Cubic curves from instanton counting

0.35 MB·English

by  Sergey Shadchin

hep-th/0511132
Cubic curves from instanton counting
Sergey Shadchin
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INFN, Sezione di Padova & Dipartimento di Fisica “G. Galilei”
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Universita` degli Studi di Padova, via F. Marzolo 8, Padova, 35131, ITALY
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1 email: [email protected]

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0 Abstract
/
h
t
-
p
e We investigate the possibility to extract Seiberg-Witten curves from the formal series for the
h
: prepotential,whichwasobtainedby the Nekrasovapproach. Amethod formodels whose Seiberg-
v
i
X Witten curves are not hyperelliptic is proposed. It is applied to the SU(N) model with one
r symmetric or antisymmetric representations as well as for SU(N ) SU(N ) model with (N ,N )
a 1 × 2 1 2
or (N ,N ) bifundamental matter. Solution are compared with known results. For the gauge
1 2
group product we have checked the instanton corrections which follow from our curves against
direct instanton counting computations up to two instantons.
Contents
1 Introduction 1
2 Instanton counting for group product 3
2.1 SU(N ) SU(N ) case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 2
×
2.2 Instanton corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Thermodynamical limit for group product . . . . . . . . . . . . . . . . . . . . . . . 6
3 Product equations 8
3.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Symplectic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 Getting cubic curves 11
4.1 Antisymmetric matter, special case . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Symmetric matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.3 Antisymmetric matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.4 Curves for group product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Discussion 17
5.1 M-theory curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Instanton corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
A Instanton counting 21
A.1 Equivariant index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
A.2 Partition function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A.3 Thermodynamical limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
B About double ramification points 29
1 Introduction
String theory can shed some light to the strong coupling regime in the supersymmetric gauge
theories. In particular the =2 super Yang-Mills theory is believed to be described by the type
N
IIA superstrings NS5-D4 branes setup [39]. The Coulomb branch of this theory in the low-energy
1
sectorisdescribedby acomplex functiononthe modulispaceofthe theory,knownasprepotential
[33]. A very elegant construction for this function was proposed by Seiberg and Witten [34, 35].
This construction includes a Riemann surface (the Seiberg-Witten curve) and a differential λ(x)
defined on this surface. The prepotential (a) can be defined indirectly with the help of relations
F
(41). The crucial observation made in [39] is that the NS5-D4 type IIA setup can be lifted to
M-theory there it becomes a single object, M5-brane, wrapped around a two-dimensional space,
which can be associated with the Seiberg-Witten curve.
Thispointofviewyieldstothesolutionsfornumerousmodels,suchasmodelswithgaugegroup
product [39, 18] symplectic and orthogonal group [23, 22, 4], and symmetric and antisymmetric
representation for unitary group [22].
Threeyearsagoanotherwaytosolve =2superYang-MillstheorywasproposedbyNekrasov
N
[30]. It is based on the localization technique, which, together with a certain deformation of the
theory, gives the direct access to the prepotential after the explicit summation over the instanton
contributions. This method (instanton counting) yields the prepotentialalreadyas a series on the
dynamicallygeneratedscale,withouthardcycleintegrationoftheSeiberg-Wittentheory. However
to study such effects as confinement or monopole condensation we have to know how to continue
the prepotential beyond the convergence radius of proposed series. Seiberg-Witten theory can
solve this problem. Therefore we have faced with the question how to extract the Seiberg-Witten
geometryfromtheseriesfortheprepotential. AlsohavingfoundtheSeiberg-Wittencurvewegain
an independent test of solutions, obtained by other methods. In particular we get a test for the
M-theory.
In [31] this problem was solved by the conformal map method, and the curves extracting
technology was generalized in [32, 37, 36] to other groups and matter content. It was shown that
the instantoncounting defines some singularequations(saddlepointequations)whichenable us to
find the Seiberg-Witten curve and differential.
Conformal map method allows, however, to find the Seiberg-Witten curves (up to some rare
exceptions)onlyinthecasewhencurvesarehyperelliptic. Formoregeneralsituationsitisnotclear
how to apply it. Thus pragmatically we need just another method to solve saddlepoint equations
in order to get more Seiberg-Witten curves.
In this paper we proposesucha method. It workswellfor the cubic curves,and, probably,can
begeneralizedtoothercaseswhentheSeiberg-Wittencurveisgivenbyafinitedegreepolynomial.
We consider the SU(N) model with symmetric or antisymmetric matter. Also, to elaborate more
examplesandcheckthecurvepredictionswiththeinstantoncountingpredictionswehavedescribe
2
the instanton counting for the gauge group product.
The paper is organized as follows. In Section 2 we describe the generalization of the Nekrasov
approachforthe gaugegroupproduct. Alsowecomputeone-andtwo-instantoncorrectionstothe
prepotential for SU(N ) SU(N ) model with one bifundamental matter representation of type
1 2
×
(N ,N ) or (N ,N ). In Section 3 we propose a method which can be used to solve saddlepoint
1 2 1 2
equations. As an illustration we apply it to the hyperelliptic curve models. In Section 4 we solve
these equation for the symmetric and antisymmetric representations of SU(N) as well as for the
SU(N ) SU(N ) case with two types of bifundamental matter. In Section 5 we discuss obtained
1 2
×
results and check curve predictions against the instanton counting predictions.
Acknowledgments. I am very grateful to Nikita Nekrasov for numerous fruitful discussions. I
would like to thank Ivan Kostov for his patient explanation of the singular equations structure.
Also I thank all my colleagues of Dipartimento di Fisica to give me an opportunity to present my
workassoonasitbecamespresentable. Thisworkwaspartiallysupportedbythe EUMRTN-CT-
2004-005104grant “Forces Universe” and by by the MIUR contract no. 2003023852.
2 Instanton counting for group product
In this section we generalize the Nekrasov approach for the case when the gauge group is a direct
product of simple classical groups. We will mostly use notations introduced in the Appendix A,
whichcontainsabriefsurveyoftheinstantoncountingmethods. Itis notourambitiontoexhaust
all possible cases. For illustrative purposes it is sufficient to consider the product of two unitary
groups. Generalization to other classical groups with richer matter content is straightforward
[37, 36].
2.1 SU(N ) SU(N ) case
1 2
×
We consider the simplest case G = SU(N ) SU(N ). When we deal with the product of two
1 2
×
groups the general expression for the partition function, which generalizes (29) and (30), depends
on two dynamically generated scales Λ and Λ :
1 2
1
1 =Zpert(a,m,Λ ,Λ ;ε)Zinst(a,m,Λ ,Λ ;ε)=exp (a,m,Λ ,Λ ;ε),
a 1 2 1 2 1 2
h i ε ε F
1 2
3
where
∞ ∞ k1 dφ k2 dϕ
Zinst(a,m,Λ ,Λ ;ε)= qk1qk2 i jz (a,m,φ,ϕ;ε). (1)
1 2 1 2 2πi 2πi k1,k2
kX1=0kX2=0 I iY=1 I jY=1
Here q1 = e2πiτ1(Λ1) = Λβ1e2πiτ0(1) and the same for q2. β1 and β2 can be computed using the
matter content of the theory. Now the prepotentialand the partitionfunction depend on two sets
of Higgs vacuum expectations: a ,...,a and b ,...,b . Also the integration is performed over
1 N1 1 N2
twodualgroupmaximaltorus Lie algebras(φ and ϕ). It reflects the fact thatin the caseofgauge
group product the total dual group is also the product of corresponding dual groups.
If there is no matter multiplets in a “mixed” representation (which has a non-trivial charge
whichrespecttobothgroups)thenthe instantonpartitionfunction(1)factorizesandthereareno
new effects.
We consider the simplest non-trivial example of such a “mixed” representation, the bifunda-
mentalone. Therearetwotypesofbifundamentals: (N ,N ),whichwillbereferredinthatfollows
1 2
as “+” and (N ,N ) which will be referred as “ ”. Let us study both of them.
1 2
−
First we are going to find the equivariant index for the Dirac operator. It can be done at the
same way as (28). We have (using (26) and (27))
1 N1 N2 k1 N2
Ind± = eial±ibp eiφi±ibp−iε+
q (eiε1 1)(eiε2 1) −
− − l=1p=1 i=1p=1
XX XX
k2 N1 N1 N2
e±iϕj+ial−iε++(e−iε1 1)(e−iε2 1) eiφi±iϕj.
− − −
j=1l=1 i=1j=1
XX XX
The integrand of the partition function is (recall that we have shifted the masses by ε , see
+
−
(33))
1 D (M ε )D (M +ε ) k1 k2
z± (a,b,φ,ϕ,M;ε)= ± − − ± − ( φ M) ( ϕ M) (2)
k1,k2 k1!k2!D±(M ε+)D±(M +ε+) P2 ∓ i∓ P1 ∓ j −
− i=1 j=1
Y Y
where
k1 k2 N1 N2
D (x)= (φ ϕ +x), (x)= (x a ), (x)= (x b ).
± i j 1 l 2 p
± P − P −
i=1j=1 l=1 p=1
YY Y Y
When k = 0 or k = 0 we have D (x) = 1. Note that the integrand is invariant under the
1 2 ±
4
following transformation
1 2, M M. (3)
↔ ↔±
It means that z± (a,b,φ,ϕ,M;ε)=z± (b,a,ϕ,φ, M;ε).
k1,k2 k2,k1 ±
2.2 Instanton corrections
Using the instanton counting strategy we can compute some instanton corrections for the group
product. With the two-instanton accuracy we have for both types of the bifundamental the fol-
lowing expression for the partition function (1):
Zinst =1+q Z± +q Z± +q2Z± +q2Z± +q q Z± +...,
± 1 1,0 2 0,1 1 2,0 2 0,2 1 2 1,1
where
k1 dφ k2 dϕ
Z± = i jz± (φ,ϕ) (4)
k1,k2 2πi 2πi k1,k2
I i=1 I j=1
Y Y
(in this section we do not display such arguments of z (φ,ϕ) as a, b, M, etc).
k1,k2
Considerthemodelwithonebifundamentalmattermultiplet(N ,N )(“+”)or(N ,N )(“ ”)
1 2 1 2
−
and also N(1) fundamental matter of SU(N ) with masses m(1), f =1,...,N(1), and N(2) funda-
f 1 f f f
mental matter of SU(N ) with masses m(2). In this model we have β = 2N N N(1) and
2 f 1 1− 2 − f
(2)
β =2N N N . Using formulae for z (φ) for the fundamental and adjoint representations
2 2− 1− f k
[37, 36] (which can be obtained from (27) and (28) with the help of (31)) as well as (2) we gain
functions z (φ,ϕ). Plugging them into (4) we can perform the integration and obtain the in-
k1,k2
stantoncorrections. With two-instantonaccuracythe resultis (in fact,Z andZ arethe same
1,0 2,0
for one unitary groupmodel with specific fundamental matter content. Therefore we can take the
expression from [30])
N1
~2Z± = S±(0),
1,0 − l
l=1
X
N1 N2 ~2
~4Z± = S±(0)T±(0) 1
1,1 l p − (a b +M)2
l=1p=1 (cid:18) l± p (cid:19)
XX
1 N S±(0)S (0)± 1 N
~4Z± = l m + S±(0) S±(~)+S±( ~) .
2,0 2lX6=m 1− (al−~a2m)2 2 4Xl=1 l (cid:16) l l − (cid:17)
(cid:16) (cid:17)
5
where
Q (x) ( x M) Q (a +x) ( a x M))
S±(x)= 1 P2 ∓ ∓ , S±(x)= 1 l P2 ∓ l∓ ∓ ,
2(x) l (a a +x)2
P1 l6=m l− m
Q (x) ( x M) Q (b +x) ( b x M))
T±(x)= 2 P1 ∓ − , T±(x)= 2 Qp P1 ∓ p∓ − ,
2(x) p (b b +x)2 (5)
P2 p6=q p− q
N(1) N(2) Q
f f
Q (x)= x+m(1) , Q (x)= x+m(2)
1 f 2 f
fY=1(cid:16) (cid:17) fY=1(cid:16) (cid:17)
S±(x) and T±(x) are referred as residue functions [13, 12], S±(x) and T±(x) being their
l p
“residues”. T±(x) can be obtained from S±(x) after the transformation (3). Note that to com-
p l
pute Z± we have used the fact that for the particular choice (5) the following identities hold
1,1
T±( a M)=S±( b M)=0.
l p
∓ ∓ ∓ −
Corresponding series for the instanton part of the prepotential is
inst =q ± +q ± +q2 ± +q2 ± +q q ± +...,
F± 1F1,0 2F0,1 1F0,2 2F2,0 1 2F1,1
where
N1
± = S±(0),
F1,0 l
l=1
X
± = N1 Sl±(0)Sm±(0) 1 N1 S±(0)S±′′(0), (6)
F2,0 − (a a )2 − 4 l l
l m
l6=m − l=1
X X
N1 N2 S±(0)T±(0)
± = l p .
F1,1 (a b +M)2
l p
l=1p=1 ±
XX
One can easily check that the Seiberg-Witten prepotential is also invariant under (3). Using this
observation we can restore ± and ± .
F0,1 F0,2
2.3 Thermodynamical limit for group product
Letusdescribeinsomedetailsthepassagetothethermodynamicallimitε ,ε 0. Wegeneralize
1 2
→
the results announced in Section A.3.
1
The double sum (1) is dominated by a single term with k k . Since now we have
1 2
∼ ∼ ε ε
1 2
two dual groups it is natural to introduce two profile functions:
N1 k1 N2 k2
f (x)= x a 2ε ε δ(x φ ), f (x)= x b 2ε ε δ(x ϕ ).
1 l 1 2 i 2 p 1 2 j
| − |− − | − |− −
l=1 i=1 p=1 j=1
X X X X
The Hamiltonian for the bifundamentals “+” and “ ” (the “interaction term”) is given by (we
−
6
have taken into account that the matter is described by fermionic functions, therefore the sign is
changed)
1
H±[f ,f ]= dxdyf′′(x)k(x y+M)f′′(y). (7)
1 2 4 1 ± 2
Z
As in the single group case, the dependence on Λ and Λ is introduced via the following term
1 2
in the total Hamiltonian of the model:
πi
+ τ (Λ ) dxf′′(x)x2+τ (Λ ) dxf′′(x)x2 .
2 1 1 1 2 2 2
(cid:18) Z Z (cid:19)
In our situation when minimizing the free energy of the theory we obtain a couple of equations
instead of single one (38):
1 δH[f ,f ]
1 2
=ξ +tτ (Λ ), t γ, l =1,...,N ,
πi δf′(t) l 1 1 ∈ l 1
 1 (8)
π1iδHδf[f′1(,t)f2] =ηp+tτ2(Λ2), t∈δp, p=1,...,N2,
2

where γ and δ are cuts for two groups, and ξ, η are certain constants, in general different for
l p l p
1
differentcuts. τ (Λ )=τ(1)+ lnΛβ1 andthesamefor1 2. Inordertosolvetheseequations
1 1 0 2πi 1 ↔
we introduce the primitives of the profile function resolvents as follows
1
F (z)= dxf′′ (x)ln(z x).
1,2 4πi 1,2 −
Z
Thenthe equations(8)canberewrittenasdifference equationsforthese functions. The prepoten-
tial is defined indirectly by formulae which generalize (41):
∂
zF′(z)dz =a , 2πi zF′(z)dz = F =al
1 l 1 ∂a D
IAl IBl l
∂
zF′(z)dz =b , 2πi zF′(z)dz = F =bp .
2 p 2 ∂b D
ICp IDp p
The unusual property is that now we have a couple of Seiberg-Witten differentials:
λ (z) = zF′ (z)dz. It seems to be in the opposition with Seiberg-Witten consideration, but
1,2 1,2
as we shall see soon, in our examples λ (z) and λ (z) are not independent, and therefore we have
1 2
only one differential.
7
3 Product equations
The difference equations for F (z) which follow from (8) define the Seiberg-Witten curve as well
1,2
as the Seiberg-Witten differential [31, 37, 36]. The conformal map method, proposed in [31], is
powerfulenoughwhenwe dealwithhyperellipticcurves. Itcorrespondstothe Yang-Millstheories
withsomefundamentalmattermultiplets. Solutionsofmoregeneraldifferenceequationscannotbe
obtained likewise (except for some very particular exceptions). In this section we propose another
method which allows to find solutions for more general models. Namely having exponentiated a
differenceequationweobtainaproductequation. ItturnsoutthattheVietatheoremtogetherwith
the simplicity principle allows to determine completely particular solutions of product equations.
We did not try to prove the unicity of obtained solutions. Instead we have found them for various
models and checked for consistency.
3.1 An example
To exhibit the idea let us considerin some details the simplest case: the Yang-Mills theory for the
group SU(N) with N <2N fundamental matter multiplets. The difference equation constructed
f
withthehelpoftheTable2isgivenbytheexpression(40). Therunningcomplexcouplingconstant
1
in this example is τ(Λ) = τ + lnΛ2N−Nf. Since our theory is not conformal we can neglect
0
2πi
1
the first term and put simply τ(Λ)= lnΛ2N−Nf. In that follows we will basically drop τ .
0
2πi
IntheSeiberg-WittentheoryoneworksnotwithF(z),butratherwithitsexponenty(z)defined
in (42). Let us rewrite the difference equation in terms of this function. Taking the exponent we
obtain the product equation:
y+(t)y−(t)=qQ(t), t γ , (9)
l
∈
Nf
where q =2πiτ =Λ2N−Nf is the instanton counting parameter and Q(z)= (z+mf).
f=1
Y
Note that the properties of the profile function (35) imply that
N 1
F(z)= lnz+O (10)
2πi z2
(cid:18) (cid:19)
when z . Therefore in this limit we have y(z) = zN +O(zN−2). Such a behavior can take
→ ∞
place if y(z) is a solution of an algebraic equation with z-dependent coefficients. Suppose this is
the case. Let the degree of this polynomial be n. It follows, that y(z) is one of its roots, which
we denote as y (z),...,y (z). This equation defines an algebraic curve. Suppose as well that this
1 n
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curvepossessonlydoubleramificationpoints. Thatis,ataparticularvalueofz no morethantwo
roots can coincide. This statement is justified in Appendix B.
Let us come back to (9). Without loss of generality we can suppose that y+(t) is what we
1
mean by y+(t) and y+(t) = y−(t) is y−(t) when t γ. Therefore for our model we find that
2 1 ∈ l
y+(t)y+(t)=qQ(t) onthe cuts. Since y (z)and y (z)are holomorphicfunctions, we can continue
1 2 1 2
theproductequationfromcutstothe wholedomainoftheiranalyticity. Thereforethesefunctions
satisfyy (z)y (z)=qQ(z). Thesimplestequationwithatleasttworootsisthequadraticequation.
1 2
With the help of the Vieta theorem we conclude that the desired equation for y(z) looks like
y2(z) P(z)y(z)+qQ(z)=0,
−
where P(z)is a polynomialofz. Further analysis showsthat the cuts appearsaroundthe zerosof
the polynomial P(z), and therefore for the SU(N) model we should use a degree N polynomial.
N
Conventionally it is written as P(z) = (z α ) with some parameters α which are related to
l l
−
l=1
Y
the Higgs expectation values via (41). The condition (10) shows that for y(z) which defines F(z)
we should take the following root (the branch of the square root is defined in such a way that
√1+2z 1+z):
≈
P(z) 4qQ(z)
y(z)= 1+ 1 .
2 s − P2(z) !
3.2 Symplectic case
In that follows it will be useful to discuss some aspects of Sp(N) models with N fundamental
f
matter. The difference equation for the primitive of the profile function resolvent can be deduces
form the Table 2. We get
2 1 Nf
F+(t)+F−(t)+ ln t ln t+m +ln t+m =2τ, t γ,
f f l
πi | |− 4πi | | |− | ∈
fX=1(cid:16) (cid:17)
where 2πiτ = lnΛN+1−Nf. As it was shown in [32] the profile function for the this case is
symmetric. It follows that y(z) = y( z). In order to absorb the second term we redefine the
−
profile function as follows: f(x) = f(x)+2x. According to the profile function redefinition we
| |
also introduce
e 1 1
F(z)= dxf′′(x)ln(z x)=F(z)+ lnz,
4πi − πi
Z
ey(z)=exp2πiF(ze)=y(z)z2.
e e
9