nLab integer Heisenberg group (changes)

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Context

Group Theory

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Idea

The integer Heisenberg groups or discrete Heisenberg groups are non-abelian discrete central extensions of the integer product groups $\mathbb{Z}^{2g}$ , $g \in \mathbb{N}$ , analogous to how the ordinary Heisenberg groups are non-abelian central extensions of $\mathbb{R}^{2g}$ . Generally there are analogous $R$ -Heisenberg groups for any commutative ring $R$ .

In fact, for even multiples of the basic such extension, integer Heisenberg groups are discrete subgroups of ordinary $\mathbb{R}$ -Heisenberg groups covering a lattice inclusion $\mathbb{Z}^{2g} \hookrightarrow \mathbb{R}^{2g}$ (cf. Prop. \ref{TwiceHeisenbergExtensionInSymplecticForm} below).

In physics, specifically in quantum field theory, these even-weight integer Heisenberg groups are closely related to quantum abelian Chern-Simons theory (spelled out below, following SS25, see also at 2-Cohomotopy moduli of surfaces).

Definition

Integer Heisenberg group extensions of $\mathbb{Z}^2$

We discuss the restriction of the basic $\mathbb{R}$ -Heisenberg group $H_3$ (see there) along the inclusion $\mathbb{Z}^2 \hookrightarrow \mathbb{R}^2$ and the mod- $n$ quotient groups of that.

In the following, we denote cyclic groups, for $n \in \mathbb{N}$ , uniformly by

\mathbb{Z}_{k} \;\coloneqq\; \left\{ \begin{array}{lcl} \mathbb{Z} &\vert& k = 0 \\ \mathbb{Z}/(k) &\vert& \text{otherwise} \mathrlap{\,.} \end{array} \right.

Recall (see there) the equivalence of central extensions of a discrete group $G$ by an abelian group $A$ (for the present case equipped with the trivial $G$ -action) with the degree=2 group cohomology $H^2(G;A)$ .

\begin{proposition} For $k \in \mathbb{N}$ , the central group extensions of $\mathbb{Z}^2$ by $\mathbb{Z}_k$ are all integral multiples of the following elementary Heisenberg extension:

(1)

\left\{ \begin{array}{l} \big(a,b,[n]\big) \,\in\, \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}_{k} \,, \\ \big( a,b,[n] \big) \cdot \big( a',b',[n'] \big) \;=\; \big( a + a', b + b', [ n + n' + {\color{red} a \, b' } ] \big) \end{array} \right\} \,.

\end{proposition} \begin{proof} The central extensions in question

1 \to \mathbb{Z}_k \xrightarrow{cntrl} H \longrightarrow \mathbb{Z}^2 \to 1

are classified by the degree=2 group cohomology of $\mathbb{Z}^2$ with coefficients in $\mathbb{Z}_k$ (equipped with the trivial $\mathbb{Z}^2$ -action).

That second group cohomology is:

(2)

\begin{array}{ccl} H^2_{Grp}(\mathbb{Z}^2;\, \mathbb{Z}_k) &\simeq& H^2(B \mathbb{Z}^2;\, \mathbb{Z}_k) \\ &\simeq& H^2(T^2;\, \mathbb{Z}_k) \\ &\simeq& H^2(S^1_a \vee S^1_b \vee S^2;\, \mathbb{Z}_k) \\ &\simeq& H^2(S^2;\, \mathbb{Z}_k) \\ &\simeq& \mathbb{Z}_k \mathrlap{\,,} \end{array}

where we used, in order of the appearance:

that group cohomology is ordinary cohomology of the domain group’s classifying space;
that the classifying space of the integers is the circle and that this respects Cartesian products;
that the stable homotopy type of the 2-torus is the wedge sum of two circles with a 2-sphere (see there).

The generator of this cohomology group must be the group 2-cocycle

\alpha \,\in\, H^2_{Grp}\big( \mathbb{Z}^2 ;\, \mathbb{Z}_k \big)

given by

(3)

\begin{array}{ccc} \mathbb{Z}^2 \times \mathbb{Z}^2 &\xrightarrow{\; \alpha \;}& \mathbb{Z}_{n} \\ \big( (a,b), (a',b') \big) &\mapsto& [ a b'] \mathrlap{\,,} \end{array}

whose cocycle condition $[a b'] + [(a+a')b''] = [a(b'+b'')] + [a' b'']$ is evidently satisfied due to the distributive law in the ring $\mathbb{Z}$ and which is clearly not divisible. \end{proof}

\begin{example} In the integer Heisenberg group (1)

the inverse elements are given by

$\big(a ,\, b ,\, [n]\big)^{-1} \;=\; \big( -a ,\, -b ,\, [-n + a b] \big)$
the basic group commutators give

(4) $\begin{array}{ccl} \big[ (1,0,0), (0,1,0) \big] &\equiv& (1,0,0) \cdot (0,1,0) \cdot (1,0,0)^{-1} \cdot (0,1,0)^{-1} \\ &=& (1,1,1) \cdot (-1,0,0) \cdot (0,-1,0) \\ &=& (0,1,1) \cdot (0,-1,0) \\ &=& (0,0,1) \mathrlap{\,.} \end{array}$

\end{example}

\begin{remark} \label{BasicIntHeisenbergAsUpperTriangularMatrices} The basic integer Heisenberg group extension (1), at $k = 0$ , is isomorphic to the matrix group of $3 \times 3$ upper triangular matrices with integer entries and all “1”s on their diagonal, since matrix multiplication gives:

\left[ \begin{matrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix} \right] \cdot \left[ \begin{matrix} 1 & a' & c' \\ 0 & 1 & b' \\ 0 & 0 & 1 \end{matrix} \right] \;=\; \left[ \begin{matrix} 1 & a + a' & c + c' + a b' \\ 0 & 1 & b + b' \\ 0 & 0 & 1 \end{matrix} \right]

In this matrix form the basic integer Heisenberg group $H_3(\mathbb{Z})$ is commonly known in the pure group theory-literature (e.g. Dummit & Foote 2006 p 35, Budylin 2014) where the higher level extensions typically do not find attention. \end{remark}

But the relation to the real Heisenberg group (in both senses of “real”) is made by the second multiple of this basic extension (cf. Gelca & Uribe 2010 p. 7, Gelca & Hamilton 2012 Def 2.4, Gelca & Hamilton 2015 Def 2.3):

\begin{proposition} \label{TwiceHeisenbergExtensionInSymplecticForm} Twice the basic integer Heisenberg extension (3) is isomorphic to

(5)

\left\{ \begin{array}{l} \big(a, b, [n]\big) \,\in\, \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}_{k} \,, \\ \big( a, b, [n] \big) \cdot \big( a', b', [n'] \big) \;=\; \big( a + a' ,\, b + b' ,\, [n + n' + {\color{red} a \cdot b' - a' \cdot b}] \big) \end{array} \right\} \mathrlap{\,.}

\end{proposition} \begin{proof} It is sufficient to see that the second red summand in (5) is a group 2-cocycle cohomologous to the first red summand, which means equivalently that the 2-cocycle

\big( (a, b), (a', b') \big) \;\mapsto\; a \cdot b' + a' \cdot b

has a coboundary: This coboundary is readily found to be

(a, b) \;\mapsto\; - a \cdot b \,,

since

- a \cdot b - a' \cdot b' \,+\, (a + a') \cdot (b + b') \;=\; a \cdot b' + a' \cdot b \,.

\end{proof}

Properties

Basic properties

\begin{prop} The multiple=2 discrete Heisenberg group $\mathbb{Z}_{k} \to \widehat{\mathbb{Z}^{2}} \to \mathbb{Z}^2$ (5) is the following quotient group of the product group $F(a,b) \times \mathbb{Z}_{k}$ with the free group $F(a,b)$ on two generators:

(6)

\widehat{\mathbb{Z}^2} \,\simeq\, \big( F(a,b) \times \mathbb{Z}_{k} \big) \big/ \big( a \cdot b = [2] \cdot b \cdot a \big) \,.

\end{prop}

\begin{proof} The map

\begin{array}{ccc} \frac{ F(a,b) \times \mathbb{Z}_{k} }{ a \cdot b = [2] \cdot b \cdot a } &\xrightarrow{\;}& \widehat{\mathbb{Z}^2} \\ [a] &\mapsto& (1,0,0) \\ [b] &\mapsto& (0,1,0) \\ [n] &\mapsto& (0,0,[n]) \end{array}

is clearly a bijection on underlying sets, and is a group homomorphism since the quotient relation (6) is respected in $\widehat{\mathbb{Z}^2}$ , by (4). \end{proof}

(cf. also arXiv:2203.08030, p 21)

Linear representations

\begin{example} For $k \in \mathbb{Z}$ , linear representations of twice the $\mathbb{Z}_{2k}$ -Heisenberg extension of $\mathbb{Z}^2$ (5)

(7)

\begin{array}{ccc} \widehat{\mathbb{Z}^2} & \xrightarrow{\; W \;} & \mathrm{Aut}\big( \mathscr{H}_1 \big) \end{array}

are given by:

\mathscr{H}_1 \;\coloneqq\; Span_{\mathbb{C}}\Big( {\vert 0 \rangle} ,\, {\vert 1 \rangle} ,\, \cdots ,\, {\vert k-1 \rangle}, \Big)

\begin{array}{ccccccl} W_a & \coloneqq & W(1,0,0) &\colon& {\big\vert n \big\rangle} &\mapsto& \zeta^{2n} {\big\vert n \big\rangle} \\ W_b &\coloneqq& W(0,1,0) &\colon& {\big\vert n \big\rangle} &\mapsto& {\big\vert (n + 1) \,mod\, k \big\rangle} \\ \zeta &\coloneqq& W(0,0,1) &\colon& {\big\vert n \big\rangle} &\mapsto& \zeta {\big\vert n \big\rangle} \mathrlap{\,,} \end{array}

for $\zeta$ a $2k$ th root of unity.

For instance:

(8)

W_a \cdot W_b \cdot \zeta^{-1} \;=\; W(1,1,0) \;=\; W_b \cdot W_a \cdot \zeta^{+1} \,.

\end{example} Notation: In the following we write “ $[n]$ ” for “ $n \,mod\, k$ ” and “ $\sum_{[n]}$ ” for “ $\sum_{n =0}^{k-1}$ ”. \begin{proof} To see that this is a linear representation, by (6) it is sufficient to check that the basic group commutator is represented, in that

(9)

W_a \cdot W_b \,=\, \zeta^2 \, W_b \cdot W_a \mathrlap{\,,}

which is evidently the case, since

\begin{array}{ccl} W_a \cdot W_b {\big\vert [n] \big\rangle} &=& W_a \cdot {\big\vert [n+1] \big\rangle} \\ &=& \zeta^{2(n+1)} \, {\big\vert [n+1] \big\rangle} \mathrlap{\,,} \end{array}

while

\begin{array}{ccl} W_b \cdot W_a {\big\vert [n] \big\rangle} &=& \zeta^{2n} \, W_b {\big\vert [n] \big\rangle} \\ &=& \zeta^{2n} \, {\big\vert [n + 1] \big\rangle} \mathrlap{\,.} \end{array}

\end{proof}

For more on linear representations of the level $=1$ integer Heisenberg group, see Floratos & Tsohantjis 2022.

\begin{remark}\label{RelationToChernSimonsStates} (relation to quantum states and quantum observables of abelian Chern-Simons theory) \linebreak The complex dimension of the representation $\mathscr{H}_1$ (7) is that expected for the space of quantum states of abelian Chern-Simons theory on a 2-torus (cf. Manoliu 1998a p 40, Gelca & Uribe 2010 Prop. 2.2)

(10)

dim(\mathscr{H}_1) \;=\; K \,,

and the group commutator-relation (9)

W_a \cdot W_b = \zeta^2 \, W_b \cdot W_a

of linear operators acting on this irrep (7) reflects the characteristic commutator-relation of Wilson loop-quantum observables in abelian Chern-Simons theory on a 2-torus (cf. Tong 2016 (5.28), p 166). \end{remark}

Modular automorphisms

Moreover, the integer Heisenberg group at leve 2 knows about the modular group acting on its irrep (7), hence on the space of quantum states $\mathscr{H}_{T^2}$ from Rem. \ref{RelationToChernSimonsStates}, hence about the “modular functor” of abelian Chern-Simons theory:

\begin{definition} (modular action on the integer Heisenberg group) \linebreak Since the colored summand in (5) is the canonical symplectic form on $\mathbb{Z}^2$ , the integer symplectic group in dimension 2, hence the modular group

\Big\{ g \in \mathrm{GL}_2(\mathbb{Z}) \,\Big\vert\, \underset{ \mathrm{det}(g) \cdot (a b' - a' b) }{ \underbrace{ (g_{1 1} a + g_{12} b) (g_{2 1} a' + g_{22} b') - (g_{11} a' + g_{12}b') (g_{21} a + g_{22} b) } } = a b' - a'b \Big\} \;\simeq\; \mathrm{SL}_2(\mathbb{Z}) \,,

acts by evident group automorphism on (5):

(11)

\begin{array}{ccc} \mathrm{SL}_2(\mathbb{Z}) \times \widehat{\mathbb{Z}^2} &\xrightarrow{}& \widehat{\mathbb{Z}^2} \\ \Big( g, \big(a,b,[n]\big) \Big) &\mapsto& \Big( g_{11}a + g_{12}b ,\, g_{21}a + g_{22}b ,\, [n] \Big) \mathrlap{\,.} \end{array}

\end{definition}

\begin{remark} \label{GenerationOfModularGroup} Recall (from there) that the modular group $SL_2(\mathbb{Z})$ is the subgroup of the general linear group generated by the two elements

(12)

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{\,.}

and presented via these generators subject to the following relations:

(13)

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 \,.

\end{remark}

\begin{proposition} \label{ModularEquivariantHeisenbergRepresentation} For even $k$ (10) there exists a linear representation

(14)

\begin{array}{ccc} \mathrm{SL}_2(\mathbb{Z}) \times \mathscr{H}_1 &\xrightarrow{\;}& \mathscr{H}_1 \mathrlap{\,,} \end{array}

of the modular group on the underlying complex vector space of (7), which intertwines the action (7) of $\widehat{\mathbb{Z}^2}$ on $\mathscr{H}_1$ , with its automorphic images under the modular group action (11) on the Heisenberg group, in that:

(15)

m(W) \cdot m\Big( {\big\vert [n] \big\rangle} \Big) \;=\; m\Big( W \cdot {\big\vert [n] \big\rangle} \Big) \,, \;\;\;\;\; \forall \left\{ \begin{array}{ccc} m &\in& SL_2(\mathbb{Z}) \\ W &\in& \widehat{\mathbb{Z}^2} \\ {\left\vert [n] \right\rangle} &\in& \mathscr{H}_1 \mathrlap{\,.} \end{array} \right.

In other words, (14) enhances $\mathscr{H}_1$ to a representation of the semidirect product group $\widehat{\mathbb{Z}^2} \rtimes \mathrm{SL}_2(\mathbb{Z})$ with operation

(W,m)\cdot (W',m') \;\coloneqq\; \big( W \cdot m(W'),\, m \cdot m' \big) \,.

This representation (14) is just the modular action known from abelian Chern-Simons theory at even level $k$ (cf. Wen 1990 (5.3), Manoliu 1998a p 67, Gannon 2005 (3.1b)):

(16)

\begin{array}{ccr} S\Big( {\big\vert [n] \big\rangle} \Big) &=& \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert [\widehat{n}] \big\rangle} \\ T\Big( {\big\vert [n] \big\rangle} \Big) &=& \underset{ \eqqcolon 1/c_k }{ \underbrace{ e^{ - \pi \mathrm{i}/12 } } } \, e^{ \tfrac{\pi \mathrm{i}}{k} n^2 } {\big\vert [ n ] \big\rangle} \end{array}

\end{proposition} \begin{proof} To see that (16) is indeed a representation of the modular group, we need to check that the relations (13) are satisfied:

First we find

\begin{array}{rcl} S^2 \Big({\big\vert [n] \big\rangle}\Big) &=& S \Big( \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert[\widehat{n}]\big\rangle} \Big) \\ &=& \sum_{ [\widehat{\widehat n}] } \, \underset{ \delta_0\big( [ n + \widehat{\widehat{n}} ] \big) }{ \underbrace{ \tfrac{1}{k} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, (n + \widehat{\widehat{n}}) }{ k } } } } \, {\big\vert[\widehat{\widehat{n}}]\big\rangle} \\ &=& {\big\vert[-n]\big\rangle} \end{array}

(where under the brace we evaluated the sum of roots of unity), which immediately implies the relation $S^4 = \mathrm{id}$ and thereby, with

T \circ S \Big( {\big\vert [n] \big\rangle} \Big) \;=\; \tfrac{1}{\sqrt{\vert k \vert} c_k} \displaystyle{ \sum_{ [\widehat n] } } \, e^{ \tfrac{ \pi \mathrm{i} }{k} ( \widehat{n}^2 + 2\widehat{n} \, n ) } {\big\vert[\widehat{n}]\big\rangle} \,,

also the relation $S^2 (T S) = (T S) S^2$ . It just remains to show that $(T S)^3 = \mathrm{e}$ or equivalently that $S T S \,=\, T^{-1} S^{-1} T^{-1}$ . Direct computation yields:

\begin{array}{l} T^{-1} S^{-1} T^{-1} {\vert n \rangle} \;=\; T^{-1} S^{-1} e^{ - \tfrac{\pi \mathrm{i}}{k} n^2 } {\vert n \rangle} \\ \;=\; T^{-1} \tfrac{1}{\sqrt{k}} \sum_{\widehat{n}} e^{ \tfrac{\pi \mathrm{i}}{k} ( - n^2 - 2 \widehat{n} n ) } {\vert \widehat{n} \rangle} \\ \;=\; \tfrac{1}{\sqrt{k}} \sum_{\widehat{n}} e^{ \tfrac{\pi \mathrm{i}}{k} ( - n^2 - 2 \widehat{n} n - \widehat{n}^2 ) } {\vert \widehat{n} \rangle} \\ \;=\; \tfrac{1}{\sqrt{k}} \sum_{\widehat{n}} e^{ - \tfrac{\pi \mathrm{i}}{k} (\widehat{n} + n)^2 } {\vert \widehat{n} \rangle} \end{array} \;\;\;\;\;\;\;\;\;\;\;\;\; \text{and} \;\;\;\;\;\;\;\;\;\;\;\;\; \begin{array}{l} S T S {\big\vert n \big\rangle} \;=\; S T \frac{1}{\sqrt{\vert k \vert}} \sum_{ \widehat n } \, e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert \widehat{n} \big\rangle} \\ \;=\; S \tfrac{1}{\sqrt{k}} \sum_{ \widehat n } \, e^{ \tfrac{\pi \mathrm{i}}{k} ( 2 \widehat{n} \, n + \widehat{n}^2 ) } {\big\vert \widehat{n} \big\rangle} \\ \;=\; \frac{1}{k} \sum_{ \widehat n ,\, \widehat{\widehat{n}} } \, e^{ \tfrac{\pi \mathrm{i}}{k} ( 2 \widehat{n} \, n + \widehat{n}^2 + 2 \widehat{\widehat{n}} \widehat{n} ) } {\big\vert \widehat{\widehat{n}} \big\rangle} \\ \;=\; \frac{1}{\sqrt{k}} \sum_{ \widehat{\widehat{n}} } \underbrace{ \tfrac{1}{\sqrt{k}} \sum_{ \widehat n } \, e^{ \tfrac{\pi \mathrm{i}}{k} (\widehat{n} + (n + \widehat{\widehat{n}}))^2 } } e^{ - \tfrac{\pi \mathrm{i}}{k} (n + \widehat{\widehat{n}})^2 } {\big\vert \widehat{\widehat{n}} \big\rangle} \,, \end{array}

where the term over the brace is in fact constant in $n$ and $\widehat{\widehat{n}}$ by the assumption that $k$ is even, because this implies that the summands are $k$ -periodic:

(17)

e^{ \tfrac{\pi \mathrm{i}}{k} (n + k)^2 } \;=\; e^{ \tfrac{\pi \mathrm{i}}{k} (n^2 + 2 n k + k^2) } \;=\; e^{ \tfrac{\pi \mathrm{i}}{k} n^2 } \underset{ 1\;if\;k\;even }{ \underbrace{ e^{ \pi \mathrm{i}(2n + k) } } } \;=\; e^{ \tfrac{\pi \mathrm{i}}{k} n^2 } \,.

This means that the last relation holds if the normalization factor $c_k$ is indeed fixed, as shown in (16), to this quadratic Gauss sum, which evaluated to (see there)

(18)

c_k \;=\; \Big( \tfrac{1}{\sqrt{k}} \textstyle{ \sum_{ \widehat n } } \, e^{ \tfrac{ \pi \mathrm{i} }{k} \widehat{n}^2 } \Big)^{1/3} \;=\; \big(\sqrt{\mathrm{i}}\big)^{1/3} \;=\; e^{ \pi \mathrm{i}/12 } \,.

Finally to see that also the semidirect product of these two groups is represented in that (15) holds:

We may explicitly check this for $m \in \{S,T\}$ any one of the modular generators (12) by unwinding the above definitions:

\begin{array}{ccl} S(W_a) \cdot S\Big( {\big\vert [n] \big\rangle} \Big) &\equiv& W_b^{-1} \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert [\widehat{n}] \big\rangle} \\ &=& \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ (\widehat{n} + 1) \, n }{ k } } {\big\vert [\widehat{n}] \big\rangle} \\ &=& e^{ 2 \pi \mathrm{i} \tfrac{n}{k} } \, S\Big( {\big\vert [n] \big\rangle} \Big) \\ &=& S\Big( W_a {\big\vert [n] \big\rangle} \Big) \mathrlap{\,,} \end{array}

\begin{array}{ccl} S(W_b) \cdot S\Big( {\big\vert [n] \big\rangle} \Big) &\equiv& W_a \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert [\widehat{n}] \big\rangle} \\ &=& \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{2 \pi \mathrm{i} \tfrac{\widehat{n}}{k}} e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert [\widehat{n}] \big\rangle} \\ &=& S\Big( W_b {\big| [n] \big\rangle} \Big) \mathrlap{\,,} \end{array}

\begin{array}{ccl} T(W_a) \cdot T\Big( {\big\vert [n] \big\rangle} \Big) &\equiv& W_a \, e^{ \mathrm{i} \pi \tfrac{n^2}{k} } {\big\vert [n] \big\rangle} \\ &=& e^{2 \pi \mathrm{i} \tfrac{n}{k}} e^{ \mathrm{i} \pi \tfrac{n^2}{k} } {\big\vert [n] \big\rangle} \\ &=& T\Big( W_a \, {\big\vert [n] \big\rangle} \Big) \mathrlap{\,,} \end{array}

and finally, using (8):

\begin{array}{ccl} T(W_b)\cdot T\Big( {\big\vert [n] \big\rangle} \Big) &\equiv& W_b \, W_a \, e^{\pi \mathrm{i} \tfrac{1}{k}} \, e^{\pi \mathrm{i} \tfrac{n^2}{k}} {\big\vert [n] \big\rangle} \\ &=& e^{\pi \mathrm{i} \tfrac{ n^2 + 2n + 1 }{k}} {\big\vert [n + 1] \big\rangle} \\ &=& e^{\pi \mathrm{i} \tfrac{ (n+1)^2 }{k}} {\big\vert [n + 1] \big\rangle} \\ &=& T\Big( W_b {\big\vert [n] \big\rangle} \Big) \mathrlap{\,.} \end{array}

This completes the proof. \end{proof}

As quotients of surface braid group

The level $=2$ Heisenberg extensions of $\mathbb{Z}^{2g}$ by the cyclic group $C_{2(n+g-1)}$ arise as quotient groups of surface braid groups $Br_n(\Sigma_{g})$ of the surface of genus $g$ for $n \geq 3$ [[Bellingeri, Gervais & Guaschi 2008 (10)](#BellingeriGervaisGuaschi08)].

Under the identification of these level $=2$ Heisenberg groups with the fundamental groups $\pi_1$ of 2-Cohomotopy cocycle spaces (see there) this quotient map is the image under $\pi_1(-)$ of the cohomotopy charge map/scanning map

Conf_n(\Sigma) \longrightarrow Map_{n+g-1}(\Sigma, S^2)

from the configuration space of $n$ points of the surface.

This observation is due to Martin Palmer (private communication, April 2025). It may be seen from the presentation given by BGG08 (10): Each Artin generator is sent to the central generator and the loop around the $a_i$ ( $b_i$ ) cycle is sent to the corresponding element $a_i$ ( $b_i$ ).

The analogous statement of surfaces with one boundary connected component is Blanchet, Palmer & Shaukat 2021 Cor. 8, 2025 Prop. 1.1

\linebreak

Related entries

Heisenberg group

Heisenberg manifold

2-Cohomotopy moduli of surfaces

Literature

In group theory

Generally on the integer/discrete Heisenberg group in the algebra and group theory literature

Soo Teck Lee, Judith A. Packer: The Cohomology of the Integer Heisenberg Groups, Journal of Algebra 184 1 (1996) 230-250 [[doi:10.1006/jabr.1996.0258](https://doi.org/10.1006/jabr.1996.0258)]

(concerning its group cohomology)
David S. Dummit, Richard M. Foote Ex. 11 on p. 35: Abstract Algebra, Wiley (2003) [[ISBN:978-0-471-43334-7](https://www.wiley.com/en-us/Abstract+Algebra%2C+3rd+Edition-p-9780471433347), pdf]
Roman Budylin: Conjugacy classes in discrete Heisenberg groups, Sbornik: Mathematics 205 8 (2014) 1069–1079 [[arXiv:1405.5499](https://arxiv.org/abs/1405.5499), doi:10.1070/SM2014v205n08ABEH004410]
Daniel Bump, Persi Diaconis, Angela Hicks, Laurent Miclo, Harold Widom: An Exercise (?) in Fourier Analysis on the Heisenberg Group, Annales de la Faculté des sciences de Toulouse: Mathématiques, Serie 6, Volume 26 (2017) no. 2, pp. 263-288 [[arXiv:1502.04160](https://arxiv.org/abs/1502.04160), numdam:AFST_2017_6_26_2_263_0]
Cornelia Druţu, Michael Kapovich (appendix by Bogdan Nica): Geometric group theory, Colloquium Publications 63, AMS (2018) [[ISBN:978-1-4704-1104-6](https://bookstore.ams.org/coll-63/), pdf]
Jayadev S. Athreya, Ioannis Konstantoulas: Lattice deformations in the Heisenberg group, Groups, Geometry and Dynamics 14 3 (2020) 1007–1022 [[arXiv:1510.01433](https://arxiv.org/abs/1510.01433), doi:10.4171/ggd/572]
Uri Bader, Vladimir Finkelshtein, §5 of: On the horofunction boundary of discrete Heisenberg group, Geom Dedicata 208 (2020) 113–127 [[arXiv:1904.11234](https://arxiv.org/abs/1904.11234), doi:10.1007/s10711-020-00513-x]
Ruiwen Dong, p 9 of: Recent advances in algorithmic problems for semigroups, ACM SIGLOG News 10 4 (2023) 3-23 [[arXiv:2309.10943](https://arxiv.org/abs/2309.10943), doi:10.1145/3636362.3636365]

On (invertibility in) the group algebra:

Martin Göll, Klaus Schmidt, Evgeny Verbitskiy: A Wiener Lemma for the discrete Heisenberg group: Invertibility criteria and applications to algebraic dynamics, Monatsh Math 180 (2016) 485–525 [[doi:10.1007/s00605-016-0894-0](https://doi.org/10.1007/s00605-016-0894-0), arXiv:1603.08225]

On the representation theory with an eye towards quantum information theory:

Emmanuel G. Floratos, I. Tsohantjis: Complete set of unitary irreps of Discrete Heisenberg Group $H W_{2^s}$ , Int. J. Theor. Phys. 64 2 (2025) 22 [[arXiv:2210.04263](https://arxiv.org/abs/2210.04263), doi:10.1007/s10773-025-05888-9]

On the automorphism group:

D.V. Osipov: Discrete Heisenberg group and its automorphism group, Mathematical Notes 98 1 (2015) 185-188 [[arXiv:1505.00348](https://arxiv.org/abs/1505.00348), doi:10.1134/S0001434615070160]

As quotient groups of surface braid groups:

Paolo Bellingeri, Sylvain Gervais, John Guaschi: Lower central series of Artin–Tits and surface braid groups, Journal of Algebra 319 4 (2008) 1409-1427 [doi:10.1016/j.jalgebra.2007.10.023]

In modular/Chern-Simons theory

In relation to $U(1)$ -current algebra (WZW-model):

Ludwig D. Fadeev: Discrete Heisenberg-Weyl Group and modular group, Lett Math Phys 34 (1995) 249–254 [[doi:10.1007/BF01872779](https://doi.org/10.1007/BF01872779)]

As describing the phase space of abelian Chern-Simons theory on closed Riemann surfaces (and its relation to skein relations and Riemann's theta functions):

Răzvan Gelca, Alejandro Uribe: From classical theta functions to topological quantum field theory, in: The Influence of Solomon Lefschetz in Geometry and Topology: 50 Years of Mathematics at CINVESTAV, Contemporary Mathematics 621, AMS (2014) 35-68 [[arXiv:1006.3252](http://arxiv.org/abs/1006.3252), doi;10.1090/conm/621, ams:conm-621, slides pdf, pdf]
Răzvan Gelca, Alastair Hamilton: Classical theta functions from a quantum group perspective, New York J. Math. 21 (2015) 93–127 [[arXiv:1209.1135](http://arxiv.org/abs/1209.1135), nyjm:j/2015/21-4]
Răzvan Gelca, Alastair Hamilton: The topological quantum field theory of Riemann’s theta functions, Journal of Geometry and Physics 98 (2015) 242-261 [[doi:10.1016/j.geomphys.2015.08.008](https://doi.org/10.1016/j.geomphys.2015.08.008), arXiv:1406.4269]

The above discussion of irreps and modular automorphisms related to abelian Chern-Simons theory follows:

Hisham Sati, Urs Schreiber: Engineering of Anyons on M5-Probes via Flux-Quantization, Lecture Notes [[arXiv:2501.17927](https://arxiv.org/abs/2501.17927)]
Hisham Sati, Urs Schreiber: Understanding Topological Quantum Gates in FQH Systems via the Algebraic Topology of exotic Flux Quantization

Heisenberg homology

Concerning the construction of linear representations of surface-mapping class groups on the homology of configuration spaces of points, twisted by linear representations of the integer Heisenberg group (“Heisenberg homology”):

Christian Blanchet, Martin Palmer, Awais Shaukat, Heisenberg homology on surface configurations [arXiv:2109.00515]
Awais Shaukat, Christian Blanchet, Weakly framed surface configurations, Heisenberg homology and Mapping Class Group action, Archiv der Mathematik 120 (2023) 99–109 [doi:10.1007/s00013-022-01793-3, arXiv:2206.11475]
Christian Blanchet, Martin Palmer, Awais Shaukat: Action of subgroups of the mapping class group on Heisenberg homologies, Contemporary Mathematics 813, AMS (2025) [doi:10.1090/conm/813, arXiv:2306.08614]
Christian Blanchet: Heisenberg homology of ribbon graphs, in: Essays in Geometry, IRMA Lectures in Math. and Theor. Physics 34, EMS (2023) [doi:10.4171/irma/34/10, arXiv:2401.04477]

and relation to abelian Chern-Simons theory:

Aleksei Andreev, Anna Beliakova, Christian Blanchet: Abelian TQFTS and Schrödinger local systems [arXiv:2306.10725]

Review:

Martin Palmer: Representations of the Torelli group via the Heisenberg group, talk at Workshop for Young Researchers in Mathematics – 10th ed. (2021) $[$ pdf, pdf]
Martin Palmer, Mapping class group representations via Heisenberg, Schrödinger and Stone-von Neumann, talk at CQTS (March 2022)
Christian Blanchet: Heisenberg homologies of surface configurations, talk at Geometric/Topological Quantum Field Theories Workshop @ CQTS NYUAD (2023) [slides:pdf, video:YT]

Last revised on February 19, 2026 at 07:45:30. See the history of this page for a list of all contributions to it.