## Friday, July 29, 2011

### Higgs Bundles and Arithmetic

Now that fairy fields are unofficially dead, we can resurrect the word Higgs in a mathematical context! At the latest KITP conference, the mathematician Hausel gives a mesmerising talk about moduli spaces of Higgs bundles, encompassing a theorem for the Langlands correspondence. His appearance at a physics conference is explained by a stringy description of his conjecture on polynomial invariants for moduli spaces.

The stringy connection creates what they call (quote) quantum motivic Pandharipande-Thomas invariants, associated to D6 branes wrapping a Calabi-Yau $3$-fold. But Hausel seemed happier sticking to Chern-Simons theories with a finite gauge group, which is to say that the talk focuses on $GL_{n}(F)$, for $F$ a finite field, and the character variety given by $2g$ matrix generators for a curve of genus $g$, in terms of a root of unity $\textrm{exp}(2 \pi i d/n)$ for $d$ coprime to $n$. For $g = 1$ the Stone von Neumann theorem tells us what this variety is, namely the cotangent bundle of $\textrm{Jac}(C)$. By fancy algebraic geometry, the character variety is like the moduli space of semistable rank $n$ degree $d$ Higgs bundles.

Higgs bundles go back to Hitchin (who now seems to like twistors and octonions). The technical part of the talk stresses the importance of finding a suitable filtration for cohomology. On the character variety side, mixed Hodge polynomials $H(X,q,t)$ give Hausel's $E$ polynomials $E(X,q)$, which have nice motivic properties, meaning that one just ends up counting points for varieties over finite fields.

Perhaps the physics of the real world actually has more to do with the simple arithmetic picture than the stringy one?

1. Cumrun Vafa just gave a talk ("M+W=T") which has opened my mind to new ways in which braids might feature in old-fashioned M-theory. To this point I had thought solely in terms of knotted Wilson loops, as in Witten's reconstruction of Khovanov homology. But what Vafa talks about are situations where M5-branes are wrapped on the branched cover of a manifold (he talks about 2-manifolds and 3-manifolds, so there are extra compact dimensions also wrapped by the M5-brane), and lines between the branch points define places where an M2-brane can end. i.e. just as an open string is a line, the ends of which are points on D-branes, an open M2-brane is a tube (or other 2-manifold with boundary), the boundary of which is a closed 1-manifold inside an M5-brane.

So if we are talking about M5-branes in which 2 of the spacelike worldvolume dimensions are compact, we are left with a 3-space in the M5 worldvolume, in which M2-brane boundaries form "tensionless strings" which can form knots and links.

Vafa is talking about something rather simpler, and he says (at 49 minutes) that in the case under consideration, you can't interlace the branch lines. Still, it makes it more plausible than ever that there are geometric backgrounds for "ordinary" M-theory which have a dual description resembling your own work.

So far I see three ways in which braids might make an appearance in 11-dimensional M-theory. They might be timelike, they might be spacelike in the large (noncompact) directions, or they might be spacelike in the compact dimensions.

Now to look at Hausel's talk...

2. PDFs for "Penrose 80" are now up, and in David Skinner's talk on holomorphic linking, I find this statement:

"It has long been the twistor theorist’s ambition that by trading topological invariance for holomorphic invariance in twistor space, one would be able to use linking ideas to encode the dynamics of interacting QFTs purely in terms of twistor geometry."

So this suggests that the way past Vafa's restriction is to work in twistor space. And I find that a student of Hitchin's long ago constructed a Twistor Higgs! He doesn't call it that; but he says that the cotangent bundle of that Jacobian is the twistor transform of the corresponding Higgs bundle.

Meanwhile, Martijn Wijnholt's talk at Strings 2011 covers the history of Higgs bundles in string phenomenology.

3. OK, Mitchell, but note that the true motivation for twistors is the cohomological nature of physical solutions. And I must stress again that we are talking about emergent geometry, so wanting timelike or spacelike things from the start is misguided. Anyway, kneemo should have M theory sorted out soon enough.

4. Marni, have you seen the recent talk about Eric Verlinde's M-Theory?

5. I usually watch Verlinde's talks online, and mention them here.