From Wikipedia, the free encyclopedia
Secondary characteristic classes of 3-manifolds
In
mathematics , the Chern–Simons forms are certain secondary
characteristic classes .
[1] The theory is named for
Shiing-Shen Chern and
James Harris Simons , co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from which the theory arose.
[2]
Definition
Given a
manifold and a
Lie algebra valued
1-form
A
{\displaystyle \mathbf {A} }
over it, we can define a family of
p -forms :
[3]
In one dimension, the Chern–Simons
1-form is given by
Tr
A
.
{\displaystyle \operatorname {Tr} [\mathbf {A} ].}
In three dimensions, the Chern–Simons 3-form is given by
Tr
F
∧
A
−
1
3
A
∧
A
∧
A
=
Tr
d
A
∧
A
+
2
3
A
∧
A
∧
A
.
{\displaystyle \operatorname {Tr} \left[\mathbf {F} \wedge \mathbf {A} -{\frac {1}{3}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right]=\operatorname {Tr} \left[d\mathbf {A} \wedge \mathbf {A} +{\frac {2}{3}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right].}
In five dimensions, the Chern–Simons 5-form is given by
Tr
F
∧
F
∧
A
−
1
2
F
∧
A
∧
A
∧
A
+
1
10
A
∧
A
∧
A
∧
A
∧
A
=
Tr
d
A
∧
d
A
∧
A
+
3
2
d
A
∧
A
∧
A
∧
A
+
3
5
A
∧
A
∧
A
∧
A
∧
A
{\displaystyle {\begin{aligned}&\operatorname {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]\\[6pt]={}&\operatorname {Tr} \left[d\mathbf {A} \wedge d\mathbf {A} \wedge \mathbf {A} +{\frac {3}{2}}d\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} +{\frac {3}{5}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right]\end{aligned}}}
where the curvature F is defined as
F
=
d
A
+
A
∧
A
.
{\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} .}
The general Chern–Simons form
ω
2
k
−
1
{\displaystyle \omega _{2k-1}}
is defined in such a way that
d
ω
2
k
−
1
=
Tr
(
F
k
)
,
{\displaystyle d\omega _{2k-1}=\operatorname {Tr} (F^{k}),}
where the
wedge product is used to define Fk . The right-hand side of this equation is proportional to the k -th
Chern character of the connection
A
{\displaystyle \mathbf {A} }
.
In general, the Chern–Simons
p -form is defined for any odd p .
[4]
Application to physics
In 1978,
Albert Schwarz formulated
Chern–Simons theory , early
topological quantum field theory , using Chern-Simons forms.
[5]
In the
gauge theory , the
integral of Chern-Simons form is a global geometric invariant, and is typically
gauge invariant modulo addition of an integer.
See also
References
^ Freed, Daniel (January 15, 2009).
"Remarks on Chern–Simons theory" (PDF) . Retrieved April 1, 2020 .
^ Chern, Shiing-Shen; Tian, G.; Li, Peter (1996).
A Mathematician and His Mathematical Work: Selected Papers of S.S. Chern . World Scientific.
ISBN
978-981-02-2385-4 .
^
"Chern-Simons form in nLab" . ncatlab.org . Retrieved May 1, 2020 .
^ Moore, Greg (June 7, 2019).
"Introduction To Chern-Simons Theories" (PDF) . University of Texas . Retrieved June 7, 2019 .
^ Schwartz, A. S. (1978). "The partition function of degenerate quadratic functional and Ray-Singer invariants". Letters in Mathematical Physics . 2 (3): 247–252.
Bibcode :
1978LMaPh...2..247S .
doi :
10.1007/BF00406412 .
S2CID
123231019 .
Further reading