Non-uniqueness of the supersymmetric extension of the O(3) $\sigma$-model

P u b l i s h e d f o r SISSA b y S p r i n g e r R e c e i v e d: June 1, 2017

R e v i s e d: September 25, 2017

A c c e p t e d: November 15, 2017

P u b l i s h e d: November 22, 2017

Non-uniqueness of the supersymmetric extension of the O (3) σ - model

Jo se M . Q ueiruga

" 61

aInstitute for Theoretical Physics, Karlsruhe Institute of Technology (KIT), Wolfgang-Gaede-Strafie 1, D-76131 Karlsruhe, Germany

bInstitute for Nuclear Physics, Karlsruhe Institute o f Technology (KIT),

Hermann-von-Helmholtz-Platz 1, D-763ĄĄ Eggenstein-Leopoldshafen, Germany cInstitute of Physics, Jagiellonian University,

Lojasiewicza 11, Kraków, Poland

E -m ail: jo se .q u e iru g a @ k it.e d u , w e r e s z c z y n s k i@ th .if.u j.e d u .p l

A b s t r a c t : W e study the supersym m etric extensions o f the O (3) a-m odel in 1 + 1 and 2 + 1 dimensions. W e show that it is possible to construct non-equivalent supersym m etric ver­

sions o f a given m odel sharing the same bosonic sector and free from higher-derivative terms.

K e y w o r d s : Superspaces, E xtended Supersym m etry

A r X i v e P r i n t : 1703.07343

1 Corresponding author.

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Contents

1 In trod u ction 1

2 T h e O (3) nonlinear ^ -m o d e l 3

3 A bosonic tw in for the O (3 ) ^ -m o d e l 5

4 A d d in g a potential 7

5 T h e bosonic tw in and ferm ion zero -m o d es 9

6 T h e O (3 ) ^ -m o d e l in 2 + 1 dim ensions 10

6.1 T he quartic fermionie terms 12

7 H igh er-derivative term s in the N = 2 bosonic tw ins 13

8 S u m m ary 14

1 Introduction

T he bosonic nonlinear O (n ) a-m odel is probably one o f the most studied examples of m odels where the field space (target space) possesses a nonlinear structure. From a physical point o f view , one o f the main m otivations for the investigation o f the lower-dimensional nonlinear a-m odels is that they share many similarities with the four-dim ensional gauge theories [1, 2] . Besides, these m odels are simpler that their four-dim ensional counterparts and therefore, they constitute a g o o d laboratory for testing theoretical ideas.

If one considers the supersym m etric version o f these m odels, a rich m athem atical struc­

ture arises. For exam ple, the relation between the amount o f supersym m etry ( N = 1 , 2 , 4 . . . ) and the geom etrical structure o f the target space m anifold has been established in [3- 5] . It is also well-known that in a number o f field theories, the classical solutions can be com puted by solving a first-order equation rather than the second-order equation obtained from the variation o f the Lagrangian. W hen this happens, the solutions satisfying the first-order equation saturate a low er-bound for the energy (B ogom oln y bou n d). It has been pointed out [6- 8] that this phenom enon has a close relationship with supersymmetry. In a theory with N = 1 supersymmetry, if a classical topologically nontrivial solution satisfies a first- order equation there exist and extra ( N = 2) supersymmetry. This peculiarity allows in certain cases to determ ine the exact mass spectrum [7] . M oreover, due to the fact that SU SY relates bosonic and ferm ionic states, one can obtain, for exam ple ferm ionic solutions in terms o f bosonic ones w ithout solving the corresponding D irac equation.

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A nother o f the fundam ental bridges between supersym m etry and geom etry has been discovered in [9 , 10] . In these works, the relation between ferm ionic and bosonic zero- m odes (via the W itten index) in the nonlinear a-m odels and topological invariants o f the underlying manifolds has been established.

It is the goal o f this work to study m ore general form s o f the supersym m etric nonlin­

ear a-m odels with special emphasis on the O (3 ) m odel. T he classical approaches to the supersym m etric version o f these m odels can be realized in tw o languages, namely N = 1 or N = 2 superspace. In the first case, all the inform ation is encoded in terms o f N = 1 real superfields ( $ k) (which can be com bined eventually in com plex superfields). The Lagrangian takes the following form

LN= 1 = ^ d2e 9lj ( $ k ) D a V D a & , (1.1)

where 9ij ( $ k) corresponds to the metric on the target-space m anifold M T , in its bosonic restriction. In the N = 2 language, the Lagrangian ( 1.1) is even simpler

LN= 2 = y d2ed 2e K ( $ k, $ k f), (1.2)

where K ( $ k, $ k t) is the so-called Kahler potential (the metric on M t can be obtained as the second derivative o f the Kahler potential, gi,j(^ fc, 4>k ^) = * & tK ( $ k, $ k^ 0=0=0).

T he superfields $ k are chiral superfields and verify the constraint D a $ k = 0. It is well- known that when M T is a Kahler manifold, the action ( 1.1) possesses an extra supersym ­ m etry [3- 5] . For an appropriate choice o f K one has L = 1 = L = 2. Or in other words, once one has a Kahler a-m odel (i.e. a a-m odel with a Kahler target space m anifold), the classical form ulations (1.1) and (1.2) lead irreparably to extra supersymmetry.

T he main aim o f the current work is to analyze the existence (and properties) o f non­

equivalent SU SY extensions, which we call bosonic tw ins, 1 for a given two-dim ensional target-space m anifold (for exam ple O (3 ) = S2). W e will see through this work that it is possible to construct supersym m etric versions o f the nonlinear a-m odels with N = 1 SU SY which d o not allow for extra supersymmetry. Further, we will show that in fact, it is possible to generate an infinite fam ily o f well-behaved SU SY extensions (in the sense that they do not possess higher-derivative term s) labeled by an arbitrary function. W e will also show that, due to the constraints im posed by supersymmetry, and despite o f the m odification o f the ferm ionic sector, certain solutions o f the D irac equation (ferm ionic zero-m odes) will remain invariant (w ith respect to (1.1) and (1.2) ).

This work is organized as follows. In section 2 , we review the SU SY nonlinear O (3 )- m odel in terms o f real fields and in its C P 1 form ulation. In section 3 , we introduce the deform ation term which allows for the generation o f a new ferm ionic sector for general nonlinear a-m odels. In section 4 , we describe the nonlinear a-m odel with potential in 1 + 1 dimensions and determ ine the ferm ionic zero-m odes from SUSY. In section 5, we determine

1The term twin-like model was previously used in the literature [27- 30] to refer to couples o f theories sharing the same topological defect solution with the same energy. Here it is used is a different sense: pairs o f supersymmetric theories sharing in the same bosonic sector.

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the full on-shell action with the deform ation term and discuss the fermionie zero-m odes. In section 6, we describe the pure O (3 )-m od el in 2 + 1 dimensions (a potential is not allowed) and study ferm ionic zero-m odes and some peculiarities o f the quartic ferm ionic Lagrangian in the presence o f the deform ation term. In section 7, we describe some properties o f the b osonic twin with N = 2 SUSY. Finally, section 8 is devoted to our summary.

2 The

O (

This section is intended for a review o f the SU SY O (3 ) nonlinear a-m odel in tw o form ula­

tions, the O (3 ) and the C P 1. In the first form ulation the m odel can be written in terms o f three real scalar fields 0 a satisfying one constraint

3

E 0 a 0 a = 1. (2.1)

“ =1

In its non-SU SY version can be written simply as follows

1 f 3

S = - - d2xd ^ 0 aO^0a, E 0 “ 0 “ = 1. (2.2)

^ a=1

T he N = 1 supersym m etric extension o f ( 2.2) is well-known [6]. It only involves three real superfields and the supersym m etric generalization o f ( 2.1) , we have

1 f 3

S = — d2x d 29 D a ^ aD a ^ a, E $ “ $ “ = 1, (2.3)

2 ' a=1

where $ a = 0 a + — d2F a are three real superfelds. T he supersym m etric action ( 2.3) can be expanded is com ponents as follows

S = 1 ^ d2x [~ d ^ 0 ad^0a + # aada3^ + F aF aj . (2.4)

T he supersym m etric invariant constraint in ( 2.3) yields to three constraints in the co m p o ­ nent fields (sum is understood in the repeated indices),

0 a0 a = 1, (2.5)

0 a< = 0, (2.6)

F a0 a = 1 ^ aa< . (2.7)

Taking into account ( 2.5) - (2.7) we can eliminate the auxiliary fields from the action ( F k = 1 0 k^ r a ). T he resulting on-shell action can be written as follows

S = 2 J d2x ( —c r 0 ac)^0a + # aad / + 8 ( ^ aa< ) 2) . (2.8)

This expression is reasonably simple, but we have to take into account the constraints ( 2.5) and ( 2.6) . W e can instead, solve explicitly the constraints ( 2.5) - ( 2.7) and rewrite the

It is easy to verify that, in terms of the new complex fields the constraints ( 2.5)- (2.7) are automatically satisfied. If we substitute ( 2.9) in (2.3) , we get

S = - 1 J d2xd 26»g($, $ t)Da$ tDa$ , (2.19)

where g ($ , $ t) = 1/(1 + $ t $ ) 2 is the C P 1 metric. The action (2.19) constitutes the N = 1 C P 1 formulation of the model. Standard calculations lead to the following expression

S = - 1 J d2x (# (0 , 0) (ida/30 130a + 1 0 ° ¾ 0 3 + da!30da f0 - 2 F p ) + d ^ g ( 0 , 0')0a0a0303 - id^g(0, 0 )d7a0 070a - 1 9 ^ (0 , 0 )d7a0 070a

-d ^ g (0 , 0 ) F 0^a0^a - d ^ g (0 ,0 ) F 0^a0^a) . (2.20) We can eliminate the auxiliary field from its equation o f motion

F = - 0 ^ 0 0 0a0a, F = - ^ 0 7 7 0 00a0a. (2.21)

2g (0 ,0 ) 2g (0 ,0 )

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action in terms of a complex superfield. To proceed, we can use a SUSY analogous of the stereographic projection

$ “

$ + $

> - i($ - $

) >1 - $ t $

> (2.9) where $ is a complex scalar superfield. We get in components

0“ =

1

0 + 0 - i ( 0

1 - 00

> (2.10) (1 + 00)

= (T+ W + ^{“ +} ^ - ( 1 + 0 0 ) x+ ' (2'11)

= ( I T S ) ( - i ( f c - ^ + i + + 0 : 0 1 x+ ■ (2-12)

= (

+1

X

(+

000)x

• (2

3)

F + T - 1 -

F 1 = 71+000) - (0 + 0 )H - 7 7 + 0 0 0 2 7 a(^ « + + « ) ’ (2.14)

(1 + 00) (1 + 00)

F — F i

F = - i (1+114 ) + i(0 - 0)H + ( 1 + .00)2 x “ (0 “ - '0a)' (2'15)

F 3 = - £ 0 + F 0 0 + P - (1 - 0 0 )H + 7 ^ + W ^ (2.16) where

Xa = + 0 + + 0, (2.17)

H = (

F 0 + F 0 + ^

- (

+

x “ x». (2.18)

A fter substituting ( 2.21) in (2.20) we can write the action in a geom etrical way

S o (3) = - 1 J d2x g (< ,< ) ( d ° 3 <pdap< + i ^ aD a/3t 3 + ip ja D appi3 ) + R t “ t a t 3t p , (2.22)

where D ap = dap — r ^ dap < and R is the Riem ann tensor for C P h

3 A bosonic twin for the

O(

Obviously, actions ( 2.3) and (2.19) are equivalent. T h ey represent the same classical field theories but expressed by means o f different target-space variables. Hence, their bosonic and ferm ionic sectors coincide and are related by the transform ations ( 2.10) - ( 2.16) .

A nother question is whether it is possible to construct different, physically n ot equiva­

lent supersym m etric extensions o f a given bosonic m odel. In other words, we want to con ­ struct actions which are: 1) invariant under the sypersym m etric transform ations; 2) share the same bosonic sector; 3) but differ as far as the ferm ionic part is considered.

This question has been answered affirmatively in the literature. For exam ple in [12- 14]

different supersym m etric extensions o f the baby Skyrme m odel were proposed. However, the ferm ionic part o f the supersym m etric m odels contains potentially dangerous higher- derivative terms. A different proposal was made in [15, 16] (in four dimensions) and in [17- 19] (in three dim ensions) with N = 1 and N = 2 SUSY. In these cases, the ferm ionic part o f the non-equivalent supersym m etric extensions suffers from the appearance o f derivative terms involving the auxiliary field. This may prom ote the auxiliary field to a dynam ical one. Furthermore, all these SU SY extensions reproduce the bosonic m odel only on-shell, i.e., once we eliminate the auxiliary degrees o f freedom from the action.

Specifically, our aim is to analyse the possibility o f the existence o f non-equivalent SU SY extensions o f a given (boson ic) m odel with an additional condition, that 4) no higher- derivative terms in the ferm ionic sector are allowed. Here a com m ent is relevant. W e use a standard, “naive” derivative counting: a term in an action is an n-derivative term if it ex­

plicitly contains n spacetim e derivatives. Then, for example, the usual kinetic (quadratic) term for the bosonic (com plex scalar) field, d^(pd^< is a two-derivative term, while the ferm ionic kinetic term, p)a d<p t p , is a one-derivative term. O f course, terms which d o not contain spacetim e derivatives, for exam ple << or are zero-derivative terms. Such a derivative counting underlines the dynam ical (usual or unusual) character o f derivatives terms. W ith this natural counting we understand that an action is free o f higher-derivative terms if it has at m ost tw o derivatives acting on bosons or one acting on fermions.

On the other hand, there exists another derivative-like counting which rather reflects di­

m ensionality o f fields in considered terms. In this approach one assigns derivative-like (dim ension) numbers as follows [<] = 0, [d] = 1, [t] = 1 /2 . Therefore, in this counting each ferm ion is often treated as “ 1/2-derivative term ” . Then, nonstandard terms containing only two explicit derivative can be o f higher dim ensionality (higher-derivative number).

For exam ple, d^<pd^< would be a three-derivative term. From now on, we will only consider the first natural counting.

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It turns out that in 2 and 3 dimensions and with N = 1 SU SY there is not to o much freedom . Let us look for terms with trivial (em pty) bosonic sector. Then, such terms can be added to a SU SY action (for exam ple ( 2.19) ) w ithout any deform ation o f the original b osonic sector. A necessary condition for such a term is that it requires at least four odd operators (in the number o f superderivatives). M oreover, the degree o f each operator (the number o f superderivatives) cannot be greater than one, otherwise we will generate higher­

tim e derivatives — a possibility which we excluded from the very beginning. A t the end only one com bination remains

L d = d20 H ( $ , ^ ) D a ^ D ^ D g ^ D g ¢ , (3.1)

where the dimensionless function H f($, $ t ) is arbitrary and depends only on the superfields but not on derivatives and A is a coupling constant o f dim ension - 1 . From now on, we will use the function H ( ¢ , $ t) = AHf($, $ t ) by absorbing the coupling constant in the definition o f the dimensionfull function H ( ¢ , $ t). Here the target-space m anifold M t verifies dim e = 1. If dim e > 1 more com binations are allowed. N ote that the com bination D a ¢ D a ¢ D g ¢^- D g ^ is proportional to ( 3.1) since

D ^ D g ¢ = 1 C ga D Y ¢ D 7 ¢ . (3.2)

As we will see, the addition o f these terms to a given SU SY m odel does not change the b osonic sector. In this sense we say that they generate “bosonic twins” . T he expansion in com ponents o f ( 3.1) leads to

L d = ( - H g F - H g F ) VVV V

+ 2 H | ( id ag V gVa + iV a dagVg + dga4>3ga ^ - 2 F F ) VV

+ ^ ^ - 4 Oga4>dga4> + 1F ^ VV + h.c.^

+ F F VV + { - i d ga(j)F'Va'Vg + h .c .j + dYa 4>dYg 0V a V g| (3.3)

and, as expected, no explicit higher-derivatives appear in the action and L d|^= 0 = 0. On the other hand, the presented deform ation is based on terms which have upper introduced

“ higher field dim ensionality” (strictly speaking these new terms are o f dim ension 3). In this sense, they go beyond the usual quadratic contribution.

It is also im portant to note that the fact that the superfield ¢^- is com plex is cru­

cial — otherwise these new terms vanish. This results in the following observation: this construction o f the twins trivializes if we consider a single real scalar bosonic model.

Specifically, we take the O (3 ) m odel in the C P 1 form ulation ( 2.19) and add ( 3.3) . Then we obtain a new m odel sharing the off-shell bosonic sector with the original one. Let us analyze the on-shell action in detail. W e first need to eliminate the auxiliary field F from (2.19) + ( 3.3) . W e obtain

F = - 2g VV + 2g id ag W a t g - H ( V V ) 2 - (V V )2 . (3 .4)

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The first term in ( 3.4) corresponds to the original O ( 3) m odel while the others are originated by ( 3.3) . After substituting (3.4) in ( 2.19) + (3.3) we get

Lo(3) = 1 g ( ( , ¢ ) ( - 2 d ^ d ^ + i p a D af p 3 + ip>aD af p f )

- H (¢ , ¢ ) ( + d ^ (d ^ (p ip - 2d7“ ( d 7f (p tar p ^ + . . . (3.5)

where the dots stand for quartic fermionie terms. T he first line in ( 3.5) corresponds to the original SU SY C P 1 m odel, while the second one is originated from the deform ation ( 3.3) .

W e underline that the new part o f the action (3.1) is the unique term which contains only ferm ionic sector and therefore leave the bosonic part unchanged. All other possible terms / d20 O $ , generated by an arbitrary operator O contribute b oth to the ferm ionic as well as the bosonic part o f the m odel, with an exception for the trivial operator O = 1 which leads to a purely bosonic part i.e., prepotential.

4 Adding a potential

It is a well-known fact that in tw o dimensions one can add a prepotential term to the action w ithout spoiling the N = 1 SUSY. For future purposes we will restrict the potential to be holom orphic and antiholom orphic functions o f the superfields. Then, a general SUSY non-linear a-m odel on com plex one-dim ensional manifolds (fixed by a particular choice o f the metric function g) reads

S& = - 1 J d2x d 20 g ($ , $ ) D “ $ tD a $ , (4.1)

SP =

J

where W ( $ ) defines the prepotential part. E xpanding the potential term in com ponents we find

Sp =

J

(

'

"

^

Let us start with the purely bosonic sector. T he static energy functional can be w ritten as

(

'

'\

E = J dx ( g ( ( +---J , (4.4)

where ( = dx ( and W ' = d ^ W . W e can use the B ogom oln y trick and rewrite the energy integral by com pleting the square

E =

J

^

(

'

-

-

'

'

E > 2

/

^'

^'

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Obviously, the bound in saturated when the field obeys the B ogom oln y equation (where we assume that the prepotential allows for static solitonic solutions)

• W '

0 = — . (4.7)

g

It is easy to verify that the solutions o f ( 4.7) o b ey the second order equation from ( 4.1) . Now we will analyze the ferm ionic sector. A fter elim inating the auxiliary field from the action ( 4.1) + (4.2) we get

L ap = g ^ —d^0d^0 — ^ - d j ¢)3¢ a — 2 ^ d a P — — — +

+ 1 w " ^ a + 2 — w * — — ' — — g g ¢ ^ +

+ d«3 0 — 2 ^ ^ 3 da3 0 + O (¢ 2 ^^2 ). (4 .8)

In the next step we explicitly express the spinors in chiral com ponents ( ¢ + , ^ ) . T he static ferm ionic part o f the Lagrangian in these new variables takes the following form

L l p = —i 2 + ^ + ¢ - ¢ - — V’- V -

+ i W '— ¢ + ¢ - + i W ' ^ ¢ + ^ — i W 'V + ¢ - — i W ''¢ + ¢ _ g - g - - -

— ¢ - ¢ - 0 ' + 2g^ ¢ + ¢ + 0 ' — 2 ^ - ^ 0 + ^ g ^ V H ^ 0 ' + 0 (¢ 2 ¢ 2 ). (4 .9)

Here, tw o observations are in order. First, the Lagrangian is invariant under the N = 1 supersym m etry transform ations

$0 = —( a ¢ a , (4 .10)

$■0« = —e3 (C« 3F — ida30 ) , (4 .11)

5 F = —ea id« 3 ¢^, (4.12)

where e is a real spinor. Second, the requirement that the theory is invariant under the N = 2 supersym m etry can be achieved by prom oting e to a com plex ob ject. This is equiva­

lent to say that we have the transform ations ( 4.10)- ( 4.12) with real param eter followed by a phase rotation for the fermions. In terms o f the chiral com ponents in the Lagrangian ( 4.9) it means

¢ + ^ e + ^ i , ¢ + ^ e ^ i . (4.13) T he substitution o f ( 4.13) in ( 4.9) leaves the Lagrangian invariant im plying that the m odel has an extra supersymmetry. This can be confirm ed directly by rewriting ( 4.8) in the N = 2 SU SY language. Namely,

L n=2 = j d2gd20 K ( $ t , $ ) + ^ d W ( ¢ ) + h .c ^ , (4.14)

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where K is the Kahler potential. Now, we will consider the fermionie (zero-m ode) equation obtained from ( 4.9)

( 9 ¾ + 9*0 ' — '' - f — ' \ ( 0 + \ = 0 ( 4 15)

\ — '' - 9f — ' gdx + 9 $ ) { $ - ) ■ ( . ) On the other hand from ( 4.11) we find

f W + \ = _ i f F + e- i a 0 ' F - e- i a 0 '\ f V\ lg ) 2 y - f i - e - i a F e - i a F - 0 ' j , ( . ) where

n = 1 (e2 + eiae1) , £ = 2 (e2 - eiae 0 ■ (4.17) It is straightforward to see that solutions o f the B ogom olny equation ( 4.7) are preserved by supersym m etry transform ations with n = 0 ,£ = 0. This has a consequence that the zero-m ode equation ( 4.15) is autom atically satisfied for

0 + = - ie - i a 0'n, 0 - = i0 n - (4.18) Therefore, the fermions are param etrized by only one real constant n (see for exam ple [20]).

As a consequences only 1 /2 o f supersym m etry is preserved (in N = 1). If the theory has a hidden extended supersym m etry only 1 /4 o f the ( N = 2) supersym m etry is preserved.

The connection between solitons and ferm ionic zero-m odes in SU SY theories has been extensively discussed in the literature [8 , 21] .

5 The bosonic twin and fermion zero-modes

As we have seen before, the introduced deform ation term does not m odify the bosonic sector o f the original action while it nontrivially contributes to the ferm ionic sector even at quadratic order. Furthermore, the auxiliary field also gets contributions at second order in the spinors

I—' H —/ ' I—' 9

F = --- 2 i - 3 a? 0 0 a 0 ? - 2 - ^ - 0 a0 a + 2 H — 0 a0 a - ^ 0 a0 a + O ( 0 20 2). (5.1)

9 9 92 9 29

A fter eliminating the auxiliary field, the full O (3 ) sigma m odel action with the potential and the deform ation term included is

LaPd = 9 ^ - ¾ 0 ^ 0 - 2 ,9 ^ 0 ? 0 a - -2-00¾ 0 ? ^ - — g — +

+ 1 — ''0 a0 a + 2 — " 0 a0 a - — ' f 0 a0 a - — ' J 0 a0 a +

+ ^ 0 0 0 3 da30 - ^ 9 , 0 0 0 ^9J30

( — '2 ——'2 - - — ' — '2 - — ' « + - — r 0 a0 a + 92 9 2 0 a0 a - 2 ---0 a0 a + 2i — 0 a? 0 a 0 ?92 9

- 2 i — da? 0 a 0 ? + 4 c r 0 3 ^ 0 a0 a - 3 9 0 d ^ 0 a0 a 9

- d ^ 0 d ^ 0 0 a0 a + 2dl a 0dlfi 0 0 a 0 ? ^ + O ( 0 20 2). (5.2)

W e write the Lagrangian ( 5.2) in terms o f the chiral spinors (0 + , 0 - ) using the following replacements

0 a0 a ^ 2 i 0 + 0 - , (5.3)

0 a0 a ^ 2 i 0 + 0 - , (5.4)

0 a0 a ^ i {0 ) - 0 + - 0 ) + 0 - ) . (5.5) T he invariance under N = 2 SU SY requires that ferm ion-num ber violating terms are absent from the action. However, the deform ation introduces tw o such terms: one proportional to 0 a0 a and the last term in ( 5.2) proportional to dYa4dY3 4 0 a0 3 . This means that they do not respect the sym m etry ( 4.13) . As a consequence, the deform ation term breaks the original N = 2 o f the a -m odel to N = 1 SUSY.

A lthough by construction our deform ation prescription does not m odify the bosonic sector, which in particular means that the B P S sector remains unchanged, the linearized equation for the fermions contains a nontrivial contribution proportional to H (the ar­

bitrary function in the deform ation part). As we explained in the previous section the ferm ionic zero-m odes are connected with the B P S sector via the supersym m etric trans­

form ations. Therefore, one expects a relation between the ferm ionic zero-m ode equations in deform ed m odels (different H ). Indeed, we find that in the background o f a solution satisfying ( 4.7) the zero-m ode equation including the deform ation term reduces to ( 4.15) . M oreover, the ferm ion-num ber violating terms are effectively absent from the action, re­

covering in some sense the original N = 2 supersymmetry. T o prove this we consider the zero m ode equation in the deform ed m odel

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( ^ ¾ + 9*4' - 2 H ( W 4 ' - W 4 -) ) ¢ +

+ ( i t " ' - It " y + 2H ( « -2 - W -2) ) ¢ - / , - , W 'W '4

- 2 H l(f> (f>---2— J ¢ - = 0. (5-6)

Obviously, all terms proportional to H vanish in the B ogom olny sector, that is, when 4 ' = -Wg- leading to the equation (4.15) . N ote also that, eq. ( 5.6) is equivalent to ( 4.15) only for a = 0, i.e. not for the continuous fam ily o f B P S equations. This is because the deform ation term breaks the sym m etry 4 ^ e ia4 which is present in the original model.

6 The

O (

In this section we consider the O (3 ) a-m odel in 2 + 1 dimensions. This obviously has a nontrivial im pact on the solitonic structure o f the m odel. First o f all, static solutions can be treated as maps 4 : R 2 U {to} = S2 ^ S2, where a single asym ptotic value o f the bosonic field is assumed. This allows for a one point com pactification o f the base space. Now, maps between two-spheres are classified by a pertinent topological index

n2(S2) = Z. (6.1)

Secondly, due to the D errick's theorem there is no stable, finite energy static solutions if a potential term is present. N ote that one can keep the potential part if another (higher- derivative term ) is simultaneously introduced. This will be analyzed in the next sections.

Once we neglect the potential, the resulting pure O (3 ) m odel is a B P S theory. This means that the static energy functional is bounded from below by the topological charge

E * 4 / , (6.2)

while the bound is saturated for solutions o f the B ogom oln y equations, which in this case are just the C auchy-Riem ann equations

dx 4 = ± id y 4, dx 4 = T id y 4. (6.3) Hence, the B P S sector (solutions o f the B ogom olny equations) is constituted by holom or- ph ic/an tih olom orp h ic functions

4 = 4( z) , z = x + iy or 4 = 4 ( f ) , f = x - iy. (6.4) T he ferm ionic zero-m odes in the B PS sector can be obtained using the supersym m etric transform ation for the ferm ionic degrees o f freedom . From ( 4.11) we have

( # + \ ( - i dz4 - i d z A ( A (6 5)

( ¢ ¢ 4 = ( - 9 - 4 9 ,4 J 4 4 (6 ' 5) where

n = 2 (gl - ie2) , £ = 2 (gl + ig2) . (6 .6) The Dirac equation from ( 3.5) can be written as follows

g (dzX + dzXc) + dtpg (dz 4 x + d - 4 x c) = 0, (6.7) where x = ¢ + + ^ - and x c = ¢ + - ^ - . Like in the one-dim ensional case, in the background o f a B P S solution the fermions depends only on one real parameter. Let us take dz 4 = 0, we have from ( 6.5)

X = - 2 i d - 4n, Xc = 0. (6.8)

These solutions autom atically satisfy ( 6.7) since d ,d z4 = 0. Due to the supersym m etry o f the m odel we can express the ferm ionic zero-m odes through holom orph ic/an tih olom orph ic derivatives o f the B P S solutions.

Now, let us see what happens if we add the deform ation term. O f course, as in the one-dim ensional case the B P S sector does not change. On the other hand, the linearized ferm ionic equation gets extra terms

g (dzX + d -Xc) + d^g (d z4x + d -4 x c) - y ( | d- 4|2 + |dz4|2) ( x - Xc) H

- H d z 4 d -4 (X - Xc) + y ( ld z 4 | 2 - |dz4|2) ( x + Xc) = 0. (6.9)

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However, in the background o f B P S solutions, for exam ple dz0 = 0 ^ x c = 0, solutions o f ( 6.9) are o f the form

X = —2idz0n. (6.10)

Therefore we observe the same effect. In the background o f the B PS solutions the ferm ionic zero-m odes are independent o f the deform ation terms. Besides, all fermion- number violating terms in the action are absent once we substitute the B P S equation ( 6.3) , im plying an “on-shell” restoration o f the N = 2 SUSY.

6 .1 T h e quartic ferm ionic term s

T he analysis o f the previous section was based on the linearized ferm ionic sector, where we did not take into account the higher ferm ionic contributions. In the original a-m odel only a quartic term appears accom panying the Riem ann tensor o f the target space manifold.

Once we add the deform ation term the situation is more com plicated. T he quartic action in fermions for the pure deform ed a-m odel can be written as

L = H (id « 3 V 3W + ida3V 3V “ V 2) + ( R - 2 H 2 W . (6.11)

In the pure undeform ed a-m odel the term proportional to R cannot be eliminated un­

less the target m anifold is trivial. In our case, extra ferm ionic terms involving derivatives appear and cannot be eliminated unless H = 0 (the first tw o terms in ( 6.11) ). On the contrary, for a special choice o f the function H , the term proportional to V 2V 2 can be elim­

inated in the background o f the B P S solutions. Let us take, for exam ple the holom orphic solution dz0 = 0. T he last term in ( 6.11) ca be rewritten as

( R - H 2 V)2 V 2. (6.12)

Let F (z) be a holom orphic function such that F ( z ) = dz0 (z ), where 0 (z ) is a particular B P S solution 0 = f (z ). Now F ( z ) defined as the holom orphic derivative o f a particular solution 0 (z ) can be written as a function o f 0 itself for this particular solution

F ( z ) = dz0 (z ) ^ F ( 0 ) = f / ( f - 1 ( 0 ) ) (6 .13) where the prime indicates differentiation with respect to its argument. T he choice

H 2 = R g , (6.14)

F ( 0 ) F ( 0 ) , ( )

eliminates ( 6.12) and leads to the following quartic contribution

L 41 BPS = F ( 0 ^ (0 ) ( i d« 3 ^ 2 + id«3 V “ V2) . (6 .15)

W e want to emphasize the fact that the quartic term ( 6.12) is only eliminated in the background o f the B P S solution for which H was constructed.

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7 Higher-derivative terms in the N = 2 bosonic twins

W e can work directly in the N = 2 language to suggest why N = 2 bosonic twins cannot exist. Let us start with the N = 2 version o f the O (3 ) a-m odel (or more precisely the C P 1 a -m od el). W e have

LNpi2 = / d20d20 lo g ( i + . (7.1)

W e need first to saturated the Grassmann integration to generate a term w ithout bosonic sector. T h e key term is given now by the fourth derivative term

L4 = y d20d20 D “ $ f . (7.2)

B y m ultiplying this term with ferm ionic ob jects we could in principle construct pure ferm ionic actions, and as we did before, we could construct inequivalent SUSY exten­

sions o f a given bosonic m odel. In this case there are tw o terms verifying the properties described above

T6 = K e F a $ D $ t D a $ D /3 $ tD ^ ¢ , (7.3)

T8 = ( P a $ D $ t D a $ D /3 2 . (7.4)

But now the Lagrangian ( 7.2) possesses nontrivial bosonic sector, namely

L4 « ( ( d ^ ) 2(dv^ ) 2 - 2 F P d ^ d ^ + ( F F ) 2) , (7.5) and T6 and T8 verify

T6|0,0=o = T8|0,0= o = 0. (7 .6) This leads to T6 = T8 = 0 and therefore, N = 2 SU SY does not allow for the extra extensions if the dim ension o f the target space m anifold ( M T) is tw o (or in other words, in the target m anifold can be constructed in terms o f only one chiral superfield). Since, in order to have N = 2 SUSY, M t must be Kahler and therefore dim M t = 3. This implies that M t must be Kahler and dim M t > 4. T he first consequence is that terms o f the form ( 3.1) cannot be extended to N = 2 (as we verified explicitly in the previous section).

Let us assume that we have a N = 1 m odel built in terms o f two com plex superfields (i.e. dim M T = 4). W e can construct N = 2 pure ferm ionic terms is this case. It is possible to generate four different terms with six derivatives and six superfields, namely

(D $ 1)2 ( P T 1^ 2 (D $ 2)2 + h.c., (7.7)

(F $ 1)2 ( D $ 2t) 2 (F $ 2)2 + h.c., (7.8)

(D $ 1D $ 2)2 (f T 1^ 2 + h.c., (7.9)

(D $ 1D $ 2)2 ( D $ 2t) 2 + h.c., (7.10)

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and three different terms involving eight derivatives and eight superfields

( D $ 1)2 (d S 1^ 2 ( D $ 2)2 ( u $ 2t) 2 , (7.11) (7.12) (7.13)

|D $1D $ 2|4,

(D $1D$ 2 ) 2 ( JD$ 1 t) 2 ( D$ 2 t ) 2 + h.c.,

where ( D$ ) 2 = D a $ D a $ , etc. A fter expanding these terms in com ponents we get

2 0 V V ( ¾ 4 W - ) 0 i0 j C La,

^ a ^ 0 j (0 ^ 4 iU 4 j - d^4iU 4 j ) 0ji0 j 0ji0 j C Lg.

(7.14)

(7.15)

Thus, bosonic twin cannot exist with N = 2 SU SY unless we allow for higher-derivative terms in the ferm ionic sector. A nother possible argument to suggest why N = 2 defor­

m ation terms w ithout higher-derivatives cannot exist may be explained by a dimensional argument. T he minimum number o f superderivatives to generate a pure ferm ionic term with tw o supersymmetries is six ( 7.3) . Since [D a] = [dd] = 1 /2 in d = 2, after Grassmann integration the resulting Lagrangian has dim ension five, leading to at least three spacetime derivatives in each term.

8 Summary

In this work we have constructed new supersym m etric versions o f the nonlinear a -m odel with two-dim ensional target-space m anifold. This construction is based on the addition o f a pure ferm ionic term (supersym m etric invariant and with vanishing bosonic sector) which is independent o f the m odel apart from its field content. A fter the expansion in com ponents we have shown that the deform ation term is well-behaved in the sense that it does not contain higher-derivative terms. If the absence o f higher-derivative contributions is im posed it turns out that such a term is unique (for dim M T = 2) up to an overall function depending on the superfields but not on derivatives. Besides, it does not contain derivatives acting on the auxiliary field (although the appearance o f the derivative o f the auxiliary field not always implies that F becom es dynam ical — it leads usually to the generation o f higher-derivative terms for the physical fields [19]).

T he inclusion o f the deform ation term to the SU SY nonlinear a -m odel has the two main effects:

1. Since it does not m odify the bosonic sector the new action constitutes a new super- sym m etric extension o f the original a -m odel with properly deform ed ferm ionic sector.

2. T he deform ation term is strictly N = 1, i.e. there is no hidden extended supersym m e­

try. This implies that our supersym m etric versions o f the a-m odel are strictly N = 1 (note that the usual form ulations o f the a-m odel are im plicitly or explicitly N = 2).

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These properties lead to interesting consequences. First o f all, it follows trivially from the preservation o f the bosonic part that the B P S sector is also not m odified. On the other hand, the ferm ionic sector receives nontrivial contributions even at the quadratic order.

This might suggest that the relationship between B P S solutions and ferm ionic zero-m odes is broken by the deform ation term. However, as we proved in the paper, everything combines is such a way that the ferm ionic zero-m ode equation remains unaltered (w .r.t. the original CT-model) in the background o f the B PS solutions. A t the same time, all ferm ion-num ber violating terms generated by the deform ation disappear leading to an “on-shell restoration”

o f the original N = 2 SUSY. Needless to say that, for general bosonic solutions (non B PS) the solutions o f the D irac equation in the deform ed m odel have no relation with the original solutions (and the ferm ion-num ber violating terms are not elim inated).

Secondly, the standard SU SY CT-model contains a quartic ferm ionic coupling propor­

tional to the Riem ann tensor on M t. This term cannot be eliminated in a supersym m etric invariant way unless M t is flat. If we add the deform ation part then new quartic ferm ionic terms show up. T h ey contain a nontrivial contribution from the bosonic field as well. This can lead to an effective elim ination o f this quartic term, in the background o f a B P S solu­

tion, if an appropriate choice o f the arbitrary function H is made. Nonetheless, the quartic ferm ionic part does not trivialize since a derivative ferm ionic term remains. For CT-models adm itting a potential (for exam ple in 1 + 1 dim ensions) the elim ination o f the Riem ann tensor term can be achieved by a suitable choice o f the prepotential.

Thirdly, the deform ation term can be added to any supersym m etric version o f a m odel based on the same bosonic d.o.f. as O (3 ) CT-model i.e., a three com ponent unit iso-vector 4. This concerns for exam ple the baby Skyrme m odel [2 2] and the B P S baby Skyrme m odel [23] (a lower-dimensional counterpart o f the B P S Skyrme m odel [24]). Therefore our construction provides a new set o f well behaved N = 1 SU SY versions o f these topologically nontrivial m odels [12]- [17] . It would be very desirable to analyze the ferm ionic sector o f these new extensions in detail with a particular focus on the ferm ionic zero-m odes, however, this issue is beyond thee scope o f the paper.

Finally, we have analyzed the bosonic twins with extended supersymmetry. It turns out that if one imposes the higher-derivative restriction such m odels cannot exist. M oreover, using the connection between N = 1 SU SY in four dimensions and extended SU SY in three dim ension (via dim ensional reduction), one can lift this non-existence to the four­

dim ensional case.

The proposed deform ation o f the N = 1 O (3 ) CT-model choses only one very specific term o f all possible term generated by operators o f dim ension 2. Obviously, in the quan­

tum version o f the m odel such terms also contribute and should be m andatory taken into account. W ell-posedness o f such general deform ations o f the O (3 ) CT-model as well as prop­

erties o f the resulting solitons are very interesting problem s, which however go beyond the scope o f the present work.

There are several straightforward directions in which presented analysis can be further investigated.

As we have already noticed our deform ation applies for any field theory with the unit, three com ponent iso-vector field. It would be interesting to present a com plete and

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system atic classification o f all supersym m etric extensions o f the pertinent bosonic models (O (3 ) m odel, the baby Skyrme and the baby B PS Skyrme m odels) also with the case when high-derivative terms are taken into account [14] .

A nother possibility is to repeat our construction for models with higher-dimensional target-space and (or) in higher dimensions. This would include supersym m etric O (n ) a -m odel and especially the Skyrme m odel, where some developm ents have been recently m ade [18, 25, 26] .

T he last very interesting issue is related to a widely known fact that the nonlinear O (3 ) a-m odel in two dimensions (and also higher-dimensional target-space generalizations) is an integrable field theory w ith a zero-curvature form ulation. It would be desirable to understand the fate o f the integrability in twin supersym m etric extensions. Obviously, the theories remain integrable in their bosonic part but probably not necessarily when the fermions are included.

Acknowledgments

A W was supported by N C N grant 2 0 1 2 /0 6 /A /S T 2 /0 0 3 9 6 . W e thank W ojtek Zakrzewski, Stefano Bolognesi and Christoph A dam for discussion.

O p en A ccess. This article is distributed under the terms o f the Creative Com m ons A ttribution License ( C C -B Y 4.0) , which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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