So Matsuura (Department of Physics Hiyoshi, - - PowerPoint PPT Presentation

球面上の超対称ゲージ理論の数値実験

Numerical Experiment of Supersymmetric Gauge Theory on 2-Sphere

So Matsuura

(Department of Physics Hiyoshi, Keio University)

2019/9/10 1

Based on work with K. Ohta, T. Misumi and S. Kamata

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Continuum theory

2019/9/10 島根大学松江キャンパス 2

Starting point 2D 𝑂 = (2,2) SYM theory 𝐵(, 𝜚, * 𝜚 bosons: fermions: global symmetries 𝑉 1 - transformation 𝑉 1 . transformation SUSY transformation

2019/9/10 島根大学松江キャンパス 3

Derivatives on a curved background spin connection:

𝜖( → ∇(

𝜕34 ≡ 𝜕

Special background or Topological twisting

𝐶

( = 𝜕(

𝜖( + 𝑗 2 𝐶

( − 𝜕( 𝜏4 𝜔 → 𝜖(𝜔

𝜖( + 𝑗 2 𝐶

( + 𝜕( 𝜏4

* 𝜔 → 𝜖( + 𝑗𝜕(𝜏4 * 𝜔

: scalar : vector

Derivatives on a curved background + background 𝑉 1 . field 𝐶

(

Seiberg 2011, 2012

2019/9/10 島根大学松江キャンパス 4

Continuum action Natural renaming in this background

: scalar : vector

𝑉 1 - symmetry 𝑉 1 . symmetry

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SUSY transformation of the action (1) scalar SUSY transformations:

̅ 𝜗( = 0 𝜗? = 𝜗?(𝑦) preserve for 𝜗? = const.

(2) vector SUSY transformations:

𝜗? = 0 ̅ 𝜗( = ̅ 𝜗((𝑦) preserve iff ̅ 𝜗( is covariantly const.

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Continuum action in Q-exact form Observation toward discretized theory Bosonic fields on lattice scalar 𝜚(𝑦) vector A((𝑦) field tensor 𝐺

(C(𝑦)

site variable link variable face variable requirement (1) assign bosons on the lattice corresponding to their vector structure (2) keep Q-symmetry

s t u

7 7 2019/9/10 島根大学松江キャンパス

Fields on lattice Discretized action cf) Sugino 2003

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Topological information is preserved on lattice

(1) The same localization technique with the continuum theory works also on lattice

non-trivial fixed point equations Euler characteristic !

1-loop contribution is exact

𝑎 ∼

(2) The 𝑉 1 - anomaly appears in the measure

→

• D

F

• ei(NS−NL+NF )(N 2−1)α

: U(1)R neutral

D B = (

NS

• s=1

DΦsD¯ Φs)(

NL

• l=1

DUl)(

NF

• f=1

DYf)

D F = (

NS

• s=1

Ds)(

NL

• l=1

Dl)(

NF

• f=1

Df) 𝑉 1 - symmetry

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Natural question : Can we take the continuum limit ?

YES!

We have to check it non-perturbatively.

(It gives a “definition” of the SUSY theory on curved background!) from power counting point of view (2) There is no Q, 𝑉 1 - and gauge-invariant radiative correction which spoils the geometry. It is expected that the continuum theory will be obtained by simply taking 𝑏 → 0.

✓ap−4 g2 + c1pp−2 + c2apg2 + · · · ◆ Z d2x√gOp(x)

tree 1-loop 2-loop

O ∼ B or B2

(1) Tree level continuum limit reproduces the continuum action:

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What should we check?

The 𝑉 1 . symmetry and G 𝑅-symmetry are broken by discretization.

They must be restored in the continuum limit.

fact These symmetries are related with each other in the continuum limit.

It is sufficient to check the 𝑉 1 . symmetry dJQ = 0

dJ ˜

Q = 0

Model

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1 2 M 3

・・・

M-1

(M,N)-polygon decomposition of 𝑇4

MN/2 M(N-1)+1 M(N-1)+2

parameter tuning 𝑒𝑡4 = 𝑆4(𝑒𝜄4 + cos4 𝜄 𝑒𝜒4 ) (− ⁄ 𝜌 2 < 𝜄 < ⁄ 𝜌 2)

Anomaly-phase-quench method

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vev in the continuum theory phase quench method in usually used in Monte Carlo method

U(1) charge: ZERO

NOT A GOOD APPROXIMATION

U(1) charge: (N 2 − 1)χh

philosophy of the phase quench Ignore only the artificial phase coming from the discretization Observation

1.U(1)R phase 2.lattice artifact

θA θ

We should ignore only θ

Pf(D) = |Pf(D)|eiθA+iθ

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Compensator

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A : an operator with

QA = 0

• [A] = −(N 2 − 1)χh

A ≡ |A|e−iθA

anomaly-phase-quench method

= 1 Zq

• D

BO|Pf(D)|eiθA

Kamata-Misumi-Ohta-S.M. 2016 trace type determinant type Izykson-Zuber type

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Trivial WT identity for 𝑅-symmetry

2019/9/10 島根大学松江キャンパス 14

basic relation (for 𝜈 = 0)

without compensator

ダメダメ

with trace compensator

compensator 超大事

Identity 1 from 𝑉 1 . symmetry

2019/9/10 島根大学松江キャンパス 15

Site action Face action 𝑉 1 . doublet 𝑅-exact

→ 1 2NF (N 2

c − 1)

(a → 0)

Identity 2 from 𝑉 1 . symmetry

2019/9/10 島根大学松江キャンパス 16

𝑉 1 . doublet 𝑅-exact

+ → 𝑏 → 0

We should check 𝑉 1 . WT identity

2019/9/10 島根大学松江キャンパス 17

In the continuum theory For a 𝑉 1 . invariant operator 𝒫(𝑦),

= 0

problem 1 There are infinitely many ways to construct composite operators on the lattice. How to define Rotation and Divergence on lattice? problem 2

Hint: exact 𝑉 1 - current on the lattice

2019/9/10 島根大学松江キャンパス 18

continuum δScont = 1 2g2 Z d2x√g(2i∂µθ)Tr n 2i(φDµ ¯ φ − ¯ φDµφ) + 2Eµνλνχ + λµη

• <latexit sha1_base64="Y4bEnfocCzRsjz+rK/nA0zHrY=">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</latexit>

lattice

2iθ 2g2 @ X

l∈<·,s0>

− X

l∈<s0,·>

1 A αlTr

• Φs(l)Ul ¯

Φt(l)U −1

l

− ¯ Φs(l)UlΦt(l)U −1

l

• <latexit sha1_base64="8YrlA5kuQNwNQm5jZ7Y+0uRWtOw=">ADMnichVHLa9RAGP8SXzU+upF8BJcKltol9lUESk1IvH7WPbSqcNk+nsZujkwWR2oYb8A/4DHjxZEBGv/gdevHj0UG/iSfRWwYsHv0iVetjQma+T2+75uZIFUyM4TsW/ax4ydOnpo47Zw5e+78ZOPCxdUsGWouejxRiV4PWCaUjEXPSKPEeqoFiwIl1oKdu2N+bSR0JpN4xeymYjNig1j2JWcGIb/xhvY147knqQmFYUXuDXy5RWuQ5Xom5ZLs2Hk54rK2L1N+XZiZjKf3Cnc2Z8JhGZKEgmq5SA0w5lKg2Zr9yc6shd0cjUGbuh9POspaYLt+crGjBdQeYHtJXPdrCe0hV6lL+J6lTF/UbTdIm5XCPBp06aEI9uknjGVDYhgQ4DCECATEYjBUwyPDbgA4QSBHbhBwxjZEseQEFOgdokqgiG6g/MAdxs1GuN+nDMr3RyrKPw1Ol2YIm/Jc3JAXpMX5CP59tdceZlj3MsurkHlFak/+fDy8tf/uiJcDYSHrn/2bKAPN8teJfaelsj4FLzyjx48Oli+tTSVXyN75BP2/4Tsk1d4gnj0hT9dFEuPwcEH6Px+3UeDVa/dmWt7i9eb8wv1U0zAFbgKLbzvGzAP96ALPeDWnHXfCixuv7Tf2e/tD5XUtmrPJfhl2J+/A+Gj06c=</latexit>

2iθ 2g2 @ X

l∈<·,s0>

− X

l∈<s0,·>

1 A αlTr ✓ i 2λlUlηt(l)U −1

l

+ λlλlUl ¯ Φt(l)U −1

l

◆

𝜄W ≡ 𝜄𝜀W,WY

2i✓ 2g2 @ X

l∈<s0,·>

X

f∈Fl

− X

f=s0

X

l∈Lf

1 A (i↵ff✏f,l) ⇢ 1 2B(Uf)Tr ⇣ fXf,llYf,l + fY †

f,llX† f,l

⌘

− 1 ✏2 1 2B(Uf)2 Tr ⇣ f(Uf − U −1

f )

⌘ Tr ⇣ Xf,llYf,l − Y †

f,llX† f,l

⌘

+ +

𝜀𝑇Z?[ = This correspondence will be a guiding principle to construct “𝑽 𝟐 𝑾 current”, rotation and divergence on the lattice. cf) Ohta-san’s talk

ROTATION DIVERGENCE DIVERGENCE

• n going…

19

• We explicitly discretize 2-sphere.
• Q-symmetry is preserved as expected.
• The relations expected from the 𝑉 1 . symmetry seems to hold in the continuum limit.
• 𝑉 1 . WT identity, restoration of 𝑇4 geometry
• Other discretization (fullerene-like discretization?)
• N=(4,4) and N=(8,8) theory
• Simulation of 4D N=4 SYM by hybrid method
• Theory with matters
• Connection to Dynamical triangulation? (supersymmetric dynamical triangulation?)

Summary and Future Works

2019/9/10

cf) Hanada-Sugino-S.M. 2012 Kamata-Ohta-Misumi-S.M. to be appeared

島根大学松江キャンパス