b

e thetituents.eriodic

Physics Letters B 626 (2005) 139–146

www.elsevier.com/locate/physlet

Doubly periodic instanton zero modes

C. Forda, J.M. Pawlowskib

a School of Mathematics, Trinity College, Dublin 2, Irelandb Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany

Received 24 June 2005; received in revised form 8 August 2005; accepted 16 August 2005

Available online 6 September 2005

Editor: N. Glover

Abstract

Fermionic zero modes associated with doubly periodicSU(2) instantons of unit charge are considered. In cases wheraction density exhibits two ‘instanton cores’ the zero mode peaks on one of four line-segments joining the two consWhich of the four possibilities is realised depends on the fermionic boundary conditions: doubly periodic, doubly anti-por mixed. 2005 Elsevier B.V. All rights reserved.

esonsof

tonses

owwo.ionin-

nts.epaern

nshinlo-ultsre--t toas

rd-

en-

In this Letter we consider fermionic zero modfor the recently discussed doubly periodic instant[1,2]. Two complementary constituent descriptionsthese objects were provided; charge one instancan be built out of two overlapping instanton coror two static monopoles. Explicit computations shthat in square tori the action density peaks at tpoints in T 2 × R2 in accord with the core pictureFor elongated tori with high aspect ratios the actdensity is concentrated in two tubes which can beterpreted as the worldlines of monopole constitueThe basic properties of these monopoles (spatial sration and mass ratio) follow much the same patt

E-mail addresses:ford@maths.tcd.ie(C. Ford),j.pawlowski@thphys.uni-heidelberg.de(J.M. Pawlowski).

0370-2693/$ – see front matter 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2005.08.094

-

as found for the monopole constituents of caloro[3–6]. Here we compute the zero mode density wita two-dimensional slice including the constituentcations for various boundary conditions. These resare compared with the action density calculationsported in Refs.[1,2]. Of particular interest are the localisation properties of the zero modes with respecthe instanton core constituents and their evolutionthe aspect ratio, or temperature, is increased.

To begin we recall some basic definitions regaing gauge fields onT 2 × R2. A doubly-periodic gaugepotential is understood to be an anti-hermitian pottial defined throughoutR4 which is periodic modulogauge transformationsU1 andU2 in two directions:

Aµ(x0, x1 + L1, x2, x3)

= U1(Aµ(x0, x1, x2, x3) + ∂µ

)U−1,

1

.

140 C. Ford, J.M. Pawlowski / Physics Letters B 626 (2005) 139–146

rehenatf-he

erohe

ses

tion

en-in

ngo

fol-

-

e paonf the-

hea-e

yion.po-

de-

is a

me-

le

two

nd

by

o

blytons

–-vedn-ic

erge.

Aµ(x0, x1, x2 + L2, x3)

(1)= U2(Aµ(x0, x1, x2, x3) + ∂µ

)U−1

2 .

In general, the transition functionsU1 andU2 arex-dependent. However, we will work in a gauge whethey are constant commuting group elements. TtrU1 and trU2 are the two holonomies (assuming thA1 andA2 vanish at infinity). We specialise to seldualSU(2) potentials in the one-instanton sector. TWeyl operator

(2)D†(A) = −σ †µ(∂µ + Aµ)

with σ †µ = (1,−iτ1,−iτ2,−iτ3) andτi are Pauli ma-

trices, is expected to have a single fermionic zmode. To specify the periodicity properties of tfermionic zero mode two phases are required:

Ψ (x0, x1 + L1, x2, x3; z)= eiz1L1U1Ψ (x0, x1, x2, x3; z),

Ψ (x0, x1, x2 + L2, x3; z)(3)= eiz2L2U2Ψ (x0, x1, x2, x3; z).

To make contact with the Nahm formalism the phaare parametrised by dimensionful coordinatesz1 andz2 rather than angles; these have the interpretaas coordinates of the dual torus,T 2, since the re-placementsz1 → z1 + 2π/L1 z2 → z2 + 2π/L2leave the boundary conditions unchanged. Such geral boundary conditions have also been studieda lattice context[7]. The choicez1 = z2 = 0 leadsto periodic fermions whilez1 = π/L1, z2 = π/L2provides anti-periodic solutions. Another interesticase is whenz1 and z2 are correlated with the twholonomies.

The transition functions can be parametrised aslows:

(4)U1 = e−iω1L1τ3, U2 = e−iω2L2τ3,

whereω = (ω1,ω2), like z = (z1, z2), can be considered an element of the dual torus. LikeR4 instantonsand calorons the charge-one solutions have a scalrameter,λ, which can be thought of as the instantsize. The scale parameter fixes another property oinstanton namely itsflux κ ; asymptotically the instanton has the form

(5)Aµ(x) ∼ aµ(x0, x3)τ3,

-

whereaµ is aU(1) self-dual potential inR2. The fluxis defined through

(6)κ = limR→∞

i

2π

∫C(R)

(a0 dx0 + a3 dx3),

whereC(R) is a circle of radiusR in thex0–x3 plane.The sign ofκ is ambiguous since the signs of taµ can be flipped via a constant gauge transformtion (a Weyl reflection). Moreover, we may assumthatκ lies between 0 and 1 sinceκ can be changed ban integer amount via a smooth gauge transformatThe asymptotic flux can also have non-zero comnents in the compactx1 andx2 directions, i.e.,a1 anda2 need not be zero. Ifa1 = a2 = 0 the charge-oneinstanton has a radial symmetry; the action density

pends onx1, x2 andr =√

x20 + x2

3 only. In this casethe action density decays exponentially and theresimple relation between the scale parameterλ and theflux

(7)κ = πλ2

L1L2.

These special radial solutions have seven paraters; the fluxκ (or equivalently the sizeλ) the twoholonomies and four translations inT 2 × R2. To-gether theκ → −κ ambiguity and theκ ≡ κ + 1equivalence imply that the fluxesκ and 1− κ arephysically indistinguishable. This gives two possibinstanton sizes

λ1 = √κL1L2/π, λ2 = √

(1− κ)L1L2/π.

In [1] it was argued that the instanton possessesinstanton coreconstituents with sizesλ1 andλ2. Tak-ing xµ = 0 as the position of the first core the secois centred atx1 = L1L2ω2/π , x2 = −L1L2ω1/π ,x0 = x3 = 0, i.e., the core separation is fixedthe holonomies. For square tori (L1 = L2) explicitcalculations of the action density clearly show twinstanton-like peaks at the expected locations.

A major difference between the caloron and douperiodic case is the absence of charge-one instanonT 2 × R2 for trivial holonomy, i.e., the HarringtonShepard one-caloron[8] has no doubly periodic counterpart. The Harrington–Shepard solution was deriby summing periodic copies of the harmonic potetial entering the ’t Hooft ansatz. In the doubly periodcase the corresponding double sum does not conv

C. Ford, J.M. Pawlowski / Physics Letters B 626 (2005) 139–146 141

o-

ap-

i-tex.tan-

in-

antant

s is-of

tionuld

ayiseseat

ep-

ord-l

eole.ks

elo-de-

n).be-

o-

owave

he

rlytede

onjoin-

,

In the core picture trivial holonomy translates into cincident core locations. For smallω the two coresmerge into a single BPST like peak whose sizeproaches zero asω → 0.

In the special caseκ = 12 the two cores are ident

cal. Here the flux can be interpreted as a center vorThese solutions are decompactified four-torus instons of unit charge (theL0,L3 → ∞ limit of an SU(2)

instanton onT 4 with periodsL0, L1, L2 andL3). Thecenter vortex is a remnant of a torus twist,Z03 = −1,see Ref.[9]. Because of the exponential decay, theκ =12 solutions are expected to approximate four torusstantons with large but finiteL0 andL3 extremely well(see also[10]). These doubly periodic instantons cbe seen as the opposite extreme to ’t Hooft’s conscurvature solutions which exist whenL1L2 = 2L0L3.An analytic interpolation between these two regimestill lacking (see, however,[11]). In the absence of analytic solutions a constituent description (in termscores, monopoles or otherwise) as well as informaconcerning the moduli-spaces and their metrics worepresent a considerable advance.

If one period is much larger than the other, sL1 � L2, the core picture fails; the action densityconcentrated around two monopole worldlines. Thmonopole constituents follow a similar pattern to thobserved for charge one-calorons;ω2 determines themass ratio of the two monopoles and their spatial saration isπλ2/L2 = κL1. The caloron zero mode[12]localises to one of the monopole constituents accing to the value ofz. As z passes through a critica

value (wherez is correlated with the holonomy) thzero mode switches its support to the other monopIf z is exactly at a critical value the zero mode peaat both monopole locations. Furthermore, these dcalised zero modes do not decay exponentially (thecay is sufficiently fast to give a normalisable solutioIn the doubly periodic case we have to distinguishtween the core (L1 ≈ L2) and monopole (L1 � L2or L2 � L1) regimes. IfL1 � L2 the fermionic zeromode is expected to be caloron-like in that it will lcalise to one monopole for−ω2 < z2 < ω2 and theother forω2 < z2 < −ω2 + 2π/L2. If z2 = ±ω2 thezero mode will see both monopoles (assumingz1 �=±ω1, since as we shall argue(z1, z2) = ±(ω1,ω2)

are very special cases). What is less obvious is hthe zero mode behaves in the core regime. We hcomputed the zero mode densityΨ †(x; z)Ψ (x; z)within the two-dimensional slicex0 = x3 = 0 forvarious choices ofκ , ω and z. When L1 = L2, thezero mode localises to one of four lines joining tcores.

Consider the caseL1 = L2 = 1, κ = 12, ω1 = ω2 =

12π . Here the two (equal sized) cores are particulawell resolved in the action density. They are locaat the origin(x1, x2) = (0,0), and in the centre of thtorus (x1, x2) = (1

2, 12). For z1 = z2 = π , the doubly

anti-periodic case, the zero mode is not localiseda single core but smeared around a line-segmenting the core at the corner(x1, x2) = (0,1) and core inthe middle of the torus(x1, x2) = (1

2, 12), see the left

plot in Fig. 1. Taking insteadz = 0, the periodic case

Fig. 1. Normalised zero mode densityΨ †Ψ and action density− 12 trF2 for x0 = x3 = 0, κ = 1

2 , ω = π2 (1,1) andz = π(1,1) (anti-periodic).

142 C. Ford, J.M. Pawlowski / Physics Letters B 626 (2005) 139–146

Fig. 2. Zero mode densityΨ †Ψ for x0 = x3 = 0, κ = 12 andz = (0,0) (periodic) andz = (0, π

2 ) (periodic inx1 and anti-periodic in 2x2).

nte in

greevenen

the

ge-

eric

ite

ton

desra-

the

ible

iesostillthe

erthe

zeroen-

ree

es

osele

eod-

not

yields a similar line-localisation but with a differepairing arrangement; it stretches between the corthe middle and that at(x1, x2) = (1,0), see the leftplot in Fig. 2.

The zero modes densities inFigs. 1 and 2(leftplots) can be mapped into each other by a 180 derotation. The other two pairing arrangements are giby z = (0,π), where the zero mode stretches betwethe cores at(x1, x2) = (0,0) and(x1, x2) = (1

2, 12) and

z = (π,0), where the zero mode stretches betweencores at(x1, x2) = (1,1) and (x1, x2) = (1

2, 12). The

transition between two of the four possible arranments is a smooth one. For example, takingz = (0, π

2 )

generates a superposition of two of the four genpairings, namely, betweenz = (0,π) andz = (0,0);see the right plot inFig. 2.

The transition from the above situation to the fintemperature case is accessed by increasingL1 (or L2)starting fromL1 = L2. This was done in[2] for theaction density showing the crossover from instancores to monopole constituents. InFig. 3 we showthe corresponding zero mode densities for zero mowith anti-periodic boundary conditions, and aspecttiosa = L1/L2 = 3

2,2,3. The equal length casea = 1is already shown inFig. 1. If a > 1 the zero modepeaks at one of the monopole worldlines. In factmonopole structure emerges in the zero modesbeforeit is visible in the action density; thea = 3

2 zero modealready has some resemblance to the largea monopolelike regime whereas (stretched) cores are still visin the action density.

In Fig. 4 we show zero mode and action densitfor the caseκ = 5

8. In thea = 1 core regime the zermode density peaks at the smaller core but onecan see a preferred line joining the smaller core atcentre to the larger core at the corner(x1, x2) = (0,1).Here the evolution to the monopole regime is slowdue to the presence of a smaller core. But as withκ = 1

2 case the monopole structure appears in themode density before it can be seen in the action dsity.

A further interesting case concerns the limit whez approaches±ω. If κ �= 1

2 the zero mode does localisto a single core asz approachesω or −ω. However,atz = ω the zero mode is not normalisable. IfΨ (x, z)

is normalised then for a fixedx it tends to zero asztends toω. Whenκ = 1

2 the zero mode also becomnon-normalisable asz approachesω but during the ap-proach the pair structure survives, i.e., however, clz is toω the zero mode will not be localised to a singcore. Whenz = ω the equationD†(A)Ψ = 0 has twosolutionsΨ I(x) andΨ II (x) which are supported at thfirst and second core, respectively. These ‘zero mes’ are smooth but not normalisable. Like thez = ω

caloron zero mode they decay algebraically—butfast enough to be normalisable. Asz approachesω thezero mode has the form

Ψ (x; z) ∼ |z − ω|κΨ I(x)

(8)+ |z − ω|−κ(z1 + iz2 − ω1 − iω2)Ψ

II (x).

C. Ford, J.M. Pawlowski / Physics Letters B 626 (2005) 139–146 143

Fig. 3. Zero mode densityΨ †Ψ for x0 = x3 = 0, κ = 12 andω = π

2 (a−1,1) anda = 32 ,2,3 (anti-periodic).

dd.llerirsity

ne

ted

es

As z approachesω theΨ II contribution is suppresseif κ < 1

2 and for κ > 12 the first term is suppresse

Accordingly, the zero mode is localised at the smacore. If κ = 1

2 neither core is favoured and the palocalisation persists. Note that the zero mode dendepends on the phase ofz1 + iz2 − ω1 − iω2 so thatthe form ofΨ †Ψ for z ∼ ω depends on the directioof approach toω. This is why all four pairings can bseen in thez → ω limit whenκ = 1.

2

The plots presented in this Letter were generausing an explicit formula forΨ (x; z) valid in the two-dimensional slicex0 = x3 = 0. To conclude we outlinethe derivation of this formula. The construction hingon the fact that theSU(2) gauge potential,Aµ(x), hasa simple abelian Nahm transform,A(z), with compo-nents[2]

A1(z) = −i∂z2φ(z), A2(z) = i∂z1φ(z),

144 C. Ford, J.M. Pawlowski / Physics Letters B 626 (2005) 139–146

Fig. 4. Zero mode densityΨ †Ψ for x0 = x3 = 0, κ = 58 andω = π

2 (a−1,1) anda = 1,3 (anti-periodic).

pt

et-,

as

-

f

(9)A0(z) = A3(z) = 0,

whereφ(z) is doubly-periodic and harmonic exceat two flux singularities inT 2. This is the form ofthe Nahm potential associated with radially symmric one-instantons onT 2 × R2. In the non-radial casewhich we do not consider here,A0 and A3 are non-zero (see also[13]). The instanton can be expresseda Nahm transform ofA(z):

(10)Apqµ (x) =

∫T 2

d2zψp †(z;x)∂

∂xµψq(z;x),

whereψp(z;x) (p = 1,2) are orthonormal and periodic (with respect toz1 → z1 + 2π/L1 and z2 →z2 + 2π/L2) zero modes of the Weyl operator

(11)D†x(A) = −σ †

µDµx (A),

whereDµx (A) = ∂/∂zµ + Aµ(z) − ixµ for µ = 1,2

and Dµx (A) = Aµ(z) − ixµ for µ = 0,3. The Nahm

zero modes can be written in the form

ψ1(z;x) = Dx(A)

(ϕ(1)(z;x)

0

),

(12)ψ2(z;x) = Dx(A)

(0

ϕ(2)(z;x)

),

where theϕ(p) are specific singular solutions of theT 2

Laplace equation

(13)(Dµ

x (A))2

ϕ(z;x) = 0.

The fermionic zero modeΨ (x; z) can be written in asimilar fashion to the Nahm zero modes

Ψ (x; z) = D(A)

(Φ(1)(x; z)

0

)

(14)= D(A)

(0

Φ(2)(x; z))

,

whereΦ(1) andΦ(2) are specific singular solutions otheT 2 × R2 Laplace equation

(15)

(∂

∂xµ

+ Aµ(x)

)2

Φ(x; z) = 0.

In fact,oneof the components ofΦ(p) is ϕ(p)

Φ(1)(x; z) =(

ϕ(1)(z;x)

ϕ(1)(z;x)

),

C. Ford, J.M. Pawlowski / Physics Letters B 626 (2005) 139–146 145

t

ex-

-n

-

ocom--

-

the

ivenero

e

on-

th

(16)Φ(2)(x; z) =(

ϕ(2)(z;x)

ϕ(2)(z;x)

).

The other components,ϕ(1) andϕ(2), can be obtainedfrom the requirement thatΦ(1) and Φ(2) generatethe same zero mode, i.e., Eq.(14). This requiremenamounts to four first order PDEs forϕ(1) andϕ(2). Theintegrability condition for these equations can bepressed as another Laplace-type equation

(17)

(∂

∂xµ

+ ABµ(x)

)2 (ϕ(1)(z;x)

ϕ(2)(z;x)

)= 0,

whereABµ(x) is anSU(1,1) self-dual potential which

is related toAµ(x) by a simple Bäcklund-type transformation; more details of this structure will be giveelsewhere. The fermionic zero mode(14) has the nor-malisation

(18)∫

T 2×R2

d4x Ψ †(x; z)Ψ (x; z) = 4L1L2.

The Nahm zero modes(12) lead to an instanton potential

Ax‖ = −τ3

2∂x‖ logρ − 2πi(τ1 − iτ2)κρ∂x⊥

ν∗

ρ,

(19)Ax⊥ = −τ3

2∂x⊥ logρ + 2πi(τ1 − iτ2)κρ∂x‖

ν∗

ρ,

and Ax‖ = −A†x‖ , Ax⊥ = −A

†x⊥ , whereρ(x) is real

and periodic andν(x) is complex and periodic up ta constant phase. Here we have used two sets ofplex coordinates forT 2 × R2; in the compact directions x‖ = x1 + ix2, x‖ = x1 − ix2, and in the transverse non-compact directionsx⊥ = x0 + ix3, x⊥ =x0 − ix3. Derivatives and potentials are defined as

∂x‖ = 1

2(∂x1 − i∂x2), Ax‖ = 1

2(A1 − iA2),

and similarly for the other coordinates. Inserting(19)into (14) one can express the components ofΨ (x; z)without reference to theϕ(p). Two components are obtained using theΦ(2) representation

(20)Ψ11 = 2i√

ρ ∂x‖ϕ(2)

√ρ

, Ψ21 = 2√

ρ ∂x⊥ϕ(2)

√ρ

,

and the remaining two components derive fromΦ(1) representation

(21)Ψ12 = 2√

ρ ∂x⊥ϕ(1)

√ρ

, Ψ22 = 2i√

ρ ∂x‖ϕ(1)

√ρ

.

-

Here we have written the zero mode componentsΨαp

whereα is a spinor index andp an SU(2) color in-dex. The representation of the caloron zero mode gin [12] has the same derivative structure. The zmode density is

Ψ †Ψ = ρ

L1L2

(∣∣∣∣∂x‖ϕ(1)

√ρ

∣∣∣∣2

+∣∣∣∣∂x‖

ϕ(2)

√ρ

∣∣∣∣2

(22)+∣∣∣∣∂x⊥

ϕ(1)

√ρ

∣∣∣∣2

+∣∣∣∣∂x⊥

ϕ(2)

√ρ

∣∣∣∣2)

.

Here we have rescaled so that the integral ofΨ †Ψ overT 2 × R2 is unity. Eq.(22) was used to generate thfigures.

We have writtenAµ(x), ψ(p)(z;x) and Ψ (x; z)in terms of the auxiliary objectsρ(x), ν(x) andϕ(p)(z;x). These can be expressed in terms of ctributions to the inverse ofD†

x(A)Dx(A) which hasthe form[2](D†

x(A)Dx(A))−1

(z, z′)

= 1

2(σ0 + iσ3)e

−φ(z)K+(z, z′;x)e−φ(z′)

(23)+ 1

2(σ0 − iσ3)e

φ(z)K−(z, z′;x)eφ(z′).

The key formulae are

ρ(x) = K+(−ω,−ω;x) = K−(ω,ω;x),

(24)ν(x) = K+(ω,−ω;x),

and

ϕ(1)(z;x) = eφ(z) K−(z,ω;x)√ρ

,

(25)ϕ(2)(z;x) = e−φ(z) K+(z,−ω;x)√ρ

.

In [2] explicit forms for theK± functions were givenfor the two-dimensional slicex⊥ = 0. Although thezero mode formulae involvesx⊥-derivatives they donot contribute ifx⊥ = 0.

Using(20) and(21) the components of the smoonon-normalisable zero modesΨ I(x) and Ψ II (x) canbe recovered. Forκ < 1

2 andz close toω, Ψ (x; z) ∼|z − ω|κΨ I(x) where

Ψ I11 = 2ic

√ρ∂x||

ν, Ψ I

21 = 2c√

ρ∂x⊥ν

,

ρ ρ

146 C. Ford, J.M. Pawlowski / Physics Letters B 626 (2005) 139–146

he

ted

antrtsnd

p-

ep-

83-eth-

p-

ep-

ep-

10,

7.

8)

009

aal,

G/

71,

(26)Ψ I12 = c

√ρ

2πκ∂x⊥

1

ρ, Ψ I

22 = ic√

ρ

2πκ∂x||

1

ρ,

wherec is defined bye−φ(z) ∼ c|z − ω|κ for z closeto ω. For large|x⊥| we have[2] ρ ∝ |x⊥|−2κ (ν de-cays exponentially) implying thatΨ I †Ψ I ∝ |x⊥|2(κ−1)

which is indeed too slowly decaying to normalise tsolution.

Acknowledgements

We are grateful to DIAS, Dublin, and the Institufor Theoretical Physics, University Tübingen, for kinhospitality. This work has been supported by a grfrom the Ministry of Science, Research and the Aof Baden-Württemberg (Az: 24-7532.23-19-18/1) athe DFG under contract GI328/1-2.

References

[1] C. Ford, J.M. Pawlowski, Phys. Lett. B 540 (2002) 153, heth/0205116.

[2] C. Ford, J.M. Pawlowski, Phys. Rev. D 69 (2004) 065006, hth/0302117.

[3] W. Nahm, Selfdual monopoles and calorons, BONN-HE-16, Presented at 12th Colloquium on Group-Theoretical Mods in Physics, Trieste, Italy, 5–10 September, 1983.

[4] K.M. Lee, C.H. Lu, Phys. Rev. D 58 (1998) 025011, heth/9802108.

[5] T.C. Kraan, P. van Baal, Nucl. Phys. B 533 (1998) 627, hth/9805168.

[6] F. Bruckmann, P. van Baal, Nucl. Phys. B 645 (2002) 105, hth/0209010.

[7] C. Gattringer, S. Solbrig, hep-lat/0410040;C. Gattringer, R. Pullirsch, Phys. Rev. D 69 (2004) 0945hep-lat/0402008.

[8] B.J. Harrington, H.K. Shepard, Phys. Rev. D 16 (1977) 343[9] G. ’t Hooft, Commun. Math. Phys. 81 (1981) 267.

[10] A. Gonzalez-Arroyo, A. Montero, Phys. Lett. B 442 (199273, hep-th/9809037.

[11] M. Garcia Perez, A. Gonzalez-Arroyo, C. Pena, JHEP 0(2000) 033, hep-th/0007113.

[12] M. Garcia Perez, A. Gonzalez-Arroyo, C. Pena, P. van BPhys. Rev. D 60 (1999) 031901, hep-th/9905016.

[13] M. Jardim, Commun. Math. Phys. 216 (2001) 1, math.D9909069;I. Frenkel, M. Jardim, Commun. Math. Phys. 237 (2003) 4math.QA/0205118.