Higher Structures and Current Algebras

Transcription

1 Higher Structures and Current Algebras Noriaki Ikeda Ritsumeikan University, Kyoto Higher Structures in String Theory and M-Theory Sendai 2016

2 1. Introduction We discuss current algebras with higher algebroid structures. Alekseev,Strobl 05, Bonelli,Zabzine 05, Ekstrand,Zabzine 09, NI,Koizumi 11, Hekmati,Mathai 12, etc. We explain Supergeometric technique is to understand, compute and clarify current algebras. 1

3 Purposes and Applications Physics Topological and nonperturbative aspects of string and M-theory Find new symmetries and new physics Math Analysis and geometric applications of higher-algebroids and highergroupoids 2

4 Plan of Talk Alekseev-Strobl type current algebras Super symplectic geometry Generalizations to higher dimensions Quantization problems 3

5 2. Generalized Current Algebras on Loop Space X 2 = S 1 R with a local coordinate σ on S 1. Alekseev, Strobl 04 x i (σ): S 1 M, p i (σ): canonical momentum. (dim[x] = 0 and dim[p] = 1, dim[ σ ] = 1). {x i, x j } P B = 0, {x i, p j } P B = δ i jδ(σ σ ), {p i, p j } P B = 0. The Poisson bracket can be twisted by a closed 3-form H as {x i (σ), x j (σ )} P B = 0, {x i (σ), p j (σ )} P B = δ i jδ(σ σ ), {p i (σ), p j (σ )} P B = H ijk (x) σ x k δ(σ σ ). 4

6 A generalization of currents on a target space T M T M: J 0(f) (σ) = f(x(σ)), J 1(X+α) (σ) = α i (x(σ)) σ x i (σ) + X i (x(σ))p i (σ), where f(x(σ)) is a function, X + α = X i (x) i + α i (x)dx i Γ(T M T M). dim[j 0(f) ] = 0 and dim[j 1(X+α) ] = 1. {J 0(f) (σ), J 0(g) (σ )} P B = 0, {J 1(X+α) (σ), J 0(g) (σ )} P B = ρ(x + α)(σ)j 0(g) (x(σ))δ(σ σ ), {J 1(X+α) (σ), J 1(Y +β) (σ )} P B = J 1([X+α,Y +β]d )(σ)δ(σ σ ) + X + α, Y + β (σ ) σ δ(σ σ ), 5

7 where X, Y Γ(T M), α, β Γ(T M), X + α, Y + β = ι X β + ι Y α = X i β i + Y i α i, ρ(x + α) = X = X i (x) x i, [X + α, Y + β] D = [X, Y ] + L X β i Y dα + i X i Y H. The algebra of these operations is the standard Courant algebroid. Liu,Weinstein,Xu 97 Anomaly cancellation condition X + α, Y + β = 0. 6

8 Dirac Structure This condition is satisfied on the Dirac structure of T M T M. Definition 1. satisfies If a subbundle L of the Courant algebroid E E e 1, e 2 = 0 (isotropic), [e 1, e 2 ] D Γ(L) (close), for e 1, e 2 Γ(L), and rank(l) = 1 2rank(E), L is called the Dirac structure. 7

9 Our Purposes Generalizations to higher dimensions. Find fundamental structures behind current algebras. 8

10 3. Supergeometry The Courant algebroid has the supergeometric construction corresponding to the BRST-BFV formalism. A graded manifold M = (M, O M ) on a smooth manifold M is a ringed space which structure sheaf O M is Z graded commutative algebras over M, locally isomorphic to C (U) S (V ), where U is a local chart on M, V is a graded vector space and S (V ) is a free graded commutative ring on V. Grading is called degree. We denote O M = C (M). If degrees are nonnegative, a graded manifold is called a N-manifold. 9

11 Definition 2. A following triple (M, ω, Q) is called a QPmanifold (a differential graded symplectic manifold) (Symplectic NQ-manifold) of degree n. Schwarz 92 M: N-manifold (nonnegatively graded manifold) ω: P-structure A graded symplectic form of degree n on M. Q: Q-structure (a homological vector field) A graded vector field of degree +1 such that Q 2 = 0, is a symplectic vector field, L Q ω = 0. 10

12 Note: If degree n 0, there exists a Hamiltonian function (a homological function) Θ C (M) of degree n + 1 such that ι Q ω = δθ. Q 2 = 0 is equal to the classical master equation, {Θ, Θ} = 0. The QP-manifold triple can be replaced to (M, ω, Θ). 11

13 Derived bracket construction (supergeometric construction) of the (standard) Courant algebroid Roytenberg 99 We take n = 2. An N-manifold is M = T [2]T [1]M = (M, O M ). A local coordinate of T [1]M is (x i, q i ) of degree (0, 1) and the conjugate coordinate is (ξ i, p i ) of degree (2, 1). P-structure is a canonical graded symplectic form, ω = δx i δξ i + δq i δp i. 12

14 {, } of degree 2 Q-structure the Hamiltonian function of degree 3, Θ = ξ i q i + 1 3! H ijk(x)q i q j q k. Impose the classical master equation, {Θ, Θ} = 0 dh = 0. This derives that H is a closed 3-form. 13

15 Functions on graded manifold O M = C (M) We decompose C (M) = i 0 C i(m), where C i (M) is functions of degree i. degree 0: f(x) C 0 (M) C (M), degree 1: α i (x)q i + X i (x)p i C 1 (M) α + X = α i (x)dx i + X i (x) i Γ(T M T M). i.e. C 1 (M) Γ(T M T M), (C 2 (M) Γ( 2 (T M T M) T M), etc.) Note : C 0 (M) C 1 (M) make a closed subalgebra by the derived bracket {{, Θ}, }. (Count degree!) 14

16 Operations The operations on E is defined by graded Poisson brackets and derived brackets. For f, g C 0 (M), e, e 1, e 2 C 1 (M), 15

17 Poisson brackets C 0 C 0, 0 = {f, g} C 1 C 0, 0 = {e, f} C 1 C 1, e 1, e 2 = {e 1, e 2 } (inner product) Derived brackets C 0 C 0 0, 0 = {{f, Θ}, g} C 1 C 0 C 0, ρ(e)f = {{e, Θ}, f} (anchor map) C 1 C 1 C 1, [e 1, e 2 ] D = {{e 1, Θ}, e 2 } (Dorfman bracket) 16

18 4. QP Structure of 2D Current Algebras NI, Xu 13 Construction of Poisson brackets from supergeometric data First we derive the following Poisson bracket from supergeometry. {x i, x j } P B = 0, {x i, p j } P B = δ i jδ(σ σ ), {p i, p j } P B = H ijk (x) σ x k δ(σ σ ). Recall supergeometric data of the standard Courant algebroid. 1, M = T [2]T [1]M 17

19 2, ω = δx i δξ i + δq i δp i 3, Θ = ξ i q i + 1 3! H ijk(x)q i q j q k. Observation in the target space computation. For (x i, p i ) T [1]M, the graded target space, the derived bracket is {{x i, Θ}, x j } = 0, {{x i, Θ}, p j } = δ i j, {{p i, Θ}, p j } = H ijk (x)q k 18

20 Canonical Transformation and Lagrangian Submanifold Definition 3. Let α C (M) be a function of degree n. A canonical transformation e δ α is defined by f = e δ α f = f + {f, α} {{f, α}, α} +. eδ α is also called twisting. If {Θ, Θ} = 0, {e δ α Θ, e δ α Θ} = e δ α {Θ, Θ} = 0 for any twisting. Let L be a Lagrangian submanifold of M and pr : M L be a natural projection. Theorem 1. Let Θ be a homological function such that {Θ, Θ} = 19

21 0. Then {f, g} L pr {{pr f, Θ}, pr g} is a graded Poisson bracket on L, where f, g C (L). 20

22 Superfields Next, we consider the mapping space. Take the supermanifold X = T [1]S 1 with local coordinates (σ, θ). Local coordinates on Map(T [1]S 1, T [2]T [1]M) are superfields, x i (σ, θ) : T [1]S 1 M of degree 0 q i (σ, θ) Γ(T [1]S 1 x (T x [1]M)) of degree 1 and canonical conjugates ξ i (σ, θ) Γ(T [1]S 1 x (T x[2]m)) of degree 2, p i (σ, θ) Γ(T [1]S 1 x (T q [2]T x [1]M)) of degree 1. 21

23 AKSZ Construction 97 Alexandrov, Kontsevich, Schwartz, Zaboronsky The AKSZ construction is, by definition, a procedure to construct a QP-manifold structure on a mapping space of two graded manifolds, Map(X, M). (X, D, µ): X = T [1]X, where X is an n 1 dimensional manifold. D is a differential on X. µ is a D-invariant nondegenerate Berezin measure. (M, ω, Q): A QP-manifold of degree n 22

24 An evaluation map ev : X M X M is defined as ev : (z, Φ) Φ(z), where z X and Φ M X. A chain map µ : Ω (X M X ) Ω (M X ) is defined as an integration on X, µ F = X µ F where F Ω (X M X ). µ ev : Ω (M) Ω (M X ) is called a transgression map. P-structure (BV antibracket) We define the graded symplectic form on Map(X, M) as ω = µ ev ω = X dσdθ(δx i δξ i + δq i δp i ), 23

25 Q-structure (BV action) Q = {S, } = {S 0, } + {S 1, }(= ˆD + ˆQ), S = S 0 + S 1 = ι ˆDµ ev ϑ + µ ev Θ = dσdθ ( ξ i dx i + p i dq i) + (ξ i x i + 13! ) H ijk(x)q i q j q k X, where ω = δϑ. S 0 is a Hamiltonian for D and S 1 is a Hamiltonian for Q. S satisfies the classical master equation, {S, S} = 0. 24

26 The transgression induces the derived bracket on superfields. {{x i (σ, θ), S}, x j (σ, θ )} = 0, {{x i (σ, θ), S}, p j (σ, θ )} = δ i jδ(σ σ )δ(θ θ ), {{p i (σ, θ), S}, p j (σ, θ )} = H ijk (x)q k (σ, θ)δ(σ σ )δ(θ θ ). Note (Map(T [1]X, M), ω, Q = {S, }) is of degree 1. {, } is of degree 1. Therefore, 25

27 Twisting The derived bracket pr {{, Θ}, } induce the graded Poisson bracket on the Lagrangian submanifold L spanned by (x i, p i ) with the symplectic form, ω L = δx i δp i, The Liouville 1-form on the Lagrangian submanifold L 0 = Map(X, L) is α 0 = ι ˆDµ ev ϑ L = X µ p i dx i. 26

28 Twisting by α 0 gives rise to the transformation q k q k dx k. If we reduce to the canonical Lagrangian submanifold L 0 defined by ξ i = q i = 0, we obtain a normal Poisson bracket, and {, } P B = pr e δ α 0{{, S1 }, }. {x i (σ, θ), x j (σ, θ )} P B = 0, {x i (σ, θ), p j (σ, θ )} P B = δ i jδ(σ σ )δ(θ θ ), {p i (σ, θ), p j (σ, θ )} P B = H ijk (x(σ, θ))dx k (σ, θ)δ(σ σ )δ(θ θ ). 27

29 Physical Fields We expand the superfields to component fields by the local coordinate θ on T [1]S 1, Φ(σ, θ) = Φ (0) (σ) + θφ (1) (σ). The degree zero component in the expansion is the physical field (and degree nonzero components are ghost fields). Physical fields are x i (σ) = x (0)i (σ) and p i (σ) = p (1) i (σ). The Poisson brackets of the physical canonical quantities are degree 28

30 zero components: {x i (σ), x j (σ )} P B = 0, {x i (σ), p j (σ )} P B = δ i jδ(σ σ ), {p i (σ), p j (σ )} P B = H ijk (x) σ x k (σ)δ(σ σ ). 29

31 Construction of currents from supergeometric data C (M) = i 0 C i(m), where C i (M) is functions of degree i. We take C 0 C 1 as space of currents. Note : C 0 (M) C 1 (M) make a closed subalgebra by the derived bracket {{, Θ}, }. j 0(f) = f(x) C 0 (M) C (M), j 1(X+α) = α i (x)q i + X i (x)p i C 1 (M) α + X = α i (x)dx i + X i (x) i Γ(T M T M). i.e. C 1 (M) Γ(T M T M). 30

32 Currents are identified with twisted functions on the Lagrangian submanifold of we apply the transgression map to j 0 and j 1. Then, we twist them by α 0 and finally restrict the resulting functions to the Lagrangian submanifold defined by ξ i = q i = 0. The corresponding currents are J (0)(f) (ϵ (1) ) = pr e δ α 0µ ϵ (1) ev j (0)(f) = T [1]S 1 µϵ (1) f(x), J (1)(X+α) (ϵ (0) ) = pr e δ α 0µ ϵ (0) ev j (1)(u,α) = µϵ (0) ( α i (x)dx i + X i (x)p i ), T [1]S 1 31

33 where ϵ (i) = ϵ (i) (σ, θ) is a test function of degree i on the super circle X = T [1]S 1. The integrands of degree zero components of J 0 and J 1 are J (0)(f) (σ) = f(x(σ)), J (1)(X+α) (σ) = α i (x) σ x i (σ) + X i (x)p i (σ), which are the correct AS currents. We compute the Poisson algebra of these supergeometric currents 32

34 from the Poisson brackets of canonical quantities (x i, p i ), {J 0(f) (ϵ), J 0(g) (ϵ )} P B = 0, {J 1(X+α) (ϵ), J 0(g) (ϵ )} P B = ρ(x + α)j 0(g) (ϵϵ ), {J 1(X+α) (ϵ), J 1(Y +β) (ϵ )} P B = J 1([X+α,Y +β]h )(ϵϵ ) + µ dϵ (0) ϵ T [1]S 1 (0) X + α, Y + β (x), where J 0(g) = T [1]S 1 µϵ (1) g(x), J 1(Y +β) = T [1]S 1 µϵ (0) ( β i (x)dx i + Y i (x)p i ). The AS current algebra is given by the physical components of the supergeometric currents, i.e., degree zero 33

35 components, {J 0(f) (σ), J 0(g) (σ )} P B = 0, {J 1(X+α) (σ), J 0(g) (σ )} P B = ρ(x + α)j 0(g) (x(σ))δ(σ σ ), {J 1(X+α) (σ), J 1(Y +β) (σ )} P B = J 1([X+α,Y +β]h )(σ)δ(σ σ ) + (X + α), (Y + β) (σ ) σ δ(σ σ ). This coincides with the generalized current algebra described in Section 2. 34

36 Righthand side of the current algebra The current terms are twist of the pullback of {{j a, Θ}, j b }, where a, b = 0, 1. The anomaly terms are the pullback of {J a, J b }. Two terms are derived from the derived bracket on the mapping space, and twisting. {{J a, S 1 + S 0 }, J b }, 35

37 5. BFV-AKSZ Formalism of Current Algebras Data (X, D, µ) : Here X = T [1]Σ, where Σ R is an n dimensional manifold. D is a differential on X. µ is a D-invariant nondegenerate Berezin measure. (M, ω, Θ, L): A QP manifold of degree n and a Lagrangian submanifold. Assume {, } L = pr {{, Θ}, } is nondegenerate. We consider the induced symplectic structure ω L {, } L on L. defined from 36

38 Twisting by canonical 1-form on Lagrangian submanifold Let ϑ L be the canonical 1-form for ω L such that ω L = δϑ L. We define the function α 0 on the mapping space, Poisson bracket α 0 = ι ˆDµ ev pr ϑ s. {, } P B = pr e δ α 0{{, S}, }. is a graded Poisson bracket of degree 0, therefore, a normal Poisson bracket. 37

39 Current functions on target space M We consider a closed subalgebra of the structure sheaf C (M) not only under the Poisson bracket {, }, but also under the derived bracket {{, Θ}, }: C (n 1) (M) = n 1 i=0 C i(m) = {f C (M) f n 1} Currents The pullbacks of j i C (n 1) (M) to Map(T [1]Σ n 1, L), is a BFV current J i (ϵ) = pr e δ α 0µ ϵ ev j i, 38

40 where ϵ is a test function on T [1]Σ n 1 of degree n 1 i. CA (n 1) (X, M) = {J i }. 39

41 Supergeometic (BFV-AKSZ) Current Algebras The Poisson algebra on the space CA (n 1) (X, M) is a current algebra. Theorem 2. NI, Xu 13, (NI, Strobl) For currents J j1 and J j2 associated to current functions j 1, j 2 C (n 1) (M) respectively, the commutation relation is given by {J j1 (ϵ 1 ), J j2 (ϵ 2 )} P B = pr e δ α 0{{µ ϵ 1 ev j 1, S 1 + S 0 }, µ ϵ 2 ev j 2 } = J [j1,j 2 ] D (ϵ 1 ϵ 2 ) pr e δ α 0ι ˆDµ (dϵ 1 )ϵ 2 ev {j 1, j 2 } 40

42 where ϵ i are test functions for j i and [j 1, j 2 ] D = {{j 1, Θ}, j 2 } is the derived bracket. Due to the second term in the equation, it fails to be a Poisson algebra. The anomalous terms vanish if j 1 and j 2 commute, {j 1, j 2 } = 0. Therefore we have Corollary 1. If anomaly terms vanish for current functions on a subspace of commutative functions of C (n 1) (M) under the graded Poisson bracket {, }. 41

43 Physical Currents For a superfield, introduce the second degree, the form degree deg f, which is the order of θ µ. gh f = f deg f is called the ghost number. We define a physical current J cl = J ghf=0. Theorem 3. The ghost number zero components of the supergeometric current algebra gives the physical current algebra: {J j1,cl(ϵ 1 ), J j2,cl(ϵ 2 )} P.B = ( J [j1,j 2 ] D (ϵ 1 ϵ 2 ) pr e δ Ssι ˆDµ (dϵ 1 )ϵ 2 ev {j 1, j 2 } b gh=0 ), 42

44 6. Results Many known current algebras are included as a special case, such as Lie algebras (gauge currents), Kac-Moody algebras, Alekseev-Strobl types (algebroids), topological membranes, L -algebra, etc.,except for energy-moment tensors. We have constructed new current algebras with algebroid structures, which are generalizations of Alekseev-Strobl, Bonelli- Zabzine. Anomaly cancellation conditions are characterized in terms of supergeometry as a Lagrangian submanifold in the target space. 43

45 Noether currents of AKSZ sigma models are nontrivial examples of this current algebra. Duality between standard and Poisson Courant algebroid and Poisson Courant algebroid is reinterpreted in terms of the current algebra. Bessho, Heller, NI, Watamura 16 44

46 7. Membrane Current Algebras As one example, we construct a new current algebra of Alekseev- Strobl type on a dimensional manifold X 3 = Σ 2 R. Target space 1, Consider a QP manifold of degree 3, M = T [3]L, where L = T [2]E[1]. 45

47 QP Manifold of degree 3 Let us consider the classical phase space (x I (σ), qµ A (σ)), Map(Σ 2, T M) Map(T Σ 2, T E). This is the phase space for the physical models in 3 dimensions. We introduce T [3]M = T [3]T [2]E[1], a graded symplectic manifold of degree 3. Take local coordinates (x I, q A, p I ) of degree (0, 1, 2) on T [2]E[1], and conjugate Darboux coordinates (ξ I, η A, χ I ) of degree (3, 2, 1) on T [3]. 46

48 A graded symplectic structure on T [3]M is ω b = δx I δξ I k AB δq A δη B + δp I δχ I, where k AB is a fiber metric on E. We take the following Q-structure function Θ = χ I ξ I k ABη A η B + 1 4! H IJKL(x)χ I χ J χ K χ L. {Θ, Θ} = 0 if H is a closed 4-form and dk = 0. A Lagrangian submanifold L = T [2]E[1] spanned by (x I, q A, p I ). 47

49 Lie 3-Algebroid NI, Uchino 10, Grützmann 10 Let F = E T M and F = E T M. C 0 (M) = C (M). C 1 (M) = Γ(F ) which is spanned by degree one basis q A, χ I. C 2 (M) = Γ(F 2 F ) which is spanned by degree two basis p I, η A, q A q B, q A χ I, χ I χ J, etc. Let f C 0 (M), t, t 1, t 2, C 1 (M) and s, s 1, s 2, C 2 (M). A graded Poisson bracket induces a natural pairing Γ(F ) Γ(F 2 F ) C, and a bilinear 48

50 form on Γ(F 2 F ) Γ(F 2 F ) Γ(F ), t, s = {t, s}, s 1, s 2 = {s 1, s 2 }. Derived brackets C 1 C 1 C, a bilinear symmetric form on ΓF, (t 1, t 2 ) = {{t 1, Θ}, t 2 }. C 2 C 0 C 0, an anchor map ρ : F 2 F T M is ρ(s)f(x) = {{s, Θ}, f(x)}. 49

51 C 2 C 1 C 1, a Lie type derivative L : (F 2 F ) F F is L s t = {{s, Θ}, t}. C 2 C 2 C 2, a higher Dorfman bracket on ΓF 2 F, [s 1, s 2 ] 3 = {{s 1, Θ}, s 2 }. Definition 4. We impose {Θ, Θ} = 0. then (E,,, (, ), ρ, L, [, ] 3 ) is called a Lie 3-algebroid. 50

52 Superfields (x I (σ, θ), q A (σ, θ), p I (σ, θ)) and (ξ I (σ, θ), η A (σ, θ), χ I (σ, θ) are corresponding superfields. Poisson brackets Derived bracket {{x I, Θ}, p J } = δ I J, {{q A, Θ}, q B } = k AB, {{p I, Θ}, p J } = 1 2 H IJKL(x)χ K χ L. 51

53 A canonical 1-form is α 0 = X µ (δxi δp I k ABδq A δq B ). These derive the Poisson brackets for canonical conjugates {x I (σ, θ), p J (σ, θ )} P B = δ I Jδ 2 (σ σ )δ 2 (θ θ ), {q A (σ, θ), q B (σ, θ )} P B = k AB δ 2 (σ σ )δ 2 (θ θ ), {p I (σ), p J (σ )} P B = 1 2 H IJKLdx K dx L δ 2 (σ σ )δ 2 (θ θ ), 52

54 Currents Let us consider the space of functions of degree equal to or less than 2. C 2 (T [3]T [2]E[1]) = {f C (T [3]T [2]E[1]) f 2}. General functions of C 2 (T [3]T [2]E[1]) of degree 0, 1 and 2 are J (0)(f) = f(x), J (1)(a,u) = a I (x)χ I + u A (x)q A, J (2)(G,K,F,B,E) = G I (x)p I + K A (x)η A F AB(x)q A q B B IJ(x)χ I χ J + E AI (x)χ I q A. 53

55 Next we pullback and twist current functions with canonical 1-form, α 0 = T [1]Σ 2 µ ( p I dx I k ABq A dq B). Currents of degree 0, 1 and 2 are J (0)(f) = J (1)(a,u) = J (2)(G,K,F,B,E) (σ, θ) = µϵ (2) f(x), T [1]Σ 2 µϵ (1) (a I (x)dx I + u A (x)q A ), T [1]Σ 2 µϵ (0) (G I (x)p I + K A (x)dq A T [1]Σ F AB(x)q A q B B IJ(x)dx I dx J + E AI (x)dx I q A ). 54

56 The current algebra is as follows: {J (0)(f) (ϵ), J (0)(f )(ϵ )} P.B. = 0, {J (1)(u,a) (ϵ), J (0)(f )(ϵ )} P.B. = 0, {J (2)(G,K,F,H,E) (ϵ), J (0)(f )(ϵ )} P.B. = G I J (0)(f ) x I (ϵϵ ), {J (1)(u,a) (ϵ), J (1)(u,a )(ϵ )} P.B. = µϵ (1) ϵ (1) ev k AB u A u B, T [1]Σ 2 {J (2)(G,K,F,B,E) (ϵ), J (1)(u,a )(ϵ )} P.B. = J (1)(ū,ᾱ) (ϵϵ ) µ(dϵ (0) )ϵ (1) (GI α I k AB K A u B), T [1]Σ 2 {J (2)(G,K,F,B,E) (ϵ), J (2)(G,K,F,B,E )(ϵ )} P.B. = J (2)( Ḡ, K, F, B,Ē) (ϵϵ ) µ(dϵ (0) )ϵ [ (0) (G J B JI + G J B JI + k AB (K A E BI + E AI K B))dx I T [1]Σ 2 +(G I E AI + G I E AI + k BC (K B F AC + F AC K B))q A]. 55

57 Here ᾱ = (i G d + di G )α + E dk, u, ū = i G du + F, u, Ḡ = [G, G ], K = i G dk i G dk + i G E + F, K, F = i G df i G df + F, F, B = (di G + i G d)b i G db + E, E + K, de dk, E + i G i G H, Ē = (di G + i G d)e i G de + E, F E, F + df, K dk, F, where all the terms are evaluated by σ. Here [, ] is a Lie bracket on T M, i G is an interior product with respect to a vector field G and, is the graded bilinear form on the fiber of E with respect to the metric k AB. 56

58 The condition vanishing anomalous terms, G I α I kab K A u B = 0, G J B JI + G J B JI + k AB (K A E BI + E AIK B ) = 0 and GI E AI + G I E AI + k BC (K B F AC + F ACK B ) = 0 is equivalent that J (i) s are commutative. The classical current algebra is the ghost number zero component 57

59 of superfields: {J (0)(f) cl (ϵ), J (0)(f ) cl (ϵ )} P.B. = 0, {J (1)(u,a) cl (ϵ), J (0)(f ) cl (ϵ )} P.B. = 0, {J (2)(G,K,F,H,E) cl (ϵ), J (0)(f ) cl (ϵ )} P.B. = G I J (0)(f ) cl x I (ϵϵ ), {J (1)(u,a) cl (ϵ), J (1)(u,a ) cl (ϵ (1) )} P.B. = (1) ϵ cl (1) (1) ev k AB u A u B cl, Σ 2 ϵ cl {J (2)(G,K,F,B,E) cl (ϵ), J (1)(u,a ) cl (ϵ )} P.B. = J (1)(ū,ᾱ) cl (ϵϵ (1) ) dϵ cl (0) ϵ cl (1) (GI α I k AB K A u B) cl, Σ 2 {J (2)(G,K,F,B,E) cl (ϵ), J (2)(G,K,F,B,E ) cl (ϵ )} P.B. = J (2)( Ḡ, K, F, B,Ē) cl (ϵϵ ) [ dϵ cl (0)ϵ cl (0) (G J B JI + G J B JI + k AB (K A E BI + E AI K B)) cl dx I Σ 2 +(G I E AI + G I E AI + k BC (K B F AC + F AC K B)) cl q A], 58

60 where ϵ cl (i) (i) = 1 i! dxµ 1 dx µ iϵ (i)µ1 µ i (σ) and q A = dx µ q A µ. Note: This current algebra includes the Chern-Simons theory with matter, the Courant sigma model, etc. 59

61 . Conclusions and Future Outlook We have proposed a new formulation of current algebras based on the BFV-AKSZ formalism. A derived bracket and a canonical transformation on a Lagrangian submanifold of a graded symplectic manifold derive a current algebra.. Outlook Generalizations to currents with higher derivatives Zabzine 11 Nonassociative generalizations, Nambu-Poisson structure Applications to string theory and M-theory Ekstrand, 60

62 Quantization problem Deformation, Geometric, Path integral anomaly terms in current algebras. Poisson vertex algebras and vertex algebras Wakimoto 10, Ekstrand, Heluani, Zabzine 11, Hekmati, Mathai 12 higher structure WZW models? Li 02, Sole, Kac, BV Algebras in String Field Theory Hata, Zwiebach 93 Higher groupoids and higher category 61

63 Thank you! 62