FIAN/TD-19/95

ITEP/TH-4/96

Exact solutions to quantum field theories and integrableequations 1

A.Marshakov 2

Theory Department, P. N. Lebedev Physics Institute , Leninsky prospect, 53, Moscow, 117924, Russia

and

ITEP, Moscow 117259, Russia

Abstract

The exact solutions to quantum string and gauge field theories are discussed and the existing arguments

in favour of possibility to describe these solutions in the framework of integrable systems are presented. In

particular I consider in detail some examples of appearence of solutions to the first-order integrable equations

of hydrodynamical type and discuss their origins and possible generalizations.

1talk given at: Singular limits of nonlinear dispersive waves, September 4-9 1995, Zvenigorod, Russia and Supersymmetries and

quantum symmetries, September 25-30, Dubna, Russia

2E-mail address: mars@lpi.ac.ru

1 Introduction

In this note I will try to describe some recent advances in modern quantum field theory related with the ap-

pearence of several examples of the exact quantum solutions. Amazingly enough it turns out that a generic

adequate language for the description of the exactly solved quantum theories can be found in the framework

of classical integrable systems and some essential role is played by rather simple first order equations of hydro-

dynamical type or so called Whitham equations on moduli spaces of complex structures of Riemann surfaces

appearing when solving (quasi)periodic problems for the systems of KP or Toda type.

The original problem in quantum field theory defined by ”bare” action functional

S =

∫dDxL(ϕ, ∂µϕ, . . . ; gk) (1)

is to compute the exact correlation function

〈Φk1 . . .Φkn〉 =

∫Dϕe−

1hSΦk1(ϕ, ∂µϕ, . . .) . . .Φkn(ϕ, ∂µϕ, . . .) ≡ Fk1,...,kn(g, u) (2)

as a function of coupling constants gk, quantum characteristics of physical operators k1, . . . , kn (in general,

of course, not being just a discrete set of numbers) and possible parameters of classical solutions uα or moduli

of the theory.

Unfortunately almost never the integral in (2) can be computed exactly. Up to now the only working

methods in quantum field theory are given by perturbation theory and lattice regularization of the problem (2)

resulting usually in some approximations being quite far from the exact answer. However, sometimes it turns

out that it is possible to propose the exact answer not computing the path integral directly and there exists a

beleif that such answers are nevertheless true though they are usually obtained not in a mathematically strict

way.

First of all this is the case of some models of string theory. Using an elegant discretization which is quite

far from an original D = 2 Lagrangian it has been possible to propose exact solutions to the nonperturbative

quantum string correlators in the technique of matrix models. An important point is that the answer was

proposed in the form of a solution to the hierarchy of classical integrable equations. The most advanced results

were obtained for so called topological 2D field theories where it has turned to be possible to get the solutions

explicitely. 1

Second, there are examples of the exact solutions in 4D supersymmetric gauge theories [2, 3] deduced from

the investigation of duality properties and an assumption of complex analiticity of quantum answers as functions

of coupling constants and/or moduli parameters. An important observation is [4] that the proposed exact answer

is again a particular solution to certain differential equation.

Below I will try to pay attention to this universal property of the exact quantum field theory solutions.

It will be demonstrated that various exact quantum answers to different theories can be obtained as different

solutions almost to the same classical problem of (integrable) differential equations. Unfortunately it is still

unclear what is the concrete mechanism of arising these equation in quantum field theory.

1There exists a big set of review papers devoted to this subject. I present here only few of them where the approach to 2D

problems is mostly closed to that developed below [1] (see also references therein).

1

The main hope is that the (first - order) equations of hydrodynamical type are generalizations of the

renormalization-group technique of the perturbation theory. Indeed it is known that in the perturbation the-

ory the scale dependence of quantum correlation functions is related with their dependence on the coupling

constants by means of the first order equation(d

d logµ−∑

βi(g)∂

∂gi

)F (g;µ) = 0 (3)

Naively in the exact solution there is no scale dependence, so the resulting equation could be of the form∑βi(g)

∂F∂gi

= 0, or the derivatives over µ could be replaced by the derivatives over moduli uα. Even accepting

this it is necessary to stress that there is no known way to define β-function beyond the perturbation theory

(i.e. not using the scale dependence βi(g) = ∂gi∂ log µ). One can also hope to find integrable structures in the

nonperturbative calculations investigating the moduli spaces of classical solutions (like instantons in gauge

theories or 2D σ-models) and this way may also lead towards the Hitchin-type constructions.

In what follows I will concentrate mostly on two examples of quantum exact solutions. The first one is the

simplest case of D = 2 topological string theory – pure topological gravity where the exact generating function

of all quantum correlators is known [5]. In this example the correlators Fk1...kn are just integer numbers – so

called intersection indices on moduli spaces of complex structures of Riemann surfaces, i.e. there are no moduli

parameters and of course this is the most well-known and less interesting physically example. The second

(almost a realistic) example is supersymmetric Yang-Mills theory. It is not yet possible to get all quantum

answers in the latter case but what was proposed recently in [2] is the exact nonperturbative dependence of the

coupling constants τ = θ+ ig2 = τ(u) where

L =1

g2TrF 2

µν + iθTrFµνFµν + . . . (4)

on the moduli of the exact solution uα – vacuum values of the Higgs field as well as the allowed massive

spectrum of the theory

M = |na(u) +maD(u)| =

∣∣∣∣(n ∮A

+m

∮B

)dS

∣∣∣∣ (5)

and the effective action of the ”light modes” determined by

aD =∂F

∂a

τ =∂aD∂a

=∂2F

∂a2

(6)

Now we will see that the above formulas can be considered as different solutions of the same problem concerning

classical integrable equations.

There are two main questions concerning the appearence of the integrable equations. First one relates the

effective action of a quantum field theory with the fundamental object in the theory of integrable systems

F = logT – with the τ -function of an integrable hierarchy and the data (5) are expressible in terms of the

generating differential dS of the Whitham hierarchy [4]. Second question arises since the Whitham equations

have lots of solutions while the interesting ones appear in the vicinity of a finite-gap solution of some integrable

system of a KP/Toda-type determined by a spectral curve. Hence, it is interesting to learn what kind of

integrable systems arise in the context of quantum field theory and discuss the particular solutions determined

by their spectral curves.

2

2 Whitham equations

Let us briefly remind the most general definition of the Whitham hierarchy due to Krichever [6]. One has a

local system of functions ΩA on one-dimensional complex curve and the corresponding set of parameters tA so

that it is possible to introduce a 1-form in the space with co-ordinates λ, t where λ is a local parameter on a

curve 2

ω =∑

ΩAδtA (7)

The Whitham equations are

δω ∧ δω = 0 (8)

where

δω = ∂λΩAδλ ∧ δtA + ∂BΩAδtA ∧ δtB (9)

so that one needs to check the independent vanishing of the two different terms – δt4 and δt3δλ. The second

term gives ∑∂λΩA∂BΩC = 0 (10)

for the antisymmetrized sum or introducing some explicit co-ordinates tA0 ≡ x and ΩA0(z, t) ≡ p(z, t) is a local

co-ordinate adjusted to the fixed choice of parameters tA

∂AΩB − ∂BΩA + ΩA,ΩB = 0

ΩA,ΩB ≡∂ΩA∂x

∂ΩB∂p−∂ΩB∂x

∂ΩA∂p

(11)

In fact (11) strongly depend on the choice of the local co-ordinate p. The equations (8), (10) and (11) are

defined only locally thus having huge amount of solutions.

A possible way to get ”globalized picture” is related with the ”modulation” of parameters of the finite-gap

solutions of integrable systems of KP-Toda type. The KP/Toda-type τ -function associated with a given spectral

Riemann curve is

T ti = e∑

tiγijtjϑ(Φ0 +

∑tiki

),

ki =

∮B

dΩi(z),(12)

where ϑ is a Riemann theta-function and dΩi(z) are meromorphic 1-differentials with poles of the order i+ 1

at a marked point z0. They are completely specified by the normalization relations∮A

dΩi = 0 (13)

and the asymptotic behaiviour

dΩi(z) =(ξ−i−1 + o(ξ)

)dξ (14)

where ξ is a local coordinate in the vicinity of z0. The moduli uα of spectral curve are invariants of KP flows,

∂uα∂ti

= 0, (15)

2Actually it might be better to write

ω = ω + Ωλdλ

and consider Ωλ = 1 as a sort of ”gauge condition”.

3

The way the moduli depend on ti after the ”modulation” is defined by the Whitham equations (11). For

z = z(λ, t) so that ∂iz = Ωi, z they acquire more simple form

∂dΩi(z)

∂tj=∂dΩj(z)

∂ti. (16)

These equations imply that

dΩi(z) =∂dS(z)

∂ti(17)

and the equations for moduli, following from (16), are:

∂uα∂ti

= vαβij (u)∂uβ∂tj

(18)

with some (in general complicated) functions vαβij .

In the KdV (and Toda-chain) case all the spectral curves are hyperelliptic, and for the KdV i takes only

odd values i = 2j + 1, and

dΩ2j+1(z) =Pj+g(z)

y(z)dz, (19)

the coefficients of the polynomials Pj being fixed by normalization conditions (13), (14) (one usualy takes

z0 = ∞ and the local parameter in the vicinity of this point is ξ = z−1/2). In this case the equations (18) can

be diagonalized if coordinates uα on the moduli space are taken to be the ramification points:

vαβij (u) = δαβdΩi(z)

dΩj(z)

∣∣∣∣z=uα

(20)

Now, an important point is that the differential dS(z) (17) can be constructed for a generic finite-gap solution,

i.e. for a solution determined by a (spectral) complex curve [7]. In general, it is a meromorphic differential

whose singularities are determined by physical properties such as coupling to matter and gravity. In what

follows several known examples of exact solutions will be discuss and the corresponding generating differentials

(17) will be presented.

3 Spherical solutions and topological gravity

Now let us turn to the simplest nontrivial solutions of the systems (11), (16) related with 2D topological string

theories. Locally they are described by the formulas

z(λ, t)p = λp + up−2(t)λp−2 + ...+ u0(t) (21)

so that

Ωi(λ, t) = z(λ, t)i+

Ωi(z, t) = zi + O(zin−1)

(22)

and in the p = 2 example

z2 = λ2 + U(x, t)

Ωi(λ, t) = z(λ, t)i+ =(λ2 + U(x, t)

)12

+

Ωi(z, t) = zi + O(zi2−1)

(23)

4

Now, if one looks for some particular solutions (trying for example globalize the picture) it might be possible to

interpret λ and z as some ”global” co-ordinate. In the simplest case (21) λ can be taken as a global co-ordinate

on sphere with one marked point (where λ =∞).

For some choice of parameters tk this generic picture gives Whitham solutions coming from the KP/Toda

equations in the following way. For example, starting with the ”zero-gap” solution to the KdV hierarchy

U(x, t) = u = const(∂2t1

+ u)

Ψ = z2Ψ

∂tkΨ =(∂2t1 + u

) k2

+Ψ

Ψ(λ, t) = e∑

k>0tkz

k(λ)+

(24)

one comes to

Ω1 = log Ψ(λ, t)t1 = z(λ)+ = λ =√z2 − u ≡ p(z)

Ωi = log Ψ(λ, t)ti = zi(λ)+ = λ3 +3

2uλ =

(z2 − u

) 32 +

3

2u√z2 − u

(25)

i.e. the formulas (22), (23). The generating differential (17) is now

dS(z) =∑

tkdΩk(z) (26)

and the solution can be found in terms of the ”periods”

tk =1

kres∞

(z−kdS

)(27)

For example, in the KdV case the third equation

∂t3Ω1(z) = ∂t1Ω3(z) (28)

is equivalent to the Hopf or dispersionless KdV

ut3 −3

2uut1 = 0 (29)

with the generic solution

t1 +3

2t3u+ P (u) = 0

P (u) ∼∑

t2n+1un

(30)

Now, for the purposes of quantum field theory one needs to take from (30) the solution of the Whitham equation

which is exact solution of the ”full” KdV – it is

u = −2

3

t1

t3(31)

The generating differential (17) is

dS|t3= 23 ,tk 6=3=0 = λd

(z2(λ)

)= −(λ2 + u)dλ+ d(. . .) (32)

Then the solution to the linear problem (∂2t1 −

2

3

t1t3

)Ψ = z2Ψ (33)

5

gives

Ψ(t1, z)|t3= 23

= Ai(t1 + z2

)φi(z) ∼

∂i−1Ψ(z, t)

∂ti−11

=√

2ze−23 z

3

∫dxxi−1e−

x3

3 +xz2(34)

The determinant formulaT (t + T )

T (t)=

detφi(zj)

∆(z)(35)

results in the τ -function of the whole hierarchy in Miwa co-ordinates Tk = − 1k

∑z−kj . The decomposition of

logT (T ) gives the correlators (2) of two-dimensional topological gravity.

4 Elliptic curves and the Gurevich-Pitaevsky solutions

Let us now turn to the global solutions related with higher genus Riemann surfaces and first discuss the elliptic

case. Now, in contrast to the previous example the Whitham times are nontrivially related to the moduli of the

curve. From a standard definition (see [6]) in the case of the curve y2 =∏3i=1(x− ui) one has

tk =2

k(2 − k)res(x1−k2 dy) (36)

or substituting x = ℘(ξ) + c and y = 12℘′(ξ), where ℘(ξ) is the Weierstrass ℘-function, we find

tk = −2

k(2− k)res(

(℘(ξ) + c)1− k2℘′′(ξ)dξ)

=

= −6

k(2 − k)Res

dξ

ξ6−k

(1 + cξ2 +

g2

20ξ4 + ...

)1−k2(

1 +g2

60ξ4 + ...

)=

=2

5δk,5 − cδk,3 +

(3

4c2 −

1

4g2

)δk,1 +O(t−k)

(37)

so that all negative times are dependent t−k = t−k(t1, t3).

The elliptic (one-gap) solution to KdV is

U(t1, t3, . . . |u) =∂2

∂t21log T (t1, t3, . . . |u) =

= U0℘(k1t1 + k3t3 + . . .+ Φ0|ω, ω′) +

u

3

(38)

and

dp ≡ dΩ1(z) =z − α(u)

y(z)dz,

dQ ≡ dΩ3(z) =z2 − 1

2uz − β(u)

y(z)dz.

(39)

Normalization conditions (13) prescribe that

α(u) =

∮Azdzy(z)∮

Adzy(z)

and β(u) =

∮A

(z2− 12uz)dz

y(z)∮A

dzy(z)

. (40)

The simplest elliptic example is the first GP solution with the underlying spectral curve

y2 = (z2 − 1)(z − u) (41)

6

specified by a requirement that all branching points except for z = u are fixed and do not obey Whitham

deformation. The generating differential corresponding to (41) is given by

dS(z) =

(t1 + t3(z +

1

2u) + . . .

)z − u

y(z)dz =

tk>1=0t1z − u

y(z)dz (42)

and it produces the simplest solution to (16) coming from the elliptic curve [8]. From (42) one derives:

∂dS(z)

∂t1=

(z − u− (

1

2t1 + ut3)

∂u

∂t1

)dz

y(z),

∂dS(z)

∂t3=

(z2 −

1

2uz −

1

2u2 − (

1

2t1 + ut3)

∂u

∂t3

)dz

y(z),

. . . ,

(43)

and comparison with explicit expressions (39) implies:

(1

2t1 + ut3)

∂u

∂t1= α(u)− u,

(1

2t1 + ut3)

∂u

∂t3= β(u) −

1

2u2.

(44)

In other words, this construction provides the first GP solution to the Whitham equation

∂u

∂t3= v31(u)

∂u

∂t1, (45)

with

v31(u) =β(u) − 1

2u2

α(u)− u=dΩ3(z)

dΩ1(z)

∣∣∣∣z=u

, (46)

which can be expressed through elliptic integrals [8].

The elliptic solution with all moving branch points is given instead of (42) by

y2 =3∏i=1

(z − ui)

dS = y(z)dz

(47)

and all ui are some functions of the Whitham times. Then one has to fulfil

dp =∂

∂t1dS = ydz

3∑i=1

viz − ui

=z − α(u)

ydz

dQ =∂

∂t3dS = ydz

3∑i=1

wiz − ui

=z2 −

(12

∑ui)z − β(u)

ydz

(48)

where

vi = −1

2

∂ui∂t1

wi = −1

2

∂ui

∂t3

(49)

and α(u) and β(u) are defined as before. Now what one gets is a simple linear system of the equations

Rij(u)vj = Vi Rij(u)wj = Wj (50)

with

Rij(u) =

1 1 1

u2 + u3 u1 + u3 u1 + u2

u2u3 u1u3 u1u2

(51)

7

and

V T = (0,−1,−α(u)) WT = (1,1

2(u1 + u2 + u3),−β(u)) (52)

The solution of linear system looks similiar to (45), for example:

w1

v1=u2

1 −12(u1 + u2 + u3) − β(u)

u1 − α(u)(53)

etc. The second GP solution 3

y2 =3∏i=1

(z − ui)

dS = (z − e)ydz

(54)

can be obtained from elliptic curve with a marked point. Now, one might expect that a ”modulated” τ -function

would still have a form (12) is

T ti = eF(t)ϑ (S(t)) (55)

so that

ki(t) =∂S

∂ti(56)

and the poles of the ”effective” potential

u(t) =∂2

∂t21log T (57)

can be identified with the massive excitations (5).

5 Toda-type curves

The above procedure can be also applied to the finite-gap solutions determined by more general types of algebraic

curves. The special case of the Toda-chain curves [17]

y2 = Pn(z)2 − 1

2Pn(z) = w + w−1 2y = w −w−1

(58)

was notices [4] to correspond to the N = 2 SYM exact solutions with the gauge group SU(n = Nc)4. The

generating functional for the Toda-type theories is given by

dS = dz log(Pn(z) +

√Pn(z)2 − 1

)dΩj(z) =

∂dS

∂tj=

dz√Pn(z)2 − 1

∂Pn(z)

∂tj

(59)

and corresponds to a rather specific Whitham solution. Indeed, for the SL(2) case the generating differential

(42) had a simple zero only in one of the branch points z = u, which means that the only branch point is

3corresponding to the Yang-Lee edge singularity or (2,5) conformal minimal model interacting with 2D gravity4In fact one can show that the elliptic curve (41) is a particular case in the class of curves discussed below. Indeed if one takes

w +w−1 = 2P1(z) = 2z

and considers the double cover

z = u+ x2

one gets (41) after the change of variables ynew = yx [12, 16].

8

influenced by the Whitham evolution. Now, it is easy to see that the zeroes of generating differential are at the

points

Pn(z) = 1 (60)

i.e. the Whitham dynamics is nontrivial only for the half of the branch points (y = 0 in (58)). In the special case

of n = 1 after the change of variables discussed in the footnote above the generating differential (59) acquires

the form of (42).

Now let us turn to the question what are the most general possible type of an integrable system appearing

in the formulation of the exact quantum field theory solution. The pure Yang-Mills theory corresponds to

the Toda-chain (KdV) systems being the simplest ones among integrable models of this type. Their existing

generalizations can be possible looked for in two different directions. One and the most obvious is towards the

Calogero-Moser particle systems and this is the case discussed from various points of view in [12, 15, 16, 14]. This

line ends up at the free N = 4 supersymmetric Yang-Mills theory which is described in terms of holomorphic

generating differential dS 5.

Another question is coupling of the YM theory with real matter, i.e. belonging to the fundamental repre-

sentation of the gauge group. Such theories are described by the following curves [2, 18]

y2 = Pn(z)2 − Pm(z) (61)

where m = Nf is the number of flavours or matter multiplets, which correspond to rather specific integrable

systems 6. In principle, an integrable system corresponding to a generic curve can be constructed by standard

methods of finite-gap integration [7].

A generic integrable system of this type is determined by a (genus g) spectral curve, a meromorphic generating

differential dS and a set of g given points P1, . . . , Pg or initial conditions. The symplectic structure is defined

by a (closed) form

Ω =

g∑α=1

∂

∂uα

(dS

dz

)∣∣∣∣zα=z(Pα)

δuα ∧ δzα (62)

where as usual the moduli uα for the hyperelliptic cases (47), (54), (58) and (61) can be taken as co-ordinates

of the branch points. Any two functions of moduli uα commute with respect to the Poisson bracket induced

by (62) thus giving the set of g independent Hamiltonians. The integrals

θα =

g∑β=1

∫ zα

z0

∂dS

∂uβ(63)

are the co-ordinates on the Jacobian of a spectral curve if ∂dS∂uβ= dωβ are the holomorphic differentials.

5The fact that the N = 4 SYM theories are described in terms of holomorphic differentials can be easily understood assuming

the connection of the Whitham hierarchy with the renormalization group approach. Indeed, on one hand the N = 4 theory has

zero β-functions and on the other hand it correspond to a trivial Whitham modulation. Since the holomorphic differentials have

no singularities on spectral curve there is no natural way to introduce higher times of the KP/Toda type which is again consistent

with the conformal invariance or trivial interaction with the ”scale part” of metrics in N = 4 theory.

6such as so called Goryachev-Chaplygin top for n = 3 and massless m = 2

9

The spectral curves (61) can be described by an equation

det (T (z)− w) = w2 −TrT (z)w + det T (z) = 0

2y = w − w−1

y2 = (TrT (z))2 − detT (z) = Pn(z)2 − Pm(z)

(64)

where T (z) is a monodromy matrix. The integrable system can be restored by constructing the Baker-Akhiezer

function on a curve with marked points at the singularities of the monodromy matrix. In cases of (58), (61) the

singular points are P± = (∞,±). The modulation of the integrals of motion again gives rise to the Whitham

equations generated by

dS = dz

(log(Pn(z) +

√Pn(z)2 − Pm(z)

)−

1

2logPm(z)

)(65)

6 String equation. KdV case

Now I will make few remarks concerning the way how in all the cases above a specific solution to the Whitham

equations was distinguished. In general a particular solution to the Whitham hierarchy should be a solution to

some ”string equation” which has a clear interpretation in terms of the spectral function for the auxiliary linear

problem.

Let us consider the Lax equation for KdV

(∂2x + U(x)

)Ψ(E, x) = EΨ(E, x) (66)

as a Schrodinger equation. Then the partition function is

Z(t) = Tre−tH = Tre−t(∂2x+U) =

=

∫x(0)=x(t)

Dxe

∫ t0x2+U(x)

(67)

The Laplace transform of (67) gives

ρ(E) =

∫ ∞0

dte−tEZ(t) = Tr1

∂2x + U(x) −E

=

∫dx∑n≥0

Rn[U(x)]

En+ 12

(68)

Now, using that

Rn[U(x)] =∂2

∂x∂t2n−1log T (x = t1, t3, ...) (69)

one can rewrite (69) as

ρ(E) =∑n≥0

1

En+ 12

∫dx

∂2

∂x∂t2n−1logT ∼

∼∑n≥0

1

En+ 12

∂

∂t2n−1logT (tk −

1

kEk2

)

(70)

Using the string equation formula (70) can be rewritten as

ρ(E) =∂

∂Elog T (tk −

1

kEk2

) ∼ zq

z =√E

(71)

10

i.e. the solutions to string equations are characterized by growing at infinity spectral function and the order of

the singularity at infinity gives a particular critical point.

Let us turn directly to the interpretation of Whitham equations from the point of view of quantum effective

theory. The sense of the Whitham deformation [6] is variation of the spectrum of the auxiliary linear problem

(66). One of the useful languages is using fermionic technique when fermions live on a given spectral curve.

The path integral for the ”spectral” fermions is∫Dψ∗Dψe

∫ψ∗(∂2

x+u(x)−E)ψ =

= det(∂2x + u(x)−E

) (72)

Now using the relation

log det(∂2x + U(x)−E

)= Tr log

(∂2x + U(x)−E

)(73)

and taking into account (68) one finds

∂

∂Elog det

(∂2x + U(x)− E

)= Tr

1

∂2x + U(x)−E

= ρ(E) (74)

and after integration the fermionic effective action

log det(∂2x + U(x)−E

)=

∫ E

dερ(ε) = S(E) (75)

coincides with the ”Whitham effective action” S(E) =∫ E

dS in the sense of [6, 19].

Such fermionic language is well-known in the solid state physics where the total energy can be divided into

fermionic and potential pieces

Etot =∑

fermions

E +Epot [U(x)] (76)

The measure of evaluating the sum in (76) is determined by a quasimomentum so that

Eferm =

∫Edp (77)

Now, the local extremum of (76) corresponds to the choice of a certain critical point if one takes

Epot [U(x)] =∑k≤q

tk

∫dxRk [U(x)] (78)

which gives a relation of the (36) type.

7 Conclusion

In this note I have tried to demonstrate the role played by the integrable equations in the nonperturbative

approach to quantum string and supersymmetric gauge field theories. Using the general phenomenon that

all known exact nonperturbative solutions can be described in terms of spectral curves and corresponding

solutions to integrable systems one might hope that arising integrable equations of hydrodynamical type can

be interpreted in terms of generalization of renormalization group beyond the perturbation theory. Amazingly

enough this construction does not depend too much upon the ”bare” quantum field theory, thus the language

of integrable systems seems to be adequate and universal for studying nonperturbative phenomena.

11

8 Acknowledgments

I am deeply indebted to V.Fock, A.Gerasimov, A.Gurevich, S.Kharchev, A.Mironov, N.Nekrasov, S.Novikov,

A.Zabrodin and especially to A.Gorsky, I.Krichever and A.Morozov for illuminating discussions. I am grateful

to J.Ambjorn and A.Niemi for warm hospitality at Niels Bohr Institute and Institute of Theoretical Physics

at Uppsala University where this work has been started and to the organizers of the conferences where these

results were presented. The work was partially supported by the RFFI grant 95-01-01106, INTAS grant 93-0633

and by NFR-grant No F-GF 06821-305 of the Swedish Natural Science Research Council.

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13