A-infinity GL(N)-equivariant matrix integrals-III. Tree-level calculations and variations of nc-Hodge structure on complex projective manifolds.

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A-infinity GL(N)-equivariant matrix integrals-III.

Tree-level calculations and variations of nc-Hodge

structure on complex projective manifolds.

Serguei Barannikov

To cite this version:

Serguei Barannikov. A-infinity GL(N)-equivariant matrix integrals-III. Tree-level calculations and

variations of nc-Hodge structure on complex projective manifolds.. D-branes, Effective Actions and

Homological Mirror Symmetry, Jun 2010, Vienna, Austria. �hal-00494633�

A

in…nity GL

(

N

)

equivariant matrix integrals-III.

Tree-level calculations and variations of nc-Hodge

structure on complex projective manifolds.

Serguei Barannikov

IMJ, CNRS

Tree level BV on complex projective manifold (g=0

calculations)

∑

Tree level BV on complex projective manifold (g=0

calculations)

∑

M- smooth projective/C, c1(TM) =0, ̟02Γ(M, KM),!BV operator on

Ω0, _{(M, Λ T),}

Tree level BV on complex projective manifold (g=0

calculations)

∑

M- smooth projective/C, c1(TM) =0, ̟02Γ(M, KM),!BV operator on

Ω0, _{(M, Λ T),}

(∆α) `̟0=∂(α`̟0)

(tree level) BV equation on Ω0, (M, Λ T)

u∆γ+∂γ+1

2[γ, γ] =0,

Tree level BV-equation on complex projective manifold

(tree level) BV equation on Ω0, (M, Λ T)

u∆γ+∂γ+1

Tree level BV-equation on complex projective manifold

(tree level) BV equation on Ω0, (M, Λ T)

u∆γ+∂γ+1

2[γ, γ] =0, the …rst order term γ₀2Ω0, ₍M, Λ T)

∂γ₀+1

2[γ0,γ0] =0

Tree level BV-equation on complex projective manifold

(tree level) BV equation on Ω0, (M, Λ T)

u∆γ+∂γ+1

2[γ, γ] =0, the …rst order term γ₀2Ω0, ₍M, Λ T)

∂γ₀+1 2[γ0,γ0] =0 γ0 A∞-deformations of DbCoh(M) The component γcl₀ 2Ω0,1₍M, T) ∂γcl₀ +1₂[γcl₀,_γcl

Noncommutative periods map (B[5])

u∆γ+∂γ+1

2[γ, γ] =0,, (u∆+∂)exp( 1 uγ) =0

Noncommutative periods map (B[5])

Noncommutative periods map (B[5])

Ω₍t, u) = Z Mruexp( 1 uγ) `̟0 Ω₍t, u) 2 H_DR(M)((u))bOMΛT t 2 MΛT, T0MΛT =H (M, Λ T)

Noncommutative periods map and g=0 Gromov-Witten of

the mirror

For γW, normalized using a …ltration W opposite to FHodge([B5],1999):

∂2 ∂ti_∂tjΩ W _=u 1_Ck ij(t) ∂ ∂tkΩ W

Noncommutative periods map and g=0 Gromov-Witten of

the mirror

For γW, normalized using a …ltration W opposite to FHodge([B5],1999):

∂2 ∂ti_∂tjΩ W _=u 1_Ck ij(t) ∂ ∂tkΩ W

Semi-in…nite (Noncommutative) Hodge structures ([B5]).

The class[exp(γW)̟] =Ω₍t, u)is obtained as intersection of moving subspace

Ω₍t, u) = L(t) \ (A¢ne space(W )) L(t) H_DR(M)((u))bOMΛT

Semi-in…nite (Noncommutative) Hodge structures ([B5]).

The class[exp(γW)̟] =Ω₍t, u)is obtained as intersection of moving subspace

Ω₍t, u) = L(t) \ (A¢ne space(W )) L(t) H_DR(M)((u))bOMΛT

The semi-in…nite subspaceL(t), t 2 MΛT, is de…ned for arbitrary projective

manifold/C

L(γ₀): _[ru(exp1

uiγ0)(ϕ0+uϕ1+. . .)], ϕi 2ΩDR

Semi-in…nite (Noncommutative) Hodge structures ([B5]).

The class[exp(γW)̟] =Ω₍t, u)is obtained as intersection of moving subspace

Ω₍t, u) = L(t) \ (A¢ne space(W )) L(t) H_DR(M)((u))bOMΛT

The semi-in…nite subspaceL(t), t 2 MΛT, is de…ned for arbitrary projective

manifold/C L(γ₀): _[ru(exp1 uiγ0)(ϕ0+uϕ1+. . .)], ϕi 2ΩDR where γ₀2Ω0, ₍M, Λ T), ∂γ₀+1₂[γ₀,γ₀] =0 uL(t) L(t), ∂ ∂tL(t) u 1_L(t) ∂ ∂uL(t) u 2_{L(t), L(u) L(uj} u=u 1)

implies tt equations, remarkablyD∂

∂u modules over A

1 _{with similar}

properties appeared in 70s in works of Birkho¤, Malgrange, K.Saito and M.Saito

Semi-in…nite (Noncommutative) Hodge structures ([B5]).

The class[exp(γW)̟] =Ω₍t, u)is obtained as intersection of moving subspace

Ω₍t, u) = L(t) \ (A¢ne space(W )) L(t) H_DR(M)((u))bOMΛT

The semi-in…nite subspaceL(t), t 2 MΛT, is de…ned for arbitrary projective

manifold/C L(γ₀): _[ru(exp1 uiγ0)(ϕ0+uϕ1+. . .)], ϕi 2ΩDR where γ₀2Ω0, ₍M, Λ T), ∂γ₀+1₂[γ₀,γ₀] =0 uL(t) L(t), ∂ ∂tL(t) u 1_L(t) ∂ ∂uL(t) u 2_{L(t), L(u) L(uj} u=u 1)

implies tt equations, remarkablyD∂

∂u modules over A

1 _{with similar}

properties appeared in 70s in works of Birkho¤, Malgrange, K.Saito and M.Saito

Over mod space of complex structures it reduces to VHS

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

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 (At) HP(At)

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 (At) HP(At)

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

uHC (A) HC (A) ∂ ∂uHC (A) u 2_{HC (A)} ∂ ∂tHC (At) u 1_{HC (A} t), ∂

∂t Getzler ‡at connection on HP(At)

where

rk_C[[u]]HC (A) =rk_C((u))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 (At) HP(At)

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

uHC (A) HC (A) ∂ ∂uHC (A) u 2_{HC (A)} ∂ ∂tHC (At) u 1_{HC (A} t), ∂

∂t Getzler ‡at connection on HP(At)

where

rk_C[[u]]HC (A) =rk_C((u))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,

Summation over trees, BCOV lagrangian and BV cyclic

operad.

For W =FHodge_{, γ(t) =Σγ}

ata+∆α(t), ∆(γ) =0

Summation over trees, BCOV lagrangian and BV cyclic

operad.

For W =FHodge_{, γ(t) =Σγ}

ata+∆α(t), ∆(γ) =0

C_kij(t) =∂3_kijFBCOV(t)

FBCOV(t) =

Z ₁

2∂α(t) ^∆α(t) + 1

3!γ(t) ^γ(t) ^γ(t) -critical value of the BCOV Kodaira-Spencer lagrangian

Summation over trees, BCOV lagrangian and BV cyclic

operad.

For W =FHodge_{, γ(t) =Σγ}

ata+∆α(t), ∆(γ) =0

C_kij(t) =∂3_kijFBCOV(t)

FBCOV(t) =

Z ₁

2∂α(t) ^∆α(t) + 1

3!γ(t) ^γ(t) ^γ(t) -critical value of the BCOV Kodaira-Spencer lagrangian

Summation over trees, BCOV lagrangian and BV cyclic

operad.

For W =FHodge_{, γ(t) =Σγ}

ata+∆α(t), ∆(γ) =0

C_kij(t) =∂3_kijFBCOV(t)

FBCOV(t) =

Z ₁

2∂α(t) ^∆α(t) + 1

3!γ(t) ^γ(t) ^γ(t) -critical value of the BCOV Kodaira-Spencer lagrangian

FBCOV _{is equal to the summation over trees with}

Summation over trees, BCOV lagrangian and BV cyclic

operad.

For W =FHodge_{, γ(t) =Σγ}

ata+∆α(t), ∆(γ) =0

C_kij(t) =∂3_kijFBCOV(t)

FBCOV(t) =

Z ₁

2∂α(t) ^∆α(t) + 1

3!γ(t) ^γ(t) ^γ(t) -critical value of the BCOV Kodaira-Spencer lagrangian

FBCOV _{is equal to the summation over trees with}

edges!X ="∆∂ 1"

Summation over trees, BCOV lagrangian and BV cyclic

operad.

For W =FHodge_{, γ(t) =Σγ}

ata+∆α(t), ∆(γ) =0

C_kij(t) =∂3_kijFBCOV(t)

FBCOV(t) =

Z ₁

2∂α(t) ^∆α(t) + 1

3!γ(t) ^γ(t) ^γ(t) -critical value of the BCOV Kodaira-Spencer lagrangian

FBCOV _{is equal to the summation over trees with}

edges!X ="∆∂ 1"

vertices!product tensor on Ω0, (M, Λ T)

Summation over trees, BCOV lagrangian and BV cyclic

operad.

For W =FHodge_{, γ(t) =Σγ}

ata+∆α(t), ∆(γ) =0

C_kij(t) =∂3_kijFBCOV(t)

FBCOV(t) =

Z ₁

2∂α(t) ^∆α(t) + 1

3!γ(t) ^γ(t) ^γ(t) -critical value of the BCOV Kodaira-Spencer lagrangian

FBCOV _{is equal to the summation over trees with}

edges!X ="∆∂ 1"

vertices!product tensor on Ω0, (M, Λ T)

legs!elements of H (Λ T)

Meaning of this : the tree-level Feynman transform of the operad of BV algebras / ∆=0= operad of H (M0,n)

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.