nLab SL(2,Z)

Idea

The special linear group in dimension 2 over the integers, $SL_2(\mathbb{Z})$, sometimes called the modular group (cf. Stein 2007 Def. 1.1) in its role as the mapping class group of the torus.

Properties

Presentations

Proposition

As a subgroup of the general linear group of invertible matrices, $SL_2(\mathbb{Z}) \subset GL_2(\mathbb{Z})$ is generated by the two elements $S,T \in SL_2(\mathbb{Z})$ given by

(1)$$S \;\coloneqq\; \left[ \begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array} \right] \;\;\;\;\; \text{and} \;\;\;\;\; T \;\coloneqq\; \left[ \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array} \right] \mathrlap{\,.}$$

Moreover, $SL_2(\mathbb{Z})$ is presented via these generators subject to the following relations:

$$SL_2(\mathbb{Z}) \;\simeq\; \big\langle S, T \,\big\vert\, S^4 = \mathrm{e} ,\, (T S)^3 = \mathrm{e} ,\, S^2 (T S) = (T S)S^2 \big\rangle \,.$$

For the first statement, an early reference (without proof) is Serre 1973 Thm. VII.2 p 78, detailed proofs are spelled out in Conrad Thm 1.1. The second statement is made by Blanc & Déserti 2012 §2.1 with reference to Newman 1972 (proof?) and by Wehler 2021 Rem. 2.17(1) following Koecher & Krieg 2007 Remark on p. 126.

Alternatively, $SL_2(\mathbb{Z})$ is presented via the generators

$$S \;\coloneqq\; \left[ \begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array} \right] \;\;\;\;\; \text{and} \;\;\;\;\; T \;\coloneqq\; \left[ \begin{array}{rr} 1 & 0 \\ 1 & 1 \end{array} \right] \mathrlap{\,.}$$

as

$$SL_2(\mathbb{Z}) \;\simeq\; \big\langle S\,, T \,\big\vert\, S^4 = \mathrm{e} ,\, (S T)^3 = S^2 \big\rangle$$

(Serre 1980 p. 81, Rubinstein & Gardiner 1979, Kassel & Turaev 2008 (A.2))

Spin-structure preserving subgroups

The torus admits exactly 4 distinct spin structures (see there)

$$\big\{ pp, pa, ap, aa \big\} \;\; \simeq \;\; H_1\big(T^2;\, C_2\big)$$

(with “periodic” or “antiperiodic” boundary conditions for fermions along two basis 1-cycles, cf. Alvares-Gaumé, Moore & Vafa 1986 §2).

Under its canonical action, $SL_2(\mathbb{Z})$ fixes the “trivial” structure $pp$ (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)

$$S \;\colon\; \left\{ \begin{array}{rcl} pp &\mapsto& pp \\ ap &\mapsto& pa \\ pa &\mapsto& ap \\ aa &\mapsto& aa \mathrlap{\,,} \end{array} \right. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; T \;\colon\; \left\{ \begin{array}{rcl} pp &\mapsto& pp \\ ap &\mapsto& ap \\ pa &\mapsto& aa \\ aa &\mapsto& pa \mathrlap{\,.} \end{array} \right.$$

Hence the subgroup of $SL_2(\mathbb{Z})$ which stabilizes the $aa$-spin structure (the “$aa$-spin mapping class group of the torus”) is that generated from $S$ and the square of $T$:

$$\Gamma_\theta \;\coloneqq\; \langle S, T^2\rangle_{SL_2(\mathbb{Z})} \subset SL_2(\mathbb{Z}) \,.$$

(This is a congruence subgroup of index=3, cf. Conrad, Coe. 3.6 & Ex. 3.7).

Its quotient group by the center $\pm I$ is well-known (but canonical reference?) to be freely generated by these two elements (cf. Bruillard et al 2017 Thm. 2.7)

$$\Gamma_\theta / (\pm I) \;\simeq\; \langle S, T^2\rangle \,.$$

This implies [MO:a/490283] that the subgroup itself has the following presentation:

$$\Gamma_{\theta} \;\simeq\; \big\langle S, T^2 \,\big\vert\, S^4 = [S^2, T^2] = \mathrm{e} \big\rangle \,.$$

Related concepts

• modular form

• Moebius group

• mapping class group

• S-duality, U-duality

References

General

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 $SL_2(Z)$ and $PSL_2(Z)$, 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, $SL_2(\mathbb{Z})$ [pdf, pdf]

• W. J. Harvey: Introductory Lectures on $SL(2,Z)$ 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 $SL(2,\mathbb{Z})$ 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:

• Filippo Callegaro, Fred Cohen, Mario Salvetti: The cohomology of the braid group $B_3$ and of $SL_2(\mathbb{Z})$ with coefficients in a geometric representation, in: Configuration Spaces, CRM Series. Edizioni della Normale, Pisa (2012) [arXiv:1204.5390, doi:10.1007/978-88-7642-431-1_8]

See also:

• Groupprops: Special linear group:SL(2,Z)

Representation theory

On the representation theory of $SL_2(\mathbb{Z})$.

General

• 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 $SL_2(\mathbb{Z})$, Proceedings of the AMS, 151 4 (2023) 1415-1431 [arXiv:2203.15701, doi:10.1090/proc/16205]

• Andrew Putman: The representation theory of $SL_n(\mathbb{Z})$ [pdf, pdf]

Modular data of CFTs

The $SL(2,\mathbb{Z})$-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 $U(1)$ 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:

• Samuel Nathan Wilson: $SL(2,\mathbb{Z})$ Representations and 2-Semiregular Modular Categories, PhD thesis, Louisiana State U. (2023) [10.31390gradschool_dissertations.6094]

Congruence subgroups

On subgroups of $SL_2(\mathbb{Z}) \,\simeq\, MCG(T^2)$ preserving a spin structure on the torus:

On the context of 2d CFT:

• Luis Alvarez-Gaumé, Gregory Moore, Cumrun Vafa, §2 of: Theta functions, modular invariance, and strings, Communications in Mathematical Physics 106 1 (1986) 1-4 [doi:10.1007/BF01210925, euclid:cmp/1104115581]

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]