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
Introduction Hopf-Algebraic Treatment Construction of QFT Summary Motivation Why Supersymmetry on a Lattice (Ultimate Goal)? Nonperturbative/Constructive/Strong-coupling formulation of 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
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. Leibniz rule failure J. Saito (Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 3 / 20
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
Introduction Hopf-Algebraic Treatment Construction of QFT Summary Leibniz Rule Failure Leibniz Rule Failure of “Derivative” Op. Superalgebra contains momentum op.: { Q A , Q B } = P AB = iγ µ ∂ µ . On the lattice, ∂ µ → ∂ lat µ : “derivative” on the lattice? Natural candidate ∂ lat = ∂ + µ : finite difference op. would obey µ slightly modified Leibniz rule: [Dondi–Nicolai, Fujikawa, . . . ] ∂ + µ ( ϕ · ϕ ′ )( x ) = ∂ + µ ϕ ( x ) · ϕ ′ ( x ) + ϕ ( x + a ˆ µ ) · ∂ + µ ϕ ′ ( 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
Introduction Hopf-Algebraic Treatment Construction of QFT Summary Leibniz Rule Problem Problem Due to the Grassmann-odd nature, supercharge would obey the exact Leibniz rule even on the lattice Q A ( ϕ · ϕ ′ )( x ) = Q A ϕ ( x ) · ϕ ′ ( x ) + ( − 1) | ϕ | ϕ ( x ) · Q A ϕ ′ ( x ) . Simple realization of superalgebra on the lattice { Q A , Q B } = iγ µ ∂ lat µ isn’t possible. J. Saito (Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 5 / 20
Introduction Hopf-Algebraic Treatment Construction of QFT Summary 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 operator = ⇒ 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
Introduction Hopf-Algebraic Treatment Construction of QFT Summary Deformed-Algebra Approach Deformed Leibniz Rule for Supercharges Let us introduce the deformed rule A ( ϕ · ϕ ′ )( x ) = Q lat A ϕ ( x ) · ϕ ′ ( x )+( − 1) | ϕ | ϕ ( x + a A ) · Q lat A ϕ ′ ( x ) . Q lat 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
Introduction Hopf-Algebraic Treatment Construction of QFT Summary Deformed-Algebra Approach Deformed Leibniz Rule for Supercharges Let us introduce the deformed rule A ( ϕ · ϕ ′ )( x ) = Q lat A ϕ ( x ) · ϕ ′ ( x )+( − 1) | ϕ | ϕ ( x + a A ) · Q lat A ϕ ′ ( x ) . Q lat 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
Introduction Hopf-Algebraic Treatment Construction of QFT Summary Deformed-Algebra Approach Deformed Leibniz Rule for Supercharges Let us introduce the deformed rule A ( ϕ · ϕ ′ )( x ) = Q lat A ϕ ( x ) · ϕ ′ ( x )+( − 1) | ϕ | ϕ ( x + a A ) · Q lat A ϕ ′ ( x ) . Q lat 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
Introduction Hopf-Algebraic Treatment Construction of QFT Summary Plan of Talk Introduction 1 Hopf-Algebraic Treatment of Lattice Superalgebra 2 Construction of QFT with the Hopf–Algebraic Supersymmetry 3 Summary & Discussion 4 J. Saito (Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 8 / 20
Introduction Hopf-Algebraic Treatment Construction of QFT Summary 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
Introduction Hopf-Algebraic Treatment Construction of QFT Summary Leibniz Rule = ⇒ Coproduct Leibniz Rules = ⇒ Coproduct Specifying Leibniz rules amounts to determining coproducts: Q lat A ( ϕ · ϕ ′ )( x ) = Q lat A ϕ ( x ) · ϕ ′ ( x ) + ( − 1) | ϕ | ϕ ( x + a A ) · Q lat A ϕ ′ ( x ) ⇓ A ⊲ ( ϕ · ϕ ′ )( x ) = m A ) ⊲ ( ϕ ⊗ ϕ ′ ) Q lat ∆( Q lat � � ( x ) , ∆( Q lat A ) = Q lat l + ( − 1) F T a A ⊗ Q lat A ⊗ 1 A , where m ( ϕ ⊗ ϕ ′ ) = ϕ · ϕ ′ , T a A ⊲ ϕ ( x ) = ϕ ( x + a A ) , ( − 1) F ⊲ ϕ ( x ) = ( − 1) | ϕ | ϕ ( x ) . J. Saito (Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 10 / 20
Introduction Hopf-Algebraic Treatment Construction of QFT Summary Coproducts Formulae Coproducts l + ( − 1) F T a A ⊗ Q lat ∆( Q lat A ) = Q lat A ⊗ 1 A , ∆( P lat µ ) = P lat µ ⊗ P lat ⊗ 1 l + T a ˆ µ , µ = ( − 1) F ⊗ ( − 1) F . ( − 1) F � � ∆( T b ) = T b ⊗ T b , ∆ Cf. Coproducts for the Normal Leibniz Rules l + ( − 1) F 1 ∆( Q A ) = Q A ⊗ 1 l ⊗ Q A , ∆( P µ ) = P µ ⊗ 1 l + 1 l ⊗ P µ , = ( − 1) F ⊗ ( − 1) F . ( − 1) F � � ∆( T b ) = T b ⊗ T b , ∆ J. Saito (Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 11 / 20
Introduction Hopf-Algebraic Treatment Construction of QFT Summary Consistency Products of More Fields Associativity = ⇒ coassociativity: ( ϕ 1 · ϕ 2 ) · ϕ 3 = ϕ 1 · ( ϕ 2 · ϕ 3 ) ⇓ Q lat A ⊲ ( ϕ 1 · ϕ 2 ) · ϕ 3 = Q lat A ⊲ ϕ 1 · ( ϕ 2 · ϕ 3 ) ⇓ (∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆ . This holds for our explicit formulae. J. Saito (Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 12 / 20
Introduction Hopf-Algebraic Treatment Construction of QFT Summary Counit Consistency Trivial Representation = ⇒ Counit c ∈ C : constant field, Q A ⊲ c ≡ ǫ ( Q A ) c Consistency ϕ = 1 · ϕ = ϕ · ⇓ � � Q A ⊲ ϕ = Q A ⊲ 1 · ϕ ) = Q A ⊲ 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
Introduction Hopf-Algebraic Treatment Construction of QFT Summary Counit & Antipode Formulae Counits ǫ ( Q lat ǫ ( P lat ( − 1) F � � A ) = 0 , µ ) = 0 , ǫ ( T b ) = 1 , = 1 . ǫ These satisfy the previous consistency conditions. Antipodes a A · ( − 1) F · Q lat S ( Q lat A ) = − T − 1 A , µ ) = − T − 1 S ( P lat µ · P lat µ , a ˆ S ( T b ) = T − 1 ( − 1) F � = ( − 1) F . � , S b J. Saito (Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 14 / 20
Introduction Hopf-Algebraic Treatment Construction of QFT Summary Superalgebra on the Lattice Hopf-Algebraic Superalgebra A consistent superalgebra on the lattice can be introduced as a Hopf algebra, with the algebraic structure B } = 2 τ µ { Q lat A , Q lat AB P lat µ , [ Q lat A , P lat µ ] = [ P lat µ , P lat ν ] = 0 , [ Q lat A , T b ] = [ P lat µ , T b ] = [ T b , T c ] = 0 , A , ( − 1) F } = [ P lat A , ( − 1) F ] = [ T b , ( − 1) F ] = 0 , { Q lat plus the algebra maps ∆ , ǫ, S . J. Saito (Hokkaido U.) Lattice SUSY w/ a Deformed Superalgebra 15 / 20
Recommend
More recommend
