April 2008

Monthly Archive

April 29, 2008

Recent Talk - Enxin Wu on Algebra Deformations

Posted by Jeffrey Morton under algebra, cohomology, quantization, talks
1 Comment 

I’m going up to Ottawa for a few days, in part to talk about spans and groupoids (basically, some cross section of the material in these posts here) at a conference put on by the Ottawa U math department, primarily for grad students and postdocs in the general vicinity. This is nice - gives me a chance to visit my parents and friends there (the fraction of my life I lived in Ottawa is now creeping down toward a mere third, but it probably has as strong a claim to “home” as anywhere). May is also one of the most tolerable months to be there. One of the grad students in our department is also going. Enxin Wu recently decided to start working with Dan Christensen too, so probably in future we’ll have various things to talk about. Last week, he gave a seminar talk on algebra deformation that was a long version of the one he’ll be giving in Ottawa.

Enxin is one of those guys who seems to really understand - it’s tempting to say grok- algebra, which I always find impressive. I’m a predominantly visual thinker, and the kind of symbolic computations common in algebra always seem a little mysterious to me at first until I can find a picture, or at least practice them a lot. Lie groups, for instance, make some sense to me - you can picture rotation groups, or at least keep a geometric picture of a manifold in mind. Lie algebras, being infinitesimal versions of Lie groups, are also not so hard to visualize. General associative algebras? Harder.

The talk was about associative algebras, to give some background on deformation, but the things whose deformations Enxin has been thinking about are A_{\infty}-algebras (see this brief intro, for instance), an “invention” of Stasheff. The talk was about deformation of these algebras - the kind of deformation that pertains to deformation quantization. This has been studied by Kontsevich. Deformation quantization has to do with replacing things valued in some algebra A by new things, valued in the bigger algebra A[[t]] of formal power series in t with coefficients in A, so that the original structure you started with is just the constant part that appears when you set t=0. (The term “quantization” applies when you consider algebras of functions on a manifold, with a Poisson bracket - in other words, algebras of observables of a physical system).

Some of the main results have to do with the Hochschild cohomology for some complex associated to the algebra you start with, and the fact that this cohomology classifies obstructions to the deformation. I expected to get lost in a maze of notation - and there certainly is a lot - but as it turns out, I had some mental pictures to attach to these things, because related things came up a few years ago in the quantum gravity seminar at UCR (week 8 on that page especially), which provides a few pictures that helped a lot. Diagrammatic notation makes algebra a lot more comprehensible to me.

So let’s get more specific.

The point is to replace a multiplication operator m : A \otimes A \rightarrow A with a power series whose coefficients are “multiplication” operators. That is, a deformation of an associative algebra (A,m) (where m : A \otimes A \rightarrow A is the multiplication for A) is (A[[t]],m_t), where the new multiplication m_t is defined (by linearity) by its action on elements of A, which works like this:

m_t(a,b) = \sum_{i=0}^{\infty} {\alpha_i}(a,b){t^i}

for some operators \alpha_i : A \otimes A \rightarrow A. Then there are a bunch of conditions on the \alpha that are needed to make m_t associative. There’s one condition for each power of t, since the coefficients in the associator should be zero:

\sum_{i+j=n\\i,j>0} \alpha_i( (\alpha_j \otimes 1) - (1 \otimes \alpha_j)) = 0

The n=0 condition just says that \alpha_0 is associative - so it’s the m from the original algebra, which you get back when t=0.

Then given an algebra A, you can create the deformation category \mathcal{D} of A whose objects are its deformations. The morphisms are continuous algebra homomorphisms that get along with the multiplication operations. It turns out that since formal power series with nonzero n=0 term are invertible (a consequence of the Lagrange theorem) this \mathcal{D} is actually a groupoid. Then the question is to classify the isomorphism classes of deformations - that is, \Pi_0(\mathcal{D}). One can easily imagine that there might be no nontrivial deformations of some algebra - that is, every one is isomorphic to the deformation where all the \alpha_i are trivial except \alpha_0 = m. So when does this happen? More generally, how can one classify the deformations up to isomorphism?

The answer has to do with Hochschild cohomology, which is related to a complex you can make from A. Taking C^n(A) = hom(A^{\otimes n},A), the space of n-ary multilinear operations on A, you build this complex:

0 \stackrel{d_0}{\longrightarrow} C^0(A) \stackrel{d_1}{\longrightarrow} C^1(A) \stackrel{d_2}{\longrightarrow} \dots

where the differential maps are d_n : C^n(A) \rightarrow C^{n+1}(A) defined by an alternating sum:

d(f)(a_1, \dots, a_n) = a_1 f(a_2, \dots, a_{n+1}) + \sum_{i=1}^{n} (-1)^i f(a_1, \dots, a_i a_{i+1}, \dots, a_{n+1}) + (-1)^{n+1} f(a_1, \dots,a_n) a_{n+1}

(Intuitively: there are too many arguments, so you start with the extra one on the left, push it into the middle as a “lump under the rug” where two arguments are combined, and push the lump all the way to the right. To ensure that d^2 = 0, you do this with alternating signs. This kind of algebraic manipulation is the kind of thing I can do, and clearly works, but I don’t exactly grok.)

Then you take the Hochschild cohomology groups in the standard cohomology way: HH^i = \frac{ker(d_{i+1})}{Im(d_i)}. A cohomology class in one of these groups is a class of multilinear maps from n copies of A to A (up to a factor which is d_n of something). As usual with cohomology, they describe obstructions to something - to exactness. Exactness, in this setting, would mean that A has no interesting deformations at the n^{th} level.

What does “level” mean here? Well, for example, at level 2 we’re talking about maps A \otimes A \rightarrow A, such as the multiplication map. In fact, we have d_3(m) = 0 for an associative algebra - you can check that d(m) is twice the associator a_1(a_2a_3) - (a_1a_2)a_3, which is zero. So m is a cochain. Is it a coboundary? Sure - it’s d_2(1). So m is in the trivial class in HH^2(A). The point then is that it turns out that if this is the only class - if HH^2(A) = 0 - then there are no interesting deformations of the multiplication of A in the sense described above. The groupoid $\mathcal{D}$ has just one object. (One thing that occurs to me is that this makes it a group - which group is something Enxin didn’t discuss. My algebra instincts aren’t quite up to answering that off the top of my head.) For example, if A = \mathbb{C} (as an algebra over \mathbb{R}), there are no nontrivial deformations: HH^2(\mathbb{C}) = 0.

What do the other levels mean? Really, this is where you’d want to look at the generalization from associative algebras to A_{\infty}-algebras. Whereas for an associative algebra A, the associator $a(x,y,z) = x(yz) - (xy)z$ is zero, in general an A_{\infty}-algebra will have an associator map a : A^{\otimes 3} \rightarrow A (that is, a \in C^3 in the complex above), which might not be zero, but which is d_3(m).

This is the beginning of a story relating A_{\infty}-algebras to weak \infty-categories: a bicategory, for example, has an associator for composition of morphisms. In a bicategory, you expect the associator to satisfy a certain identity - the Pentagon identity - but in general you’d just ask for a “pentagonator” (something in C^4), and so on (this is where those seminar notes above help me think in pictures, by the way). An A_{\infty}-algebra is a vector space equipped with maps at all these levels - described by Stasheff’s associahedra - satisfying some relations. The general story of deformation relates the Hochschild cohomology groups at different levels to deformations of A_{\infty}-algebras. Enxin didn’t go into this in his talk, but he did say a little something about the next level:

An infinitesimal deformation of A is a deformation not in A[[t]], but in the quotient A[[t]]/(t^2=0). This only needs two maps, \alpha_0 , \alpha_1. The third Hochschild cohomology measures obstructions to extending an infinitesimal deformation to a full deformation in A[[t]] - if HH^3(A) = 0, then any infinitesimal deformation can be extended to a full deformation.

All in all, I thought the talk was interesting - it tied in much more closely to things I already knew about TQFTs and higher categories than I’d expected. I’ll be really impressed if he can condense it into a 25-minute version…

 

April 11, 2008

Categorification and Matter

Posted by Jeffrey Morton under category theory, philosophical, physics, tqft
[3] Comments 

First, the obligatory excuse found in most sporadic blogs: I haven’t taken the time to write anything here recently. I was busy for a while, between the trip to UC Davis to speak (giving a form of this talk) at the “Strings and Gravity” seminar there, and then catching up on teaching - the end of the term is coming up. There: now that’s out of the way.

Right now I want to say something a bit broader than I have been doing - somewhere between “intuitive justification” and “philosophy”. The motivation is that whenever I talk about ETQFT’s and how to see them as introducing matter into quantum gravity, there’s always some puzzlement about this “categorification” business. To people who think a lot about category theory, it may seem natural, but many of those interested in physical questions don’t fall in this category, and the whole idea of “categorifying” a theory seems like a weird, arbitrary imposition.

So talking to these different audiences has forced me to think about how to give an intuitive account of why this might be a good idea. Ideally this will not be so precise as to be incomprehensible, or so vague as to be useless. In reality, this will be at best a rough sketch of such a justification.

Stuff, Structure, and Properties

One aspect of the relationship which I wanted to comment on, one that almost seems like a pun, is the trichotomy which John Baez and Jim Dolan like to use in describing mathematical, um, widgets (I would use the more standard term “objects”, or maybe “structures”, but both of these words have technical meanings in the following) in categorical terms. This is the distinction between “stuff”, “structure”, and “properties”. (More details here and via subsequent links - some of which shows up in my first paper). Almost any usual mathematical widget can be broken down this way: (1) they consist of some “stuff”, often in the form of some sets; (2) the stuff is equipped with “structure”, often described by some functions; (3) the structure satisfies some “properties”, often expressed as equations.

For example: a group is (1) a set G of elements, equipped with (2) a group operation (expressed as a function m : G \times G \rightarrow G), and a special identity element (picked out by a function from the one-element set, 1 : \star \rightarrow G), and an inverse for each element (given by an inverse function inv : G \rightarrow G. These satisfy (3) the group axioms, which are some equations involving expressing some properties - associativity, the properties of 1 and inverses.

In this case, the structure live inside the category of sets and functions - but similar things could be said in any other category. For instance, in the category of topological spaces and continuous functions, the same setup gives the definition of a topological group, likewise divided into “stuff” (objects, in this case topological spaces), “structure” (some morphisms), and “properties” (equations between morphisms).

Widgets which live in an n-category of some kind have more of these layers - such a widget will be specified by one or more objects, equipped with specified morphisms and 2-morphisms, satisfying some equations. A monoidal category, for instance, is this kind of widget: it has a category worth of “elements”, equipped with a monoidal operation given as a functor, equipped in turn with specified 2-isomorphisms such as the “associator”, which satisfies some equations such as the Pentagon identity. There are now FOUR levels to specify. I think it was Jim Dolan who came up with the following way of extending the “stuff/structure/properties” terminology (his explanation).

The highest level - equations - always deserves the name “properties”, since they either hold, or don’t (at least, there’s a truth value associated to them - but let’s not worry about multiple-valued logics). By analogy, this suggests the data for our widget given by the n-morphisms in the n-category where it lives should be called “structure”. The (n-1)-morphisms (which are the objects in a 1-category) should be called “stuff”.

For the (n-2), (n-3), and generally k-morphisms, Jim introduces the prefix “eka”, as in “eka-stuff”, which follows Mendeleev’s nomenclature for elements predicted by his form of the periodic table of elements which were heavier than known ones. This nomenclature in turn comes from the Sanskrit “eka”, meaning “one” - the new elements were one level lower on the periodic table.

So specifying a widget in a 2-category involves “eka-stuff/stuff/structure/properties”. This is suggestive, in that it seems as if categorification - adding a new level - is like digging out a new sub-basement beneath a house. First “eka-stuff”, then “eka-eka-stuff”, and so on, to “ekak-stuff”. Since, in many versions of n-category, given two objects x and y, the totality of morphisms hom(x,y) form an (n-1)-category, this is somewhat correct: there is an (n-1)-categorical structure describing each hom(x,y).

(The periodic-table analogy, I suppose, is meant to imply that the best-understood layer is the layer of equations - which describe properties. This opposes what is probably the more common intuition people have when first encountering higher categories, that we know what “objects” are, but find “higher morphisms” confusing. But when writing things concretely, it’s the highest-level morphisms which look most familiar, like functions.)

A key point here is that “stuff having structure satisfying properties” is a fairly intuitive framework for talking about things. Categorification gives us a more nuanced layering. It may seem odd to speak of “eka-stuff equipped with stuff equipped with structure satisfying properties” (even worse if you want to be consistent, and say “equipped with” instead of “satisfying”). But now the second layer - stuff, refers to 1-morphisms. Here is a layer which has some aspects we associate with “structure”: it describes relations between the eka-stuff (objects). On the other hand, it also has aspects we associate with “stuff” (it can be equipped with its own structure). When would one want something that is on the one hand something like a relational attribute between things (structure), and on the other hand something like an object in its own right (stuff).

One answer: to describe space. As a good Leibnizian, I prefer to think of space relationally: it describes how objects are situated in terms of structural relationships. On the other hand, General Relativity tells us that if we think about space, rather than spacetime, we need to describe it as having dynamics which satisfy some property. From this point of view, space is like material stuff that changes over time, according to some differential equation (classically, at least).

Matter = Stuff?

Now, part of the point of applying extended TQFT ideas to gravity is that the categorification introduces matter into the formerly empty background of topological gravity - in particular, the state of a bit matter is described by looking at the boundary conditions on a codimension-2 surface in spacetime (or codimension-1 surface in space) surrounding it. The “pun” I alluded to above is the idea that introducing matter amounts to introducing a new layer of “stuff”. Adding matter means adding “stuff”…

The pun isn’t quite dead on, however, because in the ETQFT setup, adding matter is actually adding “eka-stuff”: digging out a sub-basement on which the “stuff” of geometrized space and its dynamics can rest.

So how does the periodic table of stuff/structure/properties relate to an extended TQFT? To start with, consider the case of an ordinary TQFT in 2 dimensions. It’s well known that such TQFT’s correspond to commutative Frobenius algebras (though see e.g. this paper by Aaron Lauda and Hendryk Pfeiffer, where they explain this, and a generalization of it). That is, a TQFT defines an object with (1) Stuff: a vector space, equipped with (2) Structure: unit, counit, multiplication, and comultiplication maps, satisfying (3) Properties: a bunch of axioms, including the Frobenius relation, commutativity, and algebra axioms like associativity.

The key thing is that this correspondence comes from the fact that a 2D TQFT is a functor into \mathbf{Vect} from the category \mathbf{2Cob}, which happens to be a symmetric monoidal category freely generated by one object (the circle), and some morphisms (corresponding to four cobordisms: the cap, cup, “pair of pants”, and “inverted pair of pants”), subject to just the topological relations making the circle with these maps into a “Frobenius object”. (Since the cobordisms are only defined up to diffeomorphism).

Then any actual “physical” setting will look like: a bunch of circles, say n of them, connected to another bunch of circles, say m of them, by some cobordism. We could call this a “string world sheet” (although not in the sense of string theory, exactly, since over there one typically has conformal structure on the cobordisms too, and talks about a CFT, not a TQFT, living on the sheet). In general, the cobordism will be an n+m-punctured, genus-g torus (with orientations that distinguish the n inputs from the m outputs). So if the dynamics of the “physical” world are described by a TQFT corresponding to Frobenius algebra F, this topology will mean the space of states of the world is given by F^{\otimes n} at the beginning and F^{\otimes m} at the end (this is “stuff”). A state evolves through “time” by the morphism (”structure”) corresponding to the cobordism C - a particular combination of multiplication and comultiplication maps for the

In a theory of gravity without matter, we can see three levels as well - “slices” of space with some geometric information, connected by spacetimes with geometric information, which satisfy some equations. In particular, the geometric information on spacetime has to satisfy Einstein’s equation, if we’re talking about the classical world, or some sort of Hamiltonian constraint in (some approaches to) quantum gravity. In any case, it must have some property to be admissible. So this suggests the classifications: “space geometry” - stuff; “spacetime geometry” - structure; “dynamical laws” - properties.

Categorification suggests adding to this list: “matter/boundary conditions” - eka-stuff. That is, the eka-stuff in a specific physical setting will be a “2-space of states” for matter as measured at a particular boundary. In a 3D ETQFT, for instance, the boundaries to space will be unions of circles (just as in a 2D TQFT), so this will be generated by a 2-space of states for a circle. The circle could be thought of as the boundary around a single excised particle, but in fact that only covers the irreducible 2-states: in general, it’s a boundary around some region containing a system. Space geometry relates such boundaries to each other: it is “stuff” relating the “eka-stuff”. That stuff (space geometry), in turn, can be equipped with structure - maps associated to a spacetime topology, which describe how it evolves in “time” (though a-priori there’s no special time direction - the “stuff” could equally well describe the world-sheet of the system boundary, and the structure describing how that evolution extends outward spatially).

It seems to me there’s a lot here, but to really say it properly would require being much more technically precise than I’m up to at the moment. So that’s about all I have to say about that.