The special geometry of calibrated cycles, closely related to mirror symmetry among Calabi–Yau 3-folds, is itself a real form of a new subject, which we call slightly deformed algebraic geometry. On the other hand, both of these geometries are parallel to classical gauge theories and their complexifications. This article explains this parallelism, so that the appearance of invariants of new typ...

We investigate the properties of the QCD string in the Euclidean SU(N) pure gauge theory when the space-time dimensions transverse to it are periodic. From the point of view of an effective string theory, the string tension σ and the low-energy constants ck of the theory are arbitrary functions of the sizes of the transverse dimensions L⊥. Since the gauge theory is linearly confining in D = 2, ...

Numerical relativity describes a discrete initial value problem for general relativ-ity. A choice of gauge involves slicing space-time into space-like hypersurfaces. This introduces past and future gauge relative to the hypersurface of present time. Here, we propose solving the discretized Einstein equations with a choice of gauge in the future and a dynamical gauge in the past. The method is i...

We discuss reality conditions and the relation between spacetime diffeomorphisms and gauge transformations in Ashtekar’s complex formulation of general relativity. We produce a general theoretical framework for the stabilization algorithm for the reality conditions, which is different from Dirac’s method of stabilization of constraints. We solve the problem of the projectability of the diffeomo...

An SO(4) gauge invariant model with extended field transformations is examined in four dimensional Euclidean space. The gauge field is (A µ) αβ = 1 2 t µνλ (M νλ) αβ where M νλ are the SO(4) generators in the fundamental representation. The SO(4) gauge indices also participate in the Euclidean space SO(4) transformations giving the extended field transformations. We provide the decomposition of...

The investigation of topological properties of the gauge field in a two-dimensional Higgs model can help in understanding anomalous fermion number violation. 1. A simple model for fermion number violation Fermion number, which is the sum of baryon number and lepton number (B+L), is not conserved in the Standard Model 1. This is due to the anomaly in the fermion current. The lattice formulation ...

An SO(4) gauge invariant model with extended field transformations is examined in four dimensional Euclidean space. The gauge field is (A) = 1 2 t(M) where M are the SO(4) generators in the fundamental representation. The SO(4) gauge indices also participate in the Euclidean space SO(4) transformations giving the extended field transformations. We provide the decomposition of the reducible fiel...

We give a framework to describe gauge theory on a certain class of commutative but nonassociative fuzzy spaces. Our description is in terms of an Abelian gauge connection valued in the algebra of functions on the cotangent bundle of the fuzzy space. The structure of such a gauge theory has many formal similarities with that of Yang-Mills theory. The components of the gauge connection are functi...

We study some topological aspects of non-abelian gauge theories intimately connected to the Lie algebras of the gauge groups and the homotopy theory in the generalized gauge orbit space. The physics connection to the non-perturbative solution to strong CP problem as originally proposed by the author is also discussed. Some relevant topological formulas are also given and discussed. A result fro...

We identify the spectral curve of pure gauge SU(2) Seiberg-Witten theory with the Weierstrass curve C/L ∋ z 7→ (1, ℘(z), ℘(z)) and thereby obtain explicitely a modular form from which the moduli space parameter u and lattice parameters a, aD can be derived in terms of modular respectively theta functions. We also discuss its relationship with the c = −2 triplet model conformal field theory.