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 K(f )). In the quadratic case, the Basic Dicho t-

omy holds: the Julia set (and the filled Ju lia set) is

either connected or a Cantor set .

Bo¨ ttcher Coordinate

Let f = z

þ a

d1

þþa

be a monic polyno-

mial of degree d  2. Then 1 is a superattracting

fixed point, and hence there is a univalent function

B(z) = B

(z) near 1 satisfying the Bo¨ ttcher equation

B(fz) = B(z)

(the Bo¨ ttcher coordinate near 1).

Moreover, B(z)  z as z !1since f is monic.

If J(f ) is connected, B(z) can be analytically

extended to the whole basin of 1, and provides us

with the Riemann map CnK(f ) !Cn



D. Otherwise,

B(z) extends to a conformal map from some

invariant domain 

whose boundary contains a

critical point onto Cn



, where R = R

> 1.

The B-preimage of a straight ray {re

2i

:0< r < 1}

is called the external ray R



of angle .TheB-preimage

of a round circle {re

2i

:0 <1} is called the

equipotential E

of level t = log r. External rays and

equipotentials form two orthogonal f-invariant folia-

tions. We let R



(t) = R



Combinatorial Equivalence

Assume now that J(f) is connected. One says that an

external ray R



lands at some point z 2 J(f )if



(t) !z as t !0. Any external ray of rational

angle  = q=p with odd p lands at some repelling or

parabolic periodic point of period dividing p

(Douady and Hubbard 1982). Vice versa, any

repelling or parabolic point is a landing point of at

least one rational ray as above (Douady 1990s).

Let us consider the following equivalence relation

on the set of rational numbers with odd denomi-

nators: two such numbers  and 

are equivalent if

the corresponding rays R



and R



land at the same

point z 2 J(f ). Two polynomials f and

f with

connected Julia set are called combinatorially

equivalent if the corresponding equivalence relations

coincide. Notice that topologically equivalent poly-

nomials are combinatorially equivalent.

Parameter Phenomena

Spaces of Rational Functions

Let Rat

stand for the space of rational functions

of degree d. As an open subset of the complex

projective space CP

2dþ1

, it is endowed with the

natural topology and complex structure.

Hyperbolic Maps and Fatou’s Conjecture

Hyperbolic maps form an important and best-

understood class of rational maps (compare with

Hyperbolic Dynamical Systems). A rational map f is

called hyperbolic if one of the following equiva lent

conditions holds:



All critical points of f converge to attracting

cycles;



The map is expanding on the Julia set:

jDf

ðzÞj  C

; z 2 Jðf Þ

where C > 0, >1.

For instance, the maps z 7!z

þ " for small

", z 7!z

 1, and z 7!z

þ c for c 2 Rn[2, 1=4]

are hyperbolic. It is easy to see from the first

definition that hyperbolicity is a stable property,

that is, the set of hyperbo lic maps is open in the

space Rat

of rational maps of degree d. One of the

central open problems in holomorphic dynamics is

to prove that this set is also dense. This problem is

known as Fatou’s conjecture.

Postcritically Finite Maps and Thurston’s Theory

A rational map is called postcritically finite if the

orbits of all criti cal points are finite. In this case, any

critical point c is either a superattracting periodic

point, or a repelling preperiodic point (i.e., f

c is a

repelling periodic point for some n). If all critical

points of f are preperiodic, then J(f ) =

Important examples of postcritically finite maps with

J(f ) =

C come from the theory of elliptic functions.

Namely, let P



: C=



C be the Weierstrass

P-function , where 



is the lattice in C generated by

1and,Im>0. It satisfies the functional equation



(nz) = f

, n

(P( z)), where f

, n

is a rational function.

These f unctions called Latte´s examples possess the

desired properties. (For some special lattices, n can be

selected complex: the corresponding maps are also

called Latte´s.)

More generally, one can consider postcritically

finite topological branched coverings f : S

.Two

such maps, f and g, are called Thurston combina-

torially equivalent if there exist homeomorphisms

h, h

:(S

, O

) !(S

, O

) homotopic rel O

(and

hence coinciding on O

)suchthath

 f = h  g.

A combinatorial class is called realizable if it

contains a rational function. Thurston (1982) gave a

combinatorial criterion for a combinatorial class to

be realizable. If it is realizable, then the realization is

unique, except for Latte´s examples (Thurston’s

Rigidity Theorem).

656 Holomorphic Dynamics

Structural Stability and Holomorphic Motions

A map f 2 Rat

is called J-stable if for any maps

g 2 Rat

sufficiently close to f, the maps f jJ(f ) and

g jJ(g) are topologically conjugate, and moreover,

the conjugacy h

: J(f ) !J(g) is close to id. Thus, the

Julia set J(f ) moves continuously over the set of

J-stable maps. The following result proves a weak

version of Fatou’s conjecture:

Theorem 4 (Lyubich and Man

e´-Sad-Sullivan

1983). The set of J-stable maps is open and dense

in Rat

. Moreov er, the set of unstable maps is the

closure of maps that have a pa rabolic periodic point.

A map f 2 Rat

is called structurally stable if for

any maps g 2 Rat

sufficiently close to f, the maps f

and g are topologically conjugate on the whole

sphere, and moreover, the conjugacy h

C !

C is

close to id. The set of structurally stable maps is also

open and dense in Rat

(Man

e´-Sad-Sullivan).

The proofs make use of the theory of holomorphic

motions developed for this purpose but having much

broader range of applications in dynamics and

analysis. Let X be a subset of

C, and let h



: X !

be a family of injections depending on parameter

 2  in some complex manifold with a marked

point 



. Assume that h





= id and that the functions

 7!h



(z) are holomorphic in  for any z 2 X. Such

a family of injections is called a holomorphic

motion.

A holomorphic motion of any set X over 

extends to a holomorphic motion of the whole

sphere

C over some smaller manifol d 

  (Bers–

Royden, Sullivan–Thurston 1986). If h



is a holo-

morphic motion of an open subset of the sphere,

then the maps h



are quasiconformal (Man

-Sad-

Sullivan). These statements are usually referred to as

the -lemma.

If =D, then the holomorphic motion of a set

X 

C extends to a holomorphic motion of

C over

the whole disk D (Slodkowsky 1991).

Fundamental Conjectures

The above rigidity and stability results led to the

following profound conjectures:

QC Rigidity Conjecture If two rational maps are

topologically conjugate, then they are quasiconfor-

mally conjugate.

Let us consider the real projective tangent bundle

PT over

C, with a natural action of the map f.

A measurabl e invariant line field on the Julia set is

an invariant measurab le section X !PT over an

invariant set X  J(f ) of positive Lebesgue measure.

In other words, it is a family of tangent lines L



, z 2 X, such that Df (L

) = L

. Note that such a

field can exist only if J(f ) has positive Lebesgue

measure.

No Invariant Line Fields Conjecture Let us con-

sider two rational maps, f and

f , that are not Latte´s

examples. If they are quasiconformally conjugate

and the conjugacy is conformal on the Fatou set,

then they are conjugate by a Mo¨ bius transformation.

Equivalently, if f is not Latte´s, then there are no

measurable invariant line fields on J(f ).

This conjecture would imply Fatou’s conjecture.

Mandelbrot Set

Let us consider the quadratic family f

: z 7!z

þ c.

(Note that any quadratic polynomial is affinely

conjugate to a unique map f

.) The Mandelbrot set

classifies parameters c according to the Basic

Dichotom y of the subsec tion ‘‘Mor e pro perties of

the Julia set’’:

M ¼fc : Jðf

Þ is connectedg¼fc : f

ð0Þ7!1g

Note that 

(0) is a polynomial in c of

degree 2

n1

, and these polynomials satisfy a

recursive relation 

nþ1

= 

þ c. Moreover, M =

{c : j

(c)j2, n 2 Z

}, which gives an easy way to

make a computer image of M (see Figure 5 ).

A distinguished curve seen at the picture of M is

the main cardioid C= {c = e

2i

 e

4i

=4},  2 R=Z.

For such a c = c( ) 2C, the map f

has a neutral

fixed point 

with rotation number . For c inside

the domain H

bounded by C, f

has an attracting

fixed point 

, and the Julia set J(f

) is a Jordan

curve (see Figure 1).

Figure 5 The Mandelbrot set.

Holomorphic Dynamics 657

At the cusp c = 1=4 = c(0) of the main cardioid,

the map f

has a parabolic point with multiplier 1.

This point is also called the root of C. Other

parabolic points c = c(q=p)onC are bifurcation

points: if one crosses C transversally at c, then the

fixed point 

‘‘gives birth’’ to an attracting cycle of

period p. This cycle preserves its ‘‘attractiveness’’

within some component H

q=p

of int M attached to C.

On the boundary of H

q=p

, the above attracting

cycle becomes neutral, and similar bifurcations

happen as one crosses this boundary transversally,

etc. In this way we obtain cascades of bifurcations

and associated necklaces of components of int M.

The most famous one is the cascade of doubling

bifurcations that occur along the real slice of M.

Components of int M that occur in these bifurcation

cascades give examples of hyperbolic components of

int M. More generally, a component H of int M is

called hyperbolic of period p if the maps f

, c 2 H,

have an attracting cycle of peri od p. Many other

hyperbolic components becom e visible if one begins

to zoom-in into the Mandelbrot set. Some of them

are satellite, that is, they are born as above by

bifurcation from other hyperbolic components.

Others are primitive. They can be easily distin-

guished geometrically: primitive components have a

cusp at their root, while satellite compon ents are

bounded by smooth curves.

Given a hyperbolic component H, let us consider the

multiplier (c), c 2 H, of the corresponding attracting

cycle, as a function of c 2 H. The function  univalently

maps H onto the unit disk D (Douady and Hubbard

1982). Thus, there is a single parameter c

2 H for

which (c

) = 0, so that f

has a superattracting cycle.

This parameter is called the center of H.

Nonhyperbolic components of int M are called

queer. Conjecturally, there are no queer compo-

nents. This conjecture is equivalent to Fatou’s

conjecture for the quadratic family.

The boundary of M coincides with the set of

J-unstable quadratic maps (see the subsection

‘‘Struct ural stabili ty and holom orphic motions’’).

Connectivity and Local Connectivity

Theorem 5 (Douady and Hubbard 1982). The

Mandelbrot set is connected.

The proof provides an explicit uniformization

: CnM !Cn



D. Namely, let B

: 

!CnD

, c 2

CnM, be the Bo¨ ttcher coordinate near 1. The n

(c). This remarkable formula explains the

phase-parameter similarity between the Mandelbrot

set near a parameter c 2 M and the corresponding

Julia set J (f

) near the critical value c.

The following is the most prominent open

problem in holomorphic dynamics:

MLC Conjecture The Mandelbrot set is locally

connected.

If this is the case, then the inverse map R

1

extends to the unit circle T, and the Mandelbrot

set can be represented as the quotient of T modulo

certain equiva lence relation that can be explicitly

described. Thus, we would have an explicit topolo-

gical model for the Mandelbrot set (Douady and

Hubbard, Thurston).

The MLC conjecture is equivalent to the follow-

ing conjecture:

Combinatorial Rigidity Conjecture If two quadratic

maps f

and f

with all periodic points repelling are

combinatorially equivalent, then c = c

In turn, this conjecture would imply, in the

quadratic case, the above fundamental conjectures.

For a progress towards the MLC conjecture (see

Universality in Mathematical Physics).

Parabolic Implosion

Parabolic maps f

: z 7!z

þ c

are unstable in a

dramatic way. In particular, the Julia set J(f

) does

not depend continuously on c near c

. Instead, J (f

)

tends to fill in a good part of int J(f

). This

phenomenon called parabolic implosion has been

explored by Douady, Lavaurs, Shishikura, and many

others.

Geometric Aspects

Area

One of the basic problems in holomorphic dynamics

is whether a Julia set that does not coincide with

can have positive area. It would give an example of

‘‘observable chaos’’ that occurs on a topologically

small set. It is also related to the No Invariant Line

Fields Conjecture.

Maps with strong hyperbolic properties have zero

area Julia set. A rational map f is called Collet–

Eckmann if there exist constants C > 0and>1

such that:

jDf

ðfcÞj  C

; n 2 N

for all critical points c. If f is a Collet–Eckmann map

with J( f ) 6¼

C, then area J(f ) = 0 (Przytycki and

Rohde 1998) (see Universality and Renormalization

for more examples). On the other hand, A Douady

has set up a compelling program of constructing a

Cremer quadratic polynomial f : z 7!e

2i

z þ z

whose Julia set would have positive area. Buff and

658 Holomorphic Dynamics

Cheritat have recently announced that they have

completed the program, thus constructing the first

example of a Julia set of positive area. (It makes use

of a renormalization theorem for parabolic implo-

sion recently announced by Shishikura.)

In the parameter plane, it would be interesting to

know whether the boundary of the Mandelbrot set

has zero area.

Hausdorff Dimension

Hausdorff dimension (HD) gives us a furt her

refinement of fractal sets of zero area. Any Julia

set has positive HD. If f is a polynomial with

connected Julia set, then HD( J(f )) > 1 unless f is

affinely conjugate to z 7!z

or a Chebyshev poly-

nomial (Zdunik 1990). If f is a Collet–Eckmann map

with J(f ) 6¼

C, then HD J(f ) < 2 (Przytycki–Rohde

1998). On the other hand, in the quadratic case

: z 7!z

þ c,HD(J(f

)) = 2 for a generic parameter

c 2 @M. The corresponding parameter result is that

HD(@M) = 2 (Shishikura 1998). It is based on the

parabolic implosion phenomenon.

Conformal Measure

Let   0. A Borel measure  on

C is called

-conformal if

ðfXÞ¼

jDf j



d

for any measurable set X such that f jX is inj ective.

Theorem 6 (Sullivan 1983). Any rational map f

has a  -conformal measure with  2 (0, 2] supported

on J(f ).

This is a dynamical measure that captures well

geometric properties of J(f ). For instance, for Collet–

Eckmann maps,  = HD(J(f )), and  is equivalent to

the Hausdorff measure on J(f ) in dimension .

The hyperbolic dimension,HD

hyp

of J(f ) is the

supremum of HD(X) over all compact invariant

hyperbolic subsets of J(f ). Denker and Urbanski

(1991) proved that HD

hyp

(J(f )) is equal to the

smallest exponent  of all -conformal measures

supported on J(f )(see Universality and

Renormalization).

Measure of Maximal Entropy

An f-invariant measure  is called balanced if

(fX) = d(X) for any measurable set X such that

f jX is injective (where d = deg f ).

Theorem 7 (Brolin 1965, Lyubich 1982). Any

rational map f has a unique balanc ed measure .

Moreover, preimages of any point z except at most

two are equidistributed with respect to  (meaning

that the probability measures 

n, z

that assign mass 1

to every f

-preimage of z converge weakly to  as

n !1).

For polynomials, the balanced measure coincides

with the harmonic meas ure on J(f ) (Brolin). (The

latter is the charge distribution on the conductor J(f )

generated by the unit charge placed at 1 .) In

general, the balanced measure is the unique measure

of maximal entropy of f, and moreover, periodic

points are equidistributed with respect to 

(Lyubich).

Measure of maximal entropy is supported on a

relatively small measurable set: its HD is strictly less

than HD(J(f )), unless f is conformally equivalent to

z 7!z

, a Chebyshev polynomial, or a Latte´s example

(Zdunik 1990). In the polynomial case, it is

supported on a set of HD at most 1 (Manning 1984).

In complex analysis, there has been an extensive

study of fractal properties of harmonic meas ures,

providing insights at the balanced measure  and the

other way around (Carleson, Makarov, Jones,

Binder, Smirnov, ...)

See also: Fractal Dimensions in Dynamics; Geometric

Analysis and General Relativity; Geometric Flows and

the Penrose Inequality; Geometric Phases; Polygonal

Billiards; Renormalization: General Theory;

Renormalization: Statistical Mechanics and Condensed

Matter; Universality and Renormalization.

Further Reading

Space limitations prohibit the inclusion of references to all the

results quoted in this article. Where an author is quoted in the

text, the reader can find the corresponding reference in one of the

books listed in this section and in the MatSciNet.

Buff X and Hubbard J Dynamics in One Complex Variable.

Ithaca, NY: Matrix Editions (to appear).

Carleson L and Gamelin W (1993) Complex Dynamics. Berlin:

Springer.

Eremenko A and Lyubich M (1990) The dynamics of an alytical

transformations. Leningrad Mathematic al Journal 1:

563–633.

Milnor J (1999) Dynamics in One Complex Variable: Introduc-

tory Lectures. Braunschweig: Vieweg and Sohn.

Pilgrim KM (2003) Combinatorics of Complex Dynamical

Systems, Lecture Notes in Mathematics no 1827. Berlin–

Heidelberg–New York: Springer-Verlag.

Przytycki F and Urbanski M, Fractals in the Plane – the Ergodic

Theory Methods. (www.math.unt.edu/uranski/book1.html)

Cambridge: Cambridge University Press (to appear).

Tan Lei (ed.) (2000) The Mandelbrot Set, Theme and Variations,

London Math. Soc. Lecture Note Ser., vol. 274. Cambridge:

Cambridge University Press.

Holomorphic Dynamics 659

Holonomic Quantum Fields

J Palmer, University of Arizona, Tucson, AZ, USA

Introduction

The term, holonomic field, was coined by Sato,

Miwa, and Jimbo (SMJ) in 1978 and the subject was

investigated by them in a series of five long papers

and many shorter notes in the period from 1978–81

(Sato et al. 1979a, 1979b, 1979c, 1980, Tracy

and Widom 1994). The term refers to a special class of

two-dimensional interacting quantum field theories

whose n point correlations can be expressed in terms

of the solution to a holonomic system of differential

equations. A holonomic system is an overdetermined

system of differential equations with only a finite-

dimensional family of solutions. There is a sense in

which these interacting systems with infinitely many

degrees of freedom have a finite-dimensional substrate

(at the level of n point functions for fixed n). After

developing their theory, SMJ realized that such

quantum fields made an earlier appearance in work

of Thirring and Federbush. The models considered by

Thirring and Federbush are self-interacting fermionic

systems whose nonlinear classical field equations have

solutions that are an explicit nonlinear transformation

of solutions to the free field equations. This inspired

the idea of trying to study these models by ‘‘quantiz-

ing’’ the nonlinear transformation. Expressions were

obtained for the correlations and S-matrix but the

connection with deformation theory was not under-

stood until the SMJ work.

In what follows we will sketch the SMJ theory and

discuss some of its offshoots. There is one circum-

stance that it might help the reader to be aware of

even though it will be mostly glossed over. Quantum

fields in one space and one time dimension have

correlations which transform under the symmetries of

spacetime with metric signature (1,1). Since the work

of Osterwalder and Schrader, it is customary to pass

back and forth between this Minkowski regime and

the Schwinger functions obtained by analytically

continuing the n point functions to pure imaginary

values for the time variable where they possess the

rotational symmetries associated with a positive-

definite metric. The Ising model, which we take up

next, is naturally considered in the Euclidean domain

where the correlations have an interpretation in

statistical mechanics as the expected value of a

product of random variables. Ultimately, the SMJ

deformation analysis is done in the Euclidean

domain.

The Two-Dimensional Ising Model

The SMJ theory was inspired by, and provides an

attractive sett ing for, an earlier result of Wu,

McCoy, Tracy, and Baruch (WMTB), concerning

the spin–spin scaling functions of the two-dimen-

sional Ising model (Wu et al. 1976). Since the Ising

model is the example with the most direct signifi-

cance for physics, we will take some time to explain

the WMTB result and to sketch the way in which it

fits into the SMJ theory.

The Ising mod el is a statistical model of magnest-

ism on a lattice that incorporates ferromagnetic

interactions of nearest-neighbor spins. In the 1920s,

Ising solved the model for the one-dimensional

lattice and showed that there was no phase transition

in the infinite volume limit. Interest in the two-

dimensional model intensified dramatically following

Onsager’s calculation of the specific heat in the

infinite volume limit (see Palmer and Tracy (1981)

and references within). His formula for the specific

heat was the first instance of a thermodynamic

quantity in a nearest-neighbor model which exhibits

the sort of discontinuity in temperature dependence

expected at a phase transition. For many years, the

Ising model served as a testbed for the now accepted

notion that the infinite volume limit of Gibbsian

statistical mechanics provides a suitable setting for

the study of phase transitions.

A configuration for the Ising model on a finite

subset, , of the integer lattice, Z

,isamapC:

 !{þ1, 1}, which assigns to each site on the

lattice either an up spin (þ1) or a down spin (1).

The energy function of the Ising model, E



(), is

defined by



ðÞ¼J

hi;ji2

ðiÞðjÞ

for J > 0 and a spin configuration  is a sum over

pairs of nearest-neighbor sites i, j in  (boundary

terms require special consideration). This energy

function tends to favor spin configurations, ,in

which the nearest-neighbor spins are aligned in the

sense that the Boltzmann weight, e

E



()=kT

, is larger

for such configurations. In the Gibb s ensemble,

which is expected to describe systems in equilibrium

at temperature T, the configuration  occurs with a

probability proportional to the Boltzmann weight.

The factor k which appears is a conversion factor

between thermal and kinetic energy called the

Boltzmann constant. It is clear from the formula

for the Boltzmann weight that small temperatures

(near 0) tend to accentuate the difference in

660 Holonomic Quantum Fields

statistical weights assigned to configurations with

different energies, and large temperatures tend to

wash out the difference in statistical weights

associated to configurations with different energies.

Remarkably, there is a sharp critical temperature

0 < T

< 1 so that for T < T

the propensity for

order built into the energy triumphs in the infinite

volume limit  "Z

, and for T > T

the randomness

or disorder associated with high temperatures

governs the infinite volume behavior. More specifi-

cally, if T < T

and the infinite volume limit is taken

with plus spins assigned to the boundary of , the

system exhibits a residual magnetism (there is a

positive expected value, hi, for the spin per site).

This infinite volume plus state is the quintessential

example of symmetry breaking – the spin flip

symmetry possessed by the bulk energy is broken

below T

in the thermodynamic limit. For T > T

the spin per site is 0 no matter what boundary

conditions are imposed on the infinite volume limit.

Pure equilibrium states both above and below T

exhibit clustering in the thermodynamic limit

(uniqueness for the ground state in field theory).

This is the tendency of spin variables (a) and (b)

at sites a, b 2 Z

to become statistically independent

as the distance ja  bj tends to 1. In such a pure

state the two-point function, which is the expected

value of the product of spin variables, h(a)(b)i,

will tend to the square hi

both below (hi 6¼ 0)

and above (hi= 0) the critical temperature T

ja  bj!1. To leading order, this clustering takes

place at an exponential rate, e

jabj=(T)

, for a

function (T) called the correlation length. The

correlation length (T) !1 as T !T

. The scaling

limit (from below T

) of the spin–spin correlation is

the leading-order correction to the clustering beha-

vior of the correlations when these correlations are

examined at the scale of the correlation length. It is

the limit

hðaÞðbÞi ¼ lim

T"T

hððTÞaÞððTÞbÞi

hi

where the correlations on the right-hand side are

thermodynamic correlations on the lattice at tem-

perature T. Since hi

tends to 0 as T !T

, the

normalization by hi

1

on the right produces an

‘‘infinite wave function renormalization’’ in the limit.

Equivalently, one may think of this continuum

limit being achieved by letting the lattice spacing

shrink to 0 as T approaches T

so that the

correlation length stays fixed on the new scale. The

scaling limit from above T

turns out to be different

from the scaling limit from below T

and since

hi

= 0 for T > T

, it is defined by a different wave

function renormalization. The resulting asymptotics

are expected to capture quite a lot about what is

interesting in the behavior of the correlations near

the phase transition. In the late 1970s, the scaling

behavior in this model was also a pro totype for the

emerging connection between renormalization group

ideas in quantum field theory and statistical

mechanics.

Wu et al. (1976) showed that the two-point

scaling function, h(0)(x)i, is a function of r = jxj

and can be written as

hð0ÞðxÞi¼

coshð =2Þ

exp

sinh





sds

ðT "T

sinhð =2Þ

exp

sinh





sds

ðT #T

½1

where = (r) satisfies the differential equation,



sinhð2 Þ

The substitution  = e



transforms this differential

equation into a Painleve´ equation of the third kind.

This was used by McCoy, Tracy, and Wu (see

Palmer and Tracy (1981) and references within) to

study the short-distance behavior, r !0, of the

scaling functions – behavior which is far from

manifest in the infinite series expansions obtained

for the scaling functions.

Deformation Theory

Sato, Miwa, and Jimbo showed that there was a

class of quantum field theories that included the

scaling limits of the Ising model which have the

property that the n-point correlations are ‘‘tau

functions’’ for monodromy-preserving deformations

of the Dirac equation in two dimensions. The two-

dimensional (Euclidean) Dirac operator is

D¼

m 2@

2



@ m



with

@ ¼

 i



;



@ ¼

þ i



Holonomic Quantum Fields 661

the usual complex derivatives acting on smooth

functions on R

. The monodromy-preserving defor-

mations mentioned above are families of (multivalued)

solutions w(x)to

Dw ¼ 0 ½2

which are branched at points a

2 R

(j = 1, 2, ..., n)

and change by a factor e

2i‘

as x makes a small

circuit about a

. SMJ (1979b) show that the L

)

(square-integrable) solutions w(x) of the Dirac

equation with this prescribed branching behavior

comprise an n-dimensional subspace of L

). The

constants e

2i‘

are called the monodromy of the

solutions and the term ‘‘deformation’’ in the descrip-

tion refers to the fact that the monodromy constants

do not change as the branch points a

are varied.

SMJ show that it is possible to choose a basis

(x, a) = w

(x, a

, ..., a

)(j = 1, ..., n) so t hat the

vector

Wðx; aÞ :¼½w

ðx; aÞ; w

ðx; aÞ; ...; w

ðx; aÞ

becomes a section for a flat (Dirac compatible)

connection in the (x, a) variables. That is,

x;a

W ¼ ðx; aÞW

where d

x, a

is the exterior derivative in the x and a

variables and  is a matrix-valued 1-form that

satisfies the zero curvature condition,

x;a

 ¼  ^ 

They also introduced the notion of a tau function,

(a), for such deformations. The logarithmic deri-

vative d

log (a) = !,where! is a 1-form on

n{a

, a

, ..., a

} expressed in t erms of the m atrix

elements of . The 1-form ! introduced by SMJ is

showntobeclosedwhen satisfies the zero

curvature condition above. The scaling limit of

the Ising model is related to the situation for

monodromy multipliers 1 and when the scaling

limits of correlations are identified as suitable

-functions in this case, the WMTB result emerges

when the nonlinear zero curvature condition is

identified with a Painlev e´ equation.

The connection between the deformation theory

and quantum field theory is developed in the

computationally intensive paper SMJ (1980).Exten-

sive use is made of local operator product expan-

sions, analytic continuation, and formal series

expansions that are infinite-dimensional analogs of

Wick-type theorems for finite-dimensional spin repre-

sentations (developed by SMJ (1978)). One can get

a feeling for the source of the connection by recalling

that in one of the ‘‘exact solutions’’ of the two-

dimensional Ising model the spin operators, (a),

are identified as elements of an infinite spin

representation of the orthogonal group and are

characterized by their linear action on Fermionic

variables (Palmer and Tracy 1981). In the physics

literature, the (a) are referred to as Bogoliubov

transformations. In the scaling limit the associated

representation space is the home to a free Fermi field

(x), an operator-valued solution to the Dirac

equation. Of course, (x) has components

(x)but

for simplicity we will suppress such details in the

mostly schematic discussion that follows. For coin-

cident second coordinates x

= a

the Fermi field (x)

and (a) satisfy the commutation relation

ðaÞ ðxÞ¼sgnðx

a

Þ ðxÞðaÞ½3

which is a surviving remnant of the linear action of

(a) on lattice fermi ons. In the transfer matrix

formalism, which is natural for statistical mechanics,

translation in the ‘‘space’’ variable x

is unitary, but

translation in the ‘‘time’’ variable, x

, is governed by

the transfer matrix, the generator of a contractive

semigroup. Because of this, the quantities that are

well behaved in this formalism are ‘‘time-ordered

vacuum expectations’’; these involve only ‘‘positive’’

powers of the transfer matrix. Let T denote the

‘‘time’’-ordering operator; a sequence of operators

depending on coordinates in R

is reordered follow-

ing T so that the second coordinates appear in

increasing order from left to right. Sign changes are

incorporated whenever it is necessary to exchange

Fermi type operators like



(x) and (y) to put them

in the correct order. In the Euclidean setting (pure

imaginary time) it is well known that

Gðx; yÞ¼hT



ðxÞ ðyÞi

is a Green function for the Dirac operat or D (the

distribution kernel for D

1

This observation and [3] suggests that the hybrid

vacuum expectation

Gðx; y; aÞ¼



ðxÞ ðyÞða

Þða

Þi

hT ða

Þða

Þi

should be the Green function for a Dirac operator

with a domain containing ‘‘functions’’ branched at

the points a

having ‘‘monodromy’’ 1there.Itis

possible to recast the SMJ analysis so that a Dirac

operator, D(a), on a suitable vector bundle with base

n{a

, ..., a

} becomes the central player (see

Palmer et al. (1994) and references therein). The

data for the vector bundle includes the factors e

2i‘

incorporated in transition functions for the bundle.

The -function becomes an infinite determinant (or

Pfaffian in the Ising case)

ðaÞ¼det DðaÞ½4

662 Holonomic Quantum Fields

in the Segal–Wilson sense (see Palmer et al. (1994)

and references therein). The Green function

G(x, y; a) has a finite-rank derivative,

Gðx; y; aÞ¼

ðx; aÞs

ðy; aÞda

þ u

ðx; aÞv

ðy; aÞd



½5

which is the key result in this version of the SMJ

analysis (this observation appears in SMJ (1980) but

does not have a central role there). The ‘‘wave

functions’’ r, s, u,andv are closely related to the

wave functions w

described above. Equation [5]

is both the source of the deformation equations for

r, s, u, and v which arise from d

G = 0 coupled with

the rotational and translational symmetries of the

Green function, and also of the expression for

log (a) in terms of data associated with the

deformation theory. A ‘‘transfer matrix’’ calculation

of the determinant allows one to make the connection

with the scaling limits of lattice fields including the

Ising model (see Palmer et al. (1994) and references

therein).

The short-distance behavior of the two-point

function for the Ising model scaling functions has

been rigorously calculated by Tracy and later by

Tracy and Widom (see Harnad and Its (2002) and

references therein). A less detailed analysis of the

short-distance behavior of the n point functions that

uses the deformation analysis of the correlations in a

crucial way can be found in Palmer (2000).

The Riemann–Hilbert Problem

In SMJ (1979b), a ‘‘massless’’ version of holonomic

fields is developed. This concerns monodromy-

preserving deform ations of the Cauchy–Riemann

operator



@. The techniques used to study this lead

back to the Riemann–Hilbert problem – the problem

of determining a linear differential equation in the

complex plane with rational coefficients and pre-

scribed monodromy at the poles of the coefficients.

More specifically, suppose one is given n distinct

points {a

, ..., a

}inP

, the Riemann sphere, and

a base point a

distinct from the a

, j 6¼ 0. Let 

denote a simple closed curve based at a

which

winds counterclockwise once around a

but has

winding number 0 for the other points a

, k 6¼ j.

Choose n invertible p  p matrices M

which satisfy

the single condition M

M

= I. Then, the

homotopy classes of the curves 

are the generators

for the fundamental group of the punctured

sphere P

n{a

, ..., a

} with base point a

and the

map which sends 

! M

determines a representa-

tion of the fundamental group. One version of the

Riemann–Hilbert problem is to find p  p complex

matrices A

for j = 1, ..., n so that the linear

differential equation

z  a

Y ½6

has monodromy representation given by 

This means that the fundamental solution Y(z)

defined in a neighborhood of z = a

and normalized

so the Y(a

) = I (the identity) will become the

fundamental solution Y(z)M

1

after analytic con-

tinuation around the curve 

. This form of the

problem does not always have a solution but when

it does, it is interesting t o consider deformations

a !A

(a) that preserve the monodromy multipliers

. Such monodromy-preserving deformations

were first considered by Schlesinger in 1912 (see

Palmer and Tracy (1981) and references therein)

and he discovered that the coefficients A

must

satisfy nonlinear differential equations that, for

= 1,canbewrittenas

k6¼j

 A

 a

dða

 a

SMJ introduced -functions associated with these

deformations and they gave these -func tions a

quantum field theory interpretation as n point

functions. Eventually this theory was extended to

include the Birkhoff generalization of the Riemann–

Hilbert problem, a generalization which incorpo-

rates the additional information needed to fix local

holomorphic equivalence at higher-order poles (formal

asymptotics and Stokes’ multipliers) (Jimbo and

Miwa 1981, Sato et al. 1978). Roughly speaking, the

problem is to reconstruct a global connection with

specified singularities from its local holomorphic

equivalence data and its global monodromy represen-

tation. Thinking of the differential equation [6] as a

holomorphic connection proved very helpful in a

geometric reworking of the SMJ analysis given

by Malgrange (1983a, 1983b) who showed that the

zeros of the -function occurred at points where a

suitably defined Riemann–Hilbert problem fails to

have a solution (see also Palmer (1999) references

within). The mathematical significance of massless

holonomic quantum fields as (quantized) singular

elements of a gauge group is apparent from the SMJ

work and later work of Miwa but the possibility of

interesting physics in these models does not seem to

have been much investigated at this time. These

quantum fields are also conformal fields; however, a

comprehensive integration into the highly developed

formalism of conformal field theory on compact

Riemann surfaces has not currently been developed

Holonomic Quantum Fields 663

(an analog of [5] should survive on compact Riemann

surfaces but the deformation analysis of the correla-

tions is likely limited to symmetric spaces).

Further Developments

This work on massless holonomic fields and the

connection with the Riemann–Hilbert problem is

doubtless the aspect of holonomic fields with the

most ‘‘spin offs’’ in the mathematics and physics

literature. These include an analysis of the delta-

function gas done by Jimbo, Miwa, Mori, and Sato

in 1981, random matrix models first looked at by

Jimbo, Miwa, Mori, and Sato and later system-

atically investigated by Tracy and Widom (1994),

the deformations of line bundles on Riemann

surfaces that led to KdV in the work of Segal and

Wilson (1985), which emerged from work of Sato,

Miwa, Jimbo and collaborators, the analysis of

Painleve´ equations starting with work of McCoy,

Tracy and Wu (see Palmer and Tracy (1981) and

references within) and more systematically devel-

oped by Its and Novokshenov (1986), and the

revival of interest in monodromy-preserving defor-

mations (Harnad and Its 2002).

Holonomic fields are related to free fields in a

well-understood way and it is natural to study them

in situations where free fields make sense. In

particular, they are an interesting testbed for the

nonperturbative investigation of the influence of

geometry (or curvature) on quantum fields. In Palmer

et al. (1994),thedeformationanalysisof-functions

for holonomic fields is carried out for the Poincare´

disk. The two-point functions are shown to be

expressible in terms of solutions to the family of

Painleve´ VI equations. A quantum field t heory

interpretation of these -functions is given by

Doyon and there are natural analogs of the scaling

limit of the Ising model on the Poincare´diskas

well. The role of ‘‘spacetime’’ sym metries in the

deformation theory suggests that such analysis will

be limited to symmetric spac es. In addition to the

plane a nd the Poincare´ disk, the cylinder, the

sphere, and the torus round out the possibilities i n

two dimensions. Lisovyy has recently worked out

the analysis for the cylinder, which is important for

the study of thermodynamic correlations. It should

be possible to recast t he analysis of the continuum

Ising model on the torus (Zuber a nd Itzykson 1977)

in deformation theoretic terms. It does not appear

that the holonomic fields associated with the Dirac

operator for the constant curvature metric on the

2-sphere have been studied yet.

See also: Deformation Theory; Integrable Systems:

Overview; Isomonodromic Deformations;

Riemann–Hilbert Problem; Two-Dimensional Ising

Model.