Holomorphic Chern-Simons Theory Research Articles

Marginal deformations of [formula omitted] SYM from open/closed twistor strings

We investigate how the marginal deformations of N = 4 supersymmetric Yang–Mills theory (analysed in particular by Leigh and Strassler) arise within B-model topological string theory on supertwistor space C P 3 | 4 . This is achieved by turning on a certain closed string background in the fermionic directions. Through a specific open/closed correlation function, this mode induces a correction to holomorphic Chern–Simons theory, corresponding to the self-dual part of the chiral operator added on the gauge theory side. The effect of the deformation is interpreted as non-anticommutativity between some of the odd coordinates of C P 3 | 4 . Motivated by this, we extend the twistor formalism for calculating MHV amplitudes in N = 4 SYM to these N = 1 theories by introducing a suitable star product between the wavefunctions. We check that our prescription yields the expected results to linear order in the deformation parameter.

Topological symmetry of forms, [formula omitted] supersymmetry and S-duality on special manifolds

We study the quantization of a holomorphic 2-form coupled to a Yang–Mills field on special manifolds in various dimensions, and we show that it yields twisted supersymmetric theories. The construction determines ATQFTs (Almost Topological Quantum Field Theories), that is, theories with observables that are invariant under changes of metrics belonging to restricted classes. For Kähler manifolds in four dimensions, our topological model is related to N = 1 Super Yang–Mills theory. Extended supersymmetries are recovered by considering the coupling with chiral multiplets. We also analyse Calabi–Yau manifolds in six and eight dimensions, and seven-dimensional G 2 manifolds of the kind recently discussed by Hitchin. We argue that the 2-form field could play an interesting rôle in the study of the conjectured S-duality in topological strings. We finally show that in the case of real forms the partition function of our topological model in six dimensions is related to the square of that of the holomorphic Chern–Simons theory, and we discuss the uplift to seven dimensions and its relation to the topological M theory.

The topological B-model on a mini-supertwistor space and supersymmetric Bogomolny monopole equations

In the recent paper hep-th/0502076, it was argued that the open topological B-model whose target space is a complex (2|4)-dimensional mini-supertwistor space with D3- and D1-branes added corresponds to a super Yang-Mills theory in three dimensions. Without the D1-branes, this topological B-model is equivalent to a dimensionally reduced holomorphic Chern-Simons theory. Identifying the latter with a holomorphic BF-type theory, we describe a twistor correspondence between this theory and a supersymmetric Bogomolny model on R^3. The connecting link in this correspondence is a partially holomorphic Chern-Simons theory on a Cauchy-Riemann supermanifold which is a real one-dimensional fibration over the mini-supertwistor space. Along the way of proving this twistor correspondence, we review the necessary basic geometric notions and construct action functionals for the involved theories. Furthermore, we discuss the geometric aspect of a recently proposed deformation of the mini-supertwistor space, which gives rise to mass terms in the supersymmetric Bogomolny equations. Eventually, we present solution generating techniques based on the developed twistorial description together with some examples and comment briefly on a twistor correspondence for super Yang-Mills theory in three dimensions.

Topological B-Model on Weighted Projective Spaces and Self-Dual Models in Four Dimensions

It was recently shown by Witten on the basis of several examples that the topological B-model whose target space is a Calabi-Yau (CY) supermanifold is equivalent to holomorphic Chern-Simons (hCS) theory on the same supermanifold. Moreover, for the supertwistor space P3|4 as target space, it has been demonstrated that hCS theory on P3|4 is equivalent to self-dual = 4 super Yang-Mills (SYM) theory in four dimensions. We consider as target spaces for the B-model the weighted projective spaces WP3|2(1,1,1,1|p,q) with two fermionic coordinates of weight p and q, respectively — which are CY supermanifolds for p+q = 4 — and discuss hCS theory on them. By using twistor techniques, we obtain certain field theories in four dimensions which are equivalent to hCS theory. These theories turn out to be self-dual truncations of = 4 SYM theory or of its twisted (topological) version.

Sigma models for bundles on Calabi–Yau: A proposal for Matrix string compactifications

We describe a class of supersymmetric gauged linear sigma-model, whose target space is the infinite-dimensional space of bundles on a Calabi–Yau 3- or 2-fold. This target space can be considered the configuration space of D-branes wrapped around the Calabi–Yau. We propose that this model can be used to define matrix string theory compactifications. In the infrared limit the model flows to a superconformal non-linear sigma-model whose target space is the moduli space of BPS configurations of branes on the compact space, containing the moduli space of semi-stable bundles. We argue that the bulk degrees of freedom decouple in the infrared limit if semi-stability implies stability. We study topological versions of the model on Calabi–Yau 3-folds. The resulting B-model is argued to be equivalent to the holomorphic Chern–Simons theory proposed by Witten. The A-model and half-twisted model define the quantum cohomology ring and the elliptic genus, respectively, of the moduli space of stable bundles on a Calabi–Yau 3-fold.

The topological B model as a twisted spinning particle

The B-twisted topological sigma model coupled to topological gravity is supposed to be described by an ordinary field theory: a type of holomorphic Chern-Simons theory for the open string, and the Kodaira-Spencer theory for the closed string. We show that the B model can be represented as a particle theory obtained by reducing the sigma model to one dimension, and replacing the coupling to topological gravity by a coupling to a twisted one-dimensional supergravity. The particle can be defined on any Kähler manifold—it does not require the Calabi-Yau condition—so it may provide a more generalized setting for the B model than the topological sigma model. The one-loop partition function of the particle can be written in terms of the Ray-Singer torsion of the manifold, and agrees with that of the original B model. After showing how to deform the Kähler and complex structures in the particle, we prove the independence of this partition function on the Kähler structure, and investigate the origin of the holomorphic anomaly. To define other amplitudes, one needs to introduce interactions into the particle. The particle will then define a field theory, which may or may not be the Chern-Simons or Kodaira-Spencer theories.