Gauge-invariant discrete models of Yang-Mills equations

Abstract

ABSTRACT. Two discrete models of Yang-Mill equations are constructed in the space ~ for some matrixvalued Lie group. A gauge-invariant discrete model is examined.KEY WORDS: Yang-Mills equations, discrete model, gauge invariance, gauge transformation.We construct two discrete analogs of Yang-Mills equations in the space Rn for some matrix-valued Lie group G. We define a discrete analog of the operator of exterior covariant differentiation d~ by using the formalism described in [1] and apply this operator to construct a model satisfying the gauge invariance principle. To specify a more exact discrete analog of the exterior differentiation operator d, we consider another model of Yang-Mills equations, which formally coincides with the previous one, but the equations are not gauge-invariant. The author of [2] considered two-dimensional models of the classical Yang-MiUs equations and passed to discrete analogs, which, however, neglected the invariance requirements. Our approach is close to that of [2], but removes this shortcoming.If the initial manifold is of even dimension (n = 4), then the Yang-Mills theory can be regarded as a nonlinear generalization of Hodge theory [3]. A similar result is obtained in this paper for discrete models.w Connection, Curvature, and Gauge Transformations Let M be a smooth Riemannian manifold. Consider a product bundle P = M x G over this manifold, where G is a matrix-vedued Lie group. Generally, all assertions made in this section are valid for an arbitrary vector bundle P, but for constructing the discrete model, we only use the product bundle. Both the Lie group G and the manifold M are differentiable structures, hence, the bundle P is a differentiable manifold. Therefore, we can consider the tangent bundle TP and the cotangent bundle T*P over P.A connection is specified by a 1-form w E T*P taking values in the Lie algebra g of the group G (see, e.g., [4, p. 176]). Let (x, g), x E M, and g E G be local coordinates of the bundle P. Then w has the form = g-dg + g-lAg, where A = Z A (x) odx"a~lz and the A~ are the basis of the Lie algebra 9-The differential 1-form A, which takes values in 9, is called a connection form and the functions A~(x), connections. Consider a transformation of the coordinates of the bundle P with respect to which the 1-form w is invariant. Let (x, g) be mapped into (x', g') under this transformation. We shall consider only transformations that do not change the coordinate covering of the manifold, i.e., such that x = x' andThe invariance of the 1-form w means thatAny transformation (2) satisfying condition (3) induces a certain transformation law for the connection form A. Taking into account the fact that dg r = dhg + hdg and dhh -1 + hdh -1 = 0, we obtain A' = hdh -1 + bah -1.

Instanton-anti-instanton solutions of discrete Yang-Mills equations

“…We develop discrete models of some objects in differential geometry, including the Hodge star operator, the differential and the exterior multiplication, in such a way that they preserve the geometric structure of their continual analogs. We continue the investigations which were originated in [3], [6]- [8]. The geometrical discretisation techniques used here extend those introduced in [2] and [6].…”

“…The geometrical discretisation techniques used here extend those introduced in [2] and [6]. A combinatorial model of R 4 based on the use of the double complex construction is taken from [8].…”

Self-Dual and Anti-Self-Dual Solutions of Discrete Yang-Mills Equations on a Double Complex

“…To obtain the same result for our discrete model we need to construct a suitable combinatorial analog of the 4-sphere. It would be interesting to connect the above constructions with discrete model of S 4 described in [21]. This connection must be investigated and we hope to treat its further in future work.…”

“…Note that the definition of inner product in the double complex and a discrete analog of the Yang-Millls actions (2.11) can be found in [21].…”

“…More generally, if we assume that the gauge transformation g is an arbitrary quaternionic 0-form, then we take the imaginary part of g −1 ∪ A ∪ g + g −1 ∪ d c g in (4.13). For a deeper discussion of gauge invariant discrete models of the Yang-Mills theory we refer the reader to [19,21]. An arbitrary discrete 2-form F ∈ K(4) can be written as follows…”