nLab integral Morava K-theory

Idea

A localization of complex bordism that maps to ordinary Morava K-theory.

Definition

For instance (Kriz-Sati 04, p. 41/385, Buhné, def. 4.2.4).

Properties

Orientation

The orientation of integral Morava K-theory is discussed in (Sati-Kriz 04, Buhné 11). It is essentially given by the vanishing of the seventh integral Stiefel-Whitney class $W_7$.

Notice that this is in analogy to how orientation in complex K-theory is given by the vanishing third integral Stiefel-Whitney class $W_3$ (spin^c-structure).

Precisely:

Proposition

Let $X$ be a connected closed manifold of dimension 10 with spin structure. This is generalized oriented in second integral Morava K-theory $\tilde K(2)$ (for $p = 2$) precisely if its seventh integral Stiefel-Whitney class vanishes, $W_7(X) = 0$.

This is (Buhné 11, prop. 8.1.13), following (Sati-Kriz 04).

Proposition

Let $X$ be a connected closed manifold of dimension 10 with spin structure. This is generalized oriented in second Johnson-Wilson cohomology theory (Morava E-theory) $E(2)$ (for $p = 2$) precisely if its seventh integral Stiefel-Whitney class vanishes, $W_7(X) = 0$.

This is (Buhné 11, cor. 8.1.14), following (Sati-Kriz 04).

Role in string theory / M-theory

See at Diaconescu-Moore-Witten anomaly.

Related concepts

• chromatic homotopy theory

chromatic level	generalized cohomology theory / E-∞ ring	obstruction to orientation in generalized cohomology	generalized orientation/polarization	quantization	incarnation as quantum anomaly in higher gauge theory
1	complex K-theory $KU$	third integral SW class $W_3$	spinᶜ structure	K-theoretic geometric quantization	Freed-Witten anomaly
2	EO(n)	Stiefel-Whitney class $w_4$
2	integral Morava K-theory $\tilde K(2)$	seventh integral SW class $W_7$			Diaconescu-Moore-Witten anomaly in Kriz-Sati interpretation

References

• Igor Kriz, Hisham Sati, M-theory, type IIA superstrings, and elliptic cohomology, Adv. Theor.

  Math. Phys. 8 (2004), no. 2, 345–394 (arXiv:hep-th/0404013)

• Lukas Buhné, Properties of Integral Morava K-Theory and the Asserted Application to the Diaconescu-Moore-Witten Anomaly, Diploma thesis, Hamburg (2011)