By George A. Anastassiou, Visit Amazon's Anastasios Mallios Page, search results, Learn about Author Central, Anastasios Mallios,
Differential geometry, within the classical experience, is built throughout the concept of soft manifolds. smooth differential geometry from the author’s viewpoint is utilized in this paintings to explain actual theories of a geometrical personality with out utilizing any suggestion of calculus (smoothness). as an alternative, an axiomatic therapy of differential geometry is gifted through sheaf concept (geometry) and sheaf cohomology (analysis). utilizing vector sheaves, as opposed to bundles, in line with arbitrary topological areas, this special approach as a rule furthers new perspectives and calculations that generate unforeseen power applications.
Modern Differential Geometry in Gauge Theories is a two-volume learn monograph that systematically applies a sheaf-theoretic method of such actual theories as gauge conception. starting with quantity 1, the focal point is on Maxwell fields. all of the easy recommendations of this mathematical strategy are formulated and used thereafter to explain undemanding debris, electromagnetism, and geometric prequantization. Maxwell fields are absolutely tested and categorized within the language of sheaf conception and sheaf cohomology. carrying on with in quantity 2, this sheaf-theoretic technique is utilized to Yang–Mills fields in general.
The textual content features a wealth of certain and rigorous computations and should entice mathematicians and physicists, besides complex undergraduate and graduate scholars, attracted to functions of differential geometry to actual theories comparable to basic relativity, basic particle physics and quantum gravity.
Read Online or Download Modern Differential Geometry in Gauge Theories: Maxwell Fields, Volume I (Progress in Mathematical Physics) PDF
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Extra resources for Modern Differential Geometry in Gauge Theories: Maxwell Fields, Volume I (Progress in Mathematical Physics)
VI; p. 29, Section 7]. On the other hand, frequent use of the above material is made in subsequent chapters of this treatise, including Part II. See the following Section 6 for an important notion concerning physical applications, still based on the space of A-connections as above (cf. 1 in the sequel). 6 Related A-Connections. Moduli Space of A-Connections 33 6 Related A-Connections. Moduli Space of A-Connections By referring to the type of A-connections considered already by the title of the present section, we ﬁrst remark that this actually has, as we shall see presently below, a special bearing, for instance, on the so-called gauge equivalent (A-)connections.
68) on X . 69) . 67). 71) ˜ αβ ), δ(ω(α) ) := ∂(g which also will be of use in the sequel. So the last relation will be applied in Part II of the present treatise by further looking at the cohomological classiﬁcation of Yang–Mills ﬁelds (see Chapt. I; Section 9). 71) as well. On the other hand, further considerations of the group of gauge transformations of a given vector sheaf E on X will also be applied in Part II of this study (cf. Chapt. 1) in conjunction with metric notions, still referring to Yang–Mills ﬁelds.
See [VS: Chapt. IV; p. 1]. 7) Conn A (E) = D + Ω(EndE)(X ), with D a given A-connection of E. 2) u ∈ Ω(EndE)(X ). 1) Conn A (E) (cf. 3)), is an afﬁne space, modeled on the A(X )-module Ω(EndE)(X ). Especially, if we consider a line sheaf L on X , since in that case one has (see [VS: Chapt. II; p. 7) the following relation, pertaining to the set (in point of fact, afﬁne space, cf. 11) Conn A (L) = D + Ω(X ), where D stands, of course, for a given A-connection D of L. 13) where ω ∈ Ω(X ), that is, a “1-form” on X .