Chern-Simons form

The Chern-Simons forms are in the definition of secondary characteristic classes used differential forms , which in the mathematics in differential geometry and differential topology occur in various contexts, in particular in gauge theories . The Chern-Simons 3 form defines the action functional of the Chern-Simons theory . They are named after Shiing-Shen Chern and James Harris Simons , the authors of Characteristic Forms and Geometric Invariants published in 1974 .

definition

Let M be a Riemannian manifold . The Riemann connection

{\ displaystyle A \ in \ Omega ^ {1} (P (M), {\ mathfrak {g}} l (n))}

is a Lie algebra valued 1-form on the frame bundle . ${\ displaystyle P (M)}$

The Chern-Simons-1 form is defined by

{\ displaystyle {\ rm {Tr}} [\ mathbf {A}]}

,

where Tr denotes the trace of matrices.

The Chern-Simons 3 form is defined by

{\ displaystyle {\ rm {Tr}} \ left [\ mathbf {F} \ wedge \ mathbf {A} - {\ frac {1} {3}} \ mathbf {A} \ wedge \ mathbf {A} \ wedge \ mathbf {A} \ right].}

The Chern-Simons 5 form is defined by

{\ displaystyle {\ rm {Tr}} \ left [\ mathbf {F} \ wedge \ mathbf {F} \ wedge \ mathbf {A} - {\ frac {1} {2}} \ mathbf {F} \ wedge \ mathbf {A} \ wedge \ mathbf {A} \ wedge \ mathbf {A} + {\ frac {1} {10}} \ mathbf {A} \ wedge \ mathbf {A} \ wedge \ mathbf {A} \ wedge \ mathbf {A} \ wedge \ mathbf {A} \ right]}

where the curvature is defined by ${\ displaystyle \ mathbf {F}}$

{\ displaystyle \ mathbf {F} = d \ mathbf {A} + \ mathbf {A} \ wedge \ mathbf {A}.}

The general Chern-Simons form is defined such that ${\ displaystyle \ omega _ {2k-1}}$

{\ displaystyle d \ omega _ {2k-1} = {\ rm {Tr}} \ left (\ mathbf {F} ^ {k} \ right),}

where is defined by the outer product of differential forms. ${\ displaystyle \ mathbf {F} ^ {k}}$

If there is a parallelizable 2k-1 -dimensional manifold (e.g. an orientable 3-manifold), then there is a cut and the integral of over the manifold is a global invariant, which is well-defined modulo the addition of whole numbers. (For different cuts the integrals only differ by whole numbers.) The invariant defined in this way is the Chern-Simons invariant ${\ displaystyle M}$ ${\ displaystyle s: M \ rightarrow P (M)}$ ${\ displaystyle s ^ {*} \ omega _ {2k-1}}$ ${\ displaystyle M}$

{\ displaystyle cs (M) \ in \ mathbb {R} / \ mathbb {Z}}

.

General definition of principal bundles and invariant polynomials

Let be a Lie group with Lie algebra and an invariant polynomial . ${\ displaystyle G}$ ${\ displaystyle {\ mathfrak {g}}}$ ${\ displaystyle f \ in I ^ {k} ({\ mathfrak {g}})}$

Each invariant polynomial corresponds to a Chern-Simons form of principle bundles as follows. ${\ displaystyle f}$ ${\ displaystyle G}$

Be a principal bundle with a structure group . Choose a form of connection and use to designate its form of curvature . Then the Chern-Simons form is defined by ${\ displaystyle \ pi: P \ rightarrow M}$ ${\ displaystyle G}$ ${\ displaystyle \ omega \ in \ Omega ^ {1} (P, {\ mathfrak {g}})}$ ${\ displaystyle \ Omega \ in \ Omega ^ {2} (P, {\ mathfrak {g}})}$ ${\ displaystyle Tf \ in \ Omega ^ {2k-1} (P, \ mathbb {R})}$

{\ displaystyle Tf = \ sum _ {i = 0} ^ {k-1} A_ {i} f (\ omega \ wedge \ left [\ omega, \ omega \ right] ^ {i} \ wedge \ Omega ^ { ki-1})}

with . ${\ displaystyle A_ {i}: = (- 1) ^ {i} {\ frac {k! (k-1)!} {2 ^ {i} (k + i)! (k-1-i)! }}}$

In the case of flat bundles, this formula is simplified to . ${\ displaystyle (-1) ^ {k-1} {\ frac {k! (k-1)!} {2 ^ {k-1} (2k-1)!}} f (\ omega \ wedge \ left [\ omega, \ omega \ right] ^ {k-1})}$

The equation applies

{\ displaystyle dTf = f (\ Omega, \ ldots, \ Omega)}

,

in the case of flat bundles . ${\ displaystyle dTf = 0}$

As is well known, every characteristic class corresponds to an invariant polynomial, see Chern-Weil theory . If so , then according to Chern-Weil theory the corresponding characteristic class vanishes in real cohomology. The form is closed in this case and initially defines a class in the cohomology of . Withdrawal by means of a cut defines a cohomology class of which is well defined modulo of integers. The cohomology class defined in this way fits into the Bockstein sequence ${\ displaystyle u \ in H ^ {*} (BG, \ mathbb {Z})}$ ${\ displaystyle f (\ Omega, \ ldots, \ Omega) = 0}$ ${\ displaystyle u_ {f}}$ ${\ displaystyle Tf}$ ${\ displaystyle P}$ ${\ displaystyle M}$ ${\ displaystyle H ^ {*} (M, \ mathbb {R} / \ mathbb {Z})}$