Spontaneous supersymmetry breaking by large-N matrices

Motivated by supersymmetry breaking in matrix model formulations of superstrings, we present some concrete models, in which the supersymmetry is preserved for any finite N, but gets broken at infinite N, where N is the rank of matrix variables. The models are defined as supersymmetric field theories coupled to some matrix models, and in the induced action obtained after integrating out the matrices, supersymmetry is spontaneously broken only when N is infinity. In our models, the large value of N gives a natural explanation for the origin of small parameters appearing in the field theories which trigger the supersymmetry breaking.     

Supersymmetric renormalization group flows

The functional renormalization group equation for the quantum effective action is a powerful tool to investigate non-perturbative phenomena in quantum field theories. We discuss the application of manifest supersymmetric flow equations to the N = 1 Wess-Zumino model in two and three dimensions and the linear O(N) sigma model in three dimensions in the large-N limit. The former is a toy model for dynamical supersymmetry breaking, the latter for an exactly solvable field theory.     

Quantum theory of compact and non-compact σ models

We discuss quantization of SO(N + 1) σ models and CP^N models, and of certain non-compact counterparts, SO(N, 1) and QP^N respectively, of these, both in canonical operator formalism and the covariant path integral formulation, showing the equivalence of the two approaches. We discuss also a class of supersymmetric σ models formulated in d ? 3 dimensions and apply the results to the SO(N + 1) and SO(N, 1) cases. This allows us to calculate the Witten index in each case. For SO(2l + 1,1) we thereby find supersymmetry breaking. However, for SO(2l, 1), we find supersymmetry is unbroken. Moreover, there is no unique ground state, invariant under SO(2l, 1), rather an infinite multiple of zero energy states, carrying a unitary irreducible representation of the non-compact SO(2l, 1) group. We discuss also field theoretic aspects of the models in d ? 2 dimensions, stressing differences of the non-compact to the compact cases. These include infrared instead of ultraviolet asymptotic freedom, lack of an energy gap, failure (in the QP^N case) of the auxiliary vector field to become dynamical. A further conclusion that is argued concerns the absence of a consistent particle interpretation for the QP^N model in exactly two dimensions. For d > 2 the non-compact symmetry of QP^N is broken down to the compact subgroup.     

Grand unification with local supersymmetry

We study general conditions for obtaining spontaneous breaking of local supersymmetry in N = 1 supergravity coupled to supersymmetric matter. We consider in particular the coupling of N = 1 supergravity to grand unified theories like SU(5) and study the conditions which must be met in order to obtain a realistic model. Specific models are built in which local supersymmetry is broken at a scale √M_Wm_p ～ 10¹⁰ GeV. This breaking of supersymmetry is only detected at low energies through soft terms breaking explicitly the global supersymmetry. These soft terms (scalar masses, gaugino masses and trilinear scalar couplings) are renormalized at low energies according to the renormalization group. The (mass)² of the Higgs doublet evolve towards negative values at low energies giving rise to SU(2) × U(1) breaking as a radiative effect of local supersymmetry breaking. We finally point out the possible relevance of non-renormalizable superpotentials for the problem of fermion masses.     

Topological matter in four dimensions

Topological models involving matter couplings to Donaldson-Witten theory are presented. The construction is carried out using both the topological algebra and its central extension, which arise from the twisting of N = 2 supersymmetry in four dimensions. The framework on which the construction is based is constituted by the superspace associated to these algebras. The models show new features of topological quantum field theories which could provide either a mechanism for topological symmetry breaking, or the analog of two-dimensional mirror symmetry in four dimensions.     

Instanton effects in N = 2 supersymmetric quantum mechanics

Supersymmetric quantum mechanics with several bosonic and fermionic dynamic variables is considered. Two different N = 2 supersymmetric models involving instantons are discussed in detail. Instantons fail to break supersymmetry in one of the models considered. The vacuum state is degenerate in this model which generally results in spontaneous breaking of internal left-right symmetry. In another model supersymmetry is destroyed dynamically due to special complex instanton solutions. Possible implications for SUSY field theories are discussed.     

Dynamical supersymmetry breaking and gauge anomalies

Some aspects of supersymmetric gauge theories and discussed. It is shown that dynamical supersymmetry breaking does not occur in supersymmetric QED in higher dimensions. The cancellation of both local (perturbative) and global (non-perturbative) gauge anomalies are also discussed in supersymmetric gauge theories. We argue that there is no dynamical supersymmetry breaking in higher dimensions in any supersymmetric gauge theories free of gauge anomalies. It is also shown that for supersymmetric gauge theories in higher dimensions with a compact connected simple gauge group, when the local anomaly-free condition is satisfied, there can be at most a possibleZ ₂ global gauge anomaly in extended supersymmetricSO(10) (or spin (10)) gauge theories inD=10 dimensions containing additional Weyl fermions in a spinor representation ofSO(10) (or spin (10)). In four dimensions with local anomaly-free condition satisfied, the only possible global gauge anomalies in supersymmetric gauge theories areZ ₂ global gauge anomalies for extended supersymmetricSP(2N) (N=rank) gauge theories containing additional Weyl fermions in a representation ofSP(2N) with an odd 2nd-order Dynkin index.     

Non-perturbative effects in supersymmetric sigma models

A scheme for taking account of crucial non-perturbative effects in a quantum field theory ahead of developing perturbation series for it is extended here from bosonic to supersymmetric sigma models in two dimensions. The scheme writes field products in the lagrangian in terms of suitably defined normal ordered products and VEVs of field products. The exact values of the latter can be inferred directly from the symmetry and supersymmetry Ward identities of the theory, so that a lagrangian with scale breaking effects explicitly treated, is available for use in perturbation theory. The supersymmetric sigma model on the manifold S^N is used to illustrate many aspects of the scheme.     

The (2+1)-dimensional supersymmetric CPN−1 model and the central charge

The supersymmetry algebra is examined for the (2+1)-dimensional supersymmetric CP^N?1 model, on the basis of the observation of Witten and Olive in (1+1) and (3+1) dimensions. We then demonstrate that also in this (2+1)-dimensional model the usual supersymmetry algebra is modified by the appearance of the topological numbers of the solitons, which are nothing but the instantons in (1+1) dimensions, as central charges. To obtain the model, we begin by constructing the supersymmetric model in (3+1) dimensions. Then it is reduced to (2+1) dimensions by means of the dimensional reduction technique. We observe that the (2+1)-dimensional supersymmetric CP^N?1 model thus obtained admits an O(2) extended supersymmetry.     

Dynamical breakdown of supersymmetry and chiral symmetry in 1+1 dimensions

Dynamical breakdown of supersymmetry and chiral symmetry is studied through a 1+1 dimensional field theoretical model in the large-N limit in 1/N expansion. The study is based on the calculation of bilinear field condensates in an effective potential approach. It is shown that the condition for supersymmetry breaking is related to the values of the parameters in the model and chiral symmetry breaking exists in most cases.     

A classification of compactifying solutions for d =11 supergravity

Supergravity in eleven dimensions is known to have classical solutions of the type (anti-de Sitter space-time) × (7-dimensional Einstein space). We give a list of all homogeneous 7-manifolds which admit an Einstein metric. Known solutions are reviewed, with some emphasis on the SU(3) × SU(2) × U(1) compactifications. Their topology is discussed in detail.The list includes three new solutions, with symmetry groups SU(3) × SU(2), SO(5) and SO(5) × U(1). The first solution has no supersymmetry, while the second and third yield respectively N = 1 and N = 2 supersymmetry in four dimensions. The last two solutions may be extended to solutions with nonzero internal photon curl, breaking all supersymmetry.The existence of a spin structure on homogeneous manifolds $\text{G}\text{H}$ is discussed and related to topological properties of $\text{G}\text{H}$. As an illustration, we treat the coset spaces SU(m + 1) × SU(n + 1)/SU(m) × SU(n) × U(1), which include the spaces with SU(3) × SU(2) × U(1) symmetry.     

Anti-de Sitter BPS black holes in N=2 gauged supergravity

Electrically charged solutions breaking half of the supersymmetry in Anti-de Sitter four dimensional N=2 supergravity coupled to vector supermultiplets are constructed. These static black holes live in an asymptotic AdS₄ space time. The Killing spinor, i.e., the spinor for supersymmetry variation is explicitly constructed for these solutions.     

Spontaneous breaking of supersymmetry through dimensional reduction

A general technique for deriving consistent theories with spontaneously broken supersymmetry and massive gravitinos is illustrated by exploiting the chiral invariance of N = 1 supergravity in four dimensions to construct a theory with broken N = 2 supersymmetry in three dimensions.     

Quantum vortex strings: A review

The quantum worldsheet dynamics of vortex strings contains information about the 4d non-Abelian gauge theory in which the string lives. Here I tell this story. The string worldsheet theory is typically some variant of the CP^N-1 sigma-model, describing the orientation of the string in a U(N) gauge group. Qualitative parallels between 2d sigma-models and 4d non-Abelian gauge theories have been known since the 1970s. The vortex string provides a quantitative link between the two. In 4d theories with N=2 supersymmetry, the exact BPS spectrum of the worldsheet coincides with the bulk spectrum in 4d. Moreover, by tuning parameters, the CP^N-1 sigma-model can be coaxed to flow to an interacting conformal fixed point which is related to the 4d Argyres-Douglas fixed point. For theories with N=1 supersymmetry, the worldsheet theory suffers dynamical supersymmetry breaking and, more interestingly, supersymmetry restoration, in a way which captures the physics of Seiberg’s quantum deformed moduli space.     

Supersymmetrical string theories

A supersymmetrical light-cone-gauge string action is presented. It provides a basis for understanding the previously- studied supersymmetrical dual string theory as well as two new closed-string theories that have extended supersymmetry in ten dimensions, corresponding to N=8 supersymmetry in four dimensions.     

Dynamical supersymmetry breaking in a finite field theoretical model

We present a supersymmetric field theory in two or three space-time dimensions with an internal symmetry of the O(N) type. In the large-N limit the model is finite and supersymmetry is spontaneously broken. The fields representing the order parameters of the broken supersymmetry phase acquire dynamics through quantum corrections. In particular the Goldstone fermion is a zero-mass fermionic bound state.     

Use of compensators to break supersymmetry in an N = 1 supergravity theory

We consider an N = 1 supergravity theory with multiple compensators and show that supersymmetry is broken by a solution to the equation of motion of a compensator. When a chiral scalar superfield is coupled to supergravity, we discuss various aspects of supersymmetry breaking and show that the super-Higgs-Kibble effect is operative. Possible applications of this mechanism of supersymmetry breaking in model building and extended supergravity theories are indicated.     

From the Dirac operator to Wess-Zumino models on spatial lattices

We investigate two-dimensional Wess-Zumino models in the continuum and on spatial lattices in detail. We show that a non-antisymmetric lattice derivative not only excludes chiral fermions but in addition introduces supersymmetry breaking lattice artifacts. We study the non-local and antisymmetric SLAC derivative which allows for chiral fermions without doublers and minimizes those artifacts. The supercharges of the lattice Wess-Zumino models are obtained by dimensional reduction of Dirac operators in high-dimensional spaces. The normalizable zero modes of the models with N=1 and N=2 supersymmetry are counted and constructed in the weak- and strong-coupling limits. Together with known methods from operator theory this gives us complete control of the zero mode sector of these theories for arbitrary coupling.     

The parallelizing S7 torsion in gauged N=8 supergravity

Using the parallelizing S⁷ torsion as an ansatz we investigate two solutions of gauged N=8 supergravity with SO(7) invariance. Supersymmetry is uniformly broken. We calculate the masses for these solutions which are both unstable. Certain apparent discrepancies with the results obtained by spontaneous compactification of d=11 supergravity are discussed. We establish that the compactification on the parallelized S⁷ has an SO(7) invariance and clarify the issue of supersymmetry breaking. The lack of stability in d=4 indicates that this d=11 solution is unstable.     

Dynamical supersymmetry breaking in two-dimensional N = 1 supergravity theories

We investigate a possible dynamical mechanism for spontaneous supersymmetrybreaking in N = 1 supergravity theories in 1 + 1 space-time dimensions. It will be shown that supersymmetry is never broken at the tree level, but it can be broken for a certain class of models by quantum effects due to trace anomalies of the energy-momentum tensor and the supercurrent.