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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

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