FQFT and cohomology
Types of quantum field thories
The spinning relativistic particle is a variant of the plain relativistic particle which has an “internal degree of freedom” called spin: it is a spinor , a fermion. Examples that appear in the standard model of particle physics are electrons, and quarks.
As a 1-dimensional sigma-model, the spinning relativistic particle is like the relativistic particle but with fermion fields on the worldline. This worldline action always happens to have worldline supersymmetry, entirely independent of whether there is any supersymmetry on target spacetime.
we discuss how spinning particles automatically have supersymmetry in their worldline formalism. See the references below for more. For more on this see also at string theory FAQ – Does string theory predict supersymmetry?.
In the Polyakov action-formulation the action functional of the relativistic particle sigma model on the 1-dimensional worldline is actually 1-dimensional gravity coupled to “worldline matter fields”, where the latter are the embedding fields into the target space.
It turns out that the generalization of this 1-dimensional gravity action to supergravity yields the action functional that describes ordinary Dirac spinors – spinning particles like electrons – propagating on target space . See the references on worldline supersymmetry below.
[D,D] = 2 D^2 = H\,,\;\;\; [H,H] = 0\;,\;\; [D,H] = 0
characterize the translation super-Lie algebra.
For appreciating this fact it is important to keep the ingredients of sigma-model theory sorted out correctly: a supersymmetric theory on the worldline describes a spinning particle on some spacetime coupled to some background gauge fields. That background geometry need not have a “global supersymmetry” (a covariant constant spinor), hence under second quantization the perturbation theory on target space induced by the worldline theory need not have any global supersymmetries (in particular no superpartners to the effective particle excitations). What will happen, though, is that the full target space theory induced under second quantization will be a supergravity theory on target space. Some of its solutions may have covariantly constant spinors (and hence global supersymmetry), but generically they will not, just like the generic solution to ordinary Einstein equations does not have a Killing vector.
|partition function in -dimensional QFT||index/genus in cohomology theory|
|0||push-forward in ordinary cohomology: integration of differential forms|
|1||spinning particle||K-theory index|
|endpoint of 2d Poisson-Chern-Simons theory string||space of quantum states of boundary phase space/Poisson manifold|
|endpoint of type II superstring||D-brane charge|
|2||type II superstring||elliptic genus|
|heterotic string||Witten genus|
Discussion of worldline dynamics of spinning particles in background fields is for instance in
J.W. van Holten, Relativistic Dynamics of Spin in Strong External Fields(arXiv:hep-th/9303124)
A. Pomeranskii, R A Sen’kov, I.B. Khriplovich, Spinning relativistic particles in external fields Acta Physica Polonica B Proceedings Supplement Vol. 1 (2008) (pdf)
F.A. Berezin and M.S. Marinov, Ann. Phys. (NY) 104 (1977), 336
L. Brink, P. Di Vecchia and P. Howe, Nucl. Phys. B118 (1977), 76
R. Casalbuoni, Phys. Lett. B62 (1976), 49
A. Barducci, R. Casalbuoni and L. Lusanna, Nuov. Cim. 35A (1976), 377 Nucl. Phys. B124 (1977), 93; id. 521
An argument that for arbitrary backgrounds the spinning particle’s worldline action is supersymmetric is given in
Textbook surveys of worline supersymmetry include
the beginning of section 14.1.1 in
also p. 194 of
There, first exercise IIIB1.3 gives the action expanded out in all components, and then the following exercise IIIB1.4 gives the reformulation in superfield formalism? which makes manifest that the action is invariant under worldline supersymmetry.