. We also give some new insights for ellipses of \Rset^2.Comment: 50 pages, 13 figure

Power Series Representations for Bosonic Effective Actions

We develop a power series representation and estimates for an effective action of the form $$\ln\frac{\int e^{f(\phi,\psi)}d\mu(\phi)}{\int e^{f(\phi,0)}d\mu(\phi)}$$ Here, f(φ,ψ) is an analytic function of the real fields φ(x),ψ(x) indexed by x in a finite set X, and d μ(φ) is a compactly supported product measure. Such effective actions occur in the small field region for a renormalization group analysis. The customary way to analyze them is a cluster expansion, possibly preceded by a decoupling expansion. Using methods similar to a polymer expansion, we estimate the power series of the effective action without introducing an artificial decomposition of the underlying space into boxe

Non-Gorenstein isolated singularities of graded countable Cohen-Macaulay type

In this paper we show a partial answer the a question of C. Huneke and G. Leuschke (2003): Let R be a standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay representation type, and assume that R has an isolated singularity. Is R then necessarily of graded finite Cohen-Macaulay representation type? In particular, this question has an affirmative answer for standard graded non-Gorenstein rings as well as for standard graded Gorenstein rings of minimal multiplicity. Along the way, we obtain a partial classification of graded Cohen-Macaulay rings of graded countable Cohen-Macaulay type.Comment: 15 Page

Equivalences between GIT quotients of Landau-Ginzburg B-models

We define the category of B-branes in a (not necessarily affine) Landau-Ginzburg B-model, incorporating the notion of R-charge. Our definition is a direct generalization of the category of perfect complexes. We then consider pairs of Landau-Ginzburg B-models that arise as different GIT quotients of a vector space by a one-dimensional torus, and show that for each such pair the two categories of B-branes are quasi-equivalent. In fact we produce a whole set of quasi-equivalences indexed by the integers, and show that the resulting auto-equivalences are all spherical twists.Comment: v3: Added two references. Final version, to appear in Comm. Math. Phy

Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems

In this paper we show that, if an integrable Hamiltonian system admits a nondegenerate hyperbolic singularity then it will satisfy the Kolmogorov condegeneracy condition near that singularity (under a mild additional condition, which is trivial if the singularity contains a fixed point)Comment: revised version, 11p, accepted for publication in a sepecial volume in Regular and Chaotic Dynamics in honor of Richard Cushma

Generalized r-matrix structure and algebro-geometric solution for integrable systems

The purpose of this paper is to construct a generalized r-matrix structure of finite dimensional systems and an approach to obtain the algebro-geometric solutions of integrable nonlinear evolution equations (NLEEs). Our starting point is a generalized Lax matrix instead of usual Lax pair. The generalized r-matrix structure and Hamiltonian functions are presented on the basis of fundamental Poisson bracket. It can be clearly seen that various nonlinear constrained (c-) and restricted (r-) systems, such as the c-AKNS, c-MKdV, c-Toda, r-Toda, c-Levi, etc, are derived from the reduction of this structure. All these nonlinear systems have {\it r}-matrices, and are completely integrable in Liouville's sense. Furthermore, our generalized structure is developed to become an approach to obtain the algebro-geometric solutions of integrable NLEEs. Finally, the two typical examples are considered to illustrate this approach: the infinite or periodic Toda lattice equation and the AKNS equation with the condition of decay at infinity or periodic boundary.Comment: 41 pages, 0 figure

Cohomological Characterization of Vector Bundles on Grassmannians of Lines

We introduce a notion of regularity for coherent sheaves on Grassmannians of lines. We use this notion to prove some extension of Evans-Griffith criterion to characterize direct sums of line bundles. We also give a cohomological characterization of exterior and symmetric powers of the universal bundles of the Grassmannian.Comment: 10 pages, no figure

Quantum affine Cartan matrices, Poincare series of binary polyhedral groups, and reflection representations

We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a,b,h;p,q,r) that one usually associates with such a group and hence with a simply-laced Coxeter-Dynkin diagram have a meaningful definition for the non-simply-laced diagrams, too, and as a byproduct we extend Saito's formula for the determinant of the Cartan matrix to all cases. Returning to invariant theory we show that for each irreducible representation i of a binary tetrahedral, octahedral, or icosahedral group one can find a homomorphism into a finite complex reflection group whose defining reflection representation restricts to i.Comment: 19 page