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Can anyone please suggest some good literature or references for understanding Gauge theory from the perspective of a mathematician (from the point of view of differential geometry)? Being a Mathematics PhD student(differential geometry) I want the introduction mathematically as rigorous as possible.

Specially I want to understand the space of connections and the action of Gauge group on it.

My background is following:

I have read Fibre bundles(principle G-bundles), Theory of Connections( Connection as distribution and as a Lie algebra valued 1-form, Local connection forms, Holonomy Theorem(Ambrose and singer), flat connections and some Affine Connections from Foundation of Differential Geometry(Kobayashi and Nomizu) Vol-1, Characteristic Classes(Chern classes) from Foundation of Differential Geometry(Kobayashi and Nomizu) Vol-2, general theory of fibre bundles(Milnor's classification for principle bundles) from Fibre Bundles Dale Husemoller and I have some idea about basic Algebraic topology, cohomology theory and Category theory(at level of a Masters student in Mathematics).

Most of the literature I came across in the internet I felt they were written from the perspective of a Physicist otherwise some research papers (which I felt too advanced for me now). Being just a 1st year PhD student(differential geometry) it would be really helpful for me if someone can refer (keeping in mind of my Mathematical background) a humble but Mathematically very rigorous literature in Gauge theory where action of Gauge group on the space of Connections is discussed in details.

Thanks in advance.

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1 Answer 1

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Some classic references (maybe it has been written something better since their publication I don't know) :

  • about Instantons over 4D manifolds, Donaldson & Kronheimer's "The Geometry of four manifold" and Uhlenbeck & Freed's "Instantons and four manifolds"

  • about Seiberg-Witten, J.W. Morgan's "The Seiberg-Witten Equations And Applications To The Topology Of Smooth Four-Manifolds"

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