1 1

A SUSY lattice for 4 dimensions

Tomohisa Takimi (TIFR)

Ref) [T.T, JHEP08(2012)069[arXiv:1205.7038]]

17th December 2012@IMS

2

1. Introduction

2

Non-perturbative numerical study of SUSY theories.

Lattice formulation

3

1. Introduction

3

Non-perturbative numerical study of SUSY theories.

Lattice formulation

lattice construction of SUSY field theory has been

difficult for long time.

4

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

5

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

6

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.

7

So it has been difficult issue.

8

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] )

Extended SUSY action = TFT action

9

(we discard other part of SUSY)

We select only the TFT scalar charge

[Witten 1988, Marcus 1994]

Extended SUSY action = TFT action

10

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]

Extended SUSY action = TFT action

11

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]

12

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.

13

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

14

4. Toward a useful formulation for the 4-dimensional theory

[Hanada, Matsuura, Sugino 2010]

[Hanada 2011] [Hanada, Matsuura, Sugino 2011]

[T.T 2012]

15

New 4-d lattice theory = (2-d lattice) ×(Fuzzy S2)

• 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

16

*2- advantages

17

4-1. N=4 SUSY case [Hanada, Matsuura, Sugino 2010]

[Hanada 2011]

18

(With partially preserved SUSY)

2–d lattice ×Fuzzy Sphere

SUSY is fully recovered by the virtue of

super-renormalizability of 2-dimensions.

2–d lattice ×Fuzzy Sphere

We do not suffer from the SUSY breaking

divergence due to the FULLY PRESERVED

SUSY.

21

4-2. N=2 SUSY case

[Hanada, Matsuura, Sugino 2011]

[T.T 2012]

22

In this case, it is difficult to construct the

2-d lattice part with preserved SUSY

in usual manner.(usual deconstruction method)

23

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]

24

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 ?

25

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 !!

26

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

27 27

1-d momentum cutoff ×1-d lattice×Fuzzy S2

28

1-d momentum cutoff ×1-d lattice×Fuzzy S2

R1×1-d lattice×Fuzzy S2

No UV divergences !!

29

Later steps are analogous to N=4 case

R1×1-d lattice×Fuzzy S2

SUSY is fully recovered automatically

FULL SUSY suppress the SUSY breaking UV

divergence.

5. SUMMARY AND DISCUSSION

32

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.

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]

Thank you so much

35