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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.
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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)
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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
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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.
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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)
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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)
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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)
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...
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.
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\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)
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� 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)
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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)
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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)
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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)
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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)
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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)
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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
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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)
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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)
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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)
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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
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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
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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.
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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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