Quantum (matrix) geometry and quasi-coherent states

Now we associate to any matrix configuration two unique tensors on ${\tilde {\mathbb{R}}}^{D}$: the would-be symplectic form ω_ab and quantum metric g_ab . Since $\vert x\rangle \in \mathcal{H}$, the bundle $\mathcal{B}$ over ${\tilde {\mathbb{R}}}^{D}$ naturally inherits a metric and a connection. We can define a connection one-form A via

$$\begin{equation}P{\circ}d\vert x\rangle =\vert x\rangle \mathrm{i}\enspace A,\quad \mathrm{i}\enspace A{:=}\langle x\vert d\vert x\rangle \in {{\Omega}}^{1}\left({\tilde {\mathbb{R}}}^{D}\right),\end{equation} \tag{ 10 }$$

where P = |x⟩⟨x| is the projector on E_x . Here A is real because

$$\begin{equation}{\left(\langle x\vert d\vert x\rangle \right)}^{{\ast}}=d\left(\langle x\vert \right)\vert x\rangle =-\langle x\vert d\vert x\rangle ,\end{equation} \tag{ 11 }$$

and transforms like a U(1) gauge field

$$\begin{equation}\vert x\rangle \to {\text{e}}^{\text{i}{\Lambda}\left(x\right)}\vert x\rangle ,\qquad {A}_{a}\to {A}_{a}+{\partial }_{a}{\Lambda}.\end{equation} \tag{ 12 }$$

In particular, we can parallel transport |x⟩ along a path γ in ${\tilde {\mathbb{R}}}^{D}$. This connection is analogous to a Berry connection. It is encoded in the inner product

$$\begin{equation}\langle x\vert y\rangle = :\enspace {\text{e}}^{\text{i}\varphi \left(x,y\right)-D\left(x,y\right)},\end{equation} \tag{ 13 }$$

which defines a distance function D(x, y) and a phase function φ(x, y) which satisfy

$$\begin{align}\hfill D\left(x,y\right)& =D\left(y,x\right){\geqslant}0,\qquad D\left(x,y\right)=0{\Leftrightarrow}x=y\hfill \\ \hfill \varphi \left(x,y\right)& =-\varphi \left(y,x\right).\hfill \end{align} \tag{ 14 }$$

The phase clearly depends on the particular section |x⟩ of the bundle $\mathcal{B}$, while D(x, y) does not. To understand these two functions, we differentiate (13) w.r.t. y

$$\begin{equation}\langle x\vert {d}_{y}\vert y\rangle {\vert }_{y=x}=\mathrm{i}{d}_{y}\varphi \left(x,y\right){\vert }_{y=x}-{d}_{y}D\left(x,y\right){\vert }_{y=x}.\end{equation} \tag{ 15 }$$

Comparing with (10) we conclude

$$\begin{align}\hfill \mathrm{i}{d}_{y}\varphi \left(x,y\right){\vert }_{y=x}& =\mathrm{i}\enspace A=\mathrm{i}\enspace {A}_{a}\enspace \mathrm{d}{x}^{a}\hfill \\ \hfill {d}_{y}D\left(x,y\right){\vert }_{y=x}& =0.\hfill \end{align} \tag{ 16 }$$

Hence the phase φ(x, y) encodes the connection A. For a contractible closed path γ = ∂Ω in ${\tilde {\mathbb{R}}}^{D}$, the change of the phase of |x⟩ along γ is hence given by the field strength via Stokes theorem

$$\begin{equation}{\oint }_{\gamma }A={\int }_{{\Omega}}\mathrm{d}\enspace A.\end{equation} \tag{ 17 }$$

If the connection is flat, the phase φ(x, y) can be gauged away completely.

To proceed, consider the gauge-invariant Hermitian D × D matrix defined by

$$\begin{align}\hfill {h}_{ab}& =\left(\left({\partial }_{a}+\mathrm{i}{A}_{a}\right)\langle x\vert \right)\left({\partial }_{b}-\mathrm{i}{A}_{b}\right)\vert x\rangle {\vert }_{\xi }={h}_{ba}^{{\ast}}\hfill \\ \hfill & =\left({\partial }_{{x}^{a}}+\mathrm{i}{A}_{a}\right)\left({\partial }_{{y}^{b}}-\mathrm{i}{A}_{b}\right){\text{e}}^{\text{i}\varphi \left(x,y\right)-D\left(x,y\right)}{\vert }_{\xi }\hfill \\ \hfill & {=:}\frac{\mathrm{i}}{2}{\omega }_{ab}+{g}_{ab}\hfill \end{align} \tag{ 18 }$$

at some reference point $\xi \in {\tilde {\mathbb{R}}}^{D}$, which decomposes into the real symmetric and antisymmetric tensors g_ab and ω_ab . The symmetric part

$$\begin{align}\hfill 2{g}_{ab}& =\left(\left({\partial }_{a}+\mathrm{i}{A}_{a}\right)\langle x\vert \right)\left({\partial }_{b}-\mathrm{i}{A}_{b}\right)\vert x\rangle +\left(a{\leftrightarrow}b\right)\hfill \\ \hfill & =\left({\partial }_{a}\langle x\vert \right){\partial }_{b}\vert x\rangle -{A}_{a}{A}_{b}+\left(a{\leftrightarrow}b\right)\hfill \end{align} \tag{ 19 }$$

(using(10)) is the pull-back of the Riemannian metric ⁵ on $\mathcal{H}$ (or equivalently of the Fubini–Study metric on $\mathbb{C}{P}^{N-1}$) through the section |x⟩. The antisymmetric part of h_ab encodes a two-form

$$\begin{align}\hfill \mathrm{i}{\omega }_{ab}& =\mathrm{i}\left({\partial }_{a}{A}_{b}-{\partial }_{b}{A}_{a}\right)=\left({\partial }_{a}\langle x\vert \right){\partial }_{b}\vert x\rangle -\left({\partial }_{b}\langle x\vert \right){\partial }_{a}\vert x\rangle \hfill \\ \hfill \mathrm{i}\omega & =\frac{\mathrm{i}}{2}{\omega }_{ab}\enspace \mathrm{d}{x}^{a}\wedge \mathrm{d}{x}^{b}\enspace =\mathrm{d}\langle x\vert \wedge \mathrm{d}\vert x\rangle =d\left(\langle x\vert d\vert x\rangle \right)=\mathrm{i}\mathrm{d}A\hfill \end{align} \tag{ 20 }$$

which is the U(1) field strength of the connection A and therefore closed,

$$\begin{equation}\omega =\mathrm{d}A,\qquad \mathrm{d}\omega =0.\end{equation} \tag{ 21 }$$

It follows that the expansion of φ(x, y) to bilinear order in x and y (setting ξ = 0) is

$$\begin{equation}\varphi \left(x,y\right)={A}_{a}\left({x}^{a}-{y}^{a}\right)+\frac{1}{2}{\omega }_{ab}{x}^{a}{y}^{b}+\cdots \end{equation} \tag{ 22 }$$

dropping all terms O(x²) or O(y²) and higher. Similarly, the expansion of D(x, y) is given by

$$\begin{equation}D\left(x,y\right)=\frac{1}{2}{\left(x-y\right)}^{a}{\left(x-y\right)}^{b}{g}_{ab}+\cdots \end{equation} \tag{ 23 }$$

since D(x, x) = 0 and D(x, y) ⩾ 0. In fact viewing $\mathcal{B}/U\left(1\right)$ as subset of $\mathbb{C}{P}^{N-1}$, we can use the well-known formula

$$\begin{equation}{\mathrm{cos}}^{2}\left(\gamma \left(x,y\right)\right)={\mathrm{e}}^{-2D\left(x,y\right)},\end{equation} \tag{ 24 }$$

where γ(x, y) is the geodesic distance squared between |x⟩ and |y⟩ in the Fubini–Study metric on $\mathbb{C}{P}^{N-1}$. Combining (13) and (23), we learn that the quasi-coherent states are localized within a region of size

$$\begin{equation}{L}_{\text{coh}}^{2}={\Vert}{g}_{ab}{{\Vert}}^{-1}\end{equation} \tag{ 25 }$$

denoted as coherence scale. The |x⟩ are approximately constant below this scale due to (13). The relation with the uncertainty of X^a will be given in (67). Therefore g_ab will be denoted as quantum metric. We will see in section 5.2 that there is a different, effective metric which governs the low-energy physics on such quantum spaces in Yang–Mills matrix models. However, the intrinsic structure of the underlying quantum space is best understood using a more abstract point of view developed in section 3.

We will see that ω typically arises from a symplectic form on an underlying space $\mathcal{M}$. Therefore ω will be denoted as would-be symplectic form. Since it is the curvature of a U(1) bundle, its flux is quantized for every two-cycle S² in ${\tilde {\mathbb{R}}}^{D}$ as

$$\begin{equation}\underset{{S}^{2}}{\int }\frac{1}{2\pi }\omega =n,\quad n\in \mathbb{Z}.\end{equation} \tag{ 26 }$$

This arises using (17) as consistency condition on the U(1) holonomy for the parallel transport along a closed path γ on S². In more abstract language, ${c}_{1}=-\frac{1}{2\pi }\omega$ is the first Chern class of $\mathcal{B}$ viewed as line bundle, which is the pull-back of the first Chern class (or symplectic form) of $\mathbb{C}{P}^{N-1}$ via the section |x⟩. The bundle $\mathcal{B}$ is trivial if these numbers vanish for all cycles S², hence if ${H}^{2}\left({\tilde {\mathbb{R}}}^{D}\right)$ vanishes.

Assume that |x⟩ is a local section of the quasi-coherent states, with

$$\begin{equation}{H}_{x}\vert x\rangle =\lambda \left(x\right)\vert x\rangle .\end{equation} \tag{ 27 }$$

Using Cartesian coordinates x^a on ${\mathbb{R}}^{D}$, we observe that

$$\begin{equation}{\partial }_{a}{H}_{x}=-\left({X}_{a}-{x}_{a}\mathbb{1}\right).\end{equation} \tag{ 28 }$$

Thus differentiating (27) gives

$$\begin{equation}\left({H}_{x}-\lambda \left(x\right)\right){\partial }_{a}\vert x\rangle =-{\partial }_{a}\left({H}_{x}-\lambda \left(x\right)\right)\vert x\rangle =\left({X}_{a}-{x}_{a}+{\partial }_{a}\lambda \right)\vert x\rangle .\end{equation} \tag{ 29 }$$

Since lhs is orthogonal to ⟨x|, it follows that

$$\begin{equation}0=\langle x\vert \left({X}_{a}-{x}_{a}+{\partial }_{a}\lambda \right)\vert x\rangle \end{equation} \tag{ 30 }$$

so that the expectation value or symbol of the basic matrices X_a is given by

$$\begin{equation}{\mathbf{x}}_{\mathbf{a}}=\langle x\vert {X}_{a}\vert x\rangle ={x}_{a}-{\partial }_{a}\lambda .\end{equation} \tag{ 31 }$$

Furthermore, (29) gives (in the non-degenerate case under consideration)

$$\begin{align}\hfill {\partial }_{a}\vert x\rangle & =\vert x\rangle \langle x\vert {\partial }_{a}\vert x\rangle +{\left({H}_{x}-\lambda \right)}^{-1}\left({X}_{a}-{x}_{a}+{\partial }_{a}\lambda \right)\vert x\rangle \hfill \\ \hfill \left({\partial }_{a}-\mathrm{i}{A}_{a}\right)\vert x\rangle & ={\left({H}_{x}-\lambda \right)}^{-1}\left({X}_{a}-{x}_{a}+{\partial }_{a}\lambda \right)\vert x\rangle \hfill \end{align} \tag{ 32 }$$

using (16). Even though the ${\left({H}_{x}-\lambda \right)}^{-1}$ term is well-defined here, it is better to replace ${\left({H}_{x}-\lambda \right)}^{-1}$ with an operator that is well-defined on $\mathcal{H}$. This is achieved using the 'reduced resolvent'

$$\begin{equation}{\left({H}_{x}-\lambda \left(x\right)\right)}^{\prime -1}{:=}\left(\mathbb{1}-{P}_{x}\right){\left({H}_{x}-\lambda \left(x\right)\right)}^{-1}\left(\mathbb{1}-{P}_{x}\right),\quad {P}_{x}{:=}\vert x\rangle \langle x\vert \end{equation} \tag{ 33 }$$

which satisfies

$$\begin{align}\hfill \left({H}_{x}-\lambda \right){\left({H}_{x}-\lambda \right)}^{\prime -1}& =\mathbb{1}-{P}_{x}={\left({H}_{x}-\lambda \right)}^{\prime -1}\left({H}_{x}-\lambda \right),\hfill \\ \hfill {\left({H}_{x}-\lambda \right)}^{\prime -1}\vert x\rangle & =0.\hfill \end{align} \tag{ 34 }$$

Then we can write (32) as

$$\begin{equation}\enspace \left({\partial }_{a}-\mathrm{i}{A}_{a}\right)\vert x\rangle ={\mathcal{X}}_{a}\vert x\rangle ,\end{equation} \tag{ 35 }$$

where

$$\begin{equation}{\mathcal{X}}_{a}={\left({H}_{x}-\lambda \right)}^{\prime -1}\left({X}_{a}-{x}_{a}+{\partial }_{a}\lambda \right).\end{equation} \tag{ 36 }$$

Moreover, we note

$$\begin{equation}\langle x\vert {\mathcal{X}}_{a}\vert x\rangle =0.\end{equation} \tag{ 37 }$$

Hence ${\mathcal{X}}_{a}$ generates the gauge-invariant tangential variations of |x⟩, which take value in the orthogonal complement of |x⟩. This will be the basis for defining the quantum tangent space in section 3. The local section |x⟩ over ${\tilde {\mathbb{R}}}^{D}$ can now be written as

$$\begin{equation}\vert x\rangle =P\enspace \mathrm{exp}\left({\int }_{\xi }^{x}\left({\mathcal{X}}_{a}+\mathrm{i}{A}_{a}\right)\mathrm{d}{x}^{a}\right)\vert \xi \rangle \end{equation} \tag{ 38 }$$

near the reference point $\xi \in {\tilde {\mathbb{R}}}^{D}$. Here P indicates path ordering, which is just a formal way of writing the solution of (35). This is independent of the chosen path, since [∂_a , ∂_b ] = 0.

Since the derivatives of |x⟩ are spanned by the ${\mathcal{X}}^{a}\vert x\rangle$, the U(1) field strength ω_ab and the quantum metric g_ab should be related to algebraic properties for the ${\mathcal{X}}^{a}$. Indeed, starting from (18)

$$\begin{equation}{h}_{ab}=\langle x\vert {\mathcal{X}}_{a}^{{\dagger}}{\mathcal{X}}_{b}\vert x\rangle =\frac{\mathrm{i}}{2}{\omega }_{ab}+{g}_{ab},\end{equation} \tag{ 39 }$$

we obtain

$$\begin{equation}\mathrm{i}{\omega }_{ab}={h}_{ab}-{h}_{ba}=\langle x\vert \left({\mathcal{X}}_{a}^{{\dagger}}{\mathcal{X}}_{b}-{\mathcal{X}}_{b}^{{\dagger}}{\mathcal{X}}_{a}\right)\vert x\rangle \end{equation} \tag{ 40 }$$

and

$$\begin{equation}2{g}_{ab}={h}_{ab}+{h}_{ba}=\langle x\vert \left({\mathcal{X}}_{a}^{{\dagger}}{\mathcal{X}}_{b}+{\mathcal{X}}_{b}^{{\dagger}}{\mathcal{X}}_{a}\right)\vert x\rangle .\end{equation} \tag{ 41 }$$

This provides a first link between the geometric and algebraic structures under consideration. Furthermore, is useful to define the following Hermitian tensor (similar as in [15])

$$\begin{align}\hfill {P}_{ab}\left(x\right)& {:=}\langle x\vert \left({X}_{a}-{x}_{a}+{\partial }_{a}\lambda \right){\left({H}_{x}-\lambda \right)}^{\prime -1}\left({X}_{b}-{x}_{b}+{\partial }_{b}\lambda \right)\vert x\rangle ={P}_{ba}{\left(x\right)}^{{\ast}}\hfill \\ \hfill & =\langle x\vert \left({X}_{a}-{x}_{a}+{\partial }_{a}\lambda \right){\mathcal{X}}_{b}\vert x\rangle \hfill \\ \hfill & =\langle x\vert {X}_{a}{\mathcal{X}}_{b}\vert x\rangle ,\hfill \end{align} \tag{ 42 }$$

where (37) is used in the last step. Its symmetric part is obtained by taking derivatives of (31)

$$\begin{align}\hfill {\partial }_{b}{\mathbf{x}}_{\mathbf{a}}\left(x\right)& ={\partial }_{b}{x}_{a}-{\partial }_{b}{\partial }_{a}\lambda \hfill \\ \hfill & ={\partial }_{b}\left(\langle x\vert {X}_{a}\vert x\rangle \right)=\langle x\vert \left({X}_{a}\left({\mathcal{X}}_{b}+\mathrm{i}{A}_{b}\right)+{\left({\mathcal{X}}_{b}+\mathrm{i}{A}_{b}\right)}^{{\dagger}}{X}_{a}\right)\vert x\rangle \hfill \\ \hfill & ={P}_{ab}+{P}_{ba}\hfill \end{align} \tag{ 43 }$$

lowering indices with δ_ab ; for the antisymmetric part see (69). This will be recognized as projector on the embedded quantum space in (91), as obtained in the semi-classical limit in [15].

We would like to define a class $Loc\left(\mathcal{H}\right)\subset End\left(\mathcal{H}\right)$ of almost-local operators which satisfy

$$\begin{equation}{\Phi}\vert x\rangle \approx \vert x\rangle \langle x\vert {\Phi}\vert x\rangle ={P}_{x}{\Phi}\vert x\rangle \enspace =\vert x\rangle \phi \left(x\right)\quad \forall \enspace x\in {\tilde {\mathbb{R}}}^{D},\end{equation} \tag{ 44 }$$

where ϕ(x) = ⟨x|Φ|x⟩ is the symbol of Φ, and P_x = |x⟩⟨x| is the projector on the quasi-coherent state |x⟩. The question is how to make the meaning of ≈ precise, without considering some limit as in [12]. We should certainly require that Φ|x⟩ ≈ |x⟩ϕ(x) in $\mathcal{H}$ for every x, but it is not obvious yet how to handle the dependence on x, and how to specify bounds. The guiding idea is that it should make sense to identify Φ with its symbol

$$\begin{equation}{\Phi}\sim \phi \left(x\right)=\langle x\vert {\Phi}\vert x\rangle ,\end{equation} \tag{ 45 }$$

indicated by ∼ from now on. This will be made more precise in the section 3.1 by requiring that ∼ is an approximate isometry from $Loc\left(\mathcal{H}\right)$ to ${\mathcal{C}}_{\text{IR}}\left(\mathcal{M}\right)$, where ${\mathcal{C}}_{\text{IR}}\left(\mathcal{M}\right)$ is a class of 'infrared' functions on the abstract quantum space associated to the matrix configuration. The essence of almost-locality is then that the integrated deviations from classically are small compared with the classical values. With this in mind, we proceed to elaborate some consequences of (44) for fixed x without specifying bounds.

Since $\left(\mathbb{1}-{P}_{x}\right)$ is a projector, we have the estimate

$$\begin{equation}\langle x\vert {{\Phi}}^{{\dagger}}{\Phi}\vert x\rangle =\langle x\vert {{\Phi}}^{{\dagger}}{P}_{x}{\Phi}\vert x\rangle +\langle x\vert {{\Phi}}^{{\dagger}}\left(\mathbb{1}-{P}_{x}\right){\Phi}\vert x\rangle {\geqslant}\langle x\vert {{\Phi}}^{{\dagger}}{P}_{x}{\Phi}\vert x\rangle =\vert \phi \left(x\right){\vert }^{2}.\end{equation} \tag{ 46 }$$

It follows that every Hermitian almost-local operator Φ = Φ^† satisfies

$$\begin{equation}\langle x\vert {\Phi}{\Phi}\vert x\rangle \approx \langle x\vert {\Phi}\vert {x\rangle }^{2}=\vert \phi \left(x\right){\vert }^{2}\quad \forall \enspace x\in {\tilde {\mathbb{R}}}^{D},\end{equation} \tag{ 47 }$$

i.e. the uncertainty of Φ is negligible,

$$\begin{equation}\langle x\vert {\left({\Phi}-\langle x\vert {\Phi}\vert x\rangle \right)}^{2}\vert x\rangle \approx 0\quad \forall \enspace x\in {\tilde {\mathbb{R}}}^{D}.\end{equation} \tag{ 48 }$$

This means that (Φ − ϕ(x))|x⟩ is approximately zero, which in turn implies (44). Therefore almost-locality is essentially equivalent to (48), up to global considerations and specific bounds. A more succinct global version of (48) is given in (104).

We also note that for two operators ${\Phi},{\Psi}\in Loc\left(\mathcal{H}\right)$ the factorization properties

$$\begin{align}\hfill {\Phi}{\Psi}\vert x\rangle & \approx {\Phi}\vert x\rangle \langle x\vert {\Psi}\vert x\rangle \approx \vert x\rangle \phi \left(x\right)\psi \left(x\right)\hfill \\ \hfill \langle x\vert {\Phi}{\Psi}\vert x\rangle & \approx \phi \left(x\right)\psi \left(x\right)\hfill \end{align} \tag{ 49 }$$

follow formally. However this does not mean that $Loc\left(\mathcal{H}\right)$ is an algebra, since the specific bounds may be violated by the product. For some given matrix configuration, $Loc\left(\mathcal{H}\right)$ may be empty or very small. This happens e.g. for the minimal fuzzy spaces as discussed in section 6.3, and it is expected for random matrix configuration. But even in these cases, the associated geometrical structures still provide useful insights.

For interesting quantum geometries, we expect that all the X^a are almost-local, hence also polynomials P_n (X) up to some maximal degree n due to (49). $Loc\left(\mathcal{H}\right)$ can often be characterized by some bound on the eigenvalue of □ (142), or the uncertainty scale L_NC (68). However, $Loc\left(\mathcal{H}\right)$ can never be more than a small subset of $End\left(\mathcal{H}\right)$.

To see how the Poisson structure arises, define the real anti-symmetric matrix-valued function

$$\begin{equation}{\theta }^{ab}{:=}-\mathrm{i}\langle x\vert \left[{X}^{a},{X}^{b}\right]\vert x\rangle =-{\theta }^{ba}\end{equation} \tag{ 50 }$$

on ${\tilde {\mathbb{R}}}^{D}$. To relate it to the previous structures, we shall loosely follow [15], starting from the identity

$$\begin{equation}\left[{X}^{a},{X}^{b}\right]\left({X}_{b}-{x}_{b}\right)+\left({X}_{b}-{x}_{b}\right)\left[{X}^{a},{X}^{b}\right]=2\left[{X}^{a},{H}_{x}\right].\end{equation} \tag{ 51 }$$

Taking the expectation value, we obtain

$$\begin{equation}\langle x\vert \left[{X}^{a},{X}^{b}\right]\left({X}_{b}-{x}_{b}\right)\vert x\rangle +\langle x\vert \left({X}_{b}-{x}_{b}\right)\left[{X}^{a},{X}^{b}\right]\vert x\rangle =2\langle x\vert \left[{X}^{a},{H}_{x}\right]\vert x\rangle =0.\end{equation} \tag{ 52 }$$

If X^a is almost-local ⁶ , then this implies

$$\begin{equation}0\approx \langle x\vert \left[{X}^{a},{X}^{b}\right]\vert x\rangle \langle x\vert \left({X}_{b}-{x}_{b}\right)\vert x\rangle =-\mathrm{i}{\theta }^{ab}{\partial }_{b}\lambda \end{equation} \tag{ 53 }$$

using (31). In section 3.1 we will see that this implies λ ∼ const on the embedded quantum space $\tilde {\mathcal{M}}$, and P_ac + P_ca ∼ ∂_c x_a is its tangential projector.

We now define an almost-local quantum space to be a matrix configuration where all X^a as well as all [X^a , X^b ] are almost-local operators. Then they approximately commute, and we can proceed following [15]

$$\begin{align}\hfill -2\left({H}_{x}-\lambda \right)\left({X}^{a}-{x}^{a}+{\partial }^{a}\lambda \right)\vert x\rangle & =2\left[{X}^{a},{H}_{x}\right]\vert x\rangle \approx 2\left({X}_{b}-{x}_{b}\right)\left[{X}^{a},{X}^{b}\right]\vert x\rangle \hfill \\ \hfill & \approx 2\left({X}_{b}-{x}_{b}\right)\vert x\rangle \langle x\vert \left[{X}^{a},{X}^{b}\right]\vert x\rangle =2\mathrm{i}\left({X}_{b}-{x}_{b}\right)\vert x\rangle {\theta }^{ab}\hfill \\ \hfill & \approx 2\mathrm{i}\left({X}_{b}-{x}_{b}+{\partial }_{b}\lambda \right)\vert x\rangle {\theta }^{ab}\hfill \end{align} \tag{ 54 }$$

using the factorization property, (53) and (27). However the first approximation is subtle, since (X_b − x_b )|x⟩ ≈ 0. This can be justified if X^a is a solution of the Yang–Mills equations ⁷

$$\begin{equation}\left[{X}_{b},\left[{X}^{b},{X}^{a}\right]\right]=0\enspace \end{equation} \tag{ 55 }$$

which are indeed the equations of motion for Yang–Mills matrix models [5]. Then (51) implies

$$\begin{equation}2\left[{X}^{a},{H}_{x}\right]=2\left({X}_{b}-{x}_{b}\right)\left[{X}^{a},{X}^{b}\right]\end{equation} \tag{ 56 }$$

and the above steps become

$$\begin{align}\hfill -2\left({H}_{x}-\lambda \right)\left({X}^{a}-{x}^{a}+{\partial }^{a}\lambda \right)\vert x\rangle & =2\left[{X}^{a},{H}_{x}\right]\vert x\rangle =2\left({X}_{b}-{x}_{b}\right)\left[{X}^{a},{X}^{b}\right]\vert x\rangle \hfill \\ \hfill & \approx 2\mathrm{i}\left({X}_{b}-{x}_{b}\right)\vert x\rangle {\theta }^{ab}\hfill \\ \hfill & \approx 2\mathrm{i}\left({X}_{b}-{x}_{b}+{\partial }_{b}\lambda \right)\vert x\rangle {\theta }^{ab}.\hfill \end{align} \tag{ 57 }$$

The rhs is indeed orthogonal to ⟨x| due to (31), and we can conclude

$$\begin{align}\hfill -\left({X}^{a}-{x}^{a}+{\partial }^{a}\lambda \right)\vert x\rangle & \approx \mathrm{i}{\left({H}_{x}-\lambda \right)}^{\prime -1}\left({X}_{b}-{x}_{b}+{\partial }_{b}\lambda \right)\vert x\rangle {\theta }^{ab}\hfill \\ \hfill & =\mathrm{i}{\theta }^{ab}{\mathcal{X}}_{b}\vert x\rangle =\mathrm{i}{\theta }^{ab}\left({\partial }_{b}-\mathrm{i}{A}_{b}\right)\vert x\rangle \hfill \end{align} \tag{ 58 }$$

hence

$$\begin{equation}\left({X}^{a}-{x}^{a}+{\partial }^{a}\lambda \right)\vert x\rangle \approx -\mathrm{i}{\theta }^{ab}\left({\partial }_{b}-\mathrm{i}{A}_{b}\right)\vert x\rangle \end{equation} \tag{ 59 }$$

and by conjugating

$$\begin{equation}\langle x\vert \left({X}^{d}-{x}^{d}+{\partial }^{d}\lambda \right)\approx \mathrm{i}{\theta }^{dc}\left({\partial }_{c}+\mathrm{i}{A}_{c}\right)\langle x\vert .\end{equation} \tag{ 60 }$$

These relations are very useful. First, they imply the important relation

$$\begin{equation}\left[{X}^{a},\vert x\rangle \langle x\vert \right]\approx -\mathrm{i}{\theta }^{ab}{\partial }_{b}\left(\vert x\rangle \langle x\vert \right).\end{equation} \tag{ 61 }$$

Furthermore, multiplying (59) with (∂_c + iA_c )⟨x| gives

$$\begin{equation}-\mathrm{i}{\theta }^{ab}\left(\left({\partial }_{c}+\mathrm{i}{A}_{c}\right)\langle x\vert \right)\left({\partial }_{b}-\mathrm{i}{A}_{b}\right)\vert x\rangle \approx \langle x\vert {\mathcal{X}}_{c}^{{\dagger}}\left({X}_{a}-{x}_{a}+{\partial }_{a}\lambda \right)\vert x\rangle ={P}_{ac}^{{\ast}}={P}_{ca},\end{equation} \tag{ 62 }$$

and similarly from (60)

$$\begin{equation}\mathrm{i}{\theta }^{ac}\left(\left({\partial }_{c}+\mathrm{i}{A}_{c}\right)\langle x\vert \right)\left({\partial }_{b}-\mathrm{i}{A}_{b}\right)\vert x\rangle \approx \langle x\vert \left({X}_{a}-{x}_{a}+{\partial }_{a}\lambda \right){\mathcal{X}}_{b}\vert x\rangle ={P}_{ab}.\end{equation} \tag{ 63 }$$

Adding these and using (43) and (18) gives

$$\begin{equation}-{\theta }^{ab}{\omega }_{bc}\approx {\partial }_{c}{\mathbf{x}}^{\mathbf{a}}={\partial }_{c}\left({x}^{a}-{\partial }^{a}\lambda \right)\enspace \end{equation} \tag{ 64 }$$

in the semi-classical regime, as in [15]. The rhs will be recognized as tangential projector on the embedded quantum space $\tilde {\mathcal{M}}\subset {\mathbb{R}}^{D}$. Therefore the above relation states that θ^ac is tangential to $\tilde {\mathcal{M}}$, and the inverse of the would-be symplectic form ω_ab on $\tilde {\mathcal{M}}$. This implies that $\omega {\vert }_{\tilde {\mathcal{M}}}$ is indeed non-degenerate i.e. symplectic, and θ^ac is its associated Poisson structure ⁸ . Together with (50) we obtain

$$\begin{equation}\left[{X}^{a},{X}^{b}\right]\vert x\rangle \approx \mathrm{i}\left\{{x}^{a},{x}^{b}\right\}\vert x\rangle =\mathrm{i}{\theta }^{ab}\vert x\rangle \enspace \end{equation} \tag{ 65 }$$

which can be written in the notation of section 3.1 as semi-classical relation

$$\begin{equation}\left[{X}^{a},{X}^{b}\right]\sim \mathrm{i}\left\{{x}^{a},{x}^{b}\right\}=\mathrm{i}{\theta }^{ab}.\end{equation} \tag{ 66 }$$

Furthermore, taking the inner product of (59) and (60) we obtain

$$\begin{equation}{\left({\Delta}{X}^{a}\right)}^{2}=\langle x\vert \left({X}^{a}-{x}^{a}+{\partial }^{a}\lambda \right)\left({X}^{a}-{x}^{a}+{\partial }^{a}\lambda \right)\vert x\rangle ={\theta }^{ab}{\theta }^{ac}{g}_{bc}\end{equation} \tag{ 67 }$$

(no sum over a), where g_bc is the quantum metric (19). Hence the uncertainty of X^a is characterized by the uncertainty length ⁹

$$\begin{equation}{L}_{\text{NC}}^{2}{:=}{\Vert}{\theta }^{ab}{{\Vert}}^{2}{L}_{\text{coh}}^{-2}.\end{equation} \tag{ 68 }$$

We also note the relation [15]

$$\begin{equation}\mathrm{i}{\theta }^{ac}{g}_{cb}={\delta }^{a{a}^{\prime }}\left({P}_{{a}^{\prime }c}-{P}_{c{a}^{\prime }}\right)=2\mathrm{i}{\delta }^{a{a}^{\prime }}\enspace \mathrm{I}\mathrm{m}\left({P}_{{a}^{\prime }c}\right)\end{equation} \tag{ 69 }$$

which is obtained by subtracting (62) and (63); in particular, θ^ac g_cb is antisymmetric. Finally, by comparing (59) with (36) we obtain

$$\begin{equation}{\theta }^{ab}\left({\partial }_{b}-\mathrm{i}{A}_{b}\right)\vert x\rangle \approx \mathrm{i}\left({X}^{a}-{x}^{a}+{\partial }^{a}\lambda \right)\vert x\rangle =\left({H}_{x}-\lambda \right)\mathrm{i}\left({\partial }^{a}-\mathrm{i}{A}^{a}\right)\vert x\rangle ,\end{equation} \tag{ 70 }$$

which relates i(∂^a − iA^a )|x⟩ and θ^ab (∂_b − iA_b )|x⟩, up to the action of H_x − λ.

Given the quasi-coherent states, we can define a quantization map

$$\begin{align}\hfill \mathcal{Q}:\mathcal{C}\left(\mathcal{M}\right)& \to End\left(\mathcal{H}\right)\hfill \\ \hfill \phi \left(x\right)& {\mapsto}{\int }_{\mathcal{M}}\phi \left(x\right)\enspace \vert x\rangle \langle x\vert \hfill \end{align} \tag{ 92 }$$

which associates to every classical function on $\mathcal{M}$ an operator or observable in $End\left(\mathcal{H}\right)$. The integral on the rhs is defined ¹⁶ naturally via the symplectic volume form

$$\begin{equation}{\int }_{\mathcal{M}}\phi \left(x\right){:=}\frac{1}{{\left(2\pi \alpha \right)}^{n}}\underset{\mathcal{M}}{\int }{\Omega}\enspace \phi \left(x\right),\quad {\Omega}{:=}\frac{1}{n!}{\omega }^{\wedge n}\end{equation} \tag{ 93 }$$

(assuming $\mathrm{dim}\enspace \mathcal{M}=m=2n$), where the normalization factor α is defined by

$$\begin{equation}N=\mathrm{Tr}\left(\mathbb{1}\right)={\int }_{\mathcal{M}}1.\end{equation} \tag{ 94 }$$

Semi-classical considerations suggest that α ≈ 1, however this cannot hold in full generality, since the symplectic form is degenerate for the minimal fuzzy torus and the integral vanishes. It would be desirable to find sufficient conditions for α ≈ 1, and a precise statement in particular for the quantum Kähler manifolds discussed below. In any case, the trace is related to the intragel via

$$\begin{equation}\mathrm{Tr}\mathcal{Q}\left(\phi \right)={\int }_{\mathcal{M}}\phi \left(x\right).\end{equation} \tag{ 95 }$$

The map $\mathcal{Q}$ cannot be injective, since $End\left(\mathcal{H}\right)$ is finite-dimensional; the kernel is typically given by functions with high 'energy'. It is not evident in general if this map is surjective, which will be established below for the case of quantum Kähler manifolds.

We can now re-define the symbol map (4) more succinctly as

$$\begin{align}\hfill End\left(\mathcal{H}\right)& \to \mathcal{C}\left(\mathcal{M}\right)\hfill \\ \hfill {\Phi}& {\mapsto}\langle x\vert {\Phi}\vert x\rangle {=:}\phi \left(x\right).\hfill \end{align} \tag{ 96 }$$

Both sides have a natural norm and inner product, given by

$$\begin{equation}\langle {\Phi},{\Psi}\rangle =\mathrm{Tr}\left({{\Phi}}^{{\dagger}}{\Psi}\right)\quad \text{and}\quad \langle \phi ,\psi \rangle ={\int }_{\mathcal{M}}\phi {\left(x\right)}^{{\ast}}\psi \left(x\right)\end{equation} \tag{ 97 }$$

leading to the Hilbert–Schmidt norm ||Φ||_HS and the L² norm ||ϕ||₂, respectively. The symbol map can be viewed as de-quantization map, which makes sense for any quantum space in the present framework.

The concept of almost-local operators discussed in section 2.4 can now also be refined. We re-define $Loc\left(\mathcal{H}\right)\subset End\left(\mathcal{H}\right)$ as a maximal (vector) space of operators such that the restricted symbol map

$$\begin{align}\hfill Loc\left(\mathcal{H}\right)& \to {\mathcal{C}}_{\text{IR}}\left(\mathcal{M}\right)\hfill \\ \hfill {\Phi}\enspace & {\mapsto}\langle x\vert {\Phi}\vert x\rangle =:\phi \left(x\right)\hfill \end{align} \tag{ 98 }$$

is an 'approximate isometry' with respect to the Hilbert–Schmidt norm on $Loc\left(\mathcal{H}\right)\subset End\left(\mathcal{H}\right)$ and the L²-norm on ${\mathcal{C}}_{\text{IR}}\left(\mathcal{M}\right)\subset {L}^{2}\left(\mathcal{M}\right)$. We will then identify Φ ∼ ϕ. Approximate isometry means that |||ϕ||₂ − 1| < ɛ whenever ||Φ||_HS = 1 for some given $0{< }\varepsilon {< }\frac{1}{2}$, depending on the context. Then the polarization identity implies

$$\begin{equation}\langle {\Phi},{{\Psi}\rangle }_{\text{HS}}\approx \langle \phi ,{\psi \rangle }_{2},\end{equation} \tag{ 99 }$$

hence an ON basis of $Loc\left(\mathcal{H}\right)$ is mapped to a basis of ${\mathcal{C}}_{\text{IR}}\left(\mathcal{M}\right)$ which is almost ON. This defines the semi-classical regime, which can be made more precise in some given situation by specifying some ɛ. Accordingly, almost-local quantum spaces are (re)defined as matrix configurations where all X^a and [X^a , X^b ] are in $Loc\left(\mathcal{H}\right)$.

Of course some given matrix configuration may be far from any semi-classical space, in which case $Loc\left(\mathcal{H}\right)$ is trivial. However we will see that for almost-local quantum space, $Loc\left(\mathcal{H}\right)$ typically includes the almost-local operators in the sense of (44) up to some bound, and in particular polynomials in X^a up to some order. Moreover, $\mathcal{Q}$ is an approximate inverse of the symbol map (98) on $Loc\left(\mathcal{H}\right)$. Then the semi-classical regime should contain a sufficiently large class of functions and operators to characterize the geometry to a satisfactory precision.

Let us try to justify these claims. The first observation is that $\mathbb{1}\in Loc\left(\mathcal{H}\right)$, because its symbol is the constant function ${1}_{\mathcal{M}}$, and the norm is preserved due to (94). Conversely, we should show the completeness relation

$$\begin{equation}\mathcal{Q}\left({1}_{\mathcal{M}}\right)={\int }_{\mathcal{M}}\vert x\rangle \langle x\vert \enspace !{\approx }\enspace \mathbb{1}\enspace \end{equation} \tag{ 100 }$$

which is equivalent ¹⁷ to the trace identity

$$\begin{equation}\mathrm{Tr}{\Phi}={\int }_{\mathcal{M}}\langle x\vert {\Phi}\vert x\rangle \quad \forall \enspace {\Phi}\in End\left(\mathcal{H}\right).\end{equation} \tag{ 101 }$$

This is not automatic, since the integral vanishes e.g. on minimal ${T}_{2}^{2}$. We can establish the completeness relation at least formally ¹⁸ (or rather approximately) for almost-local quantum spaces. Indeed then (61) implies

$$\begin{align}\hfill \left[{X}^{a},\mathcal{Q}\left(\phi \right)\right]& \approx -\mathrm{i}{\int }_{\mathcal{M}}\phi \left(x\right){\theta }^{ab}{\partial }_{b}\left(\vert x\rangle \langle x\vert \right)\hfill \\ \hfill & =\mathrm{i}{\int }_{\mathcal{M}}{\theta }^{ab}{\partial }_{b}\phi \left(x\right)\vert x\rangle \langle x\vert \hfill \\ \hfill & =\mathcal{Q}\left(\mathrm{i}{\theta }^{ab}{\partial }_{b}\phi \right)\hfill \end{align} \tag{ 102 }$$

because the integration measure Ω (93) is invariant under Hamiltonian vector fields. In particular, $\mathcal{Q}\left({1}_{\mathcal{M}}\right)$ (approximately) commutes with all X^a , which by irreducibility implies $\mathcal{Q}\left({1}_{\mathcal{M}}\right)\propto \mathbb{1}$, and (100) follows using the trace (95).

Now assume that the completeness relation holds to a sufficient precision. Let Φ be an almost-local Hermitian operator as defined in section 2.4, with symbol ϕ. Then the trace relation gives

$$\begin{equation}{\Vert}{\Phi}{{\Vert}}_{\text{HS}}^{2}\approx {\int }_{\mathcal{M}}\langle x\vert {\Phi}{\Phi}\vert x\rangle \approx {\int }_{\mathcal{M}}\phi {\left(x\right)}^{2}={\Vert}\phi {{\Vert}}_{2}^{2}\end{equation} \tag{ 103 }$$

using (44). Therefore almost-local operators in the sense of (44) are indeed contained in $Loc\left(\mathcal{H}\right)$, up to the specific bounds. Conversely, assume that ||Φ||_HS ≈ ||ϕ||₂ for Hermitian Φ. Then the completeness relation implies

$$\begin{align}\hfill {\Vert}{\Phi}{{\Vert}}_{\text{HS}}^{2}\approx {\int }_{\mathcal{M}}\langle x\vert {\Phi}{\Phi}\vert x\rangle & \approx {\int }_{\mathcal{M}}\phi {\left(x\right)}^{2}={\Vert}\phi {{\Vert}}_{2}^{2}\hfill \\ \hfill {\int }_{\mathcal{M}}\langle x\vert \left({\Phi}-\phi \left(x\right)\right)\left({\Phi}-\phi \left(x\right)\right)\vert x\rangle & \approx 0\hfill \end{align} \tag{ 104 }$$

which implies that $\left({\Phi}-\phi \left(x\right)\right)\vert x\rangle \approx 0\enspace \forall \enspace x\in \mathcal{M}$. Hence they are approximately local in the sense of (44). In particular they approximately commute due to (49),

$$\begin{equation}{\Phi}{\Psi}\approx {\Psi}{\Phi},\qquad {\Phi},{\Psi}\in Loc\left(\mathcal{H}\right).\end{equation} \tag{ 105 }$$

Hence the above definition of $Loc\left(\mathcal{H}\right)$ is a refinement of the definitions in section 2.4, turning the local statements into global ones.

The image ${\mathcal{C}}_{\text{IR}}\left(\mathcal{M}\right)$ is typically given by functions which are slowly varying on the length scale L_coh, corresponding to the semi-classical or infrared regime. To see that $\mathcal{Q}$ is approximately inverse to the symbol map, we note that the completeness relation implies

$$\begin{equation}\vert y\rangle \approx {\int }_{\mathcal{M}}\vert x\rangle \langle x\vert y\rangle .\end{equation} \tag{ 106 }$$

This means that

$$\begin{equation}\langle x\vert y\rangle \approx {\delta }_{y}\left(x\right)\end{equation} \tag{ 107 }$$

for any $y\in \mathcal{M}$ w.r.t. the measure (93), consistent with $\vert \langle x\vert y\rangle \vert \sim {\mathrm{e}}^{-\frac{1}{2}{\left(x-y\right)}_{g}^{2}}$ (13) (23). Then

$$\begin{equation}\mathcal{Q}\left(\phi \right)\vert y\rangle \approx {\int }_{\mathcal{M}}\phi \left(x\right)\vert x\rangle \langle x\vert y\rangle \approx \phi \left(y\right)\vert y\rangle .\end{equation} \tag{ 108 }$$

for functions ϕ(x) which are slowly varying on L_coh. Therefore $\mathcal{Q}\left(\phi \right)$ is almost-local and hence $\mathcal{Q}\left(\phi \right)\in Loc\left(\mathcal{H}\right)$ for slowly varying ϕ, and moreover $\mathcal{Q}$ is approximately the inverse of the symbol map on $Loc\left(\mathcal{H}\right)$, since (108) gives

$$\begin{equation}\langle y\vert \mathcal{Q}\left(\phi \right)\vert y\rangle \approx \phi \left(y\right).\end{equation} \tag{ 109 }$$

For almost-local quantum spaces, $Loc\left(\mathcal{H}\right)$ contains in particular the basic matrices

$$\begin{equation}{X}^{a}\approx {\int }_{\mathcal{M}}{\mathbf{x}}^{\mathbf{a}}\vert x\rangle \langle x\vert .\end{equation} \tag{ 110 }$$

The approximation is good as long as the classical function x^a is approximately constant on L_coh. Moreover, (66) gives the approximate commutation relations on $\mathcal{M}$

$$\begin{equation}\left[{X}^{a},{X}^{b}\right]\sim \mathrm{i}{\theta }^{ab}=\mathrm{i}\left\{{x}^{a},{x}^{b}\right\}.\end{equation} \tag{ 111 }$$

We have seen that θ^ab is tangential to $\mathcal{M}$ and the inverse of the symplectic form ω on $\mathcal{M}$, hence {x^a , x^b } are Poisson brackets on $\mathcal{M}$. In this sense, the semi-classical geometry is encoded in the matrix configuration X^a . These observations are summarized in table 1. This provides the starting point of the emergent geometry and gravity considerations in [27, 28], which will be briefly discussed in section 5.2.

Table 1. Correspondence between almost-local operators and infrared functions on $\mathcal{M}$ for almost-local quantum spaces. The metric structure is encoded in the Laplacian □ (144).

$Loc\left(\mathcal{H}\right)\subset End\left(\mathcal{H}\right)$	∼	$\enspace {\mathcal{C}}_{\text{IR}}\left(\mathcal{M}\right)\subset {L}^{2}\left(\mathcal{M}\right)$
Φ	∼	ϕ(x) =⟨x|Φ|x⟩
Xa	∼	xa (x)
[., .]	∼	i{., .}
Tr	∼	${\int }_{\mathcal{M}}$
□	∼	e−σ □G

The above Poisson structure extends trivially to ${\tilde {\mathbb{R}}}^{D}$, which for $D{ >}\mathrm{dim}\enspace \mathcal{M}$ decomposes into symplectic leaves of ω_ab that are preserved by the Poisson structure. Functions which are constant on these leaves then have vanishing Poisson brackets, which leads to a degenerate effective metric as discussed in section 5.2.

In the UV or deep quantum regime, the above semi-classical picture is no longer justified, and in fact it is very misleading. In particular, consider string states which are defined as rank one operators built out of quasi-coherent states [29, 30]

$$\begin{equation}{\psi }_{x,y}{:=}\vert x\rangle \langle y\vert \in End\left(\mathcal{H}\right).\end{equation} \tag{ 112 }$$

They are highly non-local for x ≠ y, and should not be interpreted as function but rather as open strings (or dipoles) linking |y⟩ to |x⟩ on the embedded brane $\tilde {\mathcal{M}}$. These states provide a complete and more adequate picture of $End\left(\mathcal{H}\right)$, and exhibit the stringy nature of noncommutative field theory and Yang–Mills matrix models [29]. This means that the physical content of Yang–Mills matrix models, and more generally of noncommutative field theory, is much richer than suggested by the semi-classical limit. In particular, string states arise as high-energy excitation modes, leading to UV/IR mixing in noncommutative field theory [31]. This is a phenomenon which has no counterpart in conventional (quantum) field theory.

Now we return to the exact analysis. For any quantum manifold $\mathcal{M}$, the embedding $\mathcal{M}\to \mathbb{C}{P}^{N-1}$ induces the tangential map

$$\begin{equation}{T}_{\xi }\mathcal{M}\to {T}_{\xi }\mathbb{C}{P}^{N-1}.\end{equation} \tag{ 113 }$$

Now we take into account that $\mathbb{C}{P}^{N-1}$ carries an intrinsic complex structure

$$\begin{equation}\mathcal{J}:{T}_{\xi }\mathbb{C}{P}^{N-1}\to {T}_{\xi }\mathbb{C}{P}^{N-1},\quad \mathcal{J}v=\mathrm{i}v\end{equation} \tag{ 114 }$$

for any $v\in {T}_{\xi }\mathbb{C}{P}^{N-1}$. Accordingly, $T\mathbb{C}{P}^{N-1}\cong {T}^{\left(1,0\right)}\mathbb{C}{P}^{N-1}$ can be viewed as holomorphic tangent bundle, thus bypassing an explicit complexification of its real tangent space. With this in mind, we define the complex quantum tangent space of $\mathcal{M}$ as

$$\begin{equation}{T}_{\xi ,\mathbb{C}}\mathcal{M}{:=}{\left\langle {\mathcal{X}}_{a}\vert x\rangle \right\rangle }_{\mathbb{C}}\subset \enspace {T}_{\xi }\mathbb{C}{P}^{N-1}\cong {T}_{\xi ,\mathbb{C}}\mathbb{C}{P}^{N-1},\end{equation} \tag{ 115 }$$

which also carries the complex structure

$$\begin{equation}\mathcal{J}{\mathcal{X}}_{a}\vert x\rangle {:=}\mathrm{i}{\mathcal{X}}_{a}\vert x\rangle \in {T}_{\xi ,\mathbb{C}}\mathcal{M},\quad {\mathcal{J}}^{2}=-\mathbb{1}.\end{equation} \tag{ 116 }$$

Again, this complex tangent space is not necessarily the complexification of the real one. Using the basis ${\mathcal{X}}_{\mu }\vert x\rangle ,\enspace \mu =1,\dots ,m$ of ${T}_{\xi }\mathcal{M}$ which arises in normal embedding coordinates, there may be relations of the form

$$\begin{equation}\left(\mathrm{i}{c}^{\mu }{\mathcal{X}}_{\mu }-{J }^{\enspace \nu }{\mathcal{X}}_{\nu }\right)\vert x\rangle =0\quad \text{for}\enspace {J}^{\nu },{c}^{\mu }\in \mathbb{R},\end{equation} \tag{ 117 }$$

so that ${T}_{\xi ,\mathbb{C}}\mathcal{M}$ has reduced dimension over $\mathbb{C}$. We will see that for quantum Kähler manifolds as defined below, the complex dimension is half of the same as the real one.

Quantum Kähler manifolds. Consider the maximally degenerate case where the complex dimension of ${T}_{\xi ,\mathbb{C}}\mathcal{M}$ is given by $n=\frac{m}{2}\in \mathbb{N}$ where $m={\mathrm{dim}}_{\mathbb{R}}\enspace \mathcal{M}$. Then ${T}_{\xi }\mathcal{M}$ is stable under the complex structure operator $\mathcal{J}$

$$\begin{equation}{T}_{\xi ,\mathbb{C}}\mathcal{M}={T}_{\xi }\mathcal{M}\end{equation} \tag{ 118 }$$

so that ${T}_{\xi }\mathcal{M}$ should be viewed as holomorphic tangent space of $\mathcal{M}$. But this implies that $\mathcal{M}\subset \mathbb{C}{P}^{N-1}$ is a complex sub-manifold (i.e. defined by holomorphic equations), cf [32] or proposition 1.3.14 in [33]. Such quantum manifolds $\mathcal{M}$ will be called quantum Kähler manifolds, for reasons explained below. Indeed, all complex sub-manifolds of $\mathbb{C}{P}^{N-1}$ are known to be Kähler. Note that this is an intrinsic property of a quantum space $\mathcal{M}$, and no extra structure is introduced here: $\mathcal{M}$ either is or is not of this type ¹⁹ . We will see that this includes the well-known quantized or 'fuzzy' spaces arising from quantized coadjoint orbits ²⁰ .

Consider the quantum Kähler case in more detail. We can introduce a local holomorphic parametrization of $\mathcal{M}\subset \mathbb{C}{P}^{N-1}$ near ξ in terms of ${z}^{k}\in {\mathbb{C}}^{n}$. Then any local (!) holomorphic section of the tautological line bundle over $\mathbb{C}{P}^{N-1}$ defines via pull-back a local holomorphic section of the line bundle

$$\begin{equation}\tilde {\mathcal{B}}{:=}\bigcup\limits _{x\in {\tilde {\mathbb{R}}}^{D}}{E}_{x}\to \mathcal{M}\hookrightarrow \mathbb{C}{P}^{N-1}\end{equation} \tag{ 119 }$$

over $\mathcal{M}$, denoted by ||z⟩. This ||z⟩ can be viewed as holomorphic ${\mathbb{C}}^{N}$-valued function on $\mathcal{M}$, which satisfies

$$\begin{equation}\frac{\partial }{\partial {\bar{z}}^{k}}{\Vert}z\rangle =0,\quad {\Vert}z\rangle {\vert \xi }_{}=\vert \xi \rangle ,\end{equation} \tag{ 120 }$$

where ${\bar{z}}^{k}$ denotes the complex conjugate of z^k . Hence ||z⟩ arises from |x⟩ through a re-parametrization and gauge transformation along with a non-trivial normalization ²¹ factor; this is indicated by the double line in ||z⟩. In other words, the differential of the section

$$\begin{equation}d{\Vert}z\rangle =\mathit{d}{z}^{k}\frac{\partial }{\partial {z}^{k}}{\Vert}z\rangle \in {{\Omega}}_{z}^{\left(1,0\right)}\mathcal{M}\end{equation} \tag{ 121 }$$

is a (1, 0) one-form. Given this holomorphic one-form d||z⟩ and the Hermitian inner product on $\mathcal{H}$, we naturally obtain a (1, 1) form

$$\begin{align}\hfill \omega {:=}{\left(d{\Vert}z\rangle \right)}^{{\dagger}}\wedge d{\Vert}z\rangle & ={\omega }_{\bar{k}l}\enspace \mathit{d}{\bar{z}}^{k}\wedge \mathit{d}{z}^{l}\in {{\Omega}}_{z}^{\left(1,1\right)}\mathcal{M}\hfill \\ \hfill {\omega }_{\bar{k}l}& ={\left({d}_{k}{\Vert}z\rangle \right)}^{{\dagger}}{d}_{l}{\Vert}z\rangle \hfill \end{align} \tag{ 122 }$$

which is closed,

$$\begin{equation}\mathrm{d}\omega =-{\left(d{\Vert}z\rangle \right)}^{{\dagger}}\wedge dd{\Vert}z\rangle +{\left(dd{\Vert}z\rangle \right)}^{{\dagger}}\wedge d{\Vert}z\rangle =0\end{equation} \tag{ 123 }$$

using holomorphicity of ||z⟩. This is the Kähler form, which encodes the ω_ab in (20). The quantum metric then satisfies

$$\begin{align}\hfill 2g\left(X,Y\right)& =\left({\left(d{\Vert}z\rangle \right)}^{{\dagger}}\otimes d{\Vert}z\rangle \right)\left(X,Y\right)+\left(X{\leftrightarrow}Y\right)\hfill \\ \hfill & =\omega \left(X,\mathcal{J}Y\right)+\omega \left(Y,\mathcal{J}X\right)=2\omega \left(X,\mathcal{J}Y\right)\in {T}^{\left(1,1\right)}\hfill \end{align} \tag{ 124 }$$

which justifies the name 'quantum Kähler manifold'. Note that the metric is manifestly in T^(1,1), hence $g\left(\mathcal{J}X,\mathcal{J}Y\right)=g\left(X,Y\right)$ is automatic. In particular, the coherence length L_coh and the uncertainty scale L_NC coincide. Of course $\mathcal{M}$ is a Kähler manifold in the usual sense, but the attribute 'quantum' indicates its origin from the matrices X^a .

Now we relate this to the local generators ${\mathcal{X}}_{\mu }$ (35), (84). Introducing real coordinates z^k = z^k (x^μ ) where x^μ are the local (Cartesian) embedding coordinates introduced above, the holomorphicity relation (120) can be expressed using (35) as

$$\begin{equation}0=\frac{\partial }{\partial {\bar{z}}^{k}}{\Vert}z\rangle =\frac{\partial {x}^{\mu }}{\partial {\bar{z}}^{k}}\frac{\partial }{\partial {x}^{\mu }}{\Vert}z\rangle =\frac{\partial {x}^{\mu }}{\partial {\bar{z}}^{k}}\enspace \left({\mathcal{X}}_{\mu }+\mathrm{i}{A}_{\mu }\right){\Vert}z\rangle .\end{equation} \tag{ 125 }$$

Similarly,

$$\begin{equation}\frac{\partial }{\partial {z}^{k}}{\Vert}z\rangle =\frac{\partial {x}^{\mu }}{\partial {z}^{k}}\frac{\partial }{\partial {x}^{\mu }}{\Vert}z\rangle =\frac{\partial {x}^{\mu }}{\partial {z}^{k}}\left({\mathcal{X}}_{\mu }+\mathrm{i}{A}_{\mu }\right){\Vert}z\rangle .\end{equation} \tag{ 126 }$$

We can now introduce new generators ²² ${\mathcal{A}}^{k},{\bar{\mathcal{A}}}_{l}$ via

$$\begin{align}\hfill {\mathcal{A}}^{k}& =\frac{\partial {x}^{\mu }}{\partial {\bar{z}}^{k}}\enspace \left({\mathcal{X}}_{\mu }+\mathrm{i}{A}_{\mu }\right)\hfill \\ \hfill {\bar{\mathcal{A}}}_{k}& =\frac{\partial {x}^{\mu }}{\partial {z}^{k}}\enspace \left({\mathcal{X}}_{\mu }+\mathrm{i}{A}_{\mu }\right)\hfill \end{align} \tag{ 127 }$$

so that

$$\begin{equation}{\mathcal{A}}^{k}{\Vert}z\rangle =0,\qquad {\bar{\mathcal{A}}}_{k}{\Vert}z\rangle =\frac{\partial }{\partial {z}^{k}}{\Vert}z\rangle .\end{equation} \tag{ 128 }$$

These are clearly the analogs of the standard annihilation properties of coherent states. It is hence appropriate to denote the ||z⟩ on quantum Kähler manifolds as coherent states. Then

$$\begin{equation}{T}_{\xi ,\mathbb{C}}\mathcal{M}={\left\langle {\bar{\mathcal{A}}}_{k}{\Vert}z\rangle \right\rangle }_{\mathbb{C}}\enspace \cong {\mathbb{C}}^{n}\quad k=1,\dots ,n.\end{equation} \tag{ 129 }$$

The metric tensor and the symplectic form are then determined as usual by the Kähler form

$$\begin{equation}\mathrm{i}{\omega }_{\bar{k}l}={\left({d}_{k}{\Vert}z\rangle \right)}^{{\dagger}}{d}_{l}{\Vert}z\rangle =\langle z{\Vert}{\bar{\mathcal{A}}}_{k}^{{\dagger}}{\bar{\mathcal{A}}}_{l}{\Vert}z\rangle \end{equation} \tag{ 130 }$$

which arises from a local Kähler potential,

$$\begin{equation}{\omega }_{\bar{k}l}=-\frac{1}{2}{\bar{\partial }}_{k}{\partial }_{l}\rho \end{equation} \tag{ 131 }$$

given by the restriction of the (Fubini–Study) Kähler potential on $\mathbb{C}{P}^{N}$.

This provides a rather satisfactory concept of quantum Kähler geometry, which arises in a natural way from the complex structure in the Hilbert space. There is no need to invoke any semi-classical or large N limit. Not all quantum spaces are of this type, a counterexample being the minimal fuzzy torus ${T}_{2}^{2}$ as discussed in section 6.4. In [15], it is claimed that all quantum manifolds are Kähler in the semi-classical limit, based on (69). However this refers to a different almost-complex structure and metric which is not intrinsic. From the present analysis, there is no obvious reason why all quantum manifolds should be Kähler, even in the semi-classical limit.

Since for non-Kähler manifolds the complex tangent space ${T}_{\mathbb{C}}\mathcal{M}$ is higher-dimensional, quantum effects due to loops in Yang–Mills matrix models may be more significant, and the geometric trace formula (2.38) in [29] for string states would need to be replaced with some higher-dimensional analog. This suggests that quantum Kähler manifolds may be protected by some sort of non-renormalization theorems.

The present framework has a natural extension to spinors and Dirac-type operators. Namely, for any matrix configuration X^a , a = 1, ..., D we can consider [13, 14, 16, 17]

$$\begin{equation}{\mathrm{D/}}_{x}={{\Gamma}}_{a}\left({X}^{a}-{x}^{a}\right),\quad {x}^{a}\in {\mathbb{R}}^{D}\end{equation} \tag{ 140 }$$

acting on $\mathcal{H}\otimes {\mathbb{C}}^{s}$. Here Γ_a are the gamma matrices generating the Clifford algebra of SO(D) on the irreducible representation ${\mathbb{C}}^{s}$. D̸_x arises as off-diagonal part of the matrix Dirac operator ²³ $\mathrm{D/}={{\Gamma}}_{a}\left. \right]{X}^{a},.\left. \right]$ in Yang–Mills matrix models such as the IIB or IKKT model, for the matrix configuration extended by a point brane X^a ⊕ x^a . It describes a fermionic string stretched between the brane and the point x^a . Quite remarkably, numerical investigations [13] strongly suggest that the Dirac operator D̸_x always has exact zero modes

$$\begin{equation}{\mathrm{D/}}_{x}\vert x,s\rangle =0\enspace \end{equation} \tag{ 141 }$$

at $\mathcal{M}$, so that there is no need to introduce the lowest eigenvalue function λ(x). This can be justified rigorously for two-dimensional branes [14], and some heuristic reasons can be given also in more general cases; see [14, 16, 17] for further work. However, the presence of extra structure due to the spinors obscures the relation with the quasi-coherent states and $\mathcal{M}$ as introduced here. This is certainly an interesting topic for further research.

The considerations in this paper are motivated by Yang–Mills matrix models, whose solutions are precisely matrix configurations as considered here. Fluctuations in these models are governed by the matrix Laplacian

$$\begin{equation}\square {:=}{\delta }_{ab}\left[{X}^{a},\left[{X}^{b},.\right]\right]:End\left(\mathcal{H}\right)\to End\left(\mathcal{H}\right).\end{equation} \tag{ 142 }$$

The displacement Hamiltonian arises as off-diagonal part of the matrix Laplacian for a point or probe brane added to the matrix configuration [13], i.e. for X^a ⊕ x^a acting on $\mathcal{H}\oplus \mathbb{C}$. It describes a string stretched between the brane and the point x^a . This can also be viewed as a special case of intersecting branes [39], one brane being the point probe.

To understand the effective metric in matrix models, consider the inner derivations

$$\begin{equation}\left[{X}_{a},.\right]\sim \mathrm{i}{\theta }^{a\mu }{\partial }_{\mu }\end{equation} \tag{ 143 }$$

acting on $End\left(\mathcal{H}\right)$ resp. ${\mathcal{C}}_{\text{IR}}\left(\mathcal{M}\right)$, which are (quantizations of) Hamiltonian vector fields on $\mathcal{M}$ for almost-local quantum spaces. By considering the inner product ⟨Φ, Ψ⟩ :=  Tr([X^a , Φ^†][X_a , Ψ]) on $Loc\left(\mathcal{H}\right)$, one can then show [27] that

$$\begin{equation}\square \sim {\text{e}}^{\sigma }{\square }_{G},\end{equation} \tag{ 144 }$$

where G is the effective metric on $\mathcal{M}$ given by

$$\begin{equation}{G}^{\mu \nu }={\text{e}}^{-\sigma }\enspace {\theta }^{\mu {\mu }^{\prime }}{\theta }^{\nu {\nu }^{\prime }}{g}_{{\mu }^{\prime }{\nu }^{\prime }},\quad {\text{e}}^{-\sigma }=\frac{\vert {G}^{\mu \nu }{\vert }^{1/2}}{\vert {\theta }^{\mu \nu }{\vert }^{1/2}}\enspace \end{equation} \tag{ 145 }$$

for $\mathrm{dim}\enspace \mathcal{M}{ >}2$. This can be viewed as open-string metric, and it provides the starting point of the emergent geometry and gravity considerations in [27, 28]. In the two-dimensional case, the underlying Weyl invariance leads to a different interpretation of □, which is discussed in [40].

In the reducible case, $\mathcal{M}$ decomposes into a foliation of symplectic leaves. Then the effective metric is non-vanishing only along this foliation, i.e. it vanishes along the transversal directions. In the context of Yang–Mills matrix models, this means that fluctuation modes on such backgrounds only propagate along the symplectic leaves, so that the resulting gauge theory is lower-dimensional. This happens on any superficially odd-dimensional quantum space, or e.g. on κ Minkowski space [41] in dimensions larger than 2.

The fuzzy sphere ${S}_{N}^{2}$ [1, 2] is a quantum space defined in terms of three N × N Hermitian matrices

$$\begin{equation}{X}^{a}=\frac{1}{\sqrt{{C}_{N}}}\enspace {J}_{\left(N\right)}^{a},\quad \enspace a=1,2,3,\end{equation} \tag{ 146 }$$

where ${J}_{\left(N\right)}^{a}$ are the generators of the N-dimensional irrep of $\mathfrak{s}\mathfrak{u}\left(2\right)$ on $\mathcal{H}={\mathbb{C}}^{N}$, and ${C}_{N}=\frac{1}{4}\left({N}^{2}-1\right)$ is the value of the quadratic Casimir. They satisfy the relations

$$\begin{equation}\left[{X}^{a},{X}^{b}\right]=\frac{\mathrm{i}}{\sqrt{{C}_{N}}}{\varepsilon }^{\text{abc}}\enspace {X}^{c},\qquad \sum\limits _{a=1}^{3}{X}^{a}{X}^{a}=\mathbb{1}\end{equation} \tag{ 147 }$$

choosing the normalization (146) such that the radius is one. The displacement Hamiltonian is

$$\begin{equation}{H}_{x}=\frac{1}{2}\sum\limits _{a=1}^{3}{\left({X}^{a}-{x}^{a}\right)}^{2}=\frac{1}{2}\left(\mathbb{1}+\vert x{\vert }^{2}\right)-\sum\limits _{a=1}^{3}{x}^{a}{X}^{a},\end{equation} \tag{ 148 }$$

where $\vert x{\vert }^{2}={\sum }_{a}{x}_{a}^{2}$. Using SO(3) invariance, it suffices to consider the north pole x = (0, 0, x³) ≕ n where

$$\begin{equation}{H}_{x}=\frac{1}{2}\left(\mathbb{1}+\vert x{\vert }^{2}\right)-\vert x\vert \enspace {X}^{3}\end{equation} \tag{ 149 }$$

assuming x³ > 0 to be specific. Hence the ground state of H_x is given by the highest weight vector $\vert n\rangle {:=}\vert \frac{N-1}{2},\frac{N-1}{2}\rangle$ of the $\mathfrak{s}\mathfrak{u}\left(2\right)$ irrep $\mathcal{H}$, and the eigenvalue is easily found to be [13]

$$\begin{equation}\lambda \left(x\right)=\frac{1}{2}\left(1+\vert x{\vert }^{2}\right)-\vert x\vert \sqrt{\frac{N-1}{N+1}}.\end{equation} \tag{ 150 }$$

All other quasi-coherent states are obtained by SO(3) acting on |n⟩, hence the abstract quantum space $\mathcal{M}$ is given by the group orbit

$$\begin{equation}\mathcal{M}=SO\left(3\right)\cdot \vert n\rangle =SO\left(3\right){/}_{U\left(1\right)}\cong {S}^{2}\subset \mathbb{C}{P}^{N-1}.\end{equation} \tag{ 151 }$$

Note that the quasi-coherent states are constant along the radial lines in agreement with (77),

$$\begin{equation}\vert x\rangle =\vert \alpha x\rangle \quad \text{for}\enspace \alpha { >}0.\end{equation} \tag{ 152 }$$

The equivalence classes $\mathcal{N}$ consist of the radial lines emanating from the origin, and the would-be symplectic form ω_ab and the quantum metric g_ab vanish if any one component is radial. The minima of λ(x) on ${\mathcal{N}}_{x}$ describe a sphere with radius $\vert {x}_{0}\vert =\sqrt{\frac{N-1}{N+1}}=1+\mathcal{O}\left(\frac{1}{N}\right)$. This coincides precisely with the embedded quantum space (79)

$$\begin{equation}\tilde {\mathcal{M}}=\left\{\langle x\vert {X}^{a}\vert x\rangle \right\}=\left\{x\in {\mathbb{R}}^{3}:\enspace \vert x\vert =\sqrt{\frac{N-1}{N+1}}\right\}\cong {S}^{2}\enspace \end{equation} \tag{ 153 }$$

defined by the expectation value x^a (78), in accordance with (82). At the singular set $\mathcal{K}=\left\{0\right\}$ the Hamiltonian is ${H}_{0}={C}^{2}\mathbb{1}$, so that all energy levels become degenerate and cross. Following |x⟩ along the radial direction through the origin, it turns into the highest energy level. It is easy to see that the would-be symplectic form ω is the unique SO(3)-invariant two-form on $\mathcal{M}$ which satisfies the quantization condition (26) with n = N. Moreover, the abstract quantum space $\mathcal{M}\cong {S}^{2}\subset \mathbb{C}{P}^{N-1}$ is a quantum Kähler manifold, since the complex tangent space (115) is one-dimensional, spanned by

$$\begin{equation}{T}_{n,\mathbb{C}}\mathcal{M}={\left\langle {J}^{-}\vert n\rangle \right\rangle }_{\mathbb{C}}\enspace \end{equation} \tag{ 154 }$$

(at $\vert n\rangle \in \mathcal{M}$). This holds because |n⟩ is the highest weight state, so that

$$\begin{equation}{J}^{+}\vert n\rangle =0;\end{equation} \tag{ 155 }$$

therefore the two tangent vectors ${\mathcal{X}}^{1}\vert n\rangle ,{\mathcal{X}}^{2}\vert n\rangle \in {T}_{n}\mathcal{M}$ (84) are related by i, while ${\mathcal{X}}^{3}\vert n\rangle$ vanishes at n. Indeed, it is well-known that the coherent states on ${S}_{N}^{2}$ form a Riemann sphere, and the (quasi-) coherent states coincide with the coherent states introduced in [34].

All this holds for any N ⩾ 2. The coherence length is of order

$$\begin{equation}{L}_{\text{coh}}\approx {L}_{\mathrm{N}\mathrm{C}}\sim \frac{1}{\sqrt{N}}\enspace \end{equation} \tag{ 156 }$$

in the given normalization. Hence for sufficiently large N, the almost-local operators comprise all polynomials in X^a up to order $O\left(\sqrt{N}\right)$ (depending on some specific bound), so that ${S}_{N}^{2}$ is an almost-local quantum space. In contrast for the minimal fuzzy sphere ${S}_{2}^{2}$ with N = 2, the generators reduce to the Pauli matrices X^a = σ^a , and the (quasi)coherent states form the well-known Bloch sphere $\mathcal{M}={S}^{2}\cong \mathbb{C}{P}^{1}$. This is still a quantum Kähler manifold even though the semi-classical regime is trivial and contains only the constant functions $Loc\left(\mathcal{H}\right)=\mathbb{C}\mathbb{1}$, since the coherence length is of the same order as the entire space $\mathcal{M}$.

The above construction generalizes naturally to quantized coadjoint orbits for any compact semi-simple Lie group G with Lie algebra $\mathfrak{g}$. For any irreducible representation ${\mathcal{H}}_{{\Lambda}}$ with highest weight Λ = (n₁, ..., n_k ) labeled by Dynkin indices n_j , the matrix configuration

$$\begin{equation}{X}^{a}=c{T}^{a},\quad a=1,\dots ,D\end{equation} \tag{ 157 }$$

defines a quantum Kähler manifold $\mathcal{M}\cong G/K$. Here T^a are orthogonal generators of $\mathfrak{g}\cong {\mathbb{R}}^{D}$ acting on ${\mathcal{H}}_{{\Lambda}}$, K is the stability group of the highest weight Λ, and c is some normalization constant. Then the displacement Hamiltonian is

$$\begin{equation}{H}_{x}={C}^{2}\left(\mathfrak{g}\right)+\frac{1}{2}{x}_{a}{x}^{a}-{x}_{a}{T}^{a},\enspace \end{equation} \tag{ 158 }$$

where ${C}^{2}\left(\mathfrak{g}\right)\propto \mathbb{1}$ is the quadratic Casimir. Using G-invariance, we can assume that x is in (the dual of) the Cartan subalgebra and has maximal weight. Then |x⟩ = |Λ⟩ is the highest weight state, so that the quasi-coherent states are the group orbit $\mathcal{M}=G\cdot \vert {\Lambda}\rangle \cong G/K$ of the highest weight state with stabilizer K. This is a quantum Kähler manifold due to the highest weight property, and the quantum metric g_ab (19) and the symplectic form ω (20) are the canonical group-invariant structures on the Kähler manifold $\mathcal{M}$. For large Dynkin indices n_j ⩾ n ≫ 1, the almost-local operators comprise all polynomials in X^a up to some order $O\left(\sqrt{n}\right)$, so that $\mathcal{M}$ is an almost-local quantum space. This is essentially the well-known story of quantized coadjoint orbits, and the (quasi-) coherent states coincide with the coherent states introduced in [34], cf [42]. Perhaps less known is the fact that if some of the n_j are small, $\mathcal{M}$ can be viewed as 'oxidation' of some lower-dimensional brane, more precisely as a bundle over ${\mathcal{M}}_{0}$ whose fiber is very 'fuzzy'. For an application of such a structure see e.g. section 4.2 in [43].

This construction generalizes further to highest weight (discrete series) unitary irreducible representation of non-compact semi-simple Lie groups. A particularly interesting example is given by the 'short' series of unitary irreps of SO(4, 2) known as singletons, which lead to the fuzzy four-hyperboloids ${H}_{n}^{4}$ discussed below, and to quantum spaces which can be viewed as cosmological space–time [44].

(Minimal) fuzzy $\mathbb{C}{P}_{N}^{N-1}$. As an example we consider minimal fuzzy $\mathbb{C}{P}_{N}^{N-1}$, which is obtained using the above general construction for G = SU(N) and its fundamental representation $\mathcal{H}=\left(1,0,\dots ,0\right)$, so that $G/K\cong \mathbb{C}{P}^{N-1}$. This is the quantum Kähler manifold obtained from the matrix configuration

$$\begin{equation}{X}^{a}={\lambda }^{a}\in End\left(\mathcal{H}\right),\quad \mathcal{H}={\mathbb{C}}^{N}\end{equation} \tag{ 159 }$$

for a = 1, ..., N² − 1, where λ^a are a (Gell–Mann) ON basis of $\mathfrak{s}\mathfrak{u}\left(N\right)$ in the fundamental representation. Then $End\left(\mathcal{H}\right)\cong \left(0,\dots ,0\right)\oplus \left(1,0,\dots ,0,1\right)$ can be viewed as a minimal quantization of functions on $\mathbb{C}{P}^{N-1}$. The quantization map

$$\begin{equation}\mathcal{Q}\left(\phi \right)={\int }_{\mathbb{C}{P}^{N-1}}\vert x\rangle \langle x\vert \phi \left(x\right)\end{equation} \tag{ 160 }$$

is then the partial inverse of the symbol map, apart from the constant function:

$$\begin{equation}\mathcal{Q}\left(\langle x\vert {\Phi}\vert x\rangle \right)=c{\Phi}\quad \text{if}\;\mathrm{Tr}\left({\Phi}\right)=0\enspace \end{equation} \tag{ 161 }$$

for some c > 0. Near |Λ⟩, the quasi-coherent states |x⟩ can be organized as holomorphic sections

$$\begin{equation}{\Vert}z\rangle =\mathrm{exp}\left({z}^{k}{T}_{k}^{+}\right)\vert {\Lambda}\rangle ,\end{equation} \tag{ 162 }$$

where the ${T}_{k}^{+},\enspace k=1,\dots ,N-1$ are the rising operators of a Chevalley basis of $\mathfrak{s}\mathfrak{u}\left(N\right)$. Hence fuzzy $\mathbb{C}{P}_{N}^{N-1}$ is a quantum Kähler manifold which coincides with $\mathbb{C}{P}^{N-1}$, with Kähler form

$$\begin{equation}{\omega }_{\bar{k}l}=\frac{\partial }{\partial {\bar{z}}^{k}}\langle z{\Vert}\frac{\partial }{\partial {z}^{l}}{\Vert}z\rangle .\end{equation} \tag{ 163 }$$

Squashed $\mathbb{C}{P}_{N}^{2}$. Further quantum spaces can be obtained by projections of quantized coadjoint orbits. For example, starting from fuzzy $\mathbb{C}{P}_{N}^{2}$ with $\mathcal{H}=\left(N,0\right)$, consider the following matrix configuration

$$\begin{equation}{X}^{a}={T}^{a},\quad a=1,2,4,5,6,7\end{equation} \tag{ 164 }$$

dropping the Cartan generators T₃ and T₈ from the (Gell–Mann) basis of $\mathfrak{s}\mathfrak{u}\left(3\right)$. Then the displacement Hamiltonian can be written as

$$\begin{equation}{H}_{x}={\bar{H}}_{x}-\frac{1}{2}{\left({X}_{3}-{x}_{3}\right)}^{2}-\frac{1}{2}{\left({X}_{8}-{x}_{8}\right)}^{2},\end{equation} \tag{ 165 }$$

where ${\bar{H}}_{x}$ is the displacement Hamiltonian for $\mathbb{C}{P}_{N}^{2}$. Although the quasi-coherent states |x⟩ are not known in this case, they are close to those of $\mathbb{C}{P}_{N}^{2}$ in the large N limit, cf [13]. Indeed then the last two terms in (165) are small, and $0{< }\lambda \left(x\right){\leqslant}\bar{\lambda }\left(x\right)$ gives an upper bound for λ. This implies that the displacement is small, and

$$\begin{equation}\mathcal{M}\approx \mathbb{C}{P}^{2}\subset \mathbb{C}{P}^{N\left(N+3\right)/2}.\end{equation} \tag{ 166 }$$

Again, the concept of the abstract quantum space is superior to the notion of an embedded brane, which is a complicated self-intersecting variety in ${\mathbb{R}}^{6}$ related to the Roman surface [4].

The fuzzy four-sphere ${S}_{N}^{4}$. Now consider again the quantized coadjoint orbit of SU(4) ≅ SO(6) acting on the highest weight irrep ${\mathcal{H}}_{{\Lambda}}$ with Λ = (N, 0, 0). We have seen just that the matrix configuration using all $\mathfrak{s}\mathfrak{o}\left(6\right)$ generators ${\mathcal{M}}^{ab}=-{\mathcal{M}}^{ba}$ as in (157) would give fuzzy $\mathbb{C}{P}_{N}^{3}$, with coherent states acting on the highest weight state |Λ⟩. Now instead of using all ${\mathcal{M}}^{ab}$, consider the matrix configuration defined by the following five Hermitian matrices

$$\begin{equation}{X}^{a}={\mathcal{M}}^{a6}\in End\left({\mathcal{H}}_{{\Lambda}}\right),\quad a=1,\dots ,5.\end{equation} \tag{ 167 }$$

Using SO(5) invariance, it suffices to consider the displacement Hamiltonian at x = (0, 0, 0, 0, x₅),

$$\begin{equation}{H}_{x}=\frac{1}{2}\sum\limits _{i=1}^{4}{X}_{i}^{2}+\frac{1}{2}{\left({X}_{5}-{x}_{5}\right)}^{2}=\frac{1}{2}\left({R}^{2}+{x}_{5}^{2}\right)\mathbb{1}-{x}_{5}{X}^{5}\end{equation} \tag{ 168 }$$

since ${\sum }_{a}{X}_{a}^{2}={R}^{2}\mathbb{1}$ for ${R}^{2}=\frac{1}{4}N\left(N+4\right)$, cf [45, 46]. Now |Λ⟩ is by construction an eigenstate of X⁵ which commutes with SO(4), with maximal eigenvalue. Therefore the lowest eigenspace E_x of H_x is spanned by the orbit SO(4) ⋅ |Λ⟩ ≅ S², which spans an N + 1-dimensional complex vector space. This provides an example of a degenerate quantum space. The abstract quantum space $\mathcal{M}$ is obtained by acting with SO(5) on this S², which is easily seen to recover

$$\begin{equation}\mathcal{M}\cong \mathbb{C}{P}^{3}\subset \mathbb{C}{P}^{\mathrm{dim}\mathcal{H}-1}\end{equation} \tag{ 169 }$$

which is an equivariant S² bundle over S⁴. The E_x naturally form a SU(N + 1) bundle $\mathcal{B}$ over S⁴, and ω is replaced by an SU(N + 1) connection. Again the concept of an abstract quantum space greatly helps to understand the structure, as it resolves the degeneracy of the quasi-coherent states. Moreover $\mathcal{M}$ is clearly a Kähler manifold, and theorem 4.2 holds.

The fuzzy four-hyperboloid ${H}_{n}^{4}$. Using an analogous construction for SO(4, 2) and its singleton irreps ${\mathcal{H}}_{n}$ labeled by $n\in \mathbb{N}$, one obtains fuzzy ${H}_{n}^{4}$ [47, 48]. The corresponding matrix configuration is given by the following five Hermitian operators

$$\begin{equation}{X}^{a}={\mathcal{M}}^{a5}\in End\left({\mathcal{H}}_{{\Lambda}}\right),\quad a=0,\dots ,4.\end{equation} \tag{ 170 }$$

However, it is more appropriate here to define the displacement Hamiltonian using η_ab , so that SO(4, 1) is preserved. Then we can assume that x = (x₀, 0, 0, 0, 0), so that

$$\begin{equation}{H}_{x}=\frac{1}{2}\sum\limits _{i=1}^{4}{X}_{i}^{2}-\frac{1}{2}{\left({X}_{0}-{x}_{0}\right)}^{2}=\frac{1}{2}\left({R}^{2}-{x}_{0}^{2}\right)\mathbb{1}+{x}_{0}{X}^{0}.\end{equation} \tag{ 171 }$$

Then the resulting quasi-coherent states form an abstract quantum space $\mathcal{M}\cong \mathbb{C}{P}^{1,2}$, which is an S² bundle over H⁴. It is a Kähler manifold, and theorem 4.2 still holds in a weaker sense [47]. This in turn is the basis of the cosmological space–time solution ${\mathcal{M}}_{n}^{3,1}$ with an effective metric of FLRW type, as discussed in [44, 49].

Minimal fuzzy ${H}_{0}^{4}$. A particularly interesting example is obtained from ${H}_{n}^{4}$ for n = 0, which is not a quantized coadjoint orbit and not even symplectic. In that case E_x is one-dimensional, and one can check that ⟨x|∂_a |x⟩ = 0 = iA_a and ⟨x|[X^a , X^b ]|x⟩ = 0. Therefore the would-be symplectic form ω vanishes. The abstract quantum space is then

$$\begin{equation}\mathcal{M}={H}^{4}\end{equation} \tag{ 172 }$$

but it carries a trivial line bundle $\tilde {\mathcal{B}}$. It still satisfies the quantum Kähler ²⁴ condition (118) and theorem 4.2 should hold (using the SO(4, 1)-invariant integral) in a weaker sense. However this is not an almost-local quantum space, and there is no semi-classical regime.

The minimal fuzzy torus ${T}_{2}^{2}$ turns out to be a quantum manifold which is not Kähler, and not even symplectic. It is defined in terms of

$$\begin{equation}U=\left(\begin{matrix}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right)={X}_{1}+\mathrm{i}{X}_{2},\qquad V=\left(\begin{matrix}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{matrix}\right)={X}_{3}+\mathrm{i}{X}_{4}\end{equation} \tag{ 173 }$$

which defines four Hermitian matrices ${X}_{i}={X}_{i}^{{\dagger}}\in End\left({\mathbb{C}}^{2}\right)$. Noting that [U, U^†] = 0 = [V, V^†] and

$$\begin{align}\hfill \left(U-z\right){\left(U-z\right)}^{{\dagger}}& =\left(\begin{matrix}\hfill 1+\vert z{\vert }^{2}\hfill & \hfill -z-{z}^{{\ast}}\hfill \\ \hfill -z-{z}^{{\ast}}\hfill & \hfill 1+\vert z{\vert }^{2}\hfill \end{matrix}\right)\hfill \\ \hfill \left(V-w\right){\left(V-w\right)}^{{\dagger}}& =\left(\begin{matrix}\hfill \vert 1-w{\vert }^{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \vert 1+w{\vert }^{2}\hfill \end{matrix}\right),\hfill \end{align} \tag{ 174 }$$

where z = x₁ + ix₂ and w = x₃ + ix₄, the displacement Hamiltonian is

$$\begin{equation}{H}_{y}=\sum {\left({X}_{i}-{x}_{i}\right)}^{2}=\left(\begin{matrix}\hfill 1+\vert z{\vert }^{2}+\vert 1-w{\vert }^{2}\hfill & \hfill -z-{z}^{{\ast}}\hfill \\ \hfill -z-{z}^{{\ast}}\hfill & \hfill 1+\vert z{\vert }^{2}+\vert 1+w{\vert }^{2}\hfill \end{matrix}\right).\end{equation} \tag{ 175 }$$

The lowest eigenvalue is

$$\begin{equation}\lambda =2+\vert z{\vert }^{2}+\vert w{\vert }^{2}-\sqrt{\vert z+{z}^{{\ast}}{\vert }^{2}+\vert w+{w}^{{\ast}}{\vert }^{2}}\end{equation} \tag{ 176 }$$

and the corresponding quasi-coherent states are

$$\begin{equation}\vert x\rangle \propto \left(\begin{matrix}\hfill \sqrt{\vert z+{z}^{{\ast}}{\vert }^{2}+\vert w+{w}^{{\ast}}{\vert }^{2}}+{w}^{{\ast}}+w\hfill \\ \hfill {z}^{{\ast}}+z\hfill \end{matrix}\right)\in {\mathbb{R}}_{+}{\times}\mathbb{R}\subset {\mathbb{C}}^{2}.\end{equation} \tag{ 177 }$$

These clearly depend only on the real parts of z, w, and the normalized states describe a half circle in the upper half plane. However the two endpoints of this half-circle corresponding to (z = 1, w = −∞) and (z = −1, w = −∞) describe the same state $\vert x\rangle =\left(\begin{matrix}\hfill 0\hfill \\ \hfill {\pm}1\hfill \end{matrix}\right)$, and should hence be identified. Thus $\mathcal{M}={S}^{1}$, which is clearly not a Kähler manifold any not even symplectic.

Now consider the equivalence classes ∼ (74) on ${\mathbb{R}}^{4}\cong {\mathbb{C}}^{2}$. All points $\left(z,w\right)\sim \left({z}^{\prime },{w}^{\prime }\right)\in {\mathbb{C}}^{2}$ with the same real parts are identified, and also all $\left(z,w\right)\sim r\left(z,w\right)\in {\mathbb{R}}^{2}$ for r > 0. Among these, λ assumes the minimum λ = 1 for $\left(z,w\right)=\left(x,y\right)\in {S}^{1}\subset {\mathbb{C}}^{2}$, so that again ²⁵ $\mathcal{M}\cong {\mathbb{C}}^{2}{/}_{\sim }\cong {S}^{1}$.

Therefore the minimal fuzzy torus ${T}_{2}^{2}$ should really be considered as a fuzzy circle. This shows the existence of 'exotic' quantum spaces which are not quantized symplectic spaces, but do not have a semi-classical regime. There are also higher-dimensional such spaces as shown next, and the above example of minimal ${H}_{0}^{4}$.

Non-Kähler quantum space from ${T}_{2}^{2}{\times}{T}_{2}^{2}$. Now consider the Cartesian product of ${T}_{2}^{2}{\times}{T}_{2}^{2}$, realized through eight Hermitian matrices ${X}_{\left(1\right)}^{a},{X}_{\left(2\right)}^{a}$ acting on ${\mathbb{C}}^{4}={\mathbb{C}}^{2}\otimes {\mathbb{C}}^{2}$. All eigenstates of ${H}_{x}={H}_{x}^{\left(1\right)}+{H}_{x}^{\left(2\right)}$ are given by the product states of the two eigenstates (177) of ${T}_{2}^{2}$, so that the ground states or quasi-coherent states are given by

$$\begin{equation}\vert {x}_{\left(1\right)},{x}_{\left(2\right)}\rangle =\vert {x}_{\left(1\right)}\rangle \otimes \vert {x}_{\left(2\right)}\rangle \end{equation} \tag{ 178 }$$

over ${\mathbb{R}}^{8}$. They are again degenerate, and inequivalent states are parametrized by (x₍₁₎, x₍₂₎) ∈ S¹ × S¹. Hence the abstract quantum space is a torus $\mathcal{M}\cong {S}^{1}{\times}{S}^{1}$. The quantum tangent space is spanned by two vectors

$$\begin{equation}{T}_{\xi }\mathcal{M}=\left\langle \left({\partial }_{1}\vert {y}_{\left(1\right)}\rangle \right)\otimes \vert {y}_{\left(2\right)}\rangle ,\vert {y}_{\left(1\right)}\rangle \otimes \left({\partial }_{2}\vert {y}_{\left(2\right)}\rangle \right)\right\rangle \enspace \cong \enspace {\mathbb{R}}^{2}\end{equation} \tag{ 179 }$$

which are linearly independent from the two complexified vectors i∂₁|y₍₁₎⟩ ⊗ |y₍₂₎⟩ and i|y₍₁₎⟩∂₂ ⊗ |y₍₂₎⟩. Therefore ${T}_{\xi ,\mathbb{C}}\mathcal{M}\cong {\mathbb{C}}^{2}\cong {\mathbb{R}}^{4}$, and $\mathcal{M}$ is not a quantum Kähler manifold.

The Moyal–Weyl quantum plane is obtained for X₁ = X and X₂ = Y with $\left[X,Y\right]=\mathrm{i}\mathbb{1}$. Then $\mathrm{dim}\enspace \mathcal{H}=\infty$, but all considerations can be carried over easily. The displacement Hamiltonian

$$\begin{equation}2{H}_{x}={\left(X-x\right)}^{2}+{\left(Y-y\right)}^{2}\end{equation} \tag{ 180 }$$

is nothing but the shifted harmonic oscillator, with ground state

$$\begin{equation}{H}_{z}\vert z\rangle =\vert z\rangle \enspace \end{equation} \tag{ 181 }$$

given by the standard coherent states

$$\begin{equation}\vert z\rangle =U\left(z\right)\vert 0\rangle ,\qquad z=\frac{1}{\sqrt{2}}\left(x+\mathrm{i}y\right)\end{equation} \tag{ 182 }$$

using the identification of ${\mathbb{R}}^{2}\cong \mathbb{C}$. The translation operator is given as usual by

$$\begin{align}\hfill U\left(z\right)& =\mathrm{exp}\left(\mathrm{i}\left(yX-xY\right)\right)=\mathrm{exp}\left(z{a}^{{\dagger}}-\bar{z}a\right),\hfill \\ \hfill a& =\frac{1}{\sqrt{2}}\left(X+\mathrm{i}Y\right),\qquad {a}^{{\dagger}}=\frac{1}{\sqrt{2}}\left(X-\mathrm{i}Y\right).\hfill \end{align} \tag{ 183 }$$

|0⟩ is the ground state of the harmonic oscillator a|0⟩ = 0, and more generally

$$\begin{equation}\left(a-z\right)\vert z\rangle =0\end{equation} \tag{ 184 }$$

implies

$$\begin{equation}\langle z\vert \left(X+\mathrm{i}Y\right)\vert z\rangle =x+\mathrm{i}y.\end{equation} \tag{ 185 }$$

The derivatives (32) are found to be

$$\begin{align}\hfill \left({\partial }_{x}-\mathrm{i}\enspace {A}_{1}\right)\vert z\rangle & =-\mathrm{i}Y\vert z\rangle ={\mathcal{X}}_{1}\vert z\rangle \hfill \\ \hfill \left({\partial }_{y}-\mathrm{i}\enspace {A}_{2}\right)\vert z\rangle & =\mathrm{i}X\vert z\rangle ={\mathcal{X}}_{2}\vert z\rangle ,\hfill \end{align} \tag{ 186 }$$

where the second expressions arise from (35), which are given explicitly by

$$\begin{align}\hfill {\mathcal{X}}_{1}& =2{\left({H}_{z}-1\right)}^{\prime -1}\left(X-x\right)\hfill \\ \hfill {\mathcal{X}}_{2}& =2{\left({H}_{z}-1\right)}^{\prime -1}\left(Y-y\right).\hfill \end{align} \tag{ 187 }$$

The U(1) connection is found to be

$$\begin{align}\hfill \mathrm{i}\enspace {A}_{1}=\langle z\vert {\partial }_{x}\vert z\rangle & =-\mathrm{i}\langle z\vert Y\vert z\rangle =-\mathrm{i}y\hfill \\ \hfill \mathrm{i}\enspace {A}_{2}=\langle z\vert {\partial }_{y}\vert z\rangle & =\mathrm{i}\langle z\vert X\vert z\rangle =\mathrm{i}x\hfill \end{align} \tag{ 188 }$$

with field strength

$$\begin{equation}{F}_{12}={\partial }_{1}{A}_{2}-{\partial }_{2}{A}_{1}=2.\end{equation} \tag{ 189 }$$

Therefore (38) becomes

$$\begin{equation}\vert z\rangle =P\enspace \mathrm{exp}\left({\int }_{0}^{z}\left({\mathcal{X}}_{1}-\mathrm{i}y\right)\mathrm{d}x+\left({\mathcal{X}}_{2}+\mathrm{i}x\right)\mathrm{d}y\right)\vert 0\rangle .\end{equation} \tag{ 190 }$$

$\mathcal{M}\cong \mathbb{C}$ satisfies the quantum Kähler condition due to the constraint (X + iY)|0⟩ = 0, which states that iY|0⟩ = −X|0⟩, so that the complex tangent space ${T}_{0,\mathbb{C}}\mathcal{M}={T}_{0}\mathcal{M}$ coincides with the real one. The holomorphic coherent states are given by

$$\begin{equation}{\Vert}z\rangle ={\mathrm{e}}^{z{a}^{{\dagger}}}\vert 0\rangle ={\mathrm{e}}^{z{a}^{{\dagger}}}\enspace {\mathrm{e}}^{-\bar{z}a}\vert 0\rangle ={\mathrm{e}}^{\frac{1}{2}\vert z{\vert }^{2}}\vert z\rangle .\end{equation} \tag{ 191 }$$

They cannot be normalized, since the map z ↦ ⟨w||z⟩ must be holomorphic and hence unbounded. Thus ||z⟩ should be viewed as holomorphic section of the line bundle $\tilde {\mathcal{B}}$.

In the infinite-dimensional case, one can also consider matrix configurations associated to commutative manifolds. The simplest example is the circle S¹, which arises from the single operator

$$\begin{equation}X=-\mathrm{i}{\partial }_{\varphi }\end{equation} \tag{ 192 }$$

acting on ${\mathcal{C}}^{\infty }\left({S}^{1}\right)\subset {L}^{2}\left({S}^{1}\right)=\mathcal{H}$. The displacement Hamiltonian is

$$\begin{equation}{H}_{x}=\frac{1}{2}{\left(-\mathrm{i}{\partial }_{\varphi }-x\right)}^{2},\quad x\in \mathbb{R}.\end{equation} \tag{ 193 }$$

The quasi-coherent states for $x=n\in \mathbb{Z}$ are clearly

$$\begin{equation}\vert n\rangle ={\text{e}}^{\text{i}n\varphi },\quad {H}_{n}\vert n\rangle =0,\quad n\in \mathbb{Z}\end{equation} \tag{ 194 }$$

so that $\lambda \left(\mathbb{Z}\right)=0$. For any $x\notin \mathbb{Z}$, all eigenstates of H_x |ψ⟩ = E|ψ⟩ are given by the above states |n⟩, with eigenvalue

$$\begin{equation}{H}_{x}\vert n\rangle ={\left(-\mathrm{i}{\partial }_{\varphi }-x\right)}^{2}\enspace {\text{e}}^{\text{i}n\varphi }={\left(n-x\right)}^{2}\enspace {\text{e}}^{\text{i}n\varphi }.\end{equation} \tag{ 195 }$$

Therefore

$$\begin{equation}\vert x\rangle =\vert n\rangle ,\quad \vert n-x\vert {< }\frac{1}{2},\quad n\in \mathbb{Z}\end{equation} \tag{ 196 }$$

while for $x\in \mathbb{Z}+\frac{1}{2}$ the space E_x is two-dimensional, containing both states $\vert x{\pm}\frac{1}{2}\rangle$. Thus the abstract quantum space is the discrete lattice

$$\begin{equation}\mathcal{M}=\mathbb{Z}\subset \mathbb{R}\end{equation} \tag{ 197 }$$

and the quantum tangent space vanishes. This can be generalized to the higher-dimensional commutative torus Tⁿ with commutative and reducible matrix configuration X_μ = −i∂_μ , which also leads to a discrete quantum space without further structure. Thus classical manifolds are not well captured in the present framework. This can of course be treated by adding extra structure as in [7], but such a description is not well suited for Yang–Mills matrix models.