surface operators from topological string theory, part...

Surface Operators from Topological String Theory, Part 1

Can Kozçaz, CERNMay 3, 2011

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What you will hear today & tomorrow

TomorrowGaiotto’s construction & Alday-Gaiotto-Tachikawa conjecture

Surface operators & Alday-Gaiotto-Gukov-Tachikawa-Verlinde approach

Surface operators from topological string theory

TodayTopological string theory & Geometric engineering

The topological vertex

The refined topological vertex

Motivation

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Topological String Theory

String Theory

Mathematics

Gauge Theory

simpler

framework

numero

us

applic

ations

geometric

engineering

......

......

......

.Conformal Field

Theory

AGT,W

Motivation

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Losev-Moore-Nekrasov-Shatashvili instanton partition function of generalized SU(N) quiver theories in 4d

N = 2AGT, W

AGGTVInstanton partition function in the presence of surface operators

Toda conformal blocks of primary fields in 2d

Toda conformal blocks of primary fields with additional

degenerate fields

Topological B-model / Topological Recursion

AV, AKV DV

A-model MirrorSym.

AN!1

TOPOLOGICAL STRING THEORY

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A simpler framework to check with more confidence ideas in ST

Genus zero amplitude computes the prepotential

Higher genus amplitudes compute higher gravitational couplings

Numerous applications in Mathematics

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7

!

target space M


worldsheet of genus g,!g


! : !g !M

Consider the maps from the worldsheet to the 6D target space M

The Lagrangian density has the following form:

L =!

d4! K(!i,!i) +12

"!d2! W (!i) + c.c.

#

with the chiral field given by

!(xµ, !±, !̄±) = "(y±) + !!#!(y±) + !+!!F (y±)

andy± = x± ! i!±!̄±

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If the worldsheet is flat then the Lagrangian is invariant under N = (2, 2)supersymmetry!However, we are interested in formulating the theory on a curved Riemannian surface. The action is not invariant under SUSY transformations:

!S =!

!

"!µ"+ Gµ

! " . . .##

h dx2

unless we have covariantly constant spinors.

We need to topologica"y twist the theory

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Before twisting A-twist B-twistU(1)V U(1)A U(1)E U(1)!E U(1)!E

Q" -1 1 1 0 2Q+ 1 1 -1 0 0Q" 1 -1 1 2 0Q+ -1 -1 -1 -2 -2

The theory has Lorentz (Euclidean) symmetry, vector and axial R-symmetries:

M !E = ME + R =

!R = FV for A-twistR = FA for B-twist

QA = Q+ + Q! QB = Q+ + Q!10


The twisted supercharges QA & QB are nilpotent:

Q2A = 0 Q2

B = 0The Hilbert space is unchanged until now, let us choose the physical states to live in cohomology classes of the supercharges:

H = {|!! : Q|!! = 0}

The correlation function is independent of the worldsheet metric

if the insertions correspond to physical operator

!h!O1 . . .Os" =!

14"

" #h d2x !hµ!{Q,Gµ!}O1 . . .Os

#

= 0

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The path integral localizes on the holomorphic maps for the A model 12

Next we couple the theory to worldsheet gravity. In the 4D effective theory, we will have terms like

!d4x Fg(ti) R2

+ F 2g!2+ , g > 0

genus g amplitude self dual Riemann tensor (contr.) self dual graviphoton

!d4x (!i!jF0(ti))F+

i ! F+j , g = 0

Organize the topological string amplitudes into a generating function

F (ti) = log Z =!

g

!2g!2Fg(ti), ! = !F+"

GEOMETRIC ENGINEERING

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Geometric Engineering

What is the gauge group of the theory ?

How much supersymmetry is preserved ?

What is the matter content ?

In type IIA compactification from 10D to 4D, the internal space determines the physics. The method for constructing the 6D internal space is called geometric engineering.

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}Toric geometry (for SU(N) theories); i.e.we can use topological vertex

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K3AN!1

P1fiber

P1base

R3,1

x

y

z

Start with 10D theory, compactify on a K3 surface down to 6D

10D !" 6D

Further compactify on a sphere from 6D down to 4D

4D space

2D space

6D !" 4D

Over each point in there exists a 6D spaceR3,1

Geometric EngineeringWhat is the gauge group of the theory ?

1) Start with 10D string theory and compactify on a K3 with singularity down to 6D theory.

2) The type of singularity determines the gauge group through McKay correspondence:

finite subgroups of SU(2) simple laced Lie algebras

3) Compactify further on a Riemann surface down to 4D: genus higher than 1 leads to non-asymptotically free theory

16

!" C/Z3


How much supersymmetry is preserved ?

For remaining 2 dimensions, compactify on

P1 ! N = 2T2 ! N = 4


Consider an SU(2) theory: we fiber a over another P1P1

To get the neutral vector multiplet: decompose the RR 3-form into a 2-form dual to the fiber two cycle and a 1-form. To get vector multiplet: wrap D2 branes in two different orientations around the two cycles

W±

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To add matter we need to add another two cycle in a non-trivial way: add this extra cycle only over a single point of the base

At this single point the gauge symmetry is enhanced to SU(3), over the rest it is SU(2)

SU(3) ! SU(2)" U(1) : 8# 3 + 2 · 2 + 1

Toric geometries

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THE REFINED TOPOLOGICAL VERTEX

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The Topological Vertex

The geometries we will use to engineer supersymmetric gauge theories are toric.

Toric geometries can be represented in terms of trivalent planar graphs like the following:

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r1

r3

r2

C3 ! (z1, z2, z3)= (r1e

i!1 , r2ei!2 , r3e

i!3)

(!2, !3)

(!1)

The Topological Vertex

Divide the toric diagram into trivalent vertices

Compute the amplitude of each vertex

Glue the amplitudes with appropriate propagators to obtain the full amplitude

O(!3) "# P2

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Z(V1, V2, V3) =!

!,µ,"

C!µ" tr!V1 trµV2 tr"V3

Vi = Pexp!"

A

#

The Refined Topological VertexAccording to the Gopakumar & Vafa formalism of topological string theory

F =!

!!H2(X,Z)

"!

k=1

!

jL

(!1)2jLN jL

! e#kT!

"q#2jLk + . . . + q+2jLk

k(qk/2 ! q#k/2)2

#, q = eigs

There exists a refinement of this description due to Shatashvilli & Nekrasov derivation of Seiberg & Witten

F =!

!!H2(X,Z)

"!

k=1

!

jL,jR

e#kT!

(!1)2jL+2jRN (jL,jR)!

"(t q)#kjL + . . . + (t q)+kjL

# $%tq

&#kjR

+ . . . +%

tq

&+kjR'

k(tk/2 ! t#k/2)(qk/2 ! q#k/2)

The refined topological vertex computes these refined amplitudes!22

with denoting the degeneracy of particles with spin (jL,jR) coming from a certain curve

N (jL,jR)!

!

The Refined Topological Vertex

The refined topological vertex has the following form:

t=q corresponds to the usual topological vertex.This form of the usual topological vertex indicates a combinatorial interpretation of the usual vertex.

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C!µ"(t, q) =G!µ"(t, q)M(t, q)

=!q

t

" !µ!2+!!!22

t"(µ)

2 P"t(t!#; q, t)#

$

!q

t

" |#|+|$|"|µ|2

s!t/$(t!#q!")sµ/$(t!"t

q!#)

The Refined Topological VertexThe topological string theory partition function on a Calabi-Yau 3fold with the Euler character in the large volume limit is given by

where M is the MacMahon function, the generating function for 3D partitions

M(q) =!!

n=1

1(1! qn)n

= 1 + q + 3 q2 + 6 q3 +O(q4)

Z = M(q)!/2

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!

The Refined Topological VertexIs there a crystal interpretation of the topological vertex ?What is it ?

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C!µ"

! !µ µ

! !

The denoted region is excised from the corner, asymptotically in all 3 directions

We count the boxes in this deformed room: the orange boxes show counting

The Refined Topological VertexThe refined topological vertex is actually derived by incorporating a more detailed counting scheme:

26Slicing the 3D partition, each slice is weightedequally for the usual vertex

xy

yz

z x!

0 1 2 3 4 5 6

0

1

2

3

4

5

6

Figure 2: Slices of the 3D partitions are counted with parameters t and q depending on the

shape of !.

0 1 2 3 4 5 6

0

1

2

3

4

5

6

Figure 3: Slices of the 3D partitions are counted with parameters t and q depending on theshape of !.

0 1 2 3 4 5 6

0

1

2

3

4

5

6


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Excised region along z-

direction

M(q) =!

!

"

a!Zq|!a|

!(a + 1) ! !(a), a < 0!(a) ! !(a + 1), a " 0

u0v1

u1v2

u2v3

u3v4

u4

The slices are labeled by integers:x! y = a, with a " Z


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The following identity is essential to compute the generating functions of plane partitions

More specifically

An inner product can be defined on these vectors describing Young diagrams annihilation operatorcreation operator

!+(1)|µ! =!

!!µ

|!! and !"(1)|µ! =!

!#µ

|!!

The commutator between the creation and annihilation operator is

!+(x)!!(y) = (1! xy) !!(y)!+(x)

!

i

!+(xi)|µ! ="

!!µ

s!/µ(x)|!!

!!|!+(x)|µ" = !µ|!!(x)|!"


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The generating function (with a different weight for each slice) takes the following form

This can be written in terms of the creation and annihilation operators as

and computed using the previous identities, in addition to the following

!µ|!+(x1)!+(x2) . . .!+(xn)|!" = s!/µ(x1, . . . , xn)

G!µ"({qa}) =!

{#(a)}

"

a!Zq|#(a)|a =

!

$

"

a!Zq|$a|a

G!µ"({x±a }) =

!µ

""""""

#

uM >a>vM

!!(x+a ) . . .

#

vi+1>a>ui

!+(x!a )#

ui>a>vi

!!(x+a ) . . .

#

v1>a>u0

!+(x!a )

""""""!

$


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0 1 2 3 4 5 6

0

1

2

3

4

5

6


0 1 2 3 4 5 6

0

1

2

3

4

5

6


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0 1 2 3 4 5 6

0

1

2

3

4

5

6


0 1 2 3 4 5 6

0

1

2

3

4

5

6


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Excised region along z-

direction

We count half of the slices with one parameter q and the other half with another t

x x!! zz

yy

M(t, q) !!

!

"#

a<0

q|!a|

$%

&#

a!0

t|!a|

'

(

="#

i,j=1

(1" qi#1tj)#1

The slices are labeled by integers:x! y = a, with a " Z

!

Z!(t, q) =!

"

qP!

i=1 |"(!ti!i)|t

P!j=1 |"(!!j+j!1)|

="

(i,j)/"!

(1! qj!!i!1ti!!tj )!1

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The Refined Topological VertexLocal P1 ! P1

Zinst(Qb, Qf , t, q) =!

!1,!2

Q|!1|+|!2|b q||!t

2||2 t||!t1||2 "Z!1(t, q) "Z!t

2(t, q) "Z!2(q, t) "Z!t

1(q, t)

!!#

i,j=1

(1"Qf ti"1qj)(1"Qf qi"1tj)(1"Qf ti"1"!2,j qj"!1,i)(1"Qf qi"1"!1,j tj"!2,i)

B

F!1

!2

An example: (engineers pure SU(2) gauge theory)

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TomorrowGaiotto’s construction & Alday-Gaiotto-Tachikawa conjecture

Surface operators & Alday-Gaiotto-Gukov-Tachikawa-Verlinde approach

Surface operators from topological string theory

surface operators from topological string theory, part...

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