8.1 Introduction

Coherent states, since their inception dating back to E. Schrödinger’s paper [49], play—either in their original form or via their multifaceted generalisations—a prominent role in several issues in quantum mechanics, both foundational and applicative (we refer e.g. to the general treatises [1, 26, 39] for comprehensive and wide-ranging overviews). Among the many research lines related to standard and generalised coherent states we have the one related to Kähler geometry ([14, 20, 28, 34, 44, 58] just to pinpoint a few references) (see also [25] for an extension to compact integral symplectic manifolds).

In this survey we wish to review some specific geometric aspects of coherent states, closely related to the past and present research activity of the author, with possible novelties regarding the connection with Fisher information issues, in line with [18, 31, 32, 61]. Specifically, we adopt the framework of geometric quantum mechanics (see e.g. [6, 12, 18, 59]): it is then well known that the quantum version of the Fisher information defined in [22] coincides with the Fubini-Study metric on the projective Hilbert space representing the pure states of a quantum system [18]. We extend the computation of [18] to the case where the quantum system possesses internal degrees of freedom (described via an irreducible representation of a compact simple Lie group G, e.g. SU(2)) pointing out the Fisher information significance of the coherent state manifold, naturally provided by the Borel-Weil theorem (an instance of holomorphic geometric quantization). This ties up with the “gauge” approach to generalised Schrödinger equations discussed in [61]. We briefly revisit further geometric applications of coherent state techniques in the Sato-Segal-Wilson Grassmannian context, with application to KP-type hierarchies, referring to [43, 57, 62,63,64, 69] for full details.

Subsequently, we cursorily touch upon an application of hyperplane sections to 2d-vortex theory, based on [38].

The rest of the paper concerns a possibly novel interpretation of standard coherent states in terms of the Fourier-Mukai-Nahm theory: in this form they recently cropped up in Riemann surface braid group representations [60] and in the revisitation of the geometric quantization approach to Landau systems carried out in [19].

8.2 Geometric Quantum Mechanics

In this preliminary section we give a short review of the formalism of geometric quantum mechanics, tailored to our purposes. We refer e.g. to [6, 12, 18, 59] for more detailed treatments.

8.2.1 Fubini-Study Metric and Distance

Let us consider a finite dimensional—for the time being—complex vector space \((V, \langle \, | \, \rangle )\) (the inner product is taken to be linear in the second argument, antilinear in the first, to be definite), together with its projectivisation \({\mathbb P}(V)\): a point in \({\mathbb P}(V)\) represents a ray (complex one-dimensional subspace of V). Quantum mechanically, points in \({\mathbb P}(V)\) represents the pure states of a quantum system. Let \([ \,\,]\) denote the standard projection \(V \setminus \{0\} \rightarrow {\mathbb P}(V)\). The manifold \({\mathbb P}(V)\) is acted upon transitively by the unitary group U(V) (isometries of the inner product), with Lie algebra \({\mathfrak u}(V)\) (skew-hermitian endomorphisms of V).

The fundamental vector field \(A^{\sharp }\) associated to \(A \in \mathfrak {u}(V)\) reads (evaluated at \([v] \in {\mathbb P}(V)\), \(\parallel v \parallel = 1\))

$$\begin{aligned} A^{\sharp }|_{[v]} = | v \rangle \langle A v | + | A v \rangle \langle v | \end{aligned}$$

In view of homogeneity, these vectors span the tangent space of P(V) at each point. The (action of the) natural complex structure J reads, accordingly:

$$\begin{aligned} J|_{[v]} A^{\sharp }_{[v]} = | v \rangle \langle i A v | + | i A v \rangle \langle v | . \end{aligned}$$

The expression for the standard (i.e. Fubini-Study) metric \(g_{FS}\) and Kähler form \(\omega _{FS}\) read, respectively:

$$\begin{aligned} {g_{FS}}_{[v]}(A^{\sharp }|_{[v]}, B^{\sharp }|_{[v]}) = {\mathfrak {Re}} \{ \langle Av | Bv \rangle + \langle v | Av \rangle \langle v | Bv \rangle \} \end{aligned}$$

and

$$\begin{aligned} {\omega _{FS}}_{[v]}(A^{\sharp }|_{[v]}, B^{\sharp }|_{[v]}) = {g_{FS}}_{[v]}(J|_{[v]} (A^{\sharp }|_{[v]}, B^{\sharp }|_{[v]}) = \frac{i}{2} \langle v | [A , B] v \rangle . \end{aligned}$$

Also, notice that

$$\parallel \! A^{\sharp }_{[v]}\!\parallel _{FS} = \parallel \! A\, v - \langle v, A\, v \rangle v\!\parallel = \varDelta _{[v]}(A) $$

(the dispersion of the “observable” A in the state [v]).

Pulling everything back via the map \([\,]\), and working on a general complex separable Hilbert space H (neglecting domain issues for the operators involved), one obtains the Fubini-Study hermitian metric, written as in [18]:

$$ {\mathfrak h} (\psi , \psi ) = \frac{\langle d\psi | d\psi \rangle }{\langle \psi | \psi \rangle } - \frac{\langle d\psi | \psi \rangle \langle \psi | d\psi \rangle }{\langle \psi | \psi \rangle ^2} $$

We explicitly notice that, if \(\psi = \psi _1 \otimes \psi _2 \in H = H_1 \otimes H_2\), then (obvious notation)

$$ {\mathfrak h} (\psi , \psi ) = {\mathfrak h_1} (\psi _1, \psi _1) + {\mathfrak h_2} (\psi _2, \psi _2) $$

which is a manifestation of the Segre embedding

$$ S: \mathbb {P}(H_1) \times \mathbb {P}(H_2) \rightarrow \mathbb {P}(H_1 \otimes H_2) $$

—reading in coordinates (shorthand notation) \(((\alpha _j), (\beta _k)) \mapsto (\alpha _j\beta _k)\)—that is:

$$ S^*{\mathfrak h} = {\mathfrak h_1} + {\mathfrak h_2} $$

(slight notational abuse). One can give a closed formula for the Fubini-Study distance \(d(\,,\,)\) between two points \([\psi ]\) and \([\psi + \delta \psi ]\), having a clear geometrical interpretation (see e.g. [16]). Let \( \parallel \psi \parallel =1\). Then

$$ d ([\psi ],[\psi + \delta \psi ])= \arccos \big [ \frac{|\langle \psi + \delta \psi | \psi \rangle | }{\sqrt{\langle \psi + \delta \psi | \psi + \delta \psi \rangle }} \big ] $$

Expanding to first order we easily get

$$ d \approx \langle \delta \psi | \delta \psi \rangle - \langle \delta \psi | \psi \rangle \langle \psi | \delta \psi \rangle = \mathfrak {Re}\{ \langle d\psi | d \psi \rangle - \langle d\psi | \psi \rangle \langle \psi | d\psi \rangle \} $$

8.2.2 Calabi’s Diastasis Function

Recall that the Calabi diastasis function D on a Kähler manifold \((M, \omega )\) [14, 15, 62] is manufactured through the choice of a local Kähler potential f, fulfilling

$$ \frac{i}{2} \partial \overline{\partial } f = \omega $$

via the expression, in local complex coordinates (with a sesquiholomorphic local extension of the Kähler potential understood):

$$ D(z, w) := f(z,z) + f(w,w) - f(z,w) - f(w,z) $$

and it turns out to be a global object. The Calabi diastasis function for a projective Hilbert space reads

$$ D ([\psi _1], [\psi _2]) = 2 \log \frac{\parallel \psi _1\parallel \, \parallel \psi _2\parallel }{ | \langle \psi _1 | \psi _2 \rangle |} $$

whence (obvious notation)

$$ d = \arccos e^{-\frac{D}{2}} $$

Notice that, expanding at first order, we get

$$ \delta D = (\delta d)^2 $$

8.2.3 The Chern-Bott Connection on the Tautological Bundle

The tautological line bundle \(T \rightarrow {\mathbb P}(V)\) (assembled by associating to any point in \({\mathbb P}(V)\) the complex line (1-dimensional vector subspace on V) it represents) comes equipped with a canonical hermitian and holomorphic connection \(\nabla \) (Chern-Bott connection, see e.g. [21]), inherited, in the case at hand, from the standard inner product on V, concisely written (for \(\chi \in V\), \(\parallel \chi \parallel = 1\), with abuse of notation)

$$ \nabla \chi := d \chi \,- \!\langle \chi , d \chi \rangle \chi $$

or, in terms of covariant derivatives along fundamental vector fields \(A^{\sharp }\) on \({\mathbb P}(V)\) induced by \(A \in {\mathfrak u}(V))\):

$$ \nabla _{A^{\sharp }} \chi = A \chi - \langle \chi , A \chi \rangle \chi $$

Its curvature form \(\varOmega \) reads:

$$ \varOmega = - d \langle \chi |d\chi \rangle = -\langle d\chi |d\chi \rangle $$

The above connection has a clear (Levi-Civita type) geometrical significance (in a nutshell, one varies a ray within the ambient space and projects the result on the line again) and turns out to be ubiquitous in mathematics, see also Sects. 8.2.4, 8.3 and 8.7.

The dual bundle to \(T \rightarrow {\mathbb P}(V)\), denoted by \({\mathcal H} \rightarrow {\mathbb P}(V)\), is called the hyperplane section bundle (its Chern class \(c_1({\mathcal H}) = 1\), and it possesses non trivial holomorphic sections (hyperplane sections)). The geometry and topology of the latter bundle provide the clue to understanding all phase issues in quantum mechanics (cf. [2, 3, 6, 8, 16, 52, 59] as well).

8.2.4 The Borel-Weil Theorem and Coherent States

Let us briefly revisit the basics of the Borel-Weil(-Bott) theory (see e.g. [10, 42]) within the framework of geometric quantum mechanics (see e.g. [56] for further details). Let V be a finite dimensional complex vector space carrying an irreducible representation \(U(\cdot )\) of a compact simple Lie group G, with associated projective space \({\mathbb P}(V)\). Let \([ \,\,]\) again denote the standard projection \(V \setminus \{0\} \rightarrow {\mathbb P}(V)\). Then V becomes the total space of a complex G-homogeneous line bundle \( { \mathcal L} \rightarrow Y\), where Y is a compact Kähler manifold (embedded in \({\mathbb P}(V)\) given as

$$ Y = G/H $$

explicitly described as follows. Let \(|0\rangle \in V\) be a (regular) highest weight vector. The vectors \(U(g)|0\rangle \), \(g \in G\), are called coherent state vectors and their corresponding points \([U(g)|0\rangle ] \in {\mathbb P}(V)\) coherent states, giving rise, collectively, to the Kähler manifold Y (which is then the G-orbit of \([|0\rangle ] \in {\mathbb P}(V)\). The group H is isomorphic to the isotropy group of the vector \(|0\rangle \) under the action of G). The fibre \({ \mathcal L}_y\) at \(y \in Y\) is the complex line corresponding to \( y = [U(g)|0\rangle ]\) (for some \(g \in G\) determined up to \(h \in H\)).

The Kähler structure is inherited from the projective space one and V becomes the space of holomorphic sections of the dual bundle \({ \mathcal L} \rightarrow Y\). The latter, in turn, carries a natural hermitian metric and an ensuing Chern-Bott (-Berry) connection, depicted as before. Working out the SU(2) case leads to monopole bundles (see e.g. [61]).

8.3 Holomorphic Geometric Quantization and Coherent States

The Borel-Weil theory outlined above, together with its accompanying Lie group coherent states fits into the general setting provided by (holomorphic) geometric quantization [24, 27, 53]. We give a short outline of the theory, referring, in particular, to [58, 68] for more details.

Let \((M, \omega )\) be a (connected, compact, to be definite) Kähler manifold, with integral Kähler form. Then, in view of the Weil-Kostant theorem, there exists a holomorphic line bundle \(L \rightarrow M\), equipped with a hermitian metric \((\,,\,)\) and connection \(\nabla \) (the Chern-Bott one) compatible with it and with the holomorphic structure, with curvature form \(\varOmega _{\nabla } = -2\pi i \omega \) (so that the first Chern class \(c_1(L) = [\omega ]\)).

Under suitable positivity conditions (of Kodaira type) the space of holomorphic sections \(H^0(L)\) is non trivial (and finite dimensional, in view of compactness), and its dimension \(h^0(L)\) is indeed a topological invariant by the Riemann-Roch theorem. It is indeed a Hilbert space with scalar product \(\langle , \rangle \) given by \(\int _M(\, , \,)\) (the integration is carried with respect to the Liouville measure). In [58] the volume of M is normalised to \(h^0(L)\), in order to implement a proper semiclassical intepretation.

Also, (if \(L \rightarrow M\) is very ample), M embeds (à la Kodaira) into \({\mathbb P}(H^0(L))\). The embedding is given precisely by the coherent state map

$$ \epsilon : M \rightarrow {\mathbb P}(H^0(L)) $$

induced by the (continuous, in the holomorphic setting) evaluation map \( H^0(L) \ni s \mapsto s(m) \in L_m \), \(m\in M\). This gives rise to holomorphic sections \(s_m\) (Rawnsley’s coherent states [44]), defined up to a phase, which can be alternatively characterised as follows (see [58] and for extensions, [25]): they maximise (s, s) (m) among all normalised sections \(s \in H^0(L)\). Actually (in the so-called regular case, which holds for instance if M is homogeneous and simply connected) one has

$$ \epsilon ^* \omega _{FS} = \omega $$

namely, the geometry of M is “projectively induced”.

It is well-known that Rawnsley’s coherent states enjoy several properties of standard coherent states: for instance, they exhibit optimal semiclassical behaviour, in that they undergo a classical evolution and minimise the Heisenberg relations in a suitable sense [56].

We explicitly point out the following formula for the scalar product of two normalised coherent state wave functions, which can be employed as an Ansatz when defining a measure is problematic (cf. e.g. [37] and below):

$$ \langle \varPsi _z ,\varPsi _w \rangle = e^{f(z,w)} e^{-\frac{1}{2} f(z,z) } e^{-\frac{1}{2} f(w,w) } $$

and one can use the canonical diastatic potential.

8.4 Fisher Information and Coherent States

In this section we wish to slightly extend the Fisher information analysis of [4, 18] where the quantum Fisher information defined in [22] was shown to coincide with the Fubini-Study hermitian metric, to cover the case of vector valued wave functions, in presence of a gauge group. We shall easily abut at a Fisher theoretical significance of the coherent state manifold (which may be viewed as a receptacle for a generalized spin) yielding a connection with the investigations of [32, 46, 47, 61]. Let \(X = {\mathbb R}^3\) (endowed with Lebesgue measure) for simplicity, and \(H = L^2(X,V) \cong L^2(X)\otimes V\), consisting of wave functions \(\psi = \sqrt{\rho } e^{i S} \chi \), \(\chi \in V\) the (finite dimensional) G-representation space), Assume \( \langle \chi | \chi \rangle \equiv \chi ^{\dagger }\chi = 1\) and

$$ \langle \psi | \psi \rangle = \int _X \rho \chi ^{\dagger }\chi = \int _X \rho = 1 $$

For the sequel we also assume \(\rho \) smooth and rapidly vanishing at infinity.

Define (for a generic tensor f, depending on X and on a parameter space \(\varTheta \) given, in the present case, by the coherent state manifold Y):

$$ {\mathbb E} [f] \equiv E_{\rho }[f] := \int _X \rho \, f $$

Upon computing the pull-back \({\mathfrak h}_X(\psi , \psi )\) to \(\varTheta \) for the above \(\psi \), we get the following generalisation of the basic result of [18]:

Theorem 8.1

The full quantum Fisher information, defined via

$$ {\mathfrak g}_X = \mathfrak {Re}\{ \frac{\langle d\psi | d\psi \rangle }{\langle \psi | \psi \rangle } - \frac{\langle d\psi | \psi \rangle \langle \psi | d\psi \rangle }{\langle \psi | \psi \rangle ^2} \} $$

reads (pull-back to \(\varTheta \) understood, so the differential d acts on Y)

$$ {\mathfrak g}_X = \int _X \rho \{ \frac{1}{4} [\frac{d \rho }{\rho }]^2 + dS^2 + d\chi ^{\dagger }d\chi \} - \big \{ \int _X \rho (dS + (-i)\chi ^{\dagger }d\chi )\big \}^2 $$

or, equivalently (with the above notation)

$$ {\mathfrak g}_X = {\mathbb E} [ \frac{1}{4} (\frac{d \rho }{\rho })^2] + {\mathbb E} [dS^2] - \{{\mathbb E} [dS]\}^2 + {\mathbb E} [{\mathfrak g}_{FS} ([\chi ])]- 2 {\mathbb E} [dS] \cdot {\mathbb E} [(-i)\chi ^{\dagger }d\chi ] $$

One notices, besides the individual contributions, a “coupling” term proportional to \( \int _X \rho dS \cdot \int _X \rho (-i)\chi ^{\dagger }d\chi . \) i.e. the product of the integrated “external” and “internal” probability currents (cf. [61]).

If the internal space V is one-dimensional, one retrieves the expression given in [18]. We also remark that even if \(dS = 0\) (no “adiabatic” variation with respect to the parameter space), one gets the classical Fisher information (involving the “pilot” Weyl field, see e.g. [61]) plus the (quantum) Fisher information of the internal manifold, showing the tight entanglement among external and internal degrees of freedom. Also, the present picture matches the approach in [61] (cf. Proposition 10.2) aimed at geometrising Reginatto’s Fisher information approach to Schrödinger and Pauli equations [46, 47].

Moreover, if \(\rho \) and S do not depend on \( y\in Y= \varTheta \), and \(\chi = \chi (y)\), the coupling term vanishes, consistently with the Segre embedding formula.

8.5 A Coherent State Interpretation of the KP Hierarchy

This theory can be viewed as an infinite dimensional manifestation of holomorphic geometric quantization and Borel-Weil theory. Our account will be rather succinct, referring to the original papers [43, 57, 62,63,64, 69] for a complete treatment. Also refer to [11, 40, 66] for general background.

Recall that the Hilbert space Grassmannian (also called Sato-Segal-Wilson Grassmannian, see [42, 48, 51]) GR consists—given a polarised complex separable infinite dimensional Hilbert space \(H = H_{+} \oplus H_{-} \) (with \(H_{\pm }\) infinite dimensional as well)—of all subspaces W such that \(E_W - E_{+}\) (obvious orthoprojectors) is Hilbert-Schmidt. This definition agrees with the one given e.g. in [42], see [63].

In view of the Powers-Störmer theorem [41] such W’s or, indeed, their associated orthoprojectors \(E_W\), correspond, in turn, to gauge invariant quasifree states\(\omega _{E_W}\) of the CAR algebra associated to H which induce unitarily equivalent Gelfand-Naimark-Segal (GNS)-representations. Therefore, their cyclic vectors \(\xi _{W}\) all live in the GNS representation space \({\mathcal H}_{+} \equiv {\mathcal H}_{E_{+}}\) (\(\omega _{E_+}\) represents a sort of “Dirac sea”, exclusively filled with modes in \(H_{+}\)). Their corresponding complex one-dimensional spaces \(\langle \xi _{W} \rangle \subset {\mathcal H}_{+} \) then build up the fibres of the so-called determinant line bundle DET \(\rightarrow \) GR. The manifold GR embeds à la Kodaira into \({\mathbb P}({\mathcal H}_{+} )\) via the dual DET\({}^*\) and (manifestly invariant) Plücker equations describing this embedding have been manufactured in [63].

One has then a natural holomorphic section \(\tau _W\) of the dual \(\mathrm {DET}^*\) of the determinant line bundle, naturally associated to W, given by

$$ \tau _W ((W^{\prime }, v)) = \langle v,\xi _W \rangle , \,\,v \in \mathrm {DET}_{W^{\prime }}. $$

The assignment

$$ {\mathcal H}_{+} \ni \xi _W \mapsto \tau _W \in \varGamma _{L^2}( \mathrm {DET}^* \rightarrow \mathrm {Gr}) $$

is precisely the boson-fermion correspondence, in the language of [63, 64]. We act within the analytic category, and what is meant as \(L^2\)-holomorphic sections over an infinite-dimensional space, denoted by \(\varGamma _{L^2}\) above, are actually hyperplane sections in [64].

The section \(\tau _W\) is thus an example of coherent state: this was first observed within the KP (Kadomcev-Petviašvili) framework in [17]; in representation-theoretic terms \(\tau _W\) can be identified with Sato’s \(\tau \)-function (see [23, 30, 48] for the Japanese school formalism).

In [62], the Calabi diastasis function of Gr (or GR) was calculated as the pull-back under the Plücker embedding of the natural projective-space diastasis (induced by the Fubini-Study metric by the polarizing line bundle \(\mathrm {DET}^*\)). The formula compactly reads as follows, where a point of the embedded Grassmannian is identified by its Plücker coordinates and denoted by \([\tau ]\):

$$ D([\tau ], [\tau ^{\prime }]) = \log \frac{\parallel \!\tau \!\parallel ^2 \cdot \parallel \!\tau ^{\prime }\!\parallel ^2}{| \langle \tau ,\tau ^{\prime }\rangle |^2} $$

cf. Sect. 8.2. From Calabi’s global-rigidity theorem [15] we deduced in [43] that any isometric automorphism of the Grassmannian is projectively induced (see also [62] for further applications). The basic idea of the proof consists in using the coherent state embedding in conjunction with Wigner’s theorem on unitary/antiunitary implementation of symmetries preserving transition probabilities (see e.g. [9]).

Also notice that the boson-fermion correspondence yields the following equality between “boson” and “fermion” transition probabilities (cf. [62, 63]; here the \(\tau \)’s are normalized)

$$ |\langle \xi _{W_1}, \xi _{W_2}\rangle |^2 = |\langle \tau _{W_1}, \tau _{W_2}\rangle |^2 = \exp (-D(W_1,W_2)). $$

Again notice that in the present context, no measure is present (see however [36]).

We also recall that one can recover the so called BKP hierarchy via the Segre map of [64], and that, indeed, the latter can be used to define a new hierarchy of equations of KP-type (see [43]).

As for the Borel-Weil aspects previously mentioned, the involve suitably infinite dimensional restricted unitary and Spin\(^c\) groups, acting transitively on appropriate Grassmannians (thus matching Sato’s picture of the KP-hierarchy, see [23, 30, 48]) and on holomorphic sections of duals of determinant and Pfaffian line bundles (see again [43, 64] for details).

8.6 Order Parameters in 2-d Vortex Theory and Coherent States

We wish to recall in passing that hyperplane type sections also crop up in the vortex theory of Riemann surfaces devised in [38], where they describe the admissible vortex-antivortex configurations wave functions associated to the so called order parameter (a meromorphic function on the Riemann surface having prescribed zeros and poles (yielding a degree zero divisor \(D = D_v - D_a\)—suffixes “v” and “a” standing for vortex and antivortex, respectively), constrained by the fulfilment of the conditions stated by the Abel theorem; the order parameter is actually a semiclassical feature of the theory). Referring to [38] for a thorough treatment, we just observe that they are manufactured from the hyperplane bundle on the projective space \({\mathbb P} (H^0({\mathcal L_{D_v}}))\). The holomorphic sections of \({\mathcal L_{D_v}}\)—the holomorphic line bundle associated to \(D_v\)—correspond to the meromorphic functions hinted at above. Under the technical condition \(n > 2g -2\), an application of the Riemann-Roch theorem yields

$$ h^0({\mathcal L_{D_v}}) = n - g +1 $$

where n is the number of vortices (and antivortices) and g is the genus of the Riemann surface. Thus \(\mathrm{dim}({\mathbb P} (H^0({\mathcal L_{D_v}})) = n - g\).

8.7 The FMN-Transform, Theta Functions and Coherent States

8.7.1 Landau Levels, FMN and Standard Coherent States

In the paper [19], dealing with the theory of Landau levels for an electron confined on a plane and subject to a perpendicular constant magnetic field, we gave a geometric interpretation of the phenomenon of translational symmetry breaking upon quantization in geometrical terms, taking inspiration from the seminal paper [5] and from the Fourier-Mukai-Nahm (FMN) theory [13, 60] (and working with the lowest Landau level, see [65] for a general approach to Landau levels on Riemann surfaces).

In extreme brevity, one abuts at a family of translated oscillator Hamiltonians \(\hat{h}_{y}\) (\({y} \in {\mathbb R}^2\)) with corresponding annihilation operators (holomorphic structures) \(\bar{\partial }_{y}\) whose ground states \(\psi _{y}\) (satisfying \(\bar{\partial }_{y} \psi _{y} = 0\)) are precisely the standard coherent states (see e.g. [39]), so one ends up with a bona fide line bundle (FM line bundle) \({ \mathcal L}\rightarrow \mathbb {R}^2\) with fibres \({ \mathcal L}_{y} = \langle \psi _{y} \rangle \), which can be viewed as an index bundle [7, 13]. The latter carries a natural constant curvature connection (Nahm’s connection)—which may be computed exactly as in [60] via the standard Schrödinger representation of the harmonic oscillator—signalling an “anomaly” due to translational symmetry breaking. Of course, in this example everything is topologically trivial, unless we interpret the anomaly as a Lie algebra 2-cocycle. This is a geometric reinterpretation of the well-known fact that standard coherent states provide eigenfunctions of annihilation operators.

We give some details of the above portrait. The normalized standard coherent state wave functions read, for \(\alpha , \beta \in {\mathbb R}\), \(\xi _0 (x) = \pi ^{-1/4}e^{-\frac{x^2}{2}}\) (normalized oscillator ground state):

$$ \xi := \xi _{\alpha \beta }(x) = (U(\alpha )V(\beta )\xi _{0}) (x) = \pi ^{-1/4}\exp \left[ i \alpha x -\frac{(x-\beta )^2}{2} \right] , $$

where (\(\phi \in L^2({\mathbb R})\))

$$\begin{aligned}{}[U(\alpha ) \phi ] (x)&:= [\exp (i\alpha Q)\phi ](x) = e^{i\alpha x} \phi (x), \\ [V(\beta ) \phi ](x)&:= [\exp (i\beta P) \phi ](x) = \phi (x-\beta ), \end{aligned}$$

and one has

$$ U(\alpha ) V(\beta ) = e^{i\alpha \beta }V(\beta ) U(\alpha ), $$

i.e. the Weyl-Heisenberg Commutation Relations, namely, the integrated form of the Canonical Commutation Relations (CCR).

The Nahm connection (form) and curvature read, respectively (\(L^2\)-scalar products employed: this connection just comes from projecting the standard connection of the trivial \(L^2\)-bundle over the plane):

$$ A = \langle \xi , d \xi \rangle , $$

and

$$\varOmega = dA = d \langle \xi , d \xi \rangle = [\langle \partial _{\alpha }\xi ,\partial _{\beta } \xi \rangle - \langle \partial _{\beta }\xi ,\partial _{\alpha } \xi \rangle ] d\alpha \wedge d\beta = 2i \mathfrak {Im}\langle \partial _{\alpha }\xi ,\partial _{\beta } \xi \rangle d\alpha \wedge d\beta $$

A routine computation using, for \(\gamma >0\), \(\int e^{-\gamma x^2} x^2 dx = \frac{1}{2} \pi ^{\frac{1}{2} } \gamma ^{-\frac{1}{2} }\) and \(\int e^{-\gamma x^2} x dx = 0\) yields, as anticipated

$$ \varOmega = - i d\alpha \wedge d\beta $$

(translational anomaly). Summarising, we have

Theorem 8.2

([19]) The translational anomaly (lack of commutativity of the quantised translation operators with the Hamiltonian) is detected by the non trivial curvature of the Nahm connection of the “Fourier-Mukai” line bundle over the real plane, whose fibres are manufactured from the coherent states attached to a quantum harmonic oscillator (displaced ground states).

8.7.2 Generalised Theta Functions, FMN and Coherent States

A similar picture arises in the geometric approach to unitary Riemann surface (RS) braid group representations developed in [60], with a view to possible applications to quantum computing. Everything stems from the coincidence

“quantum harmonic oscillator ground state = (generalised) theta function”

(in the sense of [29]), based on the joint use of the Riemann-Roch-Hirzebruch theorem (see e.g. [7]) and the von Neumann uniqueness theorem for the CCR [67] first outlined in [54] (see also [60]) and passing essentially unaltered to the noncommutative setting [50]. In this case one deals with tori, both classical and noncommutative). We give a few details, extracted from [60]. Consider a projectively flat Hermitian-Einstein (HE) vector bundle \({\mathcal E} \rightarrow J(\varSigma _1)\) (the Jacobian of a Riemann surface \( \varSigma _1\) of genus \(g=1\), for simplicity) having (relatively prime) rank r and degree \(q\ge 1\) with slope (:= degree/rank) \(\nu = q/r\). This means that its Chern-Bott connection \(\nabla \) has constant curvature \(\varOmega _{\nabla } = -2\pi i \nu \cdot I_r\). Such a bundle can be built up via a construction due to Matsushima [29, 60].

Take \(H: = L^2({\mathcal E})\) namely the \(L^2\)-sections of \({ \mathcal E}\) obtained by completing its smooth sections with respect to the inner product

$$ \langle \cdot \, , \cdot \rangle := \int _{J(\varSigma _1)} h (\cdot \, , \cdot ) {\omega } $$

(h being the metric on \({\mathcal E}\), \(\omega \) the symplectic form on the Jacobian). Specifically, with respect to the standard (Darboux) symplectic coordinates \((q_1, p_1)\) on \(J(\varSigma _1)\), we have

$$ [\nabla _{ \!\frac{ \partial }{\partial q_1 } }, \nabla _{\! \frac{ \partial }{\partial q_1 } } ] = [\nabla _{ \!\frac{ \partial }{\partial p_1 } }, \nabla _{ \!\frac{ \partial }{\partial p_1} } ] = 0; \qquad [\nabla _{ \!\frac{ \partial }{\partial q_1 } } , \nabla _{\! \frac{ \partial }{\partial p_1 } }] = - 2\pi \sqrt{-1} \, \nu \, \cdot I_r $$

Notice in fact that, by periodicity and the compatibility of \(\nabla \) with h, one has

$$ 0 = \int _{J(\varSigma _1)} X \,h(\cdot \, ,\cdot ){\omega } = \int _{J(\varSigma _1)} \big [ h(\nabla _X \cdot ,\cdot ) + h(\cdot , \nabla _X\cdot ) \big ] {\omega } $$

with \(X = \partial /\partial q_1\), \(\partial /\partial p_1\), thus the operators \(\nabla _{ \!\frac{ \partial }{\partial q_1 }}\), \(\nabla _{ \!\frac{ \partial }{\partial p_1 }}\) are formally skew-hermitian. By classical functional analytic arguments they are skew-adjoint (cf. [45]). One has, for the ensuing annihilation operator

$$ A_1 \propto \nabla _{ \!\frac{ \partial }{\partial {\bar{z}}_1 } } $$

In the general case \(g \ge 1\) (with corresponding Jacobian \(J(\varSigma _g)\)) expounded in [60] we get a unitary representation of the Weyl-Heisenberg Commutation Relations (generalising [54]) with multiplicity \(q^g = \mathrm{dim}\, H^0({\mathcal E})\), the dimension of the space of holomorphic sections of \({\mathcal E}\)—a projectively flat HE-vector bundle of rank \(r^g\) and slope \(\mu = \nu g!\)—i.e. precisely the space of generalized theta functions. Actually, one gets a whole family of such bundles, parametrised by the Jacobian \(J(\varSigma _g)\). Here \(\nu = q/r\) as before (q and r are again relatively prime) and represents the statistical parameter of the corresponding RS-braid group representation. If \(q=r=1\) we recover ordinary theta functions, together with their coherent state interpretation already pointed out in [54]. Moreover, specialisation of FMN to the case \(g = 1\) leads to a possibly physically relevant “\(\nu \)-anyon—\(\nu ^{\prime }\)-anyon duality” (\(\nu ^{\prime } = 1/\nu \)) (see again [60] for details).

8.8 Concluding Remarks

  1. 1.

    The theory of (N-level) theta functions (emerging from the holomorphic geometric quantization of a torus) can be addressed via “translation symmetry breaking to \({\mathbb Z}_{N} \times {\mathbb Z}_{N}\)” ([35], and also [33, 39]).

  2. 2.

    Notice that the coherent states appearing in the FMN approach play a different role with respect to the Rawnsley coherent states: in the latter case the complex structure is fixed and coherent state wave functions are holomorphic sections of the same holomorphic line bundle; in the FMN case coherent states actually parametrise complex structures. A similar phenomenon also appears in noncommutative contexts (cf. e.g. [50, 55]). These “novel” coherent states could be usefully called Picard, or spectral, coherent states, being labelled by points of a Picard variety (parametrizing holomorphic flat line bundles) and, in turn yield spectral (i.e. Brillouin-type) manifolds (see e.g. [65]) and possibly deserve further scrutiny.

  3. 3.

    A sort of “tautological” coherent states have been worked out in [61] in the framework of a moment map approach to the Schrödinger equation and generalisations thereof, with respect to the group \(\mathrm{sDiff}({\mathbb R}^3)\) of measure preserving diffeomorphisms of \({\mathbb R}^3\), upon reinterpreting the standard wave functions as points of a Kähler manifold. We refer to [61] for extra information.