Comments on "Various Super Yang-Mills Theories with Exact Supersymmetry on the Lattice"

0.06 MB·English

by  A. Kwang-Hua Chu

Comments on ”Various Super Yang-Mills Theories with Exact
Supersymmetry on the Lattice”
A. Kwang-Hua Chu ∗
P.O. Box 30-15, Shanghai 200030, PR China
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0 Abstract
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2 We make corrections on the paper by Sugino [JHEP 0501 (2005) 016].

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a Keywords: Lattice Quantum Field Theory, Topological Field Theories.
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0 Sugino gave a presentation [1] in order to continue to construct lattice super Yang-Mills
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theories along the line discussed in the previous papers [2-3]. In his construction of N = 2,

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4 theories in four dimensions, the problem of degenerate vacua seen in [2] is resolved without
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6 affecting exact lattice supersymmetry,whileintheweak couplingexpansionsomesurplusmodes
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0 appear both in bosonic and fermionic sectors reflecting the exact supersymmetry. A slight
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modi- fication to the models is made such that all the surplus modes are eliminated in two-
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5 and three-dimensional models obtained by dimensional reduction thereof. N = 4, 8 models in
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/ threedimensions need fine-tuningof threeand one parameters respectively to obtain the desired
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l continuum theories, while two-dimensional models with N = 4, 8 donot requireany fine-tuning.
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p Some of Sugino’s results, say, in page 3 and in page 10 for the equations (2.10), (3.5) and (3.6)
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h are based on the derivations in AppendixA (Resolution of Vacuum Degeneracy). Theminimum
:
v Φ~(x) + ∆Φ~(x) = 0 is realized by the unique configuration U (x) = 1 for any choice of the
µν
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X constant r in the range r = cot(φ) : 0 < φ≤ π/(2N).
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a The present author, after following his derivations in Appendix A., would like to point out that
there are other possibilities except the unique solution U = V = 1 which Sugino claimed in [1].
In fact, Sugino missed one other condition : [u +v ]/2 = −π!
k i(k)
To easily illustrate our reasoning, we plot a schematic figure (shown as Fig. 1) of which the
possible regions A′A and B′B are related to the equation (A.9) in [1] for |cos(uk+vi(k) −φ)| ≥
2
cosφ. This equation comes from
u +v u −v
k i(k) k i(k)
cos( −φ)cos( )= cosφ, k =1,···,N, (1)
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once we impose |cos(uk−vi(k))| ≤ 1 (follow Sugino’s reasoning in [1]), where 0 < φ ≤ π/(2N),
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and
eiu1···eiuN =1, eivi(1) ···eivi(N) = 1.
There will be two solutions for N=1 : u = v = π; u = v = −π or u = v = π once
1 i(1) 2 i(2) 2 i(2)
φ= π/2. Under this case, eiu1 = eiπ = −1 in the expression of the equation (A.4) in [1].
∗Present Address: P.O.Box 39, Tou-Di-Ban, XihongRoad, Urumqi830000, PR China.
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References
[1] F. Sugino, JHEP 0412, (2004) 060 [hep-lat/0410035].
[2] F. Sugino, A lattice formulation of super Yang-Mills theories with exact supersymmetry,
JHEP 0401 (2004) 015 [hep-lat/0311021].
[3] F. Sugino, Super Yang-Mills theories on the two-dimensional lattice with exact supersym-
metry, JHEP 0403 (2004) 067 [hep-lat/0401017].
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B A
f f
f f
B'
A'
Fig. 1 Schematic plot for the possible region A′A and B′B for the equation (A.9) in [1]
for |cos(uk+vi(k) −φ)| ≥ cosφ, 0 < φ≤ π/(2N), k = 1,···,N.
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