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

▶
physics and geometry of knots quivers correspondence
SMART_READER_LITE
LIVE PREVIEW

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

Jun 19, 2023 592 likes •1.68k 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

slide-1
SLIDE 1

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

slide-2
SLIDE 2

Introduction Physics and geometry of KQ correspondence Summary

Collaboration

This talk is based on [1810.xxxxx] with: Pietro Longhi Tobias Ekholm with crucial references to [1707.02991, 1707.0417] with: Markus Reineke Marko Stošić Piotr Sułkowski

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-3
SLIDE 3

Introduction Physics and geometry of KQ correspondence Summary

Outline

1

Introduction Knots Quivers Knots-quivers correspondence

2

Physics and geometry of KQ correspondence General idea Physics Geometry

3

Summary

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-4
SLIDE 4

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Outline

1

Introduction Knots Quivers Knots-quivers correspondence

2

Physics and geometry of KQ correspondence General idea Physics Geometry

3

Summary

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-5
SLIDE 5

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Setting the stage

Ooguri-Vafa M-theory system: Spacetime Resolved conifold x R1,4 M5 brane Lagrangian submanifold LK x R1,2

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-6
SLIDE 6

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Calabi-Yau side (resolved conifold)

Topological strings ↔ Chern-Simons theory ↔ ↔ Knot theory [Witten] Partition function=HOMFLY-PT gen. series Z = PK(x,a,q) =

∞

∑

r=0

PK

r (a,q)xr

M5 brane Lagrangian submanifold LK M2 branes Holomorphic curves

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-7
SLIDE 7

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Calabi-Yau side (resolved conifold)

Topological strings ↔ Chern-Simons theory ↔ ↔ Knot theory [Witten] Partition function=HOMFLY-PT gen. series Z = PK(x,a,q) =

∞

∑

r=0

PK

r (a,q)xr

M5 brane Lagrangian submanifold LK M2 branes Holomorphic curves

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-8
SLIDE 8

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Calabi-Yau side (resolved conifold)

Topological strings ↔ Chern-Simons theory ↔ ↔ Knot theory [Witten] Partition function=HOMFLY-PT gen. series Z = PK(x,a,q) =

∞

∑

r=0

PK

r (a,q)xr

M5 brane Lagrangian submanifold LK M2 branes Holomorphic curves

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-9
SLIDE 9

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Flat side (R1,4)

T[LK]: 3d N = 2 eff. theory on R1,2 Partition function=generating function of Labastida-Mariño-Ooguri-Vafa invariants NK

r,i,j

Z = Exp

  • ∑r,i,j NK

r,i,jxraiqj

1−q2

  • M5 brane

R1,2 M2 branes BPS states BPS states are counted by LMOV invariants ⇒ LMOV integrality conjecture

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-10
SLIDE 10

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Flat side (R1,4)

T[LK]: 3d N = 2 eff. theory on R1,2 Partition function=generating function of Labastida-Mariño-Ooguri-Vafa invariants NK

r,i,j

Z = Exp

  • ∑r,i,j NK

r,i,jxraiqj

1−q2

  • M5 brane

R1,2 M2 branes BPS states BPS states are counted by LMOV invariants ⇒ LMOV integrality conjecture

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-11
SLIDE 11

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Flat side (R1,4)

T[LK]: 3d N = 2 eff. theory on R1,2 Partition function=generating function of Labastida-Mariño-Ooguri-Vafa invariants NK

r,i,j

Z = Exp

  • ∑r,i,j NK

r,i,jxraiqj

1−q2

  • M5 brane

R1,2 M2 branes BPS states BPS states are counted by LMOV invariants ⇒ LMOV integrality conjecture

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-12
SLIDE 12

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Flat side (R1,4)

T[LK]: 3d N = 2 eff. theory on R1,2 Partition function=generating function of Labastida-Mariño-Ooguri-Vafa invariants NK

r,i,j

Z = Exp

  • ∑r,i,j NK

r,i,jxraiqj

1−q2

  • M5 brane

R1,2 M2 branes BPS states BPS states are counted by LMOV invariants ⇒ LMOV integrality conjecture

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-13
SLIDE 13

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Outline

1

Introduction Knots Quivers Knots-quivers correspondence

2

Physics and geometry of KQ correspondence General idea Physics Geometry

3

Summary

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-14
SLIDE 14

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Quivers

Quiver Q = (Q0,Q1)

Q0 is a set of vertices Q1 is a set of arrows between them (loops allowed)

Matrix form: Cij gives the number of arrows between vertices i and j

If Cij = Cji we call the quiver symmetric

Quiver representations: (Q0,Q1) − → (Vector spaces, Linear maps) Example from the picture:

Q0 = {1,2} − → vector spaces V1 and V2 of dimensions d1 and d2 (dimension vector d = (d1,d2)) Q1 = {1 → 1} − → linear map V1 → V1

1 2

1

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-15
SLIDE 15

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Quivers

Quiver Q = (Q0,Q1)

Q0 is a set of vertices Q1 is a set of arrows between them (loops allowed)

Matrix form: Cij gives the number of arrows between vertices i and j

If Cij = Cji we call the quiver symmetric

Quiver representations: (Q0,Q1) − → (Vector spaces, Linear maps) Example from the picture:

Q0 = {1,2} − → vector spaces V1 and V2 of dimensions d1 and d2 (dimension vector d = (d1,d2)) Q1 = {1 → 1} − → linear map V1 → V1

1 2

1

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-16
SLIDE 16

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Quivers

Quiver Q = (Q0,Q1)

Q0 is a set of vertices Q1 is a set of arrows between them (loops allowed)

Matrix form: Cij gives the number of arrows between vertices i and j

If Cij = Cji we call the quiver symmetric

Quiver representations: (Q0,Q1) − → (Vector spaces, Linear maps) Example from the picture:

Q0 = {1,2} − → vector spaces V1 and V2 of dimensions d1 and d2 (dimension vector d = (d1,d2)) Q1 = {1 → 1} − → linear map V1 → V1

1 2

1

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-17
SLIDE 17

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Quivers

Quiver Q = (Q0,Q1)

Q0 is a set of vertices Q1 is a set of arrows between them (loops allowed)

Matrix form: Cij gives the number of arrows between vertices i and j

If Cij = Cji we call the quiver symmetric

Quiver representations: (Q0,Q1) − → (Vector spaces, Linear maps) Example from the picture:

Q0 = {1,2} − → vector spaces V1 and V2 of dimensions d1 and d2 (dimension vector d = (d1,d2)) Q1 = {1 → 1} − → linear map V1 → V1

1 2

1

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-18
SLIDE 18

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Quivers

Quiver Q = (Q0,Q1)

Q0 is a set of vertices Q1 is a set of arrows between them (loops allowed)

Matrix form: Cij gives the number of arrows between vertices i and j

If Cij = Cji we call the quiver symmetric

Quiver representations: (Q0,Q1) − → (Vector spaces, Linear maps) Example from the picture:

Q0 = {1,2} − → vector spaces V1 and V2 of dimensions d1 and d2 (dimension vector d = (d1,d2)) Q1 = {1 → 1} − → linear map V1 → V1

1 2

1

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-19
SLIDE 19

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Motivic generating series and DT invariants

Motivic generating series encodes information about moduli spaces of representations of Q PQ(x,q) =

∑

d1,...,d|Q0|≥0

(−q)∑1≤i,j≤|Q0| Ci,jdidj

|Q0|

∏

i=1

xdi

i

(q2;q2)di =Exp ΩQ(x,q) 1−q2

  • ΩQ(x,q) is a generating function of motivic

Donaldson-Thomas (DT) invariants Exp is the plethystic exponential Exp

  • xraiqj

=(1−xraiqj)−1 Exp(f +g) =Exp(f )Exp(g)

1 2

1

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-20
SLIDE 20

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Motivic generating series and DT invariants

Motivic generating series encodes information about moduli spaces of representations of Q PQ(x,q) =

∑

d1,...,d|Q0|≥0

(−q)∑1≤i,j≤|Q0| Ci,jdidj

|Q0|

∏

i=1

xdi

i

(q2;q2)di =Exp ΩQ(x,q) 1−q2

  • ΩQ(x,q) is a generating function of motivic

Donaldson-Thomas (DT) invariants (q2;q2)di is the q-Pochhammer symbol (z;q2)r :=

r−1

∏

i=0

(1−zq2i) = (1−z)

  • 1−zq2

...(1−zq2(r−1))

1 2

1

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-21
SLIDE 21

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Motivic generating series and DT invariants

Motivic generating series encodes information about moduli spaces of representations of Q PQ(x,q) =

∑

d1,...,d|Q0|≥0

(−q)∑1≤i,j≤|Q0| Ci,jdidj

|Q0|

∏

i=1

xdi

i

(q2;q2)di =Exp ΩQ(x,q) 1−q2

  • Example from the picture

PQ(x1,x2,q) = ∑

d1,d2≥0

(−q)d2

1

xd1

1

(q2;q2)d1 xd2

2

(q2;q2)d2 =Exp −qx1 +x2 1−q2

  • 1

2

1

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-22
SLIDE 22

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Outline

1

Introduction Knots Quivers Knots-quivers correspondence

2

Physics and geometry of KQ correspondence General idea Physics Geometry

3

Summary

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-23
SLIDE 23

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Knots-quivers correspondence

1 2

Knots-quivers (KQ) correspondence is an equality PK(x,a,q) = PQK (x,q)

  • xi=aai qqi −Cii x

PK(x,a,q) – HOMFLY-PT generating series of knot K PQK (x,q) – motivic generating series of respective quiver QK xi = aai qqi −Cii x is a change of variables

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-24
SLIDE 24

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Knots-quivers correspondence

1 2

Knots-quivers (KQ) correspondence is an equality PK(x,a,q) = PQK (x,q)

  • xi=aai qqi −Cii x

PK(x,a,q) – HOMFLY-PT generating series of knot K PQK (x,q) – motivic generating series of respective quiver QK xi = aai qqi −Cii x is a change of variables

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-25
SLIDE 25

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Knots-quivers correspondence

1 2

Knots-quivers (KQ) correspondence is an equality PK(x,a,q) = PQK (x,q)

  • xi=aai qqi −Cii x

PK(x,a,q) – HOMFLY-PT generating series of knot K PQK (x,q) – motivic generating series of respective quiver QK xi = aai qqi −Cii x is a change of variables

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-26
SLIDE 26

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

Knots-quivers correspondence

1 2

Knots-quivers (KQ) correspondence is an equality PK(x,a,q) = PQK (x,q)

  • xi=aai qqi −Cii x

PK(x,a,q) – HOMFLY-PT generating series of knot K PQK (x,q) – motivic generating series of respective quiver QK xi = aai qqi −Cii x is a change of variables

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-27
SLIDE 27

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

KQ correspondence - unknot example

1 2

P01(x,a,q) =

∞

∑

r=0

(a2;q2)r (q2;q2)r xr =1+ 1−a2 1−q2 x +... PQ01(x1,x2,q) = ∑

d1,d2≥0

(−q)d2

1

xd1

1

(q2;q2)d1 xd2

2

(q2;q2)d2 =1+ −qx1 +x2 1−q2 +...

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-28
SLIDE 28

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

KQ correspondence - unknot example

1 2

We can check that P01(x,a,q) = PQ01(x1,x2,q)

  • x1=a2q−1x, x2=x

1+ 1−a2 1−q2 x +... = 1+ −qx1 +x2 1−q2 +...

  • x1=a2q−1x, x2=x

Comparing with xi = aaiqqi−Ciix we get (a1,a2) = (2,0) (q1,q2) = (0,0)

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-29
SLIDE 29

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

KQ correspondence - unknot example

1 2

We can check that P01(x,a,q) = PQ01(x1,x2,q)

  • x1=a2q−1x, x2=x

1+ 1−a2 1−q2 x +... = 1+ −qx1 +x2 1−q2 +...

  • x1=a2q−1x, x2=x

Comparing with xi = aaiqqi−Ciix we get (a1,a2) = (2,0) (q1,q2) = (0,0)

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-30
SLIDE 30

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

KQ correspondence and BPS states

1 2

We can also look at the level of BPS states: Exp NK(x,a,q) 1−q2

  • = PK = PQK = Exp

ΩQK (x,q) 1−q2

  • xi=aai qqi −Cii x

NK(x,a,q) = ΩQK (x,q)

  • xi=aai qqi −Cii x

Integrality of DT invariants for symmetric quivers [Kontsevich-Soibelman, Efimov] implies LMOV conjecture

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-31
SLIDE 31

Introduction Physics and geometry of KQ correspondence Summary Knots Quivers Knots-quivers correspondence

KQ correspondence and BPS states

1 2

We can also look at the level of BPS states: Exp NK(x,a,q) 1−q2

  • = PK = PQK = Exp

ΩQK (x,q) 1−q2

  • xi=aai qqi −Cii x

NK(x,a,q) = ΩQK (x,q)

  • xi=aai qqi −Cii x

Integrality of DT invariants for symmetric quivers [Kontsevich-Soibelman, Efimov] implies LMOV conjecture

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-32
SLIDE 32

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Outline

1

Introduction Knots Quivers Knots-quivers correspondence

2

Physics and geometry of KQ correspondence General idea Physics Geometry

3

Summary

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-33
SLIDE 33

Introduction Physics and geometry of KQ correspondence Summary General idea Physics 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

slide-34
SLIDE 34

Introduction Physics and geometry of KQ correspondence Summary General idea Physics 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

slide-35
SLIDE 35

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Towards physics of KQ correspondence

Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ Physics 3d N = 2 T[LK]

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-36
SLIDE 36

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Missing element

Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ ↓ Physics 3d N = 2 T[LK]

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-37
SLIDE 37

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Missing element

Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ ↓ Physics 3d N = 2 T[LK] 3d N = 2 T[QK]

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-38
SLIDE 38

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Missing element

Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ ↓ Physics 3d N = 2 T[LK] → 3d N = 2 T[QK]

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-39
SLIDE 39

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Geometric interpretations

Knots Quivers Math HOMFLY-PT gen. series Motivic gen. series տ ր Geometric interpretations ւ ց Physics 3d N = 2 T[LK] 3d N = 2 T[QK]

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-40
SLIDE 40

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Outline

1

Introduction Knots Quivers Knots-quivers correspondence

2

Physics and geometry of KQ correspondence General idea Physics Geometry

3

Summary

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-41
SLIDE 41

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Construction of knot complement theory

We can construct 3d N = 2 knot complement theory T[MK] basing on large colour and classical limit of HOMFLY-PT polynomial [Fuji, Gukov, Sułkowski] PK

r (a,q) ¯ h→0

− →

q2r fixed exp

1 2¯ h

  • WT[MK] +O(¯

h)

  • WT[MK] is a twisted superpotential

Li2(...) ← → chiral field κ 2 log(...)log(...) ← → CS coupling

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-42
SLIDE 42

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Recall: what is T[LK]

T[LK]: 3d N = 2 eff. theory on R1,2 Partition function=generating function of Labastida-Marino-Ooguri-Vafa invariants NK

r,i,j

Z = Exp

  • ∑r,i,j NK

r,i,jxraiqj

1−q2

  • M5 brane

R1,2 M2 branes BPS states BPS states are counted by LMOV invariants ⇒ LMOV integrality conjecture

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-43
SLIDE 43

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Recall: what is T[LK]

T[LK]: 3d N = 2 eff. theory on R1,2 Partition function=generating function of Labastida-Marino-Ooguri-Vafa invariants NK

r,i,j

Z = Exp

  • ∑r,i,j NK

r,i,jxraiqj

1−q2

  • M5 brane

R1,2 M2 branes BPS states BPS states are counted by LMOV invariants ⇒ LMOV integrality conjecture

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-44
SLIDE 44

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Construction of T[LK] theory

Idea We construct T[LK] analogously to T[MK], but using PK(x,a,q) PK(x,a,q)

¯ h→0

− →

q2r→y

  • dy exp

1 2¯ h

  • WT[LK] +O(¯

h)

  • WT[LK] =

WT[MK] +logx logy Integral

dy means that U(1) symmetry

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

slide-45
SLIDE 45

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Construction of T[LK] theory

Idea We construct T[LK] analogously to T[MK], but using PK(x,a,q) PK(x,a,q)

¯ h→0

− →

q2r→y

  • dy exp

1 2¯ h

  • WT[LK] +O(¯

h)

  • WT[LK] =

WT[MK] +logx logy Integral

dy means that U(1) symmetry

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

slide-46
SLIDE 46

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Construction of T[LK] theory

Idea We construct T[LK] analogously to T[MK], but using PK(x,a,q) PK(x,a,q)

¯ h→0

− →

q2r→y

  • dy exp

1 2¯ h

  • WT[LK] +O(¯

h)

  • WT[LK] =

WT[MK] +logx logy Integral

dy means that U(1) symmetry

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

slide-47
SLIDE 47

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Construction of T[LK] theory

Idea We construct T[LK] analogously to T[MK], but using PK(x,a,q) PK(x,a,q)

¯ h→0

− →

q2r→y

  • dy exp

1 2¯ h

  • WT[LK] +O(¯

h)

  • WT[LK] =

WT[MK] +logx logy Integral

dy means that U(1) symmetry

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

slide-48
SLIDE 48

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Example: T[L01]

For the unknot we have P01(x,a,q)

¯ h→0

− →

q2r→y

  • dy exp

1 2¯ h

  • WT[L01] +O(¯

h)

  • WT[L01] = Li2 (y)+Li2
  • y−1a−2

+logx logy T[L01] 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 S2 in the conifold

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-49
SLIDE 49

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Example: T[L01]

For the unknot we have P01(x,a,q)

¯ h→0

− →

q2r→y

  • dy exp

1 2¯ h

  • WT[L01] +O(¯

h)

  • WT[L01] = Li2 (y)+Li2
  • y−1a−2

+logx logy T[L01] 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 S2 in the conifold

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-50
SLIDE 50

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Example: T[L01]

For the unknot we have P01(x,a,q)

¯ h→0

− →

q2r→y

  • dy exp

1 2¯ h

  • WT[L01] +O(¯

h)

  • WT[L01] = Li2 (y)+Li2
  • y−1a−2

+logx logy T[L01] 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 S2 in the conifold

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-51
SLIDE 51

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Recall of general idea - now we are here

Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ Physics 3d N = 2 T[LK]

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-52
SLIDE 52

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

We look for the missing element

Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ ↓ Physics 3d N = 2 T[LK]

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-53
SLIDE 53

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

We look for the missing element

Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ ↓ Physics 3d N = 2 T[LK] 3d N = 2 T[QK]

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-54
SLIDE 54

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Missing element: T[QK] theory

Idea Consider large colour and classical limit of motivic generating series! PQK (x,q)

¯ h→0

− →

q2di →yi

  • ∏

i

dyi exp 1 2¯ h

  • WT[QK] +O(¯

h)

  • WT[QK] = ∑

i

[Li2 (yi)+logxi logyi]+∑

i,j

Ci,j 2 logyi logyj, Gauge group: U(1)#vertices Matter content: one chiral for each vertex CS couplings given by Ci,j = #arrows

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-55
SLIDE 55

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Missing element: T[QK] theory

Idea Consider large colour and classical limit of motivic generating series! PQK (x,q)

¯ h→0

− →

q2di →yi

  • ∏

i

dyi exp 1 2¯ h

  • WT[QK] +O(¯

h)

  • WT[QK] = ∑

i

[Li2 (yi)+logxi logyi]+∑

i,j

Ci,j 2 logyi logyj, Gauge group: U(1)#vertices Matter content: one chiral for each vertex CS couplings given by Ci,j = #arrows

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-56
SLIDE 56

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Missing element: T[QK] theory

Idea Consider large colour and classical limit of motivic generating series! PQK (x,q)

¯ h→0

− →

q2di →yi

  • ∏

i

dyi exp 1 2¯ h

  • WT[QK] +O(¯

h)

  • WT[QK] = ∑

i

[Li2 (yi)+logxi logyi]+∑

i,j

Ci,j 2 logyi logyj, Gauge group: U(1)#vertices Matter content: one chiral for each vertex CS couplings given by Ci,j = #arrows

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-57
SLIDE 57

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Missing element: T[QK] theory

Idea Consider large colour and classical limit of motivic generating series! PQK (x,q)

¯ h→0

− →

q2di →yi

  • ∏

i

dyi exp 1 2¯ h

  • WT[QK] +O(¯

h)

  • WT[QK] = ∑

i

[Li2 (yi)+logxi logyi]+∑

i,j

Ci,j 2 logyi logyj, Gauge group: U(1)#vertices Matter content: one chiral for each vertex CS couplings given by Ci,j = #arrows

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-58
SLIDE 58

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Missing element: T[QK] theory

Idea Consider large colour and classical limit of motivic generating series! PQK (x,q)

¯ h→0

− →

q2di →yi

  • ∏

i

dyi exp 1 2¯ h

  • WT[QK] +O(¯

h)

  • WT[QK] = ∑

i

[Li2 (yi)+logxi logyi]+∑

i,j

Ci,j 2 logyi logyj, Gauge group: U(1)#vertices Matter content: one chiral for each vertex CS couplings given by Ci,j = #arrows

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-59
SLIDE 59

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Example: T[Q01]

For the unknot quiver we have PQ01(x1,x2,q)

¯ h→0

− →

q2di →yi

  • dy1dy2 exp

1 2¯ h

  • WT[Q01] +O(¯

h)

  • WT[Q01] = Li2 (y1)+Li2 (y2)+logx1 logy1 +logx2 logy2 + 1

2 logy1 logy1 T[Q01] is a U(1)(1) ×U(1)(2) gauge theory with one chiral field for each group CS level one for U(1)(1), consistent with C 01 = 1

  • 1

2

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-60
SLIDE 60

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Example: T[Q01]

For the unknot quiver we have PQ01(x1,x2,q)

¯ h→0

− →

q2di →yi

  • dy1dy2 exp

1 2¯ h

  • WT[Q01] +O(¯

h)

  • WT[Q01] = Li2 (y1)+Li2 (y2)+logx1 logy1 +logx2 logy2 + 1

2 logy1 logy1 T[Q01] is a U(1)(1) ×U(1)(2) gauge theory with one chiral field for each group CS level one for U(1)(1), consistent with C 01 = 1

  • 1

2

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-61
SLIDE 61

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Example: T[Q01]

For the unknot quiver we have PQ01(x1,x2,q)

¯ h→0

− →

q2di →yi

  • dy1dy2 exp

1 2¯ h

  • WT[Q01] +O(¯

h)

  • WT[Q01] = Li2 (y1)+Li2 (y2)+logx1 logy1 +logx2 logy2 + 1

2 logy1 logy1 T[Q01] is a U(1)(1) ×U(1)(2) gauge theory with one chiral field for each group CS level one for U(1)(1), consistent with C 01 = 1

  • 1

2

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-62
SLIDE 62

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Recall of general idea - now we are here

Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ ↓ Physics 3d N = 2 T[LK] 3d N = 2 T[QK]

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-63
SLIDE 63

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

We want the physical interpretation of KQ correspondence

Knots Quivers Math HOMFLY-PT gen. series → Motivic gen. series ↓ ↓ Physics 3d N = 2 T[LK] → 3d N = 2 T[QK]

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-64
SLIDE 64

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Duality

Physical meaning of the knots-quivers correspondence is a duality between two 3d N = 2 theories: T[LK] ← → T[QK] We already know that Z (T[LK]) = PK(x,a,q) We can calculate the partition function of T[QK] and indeed Z (T[QK]) = PQK (x,q)

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-65
SLIDE 65

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Duality

Physical meaning of the knots-quivers correspondence is a duality between two 3d N = 2 theories: T[LK] ← → T[QK] We already know that Z (T[LK]) = PK(x,a,q) We can calculate the partition function of T[QK] and indeed Z (T[QK]) = PQK (x,q)

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-66
SLIDE 66

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Duality

Physical meaning of the knots-quivers correspondence is a duality between two 3d N = 2 theories: T[LK] ← → T[QK] We already know that Z (T[LK]) = PK(x,a,q) We can calculate the partition function of T[QK] and indeed Z (T[QK]) = PQK (x,q)

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-67
SLIDE 67

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Outline

1

Introduction Knots Quivers Knots-quivers correspondence

2

Physics and geometry of KQ correspondence General idea Physics Geometry

3

Summary

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-68
SLIDE 68

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver nodes extracted from motivic generating series

Let’s look at motivic generating series restricted to |d| = 1 PQK

|d|=1(x,q) = ∑ i∈Q0

(−q)Ciixi 1−q2 Every vertex (i ∈ Q0) contributes once No interactions between different nodes What is the meaning of PQK

|d|=1(x,q)?

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-69
SLIDE 69

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver nodes extracted from motivic generating series

Let’s look at motivic generating series restricted to |d| = 1 PQK

|d|=1(x,q) = ∑ i∈Q0

(−q)Ciixi 1−q2 Every vertex (i ∈ Q0) contributes once No interactions between different nodes What is the meaning of PQK

|d|=1(x,q)?

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-70
SLIDE 70

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver nodes extracted from motivic generating series

Let’s look at motivic generating series restricted to |d| = 1 PQK

|d|=1(x,q) = ∑ i∈Q0

(−q)Ciixi 1−q2 Every vertex (i ∈ Q0) contributes once No interactions between different nodes What is the meaning of PQK

|d|=1(x,q)?

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-71
SLIDE 71

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver nodes extracted from motivic generating series

Let’s look at motivic generating series restricted to |d| = 1 PQK

|d|=1(x,q) = ∑ i∈Q0

(−q)Ciixi 1−q2 Every vertex (i ∈ Q0) contributes once No interactions between different nodes What is the meaning of PQK

|d|=1(x,q)?

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-72
SLIDE 72

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver nodes as homology generators

Let’s apply KQ change of variables PQK

|d|=1(x,q)

  • xi=aai qqi −Cii x = PK

1 (a,q)x

PK

1 (a,q) is an Euler characteristic of HOMFLY-PT

homology H (K) with set of generators G (K), so

∑

i∈Q0

(−q)Ciixi 1−q2

  • xi=aai qqi −Cii x

= ∑i∈G (K) aaiqqi(−1)Cii 1−q2 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

slide-73
SLIDE 73

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver nodes as homology generators

Let’s apply KQ change of variables PQK

|d|=1(x,q)

  • xi=aai qqi −Cii x = PK

1 (a,q)x

PK

1 (a,q) is an Euler characteristic of HOMFLY-PT

homology H (K) with set of generators G (K), so

∑

i∈Q0

(−q)Ciixi 1−q2

  • xi=aai qqi −Cii x

= ∑i∈G (K) aaiqqi(−1)Cii 1−q2 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

slide-74
SLIDE 74

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver nodes as homology generators

Let’s apply KQ change of variables PQK

|d|=1(x,q)

  • xi=aai qqi −Cii x = PK

1 (a,q)x

PK

1 (a,q) is an Euler characteristic of HOMFLY-PT

homology H (K) with set of generators G (K), so

∑

i∈Q0

(−q)Ciixi 1−q2

  • xi=aai qqi −Cii x

= ∑i∈G (K) aaiqqi(−1)Cii 1−q2 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

slide-75
SLIDE 75

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver nodes as homology generators

Let’s apply KQ change of variables PQK

|d|=1(x,q)

  • xi=aai qqi −Cii x = PK

1 (a,q)x

PK

1 (a,q) is an Euler characteristic of HOMFLY-PT

homology H (K) with set of generators G (K), so

∑

i∈Q0

(−q)Ciixi 1−q2

  • xi=aai qqi −Cii x

= ∑i∈G (K) aaiqqi(−1)Cii 1−q2 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

slide-76
SLIDE 76

Introduction Physics and geometry of KQ correspondence Summary General idea Physics 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[LK] counted by LMOV invariants

∑

i∈Q0

(−q)Ciixi 1−q2

  • xi=aai qqi −Cii x

= NK

1 (a,q)

1−q2 x ...or BPS states of T[QK] counted by DT invariants

∑

i∈Q0

(−q)Ciixi 1−q2 = ΩQK

|d|=1(x,q)

1−q2

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-77
SLIDE 77

Introduction Physics and geometry of KQ correspondence Summary General idea Physics 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[LK] counted by LMOV invariants

∑

i∈Q0

(−q)Ciixi 1−q2

  • xi=aai qqi −Cii x

= NK

1 (a,q)

1−q2 x ...or BPS states of T[QK] counted by DT invariants

∑

i∈Q0

(−q)Ciixi 1−q2 = ΩQK

|d|=1(x,q)

1−q2

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-78
SLIDE 78

Introduction Physics and geometry of KQ correspondence Summary General idea Physics 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[LK] counted by LMOV invariants

∑

i∈Q0

(−q)Ciixi 1−q2

  • xi=aai qqi −Cii x

= NK

1 (a,q)

1−q2 x ...or BPS states of T[QK] counted by DT invariants

∑

i∈Q0

(−q)Ciixi 1−q2 = ΩQK

|d|=1(x,q)

1−q2

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-79
SLIDE 79

Introduction Physics and geometry of KQ correspondence Summary General idea Physics 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 xr − → #windings around LK power ai in aai − → #wrappings around base S2 power qi in qqi − → invariant self-linking# Cii = ti − → linking# between disk and its small shift

Quiver nodes correspond to basic disks – holomorphic curves that wind around LK once (r = 1)

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-80
SLIDE 80

Introduction Physics and geometry of KQ correspondence Summary General idea Physics 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 xr − → #windings around LK power ai in aai − → #wrappings around base S2 power qi in qqi − → invariant self-linking# Cii = ti − → linking# between disk and its small shift

Quiver nodes correspond to basic disks – holomorphic curves that wind around LK once (r = 1)

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-81
SLIDE 81

Introduction Physics and geometry of KQ correspondence Summary General idea Physics 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 xr − → #windings around LK power ai in aai − → #wrappings around base S2 power qi in qqi − → invariant self-linking# Cii = ti − → linking# between disk and its small shift

Quiver nodes correspond to basic disks – holomorphic curves that wind around LK once (r = 1)

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-82
SLIDE 82

Introduction Physics and geometry of KQ correspondence Summary General idea Physics 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 xr − → #windings around LK power ai in aai − → #wrappings around base S2 power qi in qqi − → invariant self-linking# Cii = ti − → linking# between disk and its small shift

Quiver nodes correspond to basic disks – holomorphic curves that wind around LK once (r = 1)

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-83
SLIDE 83

Introduction Physics and geometry of KQ correspondence Summary General idea Physics 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 xr − → #windings around LK power ai in aai − → #wrappings around base S2 power qi in qqi − → invariant self-linking# Cii = ti − → linking# between disk and its small shift

Quiver nodes correspond to basic disks – holomorphic curves that wind around LK once (r = 1)

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-84
SLIDE 84

Introduction Physics and geometry of KQ correspondence Summary General idea Physics 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 xr − → #windings around LK power ai in aai − → #wrappings around base S2 power qi in qqi − → invariant self-linking# Cii = ti − → linking# between disk and its small shift

Quiver nodes correspond to basic disks – holomorphic curves that wind around LK once (r = 1)

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-85
SLIDE 85

Introduction Physics and geometry of KQ correspondence Summary General idea Physics 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 xr − → #windings around LK power ai in aai − → #wrappings around base S2 power qi in qqi − → invariant self-linking# Cii = ti − → linking# between disk and its small shift

Quiver nodes correspond to basic disks – holomorphic curves that wind around LK once (r = 1)

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-86
SLIDE 86

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Example: unknot quiver vertices and homology

For the unknot quiver we have P

Q01 |d|=1(x,q) = −qx1 +x2

1−q2 H (01) has two generators with degrees (a1,a2) = (2,0), (q1,q2) = (0,0) (t1,t2) = (1,0) = (C11,C22)

1 2

Generators correspond to quiver vertices and their degrees encode KQ change of variables x1 = a2q−1x, x2 = x giving P01

1 (a,q)x = −a2x +x

1−q2 = ∑i∈G (01) aaiqqi(−1)Cii 1−q2 x

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-87
SLIDE 87

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Example: unknot quiver vertices and homology

For the unknot quiver we have P

Q01 |d|=1(x,q) = −qx1 +x2

1−q2 H (01) has two generators with degrees (a1,a2) = (2,0), (q1,q2) = (0,0) (t1,t2) = (1,0) = (C11,C22)

1 2

Generators correspond to quiver vertices and their degrees encode KQ change of variables x1 = a2q−1x, x2 = x giving P01

1 (a,q)x = −a2x +x

1−q2 = ∑i∈G (01) aaiqqi(−1)Cii 1−q2 x

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-88
SLIDE 88

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Example: unknot quiver vertices and homology

For the unknot quiver we have P

Q01 |d|=1(x,q) = −qx1 +x2

1−q2 H (01) has two generators with degrees (a1,a2) = (2,0), (q1,q2) = (0,0) (t1,t2) = (1,0) = (C11,C22)

1 2

Generators correspond to quiver vertices and their degrees encode KQ change of variables x1 = a2q−1x, x2 = x giving P01

1 (a,q)x = −a2x +x

1−q2 = ∑i∈G (01) aaiqqi(−1)Cii 1−q2 x

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-89
SLIDE 89

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Example: unknot quiver vertices and BPS states

Two generators of H (01) correspond to two BPS states BPS states of T[L01] are counted by LMOV invariants N01

1 (a,q) = 1−a2

BPS states of T[Q01] are counted by DT invariants Ω

Q01 |d|=1(x,q) = −qx1 +x2

1 2

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-90
SLIDE 90

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Example: unknot quiver vertices and BPS states

Two generators of H (01) correspond to two BPS states BPS states of T[L01] are counted by LMOV invariants N01

1 (a,q) = 1−a2

BPS states of T[Q01] are counted by DT invariants Ω

Q01 |d|=1(x,q) = −qx1 +x2

1 2

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-91
SLIDE 91

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Example: unknot quiver vertices and BPS states

Two generators of H (01) correspond to two BPS states BPS states of T[L01] are counted by LMOV invariants N01

1 (a,q) = 1−a2

BPS states of T[Q01] are counted by DT invariants Ω

Q01 |d|=1(x,q) = −qx1 +x2

1 2

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-92
SLIDE 92

Introduction Physics and geometry of KQ correspondence Summary General idea Physics 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 LK a1 = 2 − → wrappings around base S2 q1 = 0 − → invariant self-linking C11 = 1 − → linking between disk and its small shift

1 2

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-93
SLIDE 93

Introduction Physics and geometry of KQ correspondence Summary General idea Physics 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 LK a1 = 2 − → wrappings around base S2 q1 = 0 − → invariant self-linking C11 = 1 − → linking between disk and its small shift

1 2

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-94
SLIDE 94

Introduction Physics and geometry of KQ correspondence Summary General idea Physics 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 LK a2 = 0 − → wrappings around base S2 q2 = 0 − → invariant self-linking C22 = 0 − → linking between disk and its small shift

1 2

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-95
SLIDE 95

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver arrows extracted from motivic generating series

Let’s look closer at motivic generating series PQ(x,q) :=

∑

d1,...,d|Q0|≥0

(−q)∑1≤i,j≤|Q0| Ci,jdidj

|Q0|

∏

i=1

xdi

i

(q2;q2)di Dimension di encodes the number of factors corresponding to i-th vertex #arrows Ci,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

slide-96
SLIDE 96

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver arrows extracted from motivic generating series

Let’s look closer at motivic generating series PQ(x,q) :=

∑

d1,...,d|Q0|≥0

(−q)∑1≤i,j≤|Q0| Ci,jdidj

|Q0|

∏

i=1

xdi

i

(q2;q2)di Dimension di encodes the number of factors corresponding to i-th vertex #arrows Ci,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

slide-97
SLIDE 97

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver arrows extracted from motivic generating series

Let’s look closer at motivic generating series PQ(x,q) :=

∑

d1,...,d|Q0|≥0

(−q)∑1≤i,j≤|Q0| Ci,jdidj

|Q0|

∏

i=1

xdi

i

(q2;q2)di Dimension di encodes the number of factors corresponding to i-th vertex #arrows Ci,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

slide-98
SLIDE 98

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver arrows extracted from motivic generating series

Let’s look closer at motivic generating series PQ(x,q) :=

∑

d1,...,d|Q0|≥0

(−q)∑1≤i,j≤|Q0| Ci,jdidj

|Q0|

∏

i=1

xdi

i

(q2;q2)di Dimension di encodes the number of factors corresponding to i-th vertex #arrows Ci,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

slide-99
SLIDE 99

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver arrows as disk intersections

Geometrically PQ(x,q) counts all holomorphic curves that can be made from basic disks according to quiver arrows Dimension vector di − → #copies of the disk #arrows Ci,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

slide-100
SLIDE 100

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver arrows as disk intersections

Geometrically PQ(x,q) counts all holomorphic curves that can be made from basic disks according to quiver arrows Dimension vector di − → #copies of the disk #arrows Ci,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

slide-101
SLIDE 101

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver arrows as disk intersections

Geometrically PQ(x,q) counts all holomorphic curves that can be made from basic disks according to quiver arrows Dimension vector di − → #copies of the disk #arrows Ci,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

slide-102
SLIDE 102

Introduction Physics and geometry of KQ correspondence Summary General idea Physics Geometry

Quiver arrows as disk intersections

Geometrically PQ(x,q) counts all holomorphic curves that can be made from basic disks according to quiver arrows Dimension vector di − → #copies of the disk #arrows Ci,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

slide-103
SLIDE 103

Introduction Physics and geometry of KQ correspondence Summary

Main messages

Physically knots-quivers correspondence is a duality between 3d N = 2 theories T[LK] and T[QK] Quiver elements can be intepreted in terms of T[QK] data as well as holomorphic disks: Physics Geometry U(1) gauge group Holomorphic disk Chern-Simons coupling Disk boundaries linking More details in the paper - coming soon!

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-104
SLIDE 104

Introduction Physics and geometry of KQ correspondence Summary

Main messages

Physically knots-quivers correspondence is a duality between 3d N = 2 theories T[LK] and T[QK] Quiver elements can be intepreted in terms of T[QK] data as well as holomorphic disks: Physics Geometry U(1) gauge group Holomorphic disk Chern-Simons coupling Disk boundaries linking More details in the paper - coming soon!

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-105
SLIDE 105

Introduction Physics and geometry of KQ correspondence Summary

Main messages

Physically knots-quivers correspondence is a duality between 3d N = 2 theories T[LK] and T[QK] Quiver elements can be intepreted in terms of T[QK] data as well as holomorphic disks: Physics Geometry U(1) gauge group Holomorphic disk Chern-Simons coupling Disk boundaries linking More details in the paper - coming soon!

  • P. Kucharski

Physics and geometry of KQ correspondence

slide-106
SLIDE 106

Introduction Physics and geometry of KQ correspondence Summary

Last message

Thank you for your attention!

  • P. Kucharski

Physics and geometry of KQ correspondence