aa r X i v : . [ m a t h - ph ] N ov Conference on Type Theory, Homotopy Theory and Univalent Foundations
Classical field theoryvia Cohesive homotopy types
Urs Schreiber
University Nijmegen ncatlab.org/nlab/show/Urs+Schreiber
In the year 1900, at the International Congress of Mathematics in Paris, David Hilbertstated his famous list of 23 central open questions of mathematics [
Hi1900 ]. Amongthem, the sixth problem (see [
Cor04 ] for a review) is arguably the one that Hilberthimself regarded as the most valuable: “From all the problems in the list, the sixth isthe only one that continually engaged [Hilbert’s] efforts over a very long period, at leastbetween 1894 and 1932.” [
Cor06 ]. Hilbert stated the problem as follows
Hilbert’s mathematical problem 6.
To treat by means of axioms, those physicalsciences in which mathematics plays an important part .Since then, various aspects of physics have been given a mathematical formulation.The following table, necessarily incomplete, gives a broad idea of central concepts intheoretical physics and the mathematics that captures them. physics maths prequantum physics differential geometry quantum physics noncommutative algebra ∞ , n )-category theoryThese are traditional solutions to aspects of Hilbert’s sixth problem. Two points arenoteworthy: on the one hand the items in the list are crown jewels of mathematics; onthe other hand their appearance is somewhat unconnected and remains piecemeal.Towards the end of the 20th century, William Lawvere, the founder of categoricallogic and of categorical algebra, aimed for a more encompassing answer that rests theaxiomatization of physics on a decent unified foundation. He suggested to(1) rest the foundations of mathematics itself in topos theory [ Law65 ];(2) build the foundations of physics synthetically inside topos theory by(a) imposing properties on a topos which ensure that the objects have thestructure of differential geometric spaces [ Law98 ];(b) formalizing classical mechanics on this basis by universal constructions(“Categorical dynamics” [
Law67 ], “Toposes of laws of motion” [
Law97 ]).
Research Perspectives CRM Barcelona, Fall 2013, vol. ??, in Trends in MathematicsCentre de Recerca Matem`atica, Bellaterra (Barcelona) Classical field theory via Cohesive homotopy types
While this is a grandiose plan, we have to note that it falls short in two respects:(1) Modern mathematics prefers to refine its foundations from topos theory to higher topos theory [ L06 ] viz. homotopy type theory [ UFP13 ].(2) Modern physics needs to refine classical mechanics to quantum mechanics and quantum field theory at small length/high energy scales [
Fe85, SaSc11 ].Concerning the first point, notice that indeed, as conjectured in [
Jo11 ] and proven by[
CiSh12 ]: Homotopy type theory is the internal language of locally Cartesian closed ∞ -categories C . Moreover [
Sh12a ]: The univalence axiom encodes the presence of the small objectclassifier in locally cartesian closed ∞ -categories C which are in fact ∞ -toposes H . Therefore our task is to: refine Lawvere’s synthetic approach on Hilbert’s sixth prob-lem from classical physics formalized in synthetic differential geometry axiomatized intopos theory to high energy physics formalized in higher differential geometry axioma-tized in higher topos theory. Specifically, the task is to add to (univalent) homotopytype theory axioms that make the homotopy types have the interpretation of differential geometric homotopy types in a way that admits a formalization of high energy physics.The canonical way to add such modalities on type theories is to add modal operators which in homotopy type theory are homotopy modalities [ Sh12b ]. The ∞ -categoricalsemantics of a homotopy modality is an idemponent ∞ -(co-)monad as in [ L06 ]. Forthese it is clear what an adjoint pair is. We say:
Definition 1 ([ ScSh12 ]) . Cohesive homotopy type theory is univalent homotopy typetheory equipped with an adjoint triple of homotopy (co-)modalities R ⊣ ♭ ⊣ ♯ , tobe called: shape modality ⊣ flat co-modality ⊣ sharp modality , such that there is acanonical equivalence of the ♭ -modal types with the ♯ -modal types, and such that R preserves finite product types.This has been formalized in HoTT-Coq by Mike Shulman, see [ ScSh12 ] for details.With hindsight one finds that this modal type theory is essentially what Lawvere wasenvisioning in [
Law91 ], where it is referred to as encoding “being and becoming”, andlater more formally in [
Law94, Law07 ], where it is referred to as encoding “cohesion”.While def. 1 may look simple, its consequences are rich. In [
Sc13a ] we show howcohesive homotopy type theory synthetically captures not just differential geometry, butthe theory of (generalized) differential cohomology (e.g. [
Bun12 ]). This is the coho-mology theory in which physical gauge fields (such as the field of electromagnetism) arecocycles. We show in [
Sc13a ] that cohesion implies the existence of geometric homotopytypes
Phases such that(1) the dependent homotopy types over
Phases are prequantized covariant phasespaces of physical field theories;(2) correspondences between these dependent types are spaces of trajectories equippedwith local action functionals ;(3) group actions on such dependent types encode the Hamilton-de Donder-Weylequations of motion of local covariant field theory;(4) the “motivic” linearization of these relations over suitable stable homotopy typesyields the corresponding quantum field theories.An exposition of what all this means is in section 1.2 of [
Sc13a ]. See [
Nui13 ] for detailson the last point. See [
Sc13b ] for a general overview. onference on Type Theory, Homotopy Theory and Univalent Foundations Specifically, cohesive homotopy type theory has semantics in the ∞ -topos H of ∞ -stacks over the site of smooth manifolds (section 4.4 of [ Sc13a ]). This contains a canon-ical line object A = R , the continuum , abstractly characterized by the fact that theshape modality exhibits (in the sense of [ Sh12b ]) the corresponding A -homotopy local-ization. Forming the quotient type by the type of integers yields the smooth circle group U (1) ≃ R / Z . This being an abelian group type means equivalently that for all n ∈ N there is a pointed n -connected type B n U (1) such that U (1) ≃ Ω n B n U (1) is the n -foldloop type. Write then θ B n U (1) := fib(fib( ǫ )) : B n U (1) −→ ♭ dR B n +1 U (1)for the second homotopy fiber of the co-unit ǫ : ♭ B n +1 U (1) −→ B n +1 U (1) of the flat co-modality. Cohesion implies that we may think of this as the universal Chern-character for ordinary smooth cohomology (section 3.9.5 in [ Sc13a ]). Hence we write
Phases := B n U (1) conn for the dependent sum of “all” homotopy fibers of θ B n U (1) (for some choiceof “all”, see section 4.4.16 of [ Sc13a ]). Then a dependent type ∇ over B U (1) conn is a prequantized phase space (see section 3.9.13 of [ Sc13a ]) in classical mechanics [
Ar89 ].An equivalence of dependent types over B U (1) conn is a Hamiltonian symplectomorphism and a (concrete) function term H : B R −→ Y B U (1) conn B Equiv( ∇ , ∇ )of the function type from the delooping of R to the delooping of the dependent productof the type of auto-equivalences of ∇ is equivalently a choice of Hamiltonian . It sendsthe (“time”) parameter t : R to the Hamiltonian evolution exp( t { H, −} ) with Hamilton-Jacobi action functional exp( i ~ S t ) [ Ar89 ]. In the ∞ -categorical semantics this is givenby a diagram in H of the following form :graph (exp ( t { H, −} )) y y tttttttttttttttt % % ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ space oftrajectories initialvalues x x qqqqqqqqqq Hamiltonianevolution & & ▼▼▼▼▼▼▼▼▼▼ X ∇ % % ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ X ∇ y y tttttttttttttttt incomingconfigurations prequantumbundle & & ▼▼▼▼▼▼▼▼▼▼ outgoingconfigurations prequantumbundle x x qqqqqqqqqq B U (1) conn exp ( i ~ S t ) =exp ( i ~ R t Ldt ) ssssssssssssss u } ssssssssssss actionfunctional ♣♣♣♣♣♣♣♣♣♣ s { ♣♣♣♣♣♣♣♣ . Here X := P B U (1) conn ∇ is the phase space itself and ∇ is its pre-quantum bundle [ FRS13a ].This statement concisely captures and unifies a great deal of classical Hamilton-Lagrange-Jacobi mechanics, as in [
Ar89 ]. Moreover, when replacing B U (1) conn herewith B n U (1) conn for general n ∈ N , then the analogous statement similarly captures n -dimensional classical field theory in its “covariant” Hamilton-de Donder-Weyl formu-lation on dual jet spaces of the field bundle (see e.g. [ Rom05 ]). This is shown in section1.2.11 of [
Sc13a ]. This is a pre-quantization of the
Lagrangian correspondences of [
We83 ]. I am grateful to Igor Khavkine for discussion of this point.
Classical field theory via Cohesive homotopy types
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