Towards U(N|M) knot invariant from ABJM theorybctp.uni-bonn.de/workshop2014/talks/kimura.pdf ·...

Towards U(N |M) knot invariantfrom ABJM theory

Taro Kimura

Institut de Physique Theorique, CEA Saclay

Mathematical Physics Laboratory, RIKEN

Based on a collaboration with B. Eynard [arXiv:1408.0010]

Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 1 / 28

http://arxiv.org/abs/1408.0010

Knot theory

QFT interpretation: Wilson loop in Chern–Simons theory[Witten]

SCS =k

4π

∫S3

Tr

(A ∧ dA+

2

3A ∧A ∧A

)

ex.) Jones polynomial: SU(2) CS w/ the fund rep loop

J(K; q) =⟨W�(K)

⟩/⟨W�( )

⟩with q = exp

(2πi

k +N

)Wilson loop: WR(K) = TrR exp

(∮K

A

)


http://inspirehep.net/record/264818

Generalizations:

HOMFLY polynomial: SU(2) → SU(N)

Colored polynomial: Tr� U → TrR U

We can generalize it in this way, but...

The expression becomes much more complicataed

What’s the systematic dependence on the rank/rep?


Matrix integral representation [Marino]

ZCS(S3; q) =1

N !

∫ N∏i=1

dxi2π

e− 1

2gsx2i

N∏i<j

(2 sinh

xi − xj2

)2

Wilson loop operator (especially for unknot)

WR( ) → TrR

ex1

. . .

exN

= sλ(R)(ex1 , · · · , exN )

Wilson loop vev ⟨WR( )

⟩CS

=⟨sλ(ex)

⟩matrix


http://arxiv.org/abs/hep-th/0207096

Another CS-like matrix model:

ABJM matrix model

ZABJM(S3; q) =1

N !2

∫ N∏i=1

dxi2π

dyi2π

e− 1

2gs(x2i−y2i ) (∆N,N (x; y))2

with ∆N,N (x; y) =

∏Ni<j

(2 sinh

xi−xj2

)(2 sinh

yi−yj2

)∏Ni,j

(2 cosh

xi−yj2

)[Kapustin–Willett–Yaakov] [Drukker–Trancanelli] [Marino–Putrov]

A “supersymmetric” CS matrix model w/ U(N |N) sym

cf. U(N |N) supermatrix model:

ZU(N |N) =

∫ N∏i=1

dxi2π

dyi2π

e− 1

gs(W (xi)−W (yi))

∏Ni<j(xi − xj)2(yi − yj)2∏N

i,j(xi − yj)2Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 5 / 28



Unknot Wilson loop vev with ABJM⟨WR( )

⟩ABJM

=⟨sλ(ex; ey)

⟩Matrix

Our goal

To construct the U(N |N) knot invariant through ABJM


Contents

1 Introduction

2 U(N |N) Wilson loop

3 Torus knot

4 Topological string

5 Summary


U(N) unknot invariant:

⟨WR( )

⟩U(N)

=

N∏i<j

q12(λi−λj−i+j) − q− 1

2(λi−λj−i+j)

q12(−i+j) − q− 1

2(−i+j)

= dimq R

Representation R → Young diagram λ = (λ1, · · · , λN )

An analogous expression for U(N |N) vev?


Unknot Wilson loop vev with ABJM⟨WR( )

⟩ABJM

=⟨sλ(ex; ey)

⟩Matrix

Supersymmetric Schur function −→ associated w/ U(N |N)

sλ(ex; ey) = StrR

(U(x)

−U(y)

)with U(x) = diag(ex1 , · · · , exN )


A representation for U(N |N):

λ =

N ×Nµ

νt

α1α2

α3α4α5

β1β2

β3β4β5

Frobenius coordinate:

λ = (α1, · · · , αd(λ)|β1, · · · , βd(λ))d(λ) = #diagonal blocks

if there is a box here, sλ(ex; ey) = 0

d(λ) ≤ N

Let’s focus on d(λ) = N :

Determinantal formula for U(N |N) Wilson loop

⟨WR( )

⟩U(N |N)

= det1≤i,j≤N

(1

q12(αi+βj+1) + q−

12(αi+βj+1)

)


⟨WR( )

⟩U(N |N)

= det1≤i,j≤N

(1

q12(αi+βj+1) + q−

12(αi+βj+1)

)

=

∏Ni<j

(q

12(αi−αj) − q− 1

2(αi−αj)

)(q

12(βi−βj) − q− 1

2(βi−βj)

)∏Ni,j

(q

12(αi+βj+1) + q−

12(αi+βj+1)

)⟨Wµ( )

⟩U(N)

⟨Wν( )

⟩U(N)

Including two U(N) invariants

⟨WR( )

⟩U(N |N)

∼⟨Wµ( )

⟩U(N)

×⟨Wν( )

⟩U(N)


U(1|1) theory:

λ =

α

β

hook rep: λ = (α|β)

⟨W(α|β)( )

⟩U(1|1)

=1

q12(α+β+1) + q−

12(α+β+1)

Factorization property⟨WR( )

⟩U(N |N)

= det1≤i,j≤N

⟨W(αi|βj)( )

⟩U(1|1)

cf. Giambelli compatibility [Borodin et al.]


http://arxiv.org/abs/math-ph/0505021

Summary

ABJM as the U(N |N) Chern–Simons matrix model

Wilson loop applied to knot theory

Determinantal formula for the U(N |N) Wilson loop:

⟨WR( )

⟩U(N |N)

= det1≤i,j≤N

(1

q12(αi+βj+1) + q−

12(αi+βj+1)

)∼⟨Wµ( )

⟩U(N)

×⟨Wν( )

⟩U(N)

U(1|1) vev as a building block


Contents

1 Introduction


3 Torus knot


5 Summary


The original matrix model only describes the unknot loop..

(P,Q) torus knot CS matrix model

Z(P,Q)CS =

∫ N∏i=1

dxi2π

e− 1

2gsx2i

N∏i<j

(2 sinh

xi − xj2P

2 sinhxi − xj

2Q

)

[Lawrence–Rozansky] [Beasley] [Kallen]

(P,Q) torus knotP

Q

(P,Q) = (3, 2)


http://dx.doi.org/10.1007/s002200050678



Torus knot: Adams operation

Adams operation:

⟨WR(KP,Q)

⟩=∑V

cVR,Q

⟨WV (K1,f )

⟩with f =

P

Q

A linear combination of theP

Q-framed unknot vevs

Schur function decomposition: sλ(xQ) =∑ν

cνλ,Qsν(x)


Torus knot: Spectral curve

Saddle point analysis provides the spectral curve

ex.) (P,Q) = (2, 3)

PQ cuts & (P +Q) sheets

This is just given by SL(2,Z) transform of the unknot curve

[Brini–Eynard–Marino]

What happens for the U(N |N) theory?



(P,Q) torus knot U(N |N) CS matrix model

Z(P,Q)ABJM =

∫ N∏i=1

dxi2π

dyi2π

e− 1

2gs(x2i−y2i ) ∆N,N

( xP

;y

P

)∆N,N

( xQ

;y

Q

)

Remark:Perturbatively equivalent to CS theory on the squashedL(2, 1) with b2 = P/Q through the analytic continuation

[Hama–Hosomichi–Lee] [Tanaka] [Imamura–Yokoyama]

We can show: Adams operation & spectral curve





Adams operation

Schur function decomposition:

sλ(xQ) =∑ν

cνλ,Qsν(x) −→ sλ(xQ, yQ) =∑ν

cνλ,Qsν(x, y)

The rep theory of U(N) & U(N |N) in a parallel way

Adams operation for U(N |N) theory⟨WR(KP,Q)

⟩U(N |N)

=∑V

cVR,Q

⟨WV (K1,f )

⟩U(N |N)


Spectral curveSolving the saddle point equation... in a complicated form

ex.) (P,Q) = (2, 3)

F3F4

F 03

F 04

F0

F3

F4

F 03

F 04

F1

F3

F4

F 03

F 04

F2

F0

F1

F2

F 00

F 01

F 02

F0

F1

F2

F 00

F 01

F 02

F3 F4

F 00

F 04

F 03

F3

F4

F 01

F 04

F 03

F3 F4

F 02

F 04

F 03

F3

F4

F 02F 0

0

F 01

F0

F1 F2 F 02 F 0

0

F 01

F0

F1F2

F 03 F 0

4

Table 1: The cuts of the functions Fk(u) and FQ+l(u) for (P, Q) = (2, 3), where Fk = Fk(u),

F 0k = Fk(u!

12(P�Q)) and F 0

Q+l = FQ+l(u!12(Q�P )). The solid and dotted lines correspond

to the cuts from the first resolvent W (1)(u) and the second resolvent W (2)(u). For example,

we can see F0 = F3, F4 under crossing the corresponding cut of W (1)(u), and F0 = F 03, F 0

4

through the cut from W (2)(u).

multiple angles of 2⇡/(PQ). The total number of the cuts is thus 2PQ. Due to the saddle

point equations they satisfy

Fk(u � i0) = FQ+l(u + i0) for W (1)(u) ,

Fk(u � i0) = FQ+l((u + i0)!12(Q�P )) for W (2)(u) ,

Fk((u � i0)!12(P�Q)) = FQ+l((u + i0)!

12(Q�P )) for W (1)(u) ,

Fk((u � i0)!12(P�Q)) = FQ+l(u + i0) for W (2)(u) .

(5.37)

This means that Fk(u � i0) = FQ+l(u + i0) under crossing the cut from the first resolvent

W (1)(u), Fk(u � i0) = FQ+l((u + i0)!12(Q�P )) for the cut from the second W (2)(u), and so

on. See Table 1 for the case with (P, Q) = (2, 3).

Using these functions we define a function

S(u, f) = S1(u, f) S2(u, f) , (5.38)

19

SL(2,Z) transform of the unknot curve


Remark 1: “Mirror” expression [Kapustin–Willett–Yaakov]

ZABJM =1

N !(2π)k

∫dNz

(2π)N

N∏i<j

tanh

(zi − zk

2k

) N∏i=1

(2 cosh

zi2

)−1N = 4 SYM with a fundamental & adjoint matter at k = 1

Trivial (P,Q) dependence: The mirror is the same

Remark 2: Torus knot in the lens space L(2, 1) via SL(2,Z)[Jockers–Klemm–Soroush] [Stevan]





Summary

Another matrix model describing torus knots:

SL(2,Z) transform of the unknot model

Basic properties hold for U(N |N) theory

Adams operation

Spectral curve


Contents

1 Introduction


3 Torus knot


5 Summary


Knot in Topological string

Brane insertion: [Ooguri–Vafa]

Z(K;x)

=⟨

det(1⊗ 1− U ⊗ e−x

) ⟩CS

WKB expansion: Z(K;x) ∼ exp

(1

gs

∫ x

p(x)dx

)with p(x) = lim

gs→0

∞∑n=0

gs

⟨TrUn

⟩CSe−nx


http://arxiv.org/abs/hep-th/9912123

Topological string for ABJM:

local P1 × P1 =

Brane partition function:

Z(K;x, y) =⟨

Sdet

(1⊗ 1− U ⊗

(e−x

e−y

))⟩ABJM

∼ exp

(1

gs

∫ x

yp(x)dx

)Another definition based on topological recursion/B-model


Contents

1 Introduction


3 Torus knot


5 Summary


Summary

U(N |N) Wilson loop vev from ABJM matrix model

det formula:⟨WR

⟩U(N |N)

= det1≤i,j≤N

⟨W(αi|βj)

⟩U(1|1)

Torus knot U(N |N) matrix model

SL(2,Z) transform of the unknot: Adams op, spectral curve

Topological string

Ooguri–Vafa construction: Brane pair-creation


Discussion

Is it really a knot invariant?

If so, another definition required:

Skein relation

CFT on the boundary

Topological recursion

Volume conjecture, AJ conjecture, knot homology, etc.


Towards U(N|M) knot invariant from ABJM theorybctp.uni-bonn.de/workshop2014/talks/kimura.pdf ·...

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