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We draw a parallel between the BV/BRST formalism for higher-dimensional ($\ge 2$) Hamiltonian mechanics and higher notions of torsion and basic curvature tensors for generalized connections in specific Lie $n$-algebroids based on homotopy Poisson structures. The gauge systems we consider include Poisson sigma models in any dimension and ``generalised R-flux'' deformations thereof, such as models with an $(n+2)$-form-twisted R-Poisson target space. Their BV/BRST action includes interaction terms among the fields, ghosts and antifields whose coefficients acquire a geometric meaning by considering twisted Koszul multibrackets that endow the target space with a structure that we call a gapped almost Lie $n$-algebroid. Studying covariant derivatives along $n$-forms, we define suitable polytorsion and basic polycurvature tensors and identify them with the interaction coefficients in the gauge theory, thus relating models for topological $n$-branes to differential geometry on Lie $n$-algebroids., Comment: 37 pages

We study twisted Courant sigma models, a class of topological field theories arising from the coupling of 3D 0-/2-form BF theory and Chern-Simons theory and containing a 4-form Wess-Zumino term. They are examples of theories featuring a nonlinearly open gauge algebra, where products of field equations appear in the commutator of gauge transformations, and they are reducible gauge systems. We determine the solution to the master equation using a technique, the BRST power finesse, that combines aspects of the AKSZ construction (which applies to the untwisted model) and the general BV-BRST formalism. This allows for a geometric interpretation of the BV coefficients in the interaction terms of the master action in terms of an induced generalised connection on a 4-form twisted (pre-)Courant algebroid, its Gualtieri torsion and the basic curvature tensor. It also produces a frame independent formulation of the model. We show, moreover, that the gauge fixed action is the sum of the classical one and a BRST commutator, as expected from a Schwarz type topological field theory., Comment: 50 pages

We study connections on higher structures such as Lie and Courant algebroids and their description as differential graded manifolds and explore the role of their basic curvature tensor and of the Atiyah cocycle in topological sigma models and higher gauge theories. The basic curvature of a connection on a Lie algebroid is a measure for the compatibility of the connection with the Lie bracket and it appears in the BV operator of topological sigma models in 2D. Here we define a basic curvature tensor for connections on Courant algebroids and we show that in the description of a Courant algebroid as a QP manifold it appears naturally as part of the homological vector field together with the Gualtieri torsion of a generalised connection. The Atiyah cocycle of a connection on a differential graded manifold is a measure of the compatibility of the connection with the homological vector field. We argue that in the graded-geometric description of higher gauge theories, the structure of gauge transformations is governed by a Kapranov L$_{\infty}[1]$ algebra, whose binary bracket is given by the Atiyah cocycle. We also revisit some aspects of derived structures and we uncover the role of the Atiyah cocycle in deriving $E$-tensors for $E$-connections on Lie and Courant algebroids from ordinary tensors on differential graded manifolds., Comment: 66 pages; added explanations and streamlined notation in sections 4.3 and 4.4, corrected equations (2.51), (4.86), (B.2) and (B.3), other minor improvements

Generalisations of geometry have emerged in various forms in the study of field theory and quantization. This mini-review focuses on the role of higher geometry in three selected physical applications. After motivating and describing some basic aspects of algebroid structures on bundles and (differential graded) Q-manifolds, we briefly discuss their relation to ($\alpha$) the Batalin-Vilkovisky quantization of topological sigma models, ($\beta$) higher gauge theories and generalized global symmetries and ($\gamma$) tensor gauge theories, where the universality of their form and properties in terms of graded geometry is highlighted., Comment: 15 pages; invited Mini-Review for the EPJ ST Special Issue: Noncommutativity and Physics

The AKSZ construction was developed as a geometrical formalism to find the solution to the classical master equation in the BV quantization of topological branes based on the concept of QP manifolds. However, the formalism does not apply in presence of Wess-Zumino terms, as demonstrated recently by Ikeda and Strobl in the simplest example of WZW-Poisson sigma models. In this contribution, we review a class of topological field theories in arbitrary dimensions, the twisted R-Poisson sigma models, which suitably generalize Poisson or twisted Poisson sigma models. Their relation to differential graded manifolds and higher geometry is discussed and we sketch how to identify the solution to the classical master equation even though the target space does not have a QP structure., Comment: 20 pages; contribution to the proceedings of the workshop on Quantum Geometry, Field Theory and Gravity at EISA, Corfu 2021

We present the construction of the classical Batalin-Vilkovisky action for topological Dirac sigma models. The latter are two-dimensional topological field theories that simultaneously generalise the completely gauged Wess-Zumino-Novikov-Witten model and the Poisson sigma model. Their underlying structure is that of Dirac manifolds associated to maximal isotropic and integrable subbundles of an exact Courant algebroid twisted by a 3-form. In contrast to the Poisson sigma model, the AKSZ construction is not applicable for the general Dirac sigma model. We therefore follow a direct approach for determining a suitable BV extension of the classical action functional with ghosts and antifields satisfying the classical master equation. Special attention is paid on target space covariance, which requires the introduction of two connections with torsion on the Dirac structure., Comment: 1+25 pages

We determine the solution to the classical master equation for a 3D topological field theory with Wess-Zumino term and an underlying geometrical structure of a twisted R-Poisson manifold on its target space. The graded geometry of the target space departs from the usual QP structure encountered in the AKSZ construction of topological sigma models, the obstruction being attributed to the presence of the Wess-Zumino 4-form. Due to the inapplicability of the AKSZ construction in this case, we set up the traditional BV/BRST formalism for twisted R-Poisson sigma models in any dimension, which feature an open gauge algebra and constitute multiple stages reducible constrained Hamiltonian systems. An unusual feature of the theories is that it exhibits non-linear openness of the gauge algebra, in other words products of the equations of motion appear in it. Nevertheless, we find the BV action in presence of the 4-form twist in 3D, namely for a specific 4-form twisted (pre-)Courant sigma model. Moreover, we provide a complete set of explicit formulas for the off-shell nilpotent BV operator for untwisted R-Poisson sigma models in any dimension., Comment: 32 pages

We revisit universal features of duality in linear and nonlinear relativistic scalar and Abelian 1-form theories with single or multiple fields, which exhibit ordinary or generalized global symmetries. We show that such global symmetries can be interpreted as generalized Killing isometries on a suitable, possibly graded, target space of fields or its jet space when the theory contains higher derivatives. This is realized via a generalized sigma model perspective motivated from the fact that higher spin particles can be Nambu-Goldstone bosons of spontaneously broken generalized global symmetries. We work out in detail the 2D examples of a compact scalar and the massless Heisenberg pion fireball model and the 4D examples of Maxwell, Born-Infeld and ModMax electrodynamics. In all cases we identify the 't Hooft anomaly that obstructs the simultaneous gauging of both global symmetries and confirm the anomaly matching under duality. These results readily generalize to higher gauge theories for p-forms. For multifield theories we discuss the transformation of couplings under duality as two sets of Buscher rules for even or odd differential forms., Comment: Contribution to the Universe special issue "Dualities and Geometry", v2: extra discussion added

We investigate physical implications of a gravitational analog of axion electrodynamics with a parity-violating gravitoelectromagnetic theta term. This is related to the Nieh-Yan topological invariant in gravity with torsion, in contrast to the well-studied gravitational Chern-Simons term quadratic in curvature, coupled via a dynamical axionlike scalar field. Axion gravitodynamics is the corresponding linearized theory. We find that potentially observable effects are over 80 orders of magnitude stronger than for its Chern-Simons counterpart and could be in reach for detection by experiments in the near future. For a near-Earth scenario, we derive corrections to the Lense-Thirring effect and compare them to data from satellite-based experiments (Gravity Probe B). For gravitational waves, we find modified dispersion relations, derive the corresponding polarization-dependent modified group and phase velocities, and compare them to data from neutron star mergers (GW170817) to derive even stronger bounds., Comment: Minor changes; now matches the published version