Graded Geometry, Tensor Galileons and Duality
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HYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY
Graded Geometry, Tensor Galileons and Duality A. Chatzistavrakidisa, *, G. Karagiannisa, **, and P. Schuppb, *** a
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Division of Theoretical Physics, Rudjer Bošković Institute, Zagreb, 10000 Croatia Department of Physics and Earth Sciences, Jacobs University, Bremen, 28759 Germany *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected] Received November 15, 2019; revised January 15, 2020; accepted February 28, 2020
Abstract—We review some fundamental aspects of mixed symmetry tensor gauge theories using a formulation based on graded geometry. In particular, we are able to construct kinetic, mass and Galileon-type higher derivative interaction terms for such fields. The resulting elegant geometric formulas allow for shared features of these theories to be highlighted and for possible interaction terms to be classified. In addition, we argue that this formalism is very useful in studying dualities. In particular, we construct a universal first order Lagrangian that may serve as the starting point for the off shell dualizations of differential form gauge theories and generalized gravitons. DOI: 10.1134/S1547477120050106
1. INTRODUCTION Geometry has played a pivotal role in the formulation and conceptual understanding of physical theories for more than a century. The gravitational interaction enjoys a geometric description in the context of general relativity, where the dynamics of spacetime is encoded in the metric defined on a pseudo-Riemannian manifold. Gauge theories for spin 1 fields also find a geometric formulation in terms of connections on principal bundles. In this note we review [1, 2] and explain the role that modern methods based on graded geometry can play in the formulation of physical theories and the unified description of certain aspects of such theories, such as higher derivative interactions and dualities. The motivation for the graded geometric approach we employ is threefold. 1.1. Mixed Symmetry Tensor Fields and Higher Derivative Interactions In recent years there has been an interest in theories whose Lagrangian includes interaction terms with higher derivatives. Here we focus on relativistic theories on Minkowski spacetime of any dimension D . Given that one wishes to preserve unitarity and avoid the propagation of ghosts, the question then becomes which are the most general interaction terms that can be added to the Lagrangian such that the corresponding field equations of the theory do not contain more than two derivatives acting on the field. Already in the 70s, this question was answered for the spin 2 metric by
Lovelock [3], and later extended by Horndeski to account for scalar-tensor theories in four dimensions [4]. More recently, the problem was solved for scalar fields [5, 6], differential p -forms [7] and mixed symmetry tensor fields with two sets of antisymmetrized indices, which generalize the graviton case [1]. The role of graded geometry is to offer an el
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