superalgebra and (synthetic ) supergeometry
Perturbative string theory is defined in terms of certain classes of 2d CFTs. Depending on which class that is, one speaks of different types of string theory.
In type II string theory the CFTs in question are $(1,1)$-supersymmetric and defined on oriented worldsheets;
In heterotic string theory the CFTs in question are $(0,1)$-supersymmetric and defined on oriented worldsheets;
In …
The effective quantum field theory of type II string theory containts –besides type II supergravity – the self-dual higher gauge theory of RR-fields and Kalb-Ramond fields.
Apart from the Weyl anomaly, which cancels for 10-dimensional target spaces, the action functional of the string-sigma-model also in general has an anomalous action functional , for two reasons:
The higher holonomy of the higher background gauge fields is in general not a function, but a section of a line bundle;
The fermionic path integral over the worldsheet-spinors of the superstring produces as section of a Pfaffian line bundle.
In order for the action functional to be well-defined, the tensor product of these different anomaly line bundles over the bosonic configuration space must have trivial class (as bundles with connection, even). This gives rise to various further anomaly cancellation conditions:
For the open type II string the condition is known as the Freed-Witten anomaly cancellation condition: it says that the restriction of the B-field to any D-brane must consistute the twist of a twisted spin^c structure on the brane.
A more detailed analysis of these type II anomalies is in (DFMI) and (DFMII).
cohomology theories of string theory fields on orientifolds
string theory | B-field | $B$-field moduli | RR-field |
---|---|---|---|
bosonic string | line 2-bundle | ordinary cohomology $H\mathbb{Z}^3$ | |
type II superstring | super line 2-bundle | $Pic(KU)//\mathbb{Z}_2$ | KR-theory $KR^\bullet$ |
type IIA superstring | super line 2-bundle | $B GL_1(KU)$ | KU-theory $KU^1$ |
type IIB superstring | super line 2-bundle | $B GL_1(KU)$ | KU-theory $KU^0$ |
type I superstring | super line 2-bundle | $Pic(KU)//\mathbb{Z}_2$ | KO-theory $KO$ |
type $\tilde I$ superstring | super line 2-bundle | $Pic(KU)//\mathbb{Z}_2$ | KSC-theory $KSC$ |
See F-theory.
By a holographic principle realized in this case as AdS/CFT correspondence (see the references there), type II string theory is supposed to be dual to 4-dimensional super Yang-Mills theory.
partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:
Physics textbook accounts include
Joseph Polchinski, String theory, volume II, section 10
Eric D'Hoker, String theory – lecture 7: Free superstrings , in part 3 of
Pierre Deligne, Pavel Etingof, Dan Freed, L. Jeffrey, David Kazhdan, John Morgan, D.R. Morrison and Edward Witten, eds. Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)
A comprehensive discussion of the (differential) cohomological nature of general type II/type I orientifold backgrounds is in
with details in
Daniel Freed, Lectures on twisted K-theory and orientifolds (pdf)
Jacques Distler, Dan Freed, Greg Moore, Spin structures and superstrings (arXiv:1007.4581)
Related lecture notes / slides include
Jacques Distler, Orientifolds and Twisted KR-Theory (2008) (pdf)
Daniel Freed, Dirac charge quantiation, K-theory, and orientifolds, talk at a workshop Mathematical methods in general relativity and quantum field theories, November, 2009 (pdf)
Greg Moore, The RR-charge of an orientifold (ppt)
Discussion of type II quantum anomalies is in
An exposition is at
Description of type II backgrounds in terms of generalized complex geometry/Courant Lie 2-algebroids is in
A holographic description of type II by higher dimensional Chern-Simons theory is discussed in