Prepotentials in N = 2 SU(2) supersymmetric Yang-Mills theory with massless hypermultiplets

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PHYSICS

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Physics Letters B 366 (1996) 165-173

Prepotentials in N = 2 W(2) supersymmetric Yang-Mills theory with massless hypermultiplets Katsushi Ito, Sung-Kil Yang Institute of Physics, University of Tsukuba, lbaraki 305, Japan

Received 3 1 July 1995

Editor: M. Dine

Abstract We calculate the prepotential of the low-energy effective action for N = 2 SU( 2) supersymmetric Yang-Mills theory with Nl massless hypermultiplets (Nf = 1, 2, 3). The precise evaluation of the instanton corrections is performed by making use

of the Pica&Fuchs equations associated with elliptic curves. The flavor dependence of the instanton effect is determined explicitly both in the weak- and strong-coupling regimes.

Low-energy effective action of N = 2 supersymmetric Yang-Mills theory is described in terms of a single holomorphic function [ 11. This type of holomorphic function, called the prepotential, plays a vital role in four-dimensional N = 2 theories [ 21. Seiberg and Witten discovered that the prepotential for the gauge group SU( 2) is completely determined by holomorphic data associated with elliptic curves [ 3,4]. On the basis of this seminal work, many non-perturbative aspects of the vacuum structure of N = 2 supersymmetric gauge theories have been revealed subsequently by extending the gauge group [ 51 and by introducing matter hypermultiplets

161. In this paper we study the quantum moduli space of N = 2 supersymmetric gauge theories with massless N = 2 hypermultiplets of quarks. The gauge group is SU( 2) and Nf hypermultiplets are all in the spin one-half representation of SU( 2). The theory is well-known to be asymptotically free for Nf 5 3. Our purpose is to calculate the prepotentials of the low-energy effective actions for all the theories with Nf = 1, 2 and 3. Let 4 be a complex scalar field in the N = 2 vector multiplet. The theory has a flat direction with nonvanishing 4, along which the gauge group SU( 2) is broken to U( 1) and all the quarks turn out to be massive. ‘Ibis is the Coulomb branch of the moduli space. For Nf 2 2 the other branch may develop when scalar fields in the hypermultiplets acquire the vacuum expectation value. This branch is called the Higgs branch where the gauge symmetry is completely broken. In what follows we will concentrate on the Coulomb branch. The quantum moduli space of the Coulomb branch is described by using the gauge invariant order parameter u = (Tr #2). The low-energy effective theory at generic points in the complex u-plane contains an N = 1 I/( 1) vector multiplet W, and an N = 1 chiral multiplet A whose scalar component is denoted as a. The effective action is then governed by a single holomorphic function 7(A). We have 0370-2693/96/$12,00 0 19% Elsevier Science B.V. All rights reserved SSDIO370-2693(95)01310-5

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Letters B 366 (1996) 165-I 73

(1) in N = 1 superspace. Let us define aD =-.

as(a)

(2)

da

Important insight in [ 31 is to recognize the pair ( aD, a) as a holomorphic section of an SL(2, Z) bundle over the punctured u-plane. The SL(2, Z) acts on (aD, a) as the quantum monodromy matrix around the singularities in the moduli space. At singularities there appear extra massless states in addition to the N = 2 U( 1) vector multiplet. It was found in [ 3,4] that the exact description of the moduli space is determined through the elliptic curves. The curves are given by y*

=2(x -

u) +

$o”x

u) -

$&;,4-N”(X -

for N,f = 0 and y*

=x2(x -

U)+l

for Nf = 1, 2, 3. Here we have put all the bare quark masses to zero and A v, is the mass scale generated by the dimensional transmutation. In a recent paper [7], Klemm et al. calculated the prepotential for Nf = 0 explicitly. Their main tool is the Picard-Fuchs equations associated with the Nf = 0 elliptic curve which have also been considered in view of special geometry [ 81. For the present purpose we shall also employ the technique of the Picard-Fuchs equation whose fundamental solutions are combined so as to yield the exact expressions for (up, a) in the N,f 2 1 theories. In writing (3) and (4) we have followed the conventions in [ 41 so that particles in the hypermultiplets have integral electric charge. Hence the effective coupling constant

is expressed as

&T(U) + i 87~

7(u) = -

77

&w’

(6)

The global symmetry acting on the u-plane is 22 for Nf = 0 and Z4-N, for Nf > 1. Notice that for Nf = 3 there is no symmetry in the u-plane. This fact distinguishes Nf = 3 theory from the others. Eq. (4) describes a double cover of the x-plane branched over ee, e+ and 03. We take a cut to run from e_ to e+ and from eo to 00. Let us start with the Nf = 1 curve. We have for large u eo

2

u,

e* N fi-

A:

(7)

Sfi’

Thus the singularity at u = 00 is not stable since eo + 00 and eh -+ 0 as u goes to infinity [ 41. Bearing this in mind we make resealing x -+ ux. We then obtain eo= $(I +[+l-‘>,

e* = i(l r05k:2/5),

(8)

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167

where o = e iTI3 and 5-2”3(2+27~+3~~)-“3,

(9)

with ~7 = At/(64u3). The singularities in the u-plane are located at u = cc and u = -3A;J/2s/s j-0, 1,2. In the N,f = 2 and 3 curves the singularities are stable. For Nf = 2 we have e& = &A;/8

e0 = 4

with

(10)

and the singularities in the u-plane are u = 00 and u = eh. For Nf = 3 we get

(11) and hence the singularities occur at u = co, u = 0 and u = $/256. The pair (a~, a) is now given by period integrals of the meromorphic one-form A,

(12) Here the cycle cr is taken to go around the cut connecting e_ and e+ counter-clockwise. ‘Ihe other cycle /? loops around the branch points at ea and e_ for Nf = 1 and at ea and e+ for Nf = 2, 3. The intersection of a! andpiscuep-1. The one-form A is determined by the differential equation [4]

Jz

dA

du - -A, 877 -

+ d(*),

(13)

where A1 is the holomorphic differential Ai = dx/y in u we find

and d( *) stands for the exact form in X. Integrating ( 13)

A_JZ2u-(4-Nl)xdx 8~ Y

(14)

for Nf > 0. It is interesting to observe the one-loop beta function coefficient 4 - Nf in A. We wish to derive the Picard-Fuchs equation for the period II = f A. For this it is convenient fist to write down the Picard-Fuchs equation for the period II 1 = $ Al. After some algebra we obtain d*I-I,

P(U) -j$-

+

4(u)~+II]

= 0,

(15)

where the coefficient functions are given by 27Af 64~ ’

= 4u2 + -

N.f = 1,

p(u)

N,f = 2,

p(u) = 4(uz - $3

9(u) = 8u,

Nf -3,

p(u) =u(4u-g),

9(u) =8u-

27Ab - 2)’

9(u) = $8~~

2.

(16)

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Letters B 366 (1996) 165-173

Combining ( 13) and ( 15) it is immediate to see that

d3rI

q(u)

d2rI +

dU3+-- P(U) dU2

1 dII --= P(U) du

o. (17)

Notice that the relation 9(u) = dp(u) /du holds for all Nf in ( 16). Consequently ( 17) reduce to the second order differential equations

p(u)~+rI=o

(18)

for Nf 2 1. We have checked by verifying this equation directly from (14) without relying on (13). Now that we have the Picard-Fuchs equations ( 18), the periods ( 12) will be expressed by particular linear combinations of the two fundamental solutions to ( 18). In order to fix the combinations we have to evaluate the contour integrals ( 12) explicitly. Let us first discuss the behavior at u = co. For Nf 2 1 the asymptotic freedom and the instanton contributions leads one to expect that [ 41

u(u)

=

;JzI;[l

+ ~~~(N,,(+“14-N”]. n=l

( 19)

Here the instantons with even instanton number 2n contribute the terms (A~,/u)~(~-~J). The amplitudes for the odd instanton contributions vanish because of the anomalous Z2 symmetry for Nf > 1 [ 41. From ( 19) one can easily read off the monodromy matrix at u = cc: Mm=

-1

(

o

4-Nf _1

>

.

We compute the lower order expansion of the integrals (12) explicitly. The results for Q(U) read

(21) where c, = -4 In&” +4ln2

- 2, c2 = 6ln2 - 2 and Q = -lIneiT + 121n2 - 2. For U(U) we get

(22) where al(l) = 3/2” and a,(2) = a,(3) = -l/2 lo . Thus our explicit results are in agreement with ( 19). The exact expressions for (aD, a) near u = CO are now obtained from the solutions to the Picard-Fuchs equations. The solutions to the Picard-Fuchs equations are written succinctly by introducing a new variable

,

c( NI) = 7]N,28(4_ ,‘,qN,-4,

(23)

which keeps the discrete Z&N, symmetry in the u-plane. Here 71 = -1, v2 = ~3 = 1. The Picard-Fuchs equations ( 18) for Nf > 1 become the hypergeometric system with singularities at z = 0,1,00 z(l--Z)~+(y-(ai+a+1,;,~-,pll=o,

.,

(24)

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169

where Ly= p-

2(4-

--I

3 - Nf ‘= 4.

Nf)’

(25)

The set of fundamental solutions to (24) near u = 00 are written as ~1, ~2: -1 1 ; I;:). 2(4 - Nj) ’ 2(4 - h’f)

w,(z) = z+V(

1

1

w2(z) -wt(z)ln-

+z*Ft(

Z

2(4-

-1 N~)‘2(4-

Nf)

A’), z

(26)

where F(a,p;y;

z) =

z , cM (Q)n(P)n n!fr)n n

n4

M (a)n(P)n “_-l 1 1 -cn-O ~-(n!j2 c(-a+r+ /?+r

F,(a,P;l;z)

r-O

hNJ

wl(z)v

a(u) = &C(Nf)Ti+

iANJ

= -

(27)

(A + n - 1) . Using this set of solutions, we are able to express a(u) and a~( u) in

and (A)n = A(A+ I)... the form

uo(u)

2 1 fr > Zn7

2&c(

j

NI)

(w2(2)

(28)

+ AN~wI(z)),

2c4-nJ’

where ANY= lnc( Nf) - (4 - N~)c,v,. In order to calculate the prepotential 3 we first express u as a series expansion in a/AN, by inverting a(u). Substituting the result into ah(u) , and then integrating (2) with respect to a we obtain the instanton expansion of F‘,

(29) The first five coefficients 3k( N,f) ~__ 1 31(N.f) = -&(,y)

are given

by

1

2b2’

1 F2(Nf)

= -

.mNf)

=

34( Nf) =

22b~(Nf ) 2 5-8b+4b2 28 b4

1 (1 -b)(l 23hc( Nf)3 1 24hC( ~,~)4

’

-2b)(-23+30b-

16b2)

1728 b6

( 1 - 2b) ( -677 + 391 Ob- 8 124b* + 84566” - 4672b4 + 1152b5) 2949 12 b8

K. Ito. S.-K. Yang/Physics

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Letfers B 366 (I 996) 165- I73

( 1 - 2b) ( 1 - b) (-7313 + 64386b - 210756b* + 3359606” - 301792b4 + 152704b5 - 36864b6) 18432000 b’O (30)

where b = 4 - Nf. One can obtain the Nf = 0 result by replacing c( N.f) with 1 in the Nf = 2 case, which recovers that in [ 7,9]. Note that the coefficients .&,+I (3) vanish for n 2 1. Our next task is to study the strong-coupling regime where the singularities in the moduli space appear as a consequence of the extra massless monopole (or dyon) states. Let us investigate the behavior near the singularity at u = ua where the monopole becomes massless. The position of the singularities is given by ug = -3Ai/2’i3 for Nf = 1, ua = Ai/8 for N,f = 2 and ua = 0 for N,f = 3. In the Nf > 1 theory the monopole belongs to the spinor representation of the global SO( 2N.f) symmetry [ 41. Hence, in the vicinity of u = ~0, the effective dual U( 1) theory consists of the N = 2 U(1) vector multiplet of magnetic photon and the k hypermultiplets of light monopoles. The multiplicity k is given by the dimension of the spinor representation of SO(2Nf), i.e. k = 2Nf- ‘. Then the perturbation calculations in the dual theory predict [4] k a 2 i-aD

InaD,

2?r

aD = cO(u

-

UO),

co$0,

(31)

from which we find the monodromy matrix at u = ~0,

M,=(!k ;)

(32)

Let us turn to the explicit evaluation of the periods ( 12) as u approaches ~0. Our results for aD (u) read y.,,(Nj)(y)?t_--1,

where vi = -3A:/2’i3,

v2 = A;/4,

v3 =

(33) J

J

A;/64, CD( 1) = -i/4, cD(2) = i/4, cD(3) = l/4, and bD( 1) = -l/24,

bD(2) = -l/8, bD(3) = 1/2. For a(u) we obtain VG a(u) = -

d(Nf)

2T

+ b(N,f) y[lny I

+d,vJ] +...

(34)

,

J

where d(1) = -3, d(2) = 2, d(3) = i, b(1) = l/4, b(2) d2 = -41n2 - 1, d3 = -21n2 - 1. Notice that

= -l/2,

b(3)

= i and di = -41n2

- 2ln3 - 1,

(35) Thus our results agree with the perturbation result (31) with k = 2N’-’ which is the right number of massless monopole states. Having checked the leading perturbative behavior we next discuss the exact determination of (aD, a) with the use of the Picard-Fuchs equations. In terms of the variable z defined in (23) the massless monopole points are at z = 1 for N,f = 1, 2 and z = 0 for Nf = 3. Define P(Z) = { :;-

zt

for N.f = 1, 2 , for Nf = 3 .

K. Ito, S.-K. Yang/Physics

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Letters B 366 (1996) 165-173

fundamental solutions to the Picard-Fuchs equation (24) near p( z ) = 0 are then obtained as WI(Z)

-p(z)F(l-

w2(z)

= WI(Z)

$1-&2;P(z)):

lnp(z)

+P(z)~

(1 -

(37)

1 - h;2;p(z)),

&+

where b = 4 - Nf and

O”(~)n(P)n n-l 1 1 ‘+c Un_, n!(2), r-O cu+r+p+r---

1

e(a*p;2;z)=

(a-l)(P-1);

1 l+r

1 -_) 2fr

Zn*

(38)

In comparison with (33) and (34) we find Q(U) -C(N.f)w~(z), a(u)

=WNf)(w2(z)

(39)

+dN,w(zh

where C(Nf)

=-(-I)

D(N.r)

=

?(4-

N,f)-

~2_;-;(N,_])_& JAN/V

&2N+C(N,‘).

(4.0)

In the dual U( 1) theory the prepotential FD is expressed as a holomorphic function of ao and takes the form ia 8rr

3D(4==L

(

n

r4

kln (

+ C NJ)

FD”(Nf)

E

(

It>--1

I ,1

.

(41)

where a = d3D/daD. The expansion coefficients 30 ,, are obtained explicitly in a similar way to the weakcoupling case. After some computation we find 2

~DI(N~)=3D

2CN.f)

3QN.f) =

(1 -2b)(-5+2b) 16b2

1

(1 -2b)(-61

+190b-92b2+8b’)

1152b4

22;( Nf)2

2 3~ 3(N.f) = ~ 5E( N,#

’

( 1 - 26) ( 1379 - 81626 + 14596b2 - 7288b3 + 960b4)

110592b6

(42)

where ?( N,f) = C( N,f)/A,v,. Let us finally discuss other singularities located at u = U* in the u-plane. The 2 x 2 quantum monodromy matrices M,, have already been obtained explicitly in [ 41. The left eigenvalue (p, 9) of IV,* gives the magnetic charge p and the electric charge 9 of the BPS dyonic state which becomes massless at u = u*. Except for the cace U* = ue (massless monopole point) we always have 9 # 0. Then, from the matrix MU*one can infer the behavior of (a~, a) near u = u*, obtaining an(u) N -J-E

2rri 9

a(u)-

-'"p

CO(U -

2mi 92

4)

ln(u - u,) f CI(U -u*),

co(u-u*)In(u-u*)

+$co

-pA)(u--u*),

(43)

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K. Ito, S.-K. Yang/ Physics Letters B 366 (1996) 16% I73

where CO,cl are non-vanishing constants and m is the ( 1,2) component of M,,. . Notice that a good coordinate nearu=u+ isobtainedaspa~+qa~co(u-u,). We have checked the asymptotic formula (43) for several cases of interest. For instance, take a singularity at u = A:/256 z U* in Nf = 3 theory where the monodromy matrix is given by 141 ( 44) and (p, q) = (2,l). In order to find the desired expression (43) one has to follow carefully how the positions of the branch points (and hence the branch cuts) in the x-plane evolve as u approaches U* when evaluating the contour integral ( 12). Letting u ----fU* + 0 along the real axis in the x-plane it is observed that the contour LYis pushed to the left of u = u* while ek approach At/128 parallel to the imaginary axis. In the limit eh --+ A$/128, therefore, the evaluation of the period a receives the contribution of the a-integral (with the opposite sign) twice in addition to the contour integral along the imaginary axis. We thus arrive at 1 a~ 2 -coelnE+cIE+c2, 2ri

1 a 2 -G2c0elne

+ (CO- 2~1)~ - 2~2,

(45)

where E = 1 - 256u/Az and ci are calculable constants. This result is in accordance with (43). Other cases are considered in a similar manner. In summary, we have calculated the low-energy prepotemial for N = 2 X/(2) supersymmetric Yang-Mills theory with massless hypermultiplets and determined the flavor dependence of the instanton corrections exactly. It will be interesting to study a similar issue when the bare quark masses are turned on. In this case the elliptic curve takes the form

(46)

y2=x3+u(u)x2+b(u)x+c(u),

where a( u), b(u) , c(u) are functions of u and the bare quark masses [ 41. This leads to the set of Picard-Fuchs equations for the period integrals of the first and second abelian differentials II, = $ dx/y and r12 = f xdx/y: P12(U) --n2, A(u) dn2

du

--

~21(u>

u,

A(u)

+ P22lU)

(47)

-Hz, A(u)

p,,(u)

=-P~~(u)

=ub2u’-4u2cu’+3bcu’-2b2b’+6ucb’+ubc’-9cc’,

P,~(u)

=2b2u’-6ucu’-ubb’+9cb’f2u2c’-6bc’,

p21(u)

=-2b”a’+8ubcu’-

18c2u’+ab2b’-4u2cbf+3bcb’-2b2c’+6acc’,

A(u)=2(-u2b2+4b3+4u3c-18abc+27c2),

(48)

and the prime ’ denotes the derivative with respect to U. A(u) is the discriminant of the elliptic curve (46). From (47) one can write down the third order differential equation for $ A, which is analogous to ( 17). According to our preliminary analysis, however, the equation will not reduce to the second order equation. Thus, one has to deal with the monodromy property of the third order differential equation.

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The work of K.I. is supported in part by University of Tsukuba Research Projects and the Grant-in-Aid for Scientific Research from the Ministry of Education (No. 07210210). The work of S.-K.Y. is supported in part by Grant-in-Aid for Scientific Research on Priority Area 231 “Infinite Analysis”, the Ministry of Education, Science and Culture, Japan.

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