Deformed Matrix Models, Supersymmetric Lattice Twists and N=1/4 Supersymmetry Page: 4 of 47
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sixteen supersymmetries. A more interesting case is a certain two-flux deformation of the
matrix model. This deformation preserves only Q = 1 out of Q = 16 supersymmetries, and
generates the V = 4 SYM on T4 in its classical continuum limit. We will benefit from the
relation of the deformed matrix models and supersymmetric lattices in the discussion of the
exact global supersymmetries of various lattice formulations.
In recent years, there has been significant progress on the non-perturbative lattice con-
struction for the supersymmetric gauge theories. Various approaches are used to construct
supersymmetric lattices with exact supersymmetry at finite lattice spacing. 2 One such
approach is the orbifold constructions which preserve a nilpotent subset of supersymme-
tries [10-12]. Also see [13-18] for related work. Catterall [19-22] and Sugino [23-27], starting
with a "topologically" twisted form of the target theories, successfully preserved a scalar
(nilpotent) subset of supersymmetries on the lattice. The relation between these three for-
mulations was not clear at first.
Motivated by the unconventional aspects of supersymmetric orbifold lattices, such as
scalars of the target theories residing on the links (rather than on sites) and fermions filling
single-valued integer spin representations (rather than being double-valued spinors), Ref. [28]
showed that all the supersymmetric orbifold lattices do indeed produce a twisted version of
the supersymmetric gauge theories in their continuum. The main point of Ref. [28] is depicted
in Fig. 1. This observation, merged the "topological" approach and supersymmetric orbifold
lattices at the conceptual level. Soon after, Catterall [29] showed that, the use of the correct
twist together with the geometrical discretization rules produce the supersymmetric lattice
actions for the orbifold lattices. More recent important work by Takimi [30], and Damgaard
et.al. [31-33] demonstrated the equivalence of these lattice formulations even at finite lattice
spacing. Ref. [34] also provided a full classification of the supersymmetric lattices that can
be obtained by orbifolding and argued for uniqueness in certain cases.
There is one other approach to lattice supersymmetry which aims to preserve all the
supersymmetries on the lattice, not only the nil-potent scalar supercharges. This approach is
also motivated by the twisted form of the supersymmetry algebra and Dirac-Kihler structure
of the fermions. 3 It is referred as link approach in D'Adda et.al. [37-40]. The claim of pre-
serving the whole set of supersymmetries is debated in Ref. [41,42], with a negative conclusion.
On the other hand, the lattice structures on the link approach can be obtained by orbifold
projections and in fact, these two approaches are also equivalent as shown in [32]. However,
some lattices of link approach constructions, which are claimed to possess all the supersymme-
tries, can be obtained by orbifold projections which preserve either no supersymmetry, or just
few scalar supercharges according to the criteria of [10,43]. In our opinion, Ref. [32] answered
this question satisfactorily by showing that whatever supersymmetry remains intact under
the orbifold projection is indeed the exact supersymmetry on the lattice. However, [44, 45]
asserted the consistency of the link approach despite Ref. [32, 41, 42]. Here, we will give an
2Alternatives formulations in which supersymmetry only emerges in the continuum are studied in, for
example, [7 9]. Also see the interpretation in 6.
3An earlier proposal of using Dirac-Kahler fermions to supersymmetric lattices appeared in [36].3
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Unsal, Mithat. Deformed Matrix Models, Supersymmetric Lattice Twists and N=1/4 Supersymmetry, article, September 24, 2008; [Menlo Park, California]. (https://digital.library.unt.edu/ark:/67531/metadc899415/m1/4/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.