Natural operations on differential forms

Navarro, José; Sancho, J. B.

doi:10.48550/arxiv.1412.0840

Cited by 2 publications

(3 citation statements)

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“…In fact, it is known that these are all the natural operations on Ω 1 when considered as a functor Mfd op → Set defined on all manifolds and all smooth maps between them (Navarro and Sancho, 2015), in the style of Scalar Field (Kinematic) 5.2. Now in d = 4 spacetime dimensions, we have seen that the natural choice for Spacetimes contains all the conformal maps as morphisms.…”

Section: Algebraic Structure From Natural Operationsmentioning

confidence: 99%

An Algebraic Approach to Physical Fields

Chen,

Fritz

2021

Preprint

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According to the algebraic approach to spacetime, a thoroughgoing dynamicism, physical fields exist without an underlying manifold. This view is usually implemented by postulating an algebraic structure (e.g., commutative ring) of scalar-valued functions, which can be interpreted as representing a scalar field, and deriving other structures from it. In this work, we point out that this leads to the unjustified primacy of an undetermined scalar field. Instead, we propose to consider algebraic structures in which all (and only) physical fields are primitive. We explain how the theory of natural operations in differential geometry-the modern formalism behind classifying diffeomorphism-invariant constructions-can be used to obtain concrete implementations of this idea for any given collection of fields.For concrete examples, we illustrate how our approach applies to a number of particular physical fields, including electrodynamics coupled to a Weyl spinor.

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Section: Algebraic Structure From Natural Operationsmentioning

confidence: 99%

An Algebraic Approach to Physical Fields

Chen,

Fritz

2021

Preprint

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“…Let f : Ω n → Ω * be a natural transformation of sheaves. In [NS15] it was shown that any assignment of differential forms ω → f (ω), which is natural with respect to pullback, is given by a polynomial in ω and its derivative dω. Hence, if we restrict f to the sheaf of closed forms, we see that f must assigns each section ω ∈ Ω n cl , a polynomial in ω.…”

Section: θ Factorizes Asmentioning

confidence: 99%

“…More general operations in a much broader context are studied in [KMS93], but the operations relevant to us are still linear (and we are interested in nonlinear ones as well); there it is shown that all operations that raise the form degree by one are multiples of the exterior derivative, and linearity follows from naturality. More recently, operations (both linear and nonlinear) acting on 1-forms (connections) were considered in [FH13], and generalized to differential forms of all degrees in [NS15]. We will make use of this for our construction of cohomology operations on closed differential forms Ω * cl in stacks.…”

Section: Introductionmentioning

confidence: 99%

Primary operations in differential cohomology

Grady

Sati

2018

Advances in Mathematics

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We characterize primary operations in differential cohomology via stacks, and illustrate by differentially refining Steenrod squares and Steenrod powers explicitly. This requires a delicate interplay between integral, rational, and mod p cohomology, as well as cohomology with U (1) coefficients and differential forms. Along the way we develop computational techniques in differential cohomology, including a Künneth decomposition, that should also be useful in their own right, and point to applications to higher geometry and mathematical physics.where B n U (1) ∇ represents the moduli stack of n-bundles equipped with connection, studied in [FSSt12][FSS13] [FSS15a]. The homotopy classes of such morphisms will in turn be describe the differential cohomology groupOne of the main goals of this paper is to characterize this group for various values of k and n. This in turn will lead to to a full characterization of primary cohomology operations in differential cohomology.Since differential cohomology operations, as we will see, involve various coefficients, we find it useful to point out the interrelations that already exist between these (we found the discussion in [FFG86] particularly useful). This should also help us develop some intuition for the full differential case. Note that for coefficients being one of Z, Z/p or Q i.e. an abelian group, then the set of all cohomology operations H m (K(G, n); G), where G and G ′ are from the above set, will also be abelian.Operations from Z/p to Q. We know that H q (K(G, n); Q) = 0 for all q > 0 when G is a finite abelian group, i.e. for us Z/p. This shows that there are nontrivial cohomology operations from Z/p-coefficients to Q-coefficients.Operations from Z to Q. We will distinguish the odd and even cases. For the first, H q (K(Z, 2n+ 1); Q) is nonzero only for q = 2n + 1, where it is equal to Q, with generator the image of the fundamental class ι under the homomorphism r : H 2n+1 (K(Z, 2n+1); Z) → H 2n+1 (K(Z, 2n+1); Q) induced by the natural embedding Z ֒→ Q. Thus, every operation from an odd-dimensional integral class to rational cohomology preserves the dimension, i.e. takes α ∈ H 2n+1 (X; Z) to λr(α) ∈ H 2n+1 (X; Q) for some fixed rational number λ corresponding to the operation. In the even case, H q (K(Z, 2n); Q) = Q[r(ι)], so that every operation from even integral cohomology to rational cohomology is given as the power α → λα k , where k ∈ Z, λ ∈ Q are determined by the operation.Operations from Q to Q. The group H n (K(Q, m); Q) can be straightforwardly calculated via e.g. the Serre spectral sequence. In this case, one has that any cohomology operation assigns to an element α ∈ H n (X; Q) the element λα k ∈ H nk (X; Q), where k ∈ Z, λ ∈ Q both fixed by the operation.Operations from H * (−) dR to H * (−) dR . On the other hand, operations in de Rham cohomology can be deduced from those on rational cohomology via the de Rham theorem. Hence, de Rham operations should be systematically characterized. The de Rham cohomology groups and the rational cohomology groups hav...

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Natural operations on differential forms

Cited by 2 publications

References 2 publications

An Algebraic Approach to Physical Fields

An Algebraic Approach to Physical Fields

Primary operations in differential cohomology

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