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Topological Quantum Mechanics on Orbifolds and Orbifold Index

Si Li, Peng Yang

Abstract

In this paper, we study topological quantum mechanical models on symplectic orbifolds. The correlation map gives an explicit orbifold version of quantum HKR map. The exact semi-classical approximation in this model leads to a geometric and quantum field theoretic interpretation of the orbifold algebraic index.

Topological Quantum Mechanics on Orbifolds and Orbifold Index

Abstract

In this paper, we study topological quantum mechanical models on symplectic orbifolds. The correlation map gives an explicit orbifold version of quantum HKR map. The exact semi-classical approximation in this model leads to a geometric and quantum field theoretic interpretation of the orbifold algebraic index.
Paper Structure (27 sections, 11 theorems, 148 equations, 4 figures)

This paper contains 27 sections, 11 theorems, 148 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A one-dimensional sigma-model with orbifold target
  3. The HKR map: classical v.s. quantum, manifolds v.s. orbifolds
  4. Relation with orbifold algebraic index
  5. Topological quantum mechanics on orbifolds
  6. Twisted sectors
  7. Topological quantum mechanics on orbifolds
  8. Twisted propagator
  9. Configuration spaces
  10. Vacuum expectation
  11. Correlation map
  12. Twisted chains
  13. Some operations on chains
  14. Free correlation map
  15. Interactive correlation map
  16. ...and 12 more sections

Key Result

Lemma 2.7

The following properties hold Here $\hat{\mathcal{O}}=\hat{\mathcal{O}}_0 \otimes \hat{\mathcal{O}}_1\otimes\cdots\otimes \hat{\mathcal{O}}_m$.

Figures (4)

  • Figure 1: An observable $\mathcal{O}_1$ passing through the $g$-twist
  • Figure 2: A $g$-twisted sector
  • Figure 3: A matrix $M_1$ passing through the $g$-twist
  • Figure 4: Three types of edges.

Theorems & Definitions (38)

  • Example 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Definition 2.8
  • Theorem 2.9
  • ...and 28 more