The n-Category Café

When Vafa remarks

very often the very act of ‘solving’ a non-trivial problem is finding the right ‘dual’ viewpoint

this opens the possibility that several competing notions of ‘dual’ may be available; there is some real trick to finding the right one; it also means that most notions of dual are going to be rather complicated, because there aren’t enough simple dualities to meet all demands made of them. There’s some heavy work that’s gone into even stating what this S-duality and mirror symetry are: even if that is simplifying the problem, there are plenty of folk who won’t see it as such. There are people for whom the allure of reading Vergil and Horace in the original isn’t sufficient impetus to study Latin!

But, what is a duality? A duality is an isomorphism, and more than that: it’s an isomorphism you construct so as to know right away what the inverse is. One might say it was half of an involution. And isomorphisms can be very good for clearing out perspective garbage, indeed. Which would you rather study this picture or this one? They both have some extraneities; but one is engineered to ellucidate, the other to impress. Another toy model: one may study real square matrices as lists of numbers, with a complicated composition rule, OR one may study them as representations of quotient rings $\mathbb{R}[X]/(P(X))$ of bounded degree. And we know which way is better!

I think most of the phenomena called “dualities” in physics are equivalences of (higher) categories. A famously well understood example are special cases of T-duality: in their incarnation as (homological) mirror symmetry this is an equivalence of $A_\infty$-categories.

As to going on about how useful such “dualities” are: this reminds me of how John (Baez) likes to say something like:

An equation $A = B$ is the more interesting the less $A$ seems to have anything to do with $B$.

Similarly then

An isomorphism $A \simeq B$ is the more interesting the less $A$ seems to have anything to do with $B$.

and so forth

An equivalence $A \simeq B$ is the more interesting the less $A$ seems to have anything to do with $B$.

I think this is really all there is to it: in physics they discovered interesting equivalences and called them dualities.

Okay, I couldn’t resist - I made a dual version of sokoban

http://ded.increpare.com/~locus/SOO0/

I tried doing Tetris first, but the rotations didn’t seem to turn into anything sensible, whereas pushing rows of blocks translates pretty naturally to the dual setting here…

Will evidently have to have a look at that book by Conway :)

Dear David,

To get a sense of duality in the sense that Vafa is using it, it might be helpful to consider some cohomological dualities, such as Poincare duality, Serre duality, or Hodge symmetry. There is particular reasons that I am bringing these examples up, both in relation to Vafa’s quote, and in relation to the more general questions raised in the post. But first, let me recall the basic facts about these dualities.

Poincare duality says (in its simplest form) that on a closed connected oriented $n$-dimensional manifold $X$, the cohomology spaces $H^i(X,\mathbf C)$ and $H^{n-i}(X,\mathbf C)$ are in natural duality as vector spaces.

If $X$ is a smooth connected projective algebraic variety of (complex) dimension $d$, and $\Omega^j$ is the $j$th exterior power of the holomorphic cotangent bundle on $X$, then Serre duality says (among other things) that $H^i(X, \Omega^j)$ and $H^{d-i}(X,\Omega^{d-j})$ are in natural duality.

Finally, one can combine these: if $X$ is a smooth connected projective algebraic variety of complex dimension $n$ (and so also a closed connected oriented manifold of dimension $n = 2d$), then $H^i(X,\mathbf C)$ admits a filtration (the Hodge filtration) whose $p$th graded piece is equal to $H^q(X,\Omega^p)$, where $q = i - p$. (I am using the letters $p$ and $q$ at this point because they are traditional.)

Poincare duality between $H^i(X,\mathbf C)$ and $H^{2d -i}(X,\mathbf C)$ then induces Serre duality between the graded pieces $H^q(X,\Omega^p)$ and $H^{d - q}(X,\Omega^{d-p}).$ And one has one additional piece of information, namely Hodge symmetry: complex conjugation on the coefficients induces a (conjugate-linear) automorphism of $H^i(X,\mathbf C)$, and this automorphism interchanges the subquotients $H^q(X,\Omega^p)$ and $H^p(X,\Omega^q)$. In particular, these two spaces have the same dimension.

The dimension of $H^q(\Omega^p)$ is traditionally denoted $h^{p,q}$, and these numbers are called the Hodge numbers of $X$. One has the identities $h^{p,q} = h^{d-p,d-q}$ (coming from Serre duality) and $h^{p,q} = h^{q,p}$ (coming from Hodge symmetry).

The first reason for discussing all this is just so that I can remark that (at least from a mathematical viewpoint) mirror symmetry is a conjecture in geometry that is postulates additional (mirror!) symmetries in the pattern of Hodge numbers (no longer just for $X$ itself, but for $X$ and its mirror dual $X'$).

My second motivation for bringing up these examples is to remark that $H^0$ is typically easier to compute than higher cohomology, so these dualities really do seem to relate (at least in some cases) something easier to something harder.

Another remark: If we consider just the case when $X$ is a complex surfaces (so $d = 2$ and $n = 4$), and the particular symmetry $h^{0,1} = h^{1,0}$. This is a particular case of a rather general theory from the modern viewpoint, but proving this was one of the major focuses of the Italian school of algebraic geometry in the early 20th century. The Italians didn’t have the general viewpoint of this result in terms of duality/symmetry, and so I think this example shows how duality can provide a framework for interpreting, elucidating, and generalizing a difficult mathematical problem, without necessarily rendering it trivial.