∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

• Chern-Weil theory

• ∞-Chern-Weil theory introduction

Ingredients

• cohomology

  • differential cohomology

• ∞-Lie theory

  • Lie integration

• ∞-Lie algebra cohomology

  • Chevalley-Eilenberg algebra, Weil algebra, invariant polynomial

Connection

• ∞-Lie algebroid valued differential forms

• ∞-connection on a principal ∞-bundle

Curvature

• curvature

  • Bianchi identity

  • curvature characteristic form

  • covariant derivative

• secondary characteristic class

• Chern-Simons form

Theorems

• Chern-Weil homomorphism

• Chern-Gauss-Bonnet theorem

differential cohomology

Ingredients

• cohomology

• differential geometry

Connections on bundles

• connection on a bundle

  • parallel transport, holonomy

• curvature

  • curvature characteristic form, Chern character

• Chern-Weil theory

  • Chern-Weil homomorphism

  • secondary characteristic class

  • differential characteristic class

Higher abelian differential cohomology

• differential function complex

• differential orientation

• ordinary differential cohomology

  • differential Thom class

  • differential characters,

  • Deligne cohomology

  • circle n-bundle with connection,

  • bundle gerbe with connection

• differential K-theory

• differential elliptic cohomology

• differential cobordism cohomology

Higher nonabelian differential cohomology

• principal 2-bundle, principal ∞-bundle

• connection on a 2-bundle, connection on an ∞-bundle

• Chern-Weil theory in Smooth∞Grpd

  • ∞-Lie algebra cohomology

• ∞-Chern-Simons theory

Fiber integration

• higher holonomy

• fiber integration in differential cohomology

  • fiber integration in ordinary differential cohomology

  • fiber integration in differential K-theory

Application to gauge theory

• gauge theory

• gauge field

  • electromagnetic field

  • Yang-Mills field

  • Kalb-Ramond field/B-field

  • RR-field

  • supergravity C-field

  • supergravity

• quantum anomaly