May 12, 2011

Making Things Simpler by Duality

Posted by David Corfield

If you have a moment in your busy day, try out the game of Jam. It shouldn’t take you long to realise that there’s something rather familiar about it. There’s a chance you may lose as you learn to play the game, but when you come to know its secret, this becomes very unlikely. As the title to this post suggests, the secret involves duality.

Something puzzling me at the moment is what to make of the way posing a problem in a dual situation may make it easier to resolve. As Vafa says in Geometric Physics:

Dualities very often transform a difficult problem in one setup to an easy problem in the other. In some sense very often the very act of ‘solving’ a non-trivial problem is finding the right ‘dual’ viewpoint.

Now my question is whether we should count these terms ‘easy’ and ‘difficult’ as concerning our psychology, or whether they concerns aspects of the situation themselves.

If I’m set the task to prove a result in projective geometry, my conversion of ‘line’ and ‘point’, ‘lies on’ and ‘contains’, ‘collinear’ and ‘concurrent’, etc., may make it easier to discover a proof for the dual result, which I can then transform back to a proof of the original. Clearly there’s a syntactic isomorphism occurring, inclining me to think any advantage of one dual over another is psychological.

What of the use of the Fourier transform to solve a differential equation? Having taken the Fourier transform of each side, noting that the transform of a derivative is a multiple of the transform of the function sought, you solve the resulting equation for the transform and can then take the inverse transform to give you the solution you require. There seems to be a genuine reason to move to the dual domain here.

Cranking things up a bit, what of S-duality?

S-duality transformation maps the states and vacua with coupling constant $g$ in one theory to states and vacua with coupling constant $1/g$ in the dual theory. This has permitted the use of perturbation theory, normally useful only for “weakly coupled” theories with $g$ less than $1$, to also describe the “strongly coupled” ($g$ greater than $1$) regimes of string theory, by mapping them onto dual, weakly coupled regimes.

Does this mean one actually carries out the transformation? Or is it that now it is known how to do perturbation theory in strongly coupled situations?

Vafa explains about Mirror symmetry how

…quantum corrections on one side has the interpretation on the dual side as to how correlations vary with some classical concept such as geometry. This allows one to solve difficult questions involved in quantum corrections in one theory in terms of simple geometrical concepts on the dual theory. This is the power of duality in the physical setup. Mathematics parallels the physics in that it turns out that the mathematical questions involved in computing quantum corrections in certain cases is also very difficult and the questions involved on the dual side are mathematically simple.

This sounds as though Vafa is attributing simplicity and difficulty to the mathematics itself. What could that mean to say a situation is simpler than its dual?

Posted at May 12, 2011 10:32 AM UTC

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Re: Making Things Simpler by Duality

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!

Posted by: some guy on the street on May 12, 2011 1:43 PM | Permalink | Reply to this

Re: Making Things Simpler by Duality

That seems to me right - one has to be clear what kind of duality we are talking about here. It seems to me clear that no simplification can occur if we are talking about a syntactic duality, i.e. a deductive equivalence. There is a minimum number of symbols one can use to construct a deduction for a given sentence and that cannot be improved upon by any perspective shift.

On the other hand there is a clear sense in which a semantic construal of duality can result in simplification: a synatcically equivalent dual perspective might prove elucidatory - but that is an irreducibly psychological notion. To say that ‘a proof is simpler’ is to say that ‘This model is closer to my intuition.’

There is also, I think, something to be said about the cases where we do believe that such an elucidatory simplification has taken place. To take up your example, to view matrices as reps of quotient rings (viewpoint A) might make more algebraic sense than lists-of-numbers (viewpoint B) but it also comes with a LOT more background theory. To make an unsubstantiated heuristic: the amount of time one has to spend to get viewpoint A is equal to the amount of brute calc work on has to do to get things via viewpoint B. As you said: ‘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.’

But some caution is needed here: this point obviously doesnt hold if you take it to the extreme because there is a clear benefit in actually learning A in the long run instead of spending your whole life on B. But it does illustrate the fact that we have not, in fact, taken any shortcuts. In short: one cannot escape the ‘syntactic base’ by dualizing, no matter how much more intuitive the semantics become. But that doesn’t really say anything other than the fact that the notion of a simplifying duality is psychological. And it especially says nothing about long-term work. If you look at duality less like a proof-accelerator and more like a blinder-expander then there is a sense of ‘simplification’ that would definitely parse if carried through.

Finally, I think it is interesting that more ‘classically’-oriented ZFC types would find the statement self-evidently misleading (because they think of deductions.) Whereas categorists and structuralists would find it self-evidently intriguing. Maybe the whole issue speaks to the psychology of our respective mathematical intuitions rather than our practice.

Posted by: Chuck on May 12, 2011 8:21 PM | Permalink | Reply to this

Re: Making Things Simpler by Duality

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.

Posted by: Urs Schreiber on May 12, 2011 8:40 PM | Permalink | Reply to this

Re: Making Things Simpler by Duality

That rings true to me. Most of the things called “dualities” in mathematics are equivalences or isomorphisms, too, although they usually have some contravariantness. So the observation is just that if you know that A is equivalent to B, then the less similar A and B appear at first sight, the more useful it is likely to be to translate across the equivalence.

Posted by: Mike Shulman on May 13, 2011 4:22 AM | Permalink | Reply to this

Re: Making Things Simpler by Duality

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

This would appear to make ‘interesting’ a time-dependent notion which varies with people’s experience, at least to the extent that ‘seems to have anything to do with’ should vary as theoretical understanding of the equivalence builds up.

This would tally with the reaction to the discovery of projective duality which struck some people as so wonderful that lengthy two-columned treatises were produced, displaying the dual results side by side. It’s altogether less of a big deal now, though interesting when you learn it.

Do we then also take Vafa’s idea of making things ‘mathematically simple’ to involve us and our experience, so that we would expect judgements of simplicity to change? Is it just a question of making a problem easier for us now by translating into a domain we happen to grasp more readily at the moment?

Mike’s comment

So the observation is just that if you know that $A$ is equivalent to $B$, then the less similar $A$ and $B$ appear at first sight, the more useful it is likely to be to translate across the equivalence

seems also to follow the line of it being down to current levels of understanding, if I’m reading the “appear at first sight” correctly.

An opposed view might take it that the discovery of dualities have acted more than merely as aids to help us psychologically limited humans relate the unknown to the known, but that there’s a particular mathematical richness to the places where dualities appear. Lawvere and Rosebrugh appear to favour this view:

Not every statement will be taken into its formal dual by the process of dualizing with respect to $V$, and indeed a large part of the study of mathematics

space vs. quantity

and of logic

theory vs. example

may be considered as the detailed study of the extent to which formal duality and concrete duality into a favorite $V$ correspond or fail to correspond.

Posted by: David Corfield on May 13, 2011 9:30 AM | Permalink | Reply to this

Re: Making Things Simpler by Duality

Yes, in general it seems that things which are very interesting when first discovered tend to become “obvious” with time. (-:

Is it just a question of making a problem easier for us now by translating into a domain we happen to grasp more readily at the moment?

That’s not what I meant to say. It’s certainly partly that, but other times things are really and truly “easier” in one domain than in another. Lawvere+Rosebrugh also use the words “not every”, indicating their agreement that sometimes duality is formal and other times deep. (Not being a philosopher, I feel no need to divide people into camps of “for” and “opposed”. (-:)

Posted by: Mike Shulman on May 14, 2011 2:17 AM | Permalink | Reply to this

Re: Making Things Simpler by Duality

Oh dear. Is that your impression of philosophy as one of setting up a debate in terms of warring camps? I think of it more as about distilling rival points of view with a view to arriving at a more subtle third position which integrates what was best about the other two. Then repeating the process.

We’ve already achieved a certain movement here with your opposing of formal to deep dualities, suggesting depth only comes in the shape of a concrete duality. But then why? What’s so important about mapping into particular entities? I guess it’s the ‘ambimorphic’ story about the ability of an entity to support different sets of properties/structure.

Now, do concrete dualities vary as to depth, and if so how?

Posted by: David Corfield on May 15, 2011 10:32 AM | Permalink | Reply to this

Re: Making Things Simpler by Duality

Is that your impression of philosophy as one of setting up a debate in terms of warring camps? I think of it more as about distilling rival points of view with a view to arriving at a more subtle third position which integrates what was best about the other two.

To be honest, I have sometimes had the impression that some philosophers, at least, are more interested in having a debate than in understanding the opposing point of view. I wasn’t accusing you of that… but I was saying that it’s not obvious to me that we even have rival points of view here that are in need of distillation. A hypothetical person might take the viewpoint that all dualities/equivalences are about translating into equivalent language we understand better at the moment, or contrariwise that all dualities/equivalences contain real mathematical richness—but I haven’t heard anyone actually espousing either of those positions.

Not to say that there isn’t value in discussing and comparing the different types of duality! But I’m not sure it’s helpful to talk about “opposed views”, as if there were people having an argument here.

We’ve already achieved a certain movement here with your opposing of formal to deep dualities, suggesting depth only comes in the shape of a concrete duality.

Hmm… well first of all, as Urs pointed out, much of what physicists call “dualities” are what a mathematician would just call “equivalences” (or even “isomorphisms”). But even of those that are honestly dualities, I wouldn’t jump to a one-for-one correspondence between “formal” and “deep”, in the intuitive senses that I was using them, and “formal” and “concrete” in the precise sense that Lawvere uses them. For one thing, if I understand correctly, Lawvere’s meanings are not mutually exhaustive. His formal duality means just “looking at the opposite category”, and his concrete duality means a particular kind of contravariant equivalence, but there’s no reason why a contravariant equivalence need be given by mapping into something. (I guess that means I think the first sentence of the nLab page on duality is misleading.)

It does often seem to be the case that such equivalences are “concrete” in that sense, though… and I don’t really know the answer to that. Maybe it’s the adjoint functor theorem in disguise?

Posted by: Mike Shulman on May 15, 2011 9:59 PM | Permalink | Reply to this

Re: Making Things Simpler by Duality

However one does it, ‘discussing and comparing the different types of duality’ sounds to me a worthwhile thing to do, especially if it casts some light on ‘depth’, ‘richness’ and ‘making easy’.

A line of thought that attracts me is the MacIntyrean one that the importance of something in a field is constituted by its appearance in the best histories of that field. It could be argued that dualities crop up disproportionately often at key moments in the history of mathematics: logical duality and the rise of algebraic treatments of logic; projective geometry and the emergence of a structural approach to geometry; Poincaré duality (the ‘théorème fondamentale’ of Analysis Situs); Stone duality; Pontryagin duality; Gelfand duality;…; Langlands; string theory/mirror symmetry/S-duality. But maybe this is a case of not thinking hard enough about key moments not involving dualities.

A further wonder is whether dualities appear more frequently than might be expected at the places where mathematics and physics impact on one another.

Posted by: David Corfield on May 16, 2011 5:19 PM | Permalink | Reply to this

Re: Making Things Simpler by Duality

Then Science history looks like a search “more interesting” or “more simple” representative in some class of theories. This search non random because of subjects’ property(memory, consciousness, interest…previous expirience …) May be equipped with a confidence in existence of search target(as axiom of supertheory).

Meanwhile the “more simple” may be formalized as a complexity(enthropy).

Posted by: Maxim Budaev on May 15, 2011 11:32 AM | Permalink | Reply to this

Re: Making Things Simpler by Duality

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 :)

Posted by: stephen lavelle on May 12, 2011 10:43 PM | Permalink | Reply to this

Re: Making Things Simpler by Duality

Cool! Bring on shooting games where people are lines and a shot sweeps out a pencil of lines, so you have to sweep yourself out of the way of the pencil.

Posted by: David Corfield on May 13, 2011 9:49 AM | Permalink | Reply to this

Re: Making Things Simpler by Duality

Heheh - I was thinking of something along those lines playing on duality a little more - you have a choice of either being a trick-shotting sniper, whose goal is to line up as many targets as possible and shoot them, or a celebrity whose goal is to be in the field of vision of as many people as possible.

How might one decategorify tic tac toe? I’ve tried to think of a couple of things, but my decategorification muscles are a bit out of shape.

Posted by: stephen lavelle on May 13, 2011 12:13 PM | Permalink | Reply to this

Re: Making Things Simpler by Duality

How might one decategorify tic tac toe?

Not sure what you mean here. We’re only talking about dualizing, right?

Taking it up a dimension might be fun.

Posted by: David Corfield on May 13, 2011 12:45 PM | Permalink | Reply to this

Re: Making Things Simpler by Duality

“Not sure what you mean here. We’re only talking about dualizing, right?”

Yeah, it was an aside. And I probably meant categorify (*really* out of shape*). It’s certain that my question wasn’t at all well-specified. And can probably be best responded to be me looking at any of the many category-theory flavoured game-theory/algorithm-related papers on the internet…[including works by some contributors here]

(turns out I don’t know how to quote stuff in comments…)

Posted by: stephen lavelle on May 13, 2011 3:14 PM | Permalink | Reply to this

Re: Making Things Simpler by Duality

The easiest way to quote is

>Put quotation here.

Then in the Text Filter choose Markdown (with itex to MathML, if you’re including symbols).

Posted by: David Corfield on May 13, 2011 3:48 PM | Permalink | Reply to this

Re: Making Things Simpler by Duality

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.

Posted by: Matthew Emerton on May 16, 2011 5:55 AM | Permalink | Reply to this

Re: Making Things Simpler by Duality

While I agree that Poincaré duality and similar are good to keep in mind when thinking about the “dualities” in “formal high energy physics”, I think it is still important to note that many – if not most – of these dualities are not in fact involutions. For instance T-duality in full generality is an integral transform along correspondences that could happily be composed to a long string of such without ever returning to the original source. So it’s not of the kind that “doing it twice returns you to the original setup”. Rather, it’s really just a chain of equivalences. Of course, as with any equivalence, one can reverse any particular one. But in that sense every equivalence is a “duality”. (Not that the above comment claimed otherwise, I just feel that this deserves emphasis in view of the general discussion).

Posted by: Urs Schreiber on May 16, 2011 5:27 PM | Permalink | Reply to this

Re: Making Things Simpler by Duality

Thanks very much for the example. It seems a very rich one with dualities interacting. When I get a moment, I’d like to understand better what’s going on, in terms of pairings, dualizing objects, adjunctions, etc.

Posted by: David Corfield on May 16, 2011 5:30 PM | Permalink | Reply to this

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