Vassiliev Invariants for Links from Chern-Simons Perturbation Theory

The general structure of the perturbative expansion of the vacuum expectation value of a product of Wilson-loop operators is analyzed in the context of Chern-Simons gauge theory. Wilson loops are opened into Wilson lines in order to unravel the algebraic structure encoded in the group factors of the perturbative series expansion. In the process a factorization theorem is proved for Wilson lines. Wilson lines are then closed back into Wilson loops and new link invariants of finite type are defined. Integral expressions for these invariants are presented for the first three primitive ones of lower degree in the case of two-component links. In addition, explicit numerical results are obtained for all two-component links of no more than six crossings up to degree four.Comment: 44 pages, LaTex, epsf.sty, 15 figure

A TQFT associated to the LMO invariant of three-dimensional manifolds

We construct a Topological Quantum Field Theory (in the sense of Atiyah) associated to the universal finite-type invariant of 3-dimensional manifolds, as a functor from the category of 3-dimensional manifolds with parametrized boundary, satisfying some additional conditions, to an algebraic-combinatorial category. It is built together with its truncations with respect to a natural grading, and we prove that these TQFTs are non-degenerate and anomaly-free. The TQFT(s) induce(s) a (series of) representation(s) of a subgroup ${\cal L}_g$ of the Mapping Class Group that contains the Torelli group. The N=1 truncation produces a TQFT for the Casson-Walker-Lescop invariant.Comment: 28 pages, 13 postscript figures. Version 2 (Section 1 has been considerably shorten, and section 3 has been slightly shorten, since they will constitute a separate paper. Section 4, which contained only announce of results, has been suprimated; it will appear in detail elsewhere. Consequently some statements have been re-numbered. No mathematical changes have been made.

Renormalization Ambiguities in Chern-Simons Theory

We introduce a new family of gauge invariant regularizations of Chern-Simons theories which generate one-loop renormalizations of the coupling constant of the form $k\to k+2 s c_v$ where $s$ can take any arbitrary integer value. In the particular case $s=0$ we get an explicit example of a gauge invariant regularization which does not generate radiative corrections to the bare coupling constant. This ambiguity in the radiative corrections to $k$ is reminiscent of the Coste-L\"uscher results for the parity anomaly in (2+1) fermionic effective actions.Comment: 10 pages, harvmac, no changes, 1 Postscript figure (now included

Trace as an alternative decategorification functor

Categorification is a process of lifting structures to a higher categorical level. The original structure can then be recovered by means of the so-called "decategorification" functor. Algebras are typically categorified to additive categories with additional structure and decategorification is usually given by the (split) Grothendieck group. In this expository article we study an alternative decategorification functor given by the trace or the zeroth Hochschild--Mitchell homology. We show that this form of decategorification endows any 2-representation of the categorified quantum sl(n) with an action of the current algebra U(sl(n)[t]) on its center.Comment: 47 pages with tikz figures. arXiv admin note: text overlap with arXiv:1405.5920 by other author

Khovanov-Rozansky Homology and Topological Strings

We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological string theory. We use this approach in various examples to predict the sl(N) knot homology groups for all values of N. We verify that our predictions pass some non-trivial checks.Comment: 25 pages, 2 figures, harvmac; minor corrections, references adde

From simplicial Chern-Simons theory to the shadow invariant II

This is the second of a series of papers in which we introduce and study a rigorous "simplicial" realization of the non-Abelian Chern-Simons path integral for manifolds M of the form M = Sigma x S1 and arbitrary simply-connected compact structure groups G. More precisely, we introduce, for general links L in M, a rigorous simplicial version WLO_{rig}(L) of the corresponding Wilson loop observable WLO(L) in the so-called "torus gauge" by Blau and Thompson (Nucl. Phys. B408(2):345-390, 1993). For a simple class of links L we then evaluate WLO_{rig}(L) explicitly in a non-perturbative way, finding agreement with Turaev's shadow invariant |L|.Comment: 53 pages, 1 figure. Some minor changes and corrections have been mad

Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial

An elementary introduction to Khovanov construction of superpolynomials. Despite its technical complexity, this method remains the only source of a definition of superpolynomials from the first principles and therefore is important for development and testing of alternative approaches. In this first part of the review series we concentrate on the most transparent and unambiguous part of the story: the unreduced Jones superpolynomials in the fundamental representation and consider the 2-strand braids as the main example. Already for the 5_1 knot the unreduced superpolynomial contains more items than the ordinary Jones.Comment: 33 page

The Sum over Topologies in Three-Dimensional Euclidean Quantum Gravity

In Hawking's Euclidean path integral approach to quantum gravity, the partition function is computed by summing contributions from all possible topologies. The behavior such a sum can be estimated in three spacetime dimensions in the limit of small cosmological constant. The sum over topologies diverges for either sign of $\Lambda$, but for dramatically different reasons: for $\Lambda>0$, the divergent behavior comes from the contributions of very low volume, topologically complex manifolds, while for $\Lambda<0$ it is a consequence of the existence of infinite sequences of relatively high volume manifolds with converging geometries. Possible implications for four-dimensional quantum gravity are discussed.Comment: 12 pages (LaTeX), UCD-92-1

Knot homology via derived categories of coherent sheaves II, sl(m) case

Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum sl(m) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu's by homological mirror symmetry.Comment: 51 pages, 9 figure