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This page describes a proposal for a formalization of the concept of quantization – supposedly a natural process that turns the higher geometric local Lagrangian data of a local prequantum field theory into the higher algebraic data of a local quantum field theory – formulated in terms of cohomology in higher differential geometry, specifically in cohesive (∞,1)-toposes.
The fundamental observation here is that is natural to
(prequantum fields) – identify spaces of trajectories of fields as correspondences of higher moduli stacks, hence correspondences in some suitably cohesive base (∞,1)-topos;
(local action functionals) – identify local action functionals as lifts of these correspondences to the slice of the base $\infty$-topos over an ∞-group of units of a suitable E-∞ ring object $E$;
(path integral) – identify the path integral quantization/partition function of these local action functionals on spaces of trajectories as the pull-push operation in twisted $E$-cohomology theory through these correspondences.
This perspective has many precursors, listed in the references below. What is described here might be thought of as a proposal for a coherent, general and natural formulation, in the spirit of “Synthetic Quantum Field Theory” formulated in the higher differential geometry of cohesive (∞,1)-toposes.
Before looking at the details, we indicate some of the characteristic aspects of motivic quantization:
Then in the main part of the text we first give an exposition and survey of the main ideas in
and then lay out the basic definitions of motivic quantization in detail in
Then we consider realizations of this theory in various
and finally we discuss a list of examples in
As our title indicates, it may be useful to note the analogy of the structures considered here to structures in motivic cohomology and in six operations-yoga. (See also at motives in physics.)
First notice the following heuristics: Just as the idea of a category of motives is to constitute a “linearization” or “abelianization” of a category of spaces, so quantization is a process that sends (non-linear) spaces of field configurations to linear spaces of quantum states. This linearization by which quantum states may be added as elements of an abelian group encodes the superposition principle and hence quantum interference, the hallmark of quantum physics.
Concretely, a correspondence
in the slice (∞,1)-topos is equivalently
a correspondence of spaces $\mathbf{Fields}_{in}$, $\mathbf{Fields}_{out}$
equipped with a cocycle $\xi \in H^{\bullet + i_{in}^\ast \chi_{in} + i_{out}^\ast \chi_{out}}(\mathbf{Fields}_{in}, \mathbf{Fields}_{out}; E)$ in bivariant $(i_{in}^\ast \chi_{in}, i^\ast_{out} \chi_{out})$-twisted $E$-cohomology on its correspondence space,
and the quantization step takes this to the equivalence class of maps of $E$-∞-modules that it induces under fiber integration in generalized cohomology:
This process is analogous to how Chow motives induce cocycles in motivic cohomology. For $E =$ KU and $\mathbf{Fields}$ a smooth manifold this was amplified in (Connes-Skandalis 84, Connes-Consani-Marcolli 05)), where the KK-category is regarded as the analog in noncommutative topology of a category of motives. This vague analogy becomes precise from the general point of view of noncommutative motives: according to (Tabuada 08, Cisinski-Tabuada 11, Blumberg-Gepner-Tabuada 10) these are characterized by the same kind of universal properties that also characterizes KK-theory, as summarized in the following table:
geometric context | universal additive bivariant (preserves split exact sequences) | universal localizing bivariant (preserves all exact sequences in the middle) | universal additive invariant | universal localizing invariant |
---|---|---|---|---|
noncommutative algebraic geometry | noncommutative motives $Mot_{add}$ | noncommutative motives $Mot_{loc}$ | algebraic K-theory | non-connective algebraic K-theory |
noncommutative topology | KK-theory | E-theory | operator K-theory | … |
Moreover, for the case that $\mathbf{Fields}$ is a Deligne-Mumford stack (in the context of Gromov-Witten theory, which is the pull-push quantization of a 2d field theory) the fact that pull-push quantization procceeds via morphisms of motives was amplified in (Behrend-Manin 95), see also for instance the introduction of (Toën 00).
Therefore it may be generally useful to refer to the process of quantization as discussed here as motivic quantization.
In view of this, notice that in the context of formal deformation quantization (which is roughly an infinitesimal (“formal”) approximation to geometric quantization as considered here) it is known that a quotient of the motivic Galois group (namely the Grothendieck-Teichmüller group) acts on the space of choices of formal deformation quantization of a Poisson manifold. (For details and pointers on this see at formal deformation quantization – Motivic Galois group action on the space of quantizations).
A priori, motivic quantization applies to topological field theory. However, we consider it in the full generality that includes boundary field theory (field theory with branes) and generally defect field theory (field theory with domain walls of arbitrary codimension). This induces boundary effects which are not purely topological but encode “physical” field theories.
Concretely, for $Z \colon Bord_n(def)^\otimes \to \mathcal{C}^\otimes$ a topological local field theory which includes a boundary condition encoded by a generator $(\emptyset \stackrel{\partial}{\to} \ast) \in Mor_1(Bord_n(def))$, then crossing with this yields an $(n-1)$-dimensional field theory $Z((-)\times (\partial))$, the corresponding boundary field theory. This however now depends on choices of orientations in generalized cohomology that go along with defining a boundary component for $Z$, and these choices constitute geometric data.
For instance, for $Z$ a 3-dimensional Chern-Simons theory, the relevant choice of orientation is induced by a choice of conformal structure on the boundary and so the boundary theory is a non-topological conformal field theory. (For details on this see at AdS3-CFT2 and CS-WZW correspondence.)
In physics this general kind of relation between $n$-dimensional topological field theories and $(n-1)$-dimensional non-topological field theories on their boundary is has come to be known as the holographic principle. See there for more background.
For the moment see at Expositional summary – Introduction below.
We see that the motivic quantization operation over a cohomology theory $E$ depends on the existence of, and the choice of, an orientation in E-cohomology. The conditions that such an orientation exist in the first place turns out to be what in physics are known as the quantum anomaly cancellation conditions.
physical theory | cohomology theory |
---|---|
quantum anomaly | obstruction to orientation for push-forward |
(geo-)metric structure of boundary field theory | choice of orientation |
Ever since the recognition of supersymmetric quantum mechanics in the 1980s, it is a familiar fact that index theory is naturally formulated in terms of superalgebra and supergeometry: indices can be identified with partition functions in supersymmetric quantum mechanics. Since push-forward in generalized cohomology is what generalizes the notion of index to the “relative case”, motivic quantization may be thought of as intrinsically based on indices/partition functions. Accordingly one may expect that supersymmetry plays not just an optional but an intrinsic role.
Indeed, one can observe the following seemingly deep relation between supersymmetry and higher algebra:
the $\mathbb{Z}_2$-grading of supersymmetry is a low-degree shadow of $\mathbb{S}$-grading, where $\mathbb{S}$ is the sphere spectrum;
every E-∞ ring is canonically $\mathbb{S}$-graded, or at least it ∞-group of units is.
This is discussed in a bit more detail at superalgebra – Abstract idea.
Accordingly, it follows that some kind of (“higher”) supersymmetry is intrinsic in motivic quantization.
A basic example is given by 2-dimensional motivic quantization over KU. A canonical smooth refinement of its ∞-group of units is given, up to the 3-coskeleton, by the smooth super ∞-groupoid of super line 2-bundles (as discussed there). Accordingly, 2-dimensional local field theories, i.e. string sigma-models are naturally refined to superstring $\sigma$-models. See below the example The charged particle at the boundary of the superstring.
But motivic quantization is not restricted to supersymmetric field theories, it is just that the higher phases are always naturally super-graded. Notably plain quantum mechanics encoded by traditional symplectic or generally Poisson phase spaces is naturally subsumed. See below the example The Poisson manifold at the boundary of 2d Chern-Simons theory.
We give here a leisurely exposition of the main ideas of motivic quantization of local prequantum field theory. We follow (Nuiten 13) in outline. For more details see there and/or skip ahead to General theory.
Some words of context. See (Nuiten 13, section 1).
$\,$
One of the modern cornerstones of Hilbert's sixth problem applied to the physics of quantum field theory is the following definition:
A topological+boundary+defect local field theory is a monoidal (∞,n)-functor
from an (∞,n)-category of cobordisms with branes and domain walls, to some symmetric monoidal (∞,n)-category.
The point of this axiom is that the higher categorical (∞,n)-functoriality of $Z$ is what encodes the locality of the field theory. This in turn encodes a fundamental property of the fundamental physics of the observable universe called causal locality : (spacelike-)separated regions of spacetime/worldvolume behave like independent subsystems.
fundamental physics | foundational mathematics | experimental bound on violation |
---|---|---|
gauge principle | homotopy theory | |
causal locality | higher category theory | $\lessapprox 10^{-17}m$ (Grigoriev 79), $\lessapprox 10^{-20}m \simeq 7 TeV$ (LHC) |
The cobordism hypothesis provides a good characterization of the space of all such $Z \colon Bord_n^\otimes \to \mathcal{C}^\otimes$. But for modelling physics there are typically more restrictions to be imposed.
In particular, for actual quantum (as opposed to prequantum or classical) field theory, the codomain $\mathcal{C}$ is to be an n-dimensional analog of a linear tensor category of modules, $R Mod^\otimes$, for some commutative ground ring $R$ (which in ordinary quantum mechanics is the complex numbers).
The linearity of $E Mod$ encodes the superposition principle of quantum physics, which says that quantum states may be added and possibly may additively cancel. This cancellation is quantum interference, the very hallmark of quantum physics.
In local quantum field theory/higher category theory/homotopy theory we choose, more generally, a ground commutative ∞-ring $E$. This comes with its symmetric monoidal (∞,1)-category of ∞-modules $E Mod^\otimes$.
A decent choice for $\mathcal{C}^\otimes$ is then $\mathcal{C}^\otimes = E Mod^{\Box^n}$, the symmetric monoidal (∞,n)-category of $n$-dimensional cubes in $E Mod$. With this used as the codomain in the definition of $Z$
the field theory assigns quantum propagator linear maps between spaces of quantum states to pieces of spacetime/worldvolume:
But even with $\mathcal{C}$ restricted to be of the form $E Mod^{\Box^n}$ or similar, the notion of $Z$ as above is still much more general than the field theories typically of interest in nature and in theory. The quantum field theories actually of interest both in nature and in theory have the special property that they arise via a process of quantization from higher geometric data given by a local prequantum field theory: a local action functional/local Lagrangian on a moduli space of fields.
This is the process to be indicated in the following:
local prequantum field theory | – quantization$\longrightarrow$ | local quantum field theory |
---|---|---|
higher geometry | –motives$\longrightarrow$ | higher linear algebra |
higher prequantum line bundle | –space of sections$\longrightarrow$ | ∞-module of quantum states |
section, wavefunction | quantum state | |
correspondences, integral kernels | –pull-push transform$\longrightarrow$ | linear maps |
field trajectories | –path integral$\longrightarrow$ | quantum propagators |
This process may be thought of as a refinement of geometric quantization from quantum mechanics to (non-perturbative) quantum field theory.
First we consider local prequantum field theory. See (Nuiten 13, section 2, Fiorenza-Rogers-Schreiber 13a, Nuiten-Schreiber 13).
$\,$
The notion of field in physics is often said to be axiomatized as a section of a fiber bundle over spacetime/worldvolume whose fibers are manifolds – called the field bundle. This is true for simple instances of scalar fields and sigma-model fields.
But it is false for gauge fields as in electromagnetism or Yang-Mills theory (and is ever more false for higher gauge fields, such as the B-field in type II supergravity or in the 6d (2,0)-superconformal QFT):
for $G$ a Lie group regarded as a gauge group, a $G$-gauge field on $X$ is a section of a kind of bundle whose fiber is the universal moduli stack $\mathbf{B}G_{conn}$ of $G$-principal connections $\nabla$. The underlying instanton sector $\mathbf{c}(\nabla)$ of the gauge field is still a section of a bundle whose typical fiber is the moduli stack $\mathbf{B}G$ of $G$, locally on a cover $U \to X$ one has:
Here $\mathbf{B}G$ is not a smooth manifold, but a smooth groupoid (a geometric stack), and a $(\mathbf{B}G)$-fiber bundle is a fiber 2-bundle, called a $G$-gerbe over $X$, which itself is a connected stack of groupoids over $X$.
Here the groupoid-nature of $\mathbf{B}G$ is a precise reflection of the gauge principle governing the gauge field.
Similarly, an abelian n-form gauge field (e.g. the B-field, the supergravity C-field) is locally given by maps into a moduli n-stack $\mathbf{B}^n U(1)_{conn}$ of circle n-bundles with connection, which is a smooth n-groupoid. A field bundle for these is hence a $U(1)$-n-gerbe.
(All this is true in the complete non-perturbative description of field theory. Often, however, field theory is discussed only in the approximation perturbation theory, where all spaces are linearized by their infinitesimal approximation. Since the Lie differentiation of a smooth higher stack is precisely an L-∞ algebra/L-∞ algebroid, and since these may be modeled on chain complexes, it follows that in perturbation theory (only) one may generally assume (mostly) that fields are indeed sections of an ordinary field bundle.)
So in order to formalize non-perturbative local prequantum field theory one needs to pair geometry with homotopy theory. Such a combination is called an (∞,1)-topos of ∞-stacks/geometric ∞-groupoids:
We consider now $\mathbf{H}$ to be a suitable such (∞,1)-topos. The example to keep in mind is
$\mathbf{H} =$ SmoothSuper∞Grpd $\coloneqq Sh_\infty(SuperManifolds) = L_{lhe} Func(Supermanifolds^{op}, KanComplexes)$
which is the context of higher differential geometry for the description of bosonic fields and of higher supergeometry (for the description of fermion fields).
More generally, for constructing the moduli ∞-stacks of higher gauge fields we need an (∞,1)-topos $\mathbf{H}$ that satisfies an axiom called differential cohesion.
Given such $\mathbf{H}$, every object in it serves as a moduli ∞-stack of fields, so we write here $\mathbf{Fields} \in \mathbf{H}$. This means that if $X \in \mathbf{H}$ is regarded as spacetime/worldvolume, then
a field on $X$ is a map
in $\mathbf{H}$,
a gauge transformation of such fields is a homotopy betwen two such maps,
and a higher gauge transformation is a higher homotopy between those.
With physical fields in hand, next we need to axiomatize trajectories of such fields.
Now a space of trajectories of fields is itself a space of fields, together with projections to the incoming and the outgoing fields configurations, hence a span-shaped diagram of the form
We say that this is correspondence (or in fact a higher relation) between $\mathbf{Fields}_{in}$ and $\mathbf{Fields}_{out}$.
For instance for
a cobordism with incoming and outgoing boundary manifolds as indicated, and for $\mathbf{Fields}$ a given moduli stack of fields as above, then forming mapping spaces (mapping stacks) yields the correspondence
which exhibits field configurations on $\Sigma$ as trajectories along which fields on $\partial_{in} \Sigma$ propagate to $\partial_{out} \Sigma$.
In the example here, $\mathbf{Fields}$ is the moduli of some sigma-model field (hence $\mathbf{Fields} = X$ a target space), then this describes a bunch of branes of shape the connected components of $\Sigma_{in}$ coming in, propagating and interacting along a worldvolume of shape $\Sigma$, and finally emerging as a collection of branes of shape the connected components of $\Sigma_{out}$. This describes a scattering process. Its quantization will be what is called the corresponding scattering amplitude (the probability amplitude for the process to take place) or n-point function or correlator.
Specifying the example further, suppose that $\Sigma$ is an $n$-sphere with $(k+1)$ disjoint $n$-balls marked, regarded as a cobordism
then For $\mathbf{Fields}$ some sigma-model fields this may be taken to encode a diagram where $k$ open $d$-branes come in, merge, and one comes out. For instance for $d = 1$ and $k = 3$ this is a cobordism that when “viewed in time direction” is an inclusion of three small intervals into one larger interval
These are of course the operations in the little k-cubes operad. Hence quantization of local boundary prequantum field theory restricted to cobordisms of this form yields what are called topological or locally constant factorization algebras.
To capture all this functorially, write then $Corr_n(\mathbf{H})^\otimes$ for the (∞,n)-category of n-fold correspondences whose objects are those of $\mathbf{H}$ and whose morphisms are correspondences between them, as above. This is symmetric monoidal by objectwise Cartesian product in $\mathbf{H}$.
One finds that:
every object of $\mathbf{H}$ is self-fully dualizable in $Corr_n(\mathbf{H})^\otimes$;
the higher trace $tr_{\Pi(\Sigma)}(\mathbf{Fields})$ of shape $\Pi(\Sigma)$ of this fully dualizable object is the moduli ∞-stack of flat fields on $\Sigma$ – this is the phase space of fields (for fields of ∞-Chern-Simons theory-type at least).
This means, by the cobordism hypothesis, that a choice of moduli ∞-stack $\mathbf{Fields} \in \mathbf{H}$ is equivalently a choice of monoidal (∞,n)-functor
which sends cobordisms to the phase space correspondences between in- and out-going flat field configurations over the boundary, given by flat trajectories over the cobordism:
In this way the moduli ∞-stack $\mathbf{Fields}$ of fields encodes the kinematics of a local prequantum field theory.
Next we define the dynamics by defining a local action functional which assigns to each trajectory a probability amplitude for that trajectory to be “physically realized”.
Traditionally, in non-localized prequantum field theory, an (exponentiated) action functional is a map of the form $\mathbf{Fields}_{traj} \longrightarrow U(1)$. For instance for $\mathbf{Fields}_{traj} \coloneqq [S^1, X]$ the smooth loop space of a manifold $X$ that is equipped with a $U(1)$-principal connection $\nabla \colon X \to \mathbf{B}U(1)_{conn}$ (an electromagnetic field), then the holonomy functional $hol_\nabla \colon [S^1, X] \to U(1)$ is the interaction term for the charged particle sigma model field theory on $X$.
To see how this is axiomatized in local field theory, notice that there is a homotopy fiber product diagram of the form
in $\mathbf{H}$, which exhibts the circle group (as a Lie group!) as the loop space object of the moduli stack of $U(1)$-principal bundles: $U(1)\simeq \Omega \mathbf{B}U(1)$. By the universal property of the homotopy fiber construction, this means that maps $\mathbf{Fields}_{traj} \longrightarrow U(1)$ are equivalent to diagrams of the form
More generally, for $[I,X]$ the smooth path space of $X$, the interaction term of the charged particle sigma model on $(X,\nabla)$ is not in general a $U(1)$-valued function on $X$, but is a section of the $U(1)$-principal bundle $\chi(\nabla) \colon X \to \mathbf{B}U(1)$ which underlies $\nabla$, pulled back to path space along the two endpoint evaluations. This means that it is a diagram of this form
where now the bottom morphisms are non-trivial, given by the background gauge field. In view of the above, it makes sense to think of this background gauge field $\nabla \colon X \to \mathbf{B}U(1)_{conn}$ itself as a “higher incarnation” or “local incarnation” of the action functional on path de-transgressed from path space back to the manifold itself.
Therefore, in general we say that a local action functional for a local prequantum field theory of dimension $n$ with field content $\mathbf{Fields} \in \mathbf{H}$ is a map in $\mathbf{H}$ of the form
hence is a circle n-bundle with connection on the universal moduli ∞-stack $\mathbf{Fields}$.
Exploring this localized higher prequantum geometry formulation (e.g. Fiorenza-Rogers-Schreiber 13a), one finds that the notion of localized action functional coincides with the notion of local Lagrangian and also coincides with this notion of higher prequantum bundle – by transgression to lower codimension:
the transgression of $\exp(i S)$ to codimension 0 is the traditional action functional $[\Sigma_n, \mathbf{Fields}] \to U(1)$.
the transgression of $\exp(i S)$ to codimension $1$ is the the traditional (off-shell) prequantum circle bundle $[\Sigma_{n-1}, \mathbf{Fields}] \to \mathbf{B}U(1)_{conn}$;
and so on, for instance if we think of $\exp(i S)$ as an ∞-Chern-Simons theory, then
the transgression of $\exp(i S)$ to codimension $n-1$ is the corresponding ∞-Wess-Zumino-Witten theory
(e.g. traditional WZW model if $\nabla$ is the universal Chern-Simons circle 3-connection; or the M5-brane worldvolume theory (supposedly the 6d (2,0)-superconformal QFT) for $\nabla$ the supergravity Lie 6-algebra L-∞ cocycle plus the universal Chern-Simons circle 7-connection and other terms (Fiorenza-Sati-Schreiber 12a, Fiorenza-Sati-Schreiber 13).
Therefore just as the (∞,1)-topos $\mathbf{H}$ itself is the home of moduli ∞-stacks of fields (kinematics) so the collection of all local action functionals on such moduli $\infty$-stacks (dynamics) forms the slice (∞,1)-topos
whose objects are maps $\left[\array{\mathbf{Fields} \\ \downarrow^{\mathrlap{exp(i S)}} \\ \mathbf{B}^n U(1)_{conn}}\right]$ and whose morphisms are diagrams of the form
in $\mathbf{H}$.
The automorphism ∞-groups in this (∞,1)-topos of local action functionals are precisely the quantomorphism ∞-groups (infinitesimally the Poisson L-∞ algebras), conaining the Heisenberg ∞-groups (infinitesimally the Heisenberg L-∞ algebras) of local prequantum observables (Fiorenza-Rogers-Schreiber 13a). Equivalently, they are ∞-groups of conserved currents.
The (∞,1)-category $Corr_n\left(\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}\right)$ of correspondences in the slice contains trajectories equipped with action functionals.
In conclusion, a local action functional $\exp(i S)$ on a species $\mathbf{Fields}$ of physical fields is a lift $\exp(i S)$ in
Such a diagram defines a local prequantum field theory (topological+boundary+defects).
This theory may have boundaries/branes. Below we find that most local quantum field theories of interest arise actually as the boundary field theories of a higher dimensional field theory and that their quantization is induced from that higher dimensional theory in a “holographic” way that generalizes the AdS3-CFT2 and CS-WZW correspondence.
If $Bord_n^\otimes$ is generated precisely from one boundary brane of top codimension, then specifying a local prequantum field theory is equivalent to specifying a boundary correspondence of the form
Such a diagram is equivalently a twisted chi-structure and such structures appear all over the place in local prequantum field theory (e.g. Fiorenza-Sati-Schreiber 09, Fiorenza-Sati-Schreiber 12) see for instance the lecture notes at twisted smooth cohomology in string theory
We now survey the linearization step. See (Nuiten 13, section 3).
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Before actually quantizing a local prequantum field theory $\left[ \array{ \mathbf{Fields} \\ \downarrow^{\mathrlap{\exp(i S)}} \\ \mathbf{B}^n U(1)_{conn}}\right]$ as above, we choose linear coefficients, given by
a choice of ground E-∞ ring $E$
(playing the role of the complex numbers in plain quantum mechanics);
a choice of ∞-group homomorphism
from the ∞-group of phases to the ∞-group of units of $E$, hence an ∞-representation of the circle n-group on $E$
(playing the role of the canonical $U(1) \hookrightarrow \mathbb{C}^\times$ in plain quantum mechanics).
Then for $X \longrightarrow \mathbf{B}^n U(1)$ modulating a circle n-bundle on $X$, the composite
modulates the associated ∞-bundle, which is an $E$-(∞,1)-module bundle.
Specifically, given the higher prequantum bundle $\exp(i S) \;\colon\; \mathbf{Fields} \to \mathbf{B}^n U(1)_{conn}$ as above, the composite
modulates the associated higher prequantum E-line bundle.
A section of $\chi$ is a higher wavefunction, hence a higher quantum state.
(At this point this looks un-polarized, but in fact we will see in the next section that the notion of polarization in higher prequantum geometry is automatic, but appears in a holographic/boundary field theory way in codimension $(n-1)$ instead of here in codimension $n$.)
Accordingly, the space of sections of $\chi$ is the higher space of quantum states in codimension 0.
If $X$ is a discrete ∞-groupoid then the space of sections has a particularly nice description, on which we focus for a bit:
The space of co-sections is the (∞,1)-colimit
This is also known as the $\chi$-twisted $E$-Thom spectrum of $X$ (Ando-Blumberg-Gepner 10).
a map $E \to E_{\bullet + \chi}(X)$ is a cycle in $\chi$-twisted $E$-generalized homology of $X$;
a map $E_{\bullet + \chi}(X) \to E$ is a cocycle in $\chi$-twisted $E$-generalized cohomology of $X$
Hence we write
Generally, for $\chi_i \colon X_i \to E Mod$ two $E$-(∞,1)-module bundles over two spaces, a map
is a cocycle in $(\chi_1, \chi_2)$-twisted bivariant $E$-cohomology.
Now given a local action functional on a space of trajectories, hence a correspondence as above, this induces an integral kernel for linear maps between sections of higher prequantum line bundles:
This is the integral kernel induced by the action functional, and acting on spaces of sections of the higher prequantum line bundle.
The linear map induced by these higher integral kernels is to be the quantum propagator. This we come to in the next section.
Notice that forming co-sections constitutes an (∞,1)-functor
Therefore forming co-sections sends an integral kernel as above to a correspondence of $E$-(∞,1)-modules:
The actual quantization/path integral as a pull-push transform map now consists in forming a dual morphism in $E Mod$ so as to turn one of the projections of such a correspondence around to produce a quantum propagator
that maps the incoming quantum states/wavefunctions to the outgoing ones.
We now survey the cohomological quantization step. See (Nuiten 13, section 4).
$\,$
What we need now for quantization is a path integral map that adds up the values of the action functional over the space of trajectories, a functor of the form
As such this will in general only exist for ∞-Dijkgraaf-Witten theory where $\mathbf{Fields}$ is a discrete ∞-groupoid and hence has a “counting measure”. This case has been considered in (Freed-Hopkins-Lurie-Teleman 09, Morton 10).
In the general case the path integral requires that we choose a suitable measure/orientation on the spaces of fields. We see below what this means, for the moment we just write
(i.e. with an ${(-)}^{or}$-superscript) as a mnemonic for a suitable (∞,n)-category of suitably oriented/measured spaces of fields with action functional. Then we may consider lifts of the action functional to measure-valued action functionals
A path integral is then to be a monoidal functor of the form
This we discuss now below. Once we have such a path integral functor, the quantization process is its composition with the given prequantum field theory $\exp(i S) \, d \mu$ to obtain the genuine quantized quantum field theory:
We realize this now by fiber integration in generalized cohomology.
While traditionally the definition of path integral is notoriously elusive, here we make use of general abstract but basic facts of higher linear algebra in a tensor (∞,1)-category (a stable and symmetric monoidal (∞,1)-category): the simple basic idea is that
Cohomological integration
Fiber integration of $E$-modules along a map is forming the dual morphisms of pulling back $E$-modules.
The choice of measure against which one integrates is the choice of identification of dual objects.
More in detail, given a monoidal category $\mathcal{C}^\otimes$ and given a morphism
in $\mathcal{C}$, a fiber integration/push-forward/index map is just
forming the dual morphism $f^\vee \colon V_2^\vee \to V_1^\vee$;
such that equivalences $V_i^\vee \simeq V_i$ exhibiting self-dual objects exist (Poincaré duality) and have been chosen (orientation).
This allows in total to have a morphism between the same objects, but in the opposite direction
That this is also the mechanism of fiber integration in generalized cohomology is almost explicit in the literature (Alexander-Whitehead-Atiyah duality), if maybe not fully clearly so. The statement is discussed explicitly in (Nuiten 13, section 4.1).
First, the basic example to keep in mind is integration in ordinary cohomology. Write $E = H R = H \mathbb{C}$ for the Eilenberg-MacLane spectrum of the complex numbers. Then for $X$ a manifold, the mapping spectrum
is the ordinary cohomology of $X$, its dual the ordinary homology, with coefficients in $R$.
For $X$ a closed manifold, Poincaré duality asserts that $H R^\bullet(X) \in H R Mod$ is essentially a self-dual object, except for a shift in degree: a choice of orientation of $X$ induces an equivalence
Using this, for $f \colon X \to Y$ a map of closed manifolds of dimension $d$, a compatible choice of orientation of both $X$ and $Y$ induces from the canonical push-forward map $f_\ast$ on homology the Umkehr map/push-forward map on cohomology, by the composition
This is ordinary integration: if $X$ and $Y$ are smooth manifolds, then $H \mathbb{R}^\bullet(X)$ is modeled by differential forms on $X$, $PD_X$ is given by a choice of volume form and $f^! = \int_{f}$ is ordinary integration of differential forms.
The shift in degree here seems to somewhat break the simple pattern. In fact this is not so, if only we realize that since we are working over spaces $X$, we should use a relative/fiberwise point of view and regard not duality in $E Mod$ itself, but in the functor categories $Func(X, E Mod)$, which is fiberwise duality in $E Mod$.
Accordingly, given an $E$-(∞,1)-module bundle
we form not just the mapping space $E^\bullet(X) = [X, E]$ as above, but form the space of sections of this bundle, which we write:
Here for $X$ a discrete ∞-groupoid
Consider now a morphism
along which we want to integrate, whith $\chi$ invertible in $Func(Y, E Mod)$: $\left(\chi^\vee\right)^\vee \simeq \chi$. $\left(\chi^\vee\right)^\vee \simeq \chi$.
Observe that we have the pair of adjoint triples of left/right Kan extensions and colimits/limits
Notice that $f^\ast$ preserves duals, but $f_!$ may not.
If $f_! f^\ast \chi^\vee$ is a dualizable object, say that a choice of twisted orientation of $f$ in $\chi$-twisted cohomology is a choice of $\beta \colon X \to E Mod$ together with a choice of an equivalence (if such exists) of the form
hence a choice of correction of $f_!$ preserving the duality of $f^\ast \chi$.
Then the $(f_! \dashv f^\ast)$ counit
induces the dual morphism
and under $\left[ p_! \left( - \right), E \right]$ this becomes
which is
This we may call the twisted fiber integration along $f$ in $E$-cohomology, or the twisted $E$-index map of $f$, induced by $(\beta, PD)$. If $\beta = 0$ then we call $PD$ anorientation_ of $f$ in $\chi$-twisted cohomology.
Notice that
Under fiber integration in twisted cohomology, the twist may change.
Grading in cohomology is just one incarnation of twist. Hence the fact that the twist changes under duality was already seen above in the ordinary case of Poincaré duality in ordinary cohomology.
For the special case that $X$ is a manifold, Atiyah duality identifies the dual cohomology spectrum with the Thom space cohomology spectrum. Then a choice of orientation amounts to a choice of Thom isomorphism, as traditionally considered.
Here we survey some examples of cohomological quantization. See (Nuiten 13, section 5).
$\,$
The traditional input for quantization is a phase space represented by a symplectic manifold. In the notation used here this data is an object $\left[\array{X \\ \downarrow^{\mathrlap{\omega}} \\ \Omega_{cl}^2 }\right]$ in the slice (∞,1)-topos over the sheaf of closed differential 2-forms.
The lack of traditional geometric quantization to act functorially on such data has become proverbial. An old proposal for how to deal with this (Hörmander 71, Weinstein 71) is to consider morphisms between phase spaces/symplectic manifolds $(X_1, \omega_1) \to (X_2, \omega_2)$ to be given by Lagrangian correspondences, hence Lagrangian submanifolds in $(X_1 \times X_2, p_1^\ast \omega_1 - p_2^\ast \omega_2 )$, or some “perturbation” thereof.
Observe that such Lagrangian correspondences are indeed correspondences in the slice (∞,1)-topos $\mathbf{H}_{/\Omega^2_{cl}}$, namely diagrams in $\mathbf{H}$ of the form
Notice that a prequantization of $(X,\omega)$ is a lift $\nabla$ in
Given such for $\omega_1$ and $\omega_2$, we may take a prequantized Lagrangian correspondence to be a factorization of the above through the curvature map $F_{(-)}$
This now yields a correspondence in $\mathbf{H}_{/\mathbf{B}U(1)}$. So we might be inclined to apply cohomological quantization to this.
Since by the degree of $\mathbf{B}U(1)$ this is going to be a 1-dimensional theory, one is inclined to linearize in $E = H \mathbb{C}$, the ordinary homology Eilenberg-MacLane spectrum.
However, ordinary cohomology does not receive a twist from $\mathbf{B}U(1)$, it receives a twist instead from flat connections $\flat \mathbf{B}U(1)$
This means we could $H\mathbb{C}$-linearize the above prequantized Lagrangian correspondence only if $\nabla$ is flat. But since $\omega$ is the curvature of $\nabla$, that is precisely not the case of interest.
What looks like a failure here right away, contains in it the seed of the holographic aspect of motivic quantization which controls all the typical examples: if we need to quantize over $\mathbf{B}U(1)$-coefficients and cannot in full codimension, then we can maybe make the system a boundary field theory of a higher dimensional system which we can quantize over $\mathbf{B}^2 U(1)$.
This turns out to be the case, indeed.
The canonical 2d Chern-Simons theory induced by a symplectic manifold, or more generally by a Poisson manifold $(X,\pi)$, is the non-perturbative version of the Poisson sigma-model. Its moduli stack of fields is the symplectic groupoid $\mathbf{2dCSFields}(X,\pi)$. This carries a canonical prequantum 2-bundle $\left[ \array{ \mathbf{2dCSFields}(X,\pi) \\ \downarrow^{\mathrlap{\exp\left(i S_{2dCS}\right)}} \\ \mathbf{B}^2 U(1) } \right] \,,$ see at extended geometric quantization of 2d Chern-Simons theory.
One finds that the original Poisson manifold is canonically a boundary brane for this 2d Chern-Simons theory, witnessed by a boundary correspondence like this:
This now is canonically $E$-linearized with $E \coloneqq$ KU the complex K-theory spectrum.
One finds that if $(X, \pi) = (X, \omega^{-1})$ is actually a symplectic manifold, then $\mathbf{2dCSFields}(X,\omega^{-1}) \simeq \ast$. In this symplectic case the above is
and encodes an ordinary prequantum line bundle $\xi$ on $X$, but now regarded as a boundary condition for the 2d Chern-Simons theory.
A choice of orientation/measure in $E = KU$ cohomology theory, hence a K-orientation, now is a choice of spin^c structure on $X$. If in the traditional geometric quantization step $(X,\omega)$ is equipped with a Kähler polarization, then the underlying almost complex structure canonically yields such (as discussed there).
With such a K-orientation chosen, the pull-push/fiber-integration/index-map quantization is then
which is the index of the prequantum line bundle. Standard (but maybe not so widely known) facts about geometric quantization (see there) say that this agrees with the traditional quantization via holomorphic sections of the Kähler polarization.
Hence we accurately recover the modern version of geometric quantization of symplectic manifolds by quantizing a boundary brane of the 2d Chern-Simons theory induced from the Poisson sigma-model. Moreover, by extension we find this way a geometric quantization of Poisson manifolds, too. Notice that this state of affairs is in complete analogy with the way the formal deformation quantization of Poisson manifolds via Kontsevich formality had been recognized by Cattaneo-Felder to be secretly given by the 3-point function of the open string in the perturbation theory of the Poisson sigma-model. Here we see a non-perturbative and non-formal (not just infinitesimal) analog of this situation.
So far this produces just the space of quantum states. But if there is a Hamiltonian action of a group $G$ on $X$, then we get instead a similar boundary correspondence, but now of the associated quotient stacks,
Now pull-push is in groupoid K-theory, hence in equivariant K-theory and lands in the representation ring of $G$. As such it produces space of quantum states equipped with an action of the the $G$-quantum observables. For more on this see below at Quantum observables and equivariant K-theory.
Passing away from symplectic manifolds: among non-symplectic Poisson manifolds of particular interest are the Lie-Poisson structures on the linear dual $\mathfrak{g}^\ast$ of a Lie algebra $\mathfrak{g}$. Here the corresponding symplectic groupoid is
the quotient stack of the coadjoint action of the Lie group $G$ on $\mathfrak{g}$.
This is in a way a “dictionary theory” that contains plenty of interesting symplectic manifolds, namely the regular coadjoint orbits $\mathcal{O} \hookrightarrow \mathfrak{g}^\ast$.
Taken together this produces a defect between boundary field theories for the 2d Chern-Simons theory induced by $\mathfrak{g}^\ast$:
The higher quantum propagator given by pull-push/index quantization through the bottom correspondence here yields the “universal orbit method”. See below at Quantization of Lie-Poisson structures.
A similar defects-of-boundaries diagram is obtained when we do consider Lagrangian correspondences (defects) between symplectic manifolds after all, but now with the latter correctly understood as themselves already being boundaries of 2d Chern-Simons theory.
Notice that all this quantizes the particle at the boundary of the open topological string given by 2d Chern-Simons theory.
Therefore it is natural to consider instead the open type II superstring on a spacetime $X$ with background gauge field a B-field $\chi \colon X \to \mathbf{B}^2 U(1)$. Then the analogous boundary condition is a D-brane with worldvolume $Q \hookrightarrow X$ equipped with a Chan-Paton gauge field $\xi$, which constitutes a boundary correspondence of the form
Here now the twisted fiber integration/index map comes out as
where $W_3(N_i Q)$ is the third integral Stiefel-Whitney class of the normal bundle of the D-brane in spacetime. Regarded as a quantum propagator this sends the $i^\ast \chi - W_3(N_i Q)$-twisted K-theory of the D-brane to the $\chi$-twisted K-theory of spacetime. One recognizes here
the condition that the domain are Chan-Paton gauge fields in $i^\ast \chi - W_3(N_i Q)$-twisted K-theory is the Freed-Witten-Kapustin anomaly cancellation condition;
the image of this higher quantum propagator is the D-brane charge.
Notice that by the above analogous discussion of the 2d Chern-Simons theory we have the slogan:
In view of this state of affairs one is to go up in dimension and quantize a 2-dimensional field theory, namely the string sigma model itself, once we canonically realize it as the boundary of a 3-dimensional field theory which is equipped with a suitable $E$-linearization.
A natural example for this is the heterotic string sigma model on the boundary of the M2-brane in the M9-brane inside the Hořava-Witten theory spacetime of 11-dimensional supergravity, see below.
The supergravity C-field is now the higher prequantum bundle and is naturally turned into a higher prequantum E-line bundle for $E =$ tmf.
The corresponding orientation now is the string^c structure induced by the supergravity C-field which is naturally implied by the flux quantization condition in 11-dimensional supergravity. On the M9-brane this induces the twisted differential string structure which is the Green-Schwarz anomaly cancellation condition in heterotic string theory. The corresponding fiber integration/index now is in (twisted) tmf the (twisted) Witten genus. This is, by Edward Witten‘s original Fields medal winning discovery, indeed the partition function of the heterotic string 2d sigma-model.
In this fashion one can now in principle climb higher up the dimensional ladder and quantize ever higher dimensional field theories, for instance by $E$-linearizing with $E$ a Morava K-theory (and hence after generalizing the formalism to A-infinity rings…)
(Next interesting along these lines might be the holographic motivic quantization analogously of the Yang monopole at the boundary of the M5-brane ending on the M9-brane. )
The earliest and by far best understood example of the holographic principle is the AdS3-CFT2 and CS-WZW correspondence between the WZW model on a Lie group $G$ and 3d $G$-Chern-Simons theory.
In (Witten 98) it is argued that all examples of the AdS-CFT duality are governed by the higher Chern-Simons theory terms on the gravity side, hence that the corresponding conformal theories are higher analogs of the WZW model: ∞-Wess-Zumino-Witten theory-type models.
In particular for AdS7-CFT6 this means that the 6d (2,0)-superconformal QFT on the M5-brane worldvolume should be a 6d-dimensional WZW model holographically related to the 7d Chern-Simons theory which appears when 11-dimensional supergravity is KK-reduced on a 4-sphere:
In (Witten 96) this is argued, by geometric quantization, for the bosonic and abelian contribution in 7d Chern-Simons theory. (The subtle theta characteristic involved was later formalized in Hopkins-Singer 02.)
In (Fiorenza-Sati-Schreiber 12a, Fiorenza-Sati-Schreiber 12b) the bosonic but non-abelian quantum correction to the 7d Chern-Simons theory induced by 11d Sugra is considered, and refined to a local action functional along the lines considered here. Therefore by (Witten 98) the corresponding ∞-Wess-Zumino-Witten theory should be the bosonic and nonabelian part of the 6d (2,0)-superconformal QFT on the M5-brane worldvolume.
To see this, what one needs is evidently a general formalization of holography for local prequantum field theory as these. How are ∞-Wess-Zumino-Witten theory-models higher holographic boundaries of ∞-Chern-Simons theory?
At the level of local prequantum field theory this is answered in (Fiorenza-Sati-Schreiber 13):
be a super L-∞ algebra L-∞ cocycle. Let
be its Lie integration in smooth super ∞-groupoids, according to (Fiorenza-Schreiber-Stasheff 10).
Observe that the smooth ∞-group $G$ has, by cohesion, a canonical higher Maurer-Cartan form
This is a cocycle in the nonabelian de Rham hypercohomology of $G$. We want an ∞-Wess-Zumino-Witten theory model with a globally defined curvature $(n+1)$-form. Therefore consider the universal solution $\tilde G$ of making $\theta$ globally well defined, hence the homotopy pullback
Then one observes that by cohesion the pasting diagram on the right of the following exists, and hence defines a local action functional $\exp(i S_{WZW})$ by the universal factorization on the left. This is the ∞-Wess-Zumino-Witten theory induced by the L-∞ cocycle $\mu$:
This uses the following general fact about how local action functionals $\mathbf{Fields} \longrightarrow \mathbf{B}^n U(1)_{conn}$ are themselves boundary conditions for what one might call universal higher topological Yang-Mills theory (Nuiten-Schreiber 13), the theory given by the local action functional
which is just the canonical inclusion of closed differential $(n+1)$-forms into the universal moduli stack of flat circle (n+1)-bundles with connection. By the universal property of ordinary differential cohomology one finds that boundary conditions for this somewhat degenerate theory are precisely differential cocycles:
One then shows (Fiorenza-Sati-Schreiber 13) that for $\mu$ the exceptional cocycles on the super Poincare Lie algebra and their higher extensions such as notably the supergravity Lie 3-algebra and supergravity Lie 6-algebra, that the ∞-Wess-Zumino-Witten theory models obtained this way reproduce the Green-Schwarz action functional “old brane scan” including for instance the heterotic string sigma-model and the M2-brane sigma-model, and also encodes the branes with tensor multiplet fields such as the D-branes and the single (abelian) M5-brane.
So now one just needs to put the pieces toghether with the nonabelian 7d Chern-Simons theory correction and apply holographic boundary motivic quantization. This however is to be disucssed elsewhere.
We give a summary of the central steps of motivic quantization of local prequantum field theory in general abstract terms of homotopy type theory, hence in the internal language of (∞,1)-toposes, following the idea of Synthetic quantum field theory. This is to bring out the sheer conceptual simplicity underlying the process.
(QFT 0) The gauge principle: The ambient theory is a homotopy type theory $\mathbf{H}$.
This encodes the gauge principle.
(QFT 1) Phases and action functionals: The homotopy type theory $\mathbf{H}$ is differentially cohesive, hence equipped with higher modalities $(\Pi \dashv \flat \dashv \sharp)$ (shape modality, flat modality, sharp modality) and $(\Re \dashv \Im \dashv \&)$ (reduction modality, infinitesimal shape modality, infinitesimal flat modality).
This induces for every choice of abelian ∞-group $\mathbb{G}$ the universal moduli $\mathbf{B}\mathbb{G}_{conn}$ of $\mathbb{G}$-principal ∞-connections.
Here $\mathbb{G}$ is a choice of phases.
We say that $\mathbf{H}$ is a context for local prequantum field theory.
The context of such, hence the the slice homotopy theory $\mathbf{H}_{/\mathbf{B}\mathbb{G}_{conn}}$, is the context of local action functionals assigning phases in $\mathbb{G}$.
A type in this context is such a local action functional $\left[\array{\mathbf{Fields} \\ \downarrow^{\mathrlap{\exp(i S)}} \\ \mathbf{B}\mathbb{G}_{conn} } \right]$. Its dependent sum to the ambient context is the moduli ∞-stack of fields, $\mathbf{Fields} \coloneqq \underset{\mathbf{B}\mathbb{G}_{conn}}{\sum} \exp(i S)$.
The automorphism ∞-groups $\mathbf{Aut}(\exp(i S))$ of these types $\exp(i S) \in \mathbf{H}_{/\mathbf{B}\mathbb{G}}_{conn}$ are equivalently
the quantomorphism ∞-groups containing the Heisenberg ∞-groups, whose Lie differentiation are the Poisson L-∞ algebras and Heisenberg L-∞ algebras;
the ∞-groups of higher conserved currents;
of the local prequantum field theory $\exp(i S)$.
(QFT 2) Superposition principle and wavefunctions: To choose a superposition principle in the context $\mathbf{H}_{/\mathbf{B}\mathbb{G}}_{conn}$ is to choose a function function $\rho \colon \mathbf{B}\mathbb{G} \longrightarrow \mathbf{B}GL_1(E)$ to the delooping of the ∞-group of units of an commutative ring type $E \in CRing_\infty(\mathbf{H})$.
Given a superposition principle $\rho$, the dependent sum of a local action functional along it is the higher prequantum E-line bundle $L \coloneqq \underset{\rho}{\sum} \exp(i S) \in \mathbf{H}_{/\mathbf{B}GL_1(E)}$.
A section of the higher prequantum line bundle is a wavefunction and the $E$-∞-module which is the space of sections $E^{\bullet + L}(\mathbf{Fields})$ is the space of quantum states.
(QFT 3) Quantization and path integral: A relation $\exp(i S_{traj}) \to \exp(i S_{in}) \times \exp(i S_{out})$ in the local prequantum field theory context $\mathbf{H}_{/\mathbf{B}\mathbb{G}_{conn}}$ is a space of trajectories equipped with probability amplitudes. Its dependent sum $L_{traj} \to L_{in} \times L_{out}$ along $\rho$ is the corresponding integral kernel, as is its image under $\Gamma$.
A choice of self-duality on the correspondence $E$-module $E^{\bullet + L_{traj}}(\mathbf{Fields}_{traj})$ is a path integral measure $d \mu_{traj}$. The obstruction to its existence is the quantum anomaly.
A choice of $d\mu_{traj}$ induces a linear function
by passing to dual morphisms. This is the quantum propagator given by pull-push path integral quantization of $\exp(i S_{traj}) d\mu_{traj}$.
Here we describe technical details of motivic quantization.
We first set up some
of higher category theory that we need. Then we introduce the three steps that constitute motivic quantum theory:
Readers already familar with the higher category theory notation that we happen to use may take the following as the lighning summary of the definition: the whole process that we describe is going to be summarized by the following diagram of monoidal (∞,n)-categories.
The left part of this diagram constitutes the defintion of a local prequantum field theory: the fields $\phi \in \mathbf{Fields}$ and the local action functional $\exp(i S)$ on them. The morphism $\int$ on the right is the map that sends by a path integral as a pull-push transform correspondences equipped with cocycles to $E$-linear maps of ∞-modules. The resulting composite
is the FQFT which is the quantization of the original prequantum field theory.
Let $\mathbf{H}$ be a cohesive (∞,1)-topos. Write $(\Pi \dashv \flat \dashv \sharp)$ for the corresponding shape modality, flat modality, sharp modality, respectively.
Write
$Grp(\mathbf{H}) \coloneqq E_1 Alg^{grp}(\mathbf{H})$ for the (∞,1)-category of ∞-group objects in $\mathbf{H}$;
$AbGrp(\mathbf{H}) \coloneqq E_\infty Alg^{grp}(\mathbf{H})$ for its full sub-(∞,1)-category of abelian ∞-groups;
$Sp(\mathbf{H})$ for the stable (∞,1)-category of spectrum objects in $\mathbf{H}$;
$CRing(\mathbf{H}) \coloneqq E_\infty Alg(Sp(\mathbf{H}))$ for the (∞,1)-category of E-∞ ring objects in $\mathbf{H}$;
$E Mod(\mathbf{H})$ for the symmetric monoidal (∞,1)-category of $E$-∞-modules in $\mathbf{H}$.
For $\mathbb{G} \in Grp(\mathbf{G}) \stackrel{forget}{\to} \mathbf{H}$ write $\mathbf{H}_{/\mathbb{G}}$ for the slice (∞,1)-topos of $\mathbf{H}$ over $\mathbb{G}$. This carries apart from its canonical structure of a cartesian monoidal (∞,1)-category $(\mathbf{H}_{/\mathbb{G}}, \times_{\mathbb{G}})$ also a second structure of a symmetric monoidal (∞,1)-category $(\mathbf{H}_{/\mathbb{G}}, \otimes_{\mathbb{G}})$, obtained using the monoid structure on $\mathbb{G}$:
This makes $(\mathbf{H}_{/\mathbb{G}}, \otimes_{\mathbb{G}})$ a monoidal (∞,1)-topos.
For $\mathcal{C}$ any (∞,1)-category with (∞,1)-pullbacks write $Corr(\mathcal{C})$ for the (∞,1)-category of correspondences in $\mathcal{C}$, whose objects are those of $\mathcal{C}$, whose morphisms are diagrams in $\mathcal{C}$ of the form
and composition is given by (∞,1)-fiber product of such diagrams.
If $\mathcal{C}$ is equipped with the stucture of a monoidal (∞,1)-category $(\mathcal{C}, \otimes)$, then $Corr(\mathcal{C})$ naturally inherits a monoidal structure, too, given by the objectwise tensor product in $\mathcal{C}$.
Notice that even if $(\mathcal{C}, \times)$ is a cartesian monoidal (∞,1)-category then the induced monoidal structure $(Corr(\mathcal{C}), \otimes)$ is not cartesian. Notice also that if $(\mathcal{C}, \otimes)$ is symmetric monoidal, then so is the induced structure on $Corr(\mathcal{C})$.
In the following we consider $Corr(\mathbf{H}_{/\mathbb{G}})$ always equipped with the monoidal (∞,1)-category structure which is induced by the non-cartesian tensor monoidal structure $\otimes_{\mathbb{G}}$ on $\mathbf{H}_{/\mathbb{G}}$.
For $\mathcal{C}$ any (∞,1)-category, write $\mathcal{C}^{\Box^n}$ for the (∞,1)-category of $n$-dimensional cube diagrams in $\mathcal{C}$. Under pasting of diagrams this is naturally an (∞,n)-category.
We write for short
This is a symmetric monoidal (∞,n)-category.
There is a pair of adjoint (∞,1)-functors
where $\mathbf{GL}_1(-)$ forms the ∞-group of units (see there) of an E-∞ ring object.
Choose now once and for all
$n \in \mathbb{N}$
$E \in CRing(\mathbf{H})$.
Write $Bord_n \coloneqq Bord_n(\{\ast\})$ for the (∞,n)-category of framed cobordisms.
By (Nuiten-Schreiber 13), following (Schreiber 08, Freed-Hopkins-Lurie-Teleman 09, section 3) we say
a physical field of a local prequantum field theory of dimension $n$ is a monoidal (∞,n)-functor
a local prequantum field theory of dimension $n$ with field content $\mathbf{Fields}$ and with phases in $\mathbf{GL}_1(E)$ is a lift $\exp(i S)$ in the diagram
More generally, let $def$ be a set of defect data, hence a set of shapes of cells. Write $Bord_n(def)$ for the (∞,n)-category of cobordisms with this defect structure, hence, by the cobordism hypothesis theorem, the symmetric monoidal (∞,n)-category with all fully dualizable objects freely generated on $def$. Then a diagram of symmetric monoidal (∞,n)-categories of the form
is a boundary/defect local prequantum field theory.
For instance for
such $\exp(i S)$ is equivalently the choice of a correspondence of the form
This is the local incarnation of the corresponding boundary condition/brane.
Or if
then $Bord_n(def)$ consists of cobordisms with two different boundary (brane) types and a defect where they meet.
Let $E$ be an E-∞ ring, and write $GL_1(E)$ for its ∞-group of units. With $\mathbf{H}$ the ambient (∞,1)-topos, write $\mathbf{H}_{/\mathbf{B}GL_1(E)}$ for the slice (∞,1)-topos over the delooping of this abelian ∞-group. This is the (∞,1)-category of spaces equipped with (∞,1)-line bundles over $E$. Consider an (∞,1)-functor
to the (∞,1)-category of (∞,1)-modules over $E$, which form $E$-modules of co-sections of $E$-(∞,1)-module bundles (generalized Thom spectra).
This is well understood for $\mathbf{H} =$ ∞Grpd in which case $\Gamma \simeq \underset{\to}{\lim} \circ i$ is the (∞,1)-functor homotopy colimits in $E Mod$ under the canonical embedding $\mathbf{B} GL_1(E) \simeq E Line \hookrightarrow E Mod$. But one can consider similar constructions $\Gamma$ for more general ambient (∞,1)-toposes $\mathbf{H}$.
For $\chi_i \colon X_i \to \mathbf{B}GL_1(E)$ two objects of $\mathbf{H}_{/\mathbf{B}GL_1(E)}$, the $(\chi_1,\chi_2)$-twisted bivariant $E$-cohomology on $(X_1,X_2)$ is
By the general discussion at twisted cohomology, following (ABG, def. 5.1) we have
for $X_2 = \ast$ the point, the above bivariant cohomology is the $\chi_1$-twisted $E$-cohomology of $X_1$;
for $X_1 = \ast$ the point, the above bivariant cohomology is the $\chi_2$-twisted $E$-homology of $X_2$;
KK-theory is a model for bivariant twisted topological K-theory over differentiable stacks (hence 1-truncated suitably representable objects in $\mathbf{H} =$ Smooth∞Grpd, see Tu-Xu-LG 03). According to (Joachim-Stolz 09, around p. 4) the category $KK$ first of all is naturally an enriched category $\mathbb{KK}$ over the category $\mathcal{S}$ of symmetric spectra and as such comes with a symmetric monoidal enriched functor
This sends an object to its operator K-theory spectrum, hence to the $E$-dual of the $E$-module of co-sections.
Generally, one may want to consider in def. the dualized co-section functor
A correspondence in $\mathbf{H}_{/\mathbf{B}GL_1(E)}$
is a morphism of “twisted $E$-motives” in that it is a correspondence in $\mathbf{H}$ between the spaces $X_1$ and $X_2$ equipped with an $(i_1^\ast \chi_1, i_2^\ast \chi_2)$-twisted bivariant $E$-cohomology cocycle $\xi$ on the correspondence space $Q$. Under the co-sections / Thom spectrum functor this is sent to a correspondence
in $E Mod$. If the wrong-way map of this is orientable in $E$-cohomology then we may form its dual morphism/Umkehr map to obtain the corresponding “index”
in $E Mod$. Identifying correspondences that yield the same “index” this way yields a presentation of bivariant cohomology by motive-like structures. This is how (equivariant) bivariant K-theory is presented, at least over manifolds, see at KK-theory – References – In terms of correspondences.
quantizes to
(…)
partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:
We discuss (twisted) ordinary homology and ordinary cohomology in terms of sections of (∞,1)-module bundles over the Eilenberg-MacLane spectrum.
Let $k$ be a commutative ring. Write
for the Eilenberg-MacLane spectrum of $k$, canonically regarded as an E-∞ ring. Write
for the (∞,1)-category of (∞,1)-modules over $H k$.
There is an equivalence of (∞,1)-categories
between the (∞,1)-category of (∞,1)-modules over the Eilenberg-MacLane spectrum $H k$ and the simplicial localization of the category of unbounded chain complexes of ordinary (1-categorical) $k$-modules.
This is the statement of the stable Dold-Kan correspondence, see at (∞,1)-category of (∞,1)-modules – Properties – Stable Dold-Kan correspondence.
Let $X$ be a topological space and write $\Pi(X) \in$ ∞Grpd for its underlying homotopy type (its fundamental ∞-groupoid). Then we say that an (∞,1)-functor
is (the higher parallel transport) a flat (∞,1)-module bundle over $X$, or a local system of $H k$-(∞,1)-modules over $X$.
(Riemann-Hilbert correspondence)
If $X$ is an oriented closed manifold, then there is an equivalence of (∞,1)-categories
between flat (∞,1)-module bundles/local systems and L-∞ algebroid representations of the tangent Lie algebroid of $X$. From right to left the equivalece is established by sending an L-∞ algebroid representation given (as discussed there) by a flat $\mathbb{Z}$-graded connection on bundles of chain complexes (via prop. ), to its higher holonomy defined in terms of iterated integrals.
This is the main theorem in (Block-Smith 09).
Write
for the (∞,1)-colimit functor.
We may think of $\Gamma$ equivalently as
forming flat sections of a flat (∞,1)-module bundle;
sending a flat (∞,1)-module bundle to its Thom spectrum (see at Thom spectrum – For (∞,1)-module bundles).
Write
for the flat (∞,1)-module bundle which is constant on the chain complex concentrated on $k$ in degree 0, the tensor unit in $[\Pi(X), (H k)Mod] \simeq [\Pi(X), L_{qi}Ch_\bullet(k Mod)]$.
For $X$ a topological space, we have a natural equivalence (with the identification of prop. understood) of the form
between the $(H k)$-(∞,1)-module of sections of the trivial $(H k)$-(∞,1)-module bundle $\mathbb{I}_X^{H k}$ and the singular chain complex of $X$ for ordinary homology with coefficients in $k$.
This is a classical basic (maybe folklore) statement. Here is one way to see it in full detail.
First notice that the (∞,1)-colimit of functors out of ∞-groupoids and constant on the tensor unit in $(H k)Mod$ is by definition the (∞,1)-tensoring operation of $(H k)Mod$ over ∞Grpd. Now if we find a presentation of $(H k)Mod$ by a simplicial model category the by the dicussion at (∞,1)-colomit – Tensoring and cotensoring – Models this (∞,1)-tensoring is given by the left derived functor of the sSet-tensoring in that simplicial model category.
To obtain this, use prop. and then the discussion at model structure on chain complexes in the section Projective model structure on unbounded chain complexes which says that there is a simplicial model category structure on the category of simplicial objects in the category of unbounded chain complexes which models $L_{qi} Ch_\bullet(k Mod)$, and whose weak equivalences are those morphisms that produce quasi-isomorphism under the total chain complex functor.
In summary it follows that with any simplicial set $(\Pi(X))_\bullet in L_{whe} sSet$ representing $\Pi(X)$ (under the homotopy hypothesis-theorem) we have
where on the right we have the coend over the simplex category of the tensoring (of simplicial sets with simplicial objects in the category of unbounded chain complexes) of the standard cosimplicial simplex with the simplicial diagram constant on the tensor unit chain complex.
The result on the right is manifestly, by the very definition of singular homology, under the ordinary Dold-Kan correspondence the chain complex of singular simplices:
If $X \in L_{whe} Top$ is a Poincaré duality space of dimension $n$, then $\Gamma(\mathbb{I}_X^{H A}) \in (H A) Mod$ is a dualizable object which is almost self-dual except for a degree twist of (itself) degree 0:
Under the identifications of prop. and prop. this is the theorem discussed at Poincaré duality in the section Poincaré duality – Refinement to homotopy theory.
See also at Dold-Thom theorem.
We discuss here aspects of “smooth” (not yet differential!) K-theory, namely the refinement of K-theory to smooth groupoids (a generalization of equivariant K-theory).
Write
for the periodic complex K-theory spectrum.
The complex K-theory spectrum $KU$ is the localization of the ∞-group ∞-ring of the circle 2-group at the Bott element
This is Snaith's theorem.
There is a canonical map
This is observed in (Nuiten 13, section 3).
The localization map
in $CRing(\infty Grpd)$ has by prop. an adjunct of the form
in $AbGrp(\infty Grpd)$. By the looping and delooping theorem and by prop. this is equivalently a map
in ∞Grpd.
Via the canonical smooth refinement $\mathbf{B}U(1)$ of the circle 2-group $B U(1)$ the construction in prop. induces a canonical smooth $E_\infty$-ring
Since the shape modality $\Pi$ preserves all (∞,1)-colimits, it follows that
hence that $\mathbf{KU}$ is indeed a smooth refinement of KU.
(Thanks to Thomas Nikolaus for discussion of this point.)
We expect that for $X \in$ Smooth∞Grpd a Lie groupoid that maps $X \to \mathbf{KU}$ represent “smooth” K-theory classes on $X$ as represented by the groupoid K-theory of $X$ or equivalently the operator K-theory of its groupoid convolution algebra. This still needs to be show. But motivated by this, we take the following as the linearization functor in K-theory:
There is a lax monoidal functor
from the category of differentiable stacks (Lie groupoids with Morita morphisms) equipped with circle 2-bundle and with proper maps between them to the opposite of the homotopy category of KU-∞-modules which factors by a strong monoidal functor through KK-theory
and which
sends a Lie groupoid $X$ with circle 2-bundle $\chi$ first its twisted groupoid convolution algebra $C_\chi(X) \in$ C*Alg and then produces the operator K-theory-spectrum of that algebra as discussd at KK-theory in the section Triangulated and KU-module structure;
sends a morphism of Lie groupoids to the Hilbert bimodule of sections of the corresponding Hilsum-Skandalis bibundle, regarded as a morphism in KK-theory.
The functor $\Gamma$ is strong monoidal at least when restricted to amenable Lie groupoids, where moreover it factors through a full inclusion of the KK-bootstrap category
The functor $C(-) \colon DiffStack_{/\mathbf{B}^2 U(1)}^{prop} \to KK$ is discussed in (Alldridge-Giansiracusa 06, Nuiten 13, section 3), based on (Tu-Xu-LG 03). The functor $KK \to Ho(KU Mod)$ is due to (DEKM 11, section 3). The statement about the factorization through the bootstrap category is (Tu 99) combined with (DEKM 11, section 3).
In particular, on a local quotient groupoid the functor of prop. reproduces the twisted groupoid K-theory defined via equivariant vector bundles in (FHT, I, section 3.2) and on general Lie groupoids the twisted groupoid K-theory of (Tu-Xu-LG 03).
Functoriality of
requires that pullback in twisted groupoid K-theory satisfies the Beck-Chevalley condition, which says that push-pull is equivalent to pull-push through the fiber product.
Here we list sufficient conditions for this to be the case
for untwisted correspondences of manifolds: (Connes-Skandalis 84)
for untwisted correspondences of equivariant manifolds: (Emerson-Meyer 08)
for twisted correspondences of moduli stacks of flat connections: (Freed-Hopkins-Teleman 07)
(…)
(…) elliptic cohomology, tmf (…) sigma-orientation, string orientation of tmf (..) elliptic genus, Witten genus (…)
We spell out and discuss examples and applications of the general method.
Here we discuss the notion of prequantized Lagrangian correspondences and how it serves to embed traditonal Hamiltonian mechanics and Lagrangian mechanics into the general context of local prequantum field theory and its motivic quantization.
A traditional notion is that of a plain Lagrangian correspondence, which is a Lagrangian submanifold of the Cartesian product of a symplectic manifold and another one, with opposite symplectic structure. As discussed there, plain Lagrangian correspondences serve to encode symplectomorphisms – hence transfomations between phase spaces in physics – via correspondences of symplectic manifolds.
But in prequantum field theory proper, and in particular with an eye towards geometric quantization, one considers the prequantization of these symplectic manifolds by lifting them to prequantum circle bundles with principal connection. The notion of prequantized Lagrangian correspondence is the refinement of that of plain Lagrangian correspondence which does properly respect and reflect this prequantization information: a prequantized Lagrangian correspondence is a Lagrangian subspace as before, but now equipped with an explicit gauge transformation between the pullbacks of the two prequantum circle bundles to the correspondence space.
We discuss below how the concept of prequantized Lagrangian correspondences neatly unifies “classical mechanics” formulated in terms of symplectic geometry of phase space with Hamiltonian flows on them with the Hamilton-Jacobi equation for such flows.
Then we show how also the notion of prequantized Lagrangian correspondence is – still naturally in the context of local prequantum field theory – further refined from the context of symplectic manifolds to that of Poisson manifolds. Specifically, this is obtained by realizing that prequantized Lagrangian correspondences are really naturally to be regarded as correspondences-of-correspondences in a 2-category of correspondences, where now the new lower-order correspondences are instead boundary field theories for a 2d Chern-Simons theory (a non-perturbative Poisson sigma-model).
We write $(X,\omega)$ for a symplectic manifold with underlying smooth manifold $X$ and symplectic differential 2-form $\omega \in \Omega^2_{cl}(X)$. In physics this models the phase space of a mechanical system.
The sigma-model describing the propagation of a particle on the real line $\mathbb{R}$ has as phase space the plane $\mathbb{R}^2 = T^\ast \mathbb{R}$ and as symplectic form its canonical volume form. Traditionally the two canonical coordinate functions on this phase space are denoted $q,p \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R}$ (called the “canonical coordinate” and the “canonical momentum”, respectively), and in terms of these the symplectic form in this example is $\omega = \mathbf{d} q \wedge \mathbf{d} p$.
Given two symplectic manifolds $(X_1, \omega_1)$ and $(X_2, \omega_2)$ (which might well be two copies of one single symplectic manifold), a symplectomorphism between them
is a diffeomorphism
of the underlying smooth manifolds, such that the pullback of the second symplectic form along $f$ equals the first,
In physics symplectomorphisms are traditionally known as canonical transformations. Or more precisely, a canonical transformation is a symplectomorphism of the special kind called a Hamiltonian symplectomorphism. This we come to below after prequantization.
For the present purpose it is useful to formulate symplectomorphisms in the language of the topos $\mathbf{H}$ of smooth spaces. This is discussed in much detail at geometry of physics in the section Smooth spaces.
In terms of this there is a smooth universal moduli space $\Omega^2_{cl}$ of closed differential 2-forms, and by the Yoneda lemma one such $\omega \in \Omega^2_{cl}(X)$ is equivalently a homomorphism of the form
This is explained in detail at geometry of physics in the section Differential forms.
In this language we have:
A symplectomorphism $f \colon (X_1, \omega_2) \longrightarrow (X_2, \omega_2)$ as above is equivalently a commuting diagram of the form
Yet another equivalent way to say this is that both $(X_1, \omega_1)$ and $(X_2, \omega_2)$ are naturally objects in the slice topos $\mathbf{H}_{/\Omega^2_{cl}}$ and that a symplectomorphism is equivalently just a morphism between them, in this slice topos.
Since this makes the two symplectic manifolds correspond to each other, it is useful to express this as in the formal sense as a correspondence. For any smooth function $f \colon X_1 \to X_2$, the natural choice of correspondence space is the graph of $f$, which we may depict as a subobject of the Cartesian product
or better as a correspondence span diagram
Traditionally one considers now the Cartesian product manifold $X_1 \times X_2$ itself as a symplectic manifold equipped with the pullback symplectic form
and observes then that the diffeomorphism $f$ being a symplectomorphism is equivalent to its graph being a Lagrangian submanifold of this product symplectic manifold. In that case the above correspondence is called a Lagrangian correspondence between $(X_1, \omega_1)$ and $(X_2, \omega_2)$.
In the language of the topos $\mathbf{H}$ of smooth spaces, this has a more evident formulation: that $graph(f)$ is an isotropic subspace equivalently means that there is a commuting diagram of smooth spaces of the following form
This in turn is equivalent to being a correspondence in the slice topos $\mathbf{H}_{/\Omega^2_{cl}}$.
An important class of symplectomorphisms are the Hamiltonian symplectomorphisms from a symplectic manifold to itself, those which are the flow of a Hamiltonian vector field on $(X,\omega)$ induced by a Hamiltonian function
Using the Poisson bracket $\{-,-\}$ induced by the symplectic form $\omega$ and identifying the derivation $\{H,-\}$ with the corresponding Hamiltonian vector field and the exponent notation $\exp(t \{H,-\})$ with the corresponding flow for parameter “time” $t \in \mathbb{R}$, we may write these as
Here we refer to Lagrangian correspondences induced from Hamiltonian symplectomorphisms as Hamiltonian correspondences.
The smooth correspondence space of a Hamiltonian correspondence is naturally identified with the space of classical trajectories
in that
every point in the space corresponds uniquely to a trajectory of parameter time length $t$ characterized as satisfying the equations of motion as given by Hamilton's equations for $H$;
the two projection maps to $X$ send a trajectory to its inital and to its final configuration, respectively.
Forming Hamiltonian correspondences consitutes a functor from 1-dimensional cobordisms with Riemannian structure to the category of correspondences in the slice topos:
since for all (“time”) parameter valued $t_1, t_2 \in \mathbb{R}$ we have a composition (by fiber product) of correspondences exhibited by the following pasting diagram:
But the reason to consider Hamiltonian symplectomorphisms instead of general symplectomorphisms is really because these give homomorphisms not just between plain symplectic manifold, but between their prequantizations. To these we turn now.
A prequantization of a symplectic manifold $(X,\omega)$ is – if it exists – a choice of circle group-principal connection $\nabla$ on $X$ whose curvature 2-form is the given symplectic form
In the topos of smooth spaces, or rather in the (2,1)-topos $\mathbf{H}$ of smooth groupoids, this means that a prequantization is a lift $\nabla$ in the diagram
where $\mathbf{B}U(1)_{conn}$ is the moduli stack of circle bundle with connection. For details on this see at geometry of physics the section Smooth homotopy types.
Then for two prequantized symplectic manifolds, it is now clear what a prequantized correspondence between them is:
A prequantization of a Lagrangian correspondence $Y \colon (X_1,\omega) \to (X_2,\omega_2)$ is a diagram in $\mathbf{H}$ of the form
hence a correspondence in the slice (2,1)-topos $\mathbf{H}_{/\mathbf{B}^2 U(1)_{conn}}$.
The natural question now is which Hamiltonian correspondences may be prequantized and what the corresponding prequantum data is. The following proposition shows that the prequantization of the Hamiltonian correspondence given by a Hamiltonian $H$ is given by the exponentiated action functional associated with $H$, namely the exponentiated integral over its Lagrangian $L$, which is its Legendre transform $L = p \frac{\partial H}{\partial p} - H$.
Of course all the ingredients in the statement and in the proof od the following proposition are classical. But the notion of prequantized Lagrangian correspondence serves to neatly unify these ingredients and give them a natural place in the context of local prequantum field theory which later naturally leads to the formulation of higher local prequantum field theory and its motivic quantization.
Consider the phase space $(\mathbb{R}^2, \; \omega = \mathbf{d} q \wedge \mathbf{d} p)$ equipped with its canonical prequantization by $\theta = p \mathbd{d}q$. Then for $H \colon \mathbb{R}^2 \to \mathbb{R}$ a Hamiltonian, and for $t \in \mathbb{R}$ a parameter (“time”), a lift of the Lagrangian correspondence $\exp(t \{H,-\})$ to a prequantized Lagrangian correspondence is given by
where
$S_t \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R}$ is the action functional of the classical trajectories induced by $H$,
which is the integral $S_t = \int_{0}^t L \, d t$ of the Lagrangian $L \,d t$ induced by $H$,
which is the Legendre transform
In particular, this induces a functor
The canonical prequantization of $(\mathbb{R}^2, \mathbf{d} q \wedge \mathbf{d} p)$ is the globally defined connection 1-form
We have to check that on $graph(\exp(t\{H,-\}))$ we have the equation
Or rather, given the setup, it is more natural to change notation to
Notice here that by the nature of $graph(\exp(t\{H,-\}))$ we can identify
and under this identification
and
It is sufficient to check the claim infinitesimally. So let $t = \epsilon$ be an infinitesimal, hence such that $\epsilon^2 = 0$. Then the above is Hamilton's equations and reads equivalently
and
Using this we compute
In summary, prop. and remark say that a prequantized Lagrangian correspondence is conceptually of the following form
By a naive counting, Lagrangian correpondences, modelling mechanics and supposed to quantize to quantum mechanics hence 1-dimensional quantum field theory, should be motivically quantized over HC, the ordinary cohomology spectrum. However, $H\mathbb{C}$ does admit twists only by $\flat \mathbf{B}U(1)$ (flat connections) but not by $\mathbf{B}U(1)_{conn}$ (general $U(1)$-principal connections with curvature), see at twisted ordinary cohomology. However the special case where the curvature $\omega$ vanishes no longer corresponds to symplectic manifolds, and even when regarded as a Poisson manifold it is not interesting.
Below we see that the quantization of symplectic manifolds and generally Poisson manifolds is instead to be regarded holographically as the boundary field theory of the Poisson-2d Chern-Simons theory. In this context, Lagrangian correspondences are to be regared as secretly being 2-dimensional correspondences of the form
Here on the right we have the corresponding symplectic groupoid. See below (…)
For more see at prequantized Lagrangian correspondence.
(…) string topology (…)
We show here how the traditional quantization of Poisson manifolds is reproduced and refined within motivic quantization. This proceeds in a holographic fashion, where the Poisson manifold is regarded as topological boundary/brane of the 2d Chern-Simons theory which is the non-perturbative version of the corresponding Poisson sigma-model, as discussed in (Fiorenza-Rogers-Schreiber 13a, section 4).
(…) extended geometric quantization of 2d Chern-Simons theory (…)
Let $(X,\pi)$ be a Poisson manifold. The Lie integration of the corresponding Poisson Lie algebroid $\mathfrak{P}$ is called the corresponding symplectic groupoid:
If the canonical 3-class of $\mathfrak{P}$ is suitably integral, then this carries a prequantum 2-bundle
Here $SymplGrpd(X,\pi)$ may be regarded as the moduli stack of (instanton sectors of) the non-perturbative Poisson sigma-model. This non-perturbative theory is the “2d Chern-Simons theory” induced by $\mathfrak{P}$.
There is a canonical inclusion $X \to SymplGrpd(X,\pi)$ and one finds that $\chi$ trivializes when restricted to this inclusion. Therefore the original Poisson manifold is a topological brane for the 2d Chern-Simons theory, with the boundary condition formally exhibited by a correspondence of the form
Here $\nabla$ is hence a kind or prequantum bundle applicable to Poisson manifolds.
Therefore we can quantize the original Poisson manifold by applying motivic quantization to this boundary condition for 2d Chern-Simons theory. The result is an element in the K-theory of the symplectic groupoid.
We discuss this now for some cases of interest. First we show how this reproduces and refines traditional geometric quantization of symplectic manifolds. Then we show how this reproduces in particular the orbit method for the construciton of irreducible representation of Lie groups from the geometric quantization of their coadjoint orbits.
We consider the special case of the above motivic quantization of Poisson manifolds where the Poisson manifold $(X,\pi)$ happens to be a symplectic manifold $(X,\omega^{-1})$.
The symplectic groupoid of a symplectic manifold is equivalent to the point
Accordingly, the boundary condition for the Poisson 2d Chern-Simons theory induced by a Poisson manifold which is symplectic is a correspondence of the form
and so is equivalently given by a prequantum bundle
Traditionally the strict deformation quantization of a symplectic manifold with a good polarization $\mathcal{P}$ is taken to be the C*-algebra of compact operators on the leaf space $X/\mathcal{P}$. In (EH 06) this is rederived as the polarized groupoid convolution algebra of the symplectic groupoid of $(X,\omega^{-1})$ naturally presented as the pair groupoid $SymplGrpd(X,\omega^{-1}) \simeq Pair(X/\mathcal{P})$.
However, once one passes from smooth manifolds to Lie groupoids, it is unnatural to essentially distinguish the pair groupoid from the point, since both are Morita equivalent. Notably, in analogy to this the algebra of compact operators is of course itself Morita equivalent to the base algebra of complex numbers.
As a consequence, the traditional description of the strict deformation quantization of symplectic manifolds in the refined perspective of (EH 06) contains a conundrum: either one unnaturally breaks the natural Morita equivalence or else one arrives at a trivial quantization.
In the following we see that this is resolved in motivic quantization. Here is is not just the symplectic groupoid itself that is quantized, but the correspondence $\ast \leftarrow X \to SymplGrp(X,\omega^{-1})$ which exhibits the original symplectic manifold as a boundary of the corresponding 2d Chern-Simons theory. While the quantization of $SymplGrpd(X,\omega^{-1})$ itself is trivial when properly regarded in higher geometry, that of $\ast \leftarrow X \to SymplGrpd(X,\omega^{-1})$ is not and in fact yields the correct quantization.
A conceptual way to understand this phenomenon is to recall that the symplectic groupoid of a Poisson manifold is effectively a stacky model for the leaf space by symplectic leafs of the Poisson manifold. Since for symplectic Poisson manifolds the leaf space is the point, accordingly so is the symplectic groupoid, up to equivalence.
In order to motivically quantize, we need an orientation in K-theory of $X$. This is precisely a spin^c structure, which in turn is induced notably from a Kähler polarization of $(X,\omega^{-1})$.
One then observes that the push-forward of $\nabla$ to the point in K-theory reproduces the traditional geometric quantization of $(X,\omega^{-1})$.
We discuss now the space of quantum states
and then the quantum observables
space of quantum states | quantum observables |
---|---|
index in K-theory | Dirac induction: index in equivariant K-theory |
$(X,\omega)$ a symplectic manifold
A K-orientation on $i$ is equivalently a spin^c structure on $X$, in particular induced from a Kähler polarization of $(X,\omega)$.
The corresponding push-forward is
and sends the prequantum bundle to the space of quantum states.
This is discussed in detail at geometric quantization – quantum states.
We now discuss how the traditional quantization of observables to quantum observables on symplectic manifolds is reproduced by motivic quantization of the symplectic manifold regarded as a brane of its Poisson-2d Chern-Simons theory.
Let $(X,\omega)$ be a symplectic manifold. Let $\nabla \colon X \to \mathbf{B}U(1)_{conn}$ be a prequantum bundle.
A naive infinitesimal prequantum observable of $X$ is just a smooth function on $X$. An exponentiated prequantum observable on $X$ is an element $O \in QuantMorph(X,\nabla)$ of the quantomorphism group of $(X,\omega)$. As observed in (Fiorenza-Rogers-Schreiber 13a), this is an equivalece in the slice
A Lie group $G$ of prequantum observables is hence a homomorphism
If $\mathfrak{g} = Lie(G)$ is the Lie algebra of $G$, then this is the Lie integration of a $\mathfrak{g}$-moment map hence a $G$-Hamiltonian action on $(X,\omega)$.
A group $G \to QuantMorph(\nabla)$ of exponentiated prequantum operators, hence an integrated $G$-Hamiltonian action is equivalently an equivariant structure $\nabla//G$ on the prequantum bundle $\nabla$ given by
On the level of de Rham cocycles essentially this was observed in (Atiyah-Bott 84, prop 6.18).
If follows that with a $G$-Hamiltonian action of prequantum operators given, the above situation refines to a brane that is given by a correspondence of action groupoids of the following form
The original orientation in K-theory extends to an orientation in equivariant K-theory if it is itself equivariant, hence if the prequantum operators in $G$ respect the polarization. This is precisely the standard condition for prequantum operators to qualify as quantum operators.
Hence if $X$ carries a $G$-equivariant spin^c structure then we have a push-forward in generalized cohomology of the form
(“Dirac induction”). Here we used on the left that the convolution algebra of an action groupoid $X//G$ is equivalently the corresponding crossed product C*-algebra (as discussed there), and the Green-Julg theorem which identifies the operator K-theory of that with the equivariant K-theory of $X$. On the right we used the same identification and the fact that hence $K(\ast//G)$ is the representation ring of the group $G$.
So the motivic quantization
of the $G$-Hamiltonian action is an element in $Rep(G)$ a space of quantum states equipped with the $G$-action of quantum observables in $G$.
A particularly interesting case of geometric quantization of symplectic manifolds is the quantization of coadjoint orbits of suitable Lie groups. Physically this is the quantization of the “internal degrees of freedom” of Wilson loop defect in Chern-Simons theory (see at orbit method – Nonabelian charged particle trajectories). Mathematically this is a process that yields all the irreducible representations of these Lie groups, as such called the orbit method.
By the above discussion of the motivic holographic quantization of symplectic manifolds, the geometric quantization of any single coadjoint orbit is naturally described by motivic quantization. But moreover, we discuss now how each such orbit boundary theory is naturally related by a defect to the 2d Chern-Simons theory which is given by the dual Lie algebra regarded as a Poisson manifold with its Lie-Poisson structure. The motivic quantization of this defect reproduces a map in twisted equivariant K-theory that appeared as (Freed-Hopkins-Teleman, part II, theorem 1.28): it is in the quantization language used here a “quantum operator” that sends boundary states of the $\mathfrak{g}^\ast//G$-2d Chern-Simons theory to states of the coadjoint orbit, respecting the action of the $G$-quantum observables. As such the quantization of the defects of the $\mathfrak{g}^\ast//G$-2d CS theory is a “universal orbit method” for $G$ in that it contains the quantization of each coadjoint orbit to each representation as the quantization of one of its defects.
$\,$
Let $\mathfrak{g}$ be a Lie algebra. Write $\mathfrak{g}^\ast$ for its dual vector space regarded as a Poisson manifold by its canonical Lie-Poisson structure.
The crucial fact that drives the following discussion is the following.
The symplectic groupoid of the Lie-Poisson manifold $\mathfrak{g}^\ast$ is the action groupoid $\mathfrak{g}^\ast //G$ for the coadjoint action
(Bursztyn-Crainic, example 4.3)
The manifold of morphisms of $(\mathfrak{g}^\ast//G)$ in the standard presentation is $\mathfrak{g}^\ast \times G$, which by left translation is naturally identified with the cotangent bundle of $G$:
The canonical prequantum 2-bundle
is given by the truncated Deligne cohomology cocycle whose 1-form component is the Liouville-Poincaré 1-form on the cotangent bundle $T^\ast G$ under the natural identification
This trivializes along the canonical inclusion $\mathfrak{g}^\ast \to \mathfrak{g}^\ast //G$ and therefore we have a boundary condition
where $\xi$ is a circle bundle with connection, the corresponding “prequantum bundle”.
Assume now that $\mathfrak{g}$ is semisimple with Killing form invariant polynomial $\langle -,-\rangle$. Let $\lambda \in \mathfrak{g}$ be a regular weight and $\mathcal{O}_\lambda \hookrightarrow \mathfrak{g}^\ast$ the corresponding coadjoint orbit.
Then the previous boundary condition for the 2d Chern-Simons theory of $\mathfrak{g}^\ast//G$ extends to a defect to the 2d CS theory of $\ast //G$ with in turn is the symplectic manifold $\mathcal{O}_\lambda$ as its brane. This is exhibited by the diagram
in $\mathbf{H}$ (where the bottom right square is an (∞,1)-pullback). All this lives is $\mathbf{H}_{/\mathbf{B}^2 U(1)}$, but we don’t try to draw this here.
Here on the left we have an equivariant symplectic case as above
Where $\xi$ now is a $G$-equivariant prequantum bundle exhibiting a moment map for a Hamiltonian action of $G$.
The motivic quantization yields the map in equivariant K-theory
to the representation ring of $G$, which sends the prequantum bundle + Hamiltonian action $\xi$ to the $G$-representation $i_! \xi$. This is known as Dirac induction. It is the cohomological formulation of the orbit method for a fixed orbit.
On the other hand the motivic quantization horizontally of the bottom correspondence of 2d Chern-Simons theories in the above diagram, which is
About this, (FHT II, (1.27), theorem 1.28) says the following.
For $G$ a compact Lie group with Lie algebra $\mathfrak{g}^\ast$, the push-forward in compactly supported twisted $G$-equivariant K-theory to the point (the $G$-equivariant index) produces the Thom isomorphism
Moreover, for $i \colon \mathcal{O} \hookrightarrow \mathfrak{g}^\ast$ a regular coadjoint orbit, push-forward involves a twist $\sigma$ of the form
and
$i_!$ is surjective
$ind_{\mathcal{O}} = ind_{\mathfrak{g}^\ast} \circ i_!$.
The trajectory space of any mechanical system carries a natural Poisson bracket: the “off-shell Poisson bracket”. When one considers this as the boundary field theory of the corresponding Poisson 2d Chern-Simons theory as above, then one finds the following relation:
the fields of the 2d bulk field theory are the sources of the 1d mechanical system. This relation is one of the characteristic properties of what is called the holographic principle.
More discussion of this formalization of the holographic principle is at
$X$ spacetime
$\chi \colon X \to \mathbf{B}^2 U(1)$ B-field
this is the background gauge field/local action functional for the type II string 2d sigma-model field theory
$i \colon Q \hookrightarrow X$ D-brane worldvolume
$\xi \in K^{\bullet + i^\ast \chi}(Q)$ Chan-Paton gauge field
Umkehr map yields
compatibility is Freed-Witten-Kapustin anomaly cancellation
We interpret the Freed-Witten-Kapustin anomaly mechanism in terms of push-forward in generalized cohomology in topological K-theory interpreted in terms of KK-theory with push-forward maps given by dual morphisms between Poincaré duality C*-algebras (based on Brodzki-Mathai-Rosenberg-Szabo 06, section 7, Tu 06):
Let $i \colon Q \to X$ be a map of compact manifolds and let $\chi \colon X \to B^2 U(1)$ modulate a circle 2-bundle regarded as a twist for K-theory. Then forming twisted groupoid convolution algebras yields a KK-theory morphism of the form
with notation as in this definition. By this proposition the dual morphism is of the form
If we redefine the twist on $X$ to absorb this “quantum correction” as $\chi \mapsto \frac{1}{\chi \otimes W_3(T X)}$ then this is
where now we may interpret $\frac{W_3(i^\ast \tau_X)}{W_3(\tau_Q)}$ as the third integral Stiefel-Whitney class of the normal bundle $N Q$ of $i$ (see Nuiten).
Postcomposition with this map in KK-theory now yields a map from the $i^\ast \chi \otimes W_3(N Q)$-twisted K-theory of $Q$ to the $\chi$-twisted K-theory of $X$:
If we here think of $i \colon Q \hookrightarrow X$ as being the inclusion of a D-brane worldvolume, then $\chi$ would be the class of the background B-field and an element
is called (the K-class of) a Chan-Paton gauge field on the D-brane satisfying the Freed-Witten-Kapustin anomaly cancellation mechanism. (The orginal Freed-Witten anomaly cancellation assumes $\xi$ given by a twisted line bundle in which case it exhibits a twisted spin^c structure on $Q$.) Finally its push-forward
is called the corresponding D-brane charge.
In the previous example The charged particle at the boundary of the superstring we saw – by analogy with the particle at the boundary of the 2d Chern-Simons theory – the motivic quantization with coefficients in KU of the 1-dimensional field which is the sigma-model of a charged particle at the intersection of a 2-dimensional field theory which is an open string sigma-model ending on a D-brane. Here we lift all these ingredients up one dimension and consider the motivic quantization with coefficients in tmf of the 2-dimensional field theory called the heterotic string sigma-model as the boundary intersection of the 3d field theory known as the M2-brane with a brane called sometimes the M9-brane or the $O9$-plane of Hořava-Witten theory.
The analog of the $\mathbb{Z}$-valued index/partition function in the K-theory of the point now is the Witten genus with values in topological modular forms
and the analog of the D-brane charge in the K-theory of the type II supergravity spacetime is
in tmf-cohomology of the 11-dimensional supergravity spacetime.
$\,,$
More in detail, the setup is the following.
Let $X$ be a $\mathbb{Z}_2$-orbifold equipped with spin structure, to be called the 11-dimensional spacetime of Hořava-Witten 11-dimensional supergravity/M-theory;
Let $\chi \colon X \to \mathbf{B}^3 U(1)$ be the (instanton sector of a) circle 3-bundle on $X$. This is to be called the supergravity C-field; by the discussion there and at (FSS 12b) this is constrained to satisfy
where $\tfrac{1}{2}\mathbf{p}_1$ denotes the first fractional Pontryagin class and $\mathbf{a}$ the second Chern class of an auxiliary E8-principal bundle.
This is to be called the background gauge field/local action functional for the M2-brane 3d sigma-model field theory with target space $X$.
Write $Q \hookrightarrow X$ for the inclusion of the locus of $\mathbb{Z}_2$-fixed points of $X$. This is to be called the “M9-brane” worldvolume or equivalently the spacetime of heterotic string theory in Hořava-Witten theory.
This data is required to constitute a higher orientifold structure on $X$ which means in particular that there is a trivialization
of the restriction of the background field to the $\mathbb{Z}_2$-fixed points; and hence by the above
By the discussion in (FSS 09) this is to be called the twisted B-field of heterotic string theory on $Q$, exhibited by the string^c structure $\tfrac{1}{2}\mathbf{p}_1(Q) \simeq 2 \mathbf{a}|Q$ on $Q$.
This data is summarized as constituting a correspondence for a boundary prequantum field theory in dimensions 2, 3 as follows:
By (ABG) we have:
Circle 3-bundles canonically twist the cohomology theory tmf in that there is a canonical morphism
Therefore we may consider the motivic quantization of the above setup with coefficients in tmf-∞-modules:
If the first fractional Pontryagin class of $Q$ indeed vanishes on $Q$, hence in the case that $2\mathbf{a}|_Q \simeq 0$, then the push-forward in generalized cohomology to the point exists in tmf
Here $\pi_0(tmf^\bullet(\ast))$ is the ring of topological modular forms. An element in here can be interpreted as the partition function of a 2d super conformal field theory in dependence of the modulus of the conformal torus worldsheet. As such the image of the above map is the partition function of the heterotic string, called the Witten genus. See there for discussion and references dealing with this relation.
More generally, if $2\mathbf{a}$ does not vanish but is the cup product square of a line bundle, then a twisted Witten genus still exists (Chen-Han-Zhang 10)).
By (ABG, (11.2)) we have:
In the above situation the push-forward in tmf goes form the untwisted tmf-cohomology of the heterotic supergravity spacetime $Q$ to the $(-2a)$-twisted tmf-cohomology of the 11-dimensional supergravity spacetime:
This perspective on the Green-Schwarz anomaly cancellation condition in heterotic string theory as an orientation in twisted tmf originates in (Sati 10).
By analogy with the above discussion of D-brane charge the image of this map might be called the $M9$-brane charge.
(…)
According to (Witten 89, around (2.20)) the regularization of ordinary $G$-Chern-Simons theory involves coupling it so an $O(n)$-Chern-Simons theory. Moreover the “framing anomaly” (see at 2-framing) requires that $\frac{1}{2}p_1$ vanishes .
We are thus in a situation similar to that of the membrane theory above.
(…)
After a pointer to some
we first list references and notes along the lines of the above discussion in
and then give an extensive list of precursors of articles on aspects of the idea of motivic quantization as laid out here in
Introductions and lecture notes on material used here include
(on basics of fundamental physics formulated naturally in higher differential geometry)
Domenico Fiorenza, Hisham Sati, Urs Schreiber, A higher stacky perspective on Chern-Simons theory (arXiv:1301.2580)
(on the formulation of local action functionals in higher differential geometry)
Twisted smooth cohomology in string theory, lectures at ESI Program on K-Theory and Quantum Fields (2012)
(on local/extended/higher structures naturally seen in the local prequantum field theory involved in string theory);
section 1 “Introduction” of
differential cohomology in a cohesive topos
(on the formulation of higher topos theory).
Synthetic Quantum Field Theory
geometric of physics – The full story in a few formal words
(a summary and survey of the axiomatics).
The discussion of motivic quantization as laid out here appears in
This is based on previous work such as
The following is a list of “precursors” of aspects of the idea motivic quantization as laid out here.
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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
Acta Math. 127 (1971)
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
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
Quantum groups from path integrals (arXiv:q-alg/9501025)
Higher algebraic structures and quantization (arXiv:hep-th/9212115)
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
Klaas Landsman, Functorial quantization and the Guillemin-Sternberg conjecture Proc. Bialowieza 2002 (arXiv:math-ph/0307059)
Klaas Landsman, Functoriality of quantization: a KK-theoretic approach, talk at ECOAS, ECOAS, Dartmouth College, October 2010 (web)
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
no. 6, 1139–1183 (1984) (pdf)
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
Mathematica 147 (2011), 1281–1320 (arXiv:0903.3717)
The description of string topology operations as an HQFT defined by pull-push transforms in ordinary homology/ordinary cohomology was originally realized in
Ralph Cohen, Veronique Godin, A Polarized View of String Topology (arXiv:math/0303003)
Hirotaka Tamanoi, Loop coproducts in string topology and triviality of higher genus TQFT operations (2007) (arXiv:0706.1276)
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 $KK \longrightarrow NCC_{dg}$ 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
Jeffrey Morton, 2-Vector Spaces and Groupoids (arXiv:0810.2361)
Jeffrey Morton, Cohomological Twisting of 2-Linearization and Extended TQFT (arXiv:1003.5603v4)
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
Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry, J. Amer. Math. Soc. 23 (2010), no. 4, 909-966, (arXiv:0805.0157)
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
The Riemann-Hilbert correspondence/de Rham theorem for $H A$-modules is established in
The refinement of operator K-theory to a functor to KU-module spectra is due to
The functoriality of twisted groupoid K-theory is discussed in
and (Nuiten 13).
The formulation of push-forward in KK-theory by postcomposition with dual morphisms is based on the observations in section 7 of
and generalized to equivariant KK-theory in
The twisted Witten genus in the presence of background gauge field hence for a twisted string structure/string^c structure was considered in
The identification of the Green-Schwarz anomaly cancellation condition of heterotic string theory as a string^c structure and the proposal that this hence is to be regarded as an orientation in twisted tmf is due to
Last revised on October 29, 2023 at 07:52:00. See the history of this page for a list of all contributions to it.