Index Theorem for the -Deformed Fuzzy Sphere
arXiv:hep-th/0609016 · doi:10.1088/1751-8113/40/13/023
Abstract
We calculate the index of the Dirac operator defined on the q-deformed fuzzy sphere. The index of the Dirac operator is related to its net chiral zero modes and thus to the trace of the chirality operator. We show that for the q-deformed fuzzy sphere, a $\uq$ invariant trace of the chirality operator gives the q-dimension of the eigenspace of the zero modes of the Dirac operator. We also show that this q-dimension is related to the topological index of the spinorial field. We then introduce a q-deformed chirality operator and show that its $\uq$ invariant trace gives the topological invariant index of the Dirac operator. We also explain the construction and important role of the trace operation which is invariant under the $\uq$, which is the symmetry algebra of the q-deformed fuzzy sphere. We briefly discuss chiral symmetry of the spinorial action on q-deformed fuzzy sphere and the possible role of this deformed chiral operator in its evaluation using path integral methods.
15 pages, minor changes
References in corpus (14)
- Monopoles and Solitons in Fuzzy Physics
- Quantized Gauge Theory on the Fuzzy Sphere as Random Matrix Model
- On Finite 4D Quantum Field Theory in Non-Commutative Geometry
- The Dirac operator on SU_q(2)
- The Fermion Doubling Problem and Noncommutative Geometry
- Chirality and Dirac Operator on Noncommutative Sphere
- The Origin of Chiral Anomaly and the Noncommutative Geometry
- Field Theory on the q-deformed Fuzzy Sphere I
- Differential Calculus on Fuzzy Sphere and Scalar Field
- Field Theory on the q-deformed Fuzzy Sphere II: Quantization
- Local index formula for SU_q(2)
- Dirac Operators on Coset Spaces
- Dirac operator on the q-deformed Fuzzy sphere and Its spectrum
- Aspects of the q-deformed Fuzzy Sphere