nLab Fourier-Mukai transform

Idea

The Fourier-Mukai transform is a categorified integral transform roughly similar to the standard Fourier transform.

Generally, for $X,Y$ two suitably well-behaved schemes (e.g. affine, smooth, complex) and with $D(X)$, $D(Y)$ their derived categories of quasicoherent sheaves, then a Fourier-Mukai transform with integral kernel $E \in D(X\times Y)$ is a functor (of triangulated categories/stable (infinity,1)-categories)

$$\Phi \colon D(X)\longrightarrow D(Y)$$

which is given as the composite of the (derived) operations of

1. pull (inverse image) along the projection $p_X\colon X\times Y \to X$

2. tensor product with $E$;

3. push (direct image) along the other projection $p_Y \colon X\times Y \to Y$

i.e.

$$\Phi(A) \coloneqq (p_Y)_\ast (E\otimes p_X^\ast A)$$

(where here we implicitly understand all operations as derived functors). (e.g. Huybrechts 08, page 4)

Hence this is a pull-tensor-push integral transform through the product correspondence

$$\array{ && X \times Y \\ & \swarrow && \searrow \\ X && && Y }$$

with twist $E$ on the correspondence space.

Such concept of integral transform is rather general and may be considered also in derived algebraic geometry (e.g. BenZvi-Nadler-Preygel 13) and lots of other contexts.

As discussed at integral transforms on sheaves this kind of integral transform is a categorification of an integral transform/matrix multiplication of functions induced by an integral kernel, the role of which here is played by $E\in D(X \times Y)$.

Indeed, the central kind of result of the theory (theorem ) says that every suitable linear functor $D(X)\to D(Y)$ arises as a Fourier-Mukai transform for some $E$, a statement which is the categorification of the standard fact from linear algebra that every linear function between finite dimensional vector spaces is represented by a matrix.

The original Fourier-Mukai transform proper is the special case of the above where $X$ is an abelian variety, $Y = X^\vee$ its dual abelian variety and $E$ is the corresponding Poincaré line bundle.

If $X$ is a moduli space of line bundles over a suitable algebraic curve, then a slight variant of the Fourier-Mukai transform is the geometric Langlands correspondence in the abelian case (Frenkel 05, section 4.4, 4.5).

Definition

Let $X$ and $Y$ be schemes over a field $K$. Let $E \in D(QCoh(O_{X \times Y}))$ be an object in the derived category of quasi-coherent sheaves over their product. (This is a correspondence between $X$ and $Y$ equipped with a chain complex $E$ of quasi-coherent sheaves).

The functor $\Phi(E) : D(QCoh(O_X)) \to D(QCoh(O_Y))$ defined by

$$F \mapsto \mathbf{R}q_*(\mathbf{L}p^*(F) \otimes^{\mathbf{L}} E),$$

where $p$ and $q$ are the projections from $X \times Y$ onto $X$ and $Y$, respectively, is called the Fourier-Mukai transform of $E$, or the Fourier-Mukai functor induced by $E$.

When $F : D(QCoh(O_X)) \to D(QCoh(O_Y))$ is isomorphic to $\Phi(E)$ for some $E \in D(QCoh(O_{X \times Y}))$, one also says that $F$ is represented by $E$ or simply that $F$ is of Fourier-Mukai type.

Properties

The key fact is as follows

Theorem (Orlov)

Let $X$ and $Y$ be smooth projective varieties over a field $K$. Let $F : D(X) \to D(Y)$ be a triangulated fully faithful functor. Then $F$ is represented by some object $E \in D(X \times Y)$ which is unique up to isomorphism.

See Orlov 2003, 3.2.1 for a proof.

Remark

Though theorem is stated there for $F$ admitting a right adjoint, it follows from Bondal-van den Bergh 2002 that every triangulated fully faithful functor admits a right adjoint automatically (see e.g. Huybrechts 08, p. 6).

Remark

It was believed that theorem should be true for all triangulated functors (e.g. Huybrechts 08, p. 5). However according to (RVdB 2015) this is not true.

Enhancements

For any quasi-compact quasi-separated $k$-schemes $X$ and $Y$ there is a canonical equivalence of stable (∞,1)-categories

$$\mathrm{Func}^L(D(X), D(Y)) \simeq D(X \times Y).$$

Related concepts

• triangulated categories of sheaves

• Grothendieck duality

References

General

• Shigeru Mukai, Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves. Nagoya Mathematical Journal 81: 153–175. (1981)

• Alexei Bondal, Michel van den Bergh. Generators and representability of functors in commutative and noncommutative geometry, 2002, arXiv

• Dmitri Orlov, Derived categories of coherent sheaves and equivalences between them, Russian Math. Surveys, 58 (2003), 3, 89-172, translation.

• Lutz Hille, Michel van den Bergh, Fourier-Mukai transforms (arXiv:0402043)

• Daniel Huybrechts, Fourier-Mukai transforms (2008) [pdf, pdf]

• Claudio Bartocci, Ugo Bruzzo, Daniel Hernandez Ruiperez: Fourier-Mukai and Nahm transforms in geometry and mathematical physics, Progress in Mathematics 276, Birkhäuser (2009) [doi:10.1007/b11801]

• Alice Rizzardo, Michel Van den Bergh, An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves (arXiv:1410.4039)

• Pieter Belmans, section 2.2 of Grothendieck duality: lecture 3, 2014 (pdf)

Banerjee and Hudson have defined Fourier-Mukai functors analogously on algebraic cobordism.

• Anandam Banerjee, Thomas Hudson, Fourier-Mukai transformation on algebraic cobordism, pdf.

Discussion of internal homs of dg-categories in terms of refined Fourier-Mukai transforms is in

• Bertrand Toën, The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007), 615–667

• Alberto Canonaco, Paolo Stellari, Internal Homs via extensions of dg functors (arXiv:1312.5619)

Discussion in the context of geometric Langlands duality is in

• Edward Frenkel, Lectures on the Langlands Program and Conformal Field Theory (arXiv:hep-th/0512172)

For a discussion of Fourier-Mukai transforms in the setting of $(\infty,1)$-enhancements, see

• David Ben-Zvi, David Nadler, Anatoly Preygel. Integral transforms for coherent sheaves, arXiv:1312.7164

For T-duality

Discussion of the effect of T-duality on RR-field flux densities (in ordinary cohomology or in topological K-theory) as a Fourier-Mukai-like transform goes back to:

• Kentaro Hori, Yaron Oz, Section 3.1 of: F-Theory, T-Duality on K3 Surfaces and $N=2$ Supersymmetric Gauge Theories in Four Dimensions, Nucl. Phys. B 501 (1997) 97-108 [arXiv:hep-th/9702173, doi:10.1016/S0550-3213(97)00361-1]

• Kentaro Hori: (1.1) and pp. 13 of: D-branes, T-duality, and Index Theory, Adv. Theor. Math. Phys. 3 2 (1999) 281-342 [arXiv:hep-th/9902102, doi:10.4310/ATMP.1999.v3.n2.a5, journal:pdf]

  (whence “Hori’s formula”)

with explicit formulation in the context if topological T-duality:

• Peter Bouwknegt, Jarah Evslin, Varghese Mathai, (1.9) in: T-Duality: Topology Change from H-flux, Commun. Math. Phys. 249 (2004) 383-415 [hep-th/0306062, doi:10.1007/s00220-004-1115-6]

Survery and Review:

• Martin Ruderer, Fourier-Mukai Transforms from T-Duality, PhD thesis, Regensburg (2015) [urn:nbn:de:bvb:355-epub-314033, pdf]

• Bjorn Andreas, Daniel Hernandez Ruiperez: Fourier Mukai Transforms and Applications to String Theory, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat 99 1 (2005) 29-77 [arXiv:math/0412328, spire:667425, journal: pdf]

For non-abelian T-duality:

• Eva Gevorgyan, Gor Sarkissian: Defects, Non-abelian T-duality, and the Fourier-Mukai transform of the Ramond-Ramond fields, J. High Energ. Phys. 2014 35 (2014) [doi:10.1007/JHEP03(2014)035, arXiv:1310.1264]

On superspace:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber: Prop. 6.4 in: T-Duality from super Lie n-algebra cocycles for super p-branes, Adv. Theor. Math. Phys 22 5 (2018) [arXiv:1611.06536, doi:10.4310/ATMP.2018.v22.n5.a3]

For spherical T-duality:

• Peter Bouwknegt, Jarah Evslin, Varghese Mathai: Spherical T-duality and the spherical Fourier–Mukai transform, Journal of Geometry and Physics 133 (2018) 303-314 [doi:10.1016/j.geomphys.2018.07.020, arXiv:1502.04444]