The Unity of Dualities

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In colloquial speech the word “duality” connotes two contrasting facets of a single entity, often at odds with one another. The concept is anthropomorphized in mythology by deities or monsters with multiple faces, like the two-faced Janus,¹ Roman god of doorways. It is also enshrined in pop culture in the double visages of Jekyll and Hyde, and in the Batman villain Harvey Dent (alias Two-Face). In physics and mathematics, the concept of “duality”² takes on a more positive connotation because of its ubiquity, utility, and power. Rather than perceiving the two opposing faces in tension with one another, they are complements; each face (which we often call a “duality frame”) exhibits something the other cannot, and in studying both faces or frames we acquire a more unified understanding of the whole. Many systems in mathematics and physics admit two (or more) duality frames, where difficult problems in the first frame become tractable in the second. When the physicist’s usual workmanlike tools become inadequate, dualities can help illuminate a new path.

Many profound dualities are Fourier transforms. (Of course, I am oversimplifying here, but to say “many profound dualities are like or analogous to Fourier transforms” undersells the conceptual through-line). In a Fourier transform, the unified entity or whole which we wish to understand is a certain function³ of interest, and it admits two duality frames. Depending on the application we have in mind, it can be expedient to represent the function in one domain (one of Janus’s faces) or as a function on another, complementary domain (the other, dual face). I call the entity $f(x)$ , a function $f$ of the real variable $x$ , in one frame and $\widehat{f}(p)$ , a function $\widehat{f}$ of the real variable $p$ , in the other. An entity that looks untamable in one frame can be exquisitely simple in the other; here, as with all dualities, it is not at all obvious that we should have found ourselves in such a happy circumstance!⁴

There is a precise way to get from one frame to the other, which we call the “duality map” or “duality transformation.” I will write out the map for those who like formulas, but if the symbols look forbidding, just think of them as a precise recipe for translating between the two representations. We start with $f(x)$ and apply a so-called “integral transform”: multiply by a suitable function called the kernel, in this case an exponential $e^{- 2\pi ipx}$ , and integrate over the variable $x$ :

Of course, duality transformations must go both ways, so we could have equally well started with $\widehat{f}(p)$ and gone the other way by applying the “inverse transform” with the inverse kernel.

The variables $x$ and $p$ live in dual spaces. Dual spaces are those which come equipped with a natural pairing, often denoted $\langle,\rangle$ , where you can input an element of a space on the right and its dual on the left and obtain a number, $\langle p,x\rangle$ . The pairing appears in the kernel function of a Fourier transform. In this example, the pairing is just ordinary multiplication of real numbers: $\langle p,x\rangle = px$ , which appears in the argument of the exponential. In other contexts, the pairing may be more involved.

Figure 1: A simple piecewise function and its Fourier transform

Now, what do these dual representations of our entity look like? Here, the complementary nature of the duality comes to the fore. A function that is concentrated, or localized, in one variable will be spread out in the dual variable (Figure 1). What’s more, a function that is perfectly concentrated at a particular point $x$ with infinite value, say at $x = a$ , will be as spread out as possible in $p$ (or vice versa); we say that the Fourier transform of this delta function $f(x) = \delta(x - a)$ is an exponential $\widehat{f}(p) = e^{- 2\pi ipa}$ , which is just the constant function 1 when $a = 0$ . A function that looks both a bit concentrated and a bit spread out in one variable will look that way in the dual variable, too: the Fourier transform of a Gaussian $f(x) = e^{- ax^{2}}$ is another Gaussian $\widehat{f}(p) = \sqrt{\pi/a}e^{- p^{2}\pi^{2}/a}$ (Figure 2).

Figure 2: A sample Gaussian and its Fourier transform

In physics, these complementary perspectives have meaning. Now we will think of $x,p$ not just as variable names but as encoding a particle’s position and momentum, respectively.⁵

Heisenberg’s uncertainty principle tells us that for pairs of complementary observables in the quantum mechanical world, of which position and momentum furnish the most basic examples, there is a fundamental limit to how accurately we may measure both simultaneously. If we measure $x$ precisely (the observable is localized in position space, or particle-like), we sacrifice knowledge of $p$ (the observable is spread out in momentum space); if instead we measure $p$ precisely then the observable becomes spread out in position space, or wave-like. We may measure both observables only if we are willing to accept inherent uncertainty in the simultaneous measurements (the observable is both particle-like and wave-like); the standard deviations of these measurements are bounded below by Planck’s constant: $\sigma_{x}\sigma_{p} \geq \hslash/2$ . What’s more, since these observables are complementary we may express the quantum state of our particle as a function⁶ of either $x$ or $p$ but never both simultaneously; however, we may apply a Fourier transform to choose the frame we prefer. When we have made a choice of coordinate $x$ and its corresponding dual, or conjugate, momentum $p$ we say that we have chosen a polarization on the phase space locally spanned by the coordinates $x$ and $p$ . In classical physics, we do not need to make this choice and, in particular, we may compute quantities that depend on all position and momentum coordinates at once. In quantum physics, the choice of splitting is mandatory, but a Fourier transform allows us to exchange the position and momentum coordinates of our chosen polarization.

Many other mathematical dualities, including important dualities arising in string theory, are basically Fourier transforms—or at least like Fourier transforms on steroids (think of Jose Canseco in his prime). Roughly, one starts with two complementary, or dual, mathematical spaces⁷ $\mathcal{A},\mathcal{A}^{\vee}$ on which one can define objects (which could be algebraic, geometric, number theoretic, etc. in nature) that we wish to study. In our example, $\mathcal{A}$ is the space spanned by the $x$ coordinate and we may study functions $f$ on it, while $\mathcal{A}^{\vee}$ is the “dual space,” spanned by the dual coordinate $p$ , on which the function $\widehat{f}$ lives.⁸ Of course, $x$ is a function on $\mathcal{A}$ and $p$ is a function on $\mathcal{A}^{\vee}$ . One then formally considers a bigger object which incorporates both of our original objects at the same time, which we write as $\mathcal{A} \times \mathcal{A}^{\vee}$ . Functions on this combined space (e.g. $\mathbb{R} \times \mathbb{R} \simeq \mathbb{R}^{2}$ ) would be functions on the entire phase space, though our quantum functions are restricted to be functions of only one variable or the other. Nevertheless, there is a special creature in the combined object that helps furnish our duality map: in our case the integral kernel, which is function of both $x$ and $p$ , given by $e^{- 2\pi ipx}$ . We also have its inverse for the inverse map. For a general such duality, we need the analogue of the integral transform with respect to a kernel in order to obtain the duality map that enables us to move freely between any object on $\mathcal{A}$ (any $f(x)$ ) and its dual on $\mathcal{A}^{\vee}$ (the Fourier transform $\widehat{f}(p)$ ), and vice versa.

Finding the duality map can be quite difficult. Even if the map is known, computing the transformation explicitly might only be feasible for special, simple classes of objects in $\mathcal{A}$ or $\mathcal{A}^{\vee}$ . More hairy still, in physics we often work with objects that are difficult to formulate mathematically. Even when we understand (part of) the mathematical description of a physical system, there can be various complications (spaces encoding physical degrees of freedom may be infinite-dimensional, have nasty singularities, or suffer other pathologies). But there is nevertheless always great utility in knowing that an entity has two faces and glimpsing them both.

A geometric duality of the Fourier ilk in which we still have an explicit command of many objects on both sides is called the Fourier-Mukai transform. It also arises in string theory in a physical duality called T-duality: it is the mathematical description of how T-duality acts on objects called D-branes. Instead of mapping between dual classes of functions, we map between geometric objects known as coherent sheaves, which generalize the functions of our previous example. Sheaves are naturally defined on spaces that mathematicians call varieties, and some varieties (such as what mathematicians call abelian varieties) admit a natural notion of dual variety.

There are sheaves that live only over a single point on a variety, much like the delta function we met earlier, called skyscraper sheaves; there are also simple one-dimensional sheaves called line bundles, which are analogous to the exponential function dual to the delta function. (Geometrically, the line bundles in question are “topologically trivial,” which you can think of as being akin to the special case of the constant function: spread out uniformly over its domain). Indeed, the Fourier-Mukai transform, when applied to simple abelian varieties called elliptic curves, maps skyscraper sheaves to line bundles, just as the Fourier transform mapped delta functions to exponentials, and the recipe is the same: take a skyscraper sheaf on a curve $\mathcal{A}$ and consider it as an object on the bigger space combining the abelian variety and its dual, $\mathcal{A} \times \mathcal{A}^{\vee}$ .⁹ Then use a special “kernel” sheaf in $\mathcal{A} \times \mathcal{A}^{\vee}$ called the Poincare line bundle, “multiply” it with the original skyscraper sheaf, and perform an “integral transform” with respect to this kernel. These ingredients are all understood quite explicitly for Fourier-Mukai transforms, though I won’t go into details.

Coherent sheaves are mathematical models for D-branes, which are charged, dynamical objects that are, like the strings themselves, part and parcel of string theory. D-branes have been indispensable ingredients in string theoretic studies of black hole microstate counting, top-down constructions of holographic systems, the discovery and elucidation of non-perturbative phenomena in quantum field theories, and much more. String theoretic dualities need D-branes to work. D-branes can have varying numbers of spatial dimensions (e.g. they can be point- or particle-like, one-dimensional or string-like, two-dimensional or membrane-like, and so on). Just as the Fourier-Mukai transform mapped a point-like sheaf (the skyscraper) into a one-dimensional sheaf (the line-bundle) on the elliptic curve, its physical counterpart T-duality maps D-branes with differing spatial extent into one another, even on more complicated varieties. For a physicist, this is a dramatic operation! You start with some point-like D-branes (imagine them as heavy charged particles) sitting on a certain space, or variety, and I come along and tell you that this is the same physical system as some string-like (imagine charged, high-tension strings) D-branes winding around a different variety. The two faces of Janus are not those of twins, but the magic of dualities is that a computation expressed in the first frame must give the same answer as its dual computation in the second frame. If we only have the computational power to produce answers in one frame, we can still learn something about the complicated physics of the dual frame by the existence of this duality map.

I spend a lot of my time thinking hard about physical dualities, including how to translate them into mathematical statements (some physicists may tell you that such work is the province of fussbudgets, but I have always found precision a good antidote to ego). I also study preexisting mathematical dualities that I believe arise in string theory, and probe the consequences of that mathematical structure for the physics. Sometimes, these mathematical dualities turn out to be Fourier transforms, though a bewildered physicist may need a lot of time, and many patient explanations from colleagues,¹⁰ to realize it. Koszul duality is the latest such mathematical duality that has begun to permeate theoretical physics. There are many avatars of Koszul duality, but it can still be viewed as a Fourier transform, although it looks a bit fancier than our previous examples. The mathematical objects at play are now dual algebras, rather than dual spaces, and the Fourier transform map is a map of representations of these algebras, instead of a map of functions or sheaves.¹¹ One appearance of Koszul duality in physics (though there are others) arises when considering boundary conditions of fields in certain quantum field theories. Boundary conditions involve choosing a polarization on a phase space, as we saw before in quantum mechanics—roughly: to get a good boundary value problem, you must set half of the degrees of freedom of your physical system to zero on the boundary, which means you lose access to half the coordinates on your phase space at the boundary. Choosing between “complementary” variables is often called choosing between “transverse polarizations” and the corresponding boundary conditions are sometimes called transverse themselves. Basic Neumann and Dirichlet boundary conditions in free theories are the prototypical examples of transverse boundary conditions. In certain systems, physics can associate algebras to boundary conditions, and the algebras one associates to transverse boundary conditions turn out to be Koszul dual. Kevin Costello recently proposed that Koszul duality is a mathematical ingredient in special examples of holography, a profound physical duality which equates theories of quantum gravity in $d$ -dimensions, and ordinary quantum field theories in $(d - 1)$ -dimensions said to live on the boundaries of the gravitational systems. Boundaries of physical systems can support surprising, rich physics! We wrote a paper fleshing out this suggestion in a concrete example of holography arising from a string theory construction (D-branes, as always, being a key ingredient), and there is a great deal more to explore and understand.

Now, I would never claim that all deep physical dualities boil down to dressing up the humble Fourier transform in increasingly abstruse garb. But I do claim that Nature reuses beautiful ingredients and simple ideas over and over again, sometimes when we least expect it. In view of that, we should keep our eyes peeled—perhaps even two sets of eyes. ■