Quantum Minimal Surfaces

TL;DR

This paper explores quantum analogues of classical minimal surfaces within Euclidean spaces and tori, aiming to extend geometric concepts into quantum settings.

Contribution

It introduces the concept of quantum minimal surfaces and discusses their properties in Euclidean and toroidal geometries.

Findings

Proposed a framework for quantum minimal surfaces.

Analyzed properties of quantum minimal surfaces in specific geometries.

Established connections between classical and quantum minimal surface theories.

Abstract

We discuss quantum analogues of minimal surfaces in Euclidean spaces and tori.

Equations

$$S(\vec{x})={\int }_{\Sigma }\sum _{1\le i<j\le d}\{x_i,x_j{\}}^2\cdot \omega$$

$$\sum _{i=1}^d\{x_i,\{x_i,x_j\}\}=0j=1,\dots ,d.$$

$$\sum _{i=1}^d[X_i,[X_i,X_j]]=0j=1,\dots ,d,$$

$$\displaystyle\underset{\!\widetilde{}}{\vec{x}}(\tilde{u},\tilde{v}):=\vec{x}\big{(}u(\tilde{u},\tilde{v}),v(\tilde{u},\tilde{v})\big{)}$$

$$J:=\left|\frac{\partial (\tilde{u},\tilde{v})}{\partial (u,v)}\right|=\sqrt{g},$$

$$X_i:={\tilde{x}}_i(\tilde{U},\tilde{V})$$

$$\displaystyle\frac{1}{\sqrt{g}}\partial_{a}\big{(}\sqrt{g}g^{ab}\partial_{b}\vec{x}\big{)}\equiv\{x_{i}\{x_{i},\vec{x}\}\}$$

$$\{f,h\}:=\frac{1}{\sqrt{g}}{ϵ}^{ab}({\partial }_{a}f)({\partial }_{b}h);$$

$$\begin{array}{rl}& \hat{\Delta }(W)=\frac{1}{2}[W,[W^{†},W]]+[Z,[Z,W]]=0\\ & \hat{\Delta }(Z)=\frac{1}{2}[W,[W^{†},Z]]+\frac{1}{2}[W^{†},[W,Z]]=0.\end{array}$$

$$\displaystyle\vec{x}_{\pm}=\big{(}a\cosh(v-v_{0})\cos(u-u_{0}),a\cosh(v-v_{0})\sin(u-u_{0}),\pm av\big{)}+\vec{x}_{0}$$

$$(g_{ab})=a^2{\cosh }^2(v-v_0)\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=\sqrt{g}\text{mathds}1.$$

$$\displaystyle\frac{d\tilde{v}}{dv}=a^{2}\cosh^{2}(v-v_{0})\qquad\tilde{v}(v)=\frac{a^{2}}{2}\Big{(}v-v_{0}+\frac{1}{2}\sinh 2(v-v_{0})\Big{)}+\tilde{v}_{0}$$

$$\displaystyle\vec{X}_{\pm}=\Big{(}a$$

$$\displaystyle a\cosh\big{(}v(\tilde{V}-\tilde{v}_{0}\mathds{1})-v_{0}\mathds{1}\big{)}\sin(\tilde{U}-u_{0}\mathds{1}),\pm av(\tilde{V}-\tilde{v}_{0}\mathds{1}))\Big{)}+\vec{x}_{0}$$

$$\displaystyle e^{i(\tilde{U}-u_{0}\mathds{1})}\big{|}n\rangle=e^{i\varphi}\big{|}n\rangle=\big{|}n-1\rangle\qquad(\tilde{V}-\tilde{v}_{0}\mathds{1})\big{|}n\rangle=-i\hbar\frac{\partial}{\partial\varphi}\big{|}n\rangle=-\hbar n\big{|}n\rangle$$

$$\displaystyle W\big{|}n\rangle=w_{n}\big{|}n-1\rangle\qquad Z^{(\pm)}\big{|}n\rangle=z_{n}\big{|}n\rangle.$$

$$2(z_n-z_{n-1}{)}^2=r_{n+1}+r_{n-1}-2r_n$$

$$r_n(z_n-z_{n-1})=r_{n+1}(z_{n+1}-z_n)=\text{const}=c$$

$$(r_{n+1}-r_n)-(r_n-r_{n-1})=\frac{2c^2}{r_n^2}$$

$$z_n=\frac{c}{r_n}+z_{n-1}.$$

$$\begin{split}&r_{n}=a^{2}\cosh^{2}\big{(}q(-\hbar n)\big{)}\\ &z_{n}=\pm a\big{(}q(-\hbar n)+v_{0}\big{)}+(\vec{x}_{0})_{3}\end{split}$$

$$2z^{\prime 2}=r^{\prime \prime }r{z}^{\prime }=D(D=\underset{\hslash \to 0}{\lim }c/\hslash ),$$

$$\begin{split}&r(p)=a^{2}\cosh^{2}q(p)\\ &z(p)=\pm a\big{(}q(p)+v_{0}\big{)}+(\vec{x}_{0})_{3}\qquad D=\pm a.\end{split}$$

$$0<r_0\le r_1\le r_0+\frac{2c^2}{r_0^2}$$

$$2(z_n-z_{n-1}{)}^2=r_{n+1}+r_{n-1}-2r_n$$

$$r_n(z_n-z_{n-1})=r_{n+1}(z_{n+1}-z_n)$$

$$r_{n+1}\ge r_n>0$$

$$r_{n-1}\ge r_n>0$$

$$r_n$$

$$z_n$$

$$z_n-z_{n-1}=\frac{c}{r_n},$$

$$r_{n+1}-r_n=r_n-r_{n-1}+\frac{2c^2}{r_{n-1}^2}>0$$

Full text

\shortdate\yyyymmdddate

Quantum Minimal Surfaces

Joakim Arnlind, Jens Hoppe, Maxim Kontsevich

Dept. of Math.

Linköping University

581 83 Linköping

Sweden

joakim.arnlind@liu.se

Institut des Hautes Etudes Scientifiques, Le Bois-Marie, 35, Route de Chartres, 91440 Bures-sur-Yvette, France

jhoppe@ihes.fr

Institut des Hautes Etudes Scientifiques, Le Bois-Marie, 35, Route de Chartres, 91440 Bures-sur-Yvette, France

maxim@ihes.fr

Abstract

We discuss quantum analogues of minimal surfaces in Euclidean spaces and tori.

1. Introduction

It is well-known that minimal surfaces in $\mathbb{R}^{d}$ can be characterized as extremal points of the so called Schild functional:

[TABLE]

where $\vec{x}:\Sigma\to\mathbb{R}^{d}$ is a map from a surface $\Sigma$ endowed with symplectic 2-form $\omega$ to Euclidean space $\mathbb{R}^{d}$, and $\{\cdot,\cdot\}$ denotes the Poisson bracket on $\Sigma$. More precisely, critical points of the Schild action are either “degenerate” maps with 1-dimensional image (i.e. components $x_{i},i=1,\dots,d$ of $\vec{x}$ are functionally dependent and all Poisson brackets $\{x_{i},x_{j}\}$ vanish identically), or the image of $\vec{x}$ is a minimal surface in $\mathbb{R}^{d}$ and the symplectic form $\omega$ is proportional (with a constant factor) to the volume form associated with the induced metric on $\Sigma$.

The Euler-Lagrange equations for the Schild action are

[TABLE]

One can call quantum minimal surface a solution of the equation

[TABLE]

where $X_{i}$ are self-adjoint operators in a Hilbert space – a matrix equation which is also of interest in the context of the bosonic BFSS [GH82] and IKKT [IKKT97] model, as well as for many other reasons (some of which we will comment on in Section 6).

In [ACH16] the classical Weierstrass representation for minimal surfaces, utilizing the existence of isothermal parameters, was generalized to a non-commutative one, yielding triples of Weyl algebra elements constituting non-commutative minimal surfaces. In [AH13], on the other hand, a quantization of the Catenoid was written as formal power-series in $\hbar$, satisfying (1.3).

One can give a general procedure for constructing solutions to (1.3) as follows: Start with an arbitrary minimal surface $\vec{x}(u,v)$ (given e.g., but not necessarily, in isothermal parametrization, $g_{ab}=\partial_{a}\vec{x}\cdot\partial_{b}\vec{x}=\sqrt{g}\delta_{ab}$); reparametrize as

[TABLE]

with the new coordinates chosen such that

[TABLE]

i.e. $\{\tilde{u},\tilde{v}\}=1$. With $[\tilde{U},\tilde{V}]=i\hbar\mathds{1}$,

[TABLE]

will then satisfy (1.3), to lowest order in $\hbar$, due to

[TABLE]

when

[TABLE]

note that (1.4) implies $\sqrt{\tilde{g}}=1$, i.e. furnishing a transformation to a parametrization where the determinant of the first fundamental form is constant ($=1$).

Let us first consider the following (non-compact) examples: Catenoids (for which, due to the rotational symmetry, one can easily obtain rather explicit expressions, resp. existence proofs to all orders) and Enneper surfaces (where, despite of, again, fairly explicit formulas one sees much of the difficulties involved concerning the general case).

Note that for $d=3$ equation (1.3), with $W:=X_{1}+iX_{2}$ and $Z:=X_{3}$, reads

[TABLE]

2. Catenoid

Let us consider the catenoids

[TABLE]

Reparametrizing $\vec{x}_{\pm}$ as $\vec{x}_{\pm}(\tilde{u}=u,\tilde{v})$ in accordance with (1.5), i.e.

[TABLE]

gives

[TABLE]

with a natural representation on the basis \big{|}n\rangle\hat{=}e^{-in\varphi} as

[TABLE]

which leads to the Ansatz

[TABLE]

Inserting (2.4) into (1.9) one obtains the recursion relations

[TABLE]

i.e. $r_{n}:=|w_{n}|^{2}$ determined by

[TABLE]

and then $z_{n}$ given via (2.6),

[TABLE]

Denoting $v-v_{0}$ by $q$ and $\tilde{v}-\tilde{v}_{0}$ by $p$ (i.e. $p=a^{2}/2(q+\frac{1}{2}\sinh 2q)$) being an odd function of $q$, resp. $q$ an odd function of $p$, and $\frac{dp}{dq}=a^{2}\cosh^{2}q$ (2.7) and (2.8) are by construction solved in the $\hbar\to 0$ limit ($n$ arbitrary but fixed) by

[TABLE]

which can easily be verified, as (2.5) and (2.6) in this limit become the differential equations

[TABLE]

which are satisfied (using $\frac{dq}{dp}=1/(a^{2}\cosh^{2}q(p))$) by

[TABLE]

Let us now consider the general solution to the recursion relations (2.5) and (2.6). First of all, since $r_{n}=|w_{n}|^{2}$ one is interested in positive solutions, i.e. solutions $(\{r_{n}\}_{n\in\mathbb{Z}},\{z_{n}\}_{n\in\mathbb{Z}})$ with $r_{n}>0$ for $n\in\mathbb{Z}$. Moreover, we are interested in non-constant solutions (noting that (2.5) and (2.6) have constant solutions), and it is easy to see that for a non-constant solution, both sequences $\{r_{n}\}$ and $\{z_{n}\}$ are necessarily non-constant. Let us start by showing that for appropriate initial conditions, one obtains a positive and non-constant solution to the recursion relations.

Proposition 2.1**.**

For $c,r_{0},r_{1},z_{0}\in\mathbb{R}$ such that $c\neq 0$ and

[TABLE]

there exists a non-constant solution to the system

[TABLE]

such that

[TABLE]

The solution is given recursively by

[TABLE]

Proof.

First, let us note that (2.13) implies that $c=r_{n}(z_{n}-z_{n-1})$ is independent of $n$. If $c\neq 0$ then $r_{n}\neq 0$ for $n\in\mathbb{Z}$, which implies that

[TABLE]

yielding (2.16) and (2.17). Inserting (2.18) into (2.12) gives (2.14) and (2.15). Conversely, if $r_{n}(z_{n}-z_{n-1})=c$ it is clear that (2.14)–(2.17) satisfy (2.12) and (2.13).

Now, assume that $n\geq 1$ and that $r_{n}\geq r_{n-1}>0$, which is true by assumption when $n=1$; let us show that if we define $r_{n+1}$ as in (2.14) then $r_{n+1}\geq r_{n}>0$. Equation (2.14) gives

[TABLE]

since $r_{n}-r_{n-1}\geq 0$ and $c\neq 0$. Moreover, it is clear that $r_{n+1}>0$ since $r_{n}>0$. Thus, we have shown that $r_{n+1}\geq r_{n}>0$, and it follows by induction that this is true for all $n\geq 0$. For $n\leq-1$ we start by defining

[TABLE]

and note that

[TABLE]

by using the assumption, and furthermore that $r_{-1}>0$ since $r_{0}>0$. Next, for $n\leq-1$ one assumes that $r_{n}\geq r_{n+1}>0$ and defines $r_{n-1}$ via (2.15), implying

[TABLE]

since $r_{n}-r_{n+1}\geq 0$. Moreover, it is clear that $r_{n-1}>0$ since $r_{n}>0$. Hence, it follows by induction that $r_{n}\geq r_{n+1}>0$ for all $n\leq-1$. ∎

Note that, for any solution to the recursion relations, one may always shift $n$ to obtain another solution with a minimum value at $r_{n_{0}}$ for arbitrary $n_{0}\in\mathbb{Z}$. The next result shows that these are indeed all possible non-constant solutions.

Proposition 2.2**.**

Let $(\{z_{n}\}_{n\in\mathbb{Z}},\{r_{n}\}_{n\in\mathbb{Z}})$ be a positive non-constant solution of (2.12) and (2.13). Then there exists $n_{0}\in\mathbb{Z}$, $c\in\mathbb{R}\backslash\{0\}$ and $\delta\in(-1,1]$ such that

1. (1)

$r_{n_{0}}=\min_{n\in\mathbb{Z}}r_{n}$, 2. (2)

$\displaystyle r_{n_{0}+1}=r_{n_{0}}+(1-\delta)\frac{c^{2}}{r_{n_{0}}^{2}}$, 3. (3)

$r_{n+1}>r_{n}$* for $n\geq n_{0}+1$,* 4. (4)

$r_{n-1}>r_{n}$* for $n\leq n_{0}$.*

Moreover, $\{z_{n}\}_{n\in\mathbb{Z}}$ is strictly increasing if $c>0$ and strictly decreasing if $c<0$.

Proof.

First, let us note that (2.13) implies that there exists $c\in\mathbb{R}$ such that

[TABLE]

for all $n\in\mathbb{Z}$. Moreover, if $z_{n}-z_{n-1}=0$ for some $n\in\mathbb{Z}$ then $c=0$, which implies that $z_{n}-z_{n-1}=0$ for all $n$ (by using that $r_{n}>0)$. Thus, since the solution is assumed to be non-constant, it follows that $c\neq 0$. Equations (2.12) and (2.13) can then be written as

[TABLE]

since $r_{n}>0$. It is clear from (2.19) that if $r_{N}-r_{N-1}\geq 0$ for some $N\in\mathbb{Z}$, then $r_{n+1}>r_{n}$ for all $n\geq N$. Similarly, if $r_{N+1}-r_{N}\leq 0$ then $r_{n-1}>r_{n}$ for all $n\leq N$.

Since the solution is non-constant, there exists $N\in\mathbb{Z}$ such that $r_{N}-r_{N-1}\neq 0$; assume that $r_{N}-r_{N-1}<0$ (the argument for $r_{N}-r_{N-1}>0$ is completely analogous). It follows from (2.19) that

[TABLE]

and, by induction

[TABLE]

implying that there exists $n_{0}\in\mathbb{Z}$ such that $r_{n_{0}}-r_{n_{0}-1}<0$ and $r_{n_{0}+1}-r_{n_{0}}\geq 0$ (since $2c^{2}/r_{n}^{2}$ is bounded from below by $2c^{2}/r_{N}^{2}>0$). By the previous argument one may also conclude that $r_{n-1}>r_{n}$ for $n\leq n_{0}$ and $r_{n+1}>r_{n}$ for $n\geq n_{0}+1$. Moreover, it follows that $r_{n_{0}}=\min_{n\in\mathbb{Z}}r_{n}$. For $n=n_{0}$, (2.19) gives

[TABLE]

implying that $r_{n_{0}}<r_{n_{0}}+2c^{2}/r_{n_{0}}^{2}$ since $r_{n_{0}}-r_{n_{0}-1}<0$, which, together with $r_{n_{0}+1}-r_{n_{0}}\geq 0$ gives

[TABLE]

Finally, it follows directly from (2.20) that $\{z_{n}\}$ is strictly increasing if $c>0$ and strictly decreasing if $c<0$. ∎

What about the general solution of (2.10):

[TABLE]

which is a catenoid (with $g(p,u)=\alpha/4$); with $\frac{2D}{\sqrt{\alpha}}=\pm a$ one gets (2.11), but the relation between $p$ and $q$ having an extra factor of $\frac{2}{\sqrt{\alpha}}$, i.e.

[TABLE]

instead of $\sqrt{g}$. Note that the discretization of (2.11) satisfies the recursion relations (2.5) and (2.6) also for $n\to\infty$ (fixed $\hbar$), which can be seen as follows: (2.2) implies that for large $p=\tilde{v}-\tilde{v}_{0}$

[TABLE]

and therefore, for large $|n|$

[TABLE]

indeed satisfying (2.7) to leading order ($c^{2}=a^{2}\hbar^{2}$).

3. Enneper surfaces

For Enneper-type surfaces the data entering the Weierstrass representation

[TABLE]

are

[TABLE]

the simplest case $(N=2)$ giving

[TABLE]

while more generally, with $s:=|z|^{2}=u^{2}+v^{2}$,

[TABLE]

Trying to solve (1.5) with the Ansatz

[TABLE]

gives

[TABLE]

Let us now restrict to $N=2$, giving

[TABLE]

which can be easily inverted to give

[TABLE]

hence yielding expressions for the inverse transformation

[TABLE]

which is needed to obtain the $X_{i}$. Defining $w=x_{1}+ix_{2}$ and $z=u+iv$, resp.

[TABLE]

one finds for this (Enneper) case

[TABLE]

satisfying

[TABLE]

while (1.3) resp.

[TABLE]

should (in leading order) be solved by

[TABLE]

with

[TABLE]

where

[TABLE]

is the number operator. Due to (cp. (3.9))

[TABLE]

Analogously,

[TABLE]

suggests

[TABLE]

due to

[TABLE]

[TABLE]

satisfies to leading (and sub-leading if $n\to\infty$, $\hbar$ fixed) order the recursion relations

[TABLE]

Inserting

[TABLE]

into $[X_{i},[X_{i},X_{j}]]=0$ gives

[TABLE]

In the classical limit the above equations are satisfied, using

[TABLE]

as reducible to $\{z,1\}=0$, $\{z^{2},1\}=0$, $\{z^{3},1\}=0$. Q-analogue of (3.25)

[TABLE]

has solutions of the form

[TABLE]

with $\sigma_{n}=|\lambda_{n}|^{2}$ satisfying

[TABLE]

with unique solution $\sigma_{0}=0<\sigma_{1}<\sigma_{2}<\cdots$.

4. Helicoid

Parametrizing the helicoid in $\mathbb{R}^{3}$ as

[TABLE]

gives $(g_{ab})=\cosh^{2}(u)\mathds{1}$ and $\sqrt{g}=\cosh^{2}(u)$. A solution to

[TABLE]

is again given by $\tilde{u}=u$ and $\tilde{v}=v/2+\sinh(2v)/4$, which implies that

[TABLE]

For the helicoid, choosing a representation of $\tilde{U}$ and $\tilde{V}$ on smooth functions as

[TABLE]

one may interpret $e^{i\tilde{U}}$ as a shift operator. Defining the following operators

[TABLE]

it follows that $\Delta(W)=\Delta(Z)=0$ is equivalent to the single equation

[TABLE]

If $w(x-\hbar)\neq 0$ one may write the above equation as

[TABLE]

5. Complex hyperbola

In contrast to the previous examples, let us now consider a surface in $\mathbb{R}^{4}$. Parametrizing $(x_{1}+ix_{2})(x_{3}+ix_{4})=\alpha\in\mathbb{R}$ ($\Leftrightarrow$ $x_{1}x_{3}-x_{2}x_{4}=\alpha$, $x_{1}x_{4}+x_{2}x_{3}=0$) as $\vec{x}(t,u)=(t\cos u,t\sin u,\tfrac{\alpha}{t}\cos u,-\tfrac{\alpha}{t}\sin u)_{t>0}$, one gets

[TABLE]

Using the general strategy, namely reparametrizing $\vec{x}(t,u)=\tilde{\vec{x}}(\tilde{t},u)$ with

[TABLE]

i.e. $\tilde{t}=\frac{1}{2}t^{2}-\frac{\alpha^{2}}{2t^{2}}+\gamma$, respectively

[TABLE]

$\tilde{x}_{1}+i\tilde{x}_{2}=t(\tilde{t})e^{iu}=e^{iu}t(\tilde{t})$, $\tilde{x}_{3}+i\tilde{x}_{4}=\frac{1}{t(\tilde{t})}e^{-iu}$, leading to (in the representation $\hat{U}=\varphi$, $\hat{V}=\hat{\tilde{t}}=-i\hbar\frac{\partial}{\partial\varphi}$ acting on e^{-in\varphi}\hat{=}\big{|}n\rangle)

[TABLE]

with $w_{n}=t(-\hbar n)$, and

[TABLE]

giving exact solutions to (1.3); we will come back to this example in Section 7.

6. Remarks and conjectures

6.1. I – Quantization

[TABLE]

[TABLE]

Example: $M=U(1)\times U(1)$, $\varphi_{1}=e^{i\theta_{1}}$, $\varphi_{2}=e^{i\theta_{2}}$ with $\theta_{1},\theta_{2}\in\mathbb{R}/2\pi\mathbb{Z}$, and $\omega=(d\theta_{1}\wedge d\theta_{2})/2\pi$, $\hbar=1/N$:

[TABLE]

6.2. II

Map $\varphi:M\to\mathbb{R}^{n}$ given by $x_{1},\ldots,x_{n}\in C^{\infty}(M,\mathbb{R})$. Functional

[TABLE]

invariant under rescaling of symplectic form $\omega\to\lambda\omega$ for $\lambda\in\mathbb{R}_{>0}$. The quantum version is given by $X_{1},\ldots,X_{n}\in\operatorname{Mat}(N\times N,\mathbb{C})$ with $X_{i}^{\dagger}=X_{i}$ (think as $X_{i}=\hat{x}_{i}$) and

[TABLE]

Put constraints to make this compact: e.g. sphere $\sum x_{i}^{2}=1$, ellipsoid $\sum(x_{i}/r_{i})^{2}=1$ (where $r_{1},\ldots,r_{n}>0$). Quantum sphere: $\sum X_{i}^{2}=\mathds{1}_{N\times N}$, quantum ellipsoid $\sum X_{i}^{2}/r_{i}^{2}=\mathds{1}_{N\times N}$.

Conjecture 6.1*.*

Critical values of $S_{q}$ with given constraint $\approx$ critical values of $S_{d}$ as $N\to+\infty$ (these are essentially squares of volumes of minimal surfaces).

Note that $d>1$ does not survive, wrong scaling.

6.3. III

Version: target = flat torus $U(1)\times\cdots\times U(1)=(S^{1})^{n}$. Map $M\to(S^{1})^{n}$ is given by $\varphi_{1},\ldots,\varphi_{n}\in C^{\infty}(M,\mathbb{C})$ with $|\varphi_{i}|^{2}=1$.

[TABLE]

Critical points = minimal surfaces in $(S^{1})^{n}$ parametrized by symplectic surface. $\omega$ = const $\cdot$ Riemannian volume for the induced metric.

Weierstrass parametrization: M is a complex curve $C$ of genus $g\geq 1$, $\alpha_{1},\ldots,\alpha_{n}\in\Omega^{1}(C)=\Gamma(C,T^{\ast}_{C})$ are holomorphic $1$-forms and $i\operatorname{Im}\alpha_{j}=d\varphi_{j}$. Constraints:

1. (1)

$\displaystyle\sum_{i=1}^{n}\alpha_{i}^{2}=0\in\Gamma(C,(T^{\ast}_{C})^{\otimes 2})$ (quadratic differentials, space of $\dim_{\mathbb{C}}=3g-3$). 2. (2)

$[\operatorname{Im}\alpha_{i}]\in H^{1}(C,2\pi\mathbb{Z})$ for all $i$

$\rightsquigarrow$ discrete countable subset of $\mathbb{R}_{\geq 0}$ of critical values (depending on $n$).

$\Phi_{1},\ldots,\Phi_{n}\in U(N)$ ($N\to+\infty$ think as $\Phi_{i}=\hat{\varphi}_{i}$). Define

[TABLE]

(see [DN01] for related considerations, including the lattice twisted Eguchi-Kawai model). Reasoning: If $\Phi\in U(N)$ is close to $\mathds{1}_{N\times N}$ then $-\operatorname{Tr}(\log\Phi)^{2}\sim\operatorname{Tr}(2-\Phi-\Phi^{-1})$ ($\Phi=e^{i\theta}$, $\theta^{2}\sim 2-2\cos\theta=2-e^{i\theta}-e^{-i\theta}$).

“Equations of motion”: Critical points $\frac{\delta S_{q}}{\delta\Phi_{i}}=0$ gives

[TABLE]

for all $i$. Multiply by $\Phi_{i}$ from the left:

[TABLE]

for all $i$.

Conjecture 6.2*.*

If $N\to+\infty$, limits of critical values of $S_{q}$ are either $+\infty$, or positive $\mathbb{Z}$-linear combinations of critical values of $S_{cl}$. Moreover, if the limit is $<\infty$ then $\left|\left|\Phi_{i}\Phi_{j}-\Phi_{j}\Phi_{i}\right|\right|\leq\frac{\operatorname{const}}{N}$.

One can also propose a rough criterium for a sequence of quantum maps to torus depending on dimension $N\to+\infty$ to “approximate” a given oriented surface $M\subset(S^{1})^{n}$ endowed with a symplectic 2-form $\omega$ :

[TABLE]

Generalizations: Fix symmetric positive $n\times n$ matrix $g=(g_{ij})_{1\leq i,j\leq n}$, $g=g^{\dagger}$ and $g>0$. $g$ gives a flat (constant) metric $\sum g_{ij}d\phi_{i}d\phi_{j}$ on $(S^{1})^{n}$. Weierstrass parametrization $\rightsquigarrow$ explicit description of critical values of $S_{cl}$ in terms of complex curves. Critical values of $S_{q}$ should approximate these for $N\to+\infty$.

6.4. IV – Calibrated geometry

For all $n\geq 0$ the Yang-Mills algebra is given by

[TABLE]

which is a $\ast$-algebra upon setting $X_{i}^{\dagger}=X_{i}$. Reason: if $X_{i}\in\operatorname{Mat}(N\times N,\mathbb{C})$ are self-adjoint the connection on the trivial $U(N)$-bundle$/\mathbb{R}^{n}$ with constant coefficients

[TABLE]

(where $x_{1},\ldots,x_{n}$ are coordinates in $\mathbb{R}^{n}$) satisfies Yang-Mills equation $\leftrightarrow$ (6.2).

[TABLE]

critical points $\Leftrightarrow\eqref{eq:YM.algebra.def}$. If $n=2m$ Hermitian YM equation; some equations of simpler form implying YM.

[TABLE]

for $i=1,\ldots,m$.

[TABLE]

Indeed: (6.3) implies that

[TABLE]

Meaning of HYM: On $\mathbb{R}^{2m}=\mathbb{C}^{m}$, $U(N)$-connection in trivial bundle with constant coefficients.

[TABLE]

Recall that by Donaldson-Uhlenbeck-Yau theorem, any stable holomorphic bundle with first Chern class 0 on compact Kähler manifold admits a unique (up to scalar) Hermitian metric whose canonically associated connection satisfies Hermitian YM equation.

Generalization:

[TABLE]

gives a Lie algebra with a central extension.

Classical limit of HYM. It looks very much reasonable that the classical limit of representations of HYM algebra (possibly with central extension as above given by $c\neq 0$, but still with $c_{ij}=0$) correspond to a special class of minimal surfaces in $\mathbb{R}^{2m}=\mathbb{C}^{m}$ which are holomorphic curves.

Recall that for a surface in any Kähler manifold the property to be a holomorphic curve (which is a first order constraint) implies that the surface is minimal (which is a second order constraint). This is the simplest case of so called calibrated geometry. Similarly, relations in HYM algebra are identities between single commutators, whereas in YM algebra we have double commutators.

Case: $m=2$, $n=4$, $c=0$ and $c_{ij}=0$. Nice Lie algebra SYM (self-dual Yang Mills in $\dim=4$ constant solutions).

[TABLE]

In [BBS17] these are called Banks-Seiberg-Shenker equations ([BSS97]), and unlike in the general case $m\geq 3$, they admit an enhanced symmetry $SO(4,\mathbb{R})\supset U(2)$,

In general, one can ask the following question: for a given finitely-generated module $M$ over $\mathbb{C}[Z_{1},\dots,Z_{m}]$ (which is the same as an algebraic coherent sheaf on $\mathbb{C}^{m}$) can one construct a pre-Hilbert space $\mathcal{M}_{\infty}$ containing $M$ and such that $M$ is dense in $\mathcal{M}_{\infty}$ for such that operators of multiplication by $Z_{i}$ extend to $M_{\infty}$ and admit hermitian conjugate, and satisfy the relation of HY algebra (possibly centrally extended by $c$). The question also makes sense when algebra $\mathbb{C}[Z_{1},\dots,Z_{m}]$ is replaced by its quantum deformation $[Z_{i},Z_{j}]=c_{ij}$.

For example, if $M=\mathbb{C}[Z_{1},\dots,Z_{m}]$ is the free module of rank 1 (i.e. a trivial rank one bundle in terms of sheaves), the pre-Hilbert space $M_{\infty}$ is the space of entire functions satisfying certain growth condition:

[TABLE]

and with the pre-Hilbert norm given by

[TABLE]

Nekrasov considered solution of quantum HYM which correspond to bundles or torsion-free sheaves in $\mathbb{C}^{m}$ (and also for modules close to free for the quantized $\mathbb{C}^{m}$). By our philosophy, his solutions do not correspond to minimal surfaces (or complex curves), as supports in the classical limit are full $2m$-dimensional.

From our perspective, the (noncompact) minimal surfaces in $\mathbb{R}^{2m}=\mathbb{C}^{m}$ which are complex algebraic curves, should correspond to solution of HYM for coherent sheaves supported on curves, e.g. quotient modules of $\mathbb{C}[Z_{1},\dots,Z_{m}]$ by the ideal generated by defining equations of a curve. The problem of constructing quantum analogs of algebraic curves in $\mathbb{C}^{m}$ was considered in [CT99], without stressing the relation to minimal surfaces.

Now we discuss possibilities to have “calibrated” quantum minimal surfaces in the compact case.

Unitary version, case $m=2$: $\Phi_{1},\Phi_{2},\Phi_{3},\Phi_{4}\in U(N)$

[TABLE]

gives a $A_{4}$-covariant group ($A_{4}$ = even permutations in $S_{4}$). Looks like $\pi_{1}(\text{3-dim manifold})$, #relations = 3 = #generators$-1$.

We expect critical points of

[TABLE]

approximate (as $N\to+\infty$) minimal surfaces in flat torus $(S^{1})^{4}$ (second order equation).

Conjecture 6.3*.*

Representations of group (6.4)

[TABLE]

($(ijkl)$ and even permutation) approximate as $N\to+\infty$ complex curves in the abelian variety \big{(}\mathbb{C}/\mathbb{Z}+i\mathbb{Z}\big{)}^{2} (first order equation).

6.5. V – Quantum curves in abelian varieties:

Here is another possibility, valid in arbitrary complex dimension $m$ and for any translationally invariant complex structure and a Kähler metric on $(S^{1})^{2m}$.

Let $\Gamma\simeq\mathbb{Z}^{2m}\subset\mathbb{C}^{m}$ be a lattice, such that the quotient torus $A=\mathbb{C}^{m}/\Gamma$ (here we use the standard Kähler metric in the coordinate space $\mathbb{C}^{m}$) contains a compact complex algebraic curve $C$, possibly singular. Notice that if $C$ is not degenerate in the sense that none of its irreducible components is not contained in a proper complex subtorus, then $A$ is in fact algebraic and is an abelian variety.

Our goal is to construct an infinite sequence of finite-dimensional Hilbert spaces with dimension $\to+\infty$, endowed with “almost-commuting” $2m$ unitary operators which approximates in some sense $C$, or, more generally, a coherent sheaf on $A$ supported on $C$.

The idea is the following. Let us pass to the universal cover $\mathbb{C}^{m}$ of $A$, then the pre-image of $C$ will be a non-algebraic curve $\widetilde{C}\subset\mathbb{C}^{m}$, invariant under shifts by $\Gamma$. Optimistically extending our previous considerations of quantization of curves to the non-algebraic case, we are lead to the following question. Construct an infinite-dimensional representation of HYM algebra

[TABLE]

covariant with respect to the action of $\Gamma$, i.e. assuming that we are also given a collection of commuting unitary operators $U_{1},\dots,U_{2m}$ corresponding to a basis of $\Gamma$, satisfying commutator relations

[TABLE]

where $a_{jk}\in\mathbb{C}$ are coefficients of generators of $\Gamma$ considered as vectors in $\mathbb{C}^{m}$.

We assume that the subspace $\mathcal{H}_{u_{1},\dots,u_{2m}}$ of the ambient Hilbert space $\mathcal{H}$ corresponding to eigenvalues $u_{1},\dots,u_{2m}\in U(1)\subset\mathbb{C}^{\times}$ of operators $U_{1},\dots,U_{2m}$, is finite-dimensional. This will be our finite-dimensional Hilbert space “approximating” $C\subset A$. The almost-commuting unitary operators acting on this space will be of the form

[TABLE]

where $\vec{b}=(b_{1},\dots,b_{m})\subset(\mathbb{C}^{m})^{*}$ belongs to the dual lattice determined by the constraint

[TABLE]

There is a natural proposal to define $\mathcal{H}$ based on Fourier-Mukai duality. Namely, Hilbert space $\mathcal{H}$ (or, more precisely, certain dense pre-Hilbert subspace $\mathcal{H}_{\infty}\subset\mathcal{H}$ corresponding to “Schwarz functions”) is a finitely generated projective module over the algebra of functions $C^{\infty}(T)$ on the torus of unitary characters $\vec{u}=(u_{1},\dots,u_{2m})$ of lattice $\Gamma$. In other words, $\mathcal{H}_{\infty}$ is the space of section of a complex vector bundle $\mathcal{E}$ on $T$, the pre-Hilbert space structure is given by a hermitian structure on $\mathcal{E}$ (and integration with respect to the Haar measure on $T$). Operators $Z_{i}$ in this presentation are first order differential operators, corresponding to the covariant derivatives in (anti)-holomorphic directions with respect to certain unitary connection on $\mathcal{E}$. The condition that we have a representation of HYM algebra, translates to the condition that we have a solution of the usual Hermitian Yang-Mills equations from differential geometry (this is a classical idea of $T$-duality for solutions of noncommutative YM equations on tori, see e.g. [CDS98]).

Notice that almost-commuting unitary operators $\exp(\sum_{j}(b_{j}Z_{j}-\bar{b_{j}}Z_{j}^{\dagger})$ introduced formally in (6.5), have geometric meaning as holonomy operators for the connection along geodesic loops in $T$.

By Donaldson-Uhlenbeck-Yau theorem, we see that $\mathcal{E}$ is a (semi-)stable holomorphic vector bundle on $T$, which as a complex manifold is the same as the dual abelian variety $A^{\vee}:=Pic_{0}(A)$. Here is our proposal: take $\mathcal{E}$ to be the Fourier-Mukai dual to the coherent sheaf $\mathcal{F}\in Coh(A)$ supported on $C\subset A$. In order to introduce a small parameter (the inverse rank of $\mathcal{E}$) we can consider $\mathcal{F}$ of the form

[TABLE]

where $\mathcal{F}_{0}$ is a coherent sheaf supported on $C$, and $\mathcal{L}$ is an ample line bundle on $A$.

6.6. VI – Quantum degree

Calibrated submanifolds (like complex curves in Kähler manifolds) have the characteristic property that their area depends only on their homology class, and for them the calibrating lower bound on the area in given homology class (BPS bound in physics), is saturated. Hence, we are lead to the question what is the “homology class” of a “quantum surface”. Here are two simple examples when one can define such a class, which is the quantum degree of a map from the quantum surface to the target surface.

We did not study yet the relation of quantum degree with quantum calibrated geometry of minimal surfaces discussed above.

1. (1)

$d=1$: Smooth map $\varphi:M\to S^{1}\times S^{1}=U(1)\times U(1)$ is given by $\varphi_{1},\varphi_{2}\in C^{\infty}(M,\mathbb{C})$ with $|\varphi_{1}|^{2}=|\varphi_{2}|^{2}=1$. The degree $\deg(\varphi)\in\mathbb{Z}$ is given by

[TABLE]

Proposition 6.4**.**

Let $\Phi_{1},\Phi_{2}\in U(N)$ ($N\to\infty$) be unitary matrices depending on $N$ and assume that there exists $c>0$ such that

[TABLE]

Then

[TABLE]

where $k\in\mathbb{Z}$ and $|k|<c/2\pi\cdot(1+o(1))$. Note that

[TABLE]

“Explanation”: Assume $\Phi_{i}=\hat{\varphi}_{i}$ for $i=1,2$.

[TABLE]

Proof.

Let us denote eigenvalues of unitary operator $\Psi:=\Phi_{1}\Phi_{2}\Phi_{1}^{-1}\Phi_{2}^{-1}$ by

[TABLE]

Assumption (6.6) means that $|u_{j}-1|\leq c/N$ for all $j$, hence $|\theta_{j}|<c/N\cdot(1+o(1))$. Obviously, we have $\det\Psi=1$, therefore $\sum_{j}\theta_{j}=2\pi k$ for some $k\in\mathbb{Z}$. Inequality $|\theta_{j}|<c/N\cdot(1+o(1))$ implies that $|k|<c/2\pi(1+o(1))$. Finally, expression on the l.h.s. of (6.7) is equal to

[TABLE]

∎ 2. (2)

Analogously, a smooth map $\varphi:M\to$ 2-dim sphere $S^{2}\subset\mathbb{R}^{3}$ is given by $x_{1},x_{2},x_{3}\in C^{\infty}(M,\mathbb{R})$ with $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1$. The degree $\deg(\varphi)\in\mathbb{Z}$ is given by

[TABLE]

Conjecture 6.5*.*

Let $X_{1},X_{2},X_{3}\in\operatorname{Mat}(N\times N,\mathbb{C})$ be self-adjoint operators (depending on $N$) such that $X_{1}^{2}+X_{2}^{2}+X_{3}^{2}=\mathds{1}_{N\times N}$, and assume that there exist $c$ such that

[TABLE]

for $i,j\in\{1,2,3\}$. Then

[TABLE]

where $k\in\mathbb{Z}$ and $|k|\leq 3/2c\cdot(1+O(1/N))$.

7. Quantized complex curves and integrable systems

As shown already in the 19th century (see e.g. [Kom97, Eis12]),

[TABLE]

defines a minimal surface in $\mathbb{R}^{4}$ for arbitrary analytic $f$. Similarly, static membranes (solutions of (1.3)), in particular a complex parabola, were considered in [CT99] (curiously without explicitly mentioning “minimal surfaces”) for which

[TABLE]

and (cp. Section 6)

[TABLE]

Before taking up that parabola example (deriving many new properties, and noting that it constitutes a discrete integrable system, cp. [Hal05]) let us first mention (cp. Section 5) the simpler (though previously unnoticed), most beautiful, example: the complex hyperbola

[TABLE]

yielding the recursion relations

[TABLE]

with $r_{n}=|w_{n}|^{2}$, Z_{1}\big{|}n\rangle=w_{n}\big{|}n-1\rangle for $n\in\mathbb{Z}$. Solving the quadratic equation

[TABLE]

gives (compare (5.4))

[TABLE]

as an exact solution of (7.3), resp. (1.3). The classical limit of this Quantum Curve is the minimal surface described in Section 5, where $c$ was taken for simplicity to be real; for complex $c=\alpha+i\beta$ a parametrization of the real 4-dimensional embedding is

[TABLE]

Let us now apply our general method, explained in the introduction, to the complex parabola (considered in [CT99]) $Z_{2}=Z_{1}^{2}$, respectively:

[TABLE]

\dot{\vec{x}}=\big{(}\cos(u),\sin(u),2r\cos(2u),2r\sin(2u)\big{)}, \vec{x}^{\prime}=\big{(}-r\sin(u),r\cos(u),-2r^{2}\sin(2u),2r^{2}\cos(2u)\big{)} implies

[TABLE]

Reparametrizing $\vec{x}(r,u)=\tilde{\vec{x}}(\tilde{r},u)$ according to (cp. (1.5))

[TABLE]

shows that, with e^{i\hat{U}}\big{|}n\rangle=\big{|}n+1\rangle\hat{=}e^{i\varphi}\big{|}e^{in\varphi}\rangle, $\hat{r}=r(\hat{\tilde{r}}=-i\hbar\frac{\partial}{\partial\varphi})$

[TABLE]

the condition

[TABLE]

resp. (cp. [CT99], eq. (25)), with $|w_{n}|^{2}=v_{n}$,

[TABLE]

should “approximately” (s.b., including the integration constant $c$) be solved by

[TABLE]

(7.13), resp.

[TABLE]

is to the first orders, indeed solved by (7.14), both for $n\hbar\to 0$ (i.e. $n<n_{0}$, $\hbar\to 0$) and $n\hbar\to\infty$ (resp. $\hbar$ fixed, $n\to\infty$) which is easily verified by inserting the approximations of

[TABLE]

i.e.

[TABLE]

resp.

[TABLE]

yielding (cp. (7.15), $\epsilon=2\hbar$)

[TABLE]

i.e. (ignoring the $\hbar^{2}/4\beta^{3}$ correction) $4\gamma+1=4\beta$ ($\Rightarrow$ $\alpha=-1/4$, which makes the $O(n^{2}\hbar^{2})$ term vanish) and

[TABLE]

resp.

[TABLE]

i.e., again,

[TABLE]

Finally note that while (7.13) ($\epsilon=2\hbar$) for $n=0$ ($v_{-1}=0=v_{-2}$) gives $v_{0}(1+v_{1})=2\hbar$, (7.15) (for $n=0$) implies $v_{0}(1+v_{1})=\delta$, hence

[TABLE]

in which case

[TABLE]

automatically vanishes at $n=-1$ (cp. the Enneper case), and gives

[TABLE]

which is $1/2$ for $\hbar=1/2$ (the small discrepancy with the in [CT99] numerically calculated value $\rho_{0}^{2}=v_{0}(\hbar=\tfrac{1}{2})\gtrapprox\tfrac{9}{16}\gtrapprox\tfrac{1}{2}$ could partly be due to the non-negligible value of $\tfrac{\hbar^{2}}{4\beta^{3}}=\frac{1}{4^{3}(\frac{3}{4})^{3}}=\frac{1}{27}$ in this case).

Note that (7.25) gives

[TABLE]

so that it does not correspond to an allowed initial value (as $v_{n}=|w_{n}|^{2}$ must necessarily be non-negative), although as we will see, $\epsilon-2\epsilon^{2}+8\epsilon^{3}+O(\epsilon^{4})$ is rather close.

Let us calculate the first few $v_{n}$’s, from

[TABLE]

for arbitrary $v_{0}=:x=:u_{0}$ (always taking $v_{-1}=0$):

[TABLE]

[TABLE]

[TABLE]

As indicated above, let $I_{n}\subset\mathbb{R}$ denote an open interval on which $v_{n}(x)$ is positive. We will now construct intervals $I_{n}\subset I_{n-1}$ such that $\bigcap_{n\in\mathbb{N}}I_{n}\neq\emptyset$, proving that there exist at least one initial condition $x_{0}$ such that $v_{n}(x_{0})>0$ for all $n\geq 0$. To this end, we assume that $c_{2n-2}<c_{2n}<c_{2n-1}$ and

[TABLE]

where the limit points are approached from inside the intervals. It follows from (7.27)– (7.29) that these conditions are met for $n=1,2$. From the recursion relation one finds

[TABLE]

implying that

[TABLE]

Hence, there exists $c_{2n+1}\in(c_{2n},c_{2n-1})$ such that $v_{2n+1}(c_{2n+1})=0$ and $v_{2n+1}(x)>0$ for $x\in I_{2n+1}=(c_{2n},c_{2n+1})$. Analogously,

[TABLE]

implies that

[TABLE]

and that there exists $c_{2n+2}\in(c_{2n},c_{2n+1})$ such that $v_{2n+2}(c_{2n+2})=0$ and $v_{2n+2}(x)>0$ for $x\in I_{2n+2}=(c_{2n+2},c_{2n+1})$. By induction, we conclude that (7.31) holds true for all $n$ and note that

[TABLE]

implying that $\bigcap_{n\in\mathbb{N}}I_{n}\neq\emptyset$. The uniqueness of the initial condition (conjectured in [CT99], based on numerical findings)

[TABLE]

probably follows similarly (cp. [CLVA16]). Writing (7.13), resp (7.26), in terms of $u_{n}$ resp.

[TABLE]

gives

[TABLE]

i.e homogenizing the recursion relations, while increasing the number of terms needed to calculate the next. Noting

[TABLE]

it is tempting to try to prove the polynomiality (in x=\langle 0\big{|}W^{\dagger}W\big{|}0\rangle)

[TABLE]

compactly, via (7.12). Having found the substantial cancellations leading to the extremely slow (“integrable”) growth of the degrees of numerator and denominator of the rational function $v_{n}(x)$ (cp. (7.30)), the “focusing” of zeroes of the $u_{n}$ (note that also those building blocks are increasing for even $N$, decreasing for odd $n$) one may wonder whether on can make this “integrability” more explicit. Indeed111We thank B. Eynard for suggesting (7.37). , writing

[TABLE]

produces polynomial functions of linearly growing degree (note the gap between $\tau_{2}$ and $\tau_{3}$), which (due to the special nature of the problem, cp. (7.30)), are products of 3 successive $u$’s:

[TABLE]

where for convenience we list the first few $u$’s, occurring in (7.27)/ (7.28)/ (7.29):

[TABLE]

The conserved quantity (analogue of (7.33) resp. (7.13)) written in terms of $\tau$’s reads

[TABLE]

and the analogue of (7.34)

[TABLE]

which is the easiest one to generate the $\tau$’s:

[TABLE]

Computer calculations show that the case of 2 monomials,

[TABLE]

similarly corresponds to integrable recursion relations.

Acknowledgments

We would like to thank P. Clarkson, T. Damour, B. Eynard, M. Hynek and F. Nijhoff for discussions.