# Explaining Mukai-Fourier transforms physically

A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality).

The basic algorithm is to find dual sets of eigenvectors/eigenfunctions parametrized by a continuous (e.g., $$\omega$$ below) or discrete index (e.g., $$n$$ below), that satisfy completeness and orthogonality relations encapsulated in Dirac delta function resolutions such as that for the FT

$$\delta(x-y)= \int_{-\infty}^{\infty}\exp(i2\pi \omega x)\exp(-i2\pi \omega y)d\omega$$

giving

$$\int_{-\infty}^{\infty}f(y)\delta(x-y)dy=f(x)=\int_{-\infty}^{\infty}\exp(i2\pi \omega x)\int_{-\infty}^{\infty}f(y)\exp(-i2\pi \omega y) dy d\omega$$

$$=\int_{-\infty}^{\infty}\exp(i2\pi \omega x)\hat{f}(\omega) d\omega,$$

or that for the eigenvectors of Sturm-Liouville differential operators over finite domains

$$\delta(x-y)=\sum_{n=0}^{\infty }\Psi_n(x)\Psi_n^*(y)$$

giving

$$f(x)=\sum_{n=0}^{\infty }\Psi_n(x)\int_{a}^{b}f(y)\Psi_n^*(y) dy,$$

or Kronecker delta resolutions such as that for the associated Laguerre functions

$$\frac{(n+\alpha)!}{n!}\delta_{mn}=\int_{0}^{\infty}x^{\alpha}e^{-x}L_{n}^{\alpha}(x)L_{m}^{\alpha}(x)dx$$

giving

$$f(x)=\sum_{n=0}^{\infty }\frac{n!L_{n}^{\alpha}(x)}{(n+\alpha)!}\hat{f}_n$$

with

$$\hat{f}_n=\int_{0}^{\infty}x^{\alpha}e^{-x}L_{n}^{\alpha}(x)f(x)\,dx.$$

The basic "physical" operation (BPO) at work here can be regarded as destructive/constructive interference; the product at a point of the value of the function (to be resolved) with the corresponding value of an eigenfunction has a negative or positive value (or phase factor) that may sum constructively or destructively with products at other points (seen as a matched filtering or correlation by replacing $$y$$ with $$x-z$$ above). Alternatively, the BPO may be viewed as projection of vectors onto a set of orthonormal axes. In addition, if the function and operations are discretized and/or the domains restricted (in one space or its dual or both, as for the DFT) aliasing (which seems analogous to the introduction of equivalence classes) is introduced and periodicity imposed.

Can you explain the machinery behind the Mukai-Fourier transform in terms of these BPOs or close analogies?

(Edit 1/16/212) Further to Carnahan's answer below:

Kapustin and Witten in "Electric-magnetic duality and the geometric Langlands Program" state, ". . . it must be possible to understand the geometric Langlands program using four-dimensional electric-magnetic duality (which leads to this particular T-duality) and branes (the natural quantum field theory setting for interpreting T-duality as a Fourier-Mukai transform). This hint was the starting point for the present paper." The T-duality is analogous to the modular/automorphic function symmetry involving reciprocals of a parameter/variable. They refer to the paper "Lectures on the Langlands Program and conformal field theory" by Frenkel as a good intro.

Frenkel notes, "It has long been suspected that the Langlands duality should somehow be related to various dualities observed in quantum field theory and string theory. Indeed, both the Langlands correspondence and the dualities in physics have emerged as some sort of non-abelian Fourier transforms. Moreover, the so-called Langlands dual group introduced by R. Langlands that is essential in the formulation of the Langlands correspondence also plays a prominent role in the study of S-dualities in physics and was in fact also introduced by the physicists P. Goddard, J. Nuyts, and D. Olive in the framework of four-dimensional gauge theory. ... The goal of these notes is two-fold: first, it is to give a motivated introduction to the Langlands Program, including its geometric reformulation, addressed primarily to physicists. I have tried to make it as self-contained as possible, requiring very little (lol) mathematical background. The second goal is to describe the connections between the Langlands Program and two-dimensional conformal field theory that have been found in the last few years. These connections give us important insights into the physical implications of the Langlands duality."

On pages 53 and 54, Frenkel presents an elaboration on the parallels, as noted by Carnahan, between the Dirac delta function-complex exponential duality central to the Fourier transform and the skyscraper sheaf-line bundle duality associated with the Fourier-Mukai transform.

Extrapolating, one would expect to find analogies to Poisson (or Dirac comb) summation, zeta function functional symmetry equations, and Euler product factorizations, as listed in Table I of "A Correspondence Principle" by Hughes and Ninham (subsumed, I suppose, by the analytic Langlands Program), and to log/exp, trace/determinant, moment/cumulant, and other dualities found in symmetric function theory, as well as 'gauge/conjugation' transformations. References to surveys sketching such associations, briefer than Frenkel's, would be appreciated, especially if linked to Fourier(Mellin)-Mukai transforms. (This is related to another MO_Q.)

• (Generalizing) There are two interesting narratives on the history of harmonic analysis that interweave developments in mathematical physics and mathematical analysis. For the short version, see Norbert Wiener's "The historical background of harmonic analysis" (ams.org/samplings/math-history/procsemi-wiener.pdf), and for the long version, G. Mackey's "Harmonic analysis as the exploitation of symmetry--a historical survey" (ams.org/journals/bull/1980-03-01/S0273-0979-1980-14783-7/…). Can someone extend the narrative to the Mukai-Fourier transform? – Tom Copeland Sep 28 '12 at 10:06
• Apropos Scott Carnahan's answer, see pg. 23 of E. Frenkel's "Langlands Program, Trace Formulas, and their Geometrization" (arxiv.org/abs/1202.2110). – Tom Copeland Sep 30 '12 at 19:18
• For more detail on the mathematical apparatus, see “Langlands Program, Field Theory, and Mirror Symmetry” by Ikeda. // For some more on the history, see the blog post "Geometric Langlands and QFT" at Not Even Wrong by Peter Woit. – Tom Copeland Jan 18 at 4:53
• Related Fourier harmonics in "An analytic version of the Langlands correspondence for complex curves" by Etingof, Frenkel, and Kazhdan arxiv.org/abs/1908.09677 – Tom Copeland Jan 18 at 18:17
• A brief intro to quantum dualities for the layman is in "Geometry and Physics" by Atiyah, Dijkgraaf,and Hitchin royalsocietypublishing.org/doi/full/10.1098/… – Tom Copeland Jan 18 at 20:06

You can think of line bundles and skyscraper sheaves as sheaf-theoretic analogues to exponentials and delta functions, respectively. The Fourier-Mukai transform on an elliptic curve takes one type of sheaf to the other (with a homological shift that I will ignore). In higher dimension, you get some mixtures of these types, from complexes of sheaves with cohomology supported on subvarieties of positive dimension and positive codimension - you can think of these as an analogue of more general distributions. Vector bundles on an abelian variety get "orthogonally decomposed" into skyscrapers on the "frequency space", i.e., the dual abelian variety.

• I like it too! is great! – Dox Sep 27 '12 at 12:23
• Can you deepen the analogy to include properties of the FT such as Parseval's relation or conversion of differentiation to multiplication? – Tom Copeland Sep 27 '12 at 23:24
• I do not know an analogue of unitarity. The relationship between linear polynomials and directional derivatives in the usual Fourier theory can be rephrased in terms of an endomorphism on the ring of differential operators on affine space. You may be able to do something similar with the derived category of coherent sheaves on the product of the abelian variety and its dual, but I don't know a concrete answer. – S. Carnahan Sep 28 '12 at 3:59
• You can't relate a "translation" (if that can be made sensible) of a skyscraper sheaf to "multiplication" by a "function" in the "dual space"? – Tom Copeland Sep 28 '12 at 9:29
• More on the analogue of Fourier convolution in a 2019 answer to mathoverflow.net/questions/9834/… – Tom Copeland Feb 15 '20 at 20:28

I'd suggest you to check the post HERE.

As you will see, the analogy enters by thinking the pullback of the sheaf $\mathcal{F}$ on the $X$ variety, i.e. $p_1^*\mathcal{F}$, as Fourier coefficients, while the sheaf $\mathcal{P}$ on the product variety $X\times Y$ plays the role of integral kernel.

Cheers.

• That post actually inspired my post to seek clarification and to probe how deeply the analogy goes. – Tom Copeland Sep 27 '12 at 6:14