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Register a new userProgress and prospects of lattice supersymmetry
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David Schaich
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Supersymmetry plays prominent roles in the study of quantum field theory and in many proposals for potential new physics beyond the standard model, while lattice field theory provides a non-perturbative regularization suitable for strongly interacting systems. Lattice investigations of supersymmetric field theories are currently making significant progress, though many challenges remain to be overcome. In this brief overview I discuss particularly notable progress in three areas: supersymmetric Yang--Mills (SYM) theories in fewer than four dimensions, as well as both minimal N=1 SYM and maximal N=4 SYM in four dimensions. I also highlight super-QCD and sign problems as prominent challenges that will be important to address in future work.
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We propose an unconventional formulation of lattice field theories which is quite general, although originally motivated by the quest of exact lattice supersymmetry. Two long standing problems have a solution in this context: 1) Each degree of freedom on the lattice corresponds to $2^d$ degrees of freedom in the continuum, but all these doublers have (in the case of fermions) the same chirality and can be either identified, thus removing the degeneracy, or, in some theories with extended supersymmetry, identified with different members of the same supermultiplet. 2) The derivative operator, defined on the lattice as a suitable periodic function of the lattice momentum, is an addittive and conserved quantity, thus assuring that the Leibnitz rule is satisfied. This implies that the product of two fields on the lattice is replaced by a non-local star product which is however in general non-associative. Associativity of the star product poses strong restrictions on the form of the lattice derivative operator (which becomes the inverse gudermannian function of the lattice momentum) and has the consequence that the degrees of freedom of the lattice theory and of the continuum theory are in one-to-one correspondence, so that the two theories are eventually equivalent. Regularization of the ultraviolet divergences on the lattice is not associated to the lattice spacing, which does not act as a regulator, but may be obtained by a one parameter deformation of the lattice derivative, thus preserving the lattice structure even in the limit of infinite momentum cutoff. However this regularization breaks gauge invariance and a gauge invariant regularization within the lattice formulation is still lacking.
We propose a lattice field theory formulation which overcomes some fundamental difficulties in realizing exact supersymmetry on the lattice. The Leibniz rule for the difference operator can be recovered by defining a new product on the lattice, the star product, and the chiral fermion species doublers degrees of freedom can be avoided consistently. This framework is general enough to formulate non-supersymmetric lattice field theory without chiral fermion problem. This lattice formulation has a nonlocal nature and is essentially equivalent to the corresponding continuum theory. We can show that the locality of the star product is recovered exponentially in the continuum limit. Possible regularization procedures are proposed.The associativity of the product and the lattice translational invariance of the formulation will be discussed.
We construct a lattice model for two-dimensional N=(2,2) supersymmetric QCD (SQCD), with the matter multiplets belonging to the fundamental or anti-fundamental representation of the gauge group U(N) or SU(N). The construction is based on the topological field theory (twisted supercharge) formulation and exactly preserves one supercharge along the line of the papers [1]--[4] for pure supersymmetric Yang-Mills theories. In order to avoid the species doublers of the matter multiplets, we introduce the Wilson terms and the model is defined for the case of the number of the fundamental matters (n_{+}) equal to that of the anti-fundamental matters (n_{-}). If some of the matter multiplets decouple from the theory by sending the corresponding anti-holomorphic twisted masses to the infinity, we can analyze the general n_{+} eq n_{-} case, although the lattice model is defined for n_{+} =n_{-}. By computing the anomaly of the U(1)_A R-symmetry in the lattice perturbation, we see that the decoupling is achieved and the anomaly for n_{+} eq n_{-} is correctly obtained.
We propose a new lattice superfield formalism in momentum representation which accommodates species doublers of the lattice fermions and their bosonic counterparts as super multiplets. We explicitly show that one dimensional N=2 model with interactions has exact Lie algebraic supersymmetry on the lattice for all super charges. In coordinate representation the finite difference operator is made to satisfy Leibnitz rule by introducing a non local product, the ``star product, and the exact lattice supersymmetry is realized. The standard momentum conservation is replaced on the lattice by the conservation of the sine of the momentum, which plays a crucial role in the formulation. Half lattice spacing structure is essential for the one dimensional model and the lattice supersymmetry transformation can be identified as a half lattice spacing translation combined with alternating sign structure. Invariance under finite translations and locality in the continuum limit are explicitly investigated and shown to be recovered. Supersymmetric Ward identities are shown to be satisfied at one loop level. Lie algebraic lattice supersymmetry algebra of this model suggests a close connection with Hopf algebraic exactness of the link approach formulation of lattice supersymmetry.
In this paper, we introduce the overlap Dirac operator, which satisfies the Ginsparg-Wilson relation, to the matter sector of two-dimensional N=(2,2) lattice supersymmetric QCD (SQCD) with preserving one of the supercharges. It realizes the exact chiral flavor symmetry on the lattice, to make possible to define the lattice action for general number of the flavors of fundamental and anti-fundamental matter multiplets and for general twisted masses. Furthermore, superpotential terms can be introduced with exact holomorphic or anti-holomorphic structure on the lattice. We also consider the lattice formulation of matter multiplets charged only under the central U(1) (the overall U(1)) of the gauge group G=U(N), and then construct lattice models for gauged linear sigma models with exactly preserving one supercharge and their chiral flavor symmetry.
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