a susy lattice for 4 dimensions - harish-chandra research...
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A SUSY lattice for 4 dimensions
Tomohisa Takimi (TIFR)
Ref) [T.T, JHEP08(2012)069[arXiv:1205.7038]]
17th December 2012@IMS
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1. Introduction
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Non-perturbative numerical study of SUSY theories.
Lattice formulation
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1. Introduction
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Non-perturbative numerical study of SUSY theories.
Lattice formulation
lattice construction of SUSY field theory has been
difficult for long time.
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Absense of inifinitesimal translation on the lattice
Very crude regularization for SUSY theory
since the SUSY is artificially broken.
Momentum in the SUSY algebra
v.s
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Absense of inifinitesimal translation on the lattice
Very crude regularization for SUSY theory
since the SUSY is artificially broken.
Momentum in the SUSY algebra
v.s
Many SUSY breaking UV divergences
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Long required computation time. (Fine-tuning problem)
To get a continuum limit and result with full
SUSY, we have to precisely compute many SUSY breaking counter terms.
Too huge computation time to finish the
computation before the end of world.
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So it has been difficult issue.
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But there are development
( deconstruction approach of [Cohen, Kaplan Katz Unsal 2003,2004,2005])
(topological field theory
[Sugino 2003, 2004,2005,2006, Catterall 2005,
D’Adda, Kanamori, Kawamoto, Nagata 2005] )
(Actually these are equivalent:
[T.T 2007, Damgaard Matsuura 2007, Catterall 2008] )
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Extended SUSY action = TFT action
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(we discard other part of SUSY)
We select only the TFT scalar charge
[Witten 1988, Marcus 1994]
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Extended SUSY action = TFT action
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We select only the TFT scalar charge
Partial Preserved SUSY on the lattice
(we discard other part of SUSY)
No translation in the algebra
[Witten 1988, Marcus 1994]
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Extended SUSY action = TFT action
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We select only the TFT scalar charge
Partial Preserved SUSY on the lattice
It suppresses the SUSY breaking UV
divergences.
(we discard other part of SUSY)
No translation in the algebra
[Witten 1988, Marcus 1994]
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2. SUSY lattice for two, three-dimensions
UV-divergence is not severe (Super renormalizable).
Partially preserved SUSY works well enough.
SUSY is automatically recovered at continuum limit.
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3. SUSY lattice for 4-dimensions
UV-divergence IS SEVERE (not Super renormalizable).
Partially preserved SUSY on the lattice
does not work well enough.
Far from full-fledged numerical simulations
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4. Toward a useful formulation for the 4-dimensional theory
[Hanada, Matsuura, Sugino 2010]
[Hanada 2011] [Hanada, Matsuura, Sugino 2011]
[T.T 2012]
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New 4-d lattice theory = (2-d lattice) ×(Fuzzy S2)
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• 2-dimensional lattice
UV divergence is so soft to easily suppress the SUSY breaking divergence by partial SUSY.
• Fuzzy Sphere
it provides descretized extra dimensional space without breaking SUSY
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*2- advantages
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4-1. N=4 SUSY case [Hanada, Matsuura, Sugino 2010]
[Hanada 2011]
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(With partially preserved SUSY)
2–d lattice ×Fuzzy Sphere
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SUSY is fully recovered by the virtue of
super-renormalizability of 2-dimensions.
2–d lattice ×Fuzzy Sphere
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We do not suffer from the SUSY breaking
divergence due to the FULLY PRESERVED
SUSY.
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4-2. N=2 SUSY case
[Hanada, Matsuura, Sugino 2011]
[T.T 2012]
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In this case, it is difficult to construct the
2-d lattice part with preserved SUSY
in usual manner.(usual deconstruction method)
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In this case, it is difficult to construct the
2-d lattice part with preserved SUSY
in usual manner.(usual deconstruction method)
2-d lattice part = 1-d lattice ×
1-d momentum cutoff
[T.T 2012]
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In this case, it is difficult to construct the
2-d lattice part with preserved SUSY
in usual manner.(usual deconstruction method)
2-d lattice part = 1-d lattice ×
1-d momentum cutoff
It will break the symmetry terribly
Is it O.K ?
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In this case, it is difficult to construct the
2-d lattice part with preserved SUSY
in usual manner.(usual deconstruction method)
2-d lattice part = 1-d lattice ×
1-d momentum cutoff
It will break the symmetry terribly
Is it O.K ? O.K !!
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Because if the continuum limit of this direction
is taken first, with keeping other direction
regularized,
this is quantum mechanics, there is
no dangerous UV divergences.
2-d lattice part = 1-d lattice ×
1-d momentum cutoff
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1-d momentum cutoff ×1-d lattice×Fuzzy S2
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1-d momentum cutoff ×1-d lattice×Fuzzy S2
R1×1-d lattice×Fuzzy S2
No UV divergences !!
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Later steps are analogous to N=4 case
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R1×1-d lattice×Fuzzy S2
SUSY is fully recovered automatically
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FULL SUSY suppress the SUSY breaking UV
divergence.
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5. SUMMARY AND DISCUSSION
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Good point
• First 4-d N=2 deconstruction SUSY lattice model free from fine-tuning.
• Easy and simple action useful for coding.
• First deconstruction lattice model free from soft SUSY breaking mass term.
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Future work, homework
• Noncommutative 4-d N=2 theory has severe UV-IR mixing, how can I take commutative limit from Fuzzy S2 smoothly ?
[Minwalla, Van Raamsdonk, Seiberg 1999]
• Do Non-perturbative corrections disturb the calculation ? Check by numerical test.
[Hanada, Matsuura,T.T in future]
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Thank you so much
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