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We revisit and extend the Durdevic theory of complete calculi on quantum principal bundles. In this setting one naturally obtains a graded Hopf-Galois extension of the higher order calculus and an intrinsic decomposition of degree 1-forms into horizontal and vertical forms. This proposal is appealing, since it is consistently equipped with a canonical braiding and exactness of the Atiyah sequence is guaranteed. Moreover, we provide examples of complete calculi, including the noncommutative 2-torus, the quantum Hopf fibration and differential calculi on crossed product algebras., Comment: 45 pages; corrected equation (206); added "Quantum Symmetries" to acknowledgements; added journal reference for [11]

In this paper we develop the theory of reduction of quantum principal bundles over projective bases. We show how the sheaf theoretic approach can be effectively applied to certain relevant examples as the Klein model for the projective spaces; in particular we study in the algebraic setting the reduction of the principal bundle $\mathrm{GL}(n) \to \mathrm{GL}(n)/P= \mathbf{P}^{n-1}(\mathbb{C})$ to the Levi subgroup $G_0$ inside the maximal parabolic subgroup $P$ of $\mathrm{GL}(n)$. We characterize reductions in the sheaf theoretic setting., Comment: 25 pages

On conformally compact manifolds we study Yang-Mills equations, their boundary conditions, formal asymptotics, and Dirichlet-to-Neumann maps. We find that smooth solutions with "magnetic" Dirichlet boundary data are obstructed by a conformally invariant, higher order boundary current. We study the asymptotics of the interior Yang-Mills energy functional and show that the obstructing current is the variation of the conformally invariant coefficient of the first log term in this expansion which is a higher derivative conformally invariant analog of the Yang-Mills energy. The invariant energy is the anomaly for the renormalized interior Yang-Mills functional and its variation gives higher conformal Yang-Mills equations. Global solutions to the magnetic boundary problem determine higher order "electric" Neumann data. This yields the Dirichlet-to-Neumann map. We also construct conformally invariant, higher transverse derivative boundary operators. Acting on interior connections, they give obstructions to solving the Yang-Mills boundary problem, determine the asymptotics of formal solutions, and yield conformally invariant tensors capturing the (non-local) electric Neumann data. We also characterize a renormalized Yang-Mills action functional that encodes global features analogously to the renormalized volume for Poincar\'e-Einstein structures., Comment: 71 pages LaTeX, improved presentation

The problem of computing quantum mechanical propagators can be recast as a computation of a Wilson line operator for parallel transport by a flat connection acting on a vector bundle of wavefunctions. In this picture the base manifold is an odd dimensional symplectic geometry, or quite generically a contact manifold that can be viewed as a "phase-spacetime", while the fibers are Hilbert spaces. This approach enjoys a "quantum Darboux theorem" that parallels the Darboux theorem on contact manifolds which turns local classical dynamics into straight lines. We detail how the quantum Darboux theorem works for anharmonic quantum potentials. In particular, we develop a novel diagrammatic approach for computing the asymptotics of a gauge transformation that locally makes complicated quantum dynamics trivial., Comment: 27 pages, many figures, LaTeX, references added

We construct a spinning particle that reproduces the propagation of the graviton on those curved backgrounds which solve the Einstein equations, with or without cosmological constant, i.e. Einstein manifolds. It is obtained by modifying the N=4 supersymmetric spinning particle by relaxing the gauging of the full SO(4) R-symmetry group to a parabolic subgroup, and selecting suitable Chern-Simons couplings on the worldline. We test it by computing the correct one-loop divergencies of quantum gravity in D=4., Comment: 11 pages, typos corrected in formulas (3.7), (3,9), (3.10) and (3.13) present in the published version, references updated

The purpose of this paper is to propose a sheaf theoretic approach to the theory of quantum principal bundles over non affine bases. We study the quantization of principal bundles G -> G/P, where G is a semisimple group and P a parabolic subgroup., Comment: Version 2. Thoroughly revised and extended version, 13 pages longer. Completed proof of main examples in Section 5. New Section 6 with examples from twists (2-cocycles). Revised introduction

We give an algebraic quantization, in the sense of quantum groups, of the complex Minkowski space, and we examine the real forms corresponding to the signatures $(3,1)$, $(2,2)$, $(4,0)$, constructing the corresponding quantum metrics and providing an explicit presentation of the quantized coordinate algebras. In particular, we focus on the Kleinian signature $(2,2)$. The quantizations of the complex and real spaces come together with a coaction of the quantizations of the respective symmetry groups. We also extend such quantizations to the $\mathcal{N}=1$ supersetting.

We discuss $\mathcal{N}=1$ Klein and Klein-Conformal superspaces in $D=(2,2)$ space-time dimensions, realizing them in terms of their functor of points over the split composition algebra $\mathbb{C}_{s}$. We exploit the observation that certain split form of orthogonal groups can be realized in terms of matrix groups over split composition algebras; this leads to a natural interpretation of the the sections of the spinor bundle in the critical split dimensions $D=4$, $6$ and $10$ as $\mathbb{C}_{s}^{2}$, $\mathbb{H}_{s}^{2}$ and $\mathbb{O}_{s}^{2}$, respectively. Within this approach, we also analyze the non-trivial spinor orbit stratification that is relevant in our construction since it affects the Klein-Conformal superspace structure., Comment: 1+31 pages; v2: one Ref. added, and other minor changes. To be published in Reviews in Mathematical Physics (2017)

Supermanifolds provide a very natural ground to understand and handle supersymmetry from a geometric point of view; supersymmetry in $d=3,4,6$ and $10$ dimensions is also deeply related to the normed division algebras. In this paper we want to show the link between the conformal group and certain types of symplectic transformations over division algebras. Inspired by this observation we then propose a new\,realization of the real form of the 4 dimensional conformal and Minkowski superspaces we obtain, respectively, as a Lagrangian supermanifold over the twistor superspace $\mathbb{C}^{4|1}$ and a big cell inside it. The beauty of this approach is that it naturally generalizes to the 6 dimensional case (and possibly also to the 10 dimensional one) thus providing an elegant and uniform characterization of the conformal superspaces., Comment: 15 pages, references added, minor changes

Relativistic particle actions are a useful tool to describe quantum field theory effective actions using a string-inspired first-quantized approach. Here we describe how to employ suitable particle actions in the computation of the scalar contribution to the one-loop gluon effective action. We use the well-known method of introducing auxiliary variables that create the color degrees of freedom. In a path integral they implement automatically the path ordering needed to ensure gauge invariance. It is known that the color degrees of freedom introduced this way form a reducible representation of the gauge group. We describe a method of projecting onto the fundamental representation (or any other chosen irrep, if desired) of the gauge group. Previously, we have discussed the case of anticommuting auxiliary variables. Choosing them to be in the fundamental representation allows to obtain, without any extra effort, also the situation in which the color is given by any antisymmetric tensor product of the fundamental. Here, we describe the novel case of bosonic auxiliary variables. They can be used equivalently for creating the color charges in the fundamental representation. In addition one gets, as a byproduct, the cases where the particle can have the color sitting in any symmetric tensor product of the fundamental. This is obtained by tuning to a different value a Chern Simons coupling, present in the model, which controls how the projection is achieved., Comment: 11 pages, to appear in the Proceedings of the Victor Villanueva Memorial Workshop

Relativistic particles with higher spin can be described in first quantization using actions with local supersymmetry on the worldline. First, we present a brief review of these actions and their use in first quantization. In a Dirac quantization scheme the field equations emerge as Dirac constraints on the Hilbert space, and we outline how they lead to the description of higher spin fields in terms of the more standard Fronsdal-Labastida equations. Then, we describe how these actions can be extended so that the propagating particle is allowed to take different values of the spin, i.e. carry a reducible representation of the Poincar\'e group. This way one may identify a four dimensional model that carries the same degrees of freedom of the minimal Vasiliev's interacting higher spin field theory. Extensions to massive particles and to propagation on (A)dS spaces are also briefly commented upon., Comment: 10 pages, updated references. To appear in Proceedings of Victor Villanueva Memorial Workshop

We study actions for massive bosonic particles of higher spins by dimensionally reducing an action for massless particles. For the latter we take a model with a SO(N) extended local supersymmetry on the worldline, that is known to describe massless (conformal) particles of higher spins in flat spacetimes of even dimensions. Dimensional reduction produces an action for massive spinning particles in odd dimensions. The field equations that emerge in a quantization a la Dirac are shown to be equivalent to the Fierz-Pauli ones. The massless limit generates a multiplet of massless states with higher spins, whose first quantized field equations have a geometric form with fields belonging to various types of Young tableaux. These geometric equations can be partially integrated to show their equivalence with the standard Fronsdal-Labastida equations. We covariantize our model to check whether an extension to curved spacetimes can be achieved. Restricting to (A)dS spaces, we find that the worldline gauge algebra becomes nonlinear, but remains first class. This guarantees consistency on such backgrounds. A light cone analysis confirms the presence of the expected propagating degrees of freedom. A covariant analysis is worked out explicitly for the massive case, which is seen to give rise to the Fierz-Pauli equations extended to (A)dS spaces. It is worth noting that in D=3 the massless limit of our model when N goes to infinity has the same field content of the Vasiliev's theory that accommodates each spin exactly once., Comment: 31 pages

Efficient methods for describing non abelian charges in worldline approaches to QFT are useful to simplify calculations and address structural properties, as for example color/kinematics relations. Here we analyze in detail a method for treating arbitrary non abelian charges. We use Grassmann variables to take into account color degrees of freedom, which however are known to produce reducible representations of the color group. Then we couple them to a U(1) gauge field defined on the worldline, together with a Chern-Simons term, to achieve projection on an irreducible representation. Upon gauge fixing there remains a modulus, an angle parametrizing the U(1) Wilson loop, whose dependence is taken into account exactly in the propagator of the Grassmann variables. We test the method in simple examples, the scalar and spin 1/2 contribution to the gluon self energy, and suggest that it might simplify the analysis of more involved amplitudes., Comment: 14 pages

We study the one loop effective action for a class of higher spin fields by using a first-quantized description. The latter is obtained by considering spinning particles, characterized by an extended local supersymmetry on the worldline, that can propagate consistently on conformally flat spaces. The gauge fixing procedure for calculating the worldline path integral on a loop is delicate, as the gauge algebra contains nontrivial structure functions. Restricting the analysis on (A)dS backgrounds simplifies the gauge fixing procedure, and allows us to produce a useful representation of the one loop effective action. In particular, we extract the first few heat kernel coefficients for arbitrary even spacetime dimension D and for spin S identified by a curvature tensor with the symmetries of a rectangular Young tableau of D/2 rows and [S] columns., Comment: 30 pages, typos corrected

We study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. We also develop a product formula for solving these asymptotic problems in general. The central tools of our approach are (i) the conformal geometry of differential forms and the associated exterior tractor calculus, and (ii) a generalised notion of scale which encodes the connection between the underlying geometry and its boundary. The latter also controls the breaking of conformal invariance in a very strict way by coupling conformally invariant equations to the scale tractor associated with the generalised scale. From this, we obtain a map from existing solutions to new ones that exchanges Dirichlet and Neumann boundary conditions. Together, the scale tractor and exterior structure extend the solution generating algebra of [31] to a conformally invariant, Poincare--Einstein calculus on (tractor) differential forms. This calculus leads to explicit holographic formulae for all the higher order conformal operators on weighted differential forms, differential complexes, and Q-operators of [9]. This complements the results of Aubry and Guillarmou [3] where associated conformal harmonic spaces parametrise smooth solutions., Comment: 85 pages, LaTeX, typos corrected, references added, to appear in Memoirs of the AMS

Quantum mechanical models with extended supersymmetry find interesting applications in worldline approaches to relativistic field theories. In this paper we consider one-dimensional nonlinear sigma models with O(N) extended supersymmetry on the worldline, which are used in the study of higher spin fields on curved backgrounds. We calculate the transition amplitude for euclidean times (i.e. the heat kernel) in a perturbative expansion, using both canonical methods and path integrals. The latter are constructed using three different regularization schemes, and the corresponding counterterms that ensure scheme independence are explicitly identified., Comment: 1+26 pages

Fefferman and Graham showed some time ago that four dimensional conformal geometries could be analyzed in terms of six dimensional, ambient, Riemannian geometries admitting a closed homothety. Recently it was shown how conformal geometry provides a description of physics manifestly invariant under local choices of unit systems. Strikingly, Einstein's equations are then equivalent to the existence of a parallel scale tractor (a six component vector subject to a certain first order covariant constancy condition at every point in four dimensional spacetime). These results suggest a six dimensional description of four dimensional physics, a viewpoint promulgated by the two times physics program of Bars. The Fefferman--Graham construction relies on a triplet of operators corresponding, respectively to a curved six dimensional light cone, the dilation generator and the Laplacian. These form an sp(2) algebra which Bars employs as a first class algebra of constraints in a six-dimensional gauge theory. In this article four dimensional gravity is recast in terms of six dimensional quantum mechanics by melding the two times and tractor approaches. This "parent" formulation of gravity is built from an infinite set of six dimensional fields. Successively integrating out these fields yields various novel descriptions of gravity including a new four dimensional one built from a scalar doublet, a tractor vector multiplet and a conformal class of metrics., Comment: 27 pages, LaTeX

We study a class of supersymmetric spinning particle models derived from the radial quantization of stationary, spherically symmetric black holes of four dimensional N= 2 supergravities. By virtue of the c-map, these spinning particles move in quaternionic Kaehler manifolds. Their spinning degrees of freedom describe mini-superspace-reduced supergravity fermions. We quantize these models using BRST detour complex technology. The construction of a nilpotent BRST charge is achieved by using local (worldline) supersymmetry ghosts to generating special holonomy transformations. (An interesting byproduct of the construction is a novel Dirac operator on the superghost extended Hilbert space.) The resulting quantized models are gauge invariant field theories with fields equaling sections of special quaternionic vector bundles. They underly and generalize the quaternionic version of Dolbeault cohomology discovered by Baston. In fact, Baston's complex is related to the BPS sector of the models we write down. Our results rely on a calculus of operators on quaternionic Kaehler manifolds that follows from BRST machinery, and although directly motivated by black hole physics, can be broadly applied to any model relying on quaternionic geometry., Comment: 46 pages, LaTeX

Spinning particle models can be used to describe higher spin fields in first quantization. In this paper we discuss how spinning particles with gauged O(N) supersymmetries on the worldline can be consistently coupled to conformally flat spacetimes, both at the classical and at the quantum level. In particular, we consider canonical quantization on flat and on (A)dS backgrounds, and discuss in detail how the constraints due to the worldline gauge symmetries produce geometrical equations for higher spin fields, i.e. equations written in terms of generalized curvatures. On flat space the algebra of constraints is linear, and one can integrate part of the constraints by introducing gauge potentials. This way the equivalence of the geometrical formulation with the standard formulation in terms of gauge potentials is made manifest. On (A)dS backgrounds the algebra of constraints becomes quadratic, nevertheless one can use it to extend much of the previous analysis to this case. In particular, we derive general formulas for expressing the curvatures in terms of gauge potentials and discuss explicitly the cases of spin 2, 3 and 4., Comment: 35 pages, added references

Higher spin fields in four dimensions, and more generally conformal fields in arbitrary dimensions, can be described by spinning particle models with a gauged SO(N) extended supergravity on the worldline. We consider here the one-loop quantization of these models by studying the corresponding partition function on the one-dimensional torus. After gauge fixing the supergravity multiplet, the partition function reduces to an integral over the corresponding moduli space which is computed using orthogonal polynomial techniques. We obtain a compact formula which gives the number of physical degrees of freedom for all N in all dimensions. As an aside we compute the physical degrees of freedom of the SO(4) = SU(2)xSU(2) model with only a SU(2) factor gauged, which has attracted some interest in the literature., Comment: 21 pages