Five-dimensional non-Lorentzian conformal field theories and their relation to six-dimensions

We study correlation functions in five-dimensional non-Lorentzian theories with an SU(1, 3) conformal symmetry. Examples of such theories have recently been obtained as Ω-deformed Yang-Mills Lagrangians arising from a null reduction of six-dimensional superconformal field theories on a conformally compactified Minkowski space. The correlators exhibit a rich structure with many novel properties compared to conventional correlators in Lorentzian conformal field theories. Moreover, identifying the instanton number with the Fourier mode number of the dimensional reduction offers a hope to formulate six-dimensional conformal field theories in terms of five-dimensional Lagrangian theories. To this end we show that the Fourier decompositions of six-dimensional correlation functions solve the Ward identities of the SU(1, 3) symmetry, although more general solutions are possible. Conversely we illustrate how one can reconstruct six-dimensional correlation functions from those of a five-dimensional theory, and do so explicitly at 2- and 3-points. We also show that, in a suitable decompactification limit Ω → 0, the correlation functions become those of the DLCQ description.

Two Lectures on the Jones Polynomial and Khovanov Homology

In the first of these two lectures I describe a gauge theory approach to understanding quantum knot invariants as Laurent polynomials in a complex variable q. The two main steps are to reinterpret three-dimensional Chern-Simons gauge theory in four dimensional terms and then to apply electric-magnetic duality. The variable q is associated to instanton number in the dual description in four dimensions. In the second lecture, I describe how Khovanov homology can emerge upon adding a fifth dimension.

NONCOMMUTATIVE ADHM CONSTRUCTION REVISITED

The Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction for noncommutative U(N) instantons is revisited. The origin of the instanton number is elucidated, including the case of the U(1) instantons. The group actions on the instantons are argued.

A NOVEL THEORY OF ELECTROWEAK INSTANTONS

Using ϕ-mapping topological current theory and the expansion theory of the δ-function, we found a more exact expression of conventional instanton and multi-instanton. We established a novel approach to instanton. It is found that the instantons arise from the symmetric phase of the Higgs field ϕ=0, the fine topological structure of the instanton number is also given.

TOPOLOGICAL CHARGE OF ADHM INSTANTON ON ${\bf R}^2_{\rm NC}\times{\bf R}^2$

We have calculated the topological charge of U (N) instantons on nondegenerate noncommutative spacetime to be exactly the instanton number k in a previous paper.1 This paper, which deals with the degenerate [Formula: see text] case, is the continuation of Ref. 1. We find that the same conclusion holds in this case, thus complete the answer to the problem of topological charge of noncommutative U (N) instantons.

Topological Charge of Noncommutative ADHM Instanton

We analytically calculate the topological charge of general noncommutative U (N) instantons using the Corrigan's identity and find that the result is exactly the instanton number k, which appears in the noncommutative ADHM construction as the dimension of the vector space V. This result coincides with the corresponding fact in the commutative case.

TOPOLOGICAL EFFECTS OF INSTANTON DUE TO DEFECTS

A new doublet variable is proposed to decompose non-Abelian gauge field for describing the topological effects of instantons due to the defects in appropriate phase of SU(2) Yang–Mills theory. It is shown that the instanton number can be directly related to the isospin defects of the doublet order parameter and contributed from topological charges of these defects. The θ-term in instanton action is found to be the delta-function form of the doublet and the Lagrangian of instantons in terms of new variables is also presented.

INSTANTONS AND CHIRAL ANOMALY IN FUZZY PHYSICS

In continuum physics, there are important topological aspects like instantons, θ-terms and the axial anomaly. Conventional lattice discretizations often have difficulties in treating one or the other of these aspects. In this paper, we develop discrete quantum field theories on fuzzy manifolds using noncommutative geometry. Basing ourselves on previous treatments of instantons and chiral fermions (without fermion doubling) on fuzzy spaces and especially fuzzy spheres, we present discrete representations of θ-terms and topological susceptibility for gauge theories and derive axial anomaly on the fuzzy sphere. Our gauge field action for four dimensions is bounded by a constant times the modulus of the instanton number as in the continuum.