Types of quantum field thories
n-category = (n,n)-category
n-groupoid = (n,0)-category
This entry lists aspects of fundamental physics from the nPOV: its description in terms of category theory and higher category theory. For a more coherent exposition starting see also at geometry of physics.
We discuss the setting in which fundamental physics takes place.
Physics is dynamics in spaces .
Higher topos theory provides the formalizations of this most fundamental aspect of physics.
A general context for geometrical spaces and processes in these spaces is a cohesive (∞,1)-topos.
In its Cahiers topos-version it contains also
In its full derived geometry-version it contains also
Every cohesive -topos is in particular a locally ∞-connected (∞,1)-topos. For these their intrinsic cohomology refines to differential cohomology in an (∞,1)-topos classifying connections on ∞-bundles. This encodes dynamics in physics: a connection on a principal -bundle is a gauge field which exerts forces. Such as:
In every such connected -topos every characteristic class gives rise to its ∞-Chern-Weil homomorphism that sends gauge fields encoded as ∞-connections on principal ∞-bundles to circle n-bundles with connection. Such as:
Under the higher parallel transport of these circle -bundles with connection, this assignment is the action functional for the ∞-Chern-Simons theory of the corresponding characteristic class. This includes as special cases various sigma-model quantum field theories such as:
on each piece with boundary a morphism between these -vector spaces encoding the propagation of states;
on each open subset the algebra of observables .
Two dual formalizations axiomatize this:
Well-understood examples of such quantum field theories include
By the above there is a fairly well-developed formalization of
This step in full generality is not yet well understood formally. For a list of literature addressing this problem see Literature on quantization.
But special aspects of quantization are quite well understood. See for instance
We look at some aspects of the above general abstract story in more detail.
The discovery of gauge theory is effectively the discovery of groupoids in fundamental physics. The notion of gauge transformation is close to synonymous to the notion isomorphism and more generally to equivalence in an (∞,1)-category .
From a modern point of view, the mathematical model for a gauge field in physics is a cocycle in (nonabelian) differential cohomology: a principal bundle with connection and its higher analogs. These naturally do not form just a set, but a groupoid and generally an ∞-groupoid, whose morphisms are gauge transformations, and higher morphisms are gauge-of-gauge transformations. The development of differential cohomology has to a fair extent been motivated by and influenced by its application to fundamental theoretical physics in general and gauge theory in particular.
Around 1850 Maxwell realized that the field strength of the electromagnetic field is modeled by what today we call a closed differential 2-form on spacetime. In the 1930s Dirac observed that more precisely this 2-form is the curvature 2-form of a U(1)-principal bundle with connection, hence that the electromagnetic field is modeled by what today is called a degree 2-cocycle in ordinary differential cohomology .
Meanwhile, in 1915, Einstein had identified also the field strength of the field of gravity as the -valued curvature 2-form of the canonical O(d,1)-principal bundle with connection on a -dimensional spacetime Lorentzian manifold. This is a cocycle in differential nonabelian cohomology: in Chern-Weil theory.
In the 1950s Yang-Mills theory identified the field strength of all the gauge fields in the standard model of particle physics as the -valued curvature 2-forms of U(n)-principal bundles with connection. This is again a cocycle in differential nonabelian cohomology.
Entities of ordinary gauge theory
It is noteworthy that already in this mathematical formulation of experimentally well-confirmed fundamental physics the seed of higher differential cohomology is hidden: Dirac had not only identified the electromagnetic field as a line bundle with connection, but he also correctly identified (rephrased in modern language) its underlying cohomological Chern class with the (physically hypothetical but formally inevitable) magnetic charge located in spacetime. But in order to make sense of this, he had to resort to removing the support of the magnetic charge density from the spacetime manifold, because Maxwell’s equations imply that at the support of any magnetic charge the 2-form representing the field strength of the electromagnetic field is in fact not closed and hence in particular not the curvature 2-form of an ordinary connection on an ordinary bundle.
In (Freed) this old argument was improved by refining the model for the electromagnetic field one more step: Dan Freed notices that the charge current 3-form is itself to be regarded as a curvature, but for a connection on a circle 2-bundle with connection – also called a bundle gerbe – , which is a cocycle in degree 3 ordinary differential cohomology. Accordingly, the electromagnetic field is fundamentally not quite a line bundle, but a twisted bundle with connection, with the twist being the magnetic charge 3-cocycle. Freed shows that this perspective is inevitable for understanding the quantum anomaly of the action functional for electromagnetism is the presence of magnetic charge.
In summary, the experimentally verified models, to date, of fundamental physics are based on the notion of (twisted) -principal bundles with connection for the Yang-Mills field and -principal bundles with connection for the description of gravity, hence on nonabelian differential cohomology in degree 2 (possibly with a degree-3 twist).
In attempts to better understand the structure of these two theories and their interrelation, theoretical physicists were led to consider variations and generalizations of them that are known as supergravity and string theory. In these theories the notion of gauge field turns out to generalize: instead of just Lie algebras, Lie groups and connections with values in these, one finds structures called Lie 2-algebras, Lie 2-groups and the gauge fields themselves behave like generalized connections with values in these.
Entities of 2-gauge theory
Notably the string is charged under a field called the Kalb-Ramond field or -field which is modeled by a -principal 2-bundle with connection, where is the Lie 2-group delooping of the circle group: the circle Lie 2-group. Its Lie 2-algebra is given by the differential crossed module which has shifted up by one in homological degree.
So far all these differential cocycles were known and understood mostly as concrete constructs, without making their abstract home in differential cohomology explicit. It is the next gauge field that made Freed and Hopkins propose (FreedHopkins, Freed) that the theory of differential cohomology is generally the formalism that models gauge fields in physics:
The superstring is charged also under what is called the RR-field, a gauge field modeled by cocycles in differential K-theory. In even degrees we may think of this as a differential cocycle whose curvature form has coefficients in the ∞-Lie algebra . Here is the abelian 2n-Lie algebra whose underlying complex is concentrated in degree on .
Entities of general gauge theory
The curvature characteristic forms / Chern characters in the abelian formulation of differential cohomology take values in abelian ∞-Lie algebras and are therefore effectively nothing but differential forms with values in a complex of vector spaces, but more generally in ∞-Chern-Weil theory on nonabelian principal ∞-bundles, the curvatures forms themselves take values in general ∞-Lie algebras, such as the string Lie 2-algebra, the supergravity Lie 3-algebra and the fivebrane Lie 6-algebra.
Apart from generalizing the notion of gauge Lie groups to Lie 2-groups and further, structural considerations in fundamental physics also led theoretical physicists to consider models for spacetime that are more general than than the notion of a smooth manifold. In string theory spacetime is allowed to be more generally an orbifold or a generalization thereof, such as an orientifold. The natural mathematical model for these generalized spaces are Lie groupoids or, essentuially equivalently, differentiable stacks .
It is noteworthy that the notions of generalized gauge groups and the generalized spacetime models encountered this way have a natural common context: all of these are examples of smooth ∞-groupoids.
There is a natural mathematical concept that serves to describe contexts of such generalized spaces: a gros (∞,1)-topos. The notion of differential cohomology in an (∞,1)-topos provides a unifying perspective on the mathematical structure encoding the generalized gauge fields and generalized spacetime models encountered in modern theoretical physics in such a general context.
We discuss classes of examples of gauge theories that have been considered. For all of these the configuration space is a space of connections on ∞-bundles over spacetime of sorts, which one might take to be the defining property of a gauge theory. But there are different types of action functionals on these configuration spaces.
and some invariant polynomial.
This is the original notion of gauge theory and might be taken to be the strict sense of the term.
Specifically for a discrete quotient of this is the gauge-field part of the standard model of particle physics.
A nonabelian cohomology version of higher Yang-Mills theory – replacing a connection on a bundle by a connection on a 2-bundle is expected to control certain 6-dimensional theories that compactify to ordinary Yang-Mills on the torus and thereby explain S-duality . See there for more details on this.
This includes ordinary Chern-Simons theory in the case that is a semisimple Lie algebra, but for general it subsumes a wide variety of types of TQFTs that are often counted as of different type than Chern-Simons theory, such as BF-theory and AKSZ theory.
the underlying Lagrangian is nothing but the ∞-Chern-Weil homomorphism
induced by the given invariant polynomial . This sends gauge fields in the form of -valued connections on ∞-bundles to the circle n-bundle with connection whose higher parallel transport is given by the Lagrangian;
the integral over is induced by postcomposition with the trunction morphism
For a semisimple Lie algebra equipped with its Killing form invariant polynomial we have that is the ordinary Chern-Simons element and the ordinary Chern-Simons theory action functional. This may be understood as the higher parallel transport of a Chern-Simons circle 3-bundle with connection.
For a semisimple Lie algebra equipped with its degree 8 invariant polynomial this yields 7-dimensional, whose action functional may be understood as the higher parallel transport of a Chern-Simons circle 7-bundle with connection.
The configuration spaces of gravity and supergravity may be identified with spaces of connections on ∞-bundles with gauge group a variant of the Poincare group. This parameterization of the configuration space of gravity is known as the first order formulation of gravity, to be contrasted with a formulation explicitly over a space of pseudo-Riemannian manifolds.
the translation group-component of is called the vielbein.
In contrast to the -Chern-Simons theory discussed above, the general abstract nature, if any, of the action functional for gravity remains somewhat inconclusive and subject of a plethora of speculations. If one passes from connections to their associated Dirac operators and interprets these as parts of a spectral triple there is the spectral action functional on the space of spectral triples. This we discuss in more detail below.
There are various higher group extensions of the Poincare group and the orthonormal group that lead accordingly to higher order variations of gravity.
lifting -connections through the smooth Whitehead tower
yields, in order of appearance,
This lift is necessary to cancel the quantum anomaly of spinning particles coupled to gravity;
This lift is necessary to cancel the quantum anomaly of heterotic superstrings;
This lift is necessary to cancel the quantum anomaly of super 5-branes.
The D'Auria-Fre formulation of supergravity explicitly describes higher dimensional supergravity theories as gauge theories with higher super Lie structure groups
Theoretical physics consists of two parts: theory and models, laws and initial conditions, axioms and phenomenology.
For instance the theory called general relativity describes the classical dynamics of gravity, but does not predict the value of the cosmological constant. Rather, for each choice of the latter does the theory predict a certain dynamics the large-scale universe.
The theory that describes the fundamental forces and particles except gravity is Yang-Mills theory. This, too, does not predict the fundamental particle species seen in experiments, but for a correct choice and identification of these, the theory does predict the dynamics of these particles, as observed in accelerators.
The total collection of these choices of fundamental particles that are observed in experiments is called the standard model of particle physics. It consists basically of
What precisely the “standard” model of particle physics is changes slightly over time, as new experimental insights are gained. Its particles were added item-by-item as they were discovered. More recently the mass of the particles called neutrinos, which was originally thought to be precisely 0, was measured to be very small, but non-vanishing.
The standard model as far as understood today exhibits a curious mixture of pattern and irregularity. This seems to suggest that it ought to have a more fundamental description in terms of a conceptually simpler structure out of which these patterns with their irregularities emerge. Since also the force of gravity is not presently included in the quantization of the standard model, it may seem plausible that this underlying structure is related to quantum gravity.
We discuss in the following some of the proposals that have been suggested for how to formalize this situation.
A fundamental relativistic particle is technically a 1-dimensional sigma-model QFT on 1-dimensional cobordisms with target the spacetime that it propagates in. Since both gravity as well as Yang-Mills fields are encoded in connections it is plausible to assume that the only background field on that the particle couples to is a connection on a -associated bundle over a -principal bundle.
This way a single -model may encode a rich multiple particle content and we shall speak of a single superparticle with different excitations or modes .
An early proposal for a single unified connection that would subsume both gravity as well as Yang-Mills forces in a phenomenologically realistic way is the Kaluza-Klein mechanism. This assumes a single Levi-Civita connection but on a pseudo-Riemannian manifold which is locally of the product form with a 4-dimensional pseudo-Riemannian manifold and a -dimensional Riemannian manifold of very small Riemannian volume. As described in more detail at Kaluza-Klein mechanism, this makes the isometry groups of appear as extra gauge group factors as seen on . As also described in more detail there, while this Ansatz does reproduce the correct general form of gravity coupled to Yang-Mills forces, in its original form it does also have some phenomenologically unviable aspects.
It was observed by Alain Connes and collaborators that the Kaluza-Klein mechanism works better when used not internal to the category Diff of smooth manifolds but in context for more general geometry: noncommutative geometry.
This more general geometry turns out to model exactly the most general -model backgrounds for a 1-dimensional FQFT: because such is algebraically specified by
This data is that of spectral triple, which is well known to enocode Riemannian noncommutative geometry (rather: spectral geometry ). It is therefore natural to search for a Kaluza-Klein ansatz in spectral geometry that would produce the standard model context.
It turns out that the realistic model has -theory dimension .
By the result at (1,1)-dimensional Euclidean field theories and K-theory this -theory dimension is precisely the intrinsic dimension of target space as seen by the superparticle. Moreover, is precisely the dimension for whic the 2-dimensional super-CFT sigma-model is critical and hence allows to lift the 1-dimensional superparticle described here to string theory.
This means that Connes’ spectral triple whose particle spectrum reproduces the standard model of particle physics has a chance of being the point particle limit or decategorification of the kind of 2-spectral triple – a 2-dimensional superconformal field theory – of the kind that is considered in string theory. If so the lift of Connes’ model to the corresponding element in the moduli space of 2-spectral triples called the landscape of string theory vacua might provide, via the second quantization of the latter, a (perturbative) quantization of the spectral action of the former.
We list various references related to higher category theory and fundamental physics.
For general (formal) accounts of physics see also the references at books and reviews in mathematical physics, such as
the authors try to motivate and introduce some basic concepts of category theory for an audience familiar with standard physics and in particular with quantum mechanics. The article focuses towards the end on monoidal categories, their description in terms of string diagrams and quantum mechanics in terms of dagger-compact categories.
More details on the use of string diagrams in dagger-categories for the description of quantum mechanics is is
A historical introduction to some aspects of n-categories (for low ) in physics can be found here:
The following book-to-be aims to give picture of the present state of the art of describing the category-theoretic structure of the universe, as far as fundamental physics is concerned
An unusually comprehensive collection of detailed discussion of bundles, higher bundles, cohomology, characteristic classes, etc. with an eye on applications in physics is
Kontsevich’s 1994 ICM article, is one of the seminal papers of the 1990s. This paper invented the categorical side of mirror symmetry (homological mirror symmetry), discovered D-branes (before physicists realized their role — and directly inspiring many physicists) and the fact that they naturally form dg-categories and -category, and thus led to a deluge of papers involving category theory and higher category theory in close relation with mathematical physics.
In particular the Moore--Segal paper should be seen in the light of this development. On a similar note, roughly contemporary with Moore–Segal (which is arXiv:math/0609042 though developed earlier) are the works of Kevin Costello on TCFT (arXiv:math/0412149) and Kontsevich–Soibelman (arXiv:math/0606241 — some of the results were lectured on in various places by Kontsevich in 2003 and in particular helped inspire Costello) proving a much stronger result, which is the TCFT (or equivalently differential graded or -) version of the classification of open-closed 2d TQFTs. These papers were directly motivated by homological mirror symmetry and topological string theory, and have greatly influenced work in areas such as string topology which you mention and the cobordism hypothesis (Hopkins-Lurie started from Costello’s paper and abstracted the argument, before the general argument in n-dimensions came around).
(in parts taken from here)
For the moment see the references at FQFT for more.
Literature on the relation between spectral triples and 1-dimensional super-QFT is to appear shortly. Spectral background for the standard model have been considered here:
The first proposal for such compactifications, which already came very close to the standard model, is decades old by now.
This is reviewed in
As far as I can tell, at that time the Yang-Mills terms were stilll included by hand. The main point of the spectral approach was to realize that it could nicely explain the Higgs boson and its Yukawa coupling terms to the fermions. It did (and does) so by realizing the Higgs boson as an internal part of an ordinary minimally coupled connection 1-form - an internal gauge boson.
A few years later Connes apparently realized that also the gravitational and gauge kinetic Yang-Mills terms had an inherent operator-theoretic formulation, namely the spectral action principle.
This is indicated in
and fully formulated in
both of which start by presenting the general spectral idea and then working out how to realize the standard model in detail.
A useful discussion of the details of the realization of the standard model in this approach is
This is in particular designed to take the ordinary physicst by the hand and introduce him or her gently to the operator-theoretic spectral description by motivating these by the structure of the standard model.
So already over ten years ago people had a pretty good idea that and how the standard model action has an elegant descritption as a spectral action.
The only problem was: this description was wrong. But only in the sense in which ideas in physics tend to be wrong - not entirely wrong but not quite right.
Namely, instead of containing the fermionic particle content of the standard model, these spectral models produced four copies of every fermion in the standard model. A slight overkill.
By a remarkable synchronicity, it seems that there was no progress on this aspect for about ten years, and now two preprints appear almost simultaneously, presenting the solution:
And the modification needed to get this solution is rather tiny, just a small change in the real structure of the spectral triple from ten years ago.
A survey is also at
For the moment see the references at AQFT for more.
and its companion
More formal aspects along these lines appear in
An indication of a full formalization of what that may mean for discrete QFTs such as Dijkgraaf-Witten theory is in
based on work in
Motivated by this, the theory of differential cohomology itself was developed in
and shown to encode subtle quantum anomaly-cancellation phenomena in gauge theory. A systematic formal description of the nature of these quantum anomalies and further discussion of applications on differential cohomology in physics is in
This article starts with an introduction on basic electromagnetism and points out that already there, in the presence of magnetic charge, a careful analysis of quantum anomalies shows that there are higher gauge theoretic effects even in this familiar theory.
The observation that any formalization of physics ought to take place in a suitable topos has been promoted early on by Bill Lawvere. Lawvere started out as a student in continuum mechanics and his search for a formalization of physics made him end up being one of the most general abstract thinkers in category theory.
Lawvere thought about classical physics governed, as it is since Newton, by the theory of differential equations. He proposed that differential geometry is something that takes place inside a smooth topos – a topos with a suitable notion of infinitesimal objects – an approach now known as synthetic differential geometry.
which is recalled in
he sketched an outline both of synthetic differenial geometry in general and its application to the formulation of the differential equations in classical dynamics.
In this spirit he to search for category theoretic formalizations of fundamental ( metaphysical in the good sense of the word) ontological concepts. A collection of such formalizations is indicated in
On page 14 there appears the hint of the general abstract definition of a locally connected topos as a general abstract gros topos context for geometric spaces. This idea was then refined to the concept of a cohesive topos in
In this spirit a generalization of formalization of physics in (∞,1)-topos is discussed in
Independent of these developments is another proposal for a role of topos theory in quantum mechanics initiated by Chris Isham. The motivation here is the observation that the Kochen-Specker theorem – which may be used to formalize the way in which quantum mechanics fails to secretly a theory of classical mechanics – has a natural interpretation in terms of the internal logic of the presheaf topos on the category of commutative sub-algebras of a given bounded operator algebra.
This observation led Isham to propose a fundamental role of these presheaf toposes in quantum mechanics.
This idea was further refined in
The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C-star-algebra of observables induces a topos in which the amalgamation of all of its commutative subalgebras comprises a single commutative -algebra. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum in , which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a -algebra (which is the noncommutative notion of a space). In this setting, states on become probability measures (more precisely, valuations) on , and self-adjoint elements of define continuous functions (more precisely, locale maps) from to Scott’s interval domain. Noting that open subsets of correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by is essentially turned into a classical theory, internal to the topos .
See Bohr topos for more.
In the context of phenomenology, higher category theory is currently applied mostly for 2-categories in the context of 2-dimensional TQFT (modelling phenomena in solid state physics) and CFT (modelling critical surface phenomena in statistical physics).
In the quantum hall effect and related effects are described to some extent by Chern-Simons theory, and Chern-Simons theory comes with a rich higher categorical framework. Notably there is now a fully extended TQFT description of Chern-Simons theory down to the point, hence in terms of (infinity,3)-funcots, see here.
Closely related to this, several approaches at realizing quantum computing rely on TQFT methods and are treated with methods from higher category theory. For instance the notion of a blob n-category orginates in investigations of quantum computing.
Indeed, if one counts computing as “experimental” or “phenomenology” – at least as related to the tangible world – then category theory is ubiquituous, see at computational trinitarianism for the general idea and further pointers. For instance the relation between type theory and category theory (see there) which underlies the formal understanding of modern computer science, is an equivalence of 2-categories in 2-category theory. More recently, computing in the guise of homotopy type theory is closely related to (infinity,1)-category theory. This carries in it the prospect of serving as the very foundations of mathematics and computer science. And hence, in effect, also of theoretical physics (see Schreiber-Shulman 2012 above).