String Lie bialgebra in manifolds Joint work with Dennis Sullivan

The Goldman-Turaev Lie bialgebra of curves on surfaces

The Goldman bracket

The Turaev cobracket

Goal Define the bracket, the cobracket and more operations in the reduced equivariant homology of the free loop space of M

of the free loop space of M Goal Define the bracket, the cobracket and more operations in the reduced equivariant homology of the free loop space of M Remark: The vector space with basis free homotopy classes of curves in S(mod small loops) is (reduced equivariant) H0(free loop space of S)

Step 1: An order zero diagram

Step 1: A diagram A finite union of colored directed circles

Step 1: A diagram A finite union of colored directed circles Labeled finite subset of the union of circles F={a, b, c, d, e, f, g}

Step 1: A diagram A finite union of colored directed circles Labeled finite subset of the union of circles F={a, b, c, d, e, f, g} with partition into parts {a, b},{ c, d},{ e, f, g}

Step 1: A diagram A finite union of colored directed circles Labeled finite subset of the union of circles with partition into parts {a, b},{ c, d},{ e, f, g} n-prongs joining Point in each part

Step 1: A diagram A finite union of colored directed circles Labeled finite subset of the union of circles with partition n-prongs joining Point in each part Cyclic order of each n-prong

Remark: A diagram is equivalent to a ribbon surface

Step 2:

Step 2: A bundle Bundle with fiber a union of colored directed circles

Step 2: A bundle Bundle with fiber a unionof colored directed circles When the base B is a cycle, such a bundle can be used to describe cycles and appropiate homology classes of cycles will be homology classes in equivariant homology of the free loop space of M

Step 2: A bundle A map from the total space of the bundle into M

Step 3: The configuration bundle

Step 3: The configuration bundle Consider the bundle with fiber all isotopic configurations of the points {a,b,c,d,e,f,g,e} as in the diagram.

Step 4: Recoupling

Step 4: Recoupling

(not all circle diagrams yield a single circle)

Step 4: Consider the bundle made by recoupling Conf(E)

One no longer has a map from the total space to M Conf(E)

Step 5: Restrict B M Consider the image of each fiber Points move around (configuration) Consider the image of each fiber

Step 5, example of the image of a fiber

Step 5 Compare the image of a fiber with the circle diagram

Step 5 Compare the image of a fiber with the circle diagram

Step 5: Restrict M B M Points move around (configuration)

Input OUTPUT Add arrows to picture

Recall: A diagram is equivalent to a ribbon surface

A general closed string operation Conf(E) B Restrict

So far, so good but… With the above procedure we do not necessarily get a well defined operator.

A general closed string operation Compact? Transversal? Restrict Orientable?

The bracket

The cobracket

Theorem The bracket and cobracket are well defined on the reduced equivariant homology of the free loop space of an oriented smooth manifold and satisfy the identities of a Lie bialgebra. More complicated graphs give well defined operators.

Identities of a Lie bialgebra 1 Lie bracket [a,b]=-[b,a], [,] f=-[] [[a,b],c]+[[b,c],a]+[[c,a],b]=0, ([,]id)[,](id+t+t2)=0 f is the flip of factors, t is a cyclic permutation skew symmetry Jacobi

Identities of a Lie bialgebra: Lie cobracket (id+t+t2)( id)  =0 f is the flip of factors, t is a cyclic permutation Coskew symmetry Cojacobi

Identities of a Lie bialgebra: Compatibility [a,b]=[(a),b]+[a, (b)] Where [x,yz]=[x,y] z+y [x,z]

Identities of a Lie bialgebra [a,b]=-[b,a], [,] f=-[] [[a,b],c]+[[b,c],a]+[[c,a],b]=0, ([,]id)[,](id+t+t2)=0 f =-  (id+t+t2)( id)  =0 [a,b]=[(a),b]+[a, (b)] f(ab)=b a, t(abc)= bca [x,yz]=[x,y] z+y [x,z]

The bracket Step 1

The bracket - step 2 Consider a bundle with fiber two (colored, directed) circles E B

The bracket - step 2 Consider a bundle with fiber two (directed colored) circles and a map from E to a manifold M M E B

The bracket - step 3 Consider the associated bundle with fiber all possible configurations of pairs of points (one on each circle) Conf(E) B

The bracket, step 4 Form the recoupling bundle

The bracket- step 5 Pass to the locus L where the map sends the two points to the same point in M. L is a subset of Conf(E) Conf(E)

The bracket - step 5 The output of the operation is a circle bundle produced by cutting and pasting. and a map from E” to M E’’ M Locus

The bracket Step 1

The bracket - step 2 Consider a bundle with fiber two (labeled) circles And a map from E to a manifold M M E B B is a cycle, e.g.,an oriented closed manifold

The bracket - step 3 Consider the associated bundle with fiber all possible configurations of pairs of points (one at each circle). COMPACT ORIENTED (the points are labeled and the base is oriented) Conf(E) a b B

The bracket, step 4 Form the recoupling bundle

The bracket - step 5 Pass to the locus L where the map sends the two points to the same point in M. Since the locus is the preimage of a diagonal with normal bundle, then its is compact and oriented (mod transversality) .

The bracket - step 5 The output of the operation is a circle bundle produced by cutting and pasting. and a map from E” to M E” M Locus

Identities of the bracket [a,b]=-[b,a], [,] f=-[] [[a,b],c]+[[b,c],a]+[[c,a],b]=0, ([,]id)[,](id+t+t2)=0 f is the flip of factors, t is a cyclic permutation

Skew symmetry [a,b]=-[b,a]

Jacobi identity Orientations are delicate.

Compactifying

The cobracket diagram Step 1

The cobracket - step 2 Consider a circle bundle and a map from E to a manifold M M a E B

The cobracket - step 3 Consider the associated bundle of all possible configurations of pairs of (labeled) points in a directed circle a b Conf(E) B

The cobracket - step 3 Take the locus where the map sends the two points to the same point.

The cobracket, Step 4 Form the recoupling bundle

The cobracket - step 5 Consider the bundle with base the locus, and fiber the two (colored directed) circles produced by cutting the circle at the intersection point. (Restriction of the recoupling bundle)

The cobracket diagram Step 1

The cobracket - step 2 non compact Consider the associated bundle with fiber all configurations of pairs of points Oriented (since the points are labeled) but non compact a a b Conf(E) B

Conf(E) is not compact

(Quotient by the subspace of families of very small loops). equivariant homology Reduced (Quotient by the subspace of families of very small loops).

Conf(E) is not compact but

The cobracket - step 3 relatively compact The associated bundle with fiber all configurations of pairs of points in a circle is Yields an oriented and Cycle relatively compact a b Conf(E) B

The cobracket, Step 4 Form the recoupling bundle

The cobracket - step 5 Restrict: The output of the operation is a bundle produced by cutting and pasting. and a map from E” to M E” a b M Locus

Identities of a Lie bialgebra: Lie cobracket (id+t+t2)( id)  =0 f is the flip of factors, t is a cyclic permutation Coskew symmetry Cojacobi

The Lie bracket passes to the reduced equivariant homology [cycle, very small loop]=0

Identities of a Lie bialgebra: Compatibility [a,b]=[(a),b]+[a, (b)] Where [x,yz]=[x,y] z+y [x,z]

Compatibility of the Lie Bialgebra

Two of the terms cancel

The equation holds

Generalizing the bracket, Consider the diagram

Schema of a 5 to 1 operation Step 1: The diagram Step 2: The bundle Step 3: The configuration bundle Step 4: The recoupling bundle Step 5: Restriction

Identities of our n to 1 operations These operations form a gravity algebra

The gravity algebra identities Theorem (Sullivan, C.) The ordinary homology of the free loop space of a manifold has a BV structure. Then by Getzler (alg-geom 9411004), the equivariant homology of the free loop space has a gravity algebra structure.

General order zero diagram Transversality Orientation Compactness

A remark about orientation We’d like to have a definition of orientation for which the flip of different factors is orientation preserving when each vector space is given an orientation.

Definition of orientation The graded line functor goes from finite dimensional real vector spaces to Z/2 graded vector spaces. V topV in degree 0 if dimV even 1 if dimV odd An orientation of V is a generator of the graded line.

Consequences of our definition of orientation: The flip of different factors is orientation preserving If F is a finite set then the direct sum of labeled oriented vector spaces Vi has a canonical orientation If two of the factors of an exact sequence are oriented then the third factor has a canonical orientation

The bracket - step 3 Consider the associated bundle of all possible configurations of pairs of points (one at each circle). COMPACT ORIENTED (the points are labeled and the base is oriented) Conf(E) a b B

The orientation of the associated bundle has contributions from The orientation of B The orientation M The labeling of the finite set.