Todd class



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).

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


Named after John Arthur Todd

Reviews include

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

  • Wikipedia, Todd class

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 March 21, 2014 at 08:36:27. See the history of this page for a list of all contributions to it.