Lattice Supersymmetry with a Deformed Superalgebra Jun Saito - - PowerPoint PPT Presentation

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Lattice Supersymmetry with a Deformed Superalgebra Jun Saito - - PowerPoint PPT Presentation

Dec 05, 2022 135 likes •377 views

Introduction Hopf-Algebraic Treatment Construction of QFT Summary Lattice Supersymmetry with a Deformed Superalgebra Jun Saito (Hokkaido Univ.) in collaboration with Alessandro DAdda (INFN, Turin Univ.) Noboru Kawamoto (Hokkaido Univ.)

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Introduction Hopf-Algebraic Treatment Construction of QFT Summary

Lattice Supersymmetry with a Deformed Superalgebra

Jun Saito (Hokkaido Univ.) in collaboration with Alessandro D’Adda (INFN, Turin Univ.) Noboru Kawamoto (Hokkaido Univ.)

YITP Workshop “Development of Quantum Field Theory and String Theory”, July 10, 2009

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 1 / 20

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Introduction Hopf-Algebraic Treatment Construction of QFT Summary

Motivation

Why Supersymmetry on a Lattice (Ultimate Goal)? Nonperturbative/Constructive/Strong-coupling formulation

  • f SUSY QFT with the 1st principle calculations

Rigid regularization scheme independent of perturbation Numerical simulations Possible Applications? Gauge/gravity duals SUSY breaking, phenomenology beyond SM

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 2 / 20

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Difficulties

Symmetries on a Lattice: Always Nontrivial Poincar´ e invariance = ⇒ Discretized version is enough Gauge symmetry = ⇒ Wilson’s link formulation Chiral symmetry = ⇒ Ginsparg–Wilson fermion, etc. Supersymmetry = ⇒ Lattice version as well?? Immediate Obstacles for SUSY on a Lattice Doubling phenomena = ⇒ mismatch of fermion & boson d.o.f. Leibniz rule failure

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 3 / 20

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Introduction Hopf-Algebraic Treatment Construction of QFT Summary

Difficulties

Symmetries on a Lattice: Always Nontrivial Poincar´ e invariance = ⇒ Discretized version is enough Gauge symmetry = ⇒ Wilson’s link formulation Chiral symmetry = ⇒ Ginsparg–Wilson fermion, etc. Supersymmetry = ⇒ Lattice version as well?? Immediate Obstacles for SUSY on a Lattice Doubling phenomena = ⇒ mismatch of fermion & boson d.o.f. = ⇒ avoided with extended SUSY, or G–W fermions, etc. Leibniz rule failure = ⇒ more crucial

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 3 / 20

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Leibniz Rule Failure

Leibniz Rule Failure of “Derivative” Op. Superalgebra contains momentum op.: {QA, QB} = PAB = iγµ∂µ. On the lattice, ∂µ → ∂lat

µ : “derivative” on the lattice?

Natural candidate ∂lat

µ

= ∂+µ: finite difference op. would obey slightly modified Leibniz rule: [Dondi–Nicolai, Fujikawa, . . . ] ∂+µ(ϕ · ϕ′)(x) = ∂+µϕ(x) · ϕ′(x) + ϕ(x + aˆ µ) · ∂+µϕ′(x). No-go theorem: no local “derivative” on the lattice can obey the exact Leibniz rule. [Kato–Sakamoto–So]

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 4 / 20

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Leibniz Rule Problem

Problem Due to the Grassmann-odd nature, supercharge would obey the exact Leibniz rule even on the lattice QA(ϕ · ϕ′)(x) = QAϕ(x) · ϕ′(x) + (−1)|ϕ|ϕ(x) · QAϕ′(x). Simple realization of superalgebra on the lattice {QA, QB} = iγµ∂lat

µ

isn’t possible.

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 5 / 20

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Leibniz Rule Problem

Solutions? Give up the exact algebra = ⇒ fine-tune problem in general.

[Curuci–Veneziano, . . . ]

Keep only a subalgebra which doesn’t contain the momentum

  • perator

= ⇒ works without fine-tuning in low dimensions.

[Kaplan et. al., Catterall et. al., Sugino, . . . ]

= ⇒ also manageable in four dimensions? [Elliott–Giedt–Moore, . . . ] Our Approach Deform the Leibniz rule for the supercharge.

[D’Adda–Kawamoto–Kanamori–Nagata, Arianos–D’Adda–Feo–Kawamoto–J. S.]

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 6 / 20

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Deformed-Algebra Approach

Deformed Leibniz Rule for Supercharges Let us introduce the deformed rule Qlat

A (ϕ·ϕ′)(x) = Qlat A ϕ(x)·ϕ′(x)+(−1)|ϕ|ϕ(x+aA)·Qlat A ϕ′(x).

This extends the notion of Lie superalgebra. Really a symmetry of a QFT?

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 7 / 20

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Deformed-Algebra Approach

Deformed Leibniz Rule for Supercharges Let us introduce the deformed rule Qlat

A (ϕ·ϕ′)(x) = Qlat A ϕ(x)·ϕ′(x)+(−1)|ϕ|ϕ(x+aA)·Qlat A ϕ′(x).

This extends the notion of Lie superalgebra. = ⇒ rigourous treatment: Hopf algebra. Really a symmetry of a QFT?

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 7 / 20

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Deformed-Algebra Approach

Deformed Leibniz Rule for Supercharges Let us introduce the deformed rule Qlat

A (ϕ·ϕ′)(x) = Qlat A ϕ(x)·ϕ′(x)+(−1)|ϕ|ϕ(x+aA)·Qlat A ϕ′(x).

This extends the notion of Lie superalgebra. = ⇒ rigourous treatment: Hopf algebra. Really a symmetry of a QFT? = ⇒ QFT with mildly generalized statistics and corresponding Ward–Takahashi identities. [Oeckl, Sasai–Sasakura]

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 7 / 20

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Plan of Talk

1

Introduction

2

Hopf-Algebraic Treatment of Lattice Superalgebra

3

Construction of QFT with the Hopf–Algebraic Supersymmetry

4

Summary & Discussion

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 8 / 20

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Hopf Algebra

Hopf Algebra Hopf Algebra H

  • Algebra

associative product · : H ⊗ H → H unit η : C → H + Coalgebra coassociative coproduct ∆ : H → H ⊗ H counit ǫ : H → C + Antipode S : H → H

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 9 / 20

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Leibniz Rule = ⇒ Coproduct

Leibniz Rules = ⇒ Coproduct Specifying Leibniz rules amounts to determining coproducts: Qlat

A (ϕ · ϕ′)(x) = Qlat A ϕ(x) · ϕ′(x) + (−1)|ϕ|ϕ(x + aA) · Qlat A ϕ′(x)

⇓    Qlat

A ⊲(ϕ · ϕ′)(x) = m

  • ∆(Qlat

A ) ⊲(ϕ ⊗ ϕ′)

  • (x),

∆(Qlat

A ) = Qlat A ⊗ 1

l + (−1)FTaA ⊗ Qlat

A ,

where m(ϕ ⊗ ϕ′) = ϕ · ϕ′, TaA ⊲ ϕ(x) = ϕ(x + aA), (−1)F ⊲ ϕ(x) = (−1)|ϕ|ϕ(x).

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 10 / 20

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Coproducts Formulae

Coproducts ∆(Qlat

A ) = Qlat A ⊗ 1

l + (−1)FTaA ⊗ Qlat

A ,

∆(P lat

µ ) = P lat µ

⊗ 1 l + Taˆ

µ ⊗ P lat µ ,

∆(Tb) = Tb ⊗ Tb, ∆

  • (−1)F

= (−1)F ⊗ (−1)F.

  • Cf. Coproducts for the Normal Leibniz Rules

∆(QA) = QA ⊗ 1 l + (−1)F1 l ⊗ QA, ∆(Pµ) = Pµ ⊗ 1 l + 1 l ⊗ Pµ, ∆(Tb) = Tb ⊗ Tb, ∆

  • (−1)F

= (−1)F ⊗ (−1)F.

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 11 / 20

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Consistency

Products of More Fields Associativity = ⇒ coassociativity: (ϕ1 · ϕ2) · ϕ3 = ϕ1 · (ϕ2 · ϕ3) ⇓ Qlat

A ⊲(ϕ1 · ϕ2) · ϕ3 = Qlat A ⊲ ϕ1 · (ϕ2 · ϕ3)

⇓ (∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆. This holds for our explicit formulae.

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 12 / 20

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Counit Consistency

Trivial Representation = ⇒ Counit c ∈ C: constant field, QA ⊲ c ≡ ǫ(QA)c Consistency ϕ = 1 · ϕ = ϕ· ⇓ QA ⊲ ϕ = QA ⊲

  • 1 · ϕ) = QA ⊲
  • 1 · ϕ)

⇓ (ǫ ⊗ id) ◦ ∆ = (id ⊗ ǫ) ◦ ∆ = id Counit ǫ has to be determined to satisfy this consistency.

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 13 / 20

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Counit & Antipode Formulae

Counits ǫ(Qlat

A ) = 0,

ǫ(P lat

µ ) = 0,

ǫ(Tb) = 1, ǫ

  • (−1)F

= 1. These satisfy the previous consistency conditions. Antipodes S(Qlat

A ) = −T −1 aA · (−1)F · Qlat A ,

S(P lat

µ ) = −T −1 aˆ µ · P lat µ ,

S(Tb) = T −1

b

, S

  • (−1)F

= (−1)F.

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 14 / 20

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Superalgebra on the Lattice

Hopf-Algebraic Superalgebra A consistent superalgebra on the lattice can be introduced as a Hopf algebra, with the algebraic structure {Qlat

A , Qlat B } = 2τ µ ABP lat µ ,

[Qlat

A , P lat µ ] = [P lat µ , P lat ν ] = 0,

[Qlat

A , Tb] = [P lat µ , Tb] = [Tb, Tc] = 0,

{Qlat

A , (−1)F} = [P lat A , (−1)F] = [Tb, (−1)F] = 0,

plus the algebra maps ∆, ǫ, S.

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 15 / 20

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Noncommutative Representation?

“Commutative” Representation & Statistics Hopf algebra is generally represented on a noncommutative thus nonlocal field space. In fact, we can reduce the noncommutativity to commutativity up to a generalized statistics (braiding) by adding a grading structure. ϕ ⊗ ϕ′ Ψ → ϕ′ ⊗ ϕ ↓ ↓ QA(ϕ ⊗ ϕ′)

Ψ

→ QA(ϕ′ ⊗ ϕ).

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 16 / 20

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Generalized Statistics

Generalized Statistics General braiding formula Ψ

  • ϕA0···Ap(x) ⊗ ϕ′

B0···Bq(y)

  • = (−1)pqϕ′

B0···Bq

  • y +

p

  • i=1

(al

Ai − ar Ai)

  • ⊗ ϕA0···Ap
  • x −

q

  • i=1

(al

Bi − ar Bi)

  • ,

where ϕA0···Ap := Qlat

Ap · · · Qlat A1ϕA0,

ϕA0 := φ.

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 17 / 20

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Quantization

Braided QFT

[Oeckl]

The braiding structure allows us to construct a QFT perturbatively with a formal path integral quantization.

  • Cf. Fock space representation & deformed CCR.

Braided Functional Derivative δ δϕ(x)(ϕ1 · ϕ2) = δ δϕ(x)ϕ1 · ϕ2 + (−1)|ϕ||ϕ1|Tϕϕ1 · δ δϕ(x)ϕ2 Path Integral

  • δ

δϕ(x) = 0

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 18 / 20

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Ward–Takahashi Identity

Correlation Functions The formal expression is enough to define & compute Z =

  • e−S,

ϕ1 · · · ϕn = 1 Z

  • ϕ1 · · · ϕne−S.

Ward–Takahashi Identity

[Sasai–Sasakura]

The Hopf-algebraic supersymmetry is expressed by the corresponding Ward–Takahashi identities a ⊲ϕ1 · · · ϕn = ǫ(a)ϕ1 · · · ϕn.

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 19 / 20

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Summary and Discussion

Summary We can introduce a Hopf algebra on a lattice as a lattice-deformed superalgebra. A consistency requires that the fields representing the Hopf algebra acquire a generalized statistics. The corresponding QFT can be constructed at least perturbatively. Discussion Nonperturbative/“simulationable” path integral definition? Gauge theory extension, & its strong coupling expansion? Connection with the other regularization approaches?

  • J. Saito

(Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 20 / 20