Topological Correlation Functions Research Articles

A one-dimensional theory for Higgs branch operators

We use supersymmetric localization to calculate correlation functions of half-BPS local operators in 3d mathcal{N}=4 superconformal field theories whose Lagrangian descriptions consist of vectormultiplets coupled to hypermultiplets. The operators we primarily study are certain twisted linear combinations of Higgs branch operators that can be inserted anywhere along a given line. These operators are constructed from the hypermultiplet scalars. They form a one-dimensional non-commutative operator algebra with topological correlation functions. The 2- and 3-point functions of Higgs branch operators in the full 3d mathcal{N}=4 theory can be simply inferred from the 1d topological algebra. After conformally mapping the 3d superconformal field theory from flat space to a round three-sphere, we preform supersymmetric localization using a supercharge that does not belong to any 3d mathcal{N}=2 subalgebra of the mathcal{N}=4 algebra. The result is a simple model that can be used to calculate correlation functions in the 1d topological algebra mentioned above. This model is a 1d Gaussian theory coupled to a matrix model, and it can be viewed as a gauge-fixed version of a topological gauged quantum mechanics. Our results generalize to non-conformal theories on S3 that contain real mass and Fayet-Iliopolous parameters. We also provide partial results in the 1d topological algebra associated with the Coulomb branch, where we calculate correlation functions of local operators built from the vectormultiplet scalars.

On the B-twisted topological sigma model and Calabi–Yau geometry

We provide a rigorous perturbative quantization of the B-twisted topological sigma model via a first-order quantum field theory on derived mapping space in the formal neighborhood of constant maps. We prove that the first Chern class of the target manifold is the obstruction to the quantization via Batalin–Vilkovisky formalism. When the first Chern class vanishes, i.e. on Calabi–Yau manifolds, the factorization algebra of observables gives rise to the expected topological correlation functions in the B-model. We explain a twisting procedure to generalize to the Landau–Ginzburg case, and show that the resulting topological correlations coincide with Vafa’s residue formula.

Duality in twisted [formula omitted] = 4 supersymmetric gauge theories in four dimensions

We consider a twisted version of the four-dimensional N = 4 supersymmetric Yang-Mills theory with gauge groups SU(2) and SO(3) and bare masses for two of its chiral multiplets, thereby breaking N = 4 down to N = 2. Using the wall-crossing technique introduced by Moore and Witten within the u-plane approach to twisted topological field theories, we compute the partition function and all the topological correlation functions for the case of simply connected spin four-manifolds of simple type. By including 't Hooft fluxes, we analyse the properties of the resulting formulae under duality transformations. The partition function transforms in the same way as the one first presented by Vafa and Witten for another twist of the N = 4 supersymmetric theory in their strong coupling test of S-duality. Both partition functions coincide on K3. The topological correlation functions turn out to transform covariantly under duality, following a simple pattern which seems to be inherent in a general type of topological quantum field theories.

Mirror symmetry and the moduli space for generic hypersurfaces in toric varieties

The moduli dependence of (2,2) superstring compactifications based on Calabi-Yau hypersurfaces in weighted projective space has so far only been investigated for Fermat-type polynomial constraints. These correspond to Landau-Ginzburg orbifolds with c = 9 whose potential is a sum of A-type singularities. Here we consider the generalization to arbitrary quasi-homogeneous singularities at c = 9. We use minor symmetry to derive the dependence of the models on the complexified Kähler moduli and check the expansions of some topological correlation functions against explicit genus zero and genus one instanton calculations. As an important application we give examples of how non-algebraic (“twisted”) deformations can be mapped to algebraic ones, hence allowing us to study the full moduli space. We also study how moduli spaces can be nested in each other, thus enabling a (singular) transition from one theory to another. Following the recent work of Greene, Morrison and Strominger we show that this corresponds to black hole condensation in type II string theories compactified on Calabi-Yau manifolds.

Non-scale-invariant topological Landau-Ginzburg models

The Landau-Ginzburg formulation of two-dimensional topological sigma models on the target space with positive first Chern class is considered. The effective Landau-Ginzburg superpotential takes the form of logarithmic type which is characteristic of supersymmetric theories with the mass gap. The equations of motion yield the defining relations of the quantum cohomology ring. Topological correlation functions in the CP n−1 and Grassmannian models are explicitly evaluated with the use of the logarithmic superpotential.

Topological correlation functions in Minkowski spacetime

We consider a class of non-unitary Toda theories based on the Lie superalgebras A (1) ( n, n) in two-dimensional Minkowski spacetime, which can be twisted into a topological sector. In particular we study the simplest example with n = 1 and real fields, and show how this theory can be treated as an integrable perturbation of the A(1, 0) superconformal model. We construct the conserved currents and the vertex operators which are chiral primary fields in the conformal theory. We then define the chiral ring of the A (1)(1, 1) Toda theory and compute topological correlation functions in the twisted sector. The calculation is performed using a N = 2 off shell superspace approach.

The N matrix model and gauged WZW models

It has been shown that the N matrix model two-dimensional gravity is related to certain topological field theories obtained by twisting the N = 2 minimal models. In this paper, the latter theories are studied by realizing them as gauged WZW models. This leads to an algebrogeometric description of the topological correlation functions from which many of their standard properties can be recovered. We find that in a certain sense the model at level k, with k analytically continued to −3, is equivalent to the Penner model (which computes the Euler characteristic of the moduli space of Riemann surfaces). We also gain better understanding of formulas of Lerche, Vafa, and Warner; Gepner; and Spiegelglas.

Topological Landau-Ginzburg matter at c=3

The topological correlation functions, their prepotential, and the Landau-Ginzburg potential are computed for the N=2 supersymmetric, c=3, matter model that is a tensor product of three c=1 models.