## Mathematical -physics

Nomad Institute for Quantum Gravity is dedicated to the exploration of several areas of mathematical-physics, which are closely related to field theories and their quantizations.

• Multisymplectic Geometry
• Higher Lie Structures
• Higher Gauge Theory
• Differential Geometry
• Classical and Quantum Field Theory
• Functional Analysis
• Algebraic Quantum Field Theory
• Covariant Phase Space
• Algebraic Geometry-Topology
• Non Commutative Geometry
• Quantum Gravity

Multisymplectic geometry

• Multisymplectic Geometry is a generalization of symplectic geometry for field theory. In the canonical approach to a standard field theory, the canonical variables are defined on space-like hypersurfaces. Space and time are treated asymetrically, thus we have a non-covariance scheme. The main idea is to construct a Hamiltonian description of classical fields theory compatible with Principles of Special Relativity, General Relativity and String Theories - and more generally any effort towards understanding gravitation. Since space-time should merge from the dynamics, we need a description without any space-time/field splitting a priori and a landscape where there is no space-time structure given a priori. Space-time coordinates should instead merge out from the analysis of what are the observable quantities and from the dynamics. The universal Hamiltonian formalism allows a complete democracy between time, space and internal variables: these are the revendications of Kaluza-Klein or supergravity theory. The distinction between these variables should be a consequence of dynamical equations.
• We propose a bird’eye view on multisymplectic landscape, from the roots - Weyl, Carathéodory, DeDonder, Cartan, Lepage, Dedecker to more modern considerations.

Higher Lie Structures

Higher Gauge Theory

Differential Geometry

Classical and Quantum Field Theory

Functional analysis - Micro local analysis

Note also the related work of Nguyen Viet Dang, in particular:

• Renormalization of quantum field theory on curved space-times, a causal approach arXiv:1312.5674v2
• Extension of distributions, scalings and renormalization of QFT on Riemannian manifolds arXiv:1411.3670v1
• The extension of distributions on manifolds, a microlocal approach arXiv:1412.2808v1

Algebraic Quantum Field Theory

• AQFT is connected to the study of formal functional methods and is to be understood, from a physical standpoint as motivated by the needs of QFT (the path integral approach).
• (AQFT) is connected to the study of formal functional methods and is to be understood, from a physical standpoint as motivated by the needs of (QFT) (the path integral approach). A further question concerns the possibility of formulating a consistent axiomatic Quantum Field Theory within arbitrary curved spacetimes. We mention this approach in order to emphasize a crucial point about causal structure since (AQFT) is based upon two main principles: covariance and locality. Indeed, in (QFT) the causal structure is fixed and we have a preferred notion of causality: the existence of a non-dynamical, Minkowski background metric. In (GR) the situation is drastically di erent. Since no prior geometry is given, what is the meaning of such relation?

Covariant Phase Space

• (CPS) share many features with (MG). The main idea is that we are not working on ordinary phase space but rather on the space of solutions of a Hamiltonian dynamical problem, namely a functional space
• (CPS) share many features with (MG). The main idea is that we are not working on ordinary phase space but rather on the space of solutions of a Hamiltonian dynamical problem, namely a functional space. As noticed by Zuckerman and also though the work of Goldschmidt-Sternberg, Crnkovic-Witten the key observation is to de ne a canonical pre-symplectic structure on such functional space. The same fundamental mathematical entity actually present in both approaches, (MG) and (CPS), clearly manifested in the $n$-phase space notion. We should underline that the invariant language provided crucial insight and merge out in the theory of integral invariants - Poincare, Cartan -. Within this approach to dynamical principles - what we may call ''Cartan principle of dynamics'' - we find a deep connection between (MG) and (CPS). This relation may be seen as a modern continuation of T. De Donder attempt in seeking to extend the notion of integral invariants to variational problems with several variables.

Algebraic Geometry — Algebraic Topology

Non Commutative Geometry

Quantum Gravity

• The project is dedicated to the exploration of some ingredients of Mathematical Physics that are revelent for a possible theory of Quantum Gravity.
• More precisely it concerns the construction of an adequate alphabet which would permit the building of a bridge between the language of differential geometry - and its subdomain Riemaniann geometry - which forms the framework of Einstein's theory of General Relativity, and the algebraic symbolism forming the framework of Quantum Fields Theory. In pursuing this aim, we are led to consider the status, amongst others, of the following fundamental notions: symmetry, observable, space-time, matter, relativity principle... The project is mainly focused on a (amongst others) road to Quantum Gravity that gives particular attention to the following intertwined topics: Dynamical Field Theory, n-plectic Geometry, n-plectic Gravity and Generalized Relativity. This are the main ingredients for drawing this map to the problem of the Quantization of Gravity.