Homepage for Dan Li

"To see a World in a Grain of Sand, and a Heaven in a Wild Flower.
Hold Infinity in the palm of your hand, and Eternity in an hour."
---- William Blake

Topological insulators Q&A

  • Q: What am I doing?
        A: I am chasing a squirrel called topological insulator in the garden of modern geometry and topology.
  • Q: What is a topological insulator?
        A: It is a new material characterized by a topological Z/2 invariant.
  • Q: What is the physical meaning of the Z/2 invariant?
        A: a parity anomaly or a geometric Berry phase as a holonomy
  • Q: What is the relevant symmetry?
        A: Time reversal symmetry, as a real structure in KR-theory
  • Q: Why it is interesting for a mathematician?
        A: The parity anomaly has an interpretation as a mod 2 index theorem.
  • Q: What is the relevant topology?
        A: K-theory, in fact KR-theory
  • Q: Why K-theory?
        A: K-theory classifies topological band theory.
  • Q: What is the effective field theory?
        A: CS/WZW, as a topological quantum field theory (TQFT)
  • Q: What is the geometry of this TQFT?
        A: a holonomy in a bundle gerbe
  • Q: What is the topology of this TQFT?
        A: a Pfaffian line bundle (or a real twisted K-theory?)
  • Q: Why noncommutative geometry (NCG)?
        A: NCG is a modern framework to study index theory using operator algebras.
  • Q: What is the bulk-boundary correspondence in physics?
        A: CS/WZW duality as an instance of the holographic principle
  • Q: What is the bulk-boundary correspondence in math?
        A: a correspondence identifying the Z/2 invariant derived from (e.g. K-theories of) the bulk and boundary
  • Q: What is the geometry of the bulk-boundary correspondence?
        A: a KK-cycle realizing the mod 2 index theorem
  • Q: What is the roadmap to chase it?
        A: geometric phase --> index theory and K-theory --> TQFT --> bundle gerbe and twisted K-theory --> KK-theory and NCG
  • Interested readers see Survey

    Research interests

  • Noncommutative geometry
  • Index theory and K-theory
  • Geometry and Topology
  • Mathematical Physics
  • Periods and Motives
  • Functional Analysis
  • More details see Research Statement

    Publications

    Book Chapters

    1. Index theory and noncommutative geometry of topological insulators
      to appear in Advanced Topological Insulators, WILEY-Scrivener Publisher, USA

    Preprints

    1. Bott--Kitaev periodic table and index theory arXiv
    2. Noncommutative topological Z/2 invariant(with R. Kaufmann and B. Kaufmann) arXiv
    3. The Stiefel--Whitney theory of topological insulators (with R. Kaufmann and B. Kaufmann) arXiv

    Submitted

    1. Topological insulators and K-theory (with R. Kaufmann and B. Kaufmann) arXiv
    2. Noncommutative Chern-Simons theory on the quantum 3-sphere arXiv

    Published

    1. Renormalization group flow, entropy and eigenvalues, Lett. Math. Phys., 2017 pdf
    2. Notes on topological insulators (with R. Kaufmann and B. Kaufmann), Rev. Math. Phys., 2016 pdf
    3. The Ponzano-Regge model and parametric representation, Commun. Math. Phys., 327(1), 243-260, 2014 pdf
    4. Harper operators, Fermi curves, and Picard-Fuchs equations, Lett. in Math. Phys., 104(5), 613-624, 2014 pdf
    5. The algebraic geometry of Harper operators J. Phys. A: Math. Theor. 44, 405204, 2011 pdf
    6. On coarse homotopy, J. Fudan Univ. Nat. Sci., 46(2), 2007