Seiberg-Witten theory, matrix model and AGT relation
aa r X i v : . [ h e p - t h ] J un YITP-10-40October 26, 2018
Seiberg-Witten theory, matrix model and AGT relation
Tohru Eguchi and Kazunobu Maruyoshi
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
AbstractWe discuss the Penner-type matrix model which has been proposed to explain the AGTrelation between the 2-dimensional Liouville theory and 4-dimensional N = 2 superconformalgauge theories. In our previous communication we have obtained the spectral curve of the ma-trix model and showed that it agrees with that derived from M-theory. We have also discussedthe decoupling limit of massive flavors and proposed new matrix models which describe Seiberg-Witten theory with flavors N f = 2 ,
3. In this article we explicitly evaluate the free energy ofthese matrix models and show that they in fact reproduce the amplitudes of Seiberg-Wittentheory.
Introduction
Recently a very interesting relation between the Nekrasov partition function of N = 2 conformalinvariant SU (2) gauge theory and the conformal block of the Liouville field theory was proposed[1]. It seems that this is the first example of a precise mathematical relationship betweenquantum field theories defined at different space-time dimensions. There have been variousattempts at checking this AGT relation at lower instanton numbers by direct evaluation ofLiouville correlation functions [2, 3, 4, 5, 6]. There have also been attempts at proving therelation by comparing the recursion relation satisfied by the descendants of the conformalblocks and Nekrasov’s partition function [7, 8, 9, 10].On the other hand, a Penner type matrix model has been proposed to interpolate betweenthe Liouville theory and gauge theory [11] and provide an explanation for the AGT relation.In a previous communication [12] we have studied this matrix model and also proposed modelsfor asymptotically free theories obtained by decoupling some of massive flavors. We haveshown that the spectral curves of these matrix models reproduce those based on the M-theoryconstruction and their free energies satisfy the scaling identities known in the SU (2) Seiberg-Witten theory. (See also [13, 14] for A r quiver matrix model).In this paper we would like to evaluate the free energies of these matrix models in thelarge N limit explicitly and show that they in fact exactly reproduce the amplitudes of SU (2)Seiberg-Witten theory.In section 2 we first describe the general properties of matrix models. In section 3 wecompute the free energies: we integrate the Seiberg-Witten differential of the matrix modeland evaluate the filling fraction in terms of the parameters of the spectral curve. We theninvert this relation and derive the free energy. We present the computation for SU (2) gaugetheory with two, three and four flavors and show that they all reproduce the amplitudes ofSeiberg-Witten theory. In section 4 we discuss decoupling limits of some quiver gauge theories.Section 5 is devoted to conclusion and discussion.Note: in our convention the free energy of the matrix model F m is off by a factor 4 fromthat of gauge theory. Thus we will check the agreement 4 F m = F gauge throughout this paper. SU (2) gauge theories and matrix models It has been proposed that the Nekrasov partition function for N = 2, SU (2) gauge theorywith four flavors (summarized in appendix A) coincide with the four-point conformal block of1iouville theory [1]: Z SU (2)inst = Z CFT ≡ h V ˜ m ∞ ( ∞ ) V ˜ m (1) V ˜ m ( q ) V ˜ m (0) i . (2.1)Here V ˜ m is the vertex operator, Q = b + 1 /b and the central charge of the Liouville theory is c = 1 + 6 Q .In order to relate the Liouville theory to matrix model, we consider the Dotsenko-Fateevintegral representation of the four-point conformal block in terms of the free field φ ( z ) [15]: Z DF = *(cid:18)Z dλ I : e bφ ( λ I ) : (cid:19) N V ˜ m ∞ ( ∞ ) V ˜ m (1) V ˜ m ( q ) V ˜ m (0) + , (2.2)where the vertex operator V ˜ m i ( z i ) is given by : e ˜ m i φ ( z i ) : and we have introduced the N -foldintegration of screening operators. OPE of the scalar field is given by φ ( z ) φ ( ω ) ∼ − z − ω ).Momentum conservation condition relates the external momenta and the number of integrals as P i =0 ˜ m i + ˜ m ∞ + bN = Q . We redefine the momenta as ˜ m i = im i g s for i = 0 , ∞ and ˜ m i = im i g s + Q for i = 1 , X i =0 im i + im ∞ + 2 bg s N = 0 . (2.3)As pointed out in [16, 17] and recently in [11] in the context of the AGT relation, the Dotsenko-Fateev representation may be identified as the β -deformation of a one matrix integral Z DF = q m m g s (1 − q ) m m g s N Y I =1 Z dλ I ! Y I In this section, we will evaluate the planar free energy of the matrix models introduced above.In [12], we have shown that the free energy of these models satisfies the identities known inSeiberg-Witten thery [18, 19, 20]. Here, we will evaluate the free energies explicitly and comparethem with the instanton expansions of the prepotentials at lower orders. The computation is abit simpler than in the Seiberg-Witten theory where both the A and B cycle integrals have tobe computed [21, 22]. Here we only have to compute the A integral.We first consider the matrix model for SU (2) gauge theory with N f = 2 in next subsection.Then, we will analyze the cases of N f = 3 and 4 theories in turn. SU (2) gauge theory with N f = 2 The matrix model action corresponding to the SU (2) gauge theory with N f = 2 is given by(2.11). For simplicity, we will omit the subscript 2 of the dynamical scale Λ below. There aretwo saddle points determined by the classical equation of motion: W ′ ( z ) = µ z − Λ2 (cid:18) − z (cid:19) = 0 . (3.1)These lead to the two-cut spectral curve.The planar loop equation reads as usual R ( z ) = − (cid:16) W ′ ( z ) − p W ′ ( z ) + f ( z ) (cid:17) , (3.2)where the resolvent is defined by R ( z ) = h X I g s z − λ I i . (3.3)The function f is given by f ( z ) = 4 g s h X I W ′ ( z ) − W ′ ( λ I ) z − λ I i = c z + c z . (3.4)Coefficients c and c are defined as c = − g s *X I (cid:18) µ λ I + Λ2 λ I (cid:19)+ = − g s N Λ , c = − g s *X I Λ λ I + . (3.5)In the formula for c we have used the equations of motion h P I W ′ ( λ I ) i = 0.Then, the spectral curve x = (2 h R ( z ) i + W ′ ( z )) = W ′ ( z ) + f ( z ) is given by x = Λ z + µ Λ z + 1 z (cid:18) µ + c − Λ (cid:19) + µ Λ z + Λ . (3.6)4his is similar to the curve obtained in [23]. The differential one form is identified with λ m = xdz which has double poles at t = 0 and ∞ with residues µ and µ . Note that the parameter c corresponds to the variable u in Seiberg-Witten theory.We evaluate the filling fraction as g s N = 14 πi I C λ m ( c ) , (3.7)where C is a cycle around one of the cuts in the curve. This integral is identified with theCoulomb moduli a in the gauge theory and we invert the above relation to solve the unknownparameter c .Let us compute the free energy of our model defined by e F m /g s = N Y I =1 Z dλ I ! Y I 12Λ ( c − µ + µ ) . (3.10)Therefore, we obtain Λ ∂F m ∂ Λ = 14 (2 c − µ + µ ) . (3.11)Our remaining task is to determine c in terms of g s N by using (3.7), and this leads to theexplicit form of the free energy.To derive c , let us consider the derivative of (3.7) with respect to c :4 πi ∂ ( g s N ) ∂c = I C dz p P ( z ) , (3.12)where P is the polynomial of degree 4: P ( z ) = z + 4 µ Λ z + 4Λ ( µ + c − Λ z + 4 µ Λ z + 1 . (3.13)It is easy to transform this polynomial so that (3.12) becomes the standard elliptic integral ofthe first kind. In the following, we set A = µ + c − Λ and express the result in terms of A .5or simplicity, let us consider the equal mass case: µ = µ = m in the following. In thiscase, by the transformation z = t − t +1 and rescaling of t , the integrand of the right hand side of(3.12) can be brought to the standard form √ p S + (Λ + 4 m Λ + 2 A ) dt p (1 − t )(1 − k t ) , (3.14)where k = S − /S + and S ± = 1Λ + 4 m Λ + 2 A (cid:16) − + 2 A ± Λ √ − A + 16 m (cid:17) . (3.15)Then, we can identify the integral (3.12) in terms of the hypergeometric function:4 πi ∂ ( g s N ) ∂A = 2 √ p S + (Λ + 4 m Λ + 2 A ) Z /k dt p (1 − t )(1 − k t )= √ πi p S + (Λ + 4 m Λ + 2 A ) F ( 12 , , 1; 1 − k ) . (3.16)where we have used R /k dt √ (1 − t )(1 − k t ) = iK ′ ( k ) = iK ( k ′ ) with k ′ = 1 − k . We express theright hand side as a small Λ expansion which corresponds to the instanton expansion in gaugetheory. (Note that k = 1 + O (Λ).) After integrating over A , we obtain2 g s N = √ A − m A Λ − A − Am + 15 m A Λ − m (3 A − Am + 21 m )256 A Λ − A − A m + 294 A m − Am + 1001 m )16384 A Λ + O (Λ ) ! . (3.17)Then, we invert this equation and solve for A : A = a + m a Λ + a − m a + 5 m a Λ + m (5 a − m a + 9 m )64 a Λ + 5 a − m a + 1638 m a − m a + 1469 m a Λ + O (Λ ) , (3.18)where we have introduced a = 2 g s N . Finally, we substitute this into (3.11) and integrate byΛ to obtain4 F m = 2 (cid:0) a − m (cid:1) log Λ + a + m a Λ + a − m a + 5 m a Λ + m (5 a − m a + 9 m )192 a Λ + 5 a − m a + 1638 m a − m a + 1469 m a Λ + O (Λ ) . (3.19)This agrees with the U (2) gauge theory prepotential with ~a = ( a, − a ) obtained from theNekrasov partition function (A.16) or from the Seiberg-Witten theory [22]. (The first term isthe one-loop part and the others are the instanton part.) Together with the prefactor e − Λ228 g s wesee that the full free energy is the same as that of SU (2) gauge theory.6 .2 SU (2) gauge theory with N f = 3 Next, let us consider the matrix model corresponding to the gauge theory with N f = 3. Thematrix model action is given by (2.9). We will omit the subscript 3 of the dynamical scaleΛ from now on. As in the previous subsection, there are two saddle points in the classicalequation of motion. In the planar limit, the loop equation leads to the spectral curve x ( z ) = W ′ ( z ) + f ( z ) where f ( z ) is written as f ( z ) = c z + c z − c z , (3.20)with coefficients c = − g s *X I (cid:18) µ λ I + Λ2 λ I (cid:19)+ , c = − g s *X I m λ I − + , c = − g s *X I Λ λ I + . (3.21)We can easily see that c + c = 0 due to the equations of motion.The one form defined by λ m ≡ x ( z ) dz has a double pole at z = 0 and a simple pole at z = 1 and ∞ with residues µ , m and m ∞ , respectively. The residue at z = ∞ gives a furtherconstraint on c i : c + c = m ∞ − ( µ + m ) . (3.22)This condition together with the relation c + c = 0 leaves only one of the parameters inde-pendent. Let us choose c to be independent.It is then related to the filling fraction by the integral4 πig s N = I C λ m ( c ) . (3.23)For completeness, let us write down here the explicit form of the curve x = P ( z )4 z ( z − with P ( z ) = 4 m ∞ z + 4(( µ + m )Λ + m − µ − m ∞ − c ) z +(Λ − µ + 4 µ − m + 4 c ) z − − µ ) z + Λ . (3.24)It is convenient to introduce the notation B as B = c − µ Λ + µ . (3.25)The polynomial is then rewritten as P ( z ) = 4 m ∞ z + 4(Λ m + m − m ∞ − B ) z + (Λ − µ + m ) + 4 B ) z − − µ ) z + Λ . (3.26)7et us consider the free energy of this matrix model. From the definition, its derivative inΛ is written as ∂F m ∂ Λ = − g s *X I λ I + = c 4Λ = 14Λ ( B + µ Λ − µ ) . (3.27)In order to determine B we take a derivative of (3.23) with respect to B :4 πi ∂ ( g s N ) ∂B = − I C dz p P ( z ) . (3.28)For simplicity, we consider the case where µ = m and m = m ∞ = 0 in what follows. In thiscase, P becomes a polynomial of degree 3: P ( z ) = ( z − − Bz + (Λ − m ) z − Λ ) . (3.29)After a change of variable (first shifting z → z − p and then rescaling as z = Qt ), we obtain P ( z ) → − BQ (1 + p ) × t (1 − t )(1 − k t ) , (3.30)where k = Q p , p = 12 (cid:18) − Λ4 B (Λ − m ) + Q (cid:19) , Q = Λ4 B p (Λ − m ) − B. (3.31)As a result, (3.28) becomes4 πi ∂ ( g s N ) ∂B = − p − B (1 + p ) Z dt p t (1 − t )(1 − k t )= − π p − B (1 + p ) F ( 12 , , k ) . (3.32)By expanding the hypergeometric function and then integrating over B , we obtain2 g s N = √ B m Λ4 B − B ( B + 3 m )Λ + m B (5 m + B )Λ − B (3 B + 30 m B + 175 m )Λ − m B (9 B + 70 m B + 441 m )Λ − B (5 B + 105 m B + 735 m B + 4851 m )Λ + O (Λ ) ! . (3.33)We invert this equation for B , B = a − m Λ2 + m + a a Λ + a − a m + 5 m a Λ + m a (9 a + 70 m a + 441 m )Λ + m a (185 a + 1946 m a + 15885 m )Λ + O (Λ ) , (3.34)8here we have defined a = 2 g s N . Finally, by substituting this into (3.27), we obtain4 F m = ( a − m ) log Λ + m Λ2 + m + a a Λ + a − m a + 5 m a Λ + m × a + 70 m a + 441 m a Λ + m × a + 1946 m a + 15885 m a Λ + O (Λ ) . (3.35)Term with log Λ is the one-loop contribution. Remaining terms agree precisely with the prepo-tential obtained from the Nekrasov partition function (A.13). SU (2) gauge theory with N f = 4 We now consider the matrix model with the original action (2.5). The planar loop equation R ( z ) = − (cid:16) W ′ ( z ) − p W ′ ( z ) + f ( z ) (cid:17) involves a function f ( z ) which now has a form f ( z ) = P i =0 c i z − q i . Parameters { c i } are given by c = − g s m h X I λ I i , c = − g s m h X I λ I − i , c = − g s m h X I λ I − q i . (3.36)By studying the behavior of loop equation at large z we find that the parameters obey X i =0 c i = 0 , c + qc = m ∞ − ( X i =0 m i ) . (3.37)By eliminating c and c , the spectral curve becomes x = P ( z ) z ( z − ( z − q ) , (3.38)where P is the following polynomial of degree 4 P ( z ) = m ∞ z + (cid:16) − (1 + q )( m ∞ + m ) + (1 − q )( m − m ) − m ( qm + m ) + qc (cid:17) z + (cid:16) qm ∞ + (1 + 3 q + q ) m + (1 − q )( m − qm ) + 2(1 + q ) m ( qm + m ) − (1 + q ) qc (cid:17) z + (cid:16) − q (1 + q ) m − q m m − qm m + q c (cid:17) z + q m . (3.39)The meromorphic one form xdz has simple poles at z = 0 , , q and z = ∞ with residues m , m , m and m ∞ .Again, we consider the derivative of the free energy: ∂F m ∂q = g s m (cid:28) tr 1 q − M (cid:29) = m R ( z ) | z = q . (3.40)9his can be easily computed by expanding the resolvent at z = q , R ( z ) = c m + O ( z − q ).Then, we obtain a simple expression for the free energy ∂F m ∂q = c − q ) ( X i =0 m i ) − m ∞ − c ! . (3.41)In the last equality we used the relation (3.37).In what follows, we consider the simple case where all the hypermultiplet masses are equalto m : i.e. m = m ∞ = 0 and m = m = m . In this case, the polynomial is reduced to degree3: P ( z ) = Cz ( z − z + )( z − z − ), where we have introduced C ≡ c q and z ± = 12 q − (1 − q ) m C ± (1 − q ) r − q ) m C + (1 − q ) m C ! . (3.42)By taking the C derivative of xdz , the holomorphic one form becomes ∂∂C xdz = 12 √ Cz + dz p z (1 − z )(1 − k z ) , k = z − q . (3.43)The remaining calculation is similar to those considered in the previous subsections. That is,we first evaluate the period integral of the above one form. Then by expanding in m C andintegrating over C , we obtain2 ig s N = √ C (cid:18) h ( q ) − h ( q ) m C − h ( q )3 m C − h ( q )5 m C + O ( m C ) (cid:19) , (3.44)where h i ( q ) are the expansion coefficients of the period integral in m C and depend only on q . h ( q ) is for the theory with massless flavors: h ( q ) = 1 + 14 q + 964 q + 25256 q + 122516384 q + O ( q ) . (3.45)Lower order expansions of h , h and h are given by h ( q ) = 12 + 18 q + 1128 q + 1512 q + 2532768 q + O ( q ) ,h ( q ) = 38 + 2732 q + 27512 q + 32048 q + 27131072 q + O ( q ) ,h ( q ) = 516 + 12564 q + 11251024 q + 1254096 q + 125262144 q + O ( q ) . (3.46)Solving for C , we obtain C = a h ( q ) + 2 h ( q ) h ( q ) m a + 2 h ( q ) h ( q ) − h ( q ) m a + 10 h ( q ) h ( q ) − h ( q ) h ( q ) h ( q ) + 2 h ( q ) h ( q )5 m a + . . . ! , (3.47)10here a = 2 ig s N . By substituting the above expression into (3.41) and integrating over q , wefinally obtain the N f = 4 free energy4 F m = ( a − m ) log q + a + 6 a m + m a q + 13 a + 100 m a + 22 m a − m a + 5 m a q + 23 a + 204 m a + 51 m a − m a + 45 m a − m a + 9 m a q + O ( q ) . (3.48)This perfectly agrees with the instanton partition function (A.9).Finally, we make a brief comment on the massless theory. In this case, the expression for C simplifies and becomes C = a /h ( q ) where h ( q ) is (3.45). Thus, it is easy to derive4 F m = a (cid:18) log q − log 16 + 12 q + 1364 q + 23192 q + 270132768 q + 505781920 q + O ( q ) (cid:19) , (3.49)where we have added the one-loop contribution − a log 16. Note that this can be written as4 F m = a log q IR where q and q IR = e πiτ IR are related by q = ϑ ( τ IR ) ϑ ( τ IR ) = 16 q IR − q + 704 q − q + 11488 q + . . . . (3.50)as already discussed in [25, 26, 1, 27, 7, 12]. Thus the theory appears classical in terms of IRcoupling constant τ IR . In this section, we study matrix models which describe N = 2 SU (2) quiver gauge theories.First of all, we consider a matrix model describing SU (2) linear quiver gauge theory whereeach gauge group has a vanishing beta function [28]. Then by taking its decoupling limit, wepropose models for asymptotically free gauge theories in subsection 4.1.According to the AGT conjecture, SU (2) n − linear quiver gauge theory is related to the n -point conformal block of the Liouville theory, which is represented by the trivalent graph [29]as in Fig 1. As seen in section 2, the Dotsenko-Fateev representation of the conformal blocksuggests a matrix model with the following action [11]: W ( M ) = n − X i =0 m i log( M − t i ) , (4.1)where t = 0 and t = 1. Other parameters t i = Q i − k =1 q k ( i = 2 , . . . , n − 2) describe complexstructure of the n -punctured sphere. Note that we also have the prefactor as in (2.4)From the gauge theory perspective, the parameters q k are related to the gauge couplingconstants q k = e πiτ k of the gauge group SU (2) n − . For n = 4, this reduces to the matrix11 (cid:0) Figure 1:model which we studied in subsection 3.3. Parameters m and m n − are related to the massparameters of two hypermultiplets in the fundamental representation of the SU (2) at one endof the quiver: m n − = ( µ + µ ) and m = ( µ − µ ). Also, the masses of the bifundamentalhypermultiplets are identified with m i ( i = 2 , . . . , n − m and m ∞ as m = ( µ + µ )and m ∞ = ( µ − µ ). The mass parameter m ∞ is introduced by the following condition: n − X i =0 m i + m ∞ + 2 g s N = 0 . (4.2)The critical points are determined by the equation of motion n − X i =0 m i λ I − t i + 2 g s X J ( = I ) λ I − λ J = 0 . (4.3)If we ignore the second term, we obtain n − e p ( p = 1 , , . . . , n − N p ( p = 1 , , . . . , n − 2) be the number of the matrix eigenvalues which are at the critical point e p . We take the large N limit with mass parameters { m i } and filling fractions { ν p ≡ g s N p } being kept fixed. Since this is one matrix model, the loop equation is still the same as in theprevious cases (3.2) f ( z ) ≡ g s tr (cid:28) W ′ ( z ) − W ′ ( M ) z − M (cid:29) = n − X i =0 c i z − t i ≡ Z ( t ) Q n − i =0 ( z − t i ) . (4.4)We note that a polynomial Z ( t ) is of degree n − 3, since the leading term vanishes due toequations of motion.Finally, we define the meromorphic one form λ = x ( z ) dz as x ( z ) ≡ (2 h R ( z ) i + W ′ ( z )) = W ′ ( z ) + f ( z ) . (4.5)12 .1 Matrix model for asymptotically free quiver gauge theory The matrix model corresponding to asymptotically free quiver gauge theory can be obtainedby taking the decoupling limit as in section 2. Only possible limits which does not spoil thecondition (4.2) is the case where µ (= m − m ∞ ) or µ (= m n − − m ) is taken to infinity.For the sake of illustration, let us consider the n = 5 case with the action W ( z ) = X i =0 m i log( z − t i ) , (4.6)where t = q and t = q q . This corresponds to SU (2) × SU (2) quiver gauge theory whosegauge coupling constants are q and q . We first take a limit µ → ∞ with µ q = ˜Λ fixed. Inthis limit, we obtain W ( z ) → µ log z + X i =1 , m i log( z − t i ) − q ˜Λ2 z . (4.7)It is natural to anticipate that this matrix model corresponds to the quiver theory of onefundamental matter coupled to the second gauge group SU (2) and two fundamental multipletsare coupled to the first gauge group. The relation of the mass parameters (4.2) becomes µ + P i =1 , m i + m ∞ + 2 g s N = 0 in this limit.By further taking the limit µ → ∞ with µ q = Λ fixed, we obtain from (4.7) W ( z ) → µ log z + m log( z − − Λ z − ˜Λ2 z , (4.8)where we have also rescaled z → q z . The relation of the mass parameters (4.2) becomes µ + m + µ + 2 g s N = 0 . (4.9)This matrix model is expected to describe SU (2) × SU (2) quiver gauge theory with eachgauge factor coupled to one hypermultiplet. Both of the gauge factors have nonvanishing betafunctions and the theory is asymptotically free.It is possible to generalize this construction to the case with n > 5. A decoupling limit ofa hypermultiplet at the last end of the quiver is µ → ∞ with µ q n − = ˜Λ fixed. Also, anotherdecoupling limit of a hypermultiplet at the first end of the quiver is µ → ∞ with µ q = Λfixed. By taking these limits, we finally obtain W ( z ) = µ log z + m log( z − 1) + n − X i =3 m i log z − i − Y k =2 q k ! − Λ z − ˜Λ2 z n − Y k =2 q k ! , (4.10)with the following relation for the mass parameters: µ + n − X i =2 m i + µ + 2 g s N = 0 . (4.11)13 Conclusion and discussion In this paper we have studied the matrix model proposed to explain the AGT relation andinterpolate the Liouville and N = 2 SU (2) gauge theories. We have explicitly evaluated thefree energy of the matrix models describing SU (2) gauge theory with N f = 2 , , N limit and it is very important to see if our results can be generalizedand reproduce full Nekrasov partition functions. There is already an interesting work in thisdirection [30, 31] and we hope that we can report further results in future publications. Acknowledgements K.M. would like to thank K. Hosomichi, H. Itoyama and F. Yagi for discussions and comments.He also would like to thank Ecole normale Superieure, SISSA and University of Amsterdamfor warm hospitality during part of this project. Research of T.E. is supported in part by theproject 19GS0219 of the Japan Ministry of Education, Culture, Sports, Science and Technol-ogy. Research of K.M. is supported in part by JSPS Bilateral Joint Projects (JSPS-RFBRcollaboration). AppendixA Nekrasov partition function The instanton partition function of N = 2 U (2) gauge theory with N f = 4 is expressed as asum over all possible Young tableaus parametrized as Y = ( λ ≥ λ ≥ . . . ) where λ ℓ is theheight of the ℓ -th column [24, 1]: Z inst = X ( Y ,Y ) q | ~Y | Z vector ( ~a, ~Y ) Z antifund ( ~a, ~Y , µ ) Z antifund ( ~a, ~Y , µ ) Z fund ( ~a, ~Y , − µ ) Z fund ( ~a, ~Y , − µ ) . (A.1)14ere Z vector ( ~a, ~Y ) = Y i,j =1 , Y s ∈ Y i (cid:0) a ij − ǫ L Y j ( s ) + ǫ ( A Y i ( s ) + 1) (cid:1) − × Y t ∈ Y j (cid:0) a ji + ǫ L Y j ( t ) − ǫ ( A Y i ( t ) + 1) + ǫ + (cid:1) − ,Z fund ( ~a, ~Y , µ ) = Y i =1 , Y s ∈ Y i ( a i + ǫ ( ℓ − 1) + ǫ ( m − − µ + ǫ + ) ,Z antifund ( ~a, ~Y , µ ) = Y i =1 , Y s ∈ Y i ( a i + ǫ ( ℓ − 1) + ǫ ( m − 1) + µ ) , (A.2)and ǫ + = ǫ + ǫ and a ij = a i − a j . For a box s at the coordinate ( ℓ, m ), the leg-length L Y ( s ) = λ ′ m − ℓ and the arm-length A Y ( s ) = λ ℓ − m where λ ′ m is the length of the m -th row.The minus signs of the masses in Z fund are due to the convention.In order to derive the expression for SU (2) gauge theory, we set the Coulomb moduli as ~a = ( a, − a ) which gives Z vector ( a, ~Y ) = Y i =1 , Y s ∈ Y i (cid:0) aδ ij − ǫ L Y j ( s ) + ǫ ( A Y i ( s ) + 1) (cid:1) − × Y t ∈ Y j (cid:0) − aδ ij + ǫ L Y j ( t ) − ǫ ( A Y i ( t ) + 1) + ǫ + (cid:1) − ,Z fund ( a, ~Y , µ ) = Y i =1 , Y s ∈ Y i ( aδ i + ǫ ( ℓ − 1) + ǫ ( m − − µ + ǫ + ) ,Z antifund ( a, ~Y , µ ) = Y i =1 , Y s ∈ Y i ( aδ i + ǫ ( ℓ − 1) + ǫ ( m − 1) + µ ) , (A.3)where we define δ = +1 and δ = − 1, and δ ij = i = j, i = 1 and j = 2 , − i = 2 and j = 1 . (A.4)Then, the SU (2) and U (2) partition functions are related by the U (1) factor as pointed out in[1]: Z inst | ~a =( a, − a ) = f U (1) Z SU (2)inst , f U (1) = (1 − q ) µ µ ǫ ǫ ( ǫ + + µ µ ) . (A.5)Note that this expression differs by a minus sign in front of ( µ + µ ) from the one of [1]. Asargued in [1], the SU (2) partition function is invariant under “flips”. These flips are reducedin the self-dual case ǫ = − ǫ = ~ to a → − a, µ ± µ → − ( µ ± µ ) , µ ± µ → − ( µ ± µ ) . (A.6)15auge theory prepotential can be obtained in the limit where the deformation parametersgo to zero (with a fixed ratio ǫ /ǫ ): F inst = lim ǫ , → ( − ǫ ǫ ) log Z inst . (A.7)In the self-dual case, SU (2) gauge theory prepotential is written as F SU (2)inst = lim ~ → ~ (log Z inst − log f U (1) )= F inst + 12 ( µ + µ )( µ + µ ) log(1 − q ) . (A.8)To compare with the free energy of the matrix model, we present an expansion of F inst for theequal mass case µ i = m F inst = a + 6 m a + m a q + 13 a + 100 m a + 22 m a − m a + 5 m a q (A.9)+ 23 a + 204 m a + 51 m a − m a + 45 m a − m a + 9 m a q + O ( q ) . A.1 U (2) gauge theory with N f = 3 Let us consider Nekrasov partition function of the theory with N f = 3. This can be obtainedfrom the above partition function by taking a limit µ → ∞ with µ q ≡ Λ fixed. In the k -instanton part the only factor which contains µ is Z fund ( a, ~Y , − µ ) = Y i =1 , Y s ∈ Y i ( aδ i + ǫ ( ℓ − 1) + ǫ ( m − 1) + µ + ǫ + ) . (A.10)When combined with k -instanton factor q k , this gives the leading contribution Λ k and the othercontributions are suppressed in the limit. Therefore, we obtain Z N f =3inst = X ( Y ,Y ) Λ | ~Y | Z vector ( a, ~Y ) Z antifund ( a, ~Y , µ ) Z antifund ( a, ~Y , µ ) Z fund ( a, ~Y , − µ ) . (A.11)The U (1) factor reduces to f U (1) → f U (1) ,N f =3 = exp (cid:18) − ( µ + µ )Λ ǫ ǫ (cid:19) . (A.12)In the simple case of µ = m and µ = µ = 0 which we considered in subsection 3.2, theprepotential of the gauge theory is given by F N f =3inst = 12 m Λ + a + m a Λ + a − m a + 5 m a Λ + O (Λ ) . (A.13)16 .2 U (2) gauge theory with N f = 2 We can further take a limit where µ → ∞ while keeping µ Λ ≡ Λ fixed. In this limit, thepartition function becomes: Z N f =2inst = X ( Y ,Y ) Λ | ~Y | Z vector ( a, ~Y ) Z antifund ( a, ~Y , µ ) Z fund ( a, ~Y , − µ ) , (A.14)and the U (1) factor is reduced to f U (1) → exp (cid:16) − Λ ǫ ǫ (cid:17) . SU (2) prepotential is given by F SU (2) ,N f =2inst = F N f =2inst − Λ . (A.15)For the equal mass case with µ = µ = m , lower terms of instanton expansion are given by F N f =2inst = a + m a Λ + a − a m + 5 m a Λ + m (5 a − a m + 9 m )192 a Λ + O (Λ ) . (A.16) References [1] L. F. Alday, D. Gaiotto and Y. Tachikawa, “Liouville Correlation Functions from Four-dimensional Gauge Theories,” Lett. Math. Phys. , 167 (2010) [arXiv:0906.3219 [hep-th]].[2] A. Mironov, A. Morozov and S. Shakirov, “Matrix Model Conjecture for Exact BS Periodsand Nekrasov Functions,” JHEP , 030 (2010) [arXiv:0911.5721 [hep-th]].[3] A. Mironov, A. Morozov and S. 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