This entry is about the notion in physics in the sense of field theoy? (classical/quantum field theory). For the different notion of the same name in algebra see at field.
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Types of quantum field thories
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Exotica
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In fundamental physics the basic entities that are being described are called fields, as they appear in the terms classical field theory and quantum field theory.
The basic example that probably gives the whole concept its name is the electric field and the magnetic field in the theory of electromagnetism: if we fix a coordinate chart of spacetime, then the electromagnetic field splits into the electric field and the magnetic field which are both modeled by a vector field, traditionally denoted $\vec E$ and $\vec B$, respectively, on this coordinate chart. The value $\vec E(x)$ of the vector field at a given point of spacetime is a vector that expresses the magnitude and direction of the electric force that is exerted on an electrically charged particle at $x$.
In fact more fundamentally, if we do not specify a coordinate chart, then the electromagnetic field is not in fact represented by two vector fields. Rather, its field strength is represented by a differential 2-form, hence a tensor field of rank $(0,2)$, but the the whole field as such is not a tensor field, but is a cocycle of degree-2 in ordinary differential cohomology: a circle bundle with connection.
Or for instance the field of gravity if modeled as a pseudo-Riemannian metric is a tensor field of rank $(2,0)$ – but subject to the constraint that this be pointwise non-degenerate. More fundamentally the field of gravity is instead a vielbein field.
Similar statements hold for all forces of nature, such as the force of gravity and the weak nuclear force and strong nuclear force: a configuration of these is mathematically modeled by connections. Their field strengths are rank $(0,2)$-tensor fields.
The electromagnetic field and the field of gravity are the physical fields that historically gave rise to what is now called classical field theory. But it turns out that fundamentally, in quantum physics, also all matter in physics is constituted by fields in a similar sense. Specifically, where force fields in physics are usually connections on a bundle, matter fields are sections of associated bundles.
Field theory was originally discovered as a theory of fields on spacetime. But also the physical system consisting of a single particle propagating in a fixed spacetime $X$ is described by a field theory. In this case the field is not defined on spacetime, but on the abstract worldline of the particle, say the real line $\mathbb{R}$. A configuration of the system, namely a trajectory of the particle, is then a smooth function $\phi \;\colon\; \mathbb{R}\to X$. This function may be regarded as a field on the worldline and in then called a sigma-model field. The quantum mechanics of a single particle may be equivalently thought of as a quantum field theory on the 1-dimensional worldline of the particle.
This perspective generalizes. Next one can consider fields on 2-dimensional surfaces $\Sigma_2$ which again are given by maps into some spacetime $X$. The corresponding 2-dimensional sigma-model quantum field theory is then said to describe not a particle but a string propagating in spacetime, defined on the worldsheet $\Sigma_2$, replacing the worldline of the particle. For $\Sigma$ of dimension 3 one accordingly speaks of the worldvolume of a membrane and then for $\Sigma$ of general dimension here one speaks of the worldvolume of a brane.
But there is no fundamental distinction between physical fields on spaces that are interpreted as spacetimes and those that are interpreted as worldvolumes of objects propagating in a fixed spacetime. In general these notions mix. For instance the full description of relativistic particles and relativistic strings involves a field that is really a field of gravity on the worldvolume. Conversely, theories on spacetimes that arise by Kaluza-Klein compactification of higher dimensional theories typically have “scalar moduli fields” that used to be components of the field of gravity in higher dimensions but now after compactifications become maps into some auxiliary target space, hence again sigma-model fields.
We introduce here the basic concepts of Lagrangian field theory, first for prequantum field theory and then for its deformation quantization to perturbative quantum field theory.
In full beauty these concepts are extremely general; but in this section the aim is to give a first good idea of the subject, and therefore we present for the moment only a restricted setup, notably assuming that spacetime is Minkowski spacetime, that the field bundle (see below) is an ordinary and trivial fiber bundle and that all fields are bosonic.
This does subsume what is considered in most traditional texts on the subject. In subsequent sections we will eventually discuss more general situations, notably we will eventually allow spacetime to be any globally hyperbolic Lorentzian manifold and the field bundle to be an super infinity-Lie algebroid. This is sufficient generality to capture the established perturbative BRST-BV quantization of fermions coupled to gauge fields on curved spacetimes.
Throughout we use the case of the real scalar field as an illustrative running example, which we develop alongside with the theory. The discussion of other field species that are of more genuine interest in applications is postponed to their dedicated sections below.
Thoughout, let
be a natural number and write
for Minkowski spacetime of dimension $p+1$, hence for the smooth manifold which is the Cartesian space $\mathbb{R}^{p+1}$ of dimension $p+1$ equipped with the constant pseudo-Riemannian metric $\eta$ which at the origin is given by the standard quadratic form of signature
in terms of the canonical coordinate functions
which we index starting at zero: $(x^k)_{k = 0}^p$.
We write
for the induced volume form, and we call
the canonical representative of the canonical time orientation on Minkowski spacetime.
A field configuration on a given spacetime $\Sigma$ is meant to be some kind of quantity assigned to each point of spacetime (each event), such that this assignment varies smoothly with spacetime points. For instance an electromagnetic field configuration is at each point of spacetime a collection of vectors that encode the direction in which a charged particle passing through that point will feel a force (the Lorentz force).
This is readily formalized: If
is the smooth manifold of “values” that the the given kind of field may take at any spacetime point, then a field configuration $\Phi$ is modeled as a smooth function from spacetime to this space of values:
It will be useful to unify spacetime and the space of field values into a single space, the Cartesian product
and to think of this equipped with the projection map onto the first factor as a fiber bundle of spaces of field values over spacetime
This is then called the field bundle, which specifies the kind of values that the given field species may take at any point of spacetime. Since the space $F$ of field values is the fiber of this fiber bundle, it is sometimes also called the field fiber.
(fields)
Given a spacetime $\Sigma$ and a field bundle $\array{E \\ \downarrow^{\mathrlap{fb}} \\ \Sigma }$, then a field configuration (of type specified by this field bundle) is a smooth section of this bundle, namely a smooth function of the form $\Phi \colon \Sigma \longrightarrow E$ such that composed with the projection map it is the identity function, i.e. such that $fb \circ \Phi = id$, or, diagrammatically, such that
The field configuration space is the smooth space of all these, to be denoted
This is the set of all field configurations $\Phi$ as above, and it is equipped with the structure of a smooth set by declaring that a smooth family of field configurations, parameterized over any Cartesian space $U$ is a smooth function
such that for each $u \in U$ we have $p \circ \Phi_{u}(-) = id_\Sigma$, i.e.
(trivial vector bundle as a field bundle)
In applications the field fiber $F$ is often a finite dimensional Euclidean space and equipped with the structure of a vector space. In this case the trivial field bundle with fiber $F$ is of course a trivial vector bundle.
Choosing any linear basis $(\phi^a)_{a = 1}^s$ of the field fiber, then over Minkowski spacetime we have canonical coordinates on the total space of the field bundle
If $\Sigma$ is a spacetime and if
is simply the real line, then the corresponding trivial field bundle
is the trivial real line bundle (a special case of example 1) and the corresponding field is called the real scalar field on $\Sigma$. A configuration of this field is simply a smooth function on $\Sigma$ with values in the real numbers:
Given a field bundle as above, we know what type of quantities the corresponding fields assign to a given spacetime point. Among all consistent such field configurations, some are to qualify as those that “may occur in reality” if we think of the field theory as a means to describe parts of the observable universe. Moreover, if the reality to be described does not exhibit “action at a distance” then admissibility of its field configurations should be determined over arbitrary small spacetime regions, in fact over the infinitesimal neighbourhood of any point. This means equivalently that the realized field configurations should be those that satisfy a specific differential equation, hence an equation between the value of its derivatives at any spacetime point.
In order to formalize this, it is useful to first collect all the possible derivatives that a field may have at any given point into one big space of “field derivatives at spacetime points”. This collection is called the jet bundle of the field bundle, given as def. 2 below.
Moving around in this space means to change the possible value of fields and their derivatives, hence to vary the fields. Accordingly variational calculus is just differential calculus on a jet bundle, this we consider in def. 3 below.
(jet bundle of a trivial vector bundle over Minkowski spacetime)
Given a field fiber vector space $F = \mathbb{R}^s$ with linear basis $(\phi^a)_{a = 1}^s$, then for $k \in \mathbb{N}$ a natural number, the order-$k$ jet bundle
over Minkowski spacetime $\Sigma$ of the trivial vector bundle
is the Cartesian space which is spanned by coordinate functions to be denoted as follows:
where the indices $\mu, \mu_1, \mu_2, \cdots$ range from 0 to $p$, while the index $a$ ranges from $1$ to $s$. In terms of these coordinates the bundle projection map $jb_k$ is just the one that remembers the spacetime coordinates $x^\mu$ and forgets the values of the field $\phi^a$ and its derivatives $\phi_{\mu}$. Similarly there are intermediate projection maps
given by forgetting coordinates with more indices.
The infinite-order jet bundle
is the smooth set defined so that a smooth function
from some Cartesian space $U$ is equivalently a system of ordinary smooth functions
into all the finite-order jet bundles, such that this is compatible with the above projection maps, i.e. such that
Finally jet prolongation is that function from the space of sections of the original bundle to the space of sections of the jet bundle which records the field $\Phi$ and all its spacetimes derivatives:
Smooth functions on jet bundles turn out to locally depend on only finitely many of the jet coordinates:
Given a jet bundle $J^\infty_\Sigma(E)$ as in def. 2, then a smooth function out of it
is such that around each point of $J^\infty_\Sigma(E)$ there is a neighbourhood $U \subset J^\infty_\Sigma(E)$ on which it is given by a function on a smooth function on $J^k_\Sigma(E)$ for some finite $k$.
On the jet bundle $J^\infty_\Sigma(E)$ of a trivial vector bundle over Minkowski spacetime as in def. 2 we may consider its de Rham complex of differential forms; we write its de Rham differential in boldface:
Since the jet bundle unified spacetime with field values, we want to decompose this differential into a contribution coming from forming the total derivatives of fields along spacetime (“horizontal derivatives”), and actual variation of fields at a fixed spacetime point (“vertical derivatives”):
The total spacetime derivative or horizontal derivative on $J^\infty_\Sigma(E)$ is the map on differential forms on the jet bundle of the form
which on functions $f \colon J^\infty_\Sigma(E) \to \mathbb{R}$ (i.e. on 0-forms) is defined by
and extended to all forms by the graded Leibniz rule, hence as a nilpotent derivation of degree +1.
is what remains of the full de Rham differential when the total spacetime derivative (horizontal derivative) is subtracted:
This defines a bigrading on the de Rham complex of $J^\infty_\Sigma(E)$, into horizontal degree $r$ and vertical degree $s$:
such that the horizontal and vertical derivative increase horizontal or vertical degree, respectively:
This is called the variational bicomplex.
derivatives on jet bundle
symbols | name |
---|---|
$\mathbf{d}$ | de Rham differential |
$d \coloneqq d x^\mu \frac{d}{d x^\mu}$ | (total) horizontal derivative |
$\frac{d}{d x^\mu} \coloneqq \frac{\partial}{\partial x^\mu} + \phi^a_{,\mu} \frac{\partial}{\partial \phi^a} + \cdots$ | (total) horizontal derivative along $\partial_\mu$ |
$\delta \coloneqq \mathbf{d} - d$ | (variational) vertical derivative |
$\delta_{EL} L \coloneqq \mathbf{d}L + d \Theta$ | Euler-Lagrange variational derivative |
(basic facts about variational calculus)
Given the jet bundle of a field bundle as in def. 2, then in its variational bicomplex (def. 3) we have the following:
The horizontal derivative of a spacetime coordinate function $x^\mu$ coincides with its ordinary de Rham differential
and hence this is a horizontal 1-form.
Therefore the vertical derivative of a spacetime coordinate vanishes:
In particular the given volume form on $\Sigma$ gives a horizontal $p+1$-form
Generally any horizontal $k$-form is of the form
for $f_{\mu_1 \cdots \mu_k} = f_{\mu_1 \cdots \mu_k}\left((x^\mu), (\phi^a), (\phi^a_{,\mu})\right)$ any smooth function of the spacetime coordinates and the field coordinates.
The horizontal differential of the vertical differential $\delta \phi$ of a field variable is the differential 2-form of horizontal degree 1 and vertical degree 2 given by
In words this says that “the spacetime derivative of the variation of the field is the variation of its spacetime derivative”.
Given a field bundle $E$ over a $(p+1)$-dimensional Minkowski spacetime $\Sigma$ as in example 1, then a local Lagrangian density $\mathbf{L}$ (for the field species thus defined) is a horizontal differential form of degree $(p+1)$ on the corresponding jet bundle (def. 3):
By example 3 any such Lagrangian density may uniquely be written as
with $L = L((x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots )$ a smooth function on the jet bundle.
If a Lagrangian density $\mathbf{L}$ as in def. 4, then its de Rham differential has a unique decomposition as a sum of two terms
such that $\delta_{EL}$ is a “source form”:
The map
thus defined is called the Euler-Lagrange operator and is explicitly given by
The remaining term $d \Theta$ is unique, while $\Theta \in \Omega^{p,1}(E)$ is unique only up to terms in the image of $d$. One possible choice is
where
denotes the contraction of the volume form with the vector field $\partial_\mu$.
Using $\mathbf{L} = L dvol_\Sigma$ and that $d \mathbf{L} = 0$ by degree reasons, we find
The idea now is to have $d \Theta$ pick up those terms that would appear as boundary terms under the integral $\int_\Sigma j^\infty(\Phi)^\ast \mathbf{d}L$ if we were to consider integration by parts to remove spacetime derivatives of $\delta \phi^a$.
We compute, using example 3, the total horizontal derivative of $\Theta$ from (1) as follows:
where in the last line we used that
Here the two terms proportional to $\frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a_{,\mu}$ cancel out, and we are left with
Hence $-d \Theta$ shares with $\mathbf{d} \mathbf{L}$ the terms that are proportional to $\delta \phi^a_{,\mu_1 \cdots \mu_k}$ for $k \geq 1$, and so the remaining terms are proportional to $\delta \phi^a$, as claimed:
(Euler-Lagrange equation of motion)
Given a field bundle $E$ over spacetime $\Sigma$ as in example 1 equipped with a local Lagrangian density $\mathbf{L} \in \Omega^{p+1,1}(E)$ as in def. 4 then the corresponding Euler-Lagrange equation of motion on fields $\Phi \in \Gamma_\Sigma(E)$ is the equation
where $j^\infty(\Phi) \colon \Sigma \to J^\infty(E)$ denotes the jet prolongation of $\Phi$ (def. 2), $j^\infty(E)^\ast$ the operation of pullback of differential forms along this function, and $\delta_{EL}$ is the Euler-Lagrange operator from prop. 2.
By that same proposition this equation is equivalently the differential equation
We write
for the smooth subspace of the space of all field configurations on those that solve this differential equation.
A traditional approach to formalizing the notion of physical field is to declare that the specification of a theory in physics/physical system comes with a fiber bundle $E \to X$ over the spacetime/worldvolume $X$ (or better: naturally over all spacetimes, see at Locality below) called the field bundle and that a field configuration of the system is a section of this field bundle. This is for instance the basis for the theory of the variational bicomplex, hence of BV-BRST formalism for expressing covariant phase spaces, for standard multisymplectic geometry, etc.
While this goes in the right direction, it cannot be quite the final answer, as it misses crucial properties that are demanded of a general notion of field. We now discuss these problems:
In the course of discussing the problems we also motivate and indicate their solution by a more natural notion of field moduli in higher geometry. This is then discussed in full detail in the Definition-section below.
In gauge theory specifically but in physics generally, physical fields come equipped with a notion of which fields configurations, while nominally different, are equivalent, called gauge equivalent and it is crucial to retain the information of gauge equivalences and not pass to equivalence classes of gauge equivalent fields. This means that generically for any physical theory, even if all field configurations would be represented by a section of some field bundle, many such sections are in fact to be regarded as being equivalent. Or more precisely, there should be a groupoid or ∞-groupoid of field configurations of which the sections of the field bundle only form the space of objects, while the gauge transformations form the morphisms and the higher gauge transformations of order $n$ form the n-morphisms.
To some extent this is dealt with in traditional variational calculus: after a choice of action functional on the space of field configurations, BV-BRST formalism spits out a derived L-∞ algebroid whose objects are field configurations, and whose 1-cells are infinitesimal invariances of the given action functional.
This goes in the right direction– it is the Lie differentiation of the more encompassing smooth ∞-groupoid of fields and gauge transformations – but has several problems, the main one being that this does now know about the large gauge transformations, those which are not connected to the identity (because it only sees infinitesimal data). These are important in the full quantum theory.
Famous examples of the importance of large gauge transformations appear in
Fields defined as sections of field bundles cannot capture gauge phenomena in a local way, as is necessary for a manifestly local formulation such in extended prequantum field theory, extended quantum field theory (sometimes called the “multi-tiered” formulation).
Specifically, in Yang-Mills theory for gauge group $G$, a field configuration – a gauge field configuration – is a combination of an instanton sector – modeled by the equivalence class of a $G$-principal bundle $P$ – and the “gauge potential”, modeled by a connection on this bundle (see below at Gauge fields for details). There is a fiber bundle $E(P) \to X$ such that its sections are precisely the connections on $P \to X$, and so $\coprod_{c} E(P_c) \to X$, where $c$ ranges over the instanton sectors, is a field bundle for Yang-Mills fields on $X$.
But this construction is not local: if we consider this assignment of field bundles to all suitable manifolds $X$, and if $U \to X$ is a cover of $X$, then we cannot in general obtain the field bundle on $X$ by gluing the field bundle on the cover. This is because locally every $G$-principal bundle has trivial class, so that locally there is always only a single (the trivial) instanton sector.
This failure of locality is often not recognized in the literature, since many if not most descriptions of physics restrict to trivial spacetime topology and/or restrict to perturbation theory only. A formulation accurate and encompassing enough to see this issue is AQFT on curved spacetimes. A reference that explicitly runs into this non-locality issue of the field bundle in gauge theory in this context is (Benini-Dappiaggi-Schenkel 13, Schenkel 14): the authors define a functor from spacetimes equipped with a $G$-principal bundle that assigns the algebras of observables of the corresponding Yang-Mills fields built from the field bundle of connections on the given principal bundles; and they observe that the result fails to be a local net in that the inclusion of observables of a smaller spacetime into a larger patch may fail the isotony axiom (BDS, remark 5.6). The authors then try to circumvent this by restricting to trivial instanton sectors. The fix later appears in (Benini-Schenkel-Szabo 15), where the authors then consider proper stacks of fields.
But notice that instanton sectors is a non-negligible phenomenon. For instance the very vacuum in the standard model of particle physics is a superposition of all possible instanton sectors (see at instanton in QCD for more on this). And there are field theories where the fields consist entirely of “instanton sectors” and where there is no infinitesimal information about the gauge group at all: these are theories whose gauge group is a discrete group, which includes notably Dijkgraaf-Witten theory and its higher analogy such as the Yetter model. This means that for these theories a local field bundle formalism can see nothing of the actual fields and also traditional tools applied to a global field bundle (such as traditional BV-BRST formalism) see nothing of the actual fields. All this is fixed by the formulation that we discuss below.
But this example already points to the general nature of the problem with field bundles, and also to its solution: while the instanton-component of Yang-Mills fields are not section of a bundle, they famously are sections of a stack – the “moduli stack $\mathbf{B}G$ of $G$-principal bundles”, an object in higher geometry.
The problem with the locality of the field bundle for Yang-Mills theory is solved by passing from fiber bundles to fiber ∞-bundles: in the higher differential geometry there is an object $\mathbf{B}G_{conn}$ – the moduli stack of $G$-principal connections (being the stackification of the groupoid of Lie algebra-valued forms) such that maps $X \to \mathbf{B}G_{conn}$ are equivalent to Yang-Mills fields on $X$ (even including their gauge transformations). This means that if we allow field bundles in higher geometry – fiber ∞-bundles, then that for Yang-Mills theory over $X$ is even a trivial field bundle, namely the projection
out of the product of spacetime with the moduli stack of fields.
This is a differential refinement of what is called the trivial $G$-gerbe on $X$, which is
and hence the “field bundle for instanton sectors” of Yang-Mills fields.
In summary: there cannot be a fiber bundle such that its sheaf of local sections is the sheaf of configurations of the Yang-Mills field. But there is a fiber 2-bundle whose stack of sections is the stack of configurations of the Yang-Mills field.
Judging from these examples one might be tempted to guess that the notion of field fiber bundle should simply be replaced by that of field fiber ∞-bundle. But in fact what the example rather suggests is that what matters directly is the moduli stack $\mathbf{Fields}$ of fields, which for $G$-Yang-Mills theory is simply
This perspective, which we describe in detail below also has the pleasant effect that it drastically simplifies and unifies notions of quantum field theory, for this says equivalently that if only we allow spaces in higher geometry, then Yang-Mills theory is a sigma-model quantum field theory: one whose fields are simply maps to a given target space, only that this target space here is a stack.
But there are more advantages, slightly less obvious. These we come to in the following points.
Some fields in physics are (or involve) choices of G-structure in the sense of reduction and lift of structure groups. Well-known examples include the choice of orientation and of Spin structure in field theories with fermion fields (discussed in detail in Fermions below). Often in the literature the choice of orientation and Spin structure is treated as an external parameter, but detailed analysis at least in low-dimensional examples shows that the in the full theory this is really a field configuration. For instance in path integral quantization for theories with fermions, part of the integral over all field configurations is a sum over Spin structures.
Now, a spin structure is equivalently a section of something, but again not of a principal bundle, but of an analog in higher geometry, a principal 2-bundle.
To see how this works, first recall the case of orientations, whose description as sections of the orientation bundle is familiar.
For a spacetime represented by a smooth manifold $X$ of dimension $n$, let
be the map that modulates its tangent bundle (discussed at geometry of physics - tangent bundlef physics#TangentBundle)). Consider then the following diagram, which shows lifts of this map to the classifying spaces/moduli stacks for various other groups (this is the Whitehead tower of $\mathbf{B}O(n)$):
A lift of the tangent bundle map $\tau_X$ to a map $e_X \colon X \to \mathbf{B}O(n)$ as indicated is a choice of orthogonal structure (a vielbein field, discussed in detail below in Ordinary gravity). For the present discussion assume that this is given.
The a further lift to $o_X : X \to \mathbf{B}SO(n)$ is a choice of orientation, and finally a lift to $s_X : X \to \mathbf{B}Spin(n)$ is a choice of spin structure.
Now, every hook-shaped sub-diagram in the above of the form
is a homotopy fiber sequence. By the universal property of the homotopy pullback this means that the “space” – really: homotopy type or just type, for short – of lifts of a given map $X \to \mathbf{B}G$ to a map $X \to \mathbf{B}\hat G$ is equivalently the type of trivializations of the composite $X \to \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n A$.
Now if we have an orthogonal structure $e_X : X \to \mathbf{B}O(n)$ given, then this composite map according to the above diagram is
This represents the first Stiefel-Whitney class $[w_1(\tau_X)] \in H^1(X, \mathbb{Z}_2)$ of $\tau_X$, and it classifies a $\mathbb{Z}_2$-principal bundle, hence a double cover $\hat X \to X$ and this is precisely the orientation bundle of $X$. Sections of this bundle are choices of orientation on $X$, hence are “orientation-structure fields”.
Assume then such orientation field $o_X$ is given. Then in the next step the relevant composite map is
This now represents the second Stiefel-Whitney class $[w_2(\tau_X)] \in H^2(X, \mathbb{Z}_2)$ of $X$ and classifies a $(\mathbf{B}\mathbb{Z}_2)$-principal 2-bundle
This is sometimes called the $Spin$-lifting bundle gerbe of $o_X$. A choice of Spin structure is a choice of section of this 2-bundle. Hence spin structures are parts of fields in physics which are not sections of a field 1-bundle. Again, this is faithfully captured only in higher geometry.
This is only the most famous phenomenon in a large class of similar structures of fields in field theory. Notably in higher dimensional supergravity and in string theory there are fields which are ever higher lifts through this Whitehead tower – higher spin structures, such as String structures and Fivebrane structures in the next two steps. Accordingly, these are fields which are equivalently sections of principal 3-bundles (the “Chern-Simons circle 3-bundle”) and principal 7-bundles (the “Chern-Simons circle 7-bundle”).
Comparison of the above discussions under Locality and Spin structures shows that there we had a higher-geometric field bundle of Yang-Mills fields which was hower “trivial” in the sense that it was a projection out of the product of spacetime with a moduli stack, so that a field configuration was equivalently of sigma-model-type, namely simply a map $\phi \colon : X \to \mathbf{B}G_{conn}$; whereas here the “spin-lifting 2-bundles” and its higher analogs are, in general, not of this product form, hence “Spin structure”-fields, at least superficially do not seem to be of sigma-model-type, even in higher geometry.
But a closer inspection shows that in fact both situations are entirely analogous – once we realize that here these Spin-structure fields are not really defined just on $X$, but on $X$ equipped with its orientation $o_X$. Since, by the same logic as above, also the orientation is a “field”, we may call it a background field. It serves as “background” over which spin structure fields can be considered.
In higher geometry incarnated naturally as higher topos theory, this state of affairs is naturally modeled and indeed yields again a moduli stack of spin structure fields and makes spin-structures be sigma-model-type fields, as follows:
the natural way to regard both $X$ as well as its orientation structure $o_X$ as a single object is to regard the map $X \stackrel{o_X}{\to} \mathbf{B}SO(n)$ as an object in the slice (2,1)-topos $\mathbf{H}_{/\mathbf{B}SO(n)}$. In here an object is a map of stacks into $\mathbf{B}SO(n)$, and a morphism is map of the domains of these maps together with a homotopy filling the evident triangle diagram. Notably a lift of the orientation structure $o_X$ to a spin structure $s_X$ as above, hence a diagram of the form
is equivalently a map
in $\mathbf{H}_{/\mathbf{B}SO(n)}$. This is again of the same simple form of the Yang-Mills fields on $X$, which are maps
but in the collection of stacks $\mathbf{H}$ itself, not in a slice.
The slice here encodes the presence of background fields – namely orientations in this case – whose moduli stack in turn is, in this case, $\mathbf{B}SO(n)$.
Notice that also the field of gravity has a background field in this precise sense: as metioned above, a gravitational field configuration is a lift of $\tau_X$ through $\mathbf{B}O(n) \stackrel{\mathbf{OrthStruc}_n}{\to} \mathbf{B}GL(n)$, hence a map
in the slice $\mathbf{H}_{/\mathbf{B}GL(n)}$. (Discussed in detail in Ordinary gravity below.) Hence also gravity becomes a sigma-model-type field theory in higher geometry. Notice that here it is smooth structure on $X$, as embodied in $\tau_X$, which is the background.
Now, at least for the field of gravity one can of course emulate the fields also by sections of a field bundle (while already for the second next step in the Whitehead tower, Spin structures, this is no longer the case, as we have seen). But even so, the field bundle formalism clearly misses then the relation between fields and background fields.
In particular for two reasons
Typically the presence of background fields indicates that in a more comprehensive discussion background fields are also fields that vary;
Often background fields on one space affect fields on another space.
An archetypical example for both these effects combined is 3d Chern-Simons theory with a compact, simple and simply connected gauge group $G$ in the presence of Wilson lines. This is a theory on 3-dimensional spacetime/worldvolume $\Sigma$ whose fields are $G$-gauge fields as for Yang-Mills theory above, hence given by maps $\Phi \colon \Sigma \to \mathbf{B}G_{conn}$. At the same time, this theory has a “coupling” to a 1-dimensional theory which describes particles propagating around knots $C : S^1 \to \Sigma$ in $\Sigma$ for which the restriction $\Phi|_C$ serves as the background gauge field. Specifically, a field configuration of this 1-dimensional theory is equivalently a map in the slice $\mathbf{H}_{/\mathbf{B}G_{conn}}$ which in $\mathbf{H}$ is given by a diagram of the form
for some map on the right which we discuss in detail below in Chern-Simons fields with Wilson line fields.
Here considering just these fields in the background of a fixed $\Phi|_C$ produced a 1-dimensional quantum field theory whose partition function is that “Wilson loop” observable of $\Phi|_C$. But this is not considered in isolation. The whole point of the relation of Chern-Simons theory to the Jones polynomial knot invariant of the knot $C$ is that one consider also $\Phi$ as a dynamical field, not as a fixed background. Indeed, in the full theory of Chern-Simons with Wilson loops that includes both the fields on $\Sigma$ as well as those on the knot, a field configuration is the diagram as above but regarded as the square
hence, again, a single map
but now in the arrow (∞,1)-topos $\mathbf{H}^{(\Delta^1)}$.
This subtle interplay of “bulk fields” and “defect fields” which is here captured most naturally in terms of higher geometry cannot really be expressed accurately just in terms of field bundles.
Above we have seen the generalization of field bundles to higher geometry already for traditional notions such as Yang-Mills fields and Spin-structures. But many theories considered in in theoretical physics have fields that are more “explicitly” entities in higher geometry.
For instance the higher analog of the electromagnetic field which is called the B-field or Kalb-Ramond field is a 2-connection on a principal 2-bundle. There is no way to faithfully encode this as a section of any ordinary fiber bundle. It follows that for instance also the magnetic charge anomaly (as discussed there) has no accurate description in terms of field bundles. Next the supergravity C-field is a 3-connection on a principal 3-bundle, and so on.
There is a wide variety of higher dimensional Chern-Simons theories whose fields are such higher gauge fields. In some traditional literature one sees parts of this theory be discussed by standard BV-BRST formalism applied to field bundles, namely by ignoring the non-trivial instanton sectors and pretending that a field configuration for these ∞-connections are given by globally dedined differential forms. In some special cases (for instance for spacetimes/worldvolumes of very special topology or low [[dimension]) this can be sufficient to capture everything, but in general (for instance for $U(1)$-higher dimensional Chern-Simons theory and its holographically dual self-dual higher gauge theory) it is not.
By the above, defining a physical field to be a section of some bundle goes in the right direction, but misses crucial aspects of physical fields. These problems are fixed by passing to higher geometry.
Below in Definition we discuss a natural unified formulation of the notion of physical field in terms of higher geometry (the central definition being def. 8 ) and then we spell out many Examples.
This definition turns out to be equivalent, at least under mild conditions, to a formulation where fields are sections of an associated ∞-bundle, hence a “field $\infty$-bundle”. This we discuss in Properties – Relation of fields to sections of ∞-bundles. But this is just one of several equivalent perspectives on physical fields, and not always the most transparent one. In fact, sections of higher associated bundles are best known in the literature on twisted cohomology and indeed one equivalent characterization of fields is as cocycles in twisted cohomology in the general sense of cohomology in an (∞,1)-topos. This we discuss below in Relation to twisted cohomology.
In summary we find and discuss that
fields | $\simeq$ | twisted relative cohesive cocycles |
---|
We give a general abstract definition of physical fields in
Then we consider some general abstract operations on fields in
A notion of field in physics is part of a specification of physical theory or physical model. We consider specifically the framework of prequantum field theory. Here a theory/model is specified by (or at least comes with) an action functional. The field content of the theory is part of the specification of the domain of the action functional. Therefore in def. 8 below we define action functionals and the fields relative to this notion.
We work in the following context.
Let $\mathbf{H}$ be a cohesive (∞,1)-topos. For many of the examples below it is furthermore assumed that $\mathbf{H}$ is equipped with differential cohesion. This implies in particular that there is a notion of smooth manifold internal to $\mathbf{H}$.
Fix $\mathbb{G} \in Grp(\mathbf{H})$ a group object in $\mathbf{H}$, hence a cohesive ∞-group.
For the main definition below we need the following basic notation.
For $B \in \mathbf{H}$ any object, and $X,A \in \mathbf{H}_{/B}$ two objects in the slice (∞,1)-topos, write
for the $\mathbf{H}$-valued hom object between $X$ and $A$: the dependent product over $B \to *$ of the internal hom $[X, A] \in \mathbf{H}_{/B}$.
The following defines the notion of action functional and as part of the data it defines the notion of physical field.
Given an object $\mathbf{BgFields} \in \mathbf{H}$ and given two objects, to be denoted $\Phi_X, \mathbf{Fields} \in \mathbf{H}_{/B}$, in the slice over $\mathbf{BgFields}$, then an action functional in ($\mathbf{H}, \mathbb{G})$ “on fields on $X$” is a morphism
In this context we say that
the dependent sum $X \coloneqq \underset{\mathbf{BgField}}{\sum} \Phi_X$ is the worldvolume or spacetime;
the morphism $\Phi_X \;\colon\; X \to \mathbf{BgFields}$ is the background field;
the object $\mathbf{Fields}$ is the moduli ∞-stack of fields;
the elements of $[\Phi_X,\mathbf{Fields}]_{\mathbf{H}}$, hence (see prop. 3 below) the morphisms
in $\mathbf{H}_{/\mathbf{BgFields}}$, hence the diagrams
in $\mathbf{H}$, are the fields on $X$;
a gauge transformation is a homotopy in $[\Phi_X, \mathbf{Fields}]_{\mathbf{H}}$, hence a
a higher gauge transformation is a higher homotopy in $[\Phi_X, \mathbf{Fields}]_{\mathbf{H}}$.
Definition 8 provides a unified perspective on fields from several perspectives.
On the one hand, it almost explicitly says that in higher geometry all fields are “sigma-model fields” (see below at Examples – Scalar and Sigma-model fields): if we regard the moduli ∞-stack $\mathbf{Fields}$ as the target space then fields are simply maps from their domain (when regarded as spacetime and background field) to this target space.
On the other hand, we see below in Relation to sections of ∞-bundles that from another perspective def. 8 says that all fields are sections of an associated ∞-bundle to an $\infty$-bundle modulated by the background fields. This means that in higher geometry all fields are “matter fields” (see below at Matter fields) charged under the background gauge field.
Finally, we see below in Relation to twisted cohomology that from yet another perspective def. 8 says that fields are equivalently cocycles in general twisted cohomology. This perspective is traditionally known for certain examples (see Examples – Chan-Paton gauge fields below), but we see below that it is useful in its full generality. For instance the field of gravity is in a precise sense a 0-cocycle with coefficients in the coset space $GL(n)/O(n)$ that is twisted by the tangent bundle of spacetime (which exhibits the background gauge structure for gravity: the smooth structure of spacetime). An inkling of this perspective is certainly visible in the traditional literature, notably in the generalization to type II geometry and T-duality, and here we see how this is a precise mechanism on the same conceptual footing with the twisted K-theory seen over D-branes.
It is familiar from basic examples that not every type of physical field on a spacetime/worldvolume $X$ can be pulled back (in the sense of pullback of functions) along any smooth function $f \;\colon\;Y \to X$. For instance the field of gravity, a vielbein field or pseudo-Riemannian metric, discussed below in Ordinary graviry may be pulled back only along local diffeomorphisms. More generally, one needs other properties on $f$ to pull back a given field and in fact in general one needs extra structure.
In view of def. 8 above this is immediate: by that definition a field on $X$ in general does not just depend on $X$, but in fact also on the background field structure denoted $\Phi_X$. Accordingly, it can be pulled back only along maps that also carry this background field structure along.
Since by def. 8 a physical field is a map $\phi \colon \Phi_X \to \mathbf{Fields}$ in $\mathbf{H}_{/\mathbf{BgFields}}$, it may be “pulled back” along maps of spacetime/worldvolume $f \colon Y \to X$ when these are extended to maps $\Phi_Y \to \Phi_X$ in $\mathbf{H}_{/\mathbf{BgFields}}$ of the background fields, hence to diagrams in $\mathbf{H}$ of the form
hence maps $f : X \to Y$ equipped with a choice of equivalence
between the background fields.
In standard Examples discussed below we see that this is a familar fact. For instance applied to the field of gravity (see Gravity below) it says that the gravitational field can be pulled back precisely along local diffeomorphisms, or that spin structures on oriented manifolds (see Spin structures below) can be pulled back along orientation-preserving maps. Or : for Chan-Paton gauge fields on D-branes (see Chan-Paton gauge fields) it reproduces the familiar gauge relation for the B-field on D-branes known in string theory, which is already less trivial. But the statement applies in full generality.
In def. 8 the background field $\Phi_X$ is a fixed datum of the domain (spacetime/worldvolume) on which the physical fields are defined. In some models and for some of the fields this is precisely what one needs, but in other models one may need to be able to also regard the background fields as dynamical fields and to be able to switch between these perspectives, for instance to pass to a setup where what used to be a configuration of some field is now taken to be a fixed background field for the remaining fields. We now discuss how this more general setup is naturally formulated as a generalization of def. 8.
For $\mathbf{H}$ an (∞,1)-topos and $\mathbf{Fields} \colon \underset{\mathbf{BgFields}}{\sum} \mathbf{Fields} \to \mathbf{BgFields}$ a morphism in $\mathbf{H}$, we may consider this also as an object in the arrow (∞,1)-topos $\mathbf{H}^{(\Delta^1)}$ (the (∞,1)-functor (∞,1)-category from the interval category/1-simplex to $\mathbf{H}$).
A generic object in $\mathbf{H}^{(\Delta^1)}$ here is a morphism $\iota_X$ in $\mathbf{H}$. When we think of this as a domain on which to define fields we will write this
where the subscripts are for “bulk” and for “defect” (as in QFT with defects). A field in $\mathbf{H}^{(\Delta^1)}$ which is given by a map
in $\mathbf{H}^{(\Delta^1)}$ is equivalently a diagram of the form
in $\mathbf{H}$.
This we interpret as a configuration consisting of
a bulk field configuration $\phi_{bulk}$
a defect field configuration $\phi_{def}$,
a gauge transformation that relates the restriction (or more generally: pullback to $X_{def}$) of the bulk field to the embedding (or more generally: push-forward) of the defect field into the bulk field configuration on the defect.
If $\phi_{bulk}$ is regarded as fixed, then this is equivalently a field configuration as in def. 8 defined on $X_{def}$ and for background field the composite
This “fixing of bulk fields to background fields for defect fields” we discuss in more detail below in Properties – Moduli stacks of fields.
We formalize the moduli $\infty$-stack of all bulk and boundary fields as follows
Write
for the canonical geometric morphism.
For $\iota_X ;\colon\; X_{def} \to X_{bulk}$ and $\mathbf{Fields} \;\colon\; \mathbf{Fields}_{def} \to \mathbf{Fields}_{bulk}$ morphisms in $\mathbf{H}$, we say that
is the moduli ∞-stack of bulk and boundary fields on $\iota_X$.
Several examples of this are discussed below.
The definition of fields in def. 8 is in fact the central part of a general theory of cohomology and principal ∞-bundles in higher geometry/(∞,1)-topos theory and various insights into prequantum field theory follow by making this perspective explicit. This is what we do here
The central results that underlie these identifications are in (NSS), also dcct, section 3.6.10, 3.6.11, 3.6.12, 3.6.15.
The object $[\Phi_X, \mathbf{Fields}]_{\mathbf{H}} \in \mathbf{H}$ of def. 8 we may call the moduli ∞-stack of fields. Here we discuss various properties of this object.
The ∞-groupoid of global elements of $[X, \mathbf{Fields}]_{\mathbf{H}}$ is
Using the adjunction equivalences we have
where the first line is the definition, the second is the $(\mathbf{BgFields}^* \dashv \underset{\mathbf{BgFields}}{\prod})$ adjunction-equivalence, the third is the $(\underset{\mathbf{BgFields}}{\sum} \dashv \mathbf{BgFields}^*)$-adjunction implying that $\mathbf{BgFields}^*$ preserves the terminal object, and finally the last line is the defining internal hom adjunction-equivalence.
The moduli ∞-stack of fields $[\Phi_X, \mathbf{BgFields}]_{\mathbf{H}}$ sits in a homotopy pullback diagram of the form
These relations are discussed at slice (∞,1)-topos, and (dcct, section 3.6.1).
Proposition 4 makes precise the heuristic idea that a field $\phi \in [\Phi_X, \mathbf{Fields}]_{\mathbf{H}}$ is
a configuration $\phi_X \;\colon\; X \to \underset{\mathbf{BgFields}}{\sum}\mathbf{Fields}$ on spacetime/worldvolume $X$;
together with a gauge transformation $\kappa_{\phi} \;\colon\; \Phi_X \stackrel{\simeq}{\to} \mathbf{Fields}\circ \phi_X$ between the fixed background field and the background field induced by $\phi_X$.
More generally, the moduli $\infty$-stacks of combined bulk/boundary-defect fields as in def. 10 is characterized as follows.
The moduli stack $[\iota_X, \mathbf{Fields}]_{\mathbf{H}}$ of bulk and defect fields in def. 10 sits in an (∞,1)-pullback diagram
The following proposition expresses that fixing a bulk field gives rise to a background field for the remaining defect fields
For $\phi_{bulk} \;\colon\; X_{bulk} \to \mathbf{Fields}_{bulk}$ a given bulk field, there is a natural equivalence
We discuss here how def. 8 equivalently says that fields are sections of a fiber ∞-bundle which is which is associated to a (principal) $\infty$-bundle modulated by the background field.
For simplicity of the discussion first consider the case that $\mathbf{BgFields} \simeq \mathbf{B}G$ is the delooping of an ∞-group $G \in Grp(\mathbf{H})$. (In typical applications in physics we instead have the differential refinement $\mathbf{B}G_{conn}$ (see below at Examples – Gauge fields) and the following discussion is directly generalized to that case. )
We briefly recall the central aspects of principal ∞-bundles in the (∞,1)-topos $\mathbf{H}$.
For $X \in \mathbf{H}$ any object and $x \colon * \to \X$ a global element, the homotopy pullback ((∞,1)-pullback) of the point along itse is the based loop space object
Via composition of loops, this canonically has the structure of an A-∞ algebra object such that the 0-truncation $\tau_0 \Omega_x X$ is a monoid object which is a group object. Such groupal $A_\infty$-algebra objects are group objects in $\mathbf{H}$: ∞-groups.
In fact every ∞-group is of this form. Moreover, restricted to pointed and connected objects, the looping (∞,1)-functor $\Omega$ is an equivalence of (∞,1)-categories
Its inverse $\mathbf{B}$ we call the delooping operation. Notice that this means that we can conveniently discuss all aspects of ∞-groups without ever explicitly considering A-∞ algebra structure by instead working with the pointed connected objects $\mathbf{B}G$.
Notably one finds that a $G$-principal ∞-bundle $P \to X$ in $\mathbf{H}$ is equivalently the homotopy fiber of a map $g \;\colon\; X \to \mathbf{B}G$
This construction yields an equivalence of ∞-groupoids
between $G$-principal ∞-bundles with gauge transformations between them and $G$-cocycles with coboundaries. In particular this means that $G$-principal ∞-bundles are classified by $G$-cohomology in $\mathbf{H}$
Hence if $\mathbf{BgFields} \simeq \mathbf{B}G$ then a background field on $X$ is equivalently a $G$-principal ∞-bundle on $X$.
For $G \in Grp(\mathbf{H})$ a group object in $\mathbf{H}$, the (∞,1)-functor that sends an ∞-action $\rho$ of $G$ on some $V \in \mathbf{H}$ to the corresponding associated ∞-bundle of the $G$-universal principal ∞-bundle is an equivalence of (∞,1)-categories
Moreover there is a natural equivalence
with the quotient of $V$ by the $G$-∞-action $\rho$ (this is the general abstract and version and geometric refinement of the traditional Borel construction).
For $\rho$ an ∞-action of an ∞-group $G \in Grp(\mathbf{H})$ on some $V \in \mathbf{H}$ we write
for the image of $\rho$ under the equivalence of prop. 7.
The homotopy fiber of $\overline{\rho}$ is $V$, hence we have a homotopy fiber sequence
identifying $\overline{\rho}$ with a $V$-fiber ∞-bundle over $\mathbf{B}G$. Moreover, this is the universal rho-associated ∞-bundle: for every $G$-principal ∞-bundle $P \to X$ with modulating map $g_X \;\colon\; X \to \mathbf{B}G$ there is a natural equivalence in an (∞,1)-category
in $\mathbf{H}_{/X}$.
In summary this means that an ∞-action $\rho$ of an ∞-group $G$ on any object $V \in \mathbf{H}$ is equivalently a homotopy fiber sequence of the form
Moreover, for $P \to X$ any $G$-principal ∞-bundle modulated by a map $g ;\colon\; X \to \mathbf{B}G$ and for $\rho$ an ∞-action of $G$ on some $V$, there is an (∞,1)-pullback diagram
Due to the universal property of the (∞,1)-pullback this has the following consequence which, basic as it is, is fundamental for the interpretation of fields in def. 8.
There is a canonical equivalence in an (∞,1)-category between the sections of the $\rho$-associated ∞-bundle $P \times_X G$ and maps $g_X \to \overline{\rho}$ in the slice, hence fields in the sense of def. 8 with background field $g_X$ and moduli stack $\overline{\rho}$:
This equivalence is constituted simply by forming the pasting diagram of the section with the above $(\infty,1)$-pullback
The following further central property leads in the following section to an equivalent re-formulation of this in twisted cohomology.
Every $G$-principal ∞-bundle $P \to X$ in an (∞,1)-topos $\mathbf{H}$ is locally trivizable in that there exists a 1-epimorphism $p \;\colon\; U \to X$ and an (∞,1)-pullback diagram of the form
or equivalently a diagram
Together the above facts imply that for
a section of the associated ∞-bundle, the restriction $\sigma|_U$ of the section along the trivializing cover $p \;\colon\; U \to X$ factors through $V \to V//G$:
This means that a section of an associated $V$-fiber ∞-bundle is locally a $V$-valued map, hence a cocycle in $\Omega V$-cohomology.
And so we say that globally it is a “twisted” $\Omega V$-cocycle. This leads to the following equivalent description.
We discuss here how def. 8 of fields has an entirely equivalent expression in terms of cocycles in general cohomology and in fact, if the background field is nontrivial, in twisted cohomology. This follows by direct comparison with the corresponding notions in cohomology and twisted cohomology as discussed there (NSS). The unification of notions seen this way gives a natural home for instance to the familiar observations such as that Chan-Paton gauge fields on D-branes are identified with cocycles in (differential) twisted K-theory (see Chan-Paton gauge fields below).
First observe the following
For $\phi \;\colon\; \Phi_X \to \mathbf{Fields}$ a field configuration as in def. 8 in the corresponding diagram in $\mathbf{H}$
we have the following equivalently identifications when interpreting this as a cocycle:
the object $\mathbf{Fields} \in \matbbf{H}_{/\mathbf{BgFields}}$ is the local coefficient ∞-bundle;
the homotopy fiber $V$ of $\underset{\phi \in [X, \mathbf{Fields}]}{\sum}\mathbf{Fields} \to \mathbf{BgFields}$ is the local coefficient for a cohomology theory;
the background field $\Phi_X$ is the twist
the field $\phi \colon \Phi_X \to \mathbf{Fields}$ is
a cocycle in $\Phi_X$-twisted cohomology on $X$ with local coefficients $V$
equivalently: a $\Phi_X$-twisted V-structure on $X$;
a gauge transformation is a coboundary in $\Phi_X$-twisted $V$-cohomology;
the equivalence classes of fields are the $\Phi_X$-twisted $V$-cohomology of spacetime $X$
We see several illustrations of these identifications in the list of Examples below. More generally, there is a canonical identification of physical fields in the presence of background fields and boundary/defect with twisted and relative cohomology. This we discuss below in Relation to relative cohomology.
When we refine background fields to dynamical fields as discussed above in Boundary and defect fields then the identification of fields with cocycles in twisted cohomology as discussed above in Relation to twisted cohomology accordingly generalized a bit: it becomes a combination of twisted cohomology and relative cohomology.
For consider the special case that the moduli of defect fields are trivial, hence that
is the global point inclusion into the bulk field moduli (the trivial bulk field). By prop. 5 it follows that
There is a natural equivalence
hence a homotopy fiber sequence
This identifies the equivalence classes of global points in $[\iota_X, pt_A]$ as the $\iota_X$-relative A-cohomology of $X$.
In general, if the defect fields are not trivial, the fields $\iota_X \to \mathbf{Fields}$ (hence ordinary cocycles in $\mathbf{H}^{(\Delta^1)}$) are a kind of cocycles in $\mathbf{H}$ that are a combination of relative and twisted cocycles: instead with their pullback to $X_{def}$ being equipped with a trivialization, it is equipped with a “twisted trivialization” in the sense of twisted c-structures, discussed below.
We distinguish four broad classes of examples of physical fields, according to def. 8:
The simplest type of field is a (smooth) function on spacetime $X$ with values in the real numbers or complex numbers, called a scalar field. Slightly more generally there are fields which are functions into some other vector space or more general manifold, often called linear/non-linear sigma-model fields. Often that manifold is a space of parameters of some geometry, for instance of a compact space in Kaluza-Klein compactification, in which case these fields are often called moduli fields. Scalar fields may be charged under force fields, which we turn to next.
Fields of gravity, G-structure and generalized geometry
In this case the background gauge field is a G-structure, the moduli stack of fields is a delooped ∞-group extension $\mathbf{B}\hat G \stackrel{\mathbf{Fields}}{\to} \mathbf{B}G$ and a field is a generalized vielbein exhibiting a reduction/lift of structure group.
In this case $\mathbf{Fields}$ is a moduli space $\mathbf{B}G_{conn}$ of ∞-connections on $G$-principal ∞-bundles for some gauge group $G \in Grpd(\mathbf{H})$.
In this case the background gauge field is a $G$-principal ∞-bundle $P$ for some gauge group $G \in Grp(\mathbf{H})$ and $\mathbf{Fields}$ is the univeral $\rho$-associated ∞-bundle $V//G \stackrel{\mathbf{Fields}}{\to} \mathbf{B}G$ for $\rho$ an ∞-action of $G$ on some $V \in \mathbf{H}$. A field is section of the associated $V$-fiber ∞-bundle.
Not all examples fall squarely into one of these types, some are mixtures of these. Relevant examples we discuss in
In particular the moduli stacks $\mathbf{B}G$ here are typically all differentially refined to moduli stacks $\mathbf{B}G_{conn}$ of ∞-connections so that for instance every reduction and lift of structure groups goes along with a corresponding data of the reduction of an ∞-connection. The archetypical example for this are spin connections, see the example Ordinary gravity below.
Traditionally a sigma-model field is a type of fields given simply by (smooth) functions $\Sigma \to X$ from a worldvolume $\Sigma$ to a target space $X$. Now, by the general unified definition def. 8 and as shown in the following sections, in higher geometry every type of field is of this form if we allow target space $X$ to be a general ∞-stack and functions to be maps in a suitable slice (∞,1)-topos. Nevertheless, here we start with briefly indicating those examples that are sigma-model fields also in the traditional restrictive sense of the term.
A scalar field is given simply by a function on spacetime/worldvolume, typically with values in the real numbers $\mathbb{R}$ (“real scalar field”) or the complex numbers $\mathbb{C}$ (“complex scalar field”). Hence this is the example of def. 8 with trivial background fields
(the terminal object) and with the field moduli stack being the map
from (the smooth manifold underlying) the real numbers to the point, or else
regarded as an object in $\mathbf{H}_{/*} \simeq \mathbf{H}$.
Scalar fields are, due to their simplicity, prominent in toy examples used to discuss general properties of quantum field theory. The only fundamental scalar particle observed in nature to date is the Higgs particle (or presumably so, in technicolor models it is not actually fundamental but a composite of fermion particles, discussed below), but the Higgs field is, crucially, charged under the electroweak SU(2)-gauge field. A model of relevance in phenomenology which crucially features an uncharged scalar particle is cosmic inflation. But the fundamental nature of the inflaton field is hypothetical (if it exists at all), it might well be the effective version of non-scalar fields.
The dynamics of a particle propagating in a spacetime $X$ is described by a field on the abstract worldline which is simply a smooth function $\mathbb{R} \to X$ to the target space $X$: a trajectory of the particle. The quantum mechanics that describes the dynamics of such a quantum particle is equivalently a 1-dimensional field theory on the worldline, with these maps as its physical fields.
Therefore such wordline sigma-model fields are given by the special case of def. 8 with trivial background field $\mathbf{BgFields} \simeq *$ and with $\mathbf{Fields} \;\colon\; X \to *$ regarded as an object in $\mathbf{H}_{/*} \simeq \mathbf{H}$.
More generally, the worldvolume fields of any brane sigma-model with target space $X$ are given as above.
A D-brane inclusion $Q \stackrel{\iota_X}{\to} X$ is the target space for an open string with worldsheet $\partial \Sigma \stackrel{\iota_\Sigma}{\hookrightarrow} \Sigma$: a field configuration of the open string sigma-model is a map
in $\mathbf{H}^{\Delta^1}$, hence a diagram of the form
For $X$ and $Q$ ordinary manifolds just says that a field configuration is a map $\phi_{bulk} \;\colon\; \Sigma \to X$ subject to the constraint that it takes the boundary of $\Sigma$ to $Q$. This means that this is a trajectory of an open string in $X$ whose endpoints are constrained to sit on the D-brane $Q \hookrightarrow X$.
If however $X$ is more generally an orbifold, then the homotopy filling the above diagram imposes this constraint only up to orbifold transformations, hence exhibits what in the physics literature are called “orbifold twisted sectors” of open string configurations.
The moduli stack $[\iota_\Sigma, \iota_X]$ of such field configurations is the homotopy pullback
The background gauge field for such a sigma-model is a B-field on $X$ and a Chan-Paton gauge field on $Q \hookrightarrow X$.
We discuss examples for the two classes of force fields, which are:
We discuss the example of the field of gravity, below in Gravity, and various closely related types of fields that all encode geometry in some sense, in fact all encode geometry in the sense of G-structure. This most general case we discuss in G-structure – (twisted lift of structure ∞-groups).
Let
be an ∞-group extension in $\mathbf{H}$. Then for $g_X \colon X \to \mathbf{B}G$ a given $G$-principal ∞-bundle, a lift of the structure group from $G$ to $\hat G$ is a field
If the ∞-group extension is central in that it extends to a homotopy fiber sequence of the form
then a twisted c-structure is a map
in $\mathbf{H}_{/\mathbf{B}G}$. These kinds of fields are interpreted as fields of gravity and its variants, as shown by the following examples.
For recognizing traditional constructions in this formulation, the following basic fact is important.
For $\iota \colon H \hookrightarrow G$ an subgroup inclusion of Lie groups, we have a homotopy fiber sequence
with the coset space on the left.
We discuss the formulation of the field of gravity as a special case of def. 8.
For $n \in \mathbb{N}$, let
be the delooping of the Lie-subgroup inclusion of the orthogonal group into the general linear group in dimension $n$ (the inclusion of the maximal compact subgroup).
For $X \in$ SmoothMfd $\hookrightarrow$ $\mathbf{H} \coloneqq$ Smooth∞Grpd a smooth manifold of dimension $n$, write
for the canonical map modulating the $GL(n)$-principal bundle to which the tangent bundle is canonically associated.
Morphisms
in $\mathbf{H}_{/\mathbf{B}GL(n)}$ hence diagrams
in $\mathbf{H}$ are equivalently vielbein fields $e$ exhibiting the reduction of the structure group from $GL(n)$ to $O(n)$. A gauge transformation is a local frame transformation. Hence
is the moduli stack of Riemannian metrics on $X$. Every such is a field configuration of ordinary gravity on $X$. (But see example 6 below.)
According to remark 2 such fields pull back along map $f \colon Y \to X$ that fit into a diagram
These are precisely local diffeomorphisms and indeed these are precisely the maps along which vielbein fields / pseudo-Riemannian metric fiedls pull back. (Pullback along smooth functions may not preserve the non-degeneracy of a metric tensor, but precisely the local diffeomorphisms do.)
The smooth structure on $X$ as embodied by $\tau_X$ is the background field in example 4.
Under geometric realization
the map of moduli stacks $\mathbf{B}O(n) \to \mathbf{B}GL(n)$ becomes an equivalence (a weak homotopy equivalence) of classifying space $B O(n) \stackrel{\simeq}{\to} B GL(n)$.
This means that in plain homotopy theory, hence ignoring the geometry, a lift in
is no information, up to equivalence: it always exists and exists uniquely up to a contractible space of choices. In order for such a lift really to be equivalently an orthogonal structure it needs to be taken with geometry included, hence for moduli stacks (_cohesive_ homotopy type) as above.
The analog statement is true for every delooping $\mathbf{B}H \to \mathbf{B}G$ of the inclusion $H \hookrightarrow G$ of a maximal compact subgroup into a Lie group. More examples of this kind we discuss below in Type II Gravity and generalized geometry.
The homotopy fiber of $\mathbf{OrthStruc}_n$ is the coset space $GL(n)//O(n)$, hence we have a homotopy fiber sequence of moduli stacks of the form
in $\mathbf{H}$. This means that locally, over a cover (1-epimorphism) $p \colon U \to X$ over which the tangent bundle has a trivialization $\tau_X|_U \simeq *$, the space of fields is simply $[U, GL(n)//O(n)]$.
The next example is the differential refinement of the previous one.
Let
be the canonical differential refinement of $\mathbf{OrthStruc}_n$, where now the moduli stacks of principal connections appear. For $X$ a manifold as above, let then
modulate an affine connection on the tangent bundle. A field
is now still equivalently just a vielbein, but its component
now captures the orthognal connection which in the physics literature is often called (inaccurately) the spin connection, denoted $\omega$. The vielbein then exhibits the original affine connection as that whose components are the Christoffel symbols $\Gamma$ of the Riemannian metric defined as above, a relation that is familiar from the physics literature in the form of the equation
between local connection 1-form components.
But both these examples do not fully accurately reflect the field content of gravity yet. This is because the theory of gravity is supposed to by generally covariant. This means that for two vielbein field configurations as in example 4 such that one goes into the other under a diffeomorphism of spacetime $X$, there is a gauge transformation between them.
Write $\mathbf{Diff}(X) \coloneqq \mathbf{Aut}(X) \in Grp(\mathbf{H})$ for the diffeomorphism group of the smooth manifold $X$. There is then the canonical ∞-action of the automorphism ∞-group $\mathbf{Diff}(X)$ on $X$ itself, exhibited by the universal associated ∞-bundle
This induces also an ∞-action also on the tangent bundle.
Moreover the trivial bundle
corresponds to the trivial action.
Then
is the accurate space of generally covariant fields of gravity.
In theories with fermions (discussed below) the field of gravity is more refined than just a vielbein field as above, hence an orthogonal structure on spacetime: it also involves an orientation structure and a spin structure.
To see that these structures are really all (fields) of the same kind, observe that they are the lifts through the first step of the Whitehead tower of $\mathbf{B}GL$, as shown in the following table
(all hooks are homotopy fiber sequences)
Notice, as in remark 12, that for the interpretation of the first step here it is crucial to interpret this in moduli ∞-stacks and not in classifying spaces.
Hence if a background field of gravity is assumed, $\mathbf{BgFields} \colon \mathbf{B}O(n)$, $\Phi_X = e_X$, then the moduli stack for orientation structure-fields is
in $\mathbf{H}_{/\mathbf{B}O(n)}$. And if an orientation background field is assumed, $\mathbf{BgFields} \colon \mathbf{B}SO(n)$, $\Phi_X = o_X$, then the moduli stack for spin structure-fields is
Evidently there is then also a notion of higher spin structure-fields. These appear as backgrounds when one passes from spinning particles to spinning strings and then to further “spinning” branes. This we discuss below in Higher spin structures.
Some theories involve not plain spin structures but spin-c structures on their spacetime/worldvolume (for instance in Seiberg-Witten theory or in the gauge theory over D-branes in type II string theory, see the example Chan-Paton gauge fields on D-branes below). By the discussion there, the moduli stack $\mathbf{B}Spin^c(n)$ for spin^c group-principal bundles sits in the homotopy pullback diagram
where the right vertical map represents the universal first Chern class modulo 2. In other words this says that a spin^c-structure on some $X$ is a twisted w2-structure with twist the first Chern-class of a circle bundle.
So if that circle bundle is regarded as a background field
then a $Spin^c$-structure for that underlying circle bundle is a field
in $\mathbf{H}_{/\mathbf{B}^2 \mathbb{Z}_2}$.
Or rather, if $X$ is a manifold as before and the $SO(n)$-principal bundle involved in the above is required to be a reduction of the tangent bundle $\tau_X : X \to \mathbf{B}GL(n)$ as before, then the background field for $Spin^c$-structure fields is
and the field moduli stack is
By remark 12 above the ordinary field of gravity on some manifold $X$ is equivalently a reduction of the structure group from the general linear group to its maximal compact subgroup, regarded as a field whose background field is the tangent bundle itself.
If one instead considers a variant of the tangent bundle as a background field – a generalized tangent bundle $\tau_X^{gen} : X \to \mathbf{B}G$ – with some other structure Lie group $G$, then a field with values in
in $\mathbf{H}_{/\mathbf{B}G}$ and for $H \hookrightarrow$ the inclusion of the maximal compact subgroup, may be thought of as an accordingly generized field of gravity defining a generalized Riemannian geometry.
Such fields naturally appear in theories of higher-dimensional supergravity, where related to the T-duality and more generally U-duality structure of these theories.
Notably in manifestly T-duality-equivariant type II supergravity the generalized tangent bundle on the $n$-dimensional spacetime is an $O(n,n)$-principal bundle $\tau^{gen}_X \;\colon\; X \to \mathbf{B}O(n,n)$. The corresponding maximal compact subgroup inclusion yields the moduli stack of fields
and a field $\phi \;\colon\; \tau_X^{gen} \to \mathbf{Fields}$ is equivalently the generalized vielbein for the generalized Riemannian geometry called type II geometry on $X$.
In fact also the generalized tangent bundle itself should be regarded as a field: Notice that the canonical diagonal inclusion
does not have a retraction. Write $GL(n) \hookrightarrow G_{geom} \hookrightarrow O(n,n)$ for the maximal subgroup for which a retrection to $GL(n)$ still exists, called the geometric subgroup in the context of type II geometry. Then a lift of the tangent bundle to $G_{geom}$
hence a field
is what is called a geoemtric generalized tangent bundle. These are exactly the generalized tangent bundles primarily considered in the literature (see at type II geometry for more).
In the Kaluza-Klein compactification of type II supergravity and of 11-dimensional supergravity preserving some amount of global supersymmetry this generalized type II geometry is further enhanced to various flavors of what is called exceptional generalized geometry.
Here the generalized tangent bundle has as structure group an exceptional Lie group from the $E$-series $E_{n(n)}$ (for compactification on an $n$-dimensional compact space). The moduli stack of fields is then
and a field $e_X^{gen} \;\colon\; \tau_X^{gen} \to \mathbf{Fields}$ is equivalently a generalized vielbein for exceptional generalized geometry.
By the discussion of Spin structure fields above there are evident higher analogs, obtained by climbing through the Whitehead tower of BO.
In the next step we have String structure-fields which are maps to
These appear as fields in heterotic supergravity with quantum anomaly-cancellation by the Green-Schwarz mechanism and for trivial gauge field. In the presence of a non-trivial gauge field these are further refined to Higher spin-c structrures discussed below. This field content of heterotic supergravity is discussed in more detail below in Anomaly-free heterotic supergravity fields – differential String-c structures.
Further up the Whitehead tower Fivebrane structure-fields are maps to
in $\mathbf{H}_{/\mathbf{B}String}$. These, or their twisted variants, appear in dual heterotic string theory.
As discussed above, an ordinary spin-c structure is really a spin structure which is twisted by the class of a $U(1)$-principal bundle.
Similarly the higher spin structure-fields just discussed have further twistes by background unitary bundles. For
some universal characteristic map on moduli ∞-stacks, for $\mathbf{B}^3 U(1)$ the moduli 3-stack of circle 3-bundles, hence circle 3-group-principal ∞-bundles we say that the smooth ∞-group $String^{\mathbf{c}}$ appearing in the homotopy pullback diagram
is a higher string 2-group-analog of spin^c.
In particular, if $G$ is a compact, connected and simply connected simple Lie group (such as the spin group or the special unitary group) then
(the first equivalence is discussed at Lie group cohomology as a special case of a theorem discussed at smooth infinity-groupoid). This means that there is an essentially unique map of higher moduli stacks
which maps to the generator of $H^4(BG, \mathbb{Z})$ under geometric realization of cohesive infinity-groupoids. For $G = Spin$ this is the smooth refinement of the first fractional Pontryagin class $\tfrac{1}{2}\mathbf{p}_1$, discussed further at twisted differential string structure.
Of interest in heterotic supergravity here is the case that $G = E_8 \times E_8$ (the product of the exceptional Lie group $E_8$ with itself) and $\mathbf{c}$ is twice the canonical string class.
If then $g_X \;\colon\; X \to \mathbf{B}(E_8 \times E_8)$ is the instanton sector of the gauge field in heterotic supergravity regarded as a background gauge field for the field of heterotic gravity, then with
a map
in $\mathbf{H}_{/\mathbf{B}^3 U(1)}$ is a $\mathbf{String}^{\mathbf{c}}$-structure on $X$ whose underlying gage bundle is the given $\Phi_X$.
As before for $Spin^c$-structures, in applications one in addition demands that this $\mathbf{String}^{\mathbf{c}}$-structure is indeed a refinement of the field of gravity of the theory, which means that one takes the background field to be
and the moduli $\infty$-stack of fields to be
These are precisely the instanton sectors of the fields of Green-Schwarz anomaly free heterotic supergravity, discussed further below in Anomaly-free heterotic supergravity fields. (SSS)
The term gauge field in gauge theory with respect to a gauge group $G$ refers to fields which are modeled by connections either on $G$-principal bundles or on associated bundles for these. The notion of equivalence between two such fields (hence between connections on bundles) is the original meaning of the word gauge transformation, even though that term is also used for equivalences between fields which are not modeled by connections.
We discuss the general notion of gauge fields and then various special cases and variants. The following table gives an overview over the notions involved in the concept of gauge fields
gauge field: models and components
A field configuratiotion of the electromagnetic field is a circle bundle with connection and a gauge transformation of the EM field is an equivalence of such connections.
Let therefore
be the quotient stack of the action of $U(1)$ on the sheaf of differential 1-forms, equivalently the image under the Dold-Kan correspondence of the sheaf of chain complexes given by the Deligne complex for degree-2 ordinary differential cohomology, as indicated.
This is the moduli stack for $U(1)$-principal connections. Hence setting $\mathbf{BgFields} \simeq *$ and
in def. 8 yields the type of the electromagnetic field.
More generally, let $G$ be a Lie group, and $\mathfrak{g}$ its Lie algebra. Write $\Omega^1(-,\mathfrak{g})$ the sheaf of Lie algebra valued 1-forms
The moduli stack of $G$-principal connections is the quotient stack
of the action of $G$ on it Lie algebra valued differential forms by gauge transformations
This is the moduli stack of the $G$-Yang-Mills field. Hence setting $\mathbf{BgFields} \simeq *$ and
in def. 8 yields Yang-Mills fields.
Let
be the image under the Dold-Kan correspondence of the Deligne complex for degree-3 ordinary differential cohomology. This is the moduli 2-stack for circle 2-bundle with connections.
Setting $\mathbf{BgFields} \simeq *$ and
in def. 8 yields the 2-form gauge field known as the B-field or the Kalb-Ramond field.
The B-field appears both in bosonic string theory as well as in type II superstring theory. In (Distler-Freed-Moore) it was pointed out that for the superstring the plain formulation above needs to be refined. We discuss here the natural formulation of this observation in higher supergeometry as in (FSS CSIntroAndSurvey, section 4.3).
Generally instead of the circle group $U(1)$ one can consider the multiplicative group $\mathbb{C}^\times$ of non-zero complex numbers. The canonical subgroup inclusion $U(1) \hookrightarrow \mathbb{C}^\times$ induces for each $n \in \mathbb{N}$ a canonical morphism of moduli ∞-stacks
This is not quite an equivalence of ∞-stacks but it is a shape modality-equivalence in that geometric realization of cohesive ∞-groupoids sends it to an equivalence as
For instance smooth $U(1)$-principal bundles and $\mathbb{C}^\times$-principal bundles are both classified by the universal Chern class in $H^2(-, \mathbb{Z})$ but the gauge automorphisms of the trivial $U(1)$-principal bundle form $C^\infty(-,U(1))$, while that of the trivial $\mathbb{C}^\times$-principal bundle form the larger $C^\infty(-,\mathbb{C}^\times)$.
Both versions of the moduli $n$-stack of $n$-bundles have their use. The version $\mathbf{B}^n \mathbb{C}^\times$ has the advantage that it is actually equivalent to the moduli $n$-stack of complex line $n$-bundles.
Specifically for $n = 2$ it is equivalent to the moduli 2-stack of line 2-bundles
But now in supergeometry complex super-line 2-bundle have a richer classification that plain line 2-bundle. Explicitly, let now $\mathbf{H} \colon$ SmoothSuper∞Grpd be the ambient cohesive (∞,1)-topos for higher supergeometry and write
for the 2-stack of complex super-line 2-bundle.
This has homotopy sheaves
$k$ | $\pi_k( 2 \mathbf{sLine})$ |
---|---|
0 | $\mathbb{Z}_2$ |
1 | $\mathbb{Z}_2$ |
3 | $\mathbb{C}^\times$ |
in the topos over supermanifolds. The $\pi_0(2 \mathbf{sLine}) \simeq \mathbb{Z}_2$ says that locally there is not just one super line 2-bundle, namely the standard line 2-bundle, but also its odd “superpartner”.
Under geometric realization ${\vert-\vert} \;\colon\; \mathbf{H} \to \infty Grpd \simeq L_{whe} Top$ we have the homotopy type ${\vert 2 \mathbf{sLine}\vert}$ with homotopy groups
$k$ | $\pi_k({\vert 2 \mathbf{sLine}\vert})$ |
---|---|
0 | $\mathbb{Z}_2$ |
1 | $\mathbb{Z}_2$ |
3 | $0$ |
4 | $\mathbb{Z}$ |
Since $2 \mathbf{sLine}$ is the Picard 3-group of the monoidal 2-stack $2 \mathbf{sVect}$ of super 2-vector bundles it is a supergeometric 3-group. Therefore there is the further delooping $\mathbf{B}2\mathbf{sLine}$ and hence the differential refinement $2\mathbf{sLine}_{conn}$ in the (∞,1)-pullback diagram
This is now the moduli 2-stack
for the B-field in type II string theory.
Proceeding in this fashion, let
be the image under the Dold-Kan correspondence of the Deligne complex for degree-4 ordinary differential cohomology. This is the moduli 3-stack for circle 3-bundle with connections.
Fields whose moduli stack is
are one component of the supergravity C-field.
Fundamental matter-fields as they appear in the standard model of particle physics, hence fundamental fermions such as electrons, quarks and neutrinos, are sections of a $G$-associated bundle $E \to X$ on spacetime, where $G$ is the gauge group of the force fields that interact with the matter fields and where $E \to X$ is associated to the principal bundle underlying such a force gauge field, as discussed in Gauge fields above, and, if we are talking indeed about fermions, to the spin bundle given by the gravity-spin structure field discussed in Gravity and generalized geometry above.
More precisely, fermioninc matter fields are sections of these bundles regarded in supergeometry with the fibers regarded as odd-graded. (…)
We start by discussing the general mechanism by which sections of $\rho$-associated ∞-bundles are an example of the general definition 8.
For $G \in \Grp(\mathbf{H})$ a geometric ∞-group, there is an equivalence of (∞,1)-categories
between that of ∞-actions of $G$ and the slice (∞,1)-topos of $\mathbf{H}$ over the delooping $\mathbf{B}G$ of $G$. Under this equivalence an ∞-action/∞-representation of $G$ on some $V \in \mathbf{H}$ is equivalently a homotopy fiber sequence of the form
in $\mathbf{H}$. This is the universal rho-associated ∞-bundle: for $P \to X$ any $G$-principal ∞-bundle with modulating map $g_X \;\colon\; X \to \mathbf{B}G$ the corresponding associated $V$-fiber ∞-bundle is naturally equivalent to the (∞,1)-pullback of $\overline{\rho}$ along $g$:
From this and the universal property of the (∞,1)-pullback one finds that a section of the associated ∞-bundle $P \times_G V \to X$ is equivalently a map
in $\mathbf{H}_{/\mathbf{B}G}$.
This means that such sections are fields in the sense of def. 8 where
$\mathbf{BgFields} \simeq \mathbf{B}G$;
the background field is the moduli $g_X$;
the moduli ∞-stack of fields is $\overline{\rho} \in \mathbf{H}_{/\mathbf{B}G}$.
$S$ a spin representation
in supergeometry (…)
While the term physical field probably orignates from tensor field, few fields are fundamentally given by tensor fields. Nevertheless, tensor fields, being sections of a tensor product of copies of the tangent bundle and the cotangent bundle are certainly examples of the general notion of field as in def. 8. Gere we spell this out.
For $X$ a smooth manifold, a tensor field of rank $(p,q)$ is a section of the tensor product bundle
This is canonically the associated bundle to the $GL(n)$-principal bundle which to which the tangent bundle is also canonically associated, by the given representation of $GL(n)$ on $\mathbb{R}^{n(p+q)}$.
The universal associated ∞-bundle of this representation is
$\mathbf{Fields} \;\colon\; \mathbb{R}^{n (p+q)}//GL(n) \to \mathbf{B}GL(n)$.
Hence for $\tau_X \colon X \to \mathbf{B}GL(n)$ the canonical map, the space of $(p,q)$-tensor fields on $X$ is
By passing to irreducible representations in the above, we obtain specifically the corresponding subclasses of tensor fields, such as differential forms.
The above distinction of types of physical fields into into Sigma-model fields, Force fields of gravity, Gauge force fields and Matter fields can be helpful for reasoning about fields and types of theories in physics, but is not fundamental. The general definition 8 of which all these types are examples reflects that. In fact, one way to read that definition and the above list of examples is to say that it shows that in higher geometry all types of fields are sigma-model fields: they are all given as maps $\Phi_X \to \mathbf{Fields}$ from a domain spacetime/worldvolume+background field $\Phi_X$ to a generalized target space $\mathbf{Fields}$ (the moduli ∞-stack of fields in the given slice (∞,1)-topos characterized by the nature of the background fields).
Accordingly, a general physical field in a general theory does not fall squarely into one of the above categories, but combines aspects of all of these. Here we discuss such examples.
We have seen that the moduli ∞-stack of a field of plain gravity or general geometry is a map $\mathbf{c} \colon \mathbf{B}\hat G \to \mathbf{B}G$ of moduli ∞-stack of principal ∞-bundles, regarded as an object in the slice $\mathbf{H}_{/\mathbf{B}G}$, and that the moduli ∞-stack of a plain $G$-gauge field is that of principal ∞-connections $\mathbf{B}G_{conn}$. These two concepts have an evident unification if $\mathbf{c}$ has a differential refinement $\at \mathbf{c}$ to a map of differential moduli stacks
regarded then as an object in the slice $\mathbf{H}_{/\mathbf{B}G_{conn}}$.
Since, as discussed above, a field with coefficients in $\mathbf{c}$ is equivalently a twisted $\mathbf{c}$-structure, we may calla field with coefficients in $\hat \mathbf{c}$ a twisted differential c-structure.
Moreover, we have seen that matter fields have moduli ∞-stacks coming not from a direct delooping $\mathbf{B}G \simeq *//G$ of an ∞-group $G$, but from the homotopy quotient $V//G$ of an ∞-action of $G$ on some object $V$. Combining this with differential refinement as above we consider fields whose moduli ∞-stacks are maps
This is a differential refinement of fields which are cocycles in twisted cohomology with local coefficients $V$, hence twisted differential cohomology. The above case of twisted differential c-structures is a special case of this for $V \simeq \mathbf{B}A$ the delooping of the ∞-group that extended $G$ to $\hat G$.
Some further slight variants of these combinations appear in the examples below.
We describe here a variant of the particle propagating on a spacetime $X$, where now the particle is charged under a background gauge field for a nonabelian gauge group $G$.
Let $G$ be a simple Lie group which is connected, simply connected and compact.
Let $\rho$ be an irreducible unitary representation of $G$ under which the particle is to be charged. By the Borel-Weil-Bott theorem this corresponds equivalently to a weight
Let $T \simeq G_\lambda \hookrightarrow G$ be the maximal torus which is the stabilizer subgroup that fixes this weight under the coadjoint action of $G$ on its dual Lie algebra $\mathfrak{g}^*$.
Set then in def. 8 the moduli stack of background fields to be
as in def. 13. Moreover, let the moduli $\infty$-stack of fields in $\mathbf{H}_{/\mathbf{B}G_{conn}}$ be given by the canonical map
which is induced by the defining inclusion of $T$.
For $U \in$ CartSp $\hookrightarrow \mathbf{H}$ a field $\phi \colon U \to \Omega^1(-,\mathfrak{g})//T$ is equivalently a Lie algebra valued form $A \in \Omega^1(U,\mathfrak{g})$, but a gauge transformation of such a field is constrained to be a smooth $T$-valued function $t \in C^\infty(U,T) \hookrightarrow C^\infty(U, G)$ instead of an arbitrary $G$-valued function.
With these definitions we have for $X$ a manifold that
a background gauge field $\Phi_X \;\colon\; X \to \mathbf{B}G_{conn}$ is equivalently a $G$-principal connection on $X$ (which if $G$ is assumed connected and simply connected and $\Sigma$ is of dimension at most 3 is equivalently just a Lie algebra valued form);
a field configuration is a diagram in $\mathbf{H}$ of the form
which is equivalently a differentially refined reduction of the structure group of the $G$-principal bundle underlying $\nabla$. If $\nabla$ is given by a globally defined connection form $A$ then this is equivalently just a smooth $G$-valued function $g$ on $X$ that takes $A$ to $A^g$ as indicated.
As a slight variant of prop. 13 we have
We have a homotopy fiber sequence
where on the left we have the coadjoint orbit of $\langle \lambda , -\rangle$.
This implies that if $\nabla = 0$ is the trivial background field, than fields $\phi : X \to \mathbf{Fields}$ are equivalently maps to the coadjoint orbit
Hence in this sector we have simply a sigma-model field as in Sigma-models above.
There is a canonical extended Lagrangian on $[X, \mathbf{Fields}]_{\mathbf{H}}$ with the above definitions, whose action functional, if $X = S^1$ is the closed connected 1-dimensional manifold, is that of a 1d Chern-Simons theory. The partition function of the corresponding quantum field theory is the holonomy map – the “Wilson line” – on the background gauge field connection $\nabla$.
This is discussed further at geometry of physics – Prequantum gauge theory and gravity.
We discuss the field content of 3d Chern-Simons theory for a simple, simply connected compact Lie group with Wilson loops. This is an example of Bulk fields with defect fields.
Let $\Sigma_3 \in SmthMfd \hookrightarrow \mathbf{H}$ be a smooth manifold of dimension 3, and let
be the embedding of a knot. Let the moduli of fields be
as defined in Nonabelian charged particle trajectories – Wilson lines above. Regarding the not inclusion as a defect in the 3-dimnensional manifold, a bulk-defect fiedl configuration according to def. 10 is a map
in $\mathbf{H}^{(\Delta^1)}$. This is equivalently a diagram
in $\mathbf{H}$. This in turn is equivalently
a Lie algebra valued form $A \in \Omega^1(\Sigma_3, \mathfrak{g})$ (the bulk gauge field of $G$-Chern-Simons theory)
a $g$-valued function on the circle $g \in C^\infty(S^1, G)$
which determine a background gauge field $A|_{S^1}^g$ on the knot.
Moreover a gauge transformation between two such field configurations $\kappa \;\colon\; \phi \Rightarrow \phi'$ is equivalently a gauge transformaiton of $A$ and of $A|_{S^1}$ such that together they intertwine $g$ and $g'$. In particular if the bulk field is held fixed, then such a gauge transformation is a function $t \colon S^1 \to T$ such that $g' = t g$. This means that the gauge equivalence classes of field confiurations for fixed background gauge field are labeled by maps to the coadjoint orbit $\mathcal{O}_\lambda \simeq G/T$ as above.
This are the field confugurations for 3d Chern-Simons theory (see the discussion there) with Wilson lines (FSS).
We discuss the Chan-Paton gauge fields over D-branes in type II string theory.
The extension of groups $U(1) \to U(n) \to PU(n)$ sits in a long homotopy fiber sequence of $\infty$-stacks
Let
be the differential refinement of that universal Dixmier-Douady class.
Let
be a submanifold, to be thought of as a D-brane worldvolume in an ambient spacetime $X$.
Then a field configuration of the B-field on $X$ together with a compatible rank-$n$ gauge field on the D-brane is a map
in $\mathbf{H}^{(\Delta^1)}$, hence a diagram in $\mathbf{H}$ of the form
This identifies a twisted bundle with connection on the D-brane whose twist is the class in $H^3(X, \mathbb{Z})$ of the bulk B-field.
This relation is the Kapustin-part of the Freed-Witten-Kapustin anomaly cancellation for the bosonic string or else for the type II string on $Spin^c$ D-branes. (FSS)
If we regard the B-field as a background field for the Chan-Paton gauge field, then remark 2 determines along which maps of the B-field the Chan-Paton gauge field may be transformed.
On the local connection forms this acts as
This is the famous gauge transformation law known from the string theory literature.
Let
Let
then fields are twisted differential string structures or equivalently differential $\mathbf{String}^{\mathbf{c}}$-structure with underlying gauge bundle give by $\Phi_X$, the differential refinement of the discussion in Higher spin structure above.
As in the discussion there, we implement the constraint that the string structure is on the tangent bundle $\tau_X \;\colol\; X \to \mathbf{B}GL(n)$ of the manifold by settting
and
Then a field $\phi \;\colon\; \Phi_X \to \mathbf{Fields}$ is the higher spin-connection version as discussed in Gravity above of a twisted differential string structure.
The moduli stack of these fields is that of background fields that satisfy the Green-Schwarz anomaly cancellation in heterotic supergravity. (SSS).
physical field
holographic principle in quantum field theory
bulk field theory | boundary field theory |
---|---|
dimension $n+1$ | dimension $n$ |
field | source |
wave function | correlation function |
space of quantum states | conformal blocks |
A survey of the main field species is given in
Most of the above material as of 2013 was written as part of a lecture series
A exposition specifically for gauge fields is also in
An exposition of the general formulation of fields in terms of moduli stacks in slice (∞,1)-toposes is in section 4 of
Lecture notes on fields as discussed here with applications in string theory are in
An introductory survey is also in section 1 of
For references on the tradtional formulation of physical fields by sections of field bundles as discussed above see there references there.
The formulation of physical fields as cocycles in twisted cohomology in an (∞,1)-topos as in the Definition-section above originates around
Further articles since then are listed at
In particular the general notion of fields as twisted differential c-structures appears in
and the general theory of cohomology and twisted cohomology with local coefficient ∞-bundles as referred to in Relation to twisted cohomology above as well as the theory of associated ∞-bundles as in Sections of associated ∞-bundles is laid out in
Some examples of fields in this sense are called “relative fields” in
Discussion of AQFT with stacky fields and co-stacky local nets of observables is in
The supergeometric nature of the B-field in type II string theory had been pointed out in
with a more detailed account in
The formulation of this in smooth super infinity-groupoids is (FSS, section 4.3).