Bott–Chern cohomology of solvmanifolds

We study conditions under which sub-complexes of a double complex of vector spaces allow to compute the Bott–Chern cohomology. We are especially aimed at studying the Bott–Chern cohomology of special classes of solvmanifolds, namely, complex parallelizable solvmanifolds and solvmanifolds of splitting type. More precisely, we can construct explicit finite-dimensional double complexes that allow to compute the Bott–Chern cohomology of compact quotients of complex Lie groups, respectively, of some Lie groups of the type \(\mathbb {C}^n\ltimes _\varphi N\) where N is nilpotent. As an application, we compute the Bott–Chern cohomology of the complex parallelizable Nakamura manifold and of the completely solvable Nakamura manifold. In particular, the latter shows that the property of satisfying the \(\partial \overline{\partial }\)-Lemma is not strongly closed under deformations of the complex structure.     

Chern–Simons Invariants of Torus Links

We compute the vacuum expectation values of torus knot operators in Chern–Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus knots and links, and we obtain an analogous formula for Kauffman invariants. We also derive a formula for cable knots. We use our results to test a recently proposed conjecture that relates HOMFLY and Kauffman invariants.     

Localization of Chern–Simons type invariants of Riemannian foliations

We prove an Atiyah–Bott–Berline–Vergne type localization formula for Killing foliations in the context of equivariant basic cohomology. As an application, we localize some Chern–Simons type invariants, for example the volume of Sasakian manifolds and secondary characteristic classes of Riemannian foliations, to the union of closed leaves. Various examples are given to illustrate our method.     

One Loop Approximation of the Chern–Simons Integral

It is proven that the one loop approximation of the Wilson line integral in a perturbative SU(2) Chern–Simons theory is localized around the critical point in the large level.     

Chern Classes of Gauss–Manin Bundles of Weight 1 Vanish

We show that the Chern character of a variation of polarized Hodge structures of weight one with nilpotent residues at dies up to torsion in the Chow ring, except in codimension 0.     

Existence of charged vortices in a Maxwell–Chern–Simons model

In this paper, we prove the existence of charged vortex solitons in a Maxwell–Chern–Simons model. We establish the main existence theorem by a constrained minimization method applied on an indefinite action functional which is induced from the original field-theoretical Lagrangian. We also show that the solutions obtained are smooth.     

Chern–Simons invariants and isometric immersions of warped products

In this paper, we calculate the Chern–Simons invariants on some 3-manifolds (e.g., Berger Sphere, warped product 3-manifolds) which obtain particular features in physics. We present the condition such that Berger sphere and warped product 3-manifolds are locally conformally flat. We also give a sufficient and necessary condition such that the warped product 3-manifolds can be isometrically immersed in ${\mathbb{R}^4}$ . The latter condition is different from those in the earlier works of others.     

Chern–Simons invariants and isometric immersions of warped products

In this paper, we calculate the Chern–Simons invariants on some 3-manifolds (e.g., Berger Sphere, warped product 3-manifolds) which obtain particular features in physics. We present the condition such that Berger sphere and warped product 3-manifolds are locally conformally flat. We also give a sufficient and necessary condition such that the warped product 3-manifolds can be isometrically immersed in \mathbbR⁴{\mathbb{R}^4} . The latter condition is different from those in the earlier works of others.     

A stochastic Gauss–Bonnet–Chern formula

We prove that a Gaussian ensemble of smooth random sections of a real vector bundle \(E\) over compact manifold \(M\) canonically defines a metric on \(E\) together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet theorem stating that if the bundle \(E\) and the manifold \(M\) are oriented, then the Euler form of the above connection can be identified, as a current, with the expectation of the random current defined by the zero-locus of a random section in the above Gaussian ensemble.     

Analytical Issues in the Construction of Self-dual Chern–Simons Vortices

In this note we discuss the solvability of Liouville-type systems in presence of singular sources, which arise from the study of non-abelian Chern Simons vortices in Gauge Field Theory and their asymptotic behaviour (for limiting values of the physical parameters). This investigation has contributed towards the understanding of singular PDE ’s in Mean Field form, in connection to surfaces with conical singularities, sharp Moser–Trudinger and log(HLS)-inequalities, bubbling phenomena and point-wise profile estimates in terms of Harnack type inequalities. We shall emphasise mostly the physical impact of the rigorous mathematical results established and mention several of the remaining open problems.     

Asymptotic Behavior of the Chern–Simons Higgs 6-th Theory

We analyze the asymptotic behavior of solutions of the Chern–Simons Higgs 6-th model introduced by Hong–Kim–Pac and Jackiw–Weinberg.     

Bases of Riemann–Roch spaces from Kummer extensions and algebraic geometry codes

We construct explicit bases of Riemann–Roch spaces from Kummer extensions and algebraic geometry codes with good parameters. This correspondence is a generalization of a work of Maharaj, Matthews, and Pirsic.     

Gauss–Bonnet–Chern theorem on moduli space

In this paper, we prove a Gauss–Bonnet–Chern type theorem in full generality for the Chern–Weil forms of Hodge bundles. That is, the Chern–Weil forms compute the corresponding Chern classes. This settles a long standing problem. Second, we apply the result to Calabi–Yau moduli, and proved the corresponding Gauss–Bonnet–Chern type theorem in the setting of Weil–Petersson geometry. As an application of our results in string theory, we prove that the number of flux vacua of type II string compactified on a Calabi–Yau manifold is finite, and their number is bounded by an intrinsic geometric quantity.     

Explicit construction of a Chern–Moser connection for CR manifolds of codimension two

In the present paper we suggest an explicit construction of a Cartan connection for an elliptic or hyperbolic CR manifold M of dimension six and codimension two, i.e. a pair , consisting of a principal bundle over M and of a Cartan connection form over P, satisfying the following property: the (local) CR transformations are in one to one correspondence with the (local) automorphisms for which . For any , this construction determines an explicit monomorphism of the stability subalgebra Lie (Aut(M)_x) into the Lie algebra of the structure group H of P. Mathematics Subject Classification (2000) Primary 32V05, Secondary 53C15, 53A55