quantum gravity and sc at high tc

String-like excitations toe ee ee a Particles with spin and path integral representation Spin factor and fermionic integral 2D surfaces and strings Fermionic determinants Dynamical

Ef Brézin and J Zinn-Justin, eds

Les Houches, Session XLIX, 1988

Champs, Cordes et Phénoménes Critiques

/Fields, Strings and Critical Phenomena

(C) Elsevier Science Publishers B.V., 1989

1.3 Relation with statistical mechanics

1.4 String-like excitations toe ee ee a

Particles with spin and path integral representation

Spin factor and fermionic integral

2D surfaces and strings

Fermionic determinants

Dynamical gauge fields

Two-dimensional guantum gravity

More on 2D quantum gravity

High 7, superconductivity

10 Anomalous dimensions of operators

307

308 308 308 dll 314 315 324 328 333 339 349 351 308 363

1 Introduction

1.1 Aim of the lectures

This series of lectures will try to explain some inter-relations between

different parts of physics and to discuss some ideas that provide a link

between them

In doing that we shall meet many unclear points and open problems

and the aim is to discuss them This first lecture is intended as an

overview of the problems we shall cover

1.2 Elementary excitations

One of the most important concepts of this century’s physics is that of

elementary excitations An example is given by oscillations in a crystal

described as phonons They arise as a collective effect of the electrons

of the atoms in the crystal; nevertheless they have their own individual

character

The crudest description of an elementary excitation is to say that it is

a point-like object propagating in space-time For a more quantitative

description, let us introduce a space time point X# at which we have the

elementary excitation; the probability P of finding it at another point

X'" is given, according to the rules of Quantum Mechanics, by

Pxx: being a particular path, and Š the action At this point one may

argue why these fundamental rules actually work; for example there is

no clear understanding of why what happens is a consequence of what

could have happened

308

However for the time being we shall take (1.1) and (1.2) as axioms

Let us now discuss the possible candidates for an action in (1.2) In

order to do that we need a more precise description of a path Since we

start considering structureless particles, we may approximatively think

of a path as a very thin object without structure, and, in this approxi-

mation, we can parametrize it introducing some function

such that

A physical requirement for the action is one dimensional general co-

variance, that is the action should be invariant under the following trans-

(one may check that if we drop requirement (1.6), we lose general co-

variance; in fact one has

i= [iF ivetampa= [avi Psen(S) 4 1)

Of course we can envisage terms with higher derivatives, and we have

many possibilities for doing that; those terms will depend on the extrinsic

geometry of the path, that is they will depend on the geometry of the

space in which the path is embedded

For example, a possible term of this kind would be of the form

but, since for a one dimensional path the only intrinsic quantity we can

consider is its length, J involves already extrinsic geometrical properties

310 A Polyakov

On the other hand a term proportional to L* would violate locality

on the path

What we usually do in physics is to start with the simplest case (1.7)

and then check its stability against these more complicated terms in the

scale domain in which we are interested This will be done later

Let us now see what changes can occur if we consider spinning parti-

cles For simplicity we shall confine ourselves to the case of closed paths

The amplitude is now given by

F(X,X) = }> exp[iS(Pxx)]®(Px x) (1.9)

Pxx

where ®(Pxx) is a spin factor which depends on the number of dimen-

sions of space time For example, in the case of a spin 1/2 particle

moving in 2 dimensions, the spin factor ® is given by

where 1 is the number of self-intersections of the path, and therefore we

see that in this case the spin factor is a topological invariant

An interesting question one may ask concerns the time signature of the

paths One could choose to consider only propagations forward in time

(this is permitted by Lorentz invariance) Indeed, if the propagation

takes place inside the light cone, so that

However a real elementary excitation propagates in all directions In

fact propagation backward in time corresponds to antiparticles But

why are there antiparticles, since Lorentz invariance allows us to easily

A possible answer to this question is that the existence of antiparti-

cles is a hint of an underlying euclidean structure of space time In this

respect it is interesting to note that every formula of our theories de-

scribing nature can be correctly written in euclidean space by means of

a Wick rotation If one postulates that continuation to euclidean space

is possible one is forced to sum over paths going forward and backward

in time (since they are no longer different notions) As a matter of fact,

nature seems to be organized in this way; the CPT’ theorem expresses

the necessity of considering paths that go backward in time We con-

clude that the CPT theorem and the existence of antiparticles indicate

the intrinsically euclidean signature of space time

In this framework one may think that the Minkowskian metric is just

a way of interpreting our experience, but maybe this problem is related

to some dynamical effect similar to spontaneous symmetry breaking We

shall see in the following an example in two dimensions about that

1.3 Relation with statistical mechanics

Let us now establish a contact between sum over paths and classical

statistical mechanics

312

As an example we shall discuss the two dimensional Ising model

It_is defined on a square lattice with spins oj = +1 pointing up or

down, and its energy is given by

Formula (1.13) closely resembles the sum over paths considered above:

in both cases we sum over all possible configurations

formalism

If in our model we take £ to be very large, we end up in a state in

which all spins are up or down Decreasing § will reverse some of the

spins: surrounding domains with equal spin, we obtain a certain number

of “drops” in the system (see fig 3)

where L is the perimeter of the drops

The free energy F' is then given by

F = À ` [exp(—28L(p))] ®(p) (1.15)

where we have included a correction factor ® which counts in the right way the possible intersections of closed paths P For a given path P we have

&(P) = (-1)") (1.16)

where v(P) is the number of self intersections of the path We see that

we ended up with the same formula (in euclidean space) as in the case

of the spinning particle (in Minkowski space)

Here is a simple example of why (1.16) gives correct counting Con- sider the configuration shown in fig 4

314 A Polyakov

1.4 String-like excitations

Why do we have to consider string-like excitations in physics? In statis-

tical mechanics, for example the three dimensional Ising model can be

formulated as a sum over random surfaces and not over paths

In particle physics a reason for doing that comes from gluon-dynamics:

color flux lines are compressed into strings, and therefore it is tempting

to reformulate QCD in terms of strings that span some surfaces

Transition amplitudes are now given by sums over all possible surfaces

connecting two boundaries C’ and C> (see fig 6)

Fig 6

F(Cy | C2) = S— exp[iS(Acies)] (1.17)

A c1€2

The simplest choice for S is the Nambu action which is proportional to

the area of the boundaries

This is also motivated by the fact that in QCD, for example, the energy

of the flux string is such that

where L is the length of the string Therefore the action must be

S ~ ET ~ LT = Area of the world sheet Indeed the choice of the area gives us the nuinimum number of derivatives

in the action

As before we have to include spin factors in (1.17), and this constitutes

a possible way of approaching the problem of string-like excitations

Another way of doing that is more phenomenological; we can write

down all possible actions and then try to find their universality classes

(like we do in conformal field theory in two dimensions)

A third interesting reason for studying string-like excitations is that

for some critical value of the space time dimension we have a massless

spin two particle in the spectrum This has enormous implications; a

massless graviton forces general covariance not only on the world sheet

but also in the space time

2 Particles with spin and path integral representation

This lecture discusses in more detail the propagation of particles with

spin and their path representation We shall present a geometrical de-

scription for the propagation of particles and then establish the connec-

tion with conventional methods

This will be useful for the discussion of the spinning string and of

particle statistics relevant for superconducting states

We shall explicitly find the spin factors introduced in the last lecture

The first question we shall try to answer is: what are the typical fractal

properties of path swept by point-like objects? Given a closed path of

length £ and size R (see fig 7) its fractal properties are described by

J LD Z1 j O(UWGKOĐ

particle, it turns out h = 2 For spinning particles we shall find a dif-

ferent value of h, namely h = 1 The reason why we are interested in

the computation of fh is that many physical quantities, like the specific

heat of the Ising model, depend on it Let us start considering a spinless

particle whose action is given by (1.7): then the amplitude F(X | X')

for this particle is

r(x |x'y= Í lSmm]| e—“o Jạ V#?()dr (2.2)

where the symbol [Dz(r)/Df(r)| indicates that we are integrating over

orbits of the gauge group

#(r) + f(r)e(f(r)) (2.3)

Let us rewrite (2.2) in a more geometrical way that can easily be gen-

eralized to the case of strings, by introducing the induced metric tensor

Note that this choice does not introduce ghosts At this point we may

replace the integration measure by { dZ, and write

Then the amplitude becomes

F(X | X’) -[- abe! | Ded (e?~1) 5

xx [eas

(2.9) where the last 6-function expresses the constraint that all the paths we

are considering have fixed endpoints The following step is to perform a

Fourier transformation:

F(X | X') > F(p) = I dle" (e pf, e(s)de ) (2.10)

(>1 nas) = J® ô(e? — 1)eP'J, os) de (2.11)

In this way we have obtained a form of the amplitude which is suitable

for manipulations

So far we have considered a situation in which the velocities at different

points along the path are completely uncorrelated, but we could have

added other terms to the action which may spoil this feature

A possible choice is to consider the following term

d 2

f= (<=) | (2.12)

This corresponds to a non linear o-model: (2.11) now reads

(me - [ oss) = [ De 5(e* — 1)

Since it turns out that (e) = 0, we may start considering the two-point

function: inserting a complete set of states we have

(0 | ea(s1 )ea(s2) 10) = 3 „(0 | ca |) Án |eg |0)e~hl#tT*2|, (2.16)

Tt

Taking the ground state | 0 > means looking at the limit L —+ oo

In the limit | s; — s2 | 00, ie the JR domain, and since we have

a mass gap, this sum is dominated by the first non zero term, and we

| s1 — 82 | co

Since [ e,(s) ds is a sum over practically independent variables, we ex-

pect a Gaussian distribution for it

since higher states contribution is exponentially damped

Hence:

L— oo

We can then conclude that in the large ZL limit the gap in the spectrum

implies the following form of the Fourier transform of the amplitude

We can also notice that in (2.19) the length of the path L is related

to pŸ : this means that

for the case of spinless particles Let us consider now the case of a

spinning particle The spin of a massive particle may be defined in D

dimensions (D > 2) as follows We take the rest frame of the particle

and a straight path along the time direction

x?

Fig 8.

¿U

This describes a particle at rest The wave function of a spinning

particle transforms according the spinor representation of the rotation

group O(D — 1) acting on the remaining dimensions, whose generators

are given as usual by

In order to generalize this definition to curved paths we introduce a

frame at each point of the path (see fig 9) and then we have for the

The ordered exponential is a product of infinitesimal rotations so that

this formula fits our intuition One can show that it is possible to write

®(P) in the following form:

L %(P)= m Pexp | sas Whours) el (2.27)

The ordered exponential is a product of infinitesimal rotations so that

$(P) in the following form:

h

$(P) = 11} Peso | “us whey (s) os (2.27)

where

(0uy(S) = #[u3„] uạU =1, D, (2.28)

and we are working with #? = 1

$(P ) = exp x 5h, Ƒ da as s (2.29)

Now we may ask what are the consequences of introducing ® into the

path integral for our non linear o-model in two dimensions: this amounts

to long range correlations

In two dimension the generating functional can be expressed in term

of the angular variable a:

z~ [Dosen] -2 [' ($2) a] aan

Written in this form we may read off the energy levels, which are those

of a plane rotator

so that without the spin factor we have a gap in the spectrum

If we now include ®, we find

z~ [ Des)e| -+ [ (2) ds +2 xf “Eas (2.32)

where 3/27 is the spin of the particle

The energy levels are then modified in the following way

v

Now we may think of the velocities e, as Pauli matrices o, acting on

the two degenerate ground states: therefore we obtain

(ec ƒ vee) ~„ ei(2:p)E (2.36)

and the amplitude F(p) becomes

Having completed the analysis for the spinless and spinning particle

in D = 2, let us now pass to consider the three dimensional case

The C;¿(s) defined in (2.25) are given by the torsion of the path, and

r= | O(s)ds= | as [ due [5° x SI (2.43)

Note that if we change the interpolation function from e; to e2 the

corresponding integrals J; and J2 are such that

and this is equivalent to the Dirac quantization condition for a magnetic

monopole,

Therefore we have a well defined quantum problem, that is a charged

The generating functional is of the form

/ De 6(e” — 1) exp (-= / ) exp ( / C(e) ds) (2.45) and the energy levels are again given by

E, = yol(1 + 1) with 1 > 1/2 (2.46) The eigenfunctions 7 are given by generalized spherical functions

~Djlj (6) m=—lÏ, ,L 1> 1/2 (2.47)

and we see that the ground state 7 = 1/2 is twice degenerate

As in the two dimensional case the vector e can be thought of as Pauli

matrices o acting on the two ground states:

eD!1⁄2 m,1/2 (e) ~ Oram! Daur 1 /2(€) + ree, (2.48)

Therefore in the yo — oo limit we find

(Ca, ($1 ea, ($2) +°*) ~ [ deste? — ret C9 1 (51 Jeag (82): `

~ Tr(ØœiØa; '*)- (2.49)

A more direct way of proving (2.49) (which is essentially a version of geo-

metric quantization) is to use the equation of motion and Ward identities

for €4 (s) coming from (2.43) and compare them with the properties of

the representation matrices og

Thus we can show that the introduction of the spin factor reproduces

the Dirac propagator; as before this gives a fractal dimension 1 to the

path.

324 A, Polyakov

3 Spin factor and fermionic integral

In the last lecture we have derived an expression for the spin factor in

terms of a path ordered exponential However this expression is not easy

to manipulate In addition we are looking for an expression generalizable

to the case of strings Therefore the aim of this lecture will be to derive

the spin factor in terms of a fermionic functional integral, which will

turn out to have a 1-dimensional supersymmetry

The starting point for this program is the following formula

{ Detar lu + Cu(9) } =n Pexp | Ci;(s) Di; asl = $(P)

0

(3.1) where ? = Ì, , 2 — 1

One way of proving (3.1) is to compute the eigenvalues of the oper-

ator in curly brackets, choosing an appropriate regularization and then

to calculate the determinant, Here we shall use another method We

shall consider a fermionic functional representation of the determinant,

where ¢ ’s are anticommuting Grassmann variables and satisfy antiperi-

odic boundary conditions ¢;(L) = —¢,(0) To show (3.1) we expand the

r.h.s, of (3.2) in powers of C;;; by using the Wick theorem and the fact

Thus we see that there is a certain correspondence between fermionic

fields ¢; and 7; matrices; in fact the computation of a correlation function

of ¢; is the same as taking the trace of y matrices

As a last remark notice that for periodic boundary condition on the

fermion fields, the result is

( óti(51)Ó7](s1)Ó[k(52)Ón(s2) )p = TY [œs Đi Đm] (3.5)

What we need now is a formula that can be interpreted physically

Let us briefly discuss the use of Lagrange multipliers in the functional

The function 6(e? ~ 1) can be represented as an integration over a La-

grange multiplier A(s):

At this point one can show the following facts:

1) in the D — oo limit, the Lagrange multiplier can acquire a non

vanishing vacuum expectation value

0

which can be rewritten as

L Z= | Dx exp ~const | x’ ds (3.11)

0

326 A Polyakov

Let us come back to the functional integral of spinning particles and use

the fermionic representation of the spin factor

Using (3.1), (3.2) and (3.11) we obtain

F= | dLexp—pioL | DxDé; oso( =5 Ỉ x? is)

x exp | (9:0; + Ci; 9i9;) (9.12)

We can rewrite (3.12) in terms of Neveu-Schwarz fermions #, =

This allows us to rewrite (3.12) in the form

F= [~ dL exp —poL | PxDuê (ụ -%)exp — f (x? - 4-4) ds,

3.16

The sign ambiguity, mentioned before, comes from the global angele

in the ~-integral appearing when we change the basis in (3.13) — this

rotation can change the sign of functional integral

Definition (3.13) and constraint (3.14) tell us that the field ~ is like +

matrices attached to each 1 point of the path ; and orthogonal to it

that it is ; possible to eliminate degrees of freedom which are parallel to

the path, and to consider only orthogonal ones We shall show in a

moment how to enforce supersymmetry in the action in (3.16)

First we implement the constraint (3.14) by using a Lagrange multi-

plier ¥ (the gravitino field) and we obtain

| | S= [os Fa — <i + ¥(b +x) + wok (3.17)

Then we reintroduce the metric, which we have eliminated in (2.6) in

order to have arbitrary gauge In addition it is convenient to introduce

an einbein (7) such that

g(r) = h?(r) (3.18)

Finally we find the covariant form of the action (3.17) to be:

S=5 / ds atx? —~p-pt+h Xy- X+uoh] (3.19)

For fo = 0 this action is invariant under the following local supersym- metry transformation:

For fo # 0 the term jioh breaks supersymmetry We can restore it by

introducing a new fermionic field 7s in the following form:

h = 1 for a spinning particle

But now.the gravitino 4, originally introduced as a Lagrange multi- plier, can enter the game and make strange things happen In fact if we choose 1 = 0 (which corresponds to the superconformal gauge in string

theory) we recover the bosonic case and find fractal dimension h = 2

The answer to this paradox is quite interesting; we have to recall that

we started in an euclidean space, which means

IL5 44, f OLYaROV

But the transformation of 4 violates this condition, so that the integra-

tion domain over the metric is not supersymmetric

So we see that we don’t really have gauge freedom, and this gives rise

to an anomaly which produces the wrong Hausdorff dimension

This is a non perturbative anomaly, which can also be relevant in

string theory: we usually take a metric tensor on the world sheet, gas,

to have Euclidean signature But that violates supersymmetry, since

positivity of gg, is not preserved It may have important consequences

for non-critical strings and for higher loops of critical strings In princi-

ple, one can choose a supersymmetric region of integration

h(r) + Hà *?ủs — tbs ths > 0

and this will give a correct answer for the integral

4, 2D surfaces and strings

In this lecture we shall generalize our considerations about the point

particle to the case of two dimensional surfaces and strings To do that

we start describing the extrinsic geometry of the string

At a point € on the two-dimensional surface we define the tangent

plane to be spanned by the vectors e,(€), a = 1,2 The vectors normal

to the tangent plane are denoted by nj,(€), 2 = 1, , D—2 (see fig 10)

where A, B and C are 1-forms on the two dimensional manifold

This construction can be repeated at each point of the surface, thus having a set of tangent planes which form the Grassmannian manifold

On SO(D) 2,2 ~ $0(2) x SO(D — 2)’ (4.2)

We now introduce some fermion fields ¢;, 1 = 1, ,D — 2, which are perpendicular to the surface and we give their Dirac action when coupled

to 2 dimensional gravity:

Sp — / dˆ£ 99: Va«3" (2 + = Aa? )6 + c ?; (4.3)

where Vag, @=a=1, 2, is the zweibein defined by

and we used the fact that

Age” [y"*,7"] = Aay’ (4.5)

We postulate that the spin factor is given precisely by

8(P) = / Dạ, e~ S9) | (4.6)

and we shall show in a moment that it is possible to reproduce the heterotic string from this expression (for the spinning string this is still

an open problem)

The action for the heterotic string is given by

Su = f Pe JGlOpXO_X + X (4+ O4X) + yO] (48)

Eliminating the Lagrange multiplier V leads to the constraint

by -04X = 0 (4.8)

To obtain (4.3) from (4.7) we define

~ g4 log A2/q?

However, in general, the role of ¢-fluctuations is not yet clear

Before we proceed, we should also mention that it is possible to cal-

culate the critical dimension d = 10 from the action (4.3) even though

there is no supersymmetry in this action and therefore no fermionic ghost contributes

Let us now compute the fractal dimension for the bosonic string The generalization of the term proportional to the length in the point-particle case is the area as given by the Nambu-Goto action

(C+ (9) + (—4))

where |

and jlo is the bare surface tension

As before we have to check the possibility of adding other terms to the action which depend on the extrinsic geometry

An obvious choice is

S' = 5 / Jag? (qn' + CUn;)(Ogn' + CH ng) (4.14)

where C*™* was defined in (4.1)

This term can be interpreted as a non linear o-model, where the nỉ

span the Grassmann manifold (4.2)

Therefore we can rewrite S$’ in the form

1 S= 3% | Via? Oatandptan (4.15)

0 where

or equivalently

A being the covariant laplacian in the metric g

Now the question is whether there are short or long distance correla- tions in this system, which in turn imply the existence or non existence

of a mass gap in the spectrum

We can also say that long range correlations between normal fields

forces the surfaces to be quite regular, since the n,’s have to fluctuate

coherently over distant points On the other hand short range correla- tions can produce irregularities on the surface and may eventually lead

to h = oo

We now try to discuss this point in a more precise but still qualitative way, and we start by introducing the concept of surface tension o(jio ) Let us consider a closed contour C: in the limit in which its size goes

to infinity, we expect that the amplitude (1.17) for C) = C, = C goes like

where Ảmin(C ) is the minimal area

As a function of j49,0 is expected to behave in a “critical” way

Z(Ma) ~ (Mạ — Moeut)" (4.19)

where the index v is related to the Hausdorff dimension h:

However one can see that, in the case of short range correlations, o( {19 )

does not actually go to zero but has some constant value In fact if we introduce some Lagrange multiplier in the action S$ + S’, we have

S+S'= uo | eva | d?E\°F /G(AyXOgX —~ Jap)

+ = / (A(g)X)* /g 7, (4.21)

332 A Polyakov

and, as in the particle case, the short range regime is characterized by

the appearance of a non zero vacuum expectation value of Az8

(A*#) = const g*# # 0 (4.22)

so that, as before, we are left with the effective action

Sef = const | vast? 0X ‘OX + po [va (4.23)

The value of this constant is precisely o(19) because it is just the first

term in (4.23) that gives the minimal area in the limit of large contours

(4.18)

We also notice that ø(o) plays the role of a mass since it appears in

K}?+ơ

in a 1/D approximation One can actually show, then, that o is inde-

pendent of jo, and of the form

ơ ~e_ const/1o, (4.25)

It is important to stress that this constant value of o cannot be put to

zero with any renormalization of the parameters In fact the study of the

dependence of 7 on jig involves the computation of the Øổ -function for the

non-linear o model Result (4.25) is indeed typical of an asymptotically

free theory, as one could learn from the following example

Let us consider the following 2-dimensional fermionic model

£ = )ôu + gi(09)Ÿ (4.26)

Here we have asymptotic freedom, and a mass arises dynamically as

Even if we introduce a mass term of the form pow? in (4.26) and try

to fine-tune the parameters, it is impossible to obtain mppys = 0 in the

same way as it happens with the surface tension In this respect we

see that the bosonic string is quite pathological; this can be avoided by

adding spin densities to the string

Another way, which works only in four dimensions, is to add a 6-term

to the action; it represents the number of self intersections of the surface

5 Fermionic determinants

In this lecture we shall describe the calculation of fermionic determinants

for non abelian gauge theories in 2 dimensions The general setting of

1) we shall try to find the effective action, leg, that 1s some expression

of the logarithm of the determinant we want to compute;

2) we shall integrate [eq over the gauge fields

Our aim will be to investigate what kind of field theory is induced

by Teg We start by considering point 1) We shall work in a flat

2-dimensional space which is parametrized by the light cone coordinates

The fermions }, ý} of the theory are taken to be Majorana fermions

belonging to the vector representation of an internal SO(JV) group

The Dirac lagrangian for these fermions coupled to an external SO(V)

gauge potential reads:

ean) = (Det ra(Aa + Aa}? = [Dye (5.5)

Before we treat the non abelian case let us study [eq for the abelian

gauge group SO(2): this is called the Schwinger model For SO(2) we

and the gauge transformation reads

In general this is a non local non polynomial expression in A, however

in D = 2 the situation simplifies Let us calculate the first term in (5.8),

the contribution from A is

O<pr<y (5.11)

This form of [eg is not gauge invariant and we have to add a local

counterterm of the form —24+ Á_ to (5.13) in order to obtain a gauge

invariant expression:

Deg ~ 2 (q4A_ —g-A4)° ~ arte (5.14)

If one chooses a regulator that does not break the gauge symmetry, i.e a Pauli- Villars regulator, this term appears automatically The mass-term

of the Pauli-Villars fermion ¥ is of the form:

~ CA,A_ + z(4+A-) foe (5.16)

where C is a dimensionless constant

We can give a physical example of why + and — components originally uncorrelated get mixed

20D 43 if Uty

Let us consider a 1-dimensional Fermi gas with E = p*/2m All levels

are filled up to E'r (see fig 11) and we consider the action of an external

field which can create particles and holes out of the Fermi sea In the

low frequency level important part of the spectrum is concentrated near

We have left (—pr) and right (+pr) proving particles with linear spec-

trum, described by massless Dirac equation But near the bottom of

the well distinction between left and right disappears and we have one

branch of spectrum When the bottom contributes, it mixes A, and A_

in a local way This is chiral anomaly

Before we proceed to the non abelian case, let us briefly comment on

the sign of Teg The amplitude elet has a direct physical meaning;

its modulus squared is the probability for the vacuum to remain vac-

uum under the action of an external field In order to maintain this

We shall now derive the effective action for the non abelian case It is

convenient to choose the following parametrization:

A_ = —(O_h)h™

9.19

A, = —(O49)9~* 8)