Gauge Theoretical Equivariant Gromov–Witten Invariants and the Full Seiberg–Witten Invariants¶of Ruled Surfaces

Let F be a differentiable manifold endowed with an almost K?hler structure (J,ω), α a J-holomorphic action of a compact Lie group on F, and K a closed normal subgroup of which leaves ω invariant. The purpose of this article is to introduce gauge theoretical invariants for such triples (F,α,K). The invariants are associated with moduli spaces of solutions of a certain vortex type equation on a Riemann surface Σ. Our main results concern the special case of the triple

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     

Local Gromov–Witten invariants of blowups of Fano surfaces

In this paper, using the degeneration formula we obtain a blowup formula of local Gromov–Witten invariants of Fano surfaces. This formula makes it possible to compute the local Gromov–Witten invariants of non-toric Fano surfaces from toric Fano surfaces, such as del Pezzo surfaces. This formula also verified an expectation of Chiang–Klemm–Yau–Zaslow in Section 8.3 of Chiang et al. (1999) [7]     

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     

Computing genus zero Gromov–Witten invariants of Fano varieties

We present a recursive algorithm computing all the genus zero Gromov–Witten invariants from a finite number of initial ones, for Fano manifolds with generically semisimple quantum (and tame semisimple small quantum) (p,p)$(p,p)$-type cohomology, whose first Chern class is a strictly positive combination of effective integral basic divisors.     

Gromov–Witten invariants of target curves via Symplectic Field Theory

We compute the Gromov–Witten potential at all genera of target smooth Riemann surfaces using Symplectic Field Theory techniques and establish differential equations for the full descendant potential. We need to impose (and possibly solve) different kinds of Schrödinger equations related to some quantization of the dispersionless KdV hierarchy. In particular, we find explicit formulas for the Gromov–Witten invariants of low degree of P¹${\mathbb{P}}^1$ with descendants of the Kähler class.     

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.     

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.     

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$X$.     

The Toda Equations and the Gromov–Witten Theory of the Riemann Sphere

Consequences of the Toda equations arising from the conjectural matrix model for the Riemann sphere are investigated. The Toda equations determine the Gromov–Witten descendent potential (including all genera) of the Riemann sphere from the degree 0 part. Degree 0 series computations via Hodge integrals then lead to higher-degree predictions by the Toda equations. First, closed series forms for all 1-point invariants of all genera and degrees are given. Second, degree 1 invariants are investigated with new applications to Hodge integrals. Third, a differential equation for the generating function of the classical simple Hurwitz numbers (in all genera and degrees) is found – the first such equation. All these results depend upon the conjectural Toda equations. Finally, proofs of the Toda equations in genus 0 and 1 are given.     

K-theoretic Gromov–Witten invariants in genus 0 and integrable hierarchies

We prove that the genus 0 invariants in K-theoretic Gromov–Witten theory are governed by an integrable hierarchy of hydrodynamic type. If the K-theoretic quantum product is semisimple, then we also prove the completeness of the hierarchy.     

The Local Gromov–Witten Theory of $${\mathbb{C}\mathbb{P}^1}$$ and Integrable Hierarchies

In this paper we begin the study of the relationship between the local Gromov–Witten theory of Calabi–Yau rank two bundles over the projective line and the theory of integrable hierarchies. We first of all construct explicitly, in a large number of cases, the Hamiltonian dispersionless hierarchies that govern the full-descendent genus zero theory. Our main tool is the application of Dubrovin’s formalism, based on associativity equations, to the known results on the genus zero theory from local mirror symmetry and localization. The hierarchies we find are apparently new, with the exception of the resolved conifold in the equivariantly Calabi–Yau case. For this example the relevant dispersionless system turns out to be related to the long-wave limit of the Ablowitz–Ladik lattice. This identification provides us with a complete procedure to reconstruct the dispersive hierarchy which should conjecturally be related to the higher genus theory of the resolved conifold. We give a complete proof of this conjecture for genus g ≤ 1; our methods are based on establishing, analogously to the case of KdV, a “quasi-triviality” property for the Ablowitz–Ladik hierarchy at the leading order of the dispersive expansion. We furthermore provide compelling evidence in favour of the resolved conifold/Ablowitz–Ladik correspondence at higher genus by testing it successfully in the primary sector for g = 2.     

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.     

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 $${{\,\mathrm{SU}\,}}(n)$$...     

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.