### 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?

## Re: Making Things Simpler by Duality

When Vafa remarks

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

statingwhat this S-duality and mirror symetry are: even if thatis simplifyingthe 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

isa 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: onemaystudy 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!