motives in physics


Motivic cohomology


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

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Motivic structures enters quantum physics in two dual guises, related, on the one hand, to algebraic deformation quantization and, on the other hand, to geometric quantization.

Via perturbative algebraic deformation quantization

In the first case one observes that formal deformation quantization of nn-dimensional prequantum field theory amounts to choosing an inverse equivalence to the formality map from En-algebras to Pn-algebras, this is explained at Motivic Galois group action on the space of quantizations.

The automorphism infinity-group of either side therefore naturally acts on the space of formal deformation quantization choices obtained this way and one shows (conjectured by Maxim Kontsevich (Kontsevich 99), recently proven by Dolgushev) that the connected component group of this is the Grothendieck-Teichmüller group, a quotient of the motivic Galois group. Related to this in some way is Alain Connes’ “cosmic Galois groupacting on the space of renormalizations of perturbative quantum field theory.

According to Kontsevich, this explains the role of motivic periods in correlation functions and hence in scattering amplitudes (see also at amplituhedron) in perturbative field theory.

More concretely, scattering amplitudes typically are expressions in multiple zeta values and handling them involves intricate combinatorics. By just re-expressing these in terms of motivic multiple zeta values (see the rerferences) much of the combinatorics becomes more tractable (in terms of some Hopf algebra). This is how “motivic” structures are used by many practicioners. The actual motives do not play much of a role in these computations, but one makes use of the combinatorial simplification obtained by re-expressing multiple zeta values by their motivic version.

For more see the motive-related references at Feynman diagram, at motivic multiple zeta values and at motivic L-function.

Via non-perturbative geometric quantization

On the other hand, in full non-perturbative geometric quantization in its modern cohomological form as geometric quantization by push-forward one finds a “cohesive” form of actual motivic cohomology exhibited by “cohesive pure motives”. In effect, a local action functional on a moduli space of field trajectories is exhibited by a correspondence with the local action functional itself exhibited by a twisted bivariant cocycle on the correspondence space, and the motivic path integral quantization of this corresponds to the induced pull-push index transform. See the references below.


We list references

  1. on motives in perturbative algebraic quantization

  2. on motives in non-perturbative geometric quantization

On motivic structures in perturbative quantum field theory

The action of a motivic Galois group (“cosmic Galois group”) on the space of choices in deformation quantization was first observed/conjectured in

A quick and rough survey of this and vaguely related motivic structures in physics is in the appendix of

A detailed review of the motivic cosmic Galois group action on the space of renormalization procedures is in section 7 of

Discussion of motivic structure in periods in scattering amplitudes is also the lecture

  • Spencer Bloch, Dilogarithm motives appearing in physics (video); S. Bloch, H. Esnault, D. Kreimer, On motives associated to graph polynomials pdf; Spencer Bloch, Pierre Vanhove, The elliptic dilogarithm for the sunset graph, IHES preprint P-13-24, pdf
  • Conference Amplitudes and Periods, Dec 3-7, 2012, IHÉS, (has online videos of talks)

See also the material linked at

  • Conference Amplitudes and Periods, Dec 3-7, 2012, IHÉS (web)

More recent developments on motivic structures in scattering amplitudes include for instance

  • John Golden, Alexander B. Goncharov, Marcus Spradlin, Cristian Vergu, Anastasia Volovich, Motivic Amplitudes and Cluster Coordinates (arXiv:1305.1617)

On motivic structures in non-perturbative local quantum field theory

The formulation of quantization of local prequantum field theory as a passage to cohesive generalized pure motives, hence “motivic quantization” is formulated as such in the last section of

based on

The following is a list of “precursors” of aspects of the idea motivic quantization as laid out there


The formulation of geometric quantization as an index map in K-theory is attributed to Raoul Bott.

The proposal that the natural domain for geometric quantization are Lagrangian correspondences is due to

  • Alan Weinstein, Symplectic manifolds and their lagrangian submanifolds, Advances in Math. 6 (1971)

In semiclassical quantization the precursor to this are works of Maslov in 1960s (which are quite related to Hoermander’s and Sato’s work).

With the recognition of supersymmetric quantum mechanics in the 1980s, index theory (hence push-forward in generalized cohomology to the point) was understood to be about partition functions of systems of supersymmetric quantum mechanics in

  • Luis Alvarez-Gaumé, Supersymmetry and the Atiyah-Singer index theorem, Comm. Math. Phys. 90:2 (1983) 161-173, euclid

  • Ezra Getzler, Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem, Comm. Math. Phys. 92 (1983), 163-178. (pdf)

  • Ezra Getzler, A short proof of the Atiyah-Singer index theorem, Topology 25 (1986), 111-117 (pdf)

In higher analogy to this but much more subtly, the partition function of the heterotic string, hence the Witten genus, was understood to be the push-forward to the point in tmf:

The general perspective of the path integral as a pull-push transform was originally laid out, somewhat implicitly, in

and then fully explicitly in

Discussion along these lines of a pull-push quantization over KU of a 2-dimensional Chern-Simons theory-like gauge theory is in

More in detail a functorial quantization of suitable correspondences of smooth manifolds to KK-theory by pull-push has been given in (Connes-Skandalis 84) and the generalization of that to equivariant K-theory (hence to groupoid K-theory of action groupoids) is in

The point of view that the pull-push quantization of Gromov-Witten theory should be thought of as a theory of Chow motives of Deligne-Mumford stacks is expressed in

The proposal that the natural codomain for geometric quantization is KK-theory is due to

The point of view that pull-push along correspondences equipped with operator K-theory cycles in KK-theory is a K-theory-analog of motives was amplified in

The proof that the universal property that characterizes noncommutative motives is the analog in noncommutative algebraic geometry of the universal property that characterizes KK-theory in noncommutative topology is due to

  • Gonçalo Tabuada, K-theory via universal invariants, Duke Math. J. 145 (2008), no.1, 121–206.

The description of string topology operations as an HQFT defined by pull-push transforms in ordinary homology/ordinary cohomology was originally realized in

A detailed discussion and generalization to open strings and an open-closed HQFT in the presence of a single space-filling brane is in

and for arbitrary branes in

That D-brane charge and T-duality is naturally understood in terms of pull-push/indices along correspondences in noncommutative topology/KK-theory was amplified in

The general analogy between such KK-theory cocycles and pure motives is noted explicitly in

This analogy is given a precise form in

where it is shown that there is a universal functor KKNCC dgKK \longrightarrow NCC_{dg} from KK-theory to the category of noncommutative motives, which is the category of dg-categories and dg-profunctors up to homotopy between them. This is given by sending a C*-algebra to the dg-category of perfect complexes of (the unitalization of) its underlying associative algebra.

Linearization of correspondences of geometrically discrete groupoids was considered in

and applied to a pull-push quantization of Dijkgraaf-Witten theory in

following previous work by Daniel Freed and Frank Quinn.

An unpublished predecessor note on quantization of correspondences of moduli stacks of fields is

Quantization of correspondences of perfect ∞-stacks by pull-push of stable (∞,1)-categories of quasicoherent sheaves is discussed in

Linearization of higher correspondences of discrete ∞-groupoids as the quantizaton of ∞-Dijkgraaf-Witten theories is indicated in section 3 and 8 of

and in

A clear picture of fiber integration in twisted cohomology is developed in

A proposal to axiomatize perturbative prequantum field theory by functors from cobordisms to a symplectic category of symplectic manifolds and Lagrangian correspondences is in

Revised on March 9, 2017 13:35:05 by Urs Schreiber (