nLab Todd class

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

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

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

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

Properties

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

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

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

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

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

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

Relation to Thom class and Chern character

Proposition

(rational Todd class is Chern character of Thom class)

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

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

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

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

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

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

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

(Conner-Floyd 66, p. 97)

Remark

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

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

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

In particular, this means that $\partial$ is surjective, hence that every $Fr$-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 $U$-manifolds. It follows with the short exact sequence (2) that assigning to $Fr$-manifolds the Todd class of any of their cobounding $(U,fr)$-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 $\pi^s_\bullet$, this assignment coincides with the Adams e-invariant in its Q/Z-incarnation:

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

(Conner-Floyd 66, Theorem 16.2)

Related concepts

• genus

• Grothendieck-Riemann-Roch theorem

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

$d$	partition function in $d$-dimensional QFT	supercharge	index in cohomology theory	genus	logarithmic coefficients of Hirzebruch series
0			push-forward in ordinary cohomology: integration of differential forms			orientation
1	spinning particle	Dirac operator	KO-theory index	A-hat genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin \to KO$
	endpoint of 2d Poisson-Chern-Simons theory string	Spin^c Dirac operator twisted by prequantum line bundle	space of quantum states of boundary phase space/Poisson manifold	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
	endpoint of type II superstring	Spin^c Dirac operator twisted by Chan-Paton gauge field	D-brane charge	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2	type II superstring	Dirac-Ramond operator	superstring partition function in NS-R sector	Ochanine elliptic genus		SO orientation of elliptic cohomology
	heterotic superstring	Dirac-Ramond operator	superstring partition function	Witten genus	Eisenstein series	string orientation of tmf
	self-dual string		M5-brane charge
3						w4-orientation of EO(2)-theory

References

Named after John Arthur Todd.

Original articles:

• Friedrich Hirzebruch, Section 3 of: Neue topologische Methoden in der Algebraischen Geometrie, Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge, Springer 1956 (doi:10.1007/978-3-662-41083-7)

  English translation: Section 3 of Topological Methods in Algebraic Topology, Classics in Mathematics, vol 131. Springer 1995 (doi:10.1007/978-3-642-62018-8_4)

• Pierre Conner, Edwin Floyd, Sections 12, 13 of: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)

• Max Karoubi, Chapter V.4 of: K-Theory – An introduction, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 (pdf, doi:10.1007%2F978-3-540-79890-3)

Review:

• Daniel Freed, Sections 1.1, 1.2 of: Geometry of Dirac operators, 1987 (pdf, pdf)

On the Todd character:

• Pierre Conner, Larry Smith, Section 7 of: On the complex bordism of finite complexes. II, J. Differential Geom. Volume 6, Number 2 (1971), 135-174 (euclid:euclid.ijm/1256051760)

• Larry Smith, Section 1 of: The Todd character and the integrality theorem for the Chern character, Illinois J. Math. Volume 17, Issue 2 (1973), 301-310 (euclid:ijm/1256051760)

• Larry Smith, The Todd character and cohomology operations, Advances in Mathematics Volume 11, Issue 1, August 1973, Pages 72-92 (doi:10.1016/0001-8708(73)90003-0)

Review:

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

See also:

• Wikipedia, Todd class

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

• Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard Szabo, D-Branes, RR-Fields and Duality on Noncommutative Manifolds, Commun. Math. Phys. 277:643-706,2008 (arXiv:hep-th/0607020)

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

• Pokman Cheung, The Witten genus and vertex algebras (arXiv:0811.1418)