Scalar Curvature of a Levi-Civita Connection on Cuntz algebra with three generators

TL;DR

This paper introduces a differential calculus on the Cuntz algebra with three generators, establishes the existence of a Levi-Civita connection, and computes its scalar curvature for a canonical metric.

Contribution

It constructs a differential calculus on a specific Cuntz algebra and explicitly computes the scalar curvature of its Levi-Civita connection, extending noncommutative geometry methods.

Findings

Differential calculus satisfies key assumptions enabling Levi-Civita connection

Unique Levi-Civita connection exists for the algebra with the given metric

Scalar curvature computed explicitly for the canonical metric

Abstract

A differential calculus on Cuntz algebra with three generators coming from the action of rotation group in three dimensions is introduced. The differential calculus is shown to satisfy Assumptions I-IV of [1] so that Levi-Civita Connection exists uniquely for any pseudo-Riemannian metric in the sense of [1]. Scalar curvature is computed for the Levi-Civita connection corresponding to the canonical bilinear metric.

Equations

$$\nabla (\omega .a)=\nabla (\omega )a+\omega \otimes da,\omega \in {\Omega }^1(\mathcal{A}),a\in \mathcal{A}.$$

$$\mathcal{Z}(\mathcal{E})=\{e\in \mathcal{E}:a.e=e.aforalla\in \mathcal{A}\}.$$

$${\Pi }_g(\nabla )(\omega \otimes \eta )=(g\otimes \mathrm{id}){\sigma }_{23}(\nabla (\omega )\otimes \eta +\nabla (\eta )\otimes \omega ).$$

$$\nabla ={\nabla }_0+L,$$

$${\zeta }_{\mathcal{E},\mathcal{F}}(\sum _i{e}_i\otimes {\varphi }_i)(f)=\sum _i{e}_i({\varphi }_i(f)).$$

$$T_{\mathcal{E},\mathcal{F}}^L:=((u^{\mathcal{E}}{)}^{-1}\otimes {\mathrm{id}}_{\mathcal{F}}),$$

$$H(\omega \otimes \eta )=(1-P_{\mathrm{sym}}{)}_{23}(\nabla (\omega )\otimes \eta )+\omega \otimes Q^{-1}(d\eta ),$$

$$\Pi (a_0\delta a_1...\delta a_k)=a_0[\mathcal{D},a_1]...[\mathcal{D},a_k],a_0,...,a_k\in \mathcal{A}.$$

$${\Omega }_{\mathcal{D}}^k(\mathcal{A})=\Pi ({\Omega }^k(\mathcal{A}))/\Pi (\delta J_0^{k-1}).$$

$$S_i^{\ast }{S}_i=1,i=1,\dots ,n$$

$$\sum _{j=1}^n{S}_j{S}_j^{\ast }=1.$$

$${\alpha }_A(S_i{)}^{\ast }{\alpha }_A(S_i)$$

$$X_1=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& -1\\ 0& 1& 0\end{array}\right],X_2=\left[\begin{array}{ccc}0& 0& -1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right],X_3=\left[\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 0\end{array}\right],$$

$${\partial }_i(a)=\frac{d}{d\theta }{\alpha }_{\exp (\theta X_i)}(a){|}_{\theta =0},a\in {\mathcal{O}}_3,i=1,2,3.$$

$$\exp (\theta X_1)=\left[\begin{array}{ccc}1& 0& 0\\ 0& cos(\theta )& -sin(\theta )\\ 0& sin(\theta )& cos(\theta )\end{array}\right],\exp (\theta X_2)=\left[\begin{array}{ccc}cos(\theta )& 0& -sin(\theta )\\ 0& 1& 0\\ sin(\theta )& 0& cos(\theta )\end{array}\right],$$

$$\exp (\theta X_3)=\left[\begin{array}{ccc}cos(\theta )& sin(\theta )& 0\\ -sin(\theta )& cos(\theta )& 0\\ 0& 0& 1\end{array}\right].$$

$${\partial }_1(S_1)=0,{\partial }_1(S_2)=-S_3,{\partial }_1(S_3)=S_2$$

$${\partial }_2(S_1)=-S_3,{\partial }_2(S_2)=0,{\partial }_2(S_3)=S_1$$

$${\partial }_3(S_1)=S_2,{\partial }_3(S_2)=-S_1,{\partial }_3(S_3)=0.$$

$$[{\partial }_1,{\partial }_2]=-{\partial }_3,[{\partial }_2,{\partial }_3]={\partial }_1,[{\partial }_1,{\partial }_3]=-{\partial }_2.$$

$$[\mathcal{D},\pi (a)]=\sum _{i=1}^3({\partial }_i(a))\otimes {\sigma }_i.$$

$${\Omega }_{\mathcal{D}}^1({\mathcal{O}}_3)=\{\Pi (\sum _i{a}_i\delta b_i)\subset \mathcal{B}(\mathcal{H}):a_i,b_i\in {\mathcal{O}}_3\}.$$

$$\Pi (\sum _i{a}_i\delta b_i)$$

$$\Pi (S_1^{\ast }\delta S_2)$$

$${\Omega }_{\mathcal{D}}^1({\mathcal{O}}_3)={\mathcal{O}}_3\oplus {\mathcal{O}}_3\oplus {\mathcal{O}}_3.$$

$$\Pi ({\Omega }^2({\mathcal{O}}_3))={\mathcal{O}}_3\oplus {\mathcal{O}}_3\oplus {\mathcal{O}}_3\oplus {\mathcal{O}}_3.$$

$$\Pi (\sum _i{a}_i\delta b_i\delta c_i)$$

$$\Pi (\delta S_2^{\ast }\delta S_1)=\sum _{j<k}({\partial }_j(S_2^{\ast }){\partial }_k(S_1)-{\partial }_k(S_2^{\ast }){\partial }_j(S_1))=1\otimes {\sigma }_1{\sigma }_2.$$

$$\sum _i(a_i\otimes \mathbb{I})(\sum _{j=1}^3{\partial }_j(b_i)\otimes {\sigma }_j)=0$$

$$\Pi (\delta \omega )=\sum _i(\sum _{j=1}^3{\partial }_j(a_i){\partial }_j(b_i)\otimes \mathbb{I}+\sum _{j<k}({\partial }_j(a_i){\partial }_k(b_i)-{\partial }_k(a_i){\partial }_j(b_i))\otimes {\sigma }_j{\sigma }_k).$$

$$\Pi (\delta \omega )=\sum _i(\sum _{j=1}^3{\partial }_j(a_i){\partial }_j(b_i)\otimes \mathbb{I}-\sum _{j<k}(a_i[{\partial }_j,{\partial }_k](b_i)\otimes {\sigma }_j{\sigma }_k).$$

$$(\sum _i\sum _{j=1}^3{\partial }_j(a_i){\partial }_j(b_i)\otimes \mathbb{I}-\sum _i{a}_i{\partial }_1(b_i)\otimes {\sigma }_2{\sigma }_3+\sum _i{a}_i{\partial }_2(b_i)\otimes {\sigma }_1{\sigma }_3++\sum _i{a}_i{\partial }_3(b_i)\otimes {\sigma }_1{\sigma }_2).$$

Full text

††2010 Mathematics Subject Classification: 46L87, 58B34 ††Key words and phrases: Cuntz algebra, Levi-Civita connection, Scalar curvature

1 Introduction

Notion of connection and curvature is important in any form of geometry, be it classical or noncommutative. In noncommutative geometry, in last few years varied notions of curvature have been introduced. Broadly speaking there seems to be two main avenues to define curvature of a noncommutative space. One is to define directly Ricci and scalar curvature via some asymptotic expansion of noncommutative Laplacian (see for example [5], [6], [7]) and the other is to prove existence and uniqueness of Levi-Civita connection on the module of one forms and subsequently compute the curvature operator and scalar curvatures (see [8], [12], [10]). The last one is more algebraic and has the advantage of finding the curvature operator directly. But the difficulty with this approach is to prove the existence and uniqueness of Levi-Civita connection on noncommutative spaces. Recently in [1], the authors gave some sufficient conditions on the differential calculus on noncommutative spaces to ensure existence and uniqueness of Levi-Civita connection with respect to any pseudo-Riemannian metric in some sense. In a follow up paper ([2]) the authors have given examples of noncommutative spaces admitting unique Levi-Civita connection in the sense of [1].

In the context of noncommutative geometry it seems that any formulation of Levi-Civita connection invites automatically examples of noncommutative spaces accommodating such formulation leading to the existence and uniqueness theorem of Levi-Civita connection. A formulation also has to bode well in terms of computability of Ricci and scalar curvature. In this paper we introduce a differential calculus on Cuntz algebra with three generators coming from the natural action of the rotation group in three dimensions. Then we show that the differential calculus admits a unique Levi-Civita connection in the sense of [1]. We compute the Ricci curvature and scalar curvature along the lines of [1] with respect to the canonical metric. The scalar curvature turns out to be a negative constant times identity. It is worth mentioning that noncommutative geometry of Cuntz algebra has been studied in more analytic set up (see [3] for example). Absence of any faithful trace on Cuntz algebra poses considerable amount of difficulty there. This paper attempts in a sense more algebraic study of the geometry of Cuntz algebra.

2 Preliminaries

2.1 Existence and uniqueness of Levi-Civita connection on a class of modules of one-forms

We begin by recalling the definition of a differential calculus on a $\ast$-algebra ${\cal A}$ over $\mathbb{C}$.

Definition 2.1**.**

*A differential calculus on a $\ast$-algebra ${\cal A}$ is a pair $(\Omega({\cal A}),d)$ such that

(i) $\Omega({\cal A})$ is an ${\cal A}-{\cal A}$-bimodule.

(ii) $\Omega({\cal A})=\oplus_{i\geq 0}\Omega^{i}({\cal A})$, where $\Omega^{0}({\cal A})={\cal A}$ and each $\Omega^{i}({\cal A})$ is an ${\cal A}-{\cal A}$-bimodule.

(iii) There is a bimodule map $m:\Omega^{i}({\cal A})\otimes_{{\cal A}}\Omega^{j}({\cal A})\subset\Omega^{i+j}({\cal A})$ for all $i,j$.

(iv) $d:\Omega^{i}({\cal A})\rightarrow\Omega^{i+1}({\cal A})$ satisfies $d(\omega.\eta)=d\omega.\eta+(-1)^{\rm deg\omega}\omega.d\eta$ and $d^{2}=0$.

(v) $\Omega^{i}({\cal A})$ is spanned by $da_{0}...da_{i}a_{i+1}$.*

Let $(\Omega({\cal A}),d)$ be such a differential calculus on an algebra ${\cal A}$.

Definition 2.2**.**

A $\mathbb{C}$-linear map $\nabla:\Omega^{1}({\cal A})\rightarrow\Omega^{1}({\cal A})\otimes_{{\cal A}}\Omega^{1}({\cal A})$ is said to be a (right) connection on the ${\cal A}-{\cal A}$-bimodule $\Omega^{1}({\cal A})$ if

[TABLE]

Definition 2.3**.**

A connection $\nabla$ is said to be torsionless if the right linear map $T_{\nabla}=m\circ\nabla+d$ is equal to zero.

For a bimodule ${\cal E}$ over ${\cal A}$, ${\cal Z}({\cal E})$ will denote the center of the module i.e.

[TABLE]

Similarly ${\cal Z}({\cal A})$ will denote the center of an algebra ${\cal A}$. Now we are going to state a few assumptions on a given differential calculus over a $\ast$-algebra ${\cal A}$ following [1] which enables one to prove existence and uniqueness of Levi-Civita connection on the space of one forms as in [1]. In the following we shall denote the bimodule of one forms by ${\cal E}$.

Assumption I: The ${\cal A}-{\cal A}$ bimodule ${\cal E}$ is finitely generated projective right ${\cal A}$-module. Moreover the map $u^{{\cal E}}:{\cal Z}({\cal E})\otimes_{{\cal Z}({\cal A})}{\cal A}\rightarrow{\cal E}$ given by $u^{{\cal E}}(\sum_{i}e_{i}\otimes a_{i})=\sum e_{i}a_{i}$ is an isomorphism of vector spaces.

Assumption II: The right ${\cal A}$-module ${\cal E}\otimes_{{\cal A}}{\cal E}$ admits a splitting ${\cal E}\otimes_{{\cal A}}{\cal E}={\rm ker}(m)\oplus{\cal F}$ where ${\cal F}\cong{\rm Im}(m)$.

Assumption III: If we denote the idempotent projecting onto ${\rm ker}(m)$ by $P_{\rm sym}$, then the map $\sigma:=(2P_{\rm sym}-1)$ is such that $\sigma(\omega\otimes\eta)=\eta\otimes\omega$ for $\omega,\eta\in{\cal Z}({\cal E})$.

Before stating the ${\bf IV}$th and final assumption, we recall the definition of a pseudo-Riemannian metric on a differential calculus over a $\ast$-algebra ${\cal A}$ which satisfies Assumptions I-III.

Definition 2.4**.**

A pseudo-Riemannian metric $g$ on ${\cal E}$ is an element of ${\rm Hom}_{{\cal A}}({\cal E}\otimes_{{\cal A}}{\cal E},{\cal A})$ such that $g\circ\sigma=g$ and it is non degenarate in the sense that the map $V_{g}:{\cal E}\rightarrow{\cal E}^{\ast}$ defined by $V_{g}(\omega)(\eta)=g(\omega\otimes\eta)$ is an isomorphism of right ${\cal A}$-modules.

Now we are ready to state the final assumption.

Assumption IV: There is a bilinear pseudo-Riemannian metric $g\in{\rm Hom}_{{\cal A}}({\cal E}\otimes_{{\cal A}}{\cal E},{\cal A})$.

Definition 2.5**.**

(see Proposition 2.12 qnd definition 2.13 of [1]) Suppose for a $\ast$-algebra ${\cal A}$, the differential calculus satisfies assumptions ${\bf I-IV}$. Then a connection $\nabla$ on the space of one forms is said to be unitary with respect to a pseudo-Riemannian metric $g$ if $dg=\Pi_{g}(\nabla)$, where $\Pi_{g}(\nabla):{\cal E}\otimes_{{\cal A}}{\cal E}\rightarrow{\cal E}$ is defined as

[TABLE]

Recall the definition of the right ${\cal A}$-linear map $\Phi_{g}:{\rm Hom}({\cal E},{\cal E}\otimes^{\rm sym}_{{\cal A}}{\cal E})\rightarrow{\rm Hom({\cal E}\otimes^{\rm sym}_{{\cal A}}{\cal E},{\cal E})}$ (see Theorem 2.14 of [1]). We state the following theorem from [1] (see theorem 3.12 and 2.14):

Theorem 2.6**.**

For a $\ast$-algebra ${\cal A}$ such that the differential calculus over ${\cal A}$ satisfies assumptions ${\bf I-IV}$, a unique Levi-Civita connection (a connection which is torsionless and unitary) exists. Moreover for any torsionless connection $\nabla_{0}$, the Levi-Civita connection is given by

[TABLE]

where $L=\Phi_{g}^{-1}(dg-\Pi_{g}(\nabla_{0}))$.

Let us recall some definitions which will be used later in the paper. For a right ${\cal A}$ module ${\cal E}$, ${\cal E}^{\ast}$ will denote the right ${\cal A}$-module ${\rm Hom}_{{\cal A}}({\cal E},{\cal A})$. For two bimodules ${\cal E},{\cal F}$, the map $\zeta_{{\cal E},{\cal F}}:{\cal E}\otimes_{{\cal A}}{\cal F}^{\ast}\rightarrow{\rm Hom}_{{\cal A}}({\cal F},{\cal E})$ is defined as

[TABLE]

When both ${\cal E}$ and ${\cal F}$ are finitely generated, projective, $\zeta_{{\cal E},{\cal F}}$ is an isomorphism.

Let ${\cal E}$ be an ${\cal A}-{\cal A}$ bimodule satisfying assumptions $I-IV$. Then for any ${\cal A}-{\cal A}$ bimodule ${\cal F}$, the map $T^{L}_{{\cal E},{\cal F}}:{\cal E}\otimes_{{\cal A}}{\cal F}\rightarrow{\cal Z}({\cal E})\otimes_{{\cal Z}({\cal A})}{\cal F}$ given by

[TABLE]

is a right ${\cal A}$-linear isomorphism. Similarly we have the map $T^{R}_{{\cal E},{\cal F}}$ (see Proposition 4.3) which is a left ${\cal A}$-linear isomorphism between ${\cal F}\otimes_{{\cal A}}{\cal E}$ and ${\cal F}\otimes_{{\cal Z}({\cal A})}{\cal Z}({\cal E})$. Now recall the map $\rho:{\cal E}\otimes_{{\cal A}}{\cal E}^{\ast}\rightarrow{\cal E}^{\ast}\otimes_{{\cal A}}{\cal E}$ defined as $\rho:=(T^{R}_{{\cal E},{\cal E}^{\ast}})^{-1}{\rm flip}T^{L}_{{\cal E},{\cal E}^{\ast}}$, where ${\rm flip}:{\cal Z}({\cal E})\otimes_{{\cal Z}({\cal A})}{\cal E}^{\ast}\rightarrow{\cal E}^{\ast}\otimes_{{\cal Z}({\cal A})}{\cal Z}({\cal E})$ is the map ${\rm flip}(e^{\prime}\otimes\phi)=\phi\otimes e^{\prime}$. The map ${\rm ev}:{\cal E}^{\ast}\otimes{\cal E}\rightarrow{\cal A}$ is given by ${\rm ev}(\sum_{i}\phi_{i}\otimes e_{i})=\sum_{i}\phi_{i}(e_{i})$.

We also recall the map $H:{\cal E}\otimes_{{\cal A}}{\cal E}\rightarrow{\cal E}\otimes_{{\cal A}}{\cal E}\otimes_{{\cal A}}{\cal E}$ (see Lemma 4.1 of [1]) for a connection $\nabla$.

[TABLE]

where $Q:{\rm Im}(1-P_{\rm sym})\rightarrow{\rm Im}(m)$ is the isomorphism from assumption II.

3 Connes’ space of forms for Cuntz algebra with three generators

Let ${\cal A}$ be a $\ast$-algebra. We call a triple $({\cal A},{\cal H},{\cal D})$ where ${\cal A}$ is faithfully represented on ${\cal B}({\cal H})$ and ${\cal D}$ is apriori an unbounded operator, a Dirac triple if $[{\cal D},a]\in{\cal B}({\cal H})$ for all $a\in{\cal A}$. Note that a Dirac triple in our sense is called spectral triple in general. But a spectral triple comes with some kind of summability or compactness criterion. We don’t need this in the paper and that is why we are content with a Dirac triple. For a $\ast$-algebra ${\cal A}$, recall the reduced universal differential algebra $(\Omega^{\bullet}({\cal A}):=\oplus_{k}\Omega^{k}({\cal A}),\delta)$ from [4]. Given a Dirac triple $({\cal A},{\cal H},{\cal D})$ over ${\cal A}$, there is a well defined $\ast$-representation $\Pi$ of $\Omega^{\bullet}({\cal A})$ on ${\cal B}({\cal H})$ given by (see [9])

[TABLE]

Let $J^{k}_{0}=\{\omega\in\Omega^{k}({\cal A}):\Pi(\omega)=0\}$. The Connes’ space of $k$-forms is defined to be

[TABLE]

$\Pi(\delta J^{k-1}_{0})$ is a two sided ideal of $\Pi(\Omega^{k}({\cal A}))$ and is called the space of junk forms. For any element $\omega\in\Omega^{k}({\cal A})$, if we denote the image of $\Pi(\omega)$ in $\Omega^{k}_{{\cal D}}({\cal A})$ by $\overline{\Pi(\omega)}$, then it can be shown that $(\Omega^{\bullet}_{{\cal D}}({\cal A}):=\oplus_{k}\Omega^{k}_{{\cal D}}({\cal A}),d)$ where $d$ is defined as $d\overline{\Pi(\omega)}:=\overline{\Pi(\delta\omega)}$, satisfies the conditions of definition 2.1.

3.1 A Dirac triple on ${\cal O}_{3}$: Connes’ space of forms

We begin this subsection by recalling the definition of Cuntz algebra with $n$-generators. The Cuntz algebra ${\cal O}_{n}$ with $n$-generators is defined to be the universal $C^{\ast}$-algebra generated by $n$ elements $\{S_{i}\}_{i=1,\ldots,n}$ satisfying

[TABLE]

We shall limit ourselves to the Cuntz algebra ${\cal O}_{3}$ with three generators in this paper. For $A=((a_{ij}))_{i,j=1,2,3}\in SO(3)$, define $\alpha_{A}(S_{i})=\sum_{j=1}^{3}a_{ij}S_{j}$ for $i=1,2,3$. Then

[TABLE]

Similarly we have $\sum_{i=1}^{3}\alpha_{A}(S_{i})\alpha_{A}(S_{i})^{\ast}=1$ so that for a fixed $A\in SO(3)$, we have a well defined $C^{\ast}$-homomorphism $\alpha_{A}:{\cal O}_{3}\rightarrow{\cal O}_{3}$ which in turn gives a $C^{\ast}$-dynamical system $({\cal O}_{3},\alpha,SO(3))$. Now let us describe a Dirac triple arising from the above $C^{\ast}$-dynamical system. To that end recall that $SO(3)$ has three dimensional Lie-algebra $so(3)$. Let us choose three basis elements

[TABLE]

which defines three derivations on ${\cal O}_{3}$ as

[TABLE]

Note the following:

[TABLE]

[TABLE]

Then it is easy to see the actions of the derivations on the generators of ${\cal O}_{3}$. They are given by:

[TABLE]

It is clear that the derivations are actually $\ast$-derivations. The commutation relations are given by

[TABLE]

We denote the unique faithful KMS state of ${\cal O}_{3}$ by $\tau$. Let ${\cal H}$ be the Hilbert space $L^{2}({\cal O}_{3},\tau)\otimes\mathbb{C}^{3}$. Then we define a Dirac triple $({\cal O}_{3},{\cal H},{\cal D})$ where $\pi:{\cal O}_{3}\rightarrow{\cal B}({\cal H})$ is given by $\pi(a):=a\otimes\mathbb{I}$ and ${\cal D}$ is the densely defined operator $\sum_{i=1}^{3}\partial_{i}\otimes\sigma_{i}$, $\sigma_{i}$’s being $3\times 3$ Pauli Spin matrices satisfying $\sigma_{i}^{2}=\mathbb{I}$ and $\sigma_{i}\sigma_{j}=-\sigma_{j}\sigma_{i}$ for $i\neq j$. Using the derivation properties of $\partial_{i}$’s, note that for all $a\in{\cal O}_{3}$,

[TABLE]

Hence $({\cal O}_{3},{\cal H},{\cal D})$ is in deed a Dirac triple.

Proposition 3.1**.**

Connes’ space of one forms $\Omega^{1}_{{\cal D}}({\cal O}_{3})$ is a free module of rank $3$.

Proof:

Recall the differential $\delta$ on the reduced universal differential algebra $\Omega^{\bullet}({\cal O}_{3})$ as well as the well defined representation $\Pi:\Omega^{\bullet}({\cal O}_{3})\rightarrow{\cal B}({\cal H})$. We have

[TABLE]

For $a_{i},b_{i}\in{\cal O}_{3}$,

[TABLE]

which proves that $\Omega^{1}_{{\cal D}}({\cal O}_{3})\subset{\cal O}_{3}\oplus{\cal O}_{3}\oplus{\cal O}_{3}$. We shall show that $\Omega^{1}_{{\cal D}}({\cal O}_{3})={\cal O}_{3}\oplus{\cal O}_{3}\oplus{\cal O}_{3}$. To that end observe that

[TABLE]

Similarly it can be shown that $\Pi(S_{1}^{\ast}\delta S_{3})=1\otimes\sigma_{2}$ and $\Pi(S_{2}^{\ast}\delta S_{3})=1\otimes\sigma_{1}$. Hence

[TABLE]

So $\Omega_{{\cal D}}^{1}({\cal O}_{3})$ is a free module of rank $3$. Note that the freeness follows from the linear independence of $\sigma_{1},\sigma_{2},\sigma_{3}$.$\Box$

Proposition 3.2**.**

$\Omega^{2}_{{\cal D}}({\cal O}_{3})$* is also a free module of rank $3$.*

Proof:

By definition $\Omega^{2}_{{\cal D}}({\cal O}_{3})=\Pi(\Omega^{2}({\cal O}_{3}))/\Pi(\delta J_{0}^{1})$. We claim that

[TABLE]

To prove the above, let $\sum_{i}a_{i}\delta b_{i}\delta c_{i}\in\Omega^{2}({\cal O}_{3})$. Then

[TABLE]

which proves $\Pi(\Omega^{2}({\cal O}_{3}))\subset{\cal O}_{3}\oplus{\cal O}_{3}\oplus{\cal O}_{3}\oplus{\cal O}_{3}$. To prove the equality note that $\Pi(\delta(S_{1}^{\ast})\delta(S_{1}))=\sum_{j=1}^{3}\partial_{j}(S_{1}^{\ast})\partial_{j}(S_{1})\otimes\mathbb{I}=2\otimes\mathbb{I}$. Also,

[TABLE]

Similarly choosing appropriate elements from ${\cal O}_{3}$, we can establish the equality. We are left with finding the junk forms. First, we shall show that $\Pi(\delta J_{0}^{1})\subset{\cal O}_{3}\otimes\mathbb{I}$. To that end choose $\omega=\sum_{i}a_{i}\delta b_{i}$ so that $\Pi(\omega)=0$. $\Pi(\omega)=0$ implies the following:

[TABLE]

Using the linear independence of $\sigma_{j}^{\prime}s$, we get $\sum_{i}a_{i}\partial_{j}(b_{i})=0$ for $j=1,2,3$. Note that $\delta\omega=\sum_{i}\delta a_{i}\delta b_{i}$. Hence

[TABLE]

$\sum_{i}a_{i}\partial_{j}(b_{i})=0$ implies that for all $j,k=1,2,3$, $\sum_{i}\partial_{k}(a_{i})\partial_{j}(b_{i})=-a_{i}\partial_{k}\partial_{j}(b_{i})$. Hence

[TABLE]

Using the commutation relations (4), $\Pi(\delta\omega)$ reduces to

[TABLE]

But $\sum_{i}a_{i}\partial_{j}(b_{i})=0$ for $j=1,2,3$. So

[TABLE]

We shall prove that the above inclusion is actually an equality. But this is straightforeward. Pick $\omega=S_{1}^{\ast}\delta S_{1}$. Then $S_{1}^{\ast}\partial_{j}(S_{1})=0$ for all $j$ so that $\Pi(\omega)=0$. But $\Pi(\delta\omega)=\sum_{j=1}^{3}\partial_{j}(S_{1}^{\ast})\partial_{j}(S_{1})\otimes\mathbb{I}=2\otimes\mathbb{I}$. Therefore,

[TABLE]

Hence we conclude that

[TABLE]

$\Box$.

From the proof of the Proposition 3.1, it follows that $\{1\otimes\sigma_{i}:i=1,2,3\}$ is an ${\cal O}_{3}$-basis for $\Omega^{1}_{{\cal D}}({\cal O}_{3})$. We denote them by $\{e_{i}\}_{i=1,2,3}$. It is clear from the module structure that $e_{i}\in\mathcal{Z}(\Omega^{1}_{{\cal D}}({\cal O}_{3}))$. Similarly $\{1\otimes\sigma_{ij}\}_{i<j}$ is an ${\cal O}_{3}$-basis for $\Omega^{2}_{{\cal D}}({\cal O}_{3})$. We denote the basis elements by $\{e_{ij}\}_{i<j}$. With these notations we have the following

Lemma 3.3**.**

The multiplication map $m:\Omega^{1}_{{\cal D}}({\cal O}_{3})\otimes_{{\cal O}_{3}}\Omega^{1}_{{\cal D}}({\cal O}_{3})\rightarrow\Omega^{2}_{{\cal D}}({\cal O}_{3})$ is surjective and is given by

[TABLE]

Proof:

The surjectiveness of $m$ follows immediately once we prove (5). But (5) follows from

[TABLE]

$\Box$

From now on we shall denote the ${\cal O}_{3}$-bimodule $\Omega^{1}_{{\cal D}}({\cal O}_{3})$ by ${\cal E}$.

Theorem 3.4**.**

For the Connes’ differential calculus coming from the $C^{\ast}$-dynamical system $({\cal O}_{3},\alpha,SO(3))$, a unique Levi-Civita connection exists for any pseudo-Riemannian metric.

Proof:

We shall show that the Connes’ differential calculus satisfies the assumptions I-IV of subsection 2.1 and hence by Theorem 2.6, the conclusion of the above theorem will follow. So let us check the assumptions one by one.

Assumption I: ${\cal E}$ is a free module of rank $3$ with the basis elements $e_{1},e_{2},e_{3}\in{\cal Z}({\cal E})$. Hence it is finitely generated and projective as both right and left module. It is easy to see that the map $u^{{\cal E}}:\mathcal{Z}({\cal E})\otimes_{{\cal Z}({\cal O}_{3})}{\cal O}_{3}\rightarrow{\cal E}$ given by $u^{{\cal E}}(e_{i}\otimes a_{i})=\sum e_{i}a_{i}$ is an isomorphism with the inverse $u^{{\cal E}^{-1}}(\sum_{i=1}^{3}e_{i}a_{i})=\sum_{i=1}^{3}e_{i}\otimes a_{i}$.

Assumption II-III: Recall the multiplication map $m$. From Lemma 3.3, we have the following short exact sequence:

[TABLE]

$\Omega^{2}_{{\cal D}}({\cal O}_{3})$ being a free module, the above short exact sequence splits and hence we have

[TABLE]

where ${\cal F}\cong\Omega^{2}_{{\cal D}}({\cal O}_{3})$. As in [1], This splitting implies that the map $P_{sym}$ on the elements $e_{i}\otimes e_{j}$ is given by $P_{sym}(e_{i}\otimes e_{j})=\frac{1}{2}(e_{i}\otimes e_{j}+e_{j}\otimes e_{i})$ and consequently $\sigma(e_{i}\otimes e_{j})=e_{j}\otimes e_{i}$ for all $i,j=1,2,3$. Thus Connes’ differential calculus satisfies Assumptions II-III.

Assumption IV: Consider the metric $g:{\cal E}\otimes_{{\cal O}_{3}}{\cal E}\rightarrow{\cal O}_{3}$ defined by

[TABLE]

Then it is easy to see that $g$ is a pseudo Riemannian bilinear non degenerate metric such that $g\circ\sigma=g$. Hence Assumption IV is satisfied. $\Box$.

Henceforth we call the metric $g$ the canonical metric.

4 Scalar curvature for the canonical metric

In this section we shall compute the scalar curvature of ${\cal O}_{3}$ for the canonical metric $g$ given by (6).

4.1 Computation of Christoffel symbols

Let us recall from Theorem 2.6 that the Levi-Civita connection for any pseudo-Riemannian metric $g$ is given by $\nabla=\nabla_{0}+L$, where $\nabla_{0}$ is any torsionless connection (see definition 2.3) and $L=\Phi_{g}^{-1}(dg-\Pi_{g}(\nabla_{0}))$. We prove the following Lemma which will enable us to choose a torsionless connection.

Lemma 4.1**.**

[TABLE]

Proof:

Since $e_{1}=S_{2}^{\ast}d(S_{3})$, we have

[TABLE]

Similarly one can prove the other equalities.$\Box$

Proposition 4.2**.**

A choice of torsionless connection $\nabla_{0}$ is given on the ${\cal O}_{3}$-basis of the free module $\Omega^{1}({\cal O}_{3})$ by

[TABLE]

Proof:

Consequence of the Lemma 4.1 and the fact that $T_{\nabla_{0}}$ is right ${\cal O}_{3}$-linear.$\Box$

Lemma 4.3**.**

For the canonical metric $g$ and the connection $\nabla_{0}$, let $\Pi_{g}(\nabla_{0})(e_{i}\otimes e_{j})=-\sum_{m=1}^{3}T^{m}_{ij}e_{m}$. Then

[TABLE]

Proof:

Using equation (10),

[TABLE]

With similar computation one can show that $\Pi_{g}(\nabla_{0})(e_{2}\otimes e_{3})=e_{1}$ and $\Pi_{g}(\nabla_{0})(e_{1}\otimes e_{3})=e_{2}$. Using the easy to see fact that $T^{m}_{ij}=T^{m}_{ji}$ for all $i,j,m$, we have for distinct $i,j,m$, $T^{m}_{ij}=-1$ for all $i,j,m=1,2,3$ and $T^{m}_{ij}=0$ for $i\neq j$ whenever either $m=i$ or $m=j$. So we are left with proving that $T^{m}_{ii}=0$ for all $i,m$. To that end

[TABLE]

Similarly we can prove that $\Pi_{g}(\nabla_{0})(e_{i}\otimes e_{i})=0$ for $i=2,3$, finishing the proof of the lemma.$\Box$

Theorem 4.4**.**

For the Levi-Civita connection $\nabla$, if we write $\nabla(e_{i})=\sum_{j,k=1}^{3}e_{j}\otimes e_{k}\Gamma_{jk}^{i}$, for $i=1,2,3$, then we have the following:

[TABLE]

Rest of the Christoffel symbols are zero.

Proof:

$\nabla-\nabla_{0}=L$ for some $L\in{\rm Hom}({\cal E},{\cal E}\otimes^{\rm sym}_{{\cal O}_{3}}{\cal E})$ (see Theorem 2.14 of [1]). If we write $L(e_{j})=\sum_{i,m=1}^{3}e_{i}\otimes e_{m}L^{j}_{im}$ for $j=1,2,3$, then adapting the same arguments of Theorem 4.24 of [1],

[TABLE]

Hence by equation (11),

[TABLE]

[TABLE]

Hence $\Gamma^{1}_{23}=-.5,\Gamma^{1}_{32}=.5,\Gamma^{1}_{11}=\Gamma^{1}_{22}=\Gamma^{1}_{12}=\Gamma^{1}_{21}=0$. With similar computations, we can prove the rest of the equalities of (12).$\Box$

4.2 Computation of scalar curvature

Scalar curvature is obtained by contracting the Ricci tensor by the metric. Recall the maps $\rho$ and ${\rm ev}$ from Subsection 2.1. The Ricci tensor ${\rm Ric}\in{\cal E}\otimes_{{\cal O}_{3}}{\cal E}$ is given by

[TABLE]

where $\Theta$ is the curvature operator as in [1].

Lemma 4.5**.**

$R(\nabla)(e_{i})=\frac{1}{8}\sum_{k\neq i}(e_{k}\otimes e_{k}\otimes e_{i}-e_{k}\otimes e_{i}\otimes e_{k})$, for $i=1,2,3$.

Proof:

Recall the map $H$ from (3). By definition of $R(\nabla)$, we have

[TABLE]

We calculate each terms individually.

[TABLE]

Similarly $(1-P_{\rm sym})_{23}(\nabla(e_{2})\otimes e_{3})=\frac{1}{4}(e_{3}\otimes e_{3}\otimes e_{1}-e_{3}\otimes e_{1}\otimes e_{3})$. Also using the fact that $Q^{-1}(e_{i}\otimes e_{j})=\frac{1}{2}(e_{i}\otimes e_{j}-e_{j}\otimes e_{i})$ for all $i,j$, it is easy to see that

[TABLE]

Hence

[TABLE]

With similar computations, we have

[TABLE]

Combining these, we get the result of the lemma.$\Box$

Lemma 4.6**.**

[TABLE]

i.e. ${\rm Ric}=\sum_{j=1}^{3}(-.25)e_{j}\otimes e_{j}$.

Proof:

We begin by computing the curvature operator $\Theta$. To that end, recall the map $\zeta_{{\cal E},{\cal F}}$ from (1). Denoting the element of ${\cal E}^{\ast}$ which takes the value $1$ at $e_{i}$ and [math] on the other basis elements by $e_{i}^{\ast}$, we have

[TABLE]

Now recall the map $\rho:{\cal E}\otimes{\cal E}^{\ast}\rightarrow{\cal E}^{\ast}\otimes{\cal E}$. It is easy to see that $\rho(e_{i}\otimes e_{j}^{\ast})=e_{j}^{\ast}\otimes e_{i}$ for all $i,j=1,2,3$. Using this, we get

[TABLE]

Since, by definition, ${\rm ev}(e_{i}^{\ast}\otimes e_{j})=\delta_{ij}$, we have

[TABLE]

This proves the Lemma.$\Box$

Theorem 4.7**.**

Cuntz algebra with three generators with the differential calculus coming from the action of $SO(3)$ has a constant negative scalar curvature $-0.75$ for the canonical metric.

Proof:

By the definition of the scalar curvature (see after definition 4.4 in [1]),

[TABLE]

$\Box$.

Remark 4.8**.**

We have restricted our attention to Cuntz algebra with three generators purely because the computation for the general case does not seem to be that much more enlightening.

Acknowledgement: The author acknowledges support from Department of Science and Technology, India (DST/INSPIRE/04/2016/002469). He would like to thank Jyotishman Bhowmick for some fruitful discussions and both K. B. Sinha as well as Debashish Goswami for their words of encouragement.