nLab Anderson duality

Idea

The stable $\infty$-category of spectra has a dualizing object (dualizing module) on a suitable subcategory of finite spectra. This 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 2005, appendix B, in the context of constructing a quadratic refinement of the intersection pairing on ordinary differential cohomology,

• Freed 2014, 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 2014, 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 1969.

Similarly Tmf$[1/2]$ is Anderson dual to its 21-fold suspension (Stojanoska 2012).

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 2 (1975) 305-323 [MR 52 #9212]

See also:

• Jacob Lurie, section 4.2 of Representability Theorems

Examples

On the Anderson dual of the sphere spectrum (in a context of invertible extended TQFTs:

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

• Daniel 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]

On the Anderson duals of KU and of tmf:

• 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]

On the Anderson dual of Spin^c-cobordism cohomology in relation to topological K-theory:

• Fei Han, Yuanchu Li: Differential Models for Anderson Dual to Twisted -Bordism and Twisted Anomaly Map [arXiv:2510.27286]

Equivariant duality

Anderson duality in equivariant stable homotopy theory is discussed in

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