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ABJM Wilson loops in Fermi gasand topological string approach

40th Ahrenshoop Symposium, August 2. 2012

Albrecht Klemm

Klemm, Marino, Schiereck, Soroush arXiv/1207.0611

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∙ Localisation in supersymmetric gauge theories

– Wilson Loops in supersymmetric Chern-Simons theory

with matter

∙ Topological-String on local Calabi-Yau

– The Chern-Simons Gauge theory/Topological-String

large N duality

– The ABJM slice and the N 3/2 scaling law

– Open topological string and Wilson line expectation

values

∙ The Fermigas approach

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– Reinterpretation of partition function and Wilson loop

– Fermigas techniques, Fermi surface and tropical

geometry

– The all genus expansion

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Localization in supersymmetric gauge theories

∙ Supersymmetry provides nilpotent operators Q2 = 0

which localise the path integral to the fixpoints of the

fermionic transformations given by the solutions to the

e.o.m.

∙ Beside evaluating the instanton contributions it remains

to calculate the one-loop determinant around the given

vaccua

∙ This semiclassical approximation is exact and the

integration over the fiels can be done by the Atiyah-Bott

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localisation formula∫ℳ� =

∫F∈ℳ

i ∗ (�)

e(N )

Localisation calculations in supersymmetric gauge

theories on symmetric spaces with positive curvature

have been done recently also for Wilson loop correlators

Pestun 07

In 3d for sypersymmetric Chern-Simons quiver theories

with matter Kapustin, Willet, Yakov: arXiv 0909.4559. Gauge

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action (in flat space) gCS = − k4�, k ∈ ℤ

SG = gCS

∫d3xTr

(��� (A�∂�A +

2i

3A�A�A )− �†�2D�

)Coupled to chiral matter multiplet

SM =∫d3x(D��

†D��+ 34�†�+ i †D + F †F−

�†��+ �†D�− †� + i�†�† − i †��)

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Groups and matter contents given by quiver diagrams.

E.g.

bifundamental

ΦU(N ) U(N )

k −k

21i=1,.,4

ABJM showed that this theory has N = 6 susy and

KWY localised the partition function on S3 to a matrix

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model integral

ZABJM = 1N !2

∫ ∏Ni=1

d�id�j(2�)2

∏i<j sinh2

(�i−�j

2

)sinh2

(�i−�j

2

)∏i,j cosh2

(�i−�j

2

)e−

12gs(

∑i �

2i−∑j �

2j )

γ

S3

Figure 1: 1/6 Wilson loop

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W1/6R = trRP exp

∫

(iA�x

� +2�

k∣x∣M I

JCICJ

)ds ,

In ABJM theory one has three related WL’s: In the first

gauge group

⟨W 1/6R ⟩ = ⟨trR (e�i)⟩ABJM

and in the second gauge group

⟨W 1/6R ⟩ = ⟨trR (e�i)⟩ABJM

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In addition there is a more symmetric combination,

which breaks only half the super symmetry

⟨W 1/2ℛ ⟩ =

⟨Strℛ

(e�i 0

0 −e�j

)⟩ABJM

.

The simplest insertion Tren�i in the Matrix model

integral corresponds to a combination of Wilson loops

over hook representations Rn,s

W 1/6n =

n−1∑s=1

(−1)sW1/6Rn,s

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and

⟨W 1/2n ⟩ = ⟨W 1/6

n ⟩ − (−1)n⟨W 1/6n ⟩ .

Now that the relevant matrix integrals in 3d gauge

theories are defined we decribe in the next section how to

solve them in an 1/N expansion, which is exact in the t’

Hooft parameters, using

∙ Large N-Duality between the topological string and

matrix models Putrov and Marino arXiv/0912.3074, Drukker,

Putrov and Marino arXiv/1007.3837,... .

– Partition function maps to closed string partition

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function

– The Wilson-Loop amplitudes maps to open string

amplitudes

– A real slice in B-model moduli space allows to

interpolate between strong and weak coupling.

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Topological-String on local Calabi-Yau

Chain of Chern-Simons Gauge theory/Topological-String

large N dualities

∙ Chern-Simons theory on the real three manifold M3 is

dual to open topological String on the non-compact

Calabi-Yau manifold T ∗M3. Witten hep-th/9207094

∙ Large N Conifold transition Gopakumar Vafa hep-th/9207094

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N−units of G−flux

SIP

N−D7−branes

P3

I 1T S

3

1

node O(−1) x O(−1) *

∙ Lense space transition Aganagic,AK,Marino Vafa hep

th/02211098

IPS

T S* 3 / ZZ k

/ ZZ k1

PI 1PI 1/ ZZ knode

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O(−2,....−2) x ... x

....

....

....

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Toric description of the geometry as moduli space of

(N,N) = (2, 2) gauged linear � model by r chiral fields

Xi with charge

Q(k)j , j = 1, . . . , r, k = 1, . . . , 3− r

under U(1)(3−p). With x 7→ x�(k)

Q(k)

M = {ℂ[x1, . . . , xr] ∖ SR}/(ℂ∗)(3−r)

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Mirror geometry: Y ∼ �Y , � ∈ ℂ∗

W = uv +H = uv +

r∑i=1

Yi = 0,

r∏i=1

YQ(k)i

i = zk

symmetryt

t

2

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Mirror

The ABJM theory: O(−2,−2)→ ℙ1 × ℙ1

Q(1) = (−2, 1, 1, 0, 0), Q(2) = (−2, 0, 0, 1, 1) The mirror

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geometry is the genus 1 curve

H = 1 + x+ y +z1

x+z2

y= 0

with the meromorphic differentials

�k = log(y)X(k−1)dx .

The Wilson loops integrals:

−b −1/b 1/a a

C C12 B

γ

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⟨W16n ⟩g=0 =

k

2�2

∫C1�n, ⟨W

16n ⟩g=0 = (−1)n

k

2�2

∫C2�n .

⟨W12n ⟩g=0 =

k

2�2

∮

�n .

Note that

⟨W 1/2n ⟩ = ⟨W 1/6

n ⟩ − (−1)n⟨W 1/6n ⟩ .

is a relation in homology.

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The 1/N 2 ∼ g expansion:

�0(x) =

∞∑g=0

g2gs �

(g)0 (x) , �

(g)k (x) = xk�

(g)0 (x) .

Now �(g)0 (x) = Wg(x)dx and the Wg(p) fullfill the

Eynard-Oratin recursion

Wg(p, pK) =∑

i Resq=xidS(p,q)y(q)

(∑gℎ=0

∑J⊂K

Wℎ(q, pJ)Wg−ℎ(q, pK∖J) +Wg−1(q, q, pK)).

The ABJM slice is defined by �1 = �2 and because of the

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homological relation between �i = Nik = 1

4�i

∫Ci�

(0)0

exp(4�i(�1 − �2)) =z2

z1.

weak coupling

Z =0

Z =01

2

large radius

strong coupling

Orbifold

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The ABJM coupling � lives on the real slice

orbifold

Γ(2)

λ=0

λ=

8

λ=1

Real slice λ

PI 1

coupling space

strong coupling

conifold

∙ The correlations functions are analytically continued

without changing the polarisation.

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∙ In particular the g = 0 free energy at �→∞

F = g−2s F (0) =

�√

2

3k2�

32 +O(�0, e−2�

√2�)

exhibits the �32 ∼ N 3

2 scaling law.

∙ The Wilson-loops can be similarly evaluated in low genus

at at points in the coupling space.

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The Fermigas approach

In contrast the Fermigas approach Marino Putrov,

arXiv:1110.4066 sums up the genus expansion. In particular

the leading terms in the � can be calculated in various

regions in the coupling space.

The rewritting of the matrix sum in terms of correlators

in a free Fermi gas has been explained in Marcos lecture.

In particular the Hamiltonian for the ABJM theory is

given by

H(p, q, ℏ) = log(

2 coshp

2

)+ log(

(2 cosh

q

2

)+O(ℏ2)

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and the insertion of the 1/6 Wilson line with winding n is

given by

On = exp

(n(p+ q)

k

)So one needs to calculate to one body operator

⟨On⟩ = tr (On�12�21)

in an ideal Fermi gas of N particles.

The calculation is done in the semiclassical limit, which

is very complicated as the Hamiltonian has explicite ℏcorrections to all orders, which however can be

resummed up.

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One does a low temperature expansion in the grand

canonical ensemble and transforms back to the original

correlator

⟨W 1/6n ⟩ =

1

2�iZ

∫d� e−�N⟨On⟩GC ,

p!

q!

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The results are of the type

⟨W 1/2⟩ =1

4csc

(2�

k

) Ai[C−1/3

(N − k

24 −73k

)]Ai[C−1/3

(N − k

24 −13k

)]with

C =2

�2k.

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Conclusions

∙ Extension of the Fermigas formalism to Wilson Loops,

which leads to exact expressions in 1/N

∙ We expand them and compared with B-model approach,

thereby checking the Fermigas approach to Wislon loops

∙ Some properties have been confirmed by numerical

studies Hatsuda,Moriyama,Okuyama arXiv1207.4283