More talks: Gukov on Arithmetic Topology and Gauge Theory

Sergei Gukov caught me by surprise with the first talk of the morning (only because I was too lazy to read the schedule). Since everybody else was talking about categorification, I assumed he would as well. Man, was I wrong.

The actual topic was “arithmetic topology.” I’ll be honest here, this is not a field of topology that I even knew existed before this morning (and which I am still slightly suspicious of).

As far as I can tell, this means that you do topology (or number theory) pretending that there is a dictionary between 3- manifolds and number fields similar to the one between ideals and varieties (even those this is, in fact, obviously false).

In a nutshell, what Gukov was proposing is to take a TQFT and whack your favorite 3-manifold with it to get a the partition function or, more accurately, a perturbative expansion of such.

This perturpative expansion is a power series of \hbar (where \hbar may actually be denoting a coupling constant. This is about where my knowledge of mathematical physics starts to give out) and so you can look at the subfield of \mathbb{C} generated by these.

In some cases (\mathrm{SU}(2) Chern-Simons theory) this will just be \mathbb{Q}, we will simply curse our luck and move on, since no interesting arithmetic happens.

(This section of the talk contained the following classic exchange:

Gukov: …and this is the thesis of Dror Bar-Natan.

Bar-Natan (from the audience): Sergei, I must protest. I admit to writing a Ph.D. thesis, but none of this stuff is in it. )

But in other cases (\mathrm{SL}(2,\mathbb{C}) Chern-Simons theory, for example), you will actually get other complex numbers, and one could hope the number theory will be interesting.

It seems that this hasn’t been investigated all that fully, but the punch-line of the talk was a hint of more interesting things to come, though.

Conjecture: Let M be hyperbolic. Then the coefficients of the expansion of the partition function generate the same number field as the traces of all elements of \Gamma, where \Gamma\subset \mathrm{SL}(2,\mathbb{C})=\mathrm{Isom}(\mathbb{H}^3) is a subgroup of the isometry group of hyperbolic 3-space \mathbb{H}^3 such that M=\mathbb{H}^3/\Gamma.

It’s a little unclear how much this really connects in with the main body of arithmetic topology (different audience members had different opinions on the matter), but I, at least, am of the opinion it’s a interesting question. What say ye, O mathematical physicist readers?

15 thoughts on “More talks: Gukov on Arithmetic Topology and Gauge Theory

  1. Interesting. I wonder what exactly $\hbar$ actually denotes here.

    Because, of course this phenomenon — that partition functions take values in certain rings, and in rings of power series in particular — is crucial to much of the general idea of quantum field theory:

    The ring (under direct sum and tensor product) of $d$-dimensional quantum field theories parameterized over a space $X$ should form a generalized cohomology ring $H^\bullet(X)$ of a generalized cohomology theory of “chromatic filtration” $d$.

    In particular, the partition function of the QFT computes the coefficient ring of the cohomology theory, i.e. the cohomology of a point.

    (For this generalized cohomology theory to be interesting, the QFT apparently has to be super.)

    This is a theorem

    – for $d=0$ (the landscape of (concordance classes of) super ($-1$)-brane backgrounds over $X$ is the deRham cohomology spectrum of $X$ ),

    – for $d=1$ (the landscape of (concordance classes of) super 0-brane backgrounds over $X$ is the K-theory spectrum of $X$)

    and it is about half a theorem, or so, for $d=2$, where it is supposed to say something like that the landscape of concordance classes of d=2 super-Riemannian QFTs is something like the TMF spectrum.

    Coming back to the ring of possible values of the partition function:

    the partition function of the super-particle is $\mathrm{str}(e^{- t H}) \in \mathbb{Z}$, taking values in the integers. This should be thought of as the K-theory of the point: The partition function is really the decategorification of the vector space
    $ \mathrm{ker}(D) – \mathrm{coker}(D)$, where $D$ is the supercharge, i.e. $D^2 = H$ is the Hamiltonian.

    This happens to be a power series ring in no variable: in general there should be one variable per (dimension of worldvolume minus number of supersymmetries).

    By a rather similar argument, one finds that the partition function of a 2-dimensional “heterotic” SQFT, i.e. one with one worldsheet supersymmetry (instead of two) over the torus of modulus $q^{i \tau}$, is a power series in that one parameter $q$.

    In fact, it is not just any such power series, but an integral modular form. That’s actually the main reason to suspect that the full cohomology theory associated with 2-dimensional SQFTs here is TMF.

    Most of what I just said here I once tried to summarize in more detail here. It all relates to work by Stephan Stolz and Peter Teichner. More recent developments can be found in this preprint.

  2. I remember seeing this when it came out. The analogy has an excellent pedigree – Mazur, Mumford, etc. I wonder what the best current understanding is.

    I like this bit – doesn’t sound like maths:

    “Despite its limitations and inconsistencies, the dictionary can be used for translating statements from 3-dimensional topology into number theory, and vice versa, often with a surprising accuracy.”

  3. Maybe he was…

    I don’t know what Kea has in mind here, but I do agree that often the first step before trying to categorify something is finding the “natural formulation” of that something, which may involve first better understanding this uncategorified structure.

    There are usually many ways to conceive one and the same mathematical structure. Only one or very few of these ways will lend themselves to effortless categorification. If categorification is a pain, you are probably not using the right formulation to start with.

    I like to put this as a slogan:

    We know what a concept really means if we have found a way to formulate it such that it categorifies effortlessly.

    (We talked about that here.)

  4. We know what a concept really means if we have found a way to formulate it such that it categorifies effortlessly.

    This is a nice slogan, I think. My guess is that at least a part of Gukov’s motivation is physical (though not string theory, I’m afraid, Urs) along the lines of the ‘mathematical Platonist’s reality’ of generalised number theory. All hbar are essential because they have a physical interpretation in terms of cosmological scale (for one). Yes, this is a radical varying-speed-of-light kind of QG.

  5. My guess is that at least a part of Gukov’s motivation is physical (though not string theory, I’m afraid, Urs

    I hope you don’t seriously think that I will not be interested in something just because it’s not related to 2-particles. :-)

    All hbar

    I know one \hbar. What’s all \hbar?

  6. Yes, hbar is a deformation parameter, but in the approach to QG to which I refer one uses all deformation parameters and not just one, which is usually the case, in for instance the spin foam rep cat setup. Moreover the deformation parameter is closely associated with cosmological constant (something which does not exist in the ‘number theory’ QG). The connection to physical hbar is indirect but still relevant: string/brane theory (2-physics in Urs language) only accounts for one hbar.

  7. in the approach to QG to which I refer …

    … and which is or is not the one that Gukov is talking about?

    I think \hbar here is not denoting Planck’s constant, but instead refers to any deformation parameter.

    Yes, so which kind of deformation does it describe, in Gukov’s setup. Can anyone tell me? Roughly?

  8. …and which is or is not the one that Gukov is talking about?

    I cannot speak for what Gukov has in mind. I think hbar here might be a Kontsevich hbar (for deformed products etc.) Hence the link to operads, and the QG I have in mind, in which operads for 3D ‘number theory geometry’ play a crucial part.

  9. Ben Webster wrote:

    “The actual topic was “arithmetic topology.” I’ll be honest here, this is not a field of topology that I even knew existed before this morning (and which I am still slightly suspicious of).”

    Hmmm… Why suspicious? What is now called arithmetic topology has been initiated by A. Weil in late 1930s and 1940s, advanced by Grothendieck, etc. See eg.:

    Primes and Knots (2003 Johns Hopkins Univ. conference)
    http://www.mathematics.jhu.edu/jami/jami2003/prospectus.htm
    proceedings published in book form: AMS, 2006.

  10. I’ll concur that my suspicion isn’t particularly warranted. I think the problem for me is that this “correspondence” is hard to properly explain in 20 minutes of a talk. It’s a lot more subtle of an idea than, say, the Galois correspondence.

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