arXiv research

Introduces geometric quantization and Witten's quantum invariants.

New invariants derived from Kauffman bracket for 3-manifolds.

Proof confirms Witten's conjecture for a class of 3-spheres.

In this paper, we give a simple formula for the generating function of genus-2 Gromov-Witten invariants for manifolds with semisimple quantum cohomology, and use this formula to prove the genus-2 Virasoro conjecture for such manifolds.

We produce an equality between the Gromov-Witten invariants of the moduli space M of rank two odd degree stable vector bundles over a Riemann surface $Σ$ and the Donaldson invariants of the algebraic surface $Σ\times P^1$. We discuss on to how extent the Quantum cohomology of M determines its Gromow-Witten invariants. …

This paper proves a conjecture linking quantum modular forms and WRT invariants for specific graphs.

Topological quantum computers use hyperbolic knots for computations.

Study on quantum invariants from surgeries on torus knots.

Revisits Vafa-Witten theory, deriving new invariants and homologies.

Lecture notes on gauge theory for manifold invariants.

The quantum cohomology algebra of the (full) flag manifold is a fundamental example in quantum cohomology theory, with connections to combinatorics, algebraic geometry, and integrable systems. Using a differential geometric approach, we give an algorithm for computing the multiplicative structure constants of this alge…

We provide a geometric construction of the boundary states for handlebodies which we in turn use to give a geometric formula for the Witten-Reshetikhin-Turaev quantum invariants. We then analyze the asymptotics of this invariant in the special case of a three manifold given by 1-surgery on a knot and we show that if th…

New model for Witten-Reshetikhin-Turaev invariants using Lagrangian intersections.

For a gerbe $\Y$ over a smooth proper Deligne-Mumford stack $\B$ banded by a finite group $G$, we prove a structure result on the Gromov-Witten theory of $\Y$, expressing Gromov-Witten invariants of $\Y$ in terms of Gromov-Witten invariants of $\B$ twisted by various flat $U(1)$-gerbes on $\B$. This is interpreted as a…

The abstract discusses connecting quantum mechanics and algebraic index theories.

Quantum Kirwan maps between K-theories of G-varieties and GIT quotients.

We show that when $r \geq 5$ is prime, the SO(3) Witten-Reshetikhin-Turaev quantum invariants for three-manifolds at the level $r$ form a dense set in the complex plane. This confirms a conjecture of Larsen and Wang.

In these lectures we present a general introduction to topological quantum field theories. These theories are discussed in the framework of the Mathai-Quillen formalism and in the context of twisted N=2 supersymmetric theories. We discuss in detail the recent developments in Donaldson-Witten theory obtained from the ap…

The paper proves a quantum modularity conjecture for 3-manifolds.

Arithmetic Dijkgraaf-Witten theory constructs analogues in Chern-Simons TQFT.

We will announce some results on the values of quantum sl_2 invariants of knots and integral homology spheres. Lawrence's universal sl_2 invariant of knots takes values in a fairly small subalgebra of the center of the h-adic version of the quantized enveloping algebra of sl_2. This implies an integrality result on the…

We investigate the conjectural relations between the Reshetikhin-Turaev-Witten quantum SU(2) invariants and the volume of hyperbolic 3-manifolds. Given a finite set of sufficiently large positive integers, say J, we construct examples of closed hyperbolic 3-manifolds with the same invariants at all levels in J and diff…

In this paper we study vector fields on the big phase space of Gromov-Witten theory which are idempotents of the quantum product. Such vector fields can be used to simplify universal equations for higher genus Gromov-Witten invariants.

We investigate the perturbative aspects of Rozansky-Witten's 3d $σ$-model using Costello's approach to the Batalin-Vilkovisky (BV) formalism. We show that the BV quantization (in Costello's sense) of the model, which produces a perturbative quantum field theory, can be obtained via the configuration space method of reg…

We introduce the stack of r-spin maps. These are stable maps into a variety V from n-pointed algebraic curves of genus g, with the additional data of an r-spin structure on the curve. We prove that this stack is a Deligne-Mumford stack, and we define analogs of the Gromov-Witten classes associated to these spaces. We s…

We consider the SO(3) Witten-Reshetikhin-Turaev quantum invariants of random 3-manifolds. When the level r is prime, we show that the asymptotic distribution of the absolute value of these invariants is given by the standard Rayleigh distribution and independent of the choice of level. Hence the probability that the qu…

For every rational homology 3-sphere with 2-torsion only we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring), such that the evaluation of this invariant at any odd root of unity provides the SO(3) Witten-Reshetikhin-Turaev invariant at this root and at any even …

New techniques prove quantum modularity for various functions.

Rozansky and Witten proposed a 3-dimensional sigma-model whose target space is a hyperkähler manifold. They conjectured that this theory has an associated TQFT, with Hilbert spaces given by certain cohomology groups of the hyperkähler manifold. On the other hand, there is a certain modified TQFT constructed by Murakami…

In this paper, we show that the derivative of the genus-1 Virasoro conjecture for Gromov-Witten invariants along the direction of quantum volume element holds for all smooth projective varieties. This result provides new evidence for the Virasoro conjecture.

Refined invariants for 4D 2-handlebodies, linking quantum groups and cohomology.

Using mirror symmetry as described by Hori and Vafa, we compute the quantum equivariant cohomology ring of toric manifolds. This ring arises naturally in topological gauged sigma-models and is related to the Hamiltonian Gromov-Witten invariants of the target manifold.

We prove a conjecture of Hutchings and Lee relating the Seiberg-Witten invariants of a closed 3-manifold X with b_1 > 0 to an invariant that `counts' gradient flow lines--including closed orbits--of a circle-valued Morse function on the manifold. The proof is based on a method described by Donaldson for computing the S…

Paper constructs new identities linking quantum invariants and modular forms.

Using quantum field-theoretic arguments, Witten has established a relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. In this survey article, we describe the program to prove this relation using a moduli space of PU(2) = SO(3) monopoles as a cobordism between the Donaldson moduli spac…

Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or generalized spin and complex spin structures. A crucial role in our construction is play…

For a banded link $L$ in a surface times a circle, the Witten-Reshetikhin-Turaev invariants are topological invariants depending on a sequence of complex $2p$-th roots of unity $(A_p)_{p\in 2\mathbb{N}}$. We show that there exists a polynomial $P_L$ such that these normalized invariants converge to $P_L(u)$ when $A_p$ …

We construct power series invariants of rational homology 3-spheres from quantum PSU(n)-invariants. The power series can be regarded as perturbative invariants corresponding to the contribution of the trivial connection in the hypothetical Witten's integral. This generalizes a result of Ohtsuki (the $n=2$ case) which l…

Quantum modularity proved for a knot manifold.

Tropical geometry aids in computing topological quantum field theories.

Let Z^{LMO} be the 3-manifold invariant of [LMO]. It is shown that Z^{LMO}(M)=1, if the first Betti number of M, b_{1}(M), is greater than 3. If b_{1}(M)=3, then Z^{LMO}(M) is completely determined by the cohomology ring of M. A relation of Z^{LMO} with the Rozansky-Witten invariants Z_{X}^{RW}[M] is established at a p…

Infinite families of quantum modular invariants for 3-manifolds are discovered.

Overview of 3D TQFTs and 3-manifold invariants.

Efficient algorithms for WRT invariants of torus bundles using algebraic structures.

The goal of this paper is to give a new proof of a theorem of Meng and Taubes that identifies the Seiberg-Witten invariants of 3-manifolds with Milnor torsion. The point of view here will be that of topological quantum field theory. In particular, we relate the Seiberg-Witten equations on a 3-manifold with the Abelian …

We propose a gauge model of quantum electrodynamics (QED) and its nonabelian generalization from which we derive knot invariants such as the Jones polynomial. Our approach is inspired by the work of Witten who derived knot invariants from quantum field theory based on the Chern-Simon Lagrangian. From our approach we ca…

We study the asymptotic behaviour of the quantum representations of the modular group in the large level limit. We prove that each element of the modular group acts as a Fourier integral operator. This provides a link between the classical and quantum Chern-Simons theories for the torus. From this result we deduce the …

We use Bott periodicity to relate previously defined quantum classes to certain "exotic Chern classes" on $BU$. This provides an interesting computational and theoretical framework for some Gromov-Witten invariants connected with cohomological field theories. This framework has applications to study of higher dimension…