For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
for higher abelian targets
for symplectic Lie n-algebroid targets
FQFT and cohomology
Types of quantum field thories
There are two kinds of higher dimensional generalizations of ordinary 3-dimensional Chern-Simons theory that are often called “higher dimensional Chern-Simons theory” in the literature. Both are special cases of infinity-Chern-Simons theory.
Recall that for a Lie algebra (not necessarily abelian) with non-generate binary invariant polynomial , the corresponding infinity-Chern-Simons theory QFT is ordinary Chern-Simons theory in dimension 3.
More general “higher”-generalization of Chern-Simons theory to infinity-Chern-Simons theory allow to be a (nonabelian) Lie 2-algebra or more generally a (nonabelian) L-infinity algebra or fully generally a L-infinity algebroid.
The definition of higher abelian Chern-Simons theory is simple locally when certain global cohomological effects can be ignored. We first give the simple local definition and then the full global definition.
Then the simple version of abelian -dimensional Chern-Simons theory is defined as follows.
the Lagrangian is
and the action functional
Notice that generally for an -form on a closed -dimensional manifold we have
by first using integration by parts and then switching the order of the wedge factors. Therefore this kind of action vanishes identically when is even. This is the reason for the above assumption that for and hence that the Chern-Simons theory is in dimension .
In the full theory instead the configuration space is
the space of circle (2k+1)-bundles with connection (given by cocycles in degree ordinary differential cohomology). This contains the above simplified configuration space as the subspace of -connections whose underlying circle -bundle is trivial.
The action functional is given by
where now the integral is fiber integration in ordinary differential cohomology and in the integrand we have the cup product in ordinary differential cohomology of differential cocycles.
We discuss how the above definition arises as a special case of the general notion of infinity-Chern-Simons theory.
These theories are defined by
The abelian higher dimensional Chern-Simons theories in dimension are the special case of this general situation where
is the canonical quadratic invariant polynomial on this.
See (FRS, 4.1.4).
Higher dimensional abelian Chern-Simons theories appear automatically as components of systems of higher supergravity, for instance in 11-dimensional supergravity (they are automatically induced by the requirement of local supersymmetry in these higher dimensional supergravity theories).
: 7-dimensional Chern-Simons theory is related to a fivebrane model on its boundary;
The supergravity C-field is an example of a general phenomenon of higher abelian Chern-Simons QFTs in the presence of background charge. This phenomenon was originally noticed in (Witten) and then made precise in (HopkinsSinger 05). The holographic dual of this phenomenon is that of self-dual higher gauge theories, which for the supergravity -field is the 2-form theory on the M5-brane – see there for a discussion of this example. Here we discuss this effect generally, for higher abelian Chern-Simons theory in arbitrary dimension .
Fix some natural number and an oriented manifold (compact with boundary) of dimension . The gauge equivalence class of a -form gauge field on is an element in the ordinary differential cohomology group . The cup product of this with itself has a natural higher holonomy over , denoted
This is the exponentiated action functional for bare -dimensional abelian Chern-Simons theory, as discussed above.
Observe now that the above action functional may be regarded as a quadratic form on the cohomology group . The corresponding bilinear form is the (“secondary”, since is of dimension instead of ) intersection pairing
But note that from we do not obtain a quadratic refinement
of the pairing. A quadratic refinement is, by definition, a function
(not necessarily homogenous of degree 2 as is), such that the intersection pairing is reobtained from it by the polarization formula
If we took , then the above formula would yield not , but the square , given by (the exponentiation of) twice the integral.
But since the differential classes in refine integral cohomology, we cannot in general simply divide by 2 and pass from to . The integrand in the latter expression does not make sense in general in differential cohomology. If one tried to write it out in the “obvious” local formulas one would find that it is a functional on fields which is not gauge invariant. The analog of this fact is familiar from nonabelian -Chern-Simons theory with simply connected , where also the theory is consistent only at interger levels. The “level” here is nothing but the underlying integral class .
Therefore the only way to obtain a square root of the quadratic form is to shift its origin. Here we think of the analogy with a quadratic form on the real numbers (a parabola in the plane). Replacing this by for some real number means keeping the shape of the form, but shifting its minimum from 0 to . If we think of this as the potential term for a scalar field previously with rotation-symmetric dynamics about , then the new potential exhibits spontaneous symmetry breaking: its ground state is now at (and has energy there). We may say that there is a background field or background charge that pushes the field out of its free equilibrium.
To lift this reasoning to our action quadratic form on differential cocycles, we need a differential class such that for every the composite class
is even, hence is divisible by 2. Because then we could define a shifted action functional
The condition on the existence of here means equivalently that the image of the underlying integral class in cohomology with coefficients in vanishes:
Precisely such a class does uniquely exist on every oriented manifold. It is called the Wu class , and may be defined by this condition. Moreover, if is a Spin-manifold, then every second Wu class, , has a pre-image in integral cohomology, hence does exist as required above
This was the original observation in Witten96, around (3.3).
Notice that the equations of motion of the shifted action are no longer , but are now . Comparing this to the Maxwell equations, we see that here plays the role of a background charge (or rather, of the background current that underlies a background charge). We therefore think of as the exponentiated action functional for higher dimensional abelian Chern-Simons theory with background charge .
This of course only makes sense if is such that is further divisible by 2, which we will assume now. In (Hopkins-Singer 05) is discussed a way to make sense of the further division in general if one passes to a certain notion of twisted differential cohomology. One can also adopt a different perspective and interpret the condition that is further divisible by 2 precisely as a -structure (SSS3). This is a higher analog of a Spin^c structure.
With respect to the shifted action functional it makes sense to introduce the shifted field
This is simply a re-parameterization such that the Chern-Simons equations of motion again look homogenous, namely . In terms of this shifted field the action from above equivalently reads
This is the form of the action functional first given as (Witten96 (3.6)) in for the case .
In the language of twisted differential c-structures, we may summarize this sitation as follows:
in order for the action functional of higher abelian Chern-Simons theory to be correctly divisible, the images of the fields in -cohomology need to form a twisted Wu structure. Therefore the fields themselves need to constitute a twisted -structure. For this is a twisted string structure and explains for instance the quantization condition on the supergravity C-field in 11-dimensional supergravity. For that case see also the corresponding discussion at M5-brane.
Chern-Simons actions for Lie algebras but with higher-degree invariant polynomials have in particular received attention for the super Poincare Lie algebra. In this case these action functionals can be regarded as defining higher Chern-Simons supergravity.
For higher dimensional non-abelian Chern-Simons theory see for instance
Máximo Bañados, Higher dimensional Chern-Simons theories and black holes (pdf)
G Giachetta, L Mangiarotti, G Sardanashvily, Noether conservation laws in higher-dimensional Chern-Simons theory Modern Physics Letters A Volume 18(2003) (pdf)
For the formulation of abelian Chern-Simons theory by fiber integration over cup products in ordinary differential cohomology see
Enore Guadagnini, Frank Thuillier, Deligne-Beilinson cohomology and abelian links invariants SIGMA 4 (2008), 078, 30 (arXiv:0801.1445)
R. Floreanini, R. Percacci, Higher dimensional abelian Chern-Simons theory Physics Letters B Volume 224, Issue 3, 29 (1989)
The idea of describing self-dual higher gauge theory by abelian Chern-Simons theory in one dimension higher originates in
Motivated by this the differential cohomology of self-dual fields had been discussed in
More discussion of the general principle is in
An original articles includes
An introduction and survey is in
For more references see at
F. Thuillier, Deligne-Beilinson cohomology and abelian link invariants: torsion case J. Math. Phys. 50, 122301 (2009)
B. Broda, Higher-dimensional Chern-Simons theory and link invariants Physics Letters B Volume 280, Issues 3-4, 30 (1992) Pages 213-218
L. Gallot, E. Pilon, F. Thuillier, Higher dimensional abelian Chern-Simons theories and their link invariants (arXiv:1207.1270)
Below (5.11) of (Hopkins-Singer 05).
Section 4.1.4 of
Section 4.4 of
J. Gegenberg , G. Kunstatter, Boundary Dynamics of Higher Dimensional Chern-Simons Gravity (arXiv:hep-th/0010020)
J. Gegenberg , G. Kunstatter, Boundary Dynamics of Higher Dimensional AdS Spacetime (arXiv:http://arxiv.org/abs/hep-th/9905228)