Witten–Reshetikhin–Turaev Invariants of¶Seifert Manifolds

For Seifert homology spheres, we derive a holomorphic function of K whose value at integer K is the sl ₂ Witten–Reshetikhin–Turaev invariant, Z _K , at q= exp 2πi/K. This function is expressed as a sum of terms, which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the Witten–Chern–Simons path integral. The trivial connection contribution is found to have an asymptotic expansion in powers of K ⁻¹ which, for K an odd prime power, converges K-adically to the exact total value of the invariant Z _K at that root of unity. Evaluations at rational $K$ are also discussed. Using similar techniques, an expression for the coloured Jones polynomial of a torus knot is obtained, providing a trivial connection contribution which is an analytic function of the colour. This demonstrates that the stationary phase expansion of the Chern–Simons–Witten theory is exact for Seifert manifolds and for torus knots in S ³. The possibility of generalising such results is also discussed. Received: 26 October 1998 / Accepted: 1 March 1999     

Elliptic Gromov–Witten Invariants¶and Virasoro Conjecture

We study some necessary and sufficient conditions for the genus-1 Virasoro conjecture proposed by Eguchi–Hori–Xiong and S. Katz. Received: 22 August 1999 / Accepted: 7 October 2000     

Multipoint Series of Gromov–Witten Invariants of CP1

We derive explicit formulas for the multipoint series of in degree 0 from the Toda hierarchy, using the recursions of the Toda hierarchy. The Toda equation then yields inductive formulas for the higher degree multipoint series of . We also obtain explicit formulas for the Hodge integrals , in the cases i=0 and 1.     

Skein Theory and Witten–Reshetikhin–Turaev Invariants of Links in Lens Spaces

We study the behavior of the Witten-Reshetikhin-Turaev SU(2) invariants of an arbitrary link in L(p,q) as a function of the level rф. They are given by \frac1?r\frac{1}{\sqrt{r}} times one of p Laurent polynomials evaluated at e^{\frac 2 pi 4pr}e^{\frac {2 \pi i} {4pr}}. The congruence class of r modulo p determines which polynomial is applicable. If p L 0 mod 4, the meridian of L(p,q) is non-trivial in the skein module but has trivial Witten-Reshetikhin-Turaev SU(2) invariants. On the other hand, we show that one may recover the element in the Kauffman bracket skein module of L(p,q) represented by a link from the collection of the WRT invariants at all levels if p is a prime or twice an odd prime. By a more delicate argument, this is also shown to be true for p=9.     

Noncommutative Wess–Zumino–Witten actions and their Seiberg–Witten invariance

We analyze the noncommutative two-dimensional Wess–Zumino–Witten model and its properties under Seiberg–Witten transformations in the operator formulation. We prove that the model is invariant under such transformations even for the noncritical (non-chiral) case, in which the coefficients of the kinetic and Wess–Zumino terms are not related. The pure Wess–Zumino term represents a singular case in which this transformation fails to reach a commutative limit. We also discuss potential implications of this result for bosonization.     

Remarks on Chern–Simons Invariants

The perturbative Chern–Simons theory is studied in a finite-dimensional version or assuming that the propagator satisfies certain properties (as is the case, e.g., with the propagator defined by Axelrod and Singer). It turns out that the effective BV action is a function on cohomology (with shifted degrees) that solves the quantum master equation and is defined modulo certain canonical transformations that can be characterized completely. Out of it one obtains invariants.     

Chern–Simons–Rozansky–Witten topological field theory

We construct and study a new topological field theory in three dimensions. It is a hybrid between Chern–Simons and Rozansky–Witten theory and can be regarded as a topologically-twisted version of the N=4$N=4$d=3$d=3$ supersymmetric gauge theory recently discovered by Gaiotto and Witten. The model depends on a gauge group G and a hyper-Kähler manifold X with a tri-holomorphic action of G. In the case when X is an affine space, we show that the model is equivalent to Chern–Simons theory whose gauge group is a supergroup. This explains the role of Lie superalgebras in the construction of Gaiotto and Witten. For general X, our model appears to be new. We describe some of its properties, focusing on the case when G is simple and X is the cotangent bundle of the flag variety of G. In particular, we show that Wilson loops are labeled by objects of a certain category which is a quantum deformation of the equivariant derived category of coherent sheaves on X.     

Computation of Open Gromov–Witten Invariants for Toric Calabi–Yau 3-Folds by Topological Recursion,a Proof of the BKMP Conjecture

The BKMP conjecture (2006–2008) proposed a new method to compute closed and open Gromov–Witten invariants for every toric Calabi–Yau 3-folds, through a topological recursion based on mirror symmetry. So far, this conjecture has been verified to low genus for several toric CY3folds, and proved to all genus only for \({\mathbb{C}^3}\). In this article we prove the general case. Our proof is based on the fact that both sides of the conjecture can be naturally written in terms of combinatorial sums of weighted graphs: on the A-model side this is the localization formula, and on the B-model side the graphs encode the recursive algorithm of the topological recursion.One can slightly reorganize the set of graphs obtained in the B-side, so that it coincides with the one obtained by localization in the A-model. Then it suffices to compare the weights of vertices and edges of graphs on each side, which is done in two steps: the weights coincide in the large radius limit, due to the fact that the toric graph is the tropical limit of the mirror curve. Then the derivatives with respect to Kähler radius coincide due to the special geometry property implied by the topological recursion.     

Seiberg–Witten theory and matrix models

We derive a family of matrix models which encode solutions to the Seiberg–Witten theory in 4 and 5 dimensions. Partition functions of these matrix models are equal to the corresponding Nekrasov partition functions, and their spectral curves are the Seiberg–Witten curves of the corresponding theories. In consequence of the geometric engineering, the 5-dimensional case provides a novel matrix model formulation of the topological string theory on a wide class of non-compact toric Calabi–Yau manifolds. This approach also unifies and generalizes other matrix models, such as the Eguchi–Yang matrix model, matrix models for bundles over P¹${\mathbb{P}}^1$, and Chern–Simons matrix models for lens spaces, which arise as various limits of our general result.     

On the level-dependence of Wess–Zumino–Witten three-point functions

Three-point functions of Wess–Zumino–Witten models are investigated. In particular, we study the level-dependence of three-point functions in the models based on algebras su(3) and su(4). We find a correspondence with Berenstein–Zelevinsky triangles. Using previous work connecting those triangles to the fusion multiplicities, and the Gepner–Witten depth rule, we explain how to construct the full three-point functions. We show how their level-dependence is similar to that of the related fusion multiplicity. For example, the concept of threshold level plays a prominent role, as it does for fusion.     

Seiberg–Witten theory as a Fermi gas

We explore a new connection between Seiberg–Witten theory and quantum statistical systems by relating the dual partition function of SU(2) Super Yang–Mills theory in a self-dual \(\Omega \) background to the spectral determinant of an ideal Fermi gas. We show that the spectrum of this gas is encoded in the zeroes of the Painlevé \(\mathrm{III}_3\) \(\tau \) function. In addition, we find that the Nekrasov partition function on this background can be expressed as an O(2) matrix model. Our construction arises as a four-dimensional limit of a recently proposed conjecture relating topological strings and spectral theory. In this limit, we provide a mathematical proof of the conjecture for the local \({\mathbb P}^1 \times {\mathbb P}^1\) geometry.     

Asymptotic properties of the Hitchin–Witten connection

Letters in Mathematical Physics - We explore extensions to $${{\,\mathrm{SL}\,}}(n,{\mathbb {C}})$$ -Chern–Simons theory of some results obtained for $${{\,\mathrm{SU}\,}}(n)$$...     

Gromov–Witten invariants and rational curves on Grassmannians

We study the enumerative significance of the s$s$-pointed genus zero Gromov–Witten invariant on a homogeneous space X$X$. For that, we give an interpretation in terms of rational curves on X$X$.