Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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molecule) travels between collisions in a gas with

particle density . Straightforward analysis gives

  1=

,where an ‘‘effective’’ hard-core diameter

of a particle. They considered a gas with temperature

gradient in the x-direction and assumed that the gas is

(approximately) in local equilibrium with density 

and temperature T(x). Between collisions, a particle

moves a distance  carrying a kinetic energy propor-

tional to T(x)fromx to x þ =

ﬃﬃﬃ

, while in the

opposite direction the amount carried is proportional

to T(x þ 

ﬃﬃﬃ

). Taking into account the fact that the

speed is proportional to

ﬃﬃﬃﬃ

the amount of energy J

transported per unit area and time across a plane

perpendicular to the x-axisisapproximately

J  

ﬃﬃﬃﬃ

TðxÞTðx þ 

ﬃﬃﬃ



2

ﬃﬃﬃﬃ

½5

and so  

ﬃﬃﬃﬃ

independent of , in agreement with

experiment. It was clear to the founding fathers that

starting with a local equilibrium situation the process

described above will produce, as time goes on, a

deviation from LTE. They reasoned, however, that this

deviation from local equilibrium will be small when

(=T)dT=dx  1, the regime in which Fourier’s law is

expected to hold, and the above calculation should

yield, up to some factor of order unity, the right heat

conductivity. To have a more precise theory, one can

describe the state of the gas through the probability

distribution f (r, p, t)offinding aparticleinthe

volume element dr dp around the phase space point

(r, p). Here LTE means that

f ðr; p; t Þ’exp 

2mkTðrÞ



where m is the mass of the particles. If one computes

the heat flux at a point r by averaging the microscopic

energy current at r, j = v(1=2mv

), over f (r, p, t)then

it is only the deviation from local equilibrium which

makes a contribution. The result however is essentially

the same as eqn [5]. This was shown by Boltzmann,

who derived an accurate formula for  in gases by

using the Boltzmann equation. If one takes  from

experiment, the above analysis yields a value for ,the

effective size of an atom or molecule, which turns out

to be close to other determinations of the characteristic

size of an atom. This gave an evidence for the reality of

atoms and the molecular theory of heat.

Heat Conduction in Insulating Crystals

In (electrically) conducting solids, heat is mainly

transported by the conduction electron. In this case,

one can adapt the theory discussed in the previous

section. In (electrically) insulating solids, on the other

hand, heat is transmitted through the vibrations of the

lattice. In order to use the concepts of kinetic theory, it

is useful to picture a solid as a gas of phonons which

can store and transmit heat. A perfectly harmonic

crystal, due to the fact that phonons do not interact,

has an infinite thermal conductivity: in the language of

kinetic theory, the mean free path  is infinite. In a real

crystal, the anharmonic forces produce interactions

between the phonons and therefore a finite mean free

path. Another source of finite thermal conductivity

may be the lattice imperfections and impurities which

scatter the phonons. Debye devised a kind of kinetic

theory for phonons in order to describe thermal

conductivity. One assumes that a small gradient of

temperature is imposed and that the collisions between

phonons maintain local equilibrium. An elementary

argument gives a thermal conductivity analogous to

eqn [5] obtained in the last subsection for gases

(remembering, however, that the density of phonons

is itself a function of T)

  c

 ½6

where, with respect to eqn [5],  has been replaced by

, the specific heat of phonons,

ﬃﬃﬃﬃ

by c,the(mean)

velocity of the phonons, and  by c,where is the

effective mean free time between phonon collisions.

The thermal conductivity depends on the temperature

via , and a more refined theory is needed to account

for this dependence. This was done by Peierls via a

Boltzmann equation for the phonons. In collisions

among phonons, the momentum of phonons is

conserved only modulo a vector of the reciprocal

lattice. One calls ‘‘normal processes’’ those where the

phonon momentum is conserved and ‘‘Umklap pro-

cesses’’ those where the initial and final momenta

differ by a nonzero reciprocal lattice vector. Peierls’

theory may be summarized (very roughly) as follows:

in the absence of Umklap processes, the mean free

path, and thus the thermal conductivity of an insulat-

ing solid, is infinite. A success of Peierls’ theory is to

describe correctly the temperature dependence of the

thermal conductivity. Furthermore, on the basis of this

theory, one does not expect a finite thermal conduc-

tivity in one-dimensional monoatomic lattices with

pair interactions. This seems so far to be a correct

prediction, at least in the numerous numerical results

performed on various models.

Statistical Mechanics Paradigm:

Rigorous Analysis

In a rigorous approach to the above arguments, we

have to first formulate precisely the problem on a

376 Fourier Law

mathematical level. It is natural to adapt the standard

formalism of statistical mechanics to our situation. To

this end, we assume that our system is described by

the positions Q and momenta P of a (very large)

number of particles, N,withQ = (q

, ..., q

) 2



,   R

,andP = (p

, ..., p

) 2 R

.The

dynamics (in the bulk) is given by a Hamiltonian

function H(Q, P). A state of the system is a

probability measure (P, Q ) on phase space. As

usual in statistical mechanics, the value of an

observable f (P, Q) will be given by the expected

value of f with respect to the measure . In the case of

a fluid contained in a region , we can assume that

the Hamiltonian has the form

HðP; QÞ¼

i¼1

j6¼i

ðq

 q

Þþuðq

i¼1

þVðQÞ½7

where (q) is some short-range interparticle potential

and u(q

) an external potential (e.g., the interaction of

the particle with fixed obstacles such as a conduction

electron interacting with the fixed crystalline ions). If

we want to describe the case in which the temperature

at the boundary is kept different in different regions

@



, we have to properly define the dynamics at the

boundary of the system. A possibility is to use

‘‘Maxwell boundary conditions’’: when a particle hits

the wall in @



, it gets reflected and re-emerges with a

distribution of velocities



ðdvÞ¼

2ðkT



jexp 

2kT





dv ½8

Several other ways to impose boundary conditions

have been considered in the literature. The notion of

LTE can be made precise here in the so-called

hydrodynamic scaling limit (HSL), where the ratio

of microscopic to macroscopic scales goes to zero.

The macroscopic coordinates r and t are related to

the microscopic ones q and ,byr = q and t = 



,

that is, if  is a cube of macroscopic sides l, then its

sides, now measured in microscopic length units, are

of length L = 

1

l. We then suppose that at t = 0 our

system of N =  L

particles is described by an

equilibrium Gibbs measure with a temperature

T(r) = T(q): roughly speaking, the phase-space

ensemble density has the form



ðP; QÞexp 

i¼1



ðq

(



j6¼i

ðq

 q

Þþuðq

"#)

½9

where 

1

(r) = T

(r). In the limit  !0,  fixed, the

system at t = 0 will be m acroscopically in LTE with

a local temperature T

(r) (as already noted, here we

suppress the variation in the particle density n(r)).

We are interested in the behavior of a macroscopic

system, for which   1, at macroscopic times

t  0, corresponding to microscopic times

 = 



t,  = 2 for heat conduction or other diffu-

sive behavior. The implicit assumption then made

in the macroscopic description given earlier is that,

since the variations in T

(r) are of order  on a

microscopic scale, then for   1, the system will,

also at time t,beinastateveryclosetoLTE,with

atemperatureT(r, t) that evolves in time according

to Fourier’s law, eqn [1]. From a mathematical

point of view, the difficult problem is to prove that

the system stays in LTE for t > 0 when the

dynamics are given by a Hamiltonian time evolu-

tion. This requires proving that the macroscopic

system has some very strong ergodic properties, for

example, that the only time-invariant measures

locally absolutely continuous with respect to the

Lebesgue measure are, for infinitely extended

spatially uniform systems, of the Gibbs type. This

hasonlybeenprovedsofarforsystemsevolving

via stochastic dynamics (e.g., interacting Brownian

particles or l attice gases). For such stochastic

systems, one can sometimes prove the hydrodyna-

mical limit and derive macroscopic t ransport

equations for the particle or energy density and

thus verify the validity of Fourier law. Another

possibility, as we already saw, i s to use the

Boltzmann e quation. Using ideas of hydrodynami-

cal space and time scaling described earlier, it is

possible to derive a controlled expansion for the

solution of the stationary Boltzmann equation

describing the steady state of a gas coupled to

temperature reservoirs at the top and bottom. One

then shows t hat for   1,  being now the ratio

=L, the Boltzmann equation for f in the slab h as a

time-independent solution which is close to a local

Maxwellian, corresponding to LT E (apart from

boundary layer terms) with a local temperature and

density given by the solution of the Navier–Stokes

equations which incorporates Fourier’s law as

expressed in eqn [2]. The main mathematical

problem is in controlling the remainder in an

asymptotic expansion of f in power of  .This

requires that the macroscopic temperature gradient,

that is, jT

 T

j=h,whereh = L is the thickness of

the slab on the macroscopic scale, be small. Even if

this apparently technical problem could be over-

come, we would still be left with the question of

justifying the Boltzmann equation for such s teady

states and, of course, it would not tell us anything

Fourier Law 377

about dense fluids or crystals. In fact, the Boltz-

mann equation itself is really closer to a macro-

scopic than to a microscopic description. It is

obtained in a well-defined kinetic scaling limit in

which, in addition to rescaling space and t ime, the

particle density goes to zero, that is,   .

A simplified model of a crystal is characterized by

the fact that all atoms oscillate around given

equilibrium positions. The equilibrium positions

can be thought of as the points of a regular lattice

in R

, say Z

. Although d = 3 is the physical

situation, one can also be interested in the case

d = 1, 2. In this situation,   Z

with cardinality

N, and each atom is identified by its position

= i þ q

, where i 2  and q

2 R

is the displace-

ment of the particle at lattice site i from this

equilibrium position. Since interatomic forces in

real solids have short range, it is reasonable to

assume that the atoms interact only with their

nearest neighbors via a potential that depends only

on the relative distance with respect to the equili-

brium distance. Accordingly, the Hamiltonians that

we consider have the general form

HðP; QÞ¼

i2

jijj¼1

Vðq

 q

Þþ

ðq

i2

þVðQÞ½10

where P = (p

)

i2

and analogously for Q. We shall

further assume that as jqj!1 so do U

(q) and

V(q). The addition of U

(q) pins down the crystal

and ensures that exp [H(P, Q)] is integrable with

respect to dPdQ, and thus the corresponding Gibbs

measure is well defined. In this case, in order to fix the

temperature at the boundary, one can add a Langevin

term to the equation of particles on the boundaries,

that is, if i 2 @



the equation for the particle is

¼@

HðP; QÞ p

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

T



½11

where

is a standard white noise. Other thermo-

statting mechanisms can be considered. In this case

we can also define LTE using eqn [9] but we run

into the same difficulties described above – although

the problem is somehow simpler due to the presence

of the lattice structure and the fact that the particles

oscillate close to their equilibrium points. We can

obtain Fourie r’s law only by adding stochastic

terms, for example, terms like eqn [11], to the

equation of motion of every particle and assuming

that U(q) and V(q) are harmonic. These added

noises can be thought of as an effective description

of the chaotic motion generated by the anharmonic

terms in U(q) and V(q).

Just how far we are from establishing rigorously

the Fourier law is c lear from our very limited

mathematical understanding of the stationary

nonequilibrium state (SNS) of mechanical systems

whose ends are, as in the e xample of the Benard

problem, kept at fixed temperatures T

and T

Various models have been considered, for exam-

ple, models wi th Hamiltonian [10] coupled at the

boundaries with heat reservoirs described by eqns

[11]. The best mathematical results one can prove

are: the existence and uniqueness of SNS; the

existence of a stationary nontrivial heat flow;

properties of the fluctuations of the heat flow in

the SNS; t he central-limit theorem type fluctua-

tions (related to Kubo formula and Onsager

relations; and large-deviation type fluctuations

related to the Gallavotti–Cohen fluctuation theo-

rem). What is missing is information on how the

relevant quantities depend on the size of the

system, N. In this context, the heat conductivity

can be defi ned precisely without invoking LTE. To

do this, we let

J be the expectation value in the SNS

of the energy or heat current flowing from reservoir

1 to reservoir 2. We then define the conductivity



J=(AT=L), where T=L = (T

 T

)=L is the

effective temperature gradient for a cylinder of

microscopic length L and uniform cross section A,

and (T) is the limit of 

when

T !0(T

= T

= T)andL !1. The existence of

such a limit w ith  positive and finite is what one

would like to prove.

See also: Dynamical Systems and Thermodynamics;

Ergodic Theory; Interacting Particle Systems and

Hydrodynamic Equations; Kinetic Equations;

Nonequilibrium Statistical Mechanics: Dynamical

Systems Approach; Nonequilibrium Statistical

Mechanics: Interaction Between Theory and Numerical

Simulations.

Further Reading

Ashcroft NW and Mermin ND (1988) Solid State Physics.

Philadelphia: Saunders College.

Berman R (1976) Thermal Conduction in Solids. Oxford: Clarendon.

Bonetto F, Lebowitz JL, and Rey-Bellet L (2000) Fourie r’s

law: a challenge to theorists. In: Fokas A, Grigoryan A,

Kibble T, and Zegarlinsky B (eds.) Mathematical Physics 2000,

pp. 128–150. London: Imperial College Press.

Brush S (1994) The Kind of Motion We Call Heat. Amsterdam:

Elsevier.

Debye P (1914) Vortra¨ge u¨ ber die Kinetische Theorie der Wa¨ rme.

Leipzig: Teubner.

Fourier JBJ (1822) The´orie Analytique de la Chaleur, Paris:

Firmin Didot. (English translation in 1878 by Alexander

Freeman. New York: Dover, Reprinted in 2003).

Kipnis C and Landim C (1999) Scaling Limits of Interacting

Particle Systems. Berlin: Springer.

378 Fourier Law

Lepri S, Livi R, and Politi A (2003) Thermal conduction

in classical low-dimensional latt ices. Physics Reports

377(1): 1–80.

Peierls RE (1955) Quantum Theory of Solids. Oxford: Clarendon.

Peierls RE (1960) Quantum theory of solids. In: Saint-Aubin Y

and Vinet L (eds.) Theoretical Physics in the Twentieth

Century, pp. 140–160. New York: Interscience.

Rey-Bellet L (2003) Statistical mechanics of anharmonic lattices.

In: Advances in Differential Equations and Mathematical

Physics, (Birmingham, AL, 2002), Contemporary Mathe-

matics,, vol. 327, pp. 283–298. Providence, RI: American

Mathematical Society.

Spohn H (1991) Large Scale Dynamics of Interacting Particle.

Berlin: Springer.

Fourier–Mukai Transform in String Theory

B Andreas, Humboldt-Universita

t zu Berlin,

Berlin, Germany

Introduction

The Fourier–Mukai transform has been introduced

in the study of abelian vari eties by Mukai and can

be thought of as a nontrivial algebro-geometric

analog of the Fourier transform. Since its original

introduction, the Fourier–Mukai transform turned

out to be a useful tool for studying various aspects

of sheaves on varieties and their moduli spaces, and

as a natural consequence, to learn about the

varieties themselves. Various links between geome-

try and derived categories have been uncovered; for

instance, Bondal and Orlov proved that Fano

varieties, and certain varieties of general type, can

be reconstructed from their derived categories.

Moreover, Orlov proved a derived version of the

Torelli theorem for K3 surfaces and also a structure

theorem for derived categories of abelian varieties.

Later, Kawamata gave evidence to the conjecture

that two birational smooth projective varieties with

trivial canonical sheaves have equivalent derived

categories, which has been proved by Bridgeland in

dimension 3.

The Fourier–Mukai transform also enters into

string theory. The most prominent example is

Kontsevich’s homological mirror-symmetry conjec-

ture. The conjecture predicts (for mirror dual pairs

of Calabi–Yau manifolds) an equivalence between

the bounded derived category of coherent sheaves

and the Fukaya category. The conjecture implies a

correspondence between certain self-equivalences

(given by Fourier–Mukai transforms) of the deri ved

category and symplectic self-equivalences of the

mirror manifold.

Besides thei r importance for geometrical aspects

of mirror symmetry, the Fourier–Mukai transforms

have also been important for heterotic string

compactifications. The motivation for this came

from the conjectured correspondence between the

heterotic string and F-theory, which both rely on

elliptically fibered Calabi–Yau manifolds. To give

evidence for this correspondence, an explicit descrip-

tion of stable holomorphic vector bundles was

necessary an d inspired a series of publications by

Friedman, Morgan, and Witten. Their bundle con-

struction relies on two geometrical objects: a

hypersurface in the Calabi–Yau manifold together

with a line bundle on it; more precisely, they

construct vector bundles using a relative Fourier–

Mukai transform.

Various aspects and refinements of this construc-

tion have been studied by now. For instance, a

physical way to understand the bundle construction

can be given using the fact that holomorphic vect or

bundles can be viewed as D-branes and that

D-branes can be mapped under T-duality to new

D-branes (of different dimensi ons).

We survey aspects of the Four ier–Mukai trans-

form, its relative version and outline the bundle

construction of Friedman, Morgan, and Witten. The

construction has led to many new insights, for

instance, the presence of 5-branes in heterotic string

vacua has been understood. The construction also

inspired a tremendous amount of work towards a

heterotic string phenomenology on elliptic Calabi–

Yau manifolds. For the many topics omitted the

reader should consult the ‘‘Further reading’’ section.

The Fourier–Mukai Transforms

Every object E of the derived category on the

product X  Y of two smooth algebraic varieties X

and Y gives rise to a functor 

from the bounded

derived category D(X) of coherent sheaves on X to

the similar category on Y:



: DðXÞ!DðYÞ

F 7!

ðFÞ¼R^



ð



F  EÞ

where ,

 are the projections from X  Y to X

and Y, respectively, and  denotes the derived

tensor product. 

(F) is called Fourier–Mukai

transform with kernel E 2 D(X  Y)(inanalogy

Fourier–Mukai Transform in String Theory 379

with the definition of an inte gral transform with

kernel). Note that given a Fourier–Mukai functor



, 

(F) is in general a complex having homol-

ogy in several degrees even if F is a s heaf.

Furthermore, a result by Orlov states that if X

and Y are smooth projective varieties then any

fully faithful functor D(X) !D(Y) is a Fourier–

Mukai functor.

In analogy with the Fourier transform, there is a

kind of ‘‘convolution product’’ giving the composi-

tion of two such functors. More precisely, given

smooth algebraic varieties X, Y, Z, and elements E 2

D(X  Y)andG 2 D(Y  Z), we can define G  E 2

D(X  Z)by

G  E ¼ R



ð



E  



GÞ

where 

, 

are the projections from X 

Y  Z to the pairwise products giving a natural

isomorphism of functors



 

¼ 

GE

Another analogy with the Fourier transform can

be drawn. For this, assume that we have sheaves F

and G which only have one nonvanishing Fourier–

Mukai transform, the ith one 

(F)(where

D(X) !Coh(Y), F 7!H

(

(F)); cf. remarks below)

in the case of F ,andthejth one 

(G)inthecase

of G. Given such sheaves, there is the Parseval

formula

Ext

ðF; GÞ¼Ext

hþij

ð

ðFÞ; 

ðGÞÞ

which gives a correspondence between the exten-

sions of F, G and the extensions of their Fourier–

Mukai transforms. This formula can be considered

as the analog of the Parseval formula for the

ordinary Fourier transform for functions on a torus.

The Parseval formula can be proved using two

facts. First, for arbitrary coh erent sheaves E, G the

Ext groups can be computed in terms of the derived

category, namely

Ext

ðE; GÞ¼Hom

DðXÞ

ðE; G½ iÞ

Second, the Fourier–Mukai transform s of F and G in

the derived category D( X) are given by (F) =



(F)[i] and (G) =

(G)[j]. Since the Fourier–

Mukai transform is an equivalence of categories, we

have

Hom

DðXÞ

ðF; G½iÞ ¼ Hom

DðXÞ

ð

ðFÞ; 

ðGÞ½i j þ hÞ

implying the Parseval formula.

A first simple example of a Fourier–Mukai functor

can be given: let F be the complex in D(X  X)

defined by the structure sheaf O



of the diagonal

  X  X.Thenitiseasytocheckthat

D(X) !D(X) is isomorphic to the identity functor

on D(X). Moreover, if we shift degrees by n taking

F = O



[n](acomplexwithonlythesheafO



placed

in degree n), then 

: D(X) !D(X) is the degree

shifting functor G7!G[n].

As we will be interested in relative Fourier–Mukai

transforms for elliptic fibrations, let us consider the

case of a Fourier– Mukai transform on an elli ptic

curve: consider an elliptic curve E with a fixed

origin p

and identify E with

E = Pic

(E) via

f : E !

E, x 7!O

(x  p

). As kernel we take the

normalized Poincare´ line bundle P : = O

EE

( 

}  E  E  {p

}). The restriction of P to p

 E

or E  p

is isomorph ic to the trivial line bundle

O. P has the universal property which can be

expressed by 

(k(x)) = f (x), where k(x) is the

sheaf supported at a point x 2 E ; in particular,



(k(p

)) = O

and 

) = k(p

)[1], where O

is the structure sheaf of E.

Relative Fourier–Mukai Transforms

for Elliptic Fibrations

It is often convenient to study problems for families

rather than for single varieties. The main advantage

of the relative setting is that base-change properties

(or parameter dependencies) are better encoded into

the problem. We can do that for Fourier–Mukai

functors as well. To this end, we consider two

morphisms p : X !B,

p :

X !B of algebraic vari-

eties. We will assum e that the morphisms are flat

and so give nice families of algebraic varieties. We

shall define relative Fourier–Mukai functors in this

setting by means of a ‘‘kernel’’ E in the derived

category D(X 

X).

Let us make the relative setting explicit for elliptic

fibrations: an elliptic fibration is a proper flat

morphism p : X !B of schemes whose fibers are

Gorenstein curves of arithmetic genus 1. We also

assume that p has a section  : B ,!X taking values

in the smooth locus X

!B of p. The generic fibres

are then smooth elliptic curves, whereas some

singular fibers are allowed. If the base B is a smooth

curve, elliptic fibrations were studied and classified

by Kodaira, who described all the types of singular

fibers that may occur, the so-called Kodaira curves.

When the base is a smooth surface, more compli-

cated configuration of singular curves can occur and

have indeed been studied by Miranda.

First let us fix notation and setup. We denote by

 = (B) the image of the section, by X

the fiber of

p over t 2 B (we assume, in what follows, B is either

a smooth curve or surface) and by i

: X

,!X the

inclusion. Furthermore, !

X=B

is the relative dualizi ng

380 Fourier–Mukai Transform in String Theory

sheaf and ! = R







X=B

)



, where the iso-

morphism is Grothendieck–Serre duality for p. The

sheaf L= p



X=B

is a line bundle whose first Chern

class we denote by K = c

(L). The adjunction

formula for ,!X gives that 

=   p



K as cycles

on X. Moreover, we will consider elliptic fibrations

with a section whose fibers are all geome trically

integral. This means that the fibration is isomorphic

with its Weierstrass model.

From Kodaira’s classification of possible singular

fibers one finds that the components of reducib le

fibers of p which do not meet  form rational double

point configurations disjoint from . Let X !



X be

the result of contracting these configurations and let



p :



X !B be the induced map. Then all fibers of



are irreducible with at worst nodes or cusps as

singularities. In this case, one refers to



X as the

Weierstrass model of X.

The Weierstrass model can be constructed as

follows: the divisor 3 is relatively ample and, if

E= p



(3) !



!

2

!

3

and



p : P = P(E



) !B

is the associated projective bundle, there is a

projective morphism j : X !P such that j(X) =



Now special fibers of X !B can have at most

one singular point, either a cusp or a simple node.

Thus, in this case 3 is relatively very ample

and gives rise to a closed immersion j : X ,!P

such that j



(1) = O

(3), where j is locally a

complete intersection whose normal sheaf is

N(X=P) !







6

O

(9). This follow s by rela-

tive dua lity since !

P=B



P=B







5

(3), due

to the Euler exact sequence

0 ! 

P=B

! 



Eð1Þ!O

! 0

The morphism p : X !B is then a local compl ete

intersection morphism (cf. Fulton (1984)) and has a

virtual relative tangent bundle T

X=B

= [j



P=B

] 

X=P

] in the K-group K



(X). The Todd class of

X=B

is given by

TdðT

X=B

Þ¼1 

1

K þ

ð12  p

1

K þ 13p

1



  p

1

þ terms of higher degree

Now if

p :

X !B denotes the dual elliptic fibration,

defined as the relative moduli space of torsion-free

rank-1 sheaves of relative degree 0, it is known that

for t 2 B there is an isomorphism

ﬃ X

between

the fibers of both fibr ations. Since we assume that

the original fibration p : X !B has a section , then

p and

p are globally isomorphic; hereafter we

identify X ﬃ

X, where

X denotes the compacti fied

relative Jacobian of X.

Note that

X is the scheme representing the

functor which, to any scheme morphism  : S !B,

associates the space of equivalence classes of S-flat

sheaves on p

: X 

S !S, whose restrictions to the

fibers of  are torsion-free (the usual definition of

‘‘torsion free’’ is only for integral varieties, i.e.,

varieties whose local rings have no zero-divisors. In

this case, a sheaf M is torsion free if for any open

subset U, any nonzero section m of M on U and

any nonzero section a of the relevant functions

sheaf, one has a  m 6¼ 0. When the variety is not

integral (it is reducible, or nonreduced) this defini-

tion has no real meaning, then what substitutes the

notion of ‘‘torsion free’’ is the Simpson definition

of ‘‘pure of maximal dimension’’: a sheaf M is

‘‘torsion free’’ in this sense if the support of any of

its subsheaves is the whole variety (cf. Huybrechts

and Lehn (1997)), of rank 1 and degree 0; two such

sheaves F, F

are considered to be equivalent if

ﬃFp



L for a line bundle L on S (cf. Altman

and Kleiman (1980); note the Altman–Kleiman

compactification of the relative Jacobian applies to

our situation since we consider elliptic fibrations

with integral fibers). Moreover, the natural morph-

ism X !

X, x 7!I

O

((t)) is an isomorphism

(of B-schemes); here I

is the ideal sheaf of the

point x in X

Note al so that if : Y !X

is the normalization

of one of our fibers X

and z is the exceptional

divisor (the pre-image of the singular point x)then





(z)) is the maximal ideal of x.

The variety

X is a fine moduli space. This means

that there exists a coherent sheaf P on X 

X flat

over

X, whose restrictions to the fibers of

p are

torsion free, and of rank 1 and degree 0. The sheaf

P is defined, up to tensor product, by the pullback

of a line bundle on

X, and is called the universal

Poincare´ sheaf , which we will normalize by letting

j

’O

. We shall henceforth assum e that P is

normalized in this way, so that

P¼I



 



ðÞ^



ðÞq



1

where ,

 and q = p   =

p 

 refer to the diagram

X ×

and I



is the ideal sheaf of the diagonal immersion

X ,!X 

Starting with the diagram and with the kernel

given by the normalized relative universal Poincare´

sheaf P on the fibered product X 

X, we define

the relative Fourier–Mukai trans form as

 ¼ 

: DðXÞ!DðXÞ

F 7!ðFÞ¼R^



ð



F PÞ

Fourier–Mukai Transform in String Theory 381

Note that (F) can be generalized if we allow

changes in the base space B, that is, we consider

base-change morphisms g : S !B.

We close this section with some remarks:



An important feature of Fourier–Mukai functors

is that they are exact as functors of triangulated

categories. In more familiar terms, we can say

that for any exact sequence 0 !N !F!G!0

of coherent sheaves in X, we obtain an exact

sequence

!

i1

ðGÞ ! 

ðNÞ ! 

ðFÞ ! 

ðGÞ !



iþ1

ðNÞ!

where we have written =

and 

(F) =

((F)) denotes the ith cohomology sheave s of

the complexes (F).

Given a Fourier–Mukai functor 

, a complex

F in D(X) satisfies the WIT

condition (or is WIT

)

if there is a coherent sheaf G on

X such that



(F) ’G[i]inD(

X), where G [i] is the associated

complex concentrated in degree i. Furthermore,

we say that F satisfies the IT

condition if, in

addition, G is locally free.

When the kernel E is simply a sheaf Q on X 

X flat over

X, the cohomology and base-change

theorem (cf. Hartshorne (1977)) allows one to

show that a coherent sheaf F on X is IT

if and

only if H

(X, FQ



) = 0 for all  2

X and for all

j 6¼ i, where Q



denotes the restriction of Q to

X  {} and F is WIT

if and only if it is IT

The acronym ‘‘IT’’ stands for ‘‘index theorem,’’

while ‘‘W’’ stands for ‘‘weak.’’ This terminology

comes from Nahm transforms for connections on

tori in complex differential geometry.



The Parseval form ula for the relative Fourier–

Mukai transform has been proved by Mukai in

his original Fourier–Mukai transform for abelian

varieties and can be extended to any situation

in which a Fourier–Mukai transform is fully

faithful.



For physical applications, it is often convenient to

work in cohomology H



(X, Q). The passage from

D(X)toH



(X, Q) can be described as follows. We

first send a complex Z 2 D(X) to its natural class

in the K-group; we then make use of the fact that

the Chern character ch maps K(X) !CH



(X)  Q

and finally we apply the cycle map to H



(X, Q).

This passage (by abuse of notation) is often denoted

by ch : D(X) !H

even

(X, Q), it commutes with

pullbacks and transforms tensor products into

dot products. Moreover, if we substitute the

Mukai vector v(Z) = ch(Z)

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Td(X)

for the Chern

character ch(Z)thenwefindthecommutative

diagram

D(X )

D(Y

)

H *(X, Q) H *(Y, Q)

υ(E )

υυ

This can be shown using the Grothendieck–

Riemann–Roch theorem and the fact that the

power series defining the Todd class starts with

constant term 1 and thus is invertible.

Vector Bundles for Heterotic Strings

A compactification of the ten-dimensional heterotic

string is given by a holomorphic, stable G-bundle V

(with G some Lie group specified below) over a

Calabi–Yau manifold X. The Calabi–Yau condition,

the holomorphy and stability of V are a direct

consequence of the required supersymmetry in the

uncompactified spacetime. We assume that the

underlying ten-dimensional space M

is decom-

posed as M

= M

 X, where M

(the uncompac-

tified spacetime) denotes the four-dimensional

Minkowski space and X a six-dimensional compact

space given by a Calabi–Yau 3-fold. To be more

precise: supersymmetry requires that the connection

A on V satisfies

2;0

¼ F

0;2

¼ 0; F

1;1

^ J

¼ 0

where J denotes a Ka¨hler form of X. It follows that

the connection has to be a holomorphic connection

on a holomorphic vector bundle and, in addition,

satisfies the Donaldson–Uhlenbeck–Yau equation,

which has a unique solution if and only if the vector

bundle is polystable.

In addition to X and V, we have to specify a

B-field on X of field strength H. In order to get an

anomaly-free theory, the Lie group G is fixed to be

either E

 E

or Spin(32)=Z

or one of their

subgroups and H must satisfy the identity

dH ¼ tr R ^ R  Tr F ^ F

where R and F are, respectively, the associated

curvature forms of the spin connection on X and the

gauge connection on V. Also tr refers to the trace of

the composite endomorphism of the tangent bundle

to X and Tr denotes the trace in the adjoint

representation of G. For any closed four-dimen-

sional submanifold X

of the ten-dimensional space-

time M

, the 4-form tr R ^ R  Tr F ^ F must have

trivial cohomology. Thus, a necessary topological

condition V has to satisfy is ch

(TX) = ch

(V),

which simplifies to c

(TX) = c

(V) for Calabi–Yau

manifolds, V being an SU(n) vector bundle.

382 Fourier–Mukai Transform in String Theory

A physical interpretation of the third Chern class

can be given as a result of the decomposition of the

ten-dimensional spacetime into a four-dimensional

flat Minkowski space and X. The decomposition of

the corresponding ten-dimensional Dirac operator

with values in V shows that massless four-

dimensional fermions are in one-to-one correspon-

dence with zero modes of the Dirac operator D

X. The index of D

can be effectively computed

using the Hirzebruch–Riemann–Roch theorem and

is given by

indexðDÞ¼

TdðXÞchðVÞ¼

ðVÞ

equivalently, we can write the index as index(D) =

i = 0

(1)

dim H

(X, V). For stable vector bundles,

we have H

(X, V) = H

(X, V) = 0 and so the index

computes the net number of fermion generations N

gen

in the respective model.

Now it has been observed that the inclusion of

background 5-branes changes the anomaly con-

straint. Various 5-brane solutions of the heterotic

string equations of motion have been discussed in

the gauge 5-brane, the symmetric 5-brane, and the

neutral 5-brane. It has been shown that the gauge

and symmetric 5-brane solutions involve finite-size

instantons of an unbroken nonabelian gauge group.

In contrast, the neutral 5-branes can be interpreted

as zero-size instantons of the SO(32) heterotic

string. The magnetic 5-brane contributes a source

term to the Bianchi identity for the 3-for m H,

dH ¼ tr R ^ R  Tr F ^ F þ n

five-branes



ð4Þ

and integration over a 4-cycle in X gives the anomaly

constraint

ðTXÞ¼c

ðVÞþ½W

The new term 

(4)

is a current that integrates to 1 in

the directio n transverse to a single 5-brane whose

class is denoted by [W]. The class [W] is the

Poincare´ dual of an integer sum of all these sources

and thus [W] should be an integral class, represent-

ing a class in H

(X, Z). [W] can be further specified

taking by into account that supersymmetry requires

that 5-branes are wrapped on holomorphic curves

and thus [W] must correspond to the homology class

of holomorphic curves. This fact constrains [W]to

be an algebraic class. Further, algebraic classes

include negative classes; however, these lead to

negative magnetic charges, which are unphysical,

and so they have to be excluded. This constrains [W]

to be an effective class. Thus, for a given Calabi–

Yau 3-fold X the effectivity of [W ] constrains the

choice of vector bundles V.

The study of the corre spondence between the

heterotic string (on an elliptic Calabi–Yau 3-fold)

and F-theory (on an elliptic Calabi–Yau fourfold)

has led Friedman, Morgan, and Witten to introduce

a new class of vector bundles which satisfy the

anomaly constraint with [W] nonzero. As a result,

they prove that the number obtained by integration

of [W] over the elliptic fibers of the Calabi–Yau

3–fold agrees with the number of 3-branes given by

the Euler characteristic of the Calabi–Yau fourfold

divided by 24.

Fourier–Mukai Transforms and Spectral Covers

Let us now describe how the construction of vector

bundles out of spectral data (first consi dered in

Hitchin and Beauville, Narasimhan, and Ramanan)

can be easily described in the case of elliptic

fibrations by means of the relative Fourier–Mukai

transform. This construction was wid ely exploited

by Friedman, Morgan, and Witten to construct

stable vector bundles on elliptic Calabi–Yau three-

folds X, which we will summarize now.

If V !X is a vector bundle of rank n which is

semistable and of degree 0 on each fibre f of X !B,

then its Fourier–Mukai transform 

(V) is a torsion

sheaf of pure dimension 2 on X. The support of



(V) is a surface i : C ,!X, which is finite of degree

n over B. Moreover, 

(V) is of rank 1 on C and, if

C is smooth, then 

(V) = i



L is just the extension

by zero of some line bundle L 2 Pic(C). Conversely,

given a sheaf G!X of pure dimension 2 which is

flat over B, then (G) is a vector bundle on X of

rank equal to the degree of supp(G)overB.

This correspondence between vector bundles on X

and sheaves on X supported on finite covers of B is

known as the spectral cover construction. The

torsion sheaf G is called the spectral sheaf (or line

bundle) and the surface C = supp(G) is called the

spectral cover.

For the description of vector bundles on elliptic

Calabi–Yau 3-folds X it is appropriate to take i



with Chern characters given by (

,  2 H

(B, Q)

and a

, s

2 Z)

ði



LÞ¼0; ch

ði



LÞ¼n þ 





ði



LÞ¼





þ a

f; ch

ði



LÞ¼s

The characteristic classes of the rank-n vector bundle

V can be obtained if we apply the Grothendieck–

Riemann–Roch theorem to the projection :

chðVÞ¼



½^



ðchði



LÞÞchðPÞTdðT

X=B

Þ

where Td(T

X=B

) as given above.

Fourier–Mukai Transform in String Theory 383

To make sure that the construction leads to SU(n)

vector bundles we set 

= (1=2)nc

giving c

(V) = 0

and the remaining Chern classes are given by

ðVÞ¼



ð Þ þ 



ð$Þ; c

ðVÞ¼2

where

$ ¼

ðBÞ

ðn

 nÞþ







nð   nc

ðBÞÞ

and  2 H

1, 1

(C, Z) is some cohomology class

satisfying 

C

 = 0 2 H

1, 1

(B, Z). The general solu-

tion for  has been derived by Friedman, Morgan,

and Witten and is given by  = (n

 



 þ

n



(B)) and 

= 



(



  n



(B)) with

S = C \ . The parameter  has to be determined

such that c

(L) is an integer class. If n is even,

 = m(m 2 Z) and in addition we must impose

 = c

(B) modulo 2. If n is odd,  = m þ 1=2.

It remains to discuss the stability of V. The

stability depends on the properties of the defining

data C and L.IfC is irreducible and L a line bundle

over C then V will be a vector bundle stable with

respect to the polarization

J ¼ J

þ 



;>0

if  is sufficiently small. This has been proved by

Friedman, Morgan, and Witten under the additional

assumption that the restriction of V to the generic

fiber is regular and semistable. Here J

refers to

some arbitrary Ka¨ hler class on X and H

aKa¨ hler

class on the base B. It implies that the bundle V can

be taken to be stable with respect to J while keeping

the volume of the fiber f of X arbitrarily small

compared to the volumes of effective curves asso-

ciated with the base. That J is actually a good

polarization can be seen by assuming  = 0. Now we

observe that 



is not a Ka¨ hler class on X since

its integral is non-negative on each effective curve C

in X; however, there is one curve, the fiber f, where

the integral vanishes. This means that 



is on the

boundary of the Ka¨ hler cone and, to make V stabl e,

we have to move slightly into the interior of the

Ka¨ hler cone, that is, into the chamber which is

closest to the boundary point 



. Also we note

that although 



is in the boundary of the Ka¨ hler

cone, we can still define the slope 





(V) with

respect to it. Since (



)

is some positive multiple

of the class of the fiber f, semistability with respect

to 



is implied by the semistability of the

restrictions Vj

to the fibers. Assume that V is not

stable with respect to J, then there is a destabilizing

sub-bundle V

 V with 

)  

(V). But semi-

stability along the fibers says that 





) 







(V). If we had equality, it would follow that

arises by the spectral construction from a proper

subvariety of the spectral cover of V, contradicting

the assumption that this cover is irreducible. So we

must have a strict inequality 





) <





(V).

Now taking  small enough, we can ensure that



) <

(V), thus V

cannot destabilize V.

D-Branes and Homological Mirror

Symmetry

Kontsevich proposed a homological mirror symme-

try for a pair (X, Y) of mirror dual Calabi–Yau

manifolds; it is conjectured that there exists a

categorical equivalence between the bounded

derived category D(X) and Fukaya’s A