Bernard Grossman

Solutions to {Yang-Mills} Field Equations in Eight-dimensions and the Last Hopf Map

AbstractWe will show that the Hopf map

p-Adic strings, the Weil conjectures and anomalies☆

S^{15} \xrightarrow{{S^7 }}S^8

Gauged BRST symmetry and the occurrence of higher cocycles in quantum field theory

admits a sourceless, topologically non-trivial gauge field. This result will be cast in the form of a solution to eight dimensional Euclidean Yang-Mills field equations with topological chargeQ=1. This solution is Spin (9) symmetric and leads to a new generalized duality conditionF∧F=±(F∧F)*.

Dimension reduction and massless chiral fermions

Abstract We find the existence of a non-vanishing 3-cocycle in the unitary, gauge-invariant, ray representations of translations for the quantum mechanics of a charged particle in a background magnetic monopole field. The 3-cocycle can be removed by passing to the central extension of the Lie algebra of translations, or the algebra can be made associative by imposing the Dirac quantization condition.

Conformal field theory and a topological quantum theory of vortices and knots

Abstract An analogy between the Veneziano amplitude and the p-adic interpolation of the beta-function is suggested as the basis of a new p-adic quantum geometry. Relationships with the Weil conjectures, Fermat curves and anomalies are discussed.

Solutions to Gauge Field Equations in Eight-dimensions: Conformal Invariance and the Last Hopf Map

Abstract We analyze stochastic quantization in the framework of topological field theory. We consider an action which is a pure derivative in stochastic time. The fields needed to gauge fix this topological action are provided by the noise of the Langevin equation and the fermions arising from the exponentiation of a determinant. Interaction terms can be introduced by conjugation of the free BRST operator by a Morse function. The known supersymmetric action for stochastic quantization is then recovered. The formalism that we develop has applications in Yang-Mills theory.

Dimensional reduction example with three families of low mass fermions

Abstract The BRST symmetry of Yang-Mills theories can be gauged via the introduction of an anticommuting single gauge field. The construction of a local BRST operation follows, which allows an algebraic analysis of the BRST current algebra. This construction provides, in particular, a field theory interpretation of most higher cocycles which accompany the usual chiral anomaly.

Adelic conformal field theory

Abstract Dimensional reduction of a Yang-Mills theory, when the space of extra dimensions is a compact homogeneous space, can lead to families of flavor chiral fermions. Using the Atiyah-Singer theorem we show that when representation mixing is neglected, some of these families can be massless even in the presence of a non-vanishing I = 1 2 Higgs field.