On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a $\mathbb C^*$ action with compact fixed locus. Applying virtual localisation we define invariants constant under deformations.
When the vanishing theorem of Vafa-Witten holds, the result is the (signed) Euler characteristic of the moduli space of instantons. In general there are other, rational, contributions. Calculations of these on surfaces with positive canonical bundle recover the first terms of modular forms predicted by Vafa and Witten.
This article provides a summary of arXiv:1701.08899 and arXiv:1701.08902 where the authors studied the enumerative geometry of nested Hilbert schemes of points and curves on algebraic surfaces and their connections to threefold theories, and in particular relevant Donaldson-Thomas, Vafa-Witten and Seiberg-Witten theories.
We propose a definition of Vafa-Witten invariants counting semistable Higgs pairs on a polarised surface. We use virtual localisation applied to Mochizuki/Joyce-Song pairs. For $K_S\le0$ we expect our definition coincides with an alternative definition using weighted Euler characteristics. We prove this for deg $K_S<0$ here, and it is proved for $S$ a K3 surface in \cite{MT}. For K3 surfaces we calculate the invariants in terms of modular forms which generalise and prove conjectures of Vafa and Witten.
Motivated by the S-duality conjecture of Vafa-Witten, Tanaka-Thomas have developed a theory of Vafa-Witten invariants for projective surfaces using the moduli space of Higgs sheaves. Their definition and calculation prove the S-duality prediction of Vafa-Witten in many cases in the side of gauge group $SU(r)$. In this survey paper for ICCM-2019 we review the S-duality conjecture in physics by Vafa-Witten and the definition of Vafa-Witten invariants for smooth projective surfaces and surface Deligne-Mumford stacks. We make a prediction that the Vafa-Witten invariants for Deligne-Mumford surface…
We provide a definition of Vafa-Witten invariants for projective surface Deligne-Mumford stacks, generalizing the construction of Tanaka-Thomas on the Vafa-Witten invariants for projective surfaces inspired by the S-duality conjecture. We give calculations for a root stack over a general type quintic surface, and quintic surfaces with ADE singularities. The relationship between the Vafa-Witten invariants of quintic surfaces with ADE singularities and the Vafa-Witten invariants of their crepant resolutions is also discussed.
We prove a generalized Mariño-Vafa formula for Hodge integrals over $\Mbar_{g, γ-μ}(\cB G)$ with $G$ an arbitrary finite abelian group. Then we use this formula to study the local Gromov-Witten theory of an orbi-curve with cyclic stack points in a Calabi-Yau three-orbifold.
We show that \emph{vertical} contributions to (possibly semistable) Tanaka-Thomas-Vafa-Witten invariants are well defined for surfaces with $p_g(S)>0$, partially proving conjectures of \cite{TT2} and \cite{T}. Moreover, we show that such contributions are computed by the same tautological integrals as in the stable case, which we studied in \cite{L}. Using the work of Kiem and Li, we show that stability of universal families of vertical Joyce-Song pairs is controlled by cosections of the obstruction sheaves of such families.
Vafa-Witten (VW) theory is a topologically twisted version of N=4 supersymmetric Yang-Mills theory. S-duality suggests that the partition function of VW theory with gauge group SU(N) transforms as a modular form under duality transformations. Interestingly, Vafa and Witten demonstrated the presence of a modular anomaly, when the theory has gauge group SU(2) and is considered on the complex projective plane P2. This modular anomaly could be expressed as an integral of a modular form, and also be traded for a holomorphic anomaly. We demonstrate that the modular anomaly for gauge group SU(3) invo…
We study the monopole contribution to the refined Vafa-Witten invariant, recently defined by Maulik and Thomas [13]. We apply results of Gholampour and Thomas [7] to prove a universality result for the generating series of contributions of Higgs pairs with 1-dimensional weight spaces. For prime rank, these account for the entire monopole contribution, by a theorem of Thomas. We use toric computations to determine part of the generating series, and find agreement with the conjectures of Göttsche and Kool [10] for rank 2 and 3.
As an analogy to Gopakumar-Vafa conjecture on Calabi-Yau 3-folds, Klemm-Pandharipande defined Gopakumar-Vafa type invariants of a Calabi-Yau 4-fold $X$ using Gromov-Witten theory. When $X$ is holomorphic symplectic, Gromov-Witten invariants vanish and one can consider the corresponding reduced theory. In a companion work, we propose a definition of Gopakumar-Vafa type invariants for such a reduced theory. In this paper, we give them a sheaf theoretic interpretation via moduli spaces of stable pairs.
Topologically twisted $\mathcal{N} = 4$ super Yang-Mills theory has a partition function that counts Euler numbers of instanton moduli spaces. On the manifold $\mathbb{P}^2$ and with gauge group $\mathrm{U}(3)$ this partition function has a holomorphic anomaly which makes it a mock modular form of depth two. We employ the Circle Method to find a Rademacher expansion for the Fourier coefficients of this partition function. This is the first example of the use of Circle Method for a mock modular form of a higher depth.
Using the interpretation of certain generalised Donaldson-Thomas invariants, including stable pairs curve counts, as the monodromy of a flat connection on a formal principal bundle, we show that the conjectural Gopakumar-Vafa contributions of all genera to the Gromov-Witten partition function appear in the asymptotics of the corresponding flat sections. The Fourier-Laplace integrals used to produce flat sections lead naturally to the GW/DT change of variable -q = e^{i u}.
For any scheme $M$ with a perfect obstruction theory, Jiang and Thomas associate a scheme $N$ with symmetric perfect obstruction theory. The scheme $N$ is a cone over $M$ given by the dual of the obstruction sheaf of $M$, and contains $M$ as its zero section. Locally $N$ is the critical locus of a regular function. In this note we prove that $N$ is a $d$-critical scheme in the sense of Joyce. By assuming an orientation on $N$ there exists a global motive for $N$ locally given by the motive of vanishing cycles of the local regular function. We prove a motivic localization formula under the good…
We prove the KKV conjecture expressing Gromov-Witten invariants of K3 surfaces in terms of modular forms. Our results apply in every genus and for every curve class. The proof uses the Gromov-Witten/Pairs correspondence for K3-fibered hypersurfaces of dimension 3 to reduce the KKV conjecture to statements about stable pairs on (thickenings of) K3 surfaces. Using degeneration arguments and new multiple cover results for stable pairs, we reduce the KKV conjecture further to the known primitive cases. Our results yield a new proof of the full Yau-Zaslow formula, establish new Gromov-Witten multip…
Refined Vafa-Witten invariants for toric surfaces from supersymmetric localization in 5D gauge theory
Philippe Meyer Institute, Physics Department, École Normale Supérieure (ENS), Université PSL（菲利普·梅耶研究所，物理系，巴黎高等师范学院，巴黎文理研究大学）；Laboratoire de Physique Théorique et Hautes Energies (LPTHE), UMR 7589 CNRS-Sorbonne Université（理论与高能物理实验室，CNRS-索邦大学联合研究单位）；SISSA（的里雅斯特国际高等研究学校）；INFN, Sezione di Trieste（意大利国家核物理研究所，的里雅斯特分部）；Institute for Geometry and Physics, IGAP（几何与物理研究所）
We study Gauss-Manin systems of non tame Laurent polynomial functions. We focuse on Hori-Vafa models, which are the expected mirror partners of the small quantum cohomology of smooth hypersurfaces in weighted projective spaces.
The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa formula holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.
We prove a conjectural vanishing result for Gopakumar--Vafa invariants of quintic 3-folds, referred to as Castelnuovo bound in the literature. Furthermore, we calculate Gopakumar--Vafa invariants at Castelnuovo bound $g=\frac{d^2+5d+10}{10}$. As physicists showed, these two properties allow us to compute all Gromov--Witten invariants of quintic 3-folds up to genus $53$, provided that the conifold gap condition holds. We also give a bound for the genus of any one-dimensional closed subscheme in a smooth hypersurface of degree $\leq 5$, which may be of independent interest.
We regard the work of Maulik and Toda, proposing a sheaf-theoretic approach to Gopakumar-Vafa invariants, as defining a BPS structure, that is, a collection of BPS invariants together with a central charge. Assuming their conjectures, we show that a canonical flat section of the flat connection corresponding to this BPS structure, at the level of formal power series, reproduces the Gromov-Witten partition function for all genera, up to some error terms in genus 0 and 1. This generalises a result of Bridgeland and Iwaki for the contribution from genus 0 Gopakumar-Vafa invariants.
We derive a closed formula for the generating function of genus two Gromov-Witten invariants of quintic 3-folds and verify the corresponding mirror symmetry conjecture of Bershadsky, Cecotti, Ooguri and Vafa.
We provide a definition of Tanaka-Thomas's Vafa-Witten invariants for étale gerbes over smooth projective surfaces using the moduli spaces of $μ_r$-gerbe twisted sheaves and Higgs sheaves. Twisted sheaves and their moduli are naturally used to study the period-index theorem for the corresponding $μ_r$-gerbe in the Brauer group of the surfaces. Deformation and obstruction theory of the twisted sheaves and Higgs sheaves behave like general sheaves and Higgs sheaves. We define virtual fundamental classes on the moduli spaces and define the twisted Vafa-Witten invariants using virtual localization…
We study open-closed orbifold Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric Deligne-Mumford (DM) stacks (with possibly non-trivial generic stabilizers and semi-projective coarse moduli spaces) relative to Lagrangian branes of Aganagic-Vafa type. We present foundational materials of enumerative geometry of stable holomorphic maps from bordered orbifold Riemann surfaces to a 3-dimensional Calabi-Yau smooth toric DM stack with boundaries mapped into a Aganagic-Vafa brane. All genus open-closed Gromov-Witten invariants are defined by torus localization and depend on the choice …
A difference equation is proved for the Gromov-Witten potential of the resolved conifold. Using the Gopakumar-Vafa resummation of the Gromov-Witten invariants of any Calabi-Yau threefold, it is further shown that similar difference equations are satisfied by the part of the resummed potential containing the contribution of the genus zero GV invariants.
In analogy with the Gopakumar-Vafa (GV) conjecture on Calabi-Yau (CY) 3-folds, Klemm and Pandharipande defined GV type invariants on Calabi-Yau 4-folds using Gromov-Witten theory and conjectured their integrality. In a joint work with Maulik and Toda, the author conjectured their genus zero invariants are $\mathrm{DT_4}$ invariants of one dimensional stable sheaves. In this paper, we study this conjecture on the total space of canonical bundle of a Fano 3-fold $Y$, which reduces to a relation between twisted GW and $\mathrm{DT_3}$ invariants on $Y$. Examples are computed for both compact and n…
In this paper, we propose an ansatz for defining Gopakumar-Vafa invariants of Calabi-Yau threefolds, using perverse sheaves of vanishing cycles. Our proposal is a modification of a recent approach of Kiem-Li, which is itself based on earlier ideas of Hosono-Saito-Takahashi. We conjecture that these invariants are equivalent to other curve-counting theories such as Gromov-Witten theory and Pandharipande-Thomas theory. Our main theorem is that, for local surfaces, our invariants agree with PT invariants for irreducible one-cycles. We also give a counter-example to the Kiem-Li conjectures, where …
The Gopakumar-Vafa type invariants on Calabi-Yau 4-folds (which are non-trivial only for genus zero and one) are defined by Klemm-Pandharipande from Gromov-Witten theory, and their integrality is conjectured. In a previous work of Cao-Maulik-Toda, $\mathrm{DT}_4$ invariants with primary insertions on moduli spaces of one dimensional stable sheaves are used to give a sheaf theoretical interpretation of the genus zero GV type invariants. In this paper, we propose a sheaf theoretical interpretation of the genus one GV type invariants using descendent insertions on the above moduli spaces. The con…
