If You Like Mathematical Physics….

I visited the University of Pennsylvania on Tuesday, and while I was there, Ron Donagi told me about an upcoming conference at UPenn, called “String-Math 2011”.   It’s a week-long meeting, June 6-11, with a bunch of exciting people on the visitor list.   And it’s the first of a series of such conferences.  If you want to know more, the conference webpage is

http://www.math.upenn.edu/StringMath2011/

The early registration deadline is April 2nd.

In other news, like my co-bloggers, I’m teaching a class this semester:  Quantum Field Theory for Mathematicians, which aims to explain the basic ideas of quantum field theory through the study of mathematically well-defined examples.  In particular, we’re looking to see how Wilson’s “effective field theory” philosophy motivates the rigorous constructions of the path integral measures.  I’ve been putting my lecture notes online; you’re welcome to check them out if you’re interested.  (There have only been 3 lectures so far; we’re still on preliminary material.  However, you can see from the syllabus what I’m hoping to cover.)

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Hulk Flatten Puny Octahedral Axiom

Alexander Ellis has just put up a very cool post over at Concrete Nonsense, called Wood, Glue, and the Octahedral Axiom.   For those of you who don’t know, the Octahedral Axiom is the Fourth and Most Annoying Axiom for triangulated categories.  It’s a nuisance for two reasons.  First, it’s a bit aesthetically unsatisfactory.   One of the triangulated category axioms asserts that for any map f: A \to B, there exists an exact triangle

A \to B \to X \to A[1]

The “cokernel” X isn’t unique up to unique isomorphism — this is the aesthetically unsatisfying bit — so we’re forced to introduce the Octahedral Axiom, which asserts that X, while not canonical, is at least somewhat functorial.   Second, the Octahedral Axiom is usually written as (the projection onto the plane/page) of a certain partially commutative three-dimensional octahedron diagram.    It’s sort of a pain to see what the various parts of this diagram mean.

Anyways, Ellis has gotten crafty and put together a miniature  model of the Octahedral Axiom.   (The real one is kept in Paris, in the basment of IHES, and is quite large, but Ellis’ model fits in the palm of his hand.)  You should check it out.   It’s pretty sweet.

I think it might be worth mentioning, however, that it’s not necessary to use three-dimensional commutative diagrams to understand the octahedral axiom.    There’s a pretty reasonable way of writing it using just 2d diagrams.

Octahedral Axiom

If we have the following three exact triangles

3diagrams

then there exists a 4th “vertical” exact triangle X \to Y \to Z such that the following diagram commutes.

octoflat

Obviously, you can compress this more, since some of the commutative squares are trivial, but I think the content is clearer when the original three triangles are left in their original form.   What this axiom says, basically, is that if we identify X = B/A, Y = C/A, and Z = C/B, then we have C/B = (C/A)/(B/A).

I think I’ll leave it to the reader to figure out what you have to do to this diagram to make it into an octahedron.

TeXnical Aside:   If anyone knows how to make wordpress support xypic I’d appreciate it.   At the moment I make commutative diagrams by typing xypic code into the BLANK in

http://www.codecogs.com/eq.latex?BLANK

This produces a .gif which I upload.

Gromov-Witten Invariants and Topological Field Theory

A few days ago, John Mangual requested that one of us secret blogging seminarians write a post explaining what Gromov-Witten invariants are all about.  I volunteered to do this, which puts me in an awkward position.  Gromov-Witten theory is a big subject, and there are a lot of good introductions to the subject (e.g., Behrend’s Algebraic Gromov-Witten Invariants). It’s not clear that I can do much in a single blog post.

So I’m going to limit the scope of this post somewhat.  I’d like to explain in what sense Gromov-Witten theory is a topological field theory.   Caution: This may involve some rambling.    If you’re curious, come below the fold.

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Permission for Carnivals?

So, Charles over at Rigorous Trivialities recently put up a post begging for submissions to the latest installment of the Carnival of Mathematics, and this prompted Ben and I to wonder:  Why is it customary to request permission to link to posts for the Carnival of Mathematics?  [Edit:  Is it customary?] Is this just standard operating procedure for blog carnivals?

It seems a bit bizarre.  We don’t ask permission to link to posts in any other circumstance.  And it makes it harder to assemble a Carnival.  Obviously, it would be best if people actually submitted their own posts, but I don’t see much sense in not linking to a post because you don’t have explicit permission.  (Ben, I think, didn’t bother to ask anyone for permission when he assembled the SBS-hosted Carnival.)

I suppose some folks might not be comfortable with not having explicit permission.  Maybe we can do something about that in this post.  I, for one, want Carnival hosters to know:  Any post I write is fair game for the Carnival of Mathematics.  If you feel the same way, you might mention it in the comments.

What’s a Stack?

Algebraic stacks are essential to my research.  This is more acceptable now than it was twenty years ago, but it still presents a bit of a language barrier.  Most mathematicians, I think, don’t know what a stack is in the way that they know what a manifold or a scheme is.  So I want to use this post to explain what stacks are, with an eye towards their appearance in mathematical physics.  I won’t quite define them (see Vistoli’s notes for that), but I’ll get you a lot closer than Harris & Morrison do (see p. 139), hopefully close enough to be comfortable that you know what’s going on when someone says “stack”.

Let’s start by saying that a space is what you get when you start with a set and then add some geometry. Maybe make the set into a manifold, maybe make it into a scheme; you can choose your favorite category.   The elements of the set become the points of your space.

A stack is what you get when you start with a groupoid instead of a set, and then add geometry. A groupoid, remember, is a category whose morphisms are all isomorphisms.  This means that the points of a stack aren’t just elements of some set; they also come equipped with a bunch of relations, telling you which points are isomorphic to each other.

So why would anyone try to make a groupoid into a geometry?

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New Photograph

Last Friday, we had a seminar at Berkeley — or rather, at Noah’s house — featuring Mike Freedman and some quantity of beer. Mike spoke about some of the hurdles he had to overcome in writing his recent paper with Danny Calegari and Kevin Walker. One of the main results of this paper is that there is a “complexity function” c, which maps from the set of closed 3-manifolds to an ordered set, and that this function satisfies the “topological” Cauchy-Schwarz inequality.

c(A \cup_S B) \leq max \{c(A \cup_S A),c(B \cup_S B)\}

Here, A and B are 3-manifolds with boundary S. [EDIT: and equality is only achieved if A = B] This inequality looks like the sort of things you might derive from topological field theory, using the fact that Z(A \cup_S B) = \langle Z(A), Z(B) \rangle_{Z(S)}. Unfortunately, it’s difficult to actually derive this sort of theorem from any well-understood TQFT, thanks to an old theorem of Vafa’s, which states roughly, that there’s always two 3-manifolds related by a Dehn twist that a given rational TQFT can’t distinguish. Mike speculated that non-rational TQFT might be able to do the trick, but what he and his collaborators actually did was an end run around the TQFT problem. They simply proved that that the function c exists.

I tell you all this, not because I’m about to explain what c is, but to explain our new banner picture. We realized after the talk that there were a fair number of us Secret Blogging Seminarians in one place, and that we ought to take a photo.

Real Curves, Open Strings, and A-infinity Algebras

Kevin Costello gave a talk last week in one of Peter Teichner’s many seminars, explaining A_\infty-algebras with a view towards his papers on topological string theory. It was the sort of talk that might have interested a lot of people, so (with Kevin’s permission), I’m posting my .pdf scanned notes here. I’ve added some physics interpretation that Kevin didn’t make explicit.

Defining A_\infty-algebras

Kevin also gave a shorter talk (in a different Teichner seminar) about his characterization of the homotopy type of a moduli space of genus zero open string worldsheets. My notes here are less detailed, but maybe someone will enjoy them.

The Homotopy Type of a Certain Moduli Space of Open String Worldsheets

Numerology and Gravity

Richard Borcherds wrote a post a little while back in which he remarked that we shouldn’t take the Planck units very seriously, since Newton’s constant G doesn’t have quite the same stature as c and \hbar.

His argument went more or less as follows: The latter two seem to be quite fundamental, but from a modern point of view G (actually \frac{G^{-1}}{8\pi}) is just one of the coupling constants that appears in the effective Lagrangian for gravitational fields

L =  \lambda \sqrt{|g|} + \frac{G^{-1}}{16\pi}R\sqrt{|g|} + \mbox{sub-sub-leading terms}

where the coupling constant \lambda =\frac{\Lambda_c  G^{-1}}{16\pi} is basically the cosmological constant, normalized by 16\pi G.

So, G isn’t privileged in any way. It isn’t the coupling constant of the leading order term; a cursory analysis of the scaling dimensions of \lambda and G would lead us to believe that the cosmological constant term should be dominant. It isn’t even normalized nicely, what with that 8\pi \simeq 25. And while it seems quite sensible to work in units where \hbar = 1 and c = 1, we should be a little more cautious about about the meaning we assign to units where G = 1. What we’re actually doing is identifying the length scale where our non-renormalizable effective field theory description of gravity should break down.

[Updated: The computation that previously appeared here involved one of the classic blunders: not checking your units.  \lambda \simeq 10^{-86} \mbox{GeV}^{\bf{+}6}; that’s a +6, not a -6.  Thanks to Thomas Larsson for pointing that I’m an idiot.]