nLab Anderson duality

Idea

The stable (infinity,1)-category of spectra has a dualizing object (dualizing module) on a suitable subcategory of finite spectra. It is called the Anderson spectrum $I_{\mathbb{Z}}$ (Lurie, Example 4.3.9). The duality that this induces is called Anderson duality.

Examples

The Anderson dual of the sphere spectrum is discussed in

• (Hopkins-Singer 05, appendix B) in the context of constructing a quadratic refinement of the intersection pairing on ordinary differential cohomology

• in (Freed 14, section 5.1.1) in the context of invertible extended topological field theories.

The Anderson dual of KU is (complex conjugation-equivariantly) the 4-fold suspension spectrum $\Sigma^4 KU$ (Heard-Stojanoska 14, theorem 8.2). This implies that, nonequivariantly $KU$ is Anderson self-dual and the Anderson dual of $KO$ is $\Sigma^4KO$, which were both first proven by Anderson.

Similarly Tmf$[1/2]$ is Anderson dual to its 21-fold suspension (Stojanoska 12).

Related concepts

• Brown-Comenetz duality

• Spanier-Whitehead duality

References

General

Original articles include

• Donald W. Anderson, Universal coefficient theorems for K-theory, mimeographed notes, Univ. California, Berkeley, Calif., 1969 (pdf)

• Zen-ichi Yosimura, Universal coefficient sequences for cohomology theories of CW-spectra, Osaka J. Math. 12 (1975), no. 2, 305–323. MR 52 #9212

See also

• Jacob Lurie, section 4.2 of Representability Theorems

Examples

The Anderson dual of the sphere spectrum is discussed (in a context of extended TQFTs) in

• Michael Hopkins, Isadore Singer, appendix B of, Quadratic Functions in Geometry, Topology, and M-Theory, 2005

• Dan Freed, section 5.1.1 of Short-range entanglement and invertible field theories (arXiv:1406.7278)

• Daniel Freed, Michael Hopkins, section 5.3 of Reflection positivity and invertible topological phases (arXiv:1604.06527)

The Anderson duals of KU and of tmf are discussed in

• Vesna Stojanoska, Duality for Topological Modular Forms, Doc. Math. 17 (2012) 271-311 (arXiv:1105.3968)

• Drew Heard, Vesna Stojanoska, K-theory, reality, and duality (arXiv:1401.2581)

In the context of heterotic string theory:

• Yuji Tachikawa, Mayuko Yamashita, Anderson self-duality of topological modular forms, its differential-geometric manifestations, and vertex operator algebras [arXiv:2305.06196]

Equivariant duality

Anderson duality in equivariant stable homotopy theory is discussed in

• Nicolas Ricka, Equivariant Anderson duality and Mackey functor duality (arXiv:1408.1581)