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U(1) Spin Chern-Simons Theory and Arf Invariants in Two Dimensions

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U(1) Spin Chern-Simons Theory and Arf Invariants in Two Dimensions U(1) spin Chern-Simons theory and Arf invariants in two dimensions Koichi Saito (Univ. of Tokyo, Komaba) @ Strings and Fields 2020. Day 5 Poster No.18 Novenmber 20, 2020 based on arXiv:2005.03203 a work with T.Okuda (Komaba), S.Yokoyama (YITP) Koichi Saito (Univ. of Tokyo, Komaba) U(1) spin CS theory and Arf inv @ Strings and Fields 2020 1 / 21 Table of Contents 1 Introduction 2 RCFT and Chern-Simons theory 3 Derivation of partition function the CS theory on the lens space 4 Summary and Future direction Koichi Saito (Univ. of Tokyo, Komaba) U(1) spin CS theory and Arf inv @ Strings and Fields 2020 2 / 21 Introduction The bulk-edge correspondence A non-trivial relation a between D-dim boundary and corresponding (D + 1)-dim bulk D-dim boundary (D+1)-dim bulk The topological phase of the bulk theory is characterized by its boundary theory The anomaly of the boundary theory is specified by the bulk theory (anomaly inflow) Koichi Saito (Univ. of Tokyo, Komaba) U(1) spin CS theory and Arf inv @ Strings and Fields 2020 3 / 21 Introduction A familiar example of the bulk-edge correspondence. 3d U(1) Chern-Simons (CS) theory with the even level k and the 2d RCFT of a compact boson φ with R2 rational. [Moore and Seiberg, 1989] It is known that there is one to one correspondence between A charge of a wilson line in CS theory $ A primary field in RCFT The states in CS theory $ The conformal blocks in RCFT When the level k is odd, the CS theory becomes the spin theory ! the theory has to depend on the spin structure. [Dijkgraaf and Witten 1990] The problems 2d RCFT of a compact boson does not depend on the spin structure. p ±i kφ The generators of the extendedp chiral symmetry e L , which correspond to (pL; pR ) = (± k; 0) are not in the physical spectrum ! Is there any prescription to remedy the above problems? Koichi Saito (Univ. of Tokyo, Komaba) U(1) spin CS theory and Arf inv @ Strings and Fields 2020 4 / 21 Table of Contents 1 Introduction 2 RCFT and Chern-Simons theory 3 Derivation of partition function the CS theory on the lens space 4 Summary and Future direction Koichi Saito (Univ. of Tokyo, Komaba) U(1) spin CS theory and Arf inv @ Strings and Fields 2020 5 / 21 Table of Contents 1 Introduction 2 RCFT and Chern-Simons theory 3 Derivation of partition function the CS theory on the lens space 4 Summary and Future direction Koichi Saito (Univ. of Tokyo, Komaba) U(1) spin CS theory and Arf inv @ Strings and Fields 2020 6 / 21 Definition of the Arf invariant The prescription: coupling the theory TArf expressed by Arf invariant. The definition of the Arf invariant 1 X (−1)Arf(Σ,σ) ≡ (−1)q~(a) p 1 jH (Σ; Z2)j 1 a2H (Σ;Z2) Σ : 2-dim manifold, σ : spin structure of Σ whereq ~ is known as a quadratic form ( 0 (if the spin structure around A is anti-periodic); q~(a) = 1 (if the spin structure around A is periodic) and Z q~(a + b) − q~(a) − q~(b) = a [ b Σ Koichi Saito (Univ. of Tokyo, Komaba) U(1) spin CS theory and Arf inv @ Strings and Fields 2020 7 / 21 Introducing the theory TArf The properties of TArf Realized as the low-energy limit of the Kitaev chain on S1 The ground state on S1 has ( (−1)F = 1 (anti-periodic boundary condition) (−1)F = −1 (periodic boundary condition) (F : fermion number) πiArf[ρ] The partition function: Z[TArf; ρ] = e Arf[ρ]: the topological invariant dependent on the spin structure ρ On the torus the Arf invariant is given as (yellow line: Z2 gauge field) ρ = AA ρ = AP A 0 P 0 A A ρ = PA ρ = PP A 0 P 1 P P Koichi Saito (Univ. of Tokyo, Komaba) U(1) spin CS theory and Arf inv @ Strings and Fields 2020 8 / 21 Fermionizing a theory A-cycle How does Z2 gauge field act on TArf ? Z2 gauge field can be regard as a Z2 holonomy and a non-trivial holonomy changes the spin structure B-cycle ( I 0 ! do nothing s = γ 1 ! change the spin structure For example, if ρ = AA and s = (0; 1), ρ = AA s ⋅ ρ = AP A P A A Koichi Saito (Univ. of Tokyo, Komaba) U(1) spin CS theory and Arf inv @ Strings and Fields 2020 9 / 21 Partition function of RCFT The action of 2d CFT of a compact boson 1 Z S = d2x @ φ @µφ, 8π µ φ ∼ φ + 2πR (R : the radius on the target space) The relation between the level k and the radius R is r p k = pp00; R = 2 (p; p00 : coprime. ) p00 The partition function on the torus 1 X 1 ( n + wR )2 1 ( n − wR )2 2πiτ Z = q 2 R 2 q¯ 2 R 2 ; q = e ) jη(τ)j2 n;w2Z η(τ) : Dedekind η function; τ : the modulus ( n wR R + 2 = pL (left-moving momentum) n wR R − 2 = pR (right-moving momentum) Koichi Saito (Univ. of Tokyo, Komaba) U(1) spin CS theory and Arf inv @ Strings and Fields 2020 10 / 21 Fermionizing a theory If a bosonic theory Tbos has a Z2 global symmetry, by coupling TArf with Tbos and gauging the common Z2 symmetry, we have Fermionization of a bosonic theory 1 X Z[ρ = S · ρ ] = eπiArf[s·S·ρ0]Z (s); ρ = AA: 0 2 Tbos 0 s=(s1;s2) [Tachikawa 2018], [Karch, Tong, and Turner 2019]. In particular the 2 d CFT of compact boson has Z2 symmetry φ ! φ + πR Fermionization of the 2d CFT 1 X Z[ρ = S · ρ ] = (−1)(S1+s1)(S2+s2) 0 2jη(τ)j2 s=(s1;s2) X ns 1 ( n + wR )2 1 ( n − wR )2 (−1) 2 q 2 R 2 q¯ 2 R 2 n2Z 1 w2Z+ 2 s1 Koichi Saito (Univ. of Tokyo, Komaba) U(1) spin CS theory and Arf inv @ Strings and Fields 2020 11 / 21 Fermionizing a theory With a slight calculation we obtain The partition function of the modified CFT 1 Z[ρ] = × 2jη(τ)j2 2 2 8 1 (m( 2 + R )+m ¯( 2 − R )) 1 (m¯( 2 + R )+m( 2 − R )) > q 8 R 2 R 2 q¯ 8 R 2 R 2 ρ = AA ; > 2 2 > 1 m 2 + R +m ¯ 2 − R 1 m¯ 2 + R +m 2 − R X < (−1)m+m ¯ q 8 ( ( R 2 ) ( R 2 )) q¯ 8 ( ( R 2 ) ( R 2 )) ρ = AP ; × 1 2 R 2 R 2 2 1 2 R 2 R 2 2 8 (m( R + 2 )+m ¯( R − 2 )+ R ) 8 (m¯( R + 2 )+m( R − 2 )+ R ) m;m¯2 > q q¯ ρ = PA ; Z > 2 2 > 1 m 2 + R +m ¯ 2 − R + 2 1 m¯ 2 + R +m 2 − R + 2 : (−1)m+m ¯+1q 8 ( ( R 2 ) ( R 2 ) R ) q¯ 8 ( ( R 2 ) ( R 2 ) R ) ρ = PP : At ρ = A∗, the left(right)-moving momentum pL(pR ) is given as 1 2 R 2 R 1 2 R 2 R p = m + +m ¯ − ; p = m¯ + + m − L 2 R 2 R 2 R 2 R 2 R 2 p p+p00 p−p00 We solve one of the problems as (pL; pR ) = ( k; 0) ! (m; m¯) = ( 2 ; 2 ) Koichi Saito (Univ. of Tokyo, Komaba) U(1) spin CS theory and Arf inv @ Strings and Fields 2020 12 / 21 The conformal block at the odd level With the following conformal block (kn+λ)2 (S1;S2) i S1S2 S n S L (τ) = P q 2k (−1) 2 ; λ 2 + 1 : λ η(τ) n2Z Z 2 where the spin structure of the torus and (S1; S2) are related as AA $ (0; 0); AP $ (0; 1);; PA $ (1; 0); PP $ (1; 1): we have 8 (0;0) (0;0) > L (τ)L (τ) ρ = AA ; > λ !λ k−1 > (1+!)λ (0;1) (0;1) X < (−1) L (τ)L (τ) ρ = AP ; Z[ρ] = λ !λ L(1;0) (τ)L(1;0) (τ) ρ = PA ; λ=0 > λ+p00=2 !λ+p00=2 > :> (1+!)λ+1 (1;1) (1;1) (−1) Lλ+p00=2(τ)L!λ+p00=2(τ) ρ = PP : ! The partition function is expressed as the finite sum. Koichi Saito (Univ. of Tokyo, Komaba) U(1) spin CS theory and Arf inv @ Strings and Fields 2020 13 / 21 Modular matrices for odd level Under the modular transformation (S1;S2) X (T1;T2) Lλ (τ + 1) = T(S1;S2;λ)(T1;T2;µ)Lµ (τ) T1;T2,µ (S1;S2) X (T1;T2) Lλ (−1/τ) = S(S1;S2;λ)(T1;T2;µ)Lµ (τ) T1;T2,µ we obtain the modular matrices depending on the spin structure. The modular matrices for the odd level k 1 − 2πi λµ S(S ;S ;λ)(T ;T ;µ) = p δS T δS T e k ; 1 2 1 2 k 1 2 2 1 mod 2 − πi πi λ2 T = δ δ δ e 12 e k : (S1;S2;λ)(T1;T2;µ) S1T1 S1+S2+1;T2 λµ ! consistent with the results derived from the CS theory [Belov and Moore 2005],[Stirling 2008] Koichi Saito (Univ. of Tokyo, Komaba) U(1) spin CS theory and Arf inv @ Strings and Fields 2020 14 / 21 Table of Contents 1 Introduction 2 RCFT and Chern-Simons theory 3 Derivation of partition function the CS theory on the lens space 4 Summary and Future direction Koichi Saito (Univ.
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