spin geometry

string geometry

# Contents

## Idea

The 0-mode of the supercharge in a 2d SCFT (an operator in a (sheaf of) super vertex operator algebra) behaves like a higher dimensional analog of the operator $D$ in a spectral triple, hence the supercharge in supersymmetric quantum mechanics. Specifically for a sigma-model 2d SCFT induced from some target space geometry it is a higher version of a Dirac operator on that space (roughtly like what one would expect of a Dirac operator on a smooth loop space). This is called the Dirac-Ramond operator (after Pierre Ramond).

The index of the large volume limit of the Dirac-Ramond operator is what is now known as the Witten genus (but in fact the original article (Alvarez-Killingback-Mangano-Windey 87 ) appeared independently almost in parallel of Witten’s discussion).

$d$partition function in $d$-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 $M Spin \to KO$
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 $M Spin^c \to KU$
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin^c \to KU$
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

The Dirac-Ramond operator ofriginates with the early beginning of superstring models, when they were still called spinning strings. See there for references.

The Dirac-Ramond operator gained more attention in pure mathemaitcs when it was found that the large volume limit of its index, when properly construed, is a universal elliptic genus, now known as the Witten genus. See there for more references.

Articles that explicitly consider the Dirac-Ramon operator in this context include

• Orlando Alvarez, T. P. Killingback, Michelangelo Mangano, and Paul Windey, The Dirac-Ramond operator in string theory and loop space index theorems, Nuclear Phys. B Proc. Suppl., 1A:189–215, 1987. Nonperturbative methods in field theory (Irvine, CA, 1987)., also: Comm. Math. Phys. Volume 111, Number 1 (1987), 1-160 (Ecudid)

• Gregory Landweber, Dirac operators on loop space PhD thesis (Harvard 1999) (pdf)

• Orlando Alvarez, Paul Windey, Analytic index for a family of Dirac-Ramond operators, Proc. Natl. Acad. Sci. USA, 107(11):4845–4850, 2010

Last revised on March 26, 2014 at 09:02:28. See the history of this page for a list of all contributions to it.