Narrow Results

For a prime p, the gauge group of a principal U(p)-bundle over a compact, orientable Riemann surface is decomposed up to homotopy as a product of spaces, each of which is commonly known. This is used to deduce explicit computations of the homotopy groups of the moduli space of stable vector bundles through a range, answering a question of Daskalopoulos and Uhlenbeck.

In this paper, we show that the gauge group of a principal $PU(n)$-bundle over a compact Riemann surface decomposes up to homotopy as the product of factors, one of which is a corresponding gauge group for $S^2$ and the others are immediately recognizable spaces. Further, when $n$ is a prime $p$, the gauge group for $S^2$ decomposes as a product of immediately recognizable factors. These gauge groups have strong connections to moduli spaces of stable vector bundles.

Let M be a compact smooth manifold, let be a unital involutive subalgebra of the von Neumann algebra £ (H) of bounded linear operators of some Hilbert space H, let be the unital involutive algebra , let be an hermitian projective right -module of finite type, and let be the gauge group of unitary elements of the unital involutive algebra of right -linear endomorphisms of . We first prove that noncommutative geometry provides the suitable setting upon which a consistent theory of energy representations can be built. Three series of energy representations are constructed. The first consists of energy representations of the gauge group , being the group of unitary elements of , associated with integrable Riemannian structures of M, and the second series consists of energy representations associated with (d, ∞)-summable K-cycles over . In the case where is a von Neumann algebra of type II₁ a third series is given: we introduce the notion of regular quasi K-cycle, we prove that regular quasi K-cycles over always exist, and that each of them induces an energy representation.

The particular choice of the gauge group for Yang–Mills theory plays an important role when it comes to the influence of matter fields. In particular, both the chosen gauge group and the representation of the matter fields yield structural differences in the quenched case. Especially, the qualitative behavior of the Wilson potential is strongly dependent on this selection. Though the algebraic reasons for this observation is clear, it is far from obvious how this behavior can be described besides using numerical simulations. Herein, it is investigated how the group structure appears in the Dyson–Schwinger equations, which as a hierarchy of equations for the correlation functions have to be satisfied. It is found that there are differences depending on both the gauge group and the representation of the matter fields. This provides insight into possible truncation schemes for practical calculations using these equations.

We observe a new puzzling physical phenomenon in F-theory on the multisection geometry, wherein a model without a gauge group transitions to another model with a discrete ${\mathbb{Z}}_n$ gauge group via Higgsing. This phenomenon may suggest an unknown aspect of F-theory compactification on multisection geometry lacking a global section. A possible interpretation of this puzzling physical phenomenon is proposed in this note. We also propose a possible interpretation of another unnatural physical phenomenon observed for F-theory on four-section geometry, wherein a discrete ${\mathbb{Z}}_2$ gauge group transitions to a discrete ${\mathbb{Z}}_4$ gauge group via Higgsing as described in the previous literature.

In this paper, we analyze gauge groups in six-dimensional $N=1$ F-theory models. We construct elliptic Calabi–Yau 3-folds possessing various singularity types as double covers of “1/2 Calabi–Yau 3-folds,” a class of rational elliptic 3-folds, by applying the method discussed in a previous study to classify the singularity types of the 1/2 Calabi–Yau 3-folds. One to three U(1) factors are formed in six-dimensional F-theory on the constructed Calabi–Yau 3-folds. The singularity types of the constructed Calabi–Yau 3-folds corresponding to the non-Abelian gauge group factors in six-dimensional F-theory are deduced. The singularity types of the Calabi–Yau 3-folds constructed in this work consist of $A$- and $D$-type singularities.

In this study, we construct four-dimensional F-theory models with 3 to 8 U(1) factors on products of K3 surfaces. We provide explicit Weierstrass equations of elliptic K3 surfaces with Mordell–Weil ranks of 3 to 8. We utilize the method of quadratic base change to glue pairs of rational elliptic surfaces together to yield the aforementioned types of K3 surfaces. The moduli of elliptic K3 surfaces constructed in the study include Kummer surfaces of specific complex structures. We show that the tadpole cancels in F-theory compactifications with flux when these Kummer surfaces are paired with appropriately selected attractive K3 surfaces. We determine the matter spectra on F-theory on the pairs.

We consider the energy representation for the gauge group. The gauge group is the set of C^∞-mappings from a compact Riemannian manifold to a semi-simple compact Lie group. In this paper, we obtain irreducibility of the energy representation of the gauge group for any dimension of M. To prove irreducibility for the energy representation, we use the fact that each operator from a space of test functionals to a space of generalized functionals is realized as a series of integral kernel operators, called the Fock expansion.

We discuss the local structure of the net of von Neumann algebras generated by a representation of a local gauge group . Our discussion is independent of the singularity of spectral measures, which has been discussed by many authors since the pioneering work of Gelfand–Graev–Veršic. We show that, for type (S) operators U_A,b, second quantized operators with some twists, the commutativity only with those U(ψ) is sufficient for the triviality of them, where ψ belongs to an arbitrary (small) neighborhood of constant function 1. Some properties of 1-cocycles for the representation V : ψ ↦ Ad_ψ are also discussed.

The energy representation of a gauge group on a Riemannian manifold has been discussed by several authors. Y. Shimada has shown the irreducibility with the use of white noise analysis for compact Riemannian manifolds. In this paper we extend its technique to the noncompact Riemannian manifolds which have differential operators satisfying some conditions.

The purpose of this study is to examine the effect of topology change in the initial universe. In this study, the concept of G-cobordism is introduced to argue about the topology change of the manifold on which a transformation group acts. This G-manifold has a fiber bundle structure if the group action is free and is related to the spacetime in Kaluza–Klein theory or Einstein–Yang–Mills system. Our results revealed the fundamental processes of compactification in G-manifolds. In these processes, the initial high symmetry and multidimensional universe changes to present universe by the mechanism which lowers the dimensions and symmetries.

The chiral ring of classical supersymmetric Yang-Mills theory with gauge group Sp(N) or SO(N) is computed, extending previous work (of Cachazo, Douglas, Seiberg, and the author) for SU(N). The result is that, as has been conjectured, the ring is generated by the usual glueball superfield S ~ TrW_αW^α, with the relation S^h = 0, h being the dual Coxeter number. Though this proposition has important implications for the behavior of the quantum theory, the statement and (for the most part) the proofs amount to assertions about Lie groups with no direct reference to gauge theory.

Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let A_ζ denote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UA_ζ, the group of unitaries of A_ζ. The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.