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The special linear group in dimension 2 over the integers, , sometimes called the modular group (cf. Stein 2007 Def. 1.1) in its role as the mapping class group of the torus.
As a subgroup of the general linear group of invertible matrices, is generated by the two elements given by
Moreover, is presented via these generators subject to the following relations:
Alternatively, is presented via the generators
as
(Serre 1980 p. 81, Rubinstein & Gardiner 1979, Kassel & Turaev 2008 (A.2))
The torus admits exactly 4 distinct spin structures (see there)
(with “periodic” or “antiperiodic” boundary conditions for fermions along two basis 1-cycles, cf. Alvares-Gaumé, Moore & Vafa 1986 §2).
Under its canonical action, fixes the “trivial” structure (called the odd spin structure) and permutes the other 3 (the even spin structures) transitively:
The above generators act as (cf. MO:a/72770, Bonderson, Rowell, Zhang & Wang 2018 p 3)
Hence the subgroup of which stabilizes the -spin structure (the “-spin mapping class group of the torus”) is that generated from and the square of :
(This is a congruence subgroup of index=3, cf. Conrad, Coe. 3.6 & Ex. 3.7).
Its quotient group by the center is well-known (but canonical reference?) to be freely generated by these two elements (cf. Bruillard et al 2017 Thm. 2.7)
This implies [MO:a/490283] that the subgroup itself has the following presentation:
Textbook accounts:
Morris Newman: The Classical Modular Group and Related Groups, chapter VIII in: Integral Matrices, Academic Press (1972)
Jean-Pierre Serre, §VII.1 of: A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer (1973) [doi:10.1007/978-1-4684-9884-4, pdf]
Jean-Pierre Serre, p. 81 of: Trees, Springer (1980) [doi:10.1007/978-3-642-61856-7]
William Stein, chapter 1 of: Modular Forms, a Computational Approach, Graduate Studies in Mathematics 79, AMS (2007) [doi:10.1090/gsm/079]
Christian Kassel, Vladimir Turaev: Presentations of and , appendix A in: Braid Groups, Graduate Texts in Mathematics 247, Springer (2008) 311-314 [doi:10.1007/978-0-387-68548-9_8]
Notes:
Keith Conrad, [pdf, pdf]
W. J. Harvey: Introductory Lectures on and modular forms (2008) [pdf]
More on presentations:
J. H. Rubinstein, C. Gardiner, A note on a 3-dimensional homogeneous space, Compositio Mathematica 39 3 (1979) 297-299 [numdam:CM_1979__39_3_297_0]
Max Koecher, Aloys Krieg: Die Modulgruppe, §II.2 in: Elliptische Funktionen und Modulformen, Springer (2007) [doi:10.1007/978-3-540-49325-9]
Jérémy Blanc, Julie Déserti, §2.1 in: Embeddings of into the Cremona group, Transform. Groups 17 1 (2012) 21-50 [arXiv:1103.0114, doi:10.1007/s00031-012-9174-9]
Joachim Wehler, Rem. 2.17 in: Modular Forms and Elliptic Curves (2021) [pdf]
More on the group cohomology:
See also:
On the representation theory of .
Nils R. Scheithauer: The Weil Representation of formula and Some Applications, International Mathematics Research Notices, 2009 8 (2009) 1488–1545 [doi:10.1093/imrn/rnn166, pdf]
Siu-Hung Ng, Yilong Wang, Samuel Wilson: On symmetric representations of , Proceedings of the AMS, 151 4 (2023) 1415-1431 [arXiv:2203.15701, doi:10.1090/proc/16205]
Andrew Putman: The representation theory of [pdf, pdf]
The -action on the characters in 2d CFT and abelian Chern-Simons theory:
Erik Verlinde, p. 365, 369 in: Fusion rules and modular transformations in 2D conformal field theory, Nuclear Physics B 300 (1988) 360-376 [doi:10.1016/0550-3213(88)90603-7, p. 368: pdf]
Toru Gocho: The topological invariant of three-manifolds based on the Gauge theory, Proc. Japan Acad. Ser. A Math. Sci. 66 8 (1990) 237-239 [doi:10.3792/pjaa.66.237, dml:1195512360]
Louis Funar, p. 7-8 of: Theta functions, root systems and 3-manifold invariants, Journal of Geometry and Physics 17 3 (1995) 261-282 [doi:10.1016/0393-0440(94)00050-E, pdf]
Mihaela Manoliu: Quantization of symplectic tori in a real polarization, J. Math. Phys. 38 (1997) 2219–2254 [arXiv:dg-ga/9609012, doi:10.1063/1.531970]
Mihaela Manoliu, p. 67 of: Abelian Chern-Simons theory, J. Math. Phys. 39 (1998) 170-206 [arXiv:dg-ga/9610001, doi:10.1063/1.532333]
Louis Funar, thm. 1.1: Some abelian invariants of 3-manifolds, Revue Roumaine de Mathematiques Pures et Appliquees 45 5 (2000) 825-862 [pdf, pdf]
Terry Gannon: Modular Data: The Algebraic Combinatorics of Conformal Field Theory, J Algebr Comb 22 (2005) 211–250 [doi:10.1007/s10801-005-2514-2, arXiv:math/0103044]
Peter Bantay, Terry Gannon: Conformal characters and the modular representation, Journal of High Energy Physics, 2006 JHEP02 (2006) [arXiv:hep-th/0512011, doi:10.1088/1126-6708/2006/02/005]
Christoph Schweigert, around (26) in: Introduction to conformal field theory, lecture notes (2013/14) [pdf]
Jürgen Fuchs, Christoph Schweigert, Simon Wood, Yang Yang, p 7 in: Algebraic structures in two-dimensional conformal field theory, in: Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2305.02773]
Via modular tensor categories:
On subgroups of preserving a spin structure on the torus:
On the context of 2d CFT:
In the context of super-modular categories:
Paul Bruillard, Cesar Galindo, Tobias Hagge, Siu-Hung Ng, Julia Yael Plavnik, Eric C. Rowell, Zhenghan Wang, theorem 2.7 in: Fermionic Modular Categories and the 16-fold Way, J. Math. Phys. 58 (2017) 041704 [doi:10.1063/1.4982048, arXiv:1603.09294]
Parsa Bonderson, Eric C. Rowell, Qing Zhang, Zhenghan Wang, p. 3 of: Congruence Subgroups and Super-Modular Categories, Pacific J. Math. 296 (2018) 257-270 [arXiv:1704.02041, doi:10.2140/pjm.2018.296.257]
Qing Zhang, §4.1.1 in: Super-Modular Categories, PhD thesis, Texas A&M University (2019) [pdf, pdf]
Jin-Beom Bae et al., §3 in: Fermionic rational conformal field theories and modular linear differential equations, Progress of Theoretical and Experimental Physics 2021 8 (2021) 08B104 [doi:10.1093/ptep/ptab033, arXiv:2010.12392]
Last revised on March 31, 2025 at 08:03:50. See the history of this page for a list of all contributions to it.