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

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binomial relationship to m, allowing one to ‘‘read’’

this number directly. The combination of chemical

shift and scalar coupling informat ion is of profound

importance in identifying molecular structure in

chemistry.

The terms

and I

are, respectively,

the through-space dipolar interaction, H

, and the

nuclear quadrupole interaction, H

, the latter being

nonzero only for nuclear spin quantum numbers I 

1=2, for exampl e,

H. These interactions, projected

into the zeroth-order Zeeman frame, for the dipole–

dipole interaction, are



h

4

j>i



1  3cos





 3I

 I



½13

where r

is the internuclear distance and 

is the

angle made by the internuclear vector with the

magnetic field direction; while, for the quadrupole

interaction

3eV

4Ið2I  1Þh

1  3cos

ðÞ

 3I

 IðI þ 1Þ



½14

where Q is the nuclear quadrupole moment, V

the electric field gradient (assuming axial symmetry)

and 

is the angle made by the principal axis of

that gradient with the magnetic field direction. For

protons in organic matter, the internuclear dipole

interaction strength is on the order of 100 kHz, a

similar strength being found for the quadrupole

interaction of deuterons. However, in the liquid

state, these orientation-dependent interactions fluc-

tuate so rapidly that they are typically motionally

averaged to zero. Nonetheless, their fluctuations do

contribute to the relaxation process.

Liquid-state NMR can result in exceptionally

high-resolution (sub-Hz) spectra, if care is taken to

adjust the magnetic field harmonics (shims) to

produce a highly uniform Zeeman field across the

sample. The last contribution of residual inhomo-

geneities to line broadening can often be removed by

gently spinning the sample about its axis at a rate of

a few tens of hertz.

The Evolution Domain, Multiple RF

Pulses, and Multidimensional NMR

Having seen the complexity of the spin Hamilto-

nian, one may envisage experiments where the spin

coherences evolve in a much more complicated

manner. To this end, consider the case of a

molecular liquid two-spin (AX) system coupled via

the scalar spin–spin interaction. In first-order per-

turbation theory , we may represent the simp le two-

spin Hamiltonian (in the rotating frame of the

averaged Larmor frequency) as

rot

¼

B

 

B

þ JI

¼!

 !

þ JI

½15

We now write down t he density matrix i n

the rotating frame following a single 90



pulse (I

ðtÞ¼expði!

þi!

þiJI

tÞ

exp i





eqbm

ðI

þI

Þexp i





expði!

i!

iJI

tÞ

¼expði!

þi!

þiJI

tÞa

eqbm

ðI

þI

expði!

i!

iJI

tÞ

¼expði!

þi!

Þa

eqbm

 I

þI



cos



þ2 I

þI

ðÞ



sin



expði!

i!

¼a

eqbm



cos!

t þI

cos!

þI

sin!

t þI

sin!



cos



þ2 I

cos!

t I

sin!



þI

cos!

t I

sin!

tÞ

sin



½16

Detection in the rotating frame with I

þiI

gives a

signal

SðtÞa

eqbm

ðexpði!

tÞþexpði!

t ÞÞcos



½17

Fourier transformation with respect to t yields a

spectrum corresponding to two spectral lines at !

and !

, each split into a doublet of two sidebands

separated by J.

Notice that it is easier to follow the evolution of

the density matrix by simply writing down a time

sequence of behaviors under the influence of the

successive Hamiltonians. Where simultaneous terms

in the Hamiltonians commute, the order of their

operation may be set at will. Thus, the above

example becomes

þI

!



þI

!

þI



cos



þ2 I

þI

ðÞsin



!

þ!

cos!

t þI

cos!



þiI

sin!

t þiI

sin!

tÞcos



þ2 I

cos!

t iI

sin!



þI

cos!

t iI

sin!



sin



½18

Nuclear Magnetic Resonance 597

Now consider a two RF pulse scheme as shown in

Figure 4, each RF pulse being 90



. The evolution is

þ I

!



þ I

þ!

JI

cos !

þ I

cos !



þ I

sin !

þI

sin !

Þcos



þ 2 I

cos !

I

sin !

þ I

cos !

I

sin !



sin



!



I

cos !

 I

cos !

þ I

sin !

þI

sin !

Þcos



þ 2 I

cos !

þ I

sin !



þI

cos !

þ I

sin !



sin



Keeping only observable magnetization

þ!

þJI

ðI

sin !

cos !

þ I

sin !

cos !

 cos



cos



þðI

sin !

þ I

sin !

Þsin



sin



½19

If the idealized experiment is performed with two

independent time dimensions t

and t

, then detec-

tion in the rotating frame over the t

period with

þ iI

gives a signal (restricting our attention to the

quadrant of positive frequencies)

Sðt

; t

Þa

eqbm

ðexpði!

Þexpði!

Þþexpði!

expði!

ÞÞcos



cos



þa

eqbm

ðexpði!

Þexpði!

þexpði!

Þexpði!

ÞÞ

sin



sin



½20

When Fourier transformed in two dimensions with

respect to t

and t

,thepatternshowninFigure 5

results. Remarkably, while the diagonal spectrum

is the same pair of doublets seen in the figure,

this two-dimensional spectrum contains off-diagonal

antiphase peaks for scalar-coupled sites where magnet-

ization transfer has occurred.

The idea of performing NMR in two or more

dimensions was first proposed by Jean Jeener in

1971. The example outlined above, correlation

spectroscopy (COSY), is just one of an array of

coherence transfer experiments using multiple RF

pulse trains and time domain evolution of the spin

ensemble. Notice that in the COSY experiment, t

an evolution dimension during which no detection of

NMR signal occurs, while t

is the detection domain.

Indirect (scalar)

spin–spin coupling

via electron

Local chemical shift

diamagnetic shielding

ppm

3.7

3.6 3.0

ppm 6.5 6.4 6.3

ppm 6 5 4

ppm 1.4 1.2 1.0

321

J coupling

Figure 3 The proton NMR spectrum of ethanol showing three major peaks, separated by chemical shift, each split into multiplets

arising from nearby protons via the scalar coupling.

598 Nuclear Magnetic Resonance

The effect of the evolution is indelibly imprinted in

the spin system density matrix allowing later recall of

vital information concerning the interactions present

in the spin Hamiltonian. The COSY experiment

allows one to determine which spins are coupled via

their molecular orbital electrons. Other multidimen-

sional methods that rely on dipole–dipole relaxation

effects, such as NOESY, determine which spin sites

have ‘‘through-space’’ proximity.

The use of two- and higher-dimensional methods

has allowed the NMR spectra of biological macro-

molecules to be unraveled, with COSY methods used

for spectral assignment of amino acid units, and

NOESY methods used to determine any close proxi-

mities of amino acids otherwise well separated in the

sequence. Such distance information has allowed the

reconstruction of protein conformations by NMR.

The second RF pulse of Figure 4 also generates a

state of the density matrix, I

known as a double

quantum coherence, and, in the simple COSY

experiment, lost to observable magnetization. Other

RF pulse schemes can take advantage of this state,

converting it via suitable ‘‘coherence pathways’’ into

an observable. For a detailed summary of these

various NMR phenomena, readers are referred to the

book by Ernst et al. (1987).

Solid-State NMR

As with J couplings, dipolar interactions an d

quadrupole interactions (I > 1=2) are bilinear in

the spin operat ors and can be used to generate

various higher-order coh erence pathways in NMR

experiments. Unlike the simple spin–spin coupling,

they have an angular dependence. In solids, these

interactions may broaden the NMR resonance line

by tens to hundreds of kilohertz. In the case where a

probe nucleus is located at a known site in the

material (often achieved by deuteron labeling), these

Hamiltonian terms may contribute important infor-

mation about structure, and especially orientational

anisotropy. For example, the quadrupole interaction

for the spin-1 deuteron (see eqns [11] and [14])

depends as P

( cos 

) = (1=2)(3 cos 

 1) on the

angle between the external magnetic field and

the electric field gradient (generally associated with

the local molecular orbital or bond direction, and taken

here to be axially symmetric). Note that the first-order

contribution of the quadrupole interaction leads to an

unequal separation of the m = 1, 0, 1Zeeman

energy levels, resulting in a doublet NMR spectrum,

for any particular orientation, 

. Such a unique

orientation might be found in a single crystal, or in a

nematic liquid crystalline state. For a polycrystalline

material, however, the NMR spectrum has a con-

tribution from all orientations, leading to a character-

istic powder pattern. The details of

H spectral

distributions may be used to characterize the degree

of orientational order in solids and soft, anisotropic

matter.

For

C, and other spin-1/2 nuclei, dipolar

interactions (with a wide distribution of spin

spacings and internuclear vector orientation) may

severely broaden the NMR spectrum in the solid

state (see eqns [11] and [13]). Such interact ions,

along with quadrupole interactions for nucle i with

I >1=2, may be significantly reduced by modulating

the effective dipolar Hamiltonian at a rate faster

than its strength in frequency units. Two methods

are available, one (magic angle spinning or MAS)

relying on the angular terms in eqns [13] and [14],

and the other (multiple pulse line narrowing) on the

spin terms. The MAS techniq ue relies on spinning

the sample rapidly about at angle oriented at 54.4



to the magnetic field, such that the average value of

(cos 

) becomes its projection along this spinning

axis, while the projection of the spinning axis

residual is P

(cos 54.4



)  0. Multiple pulse meth-

ods rely on a successive reorientation of the spin

system such that the effective dipol ar Hamiltonian

that results from the application of the neste d

evolution operators is rendered close to zero.

90°

= – ω

–

rot

Figure 4 RF pulse scheme used for COSY experiment.

Figure 5 Schematic COSY (modulus) spectrum for an AX spin

system. Not that the (antiphase) off-diagonal peaks indicate

J-couplings between chemical-shift-separated spins.

Nuclear Magnetic Resonance 599

In practice, MAS techniques work best with

NMR where the moderate

H–

C dipolar interactions

may be removed with achievable spinning speeds (a

few tens of kilohertz). Furthermore, the larger proton

magnetization ( 

=

 4) can be transferred to the

C nuclei via Hartman–Hahn cross-polarization thus

significantly enhancing sensitivity. Such methodology

is referred to as CPMAS NMR.

The real art of solid-state NMR is in removing the

unwanted dipolar or quadrupolar interactions, but

leaving specific interactions of interest. This may be

achieved by including in the MAS experiment,

specific combinations of pulses which recouple

selected spins. Some of the most sophisticated

experiments in modern NMR are to be found in

this domain of application.

Conclusion

NMR provides exceptional structural information

concerning molecules, biomolecules as well as

molecular assemblies, liquid crystal s, soft solids,

and solids. In addition, the method provides unique

information concerning molecular dynamics,

through both relaxation methods and the direct

measurement of diffusion or flow. One spectacular

application of NMR concerns its use in imaging,

achieved by giving the Larmor frequency a spatial

tag through the use of deliberately inhomogeneous

magnetic fields. This topic is covered in the article

on Magnetic Resonance Imaging.

Further Reading

Abragam A (1961) Principles of Nuclear Magnetism. Oxford:

Clarendon.

Bloch F, Hansen WW, and Packard M (1946) Nuclear induction.

Physical Review 70: 474.

Ernst RR, Bodenhausen G, and Wokaun A (1987) Principles of

Nuclear Magnetic Resonance in One and Two Dimensions.

Oxford: Clarendon.

Goldman M (1991) Quantum Description of High-Resolution

NMR in Liquids. New York: Oxford University Press.

Hahn EL (1950) Spin echoes. Physical Review 80: 580.

Jeener J (1971) Ampere International Summer School. Yugoslavia:

Basko Polje.

McNeil EB, Slichter CP, and Gutowsky HS (1951) Slow beats in

19F nuclear spin echoes. Physical Review 84: 1245.

Mehring M (1983) Principles of High Resolution NMR in Solids.

Berlin: Springer.

Purcell EM, Torrey HC, and Pound RV (1946) Resonance

absorption by nuclear magnetic moments in a solid. In:

Physical Review 69: 37.

Rabi II and Cohen VW (1933) Measurement of nuclear spin by

the method of molecular beams: the nuclear spin of sodium.

Physical Review 46: 707.

Schmidt-Rohr K and Spiess H-W (1994) Multi-Dimensional Solid

State NMR and Polymers. London: Academic Press.

Slichter CP (1963) Principles of Magnetic Resonance. New York:

Harper and Row.

Number Theory in Physics

M Marcolli, Max-Planck-Institut fu

r Mathematik, Bonn,

Germany

Several fields of mathematics have closely been

associated to physics: this has always been the case

for the theory of differe ntial equations. In the early

twentieth century, with the advent of general

relativity and quantum mechanics, topics such as

differential and Riem annian geometry, operator

algebras, and functional analysis, or group theory

also developed a close relation to physics. In the

1990s, mostly through the influence of string theory,

algebraic geometry also began to play a major role

in this interaction. Recent years have seen an

increasing number of results suggesting that number

theory also is beginning to play an essential part on

the scene of contemporary theoretical and mathe-

matical physics. Conversely, ideas from physics,

mostly from quantum field theory and string theory,

have started to influence work in number theory.

In describing significant occurrences of number

theory in physics, we will, on the one hand, restrict

our attention to quantum physics, while, on the other,

we will assume a somewhat extensive definition of

number theory that will allow us to include arithmetic

algebraic geometry. The territory is vast and an

extensive treatment would go beyond the size limits

imposed by the encyclopedia. The choice of topics

represented here inevitably reflects the limited knowl-

edge, particular interests, and bias of the author. Very

useful references, collecting a lot of material on number

theory and physics, are the proceedings of the Les

Houches conference in 2003 (Beilinson and Manin

1986), as well as the two volumes of a previous Les

Houches conference on number theory and physics,

which took place in 1989, published by Springer in

1990 and 1992. A number theory and physics database

is presently maintained online by M R Watkins.

600 Number Theory in Physics

In the following, we have organized the material

by topics in number theory that have so far made an

appearance in physics, and for each we briefly

describe the relevant context and results. This

singles out many themes. We first discuss a class of

functions that occur in physics and their special

values that are of great number-theoretic impor-

tance. This includes the dilogarithm, the polyloga-

rithms and multiple polylogarithms, and the

multiple zeta values. We also discuss the most

important symmetry groups of number theory, the

Galois groups, and occurrences in physics of some

forms of Galois theory. We then discuss how

techniques from the arithmetic geometry of alge-

braic varieties, especially Arakelov geometry, play a

role in string theory. Finally, we discuss briefly the

theory of motives and outline its possible relation to

quantum physics. From the physics point of view, it

seems that the most promising directions in which

number-theoretic tools have come to play a crucial

role are to be found mostly in the realm of rational

conformal field theories and of noncommutative

geometry, as well as in certain aspects of string

theory.

Among the topics that are very relevant to this

theme, but that will not be touched upon in this

article, there are important sub jects such as the

theory of ‘‘arithmetic quantum chaos,’’ the use of

methods of random matrix theory applied to the

study of zeros of zeta funct ions, or mirror symmetry

and its connection to modular forms. The interested

reader can find such topics treated in other articles

of this encyclopedia and in the references mentioned

above (see Quantum Ergodicity and Mixing of

Eigenfunctions; Random Matrix Theory in Physics;

Mirror Symmetry: a Geometric Survey).

Dilogarithm, Multiple Polylogarithms,

Multiple Zeta Values

The dilogarithm is defined as

ðzÞ¼

logð1  t Þ

dt ¼

n¼1

It satisfies the functional equation Li

(z) þ Li

(1  z) =

(1)  log (z)log(1 z), where Li

(1) = (2), for (s)

the Riemann zeta function. A variant is given by

the Rogers dilogarithm L(x) = Li

(x) þ (1=2) log (x)

log (1  x). For more details, see Zagier’s paper

(Julia et al. 2005, vol. II).

The polylogarithms are similarly defined by the

series Li

(z) =

n1

. In quantum electrody-

namics, there are corrections to the value of the

gyromagnetic ratio, in powers of the fine structure

constant. The correction terms that are known

exactly involve special values of the zeta function

such as (3), (5) and values of polylogarithms such

as Li

(1=2). The series defining the polylogarithm

function Li

(z) =

n1

converges absolutely

for all s 2 C and jz j < 1 and has analytic continua-

tion to z 2 C n [1, 1). The Fermi–Dirac and

Bose–Einstein distributions are expressed in terms

of the polylogarithm function as

x

 1

dx ¼ðs þ 1ÞLi

1þs

ðe



The multiple polylogarithms are functions defined

by the expressions

;...;s

ðz

; z

; ...; z

>>n

z

n

½1

By analytic continuation, the functions

,..., s

, z

, ..., z

) are defined for all complex s

and for z

in the complement of the cut [1, 1) in the

complex plane. Multiple zeta values of weight k and

depth r are given by the expressions

ðk

; ...; k

Þ¼

>>n

n

½2

with k

2 N and k

 2. These satisfy many combi-

natorial identities and nontrivial relations over Q.

For an informative overview on the subject, see

Cartier (2002). Notice that, for the sums in [1] and

[2], a different summation convention can also be

found in the literature.

Conformal Field Theories and the Dilogarithm

There is a relation between the torsion elements in

the algebraic K-theory group K

conformally invariant quantum field theories in two

dimensions (see Nahm (2005)). There is, in fact, a

map, given by the dilogarithm, from torsion

elements in the Bloch group (closely related to the

algebraic K-theory) to the central charges and

scaling dimensions of the conformal field theories.

This correspondence arises by considering sums of

the form

m2N

QðmÞ

ðqÞ

½3

where (q)

= (q)

(q)

,(q)

= (1  q)(1  q

) 

(1  q

)andQ(m) = m

Am=2 þ bm þ h has rational

coefficients. Such sums are naturally obtained from

considerations involving the partition function of a

bosonic rational conformal field theory (CFT). In

Number Theory in Physics 601

particular, [3] can define a modular function only if all

the solutions of the equation

logðx

Þ¼logð1  x

Þ½4

determine elements of finite order in an extension

B(C) of the Bloch group, which accounts for the fact

that the logarithm is multivalued. The Rogers

dilogarithm gives a natural group homomorphism

(2i)

L :

B(C) ! C=Z, which takes values in Q=Z

on the torsion elements. These values give the

conformal dimensions of the fields in the theory.

Feynman Graphs

Multiple zeta values appear in perturbative quantum

field theory. D Kreimer (2000) developed a connec-

tion between knot theory and a class of transcen-

dental numbers, such as multiple zeta values,

obtained by quantum field-theoretic calculations as

counterterms generated by corresponding Feynman

graphs. Broadhurst and Kreimer (1997) identified

Feynman diagrams with up to nine loops whose

corresponding counterterms give multiple zeta

values up to weight 15. Recently, Kreimer showe d

some deep analogies between residues of quantum

fields and variations of mixed Hodge–Tate struc-

tures associated to polylogarithms.

Testing predictions about the standard model of

elementary particles, in the hope of detecting new

physics, requires developing effective computational

methods handling the huge number of terms involved

in any such calculation, that is, efficient algorithms for

the expansion of higher transcendental functions to a

very high order. The interesting fact is that abstract

number-theoretic objects, such as multiple zeta values

and multiple polylogarithms, appear naturally in this

context (cf., e.g., Moch et al. (2002)). The explicit

recursive algorithms are based on Hopf algebras and

produce expansions of nested finite or infinite sums

involving ratios of gamma functions and Z-sums

(Euler–Zagier sums), which naturally generalize multi-

ple polylogarithms and multiple zeta values. Such

sums typically arise in the calculation of multiscale

multiloop integrals. The algorithms are designed to

recursively reduce the Z-sums involved to simpler ones

with lower weight or depth.

Galois Theory

Given a number field K, which is an algebraic

extension of Q of some degree [K : Q] = n, there is

an associated fundamental symmetry group, given

by the absolute Galois group Gal(



K=K), where



K is

an algebraic closure of K. Even in the case of Q, the

absolute Galois group Gal(



Q=Q) is a very compli-

cated object, far from being fully understood.

One can consider an easier symmetry group,

which is the abelianization of the absolute Galois

group. This corresponds to considering the field K

the ‘‘maximal abelian extension’’ of K, which has

the property that

GalðK

=KÞ¼Galð



K=KÞ

The Kronecker–Weber theorem shows that for

K = Q the maximal abelian extension can be

identified with the cyclotomic field (generated by

all roots of unity), Q

= Q

cycl

, and the Galois

group is identified with Gal(Q

=Q) ﬃ



, where



= A



. In general, for other number fields,

one has the ‘‘class field theory isomorphism’’

 : GalðK

=KÞ!

’

where C

= A



is the group of idele classes and

the connected component of the identity in C

.In

general, however, one does not have an explicit

description of the generators of the maximal abelian

extension K

and the action of the Galois group. This

is the content of the explicit class field theory problem,

Hilbert’s 12th problem. In addition to the Kronecker–

Weber case, a complete answer is known in the case of

imaginary quadratic fields K = Q(

ﬃﬃﬃﬃﬃﬃﬃ

d

), with d > 1a

positive integer. In this case generators are obtained by

evaluating modular functions at a point  in the

upper-half plane such that K = Q( ) and the Galois

action is described explicitly through the group of

automorphisms of the modular field, through Shimura

reciprocity. For a survey of the explicit class field

theory problem and the case of imaginary quadratic

fields, see Stevenhagen (2001).

As we mentioned above, understanding the

structure of the absolute Galois group Gal(



Q=Q)is

a fundamental question in number theory. Grothen-

dieck described, in his famous proposal ‘‘Esquisse

d’un programme,’’ how to obtain an action of

Gal(



Q=Q) on an essentially combinatorial object,

the set of ‘‘dessins d’enfants.’’ These are connected

graphs (on a surface) such that the complement of

the graph is a union of open cells and the vertices

have two different markings, with the properties

that adjacent vertices have opposite markings. Such

objects arise by considering the projective line P

minus three points. Any finite cover of P

branched

only over {0, 1, 1} gives an algebraic curve defined

over



Q. The dessin is the inverse image under the

covering map of the segment [0, 1] in P

. The

absolute Galois group Gal(



Q=Q) acts on the data of

the curve and the covering map, hence on the set of

602 Number Theory in Physics

dessins. A theorem of Bielyi sho ws that, in fact, all

algebraic curves defined over



Q are obtained as

coverings of the projective line ramified only over

the points {0, 1, 1}. This has the effect of realizing

the absolute Galois group as a subgroup of outer

automorphisms of the profinite fundamental group

of the projective line minus three points. For a

general reference on the subject, see Schneps (1994).

A different type of Galois symmetry of great

arithmetic signi ficance is ‘‘motivic’’ Galois theory.

This will be discussed later in the section dedicated

to motives, where we discuss a surpri sing occurrence

in the context of pertur bative quantum field theory

and renormalization.

Quantum Statistical Mechanics and Class

Field Theory

In quantum statistical mechanics, one considers an

algebra of observables, which is a unital C



-algebra

A with a time evolution 

. States are given by linear

functionals ’ : A!C satisfying ’(1) = 1 and posi-

tivity ’(x



x)  0. Equilibrium states ’ at inverse

temperature  satisfy the Kubo–Martin–Schwinger

(KMS) condition, namely, for all x, y 2A there

exists a bounded holomorphic function F

x, y

(z)on

the strip 0 < =(z) <, which extends continuously

to the boundary, such that for all t 2 R

x;y

ðtÞ¼’ðx

ðyÞÞ

and

x;y

ðt þ i Þ¼’ð

ðyÞxÞ½5

Cases of number-theoretic interest arise when one

considers the noncommutative space of commensur-

ability classes of Q-lattices up to scaling as algebra of

observables, with a natural time evolution determined

by the covolume, as shown in the paper Quantum

Statistical Mechanics of Q-Lattices of Connes–Marcolli

(Julia et al. 2005,vol.I). A Q-lattice in R

consists of a

pair (, ) of a lattice   R

together with a

homomorp hism of abelian groups  : Q

!

Q=.TwoQ-lattices are commensurable, (

, 

) 

(

, 

), iff Q

= Q

and 

= 

mod 

þ 

The Bost–Connes system The quantum statistical

mechanical system considered by Bost and Connes

(1995) corresponds to the case of one-dimensional

Q-lattices. The partition function of the system is

the Riemann zeta function (). The system has

spontaneous symmetry breaking at  = 1, with a

single KMS state for all 0 < 1. For >1, the

extremal equilibrium states are parametrized by the

embeddings of Q

cycl

in C with a free transitive

action of the idele class group C



. At zero

temperature, the evaluation of KMS

states on

elements of a rational subalgebra intertwines the

action of



by automorphisms of (A, 

) with the

action of Gal(Q

=Q) on the values of the states.

This recovers the explicit class field theory of Q

from a physical perspective.

Noncommutative space of adele classes The alge bra

A of the Bost–Connes system is the noncommutative

algebra of functions f (r, ), for  2

Z and r 2 Q



such that r 2

Z, with the convolution product

 f

ðr;Þ¼

s2Q



:s2

ðrs

1

; sÞf

ðs;Þ½6

and the adjoint f



(r, ) = f (r

1

, r). According to the

general philosophy of Connes style noncommutative

geometry, it is the algebra of coordinates of the

noncommutative space defined by the ‘‘bad quoti-

ent’’ GL

(Q) n (A

 {1}) – a noncommutative

version of the zero- dimensional Shimura variety

Sh(GL

,{1})= GL

(Q) n(GL

) {1}). Its ‘‘dual

system’’ (in the sense of Connes’s duality of type III

and type II factors) is obtained by taking the crossed

product by the time evolution. It gives the algebra of

coordinates of the nonc ommutative space defined by

the quotient A=Q



. This is the noncommutative

space of ‘‘adele classes’’ used by Connes in his

spectral realization of the zeros of the Riemann zeta

function.

The GL

-system A generalization of the Bost–

Connes system was introduced by Connes and

Marcolli in the paper Quantum Statistical

Mechanics of Q-Lattices (Julia et al. 2005). This

corresponds to the case of two-dimensional

Q-lattices. The partition function is the product

()(  1). The system in this case has two phase

transitions, with no KMS states for   1. For >2,

the extremal KMS states are parametrized by the

invertible Q-lattices, namely, those for which  is an

isomorphism. The algebra A has an arithmetic

structure given by a rational algebra of unbounded

multipliers. This rational algebra contains modular

functions and Hecke operators. At zero temperature,

extremal KMS states can be evaluated on these

multipliers. Symmetries of (A, 

) are realized in part

by endomorphisms (as in the theory of superselec-

tion sectors) and the symmetry group acting on

low-temperature KMS states is the group of auto-

morphisms of the modular field GL

)=Q



. For a

generic set of extremal KMS

states, evaluation at

the rational algebra intertwines this action with the

action on the values of an embedding of the modular

field as a subfield of C.

Number Theory in Physics 603

The complex multiplication system In the case of an

imaginary quadratic field K = Q(), an analogous

construction is possible. A one-dimensional K-lattice is

apair(, ) of a finitely generated O-submodule  of

C,withK = K, and a homomorphism of O-modules

 : K=O!K=.TwoK-lattices are commensurable

iff K

= K

and 

= 

mod 

þ 

. Connes et al.

(Preprint 2005) constructed a quantum statistical

mechanical system describing the noncommutative

space of commensurability classes of one-dimensional

K-lattices up to scale. The partition function is the

Dedekind zeta function 

(). The system has a phase

transition at  = 1 with a unique KMS state for higher

temperatures and extremal KMS states parametrized by

the invertible K-lattices at lower temperatures. There is

a rational subalgebra induced by the rational structure

of the GL

-system (one-dimensional K-lattices are also

two-dimensional Q-lattices with compatible notions of

commensurability). The symmetries of the system are

given by the idele class group A



K, f



.Theactionis

partly realized by endomorphisms corresponding to the

possible presence of a nontrivial class group (for class

number > 1). The values of extremal KMS

states on

the rational subalgebra intertwine the action of the idele

class group with the Galois action on the values. This

fully recovers the explicit class field theory for

imaginary quadratic fields.

Conformal Field Theory and the Absolute

Galois Group

Moore and Seiberg considered data associated to any

rational conformal field theory, consisting of matrices,

obtained as monodromies of some holomorphic multi-

valued functions on the relevant moduli spaces,

satisfying polynomial equations. Under reasonable

hypotheses, the coefficients of the Moore–Seiberg

matrices are algebraic numbers. This allows for the

presence of interesting arithmetic phenomena. Through

the Chern–Simons/Wess–Zumino–Witten correspon-

dence, it is possible to construct three-dimensional

topological field theories from solutions to the Moore–

Seiberg equations.

On the arithmetic side, Grothendieck proposed in

his ‘‘Esquisse d’un programme’’ the existence of a

Teichmu¨ ller tower given by the moduli spaces M

g, n

of Riemann surfaces of arbitrary genus g and number

of marked points n, with maps defined by operations

such as cutting and pasting of surfaces and forgetting

marked points, all encoded in a family of funda-

mental groupoids. He conjectured that the whole

tower can be reconstructed from the first two levels,

providing, respectively, generators and relations. He

called this a ‘‘game of Lego–Teichmu¨ ller.’’ He also

conjectured that the absolute Galois group acts by

outer automorphisms on the profinite completion of

the tower. The basic building blocks of the tower are

provided by ‘‘pairs of pants,’’ that is, by projective

lines minus three points.

This leads to a conjectural relation between the

Moore–Seiberg equations and this Grothendieck–

Teichmu¨ ller setting (cf. Degiovanni 1994) according

to which solutions of the Moore–Seiberg equations

provide projective representations of the Teichmu¨ller

tower, and the action of the absolute Galois group

Gal(



Q=Q) corresponds to the action on the coeffi-

cients of the Moore–Seiberg matrices.

Rational conformal field theories are, in general,

one of the most promising sources of interactions

between number theory and physics, involving

interesting Galois actions, modular forms, Brauer

groups, and complex multiplication. Some funda-

mental work in this direction was done by, for

example, Borcherds and Gannon.

Arithmetic Algebraic Geometry

In this sect ion we describe occurrences in physics of

various aspects of the arithmetic geometry of

algebraic varieties.

Arithmetic Calabi–Yau

In the context of type II string theory, compactified

on Calabi–Yau 3-folds, Greg Moore considered

certain black hole solutions and a resulting dynami-

cal system given by a differential equation in the

corresponding moduli. The fixed points of these

equations determine certain ‘‘black hole attractor

varieties.’’ In the case of varieties obtained from a

product of elliptic curves or of a K3 surface and an

elliptic curve, the attractor equation singles out

an arithmetic property: the elliptic curves have

complex multiplication. The class number of the

corresponding imaginary quadratic field counts

U-duality classes of black holes with the same area.

Other results point to a relation between the

arithmetic properties of Calabi–Yau 3-folds and

conformal field theory. For insta nce, it was shown

by Schimmrigk that, in certain cases, the algebraic

number field defined via the fusion rules of a

conformal field theory as the field defined by the

eigenvalues of the integer-valued fusion matrices



 

¼ðN



can be recovered from the Hasse–Weil L-function of

the Calabi–Yau. An interesting case is provided by

the Gepner model associated with the Fermat

quintic Calabi–Yau 3-fold.

604 Number Theory in Physics

Arakelov Geometry

For K a number field and O

its ring of integers, a

smooth proper algebraic curve X ov er K determines

a smooth minimal model X

, which defines an

arithmetic surface X

over Spec(O

). The closed

fiber X

}

of X

over a prime } 2 O

is given by the

reduction mod }.

When Spec(O

) is ‘‘compactified’’ by adding the

Archimedean primes, one can correspondingly

enlarge the group of divisors on the arithmetic

surface by adding formal real linear combinations of

irreducible ‘‘closed vertical fibers at infinity.’’ Such

fibers are only treated as formal objects. The main

idea of Arakelov geometry is that it is sufficient to

work with ‘‘infinitesimal neighborhood’’ X



(C)of

these fibers, given by the Riemann surfaces obtained

from the equation defining X over K under the

embeddings  : K ,!C that constitute the Archime-

dean primes. Arakelov developed a consistent inter-

section theory on arithmetic surfaces, by computing

the contribution of the Archimedean primes to the

intersection indices using Hermitian metrics on these

Riemann surfaces and the Green function of the

Laplacian.

A general introduction to the subject of Arakelov

geometry can be found in Lang (1988). Manin

(1991) showed that these Green functions can be

computed in terms of geodesics in a hyperbolic

3-manifold that has the Riemann surfa ce X



(C)as

its conformal boundary at infinity.

The Polyakov measure A first application to

physics of methods of Arakelov geometry was an

explicit formula obtained by Beilins on and Manin

(1986) for the Polyakov bosonic string measure in

terms of Faltings’s height funct ion at algebraic

points of the moduli space of curves.

The partition function for the closed bosonic

string has a perturbative expansion Z =

g0

with

¼ e

ð22g Þ



Sðx;Þ

DxD ½7

written in terms of a compact Riemann surface  of

genus g, maps x :  ! R

, and metrics  on . The

classical action is of the form

Sðx;Þ¼



ﬃﬃﬃﬃﬃﬃ

jj





½8

Using the invariance of the classical action with

respect to the semidirect product of diffeomorphisms

of  and the conformal group, the integral is reduced

(in the critical dimension d = 26 where the con-

formal anomaly cancels) to a zeta regularized

determinant of the Laplacian for the metric on 

and an integration over the moduli space M

of genus

g algebraic curves. Beilinson and Manin gave an

explicit formula for the resulting Polyakov measure

on M

using results of Faltings on Arakelov geometry

of arithmetic surfaces. In particular, their argument

uses essentially the properties of the Faltings metrics

on the invertible sheaves d(L) given by the ‘‘multi-

plicative Euler characteristics’’ of sheaves L of

relative 1-forms. For a suitable choice of bases {

}

and {w

} of differentials and quadratic differentials,

the formula for the Polyakov measure is then of the

form (up to a multiplicative constant)

d

¼jdet Bj

18

ðdet =Þ

13



^^

3g3



3g3

½9

with  in the Siegel upper-half space, B



and the W

given by the images of the basis w

under

the Kodaira–Spencer isomorphism.

Holography In the case of the elliptic curve



, a formula of Alvarez-Gaume,

Moore, and Vafa gives the operator product expan-

sion of the path integral for bosonic field theory as

gðz; 1Þ¼log

jqj

ðlog jzj= log jqjÞ=2

j1  zj



n¼1

j1  q

zjj1  q

1

½10

where B

is the second Bernoulli polynomial.

Expression [8] is in fact the Arakelov Green function

on X

Using this and analogous results for higher genus

Riemann surfaces, Manin and Marcolli (2001)

showed that the result of Manin (1991) on Arakelov

and hyperbolic geometry can be rephrased in terms

of the AdS/CFT correspondence, or holography

principle. Expression [8] can then be written as a

combination of terms involving geodesic lengths in

the Euclidean BTZ black hole.

In the case of higher genus curves, the Arakelov

Green function on a compact Riemann surface,

which is related to the two-point correlation func-

tion for bosonic field theory, can be expressed in

terms of the semiclassical limit of gravity (the

geodesic propagator) on the bulk space of Euclidean

versions of asymptotically AdS

2þ1

black holes

introduced by K Krasnov.

Motives

There are several cohomology theories for algebraic

varieties: de Rham, Betti, e´tale cohomology. de Rham

Number Theory in Physics 605

and Betti are related by the period isomorphism, and

comparison isomorphisms relate E

tale and Betti

cohomology. In the smooth projective case, they

have the expected properties of Poincare´ duality,

Ku¨ nneth isomorphisms, etc. Moreover, E

tale coho-

mology provides interesting ‘-adic representations of

Gal(



k=k). In order to understand what type of

information, such as maps or operations can be

transferred from one to another cohomology,

Grothendieck introduced the idea of the existence of

a ‘‘universal cohomology theory’’ with realization

functors to all the known cohomology theories for

algebraic varieties. He called this the theory of

‘‘motives.’’ Properties that can be transferred between

different cohomology theories are those that exist at

the motivic level. A short introduction to motives can

be found in Serre (1992).

The first constru ctions of a category of motives

proposed by Grothendieck covers the case of smooth

projective varieties. The corresponding motives form

a Q-linear abelian category of ‘‘pure motives.’’

Roughly, objects are varieties and morphisms are

‘‘correspondences’’ given by algebraic cycles in the

product, modulo a suitable equivalence relation. The

category also contains Tate objects generated by

Q(1), which is the inver se of the pur e motive

). Grothendieck’s standard conjectures imply

that the category of pure motives is equivalent to the

category of representations Rep

of a ‘‘motivic

Galois group,’’ which in the case of pure motives is

proreductive. The subcategory of pure Tate motives

has as motivic Galois group the multi plicative group

. The situation is more complicated for ‘‘mixed

motives,’’ for which constructions were only very

recently proposed (e.g., in the work of Voevodsky).

These provide a universal cohomology theory for

more general classes of algebraic varieties. Mixed

Tate motives are the subcategory generated by the

Tate objects. There is again a motivic Galois group.

For mixed motives it is an extension of a proreduc-

tive group by a prounipotent group, with the

proreductive part coming from pure motives and

the prounipotent part from the presence of a weight

filtration on mixed motives. The multiple zeta values

appear as periods of mixed Tate motives .

Renormalization and Motivic Galois Theory

A manifestation of motivic Galois groups in physics

arises in the context of the Connes–Kreimer theory of

perturbative renormalization (for an introduction to

this topic, see Hopf Algebra Structure of Renormaliz-

able Quantum Field Theory). In fact, according to the

Connes–Kreimer theory, the Bogoliubov–Parasiuk–

Hepp–Zimmerman (BPHZ) renormalization scheme

with dimensional regularization and minimal subtrac-

tion can be formulated mathematically in terms of the

Birkhoff factorization

ðzÞ¼



ðzÞ

1



ðzÞ½11

of loops in a prounipotent Lie group G, which is the

group of characters of the Hopf algebra of Feynman

graphs. Here, the loop  is defined on a small

punctured disk around the critical dimension D, 

is holomorphic in a neighborhood of D, and 



holomorphic in the complement of D in P

(C). The

renormalized value is given by 

(D) and the

counterterms by 



(z).

The pa per of Connes and Marcolli Renormaliza-

tion, the Riemann–Hilbert Correspondence,and

Motivic Galois Theory in volume II of Julia et al.

(2005) shows that the data of the Birkhoff factor-

ization are equivalently described in terms of

solutions to a certain class of differential systems

with irregular singularities. This is obtained by

writing the terms in the Birkhoff factorization as

time-ordered exponentials, and then using the fact

that

ðtÞdt

:¼ 1 þ

n¼1

as

s

b

ðs

Þðs

Þds

 ds

is the value g(b)atb of the unique solution g (t) 2 G

with value g( a ) = 1 of the differential equation

dg(t) = g(t)(t)dt.

The singularity types are specified by physical

conditions, such as the independence of the counter-

terms on the mass scale. These conditions are

expressed geometrically throu gh the notion of

G-valued ‘‘equisingular connections’’ on a principal



-bundle B over a disk , wher e G is the

prounipotent Lie group of characters of the

Connes–Kreimer Hopf algebra of Feynman graphs.

The ‘‘equisingularity’’ condition is the property that

such a connection ! is C



-invariant and that its

restrictions to sections of the principal bundle that

agree at 0 2  are mutually equivalent, in the sense

that they are related by a gauge transformation by a

G-valued C



-invariant map regular in B; hence, they

have the same type of (irregular) singularity at the

origin.

The classification of equivalence classes of these

differential systems via the Riemann–Hilbert corre-

spondence and differential Galois theory yields a

Galois group U



= Uo G

, where U is prounipotent,

with Lie algebra the free graded Lie algebra with

one generator e

n

in each degree n 2 N. The group



is identified with the motivic Galois group of

mixed Tate motives over the cyclotomic ring

Z[e

2i=N

], for N = 3orN = 4, localized at N.

606 Number Theory in Physics