ELSEVIER

11 January 1996

Physics Letters B 366 ( 1996) 174- 180

PHYSICS LETTERS B

Light monopoles, electric vortices and instantons

J.M. Guilarte Department of Physics, Facultad de Ciencias, Universidad de Salamanca, Salamanca 37008, Spain

Received 7 July 1995; revised manuscript received 18 September 1995 Editor: L. Alvarez-Gaumt

Abstract

A Quantum Field Theory for magnetic monopoles is described and its phase structure fully analysed.

1. The strong coupling limit of N = 2 supersymmetric gauge theory in four dimensions has been studied in depth by Seiberg and Witten in a brilliant series of papers [ 11. An important outcome is that the ‘strongly coupled’ vacuum turns out to be a weakly coupled theory of light magnetic monopoles. Moreover, a version of electric-magnetic duality fitting the Montonen-Olive conjecture [2] proves to be true in the model.

In this letter we offer a brief description of the physics involved in the Seiberg-Witten action. There are two kinds of quanta: light monopoles and dual-photons. Monopole-monopole as well as dual-photon-monopole interactions are due to the vertices coming from the non-quadratic terms of the Lagrangian. This is the content of perturbation theory although different topological sectors show important non-perturbative effects in the model. We find topological defects as the solutions of a minor modification, a sign, of the monopole equations [3]. The change yields non-trivial regular solutions in R4 which, on physical grounds, we find to be either electric vortices or instantons. As a consequence, the phase structure of the theory is determined.

We start from the euclidean action

S= s

d4x{tFpvFpv + @,*+)+D,*+ + $(D,T+)+~,,D,*+}

(1)

q+ is a right-handed Weyl spinor, the light monopole field. A, is a dual electromagnetic potential. D,‘P+ = 8&q+ + igA,q+ is the covariant derivative with respect to the gauge group U( 1 )d. the dual of the electromag- netic U( 1) . The magnetic charge, g = f is the coupling constant and the tensor field is Fpy = d,A, - &A,.

Finally, Z,, = 3 [yLt, yV ] where, yp are the euclidean y-matrices in chiral representation: if CJ~ are the Pauli matrices,

0370-2693/96/$12.00 Q 1996 Elsevier Science B.V. All rights reserved SSDI 0370-2693(95)01319-9

J.M. Guilnrte/Physics Letters B 366 (1996) 174-180 175

The unusual kinetic terms in the spinor fields are possible because in euclidean space there is SO(4) symmetry: there are no linear scalars in the derivatives for Weyl spinors, (see [ 41). These are due to the duality of the monopoles with respect to the scalar fields of the parent theory, the N = 2 supersymmetric Yang-Mills-Higgs theory [ 51. Despite the existence of terms of the form j,, jpy the theory is renormalizable; the dimension of q+ is 1, a non-crucial fact because we take the model as physically relevant in the strong e, weak g, limit.

The j - F interaction term in ( 1) tells us that physically jpy is due to the electric dipolar momentum

of the monopole. We deal with a spinor representation of the Lorentz group twisted by the U( 1) group of electro-magnetic duality transformations carrying a property that we call m-spin. The m-spin, or m-helicity in the massless case, yields an intrinsic electric dipolar momentum in the magnetic monopoles and labels the representations of this strange version of the Lorentz group which presumably comes from some twisting of the original N - 2 SYMH theory.

In this model, on the other hand, all the fields are in the adjoint representation of the S/(2) gauge group. Thus, the spin of the monopole is zero meanwhile the dyonic excitations have an integer spin: the monopole field, despite its spinorial nature, must be quantized according to Bose statistics.

Perturbation theory tells us that there are two kinds of quanta: (i) Light monopoles. The momentum space propagator is

A!!(k) = k,(ap, + a,, )kv Cij = i&ijkuk k4 ’ ci4 = iai e -e4i ’

(ii) Dual-photons. The euclidean propagator is

We also read from the non-quadratic terms of the Lagrangian two kinds of vertices: - a. A quartic monopole self-interaction:

VMM = &LV&

- b. A dual-photon/monopole vertex:

VMD(k) =&, + ~~q.w~~~v~ka

Physical features: besides renormalizability, at least at the same level as scalar QED, perturbation theory is conformally invariant and the electric dipolar momentum contributes both to the MD and the MM vertices; not only the magnetic charge g and the A2 coupling are important.

The amplitude for monopole-monopole scattering at the lowest order in perturbation theory and CY = 1, the Feynman gauge, is

h&~ k2; k’;, kl>

= U;,,(k;) (gk, + A E I pvp,%WJ~/z(kd~~ U&(k:)(gk, + AI”cL~Br~~8kv)U,/2(kl)

x~(4)(kt+k~-k’l -k:) +A:U~2(k;)~,,U,,2(k2)U~,2(k~)~p~U,,2(k,)S(4)(k,+k~-k~ -k;),

if k, - k& - kzLr = kill - kl, and Ulp( k) are the plane wave spinors:

176 J.M. Guilorte / Physics Letters R 366 f I9961 174- IX0

1

u,/,(k) = ( ) kl + ik2

Ikl+ k3

2. To study the non-perturbative regime, it is convenient to implcmcnt a Bogomolny splitting [6]: &fining the self-dual part of F,, as Fzv = i( Fky + E~~~,,F~~,) we write

S= s

d”x{i(F,‘y - iA2jF,,)2+ ~(D,V+)+y,y,D,W+}

+ s

d4x{$F&j,, + i(D,?+)+Cp,D,*+ + ~~~,.~,,F,,,\lr!~~,,~+}

- J d4x{+prr&d’pa}. (2) If AZ = g = -Al, a critical point analogous to that occurring between type I and type II superconductors, (2) becomes

and re-scaling x + $.x we see that at the critical point S is almost $ times the Seiberg-Witten action plus a topoIogica1 term looking similar to the Pontryagin or second Chem number. There is a different relative sign in the first squared term; we choose the sign of the Pauli momentum interaction, the term j - F term, in ( 1) by assigning the negative m-spin projection to the monopole quanta. This amounts to a different choice of orientation in R4 with respect to the convention assured in the Seiberg-Witten theory due to the specific choice of the twisting in the N = 2 SYMH parent theory.

Solutions of the first order system,

F$ = ;‘I’$,,*+, (3a)

y,.$,*\k+ = 0, (3b)

are absolute minima of the euclidean action and therefore play an important r6le in the system. In order to find them, it is convenient to split (3a, b) into components: (3a) reads

F:2 = -$P;h - #4952) = F3+4,

F2fn=-;WP2+4;4) =F,&

G = $02 -#;#I> =F$,

where qR = ( > z: .

Also, we have for (3b)

(03 + iD4)qh + i(DI - iD2)& = 0,

(01 + iD2)& - i( D3 - iD4)& = 0.

(da)

(4b)

(&I

(5a)

(5b)

Multiplying (5a) by (01 -t i&), (5b) by (03 + iD4), subkxting, using (&, c> and integrating over all R4 we obtain

J d4x{bh 1214212 + IDz,$212 + /44212} = o; (6)

J.M. Guilarte/Physics Letters B 366 (19961 174-180

4, = D1 - iDz. D, - D3 - iD4, after a partial integration. We are then left with two possibilities:

- A. q5z J 0. Solutions with m-spin l/2 in the xs-direction ia3 (t)=;(t).

- B. 41 = 0 and D,, h + DZ,h = 0. Solutions with m-spin projection - l/2:

177

Formula (6) tells us that type A and B exclude each other: even for solitons, only one of the two polarizations is possible.

3. Searching for explicit solutions we make the following ansatz adapted to type A:

41 (x) = 91 (x1 9 x2) 3 &=A3=A4=0, Al(x) = AI(XI,XZ) 3 AZ(X) =Az(xI,x~)

The only non-zero (4)) (5) equations reduce to

Fl2(Xl*X2) = -191 (w2)12 3 (01 + i&)h (~1~x2) = 0 (7)

It is well known that system (7) is tantamount to the Liouville equation in R2, [7], and the general solution such that lim #(xi, x2) = 0, where rf = xf + x$, guaranteeing finite energy density is, see [ 81,

r, -cm

2f’(zdV2(zd tmz1) = p/(z,)12 + lf(Z,)v(z,)12 ’ Z’ =x’ + ix2’

f(z) =fo++- i_l z - z(i) ’

V(z) = f&z - zq. (i-l)

(8)

The solutions f$f”’ have infinite action,

-4 -f

S[ @Ikl ] = z lim I g2 L,T+ccJ J J dxg dxq -I- 2 f although the electric flux is finite: @E[ +\kl] = Jd2x Fi2 q yk.

Observe that k is positive and the flux is located around z,(~), the zeroes of $I( ZI ). It spreads out, however, with Ici{, the length scale of the solution, which is a free parameter due to the scale invariance of the theory. There is also freedom in choosing argq because the U( 1 )d symmetry and the moduli space of solutions is C2k: the parameters are the centers of the solitons z:~) and the modulus and phase of ci, determining the scale and phase of each individual soliton.

Solutions of type B are given by the complementary ansatz:

41 = Ai = A2 =o, #2(X) =$2(X3,x4), A3(x) =A3(~3,~4) 9 A4(x) -A4(xn,x4).

We now meet the system

&(X3,X4) = k#2(x3,x4)j2, (03 -iD4)42(x3,X4) =O (9)

again leading to the Liouville equations with the right sign to obtain non-singular finite energy density solutions:

$@(22) = 2f'(i2)V2(f2)

jv(z2)12+ lf(Z2)vqZ2)12 ' f2 =x3 -in4. (10)

178 J.M. Guilarte/Physics Letters B 366 (1996) 174-180

The euclidean action is

there is no need to change the sign of euclidean time, but the ‘euclidean’ magnetic flux

@&#Jkl] = 2 JJ dx?dxdFsd=-$k

is negative, fitting the sign of the m-spin projection. The moduli space of these solutions is also C2k. We can understand the topological origin of these solutions in the following way: consider R4 as R2@R2. Ihe

finite energy density conditions, alternatively in each R2 according to the type of solutions, plus the conformal invariance of the equations make the problem tantamount to solving the system on S2 x S2 [lo]. We have two

topological numbers: a12 = k, electric flux, and n 34 = -k, euclidean magnetic flux, labelling the topological sectors of the theory. In fact we have more: setting the x4-coordinate to be euclidean time there are two other possibilities for splitting R4 as R2 @ R2. Choosing a basis in which gI is diagonal and the Kahler form is w = dn2 A dx3 + dx4 A dxl, similar ansatzes yield

F23(x2,x3) --IdI(~2t~3)12~ (D2+iD3)$I(x2.x3) =o : typeA,

F14(x2,x4)=l~2(xl,x4)(2, (~l-iD4)d2(xITx4)=o: typeB,

or 1s2 diagonal, 04 = dx3 A dxl + dx2 A dxq,

h(xl,x3) =-lqh(x1,x3)]~, (01 +iR)qh(x1,x3) =O : type A,

F24(x2,x4) = 1+2(X2~x4))~ 1 (02 -iD4)+2(x2,x4) =o : type R ,

There are electrically charged solitons in 3 different planes, labelled by

&?i = Eijkfijk 7 fijk = -nkj 9 tQfZf,

and also magnetically charged solitons, labelled by,

Vii = ni4 , mi E Z-

although the a ‘priori’ conserved topological numbers mi will not survive the Wick rotation to real time. We can extend the theory by modifying the spinor tensor/spinor tensor interaction in the following way:

no summation in a, with

ni=(J, n2=(!).

and a = 1,2. With a re-scale of variables q+ --+ Ok*+, A, + u,Afi and xP -+ &xr, the ansatz of type A plus u2 = 0, produces the system

F12 = I - /9,12, (01 +iD2)& =O. (11)

J.M. Guilarte/Fhysics Letrers B 366 (19%) 174-180 179

Solutions of ( 11) such that lim,,,, 41 (xl, ~2) - 1 exist and have been thoroughly analysed in [ 91. They are electric vortices, flux tubes, with the flux concentrated around the zeroes of 41 (x): there is neither freedom of scale (because the theory is not conformally invariant) nor phase freedom (because U( 1 )d symmetry is broken spontaneously). The IUOdUli space iS Ck and the same ei = &ijknjk tOpOlOgiCal quantum numbers are conserved, ei E Z+.

Alternatively, the ansatz of type B and ui = 0 leads to the system:

F34= 1#2(x3,x4>12 - 1, (03 -iD4)+2(x3,x4) =o, (12)

and there are ‘euclidean’ magnetic anti-vortices of negative integer flux concentrated around the zeroes of 42. The moduli space is Ck and the topological quantum numbers mi - n4i E Z- will not be conserved because of

the euclidean time character of x4.

4. In order to study the phase structure of the model, it is convenient to have a richer set of solutions. Adding a left handed Weyl spinor 3_, the anti- monopole field with m-spin projection - l/2, a system of equations like (3a), (3b) with Ff replaced by F- arises. There are solutions of the kind (8) and (10) with electric and magnetic fluxes of opposite sign.

Similar solutions to those described above have been studied by the author in Ref. [ 111. In that case, the Hopf invariant forced flux tube pairs. Also ‘t Hooft electric and magnetic flux lines found in pure Yang-Mills in a box [ 121 share many physical features with the Seiberg-Witten solutions. We study the quantum behaviour of the last type of solitons in ‘t Hooft’s framework.

In the A4 = 0 gauge, consider the operator

B(C,t) -exp[ig( s

d3x[&x,t) s

dsAA(yW) +fi;W> s

~.#(YW)IH C c

. l-h (x, t) = -id, (x, t) + &f42(X, t) + h-41 (x, t)),

which creates an electric flux smeared by 41 along the curve C: a type A solution for which the zero of the 41 field with multiplicity es is repeated along C. Choosing C as the xs-axis, perhaps with periodic boundary conditions x3( - L3) = x3( Ls) , 8( C, t) creates an electric solitonic string of electric flux rr12 - es which is conserved for topological reasons. On coherent states of the Hilbert state; the action of fi (C, r) is

'Ihe proportionality factor is due to the fact that L? (C, t) measures magnetic flux. Quantum states related to type B solutions are more difficult to analyse because they correspond to a tunnel

effect between different vacua when the euclidean time component x4 is Wick-rotated to real time. In fact there are non-homotopically trivial gauge transformations of the form

&,(n) = ei21rga”5x3fi(X1 ,x2) , gas = k3

These act on the quantum states by means of an unitary operator:

fik3(x)lA,&h,&2),, = eiw(k3)JA,E;41 ,#de3.

The group law &k&k; = iik3+k;, w( k3) + w(ki) = w( k3 + kj) and the requirement of the same action for all the gauge homotopy classes o( k3) = o( k3 + n) yields: w( k3) = 21rt%k3. To understand the physical origin of this angle, consider the ‘smeared’ Wilson operator:

180 J.M. Guilarte/ Physics Letters B 366 (1996) 174-180

A(C,r) -exp[ig( s

d3x[a(x,t) s

d@(YW) + fGcw, I

~4J~(ywml c C

&(x,t) =-i&(x,‘) + fL@,(w) +DJ$~(xJ)),

A( C, t) creates a solution of type B, in the Ab-gauge, along C. For T long enough and C - x3-axis,

~(CJ)(AE;~W~)~~ 0; IA+~(~,T),E;~I~(x,T),~~)(,,

lim n(x,t) = Ra(x), I-+--~

limn(x,t) =&(x), k3=m3; 1-t

A( C, t) creates magnetic flux but the tunnel effect, and the associated instanton angle, means that only strips with flux parametrized by 03 are conserved.

Due to the other possible choices of splitting R4 in R2 x R2 there are quantum states characterized by (e, e), 3 integers and 3 angles, corresponding to the solutions of the Seiberg-Witten solutions. This analysis is a semi-classical one but sufficient because it is the regime within which we trust the theory. The free energy of each

(i) (ii)

(iii)

(iv)

state is thus known and the phases are as follows: ~3 = 0. Both electric and magnetic flux spread out over all the space and we are in the Coulomb phase. uf = 0. In this phase there is a constant background 8 magnetic field in the &vacua ground states. In the SPA,

(t931e-HTJB3) cc exp{e -2rLI L2/g2 KL3T cos B3}

where K is a determinantal factor accounting for the gauge and spinor fluctuations in the presence of an instanton up to the one-loop order. u; = 0. There is electric order: electric vortices of integral magnetic charge exist and we are in a phase of electric charge confinement. Note that due to the peculiar m-spin properties of the solitons only massless particles are confined. Finally, if U: 5 0, there are no solutions of the first order equations. This is the normal phase.

The author has greatly benefited from conversations with S. Donaldson and 0. Garcia- Prada on the math- ematics of the Se&erg-Witten invariants. Financial support by the DGICYT under contract PB92-0308 is acknowledged.

References

[I ] N. Seiberg and E. Witten, Nucl. Phys. B 246 ( 1994) 19, and hep-thI9408099, to appear in Nucl. Phys. B.

[2] C. Montonen and D. Olive, Phys. Len. B 72 ( 1977) 117.

[3] E. Witten, Monopoles and Four Manifolds, Math. Res. Lett. 1 ( 1994) 769.

[4] I? Ramond, Field Theory: a Modem Primer, Benjamin ( 1991).

[5] D. Olive and E. Witten, Phys. Len. B 78 ( 1978) 97.

[6] E. Bogomolny, Sov. J. Nucl. Phys. 24 ( 1976) 449.

[ 71 E. Witten, Phys. Rev. Lett. 38 ( 1977) I2 I. [S] R. Jackiw and S.Y. Pi, Phys. Rev. D 42 ( 1990) 3500.

[ 91 A. Jaffe and C. Taubes, Vortices and Monopoles, Bikhluser, 1980. [lo] The moduli space of solutions of Eqs. (7) and (9) on closed Kahler manifolds has been thoroughly studied in: S. Bradlow, Corn.

Math. Phys. I35 ( 1990) I, and 0. Garcia Prada, Corn. Math.Phys. 156 ( 1993) 527.

[ 111 J.M. Cuilarte, Len. Math. Phys. I2 ( 1986) 21.

[ 121 G. ‘t Hooft, Nucl. Phys. B I53 ( 1979) 141.