Theta representation

In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics. It gains its name from the fact that the Jacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg group. The representation was popularized by David Mumford.

The theta representation is a representation of the continuous Heisenberg group ${\displaystyle H_{3}(\mathbb {R} )}$ over the field of the real numbers. In this representation, the group elements act on a particular Hilbert space. The construction below proceeds first by defining operators that correspond to the Heisenberg group generators. Next, the Hilbert space on which these act is defined, followed by a demonstration of the isomorphism to the usual representations.

Let f(z) be a holomorphic function, let a and b be real numbers, and let ${\displaystyle \tau }$ be an arbitrary fixed complex number in the upper half-plane; that is, so that the imaginary part of ${\displaystyle \tau }$ is positive. Define the operators S_a and T_b such that they act on holomorphic functions as

${\displaystyle (S_{a}f)(z)=f(z+a)=\exp(a\partial _{z})f(z)}$

and

${\displaystyle (T_{b}f)(z)=\exp(i\pi b^{2}\tau +2\pi ibz)f(z+b\tau )=\exp(i\pi b^{2}\tau +2\pi ibz+b\tau \partial _{z})f(z).}$

It can be seen that each operator generates a one-parameter subgroup:

${\displaystyle S_{a_{1}}\left(S_{a_{2}}f\right)=S_{a_{1}+a_{2}}f}$

and

${\displaystyle T_{b_{1}}\left(T_{b_{2}}f\right)=T_{b_{1}+b_{2}}f.}$

However, S and T do not commute:

${\displaystyle S_{a}T_{b}=\exp(2\pi iab)T_{b}S_{a}.}$

Thus ${\displaystyle S}$ and ${\displaystyle T}$ together with a unitary phase form a nilpotent Lie group, the continuous real Heisenberg group, parametrizable as ${\displaystyle H=U(1)\times \mathbb {R} \times \mathbb {R} }$, where ${\displaystyle U(1)}$ is the unitary group.

A general group element ${\displaystyle U_{\tau }(\lambda ,a,b)\in H}$ then acts on a holomorphic function f(z) as

${\displaystyle U_{\tau }(\lambda ,a,b)f(z)=\lambda (S_{a}T_{b}f)(z)=\lambda \exp(i\pi b^{2}\tau +2\pi ibz)f(z+a+b\tau ),}$

where ${\displaystyle \lambda \in U(1)}$. The subgroup ${\displaystyle U(1)}$ is the center ${\displaystyle Z(H)}$ of ${\displaystyle H}$, and is also its commutator subgroup ${\displaystyle [H,H]}$. The parameter ${\displaystyle \tau }$ on ${\displaystyle U_{\tau }(\lambda ,a,b)}$ serves only to remind that every different value of ${\displaystyle \tau }$ gives rise to a different representation of the action of the group.

The action of the group elements ${\displaystyle U_{\tau }(\lambda ,a,b)}$ is unitary and irreducible on a certain Hilbert space of functions. For a fixed value of ${\displaystyle \tau }$, define a norm on entire functions of the complex plane as

${\displaystyle \Vert f\Vert _{\tau }^{2}=\int _{\mathbb {C} }\exp \left({\frac {-2\pi y^{2}}{\Im \tau }}\right)|f(x+iy)|^{2}\,dx\,dy.}$

Here, ${\displaystyle \Im \tau }$ is the imaginary part of ${\displaystyle \tau }$ and the domain of integration is the entire complex plane. Let ${\displaystyle {\mathcal {H}}_{\tau }}$ be the set of entire functions f with finite norm. The subscript ${\displaystyle \tau }$ is used only to indicate that the space depends on the choice of parameter ${\displaystyle \tau }$. This ${\displaystyle {\mathcal {H}}_{\tau }}$ forms a Hilbert space. The action of ${\displaystyle U_{\tau }(\lambda ,a,b)}$ given above is unitary on ${\displaystyle {\mathcal {H}}_{\tau }}$, that is, ${\displaystyle U_{\tau }(\lambda ,a,b)}$ preserves the norm on this space. Finally, the action of ${\displaystyle U_{\tau }(\lambda ,a,b)}$ on ${\displaystyle {\mathcal {H}}_{\tau }}$ is irreducible.

This norm is closely related to that used to define the Segal–Bargmann space.

The above theta representation of the Heisenberg group is isomorphic to the canonical Weyl representation of the Heisenberg group. In particular, this implies that ${\displaystyle {\mathcal {H}}_{\tau }}$ and ${\displaystyle L^{2}(\mathbb {R} )}$ are isomorphic as ${\displaystyle H}$-modules. Let

${\displaystyle M(a,b,c)={\begin{bmatrix}1&a&c\\0&1&b\\0&0&1\end{bmatrix}}}$

stand for a general group element of ${\displaystyle H_{3}(\mathbb {R} )}$. In the canonical Weyl representation, for every real number h, there is a representation ${\displaystyle \rho _{h}}$ acting on ${\displaystyle L^{2}(\mathbb {R} )}$ as

${\displaystyle \rho _{h}(M(a,b,c))\psi (x)=\exp(ibx+ihc)\psi (x+ha)}$

for ${\displaystyle x\in \mathbb {R} }$ and ${\displaystyle \psi \in L^{2}(\mathbb {R} )}$.

Here, h is the Planck constant. Each such representation is unitarily inequivalent. The corresponding theta representation is:

${\displaystyle M(a,0,0)\mapsto S_{ah},}$

${\displaystyle M(0,b,0)\mapsto T_{b/2\pi },}$

${\displaystyle M(0,0,c)\mapsto e^{ihc}.}$

The Heisenberg group can be used to give a unified account of theta functions in complex analysis and algebraic geometry. For ${\displaystyle \tau }$ in the upper half-plane, the standard Jacobi theta function is

${\displaystyle \vartheta (z,\tau )=\sum _{n\in \mathbb {Z} }\exp(\pi in^{2}\tau +2\pi inz).}$

It is an entire function of ${\displaystyle z}$ satisfying the transformation laws

${\displaystyle \vartheta (z+1,\tau )=\vartheta (z,\tau ),\qquad \vartheta (z+\tau ,\tau )=e^{-\pi i\tau -2\pi iz}\vartheta (z,\tau ).}$

More generally, for integers ${\displaystyle a}$ and ${\displaystyle b}$,

${\displaystyle \vartheta (z+a+b\tau ,\tau )=\exp(-\pi ib^{2}\tau -2\pi ibz)\vartheta (z,\tau ).}$

Define the subgroup ${\displaystyle \Gamma _{\tau }\subset H}$ as

${\displaystyle \Gamma _{\tau }=\{U_{\tau }(1,a,b)\in H:a,b\in \mathbb {Z} \}.}$

The preceding transformation laws say exactly that ${\displaystyle \vartheta (z,\tau )}$ is invariant under ${\displaystyle \Gamma _{\tau }}$. It can be shown that the Jacobi theta function is the unique such entire function, up to scalar multiple.¹

Thus ${\displaystyle \vartheta (z,\tau )}$ is not an ordinary function on the elliptic curve

${\displaystyle E_{\tau }=\mathbb {C} /(\mathbb {Z} +\tau \mathbb {Z} ),}$

but rather a section of a line bundle on ${\displaystyle E_{\tau }}$. In one common convention, this line bundle is obtained from ${\displaystyle \mathbb {C} \times \mathbb {C} }$ by the identifications

${\displaystyle (z,w)\sim (z+1,w),\qquad (z,w)\sim (z+\tau ,e^{-\pi i\tau -2\pi iz}w).}$

The exponential factors in the theta transformation laws are the corresponding factors of automorphy. In this sense, a theta function is a function on the universal cover whose transformation law allows it to descend as a section of a line bundle on the quotient torus.¹

The Heisenberg group appears because translations of ${\displaystyle z}$ must be accompanied by scalar factors in order to preserve these transformation laws. Thus translations in the two period directions define a projective representation, and the corresponding central extension is a Heisenberg group.

Theta functions with rational characteristics are obtained by applying Heisenberg operators to ${\displaystyle \vartheta }$. For ${\displaystyle a,b\in \mathbb {Q} }$, define

${\displaystyle \vartheta _{a,b}(z,\tau )=(S_{b}T_{a}\vartheta )(z,\tau )=e^{\pi ia^{2}\tau +2\pi ia(z+b)}\vartheta (z+a\tau +b,\tau ).}$

Equivalently,

${\displaystyle \vartheta _{a,b}(z,\tau )=\sum _{n\in \mathbb {Z} }\exp \!\left(\pi i(n+a)^{2}\tau +2\pi i(n+a)(z+b)\right).}$

Changing ${\displaystyle a}$ and ${\displaystyle b}$ by integers changes these functions only by simple scalar factors, so the characteristics are naturally considered modulo ${\displaystyle \mathbb {Z} }$.

This gives a concrete finite-dimensional form of the Heisenberg representation. If ${\displaystyle L}$ is the degree-one theta line bundle on ${\displaystyle E_{\tau }}$, then

${\displaystyle \dim H^{0}(E_{\tau },L^{\otimes N})=N.}$

A basis of this space may be chosen from theta functions with characteristics. The action of translations by ${\displaystyle N}$-torsion points, together with the necessary scalar factors, gives a finite Heisenberg group acting on ${\displaystyle H^{0}(E_{\tau },L^{\otimes N})}$. This is the finite-dimensional analogue of the Schrödinger representation.¹

More generally, one can associate to a line bundle ${\displaystyle L}$ on an abelian variety ${\displaystyle A}$ its theta group ${\displaystyle G(L)}$. Let

${\displaystyle K(L)=\{x\in A\mid t_{x}^{*}L\simeq L\},}$

where ${\displaystyle t_{x}:A\to A}$ denotes translation by ${\displaystyle x}$. The theta group consists of pairs ${\displaystyle (x,\phi )}$, where ${\displaystyle x\in K(L)}$ and ${\displaystyle \phi :L\to t_{x}^{*}L}$ is an isomorphism of line bundles. It fits into a central extension

${\displaystyle 1\longrightarrow \mathbb {G} _{m}\longrightarrow G(L)\longrightarrow K(L)\longrightarrow 1.}$

When ${\displaystyle L}$ is ample, ${\displaystyle K(L)}$ is finite, and ${\displaystyle G(L)}$ is a finite algebraic analogue of the Heisenberg group.²³

The group ${\displaystyle G(L)}$ acts naturally on the vector space of sections ${\displaystyle H^{0}(A,L)}$. This action is the algebro-geometric analogue of the Schrödinger representation of the real Heisenberg group. When ${\displaystyle L}$ is ample, the commutator in ${\displaystyle G(L)}$ induces a nondegenerate alternating pairing on ${\displaystyle K(L)}$, analogous to the symplectic form used in the construction of the ordinary Heisenberg group. A finite version of the Stone–von Neumann theorem describes the resulting irreducible representation with prescribed central character.³

David Mumford used this Heisenberg-group formalism to give an algebraic theory of theta functions and to study equations defining abelian varieties.²³