Summations over generalized ribbon Feynman diagrams and all genus Gromov-Witten invariants

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Summations over generalized ribbon Feynman diagrams

and all genus Gromov-Witten invariants

S. Barannikov

To cite this version:

S. Barannikov. Summations over generalized ribbon Feynman diagrams and all genus Gromov-Witten invariants. 2018. �hal-01915676�

Summations over generalized ribbon Feynman

diagrams and all genus Gromov-Witten

invariants.

Skoltech

Serguei Barannikov (Paris VII, NRU HSE)

yi =c1(Tp⇤i), yi 2H

2_(M¯

g ,n). Consider H⇤(M¯ g ,n) valued

generating function for products of yi

I Theorem (S.B.[2009])

Â

 di=d yd1 1 . . . yndn n

’

i=1 (2di 1)!! l2di+1 i = = 2 4

Â

[G]₂Gdec,odd g ,n (G , or(G)) 2 c(G) |Aut(G)_|e2Edge

’

_(G) 1 li(e)+lj(e) 3 5 (1)

I the sum on the right is over stable ribbon graphs of genus g with n numbered punctures, with 2d+n edges, and such that the vertices of the graph have cyclically ordered subsets of arbitrary odd cardinality

I In the simplest case, corresponding to the top degree, the cohomology H6g 6+2n(_M¯ g ,n)is 1-dimensional, the

summation is over 3-valent usual ribbon graphs and this formula then reproduces the main identity from Kontsevich proof of Witten conjecture.

This formula is a byproduct of the construction of cohomology classes from [2006b, 2009]

I I - an odd derivation acting on cyclic associative /A•

algebra A, with odd scalar product, in general I2 _{6=0 (!)}

I Theorem (S.B.[2006b, 2009]) This data !Cohomology classes in H⇤(_Mg ,n)

I Theorem (S.B.[2018]) This construction defines Cohomological Field Theory.

I Theorem (S.B.[2006b, 2009]) This construction applied to “odd matrix algebra” A=q(N),

q(N) ={[X , p] =0|X 2gl(N|N)} , where p odd

involution, I = [X,·], X- odd element !the formula (1) for all products of yi

I This formula was the first nontrivial computation of categorical all genus Gromov-Witten invariants.

Counterexample to a Theorem of Kontsevich

Another byproduct is a counterexample to the Theorem 1.3 from Kontsevich “Feynman diagrams in low-dimansional

topology”(1993)

It turns out this sum ÂGZ(G)G is NOT

Counterexample A

=

_h

1, x

_i

/x

2

=

1

I A=h1, xi/x2 _{=1, ¯x =1, b(1, x) =1, d(x) =1, d}2_=0,

w =x⌦x.

I For this data this sum ÂGZ(G)G is nonzero on the boundary

of the following ribbon graph =a generator of the dual complex:

Cellular decomposition of ¯

_M

g ,n

⇥

R

n

A stable ribbon graph is a connected graph G (recall: a graph G is a triple (Flag(G), l, s), where Flag(G)is a finite set, whose elements are called flags, l is a partition of Flag(G)_$vertices , s is an involution acting on Flag(G)_$ edges/legs ) together with:

I partitions of the set of flags adjacent to every vertex into i(v)

nonempty subsets

Leg(v) =Leg(v)(1)_t. . .tLeg(v)(i(v)), v 2Vert(G)

I fixed cyclic order on every subset Leg(v)(k) _, I a number g(v)2Z 0 such

that|Leg(v)_{| >}2(2 i(v) 2g(v)).

I Define an orientation

or(G)₂Det(_⌦v2Vert(G)(kFlag(v) kCycle(v))

I _{G_}k(G , or(G)) has natural “generalized contraction of

Cellular decomposition of ¯

_M

g ,n

⇥

R

n

A metric on the stable ribbon graph is a function

l : Edge(G)_!R>0 . Given a stable ribbon graph G w/out legs

and a metric on G one can construct by standard procedure a punctured Riemann surfaceS(G), which have double points in general.

I replace every edge by a pair of oriented strips [0, l]_{⇥ [}0+i•[

one for each flag and glue them side 0⇥ [0+i•[to l_{⇥ [}0+i•[ according to the cyclic order of the cyclically ordered subsets at each vertex. Then glue the two strips for each edge [0, l]$ [l, 0]

I identify points corresponding to 2 subsets at vertices with g(v) =0, i(v) =2 , (double points); for points with 2g(v) +i(v) >2 remplace the vertex by some Riemann surface of genus g(v), which does not contain any marked point, connected to the rest via i(v)double points.

I This gives an isomorphism of complexes

Example of the formula for products of y

i

Example: The class of y1 in H2(M¯ g ,n)is represented as linear

combination of stable (G , or(G))with|Edge(G)| =2+n. It is the coefficient in front of _l13

1’

n

Compactifications and y

i

classes

I The compactification ¯Mg ,n is a quotent of Deligne-Mumford

compactification ¯_MDM

g ,n: the natural map ¯MDM !M¯ is the

contraction which forgets complex structure on every component of singular Riemann surface which does not contain marked points.

I One can consider ribbon graph decomposition of such components. This way we get some intermediate

compactification ¯MDM _!M¯ _{0 !M}¯ _{. On the boundary of}

new stratas again some components must be contracted and so on.

I The line bundles Tpion ¯MDMg ,n are pullbacks of the line

bundles Tpion ¯Mg ,n

I hence the yi classes on ¯MDMg ,n are pullbacks of the

The Quantum Master Equation on Cyclic Cochains

I The stable ribbon graph complex is intimately related with the Quantum Master Equation on Cyclic Cochains (QMCC).

I Let V =V0 V1 be Z/2Z -graded vector space,

dimkV <• , scalar product b : V⌦2 !k[p]

I Cl _{= (} •

j=1Hom((PV⌦j), k)Z/jZ) ,

S 2Sym(Cl[1 p])[[¯h]] (symmetric products for odd b , antisymmetric for even b )

I The Quantum Master Equation on Cyclic Cochains / The noncommutative Batalin-Vilkovisky equation (S.B.[2006a])

¯hDNCS+1

2{S, S} =0 QMCC S =Âg 0,i>0¯h2g 1+iSg ,i, Sg ,i 2Symi(Cl[1 p]) .

I

{S0,1, S0,1} =0,

S0,1 - A• algebra with (even/odd) scalar product, so S

The noncommutative BV di↵erential

I (QMCC) ()DNC(exp_¯h1S) =0

I The noncommutative BV di↵erential on F =Sym(Cl[1 p])

DNC(ar1. . . arr) l_(at 1. . . att) l ₌ =

Â

p,q ( 1)#b_rptq(ar1. . . arp 1atq+1. . . atq 1arp+1. . . arr) l₊

Â

p_±1₆₌q ( 1)˜#b_rprq(. . . arp 1arq+1. . . arr) l_(a rp+1. . . arq 1) l_(a t1. . . att) l

Â

p±16=q ( 1)˜˜#b__tptq(. . . arr) l_(at 1. . . atp 1atq+1. . . att) l_(at p+1. . . atq 1) l

I signs are the standard Koszul signs taking into account that ¯

(ar1. . . arr)l =1 p+Â ¯ari , ai 2Hom(PV) . I Theorem (S.B.[2006a]) D2

(A, m)is a d-Z/2Z graded associative algebra with odd scalar

product b, dimkA<• .

H is an odd selfadjoint operator H: A!PA, H_ =H , such that Id [d, H] =P, dP =0 , P2=P. B - the image of P.

LetG be a 3-valent ribbon graph with legs, then put:

I on every vertex v !the 3-tensors of the cyclic product on A mv _{2 ((}PA)⌦3)_

I on every interieur edge e = (↵0)_! the two tensor

b_(H_uf, vf0), b__H,e 2 (PA)⌦2,

I on every leg l 2Leg(G)!element al 2 PB

I make the contraction WG( O l2Leg(G) al) =h O v2Vert(G) mv, 0 @ O e2Edge(G) b__H,e 1 A O l2Leg(G) ali

The summation over 3-valent ribbon graphs with legs

(cont’d).

I WG( O l2Leg(G) al) =h O v2Vert(G) mv, 0 @ O e2Edge(G) b__H,e 1 A O l2Leg(G) ali S =

Â

{G} ¯h1 c(G)WG I S 2Sym( •_j=1Hom((PB⌦j_{), k)}Z/jZ_)[[¯h]]

I Theorem (S.B.[2010]) The sum

over ribbon graphs S satisfy the QMCC/noncommutative Batalin-Vilkovisky equation:

¯hDNCS+

1

Let S2Sym(Cl_{[1 p])[[¯h]] be a solution to the QMCC, with} S0,1,2=0 ( dV =0 ), let G be a stable ribbon graph, then put:

I on every vertex v !the multi-cyclic tensors Sg ,iv 2

i

Sym( •j=1Hom((PV⌦j), k)Z/jZ[1 p])

I on every edge e = (↵0)! the two tensor

b_(uf, vf0), b_,e 2 (PV)⌦2,

I take the contraction WG =h N v2Vert(G) Sv g ,i, N e2Edge(G) b_,e ! i

I Theorem (S.B.[2006a]) For any S- a solution to the QMCC equation the following chain is a cycle in the stable ribbon graph complex W(S) =

Â

{G}2SRG ¯h1 c(SG)_WG[G] dgraphW(S) =0, therefore [W(S)]₂H_⇤ M⇤,⇤

More constructions of solutions to the QMCC equation

I Conjecture (S.B.[2006a]) Counting of holomorphic curves

(S, ∂S, pi)! (M, ‰ Li, H⇤(LiTLj)), with Z/2Z -graded

local systems, gives solution to the QMCC equation.

I Theorem (S.B. 2013) If A is an A• infinity algebra with the

degeneration of the Hodge to de Rham spectral sequence, then the solution to the QMCC is constructed step by step starting from {S0,1, S0,1} =0

Homotopy theory of the QMCC equation

I Theorem (S.B.[2006a]) Solutions to the QMCC are in one-to-one correspondence with the structure of algebra over the Feynman transform of k[[Sn]

I Theorem (S.B.[2006a])(The Feynman transform of

[

Associative algebra plus odd derivation I

2

₆₌

0.

I A- associative algebra , with odd scalar product

I Assume: I - an odd derivation acting on A, preserving the scalar product: ,for example I = [L,·], L2Aodd, in general I2 _{6=0 (!), 9eI, [I ,eI] =1, str([a,·]) =0 for any a2A.}

I Theorem(S.B.[2006b, 2009]) This data !Cohomology classes in H⇤(_Mg ,n)

“Odd matrix algebra” Q

(

N

)

I Example q(N), q(N) ={[X , p] =0|X 2gl(N|N)}, where

p odd involution, q(N)has odd trace otr , I = [X,_·], X-odd element X= 0 |diag(l1,. . . , ln) , ( I2 6=0 (!))

I Theorem (S.B.[2006b, 2009]) This gives the generating function for products of tautological classes yi =c1(Tp⇤i).

I Conjecture (S.B.[2009]) This construction, applied to A• algebra A=End(C), where C is a generating object of

Db(Coh(Y)), Y is the mirror dual Calabi-Yau manifold to X

!...

Evidences for the conjecture

I In the case of X =pt the construction reproduces the all-genus Gromov-Witten descendent potential of the point.

I For projective Calabi-Yau manifolds the summation over g =0 generalized ribbon graphs reproduces the summation over trees giving Frobenius manifold of the •₂ VHS defined by the HC . It coincides with the g =0 Gromov-Witten of the mirror dual for hypersurfaces.

I The construction defines a Cohomological Field Theory (S.B.[2018]).

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

[2006b] S.Barannikov Noncommutative Batalin-Vilkovisky geometry and matrix integrals. ¡¡Comptes rendus Mathematique¿¿ of the French Academy of Sciences,

arXiv:0912.5484; Preprint NI06043 Isaac Newton Institute for Mathematical Sciences (09/2006), hal-00102085 (09/2006) [2009] S.Barannikov Supersymmetry and cohomology of graph

complexes. Lett Math.Phys. doi:10.1007/s11005-018-1123-7 hal-00429963 (2009)

[2010] S.Barannikov Solving the noncommutative

Batalin-Vilkovisky equation. Letters in Math. Phys., 2013, Vol 103, 6, pp 605-628 arXiv:1004.2253 (2010)

[2018] S.Barannikov Feynman transform and cohomological field theories. hal-01804639 (2018)