arXiv subjects

Lev Rozansky

Publications and source records attributed to Lev Rozansky.

35 records · Page 2Linked to original sources

arXiv2008

Three-dimensional topological field theory and symplectic algebraic geometry I

We study boundary conditions and defects in a three-dimensional topological sigma-model with a complex symplectic target space X (the Rozansky-Witten model). We show that boundary conditions correspond to complex Lagrangian submanifolds in X equipped with complex fibrations. The set of boundary conditions has the structure of a 2-category; morphisms in this 2-category are interpreted physically as one-dimensional defect lines separating parts of the boundary with different boundary conditions. This 2-category is a categorification of the Z/2-graded derived category of X; it is also related to categories of matrix factorizations and a categorification of deformation quantization (quantization of symmetric monoidal categories). In the appendix we describe a deformation of the B-model and the associated category of branes by forms of arbitrary even degree.

hep-th↗

arXiv2007

Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial

We suggest a categorification procedure for the SO(2N) one-variable specialization of the two-variable Kauffman polynomial. The construction has many similarities with the HOMFLYPT categorification: a planar graph formula for the polynomial is converted into a complex of graded vector spaces, each of them being the homology of a Z_2 graded differential vector space associated to a graph and constructed using matrix factorizations. This time, however, the elementary matrix factorizations are not Koszul; instead, they are convolutions of chain complexes of Koszul matrix factorizations. We prove that the homotopy class of the resulting complex associated to a diagram of a link is invariant under the first two Reidemeister moves and conjecture its invariance under the third move.

math.QA↗

arXiv2006

Matrix factorizations and link homology II

To a presentation of an oriented link as the closure of a braid we assign a complex of bigraded vector spaces. The Euler characteristic of this complex (and of its triply-graded cohomology groups) is the HOMFLYPT polynomial of the link. We show that the dimension of each cohomology group is a link invariant.

math.QA↗

arXiv2004

Matrix factorizations and link homology

For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links with this polynomial as the Euler characteristic. The core of our construction utilizes the theory of matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.

math.QA↗

arXiv2003

The loop expansion of the Kontsevich integral, the null move and S-equivalence

This is a substantially revised version. The Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev invariant of knots. We introduce a second grading of the Kontsevich integral, the Euler degree, and a geometric null-move on the set of knots. We explain the relation of the null-move to S-equivalence, and the relation to the Euler grading of the Kontsevich integral. The null move leads in a natural way to the introduction of trivalent graphs with beads, and to a conjecture on a rational version of the Kontsevich integral, formulated by the second author and proven in joint work of the first author and A. Kricker.

math.GT↗

arXiv2003

The Aarhus integral of rational homology 3-spheres III: The Relation with the Le-Murakami-Ohtsuki Invariant

Continuing the work started in Part I and II of this series (see q-alg/9706004 and math.QA/9801049), we prove the relationship between the Aarhus integral and the invariant $Ω$ (henceforth called LMO) defined by T.Q.T. Le, J. Murakami and T. Ohtsuki in q-alg/9512002. The basic reason for the relationship is that both constructions afford an interpretation as "integrated holonomies". In the case of the Aarhus integral, this interpretation was the basis for everything we did in Parts I and II. The main tool we used there was "formal Gaussian integration". For the case of the LMO invariant, we develop an interpretation of a key ingredient, the map $j_m$, as "formal negative-dimensional integration". The relation between the two constructions is then an immediate corollary of the relationship between the two integration theories.

math.QA↗

arXiv1999

The Aarhus integral of rational homology 3-spheres II: Invariance and universality

We continue the work started in part I (q-alg/9706004) and prove the invariance and universality in the class of finite type invariants of the object defined and motivated there, namely the Aarhus integral of rational homology 3-spheres. Our main tool in proving invariance is a translation scheme that translates statements in multi-variable calculus (Gaussian integration, integration by parts, etc.) to statements about diagrams. Using this scheme the straight-forward "philosophical" calculus-level proofs of part I become straight-forward honest diagram-level proofs here. The universality proof is standard and utilizes a simple "locality" property of the Kontsevich integral.

math.QA↗

arXiv1999

The Aarhus integral of rational homology 3-spheres I: A highly non trivial flat connection on S^3

Path integrals don't really exist, but it is very useful to dream that they do exist, and figure out the consequences. Apart from describing much of the physical world as we now know it, these dreams also lead to some highly non-trivial mathematical theorems and theories. We argue that even though non-trivial flat connections on S^3 don't really exist, it is beneficial to dream that one exists (and, in fact, that it comes from the non-existent Chern-Simons path integral). Dreaming the right way, we are led to a rigorous construction of a universal finite-type invariant of rational homology spheres. We show that this invariant is equal to the LMO (Le-Murakami-Ohtsuki) invariant and that it recovers the Rozansky and Ohtsuki invariants. This is part I of a 4-part series, containing the introductions and answers to some frequently asked questions. Theorems are stated but not proved in this part, and it can be viewed as a "research announcement". Part II of this series is titled "Invariance and Universality" (see math/9801049), part III "The Relation with the Le-Murakami-Ohtsuki Invariant" (see math/9808013), and part IV "The Relation with the Rozansky and Ohtsuki Invariants".

q-alg↗

arXiv1998

Wheels, Wheeling, and the Kontsevich Integral of the Unknot

We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne) for the relation between the two natural products on the space of Chinese characters. The two formulas use the related notions of "Wheels" and "Wheeling". We prove these formulas "on the level of Lie algebras" using standard techniques from the theory of Vassiliev invariants and the theory of Lie algebras.

q-alg↗

arXiv1994

A Contribution of the Trivial Connection to the Jones Polynomial and Witten's Invariant of 3d Manifolds II

We extend the results of our previous paper from knots to links by using a formula for the Jones polynomial of a link derived recently by N. Reshetikhin. We illustrate this formula by an example of a torus link. A relation between the parameters of Reshetikhin's formula and the multivariable Alexander polynomial is established. We check that our expression for the Alexander polynomial satisfies some of its basic properties. Finally we derive a link surgery formula for the loop corrections to the trivial connection contribution to Witten's invariant of rational homology spheres.

hep-th↗

arXiv1994

Reshetikhin's Formula for the Jones Polynomial of a Link: Feynman diagrams and Milnor's Linking Numbers

We use Feynman diagrams to prove a formula for the Jones polynomial of a link derived recently by N.~Reshetikhin. This formula presents the colored Jones polynomial as an integral over the coadjoint orbits corresponding to the representations assigned to the link components. The large $k$ limit of the integral can be calculated with the help of the stationary phase approximation. The Feynman rules allow us to express the phase in terms of integrals over the manifold and the link components. Its stationary points correspond to flat connections in the link complement. We conjecture a relation between the dominant part of the phase and Milnor's linking numbers. We check it explicitly for the triple and quartic numbers by comparing their expression through the Massey product with Feynman diagram integrals.

hep-th↗

arXiv1994

Witten's Invariant of 3-Dimensional Manifolds: Loop Expansion and Surgery Calculus

We review two different methods of calculating Witten's invariant: a stationary phase approximation and a surgery calculus. We give a detailed description of the 1-loop approximation formula for Witten's invariant and of the technics involved in deriving its exact value through a surgery construction of a manifold. Finally we compare the formulas produced by both methods for a 3-dimensional sphere S^3 and a lens space L(p,1).

hep-th↗

arXiv1994

A Contribution of the Trivial Connection to Jones Polynomial and Witten's Invariant of 3d Manifolds I

We use the Chern-Simons quantum field theory in order to prove a recently conjectured limitation on the 1/K expansion of the Jones polynomial of a knot and its relation to the Alexander polynomial. This limitation allows us to derive a surgery formula for the loop corrections to the contribution of the trivial connection to Witten's invariant. The 2-loop part of this formula coincides with Walker's surgery formula for Casson-Walker invariant. This proves a conjecture that Casson-Walker invariant is a 2-loop correction to the trivial connection contribution to Witten's invariant of a rational homology sphere. A contribution of the trivial connection to Witten's invariant of a manifold with nontrivial rational homology is calculated for the case of Seifert manifolds.

hep-th↗

arXiv1993

A Large k Asymptotics of Witten's Invariant of Seifert Manifolds

We calculate a large $k$ asymptotic expansion of the exact surgery formula for Witten's $SU(2)$ invariant of Seifert manifolds. The contributions of all flat connections are identified. An agreement with the 1-loop formula is checked. A contribution of the irreducible connections appears to contain only a finite number of terms in the asymptotic series. A 2-loop correction to the contribution of the trivial connection is found to be proportional to Casson's invariant.

hep-th↗

arXiv1992

Reidemeister torsion, the Alexander polynomial and $U(1,1)$ Chern-Simons theory

We show that the $U(1,1)$ (super) Chern Simons theory is one loop exact. This provides a direct proof of the relation between the Alexander polynomial and analytic and Reidemeister torsion. We then proceed to compute explicitely the torsions of Lens spaces and Seifert manifolds using surgery and the $S$ and $T$ matrices of the $U(1,1)$ Wess Zumino Witten model recently determined, with complete agreement with known results. $U(1,1)$ quantum field theories and the Alexander polynomial provide thus "toy" models with a non trivial topological content, where all ideas put forward by Witten for $SU(2)$ and the Jones polynomial can be explicitely checked, at finite $k$. Some simple but presumably generic aspects of non compact groups, like the modified relation between Chern Simons and Wess Zumino Witten theories, are also illustrated. We comment on the closely related case of $GL(1,1)$.

hep-th↗

arXiv1992

R-matrix Approach to Quantum Superalgebras su_{q}(m|n)

Quantum superalgebras $su_{q}(m\mid n)$ are studied in the framework of $R$-matrix formalism. Explicit parametrization of $L^{(+)}$ and $L^{(-)}$ matrices in terms of $su_{q}(m\mid n)$ generators are presented. We also show that quantum deformation of nonsimple superalgebra $su(n\mid n)$ requires its extension to $u(n\mid n)$.

hep-th↗

arXiv1992

S and T matrices for the super $U(1,1)$ WZW model. Application to surgery and 3-manifold invariants based on the Alexander Conway polynomial

We carry on the study of the Alexander Conway invariant from the quantum field theory point of view started in \cite{RS91}. We first discuss in details $S$ and $T$ matrices for the $U(1,1)$ super WZW model and obtain, for the level $k$ an integer, new finite dimensional representations of the modular group. These have the remarkable property that some of the $S$ matrix elements are infinite. Moreover, typical and atypical representations as well as indecomposable blocks are mixed: truncation to maximally atypical representations, as advocated in some recent papers, is not consistent. The main topological application of this work is the computation of Alexander invariants for 3-manifolds and for links in 3-manifolds. Invariants of 3-manifolds seem to depend trivially on the level $k$, but still contain interesting topological information. For Seifert manifolds for instance, they coincide with the order of the first homology group. Examples of invariants of links in 3-manifolds are given. They exhibit interesting arithmetic properties.

hep-th↗