Odd Chern-Simons Theory, Lie Algebra Cohomology and Characteristic Classes

We investigate the generic 3D topological field theory within AKSZ-BV framework. We use the Batalin-Vilkovisky (BV) formalism to construct explicitly cocycles of the Lie algebra of formal Hamiltonian vector fields and we argue that the perturbative partition function gives rise to secondary characteristic classes. We investigate a toy model which is an odd analogue of Chern-Simons theory, and we give some explicit computation of two point functions and show that its perturbation theory is identical to the Chern-Simons theory. We give concrete example of the homomorphism taking Lie algebra cocycles to Q-characteristic classes, and we reinterpreted the Rozansky-Witten model in this light.Comment: 52 page

A Self-Linking Invariant of Virtual Knots

In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the bracket polynomial and then extracted from this polynomial in terms of its exponents, particularly for the case of knots. This analog of the bracket polynomial will be denoted {K} (with curly brackets) and called the binary bracket polynomial. The key to the combinatorics of the invariant is an interpretation of the state sum in terms of 2-colorings of the associated diagrams. For virtual knots, the new invariant, J(K), is a restriction of the writhe to the odd crossings of the diagram (A crossing is odd if it links an odd number of crossings in the Gauss code of the knot. The set of odd crossings is empty for a classical knot.) For K a virtual knot, J(K) non-zero implies that K is non-trivial, non-classical and inequivalent to its planar mirror image. The paper also condsiders generalizations of the two-fold coloring of the states of the binary bracket to cases of three and more colors. Relationships with graph coloring and the Four Color Theorem are discussed.Comment: 36 pages, 22 figures, LaTeX documen

Parametric Representation of Rank d Tensorial Group Field Theory: Abelian Models with Kinetic Term ∑s∣ps∣+μ\sum_{s}|p_s| + \mu

We consider the parametric representation of the amplitudes of Abelian models in the so-called framework of rank $d$ Tensorial Group Field Theory. These models are called Abelian because their fields live on $U(1)^D$. We concentrate on the case when these models are endowed with particular kinetic terms involving a linear power in momenta. New dimensional regularization and renormalization schemes are introduced for particular models in this class: a rank 3 tensor model, an infinite tower of matrix models $\phi^{2n}$ over $U(1)$, and a matrix model over $U(1)^2$. For all divergent amplitudes, we identify a domain of meromorphicity in a strip determined by the real part of the group dimension $D$. From this point, the ordinary subtraction program is applied and leads to convergent and analytic renormalized integrals. Furthermore, we identify and study in depth the Symanzik polynomials provided by the parametric amplitudes of generic rank $d$ Abelian models. We find that these polynomials do not satisfy the ordinary Tutte's rules (contraction/deletion). By scrutinizing the "face"-structure of these polynomials, we find a generalized polynomial which turns out to be stable only under contraction.Comment: 69 pages, 35 figure