Flat coordinates,topological Landau-Ginzburg models and the Seiberg-Witten period integrals

We study the Picard-Fuchs differential equations for the Seiberg-Witten period integrals in N = 2 supersymmetric Yang-Mills theory. For A-D-E gauge groups we derive the Picard-Fuchs equations by using the flat coordinates in the A-D-E singularity theory. We then find that these are equivalent to the Gauss-Manin system for two-dimensional A-D-E topological Landau-Ginzburg models and the scaling relation for the Seiberg-Witten differential. This suggests an interesting relationship between four-dimensional N = 2 gauge theories in the Coulomb branch and two-dimensional topological field theories.     

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.     

Study of N = 2 superconformal field theories in 4 dimensions

Making use of the exact solutions of the N = 2 supersymmetric gauge theories, we construct new classes of superconformal field theories (SCFTs) by fine-tuning the moduli parameters and bringing the theories to critical points. SCFTs we have constructed represent universality classes of the 4-dimensional N = 2 SCFTs.     

Semi-classical spectrum of the homogeneous sine-Gordon theories

The semi-classical spectrum of the homogeneous sine-Gordon theories associated with an arbitrary compact simple Lie group G is obtained and shown to be entirely given by solitons. These theories describe quantum integrable massive perturbations of Gepner's G-parafermions whose classical equations-of-motion are non-abelian affine Toda equations. One-soliton solutions are constructed by embeddings of the SU(2) complex sine-Gordon soliton in the regular SU(2) subgroups of G. The resulting spectrum exhibits both stable and unstable particles, which is a peculiar feature shared with the spectrum of monopoles and dyons in N = 2 and N = 4 supersymmetric gauge theories.     

Quartic mass matrix and renormalization constants in supersymmetric Yang-Mills theories

We derive the general formula for the supertrace of the quartic mass matrix in a general supersymmetric gauge theory, with arbitrary representations for the chiral multiplets. This formula clarifies the non-renormalization theorems in presence of gauge interactions and gives “extended renormalization theorems” for N = 2 and N = 4 supersymmetric Yang-Mills theories. In particular we find the known result that g_ren = g_bare for the N = 4 theory and the new result m_ren = m_bare for the N = 2 gauge interactions of massive hypermultiplets. We give arguments to the extent that the latter non-renormalization theorem persists to all orders in perturbation theory.     

Polynomial invariants for SU(2) monopoles

We present an explicit expression for the topological invariants associated to SU(2) monopoles in the fundamental representation on spin four-manifolds. The computation of these invariants is based on the analysis of their corresponding topological quantum field theory, and it turns out that they can be expressed in terms of Seiberg-Witten invariants. In this analysis we use recent exact results on the moduli space of vacua of the untwisted N = 1 and N = 2 supersymmetric counterparts of the topological quantum field theory under consideration, as well as on electric-magnetic duality for N = 2 supersymmetric gauge theories.     

Cohomological methods in conventionalN=2 superspace

The cohomological methods of Girardi, Grimm and Stora are extended to supersymmetric gauge theories formulated in conventionalN=2 superspace.     

Electric dipole moment for supersymmetric monopoles

The electric dipole moment for the monopoles that can be present in N = 2 and N = 4 supersymmetric SU(2) gauge theories, spontaneously broken by imposing a non-zero expectation value of a scalar field at infinity, is determined by considering the response to a weak external electric field. The magnetic g factor g_M = 2 which is in accord with the duality conjecture of Montonen and Olive.     

Auxiliary field-free supersymmetric gauges

We find that, in perturbation theory, non-light-cone axial gauges, N ° A^a (x) = 0, preserve the supersymmetry remaining in N = 1 supersymmetric YM theories, after imposing the Wess-Zumino gauge.     

Dynamical generation of supersymmetry

In this letter I demonstrate that supersymmetric Yang-Mills theories can be generated as the low energy behavior of strongly coupled theories which possess no supersymmetry at high energy. Explicit examples are given for which the low energy gauge group is SU(N), SO(N) or Sp(2N).     

Anomalous generation of Chern-Simons terms in D=3, N=2 supersymmetric gauge theories

Parity-invariant three-dimensional gauge theories with N=2 extended supersymmetry are studied by the heat kernel method. The parity-anomalous part of the one-loop effective action is exactly found. It is expressed in terms of the N=2 supersymmetric Chern-Simons term and is identified as a N=2 superspace Atiyah-Patodi-Singer eta-invariant.     

Thermodynamic Bethe Ansatz for N = 1 supersymmetric theories

We study a series of N = 1 supersymmetric integrable particle theories in d = 1 + 1 dimensions. These theories are represented as integrable perturbations of specific N = 1 superconformal field theories. Starting from the conjectured S-matrices for these theories, we develop the Thermodynamic Bethe Ansatz (TBA), where we use that the 2-particle S-matrices satisfy a free fermion condition. Our analysis proves a conjecture by E. Melzer, who proposed that these N = 1 supersymmetric. TBA systems are “folded” versions of N = 2 supersymmetric TBA systems that were first studied by P. Fendley and K. Intriligator.     

Structure of the exact effective action and quark confinement in MSSM QCD

An expression for the exact (beyond perturbation theory) effective action in N=1 supersymmetric gauge theories where all particles, with the exception of gauge bosons, are massive is proposed. By analyzing the form of this expression, it is shown that, in supersymmetric theories, instanton effects can lead to quark confinement. On the basis of first principles, the characteristic scale of confinement is calculated within MSSM QCD, and the result is found to be consistent with experimental data. The proposed explanation differs drastically with the dual Higgs mechanism.     

Quantum Wall Crossing in $${{\mathcal N}=2}$$ Gauge Theories

We study refined and motivic wall-crossing formulas in N=2{{\mathcal N}=2} supersymmetric gauge theories with SU(2) gauge group and N _f < 4 matter hypermultiplets in the fundamental representation. Such gauge theories provide an excellent testing ground for the conjecture that “refined = motivic.”     

Non-supersymmetric vacua and the D-flatness condition

Some N = 1 gauge theories, including SQED and N_F = 1 SQCD, have the property that, for arbitrary superpotentials, all stationary points of the potential V = F + D are D-flat. For others, stationary points of V are complex gauge transformations of D-flat configurations. As an implication, the technique to parametrize the moduli space of supersymmetric vacua in terms of a set of basic holomorphic G invariants can be extended to non-supersymmetric vacua. A similar situation is found in non-gauge theories with a compact global symmetry group.     

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.     

Chiral rings,singularity theory and electric-magnetic duality

We study in detail the space of perturbations of a pair of dual N = 1 supersymmetric theories based on an SU(N_c) gauge theory with an adjoint X and fundamentals with a superpotential which is polynomial in X. The equivalence between them depends on non-trivial facts about polynomial equations, i.e. singularity theory. The classical chiral rings of the two theories are different. Quantum mechanically there are new relations in the chiral rings which ensure their equivalence. Duality interchanges “trivial” classical relations in one theory with quantum relations in the other and vice versa. We also speculate about the behavior of the theory without the superpotential.     

Observations on the moduli space of superconformal field theories

Some aspects of the moduli space of superconformal field theories are discussed. It is helpful to consider the conformal field theory as a background for propagation of strings and to exploit the space-time interpretation. Using this point of view we show that the metric on the moduli space of N = 4 superconformal field theory with c = 6 is locally that of O(20,4)/O(20) × O(4). We also discover some properties of the moduli space of N = 2 superconformal field theories with c = 9. Particular examples of these conformal field theories are sigma models on four- and six-dimensional Calabi-Yau spaces. Therefore, we can use this technique to learn about the moduli space of these spaces. For c = 6 we recover the known moduli space of K₃. Our analysis of the c = 9 system leads to a new coupling in four dimensional supergravity. As a by-product, we prove that gauge couplings cannot depend on the moduli of N = 1 space-time supersymmetric compactifications.