Perturbative scattering amplitudes in Yang-Mills theory have many unexpected properties, such as holomorphy of the maximally helicity violating amplitudes. To interpret these results, we Fourier transform the scattering amplitudes from momentum space to twistor space, and argue that the transformed amplitudes are supported on certain holomorphic curves. This in turn is apparently a consequence of an equivalence between the perturbative expansion of = 4 super Yang-Mills theory and the D-instanton expansion of a certain string theory, namely the topological B model whose target space is the Calabi-Yau supermanifold

Perturbative Gauge Theory as a String Theory in Twistor Space

We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the category ('number of BPS states with given charge' in physics language). Formally, our motivic DT-invariants are elements of quantum tori over a version of the Grothendieck ring of varieties over the ground field. Via the quasi-classical limit 'as the motive of affine line approaches to 1' we obtain numerical DT-invariants which are closely related to those introduced by Behrend. We study some properties of both motivic and numerical DT-invariants including the wall-crossing formulas and integrality. We discuss the relationship with the mathematical works (in the non-triangulated case) of Joyce, Bridgeland and Toledano-Laredo, as well as with works of physicists on Seiberg-Witten model (string junctions), classification of N=2 supersymmetric theories (Cecotti-Vafa) and structure of the moduli space of vector multiplets. Relating the theory of 3d Calabi-Yau categories with distinguished set of generators (called cluster collection) with the theory of quivers with potential we found the connection with cluster transformations and cluster varieties (both classical and quantum).

Stability structures, motivic Donaldson-Thomas invariants and cluster transformations

The 't Hooft expansion of SU(N) Chern-Simons theory on $S^3$ is proposed to be exactly dual to the topological closed string theory on the $S^2$ blow up of the conifold geometry. The $B$-field on the $S^2$ has magnitude $Ng_s=\lambda$, the 't Hooft coupling. We are able to make a number of checks, such as finding exact agreement at the level of the partition function computed on {\it both} sides for arbitrary $\lambda$ and to all orders in 1/N. Moreover, it seems possible to derive this correspondence from a linear sigma model description of the conifold. We propose a picture whereby a perturbative D-brane description, in terms of holes in the closed string worldsheet, arises automatically from the coexistence of two phases in the underlying U(1) gauge theory. This approach holds promise for a derivation of the AdS/CFT correspondence.

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/1999/0003/0005/ATMP-1999-0003-0005-a005.pdf

On the Gauge Theory/Geometry Correspondence

It is shown how the topological string amplitudes encode the BPS structure of wrapped M2 branes in M-theory compactification on Calabi-Yau threefolds. This in turn is related to a twisted supersymmetric index in 5 dimensions which receives contribution only from BPS states. The spin dependence of BPS states in 5 dimensions is captured by the string coupling constant dependence of topological string amplitudes.

M theory and topological strings. 2.

We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kahler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum Kodaira-Spencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.

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The Topological Vertex

After a short exposition of the basic properties of the tautological ring of Mg,n, we explain three methods of detecting non-tautological classes in cohomology. The first is via curve counting over finite fields. The second is by obtaining length bounds on the action of the symmetric group n on tautological classes. The third is via classical boundary geometry. Several new non-tautological classes are found.

Tautological and non-tautological cohomology of the moduli space of curves

We prove a closed formula for the integrals of the top Segre classes of tautological bundles over the Hilbert schemes of points of a K3 surface X. We derive relations among the Segre classes via equivariant localization of the virtual fundamental classes of Quot schemes on X. The resulting recursions are then solved explicitly. The formula proves the K-trivial case of a conjecture of M. Lehn from 1999. 
The relations determining the Segre classes fit into a much wider theory. By localizing the virtual classes of certain relative Quot schemes on surfaces, we obtain new systems of relations among tautological classes on moduli spaces of surfaces and their relative Hilbert schemes of points. For the moduli of K3 sufaces, we produce relations intertwining the kappa classes and the Noether-Lefschetz loci. Conjectures are proposed.

Segre classes and Hilbert schemes of points

The virtual geometry of the moduli space of stable quotients is used to obtain Chow relations among the kappa classes on the moduli space of nonsingular genus g curves. In a series of steps, the stable quotient relations are rewritten in successively simpler forms. The final result is the proof of the Faber-Zagier relations (conjectured in 2000).

Relations in the tautological ring of the moduli space of curves

We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants of projective space are expressed as graph sums of tautological integrals over moduli spaces of stable pointed curves (generalizing Kontsevich's genus 0 formulas). Also, excess integrals over spaces of higher genus multiple covers are computed.

Localization of virtual classes

We construct a fully equivariant correspondence between Gromov‐Witten and stable pairs descendent theories for toric 3‐folds X . Our method uses geometric constraints on descendents, An surfaces and the topological vertex. The rationality of the stable pairs descendent theory plays a crucial role in the definition of the correspondence. We prove our correspondence has a non-equivariant limit. As a result of the construction, we prove an explicit non-equivariant stationary descendent correspondence for X (conjectured by MNOP). Using descendent methods, we establish the relative GW/Pairs correspondence for X=D in several basic new log Calabi‐Yau geometries. Among the consequences is a rationality constraint for non-equivariant descendent Gromov‐Witten series for P 3 .

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Rahul Pandharipande

Papers

Tautological and non-tautological cohomology of the moduli space of curves

Segre classes and Hilbert schemes of points

Relations in the tautological ring of the moduli space of curves

Localization of virtual classes

Gromov–Witten/pairs descendent correspondence for toric 3–folds