Localization for Yang-Mills Theory on the Fuzzy Sphere

We present a new model for Yang-Mills theory on the fuzzy sphere in which the configuration space of gauge fields is given by a coadjoint orbit. In the classical limit it reduces to ordinary Yang-Mills theory on the sphere. We find all classical solutions of the gauge theory and use nonabelian localization techniques to write the partition function entirely as a sum over local contributions from critical points of the action, which are evaluated explicitly. The partition function of ordinary Yang-Mills theory on the sphere is recovered in the classical limit as a sum over instantons. We also apply abelian localization techniques and the geometry of symmetric spaces to derive an explicit combinatorial expression for the partition function, and compare the two approaches. These extend the standard techniques for solving gauge theory on the sphere to the fuzzy case in a rigorous framework.     

Gauge invariance over a group as the first principle of interacting string dynamics

It is stressed that the basic principle of the standard gauge theories is the invariance under internal symmetry transformations that do not commute with translations. This concept is generalized to the case where the translation group is replaced by an arbitrarily given non-abelian group G. The generalized Yang-Mills theory, called gauge theory over G, is an attractive extension of the standard formalism. The gauge theory over the conformal group is proposed as the fundamental theory of bosonic strings. As is usual in gauge theories, the interaction is uniquely specified by the invariance properties. For strings, overlap conditions between string positions come out in a natural way. The powerful machinery of Yang-Mills theories is fully applicable to the gauge theories over groups. In particular, an example of the Higgs-Kibble mechanism is given.     

Uniformly valid amplitude expansions for non-Abelian gauge theories

The imposition of a particular gauge condition in a non-Abelian gauge theory may lead to large-r nonuniformities in an amplitude expansion when used in the construction of an approximate solution of the field equations of the theory. We show that for both the Yang-Mills theory and general relativity it is always possible to find a class of gauge conditions that do not suffer from this defect and at the same time lead to solvable equations in each order of the approximation employed. We further show how one can construct such gauge conditions by a method similar to Lighthill's method of strained coordinates. An example of such a construction is given for the Yang-Mills theory.     

Double field formulation of Yang-Mills theory

Based on our previous work on the differential geometry for the closed string double field theory, we construct a Yang-Mills action which is covariant under O(D,D) T-duality rotation and invariant under three-types of gauge transformations: non-Abelian Yang-Mills, diffeomorphism and one-form gauge symmetries. In double field formulation, in a manifestly covariant manner our action couples a single O(D,D) vector potential to the closed string double field theory. In terms of undoubled component fields, it couples a usual Yang-Mills gauge field to an additional one-form field and also to the closed string background fields which consist of a dilaton, graviton and a two-form gauge field. Our resulting action resembles a twisted Yang-Mills action.     

Two-dimensional topological gravity and intersection theory on the moduli space of holomorphic bundles

We define a two-dimensional topological Yang-Mills theory for an arbitrary compact simple Lie group. This theory is defined in terms of intersection theory on the moduli space of flat connections on a two-dimensional surface and corresponds physically to a two-dimensional reduction and truncation of four-dimensional topological Yang-Mills theory. Two-dimensional topological Yang-Mills theory defines a topological matter system and may be naturally coupled to two-dimensional topological gravity. This topological Yang-Mills theory is also closely related to Chern-Simons gauge theory in 2 + 1 dimensions. We also discuss a relation between SL (2, ) Chern-Simons theory and two-dimensional topological gravity.     

Quantization of the massive Yang-Mills field in arbitrary gauges

A general gauge formulation of the massive Yang-Mills theory is presented with emphasis on the fact that the massive Yang-Mills field is, like the massless counterpart, also a non-Abelian gauge field.     

Dynamical Breaking of Generalized Yang-Mills Theory

The dynamical breaking of a generalized Yang-Mills theory is discussed. It is shown, in terms of the Nambu-Jona-Lasinio mechanism, that the gauge symmetry breaking can be realized dynamically in the generalized Yang-Mills theory. The combination of the generalized Yang-Mills theory and the NJL mechanism provides a way to overcome the difficulties related to the Higgs field and the Higgs mechanism in the usual spontaneous symmetry breaking theory.     

Extended Connection in Yang-Mills Theory

The three fundamental geometric components of Yang-Mills theory –gauge field, gauge fixing and ghost field– are unified in a new object: an extended connection in a properly chosen principal fiber bundle. To do this, it is necessary to generalize the notion of gauge fixing by using a gauge fixing connection instead of a section. From the equations for the extended connection’s curvature, we derive the relevant BRST transformations without imposing the usual horizontality conditions. We show that the gauge field’s standard BRST transformation is only valid in a local trivialization and we obtain the corresponding global generalization. By using the Faddeev-Popov method, we apply the generalized gauge fixing to the path integral quantization of Yang-Mills theory. We show that the proposed gauge fixing can be used even in the presence of a Gribov’s obstruction.     

Dynamical Breaking of Generalized Yang-Mills Theory

The dynamical breaking of a generalized Yang-Mills theory is discussed. It is shown, in terms of the Nambu Jona-Lasinio mechanism, that the gauge symmetry breaking can be realized dynamically in the generalized Yang-Mills theory. The combination of the generalized Yang-Mills theory and the NJL mechanism provides a way to overcome the difficulties related to the Higgs field and the Higgs mechanism in the usual spontaneous symmetry breaking theory.     

The double dimensional regularization and gauge theories without ghosts

A new version of dimensional regularization is introduced as calculational technique for the recently proposed gauge theory without ghosts. The method is applied to Yang-Mills theory.     

On the Yang-Mills structure of gravitation: A new issue of the final state

The Yang-Mills approach to gravity is presented. This extension of general relativity is based on the structures of a gauge theory: the Lorentz frame bundle acts as gauge bundle, the connection on the Lorentz bundle is the basic dynamical field, and the Yang-Mills equations define the dynamics for the connection. As a consequence of these dynamics the equivalence principle will be broken on space-time regions of high curvature, while it remains strictly preserved for the solar system, for example, and for homogeneous and isotropic world models. They are explicitly used to illustrate the extension of the general relativistic dynamics.This essay received an honorable mention (1976) from the Gravity Research Foundation-Ed.     

Yang-Mills gauge theory and gravitation

We point out that Yang's and Einstein's gravitational equations can be obtained from a geometric approach of Yang-Mills gauge theory in a sourceless case, under a decomposition of the Poincaré algebra. Otherwise, Einstein's equations cannot be derived from a Yang-Mills gauge equation when sources are inserted in the rotational sector of that algebra. A gauge Lagrangian structure is also discussed.     

Expansion of EYM amplitudes in gauge invariant vector space

Motivated by the problem of expanding the single-trace tree-level amplitude of Einstein-Yang-Mills theory to the BCJ basis of Yang-Mills amplitudes, we present an alternative expansion formula in gauge invariant vector space. Starting from a generic vector space consisting of polynomials of momenta and polarization vectors, we define a new sub-space as a gauge invariant vector space by imposing constraints on the gauge invariant conditions. To characterize this sub-space, we compute its dimension and construct an explicit gauge invariant basis from it. We propose an expansion formula in this gauge invariant basis with expansion coefficients being linear combinations of the Yang-Mills amplitude, manifesting the gauge invariance of both the expansion basis and coefficients. With the help of quivers, we compute the expansion coefficients via differential operators and demonstrate the general expansion algorithm using several examples.     

Gauge fixing and renormalization in topological quantum field theory

We show how Witten's topological Yang-Mills and gravitational quantum field theories may be obtained by a straightforward BRST gauge fixing procedure. We investigate some aspects of the renormalization of the topological Yang-Mills theory. It is found that the beta function for the Yang-Mills coupling constant is not zero.     

Infrared divergences and a non-local gauge for superspace Yang-Mills theory

The usual superspace approach to supersymmetric gauge theories suffers from problems with infrared divergences which greatly complicate multiloop calculations. We eliminate these divergences by introducing a non-local gauge-fixing term. In the background field method this term leads to unusual quantum-background interactions. Functional methods are presented for dealing with these interactions. As an example we compute the two-loop Yang-Mills β-function using the background field method in superspace. We also show how a non-local gauge can be used in ordinary, non-supersymmetric Yang-Mills theory.     

Finsler and Kaluza-Klein gauge theories

A comparison of Kaluza-Klein and Finsler-type gauge theories is sketched. It is shown that the two can be related by a mapping between fiber spaces which is equivalent to a transformation from one representation of the gauge group to another. The Finsler theory lends itself to an interpretation of the mapping operators as being geometrically similar to Yang-Mills potentials. The equations of motion in this theory contain fields which are comparable to connections instead of curvatures. This gives a new geometrical framework for unified field theories.     

A Generalized Yang-Mills Model and Dynamical Breaking of Gauge Symmetry

A generalized Yang-Mills model, which contains, besides the vector part V_μ, also a scalar part S, is constructed and the dynamical breaking of gauge symmetry in the model is also discussed. It is shown, in terms of Nambu-Jona-Lasinio (NJL) mechanism, that the gauge symmetry breaking can be realized dynamically in the generalized Yang-Mills model. The combination of the generalized Yang-Mills model and the NJL mechanism provides a way to overcome the difficulties related to the Higgs field and the Higgs mechanism in the usual spontaneous symmetry breaking theory.     

A path-space formula for non-abelian gauge theories

A functional integral representation is obtained for a semigroup in an indefinite metric space giving the dynamics of a cutoff Yang-Mills theory in space-time dimensions greater than two. The Fadde'av-Popov formula is shown to arise as a stochastic integral in the Feynman gauge.     

N = 4 Yang-Mills theory on the light cone

The 10-dimensional supersymmetric Yang-Mills theory is constructed in the light-cone gauge. When the theory is dimensionally reduced to four dimensions it is shown that the corresponding N = 4 theory is conveniently described in terms of a scalar superfield. This formalism avoids the problem of auxiliary fields but is Lorentz invariant only on the mass shell. Similar formalisms in terms of scalar superfields are also sketched for the other supersymmetric Yang-Mills as well as for N = 8 supergravity.