nLab type II string theory



String theory




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.

The Type II string theory referred to here is defined on spaces in Lorentzian geometry. It is thought that Type II (and the other string theories) can also be defined in spaces with other metric signatures, such as type II* string theory. See non-Lorentzian type II string theory.


Effective QFT

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.

Anomaly cancellation

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:

  1. The higher holonomy of the higher background gauge fields is in general not a function, but a section of a line bundle;

  2. 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).

Background fields and orientifolding

cohomology theories of string theory fields on orientifolds

string theoryB-fieldBB-field moduliRR-field
bosonic stringline 2-bundleordinary cohomology H 3H\mathbb{Z}^3
type II superstringsuper line 2-bundlePic(KU)// 2Pic(KU)//\mathbb{Z}_2KR-theory KR KR^\bullet
type IIA superstringsuper line 2-bundleBGL 1(KU)B GL_1(KU)KU-theory KU 1KU^1
type IIB superstringsuper line 2-bundleBGL 1(KU)B GL_1(KU)KU-theory KU 0KU^0
type I superstringsuper line 2-bundlePic(KU)// 2Pic(KU)//\mathbb{Z}_2KO-theory KOKO
type I˜\tilde I superstringsuper line 2-bundlePic(KU)// 2Pic(KU)//\mathbb{Z}_2KSC-theory KSCKSC


Some dualities in string theory involving type II string theory:

Duality with heterotic string theory

See duality between type II and heterotic string theory.

Duality with M-theory

See duality between type IIA string theory and M-theory.

Duality with F-theory

See F-theory.

Holographic dual

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 function and elliptic genus

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



Physics textbook accounts include

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

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)

(p,q)(p,q)-Strings and S-duality

Quantum anomalies

Discussion of type II quantum anomalies is in

An exposition is at

  • Dan Freed, Lectures on K-theory and orientifolds (2012) (pdf)

Classical solutions / vacua

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

Last revised on November 1, 2023 at 22:47:09. See the history of this page for a list of all contributions to it.