nLab secondary characteristic class

Idea

In its most refined form, a secondary characteristic class is a characteristic class in ordinary differential cohomology. The term “secondary” refers to the fact that such a differential cohomology class in degree $n$ not only encodes a degree-$n$ class in integral cohomology, but in addition higher connection data in degree $(n+1)$: the data of a circle n-bundle with connection.

The refined Chern-Weil homomorphism takes values in such “secondary characteristic classes”.

But the precise meaning of the term secondary characteristic class varies a little in the literature, as follows. Historically it was first understood in more restricted senses.

1. In the strict sense of the word, a secondary characteristic class is a characteristic of a situation where an ordinary characteristic class vanishes (PetersonStein1962).

2. More specifically, a special case of this situation in differential geometry arises where the characteristic class is represented in de Rham cohomology by a curvature characteristic form. If that curvature form happens to vanish, the corresponding Chern-Simons form itself becomes closed, and now itself represents a cohomology class, in one degree lower. This is often called the corresponding Chern-Simons secondary characteristic class . Sometimes the term “secondary geometric invariants” is used for Chern-Simons forms (see for instance the review (FreedII)).

3. Using refined Chern-Weil theory the notions of curvature characteristic forms and their Chern-Simons forms are unified into the notion of cocycles in ordinary differential cohomology. The notion of Cheeger-Simons differential character was introduced to describe this unification, and it is has become tradition to call these differential characters themselves secondary characteristic classes independently of whether the corresponding ordinary characteristic class/curvature characteristic form vanishes or not (for instance (DupontKamber, Karlsson). More descriptively, this case is maybe better referred to as a differential characteristic class . See there for more details.

Higher order invariants and boundary field theory

The statement that secondary invariants are indeed secondary to primary invariants can be formalized in the language of prequantum boundary field theory by saying that (higher) topological Yang-Mills theory (which is controled by differential Chern classes) has as boundary field theory (higher) Chern-Simons theory.

As explained at boundary field theory, this statement is reflected by the existence of a universal correspondence in the slice (∞,1)-topos $\mathbf{H}_{/\flat \mathbf{B}^{n+1}U(1)}$, where $\mathbf{H}$ is Smooth∞Grpd, $\flat$ is the flat modality and $\mathbf{B}^{n+1}U(1)$ is the circle (n+2)-group:

$$\array{ && X \\ & \swarrow &\downarrow^\nabla& \searrow^{\mathrlap{F_{(\nabla)}}} \\ \ast &\leftarrow& \mathbf{B}^n U(1)_{conn} &\stackrel{F_{(-)}}{\to}& \Omega^{n+1}_{cl} \\ & \searrow &\swArrow& \swarrow_{\mathrlap{}} \\ && \flat \mathbf{B}^{n+1}U(1) } \,,$$

where the bottom right map is the canonical one in the context of differential cohesion, the outer diagram is any specified one and the factorization through the

This exhibits a secondary invariant $\nabla$ (a cocycle in ordinary differential cohomology) of degree (n+1) as a boundary condition for the theory of closed characteristic forms.

In this picture, one obtains further “higher order invariants” by successively further transgrssing and forming higher order boundaries.

If we view here, as we may $\nabla$ as the local Lagrangian of an infinity-Chern-Simons theory, then the ternary invariants are WZW terms; the quaternary invariants are Wilson loop terms. (lpqft)

Related concepts

• Chern-Simons invariant

• Cheeger-Simons class, complex volume

References

The notion in its general cohomological sense appears in

• Franklin P. Peterson, Norman Stein, Secondary characteristic classes, Annals of mathematics, Vol 76, No. 3 (1962)

The notion of Chern-Simons forms originates in

• Shiing-shen Chern, James Simons, Characteristic forms and geometric invariants, The Annals of Mathematics, Second Series, Vol. 99, No. 1 (1974) (jstor).

The special meaning in the context of Chern-Weil theory in differential geometry was established by the introduction of Cheeger-Simons differential character in

• Jeff Cheeger, James Simons, Differential characters and geometric invariants, pages 50–80 in: Geometry and Topology, Lecture Notes in Mathematics, vol 1167. Springer (1985) (pdf, doi:10.1007/BFb0075212)

Review:

• Dan Freed, Classical Chern-Simons theory II (pdf)

• Johan Dupont, Franz Kamber, Gerbes, Simplicial forms and Invariants for Families of Foliated Bundles (pdf)

• Michelle Karlsson, Characteristic classes and bounded cohomology (pdf)

Lecture notes on secondary cohomology classes in differential cohomology for flat connections is presented in

• Ulrich Bunke, Differential cohomology (arXiv:1208.3961)

Discussion in complex algebraic geometry is in

• Spencer Bloch, Hélène Esnault, Algebraic Chern-Simons theory, American Journal of Mathematics, Volume 119, Number 4, August 1997, pp. 903-952 (pdf)

Discussion of secondary classes being torsion elements:

• Alexander Reznikov, All regulators of flat bundles are torsion, Annals of Mathematics Second Series, Vol. 141, No. 2 (Mar., 1995), pp. 373-386 (14 pages) (arXiv:dg-ga/9407006, jstor:2118525)

• Alexander Reznikov, Rationality of secondary classes, J. Differential Geom., Volume 43, Number 3 (1996), 674-692. (arXiv:dg-ga/9407007, euclid:jdg/1214458328)

• Jaya N. Iyer, Carlos Simpson, Regulators of canonical extensions are torsion: the smooth divisor case (arXiv:0707.0372)

• Hélène Esnault, Algebraic Differential Characters of Flat Connections with Nilpotent Residues, In: Baas N., Friedlander E., Jahren B., Østvær P. (eds.) Algebraic Topology, Abel Symposia, vol 4. Springer 2009 (doi:10.1007/978-3-642-01200-6_5)

Discussion of secondary characteristic classes in string cobordism cohomology theory and in tmf:

• Ulrich Bunke, Niko Naumann, Secondary Invariants for String Bordism and tmf, Bulletin des Sciences Mathématiques, Volume 138, Issue 8, 2014, Pages 912-970

  (arXiv:0912.4875, doi:10.1016/j.bulsci.2014.05.002)

Discussion of secondary and higher order invariant as higher order boundary field theories to higher topological Yang-Mills prequantum field theory is in

• Urs Schreiber et al., Local prequantum field theory

Specifically the secondary Chern-Simons and quaternary WZW invariants in this context are discussed in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, A higher stacky perspective on Chern-Simons theory, in Damien Calaque et al. (eds.)

  Mathematical Aspects of Quantum Field Theories, Springer 2014