nLab
Todd class

Contents

Contents

Idea

A characteristic class.

By the Hirzebruch-Riemann-Roch theorem the index of the Dolbeault operator is the Todd genus (e.g. Gilkey 95, section 5.2 (more generally so for the Spin^c Dirac operator).

Definition

The characteristic series of the Todd genus is

xx1e x. x \,\mapsto\, \frac{x}{ 1 - e^{-x} } \,.

This means that as a formal power series in Chern classes c ic_i the Todd class starts out as

td=1+12c 1+112(c 1 2+c 2)+124c 1c 2+𝒪(deg8). td \;=\; 1 + \tfrac{1}{2} c_1 + \tfrac{1}{12} \big( c_1^2 + c_2 \big) + \tfrac{1}{24} c_1 c_2 + \mathcal{O}(deg \geq 8) \,.

Properties

Relation between Todd class and A^\hat A-genus

On an almost complex manifold M UM_{\mathrm{U}}, the Todd class coincides with the A-hat class up to the exponential of half the first Chern class:

Td(M U)=(e c 1/2A^)(M U). Td(M_{\mathrm{U}}) \;=\; \big(e^{c_1/2} \hat A\big)(M_{\mathrm{U}}) \,.

(e.g. Freed 87 (1.1.14)).

In particular, on manifolds M SUM_{S\mathrm{U}} with SU-structure, where c 1=0c_1 = 0, the Todd class is actually equal to the A-hat class:

Td(M SU)=A^(M SU). Td(M_{S\mathrm{U}}) \;=\; \hat A(M_{S\mathrm{U}}) \,.

Relation to Thom class and Chern character

Proposition

(rational Todd class is Chern character of Thom class)

Let VXV \to X be a complex vector bundle over a compact topological space. Then the Todd class Td(V)H ev(X;)Td(V) \,\in\, H^{ev}(X; \mathbb{Q}) of VV in rational cohomology equals the Chern character chch of the Thom class th(V)K(Th(V))th(V) \,\in\, K\big( Th(V) \big) in the complex topological K-theory of the Thom space Th(V)Th(V), when both are compared via the Thom isomorphisms ϕ E:E(X)E(Th(V))\phi_E \;\colon\; E(X) \overset{\simeq}{\to} E\big( Th(V)\big):

ϕ H(Td(V))=ch(th(V)). \phi_{H\mathbb{Q}} \big( Td(V) \big) \;=\; ch\big( th(V) \big) \,.

More generally , for xK(X)x \in K(X) any class, we have

ϕ H(ch(x)Td(V))=ch(ϕ K(x)), \phi_{H\mathbb{Q}} \big( ch(x) \cup Td(V) \big) \;=\; ch\big( \phi_{K}(x) \big) \,,

which specializes to the previous statement for x=1x = 1.

(Karoubi 78, Chapter V, Theorem 4.4)

Relation to the Adams e-invariant

We discuss how the e-invariant in its Q/Z-incarnation (this Def.) has a natural formulation in cobordism theory (Conner-Floyd 66), by evaluating Todd classes on cobounding (U,fr)-manifolds.

This is Prop. below; but first to recall some background:

Remark

In generalization to how the U-bordism ring Ω 2k U\Omega^U_{2k} is represented by homotopy classes of maps into the Thom spectrum MU, so the (U,fr)-bordism ring Ω 2k U,fr\Omega^{U,fr}_{2k} is represented by maps into the quotient spaces MU 2k/S 2kMU_{2k}/S^{2k} (for S 2k=Th( k)Th( k× U(k)EU(k))=MU 2kS^{2k} = Th(\mathbb{C}^{k}) \to Th( \mathbb{C}^k \times_{U(k)} E U(k) ) = MU_{2k} the canonical inclusion):

(1)Ω (U,fr)=π +2k(MU 2k/S 2k),for any2k+2. \Omega^{(U,fr)}_\bullet \;=\; \pi_{\bullet + 2k} \big( MU_{2k}/S^{2k} \big) \,, \;\;\;\;\; \text{for any} \; 2k \geq \bullet + 2 \,.

(Conner-Floyd 66, p. 97)

Remark

The bordism rings for MU, MUFr and MFr sit in a short exact sequence of the form

(2)0Ω +1 UiΩ +1 U,fΩ fr0, 0 \to \Omega^U_{\bullet+1} \overset{i}{\longrightarrow} \Omega^{U,f}_{\bullet+1} \overset{\partial}{ \longrightarrow } \Omega^{fr}_\bullet \to 0 \,,

where ii is the evident inclusion, while \partial is restriction to the boundary.

In particular, this means that \partial is surjective, hence that every FrFr-manifold is the boundary of a (U,fr)-manifold.

Proposition

(e-invariant is Todd class of cobounding (U,fr)-manifold)

Evaluation of the Todd class on (U,fr)-manifolds yields rational numbers which are integers on actual UU-manifolds. It follows with the short exact sequence (2) that assigning to FrFr-manifolds the Todd class of any of their cobounding (U,fr)(U,fr)-manifolds yields a well-defined element in Q/Z.

Under the Pontrjagin-Thom isomorphism between the framed bordism ring and the stable homotopy group of spheres π s\pi^s_\bullet, this assignment coincides with the Adams e-invariant in its Q/Z-incarnation:

(3)0 Ω +1 U i Ω +1 U,f Ω fr π s Td Td e 0 / = /, \array{ 0 \to & \Omega^U_{\bullet+1} & \overset{i}{\longrightarrow} & \Omega^{U,f}_{\bullet+1} & \overset{\partial}{ \longrightarrow } & \Omega^{fr}_\bullet & \simeq & \pi^s_\bullet \\ & \big\downarrow{}^{\mathrlap{Td}} && \big\downarrow{}^{\mathrlap{Td}} && \big\downarrow{}^{} && \big\downarrow{}^{e} \\ 0 \to & \mathbb{Z} &\overset{\;\;\;\;\;}{\hookrightarrow}& \mathbb{Q} &\overset{\;\;\;\;}{\longrightarrow}& \mathbb{Q}/\mathbb{Z} &=& \mathbb{Q}/\mathbb{Z} } \,,

(Conner-Floyd 66, Theorem 16.2)

partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

ddpartition function in dd-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpinKOM Spin \to KO
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge
3w4-orientation of EO(2)-theory

References

Named after John Arthur Todd.

Original articles:

Review:

On the Todd character:

Review:

  • Peter Gilkey, Section 5.2 of: Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem, 1995 (pdf)

See also:

Discussion of Todd classes for noncommutative topology/in KK-theory is in

Review of the Todd genus with an eye towards generalization to the Witten genus is in the introduction of

Last revised on February 22, 2021 at 07:39:27. See the history of this page for a list of all contributions to it.