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groupoid of Lie-algebra valued forms

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

\infty-Chern-Weil theory

Contents

Idea

For 𝔤\mathfrak{g} a Lie algebra, the groupoid of 𝔤\mathfrak{g}-valued forms is the groupoid whose objects are differential 1-forms with values on 𝔤\mathfrak{g}, and whose morphisms are gauge transformations between these.

This carries the structure of a generalized Lie groupoid BG conn\mathbf{B}G_{conn} , which is a differential refinement of the delooping Lie groupoid BG\mathbf{B}G of the Lie group GG corresponding to 𝔤\mathfrak{g}:

its UU-parameterized smooth families of objects are Lie algebra valued differential forms on UU. Its UU-parameterized families of morphisms are gauge transformations of these forms by GG-valued smooth functions on UU.

A cocycle with coefficients in BG conn\mathbf{B}G_{conn} is a connection on a bundle.

For more discussion of this see ∞-Lie groupoid – Lie groups.

Definition

Definition

For GG a Lie group the groupoid of Lie(G)Lie(G)-valued differential forms is as a groupoid internal to smooth spaces, the sheaf of groupoids

B¯G:=GTrivBund ():=[P 1(),BG] \bar \mathbf{B}G := G TrivBund_\nabla(-) := [P_1(-), \mathbf{B}G]

that to a smooth test space UDiffU \in Diff assigns the functor category [P 1(U),BG][P_1(U),\mathbf{B}G] of smooth functors (functors internal to smooth spaces) from the path groupoid P 1(U)P_1(U) of UU to the one-object delooping groupoid BG\mathbf{B}G.

Properties

Theorem

The groupoid B¯G\bar \mathbf{B}G is canonically equivalent to the smooth groupoid where

  • objects are smooth gg-valued 1-forms AΩ 1(U,g)A \in \Omega^1(U, g);
  • morphisms h:AAh : A \to A' are given by smooth GG-valued functions hC (U,G)h \in C^\infty(U,G) such that
    A=Ad h(A)h *θ¯ A' = Ad_h(A) - h^* \bar \theta

Here θ¯\bar \theta is the right invariant Maurer-Cartan form on GG. A common way to write this is A=Ad h(A)+hdh 1A' = Ad_h(A) + h d h^{-1}.

A proof is in SchrWalI.

Differential nonabelian cohomology

Theorem

The cohomology with coefficients in B¯G\bar \mathbf{B}G classifies GG-principal bundles connection on a bundle with connection.

More is true: there is a natural canonical equivalence of groupoids

H Diff(X,B¯G)GBund (X). \mathbf{H}_{Diff}(X, \bar \mathbf{B}G) \simeq G Bund_\nabla(X) \,.
  • There is the obvious projection

    B¯GBG. \bar \mathbf{B}G \to \mathbf{B}G \,.
  • Lifting a GG-cocycle through this projection to a differential GG-cocycle means equipping it with a connection.

For G=U(n)G = U(n) these differential cocycles model the Yang-Mills field in physics.

  • For G=U(1)G = U(1) the sheaf B¯U(1)()\bar \mathbf{B}U(1)(-) coincides with the the Deligne complex in degree 2, B¯U(1)(2) D \bar \mathbf{B}U(1)\simeq \mathbb{Z}(2)_D^\infty, as described there.

References

Details are in

The definition in terms of differential forms is def 4.6 there. The equivalence to [P 1(),BG][P_1(-), \mathbf{B}G] is proposition 4.7.

See also ∞-Chern-Weil theory introduction

Revised on January 14, 2013 20:55:23 by Urs Schreiber (203.116.137.162)