nLab type II string theory

Idea

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 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.

Properties

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 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 $KU^0$
type IIB superstring	super line 2-bundle	$B GL_1(KU)$	KU-theory $KU^1$
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 $KSC$

• intersecting D-brane models

Dualities

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:

$d$	partition function in $d$-dimensional QFT	supercharge	index in cohomology theory	genus	logarithmic coefficients of Hirzebruch series
0			push-forward in ordinary cohomology: integration of differential forms			orientation
1	spinning particle	Dirac operator	KO-theory index	A-hat genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin \to KO$
	endpoint of 2d Poisson-Chern-Simons theory string	Spin^c Dirac operator twisted by prequantum line bundle	space of quantum states of boundary phase space/Poisson manifold	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
	endpoint of type II superstring	Spin^c Dirac operator twisted by Chan-Paton gauge field	D-brane charge	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2	type II superstring	Dirac-Ramond operator	superstring partition function in NS-R sector	Ochanine elliptic genus		SO orientation of elliptic cohomology
	heterotic superstring	Dirac-Ramond operator	superstring partition function	Witten genus	Eisenstein series	string orientation of tmf
	self-dual string		M5-brane charge
3						w4-orientation of EO(2)-theory

Related concepts

• string theory

  • heterotic string theory

    • Green-Schwarz mechanism

    • dual heterotic string theory

    • Witten genus, string orientation of tmf

  • type II string theory

    • massive type IIA string theory

    Diaconescu-Moore-Witten anomaly

    type II geometry

    elliptic genus

    4d super Yang-Mills theory

  • type I string theory

  • type 0 string theory

• landscape of string theory vacua

References

General

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

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv:0906.0795, slides)

with details in

• Daniel Freed, Lectures on twisted K-theory and orientifolds (pdf)

• Jacques Distler, Dan Freed, Greg Moore, Spin structures and superstrings, Surveys in Differential Geometry, Volume 15 (2010) (arXiv:1007.4581, doi:10.4310/SDG.2010.v15.n1.a4)

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)

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

• John Schwarz, opening talk at Strings 2013, (pdf)

Quantum anomalies

Discussion of type II quantum anomalies is in

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv, slides)

• Jacques Distler, Dan Freed, Greg Moore, Spin structures and superstrings (arXiv:1007.4581)

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

• Mariana Grana, Francesco Orsi, $N=1$ vacua in Exceptional Generalized Geometry (arXiv:1105.4855)

Holography

A holographic description of type II by higher dimensional Chern-Simons theory is discussed in

• Dmitriy Belov, Greg Moore, Type II Actions from 11-Dimensional Chern-Simons Theories (arXiv:hep-th/0611020)