\documentclass{amsart} \usepackage{amssymb} % amslatex_2 \def\rest{\restriction} \def\cal{\mathcal} \def\Bbb{\mathbb} \newcommand{\hor}{\mathfrak{hor}} \newcommand{\SO}{\mathrm{SO}} \newcommand{\so}{\mathrm{so}} \newcommand{\s}{\sigma} \newcommand{\e}{e} \def\r{\rho} \newcommand{\cl}{C\!\ell} \newcommand{\BC}{\Bbb{C}} \newcommand{\Spin}{\mbox{S}} \newcommand{\JJ}{\sqcup} \newcommand{\Sum}{\displaystyle{\sum}} \newcommand{\Int}{\int} \newcommand{\ad}{\mathrm{ad}} \newcommand{\id}{\mathrm{id}} \newcommand{\hM}{{\star}_M} \newcommand{\hE}{{\star}_E} \newcommand{\nE}{\nabla_{\!E}} \newcommand{\dM}{d_M} \newcommand{\lM}{\Delta_M} \newcommand{\lE}{\Delta_E} \newcommand{\WM}{\Omega_M} \newcommand{\EW}{{\cal E}_\Omega} \renewcommand{\S}{\Sigma} \DeclareFontFamily{OMS}{cmsy}{% \fontdimen16\font=3pt \fontdimen17\font=3pt} \setlength{\headheight}{1.5\baselineskip} \def\dj{d\kern-.30em\raise1.25ex\vbox{\hrule width .3em height .03em}} \def\Dj{D\rlap{\kern-.70em\raise0.75ex \vbox{\hrule width .3em height .03em}}} \newenvironment{pf}{\proof[\proofname]}{\endproof} \newtheorem{lem}{Lemma} \numberwithin{equation}{section} \begin{document} \title[Spinor Structures]{Classical Spinor Structures on Quantum Spaces} \author{micho {\Dj}UR{\Dj}EVICH} \thanks{Clifford Algebras and Spinor Structures, Kluwer 365--377 (1995)} \address{Instituto de Matematicas, UNAM, Area de la Investigacion Cientifica, Circuito Exterior, Ciudad Universitaria, M\'exico DF, CP 04510, MEXICO\newline \indent {\it Written In}\newline \indent Facultad de Estudios Superiores Cuautitlan, MEXICO} \begin{abstract} A noncommutative-geometric generalization of the classical concept of spinor structure is presented. This is done in the framework of the formalism of quantum principal bundles. In particular, analogs of the Dirac operator and the Laplacian are introduced and analyzed. A general construction of examples of quantum spaces with a spinor structure is presented. \end{abstract} \maketitle \section{Introduction} \renewcommand{\thepage}{} Classical geometry of spinor structures is based on two extremely reach and effective mathematical concepts. As first, there is a ``local'', purely algebraic, concept of spinors, which includes Clifford algebras and spin groups. The second one is a ``global'' concept of principal bundle, specified as a covering spin-bundle of the bundle of oriented orthonormal frames (of the base space). In this paper basic concepts of classical spinor geometry will be translated into an appropriate quantum context, in accordance with general principles of non-commutative differential geometry \cite{C1,C2}. The starting idea is that in formulating the concept of a spinor structure on a quantum space, only ``global'' aspects of the classical theory should be quantized. In formalizing this we shall use a general non-commutative-geometric theory of principal bundles, presented in \cite{D1,D2,D3}. Spinor structures on quantum spaces will be represented by certain quantum principal bundles, however the structure group will be a classical spin group. In this sense, ``local aspects'' of the classical theory will be left unchanged. The reason for this is that introducing in the theory a quantum deformation of the spin group would cause the lack of internal symmetry, because of the inherent geometrical inhomogeneity of (truly) quantum groups \cite{D1}. Such a philosophy opens the possibility to generalize, in a more or less straightforward way, fundamental concepts and constructions of the classical theory (presented in \cite{Cru}, for example). The paper is organized as follows. In the next section the concept of a quantum spin manifold will be introduced, in the framework of the formalism of framed quantum principal bundles possessing classical structure groups \cite{frm1}. A formalism of general quantum principal bundles (where both the base manifold and the structure group are quantum objects) has been developed in \cite{D2}. General framed quantum principal bundles are studied in \cite{frm2}. Informally speaking, a spinor structure on a quantum space $M$ will be represented by a principal spin-bundle $P$ over $M$, endowed with an additional structure, which expresses the idea that $M$ is an ``oriented Riemannian manifold'' such that $P$ ``covers'' the bundle of oriented orthonormal frames for $M$ (however, this bundle will figure only implicitly in the theory). Further, we shall introduce the analog of the covariant derivative of the Levi-Civita connection (operating on $P$), and sketch a construction of the canonical differential calculus on $P$. Then counterparts of the Hodge *-operator, Laplace operator and invariant integration on $P$ will be constructed, in a direct analogy with classical differential geometry. Section 3 deals with ``spinor fields''. Starting from $P$ we shall introduce a space $\cal E$, interpretable as consisting of ``smooth sections'' of the corresponding ``associated'' spinor bundle $E$. A quantum counterpart of the Dirac operator will be then introduced and analyzed. In particular, it turns out that the difference between the square of the Dirac operator and the Laplacian (acting in $E$) is proportional to the (scalar) curvature of the Levi-Civita connection, as in classical geometry \cite{Li}. We shall also define the space of ``$E$-valued'' differential forms on $M$, and the quantum counterpart of the Clifford bundle. In Section 4 the theory is illustrated in a simple, but sufficiently reach class of examples of inherently ``curved'' quantum spin manifolds. Finally, in Section 5 some concluding remarks are made. \section{General Considerations} \renewcommand{\thepage}{\arabic{page}} In this section we shall introduce and analyze spinor structures on quantum spaces. As first, let us fix the notation. The Clifford algebra associated with the complex Euclidean space $\Bbb{C}_n$ will be denoted by $\cl_n$. Since in all formal considerations quantum spaces are represented ``dually'' by the corresponding functional algebras, we shall also use this description for the classical groups $\Spin(n)$ (the spin groups) and $\SO(n)$, in parallel with the standard one. A (commutative) *-algebra of polynomial functions on $\Spin (n)$ will be denoted by ${\cal S}_n$. The group structure on $\Spin (n)$ induces a Hopf algebra structure on ${\cal S}_n$, which is determined by the coproduct $\phi\colon {\cal S}_n\rightarrow{\cal S}_n\otimes{\cal S}_n$, counit $\epsilon\colon{ \cal S}_n \rightarrow\BC$ and the antipode $\kappa\colon{\cal S}_n\rightarrow{\cal S}_n$ (we follow the notation of \cite{W}, although in the classical context). In the ``dual'' picture, the elements of $\Spin (n)$ are naturally interpretable as *-characters $g\colon{\cal S}_n\rightarrow\BC$ (non-trivial hermitian multiplicative linear functionals). Furthermore $\SO(n)$ is represented by a *-Hopf subalgebra ${\cal A}_n$ of ${\cal S}_n$, consisting of polynomial functions on $\Spin(n)$ invariant under the map $g\mapsto -g$. At the level of spaces, the inclusion ${\cal A}_n\hookrightarrow {\cal S}_n$ becomes the canonical (universal covering) epimorphism $\amalg\colon \Spin (n)\rightarrow \SO(n)$. We shall denote by $u$ the canonical representation of $\Spin (n)$ in $\BC_n$ (obtained by composing $\amalg$ and the standard representation of $\SO(n)$ in $\BC_n$). Explicitly, $$ u_g(x)=gxg^{-1}$$ for each $x\in\BC_n$ and $g\in \Spin (n)$ (and $\Spin (n)$ and $\BC_n$ are embedded in $\cl_n$). Equivalently, $u$ is understandable as a map (a right ${\cal S}_n$-comodule structure on $\BC_n$) of the form $u\colon\BC_n\rightarrow\BC_n\otimes {\cal S}_n$ so that $u_g=(\id\otimes g)u$. Explicitly, $$u(\e_i)=\sum_{j=1}^n \e_j\otimes u_{ji}$$ where $u_{ji}\in{\cal S}_n$ are matrix elements of $u$ (actually $u_{ij}$ generate the *-algebra ${\cal A}_n$). Let $M$ be a quantum space represented by a non-commutative *-algebra ${\cal V}$. The elements of $\cal V$ are interpretable as ``smooth functions'' on $M$. Let $P=({\cal B},i,F)$ be a (quantum) principal $\Spin (n)$-bundle over $M$. Here, ${\cal B}$ is a *-algebra consisting of appropriate ``functions'' on the quantum space $P$, while $i\colon{\cal V} \rightarrow{\cal B}$ is a *-monomorphism playing the role of ``the dualized projection'' of $P$ on $M$. Finally, $F\colon{\cal B}\rightarrow{\cal B}\otimes{\cal S}_n$ is a *-homomorphism playing the role of the dualized right action of $\Spin (n)$ on $P$. This interpretation is formalized in the following conditions \begin{align} (\id\otimes\phi)F&=(F\otimes \id)F\\ (\id\otimes \epsilon)F&=\id. \end{align} Speaking geometrically, the space $M$ can be identified, via the ``projection map'' with the ``orbit space'' corresponding to the action $F$. Formally, it means that $i({\cal V})$ is consisting precisely of those elements $b\in{\cal B}$ which are $F$-invariant, in the sense that $F(b)=b\otimes 1$. In what follows, we shall identify the elements of $\cal V$ with their images in $i(\cal V)$. The action $F$ is ``free'' in the sense that for each $a\in{\cal S}_n$ there exist elements $q_i,b_i\in{\cal B}$ satisfying \begin{equation}\label{free} \sum_iq_iF(b_i)=1\otimes a. \end{equation} The actual action of elements of $\Spin (n)$ on ${\cal B}$ is described by *-automorphisms $$F_g=(\id\otimes g)F.$$ By definition \cite{frm1} a {\it frame structure} on the bundle $P$ (relative to $u$) is every $n$-tuple $\tau=(\partial_1,\ldots,\partial_n)$ of hermitian derivations $\partial_i\colon{\cal V}\rightarrow{\cal B}$ satisfying $$ F\partial_i(f)=\sum_{j=1}^n\partial_j(f)\otimes u_{ji} $$ for each $f\in{\cal V}$ and $i\in\{1,\ldots,n\}$ and such that there exist elements $b_{i\alpha}\in{\cal B}$ and $v_{i\alpha}\in{\cal V}$ with the property $$ \sum_{\alpha}b_{i\alpha}\partial_j(v_{i\alpha})=\delta_{ij}1 $$ for each $i,j\in\{1,\ldots,n\}$. A frame structure $\tau$ is called {\it integrable} iff there exists a system $\widehat{\tau}=(X_1,\ldots,X_n)$ of hermitian derivations $X_i\colon{\cal B}\rightarrow{\cal B}$ which extend derivations $\partial_i$ and such that the following identities hold \begin{align*} 0&=X_i\partial_j-X_j\partial_i\\ FX_i(b)&=\sum_{j=1}^n\sum_k X_j(b_k)\otimes u_{ji}c_k, \end{align*} where $\Sum_k b_k\otimes c_k=F(b)$. Speaking geometrically, frame structures formalize the idea that $M$ is an ``oriented Riemannian manifold''. Furthermore, endowed with a frame structure $\tau$, the bundle $P$ becomes a ``covering bundle'' of the bundle $P^\ast$ of oriented orthonormal frames for $M$. Starting from an integrable frame structure $\tau$ it is possible to construct \cite{frm1} the whole differential calculus on the bundle (including the calculus on $M$). The construction has several steps. As first, a graded *-algebra $\hor_P$ representing horizontal forms on $P$ can be defined as $$ \hor_P={\cal B}\otimes \BC_n^\wedge$$ where $(\,)^\wedge$ denotes the corresponding external algebra. The *-structure on $\BC_n^\wedge$ is specified by $\e_i^*=\e_i$ and extended by antilinearity and multiplicativity on the whole $\BC_n^\wedge$. Then the formula \begin{equation} \nabla(b\otimes\vartheta)=\sum_{k=1}^n X_k(b)\otimes (\e_k\wedge\vartheta) \end{equation} consistently defines a hermitian first-order antiderivation $\nabla$ on $\hor_P$. This map plays the role of the covariant derivative (induced by the Levi-Civita connection). There exists a natural action $F^\ast\colon\hor_P\rightarrow \hor_P\otimes{\cal S}_n$ of $\Spin (n)$ on $\hor_P$. It is defined by \begin{equation} F^\ast(b\otimes\vartheta)=F(b)u^\wedge(\vartheta), \end{equation} where $u^\wedge\colon\BC_n^\wedge\rightarrow\BC_n^\wedge\otimes {\cal S}_n$ is the representation induced by $u$ (and $\cal B$, $\Bbb{C}_n^\wedge$ are understood as subalgebras of $\hor_P$). Explicitly, the action of elements of $\Spin (n)$ on $\hor_P$ is given by *-automorphisms $$F^\ast_g=(\id\otimes g)F^\ast.$$ The map $F^\ast$ is a *-homomorphism and \begin{gather} (\id\otimes\phi)F^\ast=(F^\ast\otimes \id)F^\ast\\ (\id\otimes \epsilon)F^\ast=\id\\ (\nabla\otimes \id)F^\ast=F^\ast\nabla\label{nab-cov}. \end{gather} Moreover, there exists the unique map $\varrho\colon{\cal S}_n\rightarrow\hor_P$ such that \begin{equation}\label{nab2} \nabla^2(\varphi)=-\sum_k \varphi_k \varrho(c_k) \end{equation} where $F^\ast(\varphi)=\Sum_k\varphi_k\otimes c_k$. The map $\varrho$ is interpretable as the corresponding ``curvature tensor''. We have \begin{align} \varrho(1)&=0\label{varrho1}\\ F^\ast\varrho&=(\varrho\otimes \id)\ad \end{align} where $\ad\colon{\cal S}_n\rightarrow{\cal S}_n\otimes{\cal S}_n$ is the dualized adjoint action of $\Spin (n)$ on itself. In what follows it will be assumed that $\varrho$ is {\it classical}, in the sense that \begin{equation} \varrho(ab)=\epsilon(a)\varrho(b)+ \epsilon(b)\varrho(a)\label{r-der} \end{equation} for each $a,b\in\cal{A}$. Then the following properties hold \begin{gather} \varphi\varrho(a)=\varrho(a)\varphi\\ \varrho(a^*)=\varrho(a)^*\\ \varrho(\kappa(a))=-\varrho(a). \end{gather} Let $\WM$ be the *-subalgebra of $\hor_P$ consisting of $F^\ast$-invariant elements. According to \eqref{nab-cov} we have $\nabla(\WM)\subseteq\WM$. Let $\dM\colon\WM\rightarrow\WM$ be the restriction of $\nabla$ on $\WM$. This map is a hermitian first-order antiderivation. Furthermore, $$d_M^2=0$$ (according to \eqref{nab2} and \eqref{varrho1}). The elements of $\WM$ are interpretable as ``differential forms''on $M$. The map $\dM$ plays the role of the exterior derivative. A graded-differential *-algebra $\Omega_P$ describing the whole differential calculus on the bundle can be now constructed as follows. At the level of graded *-algebras, $$\Omega_P=\hor_P\mathbin{\widehat{\otimes}}L^\wedge $$ where $\widehat{\otimes}$ denotes the graded tensor product. In the above formula $L=\so(n)^*$ (where $\so(n)$ is the Lie algebra of complex antisymmetric $n\times n$ matrices, naturally understood as the Lie algebra of $\SO(n)$) and $L^\wedge$ is endowed with its natural graded-differential *-algebra structure. The differential $d\colon L^\wedge\rightarrow L^\wedge$ is explicitly given by the Mauer-Cartan formula (on the first-order elements and extended on the whole $L^\wedge$ by the graded Leibniz rule). Also, $\hor_P$ and $L^\wedge$ are naturally understandable as subalgebras of $\Omega_P$. The differential $d^\wedge\colon\Omega_P \rightarrow\Omega_P$ is specified by \begin{align} d^\wedge(\varphi) &=\nabla(\varphi)+(-)^{\partial\varphi} \sum_k\varphi_k\pi(c_k)\label{d-hor}\\ d^\wedge(\vartheta)&=R(\vartheta)+d(\vartheta)\label{d-L} \end{align} where $\vartheta\in L$ and $\varphi\in\hor_P$, while $F^\ast(\varphi)=\Sum_k\varphi_k\otimes c_k$. Next, $\pi\colon{\cal S}_n\rightarrow L$ is a natural projection, given by $\pi(a)(X)=(Xa)_\epsilon$ (here elements $X\in \so(n)$ are understood as left-invariant vector fields on $\Spin(n)$). Finally $R\colon{\cal A}\rightarrow\hor_P$ is a map given by $R\pi=\varrho$ (consistency of this definition is a consequence of \eqref{r-der}). Using the above formulas and the graded Leibniz rule the map $d^\wedge$ can be consistently extended to the whole $\Omega_P$. Let $\varpi^\wedge\colon L^\wedge\rightarrow L^\wedge\otimes {\cal S}_n$ be the *-homomorphism extending the dualized coadjoint action $\varpi\colon L\rightarrow L\otimes {\cal S}_n$ (explicitly given by $\varpi\pi=(\pi\otimes\id)\ad$). The formula \begin{equation} F^\wedge(\varphi\otimes\vartheta)=F^\ast(\varphi) \varpi^\wedge(\vartheta) \end{equation} defines a *-homomorphism $F^\wedge\colon\Omega_P\rightarrow\Omega_P \otimes {\cal S}_n$ extending the action $F^\ast$. We have \begin{gather} F^\wedge d^\wedge=(d^\wedge\otimes \id)F^\wedge\\ (F^\wedge\otimes \id)F^\wedge=(\id\otimes \phi)F^\wedge\\ \id=(\id\otimes \epsilon)F^\wedge \end{gather} The construction of $\Omega_P$ can be viewed as a special case of a general construction of differential calculus presented in \cite{D2}--Section 6. A map $\omega\colon L\rightarrow\Omega_P$ defined by $$\omega(\vartheta)=1\otimes\vartheta$$ is a regular and multiplicative connection on $P$. It is a counterpart of the Levi-Civita connection (viewed as a connection on the spin-bundle) in classical geometry. Moreover the map $R$ corresponds to the {\it curvature } of $\omega$, and $\nabla$ can be viewed as the {\it covariant derivative} associated to $\omega$. The quantum analog of the Hodge *-operator can be introduced in a direct analogy with classical geometry. The formula \begin{equation} {\star}\bigl(b\otimes (\e_{i_1}\wedge\ldots \wedge\e_{i_k})\bigr)=b\otimes (\e_{j_1}\wedge\ldots \wedge\e_{j_l}) \end{equation} where $1\leq i_1<\ldots