For a link describing the experimental fractional quantum Hall effect
and its discoverers, cf. the
1998 Nobel Prize in Physics press release. In the paper **[MM]**
below, we propose a noncommutative geometry model on the hyperbolic plane
for the fractional quantum Hall effect, extending earlier work done in
**[CHMM]**
and also building on fundamental work mainly by Bellissard and collaborators,
who established a noncommutative geometry model on the Euclidean plane
for the *integer* quantum Hall effect. For a link describing the
experimental integer quantum Hall effect
and its discoverers, cf. the
1985 Nobel Prize in Physics press release.
The hyperbolic metric plays
the role of the effective interaction of the charge carriers in the
Hamiltonian.
The Hall conductance is derived to be a cyclic 2-cocycle on the algebra
of observables and its expression resembles a generalized Kubo formula.
Under the assumption that the Fermi level is in a spectral gap of the
Hamiltonian,
we establish that the Hall conductance is an integer mutiple of orbifold
Euler characteristics of cocompact Fuchsian groups, and is therefore
topological
in character as well as fractional valued. As a consequence, the Hall
conductance
is stable under small deformations of the Hamiltonian.Thus this model can
be generalized to systems with disorder as in **[CHM]**, where the
hypothesis
that the Fermi level is in a spectral gap of the Hamiltonian can be relaxed
to the assumption that it is in a gap of extended states, which is a necessary
step in order to establish the presence of Hall plateaux. A table below
compares the experimentally observed fractions of the Hall conductance,
to the fractions arising from our model, whereas another table below lists
some new fractional Hall conductances predicted from our model. For another
interesting link on the role of hyperbolic geometry in describing structure
and properties of crystalline materials, cf. some
current research at ANU.

The latest development is the result in **[KMS]** that
the low lying energy bands of the effective Hamiltonian
in the fractional quantum Hall effect, do not contribute
to the Hall conductance, in the semiclassical limit as the
electro-magnetic coupling constant goes to zero.
This was previously known only for the integer quantum Hall
effect.
The crude physical explanation for this semiclassical vanishing theorem
for the Hall conductance is as follows. The Hall conductance for
the model operator vanishes, since it is the Hamiltonian of a
crystal with perfectly isolated atoms,
therefore there can be no current flowing through it - this
remains valid for small perturbations of the model operator
and therefore for the effective Hamiltonian.

A recent survey of the orbifold and noncommutative geometry
approach to the fractional quantum Hall effect can be found
in **[MM2]**.

February 2008 - January 2016, Editor in charge of "Global Analysis,
Noncommutative Geometry, and the Mathematics of String Theory",
* Proceedings of the American Mathematical Society*.

I am Managing Editor of the J. Geometry and Physics special issue on String geometries, Dualities and Topological Matter,

I have co-organized multiple international IGA workshops on noncommutative geometry and the fractional quantum Hall effect at Adelaide.

__References__

**[MT18]**
V. Mathai and G. C. Thiang,

**Topological phases on the hyperbolic plane,**

26 pp., [1712.02952]

**[MT172]**
V. Mathai and G. C. Thiang,

**Differential topology of semimetals,**

*Communications in Mathematical Physics*,

**355**, no. 2,(2017) 561-602.

**[MT171]**
V. Mathai and Guo Chuan Thiang,

**Global topology of Weyl semimetals and Fermi arcs,**

* J. Physics A: Mathematical and Theoretical (Letter),*

**50** no. 11 (2017) 11LT01, 11pp
**[1607.02242] **

publicity at JPhys+

**[HMT17]**
K. Hannabuss, V. Mathai and Guo Chuan Thiang,
**T-duality trivializes bulk-boundary correspondence: the noncommutative case,
**

* Letters in Mathematical Physics,*

(published online) 39 pp.

**[HMT]**
K. Hannabuss, V. Mathai and Guo Chuan Thiang,
**T-duality trivializes bulk-boundary correspondence:
the parametrised case,
**

* Advances in Theoretical and Mathematical Physics,*

20 no. 5 (2016) 1193-1226, [1510.04785]

**[MT3]**
V. Mathai and Guo Chuan Thiang,
**T-duality trivializes bulk-boundary correspondence: some higher dimensional cases
**

*Annales Henri Poincare*,

17 no. 12 (2016) 3399--3424,,
** [1506.04492] **

**[MT2]**
V. Mathai and Guo Chuan Thiang,
**T-duality trivializes bulk-boundary correspondence
**

*Communications in Mathematical Physics*,

345 no. 2, (2016) 675-701,
**[1505.05250] **

**[MT]**
V. Mathai and Guo Chuan Thiang,
**T-duality and topological insulators
**

*Journal of Physics A: Mathematical and Theoretical (Fast Track Communication)*
**48** (2015) no. 42, 42FT02, 10pp,
**[1503.01206] **

publicity at IOPSCIENCE

**[CHM2]** A. Carey, K. Hannabuss and V. Mathai,
**Quantum Hall Effect and Noncommutative Geometry,
**

*Journal of Geometry and Symmetry in Physics,*

** 6 ** (2006) 16-37,
**[math.OA/0008115]**

**[DMY]** J. Dodziuk, V. Mathai and S. Yates,

**Arithmetic properties of eigenvalues of generalized Harper operators on graphs,
**

*Communications in Mathematical Physics,*

** 262 ** no. 2 (2006) 269-297.
**[math.SP/0311315]**

**[MM2]** M. Marcolli and V. Mathai,

**Towards the fractional quantum Hall effect:
a noncommutative geometry perspective,
**

"Noncommutative Geometry and Number Theory".

Editors C.Consani, M. Marcolli,

Vieweg Verlag, Wiesbaden, 2006, pages 235-261.

**[KMS]** Y. Kordyukov, V. Mathai and M. Shubin,

**Equivalence of spectral projections in semiclassical
limit and a
vanishing theorem for higher traces in K-theory,
**

**[MM]** M. Marcolli and V. Mathai,
**Twisted index theory on good orbifolds, II: fractional quantum
numbers**,
*Communications in Mathematical Physics,*
**217** (2001) 55-87.

**[CHM]** A. Carey, K. Hannabuss and V. Mathai,
**Quantum Hall Effect on the Hyperbolic Plane in the presence of
disorder**,
*Letters in Mathematical Physics,*
**47** (1999) 215-236.

**[CHMM]** A. Carey, K. Hannabuss, V. Mathai and P. McCann,
**Quantum Hall Effect on the Hyperbolic Plane**,
*Communications in Mathematical Physics,*
**190** (1998) 629-673.

The table below compares the experimentally observed fractions
of the Hall conductance, to the fractions arising from our model.

Experimental values for the
Hall conductance |
Fractions realized as orbifold Euler
characteristics
of orbifolds with g = 1 or g = 0 in their signature |

5/3 | S(1;6,6) |

4/3 | S(1;3,3) |

7/5 | S(0;5,5,10,10) |

4/5 | S(1;5) |

5/7 | S(0;7,14,14) |

2/3 | S(1;3) |

3/5 | S(0;5,10,10) |

4/7 | S(0;7,7,7) |

5/9 | ??? |

4/9 | S(0;3,9,9) |

3/7 | ??? |

2/5 | S(0;5,5,5) |

1/3 | S(0;3,6,6) |

5/2 | S(1;6,6,6) |

The table below lists some new fractional Hall conductances predicted
from our model.

Some fractions for the Hall conductance
predicted by our model |
These fractions realized as orbifold Euler
characteristics
of orbifolds with g = 1 or g = 0 in their signature |

4/3 | S(0;3,3,3,3,3), S(1;3,3) |

2/3 | S(0;2,2,2,2,3), S(1;3) |

4/7 | S(0;7,7,7) |

1/2 | S(0;4,8,8), S(1;2) |

4/9 | S(0;3,9,9) |

2/5 | S(0;5,5,5) |

1/3 | S(0;4,4,6), S(0;2,2,2,6) |

1/4 | S(0;2,8,8) S(0;4,4,4) |

1/5 | S(0;2,5,10) |

4/21 | S(0;3,7,7) |

1/6 | S(0;2,4,12), S(0;3,3,6) |

1/8 | S(0;2,4,8) |

1/12 | S(0;2,4,6), S(0;3,3,4) |

1/24 | S(0;2,3,8) |

1/42 | S(0;2,3,7) |