Abstract:
A class of generalized Wess-Zumino models with distinct vacua is investigated. These models allow for BPS saturated vacua interpolation along one compact spatial dimension. The properties of these interpolations are studied.

Abstract:
We discuss topologically stable solitons in two-dimensional theories with the extended supersymmetry assuming that the spatial coordinate is compact. This problem arises in the consideration of the domain walls in the popular theories with compactified extra dimensions. Contrary to naive expectations, it is shown that the solitons on the cylinder can be BPS saturated. In the case of one chiral superfield, a complete theory of the BPS saturated solitons is worked out. We describe the classical solutions of the BPS equations. Depending on the choice of the Kahler metric, the number of such solutions can be arbitrarily large. Although the property of the BPS saturation is preserved order by order in perturbation theory, nonperturbative effects eliminate the majority of the classical BPS states upon passing to the quantum level. The number of the quantum BPS states is found. It is shown that the N=2 field theory includes an auxiliary N=1 quantum mechanics, Witten's index of which counts the number of the BPS particles.

Abstract:
Domain-wall solutions in four-dimensional supersymmetric field theories with distinct discrete vacuum states lead to the spontaneous breaking of supersymmetry, either completely or partially. We consider in detail the case when the domain walls are the BPS-saturated states, and 1/2 of supersymmetry is preserved. Several useful criteria that relate the preservation of 1/2 of supersymmetry on the domain walls to the central extension appearing in the N=1 superalgebras are established. We explain how the central extension can appear in N=1 supersymmetry and explicitly obtain the central charge in various models: the generalized Wess-Zumino models, and supersymmetric Yang-Mills theories with or without matter. The BPS-saturated domain walls satisfy the first-order differential equations which we call the creek equations, since they formally coincide with the (complexified) equations of motion of an analog high-viscosity fluid on a profile which is given by the superpotential of the original problem. Some possible applications are considered.

Abstract:
We discuss quantum tunneling between classically BPS saturated solitons in two-dimensional theories with N=2 supersymmetry and a compact space dimension. Genuine BPS states form shortened multiplets of dimension two. In the models we consider there are two degenerate shortened multiplets at the classical level, but there is no obstruction to pairing up through quantum tunneling. The tunneling amplitude in the imaginary time is described by instantons. We find that the instanton is nothing but the 1/4 BPS saturated ``wall junction,'' considered previously in the literature in other contexts. Two central charges of the superalgebra allow us to calculate the instanton action without finding the explicit solution (it is checked, though, numerically, that the saturated solution does exist). We present a quantum-mechanical interpretation of the soliton tunneling.

Abstract:
Non-supersymmetric multi-wall configurations are generically unstable. It is proposed that the stabilization in compact space can be achieved by introducing a winding number into the model. A BPS-like bound is studied for the energy of configuration with nonvanishing winding number. Winding number is implemented in an ${\cal N}=1$ supersymmetric nonlinear sigma model with two chiral scalar fields and a bound states of BPS and anti-BPS walls is found to exist in noncompact spaces. Even in compactified space $S^1$, this nontrivial bound state persists above a critical radius of the compact dimension.

Abstract:
We consider BPS domain walls in the four dimensional N=1 supersymmetric models with continuous global symmetry. Since the BPS equation is covariant under the global transformation, the solutions of the BPS walls also have the global symmetry. The moduli space of the supersymmetric vacua in such models have non-compact flat directions, and the complex BPS walls interpolating between two disjoint flat directions can exist. We examine this possibility in two models with global O(2) symmetry, and construct the solution of such BPS walls.

Abstract:
We study the spectrum of the domain walls interpolating between different chirally asymmetric vacua in supersymmetric QCD with the SU(N) gauge group and including N-1 pairs of chiral matter multiplets in fundamental and anti-fundamental representations. There are always "real walls" interpolating between the chirally symmetric and a chirally asymmetric vacua which are BPS saturated. For small enough masses, there are two different "complex" BPS wall solutions interpolating between different chirally asymmetric vacua and two types of "wallsome sphalerons". At some m = m_*, two BPS branches join together and, in some interval m_* < m < m_{**}, BPS equations have no solutions, but there are solutions to the equations of motion describing a non--BPS domain wall and a sphaleron. For m > m_{**}, there are no complex wall solutions whatsoever.

Abstract:
BPS spectra give important insights into the non-perturbative regimes of supersymmetric theories. Often from the study of BPS states one can infer properties of the geometrical or algebraic structures underlying such theories. In this paper we approach this problem from the perspective of persistent homology. Persistent homology is at the base of topological data analysis, which aims at extracting topological features out of a set of points. We use these techniques to investigate the topological properties which characterize the spectra of several supersymmetric models in field and string theory. We discuss how such features change upon crossing walls of marginal stability in a few examples. Then we look at the topological properties of the distributions of BPS invariants in string compactifications on compact threefolds, used to engineer black hole microstates. Finally we discuss the interplay between persistent homology and modularity by considering certain number theoretical functions used to count dyons in string compactifications and by studying equivariant elliptic genera in the context of the Mathieu moonshine.

Abstract:
We discuss the low energy effective action for the Bosonic and Fermionic zero-modes of a smooth BPS Randall-Sundrum domain wall, including the induced supergravity on the wall. The result is a pure supergravity in one lower dimension. In particular, and in contrast to non-gravitational domain walls or domain walls in a compact space, the zero-modes representing transverse fluctuations of domain wall have vanishing action.

Abstract:
We consider a six-dimensional solitonic string solution described by a conformal chiral null model with non-trivial $N=4$ superconformal transverse part. It can be interpreted as a five-dimensional dyonic solitonic string wound around a compact fifth dimension. The conformal model is regular with the short-distance (`throat') region equivalent to a WZW theory. At distances larger than the compactification scale the solitonic string reduces to a dyonic static spherically-symmetric black hole of toroidally compactified heterotic string. The new four-dimensional solution is parameterised by five charges, saturates the Bogomol'nyi bound and has nontrivial dilaton-axion field and moduli fields of two-torus. When acted by combined T- and S-duality transformations it serves as a generating solution for all the static spherically-symmetric BPS-saturated configurations of the low-energy heterotic string theory compactified on six-torus. Solutions with regular horizons have the global space-time structure of extreme Reissner-Nordstrom black holes with the non-zero thermodynamic entropy which depends only on conserved (quantised) charge vectors. The independence of the thermodynamic entropy on moduli and axion-dilaton couplings strongly suggests that it should have a microscopic interpretation as counting degeneracy of underlying string configurations. This interpretation is supported by arguments based on the corresponding six-dimensional conformal field theory. The expression for the level of the WZW theory describing the throat region implies a renormalisation of the string tension by a product of magnetic charges, thus relating the entropy and the number of oscillations of the solitonic string in compact directions.