Supersymmetry and instantons

We show that the eigenvalue equations for the fluctuation of scalars, fermions and gluon around any classical self-dual solution of the Yang-Mills theory have the same spectrum of non-zero eigenvalues. In the case of a supersymmetric Yang- Mills theory this implies that the one loop correction around any self-dual instanton is just given by a counting of the zero modes of the gluon, fermion and ghost.     

Supersymmetry and Combinatorics

We show how a recently proposed supersymmetric quantum mechanics model leads to non-trivial results/conjectures on the combinatorics of binary necklaces and linear-feedback shift-registers. Pauli’s exclusion principle plays a crucial role: by projecting out certain states/necklaces, it allows to represent the supersymmetry algebra in the resulting subspace. Some of our results can be rephrased in terms of generalizations of the well-known Witten index.     

Supersymmetry and localization

We study conditions under which an odd symmetry of the integrand leads to localization of the corresponding integral over a (super)manifold. We also show that in many cases these conditions guarantee exactness of the stationary phase approximation of such integrals. Research is partially supported by NSF grant No. DMS-9500704.     

Supersymmetry and LHC

Motivations for introducing supersymmetry in high-energy physics are discussed along with the possibility of discovering supersymmetry at the Large Hadron Collider (LHC). Various regions of the space of parameters of the minimal supersymmetric standard model are analyzed, and phenomenological manifestations of supersymmetry inherent in these regions are discussed. The discovery potential of LHC at the planned luminosity is indicated for various channels.     

Supersymmetry and 't Hooft's twisted functional integral

't Hooft's twisted functional integral is exactly calculable in the supersymmetric version of the SU(N) gauge theories, provided that one uses periodic (up to gauge transformations) boundary conditions for fermions. The solution has in a sense light magnetic fluxes and heavy electric ones, and thus, it is consistent with confinement (but not a proof of it). This is an evidence of the fact that confinement is not ruled out by the absence of vacuum condensates in these supersymmetric gauge theories.     

Dark Supersymmetry

We propose a model of Dark Supersymmetry, where a supersymmetric dark sector is coupled to the classically scale invariant non-supersymmetric Standard Model through the Higgs portal. The dark sector contains a mass scale that is protected against radiative corrections by supersymmetry, and the portal coupling mediates this scale to the Standard Model, resulting in a vacuum expectation value for the Higgs field and the usual electroweak symmetry breaking mechanism. The supersymmetric dark sector contains dark matter candidates, and we show that the observed dark matter abundance is generated for a natural choice of parameters, while avoiding the current experimental bounds on direct detection. Future experiments can probe this scenario if the dark sector mass scale is not too high.     

Supersymmetry breaking

We review the various mechanisms of supersymmetry breaking and its transmission to the observable sector. We argue that hybrid models where gauge dominates over gravity mediation, but gravity provides the main contributions to the Higgs sector masses and the neutralino mass, are able to combine the advantages and reduce the disadvantages of the two transmission mechanisms.      

Kramers Equation and Supersymmetry

Hamilton’s equations with noise and friction possess a hidden supersymmetry, valid for time-independent as well as periodically time-dependent systems. It is used to derive topological properties of critical points and periodic trajectories in an elementary way. From a more practical point of view, the formalism provides new tools to study the reaction paths in systems with separated time scales. A ‘reduced current’ which contains the relevant part of the phase space probability current is introduced, together with strategies for its computation.     

Supersymmetry and the vacuum

It is shown that, in a supersymmetric theory, the sum of all vacuum diagrams vanishes identically as a consequence of compensations among contributions involving different fields of the supermultiplet. This means that the zero-point energy-momentum density is zero to all orders in the supersymmetric interaction. In an external gravitational field no cosmological term is induced through the vacuum polarization of the matter fields. The situation when supersymmetry is spontaneously broken, or softly broken is also discussed.     

Supersymmetry and classical solutions

We obtain classical solutions to the field equations of the massless supersymmetric Wess-Zumino model and to the field equations of the interacting SU(2) gauge supermultiplet. This is done by applying finite supersymmetry transformations to the known solutions of the scalar field equation with ?⁴ interaction and the Yang-Mills field equations. The relevance of supersymmetry to the solution of classical field equations involving anticommuting fermion fields is discussed.     

Supersymmetry and superstring phenomenology

We briefly cover the early history of supersymmetry, describe the relation of SUSY quantum field theories to superstring theories and explain why they are considered a likely tool to describe the phenomenology of high energy particle theory beyond the Standard Model.     

Supersymmetry and proton decay

Using supergraph techniques it is shown that all one-loop contributions to the proton decay amplitude, which are suppressed by only one power of the unification mass and are not of fourth order in Higgs couplings, vanish in a general supersymmetric grand unification setting. Corrections to this statement due to soft supersymmetry breaking are analysed.     

Generalized Cohomologies and Supersymmetry

We show that the complex cohomologies of Bott, Chern, and Aeppli and the symplectic cohomologies of Tseng and Yau arise in the context of type II string theory. Specifically, they can be used to count a subset of scalar moduli fields in Minkowski compactification with RR fluxes in the presence of either O5/D5 or O6/D6 brane sources, respectively. Further, we introduce a new set of cohomologies within the generalized complex geometry framework which interpolate between these known complex and symplectic cohomologies. The generalized complex cohomologies play the analogous role for counting massless fields for a general supersymmetric Minkowski type II compactification with Ramond–Ramond flux.