Additional Supersymmetries Research Articles

Dualities and loops on squashed $S^3$

We consider \mathcal{N}=4𝒩=4 supersymmetric gauge theories on the squashed three-sphere with six preserved supercharges. We first discuss how Wilson and vortex loops preserve up to four of the supercharges and we find squashing independence for the expectation values of these 2/3-BPS loops. We then show how the additional supersymmetries facilitate the analytic matching of partition functions and loop operator expectation values to those in the mirror dual theory, allowing one to lift all the results that were previously established on the round sphere to the squashed sphere. Additionally, on the squashed sphere with four preserved supercharges, we numerically evaluate the partition functions of ABJM and its dual super-Yang-Mills at low ranks of the gauge group. We find matching values of their partition functions, prompting us to conjecture the general equality on the squashed sphere. From the numerics we also observe the squashing dependence of the Lee-Yang zeros and of the non-perturbative corrections to the all order large NN expression for the ABJM partition function.

Extended supersymmetry of semichiral sigma models in 4D

Briefly: Using a novel $(1,1)$ superspace formulation of semichiral sigma models with $4D$ target space, we investigate if an extended supersymmetry in terms of semichirals is compatible with having a $4D$ target space with torsion. In more detail: Semichiral sigma models have $(2,2)$ supersymmetry and Generalized K\"ahler target space geometry by construction. They can also support $(4,4)$ supersymmetry and Generalized Hyperk\"ahler geometry, but when the target space is four dimensional indications are that the geometry is restricted to Hyperk\"ahler. To investigate this further, we reduce the model to $(1,1)$ superspace and construct the extra (on-shell) supersymmetries there. We then find the conditions for a lift to $(2,2)$ super space and semichiral fields to exist. Those conditions are shown to hold for Hyperk\"ahler geometries. The $SU(2)\otimes U(1)$ WZW model, which has $(4,4)$ supersymmetry and a semichiral description, is also investigated. The additional supersymmetries are found in $(1,1)$ superspace but shown {\em not} to be liftable to a $(2,2)$ semichiral formulation.

Three-algebras inN=5, 6 superconformal Chern-Simons theories: Representations and relations

In this work we present 3-algebraic constructions and representations for three-dimensional $\mathcal{N}=5$ supersymmetric Chern-Simons theories, and show how they relate to theories with additional supersymmetries. The $\mathcal{N}=5$ structure constants give theories with $\mathrm{Sp}(2N)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(M)$ gauge symmetry, as well as more exotic symmetries known from gauged supergravity. We find explicit lifts from $\mathcal{N}=6$ to 8, and $\mathcal{N}=5$ to 6 and 8, for appropriate gauge groups.

$ \mathcal{N} = 6 $ superspace constraints, SUSY enhancement and monopole operators

We present a systematic analysis of the N=6 superspace constraints in three space-time dimensions. The general coupling between vector and scalar supermultiplets is encoded in an SU(4) tensor $W^i_j$ which is a function of the matter fields and subject to a set of algebraic and super-differential relations. We give a genuine N=6 classification for superconformal models with polynomial interactions and find the known ABJM and ABJ models. We further study the issue of supersymmetry enhancement to N=8 and the role of monopole operators in this scenario. To this end we assume the existence of a composite monopole operator superfield which we use to formulate the additional supersymmetries as internal symmetries of the N=6 superspace constraints. From the invariance conditions of these constraints we derive a system of superspace constraints for the proposed monopole operator superfield. This constraint system defines the composite monopole operator superfield analogously to the original N=6 superspace constraints defining the dynamics of the elementary fields.

Topological A-type models with flux

We study deformations of the A-model in the presence of fluxes, by which we mean rank-three tensors with antisymmetrized upper/lower indices, using the AKSZ construction. There are two natural deformations of the A-model in the AKSZ language: 1) the Zucchini model, which can be defined on a generalized complex manifold and reduces to the A-model when the generalized complex structure comes from a symplectic structure, and 2) a topological membrane model, which naturally accommodates fluxes, and reduces to the Zucchini model on the boundary of the membrane when the fluxes are turned off. We show that the fluxes are related to deformations of the Courant bracket which generalize the twist by a closed 3-from H, in the sense that satisfying the AKSZ master equation implies precisely the integrability conditions for an almost generalized complex structure with respect to the deformed Courant bracket. In addition, the master equation imposes conditions on the fluxes that generalize dH = 0. The membrane model can be defined on a large class of U(m)- and U(m) × U(m)-structure manifolds relevant for string theory, including geometries inspired by (1, 1) supersymmetric σ-models with additional supersymmetries due to almost complex (but not necessarily complex) structures in the target space. In addition we show that the model can be defined on three particular half-flat manifolds related to the Iwasawa manifold. When only the closed 3-form flux is turned on it is possible to obtain a topological string model, which we do for the case of a Calabi-Yau. We argue that deformations from the standard A-model are due to the choice of gauge fixing fermion, rather than a flux deformation of the AKSZ action. The particularly interesting cases arise when the fermion depends on auxiliary fields, the simplest possibility being due to the (2, 0)+(0, 2) component of a non-trivial b-field. The model is generically no longer evaluated on holomorphic maps and defines new topological invariants. Deformations due to H-flux can be more radical, completely preventing auxiliary fields from being integrated out.

On the restoration of supersymmetry in twisted two-dimensional lattice Yang–Mills theory

We study a discretization of = 2 super Yang–Mills theory which possesses a single exact supersymmetry at non-zero lattice spacing. This supersymmetry arises after a reformulation of the theory in terms of twisted fields. In this paper we derive the action of the other twisted supersymmetries on the component fields and study, using Monte Carlo simulation, a series of corresponding Ward identities. Our results for SU(2) and SU(3) support a restoration of these additional supersymmetries without fine tuning in the infinite volume continuum limit. Additionally we present evidence supporting a restoration of (twisted) rotational invariance in the same limit. Finally we have examined the distribution of scalar field eigenvalues and find evidence for power law tails extending out to large eigenvalue. We argue that these tails indicate that the classical moduli space does not survive in the quantum theory.

Geometry of extended null supersymmetry in M theory

For supersymmetric spacetimes in eleven dimensions admitting a null Killing spinor, a set of explicit necessary and sufficient conditions for the existence of any number of arbitrary additional Killing spinors is derived. The necessary and sufficient conditions are comprised of algebraic relationships, linear in the spinorial components, between the spinorial components and their first derivatives, and the components of the spin connection and four-form. The integrability conditions for the Killing spinor equation are also analysed in detail, to determine which components of the field equations are implied by arbitrary additional supersymmetries and the four-form Bianchi identity. This provides a complete formalism for the systematic and exhaustive investigation of all spacetimes with extended null supersymmetry in eleven dimensions. The formalism is employed to show that the general bosonic solution of eleven dimensional supergravity admitting a $G_2$ structure defined by four Killing spinors is either locally the direct product of $\mathbb{R}^{1,3}$ with a seven-manifold of $G_2$ holonomy, or locally the Freund-Rubin direct product of $AdS_4$ with a seven-manifold of weak $G_2$ holonomy. In addition, all supersymmetric spacetimes admitting a $(G_2\ltimes\mathbb{R}^7)\times\mathbb{R}^2$ structure are classified.

Open strings in the plane wave background II: superalgebras and spectra

In hep-th/0211011 we started a systematic investigation of open strings in the plane wave background. In this paper we continue the analysis by discussing the superalgebras of conserved charges, the spectra of open strings, and the spectra of DBI fluctuations around D-brane embeddings. We also derive the gluing conditions for corresponding boundary states and analyze their symmetries. All results are consistent with each other, and confirm the existence of additional supersymmetries as previously discussed. We further show that for every symmetry current one can construct a (countably) infinite number of related currents that contain more worldsheet derivatives, and discuss non-local symmetries.

Topological quantum field theory in quaternionic geometry

A topological quantum field theory on a 4 k-dimensional manifold M admitting an almost quaternionic structure is proposed. Expectation values of certain operators on M are proved to be independent of the choice of an almost quaternionic structure used in calculations and thus carry only smooth information about M. These invariants are explicitly expressed as integrals of differential forms over the instanton moduli space associated with a chosen almost quaternionic structure. When M admits a hyperKähler structure the topological field theory has three additional supersymmetries which induce three complex structures on the associated instanton moduli space proving thus that the latter is a hypercomplex manifold. In this case an analogue of the four-dimensional Donaldson map is constructed which provides a number of candidates for new invariants of 4 k-dimensional hypercomplex structures. In the case k = 1 the proposed topological theory on an almost quaternionic manifold reproduces the Witten interpretation of the four-dimensional Donaldson invariants.

Holonomy groups, complex structures andD=4 topological Yang-Mills theory

We study conditions for the existence of extended supersymmetry in topological Yang-Mills theory. These conditions are most conveniently formulated in terms of the holonomy group of the underlying manifold, on which the topological Yang-Mills theory is defined. For irreducible manifolds we find that extended supersymmetries are in 1–1 correspondence with covariantly constant complex structures. Therefore, the topological Yang-Mills theory on any Kähler manifold possesses one additional supersymmetry and on any hyper Kähler manifold there are three additional supersymmetries. The Donaldson map, which plays a crucial role in the construction of the topological invariants, is generalized for Kähler manifolds, thus providing candidates for new invariants of complex manifolds.