So this is a couple of weeks backdated.  I’ve had a pretty serious cold for a while – either it was bad in its own right, or this was just a case of the difference in native viruses between two different continents that my immune system wasn’t prepared for.  Then, too, last week was Republic Day – the 100th anniversary of the middle of three revolutions (the Liberal, the Republican, and the Carnation revolution that ousted the dictatorship regime in 1974 – and let me say that it’s refreshing for a North American to be reminded that Republicanism is a refinement of Liberalism, though how the flowers fit into it is less straightforward).  So my family and I went to attend some of the celebrations downtown, which were impressive.

Anyway, with the TQFT club seminars starting up very shortly, I wanted to finish this post on the first talks I got to see here at IST, which were on pretty widely different topics.  The first was by Ivan Smith, entitled “Quadrics, 3-Manifolds and Floer Cohomology”.  The second was a recorded video talk arranged by the string theory group.  This was a recording of a talk given by Kostas Skenderis a couple of years ago, entitled “The Fuzzball Proposal for Black Holes”.

Ivan Smith – Quadrics, 3-Manfolds and Floer Cohomology

Ivan Smith’s talk began with some motivating questions from topology, symplectic geometry, and from the study of moduli spaces.  The topological question talks about 3-manifolds $Y$ and the space of representations $Hom(\pi_1(Y),G)$ of its fundamental group into a compact Lie group $G$, which was generally $SO(3)$ or $SU(2)$.  Specifically, the question is how this space is affected by operations on $Y$ such as surgery, taking covering spaces, etc.  The symplectic geometry question asks, for a symplectic manifold $(X,\omega)$, what the “mapping class group” of symplectic transformations – that is, the group $\pi_0(Symp(X))$ of connected components of symplectomorphisms from $X$ to itself – in a sense, this is asking how much of the geometry is seen by the symplectic situation.  The question about moduli spaces asks to characterize the action of the (again, mapping class group of) diffeomorphisms of a Riemann surface on the moduli space of bundles on it.  (This space, for  $\Sigma$ with genus $g \geq 2$, look like $M_g \simeq Hom(\pi_1(\Sigma),SU(2))$ modulo conjugation.  It is the complex-manifold version of the space of flat connections which I’ve been quite interested in for purposes of TQFT, though this is a coarse quotient, not a stack-like quotient.  Lots of people are interested in this space in its various hats.)

The point of the talk being to elucidate how these all fit together.  The first part of the title, “Quadrics”, referred to the fact that, when $\Sigma$ has genus 2, the moduli space we’ll be looking at can be described as an intersection of some varieties (defined by quadric equations) in the projective space $\mathbb{CP}^5$.  Knowing this, one can describe some of its properties just by looking at intersections of curves.

In general we’re talking about complex manifolds, here.  To start with, for Riemann surfaces (one-dimensional complex manifolds), he pointed out that there is an isomorphism between the mapping class groups of symplectomorphisms and diffeomorphisms: $\pi_0(Symp(\Sigma)) \simeq \pi_0(Diff(\Sigma))$.  But in general, for example, for 3-dimensional manifolds, there is structure in the symplectic maps which is forgotten by the smooth ones – there’s still a map $\pi_0(Symp(\Sigma)) \rightarrow \pi_0(Diff(\Sigma))$, but it has a kernel – there are distinct symplectic maps that all look like the identity up to smooth deformation.

Now, our original question was what the action of the diffeomorphisms of on the moduli space $M_g$ of bundles over $\Sigma$.  An element $h$ of $\pi_0(Diff(\Sigma))$ acts (by symplectic map) on it.  The discrepancy we mentioned is that the map corresponding to $h$ will always have fixed points, but be smoothly equivalent to one that doesn’t.  So the smooth mapping class group can’t detect the property of having fixed points.  What it CAN detect, however, is information about intersections.  In particular,   as mentioned above, the moduli space of bundles over a genus 2 surface is an intersection; in this situation, there is an injective map back from the smooth mapping class group into the group of classes of symplectic maps.  So looking symplectically loses nothing from the smooth case.

Now, these symplectic maps tie into the third part of the title, “Floer Homology”, as follows.  Given a symplectic map $\phi : (X,\omega) \rightarrow (X,\omega)$, one can define a complex of vector spaces $HF(\phi)$ which is the usual cohomology of a chain complex generated by fixed points of the map $\phi$, and with a differential $\partial$ which is defined by counting certain curves.  The way this is set up, if $\phi$ is the identity so that all points are fixed points, one gets the usual cohomology of the space $X$ – except that it’s defined so as to be the quantum cohomology of $X$ (for more, check out this tutorial by Givental).  This has the same complex as the usual cohomology, but with the cup product replaced by a deformed product.  It’s an older theorem (due to Donaldson) that, at least for genus 2, the quantum cohomology of the moduli space of bundles over $\Sigma$ splits into a direct sum of rings:

$QH^*(M_2) \cong \mathbb{C} \oplus QH^*(\Sigma_2) \oplus \mathbb{C}$

So one of the key facts is that this works also with Floer homology for other maps than the identity (so this becomes a special case).  So replacing $QH^*$ in the above with $HF^*(\phi)$ for any $\phi$ (acting either on the surface $\Sigma$, or the induced action on the moduli space) still gives a true statement.  Note that this actually implies the theorem that there are fixed points in the space of bundles, since the right hand side is always nontrivial.

So at this point we have some idea of how Floer cohomology is part of what ties the original three questions together.  To take a further look at these we can start to build a category combining much of the same information.  This is the (derived) Fukaya category.  The objects are Lagrangian submanifolds of a symplectic manifold $(X,\omega)$ – ones where the symplectic form vanishes.  To start building the category, consider what we can build from pairs of such objects $(L_1,L_2)$.  This is rather like the above – we define a complex of vector spaces, which is the cohomology of another complex.  Instead of being the complex freely generated by fixed points, though, it’s generated by intersection points of $L_1$ and $L_2$.  This automatically becomes a module over $QH^*(X)$, so the category we’re building is enriched over these.

Defining the structure of this category is apparently a little bit complicated – in particular, there is a composition product $HF(L_1,L_2) \otimes HF(L_2,L_3) \rightarrow HF(L_1,L_3)$ in the form of a cohomology operation.  Furthermore, which Ivan Smith didn’t have time to describe in detail, there are other “higher” products.  These are Massey type products, which is to say higher-order cohomology operations, which involve more than two inputs.  These give the whole structure (where one takes the direct sum of all those hom-modules $HF(L_i,L_j)$ to get one big module) the structure of an $A_{\infty}$algebra (so the Fukaya category is an $A_{\infty}$-category, I suppose).  This is one way of talking about weak higher categories (the higher products give the associator for composition, and its higher analogs), so in fact this is a pretty complex structure, which the talk didn’t dwell on in detail.  But in any case, the point is that the operations in the category correspond to cohomology operations.

Then one deals with the “derived” Fukaya category $\mathcal{DF}(X)$.  I understand derived categories to be (at least among other examples) a way of taking categories of complexes “up to homotopy”, perhaps as a way of getting rid of some of this complication.  Again, the talk didn’t elaborate too much on this.  However, the fundamental theorem about this category is a generalization of the theorem above above quantum cohomology:

$\mathcal{DF}(M_2) \cong \mathcal{DF}(pt) \oplus \mathcal{DF}(\Sigma_2) \oplus \mathcal{DF}(pt)$

That is, the derived Fukaya category for the moduli space of bundles over $\Sigma_2$ is the category for the Riemann surface itself, summed with two copies of the category for a single point (which is replacing the two copies of $\mathbb{C}$).  This reduces to the previous theorem when we’re looking at the map $\phi = id$, just as before.

So the last question Ivan Smith addressed about this is the fact that these sorts of categories are often hard to calculate explicitly, but they can be described in terms of some easily-described data.  He gave the analogy of periodic functions – which may be quite complicated, but by means of Fourier decompositions, can be easily described in terms of sines and cosines, which are easy to analyze.  In the same way, although the Fukaya categories for particular spaces might be complicated, they can be described in terms of the (derived) category of modules over the $A_{\infty}$-algebras.  In particular, every category $\mathcal{DF}(X)$ embeds in a generic example $\mathcal{D}(mod-A_{\infty}-alg)$.  So by understanding categories like this, one can understand a lot about the categories that come from spaces, which generalize quantum cohomology as described above.

I like this punchline of the analogy with Fourier analysis, as imprecise as it might be, because it suggests a nice way to approach complex entities by finding out the parts that can generate them, or simple but large things you might discover them inside.

Fuzzballs

The Skenderis talk about black holes was interesting, in that it was a recorded version of a talk given somewhere else – I haven’t seen this done before, but apparently the String Theory group does it pretty regularly.  This has some obvious advantages – they can get a wider range of talks by many different speakers.  There was some technical problem – I suppose due to the way the video was encoded – that meant the slides were sometimes unreadably blurry, but that’s still better than not getting the speaker at all.  I don’t have the background in string theory to be able to really get at the meat of the talk, though it did involve the AdS/CFT correspondence.  However, I can at least say a few concrete things about the motivation.  First, the “fuzzball” proposal is a more-or-less specific proposal to deal with the problem of black hole entropy.

The problem, basically, is that it’s known that the thermodynamic entropy associated to a black hole – which can be computed in completely macroscopic terms – is proportional to the area of its horizon.  On the other hand, in essentially every other setting, entropy has an interpretation in terms of counting microstates, so that the entropy of a “macrostate” is proportional to the logarithm of the number of microstates.  (Or, in a thermal state, which is a statistical distribution, this is weighted by the probability of the microstate).  So, for example, with a gas in a box, there are many macrostates that correspond to a relatively even distribution of position and momentum among the molecules, and relatively few in which all molecules are all in one small corner of the box.

The reason this is a problem is that, classically, the state of a black hole is characterized by very few numbers: the mass, angular momentum, and electric charge.   There doesn’t seem to be room for “microstates” in a classical black hole.  So the overall point of the proposal is to describe what microstates would be.  The specific way this is done with “fuzzballs” is somewhat mysterious to me, but the overall idea makes sense.  One interesting consequence of this approach is that event horizons would be strictly a property of thermal states, in whatever underlying theory one takes to be the quantum theory behind classical gravity (here assumed to be some specific form of string theory – the example he was using is something called the B1-B5 black hole, which I know nothing about).  That’s because a pure state would have a single microstate, hence have zero entropy, hence no horizon.

Now, what little I do understand about the particular model relies on the fact that near a (classical) event horizon, the background metric has a component that looks like anti-deSitter space – a vacuum solution to the Einstein equations with a negative cosmological constant.  (This part isn’t so hard to see – AdS space has that “saddle-shaped” appearance of a hyperbolic surface, and so does the area around a horizon, even when you draw it like this.)  But then, there is the AdS/CFT correspondence that says states for a gravitational field in (asymptotically) anti-deSitter space correspond to states for a conformal field theory (CFT) at the boundary.  So the way to get microstates, in the “fuzzball” proposal, is to look at this CFT, and find geometries that correspond to them.  Some would be well-approximated by the classical, horizon-ridden geometry, but others would be different.  The fact that this CFT is defined at the boundary explains why entropy would be proportional to area, not volume, of the black hole – this being a manifestation of the so-called “holographic principle”.  The “fuzziness” that one throws away by reducing a thermal state that combines these many geometries to the classical “no-hair” black hole determined by just three numbers is exactly the information described by the entropy.

I couldn’t follow some parts of it, not having much string-theory background – I don’t feel qualified to judge whether string theory makes sense as physics, but it isn’t an approach I’ve studied much.  Still, this talk did reinforce my feeling that the AdS/CFT correspondence, at the very least, is something well-worth learning about and important in its own right.

Coming soon: descriptions of the TQFT club seminars which are starting up at IST.