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We consider spinors on the total space of a Kaluza-Klein model with fuzzy sphere fibre and geometrically realised Dirac operator on the product. We show that a single massless spinor on the product appears on spacetime as multiplets of spinors with a particular signature of differing masses and $SU(2)$ Yang-Mills charges. For example, for the reduced fuzzy sphere isomorphic to $M_2(\mathbb{C})$, a massless spinor appears as two $SU(2)$ doublets and an $SU(2)$ quadruplet in mass ratios $1:5/3:7/3$. Although such signatures do not appear to exactly match the Standard Model, the paper provides proof of concept of the approach, which can be applied to other noncommutative fibre algebras.

We construct a quantum Dolbeault double complex $\oplus_{p,q}\Omega^{p,q}$ on the quantum plane $\Bbb C_q^2$. This solves the long-standing problem that the standard differential calculus on the quantum plane is not a $*$-calculus, by embedding it as the holomorphic part of a $*$-calculus. We show in general that any Nichols-Woronowicz algebra or braided plane $B_+(V)$, where $V$ is an object in an abelian $\Bbb C$-linear braided bar category of real type is a quantum complex space in this sense with a factorisable Dolbeault double complex. We combine the Chern construction on $\Omega^{1,0}$ in such a Dolbeault complex for an algebra $A$ with its conjugate to construct a canonical metric compatible connection on $\Omega^1$ associated to a class of quantum metrics, and apply this to the quantum plane. We also apply this to finite groups $G$ with Cayley graph generators split into two halves related by inversion, constructing such a Dolbeault complex $\Omega(G)$ in this case, recovering the quantum Levi-Civita connection for any edge-symmetric metric on the integer lattice with $\Omega(\Bbb Z)$ now viewed as a quantum complex structure. We also show how to build natural quantum metrics on $\Omega^{1,0}$ and $\Omega^{0,1}$ separately where the inner product in the case of the quantum plane, in order to descend to $\otimes_A$, is taken with values in an $A$-bimodule., Comment: 28 pages, 5 figures

We study the Yang-Baxter algebras $A(K,X,r)$ associated to finite set-theoretic solutions $(X,r)$ of the braid relations. We introduce an equivalent set of quadratic relations $\Re\subseteq G$, where $G$ is the reduced Gr\"obner basis of $(\Re)$. We show that if $(X,r)$ is left-nondegenerate and idempotent then $\Re= G$ and the Yang-Baxter algebra is PBW. We use graphical methods to study the global dimension of PBW algebras in the $n$-generated case and apply this to Yang-Baxter algebras in the left-nondegenerate idempotent case. We study the $d$-Veronese subalgebras for a class of quadratic algebras and use this to show that for $(X,r)$ left-nondegenerate idempotent, the $d$-Veronese subalgebra $A(K,X,r)^{(d)}$ can be identified with $A(K,X,r^{(d)})$, where $(X,r^{(d)})$ are all left-nondegenerate idempotent solutions. We determined the Segre product in the left-nondegenerate idempotent setting. Our results apply to a previously studied class of `permutation idempotent' solutions, where we show that all their Yang-Baxter algebras for a given cardinality of $X$ are isomorphic and are isomorphic to their $d$-Veronese subalgebras. In the linearised setting, we construct the Koszul dual of the Yang-Baxter algebra and the Nichols-Woronowicz algebra in the idempotent case, showing that the latter is quadratic. We also construct noncommutative differentials on some of these quadratic algebras., Comment: 40 pages, 3 figures. arXiv admin note: text overlap with arXiv:2308.11427

The quantum double $D(G)=\Bbb C(G)\rtimes \Bbb C G$ of a finite group plays an important role in the Kitaev model for quantum computing, as well as in associated TQFT's, as a kind of Poincar\'e group. We interpret the known construction of its irreps, which are quasiparticles for the model, in a geometric manner strictly analogous to the Wigner construction for the usual Poincar\'e group of $\Bbb R^{1,3}$. Irreps are labelled by pairs $(C, \pi)$, where $C$ is a conjugacy class in the role of a mass-shell, and $\pi$ is a representation of the isotropy group $C_G$ in the role of spin. The geometric picture entails $D^\vee(G)\to \Bbb C(C_G)\blacktriangleright\!\!\!\!< \Bbb C G$ as a quantum homogeneous bundle where the base is $G/C_G$, and $D^\vee(G)\to \Bbb C(G)$ as another homogeneous bundle where the base is the group algebra $\Bbb C G$ as noncommutative spacetime. Analysis of the latter leads to a duality whereby the differential calculus and solutions of the wave equation on $\Bbb C G$ are governed by irreps and conjugacy classes of $G$ respectively, while the same picture on $\Bbb C(G)$ is governed by the reversed data. Quasiparticles as irreps of $D(G)$ also turn out to classify irreducible bicovariant differential structures $\Omega^1_{C, \pi}$ on $D^\vee(G)$ and these in turn correspond to braided-Lie algebras $\mathcal{L}_{C, \pi}$ in the braided category of $G$-crossed modules, which we call `braided racks' and study. We show under mild assumptions that $U(\mathcal{L}_{C,\pi})$ quotients to a braided Hopf algebra $B_{C,\pi}$ related by transmutation to a coquasitriangular Hopf algebra $H_{C,\pi}$., Comment: 75 pages Latex two figures

Understanding the microscopic behavior of spacetime is critical for developing a theory of quantum gravity and perhaps solving the cosmological constant problem. In this context, it has been proposed that the quantity of interest is the quantum uncertainty in the Ricci scalar and here we investigate this for a discrete baby quantum gravity model based a single square cell. We find that the averaged Ricci scalar vanishes to leading order but has UV-divergent fluctuations. While this behavior is stable under renormalization, it appears not to be under the introduction of a small cosmological constant., Comment: 5 pages

The turn of the millennium was a time of optimism about an approach to noncommutative geometry inspired by rich mathematical objects called `quantum groups' and its applications to quantum spacetime. This would model quantum gravity effects as noncommutativity of spacetime coordinates and was arguably going to solve quantum gravity itself. It took a further 20 years from that point to develop a particularly suitable formalism of `quantum Riemannian geometry', but this was largely done and has begun to be used to construct baby quantum gravity models. In this article, we obtain new results for state of the art fuzzy sphere and n-gon models in this approach. We also review what are some elements of quantum gravity that we can already see and what are the critical conceptual and mathematical elements that are still missing to more fully achieve this goal., Comment: 19 pages 6 figures, research + review (intended for a volume on 25 years into the millennium)

Using the framework of quantum Riemannian geometry, we show that gravity on the product of spacetime and a fuzzy sphere is equivalent under minimal assumptions to gravity on spacetime, an $su_2$-valued Yang-Mills field $A_{\mu i}$ and real-symmetric-matrix valued Liouville-sigma model field $h_{ij}$ for gravity on the fuzzy sphere. Moreover, a massless real scalar field on the product appears as a tower of scalar fields on spacetime, with one for each internal integer spin $l$ representation of $SU(2)$, minimally coupled to $A_{\mu i}$ and with mass depending on $l$ and the fuzzy sphere size. For discrete values of the deformation parameter, the fuzzy spheres can be reduced to matrix algebras $M_{2j+1}(C)$ for $j$ any non-negative half-integer, and in this case only integer spins $0\le l\le 2j$ appear in the multiplet. Thus, for $j=1$ a massless field on the product appears as a massless $SU(2)$ internal spin 0 field, a massive internal spin 1 field and a massive internal spin 2 field, in mass ratio $0,1,\sqrt{3}$ respectively, which we conjecture could arise in connection with an approximate $SU(2)$ flavour symmetry.

We revisit the construction of quantum Riemannian geometries on graphs starting from a hermitian metric compatible connection, which always exists. We use this method to find quantum Levi-Civita connections on the $n$-leg star graph for $n=2,3,4$ and find the same phenomenon as recently found for the $A_n$ Dynkin graph that the metric length for each outbound arrow has to exceed the length in the other direction by a multiple, here $\sqrt{n}$. We then study quantum geodesics on graphs and construct these on the 4-leg graph and on the integer lattice line $\Bbb Z$ with a general edge-symmetric metric, Comment: 29 pages latex, 5 pdf figures

We use a recent formalism of quantum geodesics in noncommutative geometry to construct geodesic flow on the infinite chain $\cdots\bullet$--$\bullet$--$\bullet\cdots$. We find that noncommutative effects due to the discretisation of the line cause an initially real geodesic flow amplitude $\psi$ (for which the density is $|\psi|^2$) to become complex. This has been noted also for other quantum geometries and suggests that the complex nature of the wave function in quantum mechanics (and the interference effects that follow) may have its origin in a quantum/discrete nature of spacetime at the Planck scale., Comment: 10 pages latex, two pdf figures

We study the quadratic algebras $A(K,X,r)$ associated to a class of strictly braided but idempotent set-theoretic solutions $(X,r)$ of the Yang-Baxter or braid relations. In the invertible case, these algebras would be analogues of braided-symmetric algebras or `quantum affine spaces' but due to $r$ being idempotent they have very different properties. We show that all $A(K,X,r)$ for $r$ of a certain permutation idempotent type are isomorphic for a given $n=|X|$, leading to canonical algebras $A(K,n)$. We study the properties of these both via Veronese subalgebras and Segre products and in terms of noncommutative differential geometry. We also obtain new results on general PBW algebras which we apply in the permutation idempotent case., Comment: 37 pages 2 figures

We introduce left and right groups of bisections of a Hopf algebroid and show that they form a group crossed homomorphism with the group $Aut(\mathcal{L})$ of bialgebroid automorphisms. We also introduce a nonAbelian cohomology $\mathcal{H}^2(\mathcal{L},B)$ governing cotwisting of a Hopf algebroid with base $B$. We also introduce a notion of coquasi-bialgebroid $\mathcal{L}$ via a 3-cocycle on $\mathcal{L}$. For the Ehresmann-Schauenburg Hopf algebroid $\mathcal{L}(P,H)$ of a quantum principal bundle or Hopf-Galois extension, we show that the group of bisections reduces to the group $\textit{Aut}_H(P)$ of bundle automorphisms, and give a description of the nonAbelian cohomology in concrete terms in two cases: $P$ subject to a `braided' commutativity condition and $P$ a cleft extension or `trivial' bundle. Next we show that the action bialgebroid $B\# H^{op}$ associated to a braided-commutative algebra $B$ in the category of $H$-crossed (or Drinfeld-Yetter) modules over a Hopf algebra $H$ is an fact a Hopf algebroid. We show that the bisection groups are again isomorphic and can be described concretely as a natural space $Z^1_{\triangleleft}(H,B)$ of multiplicative cocycles. We also do the same for the nonAbelian cohomology and for $Aut(\mathcal{L})$. We give specific results for the Heisenberg double or Weyl Hopf algebroid $H^*\# H^{op}$ of $H$. We show that if $H$ is coquasitriangular then its transmutation braided group $B=\underline{H}$ provides a canonical action Hopf algebroid $\underline{H}\# H^{op}$ and we show that if $H$ is factorisable then $\underline{H}\# H^{op}$ is isomorphic to the Weyl Hopf algebroid of $H$. We also give constructions for coquasi versions of $\mathcal{L}(P,H)$ and of the Connes-Moscovici bialgebroid. Examples of the latter are given from the data of a subgroup $G\subseteq X$ of a finite group and choice of transversal.

Following steps analogous to classical Kaluza-Klein theory, we solve for the quantum Riemannian geometry on $C^\infty(M)\otimes M_2(\mathbb{C})$ in terms of classical Riemannian geometry on a smooth manifold $M$, a finite quantum geometry on the algebra $M_2(\mathbb{C})$ of $2\times 2$ matrices, and a quantum metric cross term. Fixing a standard form of quantum metric on $M_2(\mathbb{C})$, we show that this cross term data amounts in the simplest case to a 1-form $A_\mu$ on $M$, which we regard as like a gauge-fixed background field. We show in this case that a real scalar field on the product algebra with its noncommutative Laplacian decomposes on $M$ into two real neutral fields and one complex charged field minimally coupled to $A_\mu$. We show further that the quantum Ricci scalar on the product decomposes into a classical Ricci scalar on $M$, the Ricci scalar on $M_2(\mathbb{C})$, the Maxwell action $||F||^2$ of $A$ and a higher order $||A.F||^2$ term. Another solution of the QRG on the product has $A=0$ and a dynamical real scalar field $\phi$ on $M$ which imparts mass-splitting to some of the components of a scalar field on the product as in previous work.

We construct a canonical geometrically realised Connes spectral triple or `Dirac operator' $D\!\!\!/$ from the data of a quantum metric $g\in \Omega^1\otimes_A\Omega^1$ and quantum Levi-Civita bimodule connection, at the pre-Hilbert space level. Here $A$ is a possibly noncommutative coordinate algebra, $\Omega^1$ a bimodule of 1-forms and the spinor bundle is $S=A\oplus\Omega^1$. When applied to graphs or lattices, we essentially recover a Dirac operator previously proposed by Bianconi but now as a geometrically realised spectral triple. We also apply the construction to the fuzzy sphere and to $2\times 2$ matrices with their standard quantum Riemannian geometries. We also propose how $D\!\!\!/$ can be minimally coupled to an external potential., Comment: 24 pages AMS-Latex. Some minor corrections and clarifications, simplifying the graph case

We provide a systematic treatment of boundaries based on subgroups $K\subseteq G$ with the Kitaev quantum double $D(G)$ model in the bulk. The boundary sites are representations of a $*$-subalgebra $\Xi\subseteq D(G)$ and we explicate its structure as a strong $*$-quasi-Hopf algebra dependent on a choice of transversal $R$. We provide decomposition formulae for irreducible representations of $D(G)$ pulled back to $\Xi$. We also provide explicitly the monoidal equivalence of the category of $\Xi$-modules and the category of $G$-graded $K$-bimodules and use this to prove that different choices of $R$ are related by Drinfeld cochain twists. Examples include $S_{n-1}\subset S_n$ and an example related to the octonions where $\Xi$ is also a Hopf quasigroup. As an application of our treatment, we study patches with boundaries based on $K=G$ horizontally and $K=\{e\}$ vertically and show how these could be used in a quantum computer using the technique of lattice surgery., Comment: This is a sequel to arXiv:2107.04411

We show that the Ehresmann-Schauenburg bialgebroid of a quantum principal bundle $P$ or Hopf Galois extension with structure quantum group $H$ is in fact a left Hopf algebroid $L(P,H)$. We show further that if $H$ is coquasitriangular then $L(P,H)$ has an antipode map $S$ obeying certain minimal axioms. Trivial quantum principal bundles or cleft Hopf Galois extensions with base $B$ are known to be cocycle cross products $B\#_\sigma H$ for a cocycle-action pair ($\vartriangleright$,$\sigma$) and we look at these of a certain `associative type' where $ \vartriangleright$ is an actual action. In this case also, we show that the associated left Hopf algebroid has an antipode obeying our minimal axioms. We show that if $L$ is any left Hopf algebroid then so is its cotwist $L^\varsigma$ as an extension of the previous bialgebroid Drinfeld cotwist theory. We show that in the case of associative type, $L(B\#_\sigma H,H)=L(B\# H)^{\tilde\sigma}$ for a Hopf algebroid cotwist $\varsigma=\tilde\sigma$. Thus, switching on $\sigma$ of associative type appears at the Hopf algebroid level as a Drinfeld cotwist. We view the affine quantum group $\hat{U_q(sl_2)}$ and the quantum Weyl group of $u_q(sl_2)$ as examples of associative type.

We formulate a notion of jet bundles over a possibly noncommutative algebra $A$ equipped with a torsion free connection. Among the conditions needed for 3rd-order jets and above is that the connection also be flat and its `generalised braiding tensor' $\sigma:\Omega^1\otimes_A\Omega^1\to \Omega^1\otimes_A\Omega^1$ obey the Yang-Baxter equation or braid relations. We also cover the case of jet bundles of a given `vector bundle' over $A$ in the form of a bimodule $E$ with a flat bimodule connection with its braiding $\sigma_E$ obeying the coloured braid relations. Examples include the permutation group $S_3$ with its 2-cycles calculus, $M_2(\Bbb C)$, the bicrossproduct model quantum spacetime in two dimensions and $\Bbb C_q[SL_2]$ for $q$ a 4th root of unity., Comment: 43 pages AMS latex, four pdf figures; improved the introduction, added C_q[SL_2] example, minor corrections and clarifications elsewhere

We study geodesics flows on curved quantum Riemannian geometries using a recent formulation in terms of bimodule connections and completely positive maps. We complete this formalism with a canonical $*$ operation on noncommutative vector fields. We show on a classical manifold how the Ricci tensor arises naturally in our approach as a term in the convective derivative of the divergence of the geodesic velocity field, and use this to propose a similar object in the noncommutative case. Examples include quantum geodesic flows on the algebra of 2 x 2 matrices, fuzzy spheres and the $q$-sphere., Comment: 32 pages amslatex, 3 pdf figures; minor corrections and clarifications, more background in the preliminaries

We apply a recent formalism of quantum geodesics to the well-known bicrossproduct model $\lambda$-Minkowski quantum spacetime $[x^i,t]=\imath\lambda_p x^i$ with its flat quantum metric as a model of quantum gravity effects, with $\lambda_p$ the Planck scale. As examples, quantum geodesic flow of a plane wave gets an order $\lambda_p$ frequency dependent correction to the classical geodesic velocity. A quantum geodesic flow with classical velocity $v$ of a Gaussian with width $\sqrt{2\beta}$ initially centred at the origin changes its shape but its centre of mass moves with ${\over}=v(1+{\lambda_p^2\over 2\beta}+O(\lambda^3_p))$, an order $\lambda_p^2$ correction. This implies, at least within perturbation theory, that a `point particle' cannot be modelled as an infinitely sharp Gaussian due to quantum gravity corrections. For contrast, we also look at quantum geodesics on the noncommutative torus with a 2D curved weak quantum Levi-Civita connection., Comment: 28 pages amslatex, 3 pdf figures; some small corrections and clarified the notation in a couple of places

The `spider theorem' for a general Frobenius algebra $A$, classifies all maps $A^{\otimes m}\to A^{\otimes n}$ that are built from the operations and, in a graphical representation, represented by a {\it connected} diagram. Here the algebra can be noncommutative and the Frobenius form can be asymmetric. We view this theorem as reducing any connected diagram to a standard form with $j$ beads $B$, where $j$ is the number of bounded connected components of the original diagram. We study the associated F-dimension Hilbert series $\dim_x=\sum_{j=0}^\infty x^j\dim_j$, where $\dim_j=\epsilon\circ B^j\circ 1$ are invariants of the Frobenius structure. We also study moduli of asymmetric quasispecial and `weakly symmetric' Frobenius structures and their F-dimensions. Examples include general Frobenius structures on matrix algebras $A=M_d(k)$ and on group algebras $k G$ as well as on $u_q(sl_2)$ at low roots of unity., Comment: 20 pages ams latex, several pdf graphics; significantly extended to include u_q(sl_2) and higher F-dimensions

We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry, where $G$ is a finite group. We provide projection operators for its quasiparticles content as irreducible representations of $D(G)$ and combine this with $D(G)$-bimodule properties of open ribbon excitation spaces $L(s_0,s_1)$ to show how open ribbons can be used to teleport information between their endpoints $s_0,s_1$. We give a self-contained account that builds on earlier work but emphasises applications to quantum computing as surface code theory, including gates on $D(S_3)$. We show how the theory reduces to a simpler theory for toric codes in the case of $D( \Bbb Z_n)\cong \Bbb C\Bbb Z_n^2$, including toric ribbon operators and their braiding. In the other direction, we show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$, including site actions of $D(H)$ and partial results on ribbon equivariance even when the Hopf algebra is not semisimple., Comment: 54 pages, many figures both pdf and tkz

We construct a Connes spectral triple or `Dirac operator' on the non-reduced fuzzy sphere $C_\lambda[S^2]$ as realised using quantum Riemannian geometry with a central quantum metric $g$ of Euclidean signature and its associated quantum Levi-Civita connection. The Dirac operator is characterised uniquely up to unitary equivalence within our quantum Riemannian geometric setting and an assumption that the spinor bundle is trivial and rank 2 with a central basis. The spectral triple has KO dimension 3 and in the case of the round metric, essentially recovers a previous proposal motivated by rotational symmetry., Comment: 16 pages AMSLATEX no figures; minor revision, added sample computation of a Connes spectral distance in the concluding remarks

We revisit the notion of interacting Frobenius Hopf algebras for ZX-calculus in quantum computing, with focus on allowing the algebras to be noncommutative and coalgebras to be noncocommutative. We introduce the notion of *-structures in ZX-calculus at this algebraic level and construct examples based on the quantum group u_q(sl_2) at a root of unity. We provide an abstract formulation of the Hadamard gate at this level and clarify its relationship to Hopf algebra self-duality. We then solve the problem of extending the notion of interacting Hopf algebras and ZX-calculus to take place in a braided tensor category. In the ribbon case, the Hadamard gate coming from braided self-duality obeys a modular identity. We give the example of b_q(sl_2), the self-dual braided version of u_q(sl_2)., Comment: 29 pages AMSlatex, 7 figures; correction to prop 2.7 and minor improvements

We study the quantum geometry of the fuzzy sphere defined as the angular momentum algebra $[x_i,x_j]=2\imath\lambda_p \epsilon_{ijk}x_k$ modulo setting $\sum_i x_i^2$ to a constant, using a recently introduced 3D rotationally invariant differential structure. Metrics are given by symmetric $3 \times 3$ matrices $g$ and we show that for each metric there is a unique quantum Levi-Civita connection with constant coefficients, with scalar curvature $ \frac{1}{2}({\rm Tr}(g^2)-\frac{1}{2}{\rm Tr}(g)^2)/\det(g)$. As an application, we construct Euclidean quantum gravity on the fuzzy unit sphere. We also calculate the charge 1 monopole for the 3D differential structure., Comment: 15 pages latex, 1 figure

Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these `lattice spacing' weights do not have to be independent of the direction of the arrow. We use this greater freedom to give a quantum geometric interpretation of discrete Markov processes with transition probabilities as arrow weights, namely taking the diffusion form $\partial_+ f=(-\Delta_\theta+ q-p)f$ for the graph Laplacian $\Delta_\theta$, potential functions $q,p$ built from the probabilities, and finite difference $\partial_+$ in the time direction. Motivated by this new point of view, we introduce a `discrete Schroedinger process' as $\partial_+\psi=\imath(-\Delta+V)\psi$ for the Laplacian associated to a bimodule connection such that the discrete evolution is unitary. We solve this explicitly for the 2-state graph, finding a 1-parameter family of such connections and an induced `generalised Markov process' for $f=|\psi|^2$ in which there is an additional source current built from $\psi$. We also discuss our recent work on the quantum geometry of logic in `digital' form over the field $\Bbb F_2=\{0,1\}$, including de Morgan duality and its possible generalisations., Comment: 23 pages, 5 figures

We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus $\Omega^1$ depending on the Hamiltonian $p^2/2m + V(x)$, and a flat quantum connection $\nabla$ with torsion such that a previous quantum-geometric formulation of flow along autoparallel curves (or `geodesics') is exactly Schr\"odinger's equation. The connection $\nabla$ preserves a generalised `skew metric' given by the canonical symplectic structure lifted to a certain rank (0,2) tensor on the extended phase space where we adjoin a time variable. We also apply the same approach to the Klein Gordon equation on Minkowski spacetime with a background electromagnetic field, formulating quantum `geodesics' on the relativistic Heisenberg algebra with proper time for the external geodesic parameter. Examples include a relativistic free particle wave packet and a hydrogen-like atom., Comment: 43 pages latex, 1 pdf figure. This was revised to make it much more readable, moving the Poisson geometry to the end. The harmonic oscillator example is now more substantial and we added a discussion of `Heisenberg flow' to complement the main Schroedinger wave function flow

We take a fresh look at the geometrization of logic using the recently developed tools of `quantum Riemannian geometry' applied in the digital case over the field $\Bbb F_2=\{0,1\}$, extending de Morgan duality to this context of differential forms and connections. The 1-forms correspond to graphs and the exterior derivative of a subset amounts to the arrows that cross between the set and its complement. The line graph $0-1-2$ has a non-flat but Ricci flat quantum Riemannian geometry. The previously known four quantum geometries on the triangle graph, of which one is curved, are revisited in terms of left-invariant differentials, as are the quantum geometries on the dual Hopf algebra, the group algebra of $\Bbb Z_3$. For the square, we find a moduli of four quantum Riemannian geometries, all flat, while for an $n$-gon with $n>4$ we find a unique one, again flat. We also propose an extension of de Morgan duality to general algebras and differentials over $\Bbb F_2$., Comment: 37 pages ams latex, 3 pdf figures; final version; considerably expanded to include more precise statements and algebraic proofs on the moduli of quantum geometries on small graphs

We construct canonical strongly bicovariant differential graded algebra structures on all four flavours of cross product Hopf algebras, namely double cross products $A\hookrightarrow A\bowtie H\hookleftarrow H$, double cross coproducts $A\twoheadleftarrow A {\blacktriangleright\!\!\blacktriangleleft} H\twoheadrightarrow H$, biproducts $A{\buildrel\hookrightarrow\over \twoheadleftarrow}A{\cdot\kern-.33em\triangleright\!\!\!<} B$ and bicrossproducts $A\hookrightarrow A{\blacktriangleright\!\!\triangleleft} H\twoheadrightarrow H$ on the assumption that the factors have strongly bicovariant calculi $\Omega(A),\Omega(H)$ (or a braided version $\Omega(B)$). We use super versions of each of the constructions. Moreover, the latter three quantum groups all coact canonically on one of their factors and we show that this coaction is differentiable. In the case of the Drinfeld double $D(A,H)=A^{\rm op}\bowtie H$ (where $A$ is dually paired to $H$), we show that its canonical actions on $A,H$ are differentiable. Examples include are a canonical $\Omega(\Bbb C_q[GL_2\ltimes \Bbb C^2])$ for the quantum group of affine transformations of the quantum plane and $\Omega(\Bbb C_\lambda[{\rm Poinc_{1,1}}])$ for the bicrossproduct Poincar\'e quantum group in 2 dimensions. We also show that $\Omega(\Bbb C_q[GL_2])$ itself is uniquely determined by differentiability of the canonical coaction on the quantum plane and of the determinant subalgebra., Comment: 36 pages latex, no figures

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space $X$ is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the $q$-Hopf fibration on the standard $q$-sphere. We also construct the Poisson level of the spin connection on a principal bundle.

We construct noncommutative or `quantum' Riemannian geometry on the integers $\Bbb Z$ as a lattice line $\cdots\bullet_{i-1}-\bullet_i-\bullet_{i+1}\cdots$ with its natural 2-dimensional differential structure and metric given by arbitrary non-zero edge square-lengths $\bullet_i{\buildrel a_i\over -}\bullet_{i+1}$. We find for general metrics a unique $*$-preserving quantum Levi-Civita connection, which is flat if and only if $a_i$ are a geometric progression where the ratios $\rho_i=a_{i+1}/a_i$ are constant. More generally, we compute the Ricci tensor for the natural antisymmetric lift of the volume 2-form and find that the quantum Einstein-Hilbert action up to a total divergence is $-{1\over 2}\sum \rho\Delta \rho$ where $(\Delta\rho)_i=\rho_{i+1}+\rho_{i-1}-2\rho_i$ is the standard discrete Laplacian. We take a first look at some issues for quantum gravity on the lattice line. We also examine $1+0$ dimensional scalar quantum theory with mass $m$ and the lattice line as discrete time. As an application, we compute discrete time cosmological particle creation for a step function jump in the metric by a factor $\rho$, finding that an initial vacuum state has at later times an occupancy $=(1-\sqrt{\rho})^2/(4\sqrt{\rho})$ in the continuum limit, independently of the frequency. The continuum limit of the model is the time-dependent harmonic oscillator, now viewed geometrically., Comment: Change in title as per the accepted version, improvements to the referencing; 22 pages latex, no figures

We consider functional-integral quantisation of the moduli of all quantum metrics defined as square-lengths $a$ on the edges of a Lorentzian square graph. We determine correlation functions and find a fixed relative uncertainty $\Delta a/= 1/\sqrt{8}$ for the edge square-lengths relative to their expected value $$. The expected value of the geometry is a rectangle where parallel edges have the same square-length. We compare with the simpler theory of a quantum scalar field on such a rectangular background. We also look at quantum metric fluctuations relative to a rectangular background, a theory which is finite and interpolates between the scalar theory and the fully fluctuating theory., Comment: 17 pages latex, 2 pdf figures

We study bimodule quantum Riemannian geometries over the field $\Bbb F_2$ of two elements as the extreme case of a finite-field adaptation of noncommutative-geometric methods for physics. We classify all parallelisable such geometries for coordinate algebras up to vector space dimension $n\le 3$, finding a rich moduli of examples for $n=3$ and top form degree 2, including many that are not flat. Their coordinate algebras are commutative but their differentials are not. We also study the quantum Laplacian $\Delta=(\ ,\ )\nabla{\rm d}$ on our models and characterise when it has a massive eigenvector., Comment: 37 pages latex, one figure. Corrected some of the curvature formulae in line with final version in press

We consider Hilbert's problem of the axioms of Physics at a qualitative or conceptual level. This issue is more pressing than ever as we seek to understand how both General Relativity and quantum theory could emerge from some deeper theory of quantum gravity, and in this regard I have previously proposed a `principle of self-duality' or `quantum Born reciprocity' as a key structure. Here I outline some of my recent work around the idea of quantum spacetime as motivated by this non-standard philosophy, including a new toy model of gravity on a spacetime consisting of four points forming a square., Comment: 19 pages, 5 jpg figures; this revised version to appear in a conference special issue on Hilbert's 6th problem -- based on my conference talk to a mixed audience (physics, maths and philosophy of physics)

Noncommutative or quantum Riemannian geometry has been proposed as an effective theory for aspects of quantum gravity. Here the metric is an invertible bimodule map $\Omega^1\otimes_A\Omega^1\to A$ where $A$ is a possibly noncommutative or `quantum' spacetime coordinate algebra and $(\Omega^1,d)$ is a specified bimodule of 1-forms or `differential calculus' over it. In this paper we explore the proposal of a `quantum Koszul formula' with initial data a degree -2 bilinear map $\perp$ on the full exterior algebra $\Omega$ obeying the 4-term relations \[ (-1)^{|\eta|} (\omega\eta)\perp\zeta+(\omega\perp\eta)\zeta=\omega\perp(\eta\zeta)+(-1)^{|\omega|+|\eta|}\omega(\eta\perp\zeta),\quad\forall\omega,\eta,\zeta\in\Omega\] and a compatible degree -1 `codifferential' map $\delta$. These provide a quantum metric and interior product and a canonical bimodule connection $\nabla$ on all degrees. The theory is also more general than classically in that we do not assume symmetry of the metric nor that $\delta$ is obtained from the metric. We solve and interpret the $(\delta,\perp)$ data on the bicrossproduct model quantum spacetime $[r,t]=\lambda r$ for its two standard choices of $\Omega$. For the $\alpha$-family calculus the construction includes the quantum Levi-Civita connection for a general quantum symmetric metric, while for the more standard $\beta=1$ calculus we find the quantum Levi-Civita connection for a quantum `metric' that in the classical limit is antisymmetric., Comment: 39 pages LaTex

We define and study dense Frechet subalgebras of compact quantum groups consisting of elements rapidly decreasing with respect to an unbounded sequence of real numbers. Further, this sequence can be viewed as the eigenvalues of a Dirac-like operator and we characterize the boundedness of its commutators in terms of the eigenvalues. Grotendieck's theory of topological tensor products immediately yields a Schwartz kernel theorem for linear operators on compact quantum groups and allows us to introduce a natural class of pseudo-differential operators on compact quantum groups. As a by-product, we develop elements of the distribution theory and corresponding Fourier analysis. We give applications of our construction to obtain sufficient conditions for $L^p-L^q$ boundedness of coinvariant linear operators. We provide necessary and sufficient conditions for algebraic differential calculi on Hopf subalgebras of compact quantum groups to extend to the proposed smooth structure. We check explicitly that these conditions hold true on the quantum $SU(2)$ for both its 3-dimensional and 4-dimensional calculi., Comment: 33 pages, some corrections, bibliography updated