Chern-Weil theory

In mathematics , the Chern-Weil theory is a general procedure for calculating the characteristic classes of a principal bundle from its curvature . ( Characteristic classes are cohomology classes that measure topologically how twisted a bundle is.) Historically, it arose when the higher-dimensional version of Gauss-Bonnet's theorem was proven ; it marked the beginning of “global differential geometry”, i.e. the interaction of geometry and topology . The theory is named after André Weil and SS Chern .

definition

Let be a principal bundle with a structure group , be the Lie algebra of . Chern-Weil theory defines a homomorphism ${\displaystyle \pi :P\rightarrow M}$ ${\displaystyle G}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle G}$

${\displaystyle \phi :I^{*}({\mathfrak {g}})\rightarrow H_{dR}^{*}(M)}$

from the space of the -invariant polynomials to the deRham cohomology , the so-called Chern-Weil homomorphism . ${\displaystyle Ad(G)}$ ${\displaystyle {\mathfrak {g}}}$

Any invariant polynomial is the - form ${\displaystyle f\in I^{k}({\mathfrak {g}})}$ ${\displaystyle 2k}$

${\displaystyle f(\Omega ,\ldots ,\Omega )\in \Omega ^{2k}(M)}$

assigned, where the curvature shape is a relationship of the principal bundle. That is, for is ${\displaystyle \Omega \in \Omega ^{2}(M)}$ ${\displaystyle X_{1},\ldots ,X_{2k}\in T_{p}P}$

${\displaystyle f(\Omega )(X_{1},\dots ,X_{2k})={\frac {1}{(2k)!}}\sum _{\sigma \in {\mathfrak {S}}_{2k}}\operatorname {sign} (\sigma )f(\Omega (X_{\sigma (1)},X_{\sigma (2)}),\dots ,\Omega (X_{\sigma (2k-1)},X_{\sigma (2k)}))}$.

${\displaystyle f(\Omega )}$is a closed form and is then by definition the cohomology class of this form. It can be shown that it does not depend on the context chosen. ${\displaystyle \phi (f)}$ ${\displaystyle 2k}$ ${\displaystyle \phi (f)}$

Examples

• Be . Then the curvature shape has values ​​in . The development ${\displaystyle G=GL(n,\mathbb {C} )}$ ${\displaystyle {\mathfrak {gl}}(n,\mathbb {C} )=\operatorname {Mat} (n,\mathbb {C} )}$

${\displaystyle \det \left({\frac {it\Omega }{2\pi }}+I\right)=\sum _{k}c_{k}(\Omega )t^{k}}$

defines invariant polynomials

${\displaystyle c_{k}(\Omega )\in I^{2k}({\mathfrak {g}})}$,

for example is and . The cohomology classes are the Chern classes . ${\displaystyle c_{1}(\Omega )={\frac {i}{2\pi }}Tr(\Omega )}$ ${\displaystyle c_{n}(\Omega )=({\frac {i}{2\pi }})^{n}\det(\Omega )}$ ${\displaystyle \phi (c_{1}),\ldots ,\phi (c_{n})}$

Universal Chern-Weil homomorphism

Be a Lie group and its classifying space . is not a manifold, nevertheless a Chern-Weil homomorphism can be defined for the universal -bundle . ${\displaystyle G}$ ${\displaystyle BG}$ ${\displaystyle BG}$ ${\displaystyle G}$ ${\displaystyle \pi :EG\rightarrow BG}$ ${\displaystyle \phi _{G}:I^{*}(G)\rightarrow H^{*}(BG)}$

If is a principal bundle and its classifying map, then is . ${\displaystyle \pi :P\rightarrow M}$ ${\displaystyle G}$ ${\displaystyle F:M\rightarrow BG}$ ${\displaystyle \phi =F^{*}\circ \phi _{G}}$

See also

• Because algebra

literature

• Appendix C: Connections, Curvature, and Characteristic Classes in: Milnor, John W .; Stasheff, James D .: Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1974. vii + 331 pp.
• Chapter 5 in: Candel, Alberto; Conlon, Lawrence: Foliations. II. Graduate Studies in Mathematics, 60. American Mathematical Society, Providence, RI, 2003. xiv + 545 pp. ISBN 0-8218-0881-8