Curve counting via stable pairs in the derived category

For a nonsingular projective 3-fold $X$, we define integer invariants virtually enumerating pairs $(C,D)$ where $C\subset X$ is an embedded curve and $D\subset C$ is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of $X$. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of $X$. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.Comment: Corrected typos and duality error in Proposition 4.6. 47 page

Flexible Invariants Through Semantic Collaboration

Modular reasoning about class invariants is challenging in the presence of dependencies among collaborating objects that need to maintain global consistency. This paper presents semantic collaboration: a novel methodology to specify and reason about class invariants of sequential object-oriented programs, which models dependencies between collaborating objects by semantic means. Combined with a simple ownership mechanism and useful default schemes, semantic collaboration achieves the flexibility necessary to reason about complicated inter-object dependencies but requires limited annotation burden when applied to standard specification patterns. The methodology is implemented in AutoProof, our program verifier for the Eiffel programming language (but it is applicable to any language supporting some form of representation invariants). An evaluation on several challenge problems proposed in the literature demonstrates that it can handle a variety of idiomatic collaboration patterns, and is more widely applicable than the existing invariant methodologies.Comment: 22 page

Modular invariants for group-theoretical modular data. I

We classify indecomposable commutative separable (special Frobenius) algebras and their local modules in (untwisted) group-theoretical modular categories. This gives a description of modular invariants for group-theoretical modular data. As a bi-product we provide an answer to the question when (and in how many ways) two group-theoretical modular categories are equivalent as ribbon categories

Stable pairs on nodal K3 fibrations

We study Pandharipande-Thomas's stable pair theory on $K3$ fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of Kawai-Yoshioka's formula for the Euler characteristics of moduli spaces of stable pairs on $K3$ surfaces and Noether-Lefschetz numbers of the fibration. Moreover, we investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the $K3$ fibration. In the case that the $K3$ fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants in terms of the generalized Donaldson-Thomas invariants of 2-dimensional Gieseker semistable sheaves supported on the fibers.Comment: Published versio

Stability conditions and curve counting invariants on Calabi-Yau 3-folds

The purpose of this paper is twofold: first we give a survey on the recent developments of curve counting invariants on Calabi-Yau 3-folds, e.g. Gromov-Witten theory, Donaldson-Thomas theory and Pandharipande-Thomas theory. Next we focus on the proof of the rationality conjecture of the generating series of PT invariants, and discuss its conjectural Gopakumar-Vafa form.Comment: 50 pages, to appear in Kyoto Journal of Mat

HOMFLY polynomials, stable pairs and motivic Donaldson-Thomas invariants

Hilbert scheme topological invariants of plane curve singularities are identified to framed threefold stable pair invariants. As a result, the conjecture of Oblomkov and Shende on HOMFLY polynomials of links of plane curve singularities is given a Calabi-Yau threefold interpretation. The motivic Donaldson-Thomas theory developed by M. Kontsevich and the third author then yields natural motivic invariants for algebraic knots. This construction is motivated by previous work of V. Shende, C. Vafa and the first author on the large $N$ duality derivation of the above conjecture.Comment: 59 pages; v2 references added, minor corrections; v3: exposition improved, proofs expanded, results unchanged, to appear in Comm. Num. Th. Phy

On the Rozansky-Witten weight systems

Ideas of Rozansky and Witten, as developed by Kapranov, show that a complex symplectic manifold X gives rise to Vassiliev weight systems. In this paper we study these weight systems by using D(X), the derived category of coherent sheaves on X. The main idea (stated here a little imprecisely) is that D(X) is the category of modules over the shifted tangent sheaf, which is a Lie algebra object in D(X); the weight systems then arise from this Lie algebra in a standard way. The other main results are a description of the symmetric algebra, universal enveloping algebra, and Duflo isomorphism in this context, and the fact that a slight modification of D(X) has the structure of a braided ribbon category, which gives another way to look at the associated invariants of links. Our original motivation for this work was to try to gain insight into the Jacobi diagram algebras used in Vassiliev theory by looking at them in a new light, but there are other potential applications, in particular to the rigorous construction of the (1+1+1)-dimensional Rozansky-Witten TQFT, and to hyperkaehler geometry