Instantons, three-dimensional gauge theory, and the Atiyah-Hitchin manifold

Nuclear Physics B 502 (1997) 59-93

EXSEVIER

Instantons, three-dimensional gauge theory, and the Atiyah-Hitchin manifold N. Dorey a, V.V. Khoze b, MI? Mattis c, D. Tong a, S. Vandoren a a Physics Department, University of Wales Swansea, Singleton Park, Swansea, SA2 8Pe UK b Physics Department, Centre for Particle Theory, University of Durham, Durham DHl 3L.E. UK c Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received 6 May 1997; accepted 3 July 1997

Abstract We investigate quantum effects on the Coulomb branch of three-dimensional N = 4 supersymmetric gauge theory with gauge group W(2). We calculate perturbative and one-instanton contributions to the Wilsonian effective action using standard weak-coupling methods. Unlike the four-dimensional case, and despite supersymmetry, the contribution of non-zero-modes to the instanton measure does not cancel. Our results allow us to fix the weak-coupling boundary conditions for the differential equations which determine the hyper-Kahler metric on the quantum moduli space. We confirm the proposal of Seiberg and Witten that the Coulomb branch is equivalent, as a hyper-Kahler manifold, to the centered moduli space of two BPS monopoles constructed by Atiyah and Hitchin. @ 1997 Elsevier Science B.V.

1. Introduction Recent

work by several

authors

[l-7]

has provided

exact information

about

the

low-energy dynamics of N = 4 supersymmetric gauge theory in three dimensions. In particular, an interesting connection has emerged between the quantum moduli spaces of these theories and the classical moduli spaces of BPS monopoles in W(2) gauge theory [2]. Chalmers and Hanany [4] have proposed that the Coulomb branch of the SU(n) gauge theory is equivalent as a hyper-K&ler manifold to the centered moduli space of n BPS monopoles. For n > 2 this correspondence is intriguing because the hyper-K;ihler metric on the manifold in question is essentially unknown. Subsequently, Hanany and Witten [ 61 have shown that the equivalence, for all n, is a consequence of S-duality applied to a certain configuration of branes in type IIB superstring theory. 0550-3213/97/$17.00 @ 1997 EJsevier Science B.V. All rights reserved. PII SO550-3213(97)00454-9

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

60

The case n = 2, which is the main topic of this paper, provides these ideas because been found explicitly

the two-monopole

moduli

by Atiyah and Hitchin

an important

space and its hyper-Kahler

test for

metric have

[ 81. We will refer to the four-dimensional

manifold which describes the relative separation and charge angle of two BPS monopoles as the Atiyah-Hitchin (AH) manifold. In fact these authors effectively classified all four-dimensional hyper-K&hler manifolds with an SO(3) action which rotates the three inequivalent complex structures. Correspondingly, the moduli space of the N = 4 theory with gauge group SU( 2) was analyzed by Seiberg and Witten [ 21. The effective lowenergy theory in this case is a non-linear

cT-model with the four-dimensional

branch of the SL1(2) theory as the target manifold. low-energy kinetic

theory requires

terms

Coulomb

of the

that the metric induced on the target space by the a-model

be hyper-K%hler

[9].

By virtue

branch of this theory necessarily

scheme. Seiberg and Witten compared theory with the asymptotic

The N = 4 supersymmetry

Coulomb

of its global

symmetry

fits into Atiyah and Hitchin’s

the weak coupling

behaviour

form of the metric on the AH manifold

structure,

the

classification

of the SUSY gauge in the limit of large-

spatial separation between the monopoles, Y > 1. ’ They found exact agreement between perturbative effects in the SUSY gauge theory, and the expansion of the AH metric in inverse powers of r. In fact, as we will review below, there are an infinite number of hyper-Kahler four-manifolds with the required isometries which share this

inequivalent asymptotic

behaviour.

AH manifold [ 11, Seiberg

However, Atiyah and Hitchin showed that only one of these, the

itself, is singularity-free. and Witten

proposed

Motivated

N = 4 theory should also have no singularities. manifolds Clearly

by expectations

that the Coulomb

from string theory

branch of the three-dimensional

The correspondence

between

the two

then follows automatically. the arguments

that the Coulomb

reviewed

above come close to a first-principles

branch of the SU( 2) theory is the AH manifold.

demonstration

The only assumption

made about the strong coupling behaviour of the SUSY gauge theory which is not automatically guaranteed by symmetries alone is the absence of singularities. In this paper we will proceed without this assumption. 2 One must then choose between an infinite number of possible metrics with the same asymptotic form. By virtue of the hyper-K5hler condition and the global symmetries, the components of these metrics each satisfy the same set of coupled non-linear ODE’s as the AH metric. In fact, as we review below, there is precisely a one-parameter family of solutions of these differential equations which have the same asymptotic behaviour as the AH metric to all finite orders in l/r but differ by terms of order exp( -r). Seiberg and Witten showed that the exponentially suppressed terms correspond to instanton effects in the weakly coupled SUSY gauge theory. In this paper we will calculate the one-loop perturbative and oneinstanton contributions to the low-energy theory using standard background field and semiclassical methods, respectively. The results of these calculations suffice to fix the ’ r is the separation between the monopoles in units of the inverse gauge-boson mass in the 3 + l-dimensional gauge theory in which the two BPS monopoles live. ’ In the following, we will, however, retain the weaker assumption that the moduli space has at most isolated singularities.

N. Dorey et al./Nuclear PhysicsB SO2(1997) 59-93

boundary

conditions

that the resulting

for the differential

form of the Coulomb

of the metric.

In Section

three-dimensional correspond

as follows. In Section 2, we introduce

to N = 2 SUSY Yang-Mills

the classical

which determine

the metric. We find

metric is equal to the AH metric, thereby confirming

Seiberg and Witten. The paper is organized its relation

equations

61

the properties

to the BPS monopoles

We also review

of supersymmetric

gauge theory. The field configurations

Section 3 is devoted to obtaining

of the four-dimensional

of

the model and review

theory in four dimensions.

branch and perform an explicit one-loop

3 we discuss

(3D)

the prediction

evaluation

instantons

in question

in

themselves

(4D) gauge theory. 3 Much of

the measure for integration

over the instanton

collective

coordinates. The number of bosonic and fermionic zero modes of the BPS monopole is determined by the Callias index theorem which we briefly review. Like their fourdimensional counterparts, the three-dimensional instantons have a self-duality property which, together with supersymmetry, ensures a large degree of cancellation between non-zero-modes of the bose and fermi fields. However, unlike the four-dimensional case and despite asymmetry

supersymmetry,

time, the residual the quadratic freedom.

this cancellation

of the Dirac operator in a monopole

background.

term which arises from the non-cancelling

fluctuation

operators

We apply the resulting

perturbative

is not complete

correction

of the scalar, fermion, one-instanton

to a four-fermion

measure

because

of the spectral

We calculate,

for the first

ratio of determinants

of

gauge and ghost degrees of to calculate

vertex in the low-energy

the leading

non-

effective action.

In Section 4 we show that the one-loop and one-instanton data calculated in Sections 2 and 3, together with the (super-) symmetries of the model, are sufficient to determine the exact metric on the Coulomb branch. We begin by reviewing the arguments leading to the exact solution of the low-energy theory proposed by Seiberg and Witten. We analyze the solutions of the non-linear ODE’s which determine the metric and exhibit a one-parameter determined

family of solutions

up to one-loop

leads to a different precise agreement

prediction is obtained

fold: the singularity-free series of appendices.

2. N = 4 supersymmetry

which agree with the metric on the Coulomb

in perturbation

branch

theory. We show that each of these solutions

for the one-instanton

effect calculated

only for the solution which corresponds

case. For the most part, calculational

in Section

3 and

to the AH mani-

details are relegated

to a

in 3D

2. I. Fields, symmetries and dimensional

reduction

In this section we will briefly review some basic facts about the N = 4 supersymmetric SU( 2) gauge theory in three dimensions considered in [ 21. It is particularly convenient ‘To avoid confusion with the proposed connection to monopole moduli spaces, from now on the term ‘monopole’ will always refer to the instantons of the three-dimensional SUSY gauge theory unless otherwise stated.

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-9.3

62

to obtain

this theory from the four-dimensional

theory by dimensional

reduction.

In discussing

Euclidean

N = 2 SUSY Yang-Mills

the 4D theory we will adopt the notation

and conventions of [lo]: the N = 2 gauge multiplet contains the gauge field h (m = 0, 1,2,3), a complex scalar A and two species of Weyl fermions 5 and & all in the adjoint

of the gauge group. As in [lo]

representation

fields in the SU(2)

matrix notation,

admits an SU( 2)~

x U( 1)~

$ = X”r”/2.

group of automorphisms.

Abelian factor of the R-symmetry

we use undertwidzing

The resulting

group is anomalous

for the

N = 2 SUSY algebra

In the 4D quantum

theory, the

due to the effect of 4D instantons.

Following Seiberg and Witten, the three-dimensional theory is obtained by compactifying one spatial dimension, 4 say x3, on a circle of radius R. In the following we will restrict our attention to field configurations which are independent of the compactified dimension. This yields a classical field theory in three space-time dimensions which we will then quantize. Seiberg and Wttten also consider the distinct problem of quantizing the N = 2 theory on R3 x

S’. In this approach quantum fluctuations of the fields which

depend on x3 are included

in the path integral and one can interpolate

and 4D quantum

by varying

problem

theories

here. Integrating

R. We will not consider

between

the 3D

this more challenging

over x3 in the action gives

(1) where e = g/a

defines the dimensionful

sionless 4D counterpart Compactifying

3D gauge coupling

in terms of the dimen-

g.

one dimension

breaks

the S0(4)~

N SU( 2)~ x SU(2),

group

of

rotations of four-dimensional Euclidean space-time down to S0(3)~. Following the notation of [ 21, the double-cover of the latter group is denoted SU( 2) E. The 4D gauge field G, splits into a 3D gauge field %, and a real scalar 43 such that $ m = ,u = 0, 1,2 and 4!9 = 3. h into two real scaltk: ( &,

$

It is also convenient = &Re h and $

1+6~)of SU( 2)1 (SU( 2),)

of SU;u2)E: x,” for A = 1,2,3,4.

to decompzse

= &II”&.

can be rearranged Correspondingly,

= tit,

for

the 4D complex scalar

The 4D Weyl spinors

to form four 3D Majorana the two Weyl supercharges

5,

@a N spinors of the

N = 2 theory%e reassembled as four Majorana supercharges which generate the N = 4 supersymmetry of the three-dimensional theory. Details of dimensional reduction, field definitions and our conventions for spinors in three and four dimensions are given in Appendix A. While the number of space-time symmetries decreases reducing from 4D down to 3D, the number of R-symmetries increases. First, the sum symmetry which rotates the two species of Weyl fermions and supercharges in the four-dimensional theory 4 For simplicity of presentation we choose x3 to be the compactified dimension. In practice, however, in order to flow from the standard chiral basis of gamma matrices in 4D to gamma matrices in 3D one has to dimensionally reduce in the x2 direction. To remedy this one can always reshuffle gamma matrices in 4D. See Appendix A for more details.

N. Dorey et al. /Nuclear

Physics B 502 (1997)

59-93

63

remains unbroken in three dimensions. In addition, the U( 1)~ symmetry is enlarged Unlike U( 1)~ in the to a simple group which, following [ 21, we will call sum. 4D case, this symmetry remains unbroken at the quantum level. The real scalars, 4i with i = 1,2,3, transform as a 3 of SU(2),v. The index A = 1,2,3,4 on the Majora; spinors ,$ introduced above reflects the fact that they transform as a 4 of the combined R-symmetry group, S0(4)~

N SU(~)R x sum.

2.2. The low-energy theory The three-dimensional N = 4 theory has flat directions along which the three adjoint scalars r#+acquire mutually commuting expectation values. By a gauge rotation, each of the scalds can be chosen to lie in the third isospin component: (&) = &v~T~/~. After modding out the action of the Weyl group, Vi --) -vi, the three realiarameters vi describe a manifold of gauge-inequivalent vacua. As we will see below, this manifold is only part of the classical Coulomb branch. For any non-zero value of u = (VI, ~2, v3), the gauge group is broken down to U( 1) and two of the gauge bosons acquire masses Mw = fi]u] by the adjoint Higgs mechanism. At the classical level, the global SU(2),v symmetry is also spontaneously broken to an Abelian subgroup U( 1)~ on the Coulomb branch. The remaining component of the gauge field is the massless photon uP = Tr( &Ts) with Abelian field-strength uPV.For each matrix-valued field ,X in the microscopic theory, we define a corresponding massless field in the Abelian low-energy theory: X = Tr( 5~~). Hence, at the classical level, the bosonic part of the low-energy Euclidean action is simply given by the free massless expression

(2) The presence of 3D instantons in the theory means we must also include a surface term in the action, which is analogous to the O-term in four dimensions. In the low-energy theory this term can be written as icr ss = &T

J

d3x .Pp6’CL v UP’

(3)

A dual description of the low-energy theory can be obtained by promoting the parameter u to be a dynamical field [ 111. This field serves as a Lagrange multiplier for the Bianchi identity constraint. In the presence of this constraint one may integrate out the Abelian field strength to obtain the bosonic effective action

‘B = $

d3x

J

+Q!&&

+ ~

2e2

7~(87r)2

d3x $5’ aa I(’(+ 2cc

(4)

The Dirac quantization of magnetic charge, or, equivalently, the 3D instanton topological charge, 1 k=877

J

d3x ~~~~~~~~~E Z

,

(5)

N. Dorey ei al. /Nuclear Physics B 502 (1997) 59-93

64

means that, with the normalization

given in (3), u is a periodic

variable

with period

27r. In the absence of magnetic charge (T only enters through its derivatives and the action has a trivial symmetry, c ----fu + c where c is a constant. The VEV of the a-field spontaneously Coulomb

breaks this symmetry

branch.

Modding

and provides

an extra compact

dimension

out the action of the Weyl group, the classical

for the Coulomb

branch can then be thought of as (R3 x S’)/&. It will be convenient

to write the fermionic

terms in the low-energy

the (dimensionally reduced) Weyl fermions. At the classical free kinetic terms for these massless degrees of freedom,

The Weyl fermions to a complex

the holomorphic respectively.

components

Similarly

Weyl fermions be rewritten

in 4D can be related to the 3D Majorana

basis for the S0(4)~

index. In Appendix

xz are equal to ~~$8

the anti-holomorphic

,Y,” by going

for a = 1 and a = 2,

xt are equal to the left-handed

h, and $a for B = i and ii = 2. In this basis, the effective action (6) can

as

where yIL are gamma matrices in 3D. Perturbative corrections lead to finite corrections in powers of e2/Mw. At the one-loop renormalization of the gauge coupling 2rr

fermions

A we define a basis such that

and ~~$8

components

action in terms of

level, the action contains

27r

1

low-energy

theory

(8)

Z-)e2_27rMt/ This result is demonstrated

to the classical

level several effects occur. First, there is a finite appearing in (4) and (6) :

explicitly

in Appendix

B. Second, there is a more subtle one-

in [ 21. By considering the realization of the U( 1) N symmetry in an instanton background, Seiberg and Witten showed that a particular coupling between the dual photon (T and the other scalars 4i appears in the one-loop effective action. More loop effect discussed

generally the one-loop effective action will contain vertices with arbitrary numbers of boson and fermion legs. However, as we will see in Section 4, the exact form of the low-energy effective action is essentially determined to all orders in perturbation theory by the finite shift in the coupling (8) together with the constraints imposed by N = 4 supersymmetry. In the next section we will turn our attention to the non-perturbative effects which modify this description.

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

3. Supersymmetric

instantons

65

in three dimensions

3.1. Instantons in the microscopic

theory

The four-dimensional N = 2 theory has static BPS monopole solutions of finite energy [ 12,131. In the three-dimensional theory obtained by dimensional reduction, these field configurations

have finite Euclidean

action,

S,,(k) - 1kl&

for magnetic

charge k, with

& = ( 8rr2&fw>/e 2. For each value of k, the monopole solutions are exact minima of the action which yield contributions of order exp( -Jkl&) to the partition function and Greens functions instantons

with Appendix

C),

Hence BPS monopoles

appear as

gauge theory in 3D. In this section

as well as presenting

several

general

results,

(together

we will provide

a

of these effects for the case k = 1. We begin by determining the and fermionic zero-modes of the instanton and the corresponding collective

quantitative bosonic

of the theory at weak coupling.

in the N = 4 supersymmetric

coordinates.

analysis

We then consider

the contribution

of non-zero

frequency

modes

to the

instanton measure which requires the evaluation of a ratio of functional determinants. In Section 3.2, we use these results to calculate the leading non-perturbative correction to a four-fermion vertex in the low-energy effective action. To exhibit the properties of the 3D instantons, it is particularly convenient to work with the (dimensionally reduced) fields of the four-dimensional theory and also with the specific vacuum choice vi = vSis. In this case, the static Bogomol’nyi by the gauge and Higgs components self-dual

Yang-Mills

equation

of the monopole

for the four-dimensional

equation

satisfied

can be concisely

rewritten

as a

gauge field, ti

of Section

2.1

[ 141 5 cl _* cl L!T?Wz Kit,“.

Because of this self-duality, instantons in three dimensions have many features in common with their four-dimensional counterparts. In the following, we will focus primarily on the new features which are special to instanton effects in three-dimensional theories. One important difference is that instantons in 3D are exact solutions equations

of motion

in the spontaneously

broken phase. In four dimensions,

gauge of the

instantons

are only quasi-solutions in the presence of a VEV and this leads to several complications which do not occur in the 3D case. Bosonic

zero-modes,

Sf?” = Em, of the 3D k-instanton

by solving the linearized self-duality constraint (see Appendix C),

equation

subject

configuration

to the background

are obtained field gauge

(10) ’ Of course there is no loss of generality here, for any choice of three-dimensional vacuum vi it is possible to construct an analogous four-dimensional Euclidean gauge field which is self-dual in the monopole background: one simply needs to include the component vi+i of the scalar as the ‘fourth’ component of the gauge field. However, only if we choose vi = v8ia does this correspond to the four-dimensional gauge field of Section 2.1.

N. Dorey et aLlNuclear Physics B 502 (1997) 59-93

66

where, DI{ is the adjoint gauge-covariant derivative in the self-dual gauge background, of self-duality, there is an exact correspondence between these 2.” As a consequence bosonic zero-modes and the fermionic zero-modes, which are solutions of the adjoint Dirac equation in the self-dual background. The latter is precisely the equation of motion for the 4D Weyl fermions

in the monopole

background,

(11) (12) Following Weinberg [ 151, we form the bispinor operators, A+ =$!)JDcl and A_ =JVclpCl. In general, for a self-dual background field,6 &,

A-

can have normalizable

number

of normalizable

zero-modes,

zero-modes

indices, the number of normalizable Adapting

while A+ is positive and has none. Let the of A- be q. Then correctly accounting for spinor

zero-modes,

[ 151 showed that q could be obtained -&--A- +,u

p A++P

Z_, of the gauge field & is 2q [ 151.

[ 161 to the context of BPS monopoles,

the Callias index theorem

as the p -+ 0 limit of the regularized

trace

1

(14)

defined for p > 0. It turns out that the only non-zero a surface term that can be evaluated

Weinberg

explicitly

contribution

for arbitrary

to 1(p)

k. Weinberg’s

comes from result is

2kMw ICcL)

Setting

=

(4/

,u = 0 yields

Dirac equation

(15)

+ &/2’

(11)

q =

2k. Another

has 2k independent

consequence solutions

of this analysis

is that the adjoint

while (12) has none. Although

the

index information is contained in the ,u + 0 limit of Eq. (15) we will need this result for general p in the following. The upshot of Weinberg’s index calculation is the conventional wisdom that the BPS monopole of charge k, or, equivalently, the 3D k-instanton, has 4k bosonic collective coordinates. For k = 1 these simply correspond to the three components, X,, of the instanton position in three-dimensional space-time and an additional angle, 0 E [ 0,27r], which describes the orientation of the instanton in the unbroken U( 1) gauge subgroup. In an instanton calculation we must integrate over these coordinates with the measure obtained by changing variables in the path integral. Explicitly,

6 We define standard

self-dual

and anti-self-dual

projectors,

CT’“”= fcr[‘“~l

and ~7”~ = iF@ti].

N. Dorey et al./Nuclear

Physics B 502 (1997) 59-93

67

(16) In Appendix

C, we calculate

Similarly

the two species

zero-mode

solutions

correspond instanton

the Jacobian

factors, JX = $1 and & = &t/M& k and (/I each have 2k independent

of Weyl fermions,

in the instanton-number

k backgrounds: For k = 1 these four modes

to the action of the four supersymmetry transforms

non-trivially.

can be parametrized

generators

As in the four-dimensional

in terms of two-component

under which the 3D

case, the modes in question

Grassmann

collective

coordinates

ta

and 6: as $! = ;~&r”‘~)/,&,, $1 = $$?( &Y”),fi&~. The corresponding

(17)

contribution

to the instanton

measure is

(18) In Appendix

C we find that Jt = 2&t.

As usual in any saddle-point result it is necessary

calculation,

to obtain

the leading-order

to perform Gaussian integrals over the small fluctuations

semiclassical of the fields

around the classical background. In general these integrals yield functional determinants of the operators which appear at quadratic order in the expansion around the instanton. A simplifying fluctuation

feature

operators

that holds for all self-dual

A+ or A_. This standard

for 4D Majorana

fermions

(see Appendix

connection A),

0 %I

( )

AF= pc, 0 Performing

Pf( Ak’))

is as follows. In the

the quadratic

fluctuation

(19)

’

the Grassmann

Pf(‘b)

is that each of the

for scalars, spinors gauge fields and ghosts are related in a simple

way to one of the operators chiral basis operator is

configurations

Gaussian

integration

over A yields

(20)

where det’ denotes the removal of zero eigenvalues and the superscript (0) denotes the fluctuation operator for the corresponding field in the vacuum sector. In particular we define the operator A(O) = $J (‘) p (‘) =$J’“~(o’, formed from the vacuum Dirac operators (0) and p”‘. Integrating out the I+ field yields a second factor equal to (20). For Yj the bose fields we introduce the ga& fixing and ghost terms using the 4D background gauge, ViS$, = 0, and expand the action around the classical configuration. Quadratic

68

N. Dorey et al./Nuclear Physics B 502 (1997) 59-93

fluctuation

operators

for the complex

scalar,

5, the gauge field, 2 and the ghosts, G

and 2 are iTr(A+),

4, = -D&n

= -~Tr(c+“A_cP), + 2&;!”

(21)

where Tr is over the spinor indices. The bosonic Gaussian

Combining

the fermion

non-zero-modes

and boson contributions,

to the instanton

measure

integrations

now give

we find that the total contribution

of

is given by

(23) In any supersymmetric theory the total number of non-zero eigenvalues of Bose and Fermi fields is precisely equal. This corresponds to the fact that, for any selfdual background,

the non-zero

eigenvalues

of A + and A_ are equal

[ 171. Naively,

this suggests that the ratio R is unity. As the spectra of these two operators contain a continuum of scattering states in addition to normalizable bound states, this assertion must be considered

carefully. For the continuum

and A_ to be equal, it is necessary

contributions

not only that the continuous

same range, but that the density of these eigenvalues the original

to the determinants eigenvalues

of A+ have the

should also be the same. Following

approach of ‘t Hooft [ 181 in four dimensions,

one can regulate the problem

by putting the system in a spherical box with fixed boundary conditions. In this case the spectrum of scattering modes becomes discrete and the resulting eigenvalues depend on the phase shifts of the scattering eigenstates. In the limit where the box size goes to infinity, these phase shifts determine the density of continuum eigenvalues. For a fourdimensional instanton, ‘t Hooft famously discovered that the phase shifts in question were equal for the small fluctuation operators of each of the fields. As a direct result of this, the ratio R is equal to unity in the background of any number of instantons in a 4D supersymmetric gauge theory. However, the phase shifts associated with the operators A+ and A_ are not equal in a monopole background. In the four-dimensional theory, Kaul [ 191 noticed that this effect leads to a non-cancellation of quantum corrections to the monopole mass. In our case, as we will show below, it will yield a non-trivial value of R. In fact the mismatch in the continuous spectra of A+ and A- can be seen already from the index function 2(p) . Comparing with the definition ( 14)) we see that the fact that Weinberg’s formula ( 15) has a non-trivial dependence on p and is not simply equal to 2k precisely indicates a difference between the non-zero spectra of the two operators. This observation can be made precise by the following steps. First, dividing (14) by p and then performing a parametric integration we obtain

69

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93 O”d/l s

7

Z(,u’) = Tr [log (A+ + ,u) - log (A- + ,u)] .

(24)

II The amputated

determinant

appearing

det(Ap The power of ,u appearing of A- calculated a closed formula

R=fs

Evaluating

1*

(25)

in the denominator

above. Using the relation for R:

[p2kexp

reflects the number

of zero eigenvalues

Trlog( 8) = logdet( 8) in (24),

we obtain

(26)

([$z(p’))]“2.

this formula

can combine one-instanton

+ p) 2k

in the ratio R is properly defined as

on ( 15) we obtain

the result R = (~Mw)~~.

For k = 1, we

the various factors ( 16), ( 18) and (23) to obtain the final result for the measure

JdIL’kEI’=SdPBSd~~Rexp(-S,,+irr) =$f

s

Here we have performed we are ultimately contribution

d3Xd25d2[‘exp(-SC1 the integration

interested

+iu).

(27)

over B, anticipating

in will not depend

the fact that the integrands

on this variable.

The term icr is the

of the surface term (3) which we will discuss further below.

3.2. Instanton effects in the low-energy theory Because of their long-range fields, instantons and anti-instantons have a dramatic effect on the low-energy dynamics of three-dimensional gauge theories. As they can be thought of as magnetic

charges in (3 + 1) dimensions,

3D instantons

and anti-instantons

experience a long-range Coulomb interaction. In the presence of a massless adjoint scalar, the repulsive force between like magnetic charges is cancelled. This cancellation is reflected in the existence of static multi-monopole solutions in the BPS limit. However, in the case of an instanton purely bosonic

and anti-instanton,

gauge theory, Polyakov

there is always an attractive

force. In a

[ 1 l] showed that a dilute gas of these objects

leads to the confinement of electric charge. The long-range effects of instantons and antiinstantons can be captured by including terms in the low-energy effective Lagrangian of the characteristic form

LI - exp

(

8~21~1 Iiu

e2

> ’

where 141* = 4: + q!$ + 4:. This is simply the instanton action of the previous with the VEVs vi and u being promoted to dynamical fields.

(28) section,

N. Dorey et al./Nuclear

70

In the presence action necessarily

of massless

fermions,

Physics B SO2 (1997) 59-93

the instanton

contribution

to the low-energy

couples to n fermion fields, where II is the number of zero-modes

of

the Dirac operator in the monopole background. In the three-dimensional N = 2 theory considered in [20], instantons induce a mass term for the fermions in the effective action. In fact this corresponds Coulomb

to an instanton-induced

superpotential

which lifts the

branch. In the present case, the effective action will contain an induced four-

fermion vertex due to the contribution of a single instanton [ 21. This vertex has a simple form when written in terms of the (dimensionally reduced) Weyl fermions of (6)) St = K

J

d3.x ii2tj2exp

where the coefficient

K

--

will be calculated

explicitly

below. In this case, N = 4 SUSY

does not allow the generation of a superpotential and, as we will review below, the four-fermion vertex is instead the supersymmetric completion of an instanton correction to the metric on the moduli space. Just like the vertex induced by (four-dimensional) instantons in the four-dimensional N = 2 theory, self-duality means that the above vertex contains only Weyl fermions of a single chirality. Equivalently, the vertex contains only the holomorphic components of the 3D Majorana fermions in the complex basis of (7). In the 4D case, the chiral form of the vertex signals of our dimensional theory corresponds global symmetry

the presence reduction

of an anomaly

to U( 1)N defined in Section in the 3D theory. More precisely

as QN = QR!/~. Hence each of the right-handed - l/2,

which suggests

in the U( 1)~

and choice of vacuum,

symmetry.

By virtue

the U( 1)~ symmetry

of the 4D

1, the unbroken the corresponding

subgroup

of SU( 2)~

charges are related

4D Weyl fermions carries U( 1)~ charge

that the induced vertex (29) violates the conservation

of U( 1) N

by -2 units. However, as explained

in [2], the symmetry is non-anomalous in 3D due to the presence of the surface term iu in the instanton exponent. Assigning g the U( 1) N transformation (T+(T+2a

(30)

under the action of exp( iffQN), the symmetry ever, because the VEV of u transforms

of the effective action is restored. How-

non-trivially,

the symmetry

is now spontaneously

broken and hence SU(2)N has no unbroken subgroup. An equivalent statement is that the orbit of SU(2)N on the classical moduli space is three-dimensional, a fact that will play an important role in the considerations of the next section. In the following we will be interested in the instanton corrections to the leading terms in the derivative expansion of the low-energy effective action. As in the fourdimensional theory, this restricts our attention to terms with at most two derivatives or four fermions. However, an important difference with that case is that the 3D instantons are exact solutions of the equations of motion and there is no mechanism for the VEV to lift fermion zero-modes in a way that preserves the U( 1) N symmetry. 7 It follows ’A

more

general analysis of this

issue will be presented elsewhere.

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

that the only sectors of the theory which can contribute low-energy

effective

action

are those with instanton

71

to the leading

number

(magnetic

terms in the charge)

- 1,

0 or +l. Another important difference with the 4D case is that, because there is no holomorphic prepotential in 3D, there can also be contributions to the effective action from configurations Finally

containing

we will compute

examining

the large-distance

G(4)(~,,~2,~3r~4)

of instanton/anti-instanton of the four-anti-fermion

pairs. vertex (29) by

of the correlator (31)

=(A,(Xl)Ap(XZ)~~(X3)~~(X4)),

where the Weyl fermion

fields are replaced

background.

fermions

are straightforward

(LD)

numbers

behaviour

instanton

the magnetic

arbitrary

the exact coefficient

The explicit

monopole

formulae

to extract,

by their zero-mode

values

for the zero-modes

of the low-energy

via (17)

fields. Using the formulae

limit of the fermion

zero-modes

AbD=8?T(&(X

-X)),‘&,

&” =87r(&(X

-X)),‘&.

from the long-range of Appendix

in the onebehaviour

of

C for the large-distance

we find

(32)

This asymptotic form is valid for Ix - X( > M,’ where X is the instanton position and SF(x) = Y,J~/(~~~]xI~) is the three-dimensional Weyl fermion propagator. The leading semiclassical

contribution

to the cot-relator (3 1) is given by n

Gc4) ( XI 1 x2 I x3 9 x4)

where dp”‘k”)

= I

d~“‘““A~D(x,)A~(X2)~~(X3)~~D(X4),

is the one-instanton

measure

(27).

(33)

Performing

the 6 and 5’ integrations

we obtain Gc4)(xt, x2, ~3, x4) =29?r3i’t4tvexp( -&

d3XkP’SF(xI

+ ic+)

- X),,,

I x&(x2

This result is equivalent low-energy

-

x)~~~~y’“‘&(x3

to the contribution

action (6). Our calculation

-

x),,&(~~

of the vertex (29)

shows that the coefficient

-

x)&&r.

(34)

added to the classical K

takes the value (35)

where the four powers of (2~/e2) kinetic terms in (6).

reflect our choice of normalization

for the fermion

4. The exact low-energy effective action In this section, following the arguments of [2], we will determine the exact lowenergy effective action of the N = 4 SUSY gauge theory in three dimensions. Below,

72

N. Dorey et al./Nuclear Physics B 502 (1997) 59-93

we will write down the most general effective

possible

action with at most two derivatives

the symmetries

of the model.

supersymmetry

and the global sum

differential determines

which is consistent

As we will review, the combined symmetry

restrictions

with

of N = 4

lead to a set of non-linear

ordinary

equations for the components of the hyperK5hler metric which in turn the relevant terms in the effective action. We will find a one-parameter family

of solutions of Section

ansatz for the terms in the low-energy

or four fermions

of these equations

which agree with the one-loop

2. Our main result is that the one-instanton

vertex, calculated

from first principles

Atiyah-Hitchin

manifold

We begin

by discussing

in Section 3, uniquely

the general

d. The low-energy

supersymmetric

calculation

to the four-fermion

selects the metric of the

from this family of solutions. case of a low-energy

{ 0:) {Xi} and Majorana * fermion superpartners scalars define coordinates on the quantum moduli real dimension

perturbative

contribution

non-linear

theory with scalar fields

where i = 1,. . . , d. As usual the space, M,

which is a manifold

of

effective action has the form of a three-dimensional

o-model

with M as the target manifold,

(36) where K is an overall constant included for later convenience. The kinetic terms in the action define a metric gij on the moduli space M, and Jb and Rijkl denote the corresponding Following

covariant

Dirac operator and Riemann tensor respectively. admitted and Freedman [ 91, the supersymmetries

Alvarez-GaumC

by the

above action are written in the form

q=2

The action (36) is automatically

invariant

under the N = 1 supersymmetry

parametrized

bye ‘I1 . However, N > 1 supersymmetry requires the existence of N - 1 linearly independent tensors Jlqlij which commute with the infinitesimal generators of the holonomy group H of the target. It is also necessary

that these tensors

form an su(2)

algebra.

In general the holonomy group of a d-dimensional Riemannian manifold is a subgroup of SO(d). For N = 4, the existence of three such tensors which commute with the holonomy generators imply that d must divisible by 4 and that H is restricted to be a subgroup of Sp (d/4). Such manifold of symplectic holonomy is by definition hyperKahler. The tensors J”lij (q = 2,3,4) define three inequivalent complex structures on M. An equivalent statement of the hyper-K%hler condition is to require that these complex structures be covariantly constant with respect to the metric gij. In the case d = 4, the holonomy can be chosen to lie in the Sp( 1) N SU( 2) subgroup of SO(4) generated by the self-dual tensors @, a = 1,2,3 (defined in Appendix A). * The 3D Majorana

condition

is fi = iR, see Appendix

A.

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

In fact the holonomy sufficient

condition

generators

are components

for a hyper-Kalrler

73

of the Riemann

four-manifold

Rijkl and a

tensor

is for this tensor to be self-dual. 9

Correspondingly the complex structures J “Iii(X) (q = 2,3,4) can be taken as linear combinations of the anti-self-dual SO(4) generators $. As indicated, the relevant linear combinations

appearing

to another.

Another

in (37) will vary as one moves from one point on the manifold

restriction

global SU( 2) N symmetry. either in perturbation exact quantum

space M comes from the action of the

on the moduli

As we have seen above, there is no anomaly

theory or from non-perturbative

moduli

space has an SU( 2)~

in this symmetry

effects, hence we expect that the

isometry.

Further,

because

of the non-

trivial transformation of the dual photon described above, we know that sum generically have three-dimensional orbits on M. It can also be checked explicitly the three complex Remarkably,

structures

the problem

on M introduced of classifying

four with the required isometry As discussed moduli

in Section

all manifolds

1, these are exactly the properties

was required

as a 3 of Sum. manifolds

of dimension

of the reduced or centered

of gauge group SU(2).

[ 81

Atiyah and Hitchin

with these properties and showed that there is only one manifold

which has no singularities: singularities

that

has been solved in an entirely different physical context.

space of two BPS monopoles

considered

above transform

all the hyper-K5hler

will

the AH manifold.

In the original

because of known properties

context,

of multi-monopole

the absence

of

solutions.

In

particular, the metric on the moduli space of an arbitrary number of BPS monopoles was known to be complete. This means that every curve of finite length on the manifold has a limit point (see Chapter 3 of Ref. [ 81 and references therein). In the present context,

the absence

an assumption assumption

of singularities

leads directly

to Seiberg

space has the same hyper-Klhler show that the one-instanton section,

on the quantum

about the strong-coupling

behaviour and Witten’s

proposal

metric as the AH manifold.

contribution,

that the quantum

moduli

In the following

we will

Eqs. (35) and (29), calculated

together with the results of one-loop

proof of the SW proposal

moduli space M can be taken as of the 3D SUSY gauge theory. This

without assuming

perturbation

in the previous

theory (8), provides

the absence of strong-coupling

a direct

singularities.

Following Gibbons and Manton [ 2 I], we parametrize the orbits of SO( 3) N (whose double-cover is SUN) by Euler angles 0, 4 and + with ranges, 0 < t9 6 rr, 0 < 4 < 2~ and 0 < 1/, < 2~ and introduce the standard left-invariant one-forms (71 = - sin t,bdO+ cos $ sin edc$, u2 = cos +d0 + sin ti sin 0dq5, u3 = dfi + cos 8dqb. The remaining dimension of the moduli space M, transverse is labelled by a parameter r and the most general possible isometry takes the form [22]

(38) to the orbits of S0(3)~, metric with the required

9The more. conventional choice of an anti-self-dual Riemann tensor and self-dual related to this one by a simple redefinition of the fields and the parameters elql.

complex

structures

is

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

74

gijdX’dX’ = f*(r)dr* The function

f(r)

+ a*(r)a:

depends

define the corresponding

+ b*(r)ai

on the definition

Cartesian

+ c*(~)cT:.

(39)

of the radial parameter

r. We will also

coordinates,

X=rsinBcos4, Y=rsinBsin4, Z=rcos8.

(40)

We start by identifying fields of Section under

S0(3)~.

the parameters

introduced

2.2 in a way which is consistent The triplet

(X, YZ)

transforms

above in terms of the low-energy with their transformation

as a vector of S0(3)~

properties and will be

identified, up to a resealing, with the triplet of scalar fields appearing in the Abelian classical action (4) : (X, x Z) = (&/Mw) (c#J~,&, 43 ) . This means that we are defining the parameter

r to be equal to $1. The remaining

Euler angle, +, by definition

transforms

as t+G-+ $ + (Y under a rotation through an angle cy about the axis defined by the vector (X, Y Z) . After comparing

this transformation

property

with the U( 1) N transformation

(30) of the dual photon (T, we set $ = (r/2. In the following as standard

we will refer to (X, Y Z, a)

coordinates.

The weak-coupling behaviour of the metric gij can be deduced by comparing the general low-energy effective action (36) with its classical counterpart (4), together with the one-loop correction (8). Choosing the constant K in (36) to be equal to 2e2/(7r( SV)*), the classical metric is just the flat one, S,, in the standard coordinates introduced

above. Including

functions,

a*, b* and c* are given by

the one-loop renormalization

of the coupling

(8)) the metric

a2=b2e&(Scl-2), c* N 4 + S/&t. Corrections

(41)

to the r.h.s. of the above formulae

are down by powers

of l/&t.

come from two loops and higher and

In fact, the equality

a2 = b2 persists

to all orders in

perturbation theory, reflecting the fact that U( 1)~ is only broken (spontaneously) by non-perturbative effects. To this order, the function f* is also determined to be equal to 1 - 2/&I + OU/S~& The Majorana fermions L!i appearing in (36), will be real linear combinations of the fermions xi of the low-energy action (6). In general, this linear relation will have a non-trivial dependence on the bosonic coordinates. For our purpose it will be sufficient to determine this relation at leading order in the weak-coupling limit, r = $1 + oc. In the standard coordinates we set $

=~iAu%~~~)XAa9

(42)

up to corrections of order 1/S,t. It is convenient to complexify the S0(4)~ index as in Section 2.2 and consider instead coefficients Mia and Mia = ( Mia)*. In order to reproduce the fermion kinetic term in (7) these coefficients must obey the relations

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

SijMi”Mib

aijM’“j@

=

The first condition second is required

=0

(43)

fixes the overall

normalization

of the fermions

for their kinetic term to be U( 1)~ invariant.

present in this identification The hyper-Kahler ential equations

15

is related to the R-symmetry

condition

can be formulated a, b, c and

for the functions

in (36)

while the

The remaining

freedom

of the N = 4 SUSY algebra.

as a set of non-linear

ordinary

differ-

f: (44)

together

with the two equations

obtained

of a, b and c. The

by cyclic permutation

solutions of these equations are analyzed in detail in Chapter 9 of Ref. [8] and we will adapt the analysis given there to our current purposes. The equations completely determine choice Manton

of a, b and c as functions

the behaviour

of r only once one makes a specific

f, for example the choice f = -b/r

for the function

made by Gibbons

and

[ 211. If one makes such a choice in the present context, then the corresponding

relation between

I and the weak-coupling

which cannot be determined. and, correspondingly,

parameter

$1 will receive quantum corrections

In fact we have chosen instead to define r to be equal to $1

this implies some choice for f which can only be determined

by order in perturbation

theory. This reflects an important

order

feature of the exact results for

the low-energy structure of SUSY gauge theories which is familiar from four dimensions. In these theories the constraints of supersymmetry and (in the 4D case) duality allow one to specify the exact quantum moduli space as a Riemannian manifold. However, the Lagrangian a particular

fields and couplings of the conventional weak-coupling description define coordinate system on the manifold and sometimes the relation of these

parameters

to the parameters

determined

explicitly.

between

the tree-level

of the exact low-energy

As discussed coupling

effective

in [ 241, just such an ambiguity

constant

and the exact low-energy

four-dimensional N = 2 theory with four hypermultiplets. In the light of the above discussion we will eliminate both and focus on the information parametrization. and y = c/b, dY z=

Following

Y(1 -y)(l+y-x) x(1-x)(1+x-y)’

Lagrangian

f and

arises in the relation coupling r

about the metric which is independent

Ref. [ 81 we obtain a single differential

can not be in the finite

from the equations, of the choice of

equation

for x = a/b

(45)

The solutions of this equation are described by curves or trajectories in the (x, y) plane. In the weak coupling limit &t + 00 we have u2 = b2 --+ cm and c2 + 4. Hence we must find all the solutions of (45) which pass through the point Q with coordinates x = 1, y = 0 (see Fig. 1). The relevant features of these solutions are as follows: ( 1) A particular solution which passes through Q is the trajectory x = 1. Returning to the full equations (44), we find that this corresponds to the solution

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

Fig. I. Some members of the one-pammeter family of solutions to the non-linear ODE (45) described in the text. The solution which terminates at the point P’ corresponds to the Atiyah-Hitchin manifold.

(1_;2,q

u=b= Eliminating

(46)

$1 in (41)) we find this relation

is obeyed up to one-loop

in perturbation

theory as long as we choose square roots so that UC = bc < 0. This solution the singular

Taub-NW

the low-energy

geometry

and, as discussed

theory up to non-perturbative

effective action (36) includes solution,

so the all-orders

corrections.

the sum of all corrections

theory. However, as we have commented effective

describes

in [2], this is the exact solution

of

In other words the resulting from all orders in perturbation

above, the function

f is not determined

action cannot be written explicitly

by the

in terms of the

weak coupling parameter &I. (2) There is precisely a one-parameter family of solutions passing through Q, each of which is exponentially close to the line x = 1 (y < 0) near Q. Linearizing around x = 1, y = 0 we may integrate (45) to obtain the leading asymptotic behaviour (47) where B is a constant of integration. Corrections to the r.h.s. are down by powers of y = c/b or by powers of exp( 2/y). (3) Using numerical methods to examine the behaviour of these solutions away from the point Q, one finds that there is a unique critical trajectory which originates at the point P’ with coordinates (0, -1). All other trajectories originate either at the origin (0,O) or at negative infinity (0, -w). Atiyah and Hitchin show that only the critical trajectory corresponds to a complete manifold. The critical solution can be constructed explicitly in terms of elliptic functions, its asymptotic form near Q is given in Gibbons and Manton [ 211 (see Eq. (3.14) of this reference) and agrees with (47) with a

N. Dorey et al. /Nuclear

specific value for the integration B # B,, correspond

to singular

Using the identifications

Physics B 502 (I 997) 59-93

constant

B = B,, = 16 exp( -2).

II

All trajectories

with

geometries.

(41)) the asymptotic

behaviour

(47) becomes

a-b--8qS$exp(-&I), where

q =

(48)

Hence

B/B,,.

the leading

comes with exactly the exponential When substituted contribution expected

in the metric

In principle,

(39)

from the perturbative characteristic

and the effective

relation

of a one-instanton

action (36),

a = b effect.

this term yields a

to the boson kinetic terms which also comes with the phase factor exp( fia)

for the (anti-)instanton

parametrized

deviation

suppression

by the constant

q

it is straightforward

of a semiclassical

calculation

term. Further, each member yields a different prediction to check the coefficient of the scalar propagator.

of the family of solutions

for this one-instanton

effect.

of this term against the results This would involve calculating

the Grassmann bilinear contributions to the scalar field which come from the fermion zero-modes in the monopole background. In the following, we will choose instead to extract a prediction

for the four-fermion

the result (35) of Section 3. The effective action (36) contains

vertex in (36) and compare this directly

a four-fermion

vertex proportional

with

to the Riemann

tensor. To make contact with the results of Section 3 where the vacuum is chosen to lie in the ~75sdirection

in orbit of SU( 2)~, we evaluate the Riemann

tensor corresponding

to

the metric (39) at the point on the manifold with standard coordinates (0,0,r = $1, a). This calculation is presented in Appendix D (many of the necessary results have been given previously

by Gauntlett

and Harvey in Appendix

B of [23] ). In particular,

we

evaluate the leading-order contribution to the c-dependent terms in the Riemann tensor in the weak-coupling limit: $1 + co. The result is best expressed in the complex basis with coordinates z1=

&x-iY),

In this basis, the relevant

z2=~(Z-iu). contribution

to the Riemann

(49) tensor is pure holomorphic

and

is given by R1212 = 8q& exp (-&I

+ ic+) ,

(50)

where the other pure holomorphic components are related to this one by the usual symmetries of the Riemann tensor: Rakd = -Rbacd = Rcdab = -Rabdc. The anti-instanton contribution is pure anti-holomorphic and all components of mixed holomorphy are independent of u. The above result for the Riemann tensor means that the corresponding four-fermion vertex in the non-linear a-model (36), has exactly the chiral form expected from the discussion of Section 3.2. In particular, invariance of the vertex under V( 1)~ transformations implies that the holomorphic (anti-holomorphic) components P (LP) of the target space fermions have V( 1) ,v charge -l/2 (+ l/2). Hence we complete our identification of the fermions by demanding that the transformation (42) maps fermions

N. Dorey et al. /Nuclear Physics B SO2 (1997) 59-93

78

of positive

(negative)

U( 1)~ charge to fermions

At the chosen point (O,O, Y = $1, a), in the complex

basis (49) and ( xb, x”)

f$ cvMabx; N

g

of positive

(negative)

U( 1)~ charge.

we write the relation between fermions,

(,(2”, @)

in the complex basis of (7) as

1

M,&,

(51)

where MC5 = (Mob)* and, as in (42),

corrections

to the r.h.s. are down by inverse

Mab in (43) is satisfied by choosing M = (&/Mw) $2,

powers of $1. The second condition

in (43) is automatically

as a 2 x 2 matrix, the first condition

satisfied. Regarding

where I$? E XJ( 2) is a residual degree of freedom associated with the unbroken symmetry. ‘O Finally

we calculate

tribution

the four-fermion

to the Riemann

Riemann

tensor

described

tensor

(50).

SU( 2)~

vertex which follows from the instanton After taking into account the symmetries

above and performing

a Fierz rearrangement

conof the

the resulting

vertex becomes ,c4F = ~KR,~,~(d .a') (fc

"2).

(52)

We rewrite the vertex in terms of Weyl fermions

(0’ .a’) Collecting

(fi2. .n’) = (det(M))2

using

(x’ . ,y’) (x2. x2) = (2)’

i2$2_

together the various factors, the final result for the induced four-fermi

(53) vertex

is 134F =

Comparing induced

2’n-sqMW

X2ij2 exp (-$1

+ ia) .

value (35) for the coefficient K in the instantonwe deduce that q = 1. This implies that the quantum moduli space

this with the calculated

vertex (29),

of the theory is in fact the Atiyah-Hitchin

manifold.

Acknowledgements The authors would like to thank C. Fraser, N. Manton, B. Schroers and E. Witten for useful discussions. N.D., V.V.K. and M.P.M. would like to thank the Isaac Newton Institute for hospitality while part of this work was completed. N.D. is supported by a PPARC Advanced Research Fellowship. M.P.M. is supported by the Department of Energy. lo Strictly speaking M is only restricted to lie in U(2). However,the additional phase can be reabsorbed by changing the identification 9 = u/2 made above by an additive constant.

N. Dozy et al. /Nuclear Physics B 502 (I 997) 59-93

19

Appendix A. Dimensional reduction In this appendix we present our conventions and give the details of dimensional reduction. We work in Minkowski ” space in 4D and 3D with the metric signature (+, -, -, . ..) andm,n=0,1,2,3,,u,v=0,1,2. We start with the N = 2 supersymmetric

Here ti

Yang-Mills

is the gauge field, & is the complex

their superpartners,

while

in 4D

scalar field, Weyl fermions

g and E are auxiliary

h and @ are

fields which will be integratezout.

Also $Uh = V”,cd, and $‘a = P@$F, where V,,, z = a,,,$ - i[&*, $1. Wess and Bagger [25] spinor summation conventions are used throughout and sigma-matrices in Minkowski space are, flab = (-1,7a), ernbn = (-1,-F). The three-dimensional theory is obtained by making one spatial

dimension

and decoupling

this dimension

only subtle point in this program

is the dimensional

the two-component

can be combined

Weyl spinors

all the fields independent

of

from the theory, Eq. ( 1). The reduction

of the fermions.

into the four-component

In 4D

Majorana

spinors

(A.2) in such a way that igh+i@&=i&,r$D,,r;?Fh,

(A.3)

where Ql, is the gamma matrix of the 4D theory in the standard chiral basis, (A.4) Thus,

&, and +& are the Majorana

fermions

in the chiral basis.

For the purpoyes of dimensional reduction it is more convenient to choose a different - real - basis for Majorana spinors in 4D related by a unitary transformation to the chiral basis above, l1 With the exception of this appendix and the next one, the calculations in the rest of the paper are performed in Euclidean space.

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

80

(A.51 where new (real)

two-spinors

are simply the ‘real’ and ‘imaginary’

parts of the Weyl

two-spinors,

(‘4.6) The 4D gamma matrices

in this basis are

l+(_;2 -C), r:e=(iy3 y).

rq;

Yl) ,

Now the dimensional

reduction

xa,

fa

from 4D to 3D is straightforward,

and z*, qa become

matricesxatis?ying

the Clifzrd

Majorana

- (ya,yt,y2).

spinors

one has to decouple The four real two-

in 3D and the three gamma

algebra in 3D can be read off from (A.7):

-ir3, y2 = i$. Note that the 3D Majorana

condition

that the spinor is real since the charge conjugation the Dirac conjugate

(A.7)

(xc, x1, x2, x3) + (x0,x1,x3)

the second dimension, spmors

(_yrl-:) .

rL=

(CITC= $tya

y” = r2, y’ =

is now the condition

matrix is C = r2. We have defined

in 3D as 6 = @tya. For a Majorana

spinor this means

R” = ix”.

By renaming the gamma matrices in 4D we can always choose the third&d second dimension to decouple. This will always be assumed, see footnote 4. Finally

J!J?I=

we define bosonic fields fi,2,3 and a *,

m=0,1,2

23 9

m=3

A=” -

’

no^; the

in 3D as follows:

41 + i42

(A.81

and the 4D action (A. 1) becomes

+(r~~~212+

rt29231”+

Here Z?,,p = DP ( yfi ) ,_/ with P

= P

[~3.$12)

- i[u/,

>

.

(A.9)

] and Y’,‘,~ satisfy {yp, y”} = 2gfi”.

This action can be written in a manifestly S0(4)~ 4 is the S0(4)~ s* = (x,f,r],-2)), where A = l,..., NNN self-dual and anti-self-dual ‘t Hooft q’ matrices [ 181

invariant form. First, define index. Second, introduce the

N. Dorey et al. /Nuclear

rlf4B =

EiAB

AyB=

-SBi

A ~4 B=4

SAi

With this definition,

Physics B 502 (1997)

xi+9

A=4

71 is self-dual

&I*

and 75 anti-self-dual

+ 2$7ji,zAzB

su(2)

.

with respect

algebras,

(A.lO)

to e1234 = +l.

as [vi, $1

= 0. In this

(A.ll)

* >

i
The Lagrangian

81

1,273

Moreover, they form two sets of commuting notation the action (A.9) takes the form

+

59-93

has a global SO(4) R N SU( 2)~ x SU( 2)~

symmetry.

The SU( 2)~

leaves the three scalar fields invariant, and acts on the fermions. In terms of the fourdimensional Weyl fermions ( h t) forms a doublet under SU(~)R. Rewriting this in terms of the Majorana’s

in 3D, one finds (A.12)

The SUN

group is the remnant

of a three-dimensional

D = 6 theory. The scalar fields transform f

++ exp (PkRk)’ jf

rotation

group in the N = 1,

as

,

(A.13)

kij , e123 = 1 are the standard rotation group generators. where ( Rk)‘j = ?? SU( 2)~ acts as

gAH fw ($kfiL) ,I/.

On the fermions,

(A.14)

These transformations The supersymmetry

leave the microscopic theory invariant, as one can check explicitly. transformation rules for the scalar fields are given by

They can be obtained

from a dimensional

reduction

of the SUSY rules in 4D. One can

check that SU( 2)~ and Sum are invariances of this transformation. Finally we can relate the Majorana fermions xA to the (dimensionally reduced) Weyl fermions & and fl by complexifying N

in the S0?4)~

index. We define a complex basis:

(A.16)

82

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

Appendix B. Wilsonian effective action at one loop In this appendix microscopic

we extract the one-loop

In order to calculate

the Wilsonian

fields (except the ghosts) & = gi The

effective

U( 1) Wilsonian

action from the

SU( 2) Lagrangian.

bkgd +

background

will justify

ati

effective

into a background

action, it is customary

part and a fluctuating

(B.1)

etc.

,

fields should be thought of as comprising

a gradient

to split up all

part, e.g.,

expansion;

in particular

large-wavelength

the background

VEV v which we can choose to point in the third direction

modes which

scalar field includes

the

in both colour and SU( 2) N

space: fi

bkgd = fi%3T3/2

By convention, entirely

+

“&

under an SU(2)

assigned

gauge transformation,

to the fluctuating

&kgd

and likewise derivative.

= h’(x)

&bkgd

+ ;:)

of the total field is

parts are held fixed, thus

3 (B.3)

= O*

for the fermions.

It is convenient

Here Dp = #‘ - i[ gkkgd, ] is the background covariant to specialize to the one-parameter family of “background Rt

gauges,” linear in the fluctuating Gg.fl.

the variation

parts while the background

as(x) a+;: = eabceb(X) (#&sd h(x)

03.2)

bkgd.

= -$$

c

fields, defined by the gauge-fixing

term (B.4)

(fsa)2,

~1.2.3

where 03.5) k=l,2,3

As in the usual Rf gauges, for any 5, &.r.i. is constructed to cancel out the troublesome quadratic cross term -fiv(&$~P&$ - 8&P&$) induced in the SU(2) Lagrangian by the VEV (B.2) (with an integration by parts). The corresponding action for the triplet of complex ghosts ci follows straightforwardly from Eq. (B.3):

.cghost

2rr i+ ~~~ = -c e2

sBjcl

+t.[-D2-(DPo~v~X

)+@kbkgdX((+kbkgd+8+r)X

)I’

(B.6)

N. Dorey et al. /Nuclear

using an obvious

three-vector

We will calculate

notation

Physics B 502 (1997)

59-93

83

for the adjoint fields, e.g., vP = (u:, u:, u:).

(part of) the one-loop

effective

action

for the massless

quanta

M bkgd74: bkgd, s433 bkgd’ ‘;u bkgd’ X;kgd* %kgdv ??kgd* 7jb3kgd 1. Here 4: bkgd and 4; bkgd are the Goldstone bosons associated with the spontaneous breaking of the SU( 2)~ symmetry down to U( 1) under which they transform as a doublet, whereas S& bkgd is the singlet dilaton

(cf. Eq. (B.2)).

U( l), one generically

Since these scalars have different expects their effective couplings

charges under the unbroken

to renormalize

differently

at the

loop level, hence

c )

(‘P&bkgd)2

i=l,2

(B.7) where the one-loop

numerical

constants

Ct and Cs are not necessarily

equal, and we

omit spinor and gauge fields as well as higher derivative terms. However, since our choice of gauge fixing, (B.5), respects both scale and SU(2),v invariance, Eq. (B.7) must come from an 0( 3) -invariant expression containing no explicit factors of Mw nor of 84; bkgd; only the total background fields #$bkgd can appear. In particular, explicit factors of Mw are replaced

by 112

I

Mw+P, Thus (B.7)

03.8)

.

must come from

2 +z

(Cl

e2

-

C3>e2

c

2p3

4 I bkgd&h

bkgd

+

. . .

in terms of the two possible SU(2) invariants at the two-derivative between (B.7) and (B.9) lies in the three-point and higher-point expands about the VEV. Below we shall explicitly Cl

= c3

(B.9)

i=1,2,3

level. The difference functions when one

calculate

= &

which constitutes

our one-loop

prediction.

Calculation of the effective action Extracting the one-loop Wilsonian effective action is an intricate calculation in a generic “background Rg” gauge, but for the specific value 5 = 1 we can exploit the

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

84

following observation. Let us extend the gauge field + incorporating the three scalars #i: N

to a six-dimensional

vector

(B.11)

&6D=(~.~‘W.~-~.~~~~,~-~~). With the convention

field

that all relevant field configurations

are constant

in the final three

spatial directions,

a a a 2=&T=&?=

0,

then the three-dimensional six-dimensional gauge-fixing

(B.12) N = 4 SU(2)

Lagrangian

N = 1 SYM theory. Furthermore, conditions

(B.5)

may be rewritten

is known

to be simply

that of

for the specific choice 5 = 1, the

compactly

as (B.13)

where Dy is the six-dimensional background covariant derivative defined subject to (B.12), and the metric signature is (+, -, -, -, -, -). Likewise the ghost action (B.6) simplifies

to

L ghost= $&

[-(D6D)2-D6DpoSu~~

With these simplifications exercise

(see Section

the problem

]c. is now reduced,

focus solely on terms quadratic in the fluctuating term in (B.14).

quite literally,

16.6 of [28], which we follow closely).

The resulting

Gaussian

fields, dropping

functional

determinants

to a textbook

At the one-loop

level we

for example the second may be exponentiated

in the usual way, and lead to a contribution (B.15)

rls Tr log A, c s=o,+,1 to the effective values 71 = -$

action. Here s indexes the spin, and the weight factor vs takes the for the 6D vector, 771/z = +f for the single 6D Dirac spinor formed

from the four 3D Majorana

fermions,

‘* and r]o = +l

for the complex ghosts. A, is the

Gaussian quadratic form sandwiched between the spin-s fluctuating fields in the adjoint representation of SU( 2) colour.As shown in [28] it has the universal form

A,=-~*+A”‘+A’*‘+A(,7) where in the present

3

(B.16)

model

I2 N.B. We take 171,2 = +1/2 square the 9 operator

s

following

rather than 7)1/z = +l for the spinor because in the definition of d,12 we will Ref. [ 281.

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

Here t“ is the colour generator is the generator

in the adjoint representation,

of six-dimensional

Lorentz transformations

(J~),,=i(S~S~-SES:),

85

( ta)bc = -iE,bc,

JG;=$[yp,y”],

breaking.

direction

First, we require

the background

(B.18)

Gp”=O.

We now depart from Ref. [ 281 in two obvious ways, in order to incorporate symmetry

and J/”

in the spin-s representation:

spontaneous

fields to live purely in the third

in colour space, (B.19)

Second,

rather than truncating

total background &kgdp

the expansion

of the logarithm

at second order in the

field, we instead expand about the VEV, rewriting

= fiv

a/.~, T3/2

+

Eq. (B.2)

as (B.20)

8akgdl.L

and keep terms to second order in the deviation field Sa,kgdfi but to all orders in the VEV itself. In terms of the deviation field, Eq. (B.16) becomes

A(‘) = i{P,

f%~,Dg3d,t3},

A(*) = $2) - 2MW≫5t3t3 AcJ) s

- Mkt3t3

)

(B.21)

= &&~fivt3&pY,

Here

ii(*) = sL& ~t3 sv&g$3 t3.

(B.22)

Also, thanks to (B.19))

&&‘&,,

always to the constancy

conditions

vs Trlog

c

(-

is now the abelian field strength dlPLSv~&~Ulsubject as (B.12).

(a* + M2,t3t3)

Taylor expanding

+ A(‘) + A(*) + dig)

the logarithm

then gives

- 2MWSv;&t3t3

>

S=O,;,1 =

const. +

c

qsTr(Q.

(A”’ + A(*) + dig)

- 2MWSvfkDg3d5t3t3)

s=o,~,l -6.

(A”’ + Ais)

- 2MWc@$t3t3)

. G. (A(‘) + dig’ - 2MW&;~d5

3 3 tt))*

(B.23) where corrections G=diag(

to the r.h.s. are U( 6v&,,). ’

p2-M$’

1 p2 - I+$/

‘).

’ p*

Here &7is the diagonal colour-space

matrix, (B.24)

as fOllOWS from ( t3t3),b = &b - &3&,3. Apart from the masses in the propagators (B.24), the chief effect of the spontaneous symmetry breaking in (B.23) are the terms linear and quadratic in Mw. Let us dispose of these first. Terms linear in Mw are

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

86

c rlsMwTr

+ 28 . 6v,6,D,3,,t3t3 . Q . A(‘)

- 2G . 6u;&t3t3

S=O,~,l

+26 . 6v;&t3t3 These are would-be

. 6 s Ai3)

.

tadpole contributions

and third terms here vanish because a non-vanishing

Feynman

diagram

over the Lorentz

representation,

(B.25) to the effective action. Respectively,

the second

trcolourt3t3t3 = 0 and tr,,,JsPV = 0. The first term is whose only spin dependence

giving a relative weight of 6,4,1,

comes from the trace respectively,

for the

vector, spinor and ghost loop. From the above values of vs one sees that

671+ 47~2+ 7. so that the tadpoles supersymmetry). to M&. (Since

=

0,

(B.26)

do cancel

among

the three types of loops

And the same arithmetic these mass-dependent

(a manifestation

kills the cross term in (B.23)

terms were the only potential

of

proportional

source of difference

between vtkDg3d5 z @zkgd3and the other two massless scalars, their vanishing implies that Cl = C3 in the notation of Eq. (B.7).) In fact these three arguments kill almost all of the Mow terms in (B.23)

=--- ; X

as well, leaving only

(s d3p

J

1

d3k -(2~)3v;W$kk)

c

rls

1

(27r)3 2 * p2 - Mf$/ . (p + k)2 - M; )

tr JYJ..“.

(B.27)

s=o,~.l The p integration yields i/( 47rMw) + O( k2) (the factor of 2 inside the integrand counts the two massive colours a = 1,2). The spin sum simplifies using [ 281 tr 3,“” JTspU= WPg””

(B.28)

- d”P)C(S),

where an explicit calculation So the net contribution is

using (B.18)

gives C ( 1) = 2, C(i)

= 1, and C( 0) = 0.

(B.29) where we drop terms of 0( k4). A comparison then implies 271. 3 +

2??1 ,2 - 2rMw

which is our one-loop

with the tree-level

+ Q(e2),

prediction

Cl = C3 = l/47?.

action - 9 f s( v&,) 2

(B.30)

N. Dorey et al. /Nuclear Physics B SO2 (1997) 59-93

Appendix C. Three-dimensional

87

instantons

In this appendix we set up and give the details of the I-instanton calculus in three space-time dimensions. First we consider the bosonic part of the three-dimensional Euclidean action (A.9),

To find the instanton

configuration

we employ

Bogomol’nyi’s

lower bound

approach

1121,

(C.2) where BE = ~E~,,~v&. The 3D bosonic Bogomol’nyi bound (C.2)) 6’ N B;

=o,

instanton,

f*=O,

1 a = DC’&“31

being a minimum

of Sn, saturates

(C.3)

a,

(C.4)

where (C.3) ensures the vanishing of the commutator terms in (C. 1) and the remaining components of the bosonic instanton satisfy Bogomol’nyi equation (C.4) and are given by iI31 cl0 =

43

Mwlxl

a - 1 5 > x* ’

cothMwlxl

(C.5) with the boundary cl a

43

conditions

Xa

+

The instanton SC, = $

--Mw, X

gd

as 1x1 + oo, xaxp

a

CL

action follows from (C.2), /d”xa,(-B;

CC.61

-+-x4*

.,;I

(C.6),

“) = $47rMw.

(C.7)

From the above formulae it is obvious that the bosonic components of the 3D instanton are those of the BPS (anti-)monopole in the corresponding four-dimensional theory in the 3 = 0 gauge.

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

88

The Fermi-field persymmetry later. Isolated

components

transformations one-instanton

of the instanton of the bosonic

contribution

can be determined

components

to the functional

(CS) integral

by infinitesimal

su-

and will be discussed is of the generic

form

[If31 21

=

dt%

s

s

h-wRexp[-&I

,

where dpB and dpF are the measures bosonic

and fermionic

zero eigenvalues Eq. (23).

In this appendix

of quadratic

we determine

It will prove to be particularly four-dimensional

invariant

in this language

over collective

fluctuations

determinants

in the instanton

field b

to arrange the 3D instanton of the dimensionally

to the 4D self-duality

equations

2n-

&I=-

e2

over nonbackground,

analysis

in a

equations

(C.4)

[ 141

UC1n = *UcI a fllll ll,n and the instanton

of

reduced 4D theory

along x3) is given by (A.8). The Bogomol’nyi

are equivalent

coordinates

d,ug and dpF.

convenient

way. The four-vector

(translationally

of integrations

R is the ratio of functional

zero-modes,

of the operators

CC.81

(C.9) action (C.7)

is

J

(C.10)

The 3D instanton or, equivalently, the BPS monopole, is in this way similar to the 4D flucYang-Mills instanton [ 261. The functional integration over the (x3-independent) tuations 60:~ around the instanton will be performed in the (four-dimensional) covariant background v;;sv;*

gauge,

and the bosonic za[kl_ “l

(C.11)

= 0, instanton

alf’ ma dy[kl

+

zero-modes

will be of the general form [27]

v;;na.

(C.12)

where ytk] are the zero-modes collective coordinates - the three-translations, X,, and the U( 1) rotation, 0. The second term on the right-hand side of (C.12) is necessary to keep Z,4 in the background gauge (C.11). In general, bosonic zero-modes, 5, can be written in a more compact notation,

v$,

zJ] =*

The measure

v&s,

,

2, “mzyO.

d,uB is now simply

(C.13)

[27] (C.14)

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

First, consider

translational

Z" [PI = m

-a,vi

0 +

zero-modes, q$f

a =

89

cf. (C. 12)) (C.15)

~:,a.

Their overlap is

=--a,,254

e2

Now we consider

I

d3x

v$,“v$ = S,,Scl

the U( 1) orientation

rotate the instanton

(C.5)

.

(C.16)

zero-mode.

into the unitary

To do this we need to, first, gauge

(singular)

gauge, where (C.17)

v$ &s = & &, N P3 ) and, second, allow the further global gauge transformations obviously

form a U( 1) subgroup

$’ sins = exp [ i8r3/2] The U(1)

zero-mode

zc31 = rJn

a,

$lsing

consistent

with (C.17))

& sins exp[ -i8r3/2]

(C.18)

.

in the form (C.12)

is

-t

cl sing 7 bzwT3/2) = ---& ,&S

&

z$

they

of the SU(2),

(g&

-

(C.19)

W

with the overlaps c?33=-

2?r 2r

0

Finally

dsXZa

CL3= 2

m

[31ZO [31

n*

1 2lr d3x LI~~“v~~~ = =$-2 s

Za [PI Z;L31 =0. Jd3X

e2 s

combining

(C.20)

I?,

(C.14))

(C.16)

and (C.20)

we obtain the desired expression

( 16)

for d,uua,

(C.21) Our next goal is d,uF. The one-instanton Mills in 3D has four fermion N = 4 supersymmetry they can be understood

zero-modes

transformations

solution of the N = 4 supersymmetric which can be determined

of the bosonic

in terms of the Weyl spinors

components

Yang-

by infinitesimal

(C.5).

Equivalently

of the four-dimensional

theory,

(17), AC’= $&(a”‘fl) A@ $$’ = $$&c~“~)/~“,

(IPVC’ &JlVI’ (C.22)

where the two-component Grassmann collective coordinates [a and &&are the parameters of infinitesimal N = 2 supersymmetry transformations in (dimensionally reduced) 4D

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

90

theory. Since cy = 1,2 there are two A’s and two 9’s which is equivalent

to four zero-

modes in terms of Majorana spinors in 3D. There are no anti-fermion zero-modes due to the self-duality, (C.9). The fermion collective coordinates integration measure is in general

(J-c)-2> /d/w=/-d25d25’ where the fermion Jacobian

(C.23)

is (C.24)

that Jd2[r2

and it is understood13 evaluated Comparing

with the use of (C.22)

1. The right-hand

side of (C.24)

algebra,

and gives &

is easily = 2&t.

with ( 18) we obtain

/d/-w=/d2t d2t’ (2&) Finally

3

and the a-matrix

-2 .

we need the long-distance

(C.25) asymptotics

of the fermion

These can be readily obtained

by, first, switching

of Eq. (A.4),

into the real basis, (A.7),

second,

rotating

and, finally, switching to Y’* matrices gauge, (C.17), and the large-distance

zero-modes

from u” matrices decoupling

(C.22).

in (C.22)

to r:!,

the x2 direction

in 3D. The result is then rotated into the singular limit 1x1 --+ 00 is considered (C.26)

which confirms

Eq. (32).

Appendix D. The Riemann tensor In this appendix

we calculate

the leading exponentially

suppressed

terms in the Rie-

mann tensor for the metric ds* = f*(r)dr*

+a’(r)u: + b2(r)oj + c2(r)a;

with Ui defined in (38) and the functions a, b and c satisfying (44). Following pendix B of Ref. [ 231, we define the vierbein one-forms & = OrdXi with 6’ = aal, which, using (44),

e2 = ba2,

e3 = CU3,

e4 =

fdr,

Ap-

CD.11

gives us the spin connection

l3 Note that in (C.24) we could have summed over the isospin a = 1,2,3 rather than trace over the SU( 2) matrices. This would have produced an extra factor of l/2 which then would require an alternative prescription for Grassmannian integration, s d*et$ 3 2 and the final answer for 36 would he the same.

91

N. Dorey et al. /Nuclear Physics B 502 (1997) 59-93

4 = (a’/f>m lo: = ( 1+ The curvature

4 = (b’/f)u2

9

wg = ( 1 + a’/f>a*

,

c’/f>(+s two-forms

4 = (c’/fb3

9

,

w: = ( 1+

(D.2)

b’/f)UZ.

are now found to be

RI2 =E’drAaJ

+ [-e+iE+&+2&]at

R23 =ii’drAal

+ [G~+~+Z+~&?]CT~AA~,

Aa=,

R31=~‘drA~2+[-b+F+~+2Eii]a3A~,, where 6 = a’/ f,

(D.4)

and e = c//f.

& = b//f

R34 = RI2 and cyclic, giving us a Riemann tensor e1234 = +l. Explicitly we have

The other components tensor self-dual

are determined

(D.5)

fa, -s’/fb,

R..tjkl = e?eIP#@R I , k 1 Calculating

=

-2’1 fc),

where (D.6)

dYa*

tensor at the point (0, 0, I, (T) in the standard coordinates,

the Riemann

then change to the complex Zl

(D.7)

Labelling the coordinates (zt , 22, 21, 52) by an index P = 1,2,3,4, is conveniently given in terms of the following (4 x 4) matrices:

A=

B =

A+.e+‘@ 0 0 -A-e-”

i _A_o,+”

A-e-‘@ 0

B+e’@

0

we

basis (49) with coordinates

z2=+z-iu).

+_(X-iY),

_A:e+W

by

with respect to the epsilon

Rap+ = &j Tab $6 with T = diag( -a’/

(D.3)

0

the Riemann

tensor

A-e+‘* 0 A+e-‘@

-A+e-‘$

0

-B-e-@

0

’ i

(D.8)

where A* = a f f ibc and Bk = bf f iac. The final result is

92

N. Dorey et al. /Nuclear

RPQRS=--

*I

” 4far2

A

pp

ARS

The leading exponentially

b +---B 4fbr2

suppressed

Physics B 502 (1997) 59-93

N B + %,cRS. PQ RS fc terms are proportional

CD.91 to (a - b) whose asymp-

totic form is given in (48) as a - b N -8qr2e-‘. Hence,

to compare

(D.lO)

with the instanton

factors by their leading weak coupling

calculation, behaviour.

it suffices to approximate

all other

From (41) we have

a+b=2r+O(l/r), c 2 -2 + O( l/r), f--l Expanding

+0(1/r).

the exact expression

(D.ll) (D.9)

we find that, to the required order,

Rt2t2 = 8qre-‘+iu, RfZiZ= 8qre-r-iu. As discussed in Section 4, the remaining pure (anti)-holomorphic to these by symmetries of the Riemann tensor. All components are independent

(D.12) components are related of mixed holomorphy

of (T.

References [ 1] N. Se&erg, IR Dynamics on Branes and Space-Time Geometry, hep-thI9606017, Phys. Lett. B 384 (1996) 81. [2] N. Seiberg and E. Witten, Gauge Dynamics and Compactification to Three Dimensions, hep-th/9607163. [ 31 N. Seiberg and K. Intriligator, Mirror Symmetry in Three Dimensional Gauge Theories, hep-th/9607207, Phys. Lett. B 387 (1996) 513. [ 41 G. Chalmers and A. Hanany Three Dimensional Gauge Theories and Monopoles, hep-th/9608105, Nucl. Phys. B 489 (1997) 223. [ 51 J. de Boer, K. Hori, H. Ooguri and Y. Oz. Mirror Symmetry in Three Dimensional Gauge Theories, Quivers and D-branes, hep-th/9611063, Nucl. Phys. B 493 (1997) 101. [6] A. Hanany and E. Witten, Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics, hep-th/9611230, Nucl. Phys. B 492 (1997) 152. [7] J. de Boer, K. Hori, H. Ooguri, Y. Oz and Z. Yin, Mirror Symmetry in Three Dimensional Gauge Theories, SL(2,Z) and D-bmne Moduli Spaces, hep-thI9612131, Nucl. Phys. B 493 (1997) 148. [ 81 M. Atiyah and N. Hitchin, The Geometry and Dynamics of Magnetic Monopoles (Princeton Univ. Press, Princeton, 1988) [9] L. Alvarez-Gaume and D.Z. Freedman, Comm. Math. Phys. 80 (1981) 443. [lo] N. Dorey, V.V. Khoze and M.P Mattis, Phys. Rev. D 54 (1996) 2921. [ 111 A.M. Polyakov, Nucl. Phys. B 120 (1977) 429. [ 121 E.B. Bogomol’nyi, Sov. J. Phys 24 (1977) 97. [ 131 M.K. Prasad and C.M. Sommerfield, Phys. Rev. Lett. 35 ( 1975) 760. [ 141 M. Lohe, Phys. Lett. B 70 (1977) 325. [ 151 E.J. Weinberg, Phys. Rev. D 20 ( 1979) 936. [ 161 C. Callias, Comm. Math. Phys. 62 (1978) 213. [ 171 A. D’Adda and P Di Vecchia, Phys. Lett. B 73 (1978) 162.

N. Dorey et al./Nuclear

Physics B 502 (1997) 59-93

93

[ 181 G. ‘t Hooft, Phys. Rev. D 14 (1976) 3432. [ 191 R. Kaul, Phys. Lett. B 143 (1984) 427.

[20] [21] [22] [23] [24] [25] [26] [27] [28]

I. Atlleck, J. Harvey and E. Witten, Nucl. Phys. B 206 (1982) 413. G. Gibbons and N. Manton, Nucl. Phys. B 274 (1986) 183. G. Gibbons and C. Pope, Comm. Math. Phys. 66 (1979) 267. J.P. Gauntlett and J.A. Harvey, Nucl. Phys. B 463 (1996) 287. N. Dorey, V.V. Khoze and M.P. Mattis, hep-th/9611016; Phys. Rev. D 54 (1996) 7832. J. Wess and J. Bagger, Supersymmetry and Supetgravity (Princeton Univ. Press, Princeton, 1992). A.A. Belavin, A.M. Polyakov, A. Schwartz and Y. Tyupkin, Phys. Lett. B 59 (1975) 85. C. Bernard, Phys. Rev. D 19 (1979) 3013. M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Aeld Theory (Addison-Wesley, New York, 1996).