theories with chiral fermions) in T. Reisz, Commun. Math. Phys. 117 (1988) 79,

639.

11. The generality of the doubling phenomenon for lattice fermions has been proven

by H.B. Nielsen and M. Ninomiya, Nucl. Phys. B185 (1981) 20.

234 J. Zinn-Justin

12. Wilson™s solution to the fermion doubling problem is described in K.G. Wilson in

New Phenomena in Subnuclear Physics, Erice 1975, A. Zichichi ed. (Plenum, New

York 1977).

13. Staggered fermions have been proposed in T. Banks, L. Susskind and J. Kogut,

Phys. Rev. D13 (1976) 1043.

14. The problem of chiral anomalies is discussed in J.S. Bell and R. Jackiw, Nuovo

Cimento A60 (1969) 47; S.L. Adler, Phys. Rev. 177 (1969) 2426; W.A. Bardeen,

Phys. Rev. 184 (1969) 1848; D.J. Gross and R. Jackiw, Phys. Rev. D6 (1972) 477;

H. Georgi and S.L. Glashow, Phys. Rev. D6 (1972) 429; C. Bouchiat, J. Iliopoulos

and Ph. Meyer, Phys. Lett. 38B (1972) 519.

15. See also the lectures S.L. Adler, in Lectures on Elementary Particles and Quantum

Field Theory, S. Deser et al eds. (MIT Press, Cambridge 1970); M. E. Peskin, in

Recent Advances in Field Theory and Statistical Mechanics, Les Houches 1982, R.

Stora and J.-B. Zuber eds. (North-Holland, Amsterdam 1984); L. Alvarez-Gaum´, e

in Fundamental problems of gauge theory, Erice 1985 G. Velo and A.S. Wightman

eds. (Plenum Press, New-York 1986).

16. The index of the Dirac operator in a gauge background is related to Atiyah“Singer™s

theorem M. Atiyah, R. Bott and V. Patodi, Invent. Math. 19 (1973) 279.

17. It is at the basis of the analysis relating anomalies to the regularization of the

fermion measure K. Fujikawa, Phys. Rev. D21 (1980) 2848; D22 (1980) 1499(E).

18. The same strategy has been applied to the conformal anomaly K. Fujikawa, Phys.

Rev. Lett. 44 (1980) 1733.

19. For non-perturbative global gauge anomalies see E. Witten, Phys. Lett. B117 (1982)

324; Nucl. Phys. B223 (1983) 422; S. Elitzur, V.P. Nair, Nucl. Phys. B243 (1984)

205.

20. The gravitational anomaly is discussed in L. Alvarez-Gaum´ and E. Witten, Nucl.

e

Phys. B234 (1984) 269.

21. See also the volumes S.B. Treiman, R. Jackiw, B. Zumino and E. Witten, Current

Algebra and Anomalies (World Scienti¬c, Singapore 1985) and references therein;

R.A. Bertlman, Anomalies in Quantum Field Theory, Oxford Univ. Press, Oxford

1996.

22. Instanton contributions to the cosine potential have been calculated with increasing

accuracy in E. Br´zin, G. Parisi and J. Zinn-Justin, Phys. Rev. D16 (1977) 408;

e

E.B. Bogomolny, Phys. Lett. 91B (1980) 431; J. Zinn-Justin, Nucl. Phys. B192

(1981) 125; B218 (1983) 333; J. Math. Phys. 22 (1981) 511; 25 (1984) 549.

23. Classical references on instantons in the CP (N ’1) models include A. Jevicki Nucl.

Phys. B127 (1977) 125; D. F¨rster, Nucl. Phys. B130 (1977) 38; M. L¨scher, Phys.

o u

Lett. 78B (1978) 465; A. D™Adda, P. Di Vecchia and M. L¨scher, Nucl. Phys. B146

u

(1978) 63; B152 (1979) 125; H. Eichenherr, Nucl. Phys. B146 (1978) 215; V.L. Golo

and A. Perelomov, Phys. Lett. 79B (1978) 112; A.M. Perelemov, Phys. Rep. 146

(1987) 135.

24. For instantons in gauge theories see A.A. Belavin, A.M. Polyakov, A.S. Schwartz

and Yu S. Tyupkin, Phys. Lett. 59B (1975) 85; G. ™t Hooft, Phys. Rev. Lett. 37

(1976) 8; Phys. Rev. D14 (1976) 3432 (Erratum Phys. Rev. D18 (1978) 2199); R.

Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172; C.G. Callan, R.F. Dashen

and D.J. Gross, Phys. Lett. 63B (1976) 334; A.A. Belavin and A.M. Polyakov,

Nucl. Phys. B123 (1977) 429; F.R. Ore, Phys. Rev. D16 (1977) 2577; S. Chadha,

P. Di Vecchia, A. D™Adda and F. Nicodemi, Phys. Lett. 72B (1977) 103; T. Yoneya,

Phys. Lett. 71B (1977) 407; I.V. Frolov and A.S. Schwarz, Phys. Lett. 80B (1979)

406; E. Corrigan, P. Goddard and S. Templeton, Nucl. Phys. B151 (1979) 93.

Chiral Anomalies and Topology 235

25. For a solution of the U (1) problem based on anomalies and instantons see G. ™t

Hooft, Phys. Rep. 142 (1986) 357.

26. The strong CP violation is discussed in R.D. Peccei, Helen R. Quinn, Phys. Rev.

D16 (1977) 1791; S.Weinberg, Phys. Rev. Lett. 40 (1978) 223, (Also in Mohapatra,

R.N. (ed.), Lai, C.H. (ed.): Gauge Theories Of Fundamental Interactions, 396-399).

27. The Bogomolnyi bound is discussed in E.B. Bogomolnyi, Sov. J. Nucl. Phys. 24

(1976) 449; M.K. Prasad and C.M. Sommerfeld, Phys. Rev. Lett. 35 (1975) 760.

28. For more details see Coleman lectures in S. Coleman, Aspects of symmetry, Cam-

bridge Univ. Press (Cambridge 1985).

29. BRS symmetry has been introduced in C. Becchi, A. Rouet and R. Stora, Comm.

Math. Phys. 42 (1975) 127; Ann. Phys. (NY) 98 (1976) 287.

30. It has been used to determine the non-abelian anomaly by J. Wess and B. Zumino,

Phys. Lett. 37B (1971) 95.

31. The overlap Dirac operator for chiral fermions is constructed explicitly in H. Neu-

berger, Phys. Lett. B417 (1998) 141 [hep-lat/9707022], ibidem B427 (1998) 353

[hep-lat/9801031].

32. The index theorem in lattice gauge theory is discussed in P. Hasenfratz, V. Laliena,

F. Niedermayer, Phys. Lett. B427 (1998) 125 [hep-lat/9801021].

33. A modi¬ed exact chiral symmetry on the lattice was exhibited in M. L¨ scher, Phys.

u

Lett. B428 (1998) 342 [hep-lat/9802011], [hep-lat/9811032].

34. The overlap Dirac operator was found to provide solutions to the Ginsparg“Wilson

relation P.H. Ginsparg and K.G. Wilson, Phys. Rev. D25 (1982) 2649.

35. See also D.H. Adams, Phys. Rev. Lett. 86 (2001) 200 [hep-lat/9910036], Nucl.

Phys. B589 (2000) 633 [hep-lat/0004015]; K. Fujikawa, M. Ishibashi, H. Suzuki,

[hep-lat/0203016].

36. In the latter paper the problem of CP violation is discussed.

37. Supersymmetric quantum mechanics is studied in E. Witten, Nucl. Phys. B188

(1981) 513.

38. General determinations of the index of the Dirac operator can be found in C.

Callias, Comm. Math. Phys. 62 (1978) 213.

39. The fermion zero-mode in a soliton background in two dimensions is investigated

in R. Jackiw and C. Rebbi, Phys. Rev. D13 (1976) 3398.

40. Special properties of fermions in presence of domain walls were noticed in C.G.

Callan and J.A. Harvey, Nucl. Phys. B250 (1985) 427.

41. Domain wall fermions on the lattice were discussed in D.B. Kaplan, Phys. Lett.

B288 (1992) 342.

42. See also (some deal with the delicate problem of the continuum limit when the

¬fth dimension is ¬rst discretized) M. Golterman, K. Jansen and D.B. Kaplan,

Phys.Lett. B301 (1993) 219 [hep-lat/9209003); Y. Shamir, Nucl. Phys. B406 (1993)

90 [hep-lat/9303005] ibidem B417 (1994) 167 [hep-lat/9310006]; V. Furman, Y.

Shamir, Nucl. Phys. B439 (1995) 54 [hep-lat/9405004]. Y. Kikukawa, T. Noguchi,

[hep-lat/9902022].

43. For more discussions and references Proceedings of the workshop “Chiral 99”,

Chinese Journal of Physics, 38 (2000) 521“743; K. Fujikawa, Int. J. Mod. Phys.

A16 (2001) 331; T.-W. Chiu, Phys. Rev. D58(1998) 074511 [hep-lat/9804016],

Nucl. Phys. B588 (2000) 400 [hep-lat/0005005]; M. L¨scher, Lectures given at

u

International School of Subnuclear Physics Theory and Experiment Heading for

New Physics, Erice 2000, [hep-th/0102028] and references therein; P. Hasenfratz,

Proceedings of “Lattice 2001”, Nucl. Phys. Proc. Suppl. 106 (2002) 159 [hep-

lat/0111023].

236 J. Zinn-Justin

44. In particular the U (1) problem has been discussed analytically and studied numer-

ically on the lattice. For a recent reference see for instance L. Giusti, G.C. Rossi,

M. Testa, G. Veneziano, [hep-lat/0108009].

45. Early simulations have used domain wall fermions. For a review see P.M. Vranas,

Nucl. Phys. Proc. Suppl. 94 (2001) 177 [hep-lat/0011066]

46. A few examples are S. Chandrasekharan et al, Phys. Rev. Lett. 82 (1999) 2463

[hep-lat/9807018]; T. Blum et al., [hep-lat/0007038]; T. Blum, RBC Collaboration,

Nucl. Phys. Proc. Suppl. 106 (2002) 317 [hep-lat/0110185]; J-I. Noaki et al CP-

PACS Collaboration, [hep-lat/0108013].

47. More recently overlap fermion simulations have been reported R.G. Edwards, U.M.

Heller, J. Kiskis, R. Narayanan, Phys. Rev. D61 (2000) 074504 [hep-lat/9910041];

P. Hern´ndez, K. Jansen, L. Lellouch, Nucl. Phys. Proc. Suppl. 83 (2000) 633

a

[hep-lat/9909026]; S.J. Dong, F.X. Lee, K.F. Liu, J.B. Zhang, Phys. Rev. Lett. 85

(2000) 5051 [hep-lat/0006004]; T. DeGrand, Phys. Rev. D63 (2001) 034503 [hep-

lat/0007046]; R.V. Gavai, S. Gupta, R. Lacaze, [hep-lat/0107022]; L. Giusti, C.

Hoelbling, C. Rebbi, [hep-lat/0110184]; ibidem [hep-lat/0108007].

Supersymmetric Solitons and Topology

M. Shifman

William I. Fine Theoretical Physics Institute, School of Physics and Astronomy,

University of Minnesota, 116 Church Street SE, Minneapolis MN 55455, USA

Abstract. This lecture is devoted to solitons in supersymmetric theories. The em-

phasis is put on special features of supersymmetric solitons such as “BPS-ness”. I

explain why only zero modes are important in the quantization of the BPS solitons.

Hybrid models (Landau“Ginzburg models on curved target spaces) are discussed in

some detail. Topology of the target space plays a crucial role in the classi¬cation of the

BPS solitons in these models. The phenomenon of multiplet shortening is considered. I

present various topological indices (analogs of Witten™s index) which count the number

of solitons in various models.

1 Introduction

The term “soliton” was introduced in the 1960™s, but the scienti¬c research of

solitons had started much earlier, in the nineteenth century when a Scottish

engineer, John Scott-Russell, observed a large solitary wave in a canal near

Edinburgh.

For the purpose of my lecture I will adopt a narrow interpretation of solitons.

Let us assume that a ¬eld theory under consideration possesses a few (more

than one) degenerate vacuum states. Then these vacua represent distinct phases

of the theory. A ¬eld con¬guration smoothly interpolating between the distinct

phases which is topologically stable will be referred to as soliton.1 This de¬nition

is over-restrictive “ for instance, it does not include vortices, which present a

famous example of topologically stable solitons. I would be happy to discuss

supersymmetric vortices and ¬‚ux tubes. However, because of time limitations, I

have to abandon this idea limiting myself to supersymmetric kinks and domain

walls.

In non-supersymmetric ¬eld theories the vacuum degeneracy usually requires

spontaneous breaking of some global symmetry “ either discrete or continuous.

In supersymmetric ¬eld theories (if supersymmetry “ SUSY “ is unbroken) all

vacua must have a vanishing energy density and are thus degenerate.

This is the ¬rst reason why SUSY theories are so special as far as topological

solitons are concerned. Another (more exciting) reason explaining the enormous

interest in topological solitons in supersymmetric theories is the existence of

a special class of solitons, which are called “critical” or “Bogomol™nyi“Prasad“

Sommer¬eld saturated” (BPS for short).

A birds™ eye view on the development of supersymmetry beginning from

its inception in 1971 [1] is presented in Appendix B. A seminal paper which

1

More exactly, we will call it “topological soliton”.

M. Shifman, Supersymmetric Solitons and Topology, Lect. Notes Phys. 659, 237“284 (2005)

http://www.springerlink.com/ c Springer-Verlag Berlin Heidelberg 2005

238 M. Shifman

opened for investigation the currently ¬‚ourishing topic of BPS saturated soli-

tons is that of Witten and Olive [2] where the authors noted that in many

instances (supporting topological solitons) topological charges coincide with the

so-called central charges [3] of superalgebras. This allows one to formulate the

Bogomol™nyi“Prasad“Sommer¬eld construction [4] in algebraic terms and to ex-

tend the original classical formulation to the quantum level, making it exact. All

these statements will be explained in detail below.

In high energy physics theorists traditionally deal with a variety of distinct

solitons in various space-time dimensions D. Some of the most popular ones are:

(i) kinks in D = 1+1 (being elevated to D = 1+3 they represent domain

walls);

(ii) vortices in D = 1+2 (being elevated to D = 1+3 they represent strings

or ¬‚ux tubes);

(iii) magnetic monopoles in D = 1+3.

In the three cases above the topologically stable solutions are known from the

1930™s, ™50™s, and ™70™s, respectively. Then it was shown that all these solitons can

be embedded in supersymmetric theories. To this end one adds an appropriate

fermion sector, and if necessary, expands the boson sector. In this lecture we

will limit ourselves to critical (or BPS-saturated) kinks and domain walls. Non-

critical solitons are typically abundant, but we will not touch this theme at

all.

The presence of fermions leads to a variety of novel physical phenomena

which are inherent to BPS-saturated solitons. These phenomena are one of the

prime subjects of my lecture.

Before I will be able to explain why supersymmetric solitons are special

and interesting, I will have to review brie¬‚y well-known facts about solitons

in bosonic theories and provide a general introduction to supersymmetry in

appropriate models. I will start with the simplest model “ one (real) scalar ¬eld

in two dimensions plus the minimal set of superpartners.

D = 1+1; N = 1

2

In this part we will consider the simplest supersymmetric model in D = 1+1

dimensions that admits solitons. The Lagrangian of this model is

2

‚2W ¯

1 ‚W

¯

L= ‚µ φ ‚ φ + ψ i‚ψ ’ ’

µ

ψψ , (2.1)

‚φ2

2 ‚φ

where φ is a real scalar ¬eld and ψ is a Majorana spinor,

ψ1

ψ= (2.2)

ψ2

with ψ1,2 real. Needless to say that the gamma matrices must be chosen in the

Majorana representation. A convenient choice is

γ 0 = σ2 , γ 1 = iσ3 , (2.3)

Supersymmetric Solitons and Topology 239

where σ2,3 are the Pauli matrices. For future reference we will introduce a “γ5 ”

matrix, γ 5 = γ 0 γ 1 = ’σ1 . Moreover,

¯

ψ = ψγ 0 .

The superpotential function W(φ) is, in principle, arbitrary. The model (2.1)

with any W(φ) is supersymmetric, provided that W ≡ ‚W/‚φ vanishes at some

value of φ. The points φi where

‚W

=0

‚φ

are called critical. As can be seen from (2.1), the scalar potential is related to the

superpotential as U (φ) = (1/2)(‚W/‚φ)2 . Thus, the critical points correspond

to a vanishing energy density,

2

1 ‚W

U (φi ) = =0 . (2.4)

2 ‚φ φ=φi

The critical points accordingly are the classical minima of the potential energy “

the classical vacua. For our purposes, the soliton studies, we require the existence

of at least two distinct critical points in the problem under consideration. The

kink will interpolate between distinct vacua.

Two popular choices of the superpotential function are:

m2 »

W(φ) = φ ’ φ3 , (2.5)

4» 3

and

φ