Types of quantum field thories
derived smooth geometry
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 and , respectively, on this coordinate chart. The value 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 .
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 , 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 – 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 -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 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 . A configuration of the system, namely a trajectory of the particle, is then a smooth function . 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 which again are given by maps into some spacetime . 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 , replacing the worldline of the particle. For of dimension 3 one accordingly speaks of the worldvolume of a membrane and then for 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.
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 over the spacetime/worldvolume (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 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 , a field configuration – a gauge field configuration – is a combination of an instanton sector – modeled by the equivalence class of a -principal bundle – and the “gauge potential”, modeled by a connection on this bundle (see below at Gauge fields for details). There is a fiber bundle such that its sections are precisely the connections on , and so , where ranges over the instanton sectors, is a field bundle for Yang-Mills fields on .
But this construction is not local: if we consider this assignment of field bundles to all suitable manifolds , and if is a cover of , then we cannot in general obtain the field bundle on by gluing the field bundle on the cover. This is because locally every -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 -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 of -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 – the moduli stack of -principal connections (being the stackification of the groupoid of Lie algebra-valued forms) such that maps are equivalent to Yang-Mills fields on (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 is even a trivial field bundle, namely the projection
This is a differential refinement of what is called the trivial -gerbe on , 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 of fields, which for -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.
To see how this works, first recall the case of orientations, whose description as sections of the orientation bundle is familiar.
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 ):
A lift of the tangent bundle map to a map 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.
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 to a map is equivalently the type of trivializations of the composite .
Now if we have an orthogonal structure given, then this composite map according to the above diagram is
This represents the first Stiefel-Whitney class of , and it classifies a -principal bundle, hence a double cover and this is precisely the orientation bundle of . Sections of this bundle are choices of orientation on , hence are “orientation-structure fields”.
Assume then such orientation field is given. Then in the next step the relevant composite map is
This is sometimes called the -lifting bundle gerbe of . 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 ; 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 , but on equipped with its orientation . 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 as well as its orientation structure as a single object is to regard the map as an object in the slice (2,1)-topos . In here an object is a map of stacks into , 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 to a spin structure as above, hence a diagram of the form
is equivalently a map
in . This is again of the same simple form of the Yang-Mills fields on , which are maps
but in the collection of stacks itself, not in a slice.
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 through , hence a map
in the slice . (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 , as embodied in , 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 in the presence of Wilson lines. This is a theory on 3-dimensional spacetime/worldvolume whose fields are -gauge fields as for Yang-Mills theory above, hence given by maps . At the same time, this theory has a “coupling” to a 1-dimensional theory which describes particles propagating around knots in for which the restriction serves as the background gauge field. Specifically, a field configuration of this 1-dimensional theory is equivalently a map in the slice which in 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 produced a 1-dimensional quantum field theory whose partition function is that “Wilson loop” observable of . 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 is that one consider also 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 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 .
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 -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.
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 -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||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. 3 below we define action functionals and the fields relative to this notion.
We work in the following context.
Let be a cohesive (∞,1)-topos. For many of the examples below it is furthermore assumed that is equipped with differential cohesion. This implies in particular that there is a notion of smooth manifold internal to .
For the main definition below we need the following basic notation.
The following defines the notion of action functional and as part of the data it defines the notion of physical field.
In this context we say that
the morphism is the background field;
the object is the moduli ∞-stack of fields;
in , hence the diagrams
in , are the fields on ;
Definition 3 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 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. 3 says that all fields are sections of an associated ∞-bundle to an -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. 3 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 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 can be pulled back (in the sense of pullback of functions) along any smooth function . 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 to pull back a given field and in fact in general one needs extra structure.
In view of def. 3 above this is immediate: by that definition a field on in general does not just depend on , but in fact also on the background field structure denoted . Accordingly, it can be pulled back only along maps that also carry this background field structure along.
Since by def. 3 a physical field is a map in , it may be “pulled back” along maps of spacetime/worldvolume when these are extended to maps in of the background fields, hence to diagrams in of the form
hence maps 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. 3 the background field 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. 3.
A generic object in here is a morphism in . 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 which is given by a map
in is equivalently a diagram of the form
This we interpret as a configuration consisting of
a bulk field configuration
a defect field configuration ,
a gauge transformation that relates the restriction (or more generally: pullback to ) 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 is regarded as fixed, then this is equivalently a field configuration as in def. 3 defined on 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 -stack of all bulk and boundary fields as follows
for the canonical geometric morphism.
For and morphisms in , we say that
is the moduli ∞-stack of bulk and boundary fields on .
Several examples of this are discussed below.
The definition of fields in def. 3 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
where the first line is the definition, the second is the adjunction-equivalence, the third is the -adjunction implying that preserves the terminal object, and finally the last line is the defining internal hom adjunction-equivalence.
Proposition 2 makes precise the heuristic idea that a field is
More generally, the moduli -stacks of combined bulk/boundary-defect fields as in def. 5 is characterized as follows.
The following proposition expresses that fixing a bulk field gives rise to a background field for the remaining defect fields
For simplicity of the discussion first consider the case that is the delooping of an ∞-group . (In typical applications in physics we instead have the differential refinement (see below at Examples – Gauge fields) and the following discussion is directly generalized to that case. )
Via composition of loops, this canonically has the structure of an A-∞ algebra object such that the 0-truncation is a monoid object which is a group object. Such groupal -algebra objects are group objects in : ∞-groups.
Its inverse 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 .
This construction yields an equivalence of ∞-groupoids
for the image of under the equivalence of prop. 5.
identifying with a -fiber ∞-bundle over . Moreover, this is the universal rho-associated ∞-bundle: for every -principal ∞-bundle with modulating map there is a natural equivalence in an (∞,1)-category
There is a canonical equivalence in an (∞,1)-category between the sections of the -associated ∞-bundle and maps in the slice, hence fields in the sense of def. 3 with background field and moduli stack :
or equivalently a diagram
Together the above facts imply that for
And so we say that globally it is a “twisted” -cocycle. This leads to the following equivalent description.
We discuss here how def. 3 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
we have the following equivalently identifications when interpreting this as a cocycle:
the object is the local coefficient ∞-bundle;
the field is
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. 3 it follows that
There is a natural equivalence
hence a homotopy fiber sequence
This identifies the equivalence classes of global points in as the -relative A-cohomology of .
In general, if the defect fields are not trivial, the fields (hence ordinary cocycles in ) are a kind of cocycles in that are a combination of relative and twisted cocycles: instead with their pullback to 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. 3:
The simplest type of field is a (smooth) function on spacetime 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.
In this case the background gauge field is a G-structure, the moduli stack of fields is a delooped ∞-group extension and a field is a generalized vielbein exhibiting a reduction/lift of structure group.
In this case the background gauge field is a -principal ∞-bundle for some gauge group and is the univeral -associated ∞-bundle for an ∞-action of on some . A field is section of the associated -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 here are typically all differentially refined to moduli stacks 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 from a worldvolume to a target space . Now, by the general unified definition def. 3 and as shown in the following sections, in higher geometry every type of field is of this form if we allow target space 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 (“real scalar field”) or the complex numbers (“complex scalar field”). Hence this is the example of def. 3 with trivial background fields
(the terminal object) and with the field moduli stack being the map
regarded as an object in .
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 is described by a field on the abstract worldline which is simply a smooth function to the target space : 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.
in , hence a diagram of the form
For and ordinary manifolds just says that a field configuration is a map subject to the constraint that it takes the boundary of to . This means that this is a trajectory of an open string in whose endpoints are constrained to sit on the D-brane .
If however 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.
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).
then a twisted c-structure is a map
in . 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.
with the coset space on the left.
For , let
in hence diagrams
According to remark 2 such fields pull back along map 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.)
Under geometric realization
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 of the inclusion 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 next example is the differential refinement of the previous one.
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 . The vielbein then exhibits the original affine connection as that whose components are the Christoffel symbols 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 1 such that one goes into the other under a diffeomorphism of spacetime , there is a gauge transformation between them.
Moreover the trivial bundle
corresponds to the trivial action.
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 , as shown in the following table
|smooth ∞-group||Whitehead tower of smooth moduli ∞-stacks||G-structure/higher spin structure||obstruction|
|ninebrane 10-group||ninebrane structure||third fractional Pontryagin class|
|fivebrane 6-group||fivebrane structure||second fractional Pontryagin class|
|string 2-group||string structure||first fractional Pontryagin class|
|spin group||spin structure||second Stiefel-Whitney class|
|special orthogonal group||orientation structure||first Stiefel-Whitney class|
|orthogonal group||orthogonal structure/vielbein/Riemannian metric|
|general linear group||smooth manifold|
(all hooks are homotopy fiber sequences)
in . And if an orientation background field is assumed, , , 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 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 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 -structure for that underlying circle bundle is a field
and the field moduli stack is
By remark 12 above the ordinary field of gravity on some manifold 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.
Notably in manifestly T-duality-equivariant type II supergravity the generalized tangent bundle on the -dimensional spacetime is an -principal bundle . The corresponding maximal compact subgroup inclusion yields the moduli stack of fields
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 for the maximal subgroup for which a retrection to still exists, called the geometric subgroup in the context of type II geometry. Then a lift of the tangent bundle to
hence a field
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.
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.
in . These, or their twisted variants, appear in dual heterotic string theory.
Similarly the higher spin structure-fields just discussed have further twistes by background unitary bundles. For
some universal characteristic map on moduli ∞-stacks, for the moduli 3-stack of circle 3-bundles, hence circle 3-group-principal ∞-bundles we say that the smooth ∞-group appearing in the homotopy pullback diagram
(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 under geometric realization of cohesive infinity-groupoids. For this is the smooth refinement of the first fractional Pontryagin class , discussed further at twisted differential string structure.
in is a -structure on whose underlying gage bundle is the given .
As before for -structures, in applications one in addition demands that this -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 -stack of fields to be
The term gauge field in gauge theory with respect to a gauge group refers to fields which are modeled by connections either on -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
|physics||differential geometry||differential cohomology|
|gauge field||connection on a bundle||cocycle in differential cohomology|
|instanton/charge sector||principal bundle||cocycle in underlying cohomology|
|gauge potential||local connection differential form||local connection differential form|
|field strength||curvature||underlying cocycle in de Rham cohomology|
|minimal coupling||covariant derivative||twisted cohomology|
|BRST complex||Lie algebroid of moduli stack||Lie algebroid of moduli stack|
|extended Lagrangian||universal Chern-Simons n-bundle||universal characteristic map|
be the quotient stack of the action of 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.
in def. 3 yields the type of the electromagnetic 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).
For instance smooth -principal bundles and -principal bundles are both classified by the universal Chern class in but the gauge automorphisms of the trivial -principal bundle form , while that of the trivial -principal bundle form the larger .
Both versions of the moduli -stack of -bundles have their use. The version has the advantage that it is actually equivalent to the moduli -stack of complex line -bundles.
Specifically for 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 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
Since is the Picard 3-group of the monoidal 2-stack of super 2-vector bundles it is a supergeometric 3-group. Therefore there is the further delooping and hence the differential refinement in the (∞,1)-pullback diagram
This is now the moduli 2-stack
Proceeding in this fashion, let
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 -associated bundle on spacetime, where is the gauge group of the force fields that interact with the matter fields and where 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. (…)
in . This is the universal rho-associated ∞-bundle: for any -principal ∞-bundle with modulating map the corresponding associated -fiber ∞-bundle is naturally equivalent to the (∞,1)-pullback of along :
This means that such sections are fields in the sense of def. 3 where
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. 3. Gere we spell this out.
The universal associated ∞-bundle of this representation is
Hence for the canonical map, the space of -tensor fields on is
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 3 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 from a domain spacetime/worldvolume+background field to a generalized target space (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 of moduli ∞-stack of principal ∞-bundles, regarded as an object in the slice , and that the moduli ∞-stack of a plain -gauge field is that of principal ∞-connections . These two concepts have an evident unification if has a differential refinement to a map of differential moduli stacks
regarded then as an object in the slice .
Moreover, we have seen that matter fields have moduli ∞-stacks coming not from a direct delooping of an ∞-group , but from the homotopy quotient of an ∞-action of on some object . 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 , hence twisted differential cohomology. The above case of twisted differential c-structures is a special case of this for the delooping of the ∞-group that extended to .
Some further slight variants of these combinations appear in the examples below.
as in def. 8. Moreover, let the moduli -stack of fields in be given by the canonical map
which is induced by the defining inclusion of .
With these definitions we have for a manifold that
a field configuration is a diagram in of the form
which is equivalently a differentially refined reduction of the structure group of the -principal bundle underlying . If is given by a globally defined connection form then this is equivalently just a smooth -valued function on that takes to as indicated.
As a slight variant of prop. 11 we have
This implies that if is the trivial background field, than fields are equivalently maps to the coadjoint orbit
There is a canonical extended Lagrangian on with the above definitions, whose action functional, if 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 .
This is discussed further at geometry of physics – Prequantum gauge theory and gravity.
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. 5 is a map
in . This is equivalently a diagram
in . This in turn is equivalently
a -valued function on the circle
which determine a background gauge field on the knot.
Moreover a gauge transformation between two such field configurations is equivalently a gauge transformaiton of and of such that together they intertwine and . In particular if the bulk field is held fixed, then such a gauge transformation is a function such that . This means that the gauge equivalence classes of field confiurations for fixed background gauge field are labeled by maps to the coadjoint orbit as above.
be the differential refinement of that universal Dixmier-Douady class.
in , hence a diagram in of the form
On the local connection forms this acts as
This is the famous gauge transformation law known from the string theory literature.
then fields are twisted differential string structures or equivalently differential -structure with underlying gauge bundle give by , 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 of the manifold by settting
Then a field is the higher spin-connection version as discussed in Gravity above of a twisted differential string structure.
|bulk field theory||boundary field theory|
|wave function||correlation function|
|space of quantum states||conformal blocks|
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
Lecture notes on fields as discussed here with applications in string theory are in
An introductory survey is also in section 1 of
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
with a more detailed account in