Simple Derivation of the Picard-Fuchs equations for the Seiberg-Witten Models

Transcription

1 hep-th/ IPM-97 Simple Derivation of the Picard-Fuchs equations for the Seiberg-Witten Models Mohsen Alishahiha Institute for Studies in Theoretical Phsics and Mathematics, P.O.Box , Tehran, Iran Department of Phsics, Sharif Universit of Technolog, P.O.Box , Tehran, Iran Abstract A closed form of the Picard-Fuchs equations for N = 2 supersmmetric Yang-Mills theories with massless hpermultiplet are obtained for classical Lie gauge groups. We consider an number of massless matter in fundamental representation so as to keep the theor asmptoticall free.

2 1 Introduction Resentl dualit has become a ver important tool both in supersmmetric Yang- Mills theories and string theor. Seiberg and Witten[1] have used dualit and holomorph to obtain the exact prepotential of N = 2 SYM theor with gauge group SU(2). (for review see: e.g.[2], [3] and [4]). The ke point in N = 2 SYM models was the discover of a hperelliptic curve with r complex dimensional moduli space (r is the rank of the guage group) with certain singulareties, which gives information about the low energ Willsonian effective action. The Seiberg-Witten data is a hpereliptic curve with a certain meromorphic one form (E ui,λ SW ). Indeed, the prepotential of N = 2 SYM theor in the coulomb phase can be described with the aid of a famil of complex curves with the identification of the Vacuum expectation value (v.e.v) a i and their duals a D i with the periods of the curve a i = λ SW and a D i = λ SW, (1) α i β i where α i and β i are the homolog ccles of the corresponding Riemann surface. There are two well known methods for finding the periods and thereb the prepotential. The first method is to calculate the periods directl from the above integrals. This method has been developed in[5] and [6]. The explicitl calculated the full expansion of the renormalized order parameters using the method of residues. B this method, the worked out explicitl the perturbative corrections as well as the one and two instanton contributions to the effective prepotential. On the other hand, one ma use the fact that the periods Π = (a i,a D i ) satisf the Picard-Fuchs equations. Probabl from the Picard-Fuchs equations one can obtain the prepotential in an analtic wa, which is for example, important in the instanton calculus. Recentl some of these equations have been obtained in [7] and[8]. Also in [9] we obtained a simple closed form of the Picard-Fuchs equations for Pure N =2 SYM theories for classical Lie gauge groups. The Picard-Fuchs equations, can also be obtained from the mirror smmetr in Calabi-Yau manifold[10]. In this artical we extend the results of [9] to obtain a closed form for N = 2 SYM theories with classical Lie gauge groups which have massless matter in fundamental representation. The hperelliptic curves for classical gauge groups with an number of matter in the fundamental representation are known [11],[12] and [13]. Although in some cases different hperellieptic curves have been proposed for the same gauge group and the same hpermultiplet contents, but it was shown in [14] b explicit calculations up to two instanton processes, that the corresponding effective prepotentials are the same for all these different curves. This equivalence results from the fact that the effective prepotential is unchanged under analtic reparametrizations of the classical order parameters[14]. The Seiberg-Witten data (E ui,λ SW ) for classical gauge groups with n f massless matter in fundamental representation have been proposed as follows [11],[12] and 1

3 [13] 2 = p 2 (x) G(x), λ SW = ( G 2G p p ) xdx, (2) where p = x m+ɛ u i x m+ɛ i (3) with m = r +1,,3,..., ɛ = 0 for A r series and m =2r, i =2,4,..., ɛ = 0 for B r and D r series, and m =2r, i =2,4,..., ɛ = 2 for C r series, and u i s, the Casimirs of the gauge groups. Also G =Λ 2m n f (x δ2m 1,nf ) n f fora r G =Λ 2m 2 2n f x 2+2n f G =Λ 2m+4 2n f x2n f forb r forc r (4) G =Λ 2m 4 2n f x 4+2n f ford r. Note that the D r series has an exceptional Casimir, t, of degree r, but in our notation we set u 2r = t 2. From explicit form of λ SW and the fact that the λ SW is linear dependent on the Casimirs, setting u i = i one can see i λ SW = xm+ɛ i dx + d( ), i j λ SW = x2m+2ɛ i j p(x)dx + d( ). (5) 3 The procedure of drivation of the Picard-Fuchs equations is to find proper linear combinations of xm i dx and x2m i j p(x)dx which give total drivative, then b 3 integrat from to sides and using (1) and (5), one can find second order differential equations for the periods. For example, from the second equation of (5) one can find the following identit for the periods L i,j;p,q Π=0where 2 B r and D r Cases L i,j;p,q = i j p q, i+j = p+q (6) From (2) the proposed hperelleptic curve for these gauge groups with n f m k 1 massless matter in the fundamental representation are 2 = p 2 Λ 2m 2k 2n f x 2k+2n f, (7) 2

4 where k = 1 for B r and k = 2 for D r. B direct calculation one can see that d dx (xn )=(n k n f) xn 1 (m n f k) xm+n 1 x m+n 1 i p+ (m k n 3 f i)u i p 3 (8) Now from the equation (5), we can find the second order diffrential equations for the periods (L n Π = 0) as follows L n =(k+n f n) m n+1 +(m n f k) 2 m n 1 (m k n f i)u i i m n+1. (9) Here n =2s 1ands=1,..., r 1. Note that for s = r equation (8) does not give the second order differential equation with respect to u i. So b this method we can onl obtain r 1 differential equations. An other equation can be obtained b following linear combination or D 0 =(k+n f m)d( xm+1 )+ m (m k n f + i)u i d( xm+1 i ) (10) x m i x 2m i j D 0 = λ SW ( i(i 2)u i + iju i u j p j, 3 (m n f k) 2 Λ 2m 2n f 2k x2n f +2k p)dx. (11) 3 Note that, for the case of n f = m k 1, one can not write the last term in the above equation as the form of x2m i j p(x)dx, so it onl gives the second order 3 differention equation for the periode in the case of 1 n f m k 2, which is L r =1+ i(i 2)u i i + iju i u j i j (m n f k) 2 Λ 2m 2k 2n f 2. (12) j, where for SO(m +1) 2 = m nf k m nf k 2 = m nf 2k m nf n f odd n f even (13) and for SO(m) 2 = m nf k m nf k 2 = m nf k 1 m nf k+1 n f even n f odd (14) 3

5 For the case of n f = m k 1 we should add an extra term to D 0 to cancel the last term, which is D = D 0 +Λ 2 d( xm 1 ). (15) so the last term of the (12) changes to m Λ 2 (i 1)u i 2 i. (16) Equations (9) and (12) give a complet set of the Picard-Fuchs equations for the periods for gauge groups B r and D r with an number of massless matter so as to keep the theor asmptoticll free. (n f m k 1). 3 C r Case First let us write the Picard-Fuchs equatins for the pure gauge theor. 1 The proposed curve for pure N = 2 SYM with gauge group SP(m) is [13] x 2 2 = p 2 Λ 2m+4 (17) where p = x m+2 u i x m+2 i +Λ m+2 i =2,4,..., m (18) B direct calculation one can see that d )=nxn 1 dx (xn z z xn p p, (19) z3 where z = x. So from (5) we have following second order differential for the periods L n = n m n+3 +(m+2) 2 m n+1 (m +2 i)u i i m n+3. (20) here n =2s+1 and s =1,..., r 1. As same as B r and D r cases, there is an difficult for s = r, again, we have onl r 1 differential equatins. One can see that, the last equation can be obtained b the following linear combination D 0 =(m+2)d( xm+3 )+ z (m +2+i)u i d( xm+3 i ) (m +2) 2 Λ m+2 d( x z z ) (21) which gives a second order diffrential equation for the periods m L r =1+ i(i 2)u i i + iju i u j i j +(m+2) 2 Λ m+2 (m i)u i m i+2. (22) j, 1 Here our notation for C r gauge group is difference with one in the previous work[9] i=0 4

6 here u 0 = 1. Let us return to obtain the Picard-Fuchs equations for C r gauge group with n f m + 1 massless matter in the fundamental representation. From (2) the proposed curve for this theor is 2 = p 2 λ 2(m+2 n f ) x 2n f (23) One can see that d dx (xn )=(n n f) xn 1 (m+2 n f ) xm+n 1 x m+n 1 i p+ (m +2 n 3 f i)u i p 3 (24) wich gives the following differential equations L n =(n f n) m n+3 +(m+2 n f ) 2 m n+1 (m +2 n f i)u i i m n+3. (25) As in the previous cases from this method we obtain onl r 1 differential equations. The last equation can be obtained from the following linear combination or D 0 =(n f m 2)d( xm+3 )+ m x m+2 i D 0 = λ SW ( i(i 2)u i + (m +2 n f +i)u i d( xm+3 i ) (26) j, x 2m+4 i j iju i u j p 3 (m n f k) 2 Λ m+2 2n x2n f f p)dx. (27) 3 Similar to B r and C r cases, for n f =1andn f =m+ 1, the last term in the above expression can not rewrite in the form of x2m+4 i j p pdx, so we should add an 3 extra term to D 0. For the case of 2 n f m, the above equation gives a second order differential equation L r =1+ i(i 2)u i i + iju i u j i j +(m+2 n f ) 2 Λ 2m+4 2n f 2. (28) j, where 2 = nf nf for even n f and 2 = nf 1 nf +1 for odd n f.forn f =m+1d 0 should changes to D = D 0 +Λ 2 d( xm+1 ), (29) so the last term of the equation (28) changes to Λ 2 (i 1)u i 2 i, and for n f =1 D=D 0 (m+1)2 Λ 2(m+1) d( x ), (30) u m and the last term of the equation (28) changes to m 2 (m +1) 2Λ2m+2 ( (m +1 i)u i i 2 (m+1) 2 m ) (31) u m 5

7 4 SU(m) Case Consider gauge group SU(m) withn f 2m 2 massless hpermultiplets 2 in the defining representation of the gauge group. From (2 the hperelliptic curve for this model is where 2 = p 2 (x) Λ 2m n f x n f (32) p(x) =x m u i x m i (33) B direct calculation one can see that d dx (xn )=(n n f )xn 1 (m n f )xm+n 1 p (m n f 2 i)u x m+n 1 i i p (34) 3 From (5) we can find the second order differential equation for the periods (L n Π = 0) as follow L n =( n f 2 n) m n+1 +(m n f 2 ) 2 m n 1 (m n f 2 i)u i i m n+1. (35) where n = s 1ands=2,..., r 1. As same as before, for s = r (34) does not give the second order differential equation. Moreover, here for s = 1 same difficult arise as well. So b this method we can onl find r 2 equations. Two other equations can be obtaind b considering a particular linear combination of d( xj ). Consider the following linear combination D 0 =( n f m)d(xm+1 m 2 )+ (m n f 2 + i)u id( xm+1 i ) (36) or x m i x 2m i j D 0 = λ SW ( i(i 2)u i + iju i u j p (m n f j, 3 2 )2 Λ 2m n f xnf p)dx. 3 (37) which gives the second order differential equation for the periods in the case of 0 n f 2m 4 and take the following form L r =1+ i(i 2)u i i + iju i u j i j (m n f 2 )2 Λ 2m n f 2. (38) j, 2 For n f =2m 1, becouse of Λ dependence term ((x a 0 Λ) n f. The coefficient a 0 arise from inestanton calculations), there is a difficult, which also arise in the massive case. So we pospond that, to next work[15] 6

8 where 2 = m m nf for n f m 2and 2 = 2 m l 2 and l = 1, 0,..., m 4 for m 1 n f 2m 4. For the cases of 2m 3 n f 2m 2, one should add an extra term to D 0 as follow For n f =2m 3 D=D λ3 d( x(m 2) ), (39) so L r =1+ i(i 2)u i i + iju i u j i j + 3 m 2 Λ3 (i 3 2 )u i 3 i Λ3 3. (40) For n f =2m 2 j, D=D 0 +λ 2 d( x(m 1) ), (41) so L r =1+ i(i 2)u i i + iju i u j i j +Λ 2 m (i 1)u i 2 i. (42) j, Final the last differential equation for the periods can be obtain from following linear combination (which is the analogous to d( 1 ) in pure case [9]) E 0 =(m n f 2 )d(xm )+n f 4 (n 2m 4)u 2d( xm 2 )+(m n f 2 )2 i=3 u i d( xm i ). (43) which gives second order differential equation for periods in the cases of 1 n f 2m 3. L 0 = c 2 (e 2)u (i m)e 2 u i i+1 + c i eu i i c 2 (e i)u 2 u i 3 i i=3 + (i m)e 2 u i u j i+1 j + e 2 n f i,j=2 2 Λ2m n f 2. (44) where 2 = m m nf +1 for n f m 1and 2 = 2 m l 1,l=0,..., m 3 for m n f 2m 3andalsoe=(m n f ), c 2 i = me + in. For the case of n 2 f =2m 2 one should add an extra term as follow and then the last term in equation (44) changes to E = E 0 (m 1)Λ 2 d dx (xm 2 ) (45) (1 m)λ 2 ( 3 (1 i)u i 3 i ). (46) 7

9 5 Conclusion To compare our resultes for the groups of rank r 3 withe the resultes of [8] and [7], let us for example consider SU(4) with one massless matter. From the equations (7), (10) and (18) we have L 0 = 2u u u (2u u 4 ) u 2 u (49u u 2 u 3 ) (49u 3 u Λ7 ) 44, L 1 = u u 3 34 u 4 44, L 3 = 1+4u u u (9u u 2 u 4 ) u u 2 u (24u 3 u Λ7 ) 34. (47) where ij = i j. One can check that these equations are linear combination of those of[8] To summarize, we have obtained a closed form of the Picard-Fuchs equatins for N = 2 SYM theories with classical Lie gauge groups which have massless matter in fundamental representation. Note: After completion of this work, I recived paper[16] which has considerable overlap with our work. Acknowledgement I would like to thank S. Randjbar-Daemi for uesful commends. 8

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