arX

iv:0

706.

2857

v2 [

hep-

th]

30

Oct

200

7

FIAN/TD-15/07

ITEP/TH-24/07

O n M icroscopic O rigin ofIntegrability in Seiberg-W itten T heory�

A.M arshakov

Theory Departm ent,P.N.Lebedev Physics Institute,

Institute ofTheoreticaland Experim entalPhysics,

M oscow,Russia

e-m ail: m ars@ lpi.ru, m ars@ itep.ru

W ediscussm icroscopicoriginofintegrabilityin Seiberg-W itten theory,followingm ostlytheresultsof

[1],aswellaspresenttheircertain extension and considerseveralexplicitexam ples.In particular,we

discussin m ore detailthe theory with the only switched on higherperturbation in the ultraviolet,

where extra explicit form ulas are obtained using bosonization and elliptic uniform ization ofthe

spectralcurve.

1 Introduction

Supersym m etric gauge theories have becom e recently an area,which allows the application ofthe nontrivial

m ethods ofm odern m athem aticalphysics. In particular,one is often interested in the propertiesofthe low-

energy e�ective actions, which in the theories with extended supersym m etry can be expressed in term s of

the holom orphic functions on m odulispaces ofvacua,or prepotentials. These prepotentialsobey rem arkable

properties,which can be shortly characterized by the fact,thatthey are quasiclassicaltau-functions [2],and

the low-energy e�ective theory [3]can be form ulated in term sofan integrablesystem [4].

Exact form ofthe prepotentialprovides com prehensive inform ation about the e�ective theory at strong

coupling,while atweak coupling the prepotentialcan be expanded overthe contributionsofthe gaugetheory

instantons. Asoften happensin conventionalquantum �eld theory,even each term in thisin�nite expansion,

containing the integration overthe non-com pactinstanton m odulispace,isill-de�ned. Itturnsout,however,

thatthere existsa preserving supersym m etry infrared regularization,which allowsto perform a com putation,

reducingitto a sum overthepoint-likeinstantons,whosecontributionsareparam eterized by random partitions

[5].M oreover,itturnsourthattheregularized volum eofthefour-dim ensionalspacetim ecan bere-interpreted

asa coupling constantin dualtopologicalstring theory [6],providing a new form ofthe gauge/string duality.

This duality predicts a nontrivialrelation between the deform ed prepotentials ofN = 2 supersym m etric

gaugetheoriesand thegeneratingfunctionsoftheG rom ov-W itten classes.Sim ilartothelatter[7],thedeform ed

prepotentials can be expressed in term s of the correlation functions in the theory of two-dim ensionalfree

ferm ions. These correlation functions can be identi�ed with the tau-functions ofintegrable system s,whose

ferm ionic representation isessentially di�erentfrom the conventionalone [8]. W e shallpostpone the detailed

discussion ofthisissueforthefulldeform ed prepotentialsand concentrate,following[1],theirm ain quasiclassical

asym ptotic.

�Based on the talks at ’G eom etry and Integrability in M athem aticalPhysics’,M oscow,M ay 2006;’Q uarks-2006’,R epino,M ay

2006;Twente conference on Lie groups,D ecem ber 2006 and ’Classicaland Q uantum Integrable M odels’,D ubna,January 2007.

1

2 Prelim inaries

Freeferm ions

Letusintroduce,�rst,them ain de�nitionsand notationsforthetwo-dim ensionaltheory ofasinglefreecom plex

ferm ion the actionR~ �@ on a cylinder. O ne can expand the solutionsto Dirac equation in holom orphic co-

ordinatew 2 C�:

(w)=X

r2Z+ 1

2

r w� r

�dw

w

� 1

2

;

e (w)=X

r2Z+ 1

2

e r wr

�dw

w

� 1

2

;

(2.1)

so thatthe m odesafterquantization satisfy the (anti)com m utationalrelations

f r;e sg = �rs (2.2)

Theferm ionic Fock spaceisconstructed with the help ofthe chargeM vacuum state(a Dirac sea)

jM i= � M + 1

2

� M + 3

2

� M + 5

2

:::=^

r> � M

r (2.3)

with

rjM i= 0;r> � M ; e rjM i= 0; r< � M (2.4)

and these de�nitionscorrespond to the two-pointfunction

h0j~ (z) (w)j0i=

pdzdw

z� w(2.5)

M ore conventional\Japanese" conventions(with the integer-valued ferm ionic operators i, �i,i2 Z,see e.g.

[9])can be gotfrom theseby~ r !

r+1

2

; r ! �

r+1

2

; M = � n (2.6)

Itisalso convenientto use the basisofthe so-called partition states:foreach partition k = (k1 � k2 � :::�

k‘k = 0� 0:::)oneintroducesthe state:

jM ;ki= � M + 1

2� k1

� M + 3

2� k2

:::=^

r> � M

r� ki (2.7)

and de�nesthe U (1)currentas:

J = :e :=X

n2Z

Jnw� n dw

w; Jn =

X

r2Z+ 1

2

:e r r+ n : (2.8)

O bviously

[Jn; r]= � r+ n;

h

Jn;~ r

i

= ~ r� n

[Jn; (w)]= � w n (w);

h

Jn;~ (w)

i

= wn ~ (w)

(2.9)

2

Recallthe bosonization rules:~ = :ei� :; = :e� i� :; J = i@� (2.10)

where

�(z)�(0)� � logz+ ::: (2.11)

and a usefulfact from U (N̂ ) and perm utation’s group theory: the Schur-W eylcorrespondence,which states

that

(C N̂ ) k =M

k;jkj= k

R k R k (2.12)

asSk � U (N̂ )representation.Now letU = diag�u1;:::;uN̂

�bea U (N̂ )m atrix.Then oneeasily getsusing the

W eylcharacterform ula,and the bosonization rules(2.10),that:

TrR kU = hN̂ ;kj:ei

P

N̂

n = 1�(un ) :j0i = sk(u1;:::;uN̂ )=

detuki+ N̂ � i

j

detuN̂ � ij

(2.13)

givesthe (ratio ofthe)standard Schurfunctionsforany partition k,a very nice review oftheirpropertiescan

be found in [10].In particular,from thisform ula onederives:

eJ� 1

~ jM i=X

k

m k

~kjM ;ki=

X

k

dim R k

~k k!jM ;ki (2.14)

with

m k =dim R k

k!=Y

i< j

ki� kj + j� i

j� i=

=

‘kY

i= 1

(‘k � i)!

(‘k + ki� i)!

Y

1� i< j� ‘k

ki� kj + j� i

j� i=

Q

1� i< j� ‘k(ki� kj + j� i)

Q ‘ki= 1

(‘k + ki� i)!

(2.15)

being the Plancherelm easure.Itfollowsfrom the factthatforparticularvaluesu1 = :::= uN̂= 1

~N̂

sk

�1

~N̂;:::;

1

~N̂

�

=N̂ ! 1

m k

~k(2.16)

Instantonsand Nekrasov’scom putation

In the contextofN = 2 supersym m etric gaugetheories,one usually startswith the m icroscopictheory,deter-

m ined bytheultravioletprepotentialFU V ,which can betaken perturbed byarbitrarypowersoftheholom orphic

operators

FU V = 1

2(�0 + t1)Tr�

2 +X

k> 0

tkTr�k+ 1

k+ 1� 1

2�0Tr�

2 + Trt(�) (2.17)

and quadraticF U V = 1

2��0 Tr��

2.Then oneintegratesoutthefastm odes,i.e.theperturbative uctuationswith

m om enta abovecertain scale� aswellasthenon-perturbativem odes,e.g.instantons(and uctuationsaround

them )ofallsizessm allerthen �� 1.Theresulting e�ectivetheory hasa derivativeexpansion in thepowers @2

�2 .

The leading term sin the expansion are alldeterm ined,thanksto the N = 2 supersym m etry,by the e�ective

prepotentialF (�).As� islowered alltheway down to zero,wearriveattheinfrared prepotenialF U V ! FIR .

The supersym m etry considerations suggest that the renorm alization ows ofF and �F proceed m ore or less

3

independently from each other. Thusone can sim plify the problem by taking the lim it,��0 ! i1 ,while FU V

kept �xed. In this lim it the path integralis dom inated by the gauge instantons. The setup of[5]allows to

evaluate theircontribution,aswellasthe contribution ofthe uctuationsaround the instantons,exactly.The

priceonepaysistheintroduction ofextra param etersinto theproblem ,som esortoftheinfrared cuto�,which

wedenoteby ~� 2,sinceitappearsto bea param eteroftheloop expansion in dualtopologicalstring theory [6].

In particular,for the so-called noncom m utative U (1) theory,or the theory on a single D3 brane in the

background,which preservesonly sixteen supercharges(so thatthe theory on the brane hasonly eightsuper-

charges)1,theinstanton partition function Z(a;~;t),t= (t1;t2;:::),can beshown to begiven by thesum over

the Young diagram s,i.e.overthe partitions[5,6,11]:

Z(a;t;~)=X

k

m2k

(� ~2)jkjexp

1

~2

X

k> 0

tkchk+ 1(a;k;~)

k + 1(2.18)

where m k isthe Plancherelm easure (2.15),and the Chern polynom ialschk+ 1(a;k;~)can be introduced,e.g.

via�

e~ u2 � e

� ~ u2

� 1X

i= 1

eu(a+ ~(

1

2� i+ ki)) =

1X

l= 0

ul

l!chl(a;k;~) (2.19)

If the theory has the gauge group U (N ), e.g. it is realized on the stack of N fractionalD3 branes, the

corresponding partition function isgiven by the generalization of(2.18):

Z(~a;t;~)= Zpert(~a;t;~)

X

~k

�

m (~a;~k;~)

�2(� 1)j

~kjexp1

~2

X

k> 0

tkchk+ 1(~a;~k;~)

k+ 1(2.20)

where m (~a;~k;~) is the U (N ) generalization ofPlancherelm easure [11]and Z pert(~a;t;~) is the perturbative

partition function.

Toda chain and tau-functions

Consider,�rst,the well-known form ula forthe tau-function ofToda m olecule (orthe open N-Toda chain with

co-ordinatesqn(t;a)= logZ (t;nja)

Z (t;n� 1ja)[12]),given by allprincipaln-m inors

Z(t;nja)=X

K :i1< :::< in

�K (a)2 exp

X

l;ik

tlH l(aik )= � n� n

�A � D � A

T�

(2.21)

ofthe N � N m atrix,expressed asa m atrix productwith A ij � aj� 1

i ,D ij = �ijexp(zi)Q N

k= 1;k6= ijai� akj

� 1,

where

zi =X

l

tlH l(ai)=X

l

(tlali+ :::)+ z

(0)

i (2.22)

with som eappropriately chosen "initialphases" z(0)

i .Rewriting (2.21)in the form

Z(t;nja)=X

jK j= n

Y

i2K

ezi

Q N

k= 1;k6= ijai� akj

Y

i;j2K ;i6= j

(ai� aj)2

(2.23)

weseethatthesum in (2.21)isin facttaken overthepartitions(k1 � k2 � :::� kn)with the�xed length and

kj = in� j+ 1 + j� n � 1; j= 1;:::;n (2.24)

1This theory can also be realized at a special point on the m oduli space of U (N ) gauge theory with 2N � 2 fundam ental

hyperm ultiplets.

4

Forthe particularsolution ofthe Toda chain with ai ’ i,onegetsfor(2.23)

Z(t;n)=X

jK j= n

Y

i2K

ezi

Q

i;j2K ;i6= j(i� j)2

Q

i2K(i� 1)!(N � i)!

(2.25)

Thisisa singularor"stringy" solution,presenting a collection ofparticles,m oving each with a constantspeed,

proportionalto itsnum ber.In K P/K dV-theory theanalog isu / x=t,a lineargrowing potentialofK ontsevich

m odel[17],which nevertopples.

Com paring (2.25)to (2.21),one�ndsthat

�K (a)2��ai= i

=

� Q

i;j2K(i� j)

Q

i2K(i� 1)!

� 2 Y

i2K

(i� 1)!

(N � i)!= m

2k

��‘k = n

Y

i2K

ez(0)

i (2.26)

In thelim itN ! 1 ,afterparticularchoiceofthe Ham iltonians(2.19)

H l(ai)jai= i !chl+ 1(a;i;~)

l+ 1(2.27)

and renorm alization oftheinitialphasez(0)

i ,passingfrom sum m ation overpartitionswith a �xed length ‘k = n

to a "grand-canonical" ensem ble by a sort ofFourier transform ,one gets Z(t;n) ! Z(a;t;~) the (2.18)

partition function.By (2.14),itbecom esequivalentto the following ferm ioniccorrelator

Z(a;t;~)=X

k

m2k

(� ~2)jkje

1

~2

P

k> 0tk

chk+ 1(a;k )

k+ 1 = hM je�J1~ e

1

~

P

k> 0tk W k+ 1e

J� 1

~ jM i (2.28)

wherethe m utually com m uting m odesofthe W -in�nity generatorscan be de�ned as

W k+ 1 = �~k

k + 1

I

:~

�

wd

dw+1

2

� k+ 1

�

�

wd

dw�1

2

� k+ 1!

:=

=~k

k+ 1

X

r2Z+1

2

�(� r+ 1

2)k+ 1 � (� r� 1

2)k+ 1

�: r ~ r :

(2.29)

Them atrix elem ent(2.28)isa particularnon-standard ferm ionicrepresentation ofthe tau-function,wherethe

Toda tim esare coupled to the W -generators(2.29)instead ofthe m odesofthe U (1)current(2.8),and ithas

been discussed in [7,13].

Ifonly t1 6= 0 the correlatorin (2.28)gives

Z(a = ~M ;t1;0;0;:::)= hM je�J1~ e

1

~

t1L 0eJ� 1

~ jM i= exp

�

�1

~2

�1

2t1a

2 + et1��

= exp

�

�FP

1

~2

�

(2.30)

the partition function oftopologicalstring on P1. Thisisthe only case when sum m ing overpartitionscan be

perform ed straightforwardly,using the Burnsidetheorem

X

k;jkj= k

m2k=

1

k!2

X

k;jkj= k

dim R 2k=

1

k! (2.31)

5

Baker-Akhiezerfunctions

In addition to (2.14),one can consider

~ � r eJ� 1

~ jM + 1;;i=X

k

~CkjM ;ki (2.32)

with (com puted by the W ick theorem and using the propertiesofthe Schurfunctions(2.13))

~Ck = hM ;kj~ � r eJ� 1

~ jM + 1;;i= ~M � jkj� r�

1

2

1Y

i= 1

i� ki+ r� 1

2� M

im k (2.33)

wherethe in�nite productisactually �nite

1Y

i= 1

i� ki+ r� 1

2� M

i=

1

�(r+ 1

2� M )

‘kY

i= 1

i� ki+ r� 1

2� M

i+ r� 1

2� M

(2.34)

Therefore,onegetsforthe Baker-Akhiezerfunctions

~(r)=hM je�

J1~~ � re

1

~

P

k> 0tk W k+ 1e

J� 1

~ jM + 1i

hM je�J1~ e

1

~

P

k> 0tk W k+ 1e

J� 1

~ jM i=

=~M � r�

1

2

Z(a;t;~)e

1

~2

P

k> 0

tk ~k

k+ 1

“

(r+1

2)k+ 1

� (r�1

2)k+ 1

” X

k

m2k

(� ~2)jkje

1

~2

P

k> 0tk

chk+ 1(a;k ;~ )

k+ 1

1Y

i= 1

i� ki+ r� 1

2� M

i=

= ~M � r�

1

2 exp1

~2

X

k> 0

tk~k

k+ 1

�(r+ 1

2)k+ 1 � (r� 1

2)k+ 1

��eM �

0(r+

1

2)=�(r+

1

2)

�(r+ 1

2)

�Z(a;t� �(r);~)

Z(a;t;~)

(2.35)

with a = M ~ and the shift�(r)= (�1(r);�2(r);:::)generated by

1

~2

X

k> 0

�k(r)xk+ 1

k+ 1= log

��r+ 1

2� x

~

�

��r+ 1

2

� +x

~

�0(r+ 1

2)

�(r+ 1

2)

(2.36)

In thequasiclassicalasym ptotic~ ! 0,with ~r= z,(2.35)gives

~ (z;a;t;~)� exp1

~

X

k> 0

tkzk � z(logz� 1)+ alogz+ :::

!

(2.37)

(sim ilarform ulasin the contextof�ve-dim ensionalSeiberg-W itten theory [14]wereconsidered in [15]).In the

sam eway onecan de�nethe two-pointfunction

E(r)=hM + 1je�

J1~ � r

~ � re1

~

P

k> 0tk W k+ 1e

J� 1

~ jM + 1i

hM je�J1~ e

1

~

P

k> 0tk W k+ 1e

J� 1

~ jM i�

� exp1

~2

X

k> 0

tk~k

k+ 1

�(r+ 1

2)k+ 1 � (r� 1

2)k+ 1

��e2M �

0(r+

1

2)=�(r+

1

2)

�(r+ 1

2)2

�Z(a;t� 2� �(r);~)

Z(a;t;~)�

�~! 0

expS(z;a;t)

~

(2.38)

6

with

S(z;a;t)=X

k> 0

tkzk � 2z(logz� 1)+ 2alogz+ ::: (2.39)

The asym ptotics(2.39)playsan essentialrole in the study ofthe quasiclassicalsolution. Form ula (2.38)can

be also interpreted asaverage ofthe r-th Fourierm ode ofthe \sym m etrically splitted" bi-ferm ionic operator

E(�) =Hdw

w

�we� �=2

�~ �we�=2

�, introduced in [7]. The \doubling" of the ferm ions and their sym m etric

splitting along the w-cylinderturn into the doublecovering ofz-planeby the quasiclassicalspectralcurve.

Bosonization

O n a sm allphase space (tk = 0 with k > 1) the tau-function ofP1 m odel(2.30) can be easily com puted

exploiting thefactthatitcan bepresented asa m atrix elem entoftheevolution operatorin harm onicoscillator

forthe initialand �nalcoherentstates.Indeed,afteridentifying L 0 =1

2J20 + J� 1J1 = � 1

2M 2 + 1

~2 �

y�,where

[�;�y]= ~2,on the eigenstateswith J0 � M = a

~,one getsfor(2.28)

Z(a;t1;0;0;:::)= exp

�

�t1a

2

2~2

�

h0je� �

~2 e

t1

~2�y�e�y

~2 j0i= exp

�

�t1a

2

2~2

�X

n� 0

et1n

(� ~2)nn!=

= exp

�

�1

~2

�1

2t1a

2 + et1�� (2.40)

whereindependentofthechargea part(generated by theworld-sheetinstantonsin P1 topologicalstringm odel)

is just a kernelofthe evolution operator in the holom orphic representation with �xed at the boundaries �in

and �y

out. Thisresultiscertainly exactquasiclassically,here in the \stringy norm alized" Planck constant(~2

instead of~).

Ifthe second tim e t2 isalso switched on,the partition function

Z(M ;t1;t2;0;0;:::)= hM je�J1~ e

t1L 0+ ~t2W 0eJ� 1

~ jM i (2.41)

isno longerdescribed in term sofa singlequasiparticle.Now onegets

L0 ! H 0 =X

n> 0

n�yn�n = �

y� + 2A y

A + ::: (2.42)

the system ofcoupled oscillators [�n;�ym ] = ~

2�nm with the quadratic Ham iltonian, perturbed by special,

preserving the energy due to [H 0;H I]= 0,interaction

W 0 ! H I =p2�(�y)2A + �

2Ay�+ ::: (2.43)

M orestrictly,

Z(M ;t1;t2;0;0;:::)= et(a)

~2 h0je

� �

~2 exp

1

~2(t00(a)H 0 + t2H I)e

�y

~2 j0i (2.44)

with t(a) = t1a2

2+ t2a

3

3,t00(a) � t1 + t2a. Therefore the problem reduces to the com putation ofthe m atrix

elem ents

h0je� �

~2 exp

1

~2(t00(a)H 0 + t2H I)e

�y

~2 j0i=

1X

k= 0

t2k2

(~2)2k(2k)!h�jH 2k

I j�0i= (2.45)

forthe coherentstates

�nj�i= ��n;1j�i (2.46)

7

...

Figure 1:Prepotential,asa sum ofallconnected tree diagram sin the bosonic cubic �eld theory.Each vertex

isweighted by t2 and each pairofexternallegs{ by et00(a).Thedepicted diagram m scorrespond literally to the

contribution oftruncated \bosonicBCS m odel",with the dashed linesbeing the hAA yi-\propagators".

with � = � 1

~2and �0= 1

~2expt00(a),h�j�0i= exp

�

�t00(a)

~2

�

.

Q uasiclassically,the m atrix elem ent(2.45)gives

h0je� �

~2 exp

1

~2(t00(a)H 0 + t2H I)e

�y

~2 j0i �

~! 0exp

�

�1

~2F(t00;t2)

�

(2.47)

so thatforthe prepotentiallogZ = � 1

~2 F + O (1)onegetsfrom (2.44),(2.47)

F = t(a)+ F(t00(a);t2) (2.48)

Its nontrivialpart F(t00(a);t2) can be presented as a sum over allconnected tree diagram s (see �g.1) in

the bosonic cubic �eld theory (2.45),which is encoded by appearance ofthe Lam bert function in the exact

quasiclassicalsolution.An interesting issuewould beto solvethistheory exactly,atleastquasiclassically,which

isalready notquite obviouseven forthe \truncated BCS m odel" oftwo coupled oscillators,corresponding to

the �rstterm sin (2.42),(2.43)and diagram sdepicted at�g.1.Thecorresponding classicalsystem

_� = t00� + 2t2�

yA; _�y = � t00�y � 2t2�A

y

_A = 2t00A + t2�2; _A y = � 2t00A y � t2�

y2(2.49)

can be integrated in term sofellipticfunctions,and possesses,in particular,a "kink" solution.W eshallreturn

to a detailed discussion ofthese issueselsewhere.

3 Q uasiclassicalfree energy

Theinstantoniccalculation in N = 2 extended gaugetheory (2.17)givesriseto theSeiberg-W itten prepotential

asa criticalvalueofthe functional2

F =1

2

Z

dxf00(x)

X

k> 0

tkxk+ 1

k + 1�1

2

Z

x1> x2

dx1dx2f00(x1)f

00(x2)F (x1 � x2) (3.50)

extrem ized w.r.t.second derivativeofthe pro�lefunction f00(x)=d2f

dx2with the kernel

F (x)=x2

2

�

logx �3

2

�

(3.51)

coincideswith the perturbativeprepotentialofpureN = 2 supersym m etricYang-M illstheory.Form ula (3.50)

m eansthatthequasiclassicalfreeenergy forthepartition functions(2.18)and (2.20)issaturated onto a single

2W e choose here di�erent(by a factor of2)norm alization ofthe tim e variablest com pare to [1]and correct som e m isprints.

8

\large"partition k� with thepro�lefunction fk�(x)= f(x),whereforeach partition k = k1 � k2 � :::� k‘k �

k‘k + 1 = 0;:::the pro�lefunction isde�ned by

fk(x)= jx � aj+

‘kX

i= 1

(jx � a� ~(ki� i+ 1)j� jx � a� ~(ki� i)j� jx � a� ~(1� i)j+ jx � a+ ~ij) (3.52)

(see[11,1]fordetails).In particular,one can writefor(2.19)

chl(a;k)=1

2

Z

dx f00k(x)xl �

1X

i= 1

�(a+ ~(ki� i+ 1))l� (a+ ~(ki� i))l

�(3.53)

The variationalproblem forthe functional(3.50)should be solved upon norm alization condition forf(x)and

the constraint

a = 1

2

Z

dx xf00(x) (3.54)

which can be in standard way taken into accountby adding itwith the Lagrangem ultiplier

F ! F + aD

�

a� 1

2

Z

dx xf00(x)

�

(3.55)

having a sense ofthe k = 0 term in the sum m ation in form ula (3.50). The whole setup of(3.50) is alm ost

identicalto the standard quasiclassics ofthe m atrix m odels,where the Coulom b gas kernelis replaces by a

(m ultivalued!) Seiberg-W itten function (3.51).

The extrem alequation forthe (3.50)gives

X

k> 0

tkxk �

Z

d~xf00(~x)(x � ~x)(logjx � ~xj� 1)= aD

(3.56)

on the supportIwheref00(x)6= 0.G enerally,forthe m icroscopicnon-abelian theory thissupportconsistsofa

setofseveral(disjoint)segm entsalong the realaxisin the com plex plane,where the �lling fractionsare �xed

separately with the help ofseveralLagrangem ultipliers,seebelow.Equation (3.56)m eansthat

S(z)=X

k> 0

tkzk �

Z

dxf00(x)(z� x)(log(z� x)� 1)� a

D(3.57)

isan analyticm ultivalued function on the double-coverofthe z-planewith the following properties:

� The realpartofthe m ultivalued function (3.57)vanishes

S(x)= 1

2(S(x + i0)+ S(x � i0))= 0; x 2 I (3.58)

on the cut,dueto (3.56).

� Foritsim aginary partonecan write

1

�Im S(z� i0)= �

Z 1

z

dxf00(x)(z� x)= �

8><

>:

0 ; z > x+

a� z+ f(z); x�< z < x

+

2(a� z); z < x�

(3.59)

9

� W e seefrom (3.59)thateven the di�erentialdS ism ultivalued.Indeed,onecan easily establish for

� =dS

dz= t

00(z)�

Z

dxf00(x)log(z� x) (3.60)

that

1

�Im �(z� i0)= �

8><

>:

0 ; z > x+

1� f0(z); x

�< z < x

+

2 ; z < x�

(3.61)

However,the di�erential

d� = t000(z)dz+ dz

Zdxf00(x)

z� x(3.62)

is already single-valued on the double coverofthe cut z-plane with the periodsHd� � 4�iZ,so dS is

de�ned m odulo 4�idz,and one can m ake sense ofthe periodsHdS due to

Hdz = 0. Therefore,the

exponentexp(�=2)isalready single-valued on the double coverand equalsto unity on the cut.

� In orderto considerthe asym ptotic of(3.57)in whatfollowswe shallalwayschoose a branch,which is

realalong therealaxis,i.e.takeitatrealx ! + 1 .In particular,allresiduesbelow could beunderstood

in thissense,ascoe�cientsofexpansion ofgenerally m ultivalued di�erentialatx ! + 1 .

� Taking derivativesof(3.56)in x-variable,orintegrating by parts,one can bring itliterally to the form ,

arising in the contextofm atrix m odel. However,forthe purposesofSeiberg-W itten theory one needsa

solution with di�erentanalyticproperties:in m atrix m odelstheresolventG � dS

dzdoesnothavepolesat

the branching pointswheredz = 0 (seee.g.[16]and referencestherein),which isnottruefor(3.57).

Asym ptotically from (3.57)onegets

S(z) =z! 1

� 2z(logz� 1)+X

k> 0

tkzk + logz

Z

dxxf00(x)� a

D � 2

1X

k= 1

1

kzk

Z

dxf00(x)

xk+ 1

k+ 1=

= � 2z(logz� 1)+X

k> 0

tkzk + 2alogz�

@F

@a� 2

1X

k= 1

1

kzk

@F

@tk

(3.63)

where,according to (3.50),(with convention thatres1dz

z= 1)

@F

@tk=

1

2(k+ 1)

Z

dxf00(x)xk+ 1; k > 0 (3.64)

and,due to (3.55)

aD =

@F

@a(3.65)

Thecoe�cientatthe z(logz� 1)term is�xed by norm alization

Z

I

dxf00(x)= f

0(x+ )� f0(x� )= 2 (3.66)

wherex� (in the one-cutcase)can be de�ned astwo solutionsto the equation

f(x� )= jx� � aj (3.67)

10

Using variationalequation (3.56),onecan also writeforthefunctional(3.50)thedouble-integralrepresentation

(cf.with [18])

F =1

2

Z

x1> x2

dx1dx2f00(x1)f

00(x2)F (x1 � x2)+ aaD + �0 (3.68)

expressing it in term s ofthe perturbative kernel(3.51) and extrem alshape f(x),solving (3.56). The (tim e-

dependent)constant�0 arisesin (3.68)dueto constraint(3.66)and appearsto betheconstantpartofthe�rst

prim itiveofthe function (3.57).

4 D ispersionless Toda chain

In thecaseofa singlecutletuspresentthe double coverofthe z-planey2 = (z� x+ )(z� x� )in the form

z = v+ �

�

w +1

w

�

(4.69)

with x� = v� 2� and

y2 = (z� v)2 � 4�2 (4.70)

O n thedoublecover(4.69),which isin thecaseofsinglecutjustP1 with twom arkedpointsP� ,with z(P� )= 1 ,

w � 1(P� )= 1 ,form ula (3.57)de�nesa function with a logarithm ic cutand asym ptotic behavior(3.63),odd

undertheinvolution w $ 1

wofthecurve(4.69).In term softheuniform izing variablew onecan globally write

S = � 2

�

v+ �

�

w +1

w

��

logw � 2�(log�� 1)

�

w �1

w

�

+X

k> 0

tkk(w)+ 2alogw (4.71)

where

k(w)= zk+ � z

k� ; k > 0 (4.72)

are the Laurentpolynom ials,odd underw $ 1

w. The �rstterm in (4.71)com esfrom the Legendre transform

ofthe Seiberg-W itten di�erentiald� � z dw

w.

The canonicalToda chain tim esarede�ned by the coe�cientsatthe singularterm sin (3.63)

t0 = resP+dS = � resP�

dS = 2a (4.73)

and

tk =1

kresP+

z� kdS = �

1

kresP�

z� kdS; k > 0 (4.74)

From the expansion (3.63)italso im m ediately follows,that

@F

@tk=1

2resP+

zkdS = �

1

2resP�

zkdS; k > 0 (4.75)

Form ulas(4.73),(4.74),(4.75)togetherwith (3.65)identify the generating function (3.50)with the logarithm

ofquasiclassicaltau-function,being here,in thecaseofa singlecut,a tau-function ofdispersionlessToda chain

hierarchy.

The consistency condition for(4.75)isensured by the sym m etricity ofsecond derivatives

@2F

@tn@tk=1

2resP+

(zkdn) (4.76)

11

wherethe tim e derivativesof(3.63)

0 =@S

@a=

z! P�

�

2logz�@2F

@a2� 2

X

n> 0

@2F

@a@tn

1

nzn

!

k =@S

@tk=

z! P�

�

zk �

@2F

@a@tk� 2

X

n> 0

@2F

@tk@tn

1

nzn

!

; k > 0

(4.77)

form a basisofm erom orphicfunctionswith polesatthepointsP� ,with z(P� )= 1 .Alltim e-derivativeshere

aretaken atconstantz.

Expansion (4.77)ofthe Ham iltonian functions(4.72)expressesthe second derivativesofF in term softhe

coe�cientsofthe equation ofthe curve(4.69),e.g.

0 =z! 1

2logz� 2log��2v

z�2�2 + v2

z2+ :::

1 =z! 1

z� v�2�2

z�2v�2

z2+ :::

2 =z! 1

z2 � (v2 + 2�2)�

4v�2

z�2�2(�2 + 2v2)

z2+ :::

(4.78)

Com parison ofthe coe�cientsin (4.78)gives,in particular,

@2F

@a2= log�2

;@2F

@a@t1= v (4.79)

and@2F

@t21= �2 = exp

@2F

@a2(4.80)

which becom es the long-wave lim itofthe Toda chain equationsafter an extra derivative with respectto a is

taken@2aD

@t21=

@

@aexp

@aD

@a(4.81)

forthe Toda co-ordinateaD = @F

@a.Substituting expansions(4.78)into (4.76),one getsthe expressionsforthe

so called contactterm s[19]in theU (1)case,which arethepolynom ialsofa singlevariablev with �-dependent

coe�cients.

O necan now �nd the dependence ofthe coe�cientsofthe curve(4.69)on the deform ation param eterst of

them icroscopictheory by requiringdS = 0attheram i�cation points,wheredz = 0.Thiscondition avoidsfrom

arising ofextra singularitiesatthe branch pointsin the variation ofdS w.r.t.m oduliofthe curve.Equation

dz

dlogw= �

�

w �1

w

�

= 0 (4.82)

givesw = � 1,wherenow

dS

dlogw

����w = � 1

=X

k> 0

tkdk

dlogw

����w = � 1

+ 2a� 2v� 4�log� = 0 (4.83)

Iftk = 0 fork > 1,solution to (4.83)im m ediately gives

v = a; �2 = et1 (4.84)

12

and the prepotential

F =1

2aa

D +1

2resP+

(zdS)�a2

2=1

2a2t1 + e

t1 (4.85)

Adding nonvanishing t2,one�nds

v = a�1

2t2L

�� 4t22e

t1+ 2t2a�

log�2 = t1 + 2t2a� L

�� 4t22e

t1+ 2t2a�

(4.86)

wherethe Lam bertfunction L(t)isde�ned by an expansion

L(t)=

1X

n= 1

(� n)n� 1tn

n!= t� t

2 +3

2t3 �

8

3t4 + ::: (4.87)

and satis�esto the functionalequation

L(t)eL(t) = t (4.88)

Hence,forthe prepotentialwith t1;t2 6= 0 onegets

F =1

2

a@F

@a+X

k> 1

(1� k)tk@F

@tk

!

+@F

@t1�a2

2=

=1

2aa

D +1

4

X

k> 1

(1� k)tkresP+

�zkdS�+1

2resP+

(zdS)�a2

2=

=tk = 0;k> 2

1

2aa

D +1

2resP+

(zdS)�1

4t2resP+

�z2dS��a2

2

(4.89)

wherefrom the instanton expansion can be com puted (which can be strictly gotasan expansion in param eter

q aftert1 ! t1 +1

2logq)

F = t(a)+ et00(a)+ t

22 e

2t00(a)+

8

3t42 e

3t00(a)+

32

3t62 e

4t00(a)+

+160

3t82 e

5t00(a)+

1536

5t102 e

6t00(a)+ :::= t(a)+ S(a)+

1

4S(a)S00(a)+ :::

(4.90)

with S(a) = expt00(a). Expansion (4.90) directly corresponds to sum m ing over connected tree diagram s in

bosonicm odel,presented at�g.1.

Itisalso easy to com pute the explicitform ofthe extrem alshapefornonvanishing t1;t2,which reads

f0(x)=

2

�

�

arcsin

�x � v

2�

�

+ t2

p4�2 � (x � v)2

�

v� 2�� x � v+ 2�

(4.91)

wherev = v(t)and �= �(t)aregiven by (4.86),orobey

v� a = 2t2�2; log�2 = t1 + 2t2v (4.92)

Form ula (4.91) is a direct consequence of(3.62) and directly following from (4.71) upon the relations (4.92)

expression

�(w;t1;t2;a)= 2t2y� 2logw + (t1 + 2t2v� log�2)z� v

y+ 2

a� v+ 2t2�2

y=

=(4:92)

2t2�

�

w �1

w

�

� 2logw

(4.93)

13

Note,that(4.91)staysthattheVershik-K erov\arcsin law"[20]forthelim itingshapeisdeform ed bytheW igner

sem icircledistribution.

Ifthe �rstthreeToda tim est1;t2;t3 arenonvanishing,instead of(4.91)onegets

f0(x)=

2

�

�

arcsin

�x � v

2�

�

+ (t2 + 3t3v)p4�2 � (x � v)2 +

3

2t3(x � v)

p4�2 � (x � v)2

�

v� 2�� x � v+ 2�

(4.94)

wherev and � arenow subjected to

v� a = 2(t2 + 3t3v)�2

log�2 = t1 + 2t2v+ 3t3(v2 + 2�2)

(4.95)

G enerally we obtain forthe lim itshape

f0(x)=

2

�

arcsin

�x � v

2�

�

+X

k> 1

tkQ k(x)p4�2 � (x � v)2

!

v� 2�� x � v+ 2�

(4.96)

where v and � obey som e sort of hodograph equations v � a = P v(v;�;t), log�2 = P� (v;�;t) for som e

polynom ialsPv and P�,whoseexpansion in Toda tim estcan beeasily reconstructed from thepresented above

form ulasofgeneralsolution.

5 Extended non-abelian theory

In thecaseofU (N )gaugetheory onehasto considersolution with N cutsfIig,i= 1:::;N ,which arisesafter

adding to the functional(3.50)N constraintswith the Lagrangem ultipliers

F ! F +

NX

i= 1

aDi

�

ai�1

2

Z

Ii

dx xf00(x)

�

(5.97)

i.e.solution to the integralequation

X

k> 0

tkxk �

Z

d~xf00(~x)(x � ~x)(logjx � ~xj� 1)= aDi ; x 2 Ii; i= 1;:::;N (5.98)

Now itcan be expressed in term softhe Abelian integralson the doublecover

y2 =

NY

i= 1

(z� x+i)(z� x

�i) (5.99)

which isa hyperelliptic curveofgenusg = N � 1.De�ne,asbefore:

S(z)= t0(z)�

Z

dxf00(x)(z� x)(log(z� x)� 1)� a

D (5.100)

where the integralistaken overthe whole supportI= [Ni= 1Ii,aD = 1

N

P N

j= 1aDj ,and consideritsdi�erential,

or

�(z)=dS

dz=X

k> 0

ktkzk� 1 �

Z

dxf00(x)log(z� x) (5.101)

14

satisfying

�(x + i0)+ �(x � i0)= 0; x 2 I i; i= 1;:::;N (5.102)

on each cut,and norm alized to

�(x +N)= 0;

�(x �j � i0)= �(x +

j� 1 � i0)= � 2�i(N � j+ 1); j= 2;:::;N

�(x �1 )= � 2�iN

(5.103)

Vanishing m icroscopictim es

Consider,�rst,alltk = 0 fork 6= 1,and de�ne � 2N = et1. Now � = dS

dzis an Abelian integralon the curve

(5.99)with the asym ptotic

� =P ! P�

� 2N logz� 2N log�+ O (z� 1) (5.104)

whosejum psareinteger-valued dueto (5.103),orHd� � 4�iZ.Itm eansthatthehyperellipticcurve(5.99)can

be seen also asan algebraicRiem ann surfaceforthe function w = exp(� �=2),satisfying quadraticequation

�N

�

w +1

w

�

= PN (z)=

NY

i= 1

(z� vi) (5.105)

since for the two branchesw+ = w and w� = 1

wone im m ediately �nds thattheir productw + � w� and sum

w+ + w� arepolynom ialsofz ofgiven powers(zero and N correspondingly).

Equivalently,theendsofthecutsin (5.99)arerestricted by N constraintsin such a way,thatthisequation

can be rewritten as

y2 = PN (z)

2 � 4�2N (5.106)

i.e.fx�i g arerootsofPN (z)� 2�N = 0,and

y = �N

�

w �1

w

�

(5.107)

Thegenerating di�erential(5.101)isnow

dS = � 2logwdz = � d(2zlogw)+ 2zdw

w(5.108)

just the Legendre transform ofthe Seiberg-W itten di�erentiald� � z dw

won the curve (5.105),(5.106). It

periods

ai =1

2�i

I

A i

zdw

w(5.109)

coincidewith the Seiberg-W itten integralsand the only nontrivialresiduesatin�nity give

resP+

�z� 1dS�= � resP�

�z� 1dS�= log�2N

resP+(dS)= � resP�

(dS)= 2

NX

j= 1

vj(5.110)

15

Thedi�erential(5.108)satis�esthe condition

�dS ��w

wdz =

�P (z)

ydz =

P

Nj= 1

vj= 0

holom orphic (5.111)

wherethevariation istaken atconstantco-ordinatez and constantscalefactor�.Thus,theintegrablesystem

on \sm allphasespace" issolved forthe scale�2N = et1 and the m odulivj,j= 1;:::;N ofvacua oftheU (N )

gauge theory,satisfying the equationP N

j= 1vj = a and the transcendentalequations for the Seiberg-W itten

periods(5.122).

Nonvanishing m icroscopictim es

W hen we switch on \adiabatically" the highertim es (2.17)with k > 1,the num berofcutsin (5.98)rem ains

intact,and the di�erential(5.101)can be stillde�ned on hyperelliptic curve (5.99). However,now the role of

bipoledi�erential dw

wofthe third kind isplayed by

d� = dz

X

k> 1

k(k� 1)tkzk� 2 �

Zdxf00(x)

z� x

!

=

=X

k> 1

k(k � 1)tkdk� 1 � 2N d0 � 4�i

N � 1X

j= 1

d!j

(5.112)

whered!i,i= 1;:::;N � 1arecanonicalholom orphicdi�erentialsnorm alized to theA-cycles,surrounding�rst

N � 1 cuts.The di�erentialsd k in (5.112)are�xed by theirasym ptoticatz ! 1

dk =z! 1

8<

:

kzk� 1

dz+ O (z� 2); k > 0

dz

z+ O (z� 2); k = 0

(5.113)

and vanishing A-periods I

A i

dk = 0; k � 0; 8 i= 1;:::;N � 1 (5.114)

Thenonvanishing periodsofd� are�xed by

I

A j

d� = � 2�i

Z

Ij

f00(x)dx = � 2�i

�f0(x+j )� f

0(x�j )�= � 4�i (5.115)

which justi�esthatgenerating di�erentialdS isstillde�ned m odulo 4�idz.Theonly im portantdi�erencewith

thepreviouscaseisthatintegrality oftheperiodsHd� � 4�iZ,which wasreform ulated in term sofan algebraic

equation (5.105)forthe theory on \sm allphase space",rem ainsnow a transcendentalequation,which cannot

be resolved explicitly.

Nevertheless,on thecurve(5.99)any odd underhyperellipticinvolution di�erentialcan bealwayspresented

as

d� =s(z)dz

y(5.116)

where s(z)isa polynom ialofpowerN + K � 2 in caseofnonvanishing m icroscopictim est1;:::;tK up to the

K -th order.ItshigherK coe�cientsare�xed by leading asym ptotic(t 2;:::;tK )and the residueatin�nity

resP�d� = � 2N (5.117)

16

and therestN � 1 coe�cientscan bedeterm ined from (5.115).This�xescom pletely thedi�erentiald� on the

curve(5.99)which stillrem ain to be dependentupon 2N (yetarbitrary)branch pointsfx�j g.

The generating di�erentialdS can be now de�ned in term softhe Abelian integral�(z)

dS = �dz = dz

Z z

z0

d� (5.118)

The dependence upon 2N + 1 param eters (the positions ofthe branch points in (5.99) together with z0) is

constrained by additionalto (5.115)vanishing ofthe B -periods

I

B j

d� = 0; j= 1;:::;N � 1 (5.119)

Integralrepresentation (5.101)suggestsa naturalnorm alization (5.103),i.e.

z0 = x+

N; �(z0)= �(x

+

N)= 0 (5.120)

where x+Nis the largestam ong realram i�cation points fx�j g. These conditions lead to the following form of

expansion of�(z)in the vicinity ofram i�cation points

�(z) =z! x

�

j

�(x �j )+ �

�j

q

z� x�j + :::

��j =

2s(x�j )

Q 0

k

q

(x�j � x+

k)(x�j � x

�

k)

; j= 1;:::;N

(5.121)

wherethe constants�(x �j)aregiven by (5.103).

The restN + 1 param etersareeaten by the periods

aj =1

2

Z

Ij

dxxf00(x)= �

1

4�i

I

A j

zd� =1

4�i

I

A j

dS; j= 1;:::;N � 1 (5.122)

togetherwith the residues

a = 1

2

Z

I

dxxf00(x)= � 1

2resP+

(zd�) (5.123)

and the "freeterm " orscaling factor

t1 = resP+

�z� 1�dz

�(5.124)

Recallonce m ore,thatan essentialdi�erence with the caseofvanishing tim esisthatfortk 6= 0,the exponent

exp(�)acquiresan essentialsingularityatthepointsP � ,and theconstraints(5.115),(5.119)cannotberesolved

algebraically.Theform oftheexpansion (5.121)ensuresthatvariation ofthegenerating di�erentialatconstant

z w.r.t.m oduliofthe curve(5.99)

�(dS)= � (�dz)=

=z! x

�

j

� s(x�j)�x�

j

Q 0

k

q

(x�j� x

+

k)(x�

j� x

�

k)

dzq

z� x�j

+ :::’ holom orphic(5.125)

isindeed holom orphic.

The Lagrangem ultipliers

aDi =

@F

@ai(5.126)

17

can be com puted by a standard trick.Considerequation (5.98)fori6= j and �x there x-variablesto be atthe

endsofcorresponding cuts.Then

aDi � a

Dj = Re

Z x�

i

x+

j

dS =1

2

I

B ij

dS (5.127)

or@F

@ai=1

2

I

B i

dS; i= 1;:::;N � 1 (5.128)

Forthe tim e-derivativesofprepotentialonecan write

@F

@tk=1

2resP+

�zkdS�= �

1

2(k+ 1)resP+

�zk+ 1

d��

(5.129)

6 Q uasiclassicalhierarchy and explicit results

From the expansion (5.101) in the case ofU (N ) extended theory it stillfollows that the �rst derivatives of

quasiclassicaltau-function F aregiven by (3.64)and (4.75),whileforthesecond derivativesonegets(4.76),or

@2F

@tn@tm= 1

2resP+

(zm dn)=1

2resP+ P+

(z(P )nz(P 0)m W (P;P 0)) (6.130)

where we have introduced the bi-di�erentialW (P;P 0) = dP dP 0 logE (P;P 0),with E (P;P 0) being the prim e

form ,see [21]forthe de�nitions. In the inverse co-ordinatesz = z(P )and z0= z(P 0)nearthe pointP + with

z(P + )= 1 ithasexpansion

W (z;z0)=dzdz0

(z� z0)2+ :::=

X

k> 0

dz

zk+ 1dk(z

0)+ ::: (6.131)

Thebi-di�erentialW (P;P 0)can be related with the Szeg�o kernel[21]

Se(P;P0)S� e(P;P

0)= W (P;P 0)+ d!i(P )d!j(P0)

@

@Tijlog�e(0jT) (6.132)

which,foran even characteristicse� � e,hasan explicitexpression on hyperelliptic curve(5.99)

Se(z;z0)=

Ue(z)+ Ue(z0)

2pUe(z)Ue(z

0)

pdzdz0

z� z0(6.133)

with

Ue(z)=

vuut

NY

j= 1

z� xe+

j

z� xe�

j

(6.134)

Here xe�

j

is partition ofthe ram i�cation points of(5.99) into two sets,corresponding to a characteristic e.

For exam ple, on a sm allphase space, when (5.99) turns into the Seiberg-W itten curve (5.105), there is a

distinguished partition e= E ,corresponding to an even characteristicswith

UE (z)=

s

P (z)� 2�N

P (z)+ 2�N(6.135)

18

Substituting (6.132),(6.133)into (6.130)gives

@2F

@tn@tm= 1

2resP+ P+

�z(P )m z(P 0)nSe(P;P

0)2��

� 1

2resP+

(znd!i)� resP+(zm d!j)

@

@Tijlog�e(0jT)=

= P(e)nm (xe�

j

)� 2@2F

@ai@tn

@2F

@aj@tm

@

@Tijlog�e(0jT)

(6.136)

whereforthe "contactpolynom ials" one getsfrom (6.133)

P(e)nm (xe�

j

)=1

4resP+ P+

�zkz0n

(z� z0)2

�

1+Ue(z)

2Ue(z0)+

Ue(z0)

2Ue(z)

�

dzdz0

�

(6.137)

Ifcalculated in the vicinity ofthe sm allphase space and for the particular choice ofcharacteristic (6.135),

residues(6.137)vanish forn;m < N ,and onegetsexactly the conjectured in [19]form ula

@2F

@tn@tm= �

1

2

@un+ 1

@ai

@um + 1

@aj

@

@Tijlog �E (0jT); n;m < N (6.138)

with

un = 2@F

@tn� 1=

1

nresP+

�

zn P

0N dz

y

�

=1

n

NX

l= 1

vnl =

1

nhTr�ni (6.139)

Equation (6.138) is a particular case ofthe generalized dispersionless Hirota relations for the Toda lattice,

derived in [18].

Letuspointout,thatthisderivation oftherenorm alization group equation (6.138)in [1]isalm ostidentical

to developed previously in [22]foranotherversion ofextended Seiberg-W itten theory,which can bede�ned by

generating di�erential

d~S =X

k> 0

Tkd~k =X

k> 0

TkP (z)k=N

+

dw

w(6.140)

directly on the Seiberg-W itten curve(5.105)(whose form rem ained intactby higher owsfTkg,in contrastto

thequasiclassicalhierarchy,determ ined by (5.118)),and allderivativesweretaken atconstantw.Forexam ple,

ifN = 1 and only T0,T1 do notvanish with d~1 = d1 = (z� v)dww,onegets

@

@T1(z� v)=

@�

@T1

�

w +1

w

�

(6.141)

and@d~S

@T1= d1 + T1

@�

@T1

�

w +1

w

�dw

w=

�

1+T1

�

@�

@T1

�

d1 (6.142)

which m eans,in particular,thatthe scale factor� � T 1 linearly dependson the �rsttim e,in contrastto the

exponentialdependence in form ula (4.84).Indeed,taking derivativesof(4.71)atconstantz,instead of(6.141)

onegets

@v

@t1+@�

@t1

�

w +1

w

�

+ �

�

w �1

w

�@logw

@t1= 0 (6.143)

and therefore

@S

@t1=

@

@t1

�

t11 + 2alogw � 2zlogw � 2�(log�� 1)

�

w �1

w

��

= �

�

w �1

w

�

+

+@�

@t1(t1 � 2log�)

�

w �1

w

�

+@logw

@t1

�

(t1 � 2log�)�

�

w +1

w

�

+ 2a� 2v

� (6.144)

19

i.e.form ulas(4.72),(4.77)areprovided directly by (4.84).

The choice ofextension (6.140) in [22]was m otivated rather by technicalreasons: preserving the form

ofthe Seiberg-W itten curve (5.105)with deform ing only the generating di�erential,m oreoverthat the latter

rem ained single-valued even in the deform ed theory. W e see,however,that the old choice is not consistent

with the m icroscopicinstanton theory (2.17),(2.18),(2.20),basically since the appropriateco-ordinateforthe

quasiclassicalhierarchy isz,com ing from the scalar�eld � ofthe vectorm ultipletofN = 2 supersym m etric

gauge theory. However,the corresponding quasiclassicalhierarchy isde�ned even m ore im plicitly,due to the

highly transcendentalingredientHd� � 4�iZ forthe second kind (notforthe third kind)Abelian di�erential,

and oneneedsto apply speciale�ortsto extractexplicitresults.

Instanton expansion in the extended theory

Theinstantonicexpansion F =P

k� 0Fk in the non-Abelian theory startswith the perturbativeprepotential

F0 =

NX

j= 1

t(aj)+X

i6= j

F (ai� aj) (6.145)

de�ned entirely in term softhefunctions(2.17)and (3.51).Itistotally characterized by degeneratedi�erential

(5.116)

d�0 = t000(z)dz� 2

dPN (z)

PN (z)= t

000(z)dz� 2

NX

j= 1

dz

z� vj(6.146)

(which does not depend on higher tim es), and the coe�cients ofthe polynom ialP N (z) in (5.105),(6.146)

coincidewith the perturbativevaluesofthe Seiberg-W itten periods

ai = � 1

2resvizd�0 = vi (6.147)

Theperturbativegenerating di�erentialisdS0 = �0dz,with

�0 = t00(z)� 2

NX

j= 1

log(z� vj) (6.148)

and satis�es@dS0

@aj= 2

dz

z� vj; j= 1;:::;N

@dS0

@tk= kz

k� 1dz; k > 0

(6.149)

whatgivesriseto

S0(z)= t0(z)� 2

NX

j= 1

(z� vj)(log(x � vj)� 1) (6.150)

Equations

aDj =

@F0

@aj= 2S0(aj) (6.151)

com pletely determ ine (6.145),since on thisstageone m akesno di�erencebetween vj and aj.

20

M oreover,vanishing ofthe B -periods(5.119)ofthe di�erential(6.146)

Z x�

j

x+

i

d�0 = 0 (6.152)

where x�j= aj �

pqSj + O (q2) are positions ofthe branching points ofthe curve (5.99) in the vicinity of

perturbativerationalcurve,im m ediately givesthe deviations

Si �et

00(ai)

Q

j6= i(ai� aj)

2; i= 1;:::;N (6.153)

wherethe num ericcoe�cientis�xed from com parison with theSeiberg-W itten curve(5.106)on a sm allphase

space.Theinstantonicexpansion,sim ilarly to thatoftheU (1)theory (4.90),can bedeveloped in term softhe

functions(6.153)and theirderivatives.Forexam ple,in [1]wehavechecked,that

F1 =X

l

Sl (6.154)

for U (2) gauge group and the only nonvanishing t1,t2,using instantonic expansions ofthe equations (5.99),

(5.116)and (5.129).

Ellipticuniform ization forthe U (2)theory

In thecaseofU (1)theory theproblem wassolved explicitly by construction ofthefunction (4.71)duetoexplicit

uniform ization oftherationalcurve(4.69)in term sof\global"spectralparam eterw.Thisishardly possiblefor

genericnon-Abelian theory with the hyperelliptic curve (5.99)ofgenusg = N � 1,butin the nextto rational

casewith N = 2

y2 =

Y

i= 1;2

(z� x+i )�

4Y

i= 1

(z� xi)� R(z) (6.155)

itisan elliptic curve,and thereforecan be uniform ized using,forexam ple,the W eierstrassfunctions

z = z0 +R 0(x0)=4

}(�)� R00(x0)=24= z0 +

}0(�0)

}(�)� }(�0)=

= z0 + �(� � �0)� �(� + �0)+ 2�(�0)

dz

d�= �

}0(�0)}0(�)

(}(�)� }(�0))2= y

(6.156)

whereitwasconvenientto take

z0 = x4

R0(z0)= x43x42x41 = 4}0(�0)

R00(z0)= 2(x41x42 + x41x43 + x42x43)= 24}(�0)

(6.157)

TheAbelian integralfor�(z)can benow perform ed in term softheellipticfunctions.Takeagain forsim plicity

alltk = 0,ifk > 2.Then forthe di�erential(5.116)one has

d� =2t2z

2 + s1z+ s0

ydz =

z! 1�

�

2t2dz� 4dz

z� 2a

dz

z2+ :::

�

(6.158)

21

and,aswasprom issed before,thisasym ptotic�xesthe coe�cients

s1 = � 4� t2

4X

i= 1

xi

s0 = � 2a+ 2

4X

i= 1

xi�t2

4

4X

i= 1

x2i +

t2

2

X

i< j

xixj

(6.159)

and com pletely determ ineshere the di�erential(6.158)in term softhe curve (6.155). Forthe elliptic integral

in (5.118)onecan now write

� =

Z z

z0

d� = 2t2 (�(� + �0)+ �(� � �0))+ �1 log�(�0 � �)

�(�0 + �)+ �2� (6.160)

Theconstants�1;2 areeasily recovered from com parison ofthe expansion of

d�

d�= � 2t2 (}(� � �0)+ }(� + �0))+ �1 (�(� � �0)� �(� + �0))+ �2 (6.161)

at� ! � �0 with (6.158)upon (6.156).Thejum psofa m ultivalued Abelian integral(6.160)on theellipticcurve

(6.155),are furtherconstrained to integersby (5.115)and (5.119),which can be now rewritten in the form of

transcendentalconstraintsforthe param etersofthe W eierstrassfunctions.

7 C onclusion

W e have discussed in these notes the m ain properties ofthe quasiclassicalhierarchy,underlying the Seiberg-

W itten theory,which wasderived in [1]directly from the m icroscopic setup and instanton counting. M ostof

the progress was achieved due to existence ofthe \oversim pli�ed" U (1) exam ple,naively com pletely trivial

from the pointofview ofthe Seiberg-W itten theory. However,even in thiscase the partition function ofthe

deform ed instantonic theory becom esa nontrivialfunction on the large phase space,being the tau-function of

dispersionlessToda chain,and providing a directlink to the theory ofthe G rom ov-W itten classes. The dual

Seiberg-W itten period (\the m onopole m ass")satis�esthe long wave lim itofthe equation ofm otion in Toda

chain,as a function of\W -boson m ass" and the (logarithm of) the scale factor. M uch less transparentnon-

Abelian quasiclassicalsolution isneverthelessconstructed using standard m achinery on highergenusRiem ann

surfaces.Itisalso essential,thatswitching on highertim esdeform the Seiberg-W itten curve.

The m ain issue now is what is going beyond the quasiclassicallim it. Could at least the \sim ple" U (1)

problem be solved exactly in allordersofstring coupling in m ore orlessexplicitform ? Itisalso necessary to

stress,thatthefreeferm ion (orboson)m atrix elem entsup to now wereconsidered asform alseriesin thehigher

tim es,exceptfort1 � log�. Theirknowledge asexactfunctionsatleastoft2 could provide usan interesting

inform ation aboutphysically di�erent\phases" ofthem odel.Anotherinteresting and yetunsolved problem for

the extended theory isswitching on the m atterby extrapolating the higher owsto tk �1

k

P N f

A = 1m

� k

A,what

correspondshypothetically to thetheory with N f fundam entalhyperm ultipletswith correspondingm asses.W e

hopeto return to these problem selsewhere.

Acknowledgem ents

Iam gratefulto A.Alexandrov,H.Braden,B.Dubrovin,I.K richever,A.Losev,V.Losyakov,A.M ironov,

A.M orozov and,especially,to N.Nekrasov and S.K harchev forthe very usefuldiscussions.

22

The work was partially supported by FederalNuclear Energy Agency,the RFBR grant 05-02-17451,the

grantforsupportofScienti�cSchools4401.2006.2,INTAS grant05-1000008-7865,theprojectANR-05-BLAN-

0029-01,theNW O -RFBR program 047.017.2004.015,theRussian-Italian RFBR program 06-01-92059-CE,and

by the Dynasty foundation.

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24