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Fun with F Vaughan Jones, Vanderbilt, Berkeley, Auckland August 9, - PowerPoint PPT Presentation

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Fun with F Vaughan Jones, Vanderbilt, Berkeley, Auckland August 9, 2015 Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 1 / 10 Motivation. Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 2

  1. Fun with F Vaughan Jones, Vanderbilt, Berkeley, Auckland August 9, 2015 Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 1 / 10

  2. Motivation. Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 2 / 10

  3. Hyperfinite A ∞ at A ∞ at every index the index of E 10 ∞ supertransitivity E (1) 8 A series × 2 E 8 s × 2 × 3 e i r E (1) e 7 s D × 2 × 3 × 1 E 6 E (1) 6 × 2 × 4 × 3 ×∞ ×∞ one ∞ -depth D (1) at least one unclassifiably n +2 ∞ -depth many ∞ -depth √ √ √ √ 1 1 1 1 4 2 (5 + 13) 2 (5 + 17) 2 (5 + 21) 5 6 6 index 3+ 5 √ 5 3 + 3 Peters, Morrison, Snyder, Izumi, Bigelow, Penneys, Tener, Calegari,Liu, Grossman,... Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 3 / 10

  4. Hyperfinite A ∞ at A ∞ at every index the index of E 10 ∞ supertransitivity E (1) 8 A series × 2 E 8 s × 2 × 3 e i r E (1) e 7 s D × 2 × 3 × 1 E 6 E (1) 6 × 2 × 4 × 3 ×∞ ×∞ one ∞ -depth D (1) at least one unclassifiably n +2 ∞ -depth many ∞ -depth √ √ √ √ 1 1 1 1 4 2 (5 + 13) 2 (5 + 17) 2 (5 + 21) 5 6 6 index 3+ 5 √ 5 3 + 3 Peters, Morrison, Snyder, Izumi, Bigelow, Penneys, Tener, Calegari,Liu, Grossman,... The big question: Do they all ”come from” conformal field theory? Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 3 / 10

  5. Ingredient of CFT is a representation of the Virasoro algebra, or a projective representation of Diff ( S 1 ). Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 4 / 10

  6. Ingredient of CFT is a representation of the Virasoro algebra, or a projective representation of Diff ( S 1 ). Locally (Haag-Kastler) the family of Diff ( I ) where I is an interval on S 1 . Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 4 / 10

  7. Ingredient of CFT is a representation of the Virasoro algebra, or a projective representation of Diff ( S 1 ). Locally (Haag-Kastler) the family of Diff ( I ) where I is an interval on S 1 . Definition The Thompson group T is the group of all orientation preserving homeomorphisms of S 1 which are piecewise linear (lifted to R ), smooth except at finitely many dyadic rationals ( { m 2 n : m , n ∈ Z } ), and whose slopes, when they exist, are powers of 2. Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 4 / 10

  8. Ingredient of CFT is a representation of the Virasoro algebra, or a projective representation of Diff ( S 1 ). Locally (Haag-Kastler) the family of Diff ( I ) where I is an interval on S 1 . Definition The Thompson group T is the group of all orientation preserving homeomorphisms of S 1 which are piecewise linear (lifted to R ), smooth except at finitely many dyadic rationals ( { m 2 n : m , n ∈ Z } ), and whose slopes, when they exist, are powers of 2. The Thompson group F is the subgroup of T preserving 0 and 1. Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 4 / 10

  9. Ingredient of CFT is a representation of the Virasoro algebra, or a projective representation of Diff ( S 1 ). Locally (Haag-Kastler) the family of Diff ( I ) where I is an interval on S 1 . Definition The Thompson group T is the group of all orientation preserving homeomorphisms of S 1 which are piecewise linear (lifted to R ), smooth except at finitely many dyadic rationals ( { m 2 n : m , n ∈ Z } ), and whose slopes, when they exist, are powers of 2. The Thompson group F is the subgroup of T preserving 0 and 1. Provided the end points of I are dyadic rationals, the set of elements of T that are the identity outside I is isomorphic to F . As a first shot at answering the big question, we construct ”toy” models where Diff ( S 1 ) , Diff ( I ) is replaced by T , F . Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 4 / 10

  10. Ingredient of CFT is a representation of the Virasoro algebra, or a projective representation of Diff ( S 1 ). Locally (Haag-Kastler) the family of Diff ( I ) where I is an interval on S 1 . Definition The Thompson group T is the group of all orientation preserving homeomorphisms of S 1 which are piecewise linear (lifted to R ), smooth except at finitely many dyadic rationals ( { m 2 n : m , n ∈ Z } ), and whose slopes, when they exist, are powers of 2. The Thompson group F is the subgroup of T preserving 0 and 1. Provided the end points of I are dyadic rationals, the set of elements of T that are the identity outside I is isomorphic to F . As a first shot at answering the big question, we construct ”toy” models where Diff ( S 1 ) , Diff ( I ) is replaced by T , F . Ghys,Sergiescu. Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 4 / 10

  11. Ingredient of CFT is a representation of the Virasoro algebra, or a projective representation of Diff ( S 1 ). Locally (Haag-Kastler) the family of Diff ( I ) where I is an interval on S 1 . Definition The Thompson group T is the group of all orientation preserving homeomorphisms of S 1 which are piecewise linear (lifted to R ), smooth except at finitely many dyadic rationals ( { m 2 n : m , n ∈ Z } ), and whose slopes, when they exist, are powers of 2. The Thompson group F is the subgroup of T preserving 0 and 1. Provided the end points of I are dyadic rationals, the set of elements of T that are the identity outside I is isomorphic to F . As a first shot at answering the big question, we construct ”toy” models where Diff ( S 1 ) , Diff ( I ) is replaced by T , F . Ghys,Sergiescu. F conjugate to a subgroup of Diff ([0 , 1]) ( in Homeo ( S 1 )). Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 4 / 10

  12. For this talk we will just construct ”unitary” representations of F in a knot theoretic context (Conway tangles) which define honest unitary representations using some data of a subfactor. Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 5 / 10

  13. For this talk we will just construct ”unitary” representations of F in a knot theoretic context (Conway tangles) which define honest unitary representations using some data of a subfactor. The empty tangle will be the vacuum vector Ω and the vacuum expectation value � g Ω , Ω � for g ∈ F will be an unoriented link in R 3 . Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 5 / 10

  14. For this talk we will just construct ”unitary” representations of F in a knot theoretic context (Conway tangles) which define honest unitary representations using some data of a subfactor. The empty tangle will be the vacuum vector Ω and the vacuum expectation value � g Ω , Ω � for g ∈ F will be an unoriented link in R 3 . In fact by the linearization device ”group algebra, GNS” we need only show how to associate a link to an element of F . Honest unitarity of course requires a lot more. Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 5 / 10

  15. For this talk we will just construct ”unitary” representations of F in a knot theoretic context (Conway tangles) which define honest unitary representations using some data of a subfactor. The empty tangle will be the vacuum vector Ω and the vacuum expectation value � g Ω , Ω � for g ∈ F will be an unoriented link in R 3 . In fact by the linearization device ”group algebra, GNS” we need only show how to associate a link to an element of F . Honest unitarity of course requires a lot more. So the fun starts. Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 5 / 10

  16. Bifurcating trees. Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 6 / 10

  17. Bifurcating trees. Elements of F are given by pairs of (planar rooted) bifurcating trees T with the same number of leaves. Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 6 / 10

  18. Bifurcating trees. Elements of F are given by pairs of (planar rooted) bifurcating trees T with the same number of leaves. This is because each such tree T specifies a partition of [0 , 1] into dyadic intervals thus: Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 6 / 10

  19. Bifurcating trees. Elements of F are given by pairs of (planar rooted) bifurcating trees T with the same number of leaves. This is because each such tree T specifies a partition of [0 , 1] into dyadic intervals thus: (0,1/2) Intervals of I Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 6 / 10

  20. So given a pair T 1 , T 2 as above we can get an element of F simply by mapping the intervals specified by the leaves of T 1 in the unique affine way onto those of T 2 . Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 7 / 10

  21. So given a pair T 1 , T 2 as above we can get an element of F simply by mapping the intervals specified by the leaves of T 1 in the unique affine way onto those of T 2 . We represent the Thompson group element by a diagram with T 1 on top, T 2 beneath. Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 7 / 10

  22. So given a pair T 1 , T 2 as above we can get an element of F simply by mapping the intervals specified by the leaves of T 1 in the unique affine way onto those of T 2 . We represent the Thompson group element by a diagram with T 1 on top, T 2 beneath.The simplest nontrivial example is: Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 7 / 10

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