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NON-PERTURBATIVE ASPECTS OF GLOBAL SUPERSYMMETRY
L. Girardello
To cite this version:
L. Girardello. NON-PERTURBATIVE ASPECTS OF GLOBAL SUPERSYMMETRY. Journal de
Physique Colloques, 1982, 43 (C3), pp.C3-323-C3-325. �10.1051/jphyscol:1982364�. �jpa-00221919�
JOURNAL DE PHYSIQUE
CoZZoque C3, suppZ6ment au n o 12, Tome 43, de'cembre 1982 page C3-323
N O N - P E R T U R B A T I V E ASPECTS OF G L O B A L SUPERSYMMETRY
L. Girardello
I s t i t u t o d i Fisica and INFN, MiZano, I t a l y and CERN, Geneva, SwitzerZand
Most of the remarkable properties of global supersymmetry have been investigated and understood in the perturbative framework. Recently, prompted by the important work of ~ittenl), new interest has arisen in non-perturbative aspects. It is worth- while to try and arrive at a unlfied understanding of such properties and of their connection with the general structure of the quantum field theory. We think that a natural setting for these questions is the functional integral approach. We immediately encounter here three crucial features, not unrelated, concerning the vacuum functional of exact supersymmetric theories:
1) There exists an intrinsic definition of an integral over superfields2): i.e., the functional measure is invariant with respect to all transformations of the super- fields which respect the 8 structure. The invariance of the measure has a central role in supersymmetry just similar to the Liouville measure in statistical mechanics.
2) The vacuum energy vanishes in exact globally supersymmetric theories". The vacuum functional (superpartition function)
does not need any infinite normalization. B ( @ ) is a universal measure and Z is a well-defined number whose value depends only on the specific theory. In order to give meaning to Z, the functional integral is done in a four-dimensional Euclidean box (side-length ay). Periodic boundary conditions for bosons as well as for fermions maintain the 8 structure (supersymrnetry). Z is invariant with respect to changes of the parameters (up to jump discontinuities) and changes of box size ay3) y 4 ) . What happens if Z = O? The situation is reminiscent of the Lee-Yang theorems) which states that a necessary condition for symmetry breaking is that the partition function be zero. This suggests that Z : 0 is a necessary condition for some symmetry breaking also in this case. The criterion has a meaning only because of supersymmetry and then it is natural that the symmetry which might be broken is supersymmetry.
3) Nicolai mapping6). In any globally supersymmetric theory, there exists a non- linear and, generally, non-local mapping, of the bosonic fields cpi + ~~[q] such that the bosonic part of the action becomes Gaussian with co-variance 1. The Jacobian of the mapping is just the fermionic determinant (actually Pfaffian) obtained from the integration over the fermions. There are changes in the presence of vector fields
Interplay between these three features casts new light on some aspects of supersymmetry and reveals new structures. We outline here three applications.
a) ~eformulation~)
...
of Witten index theoryL'. The Witten indexA
can be identified with Z as defined in 2). The renormalization group equations allow a further analysis of the independence of the parameters within each possible phase of the theory. But point 3) suggests a deeper meaning for 2 . 3) reads:Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982364
JOURNAL DE PHYSIQUE
= number of times that the mapping V +
5
covers the space of continuous periodic functions.A
can be interpreted as the winding number of the mapping.If the fermionic determinant has a zero eigenvalue (possible Goldstino mode) for all bosonic configurations, the measure of the image of the mapping is zero.
A
is zero and supersymmetry might be broken. On the other hand, by combining ar- guments from 1) and 2) one can see that A amounts to the number of configurations with zero bosonic action counted with their algebraic multiplicity.As an interesting by-product of this, one can argue that in supersymmetric pure Yang-Mills theories, the most likely phase is the confining onel) 7 ) (electric confinement).
b) Local Nicolai mapping and stochastic structures. The relevance of Nicolai ma~oinn
.. -
and of its exolicit construction is now auite obvious. One could also ask about the physical interpretation, if any, of the Gaussian fieldsti.
A suggestion comes from the remark by Parisi and sourlas8) that a stochastic approach to quantum scalar field theories is related to a hidden supersymmetry. Here, it will be a matter of reversing this argument, at least in certain cases.it turns out that in dimensions d : 1 (supersymmetric quantum mechanics) and d : 2 (N = 2 extended supersymmetry) the mapping can be explicitly constructed and it indeed has a stochastic interpretationg)~~~). These mappings are local, in the sense that the original bosonic fields appear with only first order space-time derivatives.
They have the general structure:
(generalized Langevin equations)
where Ti are the bosonic (here scalar) fields, <i the Gaussian fields and w[T]
is a member of a particular class of solutions of the classical (bosonic) Euclidean Hamilton-Jacobi equations for the zero energy sub-manifold, namely
(integrability condition) H being the Euclidean Hamiltonian.
In d = 1, Ti = X(t), W = V(x) and (2) is the usual Langevin equation. In d = 2, N = 2:
G b
w . f d x [ v t y ) + v c ? ) * i F a g ? ]
with <pa complex scalar field and V(T) any real analytic polynomial. The locality of the mapping requires extended supersymmetry. In d = 4, extension requires the presence of vectors and, with the exception of the (free) Abelian theoryl1),l2) and free scalar theoryg), no local mapping has been constructed.
The knowledge of an explicit, regularized, local mapping leads to a new per- turbative scheme (infra-diagrams)lO) along the lines of the classical stochastic perturbation theory13). Having integrated over the fermions, this scheme contains only the bosonic fields and hence the usual cancellations of divergences between bosonic and fermionic loops are automatic. All N = 2 theories in d = 2 which have the local mapping are ultra-violet finitelo ), 12).
L. G i r a r d e l l o C3-325
The a t t r a c t i v e c o n j e c t u r e i s t h a t t h e e x i s t e n c e o f t h e l o c a l mapping, ( i . e . , when t h e dynamics i s governed by s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s ) " f a v o u r s n f i n i t e - n e s s . The n e x t c h a l l e n g e l l ) i s t h e N = 4 t h e o r y i n d : 4.
c ) Supersymmetry on t h e l a t t i c e - . P u t t i n g supersymmetry on t h e l a t t i c e i s s t i l l a n open problem. A t t h e moment t h i s h a s been done i n d = 2 and N = 2 i n d = 4 w i t h A b e l i a n (compact) gauge group14 )
.
For t h e c a s e N = 2 i n d = 2 , i t c a n be done15) by d i s c r e t i z i n g ( i . e . , u s i n g n e x t n e i g h b o u r d e r i v a t i v e s , 2 l a W i l s o n ) t h e e x p l i c i t N i c o l a i mapping: no poten- t i a l l y d a n g e r o u s t e r m a p p e a r s i n t h e n a i v e continuum l i m i t l 6 ) . The f e r m i o n i c L a g r a n g i a n i s a u t o m a t i c a l l y r e c o v e r e d .
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