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Literature suggestion for understanding Gauge theory from the perspective of a Mathematician.

Texts on Principal Bundles, Characteristic Classes, Intro to 4-manifolds / Gauge TheoryReference request for some topics in Differential Geometry like connections, metrics, curvature etc.Difference between types of connectionsTextbook on the differential geometry of fibre bundlesRelearning differential geometryReferences for differential cohomology and secondary characteristic classesHeat kernel, connection on fibre bundles: reference requestConnection on fiber bundle of generic fiberDifferent definitions of irreducible $mathrmSU(2)$ connectionsreferences for physical gauge theory and spinors.

2

2

$begingroup$

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.

differential-geometry reference-request connections principal-bundles gauge-theory

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asked Feb 9 at 0:40

WandereradiWandereradi

838

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add a comment | 

2

2

$begingroup$

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.

differential-geometry reference-request connections principal-bundles gauge-theory

share|cite|improve this question

asked Feb 9 at 0:40

WandereradiWandereradi

838

$endgroup$

add a comment | 

$begingroup$

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.

differential-geometry reference-request connections principal-bundles gauge-theory

share|cite|improve this question

asked Feb 9 at 0:40

WandereradiWandereradi

838

$endgroup$

share|cite|improve this question

asked Feb 9 at 0:40

WandereradiWandereradi

838

share|cite|improve this question

asked Feb 9 at 0:40

WandereradiWandereradi

838

asked Feb 9 at 0:40

WandereradiWandereradi

838

asked Feb 9 at 0:40

WandereradiWandereradi

838

1 Answer
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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"

share|cite|improve this answer

edited Mar 21 at 13:22

Martin Sleziak

44.9k10122277

answered Feb 9 at 2:06

Warlock of Firetop MountainWarlock of Firetop Mountain

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3

$begingroup$

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"

share|cite|improve this answer

edited Mar 21 at 13:22

Martin Sleziak

44.9k10122277

answered Feb 9 at 2:06

Warlock of Firetop MountainWarlock of Firetop Mountain

2,407915

$endgroup$

add a comment | 

3

$begingroup$

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"

share|cite|improve this answer

edited Mar 21 at 13:22

Martin Sleziak

44.9k10122277

answered Feb 9 at 2:06

Warlock of Firetop MountainWarlock of Firetop Mountain

2,407915

$endgroup$

add a comment | 

$begingroup$

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"

share|cite|improve this answer

edited Mar 21 at 13:22

Martin Sleziak

44.9k10122277

answered Feb 9 at 2:06

Warlock of Firetop MountainWarlock of Firetop Mountain

2,407915

$endgroup$

share|cite|improve this answer

edited Mar 21 at 13:22

Martin Sleziak

44.9k10122277

answered Feb 9 at 2:06

Warlock of Firetop MountainWarlock of Firetop Mountain

2,407915

edited Mar 21 at 13:22

Martin Sleziak

44.9k10122277

edited Mar 21 at 13:22

Martin Sleziak

44.9k10122277

answered Feb 9 at 2:06

Warlock of Firetop MountainWarlock of Firetop Mountain

2,407915

answered Feb 9 at 2:06

Warlock of Firetop MountainWarlock of Firetop Mountain

2,407915