Chern Characteristic Classes – Canonical Reference

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I'm a little uncertain about the definitions for

  1. Chern roots

  2. Chern classes

  3. Chern characters

From perusing several discussions, I gather that if one correlates the nomenclature with that of symmetric polynomials/functions and their relationships to characteristic polynomials of a generic square matrix $R$ of rank $n$

$$det[\lambda I – R] = (-1)^n (r_1 \cdot r_2 \cdots r_n)\prod_{k=1}^n (1-\frac{\lambda}{r_k})$$

$$ = (-1)^n e_n(r_1,..,r_n) [ 1 – e_1(1/r_1,..,1/r_n)\lambda + e_2(1/r_1,..) \lambda^2 – … + e_n(1/r_1,..,1/r_n) \lambda^n]$$

$$= (-1)^n e_n(r_1,..,r_n) E(-\lambda)$$

$$ = (-1)^n e_n(r_1,..,r_n) \exp[\ln[\prod_{k=1}^n (1-\frac{\lambda}{r_k})]]$$

$$= (-1)^n e_n(r_1,..,r_n) \exp[-\sum_{j \geq 1}(\frac{1}{r_1^j}+\frac{1}{r_2^j}+ … +\frac{1}{r_k^j}) \frac{\lambda^j}{j}],$$

$$ = (-1)^n e_n(r_1,..,r_n) \exp[\sum_{j \geq 1} -p_j\frac{\lambda^j}{j}]$$

$$ = (-1)^n e_n(r_1,..,r_n) \exp[-\sum_{j \geq 1} trace(R^{-j})\frac{\lambda^j}{j}],$$

then

A) Chern roots $r_k$ correspond to the zeros of the characteristic polynomial, the eigenvalues $r_k$ of $ R$;

B) Chern classes $c_k$ correspond to the elementary symmetric polynomials $e_k$ that are the coefficients of the characteristic polynomials in terms of the reciprocals of the Chern roots;

$$c_k = e_k(1/r_1,..,1/r_n),$$

for example,

$$c_1 = e_1(1/r_1,…,1/r_n) = 1/r_1+1/r_2+ … + 1/r_n,$$

the trace of $R^{-1}$, and

$$c_n = e_n(1/r_1,..,1/r_n) =1 / (r_1 \cdot r_2 \cdot … r_n),$$

the determinant of $R^{-1}$,

with the total Chern class polynomial equal to $E(-\lambda)$ and the total Chern class, to $E(-1)$;

C) Chern characters $ch_j$ corresponds to $j!$ times the traces of the powers of $R^{-j}$, i.e., the power sum symmetric polynomials $p_j$ of the reciprocals of the zeros/eigenvalues of $R$; that is

$$j! \cdot ch_j(1/r_1,..,1/r_n)= p_j(1/r_1,..,1/r_n) = \frac{1}{r_1^j}+\frac{1}{r_2^j}+..+\frac{1}{r_n^j}$$

with $ch_0 =$ the dimension of the vector space under consideration.

Question: What is a standard reference defining the Chern classes, characters, and roots through which I can check my understanding and, if necessary, correct any errors and use as a reference in notes on the topic?

Using the Newton/Waring/Girard identites, or the cycle index polynomials for the symmetric groups (OEIS A036039), the elementary symmetric polynomials, or Chern classes, can be expressed in terms of the power sum polynomials, or Chern characters. Conversely, using the Faber polynomials (A263916), the power sum polynomials, or Chern characters, can be obtained from the Chern classes, or elementary symmetric polynomials. For example,

$$3!e_3(a_1,..,a_n) = 2p_3(a_1,..,a_n) -3p_2(a_1,..,a_n)p_1(a_1,..,a_n) + p_1^3(a_1,..,a_n)$$

and

$$p_3(a_1,..,a_n)= 3 e_3(a_1,..,a_n) – 3 e_1(a_1,..,a_n)e_2(a_1,..,a_n) + e_1^3(a_1,..,a_n).$$

In response to those close votes, this MO-Q by Joe Silverman and attendant comments illustrate the need to at least an introduction of Faber polynomials into a discussion of these topics to fill in a gap of knowledge even among experts in related fields of study (e.g., number theory and elliptic curves). See also Understanding a quip from Gian-Carlo Rota

Some motivation: Zanelli asserts that the following examples involve topological invariants called Chern characteristic classes and Chern-Simons forms

  • Sum of exterior angles of a polygon
  • Residue theorem in complex analysis
  • Winding number of a map
  • Poincaré-Hopff theorem (“one cannot comb a sphere”)
  • Atiyah-Singer index theorem
  • Witten index
  • Dirac’s monopole quantization
  • Aharonov-Bohm effect
  • Gauss’ law
  • Bohr-Sommerfeld quantization
  • Soliton/Instanton topologically conserved charges

Update 3/30/2021:

I don't have copies of the books mentioned by Debray. However, Tu, in Appendix B. Invariant Polynomials of "Differential Geometry: Connections, Curvature, and Characteristic Classes," denoting by $X$ a square matrix, states, "This appendix contains results on invariant polynomials
needed in the sections on characteristic classes. We discuss first the distinction between polynomials and polynomial functions. Then we show that a polynomial identity with integer coefficients that holds over the reals holds over any commutative ring with 1. This is followed by the theorem that over the field of real or complex numbers, the ring of invariant polynomials is generated by the coefficients of the characteristic polynomial of -X. Finally, we prove Newton’s
identity relating the elementary symmetric polynomials to the power sums. As a corollary, the ring of invariant polynomials over $R$ or $C$ can also be generated by the
trace polynomials." This presentation of the universality under any commutative ring (with identity) is very appealing to me along with the underlying combinatorics of the yoga of symmetric functions. (The phrase "Chern root" is not used in this textbook. I believe "characteristic polynomial" was coined by Weyl.)

Best Answer

It sounds like, in addition to the references, it would be helpful to disentangle the definitions of Chern roots, Chern classes, and Chern characters. Different mathematicians will have different perspectives; this is mine.

The first thing one defines are Chern classes. Given a complex vector bundle $E\to X$, its $k$th Chern class is a cohomology class $c_k(E)\in H^{2k}(X;\mathbb Z)$. These classes satisfy several nice properties, including:

  1. If $f\colon Y\to X$ is a map, $c_k(f^*E) = f^*c_k(E)$.
  2. The total Chern class $c(E) := c_0(E) + c_1(E) + \dots$ is multiplicative under direct sum: $c(E\oplus F) = c(E)c(F)$.
  3. $c_0(E) = 1$, and $c_k(E) = 0$ if $k > \mathrm{rank}(E)$.

There are several different constructions, but you can think of Chern classes as measuring the extent to which $E$ is nontrivial, or measuring the curvature of a connection for $E$.

A theorem called the splitting principle simplifies some calculations. It tells us that for any complex vector bundle $E\to X$, there is a space $F(E)$ and a map $f\colon F(E)\to X$ such that

  1. $f^*\colon H^*(X; \mathbb Z)\to H^*(F(E); \mathbb Z)$ is injective, and
  2. $f^*E$ is a direct sum of line bundles $L_1,\dotsc,L_r$.

In particular, $$f^*c(E) = \prod_{i=1}^r c(L_i) = \prod_{i=1}^r (1 + c_1(L_i)).$$ The Chern roots of $E$ are $r_i := c_1(L_i)$. One reason to care about them is that, since no information was lost upon pulling back to $F(E)$, one can prove theorems about Chern classes of $E$ by pulling back to $F(E)$ and computing with the Chern roots, which are simpler to manipulate. The sum formula above implies the Chern classes are symmetric functions in the Chern roots.

There are many different perspectives on the Chern character; I'll tell you one that I like. The total Chern class behaves nicely under direct sums, but poorly under tensor products. The (total) Chern character $\mathit{ch}(E)$ is a characteristic class built out of Chern classes which behaves nicely under direct sums and tensor products, in that $\mathit{ch}(E\oplus F) = \mathit{ch}(E) + \mathit{ch}(F)$ and $\mathit{ch}(E\otimes F) = \mathit{ch}(E)\otimes\mathit{ch}(F)$.

The standard reference for Chern classes and Chern roots in differential topology (as opposed to algebraic geometry) is either Bott-Tu, Differential forms in algebraic topology, part 4, or Milnor-Stasheff, Characteristic classes. However, I don't think either discusses the Chern character, and I'm not sure what the default reference is for it.

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