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

Let A = (a1, . . . , an) be a vector of integers which sum to k(2g − 2+ n). The double ramification cycle DRg,A ∈ CH (Mg,n) on the moduli space of curves is the virtual class of an Abel-Jacobi locus of pointed curves (C, x1, . . . , xn) satisfying

Logarithmic double ramification cycles

We study the higher genus equivariant Gromov–Witten theory of the Hilbert scheme of is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture [Bryan and Graber, Algebraic Geometry–Seattle 2005, Part 1, Proceedings of Symposia in Pure Mathematics, 80 (American Mathematical Society, Providence, RI, 2009), 23–42; Coates et al., Geom. Topol. 13 (2009), 2675–2744; Coates & Ruan, Ann. Inst. Fourier (Grenoble) 63 (2013), 431–478].

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Higher genus gromov–witten theory of and associated to local curves

The Katz-Klemm-Vafa conjecture expresses the Gromov-Witten theory of K3 surfaces (and K3-fibred 3-folds in fibre classes) in terms of modular forms. Its recent proof gives the first non-toric geometry in dimension greater than 1 where Gromov-Witten theory is exactly solved in all genera. 
We survey the various steps in the proof. The MNOP correspondence and a new Pairs/Noether-Lefschetz correspondence for K3-fibred 3-folds transform the Gromov-Witten problem into a calculation of the full stable pairs theory of a local K3-fibred 3-fold. The stable pairs calculation is then carried out via degeneration, localisation, vanishing results, and new multiple cover formulae.

https://www.intlpress.com/site/pub/files/_fulltext/journals/sdg/2016/0021/0001/SDG-2016-0021-0001-a007.pdf

Notes on the proof of the KKV conjecture

Degenerate contributions to higher genus Gromov-Witten invariants of Calabi-Yau 3-folds are computed via Hodge integrals. The vanishing of contributions of covers of elliptic curves conjectured by Gopakumar and Vafa is proven. A formula for degree 1 covers for all genus pairs is computed in agreement with M-theoretic calculations of Gopakumar and Vafa. Finally, these results lead to a proof of a formula in the tautological ring of the moduli space of curves previously conjectured by Faber.

Hodge integrals and degenerate contributions

In 1998, Gopakumar and Vafa [5] proposed a duality between SU(N) Chern–Simonstheory on the 3–sphere and topological string theory on the resolved conifold. Asevidence, Gopakumar and Vafa showed the large–N free energy in Chern–Simonstheory exactly matches (after a change of variables) the topological string partitionfunction on the resolved conifold.Mathematically, the topological string partition function is just the natural generatingfunction for the Gromov–Witten invariants. The resolved conifold is the total space ofthe bundle O

/pdf/on-the-rigidity-of-stable-maps-to-calabi-yau-threefolds-4a2lat7wl0.pdf

Rahul Pandharipande

Papers

Logarithmic double ramification cycles

Higher genus gromov–witten theory of and associated to local curves

Notes on the proof of the KKV conjecture

Hodge integrals and degenerate contributions

On the rigidity of stable maps to Calabi-Yau threefolds