Classical Field Theory Via Cohesive Homotopy Types

Classical field theory via Cohesive homotopy types Urs Schreiber August 2, 2015 Abstract We demonstrate how a refined formulation of classical mechanics and classical field theory that takes global effects properly into account (e.g. classical anomalies and the global descent to reduced phase spaces) is naturally given in \higher differential geometry". This is the context where smooth manifolds are refined to smooth homotopy types. We introduce and explain this higher differential geometry as we go along. At the same time we explain how the classical concepts of classical mechanics and classical field theory follow naturally from the abstract homotopy theory of correspondences in higher slice toposes. The first part of the text is meant to serve the triple purpose of being an exposition of classical mechanics for homotopy type theorists, being an exposition of geometric homotopy theory for physicists, and finally to serve as the canonical example for the formulation of a local pre-quantum field theory which supports a localized quantization process to local quantum field theory in the sense of the cobordism hypothesis. On the way we also clarify some aspects of multisymplectic field theory by observing that the Hamilton-de Donder-Weyl field equation characterizes Maurer-Cartan elements in the L1-algebra of local observables. The second part of the text discusses the relevant statements of differential cohomology in cohesive higher toposes in more detail. This text has become section 1.2 of Differential cohomology in a cohesive 1-topos, arXiv:1310.7930. This document with further material is kept online at http://ncatlab.org/schreiber/show/Classical+field+theory+via+Cohesive+homotopy+types 1 Contents 1 Introduction 3 2 Hamilton-Jacobi-Lagrange mechanics via prequantized Lagrangian correspondences 5 2.1 Phase spaces and symplectic manifolds . .5 2.2 Coordinate systems and the topos of smooth spaces . .7 2.3 Canonical transformations and Symplectomorphisms . .9 2.4 Trajectories and Lagrangian correspondences . 10 2.5 Observables, symmetries and the Poisson bracket Lie algebra . 14 2.6 Hamiltonian (time evolution) trajectories and Hamiltonian correspondences . 16 2.7 Noether symmetries and equivariant structure . 18 2.8 Gauge theory, smooth groupoids and higher toposes . 20 2.9 The kinetic action, pre-quantization and differential cohomology . 22 2.10 The classical action, the Legendre transform and Hamiltonian flows . 25 2.11 The classical action functional pre-quantizes Lagrangian correspondences . 27 2.12 Quantization, the Heisenberg group, and slice automorphism groups . 28 2.13 Integrable systems, moment maps and homomorphism into the Poisson bracket Lie algebra . 31 2.14 Classical anomalies and projective symplectic reduction . 32 3 De Donder-Weyl field theory via higher correspondences 33 3.1 Local field theory Lagrangians and n-plectic smooth spaces . 33 3.2 The kinetic action, higher prequantization and higher differential cohomology . 37 3.3 Local observables, conserved currents and their higher Poisson brackets . 39 3.4 Field equations of motion, higher Maurer-Cartan elements, and higher Lie integration . 44 3.5 Higher gauge theory, smooth 1-groupoids and homotopy toposes . 48 3.6 Higher Chern-Simons-type boundary field theory and higher Chern-Weil theory . 48 3.7 Source terms, the off-shell Poisson bracket and holography with symplectic groupoids . 48 A Homotopy toposes and Cohesive homotopy types 52 B Differential geometry via Smooth 0-types 56 B.1 Coordinate systems . 57 B.2 Smooth 0-types . 60 B.3 Differential forms . 66 C Higher geometry via Geometric homotopy types 71 C.1 Homotopy toposes of geometric 1-groupoids . 72 C.2 Higher geometric fiber bundles . 79 C.3 Bisections of higher groupoids . 83 C.4 Higher Atiyah groupoids . 83 D Differential cohomology via Cohesive homotopy types 88 D.1 Differential coefficients . 90 D.2 Differential moduli . 95 D.3 Higher quantomorphism- and Heisenberg-groupoids . 98 D.4 Higher Courant groupoids . 101 2 1 Introduction The theory of classical mechanics (e.g. [1]) is the mathematical theory of how macroscopic physical objects of the observable world (for instance dust, rocks and planets) move through space and time while interacting via forces between them. Similarly the theory of classical field theory (e.g. [21]) is the theory of how macroscopic field excitations (for instance of the electric field, the magnetic field, and the field of gravity) relate to space and time while interacting with each other and with physical objects. As the name clearly suggests, classical mechanics and classical field theory are classical topics in mathematical physics, about whose foundations, it would seem, there is hardly much more to be said. However, the observable universe is fundamentally not described by classical field theory (when including microscopic objects and microscopic field excitations) but by quantum field theory (QFT), for which classical (or maybe better: pre-quantum) field theory is but the generating input datum. But QFT has lately seen considerable refinements in the formalization of its foundations (see [50] for a recent survey), notably in the mathematical formalization of the physical principle of (causal) locality. Namely the cobordism hypthesis [33] (see [5] for a review) asserts that the locality property { of at least those quantum field theories which are either topological or else expressible as (\holographic") boundary theories of topological theories { is encoded in a universal construction in \directed homotopy theory", called (1; n)-category theory (see for instance [34, 5]). But this insight does reflect back on the formulation of the foundations of classical/pre-quantum field theory, because, as traditionally formulated, this does not support a quantization that would yield a genuinely local QFT in the sense of the cobordism hypothesis. Therefore some refined formulation of, or at least a new perspective on, classical field theory still seems to be needed, after all. This is what we are concerned with here. In the physics literature roughly this issue is well-known as the problem of \non-covariance of canonical quantization". There are various proposals for how to refine the traditonal formulation of classical field theory such as to solve this problem, of which maybe the best developed is called \multiymplectic covariant field theory" (see e.g. [49]). This approach starts out with a natural generalization of the basic notion of symplectic 2-forms to higher degree forms, but it used to be unclear about on how the further ingredients of classical mechanics are generalized, such as notably what is to refine the Poisson bracket Lie algebra of observables as one passes to local observables. In [17, FRS13b, 44] it was shown that these problems find a solution { and that multisymplectic classical field theory finds a formulation that makes it compatible with the formulation of local quantum field theory via the cobordism hypothesis { if one understands geometric pre-quantization not in the traditional context of differential geometry (see B below for the basics of that), but in the context of higher differential geometry [52], the combination of differential geometry with homotopy theory (see [45] for the classical axiomatization of homotopy theory by model categories, [32] for the formulation by homotopy toposes and [61] for the modern perspective on homotopy theory as homotopy type theory). This is the context where smooth manifolds are allowed to be generalized first to smooth orbifolds (see for instance [40]) and then further to Lie groupoids (see for instance [37]), then further to smooth groupoids and to smooth moduli stacks and finally to smooth higher stacks (see [20]), hence to smooth homotopy types and more generally to cohesive homotopy types [52]. However, in these articles the relation to traditional descriptions of classical field theory was not discussed much. The present article is meant to expose the formulation of classical field theory via cohesive homotopy types more in detail. We introduce and explain the relevant higher differential geometry as we go along. At the same time we explain how the classical concepts of classical mechanics and classical field theory follow naturally from just the abstract theory of \correspondences in higher slice toposes". Therefore the intended readership is twofold: on the one hand theoretical and mathematical physicists, and on the other hand homotopy-type heorists and homotopy type-theorists and topos theorists. Each group might take the following as an introduction to the topic from the other group's perspective. One purpose of this article is to explain how classical mechanics and classical field theory is naturally a subject in both specialities and to provide a dictionary to translate between the two points of view. To put this statement in perspective, notice that locally and in fact infinitesimally, classical mechanics 3 is to a large extent the theory of variational calculus. This also has a natural and useful formulation in homotopy theory and higher topos theory, namely the formulation that physicists know as BV-formalism (see e.g.[25]) and that mathematicians call derived intersection theory (e.g. [35]). This \derived" aspect of classical mechanics has found considerable attention in the literature in the last years (see for instance [?]). The higher global refinement that we discuss here however is complementary to this, and has not found due attention yet. Both aspects are complementary and naturally combine in \higher derived geometry", but to keep focus we do not particularly dwell much on the infinitesimal aspects here. Notice that by far not all aspects of classical mechanics are local or even infinitesimal in nature. In particular the passage to genuine quantization { as opposed to just formal deformation quantization { crucially requires global structure to be specified. In the context of geometric quantization this extra global structure is traditionally called a pre-quantization, in order to reflect that it is more structure than typically considered in classical mechanics, but still just a pre-requisite to the actual quantization step.

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