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We initiate a study of the growth and matrix-valued Hilbert series of N-graded twisted Calabi-Yau algebras that are homomorphic images of path algebras of weighted quivers, generalizing techniques previously used to investigate Artin-Schelter regular algebras and graded Calabi-Yau algebras. Several results are proved without imposing any assumptions on the degrees of generators or relations of the algebras. We give particular attention to twisted Calabi-Yau algebras of dimension d≤3, giving precise descriptions of their matrix-valued Hilbert series and partial results describing which underlying quivers yield algebras of finite GK-dimension. For d=2, we show that these are algebras with mesh relations. For d=3, we show that the resulting algebras are a kind of derivation-quotient algebra arising from an element that is similar to a twisted superpotential.

We initiate a study of the growth and matrix-valued Hilbert series of N-graded twisted Calabi-Yau algebras that are homomorphic images of path algebras of weighted quivers, generalizing techniques previously used to investigate Artin-Schelter regular algebras and graded Calabi-Yau algebras. Several results are proved without imposing any assumptions on the degrees of generators or relations of the algebras. We give particular attention to twisted Calabi-Yau algebras of dimension d≤3, giving precise descriptions of their matrix-valued Hilbert series and partial results describing which underlying quivers yield algebras of finite GK-dimension. For d=2, we show that these are algebras with mesh relations. For d=3, we show that the resulting algebras are a kind of derivation-quotient algebra arising from an element that is similar to a twisted superpotential.

We investigate the SU(3)-invariant sector of the one-parameter family of SO(8) gauged maximal supergravity that has been recently proposed. To this end, we construct the N=2 truncation of this theory and analyse its full vacuum structure. The number of critical point is doubled and includes new N=0 and N=1 branches. We numerically exhibit the parameter dependence of the location and cosmological constant of all extrema. Moreover, we provide their analytic expressions for cases of special interest. Connections with three-dimensional RG-flows are pointed out. Finally, while the mass spectra are found to be parameter independent in most cases, we show that the novel non-supersymmetric branch with SU(3) invariance provides the first counterexample to this.

We show that hierarchically small vacuum expectation values of the superpotential in supersymmetric theories can be a consequence of an approximate R symmetry. We briefly discuss the role of such small constants in moduli stabilization and understanding the huge hierarchy between the Planck and electroweak scales.

We show that hierarchically small vacuum expectation values of the superpotential in supersymmetric theories can be a consequence of an approximate R symmetry. We briefly discuss the role of such small constants in moduli stabilization and understanding the huge hierarchy between the Planck and electroweak scales.

We consider a multi-component nonlinear partial differential equation in the two­dimensional space-time R2 which unifies Schrodinger and Klein-Gordon equations in R 2• By supersymmetric methods connected with shape-invariant potentials, we show that a class of soliton-type solutions to the equation can be constructed from solutions to nonlinear equations of some types for superpotentials. Moreover we present new soliton-type solutions which are written in terms of q-deformed hyperbolic functions.