Chern-Simons functional

The Chern-Simons functional is important in differential geometry , topology and mathematical physics . In mathematics it is used to define the Chern-Simons invariant of relationships on principal bundles over 3-manifolds . Originally introduced by Chern and Simons in the theory of secondary characteristic classes, it had at least two unexpected applications, namely Witten's classification in quantum field theory with a physical-geometrical interpretation of the Jones polynomial ( topological quantum field theory ) and the interpretation of Chern -Simons-invariant of flat bundles as a complex-valued version of the hyperbolic volume .

definition

Let be a simply connected Lie group and a 3-dimensional, closed, orientable manifold . Under these conditions, each is - principal bundle trivialisierbar, so it has a cross-section . ${\displaystyle G}$ ${\displaystyle M}$ ${\displaystyle G}$ ${\displaystyle \pi :E\rightarrow M}$ ${\displaystyle s:M\rightarrow E}$

For a context

${\displaystyle \omega \in \Omega ^{1}(E,{\mathfrak {g}})}$

its Chern-Simons action functional is defined by

${\displaystyle CS(\omega ,s)={\frac {1}{4\pi }}\int _{M}\operatorname {Tr} (s^{*}(\omega \wedge d\omega +{\frac {2}{3}}\omega \wedge \omega \wedge \omega ))}$.

This definition depends a priori on the choice of a section for a gauge transformation ${\displaystyle s:M\rightarrow E}$

${\displaystyle g\in {\mathcal {G}}=C^{\infty }(M,G)}$

but applies

${\displaystyle CS(\omega ,gs)-CS(\omega ,s)=-{\frac {1}{6}}\int _{M}g^{*}\omega _{MC}\wedge \left[\omega _{MC},\omega _{MC}\right]\in \mathbb {Z} }$,

where is the mason cartan shape . ${\displaystyle \omega _{MC}}$

So you get a modulo well-defined value ${\displaystyle \mathbb {Z} }$

${\displaystyle CS(\omega )\in \mathbb {C} /\mathbb {Z} }$.

properties

Let be a closed, orientable 3-manifold and . We denote with the (infinite-dimensional) manifold of all connections on -principal bundles over . ${\displaystyle M}$ ${\displaystyle \pi _{1}G=0}$ ${\displaystyle {\mathcal {C}}_{M}}$ ${\displaystyle G}$ ${\displaystyle M}$

Then is smooth and has the following properties: ${\displaystyle CS:{\mathcal {C}}_{M}\rightarrow \mathbb {C} /\mathbb {Z} }$

• (Functoriality)

If a bundle map is over an orientation preserving diffeomorphism then holds ${\displaystyle \phi :P_{1}\rightarrow P_{2}}$ ${\displaystyle \psi :M_{1}\rightarrow M_{2}}$

${\displaystyle CS(\phi ^{*}\omega )=CS(\omega )}$

for every context . ${\displaystyle \omega }$

• (Additivity)

If there is a disjoint union, and there is a connection , then it holds ${\displaystyle M=M_{1}\cup M_{2}}$ ${\displaystyle \omega }$ ${\displaystyle M}$

${\displaystyle CS(\omega )=CS(\omega \mid _{M_{1}})+CS(\omega \mid _{M_{2}})}$.

• (Expansion of the structural group)

If there is an inclusion of simply connected, compact Lie groups, a connection on a -bundle, and the extension from to a -bundle , then holds ${\displaystyle G_{1}\rightarrow G_{2}}$ ${\displaystyle \omega _{1}}$ ${\displaystyle G_{1}}$ ${\displaystyle E_{1}\rightarrow M}$ ${\displaystyle \omega _{2}}$ ${\displaystyle \omega _{1}}$ ${\displaystyle G_{2}}$ ${\displaystyle E_{2}\rightarrow M}$

${\displaystyle CS(\omega _{1})=CS(\omega _{2})}$.

Flat connections

It applies

${\displaystyle {\frac {\delta CS}{\delta \omega }}={\frac {1}{2\pi }}\Omega }$,

where denotes the curvature of the connection . The critical points of the Chern-Simons functional are precisely the flat relationships . In particular, the Chern-Simons theory constant on is connected components of the flat space module connections on . ${\displaystyle \Omega }$ ${\displaystyle \omega }$ ${\displaystyle M\times G}$

Yoshida's theorem

Let it be a closed, orientable hyperbolic 3-manifold and its holonomy representation. Then applies to the associated flat bundle ${\displaystyle M}$ ${\displaystyle \rho :\pi _{1}M\rightarrow PSL(2,\mathbb {C} )}$ ${\displaystyle E_{\rho }}$

${\displaystyle CS(E_{\rho })=cs(M)+{\frac {i}{2\pi ^{2}}}\operatorname {vol} (M)\in \mathbb {C} /\mathbb {Z} }$,

where the Riemann Chern-Simons invariant of the Levi-Civita relationship denotes. ${\displaystyle cs(M)}$

The picture of the fundamental class below the representation defines a homology class ${\displaystyle \rho }$

${\displaystyle (B\rho )_{*}\left[M\right]\in H_{3}(PSL(2,\mathbb {C} )^{\delta };\mathbb {Z} )\simeq {\hat {B}}(\mathbb {C} )}$

in the expanded Bloch group and the Rogers dilogarithm

${\displaystyle R:{\hat {B}}(\mathbb {C} )\rightarrow \mathbb {C} /2\pi ^{2}\mathbb {Z} }$

maps to . This provides an explicit formula for the Chern-Simons invariant and an alternative proof of Yoshida's theorem. ${\displaystyle (B\rho )_{*}\left[M\right]}$ ${\displaystyle CS(E_{\rho })}$

Shallow Bundle Algorithm

Let it be a flat bundle over a closed, orientable 3-manifold with holonomy . Then the Rogers dilogarithm maps to , where denotes the canonical homomorphism. The value of can be calculated from the Ptolemaic coordinates of the representation to a triangulation of . (This approach also works for 3-manifolds with a boundary , as long as the restriction of to the fundamental groups of the boundary is unipotent.) This algorithm is implemented in the Ptolemy Module as part of the SnapPy software . ${\displaystyle E_{\rho }}$ ${\displaystyle M}$ ${\displaystyle \rho :\pi _{1}M\rightarrow SL(n,\mathbb {C} )}$ ${\displaystyle \lambda ((B\rho )_{*}\left[M\right])}$ ${\displaystyle CS(E_{\rho })}$ ${\displaystyle \lambda :H_{3}(SL(n,\mathbb {C} );\mathbb {Z} )\rightarrow {\hat {B}}(\mathbb {C} )}$ ${\displaystyle CS(E_{\rho })}$ ${\displaystyle \rho }$ ${\displaystyle M}$ ${\displaystyle \partial M}$ ${\displaystyle \rho }$

generalization

→ Main article : Chern-Simons form

In any dimension one can use Chern-Simons forms to define secondary characteristic classes .

literature

• Freed, Daniel S .: Classical Chern-Simons theory. I .: Adv. Math. 113, no. 2, 237-303 (1995). pdf II .: Houston J. Math. 28, no. 2, 293-310 (2002). pdf

Web links

• Baseilhac: Chern Simons Theory in Dimension Three
• Young: Chern-Simons Theory, Knots and Moduli Spaces of Flat Connections
• Waldorf: Lectures on gerbes, loop spaces, and Chern-Simons theory (discusses the generalization to when there is generally no cut ) ${\displaystyle \pi _{1}G\not =1}$ ${\displaystyle s}$
• Goerner: ptolemy module (software for calculating the Chern-Simons invariants of flat bundles)

Individual evidence

1. ^ Witten, Edward: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, no. 3: 351-399 (1989). pdf
2. ^ Bar-Natan, Dror: Perturbative Chern-Simons theory. J. Knot Theory Ramifications 4 (1995), no. 4, 503-547. pdf
3. ^ Yoshida, Tomoyoshi: The η-invariant of hyperbolic 3-manifolds. Invent. Math. 81: 473-514 (1985). pdf
4. ^ Neumann, Walter D .: Extended Bloch group and the Cheeger-Chern-Simons class. Geom. Topol. 8: 413-474 (2004). pdf
5. ↑ Goette, Sebastian; Zickert, Christian K .: The extended Bloch group and the Cheeger-Chern-Simons class. Geom. Topol. 11: 1623-1635 (2007). pdf
6. ^ Marché, Julien: Geometric interpretation of simplicial formulas for the Chern-Simons invariant. Algebr. Geom. Topol. 12, No. 2, 805-827 (2012). pdf  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice.  
7. ^ S. Garoufalidis, D. Thurston, C. Zickert: The complex volume of SL (n, C) -representations of 3-manifolds . pdf