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Bidimensional Supersymmetric Field Theories with Discrete Moduli Space
Annals of PhysicsPH5786 Annals of Physics 266, 6380 (1998) Article No. PH975786
Bidimensional Supersymmetric Field Theories with Discrete Moduli Space Luis J. Boya Departamento de F@ sica Teorica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
and Javier Casahorran Departamento de F@ sica Teorica, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s.n., 33007 Oviedo, Spain Received July 14, 1997; revised November 4, 1997
We consider bidimensional supersymmetric models for which the classical theory has a discrete set of inequivalent vacua. Once we introduce the concept of vacuum manifold V, defined in terms of all the vacuum field configurations, the properties of such models can be described by means of the classical moduli space Mc(V). Quantum corrections, perturbative and nonperturbative, do not lift the vacuum degeneracy so that the quantum theory as a whole exhibits non-trivial quantum moduli space Mq(V). The former structure allows us to classify the kinklike excitations into loops and links: loops interpolate smoothly between equivalent vacua while links connect vacua located at different points in Mq(V). 1998 Academic Press
I. INTRODUCTION 1. Nowadays it seems clear that supersymmetry, a new kind of symmetry which includes bosonic and fermionic degrees of freedom, represents one of the most ambitious attempts made in order to obtain a trustworthy description of all basic interactions of nature. In four dimensions supersymmetry transformations mix half-integral and integral spin states and therefore the corresponding multiplets relate fermions to bosons. This fact suggests that the new type of symmetry may provide a remarkable principle of unification in quantum field theory, thus going over the powerful constraints imposed by the Coleman-Mandula theorem [1]. The only symmetries allowed in the old schemes are those which turn out to be locally isomorphic to the direct product of an internal symmetry and the Poincare group. In this respect supersymmetry offers a way to achieve unification of space-time and internal symmetries by resorting to the so-called graded Lie groups. Despite the 63 0003-491698 25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
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unfamiliar properties of the graded Lie algebras (superalgebras in the jargon of physics) which only close under the right combination of commutation and anticommutation brackets, it soon became clear that supersymmetry could play an important role in many branches of modern theoretical physics. Returning to quantum field theories, the models based on supersymmetry are less divergent than naively expected. In essence, putting together bosonic and fermionic fields produces in fact cancellation of many infinities because the respective contributions have opposite signs As we have no experimental evidence for such a novel symmetry emerging exactly in nature, one of the most pursued goals in the past two decades has been to describe a suitable mechanism by means of which supersymmetry shows up itself broken. To start from scratch, let us recall some basic features concerning globally supersymmetric theories. First of all, since the hamiltonian H itself is the sum of squares of hermitian operators (the supersymmetry charges Q : ), the energy of any state is positive or zero [2]. In addition, the states annihilated by the action of the Q : have zero energy and represent supersymmetric vacua of the model. If there does not exist a such state invariant under supersymmetry we know that the groundstate energy is strictly positive and supersymmetry itself appears spontaneously broken. On the other hand, a non-zero vacuum expectation for a propagating elementary scalar field is consistent with exact supersymmetry and that represents the breakdown of an internal symmetry [3]. If |0) represents globally the ground states of the model, the spontaneous breaking of supersymmetry can be analyzed by means of the anticommutator of Q : with some fermionic operator X. If it happens that Q : |0) =0, the expectation value associated with ( 0| [Q : , X] |0) would evidently have to vanish. In supersymmetric theories the elementary scalar fields , can never be written in terms of anticommutators like [Q : , X] and this fact explains that they might acquire vacuum expectation values without breaking supersymmetry. As regards the derivatives + , of the physical fields themselves, it is the Lorentz invariance that prevents the emergence of vacuum expectation values which otherwise would be possible in terms of [Q, ], where represents elementary fermions. Hence [Q : , X] is an auxiliary nonpropagating scalar field F and only (F) {0 is a signal of spontaneous susy breaking. 2. When considering models in (1+1) with dimensionless scalar field ,, the potential V(,) can take a much more general form than in the conventional (2+1) or (3+1) cases as regards the naive renormalizability of the theory. In addition the model becomes automatically supersymmetric by taking advantage of the simplest multiplet of fields consistent with N=1 supersymmetry. They form the so-called chiral scalar superfield where we find a real scalar field ,, a two-component Majorana fermion and an auxiliar field F which ultimately can be eliminated. When a general choice for the superpotential function is made, we can get models where the unbroken supersymmetry appears associated with a discrete set of physically inequivalent vacua. In doing so we go beyond the well-known cases of the supersymmetric versions of sine-Gordon or , 4. The best way to describe the main
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properties of this new family of models takes advantage of the so-called vacuum manifold V, defined in principle as the set of all vacuum field configurations. In other words, we start front the scalar fields , for which the potential is zero. In this scheme different points along V are parametrized by the corresponding vacuum expectation values of the scalar field ,. On the other hand, points P, P* in V will describe equivalent physical situations whenever we can reach P* from P by means of a discrete group of symmetry G. Therefore we deal with a d=2 discrete version of the models formulated in d=4 whose vacuum structure is described by means of a continuous manifold. Next we can define the classical moduli space Mc(V) of the model as the space of equivalence classes of vacua. Two vacua belong to the same equivalence class if they can be connected by means of the action of the group G. We are dealing with bidimensional models which exhibit discrete symmetries (whose spontaneous breaking produces no Goldstone bosons). The hypothetical breaking of a continuous symmetry would yield massless bosons, which are excluded anyway in d=2 theories. In addition we expect that as a consequence of quantum corrections (perturbative and nonperturbative in principle) the vacuum degeneracy will be lifted so that ultimately the true vacuum set of the theory is selected. If this is not the case, the quantum theory as a whole exhibits a manifold of inequivalent ground states, i.e., a non-trivial quantum moduli space Mq(V). As expected the classical kink solutions interpolate between adjacent vacua and this persists upon quantization. We classify these classical solutions with finite energy into loops and links. While the loops themselves connect vacua represented by a single point in Mq(V), the links move along the quantum moduli space. We shall see how susy is realized both in the vacuum sector and in the kink sector. 3. The article is arranged in the following way. To keep the article self-contained, we review in Section 2 some well-established aspects of the bidimensional N=1 supersymmetric models based on the chiral scalar superfield. Section 3 is in fact the heart of the whole exposition since in it we carefully describe the models just exhibiting a set of physically inequivalent vacua. We define the vacuum manifold V, the classical moduli space Mc(V) and the quantum moduli space Mq(V). We study the spectrum of the particles in the vacuum sector and the structure of kink-like excitations. Important elements in the analysis of these theories like the behaviour of the Witten index or the way in which the quantum corrections do not lift the vacua degeneracy derived from classical arguments are also considered. In particular we emphasize the way in which the well-known supersymmetric extensions of sine-Gordon or , 4 represent theories with trivial quantum moduli space Mq(V). To illustrate these ideas in a simple context we resort in Section 4 to models based on well-behaved superpotentials functions (the susy extension of , 6 and , 8 models) so that the whole structure of vacua and kink-like excitations can be presented in closed form. We conclude with an outlook and various speculations.
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II. N=1 SUSY WITH A CHIRAL SCALAR MULTIPLET 1. Although a complete discussion of the bidimensional N=1 supersymmetric models based on the so-called chiral scalar multiplet is clearly out of the scope of this article, we review in this section just as much as we believe is necessary in order to understand the main properties of the theories which exhibit inequivalent ground states. DiVecchia and Ferrara [4] together with Hruby [5] were the first to introduce the supersymmetric version of the models consisting in a self-interacting scalar field in two dimensions. The behaviour of the theory is governed by the action S given by [46]
|
S= 12 [( + ,) 2 +(i# + + ) &W$ 2 &W"] d 2x,
(2.1)
where ,(x) stands for a real scalar field and (x) represents a two-component Majorana fermion. Now W$(,) is the standard superpotential function where the prime denotes as usual the derivative with respect to ,. For the generic model written in (2.1), the supersymmetric generators derive from the conserved supercurrent S + [7], i.e., S + =( j ,) # j# ++iW$(,) # +
(2.2)
so that the Q : themselves together with the momentum P + , given as usual by P + =&i + , allow us to recover the standard superalgebra associated with N=1, i.e., [Q : , Q ; ]=2(# +P + ) :;
(2.3a)
[Q : , P + ]=0,
(2.3b)
where we use # + matrices given by # 0 =i_ 1, # 1 =i_ 3. Resorting now to a formulation in terms of the chiral components of the fermionic field, i.e., \ =(1\_ 3 ) 2, the supersymmetric charges are
|
(2.4a)
|
(2.4b)
Q + = |( 0 ,+ 1 ,) + &W$(,) & | dx Q & = |( 0 ,& 1 ,) & +W$(,) + | dx
so that the superalgebra itself reads Q 2+ =P + , Q 2& =P & , Q + Q & +Q & Q + =0 if P \ =P 0 \P 1 . In any case the field equations derived from (2.1) are of the form ( + + ,)+W$(,) W"(,)+ 12 W$$$(,) =0
(2.5a)
[i# + + +W"(,)] =0.
(2.5b)
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2. At this point it proves convenient to introduce the supersymmetric version of the well-known Bogomolny condition, namely [4] (1\_ 3 ) 2=0,
d, c =\W$(, c ). dx
(2.6)
On the other hand it is the case that in terms of the constant fermionic parameter ' the field given by =[i# + + , c &W$(, c )] '
(2.7)
solves both the supersymmetric Bogomolny condition and the equations of motion whenever ' is forced to satisfy the constraint ' i =0 for i=+ or i=&. (Notice that =0 for chiral fermions \ =0.) 3. To finish this brief review, let us make some illustrative comments about the specific realization of supersymmetry itself in the topological sector. When considering theories which exhibit kink-like excitations, one finds that the standard superalgebra is no longer valid since the central charges associated with the existence of topologically conserved quantum numbers modify the results derived in the trivial sector [7]. To be precise, the supersymmetric algebra acquires central charges due to the surface terms, which although usually discarded, are actually nonvanishing for topological configurations. When keeping carefully the surface terms, one finds Q + Q & +Q & Q + =T where
|
T= 2W$(,)
, dx x
(2.8)
so we can say that Q + Q & +Q & Q + corresponds to the integral of a total divergence. When considering the existence of static kink-like configurations, the integral written in the right hand side of (2.8) is not necessarily zero but amounts to the difference between the values of 2W(,) at x=+ and x=& . The topological character of T becomes clear if one resorts to the improved topological current j +i given by j +i =2W$(,) = +& & ,
(2.9)
whose conservation law does not arise by Noether's theorem from a well-behaved symmetry of the lagrangian at issue but characterizes the large distance behaviour of the field configurations. In physical terms, it is the case that j +i provides us in essence with the same information already obtained from the analysis of the standard topological current j + written as j + == +& & , [8]. Despite the model dependent character of j +i , we use in the following such improved topological current so that the T operator itself results from integrating the j 0i component along the real line.
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Returning now to the modified algebra which includes the presence of the central charge term T, i.e., Q 2+ =P + , Q 2& =P & , Q + Q & +Q & Q + =T, we can write (Q + &Q & ) 2 +T=P + +P &
(2.10a)
(Q + +Q & ) 2 &T=P + +P &
(2.10b)
and since (Q + &Q & ) 2 0 we conclude that (P + +P & ) |T |. For a single kink-like excitation at rest with mass M we have P + =P & =M so that the existence of a central charge term translates into the physical condition M
|T | . 2
(2.11)
Before going to the quantum corrections, an explicit calculation based on the generic model written in (2.1) shows that at classical level the kink configurations saturate in fact (2.11): starting from the hypothetical solution , c(x) of (2.6) with (x)=0 (see for instance [5]), the classical mass M associated with this static kink-like configuration is M=
|
+ &
1 d, c(x) 2 dx
_\
2
1
+ +2 W$(, (x)) & dx, 2
c
(2.12)
which, once we insert the Bogomolny condition, leads to M=
}|
+
&
dW$(, c(x)) 1 dx = |T |. dx 2
}
(2.13)
Notice that (2.11) is the Bogomolny classical bound, here shown to be true at quantum level. However in this case the computation of quantum corrections to physical quantities is plagued with subtle aspects which must be handled appropriately. A naive examination of the problem would imply that the Bogomolny bound saturation holds at O(). If kink-like configurations receive no quantum mass corrections in this order, it suffices an identical assumption for the central charge term to close the argument. Unfortunately this straightforward approach to the problem underestimates the subtleties associated with the difference in density of fluctuation modes between bosons and fermions. In other words, cancellation of boson and fermion fluctuation energies, though valid for discrete modes, is not automatically true for the continuum. Going now to arbitrary choices of the superpotential W$(,) in (2.1), the use of trace formulae for linear differential operators on non-compact manifolds, as first stated by Callias et al. [9], allows us to calculate the O() correction to the classical kink mass. Finally the saturation of the Bogomolny bound at such order is shown by an explicit computation of the central charge term T (see for details [10]). It happens therefore that one of the most striking features of the theories just based on unbroken supersymmetry, the fact that the vacuum energy receives no quantum contributions in the perturbative
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regime, does not generalize to the topological sector. However, the results obtained at O() strongly suggest that it is the operator (H&T2) that receives no quantum correction. In other words, this would be the topological version of the results first derived by Zumino about the cancellation of vacuum diagrams in supersymmetric theories [11].
III. MODELS WITH DISCRETE MODULI SPACE Mq(V) 1. Along the past three years we have seen a waterfall of valuable results about the nonperturbative behaviour of supersymmetric Yang-Mills theories with or without matter, not to mention the new insights which concern duality and string models. Restricting ourselves to N=1 or N=2 supersymmetric gauge theories in four dimensions, the key results appear as a consequence of the holomorphic character of the low energy effective superpotential [12]. In principle these models exhibit a space of degenerate vacua so that they mimic the properties of the standard models just based on the spontaneous symmetry breaking phenomenon. The novelties appear when one notices that in fact the supersymmetric vacua at issue are inequivalent. Physically it means that for instance different vacua yield a well different spectra of excitations. Although the degeneracy between these states cannot be lifted perturbatively, the nonperturbative effects can produce new contributions that dramatically change the tree-order vacuum structure [13]. 2. Returning now to our bidimensional theories based on N=1 supersymmetry with chiral scalar superfield, it should be recalled that almost from the very beginning of the subject the cases considered in the literature just correspond to the susy extensions of sine-Gordon or , 4 [4, 5]. For reference, we can recall the main features concerning these well-known models in the spirit of the moduli spaces scheme. For example, if we start with the superpotential , W$(,)=2 cos , 2
(3.1)
the lagrangian density of the model enjoys the global symmetry given by , ,+2p?( p # Z), # 0. Although valuable information is available on the longestablished multisoliton sector of sine-Gordon, we restrict ourselves in the following to the physical consequences derived from the static solution , c(x) given by [8] , c(x)=2 arcsin (tanh x),
(3.2)
which smoothly interpolates between the adjacent vacua , &1 =&? and , +1 =+?. Going to the vacuum manifold V, this susy extension of sine-Gordon possesses an infinite number of vacua since V=[, v =(2v+1) ?, v # Z].
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(3.3)
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Accordingly, two generic vacua (, q , , r ) in V can always be connected by the action of the Z symmetry at issue. Therefore the quantum moduli space Mq(V), defined as usual in terms of the space of equivalence classes of vacua, i.e., Mq(V)= VZ, reduces itself to a single point. At this stage one could just as well take into account the susy version of the celebrated , 4 model. In this case we consider W$(,)=(, 2 &1)
(3.4)
so that now the symmetry at issue corresponds to , &,, # 0, i.e., Z 2 . In addition, the superpotential written in (3.4) gives rise to a static kink-like configuration , c(x) of the form , c(x)=tanh x,
(3.5)
which as expected connects the two vacua of the model. Proceeding along the same lilies than before we get trivial quantum moduli space since V=[, \ =\1]
(3.6)
so that Mq(V)=VZ 2 reduces again to a single point. To conclude this brief exposition let us remark that, being trivial the quantum moduli spaces Mq(V), the fine distinction between loops and links is out of order. 3. Now we consider the case where the superpotential W$(,) is a polynomial in , 2 so that the symmetry at issue corresponds to , &,, # 0 in the spirit of the , 4 theory. The polynomial superpotentials we are talking about can be understood as a general class of models which approximate the supersymmetric sine-Gordon theory. To be precise this approximation can be achieved by truncating the infinite-product formula for W$(,)=2 cos ,2 (see (3.1)) in order to obtain either of the following two polynomial superpotentials n
W$1(,)= ` (, 2 &c 2k )
(3.7a)
k=1 n
W$2(,)=, ` (, 2 &c 2k )
(3.7b)
k=1
for c 2m >c 2n if m>n. Notice that for n=1 the first superpotential in (3.7a) represents the susy version of , 4 while (3.7b) would correspond to , 6 [16]. One is adding more vacua for higher n and, correspondingly, the structure of the vacuum manifold V is enriched in a systematic way. Next, we discuss the way in which the vacuum degeneracy pattern associated with W$1(,) and W$2(,) allows the existence of kink-like excitations. As long as we are dealing with supersymmetric models the whole structure of these configurations relies on our ability for handling the well-known Bogomolny condition just written
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in (2.6). It may be interesting at this point to recall how the scalar component of the susy Bogomolny condition (recall that (x)=0), i.e., d, c =\W$(, c ) dx
(3.8)
has proved quite useful in this context. It should be emphasized that when we consider a physical situation with n discrete degenerate vacua, we can have 2(n&1) types of topological configurations. In addition such kink-like excitations serve to connect any two neighbouring vacua as the spatial coordinate x varies from & to + [8]. The topological configurations can be understood as interpolating classical solutions between adjacent vacua. However, the most important element in the analysis of these theories relies on the existence of a discrete degeneracy of inequivalent ground states. Classically, we find the spontaneous breaking of the Z 2 symmetry , &, since in general the vacuum expectation values of the field configurations which preserve supersymmetry are non-zero. In other words, the so-called vacuum manifold V can be described in terms of the vacuum expectation values of the scalar field ,. Accordingly, the vacuum manifold V associated with the superpotentials W$1(,) and W$2(,) written in (3.7) would correspond respectively to V1 =[,=\c k , k=1, ..., n]
(3.9a)
V2 =[,=0, ,=\c k , k=1, ..., n].
(3.9b)
With regards to V1 , it is the case that field configurations like ,=c k and ,=&c k (k=1, ..., n) will describe as usual equivalent physical situations since they are just connected by the action of the discrete symmetry , &, at issue. So we can have exact supersymmetry and breaking of the internal symmetry. Our choice in (3.7) provides however a more sophisticated realization of this scheme as far as the second vacuum manifold V2 is concerned. In fact the structure associated with V1 is enriched due to the existence of the additional vacuum ,=0 where precisely the discrete symmetry at issue is enhanced. In more physical terms, it seems plausible to remark the way in which observed supersymmetry can share the same life with both exact and broken internal symmetries depending on the ground state the model chooses between. Then we go to the classical moduli space Mc(V) defined as the space of equivalence classes of vacua, i.e., Mc(V)=VZ 2 . To sum up, in our case two vacua belong to the same equivalence class if they can be just connected by means of the action of the discrete symmetry , &,. The above discussion allows us to write that Mc1(V)=[3 k , k=1, ..., n],
(3.10)
where as expected the generic equivalence class 3 k just contains the two vacua ,=\c k . Essentially the same structure arises for V2 , the only difference between
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both classical moduli spaces being the existence in Mc2(V) of the equivalence class 3 0 associated with the vacuum ,=0 at which the Z 2 symmetry is enhanced because is unbroken. So we have Mc2(V)=[3 0 , 3 k , k=1, ..., n].
(3.11)
The above discussion sheds light on the properties of the kink-like excitations associated with W$1(,) and W$2(,). Once we carefully locate the string of vacua along the real line we notice the existence of two well different configurations. For W$1(,) itself we have a first solution just interpolating between ,=&c 1 and ,=c 1 , two points of the vacuum manifold V1 which share identical equivalence class in Mc1(V). The rest of topological excitations connect vacua located at different points in the moduli space Mc1(V). When going to W$2(,) we only find topological configurations of this second kind due to the presence of the vacuum ,=0. In the light of this distinction we classify the kink-like excitations into loops and links. To be precise, the loops interpolate smoothly between equivalent vacua while the links serve to connect vacua located at different points in the moduli space. 4. Generically it happens that quantum corrections, perturbative and nonperturbative in principle, can lift the vacuum degeneracy so that as a last resort one particular theory is selected. If certain vacuum degeneracy persists once the quantum contributions have been taken into account, the complete theory exhibits a non-trivial quantum moduli space Mq(V). From physical grounds one expects that inequivalent vacua yield bosonic and fermionic excitations with different masses for well different points in Mq(V). We can carry things further and discuss in brief the very existence of the quantum moduli space for the N=1, d=2 supersymmetric models with a single chiral scalar multiplet. First of all, it has been shown that for field expectation values which conserve supersymmetry the effective potential vanishes to all orders in perturbation theory [14]. In other words, when susy is exact we obtain no more information about the vacuum degeneracy pattern by computing higher order corrections of the effective potential than was given by the tree-order contribution. Therefore the degeneracy is not removed in principle by perturbative terms. It may be interesting at this point to remind that certain loopholes appear in connection with the non-renormalization theorems when considering susy theories which involve massless fields. In such a case the models at issue possess local quantum corrections of the form of an integral over a chiral subspace of superspace so that we can interpret the above terms as a finite renormalization of the superpotential [15]. Returning now to the models written in (3.7), it is the case that no matter what the vacuum expectation value of the scalar field, both bosonic and fermionic excitations built over a generic ground state with ,=0 or ,=\c k (k=1, ..., n) have nonzero mass. As expected for a model just exhibiting exact supersymmetry with inequivalent vacua, the mass of the boson-fermion couple changes as far as one moves along the quantum moduli space Mq(V).
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As the perturbative regime has nothing to say about the differences between Mc(V) and Mq(V), it remains to consider the way in which the vacuum degeneracy can be lifted by other effects like the tunneling phenomenon. It may be interesting at this point to recall the way in which supersymmetry is spontaneously broken in any finite volume but restored however in the infinite volume limit. When considering supersymmetric theories in a finite spatial volume we must impose boundary conditions that preserve translation invariance since otherwise susy itself appears explicitly broken. In such a case we use periodic boundary conditions (in the spatial direction) for bosons as for fermions. As expected, the tunneling suppression arises already for the familiar case of the susy extension of the , 4 model [16]. For reference, we can take into account simultaneously a couple of superpotentials W$\(,)=(, 2 \c 2 ) so that as long as we work in a finite volume supersymmetry is spontaneously broken for both W$+(,) and W$&(,). First of all, if considering W$+(,) susy appears broken as expected for a ground state with energy given by c 42. Now it suffices a conjugation of the form c 2 &c 2 to conclude that susy is spontaneously broken for either of the two superpotentials W$+(,) or W$&(,). The above picture changes dramatically however when going to the infinite volume limit. When we consider W$&(,) rather than W$+(,), the scalar field acquires a nonzero vacuum expectation value so that the fermion itself, being massive, cannot represent the Goldstone particle of the hypothetical breaking of susy. In other words, we are in need of a mechanism which serves to explain why susy is broken in a finite volume but restored in the infinite volume limit. It is customarily assumed that tunneling between the two vacua ,=&c and ,=c is feasible whenever we work in a finite volume. The physics of the tunneling phenomenon can be understood in terms of the so-called instanton solutions which in finite volume become independent of the spatial coordinate. As expected the action of such topological configurations is in fact proportional to the volume so that we encounter a picture which mimics the standard quantum mechanics in d=0+1 dimensions. However when going from the finite volume frame to the infinite limit the role of the instantons become irrelevant and the tunneling is utterly suppressed. Resorting to more mathematical terms, we can take advantage of the no-go theorems which greatly restrict the number of theories just exhibiting topological solutions [17]. For reference, it is noteworthy that there are no static non singular configurations for a pure scalar field theory described by the lagrangian density L= 12 ( + ,)( +,)&U(,)
(3.12)
with U(,) 0 except for d (number of spatial dimensions) =1. As the term instanton has come to refer to localised finite-action solutions of the classical euclidean field equations and the aforementioned theorem prohibits static non singular topological configurations for d=3, we have no instantons with finite-action as long as we consider pure scalar field theories in d=2. Now it suffices to resort to the well-grounded supersymmetric Bogomolny condition to conclude that the absence of pure scalar configurations ruins any possibility of tunneling in the
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models we are talking about. To sum up, we deal in a systematic way with physical theories whose classical moduli spaces Mc(V) cannot be modified by quantum corrections. 5. The above arguments are borne out by the behaviour of the so-called Witten's index for the models we are taking into account. To begin, let us remind that one of the most pursued goals in the last decades has been to derive patterns of dynamical breaking once the conditions under which supersymmetry is spontaneously broken at tree-order are well understood from the very beginning of the subject. In the early eighties was Witten the first to introduce a necessary condition for the aforementioned breaking based on the vanishing of a certain topological index 2 (henceforth called Witten's index) [16]. To be precise, the index itself can &n E=0 , where n E=0 be written as 2=n E=0 B F B(F ) is the number of bosonic (fermionic) zero-energy states of the theory as a whole. Accordingly, the vanishing of 2 is a necessary condition for supersymmetry breaking. The Witten's index has been calculated for the most diverse theories via the functional integral. In doing so handy expressions for 2 appear in terms of the partition function of the theory once periodic boundary conditions on all fields are carefully imposed [18]. Restricting ourselves to N=1 susy models in two dimensions with a single chiral scalar superfield, the topological character of the Witten's index is clear simply by observing that 2 is equal to zero if W$(,)t, (even) for large , while it amounts to the unity if W$(,)t, (odd ) [18]. In Summary, these field theories mimic the behaviour of supersymmetric quantum mechanics where the Witten's index follows identical pattern once we perform the dimensional reduction from d=2 to d=1. In the light of this conventional wisdom about the Witten's index we must explain the situation concerning the superpotentials W$1(,) and W$2(,) written in (3.7). On that case we are in need of a criterion which serves to classify the ground states into bosonic and fermionic. Notice that for W$1(,) we have 2=0 while W$2(,) yields 2=1. In four dimensions the bosonic states can be distinguished from the fermionic ones by the angular momentum. However, there is no angular momentum in two dimensions so that the former procedure is not feasible. The way in which supersymmetry manages itself to organize the vacuum structures is both surprising and unexpectedly subtle [16]. Before going to the models we are dealing with, let us consider a free Majorana fermion with a lagrangian density like L= 12 (i# + + &m),
(3.13)
where either sign of the mass may be considered since by chiral symmetry we can set up that the physical mass is |m|. Now we introduce the zero-momentum modes of the field , i.e.,
|
_ 1 = 1(x) dx,
|
_ 2 = 2(x) dx.
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(3.14)
BIDIMENSIONAL SUPERSYMMETRIC MODELS
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The notation used is motivated by the fact that, after quantization, the operators fulfill the well-known sigma algebra _ 21 =_ 22 =1, _ 1 _ 2 +_ 2 _ 1 =0. As expected the zero-momentum mode has no kinetic energy so that the hamiltonian reduces itself to the mass term. In doing so we find that H=&im_ 1 _ 2 or equivalently H=m_ 3 . In other words, the zero-momentum mode is empty or filled according to whether m is positive or negative. In more physical terms we say that when m is changed in sign, it is the case that the ground state gains a fermion it did not have or loss one it had. That is to say, the ground state goes from being bosonic to being fermionic when m is changed in sign. As a matter of fact, this is the rule we were in need to reconcile the standard results derived from the Witten's index with the physical information we acquire after a survey of our superpotentials W$1(,) and W$2(,). As we have 2n or 2n+1 vacua respectively, the theory as a whole contains fermions with masses m j ( j=1, ..., 2n) or ( j=1, ..., 2n+1). Next we choose a criterion by which a vacuum is considered bosonic if m j is positive so that in general the Witten's index depends on the difference between positive and negative masses m j one finds as moving along the vacuum manifold V. For W$1(,) it is the case that the change from ,=&c j to ,=c j alters the fermion mass so that the two vacua associated with the factor (, 2 &c 2j ) produce no contribution to 2. Repeating the same procedure along V1 we explain that in fact 2=0. For W$2(,) the situation is slightly different since for instance the vacuum associated with ,=0 is either bosonic or fermionic for n even or odd respectively. In any case the string of vacua corresponding to nonzero vacuum expectation value manages itself to produce the expected result 2=1 by means of a well-controlled relationship between bosonic and fermionic ground states.
IV. TWO ILLUSTRATIVE EXAMPLES 1. Although exact results are very hard to come by, we resort now to wellbehaved superpotentials so that in particular the kink-like excitations can be presented in closed form. To be precise, we restrict ourselves to superpotentials W$(,) nice enough to provide us with the supersymmetric extension of , 6 and , 8 models. For such a purpose it seems plausible to start by considering the superpotential W$(,)=,(, 2 &1),
(4.1)
which yields the potential V(,) represented in Fig. 1. To elucidate the topological sector we combine (2.6) and (4.1) (always with (x)=0) to get the kink profile [19] , c(x)=[(tanh x+1)2] 12,
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(4.2)
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BOYA AND CASAHORRAN
FIG. 1.
Potential V(,)=W$ 2(,)2 corresponding to the choice written in (4.1).
a classical configuration which has the appearance shown in Fig. 2 and serves to make the smooth interpolation between the adjacent vacua ,=0 and ,=1. Proceeding along the same lines we obtain other solutions by putting x &x and , c &, c to give the four possibilities anticipated by the survey of Fig. 1. As a matter of fact the physics of the model is governed by a vacuum manifold V given by V=[,=0, ,=\1],
(4.3)
where one finds the two configurations ,=\1 just connected by the action of the Z 2 symmetry , &, together with the vacuum ,=0 at which the aforementioned symmetry is enhanced. It can prove convenient to write now the quantum moduli space Mq(V) which is merely the space of equivalence, classes of vacua. So we have Mq(V)=[3 0 , 3 1 ],
(4.4)
where as expected 3 0 includes the vacuum ,=0 while the couple ,=\1 belongs to 3 1 . The above analysis means that in this case the topological sector reduces itself to links since no well-behaved solutions exist inside 3 0 or 3 1 .
FIG. 2.
Profile of the kink solution interpolating between ,=0 and ,=1.
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BIDIMENSIONAL SUPERSYMMETRIC MODELS
FIG. 3.
77
Potential V(,) exhibiting four vacua.
2. To sketch the supersymmetric extension of , 8 we start from W$(,)=(, 2 &1)(, 2 &2),
(4.5)
a superpotential which corresponds to potential V(,) with four minima (see Fig. 3). As regards the topological sector it is the case that the Bogomolny condition leads to e \12x =
(1+,) 2 (2&,) . (2+,)(1&,) 2
(4.6)
We can carry things further and obtain the kinks of the model just by solving the cubic equation , 3 &3,+
2(e \12x &1) =0. e \12x +1
(4.7)
The picture which derives from (4.7) is as follows. The first root reads , c1(x)=&2 cos
_
1 e \12x &1 arccos \12x 3 e +1
\
+& ,
(4.8)
where the configuration with the sign minus in the exponential represents the kink which goes from ,=&2 to ,=&1. For the second root we find , c2(x)=2 cos
_
? 1 e \12x &1 + arccos \12x 3 3 e +1
\
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+& ,
(4.9)
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FIG. 4.
Picture of the three kinks which interpolate between the adjacent zeroes of such V(,).
although we need now the sign plus to make the transition between ,=&1 and ,=1. To conclude we write the third solution, namely, , c3(x)=2 cos
_
? 1 e \12x &1 & arccos \12x 3 3 e +1
\
+& ,
(4.10)
where again we resort to the sign plus to get the kink which interpolate between ,=1 and ,=2. Anyway the profile of these topological configurations can be seen in Fig. 4. The behaviour of the antikinks comes out to be the result of the change x &x. Notice that in this case the vacuum manifold V is given by V=[,=\1, ,=\- 2]
(4.11)
so that the quantum moduli space Mq(V) reads Mq(V)=[3 1 , 3 2 ],
(4.12)
where 3 1 and 3 2 contain the vacua ,=\1 and ,=\- 2 respectively. For reference, we point out that both loops (, c2(x)) and links (, c1(x) and , c3(x)) appear in this case.
V. CONCLUSIONS AND OUTLOOK Along this paper we have considered in detail the existence of bidimensional N=1 supersymmetric models for which the quantum theory at its best exhibits non-trivial quantum moduli space Mq(V). Starting from the vacuum manifold V, defined in terms of all the vacuum field configurations which preserve supersymmetry, we introduce as usual the so-called classical moduli space Mc(V). The most charming property of the models we are dealing with is that quantum corrections
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in the most wide sense do not modify the vacuum degeneracy derived from the treeorder picture. In more physical terms it is the case that inequivalent vacua yield bosonic and fermionic excitations with different masses for well different points in Mq(V). The aforementioned stability of Mc(V) against quantum corrections relies on diverse arguments concerning both perturbative and nonperturbative regimes. The effective potential vanishes to all orders in perturbation theory for field expectation values which preserve supersymmetry. On the other hand, the well-grounded tunneling phenomenon in finite volume is utterly suppressed when going to the infinite volume limit. To explain the physics of this behaviour we resort as usual to the nogo theorems which restrict the number of theories exhibiting instanton-like configurations. The above picture is borne out by the properties of the Witten's index for bidimensional models. In addition we classify the kink-like excitations into loops and links. To finish we would like to stress the way in which the loopholes of the perturbative no-renormalization theorems can provide, at least partially, the lifting of the tree-order vacuum degeneracy. When dealing with theories which contain massless excitations the superpotentials at issue suffer finite renormalization. Once these unexpected terms are taken into account we have no longer valid arguments to identify Mc(V) with Mq(V). It remains to modify appropiately the generic superpotentials considered in this article so that the massless fields come into play. Although it is not apparent a priori how to handle these subtleties, we hope to report on the subject of the perturbative lifting in a next future.
ACKNOWLEDGMENTS We thank Professor R. Jackiw for calling our attention to Ref. [17]. The authors also express gratitude to the Comision Interministerial de Ciencia y Tecnolog@ a for financial support.
REFERENCES S. Coleman and J. Mandula, Phys. Rev. 159 (1967), 1251. J. Iliopoulos and B. Zumino, Nucl. Phys. B 76 (1974), 310. E. Witten, Nucl. Phys. B 185 (1981), 513. P. DiVecchia and S. Ferrara, Nucl. Phys. B 130 (1977), 93. J. Hruby, Nucl. Phys. B 131 (1977), 275. T. Murphy and L. O. Raifeartaigh, Nucl. Phys. B 218 (1983), 484. E. Witten and D. Olive, Phys. Lett. B 78 (1978), 97. R. Rajaraman, ``Solitons and Instantons,'' p. 33, North Holland, Amsterdam, 1982. C. Callias, Commun. Math. Phys. 62 (1978), 213; R. Bott and R. Seeley, Commun. Math. Phys. 62 (1978), 235. 10. C. Imbimbo and S. Mukhi, Nucl. Phys. B 247 (1984), 471. 11. B. Zumino, Nucl. Phys. B 89 (1975), 535. 1. 2. 3. 4. 5. 6. 7. 8. 9.
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12. N. Seiberg and E. Witten, Nucl. Phys. B 426 (1994), 19; N. Seiberg and E. Witten, Nucl. Phys. B 431 (1994), 484. 13. N. Seiberg, Phys. Rev. D 49 (1994), 6857. 14. P. West, Nucl. Phys. B 106 (1976), 219. 15. P. West, Phys. Lett. B 258 (1991), 375. 16. E. Witten, Nucl. Phys. B 202 (1982), 253. 17. R. Hobart, Proc. Phys. Soc. 82 (1963), 201. 18. L. Girardello, C. Imbimbo, and S. Mukhi, Phys. Lett. B 132 (1983), 69. 19. M. A. Lohe, Phys. Rev. D 20 (1979), 3120.
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Bidimensional Supersymmetric Field Theories with Discrete Moduli Space Luis J. Boya Departamento de F@ sica Teorica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
and Javier Casahorran Departamento de F@ sica Teorica, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s.n., 33007 Oviedo, Spain Received July 14, 1997; revised November 4, 1997
We consider bidimensional supersymmetric models for which the classical theory has a discrete set of inequivalent vacua. Once we introduce the concept of vacuum manifold V, defined in terms of all the vacuum field configurations, the properties of such models can be described by means of the classical moduli space Mc(V). Quantum corrections, perturbative and nonperturbative, do not lift the vacuum degeneracy so that the quantum theory as a whole exhibits non-trivial quantum moduli space Mq(V). The former structure allows us to classify the kinklike excitations into loops and links: loops interpolate smoothly between equivalent vacua while links connect vacua located at different points in Mq(V). 1998 Academic Press
I. INTRODUCTION 1. Nowadays it seems clear that supersymmetry, a new kind of symmetry which includes bosonic and fermionic degrees of freedom, represents one of the most ambitious attempts made in order to obtain a trustworthy description of all basic interactions of nature. In four dimensions supersymmetry transformations mix half-integral and integral spin states and therefore the corresponding multiplets relate fermions to bosons. This fact suggests that the new type of symmetry may provide a remarkable principle of unification in quantum field theory, thus going over the powerful constraints imposed by the Coleman-Mandula theorem [1]. The only symmetries allowed in the old schemes are those which turn out to be locally isomorphic to the direct product of an internal symmetry and the Poincare group. In this respect supersymmetry offers a way to achieve unification of space-time and internal symmetries by resorting to the so-called graded Lie groups. Despite the 63 0003-491698 25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
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unfamiliar properties of the graded Lie algebras (superalgebras in the jargon of physics) which only close under the right combination of commutation and anticommutation brackets, it soon became clear that supersymmetry could play an important role in many branches of modern theoretical physics. Returning to quantum field theories, the models based on supersymmetry are less divergent than naively expected. In essence, putting together bosonic and fermionic fields produces in fact cancellation of many infinities because the respective contributions have opposite signs As we have no experimental evidence for such a novel symmetry emerging exactly in nature, one of the most pursued goals in the past two decades has been to describe a suitable mechanism by means of which supersymmetry shows up itself broken. To start from scratch, let us recall some basic features concerning globally supersymmetric theories. First of all, since the hamiltonian H itself is the sum of squares of hermitian operators (the supersymmetry charges Q : ), the energy of any state is positive or zero [2]. In addition, the states annihilated by the action of the Q : have zero energy and represent supersymmetric vacua of the model. If there does not exist a such state invariant under supersymmetry we know that the groundstate energy is strictly positive and supersymmetry itself appears spontaneously broken. On the other hand, a non-zero vacuum expectation for a propagating elementary scalar field is consistent with exact supersymmetry and that represents the breakdown of an internal symmetry [3]. If |0) represents globally the ground states of the model, the spontaneous breaking of supersymmetry can be analyzed by means of the anticommutator of Q : with some fermionic operator X. If it happens that Q : |0) =0, the expectation value associated with ( 0| [Q : , X] |0) would evidently have to vanish. In supersymmetric theories the elementary scalar fields , can never be written in terms of anticommutators like [Q : , X] and this fact explains that they might acquire vacuum expectation values without breaking supersymmetry. As regards the derivatives + , of the physical fields themselves, it is the Lorentz invariance that prevents the emergence of vacuum expectation values which otherwise would be possible in terms of [Q, ], where represents elementary fermions. Hence [Q : , X] is an auxiliary nonpropagating scalar field F and only (F) {0 is a signal of spontaneous susy breaking. 2. When considering models in (1+1) with dimensionless scalar field ,, the potential V(,) can take a much more general form than in the conventional (2+1) or (3+1) cases as regards the naive renormalizability of the theory. In addition the model becomes automatically supersymmetric by taking advantage of the simplest multiplet of fields consistent with N=1 supersymmetry. They form the so-called chiral scalar superfield where we find a real scalar field ,, a two-component Majorana fermion and an auxiliar field F which ultimately can be eliminated. When a general choice for the superpotential function is made, we can get models where the unbroken supersymmetry appears associated with a discrete set of physically inequivalent vacua. In doing so we go beyond the well-known cases of the supersymmetric versions of sine-Gordon or , 4. The best way to describe the main
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properties of this new family of models takes advantage of the so-called vacuum manifold V, defined in principle as the set of all vacuum field configurations. In other words, we start front the scalar fields , for which the potential is zero. In this scheme different points along V are parametrized by the corresponding vacuum expectation values of the scalar field ,. On the other hand, points P, P* in V will describe equivalent physical situations whenever we can reach P* from P by means of a discrete group of symmetry G. Therefore we deal with a d=2 discrete version of the models formulated in d=4 whose vacuum structure is described by means of a continuous manifold. Next we can define the classical moduli space Mc(V) of the model as the space of equivalence classes of vacua. Two vacua belong to the same equivalence class if they can be connected by means of the action of the group G. We are dealing with bidimensional models which exhibit discrete symmetries (whose spontaneous breaking produces no Goldstone bosons). The hypothetical breaking of a continuous symmetry would yield massless bosons, which are excluded anyway in d=2 theories. In addition we expect that as a consequence of quantum corrections (perturbative and nonperturbative in principle) the vacuum degeneracy will be lifted so that ultimately the true vacuum set of the theory is selected. If this is not the case, the quantum theory as a whole exhibits a manifold of inequivalent ground states, i.e., a non-trivial quantum moduli space Mq(V). As expected the classical kink solutions interpolate between adjacent vacua and this persists upon quantization. We classify these classical solutions with finite energy into loops and links. While the loops themselves connect vacua represented by a single point in Mq(V), the links move along the quantum moduli space. We shall see how susy is realized both in the vacuum sector and in the kink sector. 3. The article is arranged in the following way. To keep the article self-contained, we review in Section 2 some well-established aspects of the bidimensional N=1 supersymmetric models based on the chiral scalar superfield. Section 3 is in fact the heart of the whole exposition since in it we carefully describe the models just exhibiting a set of physically inequivalent vacua. We define the vacuum manifold V, the classical moduli space Mc(V) and the quantum moduli space Mq(V). We study the spectrum of the particles in the vacuum sector and the structure of kink-like excitations. Important elements in the analysis of these theories like the behaviour of the Witten index or the way in which the quantum corrections do not lift the vacua degeneracy derived from classical arguments are also considered. In particular we emphasize the way in which the well-known supersymmetric extensions of sine-Gordon or , 4 represent theories with trivial quantum moduli space Mq(V). To illustrate these ideas in a simple context we resort in Section 4 to models based on well-behaved superpotentials functions (the susy extension of , 6 and , 8 models) so that the whole structure of vacua and kink-like excitations can be presented in closed form. We conclude with an outlook and various speculations.
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II. N=1 SUSY WITH A CHIRAL SCALAR MULTIPLET 1. Although a complete discussion of the bidimensional N=1 supersymmetric models based on the so-called chiral scalar multiplet is clearly out of the scope of this article, we review in this section just as much as we believe is necessary in order to understand the main properties of the theories which exhibit inequivalent ground states. DiVecchia and Ferrara [4] together with Hruby [5] were the first to introduce the supersymmetric version of the models consisting in a self-interacting scalar field in two dimensions. The behaviour of the theory is governed by the action S given by [46]
|
S= 12 [( + ,) 2 +(i# + + ) &W$ 2 &W"] d 2x,
(2.1)
where ,(x) stands for a real scalar field and (x) represents a two-component Majorana fermion. Now W$(,) is the standard superpotential function where the prime denotes as usual the derivative with respect to ,. For the generic model written in (2.1), the supersymmetric generators derive from the conserved supercurrent S + [7], i.e., S + =( j ,) # j# ++iW$(,) # +
(2.2)
so that the Q : themselves together with the momentum P + , given as usual by P + =&i + , allow us to recover the standard superalgebra associated with N=1, i.e., [Q : , Q ; ]=2(# +P + ) :;
(2.3a)
[Q : , P + ]=0,
(2.3b)
where we use # + matrices given by # 0 =i_ 1, # 1 =i_ 3. Resorting now to a formulation in terms of the chiral components of the fermionic field, i.e., \ =(1\_ 3 ) 2, the supersymmetric charges are
|
(2.4a)
|
(2.4b)
Q + = |( 0 ,+ 1 ,) + &W$(,) & | dx Q & = |( 0 ,& 1 ,) & +W$(,) + | dx
so that the superalgebra itself reads Q 2+ =P + , Q 2& =P & , Q + Q & +Q & Q + =0 if P \ =P 0 \P 1 . In any case the field equations derived from (2.1) are of the form ( + + ,)+W$(,) W"(,)+ 12 W$$$(,) =0
(2.5a)
[i# + + +W"(,)] =0.
(2.5b)
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2. At this point it proves convenient to introduce the supersymmetric version of the well-known Bogomolny condition, namely [4] (1\_ 3 ) 2=0,
d, c =\W$(, c ). dx
(2.6)
On the other hand it is the case that in terms of the constant fermionic parameter ' the field given by =[i# + + , c &W$(, c )] '
(2.7)
solves both the supersymmetric Bogomolny condition and the equations of motion whenever ' is forced to satisfy the constraint ' i =0 for i=+ or i=&. (Notice that =0 for chiral fermions \ =0.) 3. To finish this brief review, let us make some illustrative comments about the specific realization of supersymmetry itself in the topological sector. When considering theories which exhibit kink-like excitations, one finds that the standard superalgebra is no longer valid since the central charges associated with the existence of topologically conserved quantum numbers modify the results derived in the trivial sector [7]. To be precise, the supersymmetric algebra acquires central charges due to the surface terms, which although usually discarded, are actually nonvanishing for topological configurations. When keeping carefully the surface terms, one finds Q + Q & +Q & Q + =T where
|
T= 2W$(,)
, dx x
(2.8)
so we can say that Q + Q & +Q & Q + corresponds to the integral of a total divergence. When considering the existence of static kink-like configurations, the integral written in the right hand side of (2.8) is not necessarily zero but amounts to the difference between the values of 2W(,) at x=+ and x=& . The topological character of T becomes clear if one resorts to the improved topological current j +i given by j +i =2W$(,) = +& & ,
(2.9)
whose conservation law does not arise by Noether's theorem from a well-behaved symmetry of the lagrangian at issue but characterizes the large distance behaviour of the field configurations. In physical terms, it is the case that j +i provides us in essence with the same information already obtained from the analysis of the standard topological current j + written as j + == +& & , [8]. Despite the model dependent character of j +i , we use in the following such improved topological current so that the T operator itself results from integrating the j 0i component along the real line.
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Returning now to the modified algebra which includes the presence of the central charge term T, i.e., Q 2+ =P + , Q 2& =P & , Q + Q & +Q & Q + =T, we can write (Q + &Q & ) 2 +T=P + +P &
(2.10a)
(Q + +Q & ) 2 &T=P + +P &
(2.10b)
and since (Q + &Q & ) 2 0 we conclude that (P + +P & ) |T |. For a single kink-like excitation at rest with mass M we have P + =P & =M so that the existence of a central charge term translates into the physical condition M
|T | . 2
(2.11)
Before going to the quantum corrections, an explicit calculation based on the generic model written in (2.1) shows that at classical level the kink configurations saturate in fact (2.11): starting from the hypothetical solution , c(x) of (2.6) with (x)=0 (see for instance [5]), the classical mass M associated with this static kink-like configuration is M=
|
+ &
1 d, c(x) 2 dx
_\
2
1
+ +2 W$(, (x)) & dx, 2
c
(2.12)
which, once we insert the Bogomolny condition, leads to M=
}|
+
&
dW$(, c(x)) 1 dx = |T |. dx 2
}
(2.13)
Notice that (2.11) is the Bogomolny classical bound, here shown to be true at quantum level. However in this case the computation of quantum corrections to physical quantities is plagued with subtle aspects which must be handled appropriately. A naive examination of the problem would imply that the Bogomolny bound saturation holds at O(). If kink-like configurations receive no quantum mass corrections in this order, it suffices an identical assumption for the central charge term to close the argument. Unfortunately this straightforward approach to the problem underestimates the subtleties associated with the difference in density of fluctuation modes between bosons and fermions. In other words, cancellation of boson and fermion fluctuation energies, though valid for discrete modes, is not automatically true for the continuum. Going now to arbitrary choices of the superpotential W$(,) in (2.1), the use of trace formulae for linear differential operators on non-compact manifolds, as first stated by Callias et al. [9], allows us to calculate the O() correction to the classical kink mass. Finally the saturation of the Bogomolny bound at such order is shown by an explicit computation of the central charge term T (see for details [10]). It happens therefore that one of the most striking features of the theories just based on unbroken supersymmetry, the fact that the vacuum energy receives no quantum contributions in the perturbative
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regime, does not generalize to the topological sector. However, the results obtained at O() strongly suggest that it is the operator (H&T2) that receives no quantum correction. In other words, this would be the topological version of the results first derived by Zumino about the cancellation of vacuum diagrams in supersymmetric theories [11].
III. MODELS WITH DISCRETE MODULI SPACE Mq(V) 1. Along the past three years we have seen a waterfall of valuable results about the nonperturbative behaviour of supersymmetric Yang-Mills theories with or without matter, not to mention the new insights which concern duality and string models. Restricting ourselves to N=1 or N=2 supersymmetric gauge theories in four dimensions, the key results appear as a consequence of the holomorphic character of the low energy effective superpotential [12]. In principle these models exhibit a space of degenerate vacua so that they mimic the properties of the standard models just based on the spontaneous symmetry breaking phenomenon. The novelties appear when one notices that in fact the supersymmetric vacua at issue are inequivalent. Physically it means that for instance different vacua yield a well different spectra of excitations. Although the degeneracy between these states cannot be lifted perturbatively, the nonperturbative effects can produce new contributions that dramatically change the tree-order vacuum structure [13]. 2. Returning now to our bidimensional theories based on N=1 supersymmetry with chiral scalar superfield, it should be recalled that almost from the very beginning of the subject the cases considered in the literature just correspond to the susy extensions of sine-Gordon or , 4 [4, 5]. For reference, we can recall the main features concerning these well-known models in the spirit of the moduli spaces scheme. For example, if we start with the superpotential , W$(,)=2 cos , 2
(3.1)
the lagrangian density of the model enjoys the global symmetry given by , ,+2p?( p # Z), # 0. Although valuable information is available on the longestablished multisoliton sector of sine-Gordon, we restrict ourselves in the following to the physical consequences derived from the static solution , c(x) given by [8] , c(x)=2 arcsin (tanh x),
(3.2)
which smoothly interpolates between the adjacent vacua , &1 =&? and , +1 =+?. Going to the vacuum manifold V, this susy extension of sine-Gordon possesses an infinite number of vacua since V=[, v =(2v+1) ?, v # Z].
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Accordingly, two generic vacua (, q , , r ) in V can always be connected by the action of the Z symmetry at issue. Therefore the quantum moduli space Mq(V), defined as usual in terms of the space of equivalence classes of vacua, i.e., Mq(V)= VZ, reduces itself to a single point. At this stage one could just as well take into account the susy version of the celebrated , 4 model. In this case we consider W$(,)=(, 2 &1)
(3.4)
so that now the symmetry at issue corresponds to , &,, # 0, i.e., Z 2 . In addition, the superpotential written in (3.4) gives rise to a static kink-like configuration , c(x) of the form , c(x)=tanh x,
(3.5)
which as expected connects the two vacua of the model. Proceeding along the same lilies than before we get trivial quantum moduli space since V=[, \ =\1]
(3.6)
so that Mq(V)=VZ 2 reduces again to a single point. To conclude this brief exposition let us remark that, being trivial the quantum moduli spaces Mq(V), the fine distinction between loops and links is out of order. 3. Now we consider the case where the superpotential W$(,) is a polynomial in , 2 so that the symmetry at issue corresponds to , &,, # 0 in the spirit of the , 4 theory. The polynomial superpotentials we are talking about can be understood as a general class of models which approximate the supersymmetric sine-Gordon theory. To be precise this approximation can be achieved by truncating the infinite-product formula for W$(,)=2 cos ,2 (see (3.1)) in order to obtain either of the following two polynomial superpotentials n
W$1(,)= ` (, 2 &c 2k )
(3.7a)
k=1 n
W$2(,)=, ` (, 2 &c 2k )
(3.7b)
k=1
for c 2m >c 2n if m>n. Notice that for n=1 the first superpotential in (3.7a) represents the susy version of , 4 while (3.7b) would correspond to , 6 [16]. One is adding more vacua for higher n and, correspondingly, the structure of the vacuum manifold V is enriched in a systematic way. Next, we discuss the way in which the vacuum degeneracy pattern associated with W$1(,) and W$2(,) allows the existence of kink-like excitations. As long as we are dealing with supersymmetric models the whole structure of these configurations relies on our ability for handling the well-known Bogomolny condition just written
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in (2.6). It may be interesting at this point to recall how the scalar component of the susy Bogomolny condition (recall that (x)=0), i.e., d, c =\W$(, c ) dx
(3.8)
has proved quite useful in this context. It should be emphasized that when we consider a physical situation with n discrete degenerate vacua, we can have 2(n&1) types of topological configurations. In addition such kink-like excitations serve to connect any two neighbouring vacua as the spatial coordinate x varies from & to + [8]. The topological configurations can be understood as interpolating classical solutions between adjacent vacua. However, the most important element in the analysis of these theories relies on the existence of a discrete degeneracy of inequivalent ground states. Classically, we find the spontaneous breaking of the Z 2 symmetry , &, since in general the vacuum expectation values of the field configurations which preserve supersymmetry are non-zero. In other words, the so-called vacuum manifold V can be described in terms of the vacuum expectation values of the scalar field ,. Accordingly, the vacuum manifold V associated with the superpotentials W$1(,) and W$2(,) written in (3.7) would correspond respectively to V1 =[,=\c k , k=1, ..., n]
(3.9a)
V2 =[,=0, ,=\c k , k=1, ..., n].
(3.9b)
With regards to V1 , it is the case that field configurations like ,=c k and ,=&c k (k=1, ..., n) will describe as usual equivalent physical situations since they are just connected by the action of the discrete symmetry , &, at issue. So we can have exact supersymmetry and breaking of the internal symmetry. Our choice in (3.7) provides however a more sophisticated realization of this scheme as far as the second vacuum manifold V2 is concerned. In fact the structure associated with V1 is enriched due to the existence of the additional vacuum ,=0 where precisely the discrete symmetry at issue is enhanced. In more physical terms, it seems plausible to remark the way in which observed supersymmetry can share the same life with both exact and broken internal symmetries depending on the ground state the model chooses between. Then we go to the classical moduli space Mc(V) defined as the space of equivalence classes of vacua, i.e., Mc(V)=VZ 2 . To sum up, in our case two vacua belong to the same equivalence class if they can be just connected by means of the action of the discrete symmetry , &,. The above discussion allows us to write that Mc1(V)=[3 k , k=1, ..., n],
(3.10)
where as expected the generic equivalence class 3 k just contains the two vacua ,=\c k . Essentially the same structure arises for V2 , the only difference between
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both classical moduli spaces being the existence in Mc2(V) of the equivalence class 3 0 associated with the vacuum ,=0 at which the Z 2 symmetry is enhanced because is unbroken. So we have Mc2(V)=[3 0 , 3 k , k=1, ..., n].
(3.11)
The above discussion sheds light on the properties of the kink-like excitations associated with W$1(,) and W$2(,). Once we carefully locate the string of vacua along the real line we notice the existence of two well different configurations. For W$1(,) itself we have a first solution just interpolating between ,=&c 1 and ,=c 1 , two points of the vacuum manifold V1 which share identical equivalence class in Mc1(V). The rest of topological excitations connect vacua located at different points in the moduli space Mc1(V). When going to W$2(,) we only find topological configurations of this second kind due to the presence of the vacuum ,=0. In the light of this distinction we classify the kink-like excitations into loops and links. To be precise, the loops interpolate smoothly between equivalent vacua while the links serve to connect vacua located at different points in the moduli space. 4. Generically it happens that quantum corrections, perturbative and nonperturbative in principle, can lift the vacuum degeneracy so that as a last resort one particular theory is selected. If certain vacuum degeneracy persists once the quantum contributions have been taken into account, the complete theory exhibits a non-trivial quantum moduli space Mq(V). From physical grounds one expects that inequivalent vacua yield bosonic and fermionic excitations with different masses for well different points in Mq(V). We can carry things further and discuss in brief the very existence of the quantum moduli space for the N=1, d=2 supersymmetric models with a single chiral scalar multiplet. First of all, it has been shown that for field expectation values which conserve supersymmetry the effective potential vanishes to all orders in perturbation theory [14]. In other words, when susy is exact we obtain no more information about the vacuum degeneracy pattern by computing higher order corrections of the effective potential than was given by the tree-order contribution. Therefore the degeneracy is not removed in principle by perturbative terms. It may be interesting at this point to remind that certain loopholes appear in connection with the non-renormalization theorems when considering susy theories which involve massless fields. In such a case the models at issue possess local quantum corrections of the form of an integral over a chiral subspace of superspace so that we can interpret the above terms as a finite renormalization of the superpotential [15]. Returning now to the models written in (3.7), it is the case that no matter what the vacuum expectation value of the scalar field, both bosonic and fermionic excitations built over a generic ground state with ,=0 or ,=\c k (k=1, ..., n) have nonzero mass. As expected for a model just exhibiting exact supersymmetry with inequivalent vacua, the mass of the boson-fermion couple changes as far as one moves along the quantum moduli space Mq(V).
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As the perturbative regime has nothing to say about the differences between Mc(V) and Mq(V), it remains to consider the way in which the vacuum degeneracy can be lifted by other effects like the tunneling phenomenon. It may be interesting at this point to recall the way in which supersymmetry is spontaneously broken in any finite volume but restored however in the infinite volume limit. When considering supersymmetric theories in a finite spatial volume we must impose boundary conditions that preserve translation invariance since otherwise susy itself appears explicitly broken. In such a case we use periodic boundary conditions (in the spatial direction) for bosons as for fermions. As expected, the tunneling suppression arises already for the familiar case of the susy extension of the , 4 model [16]. For reference, we can take into account simultaneously a couple of superpotentials W$\(,)=(, 2 \c 2 ) so that as long as we work in a finite volume supersymmetry is spontaneously broken for both W$+(,) and W$&(,). First of all, if considering W$+(,) susy appears broken as expected for a ground state with energy given by c 42. Now it suffices a conjugation of the form c 2 &c 2 to conclude that susy is spontaneously broken for either of the two superpotentials W$+(,) or W$&(,). The above picture changes dramatically however when going to the infinite volume limit. When we consider W$&(,) rather than W$+(,), the scalar field acquires a nonzero vacuum expectation value so that the fermion itself, being massive, cannot represent the Goldstone particle of the hypothetical breaking of susy. In other words, we are in need of a mechanism which serves to explain why susy is broken in a finite volume but restored in the infinite volume limit. It is customarily assumed that tunneling between the two vacua ,=&c and ,=c is feasible whenever we work in a finite volume. The physics of the tunneling phenomenon can be understood in terms of the so-called instanton solutions which in finite volume become independent of the spatial coordinate. As expected the action of such topological configurations is in fact proportional to the volume so that we encounter a picture which mimics the standard quantum mechanics in d=0+1 dimensions. However when going from the finite volume frame to the infinite limit the role of the instantons become irrelevant and the tunneling is utterly suppressed. Resorting to more mathematical terms, we can take advantage of the no-go theorems which greatly restrict the number of theories just exhibiting topological solutions [17]. For reference, it is noteworthy that there are no static non singular configurations for a pure scalar field theory described by the lagrangian density L= 12 ( + ,)( +,)&U(,)
(3.12)
with U(,) 0 except for d (number of spatial dimensions) =1. As the term instanton has come to refer to localised finite-action solutions of the classical euclidean field equations and the aforementioned theorem prohibits static non singular topological configurations for d=3, we have no instantons with finite-action as long as we consider pure scalar field theories in d=2. Now it suffices to resort to the well-grounded supersymmetric Bogomolny condition to conclude that the absence of pure scalar configurations ruins any possibility of tunneling in the
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models we are talking about. To sum up, we deal in a systematic way with physical theories whose classical moduli spaces Mc(V) cannot be modified by quantum corrections. 5. The above arguments are borne out by the behaviour of the so-called Witten's index for the models we are taking into account. To begin, let us remind that one of the most pursued goals in the last decades has been to derive patterns of dynamical breaking once the conditions under which supersymmetry is spontaneously broken at tree-order are well understood from the very beginning of the subject. In the early eighties was Witten the first to introduce a necessary condition for the aforementioned breaking based on the vanishing of a certain topological index 2 (henceforth called Witten's index) [16]. To be precise, the index itself can &n E=0 , where n E=0 be written as 2=n E=0 B F B(F ) is the number of bosonic (fermionic) zero-energy states of the theory as a whole. Accordingly, the vanishing of 2 is a necessary condition for supersymmetry breaking. The Witten's index has been calculated for the most diverse theories via the functional integral. In doing so handy expressions for 2 appear in terms of the partition function of the theory once periodic boundary conditions on all fields are carefully imposed [18]. Restricting ourselves to N=1 susy models in two dimensions with a single chiral scalar superfield, the topological character of the Witten's index is clear simply by observing that 2 is equal to zero if W$(,)t, (even) for large , while it amounts to the unity if W$(,)t, (odd ) [18]. In Summary, these field theories mimic the behaviour of supersymmetric quantum mechanics where the Witten's index follows identical pattern once we perform the dimensional reduction from d=2 to d=1. In the light of this conventional wisdom about the Witten's index we must explain the situation concerning the superpotentials W$1(,) and W$2(,) written in (3.7). On that case we are in need of a criterion which serves to classify the ground states into bosonic and fermionic. Notice that for W$1(,) we have 2=0 while W$2(,) yields 2=1. In four dimensions the bosonic states can be distinguished from the fermionic ones by the angular momentum. However, there is no angular momentum in two dimensions so that the former procedure is not feasible. The way in which supersymmetry manages itself to organize the vacuum structures is both surprising and unexpectedly subtle [16]. Before going to the models we are dealing with, let us consider a free Majorana fermion with a lagrangian density like L= 12 (i# + + &m),
(3.13)
where either sign of the mass may be considered since by chiral symmetry we can set up that the physical mass is |m|. Now we introduce the zero-momentum modes of the field , i.e.,
|
_ 1 = 1(x) dx,
|
_ 2 = 2(x) dx.
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(3.14)
BIDIMENSIONAL SUPERSYMMETRIC MODELS
75
The notation used is motivated by the fact that, after quantization, the operators fulfill the well-known sigma algebra _ 21 =_ 22 =1, _ 1 _ 2 +_ 2 _ 1 =0. As expected the zero-momentum mode has no kinetic energy so that the hamiltonian reduces itself to the mass term. In doing so we find that H=&im_ 1 _ 2 or equivalently H=m_ 3 . In other words, the zero-momentum mode is empty or filled according to whether m is positive or negative. In more physical terms we say that when m is changed in sign, it is the case that the ground state gains a fermion it did not have or loss one it had. That is to say, the ground state goes from being bosonic to being fermionic when m is changed in sign. As a matter of fact, this is the rule we were in need to reconcile the standard results derived from the Witten's index with the physical information we acquire after a survey of our superpotentials W$1(,) and W$2(,). As we have 2n or 2n+1 vacua respectively, the theory as a whole contains fermions with masses m j ( j=1, ..., 2n) or ( j=1, ..., 2n+1). Next we choose a criterion by which a vacuum is considered bosonic if m j is positive so that in general the Witten's index depends on the difference between positive and negative masses m j one finds as moving along the vacuum manifold V. For W$1(,) it is the case that the change from ,=&c j to ,=c j alters the fermion mass so that the two vacua associated with the factor (, 2 &c 2j ) produce no contribution to 2. Repeating the same procedure along V1 we explain that in fact 2=0. For W$2(,) the situation is slightly different since for instance the vacuum associated with ,=0 is either bosonic or fermionic for n even or odd respectively. In any case the string of vacua corresponding to nonzero vacuum expectation value manages itself to produce the expected result 2=1 by means of a well-controlled relationship between bosonic and fermionic ground states.
IV. TWO ILLUSTRATIVE EXAMPLES 1. Although exact results are very hard to come by, we resort now to wellbehaved superpotentials so that in particular the kink-like excitations can be presented in closed form. To be precise, we restrict ourselves to superpotentials W$(,) nice enough to provide us with the supersymmetric extension of , 6 and , 8 models. For such a purpose it seems plausible to start by considering the superpotential W$(,)=,(, 2 &1),
(4.1)
which yields the potential V(,) represented in Fig. 1. To elucidate the topological sector we combine (2.6) and (4.1) (always with (x)=0) to get the kink profile [19] , c(x)=[(tanh x+1)2] 12,
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(4.2)
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FIG. 1.
Potential V(,)=W$ 2(,)2 corresponding to the choice written in (4.1).
a classical configuration which has the appearance shown in Fig. 2 and serves to make the smooth interpolation between the adjacent vacua ,=0 and ,=1. Proceeding along the same lines we obtain other solutions by putting x &x and , c &, c to give the four possibilities anticipated by the survey of Fig. 1. As a matter of fact the physics of the model is governed by a vacuum manifold V given by V=[,=0, ,=\1],
(4.3)
where one finds the two configurations ,=\1 just connected by the action of the Z 2 symmetry , &, together with the vacuum ,=0 at which the aforementioned symmetry is enhanced. It can prove convenient to write now the quantum moduli space Mq(V) which is merely the space of equivalence, classes of vacua. So we have Mq(V)=[3 0 , 3 1 ],
(4.4)
where as expected 3 0 includes the vacuum ,=0 while the couple ,=\1 belongs to 3 1 . The above analysis means that in this case the topological sector reduces itself to links since no well-behaved solutions exist inside 3 0 or 3 1 .
FIG. 2.
Profile of the kink solution interpolating between ,=0 and ,=1.
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BIDIMENSIONAL SUPERSYMMETRIC MODELS
FIG. 3.
77
Potential V(,) exhibiting four vacua.
2. To sketch the supersymmetric extension of , 8 we start from W$(,)=(, 2 &1)(, 2 &2),
(4.5)
a superpotential which corresponds to potential V(,) with four minima (see Fig. 3). As regards the topological sector it is the case that the Bogomolny condition leads to e \12x =
(1+,) 2 (2&,) . (2+,)(1&,) 2
(4.6)
We can carry things further and obtain the kinks of the model just by solving the cubic equation , 3 &3,+
2(e \12x &1) =0. e \12x +1
(4.7)
The picture which derives from (4.7) is as follows. The first root reads , c1(x)=&2 cos
_
1 e \12x &1 arccos \12x 3 e +1
\
+& ,
(4.8)
where the configuration with the sign minus in the exponential represents the kink which goes from ,=&2 to ,=&1. For the second root we find , c2(x)=2 cos
_
? 1 e \12x &1 + arccos \12x 3 3 e +1
\
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+& ,
(4.9)
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FIG. 4.
Picture of the three kinks which interpolate between the adjacent zeroes of such V(,).
although we need now the sign plus to make the transition between ,=&1 and ,=1. To conclude we write the third solution, namely, , c3(x)=2 cos
_
? 1 e \12x &1 & arccos \12x 3 3 e +1
\
+& ,
(4.10)
where again we resort to the sign plus to get the kink which interpolate between ,=1 and ,=2. Anyway the profile of these topological configurations can be seen in Fig. 4. The behaviour of the antikinks comes out to be the result of the change x &x. Notice that in this case the vacuum manifold V is given by V=[,=\1, ,=\- 2]
(4.11)
so that the quantum moduli space Mq(V) reads Mq(V)=[3 1 , 3 2 ],
(4.12)
where 3 1 and 3 2 contain the vacua ,=\1 and ,=\- 2 respectively. For reference, we point out that both loops (, c2(x)) and links (, c1(x) and , c3(x)) appear in this case.
V. CONCLUSIONS AND OUTLOOK Along this paper we have considered in detail the existence of bidimensional N=1 supersymmetric models for which the quantum theory at its best exhibits non-trivial quantum moduli space Mq(V). Starting from the vacuum manifold V, defined in terms of all the vacuum field configurations which preserve supersymmetry, we introduce as usual the so-called classical moduli space Mc(V). The most charming property of the models we are dealing with is that quantum corrections
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in the most wide sense do not modify the vacuum degeneracy derived from the treeorder picture. In more physical terms it is the case that inequivalent vacua yield bosonic and fermionic excitations with different masses for well different points in Mq(V). The aforementioned stability of Mc(V) against quantum corrections relies on diverse arguments concerning both perturbative and nonperturbative regimes. The effective potential vanishes to all orders in perturbation theory for field expectation values which preserve supersymmetry. On the other hand, the well-grounded tunneling phenomenon in finite volume is utterly suppressed when going to the infinite volume limit. To explain the physics of this behaviour we resort as usual to the nogo theorems which restrict the number of theories exhibiting instanton-like configurations. The above picture is borne out by the properties of the Witten's index for bidimensional models. In addition we classify the kink-like excitations into loops and links. To finish we would like to stress the way in which the loopholes of the perturbative no-renormalization theorems can provide, at least partially, the lifting of the tree-order vacuum degeneracy. When dealing with theories which contain massless excitations the superpotentials at issue suffer finite renormalization. Once these unexpected terms are taken into account we have no longer valid arguments to identify Mc(V) with Mq(V). It remains to modify appropiately the generic superpotentials considered in this article so that the massless fields come into play. Although it is not apparent a priori how to handle these subtleties, we hope to report on the subject of the perturbative lifting in a next future.
ACKNOWLEDGMENTS We thank Professor R. Jackiw for calling our attention to Ref. [17]. The authors also express gratitude to the Comision Interministerial de Ciencia y Tecnolog@ a for financial support.
REFERENCES S. Coleman and J. Mandula, Phys. Rev. 159 (1967), 1251. J. Iliopoulos and B. Zumino, Nucl. Phys. B 76 (1974), 310. E. Witten, Nucl. Phys. B 185 (1981), 513. P. DiVecchia and S. Ferrara, Nucl. Phys. B 130 (1977), 93. J. Hruby, Nucl. Phys. B 131 (1977), 275. T. Murphy and L. O. Raifeartaigh, Nucl. Phys. B 218 (1983), 484. E. Witten and D. Olive, Phys. Lett. B 78 (1978), 97. R. Rajaraman, ``Solitons and Instantons,'' p. 33, North Holland, Amsterdam, 1982. C. Callias, Commun. Math. Phys. 62 (1978), 213; R. Bott and R. Seeley, Commun. Math. Phys. 62 (1978), 235. 10. C. Imbimbo and S. Mukhi, Nucl. Phys. B 247 (1984), 471. 11. B. Zumino, Nucl. Phys. B 89 (1975), 535. 1. 2. 3. 4. 5. 6. 7. 8. 9.
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12. N. Seiberg and E. Witten, Nucl. Phys. B 426 (1994), 19; N. Seiberg and E. Witten, Nucl. Phys. B 431 (1994), 484. 13. N. Seiberg, Phys. Rev. D 49 (1994), 6857. 14. P. West, Nucl. Phys. B 106 (1976), 219. 15. P. West, Phys. Lett. B 258 (1991), 375. 16. E. Witten, Nucl. Phys. B 202 (1982), 253. 17. R. Hobart, Proc. Phys. Soc. 82 (1963), 201. 18. L. Girardello, C. Imbimbo, and S. Mukhi, Phys. Lett. B 132 (1983), 69. 19. M. A. Lohe, Phys. Rev. D 20 (1979), 3120.
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