We use the holomorphic anomaly equation to solve the gravitational corrections to Seiberg-Witten theory and a two-cut matrix model, which is related by the Dijkgraaf-Vafa conjecture to the topological B-model on a local Calabi-Yau manifold. In both cases we construct propagators that give a recursive solution in the genus modulo a holomorphic ambiguity. In the case of Seiberg-Witten theory the gravitational corrections can be calculated to all genus as quasimodular functions of Γ(2). In the matrix model we fix the holomorphic ambiguity up to genus two. The latter result establishes the Dijkgraaf-Vafa conjecture at that genus and yields a new method for solving the matrix model at fixed genus in closed form in terms of generalized hypergeometric functions.

https://iopscience.iop.org/article/10.1088/1126-6708/2007/09/054/pdf

Holomorphic anomaly in gauge theories and matrix models

We give a general review of extended supergravities and their gauging using the duality-covariant embedding tensor formalism. Although the focus is on four-dimensional theories, an overview of the gauging procedure and the related tensor hierarchy in the higher-dimensional models is given. The relation of gauged supergravities to flux compactifications is discussed and examples are worked out in detail.

Gauged Supergravities.

We present the exact solution of a scalar field theory defined with noncommuting position and momentum variables. The model describes charged particles in a uniform magnetic field and with an inter ...

Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces

Supersymmetric gauge theories on noncommutative superspace

The Wess-Zumino model on $\mathcal{N}=\frac{1}{2}$ nonanticommutative superspace, which contains the dimension-6 term ${F}^{3}$, is shown to be renormalizable to all orders in perturbation theory, upon adding $F$ and ${F}^{2}$ terms to the original Lagrangian. The renormalizability is possible, even with this higher-dimension operator, because the Lagrangian is not Hermitian. Such deformed field theories arise naturally in string theory with a graviphoton background.

N=1/2 Wess-Zumino model is renormalizable

Non commutative superspaces can be introduced as the Moyal-Weyl quan- tization of a Poisson bracket for classical superflelds. Difierent deformations are studied corresponding to constant background flelds in string theory. Supersymmetric and non supersymmetric deformations can be deflned, depending on the difierential operators used to deflne the Poisson bracket. Some examples of deformed, 4 dimensional lagrangians are given. For extended superspace (N > 1), some new deformations can be deflned, with no analogue in the N = 1 case.

/pdf/supersymmetry-in-noncommutative-superspaces-1hatm6pmx9.pdf

Supersymmetry in noncommutative superspaces

We discuss the geometry of the c-map from projective special Kahler to quaternionic Kahler manifolds using the twist construction to provide a global approach to Hitchin’s description. As found by Alexandrov et al. and Alekseevsky et al. this is related to the quaternionic flip of Haydys. We prove uniqueness statements for several steps of the construction. In particular, we show that, given a hyperKahler manifold with a rotating symmetry, there is essentially only a one parameter degree of freedom in constructing a quaternionic Kahler manifold of the same dimension. We demonstrate how examples on group manifolds arise from this picture.

Twist Geometry of the c-Map

We discuss the geometry of the c-map from projective special Kahler to quaternionic Kahler manifolds using the twist construction to provide a global approach to Hitchin's description. As found by Alexandrov et al. and Alekseevsky et al. this is related to the quaternionic flip of Haydys. We prove uniqueness statements for several steps of the construction. In particular, we show that given a hyperKahler manifold with a rotating symmetry, there is essentially only a one parameter degree of freedom in constructing a quaternionic Kahler manifold of the same dimension. We demonstrate how examples on group manifolds arise from this picture.

Twist geometry of the c-map

We exploit some relations which exist when (rigid) special geometry is for- mulated in real symplectic special coordinates P I = (p ¤ ,q¤), I = 1,...,2n. The central role of the real 2n × 2n matrix M(<F,=F), where F = @¤@§F and F is the holomorphic prepotential, is elucidated in the real formalism. The property M­M = ­, where ­ is the invariant symplectic form, is used to prove several identities in the Darboux formu- lation. In this setting the matrix M coincides with the (negative of the) Hessian matrix H(S) = @ 2S @PI@PJ of a certain hamiltonian real function S(P), which also provides the metric of the special Kahler manifold. When S(P) = S(U+ ¯ U) is regarded as a "Kahler potential" of a complex manifold with coordinates U I = 1 (P I + iZ I ), it provides a Kahler metric of a hyperkahler manifold, which describes the hypermultiplet geometry obtained by c-map from the original n-dimensional special Kahler structure.

/pdf/observations-on-the-darboux-coordinates-for-rigid-special-4q4au3dbpx.pdf

Observations on the Darboux coordinates for rigid special geometry

http://cds.cern.ch/record/934522/files/0603111.pdf

Oscar Macia

Papers

Supersymmetry in noncommutative superspaces

Twist Geometry of the c-Map

Twist geometry of the c-map

Observations on the Darboux coordinates for rigid special geometry

Real symplectic formulation of local special geometry