A-infinity GL(N)-equivariant matrix integrals-I

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A-infinity GL(N)-equivariant matrix integrals-I

Serguei Barannikov

To cite this version:

Serguei Barannikov. A-infinity GL(N)-equivariant matrix integrals-I. Topological String Theory,

Mod-ularity & non-perturbative Physics, Jun 2010, Vienna, Austria. �hal-00493919�

A in…nity GL

(

N

)

equivariant matrix integrals-I

Serguei Barannikov

IMJ, CNRS

21/06/2010

hal-00493919, version 1 - 21 Jun 2010

Author manuscript, published in "Topological String Theory, Modularity & non-perturbative Physics, Vienna : Austria (2010)"

The noncommutative Batalin-Vilkovisky equation and

A-in…nity algebras

U -Z/2Z graded vector space/C, l-scalar product on U of degree d₂Z/2Z, (variant: Z graded), consider

F =Symm(C_λ[1+d]), C_λ= ∞_{j =0}(U[1] j)Z/jZ

-the symmetric (resp. exterior) powers for odd (resp even) d , of cyclic tensors

([B1],2006)∆ : F !F[1],∆2 =0, de…ned via dissection-gluing of cyclic tensors, of the second order w.r.t. product of cycles.

The noncommutative Batalin-Vilkovisky equation (nc-BV) h∆S+1 2fS, Sg =0, S =_g

∑

_0,i h 2g 1+i_S g ,i, Sg ,i 2Symmi(Cλ[1+d]), nc-BV_,∆ exp(S / h) =0 fS0,1, S0,1g =0,

V =U_, and S0,1 =mA∞ is A∞ algebra structure on V with invariant

scalar product of degree d

A_∞ algebras without scalar product are included in the formalism by setting U =A A_[d], giving an A_∞ algebra with scalar product.

The noncommutative Batalin-Vilkovisky equation and

A-in…nity algebras

U -Z/2Z graded vector space/C, l-scalar product on U of degree d₂Z/2Z, (variant: Z graded), consider

F =Symm(C_λ[1+d]), C_λ= ∞_{j =0}(U[1] j)Z/jZ

-the symmetric (resp. exterior) powers for odd (resp even) d , of cyclic tensors ([B1],2006)∆ : F _!F[1],∆2 =0, de…ned via dissection-gluing of cyclic tensors, of the second order w.r.t. product of cycles.

The noncommutative Batalin-Vilkovisky equation (nc-BV) h∆S+1 2fS, Sg =0, S =_g

∑

_0,i h 2g 1+i_S g ,i, Sg ,i 2Symmi(Cλ[1+d]), nc-BV_,∆ exp(S / h) =0 fS0,1, S0,1g =0,

V =U_, and S0,1 =mA∞ is A∞ algebra structure on V with invariant

scalar product of degree d

A_∞ algebras without scalar product are included in the formalism by setting U =A A_[d], giving an A_∞ algebra with scalar product.

The noncommutative Batalin-Vilkovisky equation and

A-in…nity algebras

U -Z/2Z graded vector space/C, l-scalar product on U of degree d₂Z/2Z, (variant: Z graded), consider

F =Symm(C_λ[1+d]), C_λ= ∞_{j =0}(U[1] j)Z/jZ

-the symmetric (resp. exterior) powers for odd (resp even) d , of cyclic tensors ([B1],2006)∆ : F _!F[1],∆2 =0, de…ned via dissection-gluing of cyclic tensors, of the second order w.r.t. product of cycles.

The noncommutative Batalin-Vilkovisky equation (nc-BV) h∆S+1 2fS, Sg =0, S =_g

∑

_0,i h 2g 1+i_S g ,i, Sg ,i 2Symmi(Cλ[1+d]), nc-BV_,∆ exp(S / h) =0 fS0,1, S0,1g =0,

V =U_, and S0,1 =mA∞ is A∞ algebra structure on V with invariant

scalar product of degree d

A_∞ algebras without scalar product are included in the formalism by setting U =A A_[d], giving an A_∞ algebra with scalar product.

The noncommutative Batalin-Vilkovisky equation and

A-in…nity algebras

U -Z/2Z graded vector space/C, l-scalar product on U of degree d₂Z/2Z, (variant: Z graded), consider

F =Symm(C_λ[1+d]), C_λ= ∞_{j =0}(U[1] j)Z/jZ

-the symmetric (resp. exterior) powers for odd (resp even) d , of cyclic tensors ([B1],2006)∆ : F _!F[1],∆2 =0, de…ned via dissection-gluing of cyclic tensors, of the second order w.r.t. product of cycles.

The noncommutative Batalin-Vilkovisky equation (nc-BV) h∆S+1 2fS, Sg =0, S =_g

∑

_0,i h 2g 1+i_S g ,i, Sg ,i 2Symmi(Cλ[1+d]), nc-BV_,∆ exp(S / h) =0 fS0,1, S0,1g =0,

V =U_, and S0,1 =mA∞ is A∞ algebra structure on V with invariant

scalar product of degree d

A_∞ algebras without scalar product are included in the formalism by setting U =A A_[d], giving an A_∞ algebra with scalar product.

The noncommutative Batalin-Vilkovisky equation and

A-in…nity algebras

U -Z/2Z graded vector space/C, l-scalar product on U of degree d₂Z/2Z, (variant: Z graded), consider

F =Symm(C_λ[1+d]), C_λ= ∞_{j =0}(U[1] j)Z/jZ

-the symmetric (resp. exterior) powers for odd (resp even) d , of cyclic tensors ([B1],2006)∆ : F _!F[1],∆2 =0, de…ned via dissection-gluing of cyclic tensors, of the second order w.r.t. product of cycles.

The noncommutative Batalin-Vilkovisky equation (nc-BV) h∆S+1 2fS, Sg =0, S =_g

∑

_0,i h 2g 1+i_S g ,i, Sg ,i 2Symmi(Cλ[1+d]), nc-BV_,∆ exp(S / h) =0 fS0,1, S0,1g =0, _

A_∞ algebras without scalar product are included in the formalism by setting U =A A_[d], giving an A_∞ algebra with scalar product.

The noncommutative Batalin-Vilkovisky equation and

A-in…nity algebras

U -Z/2Z graded vector space/C, l-scalar product on U of degree d₂Z/2Z, (variant: Z graded), consider

F =Symm(C_λ[1+d]), C_λ= ∞_{j =0}(U[1] j)Z/jZ

-the symmetric (resp. exterior) powers for odd (resp even) d , of cyclic tensors ([B1],2006)∆ : F _!F[1],∆2 =0, de…ned via dissection-gluing of cyclic tensors, of the second order w.r.t. product of cycles.

The noncommutative Batalin-Vilkovisky equation (nc-BV) h∆S+1 2fS, Sg =0, S =_g

∑

_0,i h 2g 1+i_S g ,i, Sg ,i 2Symmi(Cλ[1+d]), nc-BV_,∆ exp(S / h) =0 fS0,1, S0,1g =0,

V =U_, and S0,1 =mA_∞ is A∞ algebra structure on V with invariant scalar product of degree d

A_∞ algebras without scalar product are included in the formalism by setting U =A A_[d], giving an A_∞ algebra with scalar product.

The A-in…nity equivariant matrix integrals ([B2],09/2006)

([B2],09/2006) element S₂Symm(C_λ[1+d_{]) !}matrix integral

Z

exp bS(X ,Λ)dX

X 2gl(NjN) V[1]in the odd d case, X 2q(N) V[1]in the even d case,

If S satis…es nc-BV equation then

(∆matrix+igl)exp bS(X ,Λ) =0 ,exp bS(X ,Λ)dX is gl equivariantly closed di¤erential form.

In the case of the algebra 1 1=1, - solution to nc BV for V = f1g, this is the matrix Airy integralR exp(1₆Tr(Y3) 1₂Tr(ΛY2))dY

This is the higher genus counterpart of the (nc)Hodge theory integration on CY projective manifolds, ( h∆γ+∂γ+1₂[γ, γ] =0, γ2Ω0, (M,ΛT))

S satis…es nc-BV, asymptotic expansion asΛ_!∞ -sum over stable ribbon graphs₎cohomology classes in H _(MK_{g ,n})(in H _(MK_{g ,n},_L) for odd d ) My A_∞equivariant matrix integrals de…ne an integration framework in the noncommutative (derived algebraic) geometry, particularly adobted to the equation_fmA_∞, mA_∞g =0

Z

The A-in…nity equivariant matrix integrals ([B2],09/2006)

([B2],09/2006) element S₂Symm(C_λ[1+d_{]) !}matrix integral

Z

exp bS(X ,Λ)dX

X 2gl(NjN) V[1]in the odd d case, X 2q(N) V[1]in the even d case, If S satis…es nc-BV equation then

(∆matrix+igl)exp bS(X ,Λ) =0 ,exp bS(X ,Λ)dX is gl equivariantly closed di¤erential form.

In the case of the algebra 1 1=1, - solution to nc BV for V = f1g, this is the matrix Airy integralR exp(1₆Tr(Y3) 1₂Tr(ΛY2))dY

This is the higher genus counterpart of the (nc)Hodge theory integration on CY projective manifolds, ( h∆γ+∂γ+1₂[γ, γ] =0, γ2Ω0, (M,ΛT))

S satis…es nc-BV, asymptotic expansion asΛ_!∞ -sum over stable ribbon graphs₎cohomology classes in H _(MK_{g ,n})(in H _(MK_{g ,n},_L) for odd d ) My A_∞equivariant matrix integrals de…ne an integration framework in the noncommutative (derived algebraic) geometry, particularly adobted to the equation_fmA_∞, mA_∞g =0

Z

exp bS(X ,Λ)bϕdX , ϕ2Ker(h∆_{+ f}S, g) Symm(C_λ[1+d])

The A-in…nity equivariant matrix integrals ([B2],09/2006)

([B2],09/2006) element S₂Symm(C_λ[1+d_{]) !}matrix integral

Z

exp bS(X ,Λ)dX

X 2gl(NjN) V[1]in the odd d case, X 2q(N) V[1]in the even d case, If S satis…es nc-BV equation then

(∆matrix+igl)exp bS(X ,Λ) =0 ,exp bS(X ,Λ)dX is gl equivariantly closed di¤erential form.

In the case of the algebra 1 1=1, - solution to nc BV for V = f1g, this is the matrix Airy integralR exp(1₆Tr(Y3) 1₂Tr(ΛY2))dY

This is the higher genus counterpart of the (nc)Hodge theory integration on CY projective manifolds, ( h∆γ+∂γ+1₂[γ, γ] =0, γ2Ω0, (M,ΛT))

S satis…es nc-BV, asymptotic expansion asΛ_!∞ -sum over stable ribbon graphs₎cohomology classes in H _(MK_{g ,n})(in H _(MK_{g ,n},_L) for odd d ) My A_∞equivariant matrix integrals de…ne an integration framework in the noncommutative (derived algebraic) geometry, particularly adobted to the equation_fmA_∞, mA_∞g =0

Z

The A-in…nity equivariant matrix integrals ([B2],09/2006)

([B2],09/2006) element S₂Symm(C_λ[1+d_{]) !}matrix integral

Z

exp bS(X ,Λ)dX

X 2gl(NjN) V[1]in the odd d case, X 2q(N) V[1]in the even d case, If S satis…es nc-BV equation then

(∆matrix+igl)exp bS(X ,Λ) =0 ,exp bS(X ,Λ)dX is gl equivariantly closed di¤erential form.

In the case of the algebra 1 1=1, - solution to nc BV for V = f1g, this is the matrix Airy integralR exp(1₆Tr(Y3) 1₂Tr(ΛY2))dY

This is the higher genus counterpart of the (nc)Hodge theory integration on CY projective manifolds, ( h∆γ+∂γ+1₂[γ, γ] =0, γ2Ω0, (M,ΛT))

S satis…es nc-BV, asymptotic expansion asΛ_!∞ -sum over stable ribbon graphs₎cohomology classes in H _(MK_{g ,n})(in H _(MK_{g ,n},_L) for odd d ) My A_∞equivariant matrix integrals de…ne an integration framework in the noncommutative (derived algebraic) geometry, particularly adobted to the equation_fmA_∞, mA_∞g =0

Z

exp bS(X ,Λ)bϕdX , ϕ2Ker(h∆_{+ f}S, g) Symm(C_λ[1+d])

The A-in…nity equivariant matrix integrals ([B2],09/2006)

([B2],09/2006) element S₂Symm(C_λ[1+d_{]) !}matrix integral

Z

exp bS(X ,Λ)dX

X 2gl(NjN) V[1]in the odd d case, X 2q(N) V[1]in the even d case, If S satis…es nc-BV equation then

(∆matrix+igl)exp bS(X ,Λ) =0 ,exp bS(X ,Λ)dX is gl equivariantly closed di¤erential form.

In the case of the algebra 1 1=1, - solution to nc BV for V = f1g, this is the matrix Airy integralR exp(1₆Tr(Y3) 1₂Tr(ΛY2))dY

This is the higher genus counterpart of the (nc)Hodge theory integration on CY projective manifolds, ( h∆γ+∂γ+1₂[γ, γ] =0, γ2Ω0, (M,ΛT))

S satis…es nc-BV, asymptotic expansion asΛ_!∞ -sum over stable ribbon graphs₎cohomology classes in H _(MK_{g ,n})(in H _(MK_{g ,n},_L) for odd d )

My A_∞equivariant matrix integrals de…ne an integration framework in the noncommutative (derived algebraic) geometry, particularly adobted to the equation_fmA_∞, mA_∞g =0

Z

The A-in…nity equivariant matrix integrals ([B2],09/2006)

([B2],09/2006) element S₂Symm(C_λ[1+d_{]) !}matrix integral

Z

exp bS(X ,Λ)dX

X 2gl(NjN) V[1]in the odd d case, X 2q(N) V[1]in the even d case, If S satis…es nc-BV equation then

(∆matrix+igl)exp bS(X ,Λ) =0 ,exp bS(X ,Λ)dX is gl equivariantly closed di¤erential form.

In the case of the algebra 1 1=1, - solution to nc BV for V = f1g, this is the matrix Airy integralR exp(1₆Tr(Y3) 1₂Tr(ΛY2))dY

This is the higher genus counterpart of the (nc)Hodge theory integration on CY projective manifolds, ( h∆γ+∂γ+1₂[γ, γ] =0, γ2Ω0, (M,ΛT))

S satis…es nc-BV, asymptotic expansion asΛ_!∞ -sum over stable ribbon graphs₎cohomology classes in H _(MK_{g ,n})(in H _(MK_{g ,n},_L) for odd d ) My A_∞equivariant matrix integrals de…ne an integration framework in the noncommutative (derived algebraic) geometry, particularly adobted to the equation_fmA_∞, mA_∞g =0

Z

exp bS(X ,Λ)_{bϕdX , ϕ2}Ker(h∆_{+ f}S, g) Symm(C_λ[1+d])

Noncommutative Batalin-Vilkovisky operator ([B1])

I de…ne my noncommutative BV di¤erential on Symm(C_λ[1+d])via

∆(xρ1. . . xρr)λ(xτ1. . . xτt)λ= =

∑

p,q ( 1)ε_l ρpτq(xρ1. . . xρp 1xτq+1. . . xτq 1xρp+1. . . xρr)λ+

∑

p 16=q ( 1)eεlρpρq(xρ1. . . xρp 1xρq+1. . . xρr)λ(xρp+1. . . xρq 1)λ(xτ1. . . xτt)λ

∑

p 16=q ( 1)eeεl_τpτq(xρ1. . . xρr)λ(xτ1. . . xτp 1xτq+1. . . xτt)λ(xτp+1. . . xτq 1)λ l_ρ pρq =l(xρp, xτq)

signs are the standard Koszul signs taking into account that (x_ρ1. . . xρr)λ= (1+d) +∑ xρi, xi 2V[1].

Noncommutative Batalin-Vilkovisky operator ([B1])

I de…ne my noncommutative BV di¤erential on Symm(C_λ[1+d])via

∆(xρ1. . . xρr)λ(xτ1. . . xτt)λ= =

∑

p,q ( 1)ε_l ρpτq(xρ1. . . xρp 1xτq+1. . . xτq 1xρp+1. . . xρr)λ+

∑

p 16=q ( 1)eεlρpρq(xρ1. . . xρp 1xρq+1. . . xρr)λ(xρp+1. . . xρq 1)λ(xτ1. . . xτt)λ

∑

p 16=q ( 1)eeεl_τpτq(xρ1. . . xρr)λ(xτ1. . . xτp 1xτq+1. . . xτt)λ(xτp+1. . . xτq 1)λ l_ρ pρq =l(xρp, xτq)

signs are the standard Koszul signs taking into account that (x_ρ1. . . xρr)λ= (1+d) +∑ xρi, xi 2V[1].

The cyclic tensors, invariant functions and the matrix

algebra with odd trace.

Invariant theory:

Symm(C_λ) !Symm((gl(NjNe) V[1])_)GL(N j eN )

∏

(a1. . . ak)λ !

∏

sTr(A1 . . . Ak)

This is an isomorphism in degrees N, it was at the origin of the discovery of cyclic homology, cyclic di¤erential_$Lie cohomology di¤erential of gl(V). To relate this with the nc-BV equation, one needs to solve the problem: for usual algebras (i.e. with scalar product of degree d=0) this is the wrong space: the symmetric instead of the exterior powers of cyclic tensors Solution: there must be a matrix algebra with odd trace:

tr(A1 . . . Ak) =1+ΣAi Such algebra exists:

q(N) = f[X , π] =0jX 2gl(NjN)g where π= 0 1N 1N 0 is an odd isomorphism, π 2_{= 1,} q(N) = X = A B B A

The cyclic tensors, invariant functions and the matrix

algebra with odd trace.

Invariant theory:

Symm(C_λ) !Symm((gl(NjNe) V[1])_)GL(N j eN )

∏

(a1. . . ak)λ !

∏

sTr(A1 . . . Ak)

This is an isomorphism in degrees N, it was at the origin of the discovery of cyclic homology, cyclic di¤erential_$Lie cohomology di¤erential of gl(V).

To relate this with the nc-BV equation, one needs to solve the problem: for usual algebras (i.e. with scalar product of degree d=0) this is the wrong space: the symmetric instead of the exterior powers of cyclic tensors Solution: there must be a matrix algebra with odd trace:

tr(A1 . . . Ak) =1+ΣAi Such algebra exists:

q(N) = f[X , π] =0jX 2gl(NjN)g where π= 0 1N 1N 0 is an odd isomorphism, π 2_{= 1,} q(N) = X = A B B A

The cyclic tensors, invariant functions and the matrix

algebra with odd trace.

Invariant theory:

Symm(C_λ) !Symm((gl(NjNe) V[1])_)GL(N j eN )

∏

(a1. . . ak)λ !

∏

sTr(A1 . . . Ak)

This is an isomorphism in degrees N, it was at the origin of the discovery of cyclic homology, cyclic di¤erential_$Lie cohomology di¤erential of gl(V). To relate this with the nc-BV equation, one needs to solve the problem: for usual algebras (i.e. with scalar product of degree d=0) this is the wrong space: the symmetric instead of the exterior powers of cyclic tensors

Solution: there must be a matrix algebra with odd trace: tr(A1 . . . Ak) =1+ΣAi

Such algebra exists:

q(N) = f[X , π] =0jX 2gl(NjN)g where π= 0 1N 1N 0 is an odd isomorphism, π 2_{= 1,} q(N) = X = A B B A

The cyclic tensors, invariant functions and the matrix

algebra with odd trace.

Invariant theory:

Symm(C_λ) !Symm((gl(NjNe) V[1])_)GL(N j eN )

∏

(a1. . . ak)λ !

∏

sTr(A1 . . . Ak)

This is an isomorphism in degrees N, it was at the origin of the discovery of cyclic homology, cyclic di¤erential_$Lie cohomology di¤erential of gl(V). To relate this with the nc-BV equation, one needs to solve the problem: for usual algebras (i.e. with scalar product of degree d=0) this is the wrong space: the symmetric instead of the exterior powers of cyclic tensors Solution: there must be a matrix algebra with odd trace:

tr(A1 . . . Ak) =1+ΣAi

Such algebra exists:

q(N) = f[X , π] =0jX 2gl(NjN)g where π= 0 1N 1N 0 is an odd isomorphism, π 2_{= 1,} q(N) = X = A B B A

The cyclic tensors, invariant functions and the matrix

algebra with odd trace.

Invariant theory:

Symm(C_λ) !Symm((gl(NjNe) V[1])_)GL(N j eN )

∏

(a1. . . ak)λ !

∏

sTr(A1 . . . Ak)

This is an isomorphism in degrees N, it was at the origin of the discovery of cyclic homology, cyclic di¤erential_$Lie cohomology di¤erential of gl(V). To relate this with the nc-BV equation, one needs to solve the problem: for usual algebras (i.e. with scalar product of degree d=0) this is the wrong space: the symmetric instead of the exterior powers of cyclic tensors Solution: there must be a matrix algebra with odd trace:

tr(A1 . . . Ak) =1+ΣAi Such algebra exists:

q(N) = f[X , π] =0jX 2gl(NjN)g where π= 0 1N

1N 0 is an odd isomorphism, π

2_{= 1,}

Algebra q(N) and nc-BV equation

q(N)has odd trace

otr A B

B A =tr(B) otr([X1, X2]) =0

q(N)is a simpleZ/2Z graded associative algebra The map

Symm(ΠC_{λ) !}Symm((q(N) V[1])_)Q (N )

∏

(a1. . . ak)_λ!

∏

otr(A1 . . . Ak) is an isomorphism in degrees N

Theorem([B4]): nc-BV di¤erential∆ on Symm(ΠC_λ)is identi…ed with Q(N) invariant odd BV-operator on (q(N) V[1])_ _{corresponding to the} odd a¢ ne symplectic structure de…ned by otr(X1X2) l

Corollary: tensor multiplication by q(N), gl(NjNe) !super Morita equivalence on solutions to nc-BV.

Algebra q(N) and nc-BV equation

q(N)has odd trace

otr A B

B A =tr(B) otr([X1, X2]) =0 q(N)is a simpleZ/2Z graded associative algebra

The map

Symm(ΠC_{λ) !}Symm((q(N) V[1])_)Q (N )

∏

(a1. . . ak)_λ!

∏

otr(A1 . . . Ak) is an isomorphism in degrees N

Theorem([B4]): nc-BV di¤erential∆ on Symm(ΠC_λ)is identi…ed with Q(N) invariant odd BV-operator on (q(N) V[1])_ _{corresponding to the} odd a¢ ne symplectic structure de…ned by otr(X1X2) l

Corollary: tensor multiplication by q(N), gl(NjNe) !super Morita equivalence on solutions to nc-BV.

Algebra q(N) and nc-BV equation

q(N)has odd trace

otr A B

B A =tr(B) otr([X1, X2]) =0 q(N)is a simpleZ/2Z graded associative algebra The map

Symm(ΠC_{λ) !}Symm((q(N) V[1])_)Q (N )

∏

(a1. . . ak)_λ!

∏

otr(A1 . . . Ak) is an isomorphism in degrees N

Theorem([B4]): nc-BV di¤erential∆ on Symm(ΠC_λ)is identi…ed with Q(N) invariant odd BV-operator on (q(N) V[1])_ _{corresponding to the} odd a¢ ne symplectic structure de…ned by otr(X1X2) l

Corollary: tensor multiplication by q(N), gl(NjNe) !super Morita equivalence on solutions to nc-BV.

Algebra q(N) and nc-BV equation

q(N)has odd trace

otr A B

B A =tr(B) otr([X1, X2]) =0 q(N)is a simpleZ/2Z graded associative algebra The map

Symm(ΠC_{λ) !}Symm((q(N) V[1])_)Q (N )

∏

(a1. . . ak)_λ!

∏

otr(A1 . . . Ak) is an isomorphism in degrees N

Theorem([B4]): nc-BV di¤erential∆ on Symm(ΠC_λ)is identi…ed with Q(N) invariant odd BV-operator on (q(N) V[1])_ corresponding to the odd a¢ ne symplectic structure de…ned by otr(X1X2) l

Corollary: tensor multiplication by q(N), gl(NjNe) !super Morita equivalence on solutions to nc-BV.

Algebra q(N) and nc-BV equation

q(N)has odd trace

otr A B

B A =tr(B) otr([X1, X2]) =0 q(N)is a simpleZ/2Z graded associative algebra The map

Symm(ΠC_{λ) !}Symm((q(N) V[1])_)Q (N )

∏

(a1. . . ak)_λ!

∏

otr(A1 . . . Ak) is an isomorphism in degrees N

Theorem([B4]): nc-BV di¤erential∆ on Symm(ΠC_λ)is identi…ed with Q(N) invariant odd BV-operator on (q(N) V[1])_ corresponding to the odd a¢ ne symplectic structure de…ned by otr(X1X2) l

Corollary: tensor multiplication by q(N), gl(NjNe) !super Morita equivalence on solutions to nc-BV.

Integration of polyvectors - the (equivariant) BV-formalism

S 2Symm(ΠC_{λ) !}Sb2Symm((q(N) V[1])_₎Q (N )_{, superfunction on} a¢ ne BV manifold

b

S- polyvector …eld on the even part(q(N) V[1])0

canonically de…ned, up to a sign, a¢ ne holomorphic volume element dX on (q(N) V[1])0

polyvector bS _!di¤erential form

action by the super-Lie algebra[Λ, _{] !}extension to gl equivariantly closed di¤erential form.

Integration of polyvectors - the (equivariant) BV-formalism

S 2Symm(ΠC_{λ) !}Sb2Symm((q(N) V[1])_₎Q (N )_{, superfunction on} a¢ ne BV manifold

b

S- polyvector …eld on the even part(q(N) V[1])0

canonically de…ned, up to a sign, a¢ ne holomorphic volume element dX on (q(N) V[1])0

polyvector bS _!di¤erential form

action by the super-Lie algebra[Λ, _{] !}extension to gl equivariantly closed di¤erential form.

Integration of polyvectors - the (equivariant) BV-formalism

S 2Symm(ΠC_{λ) !}Sb2Symm((q(N) V[1])_₎Q (N )_{, superfunction on} a¢ ne BV manifold

b

S- polyvector …eld on the even part(q(N) V[1])0

canonically de…ned, up to a sign, a¢ ne holomorphic volume element dX on (q(N) V[1])0

polyvector bS _!di¤erential form

action by the super-Lie algebra[Λ, _{] !}extension to gl equivariantly closed di¤erential form.

Integration of polyvectors - the (equivariant) BV-formalism

S 2Symm(ΠC_{λ) !}Sb2Symm((q(N) V[1])_₎Q (N )_{, superfunction on} a¢ ne BV manifold

b

S- polyvector …eld on the even part(q(N) V[1])0

canonically de…ned, up to a sign, a¢ ne holomorphic volume element dX on (q(N) V[1])0

polyvector bS _!di¤erential form

action by the super-Lie algebra[Λ, _{] !}extension to gl equivariantly closed di¤erential form.

Integration of polyvectors - the (equivariant) BV-formalism

S 2Symm(ΠC_{λ) !}Sb2Symm((q(N) V[1])_₎Q (N )_{, superfunction on} a¢ ne BV manifold

b

S- polyvector …eld on the even part(q(N) V[1])0

canonically de…ned, up to a sign, a¢ ne holomorphic volume element dX on (q(N) V[1])0

polyvector bS _!di¤erential form

action by the super-Lie algebra[Λ, _{] !}extension to gl equivariantly closed di¤erential form.

New integration framework in the noncommutative

(derived algebraic) geometry,

A.Connes: integration theory based on maps C_λ!di¤erential forms-"cycles" on HC (A) ,(Ω, d,R),

τ(a0, . . . , an) =

Z

a0da1. . . dan (closely related Karoubi’s nc-De Rham complex)

My nc-BV formalism is an integration framework in the noncommutative geometry, based on maps to polyvectors rather than di¤erential forms, which is particularly adobted to the equation_fmA_∞, mA_∞g =0 of derived

nc-algebraic geometry, variant: _fS, S_{g =}0

Z

exp bS(X ,Λ)bϕdX

ϕ2Ker(h∆+ fS, g +∆S), ϕ2Symm(C_λ[1 d])

Extends to non CY via V =A A_[d]etc.

Invariance with respect to A_∞ gauge transformation (and more general gauge transformation)

New integration framework in the noncommutative

(derived algebraic) geometry,

A.Connes: integration theory based on maps C_λ!di¤erential forms-"cycles" on HC (A) ,(Ω, d,R),

τ(a0, . . . , an) =

Z

a0da1. . . dan (closely related Karoubi’s nc-De Rham complex)

My nc-BV formalism is an integration framework in the noncommutative geometry, based on maps to polyvectors rather than di¤erential forms, which is particularly adobted to the equation_fmA_∞, mA_∞g =0 of derived

nc-algebraic geometry, variant: _fS, S_{g =}0

Z

exp bS(X ,Λ)bϕdX

ϕ2Ker(h∆+ fS, g +∆S), ϕ2Symm(C [1 d])

Extends to non CY via V =A A_[d]etc.

Invariance with respect to A_∞ gauge transformation (and more general gauge transformation)

New integration framework in the noncommutative

(derived algebraic) geometry,

A.Connes: integration theory based on maps C_λ!di¤erential forms-"cycles" on HC (A) ,(Ω, d,R),

τ(a0, . . . , an) =

Z

a0da1. . . dan (closely related Karoubi’s nc-De Rham complex)

My nc-BV formalism is an integration framework in the noncommutative geometry, based on maps to polyvectors rather than di¤erential forms, which is particularly adobted to the equation_fmA_∞, mA_∞g =0 of derived

nc-algebraic geometry, variant: _fS, S_{g =}0

Z

exp bS(X ,Λ)bϕdX

ϕ2Ker(h∆+ fS, g +∆S), ϕ2Symm(Cλ[1 d])

Extends to non CY via V =A A_[d]etc.

Invariance with respect to A_∞ gauge transformation (and more general gauge transformation)

New integration framework in the noncommutative

(derived algebraic) geometry,

A.Connes: integration theory based on maps C_λ!di¤erential forms-"cycles" on HC (A) ,(Ω, d,R),

τ(a0, . . . , an) =

Z

a0da1. . . dan (closely related Karoubi’s nc-De Rham complex)

My nc-BV formalism is an integration framework in the noncommutative geometry, based on maps to polyvectors rather than di¤erential forms, which is particularly adobted to the equation_fmA_∞, mA_∞g =0 of derived

nc-algebraic geometry, variant: _fS, S_{g =}0

Z

exp bS(X ,Λ)bϕdX

ϕ2Ker(h∆+ fS, g +∆S), ϕ2Symm(Cλ[1 d])

Extends to non CY via V =A A_[d]etc.

Example: V={e} and matrix Airy integral

V _{= f}e_g, e e=e, l_(e) =1,_!potential(ξ, ξ, ξ)λ

on(q ΠV)0 this gives the nonhomogenious polyvector otr(Ξ3), action by[Λ, ],Λ₂qodd, !extension of exp_3!1otr(Ξ3)dX to equivariantly closed di¤erential form exp_3!1otr(Ξ3) + ₂1otr([Λ, Ξ],Ξ)dX its highest degree component is the matrix Airy integral

Example: V={e} and matrix Airy integral

V _{= f}e_g, e e=e, l_(e) =1,_!potential(ξ, ξ, ξ)λ

on(q ΠV)0 this gives the nonhomogenious polyvector otr(Ξ3),

action by[Λ, ],Λ₂qodd, !extension of exp_3!1otr(Ξ3)dX to equivariantly closed di¤erential form exp_3!1otr(Ξ3) + ₂1otr([Λ, Ξ],Ξ)dX its highest degree component is the matrix Airy integral

Example: V={e} and matrix Airy integral

V _{= f}e_g, e e=e, l_(e) =1,_!potential(ξ, ξ, ξ)λ

on(q ΠV)0 this gives the nonhomogenious polyvector otr(Ξ3), action by[Λ, ],Λ₂qodd, !extension of exp_3!1otr(Ξ3)dX to equivariantly closed di¤erential form exp_3!1otr(Ξ3) + ₂1otr([Λ, Ξ],Ξ)dX

its highest degree component is the matrix Airy integral

Example: V={e} and matrix Airy integral

V _{= f}e_g, e e=e, l_(e) =1,_!potential(ξ, ξ, ξ)λ

on(q ΠV)0 this gives the nonhomogenious polyvector otr(Ξ3), action by[Λ, ],Λ₂qodd, !extension of exp_3!1otr(Ξ3)dX to equivariantly closed di¤erential form exp_3!1otr(Ξ3) + ₂1otr([Λ, Ξ],Ξ)dX its highest degree component is the matrix Airy integral

References:

[B1] S.Barannikov, Modular operads and Batalin-Vilkovisky geometry. IMRN, Vol. 2007, article ID rnm075. Preprint Max Planck Institute for Mathematics 2006-48 (04/2006),

[B2] S.Barannikov, Noncommutative Batalin-Vilkovisky geometry and matrix integrals. « Comptes rendus Mathematique» , presented for publication by M.Kontsevich in 05/2009, arXiv:0912.5484; Preprint NI06043 Newton Institute (09/2006), Preprint HAL, the electronic CNRS archive, hal-00102085 (09/2006)

[B3] S.Barannikov, Supersymmetry and cohomology of graph complexes. Preprint hal-00429963; (11/2009).

[B4] S.Barannikov, Matrix De Rham complex and quantum A-in…nity algebras. arXiv:1001.5264, Preprint hal-00378776; (04/2009).

[B5] S.Barannikov, Quantum periods - I. Semi-in…nite variations of Hodge structures. Preprint ENS DMA-00-19. arXiv:math/0006193 (06/2000), Intern. Math. Res. Notices. 2001, No. 23

[B6] S.Barannikov, Solving the noncommutative Batalin-Vilkovisky equation. Preprint hal-00464794 (03/2010). arXiv:1004.2253.