Witten Theory Research Articles

Global Kuranishi charts and a product formula in symplectic Gromov–Witten theory

Global Kuranishi charts and a product formula in symplectic Gromov–Witten theory

Frobenius–Schur indicators for twisted Real representation theory and two dimensional unoriented topological field theory

We construct a two dimensional unoriented open/closed topological field theory from a finite graded group π:Gˆ↠{1,−1}, a π-twisted 2-cocycle θˆ on BGˆ and a character λ:Gˆ→U(1). The underlying oriented theory is a twisted Dijkgraaf–Witten theory. The construction is based on a detailed study of the (Gˆ,θˆ,λ)-twisted Real representation theory of ker⁡π. In particular, twisted Real representations are boundary conditions of the unoriented theory and the generalized Frobenius–Schur element is its crosscap state.

Knot concordance invariants from Seiberg–Witten theory and slice genus bounds in 4-manifolds

We construct a new family of knot concordance invariants [Formula: see text], where [Formula: see text] is a prime number. Our invariants are obtained from the equivariant Seiberg–Witten–Floer cohomology, constructed by the author and Hekmati, applied to the degree [Formula: see text] cyclic cover of [Formula: see text] branched over [Formula: see text]. In the case [Formula: see text], our invariant [Formula: see text] shares many similarities with the knot Floer homology invariant [Formula: see text] defined by Hom and Wu. Our invariants [Formula: see text] give lower bounds on the genus of any smooth, properly embedded, homologically trivial surface bounding [Formula: see text] in a definite [Formula: see text]-manifold with boundary [Formula: see text].

Higher Deformation Quantization for Kapustin–Witten Theories

Higher Deformation Quantization for Kapustin–Witten Theories

Gopakumar–Vafa Type Invariants of Holomorphic Symplectic 4-Folds

Using reduced Gromov–Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi–Yau 3-folds, Klemm and Pandharipande for Calabi–Yau 4-folds, and Pandharipande and Zinger for Calabi–Yau 5-folds. We conjecture that our invariants are integers and give a sheaf-theoretic interpretation in terms of reduced 4-dimensional Donaldson–Thomas invariants of one-dimensional stable sheaves. We check our conjectures for the product of two K3 surfaces and for the cotangent bundle of P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^2$$\\end{document}. Modulo the conjectural holomorphic anomaly equation, we compute our invariants also for the Hilbert scheme of two points on a K3 surface. This yields a conjectural formula for the number of isolated genus 2 curves of minimal degree on a very general hyperkähler 4-fold of K3[2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K3^{[2]}$$\\end{document}-type. The formula may be viewed as a 4-dimensional analogue of the classical Yau–Zaslow formula concerning counts of rational curves on K3 surfaces. In the course of our computations, we also derive a new closed formula for the Fujiki constants of the Chern classes of tangent bundles of both Hilbert schemes of points on K3 surfaces and generalized Kummer varieties.

Correction to: d-elliptic loci in genus 2 and 3

Abstract We correct an error in Proposition 5.1 of the named paper, which was brought to the author’s attention by Aaron Pixton. This affects various other formulas in the paper, including Theorem 1.4, which are also corrected.

Logarithmic Gromov–Witten theory and double ramification cycles

Abstract We examine the logarithmic Gromov–Witten cycles of a toric variety relative to its full toric boundary. The cycles are expressed as products of double ramification cycles and natural tautological classes in the logarithmic Chow ring of the moduli space of curves. We introduce a simple new technique that relates the Gromov–Witten cycles of rigid and rubber geometries; the technique is based on a study of maps to the logarithmic algebraic torus. By combining this with recent work on logarithmic double ramification cycles, we deduce that all logarithmic Gromov–Witten pushforwards, for maps to a toric variety relative to its full toric boundary, lie in the tautological ring of the moduli space of curves. A feature of the approach is that it avoids the as yet undeveloped logarithmic virtual localization formula, instead relying directly on piecewise polynomial functions to capture the structure that would be provided by such a formula. The results give a common generalization of work of Faber and Pandharipande, and more recent work of Holmes and Schwarz as well as Molcho and Ranganathan. The proof passes through general structure results on the space of stable maps to the logarithmic algebraic torus, which may be of independent interest.

Virasoro Constraints for Toric Bundles

Abstract We show that the Virasoro conjecture in Gromov–Witten theory holds for the the total space of a toric bundle $E \to B$ if and only if it holds for the base B. The main steps are: (i) We establish a localization formula that expresses Gromov–Witten invariants of E, equivariant with respect to the fiberwise torus action in terms of genus-zero invariants of the toric fiber and all-genus invariants of B, and (ii) we pass to the nonequivariant limit in this formula, using Brown’s mirror theorem for toric bundles.

Quantum Curve and Bilinear Fermionic Form for the Orbifold Gromov–Witten Theory of ℙ[r

Quantum Curve and Bilinear Fermionic Form for the Orbifold Gromov–Witten Theory of ℙ[r

Quasimaps to moduli spaces of sheaves on a surface

Abstract In this article, we study quasimaps to moduli spaces of sheaves on a $K3$ surface S. We construct a surjective cosection of the obstruction theory of moduli spaces of $\epsilon $ -stable quasimaps. We then establish reduced wall-crossing formulas which relate the reduced Gromov–Witten theory of moduli spaces of sheaves on S and the reduced Donaldson–Thomas theory of $S\times C$ , where C is a nodal curve. As applications, we prove the Hilbert-schemes part of the Igusa cusp form conjecture; higher-rank/rank-one Donaldson–Thomas correspondence with relative insertions on $S\times C$ , if $g(C)\leq 1$ ; Donaldson–Thomas/Pandharipande–Thomas correspondence with relative insertions on $S\times \mathbb {P}^1$ .

Counting embedded curves in symplectic $6$-manifolds

Based on computations of Pandharipande (1999), Zinger (2011) proved that the Gopakumar–Vafa BPS invariants \mathrm{BPS}_{A,g}(X,\omega) for primitive Calabi–Yau classes and arbitrary Fano classes A on a symplectic 6 -manifold (X,\omega) agree with the signed count n_{A,g}(X,\omega) of embedded J -holomorphic curves representing A and of genus g for a generic almost complex structure J compatible with \omega . Zinger's proof of the invariance of n_{A,g}(X,\omega) is indirect, as it relies on Gromov–Witten theory. In this article we give a direct proof of the invariance of n_{A,g}(X,\omega) . Furthermore, we prove that n_{A,g}(X,\omega) = 0 for g \gg 1 , thus proving the Gopakumar–Vafa finiteness conjecture for primitive Calabi–Yau classes and arbitrary Fano classes.

Splitting of Gromov–Witten invariants with toric gluing strata

We prove a splitting formula that reconstructs the logarithmic Gromov–Witten invariants of simple normal crossing varieties from the punctured Gromov–Witten invariants of their irreducible components, under the assumption of the gluing strata being toric varieties. The formula is based on the punctured Gromov–Witten theory developed by Abramovich, Chen, Gross, and Siebert.

Positive scalar curvature and homology cobordism invariants

AbstractWe give an obstruction to positive scalar curvature metrics on 4‐manifolds with the homology described in terms of homology cobordism invariants from Seiberg–Witten theory. The main tool of the proof is a relative Bauer–Furuta‐type invariant on a periodic‐end 4‐manifold.

When Does a Three-Dimensional Chern–Simons–Witten Theory Have a Time Reversal Symmetry?

When Does a Three-Dimensional Chern–Simons–Witten Theory Have a Time Reversal Symmetry?

Gromov–Witten theory of complete intersections via nodal invariants

AbstractWe provide an inductive algorithm computing Gromov–Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov–Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov–Witten invariants, we introduce the new notion of nodal relative Gromov–Witten invariants. We then prove a nodal degeneration formula and a relative splitting formula. These results for nodal relative Gromov–Witten theory are stated in complete generality and are of independent interest.

Motivic integration and birational invariance of BCOV invariants

Bershadsky, Cecotti, Ooguri, and Vafa constructed a real-valued invariant for Calabi--Yau manifolds, which is now called the BCOV torsion. Based on it, a metric-independent invariant, called the BCOV invariant, was constructed by Fang--Lu--Yoshikawa and Eriksson--Freixas i Montplet--Mourougane. The BCOV invariant is conjecturally related to the Gromov--Witten theory via mirror symmetry. Based upon the previous work of the second author, we prove the conjecture that birational Calabi--Yau manifolds have the same BCOV invariant. We also extend the construction of the BCOV invariant to Calabi--Yau varieties with Kawamata log terminal singularities and prove its birational invariance for Calabi--Yau varieties with canonical singularities. We provide an interpretation of our construction using the theory of motivic integration.

A Gromov–Witten Theory for Simple Normal-Crossing Pairs Without Log Geometry

We define a new Gromov–Witten theory relative to simple normal crossing divisors as a limit of Gromov–Witten theory of multi-root stacks. Several structural properties are proved including relative quantum cohomology, Givental formalism, Virasoro constraints (genus zero) and a partial cohomological field theory. Furthermore, we use the degree zero part of the relative quantum cohomology to provide an alternative mirror construction of Gross and Siebert (Intrinsic mirror symmetry, arXiv:1909.07649) and to prove the Frobenius structure conjecture of Gross et al. (Publ Math Inst Hautes Études Sci 122:65–168, 2015) in our setting.

Invariant probability measures from pseudoholomorphic curves Ⅱ: Pseudoholomorphic curve constructions

In the previous work, we introduced a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odd-dimensional smooth manifolds with pseudoholomorphic curve techniques from symplectic geometry. The technique requires existence of certain pseudoholomorphic curves satisfying some weak assumptions. In this work, we appeal to Gromov–Witten theory and Seiberg–Witten theory to construct large classes of examples where these pseudoholomorphic curves exist. Our argument uses neck stretching along with new analytical tools from Fish–Hofer's work on feral pseudoholomorphic curves.

$3$-manifolds and Vafa–Witten theory

$3$-manifolds and Vafa–Witten theory

Strong Positivity for the Skein Algebras of the 4-Punctured Sphere and of the 1-Punctured Torus

The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the $${\textit{SL}}_2$$ character variety of a topological surface. We realize the skein algebra of the 4-punctured sphere as the output of a mirror symmetry construction based on higher genus Gromov–Witten theory and applied to a complex cubic surface. Using this result, we prove the positivity of the structure constants of the bracelets basis for the skein algebras of the 4-punctured sphere and of the 1-punctured torus. This connection between topology of the 4-punctured sphere and enumerative geometry of curves in cubic surfaces is a mathematical manifestation of the existence of dual descriptions in string/M-theory for the $$\mathcal {N}=2$$ $$N_f=4$$ SU(2) gauge theory.