Quantum corrections to solitons and BPS saturation
aa r X i v : . [ h e p - t h ] F e b October 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02
Quantum corrections to solitons and BPS saturation A. Rebhan
Institut f¨ur Theoretische PhysikTechnische Universit¨at Wien, A-1040 Wien, [email protected]
P. van Nieuwenhuizen
C.N. Yang Institute for Theoretical PhysicsStony Brook University, Stony Brook, NY 11794-3840, [email protected]
R. Wimmer
Laboratoire de Physique, ENS Lyon,46 all´ee d’Italie, F-69364 Lyon CEDEX 07, [email protected]
We review our work of the past decade on one-loop quantum correctionsto the mass M and central charge Z of solitons in supersymmetric fieldtheories: the kink, the vortex, and the monopoles (focussing on the kinkand the monopoles here). In each case a new feature was needed toobtain BPS saturation: a new anomaly-like contribution to Z for thekink and the N = 2 monopole, the effect of classical winding of thequantum vortex contributing to Z , surface terms contributing to M ofthe N = 4 monopole and to Z of the N = 2 and N = 4 monopoles, andcomposite operator renormalization for the currents of the “finite” N = 4model. We use dimensional regularization, modified to preserve susyand be applicable to solitons, and suitable renormalization conditions.In the mode expansion of bosonic and fermionic quantum fields, zeromodes appear then as massless nonzero modes. Contribution to “Fundamental Interactions—A Memorial Volume for Wolfgang Kum-mer”, D. Grumiller, A. Rebhan, D.V. Vassilevich (eds.)1 ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer
In the beginning of the 1970’s particle physicists became interested in soli-tons. Since Dirac’s work on the quantization of the electromagnetic fieldin the late 1920’s, particles had been associated with the Fourier modesof the second-quantized fields, and perturbation theory had been used tocompute scattering amplitudes. However, for the strong interactions thisapproach could not be used because the coupling constant is larger thanunity, and nonlinearities are essential. Thus particle physicists turned tosolitons as representations of particles in strongly interacting field theories.This changed the emphasis from properties of scattering amplitudes of twoor more solitons to properties of single solitons.
Also in the early 1970’s, the renormalizability of nonabelian gauge the-ory was proven, and supersymmetry (susy) was discovered. A natural ques-tion that arose was: are nonabelian gauge theories (and abelian gaugetheories) with solitons also renormalizable? In susy theories some diver-gences cancel, so it seemed interesting to extend the theories with solitonsto susy theories with the same solitons, and to study whether cancellationsof radiative corrections did occur. In particular, the mass of a soliton getscorrections from the sum over zero-point energies of bosons and fermions.A formal proof had been constructed that in susy theories the sum of allzero-point energies cancels, and it was conjectured that also the correc-tions to the mass of a soliton vanish in susy theories. We shall see thatthis is an oversimplification, and that the mass of solitons already receivescorrections at the one-loop level.In addition to susy, also topology became a major area of interest insoliton physics. In 1973 Nielsen and Olesen used the vortex solution, whichis a soliton in 2 + 1 dimensions based on the abelian Higgs model with acomplex scalar field, to construct topologically stable extended particles.Ginzberg and Landau had used this model to describe superconductivityin 1950, and Abrikosov had found the vortex solution in 1957. Nielsen andOlesen embedded this vortex solution into 3 + 1 dimensions, and obtainedin this way stringlike excitation of the dual resonance model of particlephysics with a magnetic field confined inside the tubes. ’t Hooft wonderedif their construction could be extended to non-abelian Higgs models, andin 1974 he and Polyakov discovered that the nonabelian Higgs model in3 + 1 dimensions with gauge group SU(2) and a real triplet of Higgs scalarscontains monopoles, which are solitonic solutions with a magnetic charge.They contain a topological number, the winding number, which prevents ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 them from decaying to the trivial vacuum. Similarly, the vortex solutionin 2 + 1 dimensions has a winding number, and even the kink (a solitonin 1+1 dimensions ) is topologically stable. There exist also nontopo-logical solitons but we shall not discuss them. In 1975 Julia and Zeeconstructed dyons, solitons in the SU(2) nonabelian Higgs model with anelectric and a magnetic charge, and soon afterwards Prasad and Sommer-feld found exact expressions for these solitons in the limit of vanishing λϕ coupling constant (the PS limit). In 1976 Bogomolnyi showed thatin all these cases of topological solitons one can write the energy densityas a sum of squares plus total derivatives. Requiring these squares to van-ish leads to first-order differential equations for solitons, the Bogomolnyiequations, which are much easier to solve than the second-order field equa-tions. He also noted that the total energy has a bound H ≥ | Z | where Z isthe contribution from the total derivatives. This bound is called the BPSbound because for monopoles in the nonabelian Higgs model it can only besaturated for vanishing coupling constant λ . For a classical soliton at rest, H is equal to its mass M , and M = | Z | . Finally in 1978 Olive and Witten noted that the total derivative terms in the Bogomolnyi expression for theenergy density are the central charges of the susy algebra of the correspond-ing susy theories. These charges are Heisenberg operators, containing allperturbative and nonperturbative quantum corrections. By using results ofthe representation theory of superalgebras in terms of physical states, theyproved that for topological solitons the BPS bound M ≥ | Z | must remainsaturated at the quantum level: M = | Z | .We shall calculate the one-loop corrections to M and Z , and show thatthey are indeed equal, but nonvanishing for susy kinks and the N = 2monopole. These calculation are not meant as a check of the proof ofOlive and Witten, but rather they are a test of whether our understandingof quantum field theory in the presence of solitons has progressed enoughto obtain saturation of the BPS bound. As we shall see, this is a non-trivial issue. The vacuum expectation values of the Higgs scalars acquirelocal corrections in the presence of solitons, boundary terms contribute to M and Z , a new anomaly-like contribution to Z yields a finite correction,and composite operators require infinite renormalization to obtain a finiteanswer in the “finite” N = 4 susy model. We shall use the backgroundfield formalism to formulate background-covariant R ξ gauges, and we shalluse the extended Atiyah-Singer-Patodi index theorem for noncompactspaces to calculate the sum over zero-point energies in the presence of soli-tons. We shall also introduce an extension of dimensional regularization ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer which preserves susy and can be used for solitons.
As we have discussed, solitons were initially proposed for describinghadrons, but when duality between electric and magnetic fields, and ex-tended dualities in supersymmetric field theories, were developed, anotherpoint of view emerged. It was conjectured by Montonen and Olive, andWitten that there exist dual formulations of field theories in which parti-cles become solitons, and solitons become particles. Modern work in stringtheory has confirmed and extended this hypothesis in an amazing way. In order to test one’s understanding of a quantum field theory, static quanti-ties should be among the first to consider. In the following we shall considertwo static quantities in one of the simplest quantum field theories with asoliton: the mass and the central charge of the susy kink at the one-looplevel. This exercise has proved to be a surprisingly subtle topic with allkinds of pitfalls. Even when the same renormalization conditions were em-ployed, different regularization methods led to contradictory results, and this confusing state of matters lasted until the end of the 1990’s, whenthe question was reopened by a work by two of us, in which it was shownthat the methods used to produce the most widely accepted result of zerocorrections in the susy case were inconsistent with the known integrabil-ity of the bosonic sine-Gordon model. Subsequently, the pitfalls of thevarious methods were sorted out, which involved the discov-ery of an anomalous contribution to the central charge guaranteeing BPSsaturation. In the following, we shall show how all this works out usingdimensional regularization adapted to susy solitons.
Mass
The mass of a soliton is obtained by taking the expectation value of theHamiltonian with respect to the ground state in the soliton sector b . Inaddition one needs the contribution from counter terms which are neededto renormalize the model. a Dimensional regularization in the context of (bosonic) solitons was employed before inRefs.
This state is often called the soliton vacuum, but this is a misnomer because vacuahave by definition vanishing energy while the soliton has a nonvanishing mass. Thevacuum is the state with vanishing energy in the sector without winding, but to avoidmisunderstanding, we shall consistently call it the trivial vacuum. ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 As Hamiltonian we take the gravitational Hamiltonian (obtained byvarying the action with respect to an external gravitational field). We writeall fields ϕ ( x, t ) as a sum of (static) background fields ϕ b ( x ) and quantumfields η ( x, t ), and only retain all terms quadratic in quantum fields. Forreal bosonic fields the Hamiltonian density of the quantum fields is of theform H = 12 ˙ η ˙ η + 12 ∂ x η∂ x η + · · · (1.1)and using partial integration yields ∂ x η∂ x η + · · · = ∂ x ( η∂ x η ) − η ( ∂ x η + · · · ). The terms − η ( ∂ x η + · · · ) are then equal to − η ¨ η if one uses thelinearized field equations for the fluctuations, and the expectation value − h soliton | η ¨ η | soliton i is equal to + h soliton | ˙ η ˙ η | soliton i . For a real (Ma-jorana) fermion there are no background fields, and the Hamiltonian densityis of the form H = ¯ ψγ ∂ x ψ + . . . . Again using the linearized field equa-tions for the fermion’s fluctuations, one finds H = i ψ † ˙ ψ since ¯ ψ = ψ † iγ and ( γ ) = −
1. Thus the one-loop quantum corrections to the mass of asoliton, M = h soliton | R H dx | soliton i , are of the generic form M (1) = R [ h ˙ η ˙ η i + i h ψ † ˙ ψ i ] dx + boundary terms Z ∂ x ( 12 h η∂ x η i )+ counter terms ∆ M . (1.2)To define the infinite and finite parts of the one-loop corrections, weneed a regularization scheme that preserves susy and is easy to work with.This singles out dimensional regularization. Usually one needs dimensionalregularization by dimensional reduction to preserve susy, but that option isnot available to us because the soliton occupies all space dimensions. Goingup in dimensions in general violates susy, but there is a way around theseobjections which combines the virtues of both approaches. In all cases weconsider, the susy action in D + 1 dimensions can be rewritten as a susyaction in D + 2 dimensions. Then going down in dimensions, we use in( D + ǫ ) + 1 dimensions standard dimensional regularization. This schemeclearly preserves susy, and it leaves enough space for the soliton.Let us see how things work out for the susy kink. The susy action (aftereliminating the susy auxiliary field) is given by L = 12 ˙ ϕ −
12 ( ∂ x ϕ ) − U −
12 ¯ ψ /∂ψ − U ′ ¯ ψψ, U = λ ϕ − µ /λ ) , (1.3) ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer where ψ is a 2-component Majorana spinor and ϕ a real scalar field. Thismodel has N = (1 ,
1) susy in 1+1 dimensions, but the same expression for L can also be viewed as an N = 1 model in 2+1 dimensions. The operator /∂ in the Dirac action and in the transformation law δψ = ( /∂ϕ − U ) ǫ is thengiven by γ ∂ ϕ + γ ∂ x ϕ + γ ∂ y ϕ .The energy density obtained from the gravitational stress tensor reads H = 12 ˙ ϕ + 12 ( ∂ k ϕ ) + 12 U + 12 ¯ ψγ k ∂ k ψ + 12 U ′ ¯ ψψ , k = 1 , . (1.4)For the classical soliton solution we set ˙ ϕ = ψ = 0 and denote ϕ b by ϕ K .The classical mass of the kink follows from the Bogomolnyi way of writingthe classical Hamiltonian as a sum of squares plus a boundary term H = 12 ( ∂ x ϕ K ) + 12 U K = 12 ( ∂ x ϕ K + U K ) − ( ∂ x ϕ K ) U K , (1.5)where U K = U ( ϕ K ). Thus the classical field equation for the soliton reads ∂ x ϕ K + U K = 0, and the classical mass is M cl = − Z + ∞−∞ dx ∂ x [ Z ϕ K ( x )0 U ( ϕ ′ ) dϕ ′ ] = 2 √ µ λ . (1.6)The kink solution is given by ϕ K ( x ) = µ √ λ tanh µx √ (where µ is thenormalized mass introduced below), but we shall not need this. Decompos-ing ϕ into ϕ K ( x ) + η ( x, y, t ) , we find for the terms quadratic in quantumfluctuations H (2) = 12 ˙ η ˙ η + 12 ( ∂ k η )( ∂ k η ) + 12 ( 12 U K ) ′′ ηη + 12 ¯ ψγ k ∂ k ψ + 12 U ′ K ¯ ψψ , (1.7)where k = 1 ,
2. Partial integration, use of the linearized field equations forquantum fields η and ψ , and substitution of ¯ ψγ = − iψ † yields H (2) = 12 ˙ η ˙ η + 12 ∂ k ( η∂ k η ) − η ¨ η + i ψ † ˙ ψ hH (2) i = h ˙ η ˙ η + 12 ∂ k ( η∂ k η ) + i ψ † ˙ ψ i . (1.8)(We shall later choose a real (Majorana) representation for the Dirac ma-trices, and then ψ is real, thus ψ † = ψ T .)To renormalize the field theory with quantum fields η and ψ , one con-siders the trivial vacuum, and chooses as background field ϕ b = µ √ λ . Thereare terms with 2, 3, and 4 quantum fields, and the terms with three η ’s, orone η and two ψ ’s, can give a tadpole loop which is divergent and needs ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 renormalization. We therefore decompose the bare mass µ into a renor-malized part µ and a counter term ∆ µ , and require that ∆ µ cancels all(finite as well as infinite) contributions of the tadpoles. The bosonic loopyields ∆ µ = 3 λ h η i while the fermionic loop yields ∆ µ = − λ h η i + = 0+ , µ = µ + ∆ µ , ∆ µ = λ Z d ǫ k (2 π ) ǫ Z − ik + m − iǫ = λ Z d ǫ k (2 π ) ǫ √ k + m . No further renormalizations are needed, so the Z factors for λ , η and ψ areall unity. This is a particular set of renormalization conditions.Having fixed ∆ µ in the trivial sector, we now return to the kink sectorand find for the mass counter term at the one-loop level∆ M = ( 2 √ λ )( µ + ∆ µ ) − M cl = m ∆ µ λ ; m = √ µ . (1.9)We must now evaluate the terms in (1.8). We do this by expanding η and ψ into modes, but as we shall see, the sum h ˙ η ˙ η + iψ † ˙ ψ i can also be extractedfrom an index theorem.The field equation for η ( x, y, t ) = φ ( x ) e ily e − iωt reads − ∂ x φ + ( 12 U K ) ′′ φ = ( ω − l ) φ . (1.10)Actually, the field operator − ∂ x + ( U K ) ′′ factorizes into ( − ∂ x + m tanh mx )( ∂ x + m tanh mx ) ≡ L † L , and this allows explicit expressionsfor the zero mode φ ( x ) (satisfying [ − ∂ x + ( U K ) ′′ ] φ = 0, with ω = 0),the bound state φ B ( x ) (with ω B = − m ), and the continuous spectrum φ ( k, x ) (with ω k = k + m ). However, we do not need explicit expressionsfor these functions. The mode expansion in 1+ ǫ spatial dimensions reads η ( x, y, t ) = Z ∞−∞ d ǫ l (2 π ) ǫ (cid:26)Z ∞−∞ dk √ π √ ω kl ( a kl φ ( k, x ) e ily e − iω kl t + a † kl φ ∗ ( k, x ) e − ily e iω kl t )+ 1 √ ω Bl ( a Bl φ B ( x ) e ily e − iω Bl t + a † Bl φ B ( x ) e − ily e iω Bl t )+ 1 √ ω l ( a l φ ( x ) e ily e − i | l | t + a † l φ ( x ) e − ily e i | l | t ) (cid:27) , (1.11) ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer where ω kl = k + l + m , ω Bl = − m + l , and ω l = l . The annihilationand creation operators a and a † satisfy the usual commutation relations,for example [ a l , a † l ′ ] = δ ǫ ( l − l ′ ). The functions φ B ( x ) and φ ( x ) arenormalized to unity, while the distorted plane waves φ ( k, x ) are normalizedsuch that they become plain waves e i ( kx + δ ( k )) for x → + ∞ and e i ( kx − δ ( k )) for x → −∞ , satisfying the completeness relation c Z ∞−∞ ( | φ ( k, x ) | − dk π + φ B ( x ) + φ ( x ) = 0 . (1.12)For the fermion we use a real (Majorana) representation of the Diracmatrices γ µ which diagonalizes the iterated field equations in 2+1 dimen-sions γ = (cid:18) − (cid:19) , γ = (cid:18) −
11 0 (cid:19) , γ = (cid:18) (cid:19) (1.13)and also makes ψ real. The field equation( /∂ + U ′ K ) ψ = 0 ; U ′ K = m tanh mx ∂ x + U ′ K ) ψ + = ( ∂ − ∂ y ) ψ − ( ∂ x − U ′ K ) ψ − = ( ∂ + ∂ y ) ψ + (cid:27) ψ = (cid:18) ψ + ψ − (cid:19) (1.15)The iterated field equation of ψ + is the same as the η field equation, whilefor ψ − we find the conjugate field operator( L † L − ∂ y + ∂ )( η or ψ + ) = 0( L L † − ∂ y + ∂ ) ψ − = 0 ) L = ∂ x + U ′ K L † = − ∂ x + U ′ K (1.16)Setting ψ ± = ψ ± ( x ) e ily − iωt , the Dirac equation yields ψ − ( k, x ) = i ( ∂ x + U ′ K ) ω kl + l ψ + ( k, x ) ; ω kl = k + l + m (1.17) c The completeness relation reads R φ ( k, x ) φ ∗ ( k, x ′ ) dk π + φ B ( x ) φ B ( x ′ ) + φ ( x ) φ ( x ′ ) = δ ( x − x ′ ), and has been rewritten in terms of ( φ ( k, x ) φ ∗ ( k, x ′ ) − e ik ( x − x ′ ) ) by bringingthe delta function to the left-hand side. It follows that φ ( k, x ), φ B ( x ) and φ ( x ) areorthonormal, for example R φ ( k, x ) φ ∗ ( k ′ , x ) dx = 2 πδ ( k − k ′ ). ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 The mode expansion of ψ in 1+ ǫ spatial dimensions is then given by ψ = (cid:18) ψ + ψ − (cid:19) = Z ∞−∞ d ǫ l (2 π ) ǫ Z ∞−∞ dk √ π (cid:26) √ ω kl (cid:20) b kl (cid:18) √ ω kl + lφ ( k, x ) √ ω kl − lis ( k, x ) (cid:19) e ily e − iω kl t + b † kl (cid:18) √ ω kl + lφ ( k, x ) ∗ √ ω kl − l ( − i ) s ( k, x ) ∗ (cid:19) e − ily e iω kl t (cid:21) + 1 √ ω Bl (cid:20) b Bl (cid:18) √ ω Bl + lφ B ( x ) √ ω Bl − lis B ( x ) (cid:19) e ily e − iω Bl t + b † Bl (cid:18) √ ω Bl + lφ B ( x ) √ ω Bl − l ( − i ) s B ( x ) (cid:19) e − ily e iω Bl t (cid:21) + 1 p | l | (cid:20) b l (cid:18) p | l | + l φ ( x )0 (cid:19) e ily e − i | l | t + b † l (cid:18) p | l | + l φ ( x )0 (cid:19) e − ily e i | l | t (cid:21)(cid:27) (1.18)where s ( k, x ) = ( ∂ x + U ′ K ) φ ( k, x ) ω k , ω k = k + m . (1.19)Several remarks are to be made • we have extracted the same factors √ ω as for the boson; • the normalization factors √ ω + l and √ ω − l are needed to satisfythe equal-time canonical anticommutation relations, as we shallcheck, { ψ ± ( x, y, t ) , ψ ± ( x ′ , y ′ , t ) } = δ ( x − x ′ ) δ ǫ ( y − y ′ ) , { ψ + ( x, y, t ) , ψ − ( x ′ , y ′ , t ) } = 0 (1.20) • we treat zero modes and nonzero modes on equal footing. In fact,the zero modes have become massless nonzero modes at the regu-larized level with energy | l | ; • there are no zero modes for ψ − , while the zero modes of ψ + haveonly positive momenta l in the extra dimensions, yielding masslesschiral domain-wall fermions, which are right-moving on the domainwall; ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer • the zero mode sector can also be written as Z ∞−∞ dl (2 π ) ǫ b l φ ( x ) e il ( y − t ) (1.21)where for positive l , b l is an annihilation operator, but for negative l a creation operator ( b , − l = b † ,l ); • the normalization factor √ ω − l for s k in ψ − is obtained as fol-lows: given that ψ + ( k, x ) is written in terms of √ ω + lφ , multiply ψ − ( k, x ) = i ( ∂ x + U ′ K ) ω + l ψ + ( k, x ) in the numerator and denominatorby √ ω − l √ ω + lψ − ( k, x ) = √ ω + l √ ω − l √ ω − l i ( ∂ x + U ′ K ) ω + l φ ( k, x ) (1.22)= √ ω − l i ( ∂ x + U ′ K ) √ ω − l φ ( k, x ) = √ ω − lis ( k, x ); • the reality of ψ is manifest. One can also write the spinors inthe terms with e iωt as √ ω + lφ and −√ ω − lis since φ ( k, x ) ∗ = φ ( − k, x ) and thus also s ( k, x ) ∗ = s ( − k, x ), which corresponds to ψ = − C ¯ ψ T where C = iγ is the charge conjugation matrix. (Therelation φ ( k, x ) ∗ = φ ( − k, x ) follows from the reflection symmetry x → − x of the action, but one can also read it off from the explicitexpression for φ ( k, x ). )Let us check that this mode expansion for ψ ± is correct by assuming thatthe annihilation and creation operators satisfy the usual anticommutators,and verifying that we obtain δ ( x − x ′ ) δ ǫ ( y − y ′ ) and zero in (1.20). Webegin with { ψ + ( x, y, t ) , ψ + ( x ′ , y ′ , t ) } = Z dk π Z d ǫ l (2 π ) ǫ (1.23) (cid:20) ω kl + l ω kl { φ ( k, x ) φ ∗ ( k, x ′ ) e il ( y − y ′ ) + φ ∗ ( k, x ) φ ( k, x ′ ) e − il ( y − y ′ ) } + { ω Bl + l ω Bl φ B ( x ) φ B ( x ′ ) + | l | + l | l | φ ( x ) φ ( x ′ ) } ( e il ( y − y ′ ) + e − il ( y − y ′ ) ) (cid:21) . Using φ ∗ ( k, x ) = φ ( − k, x ), and changing the integration variable for theterms with φ ∗ ( k, x ) from k to − k , we find that all terms factorize intoterms with ω + l times e il ( y − y ′ ) + e − il ( y − y ′ ) . The factors l in ω + l cancel bysymmetric integration, and then also the terms with ω cancel. All termsare now proportional to e il ( y − y ′ ) , and integration over l yields the required ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 δ ǫ ( y − y ′ ). One is left with Z φ ( k, x ) φ ∗ ( k, x ′ ) dk π + φ B ( x ) φ B ( x ′ ) + φ ( x ) φ ( x ′ ) (1.24)which is indeed equal to δ ( x − x ′ ).For the { ψ − , ψ − } anticommutator there are two differences: instead of φ ( k, x ) one has s ( k, x ), and there are no zero modes. One finds Z s ( k, x ) s ∗ ( k, x ′ ) dk π + s B ( x ) s B ( x ′ ) . (1.25)This is again equal to δ ( x − x ′ ), as it is the completeness relation for L L † .One can also directly check this d . For the { ψ − , ψ + } anticommutator onefinds along the same lines { ψ + ( x, y, t ) , ψ − ( x ′ , y ′ , t ) } = Z dk π Z d ǫ l (2 π ) ǫ (1.26) " p ω kl − l ω kl n φ ( k, x )( − i ) s ∗ ( k, x ′ ) e il ( y − y ′ ) + is ( k, x ) φ ∗ ( k, x ′ ) e − il ( y − y ′ ) o + p ω Bl − l ω Bl n φ B ( x )( − i ) s B ( x ′ ) e il ( y − y ′ ) + is B ( x ) φ B ( x ′ ) e il ( y − y ′ ) o . Because there are now no terms linear in l which multiply the exponents e il ( y − y ′ ) , we can change the integration variables k and l to − k and − l inhalf of the terms, and, using φ ( k, x ) ∗ = φ ( − k, x ) and s ( k, x ) ∗ = s ( − k, x ),all terms cancel.The calculation of the one-loop mass of the susy kink is now simple. Wemust evaluate M (1) = Z dx Z d ǫ y h ˙ η ˙ η + i ψ T ˙ ψ i + Z d ǫ y h η∂ x η i (cid:12)(cid:12)(cid:12)(cid:12) x = ∞ x = −∞ + mλ ∆ µ (1.27)The first term gives the sum over zero-point energies Z dxd ǫ y h ˙ η ˙ η + i ψ T ˙ ψ i = V y Z dx Z dk π Z d ǫ l (2 π ) ǫ (1.28) × ω kl (cid:20) φ ∗ ( k, x ) φ ( k, x ) − ω kl + l ω kl φ ∗ ( k, x ) φ ( k, x ) − ω kl − l ω kl s ∗ ( k, x ) s ( k, x ) (cid:21) = V y Z dx Z dk π Z d ǫ l (2 π ) ǫ ω kl | φ ( k, x ) | − | s ( k, x ) | ) d Use Eqs. (9) and (10) of Ref. together with Eq. (1.19) above. ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer where V y is the volume R d ǫ y of the extra dimensions and where only contri-butions from the continuous spectrum have remained. There is no contribu-tion from the bound state because R dx ( ϕ B ( x ) − s B ( x )) vanishes (partiallyintegrate as in (1.31), there is no boundary term because ϕ B ( x ) falls off ex-ponentially fast). There is also no contribution from the zero mode becausethe corresponding integral R dkd ǫ l l / | l | is a scaleless integral, and scalelessintegrals vanish in dimensional regularization. Note that the terms propor-tional to a single power of l (arising from the √ ω + l and √ ω − l in (1.18))drop out because they are odd in the loop momentum l ; in the calcula-tion of Z these terms will give a crucial contribution. The total derivative R dx ∂∂x R d ǫ y h η∂ x η i does not contribute because η∂ x η = ∂ x ( ηη ), and h ηη i can only depend on x as x , in which case the derivative ∂ k yields x whichvanishes for large x . e (In 3+1 dimensions one can get a contribution becausethere the measure is 4 πr ).The expression in (1.28) is what in early approaches was believed to bezero, but which is actually infinite. Combining it with the counter termcontribution in (1.27), the total mass per volume V y becomes then M (1) = Z ∞−∞ dk π Z d ǫ l (2 π ) ǫ ω kl ρ ( k ) + mλ ∆ µ (1.29)where ∆ ρ ( k ) = Z ∞−∞ dx ( | φ ( k, x ) | − | s ( k, x ) | ) (1.30)is the difference of spectral densities of ψ + and ψ − . One can use an indextheorem to compute ∆ ρ ( k ), or one can directly calculate it, usingpartial integration, Z | s ( k, x ) | dx = Z [( ∂ x + U ′ ) φ ∗ ( k, x )][( ∂ x + U ′ ) φ ( k, x )] ω k dx = [ φ ∗ ( k, x )( ∂ x + U ′ ) φ ( k, x ) ω k ] (cid:12)(cid:12)(cid:12)(cid:12) x = ∞ x = −∞ + Z ∞−∞ φ ∗ ( k, x )( − ∂ x + U ′ )( ∂ x + U ′ ) φ ( k, x ) ω k dx = 2 mk + m + Z ∞−∞ dx | φ ( k, x ) | . (1.31)We used that since φ ∗ ( k, x ) ∂ x φ ( k, x ) = ik and U ′ → ± m as x → ±∞ , theterms with φ ∗ ( k, x ) ∂ x φ ( k, x ) cancel, while the terms with U ′ add. Note e Actually, h ηη i falls off even faster then 1 /x , namely exponentially fast. ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 that ∆ ρ ( k ) is nonvanishing, because | φ ( k, x ) | of the continuous spectrumin the second line of (1.31) does not vanish as x → ±∞ . With this resultfor the difference of spectral densities we obtain M (1) = m Z dkd ǫ l (2 π ) ǫ " − √ k + l + m k + m + 1 √ k + l + m = m Z dkd ǫ l (2 π ) ǫ (cid:20) − l ( k + m ) √ k + l + m (cid:21) . (1.32)Note that the extra dimensions, needed to maintain susy at the regularizedlevel, have produced a nonvanishing correction proportional to the squareof the momentum in the extra dimensions! Using the standard formula fordimensional regularization Z d n l ( l + M ) α = π n/ ( M ) n − α Γ( α − n )Γ( α ) , (1.33)we find for the l integral Z d ǫ l l ( l + M ) = Z d ǫ l ( l + M ) − − M Z d ǫ l ( l + M ) = π ǫ ( M ) ǫ + ( − ǫǫ + 1 ) Γ( − ǫ )Γ( ) , (1.34)where M = k + m . We are left with the k integral Z dk ( k + m ) ǫ − = − ǫ for ǫ → . (1.35)The factors ǫ and ǫ cancel, and the final result is M (1) = − m π . (1.36) Central Charge
The central charge is one of the generators of the susy algebra. To con-struct the latter, we begin with the Noether current for rigid susy. If oneintegrates its time-component over space, one obtains the susy charge Q ,but it is advantageous to postpone this integration and first evaluate thesusy variation of the susy current, δj µ = − i [ j µ , ¯ Qǫ ]. Extracting ǫ , andintegrating over space yields the { Q, Q } anticommutators.In order to regularize the quantum corrections, we first construct the { Q, Q } anticommutators in 2 + 1 dimensions, and then descend to (1 + ǫ ) +1 dimensions. In 1 + ǫ dimensions, the translation generators P y in the ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer direction of the ǫ extra dimensions are still present, and they are added tothe central charge Z x which one naively finds in 1 + 1 dimensions. As weshall show, − P y + Z x is the regularized central charge. In loop calculations P y will give a finite but nonvanishing contribution. For bosons in the loop,symmetric integration over l gives a vanishing result, but for fermions inthe loop, a factor l coming from the derivative ∂∂y in P y combines withanother factor l coming from the normalization factors √ ω + l and √ ω − l of the spinors ψ + and ψ − to give a factor l . Integration over l yields thena nonvanishing contribution, h P y i ∼ Z h ψ + ∂ y ψ + + ψ − ∂ y ψ − i ∼ Z l ω ∆ ρ ( k ) = 0 . (1.37)This result has the same form as one encounters in the calculation of thechiral triangle anomaly using dimensional regularization, namely a factor l in the numerator which yields a factor n as n →
0, and a divergent loopintegral which gives a factor n . We therefore refer to the term in (1.37) asan anomaly-like contribution, or, less precisely, as an anomaly. (There isno anomaly in the conservation of the central charge current, just as thereis no anomaly in the ordinary susy current, or the stress tensor, but thereis an anomaly in the conservation of the conformal current ). The finalresult for the one-loop contributions to the regularized central charge isequal to the one-loop mass correction, and thus BPS saturation continuesto hold at the quantum level.In earlier work, not enough attention was paid to careful regularization,and extra terms, such as the occurrence of P y , were missed. Any otherregularization scheme should also lead to BPS saturation if one is carefulenough. Of course, one should specify the same renormalization conditionsin the calculation of M (1) and Z (1) ; in our case this means that we againremove tadpoles by decomposing µ into µ + ∆ µ . Let us now show thedetails for the kink.The Noether current (in 2 + 1 dimensions) is given by j µ = − /∂ϕγ µ ψ − U γ µ ψ , and with the representation in (1.13) we find for the two spinorcomponents j = ( ˙ ϕ − ∂ y ϕ ) ψ + + ( ∂ x + U ) ψ − j − = ( ˙ ϕ + ∂ y ϕ ) ψ − + ( ∂ x − U ) ψ + (1.38)We can evaluate the variation of j ± either by transforming the fields in j ± under rigid susy transformations, or by evaluating the anticommutatorswith Q ± = R j ± ( x ′ , y ′ , t ) dx ′ dy ′ . We follow the latter approach. Using the ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 equal-time canonical (anti)commutation relations[ ˙ ϕ ( x ′ , y ′ , t ) , ϕ ( x, y, t )] = 1 i δ ( x ′ − x ) δ ( y ′ − y ) { ψ ± ( x ′ , y ′ , t ) , ψ ± ( x, y, t ) } = δ ( x ′ − x ) δ ( y ′ − y ) { ψ + ( x ′ , y ′ , t ) , ψ − ( x, y, t ) } = 0 (1.39)one finds straightforwardly, after partial integration of ∂∂x ′ and ∂∂y ′ deriva-tives, { Q + , j + } = ˙ ϕ − ϕ∂ y ϕ + ( ∂ y ϕ ) + ( ∂ x ϕ ) + 2 U ∂ x ϕ + U − iψ + ∂ y ψ + − iU ψ + ψ − + iψ + ∂ x ψ − + iψ − ∂ x ψ + (1.40)The right-hand side can be written in terms of the densities of the Hamil-tonian, translation generator P y , and naive central charge Z x as follows { Q + , j + } = 2 H − P y + 2 Z x H = ˙ ϕ + ( ∂ y ϕ ) + ( ∂ x ϕ ) + U − iU ψ + ψ − + iψ + ∂ x ψ − + iψ − ∂ x ψ + − iψ + ∂ y ψ + + iψ − ∂ y ψ − − P y = − ϕ∂ y ϕ − iψ + ∂ y ψ + − iψ − ∂ y ψ − Z x = 2 U ∂ x ϕ (1.41)The sum of the last term of 2 H and − P y cancels, but we have added theseterms to obtain the complete expressions for H and P y . One can check that H and P y generate the correct time- and space- translations of ϕ , ˙ ϕ , ψ + ,and ψ − . The other susy anticommutators are given by { Q − , j − } = 2 H + 2 P y − Z x { Q + , j − } = 2 P x + 2 Z y (1.42)(1.43)where 2 Z y = 2 U ∂ y ϕ .Integrating over x and y , and using two-component spinors we obtain12 { Q, Q } = − ( γ µ γ ) P µ + ( γ γ ) Z x − ( γ γ ) Z y (1.44)where P = H , and this clearly demonstrates that Z x − P y and P x + Z y arethe regulated versions of Z x and P x , respectively.The naive central charge Z x receives no quantum corrections. This wasobserved by several authors. To demonstrate this, we expand ϕ = ϕ K + η ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer and µ = µ + ∆ µ , and find to second order in η Z x = U ∂ x ϕ = ∂ x ( Z ϕ U ( ϕ ′ ) dϕ ′ )= U K ∂ x ϕ K + ∂ x ( U K η ) + 12 ∂ x ( U ′ K η ) − ∆ µ √ λ ∂ x ϕ K (1.45)The first term yields classical BPS saturation, since it is just minus the totalderivative in (1.5). Taking the expectation value in the kink ground state,the term linear in η vanishes, and the last two terms give, after integrationover x and y , h Z (1) x i = (cid:20) m h η ( x → ∞ ) i − µ µ √ λ (cid:21) V y . (1.46)We used that U ′ K → ± m and ϕ K → ± µ/ √ λ as x → ±∞ . Recalling that µ = m/ √
2, and ∆ µ = λ h η i in the trivial vacuum, we see that h Z (1) x i vanishes. The tadpole renormalization in the trivial vacuum, and thus alsofar away from the kink, cancels the contribution from the naive centralcharge.However, we get a nonvanishing correction from P y . The bosonic fluc-tuation do not contribute h P bosy i = Z h ˙ η∂ y η i dxd ǫ y ∼ Z ωl ω | φ ( k, x ) | dkd ǫ l = 0 (1.47)due to symmetric integration over l . But from the fermions we get a non-vanishing contribution h P fermy i = Z i h ψ + ∂ y ψ + + ψ − ∂ y ψ − i dxd ǫ y = 12 Z dk π d ǫ l (2 π ) ǫ ( l ( ω + l ) | ϕ ( k, x ) | ω + l ( ω − l ) | s ( k, x ) | ω ) dxd ǫ y = 12 Z dk π d ǫ l (2 π ) ǫ l ω ( | ϕ ( k, x ) | − | s ( k, x ) | ) dxd ǫ y (1.48)This is the same expression as we found for M (1) , hence BPS saturationholds.Repeating the same calculation for N = 2 susy ϕ kinks, one finds that BPS saturation holds without anomalous contributions from h P y i , be-cause in these models the extra fields lead to a complete cancellation of∆ ρ ( k ). However, in the 1+1-dimensional N = 2 CP model with so-called twisted mass term, ∆ ρ ( k ) is instead twice the amount found in ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 the minimally susy kink models. f The appearance of an anomalous contri-bution in the N = 2 twisted-mass CP model has to do with the factthat the N = 2 CP model provides an effective field theory for confinedmonopoles of 3+1-dimensional N = 2 SU(2) × U(1) gauge theories,which in turn are related by holomorphicity to ’t Hooft-Polyakov N = 2monopoles, and for the latter we shall indeed find anomalous contributionsto the central charge in what follows. We now discuss susy monopoles in 3+1 dimensions, and study how BPS sat-uration is realized when one-loop quantum corrections are included. Fromwhat has been learned from the kink, one might expect that if one de-fines proper renormalization conditions and takes again into account ananomaly-like contribution to the central charge, BPS saturation will fol-low. This turns out to be the case for the N = 2 monopole and leads us tocorrecting once again previous results in the literature g , but, surprisinglyenough, for the N = 4 monopole in the “finite” N = 4 super Yang-Millstheory, there are divergences left in boundary contributions, and these canonly be canceled, it seems, by introducing a new concept in the studyof solitons, which was not necessary before: infinite composite operatorrenormalization of the stress tensor and the central charge current. For the N = 2 model, all surface contributions, which are individually divergent,cancel nicely.Composite operator renormalization of the stress tensor and the centralcharge is no contradiction to the lore that “conserved currents don’t renor-malize”, because that applies only to internal currents, not to spacetimeones. The stress tensor appearing in the susy algebra can be written asthe sum of an improved stress tensor, which is traceless, and “improvementterms” corresponding to Rϕ terms in the action in curved space. Whilethe improved stress tensor turns out to be finite, the non-traceless partrenormalizes multiplicatively in the N = 4 model, and just happens to befinite as well in the N = 2 case. Thus this new feature of composite opera- f Another special feature of the N = 2 twisted-mass CP model is that the nonrenormal-ization of h Z x i involves fermionic boundary terms. In contrast to the susy kink, the few explicit calculations of one-loop corrections tothe N = 2 monopole were all agreeing on a null result in a minimal renormalizationscheme. ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer tor renormalization in the N = 4 model does not upset the BPS saturationof the N = 2 model that was obtained previously without it. One could ofcourse start with the improved currents at the classical level, but this wouldchange the traditional value of the classical mass of the ’t Hooft-Polyakovmonopole. The N = 2 monopole The action of the N = 2 super Yang-Mills model in 3+1 dimensions can beobtained in a simple way by applying dimensional reduction to the actionof minimal super Yang-Mills theory in 5+1 dimensions L = − F MN − ¯ λ Γ M D M λ ; λ = (cid:18) ψ (cid:19) = − F µν −
12 ( D µ P ) −
12 ( D µ S ) − g ( S × P ) − ¯ ψγ µ D µ ψ − g ¯ ψγ ( P × ψ ) − ig ¯ ψ ( S × ψ ) . (1.49)with ψ a 4-component complex spinor and ( S × P ) a = ǫ abc S b P c . Wedecomposed A aM into ( A aµ , P a , S a ) and used a particular representation ofthe Dirac matrices in 5 + 1 dimensions. In the topologically trivial sectorwe take S as the Higgs field with vev v (and S , S the would-be Goldstonefields). In the soliton sector, the energy density for a static configurationwith nonvanishing A aj and S a can be written as H = 14 (cid:0) F aij + ǫ ijk D k S a (cid:1) − ∂ k (cid:0) ǫ ijk F aij S a (cid:1) . (1.50)Thus the Bogomolnyi equation for a monopole residing in A j and S reads F aij + ǫ ijk D k S a = 0 . (1.51)The asymptotic behavior of A j and S is given by A aj = ǫ aij x j gr + . . . , F aij = − ǫ ijk x a x k gr + . . . ,S a = x a vr − x a gr + . . . , D k S a = x a x k gr + . . . , (1.52)where the suppressed subleading terms are exponentially decreasing forlarge radius r , and the classical mass of the monopole reads M cl = 4 πvg = 4 πmg (1.53)with m = gv . ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 The susy algebra can be obtained as before by varying the time com-ponent of the Noether current and afterwards integrating over space. Oneobtains h { Q α , Q † β } = δ αβ P − ( γ k γ ) αβ P k − ( γ γ ) αβ U + i ( γ ) αβ V (1.54)where U = Z d x ∂ k (cid:20) ǫ ijk F ij · S + F k · P (cid:21) ,V = Z d x ∂ k (cid:20) ǫ ijk F ij · P − F k · S (cid:21) , (1.55)and P µ = R T µ d x so that P = H >
0. To make contact with the usualform of the susy algebra for N -extended susy,12 { Q Ai , Q Bj } = ǫ AB Z ij , Z ij = − Z ij complex12 { Q Ai , ¯ Q ˙ Bj } = δ ij ( σ µ ) A ˙ B P µ , (1.56)note that our complex Q α can be written in terms of Majorana Q jα as Q α = ( Q + iQ ) α , and Q αj = ( Q Aj , ¯ Q ˙ Aj ) in terms of two-componentspinors. Then Z = − Z = − U + iV for the N = 2 model, whereas wealready see that for the N = 4 model to be discussed below there will be 6complex (12 real) central charges. Classically only U is nonvanishing, andBPS saturation holds for the above monopole solution. i For calculating quantum corrections, we use an “ R ξ ” gauge-fixing term L fix = − ξ (cid:0) D M ( A ) a M (cid:1) = − ξ ( D µ ( A ) a µ + gP × p + gS × s ) . (1.57)We have written “ R ξ ” in quotation marks because a genuine R ξ gauge-fixing term would have a factor ξ in front of gP × p and gS × s . The aboveform is advantageous to keep the SO(5,1) symmetry of the theory prior todimensional reduction. We shall set ξ = 1 in which case the kinetic termsin the fluctuation equations become diagonal (in a genuine R ξ gauge, thisis also true for ξ = 1). h For { Q α , Q β } one finds the integral of a total derivative of a bilinear in fermions, R ∂ j ( ψ T Cγ ψ ) d x ( γ j C − ) αβ . Since h ψψ i vanishes, we shall omit this term from thealgebra. i Using a suitable representation of the Dirac matrices, the right-hand side of (1.54) takeson the form „ P + σ k P k iU + V − iU + V P − σ k P k « . For vanishing P k one obtains P ≥ U + V ,hence in general, M ≥ U + V ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer
The field equations for the fluctuations a m = { a i , s } , i = 1 , ,
3, read (cid:16) ( ∂ − ∂ − D ℓ ) δ mn − gF mn × (cid:17) a n = 0 , (1.58)where D abℓ = ( D abi , igS ab ) with D abi = ∂ i δ ab + ǫ acb A cµ and S ab = ǫ acb S c .They can be written in spinor notation as( ¯ / D / D + ∂ − ∂ )¯ /a = 0; ¯ / D / D = D m + 12 ¯ σ mn gF mn , (1.59)where ¯ /a = ¯ σ m a m , ¯ σ mn = (¯ σ m σ n − ¯ σ n σ m ) with ¯ σ m = ( ~σ, − i ) and σ m = ( ~σ, i ) in the 4-dimensional Euclidean space labeled by the index m . Furthermore, ( / D ¯ / D + ∂ − ∂ ) /q = 0; / D ¯ / D = D m (1.60)for the remaining quartet of bosonic fields q m = ( a , p, b, c ), where b, c areFaddeev-Popov ghost fields.For the spinors we find / D ψ + = ( ∂ − ∂ ) ψ − , ¯ / D ψ − = ( ∂ + ∂ ) ψ + , (1.61)where / D ab = σ m D abm = σ k D abk + igS ab , ¯ / D ab = ¯ σ m D abm = σ k D abk − igS ab . (1.62)Iterating (1.61) we have¯ / D / D ψ + = ( ∂ − ∂ ) ψ + , / D ¯ / D ψ − = ( ∂ − ∂ ) ψ − , (1.63)so the two columns of ¯ /a have the same field equations as ψ + , and the twocolumns of /q have the same field equations as ψ − , the analogous situationas we found for the susy kink.One can now construct the gravitational stress tensor T µν and considerthe terms in the Hamiltonian density T which are quadratic in quantumfields. For the bosons, there are terms of the form ∂a∂a and terms of theform a∂ a . Partially integrating the former, we can use the field equationsfor the fluctuations to obtain the following result M − loop = Z d x h a ∂ a − a j ∂ a j − p∂ p − s∂ s − b∂ c − ( ∂ b ) c + i ψ †↔ ∂ ψ i + lim r →∞
14 4 πr ∂∂r h a + a j + p + s + 2 bc − a j i , (1.64)where we used that ∂ j ∂ k h a j a k i = ∂ k h a j i and h a j a i = 0 for large r . ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 The bulk contributions give the sum over zero-point energies of all quan-tum fields. Fermions have the mode expansion ψ = (cid:18) ψ + iψ − (cid:19) = Z d ǫ ℓ (2 π ) ǫ/ Z d k (2 π ) / √ ω X(cid:26) b kl e − i ( ωt − ℓy ) (cid:18) √ ω + ℓ χ + k −√ ω − ℓ χ − k (cid:19) + d † kl e i ( ωt − ℓy ) (cid:18) √ ω + ℓ χ + k √ ω − ℓ χ − k (cid:19)(cid:27) + bound states + zero modes , (1.65)where the sum refers to the two possible polarizations of the χ k ’s. On theother hand, the mode expansion of the bosonic fields a j and s (which wecombined into /a ) only involves χ + k and the one of the quartet /q only χ − k .This leads to M (1)bulk = V y Z d x Z d k d ǫ ℓ (2 π ) ǫ ω N + | χ + k | ( x ) + N − | χ − k | ( x )) (1.66)where N + = 4 − a m and ψ + , and N − = 1 + 1 − − − q m = ( a , p, b, c ) and ψ − . The result thus involves only a difference ofspectral densities which can be evaluated by an index theorem ∆ ρ ( k ) = Z d x ( | χ + k | ( x ) − | χ − k | ( x )) = − πmk ( k + m ) . (1.67)On the other hand, all surface contributions (in the present N = 2 case)cancel, M (1)surface = lim r →∞
14 4 πr ∂∂r h a + a j + p + s + 2 bc − a j i = ( − − −
2) lim r →∞ πr ∂∂r h s i = 0 , (1.68)where we have used that the propagators of all bosonic fields become thesame for large r , since only terms of order 1 /r can contribute, whereas F aµν falls off as 1 /r . (The contribution of h a i is minus h s i because ofthe metric η µν in the canonical commutation relations of the creating andannihilation operators.) Hence, M (1)bulk is the complete, but still unrenor-malized, one-loop result.The momentum integral that we are left with to evaluate upon insertionof (1.67) into (1.66) is UV divergent, and the required counter term ∆ M comes from the renormalization of M cl . = 4 πv /g . We clearly need Z g and Z v . In the background field formalism which we have been using,one has the well-known relation Z g = Z − / A , so we could first determine Z A by requiring that all loops with two external background fields A µ arecancelled at zero external momentum (the quantum fields in these loops ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer are all massive, so there are no IR problems). We did this in Ref. evenat arbitary ξ , but one can get Z g also from the known one-loop formula ofthe β -function Z g = 1 − g { − n Maj . ferm . − n real scalars } C (SU(2)) I
2= 1 − g I for N = 2; but Z g = 1 for N = 4 , (1.69)where I ≡ Z d ǫ k (2 π ) ǫ − i ( k + m ) = Z d ǫ k E (2 π ) ǫ k E + m ) = − π ǫ + O ( ǫ ) . (1.70)The value of Z v is equal to Z S , because constant v ’s are a special caseof arbitrary background fields S (more precisely, in the trivial sector onlythe combination v + S occurs). At ξ = 1, the value of Z S is equal to Z A because all relevant background-field vertices are contained in a M D m a M which is SO(5,1) invariant. (At ξ = 1, Z S becomes ξ -dependent, while Z A is ξ -independent, because it is given by the β -function.) Since Z g Z / S = 1,the mass m = gv does not renormalize (at ξ = 1), and thus∆ M = 4 πmZ g g − πmg = 4 πmg g I = 16 πmI. (1.71)The mass correction to the N = 2 monopole is finally given by M (1) = 2 Z d x Z d k d ǫ ℓ (2 π ) ǫ √ k + ℓ + m ρ ( k ) + ∆ M = − mπ Γ( − − ǫ )(2 π ) ǫ Γ( − ) Z ∞ dk ( k + m ) − + ǫ + 16 πmI = (cid:18) −
11 + ǫ + 1 (cid:19) πmI = − mπ + O ( ǫ ) . (1.72)The one-loop corrections to the original expression in (1.55) for thecentral charge U of the N = 2 monopole cancel completely j against thecounterterms due to ordinary renormalization, but the translation opera-tor P y in the extra ǫ dimensions gives again a nonvanishing “anomalous”contribution which exactly matches M (1) , in complete analogy to the case j The first graph above (1.88) yields a divergence − g IU , but wave function renormal-ization of S and A µ in U yields a counterterm 4 g IU . This cancellation was worked outfirst in Ref., but it only works for N = 2, while N = 4 involves new issues thatwe shall discuss below. ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 of the susy kink (see Eq. 1.48) k , U (1) = P y = Z d k d ǫ ℓ (2 π ) ǫ ℓ √ k + ℓ + m ∆ ρ ( k ) = − mπ + O ( ǫ ) . (1.73)Clearly, BPS saturation holds for the N = 2 monopole at the one-looplevel. However, the finite nonvanishing correction to both the mass andthe central charge had been missed in all the literature preceding Ref., although closer inspection reveals that the commonly accepted null resultwas in conflict with the low-energy effective action of N = 2 super Yang-Mills theory obtained some time ago by Seiberg and Witten. The N = 4 monopole We now turn to the monopole in N = 4 super Yang-Mills theory in 3+1dimensions, where the naive expectation of vanishing one-loop correctionsto mass and central charge in the end turns out to be correct. However, howthis comes about is highly nontrivial, and in several ways the properties ofthe N = 4 case are opposite to the N = 2 case, with dramatic consequences.We begin by following the same steps as in the N = 2 case. The actionof N = 4 super Yang-Mills theory in 3+1 dimensions is most easily obtainedby applying dimensional reduction to the N = 1 super Yang-Mills theoryin 9+1 dimensions, yielding L = − F MN −
12 ¯ λ Γ M D M λ (1.74)= − F µν −
12 ( D µ S j ) −
12 ( D µ P j ) −
12 ¯ λ I /Dλ I + interactions . where we decomposed A aM into ( A aµ , S a j , P a j ), with 3 adjoint scalars S j and3 pseudoscalars P j , j = 1 , ,
3, instead of only one of each in the N = 2 case.The 16-component adjoint Majorana-Weyl spinor λ a has been decomposedinto four 4-component Majorana spinors λ aI with I = 1 , . . . ,
4, with afactor in front of their action because of the Majorana property. The k However, in contrast to the case of the susy kink, if one combines the integral with ∆ ρ in the mass correction (1.72) with the integral representation of the counter term ∆ M ,one does not obtain a factor ℓ in the numerator as in (1.73). ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer susy algebra reads12 { Q αI , Q βJ } = δ IJ ( γ µ C − ) αβ P µ + i ( γ C − ) αβ ( α j ) IJ Z d x U j − ( C − ) αβ ( β j ) IJ Z d x V j +( C − ) αβ ( α j ) IJ Z d x ˜ V j + i ( γ C − ) αβ ( β j ) IJ Z d x ˜ U j − Z d x (¯ λ Γ Γ P Q D R λ )(Γ P QR C − ) αβ , (1.75)where the last term is on-shell a total derivative l of the form ∂ ρ (¯ λ Γ RS λ ).Since the expectation value of this term contains the spinor trace tr(Γ RS /k ),which vanishes, we drop this term from now on. In (1.75) we have used aparticular representation of the 32 ×
32 Dirac matrices Γ M in terms of the4 × α j and β j which are proportionalto the matrices η IJ j and ¯ η IJ j which ’t Hooft introduced for the constructionof instantons. The α j and β j respresent the 6 generators of SO(4): totallyantisymmetric 4 × α ) oranti-self-dual ( β ). The indices I and J are raised and lowered with theEuclidean metric δ IJ and δ IJ , and finally α, β = 1 , . . . , U j = ∂ i ( S a j ǫ ijk F ajk ) , ˜ U j = ∂ i ( P a j F a i ) V j = ∂ i ( P a j ǫ ijk F ajk ) , ˜ V j = ∂ i ( S a j F a i ) . (1.76)In the N = 2 case, Eqs. (1.55), only the sums U + ˜ U and V + ˜ V appeared,but here they split into parts with different tensor structures, half of themwith α matrices, the other half with β ’s.We set S a = v for adjoint color index a = 3 in the topologically trivialsector, and locate the monopole inside the fields A aj and S a . For quantumcalculations we use again the background field formalism as in the N = 2case above, which now gives Z v = Z S = Z A = Z g = 1, since the β -functionfor the N = 4 model vanishes.The gravitational stress tensor yields the Hamiltonian density, which wewrite again after use of the linearized field equations for fluctuations as time l Use Γ RS Γ N D N λ = 0 = Γ RSN D N λ + Γ R D S λ − Γ S D R λ to write all terms as ¯ λ Γ RST λ .Then use ¯ λ Γ RST D N λ = ∂ N (¯ λ Γ RST λ ). ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 derivatives giving the sum over zero-point energies, and surface terms M − loop = Z d x h a ∂ a − a j ∂ a j − p j ∂ p j − s j ∂ s j − b∂ c − ( ∂ b ) c + i λ I ) T ∂ λ I i + lim r →∞
14 4 πr ∂∂r h a + a j + p j + s j + 2 bc − a j i , (1.77)where the only differences with the N = 2 model are that there are nowthree times as many scalar and pseudoscalar fields and we have four real 4-component spinors instead one one complex 4-component spinor. However,the consequences could not have been more severe. The sum of the zero-point energies (the bulk contribution in (1.77)) vanishes for the N = 4case: the fields associated with the field operator ¯ / D / D are a j , s , λ I + andyield in eq. (1.66) N + = 3 + 1 − − / D ¯ / D are a , s , s , p j , b, c , and λ I − yielding N − =1 + 1 + 1 + 3 − − − − − − −
2. On theother hand, for N = 4 the boundary terms no longer vanish, since insteadof − − − − − − M (1)surface = lim r →∞ πr ∂∂r h s a ) i . (1.78)For large r , the difference between the operators ¯ / D / D and / D ¯ / D is due to F mn ,see (1.59) and (1.60), which falls off like 1 /r , so the bosonic propagatorsin the background covariant ξ = 1 gauge have a common asymptotic form, h a aM ( x ) a bN ( y ) i ≃ η MN G ab ( x, y ) , h b a ( x ) c b ( y ) i ≃ − G ab ( x, y ) , (1.79)with G ab ( x, y ) = h x | − i − (cid:3) + m − mr ( δ ab − ˆ x a ˆ x b ) + i (cid:3) ˆ x a ˆ x b | y i . (1.80)Inserting complete sets of momentum eigenstates, the same procedureas used to calculate anomalies from index theorems yields for the r -dependent part of G aa ( x, x ) h s a ( x ) s a ( x ) i ≃ Z d ǫ k (2 π ) ǫ − i ( k + m ) + 2 ik µ ∂ µ − ∂ µ − mr → mr I (1.81)with I given in (1.70). (The overall factor 2 is due to tracing δ ab − ˆ x a ˆ x b .) ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer
Hence, we have arrived at a divergent result for the mass of the N = 4monopole, M − loop = M (1)surface = − πmI. (1.82)Ordinary renormalization, namely renormalization of the parameters in theaction, is of no help, since, as we have seen, all Z factors which helped tomake the N = 2 result finite, are unity in the N = 4 case.The solution is extra-ordinary renormalization, namely renormalizationof the stress tensor density, and also of the central charge density (and, infact, all currents of the corresponding susy multiplet) as composite opera-tors. In the literature it has been shown that the improved stress tensor does not renormalize, which we extend to the statement that noneof the improved currents in the susy multiplet renormalize. However,the currents in the susy algebra displayed above are nonimproved currents,and to construct improved currents one must add improvement terms to theunimproved currents. In order to find those for both the stress tensor andthe central charges, we go back one step and begin with our unimprovedNoether susy currents j µ , to which we add the improvement terms ∆ j µ imp , j µ imp = j µ + ∆ j µ imp = 12 Γ RS F aRS Γ µ λ a −
23 Γ µν ∂ ν ( A a J Γ J λ a ) (1.83)where we use a 10-dimensional notation in which A J , J = 5 , . . .
10, com-prises all scalars and pseudoscalars in the model. The Γ-matrices are32 × ν runs only from 0 to 3. Both j µ and j µ imp are on-shell conserved. (Use the Bianchi identity Γ RS J A J × F RS = 0).In addition, the improved (ordinary, not conformal) susy currents areon-shell gamma-traceless, Γ µ j µ imp = 0 . This can be verified by usingΓ µ Γ ρσ Γ µ = 0 and F ρ J = D ρ A J . One finds Γ µ j µ imp = 2Γ ρ J ( D ρ A J ) · λ + 2Γ J K ( A J × A K ) · λ − ν ∂ ν ( A J Γ J · λ ) which indeed vanishes on-shell,where Γ ρ D ρ λ = − Γ K A K × λ . From the susy variation of ∆ j µ imp we findthe improvement terms in T µν and the central charges. Switching to 3+1-dimensional notation, we find ∆ T impr00 = − ∂ j ( A J A J ) (1.84)∆ U impr3 = − (cid:20) U + Z i ∂ i ( ǫ ijk ¯ λα γ jk λ ) d x (cid:21) . (1.85)It is important to note that we do not start with improved Noether cur-rents at the classical level. Indeed, the standard result for the classical value ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 of the mass of a monopole is only obtained when the unimproved stress ten-sor is used, and also the classical value of the improved and unimprovedcentral charge differ. m However, even when starting with unimproved cur-rents, we have to expect improvement terms as counterterms, since we canwrite the unimproved currents as j µ = j impr µ − ∆ j impr µ and we expect the im-proved part to be finite, and the improvement terms ∆ j impr µ to renormalizemultiplicatively. Denoting the common Z factor for all improvement termsin the susy current, the stress tensor and the central charges by Z impr , thecomposite operator counterterms to mass and central charge will be givenby ∆ T = − ( Z impr − T impr00 , ∆ U = − ( Z impr − U impr , (1.86)where the overall minus sign is due to having written j µ = j impr µ − ∆ j impr µ .We shall now show by a detailed calculation that a single factor Z impr removes the divergences in mass and central charge and thus ensures BPSsaturation, but it would be interesting to check by an explicit (though labo-rious) calculation that there are no more composite operator countertermsin the renormalization of the full susy algebra, and to find the superspaceformulation from which this follows.To determine Z impr we decompose U = U = R ∂ i ( S a ǫ ijk F ajk ) d x as U impr − ∆ U impr where U impr = 23 (cid:20) U − Z i ∂ i ( ǫ ijk ¯ λα γ jk λ ) d x (cid:21) (1.87)∆ U impr = − (cid:20) U + Z i ∂ i ( ǫ ijk ¯ λα γ jk λ ) d x (cid:21) ≡ −
13 ( U + F ) . For the one-loop composite operator renormalization of U we thus haveto consider four classes of proper diagrams: graphs with the bosonic U or the fermionic F inserted in proper one-loop diagrams with externalbosonic background fields or external fermionic fields. The number ofpotentially divergent graphs with bosonic-bosonic structure is 31, withfermionic-fermionic struture it is 2, while there is one graph with U in-sertion and external fermions, and 3 graphs with F insertion and externalbosonic fields. Of the set of 31 graphs, some vanish by themselves, somevanish in the background covariant R ξ =1 gauge, some subsets of diagramsare finite, and if we only consider graphs with one external field S and oneexternal gauge field A µ , only one graph survives! m This is clear from the fact that ∆ U impr involves the bosonic term U . ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer
The set of one-loop graphs to be evaluated and their divergent contri-butions (obtained in the topologically trivial sector) is as follows (denotinggauge fields A µ by wavy lines, the scalar field S by a dashed line, andfermions λ by straight lines), U U F F F | {z } − g U I − g F I O ( ξ − − g U I to which one needs to add a term Z λ F = − g F I + O ( ξ − , (1.88)since in N = 4 theory there is nontrivial ordinary wave function renormal-ization Z λ for fermions, whereas bosonic fields do not obtain ordinary wavefunction renormalization in the gauge ξ = 1.This leads to the ( ξ -independent) composite operator counterterms,∆ U c.o.r. = 4 g ( U + F ) I, ∆ F c.o.r. = 8 g ( U + F ) I. (1.89)From this it follows that Z impr = 1 + 12 g I , while the improved part of U , U impr = U − F , is indeed finite.In the monopole background, the classical values of − U and M areequal and given by 4 πm/g without renormalization of m or g . Since U = U impr − ∆ U impr , composite operator renormalization produces thecontribution 12 g I × (+ ) × πm/g = 16 πmI to − U as well as M , whichindeed cancels the divergence obtained above in (1.82). Thus we have M (1) = − U (1) ( = 0) and BPS saturation is verified.Having found that composite operator renormalization is needed for the N = 4 case, we should of course go back to the N = 2 calculation (andalso all other one-loop calculations for solitons performed so far), and makesure that in these cases there is no new contribution that could upset theBPS saturation obtained previously. For the kink in 1+1 dimensions, it iseasy to check that there are no divergent one-loop diagrams for compositeoperator renormalization of the stress tensor (but it turns out that there isin fact a need for composite operator renormalization in the local energydensity of 3+1 dimensional kink domain walls, which does however not ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 contribute to the integrated total energy ). For the vortex in 2+1 dimen-sions, all currents are finite in dimensional regularizaton. But for the N = 2 monopole, no composite operator renormalization is needed for tworeasons: (1) the improvement terms of the central charge ∆ U N=2 , impr areproportional to the central charge U N=2 itself and (2) the central charge U N=2 is a finite operator due to ordinary renormalization. To prove thesestatements consider the central charge density for the N = 2 model U N =2 = 12 ǫ ijk ∂ i ( S a F ajk ) + ∂ i ( P a F ai ) = U − ˜ U . (1.90)The improvement terms in the susy current are given by∆ j µN =2 , impr = −
23 Γ µν ∂ ν [( P Γ + S Γ ) λ ]= − γ µν ∂ ν [( P γ + iS ) λ ] . (1.91)The susy variation of ∆ j µN =2 , impr yields ∆ U N =2 , impr . Using δP = δA =¯ λ Γ ǫ = ¯ λγ ǫ , δS = δA = ¯ λ Γ ǫ = i ¯ λǫ and δλ = Γ P Q F P Q ǫ we find∆ j N =2 , impr = γ j ∂ j [ γ λ (¯ λγ ǫ ) − λ (¯ λǫ ) + ( P γ + iS ) 12 γ ρσ F ρσ ǫ ]= γ j ∂ j [ −
14 (¯ λ O I λ )( γ O I γ − O I )+ P γ γ k F k + iS γ kl F kl ] ǫ ∼ γ ∂ j ( P F j − ǫ jkl SF kl ) ǫ, (1.92)because the terms with O I ∼ γ kl cancel in the first line. Thus ∆ U N=2 , impr is proportional to U N=2 , so there are no fermionic terms in ∆ U N=2 , impr .Both U and ˜ U produce divergence proportional to F , but their sum can-cels. Since ordinary renormalization already gave counterterms which make U N =2 finite, we do not need composite operator counterterms, so we canset Z impr = 1, leaving the results for the N = 2 monopole unchanged. The one-loop corrections to the mass and central charge of kinks, vortices(not discussed here, but treated in ), and monopoles in N = 2 and N = 4 super Yang-Mills theory satisfy the BPS bound. To obtain thisresult, we needed to carefully regularize the susy field theories, which inour choice of regularization scheme meant that we needed to take extra di-mensions into account. In these extra dimensions the modes of bosonic and ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer fermionic quantum fields had extra momenta, and the square of these extramomenta gave an extra contribution to the 1-loop central charge. In addi-tion we found that boundary terms contributed to the mass of the N = 4monopole. These boundary terms were divergent, and we needed multi-plicative composite operator renormalization of the improvement terms inthe stress tensor to obtain a finite quantum mass. The same compositeoperator renormalization was needed for the central charge. For the N = 2monopole, all boundary terms canceled, and there was no composite op-erator renormalization, but the sum over zero point energies in the bulkwas divergent, and standard renormalization counter terms cancelled thesedivergences. For the susy kink, boundary terms could not even appearbecause the classical kink solution falls off exponentially fast.We found that the 1-loop corrections to the susy kink and N = 2monopole are nonzero. In the literature it was assumed, or proofs were pro-posed, that these corrections vanish. Our results for the N = 2 monopoleagree with results based on holomorphicity by Seiberg-Witten, which alsorequire a nonvanishing correction to the mass and central charge (althoughthis was noticed only subsequently ). This raises the question whetherour results are consistent with Zumino’s general proof that the sum overzero-point energies must vanish in any susy theory. This proof is based onpath integrals and does not take into account regularization. Hence, it isnot clear that there is a disagreement. There is a way of understanding ournonvanishing result. If one encloses the kink in a large box, and imposessusy boundary conditions, one finds a spurious boundary energy which onemust subtract to obtain the true mass of the susy soliton. Dimensionalregularization by itself subtracts this spurious boundary energy.A superspace treatment of solitons would be useful, but we have foundproblems in gauge theories with a superspace R ξ gauge if solitons arepresent. A superspace treatment of the anomalies in the superconfor-mal currents of the kink has been given in collaboration with Fujikawa, see also Shizuya.
Our methods could perhaps be applied to D-branes, at least the D-branes that are solitons. Also extension to finite temperature is interesting;in fact, we have found new surprises for kink domain walls at finite tem-perature. ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 Acknowledgements
We thank Yu-tin Huang for assistance in writing up this review and ac-knowledge financial support from the Austrian Science Foundation FWF,project nos. J2660-N16 and P19958.
References
1. R. Rajaraman,
Solitons and Instantons (North-Holland, Amsterdam, 1982).2. C. Rebbi and G. Soliani (eds.),
Solitons and particles (World Scientific, Sin-gapore, 1984).3. B. Zumino, “Supersymmetry and the Vacuum”,
Nucl. Phys.
B89 , 535 (1975).4. A. D’Adda, R. Horsley and P. Di Vecchia, “Supersymmetric magneticmonopoles and dyons”,
Phys. Lett.
B76 , 298 (1978).5. H. B. Nielsen and P. Olesen, “Vortex-line models for dual strings”,
Nucl.Phys.
B61 , 45 (1973).6. V. L. Ginzburg and L. D. Landau, “On the Theory of superconductivity”,
Zh. Eksp. Teor. Fiz. , 1064 (1950).7. A. A. Abrikosov, “On the Magnetic properties of superconductors of thesecond group”, Sov. Phys. JETP , 1174 (1957).8. G. ’t Hooft, “Magnetic monopoles in unified gauge theories”, Nucl. Phys.
B79 , 276 (1974).9. A. M. Polyakov, “Particle spectrum in quantum field theory”,
JETP Lett. , 194 (1974).10. R. F. Dashen, B. Hasslacher and A. Neveu, “The particle spectrum in modelfield theories from semiclassical functional integral techniques”, Phys. Rev.
D11 , 3424 (1975).11. L. D. Faddeev and V. E. Korepin, “Quantum theory of solitons”,
Phys. Rept. , 1 (1978).12. J. L. Gervais and B. Sakita, “Extended particles in quantum field theories”, Phys. Rev.
D11 , 2943 (1975).13. T. D. Lee and Y. Pang, “Nontopological solitons”,
Phys. Rept. , 251(1992).14. B. Julia and A. Zee, “Poles with Both Magnetic and Electric Charges inNonabelian Gauge Theory”,
Phys. Rev.
D11 , 2227 (1975).15. M. K. Prasad and C. M. Sommerfield, “An exact classical solution for the ’tHooft monopole and the Julia-Zee dyon”,
Phys. Rev. Lett. , 760 (1975).16. E. B. Bogomolnyi, “Stability of classical solutions”, Sov. J. Nucl. Phys. ,449 (1976).17. E. Witten and D. Olive, “Supersymmetry algebras that include topologicalcharges”, Phys. Lett.
B78 , 97 (1978).18. H. Nastase, M. Stephanov, P. van Nieuwenhuizen and A. Rebhan, “Topolog-ical boundary conditions, the BPS bound, and elimination of ambiguities inthe quantum mass of solitons”,
Nucl. Phys.
B542 , 471 (1999). ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer
19. N. Graham and R. L. Jaffe, “Energy, central charge, and the BPS bound for1+1 dimensional supersymmetric solitons”,
Nucl. Phys.
B544 , 432 (1999).20. M. A. Shifman, A. I. Vainshtein and M. B. Voloshin, “Anomaly and quantumcorrections to solitons in two-dimensional theories with minimal supersym-metry”,
Phys. Rev.
D59 , 045016 (1999).21. A. Rebhan, P. van Nieuwenhuizen and R. Wimmer, “The anomaly in thecentral charge of the supersymmetric kink from dimensional regularizationand reduction”,
Nucl. Phys.
B648 , 174 (2003).22. A. Rebhan, P. van Nieuwenhuizen and R. Wimmer, “A new anomalous con-tribution to the central charge of the N = 2 monopole”, Phys. Lett.
B594 ,234 (2004).23. M. F. Atiyah, V. K. Patodi and I. M. Singer, “Spectral asymmetry andRiemannian Geometry 1”,
Math. Proc. Cambridge Phil. Soc. , 43 (1975).24. C. Callias, “Index theorems on open spaces”, Commun. Math. Phys. , 213(1978).25. E. J. Weinberg, “Parameter counting for multimonopole solutions”, Phys.Rev.
D20 , 936 (1979).26. E. J. Weinberg, “Index calculations for the fermion–vortex system”,
Phys.Rev.
D24 , 2669 (1981).27. A. Rebhan, P. van Nieuwenhuizen and R. Wimmer, “One-loop surface ten-sions of (supersymmetric) kink domain walls from dimensional regulariza-tion”,
New J. Phys. , 31 (2002).28. M. L¨uscher, “Dimensional regularization in the presence of large backgroundfields”, Ann. Phys. , 359 (1982).29. A. Parnachev and L. G. Yaffe, “One-loop quantum energy densities of domainwall field configurations”,
Phys. Rev.
D62 , 105034 (2000).30. C. Montonen and D. I. Olive, “Magnetic Monopoles as Gauge Particles?”,
Phys. Lett.
B72 , 117 (1977).31. E. Witten, “Dyons of charge eθ/ π ”, Phys. Lett.
B86 , 283 (1979).32. J. F. Schonfeld, “Soliton masses in supersymmetric theories”,
Nucl. Phys.
B161 , 125 (1979).33. R. K. Kaul and R. Rajaraman, “Soliton energies in supersymmetric theories”,
Phys. Lett.
B131 , 357 (1983).34. H. Yamagishi, “Soliton mass distributions in (1+1)-dimensional supersym-metric theories”,
Phys. Lett.
B147 , 425 (1984).35. C. Imbimbo and S. Mukhi, “Index theorems and supersymmetry in the soli-ton sector”,
Nucl. Phys.
B247 , 471 (1984).36. A. Uchiyama, “Nonconservation of supercharges and extra mass correctionfor supersymmetric solitons in (1+1) dimensions”,
Prog. Theor. Phys. ,1214 (1986).37. J. Casahorr´an, “Nonzero quantum contribution to the soliton mass in theSUSY sine-Gordon model”, J. Phys.
A22 , L413 (1989).38. L. J. Boya and J. Casahorr´an, “Kinks and solitons in SUSY models”,
J. Phys.
A23 , 1645 (1990).39. A. Rebhan and P. van Nieuwenhuizen, “No saturation of the quantum Bo-gomolnyi bound by two-dimensional N = 1 supersymmetric solitons”, Nucl. ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 Phys.
B508 , 449 (1997).40. A. Litvintsev and P. van Nieuwenhuizen, “Once more on the BPS bound forthe susy kink”, hep-th/0010051.41. A. S. Goldhaber, A. Litvintsev and P. van Nieuwenhuizen, “Mode regular-ization of the susy sphaleron and kink: Zero modes and discrete gauge sym-metry”,
Phys. Rev.
D64 , 045013 (2001).42. A. S. Goldhaber, A. Litvintsev and P. van Nieuwenhuizen, “Local Casimirenergy for solitons”,
Phys. Rev.
D67 , 105021 (2003).43. A. S. Goldhaber, A. Rebhan, P. van Nieuwenhuizen and R. Wimmer, “Clashof discrete symmetries for the supersymmetric kink on a circle”,
Phys. Rev.
D66 , 085010 (2002).44. M. Bordag, A. S. Goldhaber, P. van Nieuwenhuizen and D. Vassilevich, “Heatkernels and zeta-function regularization for the mass of the SUSY kink”,
Phys. Rev.
D66 , 125014 (2002).45. A. Rebhan, P. van Nieuwenhuizen and R. Wimmer, “Quantum mass andcentral charge of supersymmetric monopoles: Anomalies, current renormal-ization, and surface terms”,
JHEP , 056 (2006).46. N. Dorey, “The BPS spectra of two-dimensional supersymmetric gauge the-ories with twisted mass terms”,
JHEP , 005 (1998).47. C. Mayrhofer, A. Rebhan, P. van Nieuwenhuizen and R. Wimmer, “Pertur-bative Quantum Corrections to the Supersymmetric CP Kink with TwistedMass”,
JHEP , 069 (2007).48. M. Shifman, A. Vainshtein and R. Zwicky, “Central charge anomalies in 2Dsigma models with twisted mass”,
J. Phys.
A39 , 13005 (2006).49. A. Hanany and D. Tong, “Vortices, instantons and branes”,
JHEP , 037(2003).50. R. Auzzi et al. , “Nonabelian superconductors: Vortices and confinement inN = 2 SQCD”, Nucl. Phys.
B673 , 187 (2003).51. R. Auzzi, S. Bolognesi, J. Evslin and K. Konishi, “Nonabelian monopolesand the vortices that confine them”,
Nucl. Phys.
B686 , 119 (2004).52. M. Shifman and A. Yung, “Localization of non-abelian gauge fields on domainwalls at weak coupling (D-brane prototypes II)”,
Phys. Rev.
D70 , 025013(2004).53. M. Shifman and A. Yung, “Non-abelian string junctions as confinedmonopoles”,
Phys. Rev.
D70 , 045004 (2004).54. R. K. Kaul, “Monopole mass in supersymmetric gauge theories”,
Phys. Lett.
B143 , 427 (1984).55. C. Imbimbo and S. Mukhi, “Index theorems and supersymmetry in the soli-ton sector. 2. Magnetic monopoles in (3+1)-dimensions”,
Nucl. Phys.
B249 ,143 (1985).56. A. Rebhan, R. Sch¨ofbeck, P. van Nieuwenhuizen and R. Wimmer, “BPSsaturation of the N = 4 monopole by infinite composite-operator renormal-ization”,
Phys. Lett.
B632 , 145 (2006).57. N. Seiberg and E. Witten, “Electric-magnetic duality, monopole condensa-tion, and confinement in N=2 supersymmetric Yang-Mills theory”,
Nucl.Phys.
B426 , 19 (1994). ctober 31, 2018 17:15 World Scientific Review Volume - 9in x 6in YITP-SB-09-02 A. Rebhan, P. van Nieuwenhuizen, R. Wimmer
58. N. Seiberg and E. Witten, “Monopoles, duality and chiral symmetry breakingin N=2 supersymmetric QCD”,
Nucl. Phys.
B431 , 484 (1994).59. C. G. Callan, Jr., S. Coleman and R. Jackiw, “A new improved energy-momentum tensor”,
Ann. Phys. , 42 (1970).60. D. Z. Freedman, I. J. Muzinich and E. J. Weinberg, “On the energy-momentum tensor in gauge field theories”, Ann. Phys. , 95 (1974).61. D. Z. Freedman and E. J. Weinberg, “The energy-momentum tensor in scalarand gauge field theories”, Ann. Phys. , 354 (1974).62. J. C. Collins, “The energy-momentum tensor revisited”, Phys. Rev.
D14 ,1965 (1976).63. L. S. Brown, “Dimensional regularization of composite operators in scalarfield theory”,
Ann. Phys. , 135 (1980).64. T. Hagiwara, S.-Y. Pi and H.-S. Tsao, “Regularizations and superconformalanomalies”,
Ann. Phys. , 282 (1980).65. A. Rebhan, A. Schmitt and P. van Nieuwenhuizen, YITP-SB-09-03, to ap-pear.66. D. V. Vassilevich, “Quantum corrections to the mass of the supersymmetricvortex”,
Phys. Rev.
D68 , 045005 (2003).67. A. Rebhan, P. van Nieuwenhuizen and R. Wimmer, “Nonvanishing quantumcorrections to the mass and central charge of the N = 2 vortex and BPSsaturation”, Nucl. Phys.
B679 , 382 (2004).68. S. ¨Olmez and M. Shifman, “Revisiting Critical Vortices in Three-DimensionalSQED”,
Phys. Rev.
D78 , 125021 (2008).69. A. S. Goldhaber, A. Rebhan, P. van Nieuwenhuizen and R. Wimmer, “Quan-tum corrections to mass and central charge of supersymmetric solitons”,
Phys. Rept. , 179 (2004).70. K. Fujikawa and P. van Nieuwenhuizen, “Topological anomalies from thepath integral measure in superspace”,
Ann. Phys. , 78 (2003).71. K. Fujikawa, A. Rebhan and P. van Nieuwenhuizen, “On the nature of theanomalies in the supersymmetric kink”,
Int. J. Mod. Phys.
A18 , 5637 (2003).72. K. Shizuya, “Superfield formulation of central charge anomalies in two-dimensional supersymmetric theories with solitons”,
Phys. Rev.
D69 , 065021(2004).73. K. Shizuya, “Topological-charge anomalies in supersymmetric theories withdomain walls”,
Phys. Rev.
D70 , 065003 (2004).74. K. Shizuya, “Central charge and renormalization in supersymmetric theorieswith vortices”,
Phys. Rev.
D71 , 065006 (2005).75. K. Shizuya, “Effect of quantum fluctuations on topological excitations andcentral charge in supersymmetric theories”,
Phys. Rev.
D74 , 025013 (2006).76. J. Polchinski, “Dirichlet-Branes and Ramond-Ramond Charges”,
Phys. Rev.Lett.75