physics and geometry of knots quivers correspondence
play

Physics and geometry of knots-quivers correspondence Piotr - PowerPoint PPT Presentation

Jun 19, 2023 •592 likes •1.67k views

Introduction Physics and geometry of KQ correspondence Summary Physics and geometry of knots-quivers correspondence Piotr Kucharski Uppsala University, Sweden September 27, 2018 P. Kucharski Physics and geometry of KQ correspondence

  1. Introduction Knots Physics and geometry of KQ correspondence Quivers Summary Knots-quivers correspondence Motivic generating series and DT invariants Motivic generating series encodes information about moduli spaces of representations of Q | Q 0 | x d i P Q ( x , q ) = ( − q ) ∑ 1 ≤ i , j ≤| Q 0 | C i , j d i d j ∑ ∏ i ( q 2 ; q 2 ) d i d 1 ,..., d | Q 0 | ≥ 0 i = 1 � Ω Q ( x , q ) � 1 = Exp 1 − q 2 2 Ω Q ( x , q ) is a generating function of motivic � 1 � 0 Donaldson-Thomas (DT) invariants 0 0 ( q 2 ; q 2 ) d i is the q-Pochhammer symbol r − 1 � 1 − zq 2 � ... ( 1 − zq 2 ( r − 1 ) ) ( z ; q 2 ) r := ∏ ( 1 − zq 2 i ) = ( 1 − z ) i = 0 P. Kucharski Physics and geometry of KQ correspondence

  2. Introduction Knots Physics and geometry of KQ correspondence Quivers Summary Knots-quivers correspondence Motivic generating series and DT invariants Motivic generating series encodes information about moduli spaces of representations of Q | Q 0 | x d i ( − q ) ∑ 1 ≤ i , j ≤| Q 0 | C i , j d i d j P Q ( x , q ) = ∑ ∏ i ( q 2 ; q 2 ) d i d 1 ,..., d | Q 0 | ≥ 0 i = 1 � Ω Q ( x , q ) � 1 = Exp 1 − q 2 2 Example from the picture � 1 � 0 x d 1 x d 2 P Q ( x 1 , x 2 , q ) = ∑ ( − q ) d 2 1 2 0 0 1 ( q 2 ; q 2 ) d 1 ( q 2 ; q 2 ) d 2 d 1 , d 2 ≥ 0 � − qx 1 + x 2 � = Exp 1 − q 2 P. Kucharski Physics and geometry of KQ correspondence

  3. Introduction Knots Physics and geometry of KQ correspondence Quivers Summary Knots-quivers correspondence Outline Introduction 1 Knots Quivers Knots-quivers correspondence Physics and geometry of KQ correspondence 2 General idea Physics Geometry Summary 3 P. Kucharski Physics and geometry of KQ correspondence

  4. Introduction Knots Physics and geometry of KQ correspondence Quivers Summary Knots-quivers correspondence Knots-quivers correspondence 1 2 Knots-quivers (KQ) correspondence is an equality � � P K ( x , a , q ) = P Q K ( x , q ) � x i = a ai q qi − Cii x P K ( x , a , q ) – HOMFLY-PT generating series of knot K P Q K ( x , q ) – motivic generating series of respective quiver Q K x i = a a i q q i − C ii x is a change of variables P. Kucharski Physics and geometry of KQ correspondence

  5. Introduction Knots Physics and geometry of KQ correspondence Quivers Summary Knots-quivers correspondence Knots-quivers correspondence 1 2 Knots-quivers (KQ) correspondence is an equality � � P K ( x , a , q ) = P Q K ( x , q ) � x i = a ai q qi − Cii x P K ( x , a , q ) – HOMFLY-PT generating series of knot K P Q K ( x , q ) – motivic generating series of respective quiver Q K x i = a a i q q i − C ii x is a change of variables P. Kucharski Physics and geometry of KQ correspondence

  6. Introduction Knots Physics and geometry of KQ correspondence Quivers Summary Knots-quivers correspondence Knots-quivers correspondence 1 2 Knots-quivers (KQ) correspondence is an equality � � P K ( x , a , q ) = P Q K ( x , q ) � x i = a ai q qi − Cii x P K ( x , a , q ) – HOMFLY-PT generating series of knot K P Q K ( x , q ) – motivic generating series of respective quiver Q K x i = a a i q q i − C ii x is a change of variables P. Kucharski Physics and geometry of KQ correspondence

  7. Introduction Knots Physics and geometry of KQ correspondence Quivers Summary Knots-quivers correspondence Knots-quivers correspondence 1 2 Knots-quivers (KQ) correspondence is an equality � � P K ( x , a , q ) = P Q K ( x , q ) � x i = a ai q qi − Cii x P K ( x , a , q ) – HOMFLY-PT generating series of knot K P Q K ( x , q ) – motivic generating series of respective quiver Q K x i = a a i q q i − C ii x is a change of variables P. Kucharski Physics and geometry of KQ correspondence

  8. Introduction Knots Physics and geometry of KQ correspondence Quivers Summary Knots-quivers correspondence KQ correspondence - unknot example ∞ ( a 2 ; q 2 ) r P 0 1 ( x , a , q ) = x r ∑ ( q 2 ; q 2 ) r r = 0 = 1 + 1 − a 2 1 − q 2 x + ... x d 1 x d 2 P Q 01 ( x 1 , x 2 , q ) = ∑ ( − q ) d 2 1 2 1 1 ( q 2 ; q 2 ) d 1 ( q 2 ; q 2 ) d 2 d 1 , d 2 ≥ 0 = 1 + − qx 1 + x 2 + ... 2 1 − q 2 P. Kucharski Physics and geometry of KQ correspondence

  9. Introduction Knots Physics and geometry of KQ correspondence Quivers Summary Knots-quivers correspondence KQ correspondence - unknot example 1 2 We can check that � � P 0 1 ( x , a , q ) = P Q 01 ( x 1 , x 2 , q ) � x 1 = a 2 q − 1 x , x 2 = x � 1 + 1 − a 2 � 1 − q 2 x + ... = 1 + − qx 1 + x 2 � + ... � 1 − q 2 x 1 = a 2 q − 1 x , x 2 = x Comparing with x i = a a i q q i − C ii x we get ( a 1 , a 2 ) = ( 2 , 0 ) ( q 1 , q 2 ) = ( 0 , 0 ) P. Kucharski Physics and geometry of KQ correspondence

  10. Introduction Knots Physics and geometry of KQ correspondence Quivers Summary Knots-quivers correspondence KQ correspondence - unknot example 1 2 We can check that � � P 0 1 ( x , a , q ) = P Q 01 ( x 1 , x 2 , q ) � x 1 = a 2 q − 1 x , x 2 = x � 1 + 1 − a 2 � 1 − q 2 x + ... = 1 + − qx 1 + x 2 � + ... � 1 − q 2 x 1 = a 2 q − 1 x , x 2 = x Comparing with x i = a a i q q i − C ii x we get ( a 1 , a 2 ) = ( 2 , 0 ) ( q 1 , q 2 ) = ( 0 , 0 ) P. Kucharski Physics and geometry of KQ correspondence

  11. Introduction Knots Physics and geometry of KQ correspondence Quivers Summary Knots-quivers correspondence KQ correspondence and BPS states 1 2 We can also look at the level of BPS states: � N K ( x , a , q ) � � Ω Q K ( x , q ) �� � = P K = P Q K = Exp � Exp � 1 − q 2 1 − q 2 x i = a ai q qi − Cii x � � N K ( x , a , q ) = Ω Q K ( x , q ) � x i = a ai q qi − Cii x Integrality of DT invariants for symmetric quivers [Kontsevich-Soibelman, Efimov] implies LMOV conjecture P. Kucharski Physics and geometry of KQ correspondence

  12. Introduction Knots Physics and geometry of KQ correspondence Quivers Summary Knots-quivers correspondence KQ correspondence and BPS states 1 2 We can also look at the level of BPS states: � N K ( x , a , q ) � � Ω Q K ( x , q ) �� � = P K = P Q K = Exp � Exp � 1 − q 2 1 − q 2 x i = a ai q qi − Cii x � � N K ( x , a , q ) = Ω Q K ( x , q ) � x i = a ai q qi − Cii x Integrality of DT invariants for symmetric quivers [Kontsevich-Soibelman, Efimov] implies LMOV conjecture P. Kucharski Physics and geometry of KQ correspondence

  13. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Outline Introduction 1 Knots Quivers Knots-quivers correspondence Physics and geometry of KQ correspondence 2 General idea Physics Geometry Summary 3 P. Kucharski Physics and geometry of KQ correspondence

  14. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry KQ correspondence discussed so far Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series Physics P. Kucharski Physics and geometry of KQ correspondence

  15. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Towards physics of KQ correspondence Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ Physics P. Kucharski Physics and geometry of KQ correspondence

  16. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Towards physics of KQ correspondence Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ Physics 3d N = 2 T [ L K ] P. Kucharski Physics and geometry of KQ correspondence

  17. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Missing element Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ ↓ Physics 3d N = 2 T [ L K ] P. Kucharski Physics and geometry of KQ correspondence

  18. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Missing element Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ ↓ Physics 3d N = 2 T [ L K ] 3d N = 2 T [ Q K ] P. Kucharski Physics and geometry of KQ correspondence

  19. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Missing element Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ ↓ Physics 3d N = 2 T [ L K ] → 3d N = 2 T [ Q K ] P. Kucharski Physics and geometry of KQ correspondence

  20. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Geometric interpretations Knots Quivers Math HOMFLY-PT gen. series Motivic gen. series տ ր Geometric interpretations ւ ց Physics 3d N = 2 T [ L K ] 3d N = 2 T [ Q K ] P. Kucharski Physics and geometry of KQ correspondence

  21. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Outline Introduction 1 Knots Quivers Knots-quivers correspondence Physics and geometry of KQ correspondence 2 General idea Physics Geometry Summary 3 P. Kucharski Physics and geometry of KQ correspondence

  22. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Construction of knot complement theory We can construct 3d N = 2 knot complement theory T [ M K ] basing on large colour and classical limit of HOMFLY-PT polynomial [Fuji, Gukov, Sułkowski] � 1 �� � h → 0 ¯ � P K r ( a , q ) q 2 r fixed exp − → W T [ M K ] + O (¯ h ) 2 ¯ h � W T [ M K ] is a twisted superpotential Li 2 ( ... ) ← → chiral field κ 2 log ( ... ) log ( ... ) ← → CS coupling P. Kucharski Physics and geometry of KQ correspondence

  23. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Recall: what is T [ L K ] T [ L K ] : 3d N = 2 eff. theory on R 1 , 2 Partition function=generating function of Labastida-Marino-Ooguri-Vafa invariants N K r , i , j � � ∑ r , i , j N K r , i , j x r a i q j Z = Exp 1 − q 2 R 1 , 2 M5 brane M2 branes BPS states BPS states are counted by LMOV invariants ⇒ LMOV integrality conjecture P. Kucharski Physics and geometry of KQ correspondence

  24. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Recall: what is T [ L K ] T [ L K ] : 3d N = 2 eff. theory on R 1 , 2 Partition function=generating function of Labastida-Marino-Ooguri-Vafa invariants N K r , i , j � � ∑ r , i , j N K r , i , j x r a i q j Z = Exp 1 − q 2 R 1 , 2 M5 brane M2 branes BPS states BPS states are counted by LMOV invariants ⇒ LMOV integrality conjecture P. Kucharski Physics and geometry of KQ correspondence

  25. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Construction of T [ L K ] theory Idea We construct T [ L K ] analogously to T [ M K ] , but using P K ( x , a , q ) � 1 �� � � h → 0 ¯ � P K ( x , a , q ) − → dy exp W T [ L K ] + O (¯ h ) 2 ¯ h q 2 r → y W T [ L K ] = � � W T [ M K ] + log x log y � dy means that U ( 1 ) symmetry Integral corresponding to fugacity y is gauged We have the same matter content with just extra CS coupling with background field P. Kucharski Physics and geometry of KQ correspondence

  26. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Construction of T [ L K ] theory Idea We construct T [ L K ] analogously to T [ M K ] , but using P K ( x , a , q ) � 1 �� � � h → 0 ¯ � P K ( x , a , q ) − → dy exp W T [ L K ] + O (¯ h ) 2 ¯ h q 2 r → y W T [ L K ] = � � W T [ M K ] + log x log y � dy means that U ( 1 ) symmetry Integral corresponding to fugacity y is gauged We have the same matter content with just extra CS coupling with background field P. Kucharski Physics and geometry of KQ correspondence

  27. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Construction of T [ L K ] theory Idea We construct T [ L K ] analogously to T [ M K ] , but using P K ( x , a , q ) � 1 �� � � h → 0 ¯ � P K ( x , a , q ) − → dy exp W T [ L K ] + O (¯ h ) 2 ¯ h q 2 r → y W T [ L K ] = � � W T [ M K ] + log x log y � dy means that U ( 1 ) symmetry Integral corresponding to fugacity y is gauged We have the same matter content with just extra CS coupling with background field P. Kucharski Physics and geometry of KQ correspondence

  28. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Construction of T [ L K ] theory Idea We construct T [ L K ] analogously to T [ M K ] , but using P K ( x , a , q ) � 1 �� � � h → 0 ¯ � P K ( x , a , q ) − → dy exp W T [ L K ] + O (¯ h ) 2 ¯ h q 2 r → y W T [ L K ] = � � W T [ M K ] + log x log y � dy means that U ( 1 ) symmetry Integral corresponding to fugacity y is gauged We have the same matter content with just extra CS coupling with background field P. Kucharski Physics and geometry of KQ correspondence

  29. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Example: T [ L 0 1 ] For the unknot we have � 1 �� � � h → 0 ¯ � P 0 1 ( x , a , q ) − → dy exp W T [ L 01 ] + O (¯ h ) 2 ¯ h q 2 r → y � y − 1 a − 2 � � W T [ L 01 ] = Li 2 ( y )+ Li 2 + log x log y T [ L 0 1 ] is a U ( 1 ) gauge theory (fugacity y ) with one fundamental and one antifundamental chiral Antifundamental chiral is charged under the U ( 1 ) a global symmetry arising from S 2 in the conifold P. Kucharski Physics and geometry of KQ correspondence

  30. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Example: T [ L 0 1 ] For the unknot we have � 1 �� � � h → 0 ¯ � P 0 1 ( x , a , q ) − → dy exp W T [ L 01 ] + O (¯ h ) 2 ¯ h q 2 r → y � y − 1 a − 2 � � W T [ L 01 ] = Li 2 ( y )+ Li 2 + log x log y T [ L 0 1 ] is a U ( 1 ) gauge theory (fugacity y ) with one fundamental and one antifundamental chiral Antifundamental chiral is charged under the U ( 1 ) a global symmetry arising from S 2 in the conifold P. Kucharski Physics and geometry of KQ correspondence

  31. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Example: T [ L 0 1 ] For the unknot we have � 1 �� � � h → 0 ¯ � P 0 1 ( x , a , q ) − → dy exp W T [ L 01 ] + O (¯ h ) 2 ¯ h q 2 r → y � y − 1 a − 2 � � W T [ L 01 ] = Li 2 ( y )+ Li 2 + log x log y T [ L 0 1 ] is a U ( 1 ) gauge theory (fugacity y ) with one fundamental and one antifundamental chiral Antifundamental chiral is charged under the U ( 1 ) a global symmetry arising from S 2 in the conifold P. Kucharski Physics and geometry of KQ correspondence

  32. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Recall of general idea - now we are here Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ Physics 3d N = 2 T [ L K ] P. Kucharski Physics and geometry of KQ correspondence

  33. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry We look for the missing element Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ ↓ Physics 3d N = 2 T [ L K ] P. Kucharski Physics and geometry of KQ correspondence

  34. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry We look for the missing element Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ ↓ Physics 3d N = 2 T [ L K ] 3d N = 2 T [ Q K ] P. Kucharski Physics and geometry of KQ correspondence

  35. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Missing element: T [ Q K ] theory Idea Consider large colour and classical limit of motivic generating series! � 1 �� � � h → 0 P Q K ( x , q ) ¯ � ∏ − → dy i exp W T [ Q K ] + O (¯ h ) 2 ¯ h q 2 di → y i i C i , j W T [ Q K ] = ∑ � [ Li 2 ( y i )+ log x i log y i ]+ ∑ 2 log y i log y j , i i , j Gauge group: U ( 1 ) #vertices Matter content: one chiral for each vertex CS couplings given by C i , j = #arrows P. Kucharski Physics and geometry of KQ correspondence

  36. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Missing element: T [ Q K ] theory Idea Consider large colour and classical limit of motivic generating series! � 1 �� � � h → 0 P Q K ( x , q ) ¯ � ∏ − → dy i exp W T [ Q K ] + O (¯ h ) 2 ¯ h q 2 di → y i i C i , j W T [ Q K ] = ∑ � [ Li 2 ( y i )+ log x i log y i ]+ ∑ 2 log y i log y j , i i , j Gauge group: U ( 1 ) #vertices Matter content: one chiral for each vertex CS couplings given by C i , j = #arrows P. Kucharski Physics and geometry of KQ correspondence

  37. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Missing element: T [ Q K ] theory Idea Consider large colour and classical limit of motivic generating series! � 1 �� � � h → 0 P Q K ( x , q ) ¯ � ∏ − → dy i exp W T [ Q K ] + O (¯ h ) 2 ¯ h q 2 di → y i i C i , j W T [ Q K ] = ∑ � [ Li 2 ( y i )+ log x i log y i ]+ ∑ 2 log y i log y j , i i , j Gauge group: U ( 1 ) #vertices Matter content: one chiral for each vertex CS couplings given by C i , j = #arrows P. Kucharski Physics and geometry of KQ correspondence

  38. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Missing element: T [ Q K ] theory Idea Consider large colour and classical limit of motivic generating series! � 1 �� � � h → 0 P Q K ( x , q ) ¯ � ∏ − → dy i exp W T [ Q K ] + O (¯ h ) 2 ¯ h q 2 di → y i i C i , j W T [ Q K ] = ∑ � [ Li 2 ( y i )+ log x i log y i ]+ ∑ 2 log y i log y j , i i , j Gauge group: U ( 1 ) #vertices Matter content: one chiral for each vertex CS couplings given by C i , j = #arrows P. Kucharski Physics and geometry of KQ correspondence

  39. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Missing element: T [ Q K ] theory Idea Consider large colour and classical limit of motivic generating series! � 1 �� � � h → 0 P Q K ( x , q ) ¯ � ∏ − → dy i exp W T [ Q K ] + O (¯ h ) 2 ¯ h q 2 di → y i i C i , j W T [ Q K ] = ∑ � [ Li 2 ( y i )+ log x i log y i ]+ ∑ 2 log y i log y j , i i , j Gauge group: U ( 1 ) #vertices Matter content: one chiral for each vertex CS couplings given by C i , j = #arrows P. Kucharski Physics and geometry of KQ correspondence

  40. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Example: T [ Q 0 1 ] For the unknot quiver we have � 1 �� � � ¯ h → 0 � P Q 01 ( x 1 , x 2 , q ) − → dy 1 dy 2 exp W T [ Q 01 ] + O (¯ h ) 2 ¯ h q 2 di → y i W T [ Q 01 ] = Li 2 ( y 1 )+ Li 2 ( y 2 )+ log x 1 log y 1 + log x 2 log y 2 + 1 � 2 log y 1 log y 1 T [ Q 0 1 ] is a U ( 1 ) ( 1 ) × U ( 1 ) ( 2 ) gauge theory with one chiral field for each group 1 CS level one for U ( 1 ) ( 1 ) , consistent with � 1 � 0 C 0 1 = 0 0 2 P. Kucharski Physics and geometry of KQ correspondence

  41. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Example: T [ Q 0 1 ] For the unknot quiver we have � 1 �� � � ¯ h → 0 � P Q 01 ( x 1 , x 2 , q ) − → dy 1 dy 2 exp W T [ Q 01 ] + O (¯ h ) 2 ¯ h q 2 di → y i W T [ Q 01 ] = Li 2 ( y 1 )+ Li 2 ( y 2 )+ log x 1 log y 1 + log x 2 log y 2 + 1 � 2 log y 1 log y 1 T [ Q 0 1 ] is a U ( 1 ) ( 1 ) × U ( 1 ) ( 2 ) gauge theory with one chiral field for each group 1 CS level one for U ( 1 ) ( 1 ) , consistent with � 1 � 0 C 0 1 = 0 0 2 P. Kucharski Physics and geometry of KQ correspondence

  42. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Example: T [ Q 0 1 ] For the unknot quiver we have � 1 �� � � ¯ h → 0 � P Q 01 ( x 1 , x 2 , q ) − → dy 1 dy 2 exp W T [ Q 01 ] + O (¯ h ) 2 ¯ h q 2 di → y i W T [ Q 01 ] = Li 2 ( y 1 )+ Li 2 ( y 2 )+ log x 1 log y 1 + log x 2 log y 2 + 1 � 2 log y 1 log y 1 T [ Q 0 1 ] is a U ( 1 ) ( 1 ) × U ( 1 ) ( 2 ) gauge theory with one chiral field for each group 1 CS level one for U ( 1 ) ( 1 ) , consistent with � 1 � 0 C 0 1 = 0 0 2 P. Kucharski Physics and geometry of KQ correspondence

  43. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Recall of general idea - now we are here Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ ↓ Physics 3d N = 2 T [ L K ] 3d N = 2 T [ Q K ] P. Kucharski Physics and geometry of KQ correspondence

  44. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry We want the physical interpretation of KQ correspondence Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ ↓ Physics 3d N = 2 T [ L K ] → 3d N = 2 T [ Q K ] P. Kucharski Physics and geometry of KQ correspondence

  45. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Duality Physical meaning of the knots-quivers correspondence is a duality between two 3d N = 2 theories: T [ L K ] ← → T [ Q K ] We already know that Z ( T [ L K ]) = P K ( x , a , q ) We can calculate the partition function of T [ Q K ] and indeed Z ( T [ Q K ]) = P Q K ( x , q ) P. Kucharski Physics and geometry of KQ correspondence

  46. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Duality Physical meaning of the knots-quivers correspondence is a duality between two 3d N = 2 theories: T [ L K ] ← → T [ Q K ] We already know that Z ( T [ L K ]) = P K ( x , a , q ) We can calculate the partition function of T [ Q K ] and indeed Z ( T [ Q K ]) = P Q K ( x , q ) P. Kucharski Physics and geometry of KQ correspondence

  47. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Duality Physical meaning of the knots-quivers correspondence is a duality between two 3d N = 2 theories: T [ L K ] ← → T [ Q K ] We already know that Z ( T [ L K ]) = P K ( x , a , q ) We can calculate the partition function of T [ Q K ] and indeed Z ( T [ Q K ]) = P Q K ( x , q ) P. Kucharski Physics and geometry of KQ correspondence

  48. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Outline Introduction 1 Knots Quivers Knots-quivers correspondence Physics and geometry of KQ correspondence 2 General idea Physics Geometry Summary 3 P. Kucharski Physics and geometry of KQ correspondence

  49. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes extracted from motivic generating series Let’s look at motivic generating series restricted to | d | = 1 ( − q ) C ii x i | d | = 1 ( x , q ) = ∑ P Q K 1 − q 2 i ∈ Q 0 Every vertex ( i ∈ Q 0 ) contributes once No interactions between different nodes What is the meaning of P Q K | d | = 1 ( x , q ) ? P. Kucharski Physics and geometry of KQ correspondence

  50. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes extracted from motivic generating series Let’s look at motivic generating series restricted to | d | = 1 ( − q ) C ii x i | d | = 1 ( x , q ) = ∑ P Q K 1 − q 2 i ∈ Q 0 Every vertex ( i ∈ Q 0 ) contributes once No interactions between different nodes What is the meaning of P Q K | d | = 1 ( x , q ) ? P. Kucharski Physics and geometry of KQ correspondence

  51. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes extracted from motivic generating series Let’s look at motivic generating series restricted to | d | = 1 ( − q ) C ii x i | d | = 1 ( x , q ) = ∑ P Q K 1 − q 2 i ∈ Q 0 Every vertex ( i ∈ Q 0 ) contributes once No interactions between different nodes What is the meaning of P Q K | d | = 1 ( x , q ) ? P. Kucharski Physics and geometry of KQ correspondence

  52. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes extracted from motivic generating series Let’s look at motivic generating series restricted to | d | = 1 ( − q ) C ii x i | d | = 1 ( x , q ) = ∑ P Q K 1 − q 2 i ∈ Q 0 Every vertex ( i ∈ Q 0 ) contributes once No interactions between different nodes What is the meaning of P Q K | d | = 1 ( x , q ) ? P. Kucharski Physics and geometry of KQ correspondence

  53. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes as homology generators Let’s apply KQ change of variables � � P Q K x i = a ai q qi − Cii x = P K | d | = 1 ( x , q ) 1 ( a , q ) x � P K 1 ( a , q ) is an Euler characteristic of HOMFLY-PT homology H ( K ) with set of generators G ( K ) , so � � = ∑ i ∈ G ( K ) a a i q q i ( − 1 ) C ii ( − q ) C ii x i � ∑ � x 1 − q 2 1 − q 2 � i ∈ Q 0 x i = a ai q qi − Cii x Each vertex corresponds to the homology generator KQ change of variables is encoded in homological degrees P. Kucharski Physics and geometry of KQ correspondence

  54. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes as homology generators Let’s apply KQ change of variables � � P Q K x i = a ai q qi − Cii x = P K | d | = 1 ( x , q ) 1 ( a , q ) x � P K 1 ( a , q ) is an Euler characteristic of HOMFLY-PT homology H ( K ) with set of generators G ( K ) , so � � = ∑ i ∈ G ( K ) a a i q q i ( − 1 ) C ii ( − q ) C ii x i � ∑ � x 1 − q 2 1 − q 2 � i ∈ Q 0 x i = a ai q qi − Cii x Each vertex corresponds to the homology generator KQ change of variables is encoded in homological degrees P. Kucharski Physics and geometry of KQ correspondence

  55. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes as homology generators Let’s apply KQ change of variables � � P Q K x i = a ai q qi − Cii x = P K | d | = 1 ( x , q ) 1 ( a , q ) x � P K 1 ( a , q ) is an Euler characteristic of HOMFLY-PT homology H ( K ) with set of generators G ( K ) , so � � = ∑ i ∈ G ( K ) a a i q q i ( − 1 ) C ii ( − q ) C ii x i � ∑ � x 1 − q 2 1 − q 2 � i ∈ Q 0 x i = a ai q qi − Cii x Each vertex corresponds to the homology generator KQ change of variables is encoded in homological degrees P. Kucharski Physics and geometry of KQ correspondence

  56. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes as homology generators Let’s apply KQ change of variables � � P Q K x i = a ai q qi − Cii x = P K | d | = 1 ( x , q ) 1 ( a , q ) x � P K 1 ( a , q ) is an Euler characteristic of HOMFLY-PT homology H ( K ) with set of generators G ( K ) , so � � = ∑ i ∈ G ( K ) a a i q q i ( − 1 ) C ii ( − q ) C ii x i � ∑ � x 1 − q 2 1 − q 2 � i ∈ Q 0 x i = a ai q qi − Cii x Each vertex corresponds to the homology generator KQ change of variables is encoded in homological degrees P. Kucharski Physics and geometry of KQ correspondence

  57. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes as BPS states Generators of uncoloured homology correspond to BPS states [Gukov, Schwarz, Vafa] Quiver nodes correspond to BPS states of T [ L K ] counted by LMOV invariants � � ( − q ) C ii x i = N K 1 ( a , q ) � ∑ 1 − q 2 x � 1 − q 2 � i ∈ Q 0 x i = a ai q qi − Cii x ...or BPS states of T [ Q K ] counted by DT invariants Ω Q K | d | = 1 ( x , q ) ( − q ) C ii x i ∑ = 1 − q 2 1 − q 2 i ∈ Q 0 P. Kucharski Physics and geometry of KQ correspondence

  58. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes as BPS states Generators of uncoloured homology correspond to BPS states [Gukov, Schwarz, Vafa] Quiver nodes correspond to BPS states of T [ L K ] counted by LMOV invariants � � ( − q ) C ii x i = N K 1 ( a , q ) � ∑ 1 − q 2 x � 1 − q 2 � i ∈ Q 0 x i = a ai q qi − Cii x ...or BPS states of T [ Q K ] counted by DT invariants Ω Q K | d | = 1 ( x , q ) ( − q ) C ii x i ∑ = 1 − q 2 1 − q 2 i ∈ Q 0 P. Kucharski Physics and geometry of KQ correspondence

  59. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes as BPS states Generators of uncoloured homology correspond to BPS states [Gukov, Schwarz, Vafa] Quiver nodes correspond to BPS states of T [ L K ] counted by LMOV invariants � � ( − q ) C ii x i = N K 1 ( a , q ) � ∑ 1 − q 2 x � 1 − q 2 � i ∈ Q 0 x i = a ai q qi − Cii x ...or BPS states of T [ Q K ] counted by DT invariants Ω Q K | d | = 1 ( x , q ) ( − q ) C ii x i ∑ = 1 − q 2 1 − q 2 i ∈ Q 0 P. Kucharski Physics and geometry of KQ correspondence

  60. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes as holomorphic disks Recall: M2 branes are BPS states on the flat side and holomorphic disks on the Calabi-Yau side We can reinterpret topological data of KQ change of variables in the language of disks: power r in x r − → #windings around L K power a i in a a i − → #wrappings around base S 2 power q i in q q i − → invariant self-linking# C ii = t i − → linking# between disk and its small shift Quiver nodes correspond to basic disks – holomorphic curves that wind around L K once ( r = 1) P. Kucharski Physics and geometry of KQ correspondence

  61. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes as holomorphic disks Recall: M2 branes are BPS states on the flat side and holomorphic disks on the Calabi-Yau side We can reinterpret topological data of KQ change of variables in the language of disks: power r in x r − → #windings around L K power a i in a a i − → #wrappings around base S 2 power q i in q q i − → invariant self-linking# C ii = t i − → linking# between disk and its small shift Quiver nodes correspond to basic disks – holomorphic curves that wind around L K once ( r = 1) P. Kucharski Physics and geometry of KQ correspondence

  62. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes as holomorphic disks Recall: M2 branes are BPS states on the flat side and holomorphic disks on the Calabi-Yau side We can reinterpret topological data of KQ change of variables in the language of disks: power r in x r − → #windings around L K power a i in a a i − → #wrappings around base S 2 power q i in q q i − → invariant self-linking# C ii = t i − → linking# between disk and its small shift Quiver nodes correspond to basic disks – holomorphic curves that wind around L K once ( r = 1) P. Kucharski Physics and geometry of KQ correspondence

  63. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes as holomorphic disks Recall: M2 branes are BPS states on the flat side and holomorphic disks on the Calabi-Yau side We can reinterpret topological data of KQ change of variables in the language of disks: power r in x r − → #windings around L K power a i in a a i − → #wrappings around base S 2 power q i in q q i − → invariant self-linking# C ii = t i − → linking# between disk and its small shift Quiver nodes correspond to basic disks – holomorphic curves that wind around L K once ( r = 1) P. Kucharski Physics and geometry of KQ correspondence

  64. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes as holomorphic disks Recall: M2 branes are BPS states on the flat side and holomorphic disks on the Calabi-Yau side We can reinterpret topological data of KQ change of variables in the language of disks: power r in x r − → #windings around L K power a i in a a i − → #wrappings around base S 2 power q i in q q i − → invariant self-linking# C ii = t i − → linking# between disk and its small shift Quiver nodes correspond to basic disks – holomorphic curves that wind around L K once ( r = 1) P. Kucharski Physics and geometry of KQ correspondence

  65. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes as holomorphic disks Recall: M2 branes are BPS states on the flat side and holomorphic disks on the Calabi-Yau side We can reinterpret topological data of KQ change of variables in the language of disks: power r in x r − → #windings around L K power a i in a a i − → #wrappings around base S 2 power q i in q q i − → invariant self-linking# C ii = t i − → linking# between disk and its small shift Quiver nodes correspond to basic disks – holomorphic curves that wind around L K once ( r = 1) P. Kucharski Physics and geometry of KQ correspondence

  66. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver nodes as holomorphic disks Recall: M2 branes are BPS states on the flat side and holomorphic disks on the Calabi-Yau side We can reinterpret topological data of KQ change of variables in the language of disks: power r in x r − → #windings around L K power a i in a a i − → #wrappings around base S 2 power q i in q q i − → invariant self-linking# C ii = t i − → linking# between disk and its small shift Quiver nodes correspond to basic disks – holomorphic curves that wind around L K once ( r = 1) P. Kucharski Physics and geometry of KQ correspondence

  67. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Example: unknot quiver vertices and homology For the unknot quiver we have | d | = 1 ( x , q ) = − qx 1 + x 2 Q 01 P 1 − q 2 1 H ( 0 1 ) has two generators with degrees ( a 1 , a 2 ) = ( 2 , 0 ) , ( q 1 , q 2 ) = ( 0 , 0 ) 2 ( t 1 , t 2 ) = ( 1 , 0 ) = ( C 11 , C 22 ) Generators correspond to quiver vertices and their degrees encode KQ change of variables x 1 = a 2 q − 1 x , x 2 = x giving = ∑ i ∈ G ( 0 1 ) a a i q q i ( − 1 ) C ii 1 ( a , q ) x = − a 2 x + x P 0 1 x 1 − q 2 1 − q 2 P. Kucharski Physics and geometry of KQ correspondence

  68. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Example: unknot quiver vertices and homology For the unknot quiver we have | d | = 1 ( x , q ) = − qx 1 + x 2 Q 01 P 1 − q 2 1 H ( 0 1 ) has two generators with degrees ( a 1 , a 2 ) = ( 2 , 0 ) , ( q 1 , q 2 ) = ( 0 , 0 ) 2 ( t 1 , t 2 ) = ( 1 , 0 ) = ( C 11 , C 22 ) Generators correspond to quiver vertices and their degrees encode KQ change of variables x 1 = a 2 q − 1 x , x 2 = x giving = ∑ i ∈ G ( 0 1 ) a a i q q i ( − 1 ) C ii 1 ( a , q ) x = − a 2 x + x P 0 1 x 1 − q 2 1 − q 2 P. Kucharski Physics and geometry of KQ correspondence

  69. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Example: unknot quiver vertices and homology For the unknot quiver we have | d | = 1 ( x , q ) = − qx 1 + x 2 Q 01 P 1 − q 2 1 H ( 0 1 ) has two generators with degrees ( a 1 , a 2 ) = ( 2 , 0 ) , ( q 1 , q 2 ) = ( 0 , 0 ) 2 ( t 1 , t 2 ) = ( 1 , 0 ) = ( C 11 , C 22 ) Generators correspond to quiver vertices and their degrees encode KQ change of variables x 1 = a 2 q − 1 x , x 2 = x giving = ∑ i ∈ G ( 0 1 ) a a i q q i ( − 1 ) C ii 1 ( a , q ) x = − a 2 x + x P 0 1 x 1 − q 2 1 − q 2 P. Kucharski Physics and geometry of KQ correspondence

  70. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Example: unknot quiver vertices and BPS states Two generators of H ( 0 1 ) correspond to two BPS states BPS states of T [ L 0 1 ] are counted by LMOV invariants 1 N 0 1 1 ( a , q ) = 1 − a 2 2 BPS states of T [ Q 0 1 ] are counted by DT invariants Q 01 Ω | d | = 1 ( x , q ) = − qx 1 + x 2 P. Kucharski Physics and geometry of KQ correspondence

  71. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Example: unknot quiver vertices and BPS states Two generators of H ( 0 1 ) correspond to two BPS states BPS states of T [ L 0 1 ] are counted by LMOV invariants 1 N 0 1 1 ( a , q ) = 1 − a 2 2 BPS states of T [ Q 0 1 ] are counted by DT invariants Q 01 Ω | d | = 1 ( x , q ) = − qx 1 + x 2 P. Kucharski Physics and geometry of KQ correspondence

  72. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Example: unknot quiver vertices and BPS states Two generators of H ( 0 1 ) correspond to two BPS states BPS states of T [ L 0 1 ] are counted by LMOV invariants 1 N 0 1 1 ( a , q ) = 1 − a 2 2 BPS states of T [ Q 0 1 ] are counted by DT invariants Q 01 Ω | d | = 1 ( x , q ) = − qx 1 + x 2 P. Kucharski Physics and geometry of KQ correspondence

  73. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Example: unknot quiver vertices and holomorphic disks Two BPS states on the flat side ← → two holomorphic disks on the Calabi-Yau side For the first we have the following intepretation of topological data in KQ change of variables: r = 1 − → winding around L K → wrappings around base S 2 a 1 = 2 − 1 q 1 = 0 − → invariant self-linking C 11 = 1 − → linking between disk and its 2 small shift P. Kucharski Physics and geometry of KQ correspondence

  74. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Example: unknot quiver vertices and holomorphic disks Two BPS states on the flat side ← → two holomorphic disks on the Calabi-Yau side For the first we have the following intepretation of topological data in KQ change of variables: r = 1 − → winding around L K → wrappings around base S 2 a 1 = 2 − 1 q 1 = 0 − → invariant self-linking C 11 = 1 − → linking between disk and its 2 small shift P. Kucharski Physics and geometry of KQ correspondence

  75. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Example: unknot quiver vertices and holomorphic disks Two BPS states on the flat side ← → two holomorphic disks on the Calabi-Yau side For the second we have the following intepretation of topological data in KQ change of variables: r = 1 − → winding around L K → wrappings around base S 2 a 2 = 0 − 1 q 2 = 0 − → invariant self-linking C 22 = 0 − → linking between disk and its 2 small shift P. Kucharski Physics and geometry of KQ correspondence

  76. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver arrows extracted from motivic generating series Let’s look closer at motivic generating series | Q 0 | x d i ( − q ) ∑ 1 ≤ i , j ≤| Q 0 | C i , j d i d j P Q ( x , q ) := ∑ ∏ i ( q 2 ; q 2 ) d i d 1 ,..., d | Q 0 | ≥ 0 i = 1 Dimension d i encodes the number of factors corresponding to i -th vertex #arrows C i , j encodes interactions between vertices i & j Motivic generating series counts all objects that can be made from basic ones (nodes) according to quiver arrows P. Kucharski Physics and geometry of KQ correspondence

  77. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver arrows extracted from motivic generating series Let’s look closer at motivic generating series | Q 0 | x d i ( − q ) ∑ 1 ≤ i , j ≤| Q 0 | C i , j d i d j P Q ( x , q ) := ∑ ∏ i ( q 2 ; q 2 ) d i d 1 ,..., d | Q 0 | ≥ 0 i = 1 Dimension d i encodes the number of factors corresponding to i -th vertex #arrows C i , j encodes interactions between vertices i & j Motivic generating series counts all objects that can be made from basic ones (nodes) according to quiver arrows P. Kucharski Physics and geometry of KQ correspondence

  78. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver arrows extracted from motivic generating series Let’s look closer at motivic generating series | Q 0 | x d i ( − q ) ∑ 1 ≤ i , j ≤| Q 0 | C i , j d i d j P Q ( x , q ) := ∑ ∏ i ( q 2 ; q 2 ) d i d 1 ,..., d | Q 0 | ≥ 0 i = 1 Dimension d i encodes the number of factors corresponding to i -th vertex #arrows C i , j encodes interactions between vertices i & j Motivic generating series counts all objects that can be made from basic ones (nodes) according to quiver arrows P. Kucharski Physics and geometry of KQ correspondence

  79. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver arrows extracted from motivic generating series Let’s look closer at motivic generating series | Q 0 | x d i ( − q ) ∑ 1 ≤ i , j ≤| Q 0 | C i , j d i d j P Q ( x , q ) := ∑ ∏ i ( q 2 ; q 2 ) d i d 1 ,..., d | Q 0 | ≥ 0 i = 1 Dimension d i encodes the number of factors corresponding to i -th vertex #arrows C i , j encodes interactions between vertices i & j Motivic generating series counts all objects that can be made from basic ones (nodes) according to quiver arrows P. Kucharski Physics and geometry of KQ correspondence

  80. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver arrows as disk intersections Geometrically P Q ( x , q ) counts all holomorphic curves that can be made from basic disks according to quiver arrows Dimension vector d i − → #copies of the disk #arrows C i , j − → disk boundaries linking# Example: one pair of arrows corresponds to two bagel disk boundaries with linking# = 1 P. Kucharski Physics and geometry of KQ correspondence

  81. Introduction General idea Physics and geometry of KQ correspondence Physics Summary Geometry Quiver arrows as disk intersections Geometrically P Q ( x , q ) counts all holomorphic curves that can be made from basic disks according to quiver arrows Dimension vector d i − → #copies of the disk #arrows C i , j − → disk boundaries linking# Example: one pair of arrows corresponds to two bagel disk boundaries with linking# = 1 P. Kucharski Physics and geometry of KQ correspondence

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Knots-quivers correspondence, lattice paths, and rational knots Marko Sto si c 1 CAMGSD,

Knots-quivers correspondence, lattice paths, and rational knots Marko Sto si c 1 CAMGSD,

Knots-quivers correspondence, lattice paths, and rational knots Marko Sto si c 1 CAMGSD, Departamento de Matem atica, Instituto Superior T ecnico, Portugal 2 Mathematical Institute SANU, Belgrade, Serbia Warszawa, September 2018

756 views • 59 slides
Quivers, black holes and attractor indices Boris Pioline Conference "Quantum fields, knots

Quivers, black holes and attractor indices Boris Pioline Conference "Quantum fields, knots

Quivers, black holes and attractor indices Boris Pioline Conference "Quantum fields, knots and strings", Warsaw, 25/09/2018 based on 1804.06928 with Sergei Alexandrov (prerequisite for 1808.08479) and on earlier work 2011-15 with Jan

949 views • 37 slides
AdS/CFT Correspondence and Differential Geometry Johanna Erdmenger Max PlanckInstitut f ur

AdS/CFT Correspondence and Differential Geometry Johanna Erdmenger Max PlanckInstitut f ur

. . AdS/CFT Correspondence and Differential Geometry Johanna Erdmenger Max PlanckInstitut f ur Physik, M unchen 1 Outline 1. Introduction: The AdS/CFT correspondence 2. Conformal Anomaly 3. AdS/CFT for field theories with N = 1

275 views • 24 slides
The Geometry of Knots Brandon Shapiro 1 Shruthi Sridhar 2 1 Brandeis University

The Geometry of Knots Brandon Shapiro 1 Shruthi Sridhar 2 1 Brandeis University

The Geometry of Knots Brandon Shapiro 1 Shruthi Sridhar 2 1 Brandeis University bts8394@brandeis.edu 2 Cornell University ss2945@cornell.edu Research work from SMALL REU 2016 MathFest 2016 Shapiro, Sridhar The Geometry of Knots MathFest 2016 1

639 views • 24 slides
Mosaic Knots Mosaic Knots Quantum Knots & Quantum Knot Systems Quantum Knots &

Mosaic Knots Mosaic Knots Quantum Knots & Quantum Knot Systems Quantum Knots &

Quantum Quantum An Intuitive Overview An Intuitive Overview Knots ??? Knots ??? of the Theory of of the Theory of Quantum Knots Quantum Knots Samuel Lomonaco Samuel Lomonaco University of Maryland Baltimore County (UMBC) University of

535 views • 36 slides
13.1 Dynamic Geometry Processing I Hao Li http://cs621.hao-li.com 1 Problem Classification 2

13.1 Dynamic Geometry Processing I Hao Li http://cs621.hao-li.com 1 Problem Classification 2

Spring 2019 CSCI 621: Digital Geometry Processing 13.1 Dynamic Geometry Processing I Hao Li http://cs621.hao-li.com 1 Problem Classification 2 Correspondence Classification 3 Correspondence Classification 4 3-D Reconstruction acquisition

817 views • 69 slides
Snow Leopard Permutations, Even Knots, Odd Knots, Janus Knots, and Restricted Catalan Paths Ben

Snow Leopard Permutations, Even Knots, Odd Knots, Janus Knots, and Restricted Catalan Paths Ben

Snow Leopard Permutations, Even Knots, Odd Knots, Janus Knots, and Restricted Catalan Paths Ben Caffrey, Eric Egge*, Greg Michel, Kailee Rubin, Jon Ver Steegh Carleton College May 21, 2015 My Students Ben Greg Jon Kailee A Problem in

818 views • 61 slides
Knots and Permutations Chaim Even-Zohar , Joel Hass, Nati Linial, Tahl Nowik Knots Unknot Trefoil

Knots and Permutations Chaim Even-Zohar , Joel Hass, Nati Linial, Tahl Nowik Knots Unknot Trefoil

Knots and Permutations Chaim Even-Zohar , Joel Hass, Nati Linial, Tahl Nowik Knots Unknot Trefoil Figure-eight 3 Knot Theory How to represent knots? show that two knots are the same? tell non-equivalent knots apart? study properties of

519 views • 48 slides
The distortion of knots John Pardon Princeton University May 9, 2017 Knots Two views of the figure

The distortion of knots John Pardon Princeton University May 9, 2017 Knots Two views of the figure

The distortion of knots John Pardon Princeton University May 9, 2017 Knots Two views of the figure eight knot. Origins of knot theory can be traced back to Lord Kelvin who thought different elements corresponded to different knots (1860s). First

175 views • 16 slides
Quivers and relations via the Bruhat order of the symmetric group Daiva Pu cinskait e

Quivers and relations via the Bruhat order of the symmetric group Daiva Pu cinskait e

Quivers and relations via the Bruhat order of the symmetric group Daiva Pu cinskait e Florida Atlantic University Conference on Geometric Methods in Representation Theory Iowa November 22-24, 2014 Daiva Pu cinskait e Quivers and

1.43k views • 105 slides
Minimal and normal surfaces There is a correspondence between the theory of minimal surfaces in

Minimal and normal surfaces There is a correspondence between the theory of minimal surfaces in

Minimal and normal surfaces There is a correspondence between the theory of minimal surfaces in differential geometry and the theory of normal surfaces. We will explore the correspondence and use it to derive: Problem : 3-SPHERE RECOGNITION

1.36k views • 77 slides
Noncommutative Geometry in Physics Ali H. Chamseddine American University of Beirut (AUB) &

Noncommutative Geometry in Physics Ali H. Chamseddine American University of Beirut (AUB) &

Noncommutative Geometry in Physics Ali H. Chamseddine American University of Beirut (AUB) & Instiut des Hautes Etudes Scientifique (IHES) Frontiers of Fundamental Physics FFP14, Math. Phys. July 16, 2014 1 Introduction A Brief

584 views • 43 slides
Business Correspondence Tone! Dr Bean ( ) at Business Correspondence Tone! Tone

Business Correspondence Tone! Dr Bean ( ) at Business Correspondence Tone! Tone

Tone 2 emails Strategies for improving tone Conditional Homework https://github.com/drbean/curriculum/trunk/correspondence Mon Mar 13 09:36:25 CST 2017 Business Correspondence Tone! Dr Bean ( ) at Business Correspondence

557 views • 19 slides
Disk complexes, arc complexes, and knots Darryl McCullough University of Oklahoma William Rowan

Disk complexes, arc complexes, and knots Darryl McCullough University of Oklahoma William Rowan

Disk complexes, arc complexes, and knots Darryl McCullough University of Oklahoma William Rowan Hamilton Geometry and Topology Workshop Trinity College August 28, 2008 1 Topics: I. The tree of knot tunnels: a classification of all tunnels

465 views • 34 slides
Free fields, Quivers and Riemann surfaces Sanjaye Ramgoolam Queen Mary, University of London 11

Free fields, Quivers and Riemann surfaces Sanjaye Ramgoolam Queen Mary, University of London 11

Free fields, Quivers and Riemann surfaces Sanjaye Ramgoolam Queen Mary, University of London 11 September 2013 Quivers as Calculators : Counting, correlators and Riemann surfaces, arxiv:1301.1980, J. Pasukonis, S. Ramgoolam Introduction

922 views • 60 slides
The Teichm uller TQFT Volume Conjecture for Twist Knots Fathi Ben Aribi UCLouvain 24th

The Teichm uller TQFT Volume Conjecture for Twist Knots Fathi Ben Aribi UCLouvain 24th

Introduction Topology (triangulations) Geometry (angles) Algebra (TQFT) Analysis (asymptotics) Conclusion The Teichm uller TQFT Volume Conjecture for Twist Knots Fathi Ben Aribi UCLouvain 24th September 2020 (joint work with Fran

329 views • 22 slides
RAQIS-2016 NIKITA NEKRASOV Geneva, August 23, 2016 RAQIS2016 Geneva August 23 Nikita

RAQIS-2016 NIKITA NEKRASOV Geneva, August 23, 2016 RAQIS2016 Geneva August 23 Nikita

RAQIS-2016 NIKITA NEKRASOV Geneva, August 23, 2016 RAQIS2016 Geneva August 23 Nikita Nekrasov Plan Quantum integrable systems from gauge theories Gauge theories from branes Y , Q and Q -observables

1.29k views • 128 slides
Comparison of Hydrodynamics and Kinetic Transport Theory for p+A and A+A Collisions Carsten

Comparison of Hydrodynamics and Kinetic Transport Theory for p+A and A+A Collisions Carsten

Comparison of Hydrodynamics and Kinetic Transport Theory for p+A and A+A Collisions Carsten Greiner with Kai Gallmeister, Harri Niemi , Dirk Rischke Bormio 56 th winter meeting, january 2018 Hydrodynamics & BAMPS Initial state specific

587 views • 32 slides
Combinatorics in Mayers theory of cluster and virial expansions Quantum Many Body Systems

Combinatorics in Mayers theory of cluster and virial expansions Quantum Many Body Systems

Combinatorics in Mayers theory of cluster and virial expansions Quantum Many Body Systems Workshop - Warwick University Stephen James Tate University of Warwick March 2014 S. J. Tate Combinatorics of Mayer and Virial Expansions Context

102 views • 8 slides
Models through Tensor Networks 2019.12.04 Naoki KAWASHIMA (ISSP) Tensor Network (TN) (1)

Models through Tensor Networks 2019.12.04 Naoki KAWASHIMA (ISSP) Tensor Network (TN) (1)

TNSAA2019-2020 2019.12.04-06 Understanding Kitaev-Related Models through Tensor Networks 2019.12.04 Naoki KAWASHIMA (ISSP) Tensor Network (TN) (1) Classical statistical mechanical models are TNs --- corner

282 views • 25 slides
Introduction to the Correlation Decay Method Yitong Yin Nanjing University orkshop, July 20,

Introduction to the Correlation Decay Method Yitong Yin Nanjing University orkshop, July 20,

Introduction to the Correlation Decay Method Yitong Yin Nanjing University orkshop, July 20, 2018 Introduction to Partition Functions W Computational Aspects of Partition Functions, CIB@EPFL July 20: Introduction to the correlation decay

837 views • 51 slides
Multiplicity Fluctuations Josef Uchytil FNSPE CTU in Prague 27. 9. 2017 Josef Uchytil (FNSPE

Multiplicity Fluctuations Josef Uchytil FNSPE CTU in Prague 27. 9. 2017 Josef Uchytil (FNSPE

Multiplicity Fluctuations Josef Uchytil FNSPE CTU in Prague 27. 9. 2017 Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 1 / 25 Outline Statistical moments 1 Multiplicity fluctuations within a simple model 2

447 views • 25 slides
Bulk viscosity, spectra, and flow in heavy ion collisions Thomas Schaefer & Kevin Dusling,

Bulk viscosity, spectra, and flow in heavy ion collisions Thomas Schaefer & Kevin Dusling,

Bulk viscosity, spectra, and flow in heavy ion collisions Thomas Schaefer & Kevin Dusling, North Carolina State University Why bulk viscosity? 0.4 0.35 Ideal Gas 0.3 2 c s 0.25 Lattice 0.2 0.15 0.1 100 150 200 250 300 350 400

443 views • 22 slides
Phase Transition for the mixing time of Glauber Dynamics on Regular Trees at Reconstruction:

Phase Transition for the mixing time of Glauber Dynamics on Regular Trees at Reconstruction:

Phase Transition for the mixing time of Glauber Dynamics on Regular Trees at Reconstruction: Colorings and Independent Sets. Juan Vera Tilburg University, Netherlands J OINT WORK WITH Ricardo Restrepo , Daniel Stefankovic , Eric Vigoda

840 views • 44 slides

More recommend