The ADHM variety and perverse coherent sheaves

We study the full set of solutions to the ADHM equation as an affine algebraic set, the ADHM variety. We determine a filtration of the ADHM variety into subvarieties according to the dimension of the stabilizing subspace. We compute dimension, and analyze singularity and reducibility of all of these varieties. We also establish a connection between arbitrary solutions of the ADHM equation and coherent perverse sheaves on P²${\mathbb{P}}^2$ in the sense of Kashiwara.     

Chamber structure and wallcrossing in the ADHM theory of curves, I

ADHM invariants are equivariant virtual invariants of moduli spaces of twisted cyclic representations of the ADHM quiver in the abelian category of coherent sheaves of a smooth complex projective curve X. The goal of the present paper is to present a generalization of this construction employing a more general stability condition which depends on a real parameter. This yields a chamber structure in the ADHM theory of curves, residual ADHM invariants being defined by equivariant virtual integration in each chamber. Wallcrossing formulas will be presented in the second part of this work.     

Chamber structure and wallcrossing in the ADHM theory of curves II

This is the second part of a project concerning variation of stability and chamber structure for ADHM invariants of curves. Wallcrossing formulas for such invariants are derived using the theory of stack function Ringel-Hall algebras constructed by Joyce and the theory of generalized Donaldson-Thomas invariants of Joyce and Song. Some applications are presented, including strong rationality for local stable pair invariants of higher genus curves, and comparison with wallcrossing formulas of Kontsevich and Soibelman, and the halo formula of Denef and Moore.     

Stable quasimaps to GIT quotients

We construct new compactifications with good properties of moduli spaces of maps from nonsingular marked curves to a large class of GIT quotients. This generalizes from a unified perspective many particular examples considered earlier in the literature.     

Torus action on the moduli spaces of torsion plane sheaves of multiplicity four

We describe the torus fixed locus of the moduli space of stable sheaves with Hilbert polynomial $4m+1$ on ${\mathbb{P}}^2$. We determine the torus representation of the tangent spaces at the fixed points, which leads to the computation of the Betti and Hodge numbers of the moduli space.     

Formal Deformations,Contractions and Moduli Spaces of Lie Algebras

Jump deformations and contractions of Lie algebras are inverse concepts, but the approaches to their computations are quite different. In this paper, we contrast the two approaches, showing how to compute the jump deformations from the miniversal deformation of a Lie algebra, and thus arrive at the contractions. We also compute contractions directly. We use the moduli spaces of real 3-dimensional and complex 3 and 4-dimensional Lie algebras as models for explaining a deformation theory approach to computation of contractions. The research of the authors was partially supported by grants from the Mathematisches Forschungsinstitut Oberwolfach, OTKA T043641, T043034 and the University of Wisconsin-Eau Claire.     

1-forms over the moduli space of irreducible connections defined by the spectrum of Dirac operators

Let (P) be the moduli space of irreducible connections of a G-principal bundle P over a closed Riemannian spin manifold M. Let D_A be the Dirac operator of M coupled to a connection A of P and f a smooth function on M. We consider a smooth variation A(u) of A with tangent vector ω and denote T_ω:= (D_A(u)−f) (u=0. The coefficients of the asymptotic expansion of trace (T_ω · e^-t(D_A−f)2) near t=0 define 1-forms a^(k)_f, K=0, 1, 2, … on (P). In this paper we calculate aa⁽⁰⁾_f, a⁽¹⁾_f, a⁽²⁾_f and study some of their properties. For instance using the 1-form a⁽²⁾_f for suitable functions f we obtain a foliation of codimension 5 of the space of G-instantons of S⁴.     

The spectral curve theory for (k,l)-symmetric CMC surfaces

Constant mean curvature surfaces in $S^3$ can be studied via their associated family of flat connections. In the case of tori this approach has led to a deep understanding of the moduli space of all CMC tori. For compact CMC surfaces of higher genus the theory is far more involved due to the non abelian nature of their fundamental group. In this paper we extend the spectral curve theory for tori developed in Hitchin (1990), Pinkall and Sterling (1989) and for genus 2 surfaces (Heller, 2014) to CMC surfaces in $S^3$ of genus $g=k\cdot l$ with commuting ${\mathbb{Z}}_{k+1}$ and ${\mathbb{Z}}_{l+1}$ symmetries. We determine their associated family of flat connections via certain flat line bundle connections parametrized by the spectral curve. We generalize the flow on spectral data introduced in Heller (2015) and prove the short time existence of this flow for certain families of initial surfaces. In this way we obtain countably many $1-$parameter families of new CMC surfaces of higher genus with prescribed branch points and prescribed umbilics.     

The scale-transformation of electromagnetic theory and its applications

The scale-transformation of electromagnetic theory is investigated in detail based on the form of Maxwell equations in scale-transformation being unchanged in different coordinate systems. The relations of electromagnetic parameters in a rectangular coordinate system and in a spherical coordinate system are presented respectively. The scale-transformation invariants for electromagnetic field are derived and their physical meaning is also presented. It is indicated by simulation that the electromagnetic waves located in medium can be considered to be isotropic due to the fact that the size of propagating vector affected by the scale factors and observing azimuth is on a size of 10^-9, which provides a new approach for investigating the electromagnetic characteristics of ellipsoidal targets.     

Topological Invariants for Lens Spaces and Exceptional Quantum Groups

The Reshetikhin–Turaev invariants arising from the quantum groups associated with the exceptional Lie algebras G2, F4 and E8 at odd roots of unity are constructed and explicitly computed for all the lens spaces.     

Parametrization of the Moduli Space of Flat SL(2, R) Connections on the Torus

The moduli space of flat SL(2, R)-connections modulo gauge transformations on the torus may be described by ordered pairs of commuting SL(2, R) matrices modulo simultaneous conjugation by SL(2, R) matrices. Their spectral properties allow a classification of the equivalence classes, and a unique canonical form is given for each of these. In this way the moduli space becomes explicitly parametrized, and has a simple structure, resembling that of a cell complex, allowing it to be depicted. Finally, a Hausdorff topology based on this classification and parametrization is proposed.     

Histories and observables in covariant field theory

Motivated by DeWitt’s viewpoint of covariant field theory, we define a general notion of a non-local classical observable that applies to many physical Lagrangian systems (with bosonic and fermionic variables), by using methods that are now standard in algebraic geometry. We review the methods of local functional calculus, as they are presented by Beilinson and Drinfeld, and relate them to our construction. We partially explain the relation of these with Vinogradov’s secondary calculus. The methods present here are all necessary to understand mathematically properly, and with simple notions, the full renormalization of the standard model, based on functional integral methods. Our approach is close in spirit to non-perturbative methods since we work with actual functions on spaces of fields, and not only formal power series. This article can be seen as an introduction to well-grounded classical physical mathematics, and as a good starting point to study quantum physical mathematics, which make frequent use of non-local functionals, like for example in the computation of Wilson’s effective action. We finish by describing briefly a coordinate-free approach to the classical Batalin–Vilkovisky formalism for general gauge theories, in the language of homotopical geometry.     

Comment: The Square of the Weyl Tensor Can Be Negative

We show that the square of the Weyl tensor can be negative by giving an example:

Wave vector,local momentum and local coordinate from the perspective of information theory

In the context of informational analysis of atomic and molecular systems, the connection between local quantum observables and information measures is of interest. In this paper, analytical relationships for the imaginary part of the total local momentum (coordinate) in terms of information theoretic measures have been established. Moreover, on the basis of another scheme in which the relationship among densities of information energy, Shannon entropy and Fisher information has been proposed [M. Alipour and A. Mohajeri, Mol. Phys. 110, 403 (2012)], the general formulae for the imaginary part of the total local momentum and the corresponding variance are expressed. The presented proof may be viewed in light of the relation between the local wave vector and the information energy density. Through this study, another noteworthy application of information theory is highlighted.     

Adiabatic theory,Liapunov exponents,and rotation number for quadratic Hamiltonians

We consider the adiabatic problem for general time-dependent quadratic Hamiltonians and develop a method quite different from WKB. In particular, we apply our results to the Schrödinger equation in a strip. We show that there exists a first regular step (avoiding resonance problems) providing one adiabatic invariant, bounds on the Liapunov exponents, and estimates on the rotation number at any order of the perturbation theory. The further step is shown to be equivalent to a quantum adiabatic problem, which, by the usual adiabatic techniques, provides the other possible adiabatic invariants. In the special case of the Schrödinger equation our method is simpler and more powerful than the WKB techniques.     

The moduli space of flat SU (2) and SO (3) connections over surfaces

All the connected components of the moduli space of flat connections on SU (2) and SO (3) (trivial and non-trivial) bundles over closed oriented surfaces are determined. The symplectic structure and volumes of the non-maximal strata of the moduli space are also determined.