Schauenburg–Ng theorem

In mathematics, the Schauenbug–Ng theorem is a theorem about the modular group representations of modular tensor categories proved by Siu-Hung Ng and Peter Schauenburg in 2010. It asserts that that the kernels of the modular representations of all modular tensor categories are congruence subgroups of $\backslash text\{SL\}\_2(\backslash mathbb\{Z\})$.{{Cite journal |last1=Ng |first1=Siu-Hung |last2=Schauenburg |first2=Peter |date=2010-11-01 |title=Congruence Subgroups and Generalized Frobenius-Schur Indicators |url=https://link.springer.com/article/10.1007/s00220-010-1096-6 |journal=Communications in Mathematical Physics |language=en |volume=300 |issue=1 |pages=1–46 |arxiv=0806.2493 |bibcode=2010CMaPh.300....1N |doi=10.1007/s00220-010-1096-6 |issn=1432-0916}} Since congruence subgroups all have finite index in $\backslash text\{SL\}\_2(\backslash mathbb\{Z\})$, this implies in particular that the modular representations of all modular representations have finite image.

On physical grounds coming from conformal field theory, it has been conjectured since 1987 by Greg Moore and others that the kernel of the modular group representations should be congruence subgroups.{{Cite journal |last=Moore |first=Gregory |date=1987-01-01 |title=Atkin-Lehner symmetry |url=https://www.sciencedirect.com/science/article/abs/pii/0550321387900678 |journal=Nuclear Physics B |volume=293 |pages=139–188 |doi=10.1016/0550-3213(87)90067-8 |issn=0550-3213|url-access=subscription }}{{Citation |last=Eholzer |first=Wolfgang |title=On the classification of modular fusion algebras |journal=Communications in Mathematical Physics |date=1995-05-08 |volume=172 |issue=3 |pages=623–659 |doi=10.1007/BF02101810 |arxiv=hep-th/9408160}}{{Citation |last1=Eholzer |first1=Wolfgang |title=Modular invariance and uniqueness of conformal characters |date=1994-07-14 |last2=Skoruppa |first2=Nils-Peter|journal=Communications in Mathematical Physics |volume=174 |pages=117–136 |doi=10.1007/BF02099466 |arxiv=hep-th/9407074 }} The proof by Schauenbug and Ng came after a series of partial results by other mathematicians, which proved the theorem in special cases.{{Citation |last1=Coste |first1=A. |title=Congruence subgroups and rational conformal field theory |date=1999-09-15 |last2=Gannon |first2=T.|arxiv=math/9909080 }}{{Citation |last=Bantay |first=P. |title=The Kernel of the Modular Representation and the Galois Action in RCFT |journal=Communications in Mathematical Physics |date=2001-04-19 |volume=233 |issue=3 |pages=423–438 |doi=10.1007/s00220-002-0760-x |arxiv=math/0102149 |id=arXiv:math/0102149}}{{Cite journal |last=Xu |first=Feng |date=2006-11-01 |title=Some Computations in the Cyclic Permutations of Completely Rational Nets |url=https://link.springer.com/article/10.1007/s00220-006-0042-0 |journal=Communications in Mathematical Physics |language=en |volume=267 |issue=3 |pages=757–782 |doi=10.1007/s00220-006-0042-0 |issn=1432-0916|arxiv=math/0511662 }}

To prove their result Schauenbug and Ng introduced the notion of 'generalied Frobenius–Schur' indicators, which have since found separate applications to mathematical physics.{{cite arXiv |eprint=2208.14500 |last1=Simon |first1=Steven H. |last2=Slingerland |first2=Joost K. |title=Straightening Out the Frobenius-Schur Indicator |date=2022 |class=hep-th }}