Flavour and superunification

We consider the possibility of a non-trivial embedding of $\text{(}\text{10}\text{+}\text{5}\text{)}\text{SU}\text{(5)}$ families of spin if$\text{1}\text{2}$ left-handed fermions in a combination of irreducible massless supermultiplets of N extended supersymmetry. We demand the whole spectrum of spin $\text{1}\text{2}$ states to be anomaly free with respect to SU(N). This turns out to be a necessary condition for the absence of anomalies at the SU(5) level. We find two classes of models, with spin $\text{1}\text{2}$ fermions in SU(N) representations associated to one- and two-column Young tableaux, respectively, in which each irreducible massless multiplet occurs at most once. These two classes of models lead to a nontrivial family generation due to supersymmetry. For N = 8 extended supersymmetry, they give at most three and five families, respectively. The first class of models is more natural in the way it excludes SU(5) exotics. The same analysis is extended to the massless multiplets that can be obtained from bilinear composite fields of the (preonic) elementary fields of N extended supergravity. We prove that the generation of families requires the repetition of massless multiplets and that ($\text{10 +}\text{5}$) SU(5) families can only be generated in pairs. General properties of multilinear composite operators of the preonic fields are given and the rôle of massive representations to classify towers of operators with definite spin is pointed out.     

Spontaneous supersymmetry breaking in 1N leading order

We present a supersymmetric model, with only chiral superfields, invariant under an internal O(N) global symmetry. We show that, to leading order in $\text{1}\text{N}$, an auxiliary scalar field acquires a non-vanishing vacuum expectation value, that is, supersymmetry is spontaneously broken to this order of the approximation.     

Local supersymmetry anomaly in two dimensions

We study the local supersymmetry anomaly by constructing an N = 1 (counted by Majorana-Weyl spinors) chiral supergravity model in two dimensions. There is the local supersymmetry anomaly as well as the gravitational anomaly. We obtain the linearized forms of these anomalies by perturbation calculation. The full non-linear forms are obtained by finding a solution to the Wess-Zumino consistency condition. These anomalies can be derived from the supersymmetric extension of the Chern-Simons invariant in three dimensions.     

Symmetry breaking by instantons in supergravity

The non-zero modes of different spin operators on the background of a self-dual gravitational instanton are all related by global supersymmetry transformations and completely cancel in the one-loop term, which is determined entirely by the zero modes. We derive the number of zero modes of each spin. In an asymptotically locally Euclidean self-dual instanton there are 2τ $\text{spin}\text{-}\text{3}\text{2}$ zero modes and 3τ spin-2 zero modes, where τ is the Hirzebruch signature. Up to 3 of the spin-2 zero modes (depending on boundary conditions) may correspond to global rotations. The $\text{spin}\text{-}\text{3}\text{2}$ zero modes break the U(1) chiral symmetry and give rise to helicity-changing amplitudes. Together with the spin-2 zero modes they determine the trace anomaly or scaling behaviour. We can compare our results with the perturbation theory predictions for the axial vector current and trace anomalies in K3, the unique compact self-dual gravitational instanton, because in this case there are no boundary terms. We obtain agreement.     

Supersymmetric regulators and supercurrent anomalies

The supercurrent anomalies of the supercurrent $\text{S}$_μ of the supersymmetric Yang-Mills theory in Wess-Zumino gauge are computed using the supersymmetric dimensional regulator of Siegel. It is shown that $\text{γ}{}_{\mathrm{\mu }}\text{S}{}^{\mathrm{\mu }}\text{=0}$ and $\text{?}{}_{\mathrm{\mu }}\text{S}{}^{\mathrm{\mu }}\text{≠0}$ in agreement with an earlier calculation based on the Adler-Rosenberg method. The problem of exhibiting the chiral anomaly and a regulator for local supersymmetry suggests that the interpretation of dimensional reduction in component language is incomplete.     

Gauge internal symmetry in extended supergravity

Extended supergravity theories with global O(2) and SO(3) internal symmetry have recently been constructed, and a mechanism which implements local O(2) and SO(3) gauge invariance is given here. The introduction of a minimal gauge coupling automatically leads to a $\text{spin}\text{-}\text{3}\text{2}$ mass and a cosmological term in order to preserve local supersymmetry. Local internal symmetry for a $\text{spin}\text{-}\text{3}\text{2}$ field is related to spontaneous breakdown of global supersymmetry. Perturbation theory results which confirm the physical consistency of the system are given.     

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.     

Anomaly constraints on nonlinear sigma models

Anomalies in nonlinear sigma models can sometimes be cancelled by local counterterms. We show that these counterterms have a simple topological interpretation, and that the requirements for anomaly cancellation can be easily understood in terms of 't Hooft's anomaly matching conditions. We exhibit the anomaly cancellation on homogeneous spaces $\text{G}\text{H}$ and on general riemannian manifolds $\text{M}$. We include external gauge fields on the manifolds and derive the generalized anomaly-cancellation conditions. Finally, we discuss the implications of this work for superstring theories.     

Symmetric space scalar field theory

Some quantum field theories, such as the chiral SU(2) ? SU(2) theory, can have a dynamics invariant under a group $\text{G}$ that is realized on a vacuum which is invariant only under a subgroup $\text{H}$ of $\text{G}$. These theories may be defined by scalar fields which are coordinates for the coset manifold $\text{G}$/$\text{H}$. They are thus non-polynomial theories on a symmetric space, with the group motions in this space described by a set of Killing vectors. We show how the Lagrange function may be constructed entirely from the Killing vectors. In particular, all physical quantities may be expressed in terms of the currents formed out of the Killing vectors. The current correlation functions do not exhibit the spurious wave function renormalizations which are encountered if ordinary Green's functions are computed. We illustrate the general method by calculating one-loop counter terms in a completely invariant fashion. An Appendix describes in simple terms the general theory of symmetric spaces, which should prove useful in other contexts.     

Constructing effective actions for N = 1 supersymmetry theories: Symmetry principles and ward identities

The implications of the anomalous superconformal Ward identities for the construction of effective actions describing the low-energy dynamics of supersymmetry theories are investigated. The results apply to any N = 1 supersymmetry theory with an anomaly structure,

$\text{D}{}^{\mathrm{<mtext>\alpha </mtext><mtext>?</mtext>}}\text{V}{}_{\mathrm{<mtext>\alpha </mtext><mtext>\alpha </mtext><mtext>?</mtext>}}\text{+2D}{}_{\mathrm{\alpha }}\text{S=0}$

, where $\text{V}{}_{\mathrm{<mtext>\alpha </mtext><mtext>\alpha </mtext><mtext>?</mtext>}}$ is the supercurrent and the chiral superfield S represents the anomaly.     

Consistency in explicitly broken N = 1 supergravity with matter couplings

Using N = 1 supergravity coupled to general, gauged, N = 1 matter, we show that the addition of a wide class of explicit supersymmetry breaking terms in the N = 1 matter fields can still lead to the spin $\text{3}\text{2}$ field satisfying an equation of motion in which propagation is causal. We argue that a similar result holds in the case of N = 2 supergravity coupled to matter, plus explicit terms.     

Algebraic Bethe ansatz for the one-dimensional Hubbard model with chemical potential

The integrability and the algebraic Bethe ansatz approach for the one-dimensional (1D) Hubbard model with chemical potential are studied in the framework of the quantum inverse scattering method. We also investigate the hidden local gauge invariance for the model. It is found that the R-matrix only permits Abelian $\text{U(1) ⋇}{}_{\mathrm{s}}\text{U(1)}$ gauge transformations, and it is shown that the energy spectrum is gauge invariant whereas the eigenvectors and the Bethe ansatz equations are explicitly gauge dependent.     

A gauge invarlant approach to two-dimensional quantum chromodynamics

Quantum chromodynamics in 1 + 1 space-time is formulated in terms of gauge invariant phase operators; i.e., the Hamiltonian as well as other Poincaré generators are written in a gauge invariant hadronic language without reference to the gluon and quark fields. A systematic method for computing the $\text{1}\text{N}$ expansion is given. Both the meson and the baryon sectors are studied in this context. It is shown that no infrared divergences appear at any step of the calculations.     

Consistent supergravity

A combined spin 2 - spin $\text{3}\text{2}$ extension of general relativity is given which is both free of the usual higher spin inconsistencies and invariant under local supersymmetry transformations.     

On the geometrical aspects of electromagnetism,I

The trajectory of a charged test particle under a Lorentz force is obtained as the geodesic of a riemannian four dimensional manifold. Originally, the geodesic equation is nonlinear in some vector field A′_μ. The nonlinearity is traded in for the correct characteristic $\text{e}\text{m}$ of the test particle through a gauge condition, imposed upon A′_μ, which turns the geodesic into the fully covariant linear and gauge invariant Lorentz equation. Fitting the $\text{e}\text{m}$ ratio inside the gauge leaves F′_μν independent of $\text{e}\text{m}$ and allows its identification with the E.-M. tensor F_μν. This four dimensional approach allows the identification of the fifth coordinate used in Kaluza's geometrization |1,2|. The gauge function appears as the sum of Hamilton-Jacobi function plus an additional term, related to the “length” of the trajectory. It is this latter term which guarantees the correct “normalisation” of the $\text{e}\text{m}$ ratio.     

Auxiliary-field anomalies

In general, quantum corrections to matter-supergravity couplings uniquely determine what are acceptable auxiliary fields for N = 1 supergravity, and partially determine those for N = 2. This is because one-loop corrections produce anomalies in not only the local superscale transformations, but also in the local (Poincaré) supersymmetry transformations themselves, except for special cases: in particular, for N = 1 the $\text{n =}\text{1}\text{3}$ minimimal set of auxiliary fields is uniquely chosen.     

Instantons and condensates in supersymmetric CPN−1 models

The partition function in the multi-instanton background for the supersymmetric CP^N?1 model is calculated and is shown to be supersymmetry invariant. This is used to calculate Green's function involving N pairs of $\text{ψ}\text{ψ(x}{}_{\mathrm{i}}\text{)}$, and this turns out to be independent of the positions. A possible interpretation is that the composite field $\text{ψ}\text{ψ}$ acquires a vacuum expectation value. The results are in agreement with those obtained in the 1/N expansion.