"To see a World in a Grain of Sand, and a Heaven in a Wild Flower.
Topological insulators Q&A
A: I am chasing a squirrel called topological insulator in the garden of modern geometry and topology.
A: It is a new material characterized by a topological Z/2 invariant.
A: a parity anomaly or a geometric Berry phase as a holonomy
A: Time reversal symmetry, as a real structure in KR-theory
A: The parity anomaly has an interpretation as a mod 2 index theorem.
A: K-theory, in fact KR-theory
A: K-theory classifies topological band theory.
A: CS/WZW, as a topological quantum field theory (TQFT)
A: a holonomy in a bundle gerbe
A: a Pfaffian line bundle (or a real twisted K-theory?)
A: NCG is a modern framework to study index theory using operator algebras.
A: CS/WZW duality as an instance of the holographic principle
A: a correspondence identifying the Z/2 invariant derived from (e.g. K-theories of) the bulk and boundary
A: a KK-cycle realizing the mod 2 index theorem
A: geometric phase --> index theory and K-theory --> TQFT --> bundle gerbe and twisted K-theory --> KK-theory and NCG
Interested readers see Survey
More details see Research Statement
- Index theory and noncommutative geometry of topological insulators
to appear in Advanced Topological Insulators, WILEY-Scrivener Publisher, USA
- Bott--Kitaev periodic table and index theory
- Noncommutative topological Z/2 invariant(with R. Kaufmann and B. Kaufmann)
- The Stiefel--Whitney theory of topological insulators (with R. Kaufmann and B. Kaufmann)
- Topological insulators and K-theory (with R. Kaufmann and B. Kaufmann)
- Noncommutative Chern-Simons theory on the quantum 3-sphere
- Renormalization group flow, entropy and eigenvalues, Lett. Math. Phys., 2017
- Notes on topological insulators (with R. Kaufmann and B. Kaufmann), Rev. Math. Phys., 2016
- The Ponzano-Regge model and parametric representation, Commun. Math. Phys., 327(1), 243-260, 2014
- Harper operators, Fermi curves, and Picard-Fuchs equations,
Lett. in Math. Phys., 104(5), 613-624, 2014
- The algebraic geometry of Harper operators
J. Phys. A: Math. Theor. 44, 405204, 2011
- On coarse homotopy, J. Fudan Univ. Nat. Sci., 46(2), 2007