Quantum mechanics on moduli spaces

Nuclear Physics B 565 Ž2000. 345–362 www.elsevier.nlrlocaternpe

Quantum mechanics on moduli spaces Ian G. Moss, Noriko Shiiki Department of Physics, UniÕersity of NewcastleUpon Tyne, NE1 7RU, UK Received 7 April 1999; received in revised form 4 October 1999; accepted 11 October 1999

Abstract It has been assumed that it is possible to approximate the interactions of quantized BPS solitons by quantising a dynamical system induced on a moduli space of soliton parameters. General properties of the reduction of quantum systems by a Born–Oppenheimer approximation are described here and applied to sigma models and their moduli spaces in order to learn more about this approximation. New terms arise from the reduction procedure, some of them geometrical and some of them dynamical in nature. The results are generalised to supersymmetric sigma models, where most of the extra terms vanish. q 2000 Elsevier Science B.V. All rights reserved. PACS: 03.70.qk; 98.80.Cq

1. Introduction In most physical problems, the degrees of freedom that are of interest are only a small fraction of the total number of degrees of freedom present. Some kind of reduction of the system has been performed. The way in which this reduction proceeds from quantum field theory to a moduli space of topological solitons is the subject of this paper. Soliton solutions can give rise to particle states w1x and it has been predicted that their low-energy interactions can be reduced to a simplified quantum mechanical problem in a set of collective coordinates w2,3x. One way in which this can be used is to indicate a duality between two quantum theories, where the soliton particle states in one theory become the fundamental particles in an equivalent quantum theory. A particular example can be seen in the monopole bound states which are important for the duality between the large and small coupling constant limits of the heterotic string theory w4x. In a range of different models there are classical multiple-soliton solutions that saturate Bogomol’nyi bounds and are parameterised by moduli. When the solitons move slowly and interact they deform adiabatically and the moduli trace out a path in a moduli space. The equations governing the path are equivalent to the equations for a point particle moving on a curved space w5,6x. 0550-3213r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 Ž 9 9 . 0 0 6 5 0 - 1

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We would expect to be able to describe the interactions of quantized solitons by quantising the dynamical system induced on the moduli space. We will call this procedure truncation. However, the quantisation rules of the truncated theory are often ambiguous without additional assumptions. One natural assumption would be general covariance, reflecting the freedom to choose coordinates on the moduli space. The covariant Hamiltonian operator takes the form y 12 D q j R q . . . ,

Ž 1.

where D is the Laplacian on the moduli space, and ambiguities are reduced to the coefficients j of the Ricci scalar R and other curvature invariants w7x. These ambiguities can be further reduced by introducing supersymmetry, in which case it has been argued that j s 0 w8,9x. In this paper we describe the alternative to truncating the full theory, which is to reduce it by a Born–Oppenheimer approximation. The effect of this reduction is to integrate out fluctuations that are orthogonal to the moduli space. In a purely bosonic theory, we will see that general covariance breaks down, in the sense that a scalar potential appears on the moduli space. In the basic situation, we begin with a classical dynamical system in flat space which has a potential V with a set of degenerate minima M . We find that the original quantum theory reduces to a quantum theory on M with Hamiltonian HR s y 12 D M q U q 14 R M y 18 < k < 2 q . . .

Ž 2.

The two purely geometrical terms depend on the intrinsic curvature scalar R M and the Žtraced. extrinsic curvature k I. These geometrical terms where first written down by Maraner w10,11x. The dynamical terms denoted by U depend upon derivatives of the potential. Those depending on up to two extrinsic derivatives of the potential can be written U s 12 tr Ž v . q 161 tr Ž vy1I =v P vy1I =v . ,

Ž 3.

2

where v is a matrix, v is the hessian matrix of V and the covariant derivative is taken tangentially to M . The application of these results to sigma models is considered in Section 3. In these models the original configuration space, consisting of maps from physical space to a curved target space, is both infinite and curved. The infinities can be dealt with by regularisation. The curvature would lead in general to ambiguities even before the reduction. However, as shown in Appendix A, these ambiguities are not present if the original space is a Riemannian symmetric space. Most models of interest fall into this class. The results are generalised to supersymmetric sigma models in Section 4. The point at issue is whether the reduction leads to a superpotential on the moduli space and alters the Hamiltonian operator. We find that this does not occur in the N s 2 case and, to leading order, the theory reduces to the Hodge Laplacian acting on p y forms, HR s y 12 D M q . . .

Ž 4.

confirming previous expectations w3,8,9x. Possible developments of this work are addressed in the conclusion, where we also discuss possible corrections.

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2. Born–Oppenheimer reduction Consider a classical dynamical system in flat space which has a degenerate set of stable points M . If the potential is sufficiently steep, then energy spectrum of the quantum system typically falls into separate bands. An adiabatic approximation scheme can then be used to reduce the quantum system to a quantum system on M . We shall restrict attention to the case where the potential has an analytic expansion about its minima. We take a set of generalised coordinates Ž x a, x I . such that x I s 0 on M . In the lowest energy bands, the value of x I fluctuates over a narrow range of values determined by the eigenvalues of a matrix v , where

v 2 s E I EJ V .

Ž 5.

The expansion parameter of the approximation scheme is given by the width of an energy band divided by the smallest eigenvalue. The first step is to rewrite the Hamiltonian operator, H s y 12 D q V

Ž 6.

in the new coordinate system. We choose normal coordinates x I along the principal directions of E I E J V. It is always possible to find two mutually orthogonal sets of derivatives Eˆa and E I , where

Eˆa s Ea y N I a E I .

Ž 7.

This implies that the metric can be written g s ga b dx adx b q g I J Ž dx I q N I a dx a .Ž dx J q N J b dx b . .

Ž 8.

The metric components have normal coordinate expansions

g I J s dI J ,

Ž 9.

I

N a s ya J

I

J

a

x ,

Ž 10 .

ga b s sa b q 2 k I a b x I q k I ac k J b c x I x J , I

Ž 11 . J

where the extrinsic curvature k and the torsion a I are defined in Appendix B. These expansions are exact if the space is flat, but we will keep in mind the possibility of generalising these results to curved spaces. In the new coordinate system, the Hamiltonian becomes H s 12 < g
Ž 12 .

I

We regard the x as small and expand the Hamiltonian as a series of terms H s H0 q H1 q . . . . The potential has a series expansion V s 12 Ž v 2 . I J x I x J q 16 VI JK x I x J x K q . . . ,

Ž 13.

where v I J is a diagonal matrix with eigenvalues v I . The first few terms in the full Hamiltonian are then H0 s y 12 d I JE I E J q 12 Ž v 2 . I J x I x J , 1 2

I

1 6

I

J

Ž 14 . K

H1 s y tr Ž k . E I q VI JK x x x ,

Ž 15 .

H2 s y 12 D P D q 12 tr Ž k I k J . x IE J q 241 VI JK L x I x J x K x L .

Ž 16 .

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The dot denotes a scalar product using the intrinsic metric on M . We have introduced a covariant derivative Da , where Da sa b s 0 and Da f s Ž Ea q a J I a x JE J . f

Ž 17 .

for scalar functions f. The next step is to apply degenerate perturbation theory. We take the unperturbed harmonic oscillator Hamiltonian H0 together with a set of states f n , where H0 f n s En f n . The perturbation theory is described in Appendix C and gives a reduced Hamiltonian operator HR acting on the space of wave functions c Ž x a ., HR s ² H0 : 00 q ² H1 : 00 q ² H2 : 00 q

Ý

² H1 : 0 m Ž E0 y Em . y1 ² H1 :m0

m/0

qOŽ v

y1

..

Ž 18 .

The matrix elements defined by ² H :n m s f n) H f m Ł dx I

H

Ž 19 .

I

are allowed to be operator valued. The first term ² H0 : 00 consists of the vacuum energy of the system of unperturbed harmonic oscillators Ž14., E0 s 12 tr Ž v . .

™

Ž 20 .

The second term ² H1 : 00 vanishes because the operator is odd under the inversion x I yx I. For the next term ² H2 : 00 , with H2 given by Eq. Ž16., we set ² Da :m n s dmI n=a q ² Aa :m n ,

Ž 21 .

I

where =a is defined in Appendix B and A arises form the action of Da on the states f n in Eq. Ž19.. The matrix element of the derivative of an energy eigenstate is given by a standard identity, ² Aa :m n s

Ž w Da , H0 x . m n En y Em

.

Ž 22 .

After using Eq. Ž17. for D, and comparing with Eq. ŽB.5., the commutator produces a covariant derivative, ² Aa :m n s

½

1I 2

=a v I J ² x I x J :m n

0

Em - E n , m s n.

Ž 23 .

We can now evaluate ² D P D : 00 s D M y 18 tr Ž vy1I =v P vy1I =v . , I

Ž 24 .

I

where the covariant Laplacian D M s = P = . We also have, using ŽB.6., ²tr Ž k I k J . x EI J : 00 s 12 tr Ž k I k I . s 12 < k < 2 y 21 R M , 2

Ž 25 .

I.

where < k < s trŽ k trŽ k I .. Therefore we get ² H2 : 00 s y 12 D M y 14 < k < 2 q 14 R M q 161 tr Ž vy1I =v P vy1I =v . y1 q 321 Ý vy1 I v J VI JI J .

IJ

Ž 26 .

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The last term given in Eq. Ž18. leads to 1 8

< k < 2 y 481

2 y1 y1 Ý Ž v I q v J q v K . y1 vy1 I v J v K Ž VI JK .

IJK

y 321

y1 y1 Ý vy2 I v J v K VI J J VIK K .

Ž 27 .

IJK

Putting the terms together gives a final expression for the reduced Hamiltonian, HR s y 12 D M q U q 14 R M y 18 < k < 2 q O Ž vy1 . .

Ž 28 .

The terms are divided into purely geometrical terms and dynamical terms which are collected into U, where y1 U s 12 tr Ž v . q 161 tr Ž vy1 I =v P vy1I =v . q 321 Ý vy1 I v J VI JI J

IJ

y

1 48

Ý Ž vI q v J q vK . IJK

y1

1 y1 y1 2 vy1 I v J v K VI JK y 32

y1 y1 Ý vy2 I v J v K VI J J VIK K .

IJK

Ž 29 . The vacuum energy, which is of order v , is the dominant term in this expression. The next term depends on the gradient of the normal mode frequencies along the moduli space and includes the effects of the twisting, or torsion, of the normal mode directions. The remaining terms contain the effects of higher derivatives of the potential. The geometrical terms where discovered in the restricted situation of a surface embedded in R 3 by Jensen and Koppe w12x and more generally by Maraner w10,11x. The covariant forms for the dynamical terms beyond the obvious vacuum energy term are new, as far as we know.

3. Bosonic field theory The non-linear sigma-model in two space dimensions x m and one time dimension t has fields f i taking values in a curved target space T. We will take the Lagrangian to be L s T y V, where Ts

1 2 1

Hg

i j i j Et f Et f

d2 x,

Ž 30 .

g i j Em f Ei m f j d 2 x. Ž 31 . 2 Depending upon the topology of T , the classical field equations can have static solutions with finite energy, or topological solitons. These solutions minimise the potential energy for a given topological class. When T is a compact Kahler manifold the soliton solutions can be represented by ¨ holomorphic maps from the complex plane into T w6,13–15x. They depend on a continuous set of parameters, or moduli, which can be interpreted as the positions and charges of individual lumps. The situation here is similar to the one considered in Section 2, except that the configuration space is no longer finite and flat. The introduction of curvature leads to the Vs

H

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appearance of curvature invariants in the initial Schrodinger equation. However, we will ¨ take the target space to be a Riemannian symmetric space. The space of fields inherits this symmetry and is also a Riemannian symmetric space. In this case the curvature invariants are constants which can be absorbed into an overall phase factor. The soliton solutions f 0 will be parameterised by a set of coordinates x a belonging to the moduli space. We shall expand the Lagrangian about these solutions using standard background field methods w16x and express the result in terms of time-dependent collective coordinates x a and normal coordinates x I. This will enable us to use the results of Section 2. The tangent vector j to the geodesic from f to f 0 provides a convenient measure of the displaced field. We also introduce the vectors em with target space components em i s Em f i. The tangent vector j commutes with em , and consequently =j em s Dm j , where Dm is the gradient along em . By repeated applications of =j , we deduce that

=j n em s =j ny 2 R Ž j ,em . j ,

Ž 32 .

where RŽ X,Y . is the Riemann curvature operator. We will define

Ž u,z . s H g i j Ž f 0 . u i Õ j d 2 x.

Ž 33 .

The potential V has a series expansion in j of the form `

Vs

Ý ns0

1 n!

=j

n1 2

Ž em ,em . s V0 q V1 q V2 . . .

Ž 34.

After applying Eq. Ž32., V0 s 12 Ž em ,em . ,

Ž 35 .

V1 s Ž em , Dm j . ,

Ž 36 .

V2 s 12 Ž Dm j , Dm j . q 12 Ž em , R Ž j ,em . j . ,

Ž 37 .

V3 s 23 Ž Dm j , R Ž j ,em . j . ,

Ž 38 .

V4 s

1 6

1 6

Ž Dm j , R Ž j , Dm j . j . q Ž R Ž j ,em . j , R Ž j ,em . j . .

Ž 39 .

Integration by parts in Ž36. produces the field equation Dm em s Em Em f 0 i y G

i

jk Em f 0

j

Em f 0 k s 0.

Ž 40 .

Integration by parts on Ž37. produces the operator

D f j s yDm Dm j y R Ž j ,em . em ,

Ž 41 .

which describes fluctuations about the moduli space. In the usual analysis of the sigma model, the inverse of this propagator would be the Green function in the background field approximation. The normalised positive modes of the fluctuation operator will be denoted by u I and their eigenvalues by Ž v I . 2 . The fluctuation operator also has a set of zero-modes u a s Ea f 0 . All of the modes are parameterised by the collective coordinates x a. We define the remaining coordinates as components of the displacement vector,

j s x IuI .

Ž 42 .

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The potential then has a series expansion in the normal coordinates identical to Eq. Ž13. used in Section 2. From Eq. Ž38., VI JK s 4CŽ I J . K ,

Ž 43 .

where CI JK s Ž Dm u I , R Ž u J ,em . u K . .

Ž 44 .

These coefficients are symmetrical, since CŽ I J . K s CŽ I JK . . They are also trace-free if the zero-modes u a are included, C I I J q C a a J s 0. The kinetic energy T has a similar series expansion, where we replace em by e t and Dm j by Dt j in Eqs. Ž35. – Ž38.. However, both the x I and the x a depend on time, so that the time derivative of Eq. Ž42. implies Dt j s x˙ I u I q x˙ a x I Da u I ,

Ž 45 .

where the covariant derivative Da u i s Ea u i q u ka G

i

jk u

j

.

Ž 46 .

Also, since u a s Ea f 0 , e t s E t f 0 s x˙ a u a .

Ž 47 .

Eqs. Ž45. and Ž47. allow terms in the series expansion of the kinetic energy to be grouped into a quadratic polynomial in the generalised velocities, T s 12 ga b x˙ a x˙ b q 12 g I J Ž x˙ I q N I a x˙ a .Ž x˙ J q N J b x˙ b . .

Ž 48 .

To second order in x I,

g I J s d I J y 13 R IK JL x K x L , I

N a s ya J

I

J

a

x ,

ga b s sa b q 2 k I a b x I q k I ac k J b c x I x J y R a I b J x I x J .

Ž 49 . Ž 50 . Ž 51 .

The coefficients can be obtained by examining Eqs. Ž35. – Ž38.,

sa b s Ž u a ,u b . , k

I

Ž 52 .

Ž u a , Db u I . ,

Ž 53 .

a I J a s Ž u I , Da u J . ,

Ž 54 .

R I JK L s Ž u I , R Ž u K ,u L . u J . .

Ž 55 .

ab s

These results recover the normal coordinate expansion of Section 2, Eqs. Ž9. – Ž11., with extra terms due to the curvature of the original configuration space. We now have explicit expressions for the curvature and the extrinsic geometry of the space of soliton parameters in terms of eigenstates of the fluctuation operator. The results of Section 2 still apply, and the reduced Hamiltonian is HR s y 12 D M q U q 14 R M y 18 < k < 2 q O Ž vy1 .

Ž 56 .

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except that now there are additional terms depending on the target space curvature. A similar analysis to Section 2 shows that two extra terms 1 4

s acs b d R a b c d q 16 Ý v I vy1 J R I JI J

Ž 57 .

IJ

should be included in the potential U. Expressions involving derivatives of the eigenmodes can be rewritten using the identity

Ž u I , Da u J . s

Ž uI ,

Da , D f u J .

v J2 y v I2

.

Ž 58 .

With the fluctuation operator D f given explicitly in Eq. Ž41., the commutator becomes

Ž uI ,

Da , D f u J . s 4CŽ I J . a ,

Ž 59 .

where CI J a is defined as in Eq. Ž44.. This gives 4 k I a b s y 2 CŽ Ia. b , vI I

=a v I J s

4

v J q vI

Ž 60 .

CŽ I J . a ,

Ž 61 .

where Eq. ŽB.5. has been used. The dominant term in the reduced Hamiltonian should be the zero point energy 1 Ž v .. In general, in order to evaluate this term it will be necessary to solve the tr 2 eigenvalue problem for the fluctuation operator numerically. Once this has been done, Eqs. Ž43., Ž60. and Ž61. can then be used to evaluate the geometrical terms in the reduced Hamiltonian Ž56. in terms of integrals of eigenfunctions.

4. Supersymmetric reduction We now turn to the reduction of an N s 2 supersymmetric quantum system defined on a Riemannian symmetric space. As before, we suppose that the system has a potential V with a degenerate set of minima. The existence of two supersymmetries requires that the space is a Kahler manifold with a covariantly preserved complex structure J w17,18x. ¨ We shall suppose that the moduli space of stable points is also a Kahler manifold. ¨ The variables consist of coordinates x a and single complex component fermions c a. We also assume the existence of an operator pa , related to the momentum, and the following commutation relations

 c a , c ) b 4 s g ab , pa , x b s yi da b ,

Ž 62 . Ž 63 .

pa , c b s yi G b ga c g ,

Ž 64 .

w pa ,pb x s yiRa bgd c ) gc d involving the inverse metric g nents R abgd .

ab

, connection components G

Ž 65 . b ga

and curvature compo-

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It is possible to represent the commutator relations by covariant differential operators on a Hilbert space with the basis

C Ž x . a 1 . . . a p c ) a 1 . . . c ) a p <0: ,

Ž 66 .

where c a <0: s 0. This gives a representation in which

pa s yi Ž Ea y Ggba c ) bc g . .

Ž 67 .

The action of p is equivalent to yi=a acting on the p-forms C , where =a is the metric connection. We shall assume that the classical system has two supersymmetries generated by the supercharges Q Ž1. s c apa y c ) a Wa Ž x . , a

a

Q Ž2. s Ž Jc . pa y Ž Jc ) . Wa Ž x . .

Ž 68 . Ž 69 .

The reason for choosing this particular form for these supercharges will become apparent when we come to discuss sigma models in Section 5. We will only require the combination Q s 12 Ž Q Ž1. q iQ Ž2. . s l apa y n a Wa Ž x . ,

Ž 70 .

where the fermion fields

l s 12 Ž 1 q iJ . c , 1 2

n s Ž 1 q iJ . c )

Ž 71 . Ž 72 .

are purely holomorphic. The Hamiltonian is given Žup to a constant. by H s  Q,Q ) 4 , which can be evaluated using the commutation relations Ž65., H s y 12 = 2 q g a b Wa Wb) q iWa); b l bn ) a q iWa ; b l) bn a y 41 R abgd w l a , l) b xw l) g , ld x y 41 R a bgd w l a , l) b xw n g , n ) d x .

Ž 73 .

The moduli space lies at the minimum of the bosonic potential and is parameterised by a set of complex coordinates z a. We assume that the superpotential Wa has a series expansion in normal coordinates z I, Wa s Wa J z J q Wa JK z J z K q Wa JK L z J z K z L q . . .

Ž 74.

The normal coordinates can be chosen to ensure that the boson mass matrix is diagonal, g ab Wa J Wb)K s Ž v 2 . JK .

Ž 75 .

For the fermion mass matrix Wa ; b , we need two separate sets of basis vectors, in general, in order to put it into diagonal form. We can always choose one basis Ž e a ,e I . to coincide with the coordinate basis at z I s 0 for the fields l. Another basis Ž e aX ,e I X . then has to be used for the fields n . ŽThis procedure is similar to the independent unitary rotations of the left and right chirality fermions in the standard model.. With the choice a e I X s vy1 I W I ea ,

Ž 76 .

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the fermion mass matrix becomes diagonal, WI X J s v I J .

Ž 77 . X

a

a

The fermion fields l and n are massless and remain in the reduced quantum system. However, there is an important difference between the finite and infinite dimensional situations. In the infinite dimensional situation the basis e I X may be complete Žin which X case there are no massless n a fermions.. We shall proceed assuming this to be the case. Previously, we used a series expansion for the Hamiltonian to arrive at the reduced theory. However, as shown in Appendix C, for a supersymmetric theory we need only consider the supercharge. The supercharge Ž70. depends on the normal coordinates through the superpotential and through the connection coefficients in pa . In the basis ea s Ž e a ,e I .,

Ggb a c ) bc g ; Ggb a n bn ) g

Ž 78 .

in the limit z I s 0. The first few terms in the series expansion of the supercharge Ž70. are now X

Q0 s yi lEI I y n I WI X J z J ,

Ž 79 .

X

Q1 s yi l a Da y n I WI X JK z J z K ,

Ž 80 . X

Q2 s yi l a Rbg I a z I Ž l) blg q n bn ) g . q n I WI X JK L z J z K z L ,

Ž 81 .

where Da now includes the connection terms Ž78.. The full quantum theory can now be reduced to the moduli space by following the methods described in Appendix C. We define the fermion vacuum state <0 T : which is annihilated by two sets of annihilation operators I ly s

1

X

I s Ž l I q i n I . , ly

'2

1

X

Ž l I q in I . .

Ž 82 .

'2

As in Section 2, we build up from <0 T : a set of oscillator states f n which satisfy H0 f n s En f n , where H0 is the unperturbed Hamiltonian. In the supersymmetric case, E0 s 0. The reduced Hamiltonian is given in terms of the reduced supercharge by HR s  Q R ,Q R) 4 ,

Ž 83 .

where Q R s ² Q1 : 00 q

Ý ² Q1 : 0 n Ž E0 y En . y1 ² H1 :n0 q O Ž vy1 . .

Ž 84 .

n/0

The covariant derivatives are treated as in Section 2, that is we write ² Da : 0 n s d 0 nI =a q ² Aa : 0 n .

Ž 85 .

In the supersymmetric case we include the fermion terms from Eq. Ž78., X

X

² Aa : 0 n sI =a v I J ² z I z J : 0 n q aX I X J X a² n I n J : 0 n .

Ž 86 .

At leading order, we recover the same result that we would obtain by a trivial truncation of the theory, Q R s yi l aI =a q O Ž vy1 . .

Ž 87 .

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As discussed in Appendix A, the reduced states can be identified with antiholomorphic forms on the moduli space and the reduced Hamiltonian can be identified with the Laplacian. At this order the reduced theory is identical with the truncated theory obtained under the simplest assumptions w3x.

5. Supersymmetric sigma models The supersymmetric sigma model in two space and one time dimension has Lagrangian Ls

1

H ž yg

2

i jE

m i

f Em f j q ig i j x ig m Dm x j d 2 x ,

/

Ž 88 .

where m s 0,1,2, and x is a two-component majorana spinor. If the target space is a Kahler manifold, with complex structure J, then the model has two supersymmetries. ¨ We shall consider the reduction of this theory. It is convenient to use a complex representation for the fermion fields, with

xs

c , c)

ž /

Ž 89 .

and a complex spatial coordinate z s x q iy. There are two complex supercharges in this representation, Q Ž1. s

HŽ g

i j )i j i j c E t f y 2 g i j c Ez f

Q Ž2. s

Hž g

ij

. d2 z,

i )i Ž Jc . E t f j y 2 g i j Ž Jc . Ez f j / d 2 z.

Ž 90 . Ž 91 .

These can be used to generate the classical supersymmetry transformations by transforming to phase space and using Dirac brackets. The supercharges produce the Hamiltonian by Dirac brackets H s 2i  Q Ž1. ,Q Ž1.) 4 DB s 2i  Q Ž2. ,Q Ž2.) 4 DB . Ž1.

The combination Q s Ž Q q iQ

Ž 92 .

Ž2. .

r2 can also be used, but in this case

)

i  Q,Q 4 DB s H q ET ,

Ž 93 .

where i

j

ET s 2 g i j Ž JEz f . Ž Ez f . d 2 z.

H

Ž 94 .

The integral is a constant for fields f in the same topological class. We can still use Ž93. to generate the quantum Hamiltonian. The background field expansion used in Section 3 can be used again here to expand the supercharge Q. The bosonic fluctuations about the background f 0 are described by a vector j and a fluctuation operator D f . The fermion fields are simply parallel propagated to f 0 and decomposed into c s l q n ) , where l and n are holomorphic, i.e. Jl s i l.

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Since the complex structure J commutes with the fluctuation operator, we can choose holomorphic eigenmodes ua and use these as a basis ea . The supercharge for this theory is given by Qs

i

HŽ lp y 2 g i

i jn

i

Ez f j . d 2 z.

Ž 95 .

When the eigenmode expansions are used, we recover the expression Ž70. used in Section 4, Q s l apa y n a Wa Ž j . .

Ž 96 .

However, now we have an explicit formula for the superpotential Wa s 2 Ž ua ,exp Ž =j . e z .

Ž 97 .

in terms of the product Ž33.. The series expansion for Wa in the coordinate z I is obtained using Eq. Ž32.. The linear term is, Wa J s 2 Ž ua , Dz u )J . ,

Ž 98 .

where Dz u I s Ez u I i q Ž Ez f 0j . G

i

jk u I

k

.

Ž 99 .

According to Eq. Ž75., this implies that the fluctuation operator is

D f u I s y4Dz Dz u I

Ž 100.

and also that the fermion mass matrix is diagonalised by choosing a basis e I X s 2 v I y1 Dz u I .

Ž 101.

These vectors actually form a complete basis, that is they form a normalisable set of eigenvectors of the Žpositive. operator Dz Dz . The only massless fermions are the l a and these remain as fermions on the moduli space. This is precisely the situation considered in Section 4. The reduced theory is therefore identical to the truncated theory at leading order in the adiabatic approximation. The magnitude of the next to leading order corrections is discussed in the conclusion.

6. Conclusion The reduction of a classical dynamical system generally involves eliminating the internal forces and introducing generalised coordinates. In quantum theory, the uncertainty principle implies that the internal coordinates can never be frozen, but the reduction can be performed approximately when the internal degrees of freedom remain close to their ground state. We have considered the adiabatic reduction of various quantum systems onto a moduli space of collective coordinates. The results for a quantum mechanical system are quite general and depend on geometrical properties of the moduli space as well as the frequencies of the internal modes. The reduced Hamiltonian to leading order can be found at the end of Section 1.

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The moduli space of sigma-model solitons has been quantised by taking the continuum limit of the quantum mechanical system. In the Bosonic case, the terms in the reduced Hamiltonian operator depend on the eigenfunctions of a fluctuation operator D f . The spectrum of the fluctuation operator is continuous and the eigenvalue sums of the quantum mechanical system have to be replaced by integrals and regularised. We could have attempted to quantise the sigma-model moduli space with the instanton techniques used in quantum field theory. In this approach, based on a path integral, the field is replaced by collective coordinates x a and field fluctuations j . The fluctuations are integrated out, leaving a path integral over the coordinates x a and a Jacobian factor. The classical action S gets replaced by X

W w x a x s S w x a x q 12 log det Ž D f y E t2 . q . . . ,

Ž 102.

where the prime indicates omission of any zero eigenvalues and the dots denote contributions from higher loop Feynman diagrams. An adiabatic reduction can now be obtained by taking the time derivatives to be small and evaluating some of the higher loop terms. In practice, we have not found this approach any easier than the Hamiltonian approach. The bosonic sigma model considered in Section 3 can be generalised in various ways. An important possibility would be to include potentials of the form

F s W Ž f . d 2 x.

H

Ž 103.

The changes to the results brought about by introducing potentials are confined to the terms in Eq. Ž28. involving VI JK and VI JK L , which depend on functional derivatives of F. The supersymmetric sigma model is tightly constrained and reduces to the Hodge Laplacian acting on antiholomorphic p-forms, confirming a result that has been used previously with strong support but not derived rigorously w3,8,9x. It is here, especially, that higher order terms in the adiabatic expansion might be of interest. The Laplacian is formally O Ž v 0 . in the adiabatic expansion and the size of the corrections for the supersymmetric model are formally O Ž vy1 .. Although, in the continuum limit, there is no preferred value for v , some indication of the actual size of the expansion parameter can be obtained from specific examples and from general considerations. Results for the CP Ž1. model in 2 q 1 dimensions w24x indicate that, for a single soliton of width a , the expansion parameter of the adiabatic approximation is g 2ra , where g is the CP Ž1. model coupling constant. For a double soliton solution, the corrections to the leading order result include the backreaction of radiation by the solitons as they interact and should depend additionally on the ratio of separation to width.

Appendix A. Factor ordering When constructing a quantum theory the ordering of operators is an important issue. A minimum requirement for choosing the operator ordering is to try to retain as much of

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the symmetry of the classical system as possible. We shall consider symmetry under general coordinate transformations, supersymmetry and the symmetry of a Riemannian Symmetric space. Our first example is classical particle mechanics in curved space, described by the Hamiltonian H s 12 g ab pa pb .

Ž A.1 .

If we introduce a Hilbert space of wave functions c Ž x . and use the metric to construct an inner product on the Hilbert space, then the Hamiltonian operator H s y 12 < g
Ž A.2 .

is both covariant and consistent with the basic principals of quantum mechanics. However, many other possibilities exist, since we are free to add terms such as i j R a b x a , pb ,

Ž A.3 .

which are of order " or higher and vanish in the classical limit. If we assume that the Hamiltonian operator is a covariant second-order operator, then the form of this operator must be w7x 2

H s y 12 Ž = q A . q X .

Ž A.4 .

However, the vector field A and scalar field X cannot be inferred from the classical Hamiltonian. The quantum theory is far more restrictive if the space is a Riemannian Symmetric space. A Riemannian Symmetric space is defined by its geodesic symmetries. The geodesic symmetries about a point leave the point unchanged and map each of the geodesics through the point into itself. A Riemannian Symmetric space has these symmetries at every point and covariant derivatives of the curvature, which are odd under the reflection symmetry, must vanish w21x. In this case, both = X and A, which are also odd under the geodesic symmetry, must vanish. The Hamiltonian on a Riemannian symmetric space is therefore given by the Laplacian and a constant. Our second example is the supersymmetric particle mechanics on a Kahler manifold, ¨ originally discussed by Witten w8x. The coordinates z a and momenta pa are supplemented by a set of complex fermions l a. The commutation relations are

 la , lb 4 s g a b , pa , z

b

Ž A.5 . b

s yi d a ,

pa , l b s i G

b

ca l

c

Ž A.6 . ,

Ž A.7 .

w pa ,p b x s iR a b d c lcld

Ž A.8 . ab

b

c

and their hermitian conjugates, where g is the inverse metric, G c a l the connection components and R a b d c the Riemannian curvature. The supersymmetry is generated by a supercharge Q s l apa .

Ž A.9 .

The ordering of the operators in the last commutation relation and the supercharge have so far been chosen arbitrarily.

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The commutation relations can be represented by differential operators acting on the Hilbert space of functions C Ž z a, z a, l a ., with

pa s yi Ea ,

Ž A.10 .

pa s yi Ž Ea y Gcb a l blc . ,

Ž A.11 .

where

E la s g a b

. Ž A.12 . El b In this representation all of the operators, including Q, act covariantly on the functions C . The covariance is explicit if we regard C as an element of the exterior algebra over a basis of forms l a. In this case pl a s yi=l a, where = is the covariant derivative, and Q s i E †, where E is the antiholomorphic exterior derivative. The Hamiltonian is obtained from Q by the relation H s  Q,Q ) 4 , which by the use of Eq. ŽA.9. and the commutation relations gives H s y 12 = 2 q 12 R a b l al b ,

Ž A.13 .

where R a b is the Ricci tensor. De Alfaro et al. w9x have pointed out that alternative orderings of the two operators in the supercharge ŽA.9. would make the representation non-covariant. However, there do exist alternative expressions for the supercharge, Q s l apa y i l a Wa ,

Ž A.14 .

which have the same classical limit if Wa s O Ž " .. The condition Q 2 s 0 is satisfied if E W s 0. The simplest way to satisfy these conditions would be to make W a total derivative, Wa s i jEa Ž R b c z c ,p b

.,

Ž A.15 .

which gives additional contributions to the Hamiltonian of order ". Other contributions to the Hamiltonian may exist if the manifold has non-trivial Homology.

Appendix B. Geometrical formulae This appendix reviews some formulae used to reduce the Riemannian connection = onto a submanifold M . We construct a basis of 1-forms Ž n I,e a ., with dual basis Ž n I ,e a ., made up to include the normal forms n I and vectors e a tangential to M . The metric is then g s sa b e a m e b q hI J n I m n J ,

Ž B.1 .

where s is the metric on M . Any vector X can be decomposed into a tangential and normal parts, X I s X a e a and X H s X I n I . The extrinsic curvature k I and extrinsic torsion tensor a I J are defined by k I a b s Ž =a n I . b ,

Ž B.2 . J

a I J a s Ž =a n J . I s y Ž =a n I . .

Ž B.3 .

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The extrinsic curvature measures the expansion of the normal forms and it is always symmetric in the surface indices. The extrinsic torsion measures changes in the normals along lines drawn in the surface. The extrinsic torsion is antisymmetric if the normals are of unit length. We will define the tangential covariant derivative by its action on the basis vectors, I

= X e a s Ž =X I e a . I ,

I

= X n I s Ž =X I n I . H .

Ž B.4 .

With this definition the derivative is a metric connection for both s and h , with torsion related to a I J. We make repeated use of the following expression for the derivative of a normal directed tensor in the body of the text, I

=a v I J s Ea v I J q a I K a v K J q a J K a v IK . I

Ž B.5 . I

The curvature tensor of = will be denoted by R. The reduction of the curvature tensor is a standard procedure, and has been described in this notation in Ref. w19x. The most useful formula is Gauss equation, R a b c d sI R a b c d y hI J k I ac k J b d q hI J k I a d k J b c .

Ž B.6 . I

In the case where M is a complex submanifold of a Kahler manifold, the trace k of the ¨ extrinsic curvature tensor vanishes w20x. Geodesic normal coordinates can be constructed in the neighbourhood of the submanifold by setting x I s sj I, where s is the distance along a geodesic and j is the tangent vector to the geodesic. We use the coordinate basis Ž e a ,e I . s Ž Ea , E I ., where e I s n I on M . Since w j ,e a x s 0,

=j e a s j I k Ia b e b y j Ia I J a e J .

Ž B.7 .

Furthermore, by considering the commutator w j , s e I x, as in Ref. w22x, we get

=j n e I s

ny1 nq1

=j ny2 R Ž j ,e I . j ,

=j n e a s =j ny 2 R Ž j ,e a . j .

Ž B.8 . Ž B.9 .

These can be used to obtain the normal coordinate expansion of the metric, `

gmn s

Ý ns0

1 n!

=j n Ž em P en . .

Ž B.10 .

A typical application is given in Eq. Ž11..

Appendix C. Adiabatic approximation schemes This appendix shows how the Hamiltonian operator and other operators can be reduced by using almost degenerate perturbation theory w23x. The method is equivalent to the adiabatic, or Born–Oppenheimer, approximation scheme for the energy eigenstates. The Hamiltonian is first separated into an unperturbed part and a perturbation H s H 0 q H1 .

Ž C.1 .

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The eigenstates of H0 are then grouped into eigenspaces that cover a narrow range of eigenvalues. A projection operator P0 projects onto the lowest of these eigenspaces and P1 s 1 y P0 . The equation for the eigenstate with energy E is standard Žsee w23x for example., HR P0C s E P0C ,

Ž C.2 .

where HR s E0 q P0 H1 P0 q P0 H1 P1 Ž E y H0 y H1 .

y1

P1 H1 P0 .

Ž C.3 .

P1 H1 P0 q . . .

Ž C.4 .

The leading order terms give HR s E0 q P0 H1 P0 q P0 H1 P1 Ž E0 y H0 .

y1

For an adiabatic approximation scheme, we look for an expansion of the wave function in the form

C Ž x . s Ý c Ž x a . fn Ž x I , x a . ,

Ž C.5 .

n

where the wave function depends strongly on coordinates x I and weakly on coordinates x a. We take an unperturbed Hamiltonian H0 which commutes with the x a and define the functions f n by H 0 f n s En Ž x a . f n .

Ž C.6 .

These can be normalised with respect to a reduced product ² f , g : Hs f ) Ž x a , x I . g Ž x a , x I . d m Ž x I .

Ž C.7 .

H

with measure d m Ž x I . chosen to make H0 self-adjoint. The projection operator P0 is given by P0C s c 0 Ž x a . .

Ž C.8 .

Inserting complete sets of states into Eq. ŽC.4. gives the equation for c 0 , HR c 0 s Ec 0 ,

Ž C.9 .

where the reduced Hamiltonian operator has the perturbative expansion HR s E0 q ² H1 : 00 q Ý ² H1 : 0 m Ž E0 y Em .

y1

² H1 :m0 q . . .

Ž C.10 .

m

The matrix elements ² H1 :m n s ² f m , H1 f n : H

Ž C.11 .

are operators of the reduced theory. They also act on the En . Other operators are reduced in a similar way. Given an operator Q, we follow ŽC.3. and define Q R s P0 QP0 q P0 QP1 Ž E y H .

y1

P1 HP0 .

Ž C.12 .

In the adiabatic approximation, Q R s ² Q : 00 q Ý ² Q : 0 n Ž E0 y En . n

is an operator that acts on the states c 0 .

y1

² H1 :n0 q . . .

Ž C.13 .

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The reduced operator satisfies Q R P0C s P0 QC ,

Ž C.14 .

where C is any energy eigenstate with eigenvalue E. This reduction is not generally a homomorphism of the operator algebras, i.e. Ž QQX .R / Q R QXR in general, but if Q and QX are symmetries of the Hamiltonian, then it follows from ŽC.12. that

Ž QQX . R s Q R QXR .

Ž C.15 . )4

In particular, if Q is a supersymmetry operator with H s  Q,Q , then HR s  Q R ,Q R) 4 .

Ž C.16 .

This simplifies the problem of reducing the Hamiltonian operator.

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