Hamburg

Since I moved to Hamburg,   Alessandro Valentino and I have been organizing one series of seminar talks whose goal is to bring people (mostly graduate students, and some postdocs and others) up to speed on the tools used in Jacob Lurie’s big paper on the classification of TQFT and proof of the Cobordism Hypothesis.  This is part of the Forschungsseminar (“research seminar”) for the working groups of Christoph Schweigert, Ingo Runkel, and Christoph Wockel.  First, I gave one introducing myself and what I’ve done on Extended TQFT. In our main series We’ve had a series of four so far – two in which Alessandro outlined a sketch of what Lurie’s result is, and another two by Sebastian Novak and Marc Palm that started catching our audience up on the simplicial methods used in the theory of (\infty,n)-categories which it uses.  Coming up in the New Year, Nathan Bowler and I will be talking about first (\infty,1)-categories, and then (\infty,n)-categories.   I’ll do a few posts summarizing the talks around then.

Some people in the group have done some work on quantum field theories with defects, in relation to which, there’s this workshop coming up here in February!  The idea here is that one could have two regions of space where different field theories apply, which are connected along a boundary. We might imagine these are theories which are different approximations to what’s going on physically, with a different approximation useful in each region.  Whatever the intuition, the regions will be labelled by some category, and boundaries between regions are labelled by functors between categories.  Where different boundary walls meet, one can have natural transformations.  There’s a whole theory of how a 3D TQFT can be associated to modular tensor categories, in sort of the same sense that a 2D TQFT is associated to a Frobenius algebra. This whole program is intimately connected with the idea of “extending” a given TQFT, in the sense that it deals with theories that have inputs which are spaces (or, in the case of defects, sub-spaces of given ones) of many different dimensions.  Lurie’s paper describing the n-dimensional cobordism category, is very much related to the input to a theory like this.

Brno Visit

This time, I’d like to mention something which I began working on with Roger Picken in Lisbon, and talked about for the first time in Brno, Czech Republic, where I was invited to visit at Masaryk University.  I was in Brno for a week or so, and on Thursday, December 13, I gave this talk, called “Higher Gauge Theory and 2-Group Actions”.  But first, some pictures!

This fellow was near the hotel I stayed in:

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Since this sculpture is both faceless and hard at work on nonspecific manual labour, I assume he’s a Communist-era artwork, but I don’t really know for sure.

The Christmas market was on in Náměstí Svobody (Freedom Square) in the centre of town.  This four-headed dragon caught my eye:

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On the way back from Brno to Hamburg, I met up with my wife to spend a couple of days in Prague.  Here’s the Christmas market in the Old Town Square of Prague:

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Anyway, it was a good visit to the Czech Republic.  Now, about the talk!

Moduli Spaces in Higher Gauge Theory

The motivation which I tried to emphasize is to define a specific, concrete situation in which to explore the concept of “2-Symmetry”.  The situation is supposed to be, if not a realistic physical theory, then at least one which has enough physics-like features to give a good proof of concept argument that such higher symmetries should be meaningful in nature.  The idea is that Higher Gauge theory is a field theory which can be understood as one in which the possible (classical) fields on a space/spacetime manifold consist of maps from that space into some target space X.  For the topological theory, they are actually just homotopy classes of maps.  This is somewhat related to Sigma models used in theoretical physics, and mathematically to Homotopy Quantum Field Theory, which considers these maps as geometric structure on a manifold.  An HQFT is a functor taking such structured manifolds and cobordisms into Hilbert spaces and linear maps.  In the paper Roger and I are working on, we don’t talk about this stage of the process: we’re just considering how higher-symmetry appears in the moduli spaces for fields of this kind, which we think of in terms of Higher Gauge Theory.

Ordinary topological gauge theory – the study of flat connections on G-bundles for some Lie group G, can be looked at this way.  The target space X = BG is the “classifying space” of the Lie group – homotopy classes of maps in Hom(M,BG) are the same as groupoid homomorphisms in Hom(\Pi_1(M),G).  Specifically, the pair of functors \Pi_1 and B relating groupoids and topological spaces are adjoints.  Now, this deals with the situation where X = BG is a homotopy 1-type, which is to say that it has a fundamental groupoid \Pi_1(X) = G, and no other interesting homotopy groups.  To deal with more general target spaces X, one should really deal with infinity-groupoids, which can capture the whole homotopy type of X – in particular, all its higher homotopy groups at once (and various relations between them).  What we’re talking about in this paper is exactly one step in that direction: we deal with 2-groupoids.

We can think of this in terms of maps into a target space X which is a 2-type, with nontrivial fundamental groupoid \Pi_1(X), but also interesting second homotopy group \pi_2(X) (and nothing higher).  These fit together to make a 2-groupoid \Pi_2(X), which is a 2-group if X is connected.  The idea is that X is the classifying space of some 2-group \mathcal{G}, which plays the role of the Lie group G in gauge theory.  It is the “gauge 2-group”.  Homotopy classes of maps into X = B \mathcal{G} correspond to flat connections in this 2-group.

For practical purposes, we use the fact that there are several equivalent ways of describing 2-groups.  Two very directly equivalent ways to define them are as group objects internal to \mathbf{Cat}, or as categories internal to \mathbf{Grp} – which have a group of objects and a group of morphisms, and group homomorphisms that define source, target, composition, and so on.  This second way is fairly close to the equivalent formulation as crossed modules (G,H,\rhd,\partial).  The definition is in the slides, but essentially the point is that G is the group of objects, and with the action G \rhd H, one gets the semidirect product G \ltimes H which is the group of morphisms.  The map \partial : H \rightarrow G makes it possible to speak of G and H acting on each other, and that these actions “look like conjugation” (the precise meaning of which is in the defining properties of the crossed module).

The reason for looking at the crossed-module formulation is that it then becomes fairly easy to understand the geometric nature of the fields we’re talking about.  In ordinary gauge theory, a connection can be described locally as a 1-form with values in Lie(G), the Lie algebra of G.  Integrating such forms along curves gives another way to describe the connection, in terms of a rule assigning to every curve a holonomy valued in G which describes how to transport something (generally, a fibre of a bundle) along the curve.  It’s somewhat nontrivial to say how this relates to the classic definition of a connection on a bundle, which can be described locally on “patches” of the manifold via 1-forms together with gluing functions where patches overlap.  The resulting categories are equivalent, though.

In higher gauge theory, we take a similar view. There is a local view of “connections on gerbes“, described by forms and gluing functions (the main difference in higher gauge theory is that the gluing functions related to higher cohomology).  But we will take the equivalent point of view where the connection is described by G-valued holonomies along paths, and H-valued holonomies over surfaces, for a crossed module (G,H,\rhd,\partial), which satisfy some flatness conditions.  These amount to 2-functors of 2-categories \Pi_2(M) \rightarrow \mathcal{G}.

The moduli space of all such 2-connections is only part of the story.  2-functors are related by natural transformations, which are in turn related by “modifications”.  In gauge theory, the natural transformations are called “gauge transformations”, and though the term doesn’t seem to be in common use, the obvious term for the next layer would be “gauge modifications”. It is possible to assemble a 2-groupoid Hom(\Pi_2(M),\mathcal{G}, whose space of objects is exactly the moduli space of 2-connections, and whose 1- and 2-morphisms are exactly these gauge transformations and modifications.  So the question is, what is the meaning of the extra information contained in the 2-groupoid which doesn’t appear in the moduli space itself?

Our claim is that this information expresses how the moduli space carries “higher symmetry”.

2-Group Actions and the Transformation Double Category

What would it mean to say that something exhibits “higher” symmetry? A rudimentary way to formalize the intuition of “symmetry” is to say that there is a group (of “symmetries”) which acts on some object. One could get more subtle, but this should be enough to begin with. We already noted that “higher” gauge theory uses 2-groups (and beyond into n-groups) in the place of ordinary groups.  So in this context, the natural way to interpret it is by saying that there is an action of a 2-group on something.

Just as there are several equivalent ways to define a 2-group, there are different ways to say what it means for it to have an action on something.  One definition of a 2-group is to say that it’s a 2-category with one object and all morphisms and 2-morphisms invertible.  This definition makes it clear that a 2-group has to act on an object of some 2-category \mathcal{C}. For our purposes, just as we normally think of group actions on sets, we will focus on 2-group actions on categories, so that \mathcal{C} = \mathbf{Cat} is the 2-category of interest. Then an action is just a map:

\Phi : \mathcal{G} \rightarrow \mathbf{Cat}

The unique object of \mathcal{G} – let’s call it \star, gets taken to some object \mathbf{C} = \Phi(\star) \in \mathbf{Cat}.  This object \mathbf{C} is the thing being “acted on” by \mathcal{G}.  The existence of the action implies that there are automorphisms \Phi(g) : \mathbf{C} \rightarrow \mathbf{C} for every morphism in \mathbf{G} (which correspond to the elements of the group G of the crossed module).  This would be enough to describe ordinary symmetry, but the higher symmetry is also expressed in the images of 2-morphisms \Phi( \eta : g \rightarrow g') = \Phi(\eta) : \Phi(g) \rightarrow \Phi(g'), which we might call 2-symmetries relating 1-symmetries.

What we want to do in our paper, which the talk summarizes, is to show how this sort of 2-group action gives rise to a 2-groupoid (actually, just a 2-category when the \mathbf{C} being acted on is a general category).  Then we claim that the 2-groupoid of connections can be seen as one that shows up in exactly this way.  (In the following, I have to give some credit to Dany Majard for talking this out and helping to find a better formalism.)

To make sense of this, we use the fact that there is a diagrammatic way to describe the transformation groupoid associated to the action of a group G on a set S.  The set of morphisms is built as a pullback of the action map, \rhd : (g,s) \mapsto g(s).

pullback

This means that morphisms are pairs (g,s), thought of as going from s to g(s).  The rule for composing these is another pullback.  The diagram which shows how it’s done appears in the slides.  The whole construction ends up giving a cubical diagram in \mathbf{Sets}, whose top and bottom faces are mere commuting diagrams, and whose four other faces are all pullback squares.

To construct a 2-category from a 2-group action is similar. For now we assume that the 2-group action is strict (rather than being given by \Phi a weak 2-functor).  In this case, it’s enough to think of our 2-group \mathcal{G} not as a 2-category, but as a group-object in \mathbf{Cat} – the same way that a 1-group, as well as being a category, can be seen as a group object in \mathbf{Set}.  The set of objects of this category is the group G of morphisms of the 2-category, and the morphisms make up the group G \ltimes H of 2-morphisms.  Being a group object is the same as having all the extra structure making up a 2-group.

To describe a strict action of such a \mathcal{G} on \mathbf{C}, we just reproduce in \mathbf{Cat} the diagram that defines an action in \mathbf{Sets}:

action

The fact that \rhd is an action just means this commutes. In principle, we could define a weak action, which would mean that this commutes up to isomorphism, but we won’t be looking at that here.

Constructing the same diagram which describes the structure of a transformation groupoid (p29 in the slides for the talk), we get a structure with a “category of objects” and a “category of morphisms”.  The construction in \mathbf{Set} gives us directly a set of morphisms, while S itself is the set of objects. Similarly, in \mathbf{Cat}, the category of objects is just \mathbf{C}, while the construction gives a category of morphisms.

The two together make a category internal to \mathbf{Cat}, which is to say a double category.  By analogy with S / \!\! / G, we call this double category \mathbf{C} / \!\! / \mathcal{G}.

We take \mathbf{C} as the category of objects, as the “horizontal category”, whose morphisms are the horizontal arrows of the double category. The category of morphisms of \mathbf{C} /\!\!/ \mathcal{G} shows up by letting its objects be the vertical arrows of the double category, and its morphisms be the squares.  These look like this:

squares

The vertical arrows are given by pairs of objects (\gamma, x), and just like the transformation 1-groupoid, each corresponds to the fact that the action of \gamma takes x to \gamma \rhd x. Each square (morphism in the category of morphisms) is given by a pair ( (\gamma, \eta), f) of morphisms, one from \mathcal{G} (given by an element in G \rtimes H), and one from \mathbf{C}.

The horizontal arrow on the bottom of this square is:

(\partial \eta) \gamma \rhd f \circ \Phi(\gamma,\eta)_x = \Phi(\gamma,\eta)_y \circ \gamma \rhd f

The fact that these are equal is exactly the fact that \Phi(\gamma,\eta) is a natural transformation.

The double category \mathbf{C} /\!\!/ \mathcal{G} turns out to have a very natural example which occurs in higher gauge theory.

Higher Symmetry of the Moduli Space

The point of the talk is to show how the 2-groupoid of connections, previously described as Hom(\Pi_2(M),\mathcal{G}), can be seen as coming from a 2-group action on a category – the objects of this category being exactly the connections. In the slides above, for various reasons, we did this in a discretized setting – a manifold with a decomposition into cells. This is useful for writing things down explicitly, but not essential to the idea behind the 2-symmetry of the moduli space.

The point is that there is a category we call \mathbf{Conn}, whose objects are the connections: these assign G-holonomies to edges of our discretization (in general, to paths), and H-holonomies to 2D faces. (Without discretization, one would describe these in terms of Lie(G)-valued 1-forms and Lie(H)-valued 2-forms.)

The morphisms of \mathbf{Conn} are one type of “gauge transformation”: namely, those which assign H-holonomies to edges. (Or: Lie(H)-valued 1-forms). They affect the edge holonomies of a connection just like a 2-morphism in \mathcal{G}.  Face holonomies are affected by the H-value that comes from the boundary of the face.

What’s physically significant here is that both objects and morphisms of \mathbf{Conn} describe nonlocal geometric information.  They describe holonomies over edges and surfaces: not what happens at a point.  The “2-group of gauge transformations”, which we call \mathbf{Gauge}, on the other hand, is purely about local transformations.  If V is the vertex set of the discretized manifold, then \mathbf{Gauge} = \mathcal{G}^V: one copy of the gauge 2-group at each vertex.  (Keeping this finite dimensional and avoiding technical details was one main reason we chose to use a discretization.  In principle, one could also talk about the 2-group of \mathcal{G}-valued functions, whose objects and morphisms, thinking of it as a group object in \mathbf{Cat}, are functions valued in morphisms of \mathcal{G}.)

Now, the way \mathbf{Gauge} acts on \mathbf{Conn} is essentially by conjugation: edge holonomies are affected by pre- and post-multiplication by the values at the two vertices on the edge – whether objects or morphisms of \mathbf{Gauge}.  (Face holonomies are unaffected).  There are details about this in the slides, but the important thing is that this is a 2-group of purely local changes.  The objects of \mathbf{Gauge} are gauge transformations of this other type.  In a continuous setting, they would be described by G-valued functions.  The morphisms are gauge modifications, and could be described by H-valued functions.

The main conceptual point here is that we have really distinguished between two kinds of gauge transformation, which are the horizontal and vertical arrows of the double category \mathbf{Conn} /\!\!/ \mathbf{Gauge}.  This expresses the 2-symmetry by moving some gauge transformations into the category of connections, and others into the 2-group which acts on it.  But physically, we would like to say that both are “gauge transformations”.  So one way to do this is to “collapse” the double category to a bicategory: just formally allow horizontal and vertical arrows to compose, so that there is only one kind of arrow.  Squares become 2-cells.

So then if we collapse the double category expressing our 2-symmetry relation this way, the result is exactly equivalent to the functor category way of describing connections.  (The morphisms will all be invertible because \mathbf{Conn} is a groupoid and \mathbf{Gauge} is a 2-group).

I’m interested in this kind of geometrical example partly because it gives a good way to visualize something new happening here.  There appears to be some natural 2-symmetry on this space of fields, which is fairly easy to see geometrically, and distinguishes in a fundamental way between two types of gauge transformation.  This sort of phenomenon doesn’t occur in the world of \mathbf{Sets} – a set S has no morphisms, after all, so the transformation groupoid for a group action on it is much simpler.

In broad terms, this means that 2-symmetry has qualitatively new features that familiar old 1-symmetry doesn’t have.  Higher categorical versions – n-groups acting on n-groupoids, as might show up in more complicated HQFT – will certainly be even more complicated.  The 2-categorical version is just the first non-trivial situation where this happens, so it gives a nice starting point to understand what’s new in higher symmetry that we didn’t already know.

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