1. String Lie bialgebra in manifolds
Joint work with Dennis Sullivan

2. The Goldman-Turaev Lie bialgebra of curves on surfaces

4. The Goldman bracket

5. The Turaev cobracket

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

8. 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)

9. Step 1: An order zero diagram

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

11. 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}

12. 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}

13. 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

14. 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

15. Remark: A diagram is equivalent to a ribbon surface

16. Step 2:

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

18. 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

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

20. Step 3: The configuration bundle

21. 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.

22. Step 4: Recoupling

23. Step 4: Recoupling

24. (not all circle diagrams yield a single circle)

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

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

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

28. Step 5, example of the image of a fiber

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

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

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

32. Input
OUTPUT
Add arrows to picture

33. Recall: A diagram is equivalent to a ribbon surface

34. A general closed string operation
Conf(E) B
Restrict

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

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

37. The bracket

38. The cobracket

39. 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.

40. 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

41. 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

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

43. 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]

44. The bracket Step 1

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

46. 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

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

48. The bracket, step 4
Form the recoupling bundle

49. 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)

50. 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

51. The bracket Step 1

52. 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

53. 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

54. The bracket, step 4
Form the recoupling bundle

55. 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) .

56. 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

57. 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

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

59. Jacobi identity
Orientations are delicate.

60. Compactifying

61. The cobracket diagram Step 1

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

63. 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

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

65. The cobracket, Step 4
Form the recoupling bundle

66. 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)

67. The cobracket diagram Step 1

68. 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

69. Conf(E) is not compact

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

71. Conf(E) is not compact but

72. 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

73. The cobracket, Step 4
Form the recoupling bundle

74. 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

75. 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

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

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

78. Compatibility of the Lie Bialgebra

79. Two of the terms cancel

80. The equation holds

81. Generalizing the bracket,
Consider the diagram

82. 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

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

84. 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 ), the equivariant homology of the free loop space has a gravity algebra structure.

85. General order zero diagram
Transversality
Orientation
Compactness

86. 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.

87. 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.

88. 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

89. 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

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