nLab SL(2,Z)

Contents

Contents

Idea

The special linear group in dimension 2 over the integers, SL 2()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()GL 2()SL_2(\mathbb{Z}) \subset GL_2(\mathbb{Z}) is generated by the two elements S,TSL 2()S,T \in SL_2(\mathbb{Z}) given by

(1)S[0 1 1 0]andT[1 1 0 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()SL_2(\mathbb{Z}) is presented via these generators subject to the following relations:

SL 2()S,T|S 4=e,(TS) 3=e,S 2(TS)=(TS)S 2. 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()SL_2(\mathbb{Z}) is presented via the generators

S[0 1 1 0]andT[1 0 1 1]. 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()S,T|S 4=e,(ST) 3=S 2 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)

{pp,pa,ap,aa}H 1(T 2;C 2) \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()SL_2(\mathbb{Z}) fixes the “trivial” structure pppp (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:{pp pp ap pa pa ap aa aa,T:{pp pp ap ap pa aa aa pa. 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()SL_2(\mathbb{Z}) which stabilizes the aaaa-spin structure (the “aaaa-spin mapping class group of the torus”) is that generated from SS and the square of TT:

Γ θS,T 2 SL 2()SL 2(). \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 ±I\pm I is well-known (but canonical reference?) to be freely generated by these two elements (cf. Bruillard et al 2017 Thm. 2.7)

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

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

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

References

General

Textbook accounts:

Notes:

  • Keith Conrad, SL 2()SL_2(\mathbb{Z}) [pdf, pdf]

  • W. J. Harvey: Introductory Lectures on SL(2,Z)SL(2,Z) and modular forms (2008) [pdf]

More on presentations:

More on the group cohomology:

  • Filippo Callegaro, Fred Cohen, Mario Salvetti: The cohomology of the braid group B 3B_3 and of SL 2()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:

Representation theory

On the representation theory of SL 2()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()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()SL_n(\mathbb{Z}) [pdf, pdf]

Modular data of CFTs

The SL(2,)SL(2,\mathbb{Z})-action on the characters in 2d CFT and abelian Chern-Simons theory:

Via modular tensor categories:

Congruence subgroups

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

On the context of 2d CFT:

In the context of super-modular categories:

Last revised on March 31, 2025 at 08:03:50. See the history of this page for a list of all contributions to it.