A-infinity gl(N)-equivariant matrix integrals

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A-infinity gl(N)-equivariant matrix integrals

Serguei Barannikov

To cite this version:

Serguei Barannikov. A-infinity gl(N)-equivariant matrix integrals. Workshop on Geometry and

Physics of the Landau-Ginzburg model, May 2010, Grenoble, France. �hal-00490107�

A in…nity GL ( N ) equivariant matrix integrals

Serguei Barannikov

CNRS

02/06/2010

The noncommutative Batalin-Vilkovisky formalism

V - Z / 2 Z graded vector space, l- scalar product on V ^_ of degree d,

The noncommutative Batalin-Vilkovisky formalism

V - Z / 2 Z graded vector space, l- scalar product on V ^_ of degree d, The noncommutative Batalin-Vilkovisky equation ([B1],2005),

h ∆ S + ¹

2 f S, S g = 0 , S = ^∑ g 0 ,i h ^{2g 1+i} S _g,i where

S _{g ,i} 2 Symm ⁱ ( C _λ [ 1 d ]) ^, C _λ = ( ^∞ _{j =0} (( V [ 1 ] ^j ) ^_ ) ^{Z /j Z} )

-the symmetric/exterior, powers for odd/even d, of cyclic cochains

The noncommutative Batalin-Vilkovisky formalism

V - Z / 2 Z graded vector space, l- scalar product on V ^_ of degree d, The noncommutative Batalin-Vilkovisky equation ([B1],2005),

h ∆ S + ¹

2 f S, S g = 0 , S = ^∑ g 0 ,i h ^{2g 1+i} S _g,i where

S _{g ,i} 2 Symm ⁱ ( C _λ [ 1 d ]) ^, C _λ = ( ^∞ _{j =0} (( V [ 1 ] ^j ) ^_ ) ^{Z /j Z} ) -the symmetric/exterior, powers for odd/even d, of cyclic cochains

f S ₀ , 1 , S ₀ , 1 g = 0 ,

so S ₀ , 1 = m _A

- Calabi-Yau A ∞ algebra, (= A ∞ algebra with invariant

scalar product of degree d)

The noncommutative Batalin-Vilkovisky formalism

V - Z / 2 Z graded vector space, l- scalar product on V ^_ of degree d, The noncommutative Batalin-Vilkovisky equation ([B1],2005),

h ∆ S + ¹

2 f S, S g = 0 , S = ^∑ g 0 ,i h ^{2g 1+i} S _g,i where

S _{g ,i} 2 Symm ⁱ ( C _λ [ 1 d ]) ^, C _λ = ( ^∞ _{j =0} (( V [ 1 ] ^j ) ^_ ) ^{Z /j Z} ) -the symmetric/exterior, powers for odd/even d, of cyclic cochains

f S ₀ , 1 , S ₀ , 1 g = 0 ,

so S ₀ , 1 = m _A

- Calabi-Yau A ∞ algebra, (= A ∞ algebra with invariant scalar product of degree d)

([B2],2006) solution S ! matrix integrals Z

exp b S ( X , Λ ) dX

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

The noncommutative Batalin-Vilkovisky formalism

V - Z / 2 Z graded vector space, l- scalar product on V ^_ of degree d, The noncommutative Batalin-Vilkovisky equation ([B1],2005),

h ∆ S + ¹

2 f S, S g = 0 , S = ^∑ g 0 ,i h ^{2g 1+i} S _g,i where

S _{g ,i} 2 Symm ⁱ ( C _λ [ 1 d ]) ^, C _λ = ( ^∞ _{j =0} (( V [ 1 ] ^j ) ^_ ) ^{Z /j Z} ) -the symmetric/exterior, powers for odd/even d, of cyclic cochains

f S ₀ , 1 , S ₀ , 1 g = 0 ,

so S ₀ , 1 = m _A

- Calabi-Yau A ∞ algebra, (= A ∞ algebra with invariant scalar product of degree d)

([B2],2006) solution S ! matrix integrals Z

exp b S ( X , Λ ) dX

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

In the case of the algebra e e = e, the answer is the matrix Airy integral

R exp ( ¹ ₆ Tr ( Y ³ ) ¹ ₂ Tr ( ΛY ² )) dY

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

The asymptotic expansion as Λ ! ^∞ ) sum over stable ribbon

graphs ) cohomology classes in H (M ^K _{g ,n} ) (in H (M ^K _g,n ^, L) for odd d)

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

The asymptotic expansion as Λ ! ^∞ ) sum over stable ribbon

graphs ) cohomology classes in H (M ^K _{g ,n} ) (in H (M ^K _g,n ^, L) for odd d)

This is the higher genus counterpart of the (nc)Hodge theory integration on

CY projective manifolds, ( h ∆ γ + ∂γ + ¹ ₂ [ γ , γ ] = 0, γ 2 ^{Ω 0,} ( M , Λ T ) )

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

The asymptotic expansion as Λ ! ^∞ ) sum over stable ribbon

graphs ) cohomology classes in H (M ^K _{g ,n} ) (in H (M ^K _g,n ^, L) for odd d) This is the higher genus counterpart of the (nc)Hodge theory integration on CY projective manifolds, ( h ∆ γ + ∂γ + ¹ ₂ [ γ , γ ] = 0, γ 2 ^{Ω 0,} ( M , Λ T ) )

( ^∆ matrix + i _gl ) exp b S ( X , Λ ) = 0

) exp S b ( X , Λ ) corresponds to gl equivariantly closed di¤erential form.

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

The asymptotic expansion as Λ ! ^∞ ) sum over stable ribbon

graphs ) cohomology classes in H (M ^K _{g ,n} ) (in H (M ^K _g,n ^, L) for odd d) This is the higher genus counterpart of the (nc)Hodge theory integration on CY projective manifolds, ( h ∆ γ + ∂γ + ¹ ₂ [ γ , γ ] = 0, γ 2 ^{Ω 0,} ( M , Λ T ) )

( ^∆ matrix + i _gl ) exp b S ( X , Λ ) = 0

) exp S b ( X , Λ ) corresponds to gl equivariantly closed di¤erential form.

By setting e A = A A ^_ [ d ] I extend the formalism to nonCY A ∞ algebras,

including weak CY algebras.

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

The asymptotic expansion as Λ ! ^∞ ) sum over stable ribbon

graphs ) cohomology classes in H (M ^K _{g ,n} ) (in H (M ^K _g,n ^, L) for odd d) This is the higher genus counterpart of the (nc)Hodge theory integration on CY projective manifolds, ( h ∆ γ + ∂γ + ¹ ₂ [ γ , γ ] = 0, γ 2 ^{Ω 0,} ( M , Λ T ) )

( ^∆ matrix + i _gl ) exp b S ( X , Λ ) = 0

) exp S b ( X , Λ ) corresponds to gl equivariantly closed di¤erential form.

By setting e A = A A ^_ [ d ] I extend the formalism to nonCY A ∞ algebras, including weak CY algebras.

My A ∞ equivariant matrix integrals give an integration framework in the noncommutative (derived algebraic) geometry, particularly adobted to the equation f m _A

, m _A

g = 0

Z

exp b S ( X , Λ ) b ϕdX

ϕ 2 Ker ( h ∆ + f S , g) ^, ϕ _{g ,i} 2 Symm ⁱ ( C _λ [ 1 d ])

CY complex projective variety (g=0 calculations)

M- projective manifold/ C , c ₁ ( T _M ) = 0 h ∆ γ + ∂γ + ¹

2 [ γ , γ ] = 0 ,

γ ( h ) 2 ^{Ω 0,} ( M , Λ T )

CY complex projective variety (g=0 calculations)

M- projective manifold/ C , c ₁ ( T _M ) = 0 h ∆ γ + ∂γ + ¹

2 [ γ , γ ] = 0 , γ ( h ) 2 ^{Ω 0,} ( M , Λ T )

γ ₀ , A ∞ deformations of D ^b Coh ( M ) (deformations of the A ∞ algebra

A = Ext ( C ) , C compact generator)

CY complex projective variety (g=0 calculations)

M- projective manifold/ C , c ₁ ( T _M ) = 0 h ∆ γ + ∂γ + ¹

2 [ γ , γ ] = 0 , γ ( h ) 2 ^{Ω 0,} ( M , Λ T )

γ ₀ , A ∞ deformations of D ^b Coh ( M ) (deformations of the A ∞ algebra A = Ext ( C ) , C compact generator)

Ω ( t , h ) = Z

M exp ( γ ) ̟

γ ( t , h ) = ^Σ i γ _i h ⁱ , t 2 M ΛT , ̟ 2 ^Γ ( M , K _M )

CY complex projective variety (g=0 calculations)

M- projective manifold/ C , c ₁ ( T _M ) = 0 h ∆ γ + ∂γ + ¹

2 [ γ , γ ] = 0 , γ ( h ) 2 ^{Ω 0,} ( M , Λ T )

γ ₀ , A ∞ deformations of D ^b Coh ( M ) (deformations of the A ∞ algebra A = Ext ( C ) , C compact generator)

Ω ( t , h ) = Z

M exp ( γ ) ̟ γ ( t , h ) = ^Σ i γ _i h ⁱ , t 2 M ΛT , ̟ 2 ^Γ ( M , K _M )

for γ ^W , normalized via a …ltration W opposite to F ^Hodge ([B5]):

∂ ²

∂t ⁱ ∂t ^j Ω = h ¹ C _ij ^k ( t ) ^∂

∂t ^k Ω

CY complex projective variety (g=0 calculations)

M- projective manifold/ C , c ₁ ( T _M ) = 0 h ∆ γ + ∂γ + ¹

2 [ γ , γ ] = 0 , γ ( h ) 2 ^{Ω 0,} ( M , Λ T )

γ ₀ , A ∞ deformations of D ^b Coh ( M ) (deformations of the A ∞ algebra A = Ext ( C ) , C compact generator)

Ω ( t , h ) = Z

M exp ( γ ) ̟ γ ( t , h ) = ^Σ i γ _i h ⁱ , t 2 M ΛT , ̟ 2 ^Γ ( M , K _M )

for γ ^W , normalized via a …ltration W opposite to F ^Hodge ([B5]):

∂ ²

∂t ⁱ ∂t ^j Ω = h ¹ C _ij ^k ( t ) ^∂

∂t ^k Ω

C _kij ( t ) = ∂ ³ ( genus = 0 GW-potential of M ^mirror )

Noncommutative Hodge structures ([B5])

The class [ exp ( γ ^W ) ̟ ] = ^Ω ( t, h ) is obtained as intersection Ω ( t , h ) = L( t ) \ ( A¢ne space(W ) )

L( t ) H _DR ( M )(( h )) b O _M

Noncommutative Hodge structures ([B5])

The class [ exp ( γ ^W ) ̟ ] = ^Ω ( t, h ) is obtained as intersection Ω ( t , h ) = L( t ) \ ( A¢ne space(W ) )

L( t ) H _DR ( M )(( h )) b O _M

The semi-in…nite subspace L( t ) , t 2 M ΛT , is de…ned for arbitrary projective manifold/ C

L( t ) ^: ( exp 1

h i _γ

)( ϕ ₀ + hϕ ₁ + ^{. . .} ) ^, ϕ _i 2 ^Ω DR

Noncommutative Hodge structures ([B5])

The class [ exp ( γ ^W ) ̟ ] = ^Ω ( t, h ) is obtained as intersection Ω ( t , h ) = L( t ) \ ( A¢ne space(W ) )

L( t ) H _DR ( M )(( h )) b O _M

The semi-in…nite subspace L( t ) , t 2 M ΛT , is de…ned for arbitrary projective manifold/ C

L( t ) ^: ( exp 1

h i _γ

)( ϕ ₀ + hϕ ₁ + ^{. . .} ) ^, ϕ _i 2 ^Ω DR

h L( t ) L( t ) ^{, ∂}

∂t L( t ) h ¹ L( t )

∂

∂ h L( t ) h ² L( t ) ^, L( h ) L( h j _{h= h}

) (implies tt equations, remarkably D

∂h

modules over A ¹ with similar

properties appeared many years ago in works of Birkho¤, Malgrange, K.Saito

and M.Saito)

Noncommutative Hodge structures ([B5])

The class [ exp ( γ ^W ) ̟ ] = ^Ω ( t, h ) is obtained as intersection Ω ( t , h ) = L( t ) \ ( A¢ne space(W ) )

L( t ) H _DR ( M )(( h )) b O _M

The semi-in…nite subspace L( t ) , t 2 M ΛT , is de…ned for arbitrary projective manifold/ C

L( t ) ^: ( exp 1

h i _γ

)( ϕ ₀ + hϕ ₁ + ^{. . .} ) ^, ϕ _i 2 ^Ω DR

h L( t ) L( t ) ^{, ∂}

∂t L( t ) h ¹ L( t )

∂

∂ h L( t ) h ² L( t ) ^, L( h ) L( h j _{h= h}

) (implies tt equations, remarkably D

∂h

modules over A ¹ with similar properties appeared many years ago in works of Birkho¤, Malgrange, K.Saito and M.Saito)

Over moduli space of complex structures L( t ) = ∑

r

( F ^Hodge ) ^r h ^r [[ h ]]

Noncommutative Hodge structures ([B5]), cont’d

L( t ) corresponds via HKR and formality isomorphisms for C ( A , A ) + k [ ξ , _∂ξ ^∂ ] module C ( A ) to

L( t ) = HC ( A t ) HP ( A t )

Noncommutative Hodge structures ([B5]), cont’d

L( t ) corresponds via HKR and formality isomorphisms for C ( A , A ) + k [ ξ , _∂ξ ^∂ ] module C ( A ) to

L( t ) = HC ( A t ) HP ( A t )

Recall HP : C ( A )(( h )) , b + hB, HC ( A ) ^: C ( A )[[ h ]] , b + hB

Noncommutative Hodge structures ([B5]), cont’d

L( t ) corresponds via HKR and formality isomorphisms for C ( A , A ) + k [ ξ , _∂ξ ^∂ ] module C ( A ) to

L( t ) = HC ( A t ) HP ( A t )

Recall HP : C ( A )(( h )) , b + hB, HC ( A ) ^: C ( A )[[ h ]] , b + hB Let A be an arbitrary A ∞ algebra, the ^∞ ₂ subspace HC ( A ) ! HP ( A ) ,

hHC ( A ) HC ( A )

∂

∂ h HC ( A ) h ² HC ( A )

∂

∂t HC ( A _t ) h ¹ HC ( A _t ) ^{, ∂}

∂t Getzler ‡at connection on HP ( A _t ) where

rk _{C [[ h]]} HC ( A ) = rk _{C (( h))} HP

assumed, i.e. the degeneration of nc Hodge -to-De Rham spectral sequence,

proven (Kaledin) for A smooth and compact, Z ₊ graded, then HC HP,

Noncommutative Hodge structures ([B5]), cont’d

L( t ) corresponds via HKR and formality isomorphisms for C ( A , A ) + k [ ξ , _∂ξ ^∂ ] module C ( A ) to

L( t ) = HC ( A t ) HP ( A t )

Recall HP : C ( A )(( h )) , b + hB, HC ( A ) ^: C ( A )[[ h ]] , b + hB Let A be an arbitrary A ∞ algebra, the ^∞ ₂ subspace HC ( A ) ! HP ( A ) ,

hHC ( A ) HC ( A )

∂

∂ h HC ( A ) h ² HC ( A )

∂

∂t HC ( A _t ) h ¹ HC ( A _t ) ^{, ∂}

∂t Getzler ‡at connection on HP ( A _t ) where

rk _{C [[ h]]} HC ( A ) = rk _{C (( h))} HP

assumed, i.e. the degeneration of nc Hodge -to-De Rham spectral sequence,

proven (Kaledin) for A smooth and compact, Z ₊ graded, then HC HP,

Real structure on HP in the case of arbitrary A ∞ algebra?

Noncommutative Batalin-Vilkovisky operator ([B1])

V - Z/ 2 Z graded vector space, l- scalar product on V ^_ of degree d, F = Symm ( C _λ ( V )[ 1 d ])

-symmetric/exterior, powers for odd/even d, of cyclic cochains:

C _λ = ( ^∞ _{j =0} (( V [ 1 ] ^j ) ^_ ) ^{Z /j Z} )

Noncommutative Batalin-Vilkovisky operator ([B1])

V - Z/ 2 Z graded vector space, l- scalar product on V ^_ of degree d, F = Symm ( C _λ ( V )[ 1 d ])

-symmetric/exterior, powers for odd/even d, of cyclic cochains:

C _λ = ( ^∞ _{j =0} (( V [ 1 ] ^j ) ^_ ) ^{Z /j Z} )

De…ne the noncommutative BV di¤erential on F via

∆ ( x _ρ

. . . x _ρ

) λ ( x _τ

. . . x _τ

) λ =

= ∑

p,q

( 1 ) ^ε l _ρ

_τ

( x _ρ

. . . x _ρ

x _τ

. . . x _τ

x _ρ

. . . x _ρ

) λ +

∑

p 16=q

( 1 ) ^{e ε} l _ρ

p

ρ

( x _ρ

. . . x _ρ

p 1

x _ρ

q+1

. . . x _ρ

r

) _λ ( x _ρ

p+1

. . . x _ρ

q 1

) _λ ( x _τ

. . . x _τ

) _λ

∑

p 16=q

( 1 ) ^{ee ε l} τ

τ

( ^x ρ

. . . x _ρ

r

) λ ( ^x τ

. . . x _τ

x _τ

. . . x _τ

) λ ( ^x τ

. . . x _τ

) λ

Noncommutative Batalin-Vilkovisky di¤erential cont’d

signs are the standard Koszul signs taking into account that

( x _ρ

. . . x _ρ

) λ = ( 1 d ) + ^∑ x _ρ

, x _i 2 V [ 1 ] .

Noncommutative Batalin-Vilkovisky di¤erential cont’d

signs are the standard Koszul signs taking into account that ( x _ρ

. . . x _ρ

) λ = ( 1 d ) + ^∑ x _ρ

, x _i 2 V [ 1 ] .

∆ ² = 0

Noncommutative Batalin-Vilkovisky di¤erential cont’d

signs are the standard Koszul signs taking into account that ( x _ρ

. . . x _ρ

) λ = ( 1 d ) + ^∑ x _ρ

, x _i 2 V [ 1 ] .

∆ ² = 0

∆ = ^∆ 1 + ^∆ 2 ,

Noncommutative Batalin-Vilkovisky di¤erential cont’d

signs are the standard Koszul signs taking into account that ( x _ρ

. . . x _ρ

) λ = ( 1 d ) + ^∑ x _ρ

, x _i 2 V [ 1 ] .

∆ ² = 0

∆ = ^∆ 1 + ^∆ 2 ,

∆ ₁ di¤erential of Lie algebra on C _λ ( ! non-commutative symplectic

geometry, ribbon graph complex, open moduli space H (M g ,n ) (M.K.,1992))

Noncommutative Batalin-Vilkovisky di¤erential cont’d

signs are the standard Koszul signs taking into account that ( x _ρ

. . . x _ρ

) λ = ( 1 d ) + ^∑ x _ρ

, x _i 2 V [ 1 ] .

∆ ² = 0

∆ = ^∆ 1 + ^∆ 2 ,

∆ ₁ di¤erential of Lie algebra on C _λ ( ! non-commutative symplectic geometry, ribbon graph complex, open moduli space H (M g ,n ) (M.K.,1992))

∆ ₁ + h ∆ ₂ ! non-commutative Batalin–Vilkovisky geometry, stable ribbon

graphs, compacti…ed moduli spaces H (M ^K _g,n ) ([B1])

Noncommutative Batalin-Vilkovisky di¤erential cont’d

signs are the standard Koszul signs taking into account that ( x _ρ

. . . x _ρ

) λ = ( 1 d ) + ^∑ x _ρ

, x _i 2 V [ 1 ] .

∆ ² = 0

∆ = ^∆ 1 + ^∆ 2 ,

∆ ₁ di¤erential of Lie algebra on C _λ ( ! non-commutative symplectic geometry, ribbon graph complex, open moduli space H (M g ,n ) (M.K.,1992))

∆ ₁ + h ∆ ₂ ! non-commutative Batalin–Vilkovisky geometry, stable ribbon graphs, compacti…ed moduli spaces H (M ^K _g,n ) ([B1])

Ker ∆ ₁ + ^∆ 2 = ^{Im ∆} 1 + ^∆ 2

Solutions to nc BV equation

Conjecture ([B1]). Counting of holomorphic curves ( ^Σ, ∂ Σ, p _i ) ! ( M ,

∐ ^L i , H ( L _i ^\ L _j )) , with Z /2 Z -graded local systems, gives solution to

the nc-BV equations.

Solutions to nc BV equation

Conjecture ([B1]). Counting of holomorphic curves ( ^Σ, ∂ Σ, p _i ) ! ( M ,

∐ ^L i , H ( L _i ^\ L _j )) , with Z /2 Z -graded local systems, gives solution to the nc-BV equations.

Theorem ([B6]). Summation over ribbon graphs ! solution to the nc

Batalin-Vilkovisky equation from dg-associative algebras (summation over

trees ! A-in…nity algebra structure)

Strange associative superalgebra with odd trace and psi-classes.

V - associative algebra, odd/even scalar product

Strange associative superalgebra with odd trace and psi-classes.

V - associative algebra, odd/even scalar product

Assume: I - an odd derivation acting on V , preserving the scalar product: ,

in general I ² 6= 0 (!), 9 ^e I , [ I , e I ] = 1, str ([ a , ]) = 0 for any a 2 A.

Strange associative superalgebra with odd trace and psi-classes.

V - associative algebra, odd/even scalar product

Assume: I - an odd derivation acting on V , preserving the scalar product: , in general I ² 6= 0 (!), 9 ^e I , [ I , e I ] = 1, str ([ a , ]) = 0 for any a 2 A.

Theorem ([B2],[B3]) This data ! Cohomology classes in H (M ^K g ,n )

Strange associative superalgebra with odd trace and psi-classes.

V - associative algebra, odd/even scalar product

Assume: I - an odd derivation acting on V , preserving the scalar product: , in general I ² 6= 0 (!), 9 ^e I , [ I , e I ] = 1, str ([ a , ]) = 0 for any a 2 A.

Theorem ([B2],[B3]) This data ! Cohomology classes in H (M ^K g ,n )

Example q ( N ) , q ( N ) = f[ X , π ] = 0 j X 2 gl ( N j N )g ,where π odd

involution, q ( N ) has odd trace otr, I = [ ^Ξ, ] ^{, Ξ} - odd element

Ξ = 0 j diag ( λ ₁ , . . . , λ n ) , ( I ² 6= 0 (!))

Strange associative superalgebra with odd trace and psi-classes.

V - associative algebra, odd/even scalar product

Assume: I - an odd derivation acting on V , preserving the scalar product: , in general I ² 6= 0 (!), 9 ^e I , [ I , e I ] = 1, str ([ a , ]) = 0 for any a 2 A.

Theorem ([B2],[B3]) This data ! Cohomology classes in H (M ^K g ,n ) Example q ( N ) , q ( N ) = f[ X , π ] = 0 j X 2 gl ( N j N )g ,where π odd involution, q ( N ) has odd trace otr, I = [ ^Ξ, ] ^{, Ξ} - odd element Ξ = 0 j diag ( λ ₁ , . . . , λ n ) , ( I ² 6= 0 (!))

Theorem ([B2],[B3]) This is the generating function for products of classes

c ₁ ( T _i ) .

Strange associative superalgebra with odd trace and psi-classes.

V - associative algebra, odd/even scalar product

Assume: I - an odd derivation acting on V , preserving the scalar product: , in general I ² 6= 0 (!), 9 ^e I , [ I , e I ] = 1, str ([ a , ]) = 0 for any a 2 A.

Theorem ([B2],[B3]) This data ! Cohomology classes in H (M ^K g ,n ) Example q ( N ) , q ( N ) = f[ X , π ] = 0 j X 2 gl ( N j N )g ,where π odd involution, q ( N ) has odd trace otr, I = [ ^Ξ, ] ^{, Ξ} - odd element Ξ = 0 j diag ( λ ₁ , . . . , λ n ) , ( I ² 6= 0 (!))

Theorem ([B2],[B3]) This is the generating function for products of classes c ₁ ( T _i ) .

Similarly, with even scalar product and an odd derivation, in particular for

gl ( N j N ) and I = [ ^Ξ, ] , Ξ 2 gl ( N j N ) odd .

References:

[B1] S.Barannikov, Modular operads and Batalin-Vilkovisky geometry. IMRN, Vol. 2007, article ID rnm075. Preprint Max Planck Institute for Mathematics 2006-48 (25/04/2006),

[B2] S.Barannikov, Noncommutative Batalin-Vilkovisky geometry and matrix integrals. «Comptes rendus Mathematique» of the French Academy of Sciences, presented for publication by Academy member M.Kontsevich on 20/05/2009, 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.