remodeling the refined topological vertexhome.ku.edu.tr/~sbsg/talks/kozcaz.pdf · refined...

Istanbul 2011: Strings, Branes And Supergravity, 1 August 2011

Remodeling the Refined Topological VertexCan Kozçaz, CERN (joint work in collaboration with Bertrand Eynard, arXiv:1107.5181)

Motivation

✤ The refined topological string theory is related to microscopic derivation of Seiberg-Witten theory

✤ On the A-model side, for the refined case we know how to compute the topological string partition function

✤ On the B-model side, the refined formulation does not exist

✤ The B-model is remodelled using topological recursion which applies for any matrix model, i.e. the mirror curve is placed the spectral curve

✤ If we can reformulate the refined case as a matrix model, we find the refined mirror curve

Outline

✤ Topological String Theory

✤ Geometric Engineering & Topological Vertex

✤ Matrix Model for Plane Partitions

✤ Matrix Model for Refined Topological Vertex & Strip Geometry

✤ Example

✤ Conclusions

Topological String Theory


✤ A simpler framework to check with more confidence ideas in String Theory

✤ Genus zero amplitude computes the prepotential

✤ Higher genus amplitudes compute higher gravitational couplings

✤ Numerous applications in Mathematics


worldsheet of genus g target space M

!


Consider the maps from the worldsheet to the 6d target space M! : !g !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±)

and

y± = x± ! i!±!±


If the worldsheet is flat then the Lagrangian is invariant under supersymmetry!N = (2, 2)

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 d2x

unless we have covariantly constant spinors.

We need to topologically twist the theory


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

M !E = ME + R =

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

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

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


The twisted supercharges QA & QB are nilpotent:

Q2A = 0 Q2

B = 0

The 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

!h!O1 . . .Os" =!

14"

" #h d2x !hµ!{Q, Gµ!}O1 . . .Os

#

= 0

if the insertions correspond to physical operators.


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

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

j , g = 0

!d4x Fg(ti)R2

+ F 2g!2+ , g > 0

genus g amplitude self dual Riemann

tensor (contr.)

self dual graviphoton

Organize the topological string amplitudes into a generating function

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

F (ti) = log Z =!!

g=0

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

Geometric Engineering & Topological Vertex

Geometric Engineering

K3

AN!1

The gauge groupEncoded in the K3: the Cartan matrix of the gauge group the intersection matrix of the 2-cycles!

Asymptotical freedomFurther compactify on a Riemann surface with genus 0 or 1

Supersymmetry

P1

R1,3P1 ! N = 2

T2 ! N = 4

Gauge bosonsD2-branes wrapping 2-cycles in K3, e.g. For SU(2) two different ways of wrapping gives rise to W-bosonsand integrating the RR 3-form over the (only) 2-cycle to get a 1-form for Z!

Matter multiplets and quiversModifications on the 2d space, e.g. enhance over a single point of the base space the singularity, introduce intersecting spheres

Toric geometriesThe low energy dynamics is determined by the prepotential

Topological Vertex

The prepotential of is given by the genus g=0 amplitude, , of topological string theory F0(ti)

All genus answer of the A-model topological string theory is solved by topological vertex formalism for the toric geometries

Example:

r1 = 0, r2 != 0, r3 != 0

C3

C3 ! (z1, z2, z3)

= (r1ei!1 , r2e

i!2 , r3ei!3)

(!2, !3)

(!1)

r1

r2

r3

O(!3) "# P2

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

Z(V1, V2, V3) =!

!,µ,"

C!µ" tr!V1 trµV2 tr"V3

Vi = Pexp!"

A

#

Topological Vertex

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

!

Z = M(q)!/2 with q = eigs

with M(q) is the so-called MacMahon function, that is the generating function of 3d partitions

M(q) =!!

n=1

1(1! qn)n

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

Does a combinatorial description exist for the

topological vertex?

Topological Vertex

The answer is affirmative!

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

directions

C!µ" =G!µ"

M(q)

The generating function of 3dpartitions in the excised region

We count the boxes in this deformed room: the orange

boxes show counting

! µ

!

! µ

!

Topological Vertex

How do we obtain the generating functions?

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

M(q) =!

!

"

a!Zq|"(a)|

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

!

i

!+(xi)|µ! ="

!!µ

s!/µ(x)|!!

Transfer matrix approach

!+(1)|µ! =!

!!µ

|!! !!(1)|µ! =!

!"µ

|!!and

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 )

""""""!

$

The generating function can be writting as

Refined Topological Vertex

The motivation is based on the microscopic derivation of prepotential due to Losev-Moore-Nekrasov-Shatashvili, i.e. performed the integrals over the instanton moduli space!

LMNS deform the space-time from 4d to 6d which admits some Lie group action that can be lifted to the moduli space of instantons, hence, allows to equivariantly integrate the following integrals

Z(a, !1, !2; q) =!!

k=0

qk

"

fMk

1equivariant parameters

F inst(a, !1, !2; q) = !1!2 log Z(a, !1, !2; q)

Localization formulas can be used to perform the integrals (Duistermaat-Heckman formula). The prepotential is obtain in the limit of !1 + !2 = 0, !1,2 ! 0


The implication of this computation to the topological string theory is best understood in terms of the target space interpretation due to Gopakumar&Vafa:

F (!; q) =!

!!H2(X,Z)

"!

k=1

!

jL

(!1)2jLN (jL)! e#kT!

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

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

#, q = e#gs

!1 + !2 = 0In the limit , with !1 ! gs

Degeneracies of BPS particles in M-theory compactificationsto 5d due to M2 branes wrapping 2-cycle which is charged under

!

SO(4) = SU(2)L ! SU(2)R

!1 + !2 != 0When , the refined topological string theory partition functions reads

N (jL)! =

!

jR

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

F (!; t, q) =

!

!!H2(X,Z)

"!

n=1

!

jL,jR

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

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

#"( t

q )#njR + . . . + ( tq )njR

#

n(tn/2 ! t#n/2)(qn/2 ! q#n/2)e#nT! t = e!1 , q = e!!2

The refined topological vertex is constructed to compute this free energy


refinement

t q

Z =!

a

q|!(a)|

The refinement can be accommodated changing the weights of the slices

z z

For an arbitrary representation along the z-direction

Z! = qP!

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

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

!

a"0

t|!(a)|!

a>0

q|!(a)|

u! u" u# u$v" v# v$

Matrix Model for Plane Partitions


Plane partitions are known to be expressed as a model of self-avoiding jumping particles

!"

!!

Left Right

N N!

!!!!!

"

0

!

!

RightLeft

time

"

#$%

time s

particle position


Eynard proposed the following matrix model for the generating function of 3d partitions

Z =!

(HN )smax!smin!1

smax!1"

s=smin+1

dMs

!

(i HN )smax!smin

smax! 12"

s"=smin+ 12

dRs"

smax"

s=smin

e!Tr Vs(Ms)

!smax! 1

2"

s"=smin+ 12

e!Tr Us" (Rs" )

smax! 12"

s"=smin+ 12

eTr Rs" (Ms"+ 1

2!M

s"! 12)

This is a multi-matrix integral of normal matrices and hermitian matrices Ms Rs!

The eigenvalues encode the position of the particles at the sth slice

Imposing the jumps & non-intersecting path condition

Keeping the particles withinthe boundaries


At each time step particles should jump by

Rs+1/2Ms Ms+1 time s

particle position

! ±1/2

!(s +12)"(hi(s + 1)! hi(s)!

12) + #(s +

12)"(hi(s + 1)! hs +

12)

=! !

!dri e2!iri(hi(s+1)"hi(s))

"!(s +

12)e"i!ri + #(s +

12)ei!ri

#

probability for a particle at

slice s to go up

probability for a particle at slice s

to go down}The particles are not allowed to enter the excised region by , imposed by the potential being non-zero!

! e!Tr Us! (Rs! )

e!Tr Vs(Ms)

initial slice


Rs+1/2Ms Ms+1 time s

particle position

! Particles can not occupy the same place, or the paths can not intersect, the generating function of non-intersecting paths can be written as a determinant

!e2!irihi ! det(e2!irihj )

which can be written as a matrix integral using Itzykson-Zuber-Harish-Chandra formula

det(e2!irihj ) = !(R)!(h)!

U(N)dU e2!iRUhU†

Matrix Model for Refined Topological Vertex & Strip Geometry

Matrix Model for Refined Topological Vertexu! u" u# u$v" v# v$

} }

Vs(Ms) = ! log t Ms Vs(Ms) = ! log q Ms

Our choice of potentials

The potentials except slices are linear, i.e. most of the integrals can be performed explicitly to get

s = ui

Zi!1,i = eTr Rvi+1/2(Mui!Mui!1 )e(vi!ui!1) log t Tr Mui!1+(ui!vi!1) log q Tr Mui e(2vi!ui!ui!1)Tr Rvi+1/2/2

!det

!"t e!Rvi+1/2 ; t

""

det!"tvi!ui!1+1 e!Rvi+1/2 ; t

""

det!"eRvi+1/2 ; q

""

det!"q(ui!vi) eRvi+1/2 ; q

""

The whole integral can be written as

Z =!

(HN )N

N"

i=1

dMui

!

(iHN )NdRvi+1/2 e!Tr Vui (Mui )Zi!1,i(Mui!1 , Rvi+ 1

2, Mui)

Matrix Model for Refined Topological Vertex

Having found the matrix model, we can compute the spectral curve. Spectral curve is obtain by the saddle point of the matrix model. In general

rs!1/2(z)! rs+1/2(z) = V "s (xs(z))

xs!+1/2(z)! xs!!1/2(z) = U "s!(rs!(z)) = ! 1

2ers! (z) ! 1ers! (z) + 1

Specifically, for our model (after summing over the intermediate slices)• ui!1 < s ! vi

xvi(z)! xs(z) = !12

vi!s!

k=1

Yi(z)! tk

Yi(z) + tk=

s! vi

2+

vi!s!

k=1

tk

Yi(z) + tkYi(z) = eRi(z) , Xs(z) = t!xs(z)

In the limit log t! 0 Xs(z)Xvi(z)

! tvi!s

2 + ts!vi

2 Yi(z)1 + Yi(z)

(1 + O(log t))


• vi < s < ui

xs(z)! xvi(z) = ! 12

s!vi!1!

k=0

qkYi(z)! 1qkYi(z) + 1

=s! vi

2!

s!vi!1!

k=0

qkYi(z)qkYi(z) + 1

In the limit log t! 0!

Xs(z)Xvi(z)

"1/!

! qvi!s

2 + qs!vi

2 Yi(z)1 + Yi(z)

(1 + O(ln t))

with ! = log t/ log q

• s = uirui! 1

2(z)! rui+ 1

2(z) = V "

ui(xui(z))

Yi+1(z)Yi(z)

= tvi+1!ui qui!vi


Our Ansatz is in the following form with undetermined functions whose analyticy conditions are determined by boundary conditions

Xs(z) = Xvi(z)t

vi!s2 + t

s!vi2 Yi(z)

1 + Yi(z), ui!1 ! s ! vi

Xs(z) = Xvi(z)

!q

vi!s2 + q

s!vi2 Yi(z)

1 + Yi(z)

"!

, vi ! s ! ui

Yi+1(z)Yi(z)

= tvi+1!ui qui!vi , s = ui

Matrix Model for Strip Geometry

One of the most interesting geometries is the following strip geometry since it engineers a superconformal theory in 4d, and through AGT correspondence they are related to (chiral) 3-point function of Toda theory

5.2 Some facts about the geometric engineering of SU(N) theories

In this section we briefly review the toric geometries that engineer N = 2 SU(N)theories. We first recall that the SU(N) theory with Nf = 2N can be engineered bythe following toric geometry

Figure 4: The toric version of SU(N) with Nf = 2Nc.

The partion function for this geometry (which agrees with the Nekrasov instantonpartition) can be computed by gluing the left and right parts (each of which is a strip)along the dotted line using the refined topological vertex (see e.g. [20] for a discussion).

Each of the two strips in the above geometry is related to special case of the TN

theory. This follows from the fact that in Gaiotto’s language the SU(N) theory withNf = 2N theory is obtained via compactification on a sphere C with four punctures(two basic U(1) punctures and two full SU(N) punctures). In the weakly coupleddegeneration limit where C splits into two spheres, each sphere has one basic and twofull punctures (one full puncture comes from the degeneration of the thin neck). Eachspehere corresponds to a degenerate TN theory with one basic U(1) puncture and twofull SU(N) punctures. We will refer to this theory as !TN . Via the AGT conjecture,the TN theory is related to a (chiral) AN!1 Toda three-point function with one of thethree primary fields of a special type [4].

Let us give some details for the !T2 case. The relevant toric strip diagram for !T2 isgiven in figure 5.

The partition function for this strip is

Z "eT2

=#"

i,j=1

(1!Q1 q!!it!!j )(1!Qf q!!it!!j )(1!Q2 q!!it!!j )(1!Q1QfQ2 q!!it!!j )

(1 ! Q1Qf q!!i+1/2t!!j!1/2)(1 ! QfQ2 q!!i!1/2t!!j+1/2).

(5.12)

Clearly this partition function for the !T2 geometry does not agree with the one for theT2 geometry (5.3) (the closed topological vertex), even though in this case all threepunctures are of the same type (basic). The latter geometry treats all punctures on

24

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29

SU(N) withNf = 2N strip

In other words, knowing the spectral curve, one can recover the full ln t power series expansionof the partition function, by applying the topological recursion to the spectral curve.

We shall not enter detailed computation of symplectic invariants of the spectral curve we havejust found, but we just mention that this statement, is the refined version of ”re–modelingthe B–Model” of Marino, Bouchard, Klemm, Pasquetti [?, ?].

This seems to imply that mirror symmetry extends to the refined version of Gromov–Wittentheory.

7. Toric Geometriestoric

In this section we want to propose a crystal model for the refined strip geometry. Our modelwill resemble the bubbling picture of [11]. We will excise a region such as the one in figure 13whose lengths are related to the Kahler parameters in the strip geometry. We grow the crystalin the remaining unbounded room. Our model for this geometry requires to redefine the weightson the slices along the outer corners, each such slice is counted with t. We want to work out theexample shown in figure 13 in detail. From our appr it is clear that there is no obstruction ingeneralizing this example to a longer strip.

Q!

Q"

Q#

Q$m!

m"

n! n"

t

q

% &

'

(

)

Figure 13: (a) the definition of slices, (b) the region where the crystal grows, (c) the refined stripgeometry young

We will compute the generating function for the crystal using the transfer matrix approach.The following identities are crucial for our computation. The vertex operators !±(1) and theHamiltonian L0 satisfy

!!(1)|µ! =!

!"µ

|!!, !+(1)|µ! =!

!#µ

|!!, and qL0 |µ! = q|µ||µ!. (7.1)

We will compute the generating function in the following way: we will assign an arbitraryYoung diagram to the slices along the inner and outer edges and find the contributions with these‘boundary’ conditions. In other words we are inserting identity operators at certain places in thegenerating function to ease the computation. Later we sum over all possible Young diagramsalong these slices and match these sums to the summations over Young diagram we obtain fromthe refined topological vertex gluing. It will be obvious that the continuation of this procedure

– 26 –

Motivated by bubbling and wall constructions

Matrix Model for Strip Geometry

In other words, knowing the spectral curve, one can recover the full ln t power series expansionof the partition function, by applying the topological recursion to the spectral curve.

We shall not enter detailed computation of symplectic invariants of the spectral curve we havejust found, but we just mention that this statement, is the refined version of ”re–modelingthe B–Model” of Marino, Bouchard, Klemm, Pasquetti [?, ?].

This seems to imply that mirror symmetry extends to the refined version of Gromov–Wittentheory.

7. Toric Geometriestoric

In this section we want to propose a crystal model for the refined strip geometry. Our modelwill resemble the bubbling picture of [11]. We will excise a region such as the one in figure 13whose lengths are related to the Kahler parameters in the strip geometry. We grow the crystalin the remaining unbounded room. Our model for this geometry requires to redefine the weightson the slices along the outer corners, each such slice is counted with t. We want to work out theexample shown in figure 13 in detail. From our appr it is clear that there is no obstruction ingeneralizing this example to a longer strip.

Q!

Q"

Q#

Q$m!

m"

n! n"

t

q

% &

'

(

)

Figure 13: (a) the definition of slices, (b) the region where the crystal grows, (c) the refined stripgeometry young

We will compute the generating function for the crystal using the transfer matrix approach.The following identities are crucial for our computation. The vertex operators !±(1) and theHamiltonian L0 satisfy

!!(1)|µ! =!

!"µ

|!!, !+(1)|µ! =!

!#µ

|!!, and qL0 |µ! = q|µ||µ!. (7.1)

We will compute the generating function in the following way: we will assign an arbitraryYoung diagram to the slices along the inner and outer edges and find the contributions with these‘boundary’ conditions. In other words we are inserting identity operators at certain places in thegenerating function to ease the computation. Later we sum over all possible Young diagramsalong these slices and match these sums to the summations over Young diagram we obtain fromthe refined topological vertex gluing. It will be obvious that the continuation of this procedure

– 26 –

Z =!

!,",#,µ,$

!"|!"

i=1

qL0!+(1)|!#!!|tL0!"(1)n1"1"

i=1

tL0!"(1)|"#!"|tL0!+(1)m1"m2"1"

i=1

qL0!+(1)|##

$ !#|tL0!"(1)n2"n1"1"

i=1

tL0!"(1)|µ#!µ|tL0!+(1)|$#!$|tL0

!"

i=1

!"(1)tL0 |"# The generating function matches the vertex computation

Q1

!q

t= qm2

Q2

!q

t= tn2!n1

Q3

!q

t= qm1!m2

Q4

!q

t= tn1

Example

Example: Hexagon

Hexagonal region corresponds to the refined vertex with empty representations. The corresponding matrix model reads N

a b

Z =!

(iHN )dR eTr R(Mb!M!a) ea log t Tr M!a e(b!1) log q Tr Mb

det"!t e!R; t

#"

det (!ta+1 e!R; t)"

det"!eR; q

#"

det (!qb eR; q)"

Our Ansatz becomest!xs(z) = Xs(z) = u(z)

t!s/2 + ts/2 Y (z)1 + Y (z)

, !a " s " 0

q!xs(z) = (Xs(z))1/! = u(z)1/! q!s/2 + qs/2 Y (z)1 + Y (z)

, 0 " s " b

Example: Hexagon

Tangecy and pole cancellation for the minimum degree curve lead to the generic form

Y (z) = !C(z ! !)(z ! ")(z ! #)(z ! $)

, u(z) = rz

1! z

!1

1! w z

"!One can easily check that those 8 equations are not independent, only 7 of them are independent,and thus they allow to determine the 7 unknowns.

Plots of the corresponding arctic circle, generated by a Mathematica code, are displayed infig.11 for various values of !.

-6 -4 -2 2 4 6

-12

-10

-8

-6

-4

-2

-6 -4 -2 2 4 6

-12

-10

-8

-6

-4

-2

-6 -4 -2 2 4 6

-12

-10

-8

-6

-4

-2

-6 -4 -2 2 4 6

-12

-10

-8

-6

-4

-2

-6 -4 -2 2 4 6

-12

-10

-8

-6

-4

-2

-6 -4 -2 2 4 6

-12

-10

-8

-6

-4

-2

Figure 11: ! = 1, 2, 3, 5, 10, 50 fighexagon

5.1.1 Sending the size to !

First, let us keep a and b finite and send N "!.In that case, Y (z) needs to have only one pole and one zero. Upon a reparametrization of

z, we may choose

Y (z) = #z (5.11)

We must have (5.8), i.e.:

u(1) = 0u(0) = t!a

u(!) = t!b

u(ta) =!u(q!b) =!

(5.12)

– 20 –

! = 1 ! = 2 ! = 3

! = 5 ! = 10 ! = 50

From this data we can construct the spectral curve!

Conclusions

Conclusions

✤ The refined topological vertex and the refined strip geometries admit combinatorial interpretations, hence, can be described as a matrix model

✤ Our matrix model is a multi-Hermitian matrix model, as opposed to the so-called -deformed matrix models

✤ The spectral curves of these models can be found (by minimal assumptions) and give nice generalizations of the mirror curves to the refined case

✤ Topological recursion can be applied and implies integrability, e.g. integrability of (chiral) 3-point functions of Toda theory

!

Thank you!

remodeling the refined topological vertexhome.ku.edu.tr/~sbsg/talks/kozcaz.pdf · refined...

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