Talk presented by the second author at the Inaugural Coference of the Asia
Pacific Center for Theoretical Physics, Seoul, June 1996. The purpose of this
note is to give a resume of the Seiberg-Witten theory in the simplest possible
mathematical terms.Comment: 10 pages, LaTe

Flume, R.

O'Raifeartaigh, L.

Sachs, I.

English

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DIAS-STP-96-22Brief Resume of SeibergWitten TheoryR. Flume**, L.O Raifeartaigh* and I.Sachs***Dublin Institute for Advanced Studiese-mail physics©stp.dias.ieIntroductionAn outstanding problem in non-abelian gauge theory has been to make reliable predictions about the (non-perturbative) strong coupling region. (Another interestingproblem has ben to find solvable models in more than two dimensions). In a recentpaper [1] Seiberg and Witten (SW) have shown that the simplest non-trivial N = 2supersymmetric theory provides at least a partial answer to these problems. First,they have shown that the local part of the effective Action is governed by a singleanalytic function F of a complex variable. Second they have made an Ansatz forthe F that satisfies all the physical criteria and embodies electromagnetic duality,thus directly connecting the weak to the strong coupling regions. The correctness oftheir Ansatz is supported by some direct instanton computations [2]. The purpose ofthis note is to give a resume of the SW theory in the simplest possible mathematicalterms.1. N = 2 SupersymmetryWe begin by recalling the essentials [3] of the N = 2 supersymmetry algebra and itsAction. The algebra is{Q, Q} = ójkJP {Q, Q} = (1.1)plus the hermitian conjugate of the second relation, where i, k = 1, 2 and Z is a centralcharge. This algebra is realized on the simplest possible non-trivial supermultiplet,namelyW D {c,5,A,1;F,D} (1.2)where is a complex scalar field, / is a Dirac spinor A,1 is a gauge-field and F andD are complex and realdummy-fields respectively. This N = 2 superfield actuallyconsists of two N = 1 superfields, namelyI D {,q,F} and V D {A,1,f,D} or Wa D {F,1v,f,D} (1.3)where k and V/Wa are chiral and vector multiplets respectively, the q* Talk presented at the Inaugural Coference of the Asia Pacific Center for TheoreticalPhysics, Seoul, June 1996 and at XXIst Conference on Group Theoretical Methodsin Physics, Goslar, Germany, July 1996Volkswagen Foundation Fellow, on leave of absence from University of BonnPartially supported by Swiss Nationalfond for Scientific Researchand f fields being Weyl spinors of opposite chirality. Since A belongs to the adjointrepresentation of the gauge group G and all the fields belong to the same multipletthey must all belong to the adjoint representation of G. The simplest SW model isfor G = SU(2) and we shall concentrate on this case.2. N = 2 Super-Action.The superaction for the N = 2 superfield just described isA = Im Trfd4xd26d2() (2.1)On expanding this in terms of the N = 1 superfields it becomesA=Imfd4xd26ad2(Ae_2Y0VA) +Tofd4Xd28a(WW) (2.2)where=—- + (2.3)2T g0the parameter g0 being the usual gauge-coupling constant and 80 being the QCDvacuum-angle (not to be confused with the usual supersymmetric Grassman variables). The exponential in the first term is just the supersymmetric generalization ofthe covariant derivative. Expanding (2.2) further in terms of conventional fields weobtainA = trf d4x{tD2) + +g[,]) + g2{tj2}1 9(3.4)+trfd4x{F’F, + F’F}This Action will be immediately recognized as the standard action for a Quark-GluonHiggs system in which all the fields are in the adjoint representation and the couplingconstants are reduced to g and 9 by the supersymmetry. Thus it is not very exotic.Indeed it could be the QCD Action except for the fact that the quarks are in theadjoint and presence of the scalar field.4. Text-Book PropertiesThe Action (3.4) is actually so normal that it embodies all the properties of Quantumgauge Theory that have surfaced over the past thirty years and could even be usedas a model to teach quantum gauge theory. It might be worthwhile to list theseproperties:1. It contains a gauge-field coupled to matter2. It is asymptotically free3. It is scale-invariant, but with a scale-anomaly4. It has spontaneous symmetry breaking5. It has central charges (Z and Z)6. It admits both instantons and monoplesBecause of the supersymmetry it has some further special properties, whose significance will become clear later, namely,7. It not only has a Montonon-Olive mass formula [4] for gauge-fields and monopolesbut generalizes that formulafrom M=jv(Ne+nm) to MZ where Z=(ane±adnm) (4.1)where ne and m denote the gauge-field and monople charges respectively, andthe coefficients a and ad will be explained later.8. It is symmetric with respect to a Z4 symmetry which is the relic of the R-symmetry(&cr — e8) that survives the axial anomaly breakdown.9. It has a holomorphic structure10. It has a duality that connects the weak and strong coupling regimes12. The duality generalizes to an SL(2, Z) symmetry. In section 6 we explain theselast three concepts in a little more detail.5. Spontaneous Symmetry-BreakingFor SU(2) this concept is very simple. From the form of the Higgs potential in (3.4)we see at once that there is a Higgs vacuum for = va where v is any complexnumber and o is any fixed generator of SU(2). Furthermore, for v 0 this breaks thegauge-symmetry from SU(2) to U(1). For other gauge-groups G the correspondingstatement is that v must lie in the Cartan subalgebra of G. On the other hand thereis no spontaneous breakdown of supersymmetry Thus the full breakdown isSU(2) —+ U(1) N = 2 supersyinmetry unbroken (5.1)Indeed it is the fact that the supersymmetry is unbroken that gives the model itsnice properties, since otherwise the classical properties would not be preserved afterquantization.After the spontaneous breakdown the restriction of the N = 1 form of the classicalAction (2.2) to the massless U(1) fields takes the formA = Imf d4Xd29ad2(AA) + ToImfd4xd28a (wawa) (5.2)Since the adjoint representation of U(1) is trivial this Action is a free-field one. However, in the quantum theory this does not mean that the effective Lagrangian is alsofree because, through the quantum fluctuations, the massive fields induce interactionterm for the massless ones. The first great virtue of the SW model is that theseinteractions have a very specific form. In fact they show that, due to the N — 2supersymmetry the local part of the effective Lagrangian can only be of the formA fd4xd2ed2(A4d —AdA) +Imfd4xd2&(r(A))(WW) (53)whereAd F’(A) and -(A) = F”(A) (54)for some function F(A). Thus the effective Lagrangian is completely governed by thesingle function F(A). Note that (5.3) is very similar to the classical Action which isthe special case for which F(A) =r0A2. As we shall see, the SW solution is actuallya special Ansatz for the functional form of F(A).6. Holomorphy and DualityIt is now easy to quantify what is meant by holomorphy and duality. Holomorphyis simply the statement that F(A) depends only on A and not on A. Duality meansthat the physics described by the effective Action (5.3) is invariant with respect tothe duality transformation.() (i ) (A)) DW DW r(A) (T(A)1 (6.1)Note that the duality transformation is closely linked to the Legendre transform ofF(A) with respect A. By noting that in the free classical theory with = 0 thetransformation (6.1) reduces toEB and (6.2)we see that it is just the generalization of well-known Maxwell-Dirac (MD) duality.Thus the Action (5.3) not only generalizes MD duality but puts it into a genuinedynamical model. Furthermore, the duality generalizes to(A (p q (A and T(A) pr(A)+q (6.3)AdJ r sj AdJ rr(A)+swhere the matrix with entries (p. q, r, s) is in SL(2, Z). The integer-valuedness of thetransformation follows from the requirement that, in the perturbation theory at least,it should change the & angle only by multiples of 2w and should leave the mass-formula(4.1) form-invariant.7. Perturbative and Non-Perturbative F(A)Before going on to describe the S-W Ansatz for F(A) we consider the perturbativecontribution F(A) of F(A). This turns out to beF(A) = h42ln(A2/A2) (7.1)where A is the renormalization scale. This is evidently a classical plus a one-loopexpression but it is correct to all orders in perturbation because [3] the energy triomentum tensor is in the same multiplet as the axial current= —(8Kj + Dj)(7.2)= = —i757vaj +and in N = 2 supersymmetry this situation is protected to all orders in perturbation.It follows that the quantum correction to the energy momentum tensor are similar tothe axial anomaly. for which it is well-known that the one-loop result is exact to allorders.For the non-perturbative part of F(A) the only solid a priori pieces of informationare:Im(F”(A)) 0 F(A) 0 F(iA) = F(A) (7.3)and the fact that it vanishes for large A. The first relation from the convexity of theeffective potential, specifically from the fact that Im(F”(A)) is the coefficient of thekinetic term for the gauge-field, the second relation from 1-instanton computationsand the third relation from the residue of R-invariance that is left alter spontaneoussymmetry-breaking. F(A) has to be guessed from this apparently meagre information.The SW-Ansatz8. PreliminariesSW begin by reducing the problem to one in complex analysis by considering onlythe vacuum value A = v of the chiral scalar superfield and determining the functionalform of F(v). Afterwards, (A) can be recovered by the simple substitution F(v) —*F(v + A) F(A). This is analogous to the substitutionV(m,f,g) V(m + f + g, f + g,g) = Veff() (8.1)which is made to obtain the effective potential from the partition function P(m,f,g)of a standard renormalizable theory with a single scar field with masses m, andcoupling constants f and g.Next, they note that asymptotic freedom allows them to identify the perturbativeregion as the large scale one v —+ cc and thusr(v) — -ln(v) for v — cc (8.2)Since v is expected to be singular in the small scale (strong coupling) region one alsopostulates the existence of a universal (complex) parameter u E C normalized so thata(u) —+ u for u —* cc. Assembling all this information they reduce the problem tothe search for a function r(u) such thatIm(r(u)) 0 and r(u) ln(u) (8.3)The procedure for choosing a r(u) to satisfy (8.3) is actually rather similar that usedby Veneziano in choosing his formula for the S-matrix S(s,t,u). where s,t and u are theinvariant squares of the momenta. That is to say. instead of computing the functiondirectly from the underlying theory one uses its properties (symmetries, boundaryconditions etc.), to try to guess what it should be. Indeed duality (actually triality)plays here a role which is analogous to that played by crossing symmetry (symmetryin s,t and u) in the Veneziano case. But first one has to decide the general classof functions out of which the function r(u) should be chosen. The standard classof functions which map the upper part C of the complex plane into itself modulosubgroups of SL(2, Z) is the class of Fuchsian functions {öj and the choice will bemade out of these. So, to put the results in perspective, we digress for a moment toconsider Fuchsiam functions or maps.9. Fuchsian MapsThe Fuchsian maps T(u) are maps from C to D = C+/G where G is a subgroup ofSL(2, R) (restricted in our case to SL(2, Z). Thus they map C into fundamentaldomains D E C whose G-equivalent copies fill out C. The domains D are circularpolygons (ones whose sides may be straight lines or circles) and in our case will stretchout to infinity. In general the polygons may not be of genus zero because the sidesmay be have to be identified in a non-trivial manner. The corners of the polygoncorrespond to points on the real axis in the u-variable and if the genus of the polygonis zero u can be extended to cover the whole complex plane, which compactifiesto the Riemann sphere. Otherwise the compactification of the u-space leads to aRiemann surface of higher genus. The essential point is that these maps guaranteethat Im(r(u)) 0.9. The Schwarzian DerivativesThe Fuchsian functions r(u) are apt to be complicated but a great simplification isachieved by considering not the functions themselves but their Schwarzian derivativesr” 3 -“ 2S(r)= -- (—) (9.1)In general the main property of Schwarzian derivatives is that they are invariantwith respect to the modular transformations (6.1). However, in the case of Fuchsianfunctions they have the added advantage that they are simple meromorphic functionsof the formz=nS(r(u))=)2+ ( )} S(r(u)) u (9.2)and this is why it is convenient to use S(r) rather than r itself. One S(r) is knownthere is a simple and elegant way to recover r from it, namely to writeT = where y” + S(r(u))y = 0 (9.3)Such functions T are tailor-made for the S-W model where r(u) = a/a”. All butthe 3-parameters in (9.2) have a simple geometrical meaning. The number ri isthe number of corners on the polygon D (excluding the point at infinity) and theparameters a and a are the locations of the corners and the internal angles of thepolygon respectively. Because of the freedom of choosing the axes in u-space thenumber of independent parameters in Q(u) is 3n — 2. The boundary condition putstwo further restrictions on the on the 3’s and this reduces the number to 3n 4.This already shows that ri 2. It is clear that the real choice in choosing a Fuchsianfunction is to choose ri and then choose the 3n — 4 parameters in the meromorphicfunction Q(u).10. S-W ChoiceThe question is: which Fuchsian function to choose? The S-W choice is made byadding two further inputs to the basic conditions, namely1. Minimality: n = 2i’i 9”\ /i 02. Duality: M= (%\0 E) ÷ = (\\2 1Duality in this sense means that the monodromy matrix M1 at one of the singularitiesis the dual of the monodromy matrix M at infinity. Physically, it means that theasymptotic freedom of g for large scales is the dual of the infra-red slavery of g(asymptotic freedom of g’) for low scales. It turns out that M and M1 generate amonodromy group P2 which includes the monodromy group of the second singularity,and consists of all matrices of the formI+2( ) e,f,g,heZ (10.1)According to our previous analysis this determines a unique Fuchsian map (modulothe positions of the two singularities which are normalized to be a = ± 1) and it turnsout to be the map with c = 0 and ,3 = +1/4. Thus with SW Ansatz equation (9.3)becomeswhere “+[ 3+u20 (10.2)Y2 4 (u —1)-There is actually a simplification in this case on changing to the variable a, wherea’ y because then (10.2) becomeswhere a”+(91)a=O (10.3)and the differential equation in (10.3) is just a hypergeometric equation. So finallyr(u) is simply the ratio of the derivatives of two simple hypergeometric functions.In fact the function r’(u) is a well-known automorphic function called the ellipticmodular function. Thus when all the smoke has cleared away it turns out that theSW Ansatz is to propose that F(A) is such that F”(A) is the inverse of the ellipticmodular function!11. Uniqueness•The S-W Ansatz is certainly not the only Ansatz compatible with the basic conditionsIm(r(u)) 0 r(u) ln(u) (11.1)Indeed, even for two singularities, the Schwarzian could be(1—a2) (1_,2) l(9_a2—7)S(r(u))= 2(u - 1)2 + (u + 1)2 - 9 (1- u2)(11.2)where a,’y = 0, , -, with monodromy groups called F2, G2 and G9. The solutionwith G8 is ruled out because it is not R-invariant but G2 remains as a refiexioninvariant alternative. For more than two singularities there are many more possibilities. For example a three-singularity reffexion-invariant solution with monodromygroup SL(2, Z)/Z2 is provided by T(u) = J(u2), where J is the standard modularfunction.What distinguishes the S-B solution is that it carries the dual symmetry that wasused by S-W as input. The point is that F2 is an invariant subgroup of SL(2,Z) andSL(2,Z)/F2= F3 (11.3)where P3 is the permutation group of order 3. Mathematically the permutationgroup P3 interchanges the point at infinity and the two singularities and physically itinterchanges gauge-fields, monopoles and dyons. The original duality input emergesas the symmetry between the the gauge-field at infinity and the monople at one ofthe singularities.12. CorrectnessSince the S-W Ansatz is not unique one has to check whether it is, in fact the correctchoice. In principle this can be done by making direct computations of the nonpertubative part of the Action using instanton computations. In practice this hasbeen done [2] only for arbitrary gauge-groups in the 1-instanton configurations andfor SU(2) in the 2-instanton configurations and in these cases the SW Ansatz agreeswith direct computations.The general idea of these computations is as follows: For any Fuchsian function satisfying the boundary conditions and R-invariance we have the asymptotic expansionv A 4mF(v) = v2 + hv2ln()2+ v2 Cm () (12.1)In the direct instanton computations F(v) is supposed to be the partition function andthe contributions of the various powers in rn are supposed to come from the instantonsectors of topological charge m. The idea, therefore, is to compute the partitionfunction in an m-instanton background under the assumption that the scalar fieldhas a non-zero value v on the sphere at infinity. What one does in practice is to firstchoose a background for the gauge and fermion fields by the conditionsF=F 7D=0 (12.2)where F* denotes the dual of F. These fields are parametrized by the Sm ADHMparameters p, ii for self-dual gauge-fields, where p. in particular, denotes the size ofthe instanton, and the 8m parameters 17, for their zero-mode fermion fields. Onethen postulates that the scalar field is given byD2=[,z4J b(oc)=v (12.3)and is thus a functional of these 16n parameters. Finally one postulates that long-range part of the partition function comes only from the surface term tr(f pDrb)in the Action (3.4) and that the short range part drops out because the bosonicand fermionic contributions cancel on account of supersymmetry. In that case thepartition function evidently takes the formP(v) = f dpd(v)pdpdvdd17efd.(&ñ) (12.4)For dimensional reasons these integrals are of the form8mir2 2 A 4m Sm,t-2P(v) =J4me fp43dpd(v)e f(v) = a2(_) eem (12.5)where the em ‘s are dimensionless constants. It is these constants that are to be compared with the SW constants Cm and, as mentioned earlier, the coefficients computedup to now are in agreement. So the instanton computations provide reasonably strongsupport for the correctness of the SW Ansatz. As a check on the sensitivity of theresult we have computed c1 for the alternative n = 2 Ansatz and it is different fromthe SW one.It must be admitted that the validity of the instanton computations as describedabove is not quite clear, because the equations (12.2) and (12.3) are regarded onlyas approximations and the background configurations described by them are neithersolutions of the classical field equations nor supersymmetric-invariant. However, wehope to present a more convincing argument for the validity of the instanton computations in a later paper.I. Sachs would like to thank the Swiss National Science Foundation for partial support.References[1] N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19 430 (1994) 485(E)[2] D. Finnell and P. Pouliot, Nucl. Phys. B453 (1995) 225N. Dorey et al. Phys. Rev. D54 (1996) 2921; hep-th/9607202K.Ito and N. Sasakura, Phys. Lett. B382 (1996) 95A. Yung, hep-th/9605096F. Fucito and G. Travaglini, hep-th/9605215T. Harano and M. Sato. hep-th/9608060[3] J. Wess and R. Bagger, Supersymmetry, Princeton University Press (19.)P.West, Introduction to Supersymmetry and Supergravity World Scientific (1990)[4] C. Montonen and D. Olive, Phys. Lett. B72 (1977) 117[5] A. Erdelyi et al. Higher transcendental functions, Vol. 3, McGraw-Hill, NY (1935)Z. Nehari, conformal Mapping, McGraw-Hill, NY (1952)R.C. Gunning, Lectures on Modular Forms, Princeton University Press (1962)L. R. Ford, Automorphic Functions, Chelsea Publishing Company, NY (1951)

Brief Resume of Seiberg-Witten Theory

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