Disk Instantons, Mirror Symmetry and the Duality Web

Mina Aganagic, Albrecht Klemma, and Cumrun Vafa Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA a Institut fur¨ Physik, Humboldt Universitat¨ zu Berlin, Invaliden Straße 110, D-10115 Berlin, Germany Reprint requests to Prof. A. K.; E-mail: [email protected]

Z. Naturforsch. 57 a, 1–28 (2002); received February 3, 2002 We apply the methods recently developed for computation of type IIA disk instantons using mirror symmetry to a large class of D-branes wrapped over Lagrangian cycles of non-compact Calabi-Yau 3-folds. Along the way we clarify the notion of “flat coordinates” for the boundary theory. We also discover an integer IR ambiguity needed to define the quantum theory of D-branes

wrapped over non-compact Lagrangian submanifolds. In the large N dual Chern-Simons theory,

this ambiguity is mapped to the UV choice of the framing of the knot. In a type IIB dual description p; q involving (p; q ) 5-branes, disk instantons of type IIA get mapped to ( ) string instantons. The M-theory lift of these results lead to computation of superpotential terms generated by M2 brane

instantons wrapped over 3-cycles of certain manifolds of G2 holonomy. Key words: Supersymmetry; Open String Theory; Topological Theories; Mirror Symmetry.

1. Introduction classical considerations of the worldsheet theory. In a recent paper [6] it was shown how one can use mirror D-branes wrapped over non-trivial cycles of a symmetry in an effective way to transform the type Calabi-Yau threefold provide an interesting class of IIA computation of disk instantons to classical com- theories with 4 supercharges (such as N = 1 super- putations in the context of a mirror brane on a mirror symmetric theories in d = 4). As such, they do allow CY for type IIB strings. The main goal of this paper is the generation of a superpotential on their worldvol- to extend this method to more non-trivial Calabi-Yau ume. This superpotential depends holomorphically on geometries. the chiral fields which parameterize normal deforma- One important obstacle to overcome in generaliz- tions of the wrapped D-brane. ing [6] is a better understanding of “flat coordinates” On the other hand, F-terms are captured by topo- associated with the boundary theory, which we re- logical string amplitudes [1], and in particular the solve by identifying it with BPS tension of associated superpotential is computed by topological strings at domain walls. We also uncover a generic IR ambiguity

the level of the disk amplitude [1 - 4]. More generally given by an integer in defining a quantum Lagrangian h the topological string amplitude at genus g with D-brane. We relate this ambiguity to the choice of the

holes computes superpotential corrections involving regularizations of the worldsheet theory associated to N

the gaugino superfield W and the = 2 graviphoton the boundaries of moduli space of Riemann surfaces

R

g

2 2 h 1 2

h  W W multiplet W given by d (Tr ) ( ) [5]. with holes (the simplest one being two disks con- So the issue of computation of topological string am- nected by an infinite strip). In the context of the Large

plitudes becomes very relevant for this class of super- N Chern-Simons dual [7] applied to Wilson Loop symmetric theories. observables [4] this ambiguity turns out to be related In the context of type IIA superstrings such disk to the UV choice of the framing of the knot, which amplitudes are given by non-trivial worldsheet instan- is needed for defining the Wilson loop observable by tons, which are holomorphic maps from the disk to the point splitting [8]. CY with the boundary ending on the D-brane. Such Along the way, for gaining further insight, we con- computations are in general rather difficult. The same sider other equivalent dual theories, including the lift

questions in the context of type IIB strings involve to M-theory, involving M-theory in a G2 holonomy
0932–0784 / 02 / 0100–0001 $ 06.00 c Verlag der Zeitschrift fur¨ Naturforschung, Tubingen¨ www.znaturforsch.com

2 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web

;:::k G background. In this context we are able to transform where a =1 , and dividing by

the generation of superpotential by Euclidean M2

a

i iQ  i a

3 i

X ! e X : branes (with the topology of S ) to disk instantons (2.2)

1 G

of type IIA for M-theory on 2 holonomy manifolds P

a

Q X

The c ( ) = 0 condition is equivalent to =0. i

and use mirror symmetry to compute them! We also 1 i a relate this theory to another dual type IIB theory in The Kahler¨ structure is encoded in terms of the r , and varying them changes the sizes of various 2 and 4 a web of (p; q ) 5-branes in the presence of ALF-like

geometries. cycles. In the linear sigma model realization [9] this k

The organization of this paper is as follows: In is realized as a (2,2) supersymmetric U (1) gauge

i X

Sect. 2 we review the basic setup of [6]. In Sect. 3 we theory with 3 + k matter fields with charges given

k a

k U by Q , and with FI terms for the (1) gauge group

consider the lift of these theories to M-theory in the i a

given by r .

context of G2 holonomy manifolds, as well as to type k The mirror theory is given in terms of n + dual

IIB theory with a web of (p; q ) 5-branes in an ALF-like

i  background. In Sect. 4 we identify the flat coordinates C fields Y [10], where

for boundary fields by computing the BPS tension of i

i 2

jX j D4 brane domain walls ending on D6 branes wrap- Re(Y )= (2.3)

ping Lagrangian submanifolds. In Sect. 5 we discuss

i i

 Y i the integral ambiguity in the computation of topolog- with the periodicity Y +2 . The D-term equa- tion (2.1) is mirrored by

ical string amplitudes and its physical meaning. This

a a a a

is discussed both in the context of Large N Chern- 1 2 3+k

Y Y Q Y :::Q t ; Q + + = (2.4)

Simons / topological string duality, as well as in the 1 2 3+k

p; q

a a a

context of the type IIB theory with a web of ( ) a

r i   where t = + and denotes the -angles of

5-branes. In Sect. 6 we present a large class of exam- a the U (1) gauge group. Note that (2.4) has a three-

ples, involving non-compact CY 3-folds where the D6 dimensional family of solutions. One parameter is i

brane wraps a non-compact Lagrangian submanifold. i

! Y c trivial and is given by Y + . Let us pick In appendix A we perform some of the computations a parameterization of the two non-trivial solutions relevant for the framing dependence for the unknot by u; v . and verify that in the large N dual description this The mirror theory can be represented as a theory of UV choice maps to the integral IR ambiguity we have variations of complex structures of a hypersurface Y discovered for the quantum Lagrangian D-brane.

1 k +3

u;v Y u;v

Y ( ) ( )

xz ::: e  P u; v ; = e + + ( ) (2.5) 2. Review of Mirror Symmetry for D-branes where

In this section we briefly recall the mirror sym-

i i i i

u; v a u b v t t metry construction for non-compact toric Calabi-Yau Y ( )= + + ( ) (2.6) manifolds (specializing to the case of threefolds), in-

cluding the mirror of some particular class of (special) is a solution to (2.4) (in obtaining this form, roughly i

Lagrangian D-branes on them. speaking the trivial solution of shifting of all the Y

Toric Calabi-Yau threefolds arise as symplectic has been replaced by x; z whose product is given k

3+k

X G U

quotient spaces = C ==G, for = (1) . The by the above equation). We choose the solutions so

i i

kD

 Y i quotient is obtained by imposing the -term con- that the periodicity condition of the Y +2

straints are consistent with those of u; v and that it forms a

fundamental domain for the solution. Note that this

a a a a k a i

1 2 2 2 3+ 2 i

D Q jX j Q jX j :::Q jX j r ;b = + + a 1 2 3+k in particular requires to be integers. Even after

taking these constrains into account there still is an

;

ZZ =0 (2.1) SL(2; ) group action on the space of solutions via

1More generally we can map the generation of superpotential-

! au bv ;

u + h like terms associated to topological strings at genus g with bound-

aries to Euclidean M2 brane instantons on a closed 3-manifold with

b g h dv : ! cu 1 =2 + 1. v + M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web 3

Note that the holomorphic 3-form for CY is given by Under mirror symmetry, the A-brane maps to a

holomorphic submanifold of the Y given by

u v

dx d d ; =

1 3+k

Y u;v Y u;v

x ( ) ( )

P u; v e ::: e :

x =0= ( )= + + (2.9)

ZZ and is invariant under the SL(2; ) action. The mirror brane is one-complex dimensional, and is

parameterized by z . Its moduli space is one complex 2.1. Special Lagrangian Submanifolds dimensional parameterized by a point on a Riemann

and Mirror Branes u; v surface P ( ) = 0. The choice of the point depends

1 S In [6] a family of special Lagrangian submanifolds on c and the Wilson line around and it is possi-

of the A-model geometry was studied, characterized ble to read it off in the weak coupling limit of large

c

volume of Calabi-Yau and large parameters ,as

i ; :::; k ;

by two charges q with =1 + 3 and =12, i

subject to discussed in [6]. X

2.2. Example

q =0

i

i

O  O ! For illustration consider X = ( 1) ( 1) 1 and in terms of which the Lagrangian submanifold is P , which is also called small resolution of conifold.

given by three constrains. Two of them are given by This sigma model is realized by U (1) gauge theory

; ; ;

with 4 chiral fields, with charges Q =(1 1 1 1).

X

i

2 1 2 2 2

jX j jX j

jX j c q = (2.7) The D-term potential vanishes on +

i 3 2 4 2

jX jX j r X

j = , and is a quotient of this by

P

i i

U (1). The D-term equations can be regarded as lin- 

and the third is = 0, where  denotes the phase i

i 2

i

!jX j ear equations by projecting X , and solved of X . The worldsheet boundary theory for this class 3 of theories has been further studied in [11]. graphically in the positive octant of R (see Fig. 1).

The submanifolds in question project to the one X is fibered over this base with a fiber which is

i 3 4

U T T =U dimensional subspaces of the toric base (taking into torus of phases of X ’s modulo (1), = (1).

1 3 4

X X account the constrains (2.1), (2.7)), Note that r is the size of a minimal P at =0= .

Consider a special Lagrangian D-brane in this i

i 2

X j r b

j = + (2.8)

; ; ; q ; ; ; background with q 1 =(1 0 0 1), 2 =(0 0 1 1).

1 2 4 2

X j jX j c

This gives the two constrains j = 1 and

i a

c ;r r 2

for some fixed b (depending on ) and 3 2 4 2

X j jX j c + j = 2 in the base which determine a two R . In order to get a smooth Lagrangian subman- dimensional family of Lagrangians, but D-branes of

ifold one has to double this space (by including

P

i   = ). The topology of the Lagrangian submani-

1 1

 S  S

fold is R . There is, however, a special choice

of c which makes the Lagrangian submanifold pass X³=0 through the intersection line of two faces of the toric base. The topology of the Lagrangian submanifold X²=0 will be different in this limit. It corresponds to having X¹=0

1 R one of the S cycles pinched at a point of in the La- 4 grangian submanifold. This is topologically the same X =0

1 S as two copies of C  touching at the origin of C. In this limit we view the Lagrangian submanifold as

being made of two distinct ones intersecting over an 1

O  O ! 1 Fig. 1. X = ( 1) ( 1) P viewed as a toric fibra-

S . We can now have a deformation, which moves the 1 2 3 2 4 2

X j ; jX j ; jX j tion. The base is (j ) as generic solution two Lagrangian submanifolds independently, where to the vanishing of the D-term potential, but is bounded by

2 2

X j  the end point of each one should be a point (not nec- the j 0 hyperplane. Over the faces of the bounding essarily the same) on the base of the toric geometry hyperplanes some cycles of the fiber shrink. For example, 1 (see the example below). there is a minimal P in X which lies over the finite edge. 4 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web

Fig. 2. The special Lagrangian sub-  manifold which has the topology R

2

T c a. for generic values of i (case a) can b. degenerate (case b) and split (case c) into two Lagrangian submanifolds, when it approaches a one-dimensional edge of the toric base. The two re- sulting components have the topol- 1

ogy C  S , and can move indepen- dently, but only along one-dimen- sional edges. c.

infinity

1

P u; v t X Fig. 3. Riemann surface  : ( ) = 0 corresponding to Fig. 4. In the limit in which the size of the P in =

1 1

O  O !  O  O ! the mirror of X = ( 1) ( 1) P . is related to the ( 1) ( 1) P goes to infinity, the manifold looks 3 toric diagram of X by thickening out the one-dimensional locally like C , together with a Lagrangian D-brane. edges of the base in Figure 1. The mirror B-brane propagates on the Riemann

1

u v tu v S

topology C  are further constrained to live on the +

u; v e e e surface 0 = P ( )= + + + 1 shown in

one dimensional faces of the base. For this we need Figure 3.

c ;r for example c2 = 0, and 1 arbitrary but in the (0 ) Note that mirror map (2.3) gives the B-brane at

interval. As discussed above, this can be viewed as

c v Re(u)= 1 and Re( ) = 0 which is on the Riemann

coming from the deformation of a Lagrangian sub-

 r= > surface in the large radius limit, r 0 and 2

manifold which splits to two when it intersects the  c1 0. In other words, in the large radius limit, edges of the toric geometry and move them indepen- classical geometry of the D-brane moduli space is a dently on the edge. See Figure 2. Typically we would good approximation to the quantum geometry given be interested in varying the position of one brane, by  . keeping the other brane fixed (or taken to infinity We can also construct, as a limit, Lagrangian sub-

along an edge). 3

i t !

manifolds of C by considering the limit r + =

X 1

The mirror of is holding c 1 fixed, as shown in Figure 4. In this limit

u v

e e

the mirror geometry becomes xz = + + 1. This

v v tu

u + case was studied in detail in [12].

e e e ; xz = + + +1 2.3. Disk Amplitude

1 2 3 4

Y Y Y t obtained by solving Y + = for

2 4 Y Y , fixing the trivial solution by setting = 0, and The disk amplitudes of the topological A-model

1 3

u Y v N putting Y = and = . give rise to an = 1 superpotential in the corre-

M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web 5

; ZZ sponding type IIA superstring theory [1 - 4] where we domain for u; v given by an SL(2 ) transformation, view the D6 brane as wrapping the Lagrangian sub- but transforms as

manifold and filling the spacetime. The corresponding Z

superpotential for the mirror of the Lagrangian sub- 2 2

u ! W u acu = bdv = bcuv W ( ) ( )+ d[ 2+ 2 ] manifolds we have discussed above was computed in [6], and is given in terms of the Abel-Jacobi map

2 2

u acu = bdv = bcuv ;

= W ( )+ 2+ 2

Z

u

u v u u;

W ( )= ( )d (2.10) u

where v is defined implicitly in terms of by

u

 u; v

P ( ) = 0. Note that if we added a boundary term u

where  is some fixed point on the Riemann sur- it could have canceled this change in superpotential, u; v face P ( ) = 0 and the line integral is done on this which can be viewed as a choice of boundary con- surface. This defines the superpotential up to an ad- dition at infinity on the non-compact brane [6]. Thus dition of a constant. More physically, if we construct this IR choice is needed for the definition of the brane,

the ‘splitting’ of the Lagrangian brane over the toric and as we see it affects the physics by modifying the u

edges, we can view  as the location of one of the superpotential. As discussed in [6] the choice of the

Lagrangian halves, which we consider fixed. splitting to u; v depends on the boundary conditions

Note that, if we move the point u on the Riemann at infinity on the fields normal to the brane. Each

ZZ surface over a closed cycle and come back to the SL(2; ) action picks a particular choice of boundary same point, the superpotential (2.10) may change by conditions on the D-brane. Using the mirror symme- an overall shift which depends on the choice of the try and what A-model is computing, below we will be cycle as well as the moduli of the Riemann surface able to fix a canonical choice, up to an integer, which

(given by t’s). It is natural to ask what is the inter- we will interpret physically. pretation of this shift. This shift in superpotential can As noted above, in terms of the topological A-

be explained both from the viewpoint of type IIA model, superpotential W is generated by the disk am- and type IIB. In the context of type IIA this corre- plitudes. The general structure of these amplitudes sponds to taking the Lagrangian D6 brane over a path has been determined in [4], where it was found that whose internal volume traces a 4-dimensional cycle X 1

C4 of CY (fixing the boundary conditions at infinity).

W N n ku
: = 2 k;mexp( [ m t]) (2.11) By doing so we have come back to the same Brane n

k ;n;m

configuration, but in the process we have shifted the C RR 2-form flux. The 4-cycle 4 is dual to a 2-form Here u parameterizes the size of a non-trivial holo-

which we identify with the shift in the RR 2-form N morphic disk, and where k;m are integers capturing

flux. In the type IIA setup this process changes the the number of domain wall D 4 branes ending on the

R

k ^ k superpotential by (the quantum corrected) , D 6 brane, which wrap the CY geometry in the 2-cycle C4

as discussed in [13 - 15]. The Type IIB version of this class captured by m, and k denotes the wrapping num- involves varying the D5 brane wrapped over a 2-cycle ber around the boundary. over a path and bringing it back to the original place. In the large volume limit (where the area of 2- During this process the brane traces a 3-cycle in the cycles ending or not ending on the D-brane are large) internal Calabi-Yau which contributes the integral of the A-model picture is accurate enough. In this case the holomorphic 3-form over the 3-cycle to the we do not expect a classical superpotential as there is

superpotential. This is interpreted as shifting the RR a family of special Lagrangian submanifolds. Since

H

u v W flux of along the dual 3-cycle. Note that we can dW=d = and should be zero for any moduli

use this idea to generate fluxes by bringing in branes of the brane, we learn that v = 0 on the brane. This not intersecting the toric edge, to the edges, splitting in particular chooses a natural choice of parameteri- them on the edge and bringing it back together and zation of the curve adapted to where the brane is. In then moving it off the toric edge. The process leads particular the D-brane is attached to the line which is

to the same CY but with some RR flux shifted. classically specified by v = 0 (which can always be

The superpotential (2.10) is not invariant under dif- done). u should be chosen to correspond to the area ferent choices of parameterization of the fundamental of a basic disk instanton. However this can be done in 6 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web many ways. In particular, suppose we have one choice

of such u. Then

z1

! u nv u +

x is an equally good choice, because v vanishes on the Lagrangian submanifold in the classical limit. So even though this ambiguity by an integer is irrelevant in z2

the classical limit, in the quantum theory, since v is non-vanishing due to worldsheet instanton correc- tions, this dramatically changes the quantum answer. S³ Thus we have been able to fix the u; v coordinates up

3 1 1

n

S  S to an integer choice for each particular geometry Fig. 5. We can view S as an fibration over an of brane. We will discuss further the meaning of the interval. Near the ends of the interval it can be viewed as

1

S z a complex plane C  , where the complex plane is 1 choice of n in Sect. 3 and 4.

at one end and z2 at the other. This gives two inequivalent Later we will see that there is a further correction 3

descriptions of S in terms of a circle fibered over a disk. to what u; v are quantum mechanically. In particular, as we will discuss in Sect. 3, this arises because the M2 brane instantons wrapping around non-trivial 3- quantum area of the disk differs from the classical cycles. Some examples of Euclidean M2 brane instan-

computation which gives u. This is similar to what tons for G2 holonomy manifolds have been studied happens for the closed string theory where the pa- in [21] In fact there is a direct map from the disks

rameter t which measures the area of the basic sphere ending on L to a closed 3-cycle with the topology

is replaced by the quantum corrected area T . This is 3 of S . In order to explain this we first discuss some usually referred to as the choice of the “flat coordi- 3 topological facts about S . 3 nates” for the Calabi-Yau moduli. We can view S as

2 2

z j jz j j 1 + 2 =1 3. G2 Holonomy and Type IIB 5-brane Duals

2

z x jz j

with i complex numbers. Let = 1 . The range

Consider type IIA superstrings on a non-compact 1 1

S  S for x varies from 0 to 1. There is an of

Calabi-Yau threefold X with a special Lagrangian 3

x < x <

S which projects to any fixed with 0 1,

 X

submanifold L . Consider wrapping a D6 brane

z x

4 given by the phases of z1 and 2.At = 0, the circle

R N

around L and filling . This theory has = 1 su- x 4 corresponding to the phase of z1 shrinks and at =1 persymmetry on R and we have discussed the super-

the circle corresponding to the phase of z2 shrinks. potential generated for this theory. In this section we 3

So we can view the S as the product of an interval

would like to relate this to other dual geometries. 1 1 S with two S ’s where one shrinks at one end and 1

the other S shrinks at the other end. See Figure 5. 3

3.1. M-theory Perspective We can also view S as a disk times a circle where the circle vanishes on one boundary – this can be done D6 branes are interpreted as KK monopoles of M- in two different ways, as shown in Figure 5. theory. This means that in the context of M-theory the Now we are ready to return to our case. Consider a theories under consideration should become purely disk of type IIA. The M2 brane Euclidean instanton

1 1 S geometric. This in fact was studied in [16 - 20], where can be viewed as the disk times an S , where the it was seen that the M-theory geometry corresponds is the ‘11-th’ circle. Note that on the boundary of the to a 7 dimensional manifold with G 2 holonomy. In disk, which corresponds to the Lagrangian subman- other words we consider a 7-fold which is roughly ifold, the 11-th circle shrinks. Therefore, from our

1 1

 X  S S Y where is fibered over the CY manifold discussion above, this three dimensional space has

3 X

and vanishes over the location of the Lagrangian the topology of S .

 X submanifold L . In this context the superpoten- We have thus seen that using mirror symmetry, tials that we have computed must be generated by by mapping the type IIA geometry with a brane to M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web 7 an equivalent type IIB with a brane, and computing (1,1) the superpotential, in effect we have succeeded in transforming the question of computation of super- potentials generated by M2 brane instantons in the (0,1) context of G2 holonomy manifolds, to an applica- tion of mirror symmetry in the context of D-branes! More generally, one can in principle compute, using (1,0) mirror symmetry [22], a partition function for higher genus Riemann surfaces with boundaries. This com-

putes F-term corrections to the spacetime theory [5] p; q

R Fig. 6. A ( ) web of 5-branes which is dual to M-theory h 1

4 2 2 g 2 3

h x F W W W

given by d d g;h ( ) Tr( ) , where on C . This web is a junction involving a D5 brane which

p; q ; ; N is the gravi-photon multiplet, and W the = 1 gaug- has ( )=(10), an NS5 brane which has (0 1) and a

(1; 1) brane. M ino superfield containing the U ( ) field strength on

the worldvolume of M coincident KK monopoles (if metry requirement (balancing of the tensions) de- we wish to get infinitely many such contributions we

pending on the value of the type IIB coupling con-

! 1

need M ). It is easy to see that the topology p; q stant  . In particular each ( ) fivebrane is stretched of the corresponding M2 brane instantons is a closed

along one dimensional line segments on a 2-plane

g h 3-manifold with b =2 + 1. The case of the 1 which is parallel to the complex vector given by

ordinary superpotential is a special case of this with q

p + . An example of a configuration involving a D5 ;h g =0 =1. brane, NS 5 brane and a (1,1) 5-brane is depicted in Figure 6. 3.2. Dual Type IIB Perspective Now we recall from the previous discussion that to

get the G 2 holonomy manifold we need to consider 1 We have already given one dual type IIB theory an extra S which is fibered over the corresponding related to our type IIA geometry, and that is given CY. In other words we are now exchanging the ‘5-th’ by the mirror symmetry we have been considering. circle with the ‘11-th’ circle. So we consider going However there is another type IIB dual description down to 4 dimensions on a circle which is varying in which is also rather useful. size depending on the point in Calabi-Yau. In partic- Consider M-theory on a non-compact Calabi-Yau ular the circle (i. e. the one corresponding to the 5-th

2 T X compactified to 5 dimensions, which admits a dimension) vanishes over a 2-dimensional subspace action, possibly with fixed points. We can use the du- of 5-dimensional geometry of type IIB (it vanishes

2 1 S ality of M-theory on T with type IIB on [23] along the radial direction of the Lagrangian subman- to give a dual type IIB description for this class of ifold on the base of the toric geometry as well as on

1 2 T Calabi-Yau manifolds. Note that the complex struc- the S which is dual to the of M-theory). Indeed 2 ture of the T gets mapped to the coupling constant it corresponds to putting the IIB 5-brane web in to a of type IIB. The non-compact Calabi-Yau manifolds background of ALF geometry dictated by the loca- 2 we have been considering do admit T actions and tion of the Lagrangian submanifold in the base times 1 in this way they can be mapped to an equivalent type S , and varying the geometry and splitting the ALF IIB theory. This in fact has been done in [24] where geometry into two halves, as shown in Figure 2. In

it was shown that this class of CY gets mapped to this picture the worldsheet disk instantons of type IIA p; q type IIB propagating on the web of (p; q ) 5-branes get mapped to ( ) Euclidean worldsheet instantons, considered in [25]. In this picture the 5-branes fill the wrapping the 5-th circle and ending on the 5-branes. 5 dimensional space time and extend along one di- In particular, if we follow the map of the Euclidean in- rection in the internal space, identified with various stanton to this geometry, it is the other disk realization

3 2 S edges of the toric diagram. The choice of (p; q )5- of (see Fig. 7) .

2 T branes encodes the (p; q ) cycle of shrinking over 2 3

the corresponding edge. The 5-branes are stretched Note that a D6 brane wrapped around S is realized in type IIB

x D y as an NS 5-brane in the direction, a 5 brane in the direction along straight lines ending on one another and mak- separated in the z direction, and where the 5-th circle vanishes

ing very specific angles dictated by the supersym- along the interval in the z direction joining the two branes. 8 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web the domain-wall D-brane which is mirror brane to the D4 brane of the A-model. IIA Instanton

The D4 brane wrapping a minimal disk D is mag- netically charged under the gauge field on the D6

3;1

brane. Consider the domain wall which in the R is

at a point in x 3 and fills the rest of the spacetime. The L Bianchi identity for the gauge field-strength F on , modified by the presence of the D4 brane, says that IIB Instanton

if the B is the cycle Poincare´ dual to the boundary

 L L 3 of the disk, @D . Recall that our brane has the

Fig. 7. The M2 brane instanton with a topology of S wrap- 1

S B topology of C  ,so can be identified with C. ping over a 3-cycle of a local G2 manifold gets mapped to

3 Then the charge n of the domain wall is measured by two alternative disk projections of S , depending on which

duality we use. In one case we get the description involving Z

a type IIA string theory on a CY with the D-brane wrapped 2

z F x 1 F x 1 over a Lagrangian submanifold and in the other we get a 2n = d [ ( 3 = ) ( 3 = )]

web of IIB 5-branes in the presence of ALF like geometries. B

Z

A x 1 A x 1

4. Choice of Flat Coordinates = ds[ ( 3 = ) ( 3 = )] @B

In this section we consider the map between the v Recall that Im(u) and Im( ) map to the one forms re-

moduli of the brane between the A- and B-model. 1 1

S lated to the S cycles of the Lagrangian geometry,

As discussed in Sect. 2, the moduli of the D-branes 2 u

viewed as a cone over T . Thus Im( ) is the mirror of R in the A-model are labeled by c which measures the

the Wilson-line dsA, and the Wilson-line around @D

size of the disk instanton ending on the Lagrangian 1

@B v the dual S = is identified with Im( ). Thus we submanifold. In quantum theory c gets complexified by the choice of the Wilson line on the brane, and gets find that v jumps over the mirror domain wall by

mapped to the choice of a complex point on the mirror

! v in: v +2

type IIB geometry. The choice is characterized by the u; v choice of a point on a Riemann surface F ( )=0, The case of n = 1 is depicted in Figure 8. which we choose to be our ‘u’ variable. However it This allows us to find the tension of the mirror could be that the ‘size’ of the disk instanton receives domain wall as discussed in [6]. The BPS tension of quantum corrections, and this, as we will now discuss, the domain wall is given by ∆W , the difference of

is relevant for finding the natural (“flat”) coordinates the superpotentials on the two sides of the domain R

parameterizing the moduli space of Lagrangian D- 1 u

W u

wall. Since = v d , the tension of the BPS

u i

2 

branes. 1 R v

domain wall is simply the integral du, where

i C

First we have to discuss what we mean by the “nat- 2 u

C v ! v i

u denotes the appropriate cycle shifting +2 , ural” choice of coordinate for the A-model. This is

beginning and ending on a given u. We thus define motivated by the integrality structure of the A-model the flat coordinate expansion parameter. There is a special choice of co-

ordinates [4] on the moduli space of D-branes in terms 1 Z

v u; u: uˆ( t)= d (4.1)

of which the A-model disk partition function has inte- i

2 C ger expansion (2.11), and this is the coordinate which u

measures the tension of the D 4 brane domain walls. To summarize, we predict that the disk partition func-

u u;

There is no reason to expect this to agree with the tion (2.10), expanded in terms of uˆ = ˆ( t), and the t classical size of the disk that the B-model coordinate corresponding closed string counterpart tˆ( )– has the measures, and in general the two are not the same, integral expansion (2.11), the coefficients of which

as we will discuss below. This is what we take as the count the “net number” of D 4 brane domain walls

natural coordinates on the A-model side. ending on the Lagrangian submanifold L (for a more

The B-model and the A-model are equivalent theo- precise definition see [4, 26]). u ries, and this dictates the corresponding flat coordinate It is not hard to see that u and ˆ as defined above

on the B model moduli space. This is the tension of agree at the classical level and differ by instanton M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web 9

Im v u x³ Fig. 8. The D4 brane ending on the D6

brane is mirrored to a domain wall in i type IIB where v shifts by 2 across

it. This projects to a closed cycle on u; v the Riemann surface P ( )=0.

generated corrections. In the large radius limit, the is independent of u, by the deformation of the contour

C C

local A-model geometry in the neighborhood of the u , and only depends on the class of the contour . 3 disk D is just C – all toric vertices other than the one It thus depends only on the bulk moduli.

supporting D goawaytoinfinity. In this limit, the A cautionary remark is in order. We have talked

C

P u; v !

equation of the mirror simply becomes ( ) about the cycle u on the B-model side as a cycle

u v

F u; v e

e + +1,so on the Riemann surface ( ) = 0. In general the u; v

cycles on the Riemann surface F ( ) = 0 can be Z

1 Z 1

u v

u divided in those that lift to cycles where and come

v u u ! e i u:

uˆ = ( )d [log(1 + )+ ]d

i i C

2 C 2 back to the original values, or those that shift by an

u u

integer multiple of 2i. The cycles that come back to

This has a branch point in the u plane around which v

themselves without any shifts in u and correspond

v v ! v i C has the monodromy +2 . The contour u to closed 3-cycles in the underlying CY.Integration of receives a contribution only from the difference of vdu over those cycles correspond to computation of

values of v on the two sides of the cut, and thus for a electric and magnetic BPS masses for the underlying

u i single domain-wall uˆ = + . N = 2 theory in 4 dimensions (and are relevant for Away from the classical limit we can have sub- the computation of the “flat” coordinate for the bulk

leading corrections to the above relation that can in

t u v

field tˆ( )). However, the cycles whose or values

t u e principle be subleading in e and in . But we will shift by an integer multiple of 2i do not give rise to now argue that it is of the form closed 3-cycles in the CY (as the CY in question does

have non-trivial cycles corresponding to shifting u or

t

u O e :

uˆ = + const + ( ) i v by integer multiples of 2 ). Nevertheless, as we

discussed above such cycles are important for finding u= u In other words, we show that  ˆ = 1 is exact even the natural coordinates in the context of D-branes. away from the classical limit. This in fact is obvious Note that closed string periods which determine tˆ from the definition of (4.1) because as we change u, can also be expressed in terms of linear combinations

the change in (4.1) can be computed from the begin- v of periods where the u’s and ’s shift. Thus comput-

ning and the end of the path. But the integrands are v

ing periods where u and shift are the fundamental i the same except for the shift of v by 2 , and there-

R quantities to compute. We will discuss these in the

u  u u

fore the difference is given by  ˆ = d = .We context of examples in Section 6. u have thus shown that uˆ differs from by closed string Just as we have defined uˆ as the quantum corrected instanton corrections only. Another way to see this is tension of a domain wall, we can define vˆ aa the quan-

to note that tum corrected tension of the domain wall associated

Z Z

! u i

1 1 1 with shifting u +2 . Note that in the derivation

u
uv v: vdu u

ˆ = = ( ) d of the superpotential [6] u; v are conjugate fields of

i i i C

2 C 2 2 u u the holomorphic Chern-Simons field. Thus replacing

3

u v

u by ˆ will require replacing by the quantum cor-

uv iu v

Noting that
( )=2 due to the shift in , and dv using the fact that u is not shifting and that is 3

R To see this from the target space viewpoint it is natural to

1

v

well defined, we deduce that the ud = consider the 1+1 realization of this theory as D4 brane wrapped

C i

2 u 10 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web

rected conjugate field vˆ, and so the equation satisfied computed using the observables of the Chern-Simons by the superpotential changes to theory [4, 26, 27] These were obtained by considering

expectation values for Wilson loop observables in the

@W @W

N

! v

= v = ˆ large Chern-Simons theory, in the context of the

@ u @u ˆ large N duality of Chern-Simons / closed topologi- which is the equation we will use in Sect. 6 to com- cal strings proposed in [28]. Let us briefly recall this

pute W . setup.

3

N S Consider the SU ( ) Chern-Simons theory on . 5. Quantum Ambiguity for Lagrangian As was shown in [29], if we consider the topolog- Submanifold ical A-model on the conifold, which has the same

 3 S symplectic structure as T , and consider wrapping

We have seen that the choice of flat coordinates nat- 3 S N D3 branes on , the open string field theory liv-

urally adapted to the A-model Lagrangian D-branes N ing on the D3 brane is SU ( ) Chern-Simons the- are fixed up to an integer choice. In particular we ory where the level of the Chern-Simons theory (up

found that if u; v are complex coordinates satisfying

N g i= k N

to a shift by , i.e., s =2 ( + )) is identi- u; v P ( ) = 0, and if the brane is denoted in the classi-

fied with the inverse of the string coupling constant. u cal limit by v = 0 and classically measures the size

The large N duality proposed in [28] states that this of the disk instanton, then we can consider a new u topological string theory is equivalent to topological given by strings propagating on the non-compact CY 3-fold

1

O  O !

! u nv u + ( 1) ( 1) P , which is the resolution of the conifold, where the complexified Kahler¨ class on P1

for any n, which classically still corresponds to the

t Ng

has the size = s . In [4] it was shown how to use disk instanton action. In this section we explain why this duality to compute Wilson loop observables. The

fixing the arbitrary choice is indeed needed for a quan- 3

 S idea is that for every knot one considers a non-

tum definition of the A-model Lagrangian D-brane.  3

L  T S

compact Lagrangian submanifold such

In particular specifying the A-model Lagrangian D- 3

L \ S M L

that = . We wrap D3 branes over ,

brane just by specifying it as a classical subspace M which gives rise to an SU ( ) Chern-Simons gauge

of the CY does not uniquely fix the quantum theory,

L

theory on . In addition, bi-fundamental fields on

given by string perturbation theory. The choice of n M transforming as (N; ) arise from open strings with reflects choices to be made in the quantum theory, 3 one end on the D-branes wrapped over S and with the

which has no classical counterpart. In this section we L

other end on D-branes wrapped over . Integrating show how this works in two different ways: First we out these fields give rise to the insertion of

map this ambiguity to an UV Chern-Simons ambigu-

  

n n

ity related to framing of the Wilson Loop observables. X V trU tr

Secondly we relate it to the choice of the Calabi-Yau exp ; n geometry at infinity, and for this we use the type IIB n

5-brane web dual, discussed in Section 3.

V SU N

where U and denote the holonomies of the ( )

M

5.1. Framing Choices for the Knot and SU ( ) gauge groups around respectively. M Considering the SU ( ) gauge theory as a specta-

To see how this works it is simplest to consider the N tor, we can compute the correlations of the SU ( ) case where the D-brane topological amplitudes were Chern-Simons theory and obtain

over the Lagrangian submanifold. Then, as discussed in [4] the

R

  

n n

2 2 X

S xd  W=   U V

disk amplitude computes = d (d d ) for the (1) tr U tr

F V; t; g s

gauge theory in 1 + 1 dimension, where  is the twisted chiral exp = exp( ( )) (5.1)

n n gauge field strength multiplet whose bottom component is uˆ.In

this formulation the domain wall associated with shifting of  to

i u ! u i

 +2 is realized by +2 , whose BPS mass we have It was shown in [4] that the right-hand side can be S

denoted by vˆ. From , the change in the value of the superpotential interpreted as the topological string amplitude in the

W= u

under shifting  is given by d d ˆ which leads to the statement N

large N gravitational dual, where the D-branes

u v that dW=d ˆ = ˆ. This provides an alternative, and more physical

2 S derivation of the main formula we use for the computation of W . have disappeared and are replaced by . In this dual M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web 11

L Fig. 9. The Wilson loop observables arise

from worldsheet diagrams where some

L N

boundaries end on and branes wrap-

3 N ping S . In the large limit the holes end- 3

ing on S get “filled” and we end up with a

Riemann surface which has only the bound- L

aries associated with . In the above figure L

the outer hole is the only one ending on . 3 S³ All of the interior holes end on S and dis-

appear in the large N limit, leaving us with

a disk. geometry the M non-compact D-branes are left-over field on the knot , which is non-vanishing every- and wrapped over some Lagrangian submanifold in where on the knot. Note that if we are given a framing

1

 O ! O ( 1) ( 1) P . This Lagrangian submanifold of the knot, any other topologically distinct framing was constructed for the case of the unknot explicitly is parameterized by an integer, given by the class of

1 1 1

! S S in [4] and extended to algebraic knots in [26] (this lat- the map S , where the domain parameter-

ter construction has been recently generalized to all izes and the range denotes the relative choice of

F V ; t; g

knots [30]). Moreover ( s ) denotes the topo- the framing which is classified by the direction of the

logical string amplitude in the presence of the M vector field on the normal plane to the direction along D-branes wrapped over some particular Lagrangian the knot. The framing of the knot enters the gauge

1

 O ! submanifold in O ( 1) ( 1) P with a non- theory computation by resolving UV divergencies of

1

S F V; t; g

trivial cycle. Note that a term in ( s )of the Chern-Simons theory in the presence of Wilson

Q

b

k i the form tr V comes from a worldsheet with loops. It arises when we take the Greens function for

i=1 i b boundaries, where the -th boundary of the world- the gauge field coming from the same point on the

1 k S knot. The framing of the knot allows a point splitting sheet wraps the of the Lagrangian submanifold i times. definition of the Greens function. The left-hand side of (5.1) is computable by the We have thus seen that both sides of (5.1) have a methods initiated in [8], and in this way gives us a way quantum ambiguity that can be resolved by a choice to compute the open string topological amplitudes for of an integer. On the left hand side the ambiguity

this class of D-branes. Note in particular that the disk arises from the UV. On the right hand side the ambi-

F V ; t; g

=g guity arises from the IR (i.e. boundary conditions on s amplitude corresponds to the 1 s term in ( ). A particular case of the brane we have considered in the brane at infinity). We have checked that the two

1

 O ! O ( 1) ( 1) P corresponds to the unknot. ambiguities match for the case of the unknot, by com- This is depicted in the toric Figure 2. paring the disk amplitudes on both sides (using CS The match between the computation in this case computation of the framing dependence of the knot using mirror symmetry, and the result expected from on the left and comparing it with the mirror symmetry the Chern-Simons theory was demonstrated in [6]. computation of the knot on the right). Some aspects However as we have discussed here the disk ampli- of this computation are presented in the Appendix A. tudes have an integer ambiguity when we use mirror The computation of the disk amplitude for this case, symmetry for their computation. Thus apparently the using mirror symmetry, is presented in Section 6. right hand side of (5.1) is defined once we pick an Here let us discuss further how this match arises. integer related to the boundary conditions at infinity Consider the disk amplitude at large N corresponding on the B-brane in the type IIB mirror. If the right hand to a given knot. In the gauge theory side the compu- side of (5.1) is ambiguous, then so should the left hand tation arises from open string diagrams of a planar

side. In fact the computation of Wilson loop observ- diagram with the outer hole on the Lagrangian sub- L ables also has an ambiguity given by an integer! In manifold , and the rest of the holes ending on D-

3 S particular we have to choose a framing on the knot branes wrapping the , as shown in Figure 9. In the to make the computation well defined in the quantum large N limit, the interior holes get “filled” and we theory [8]. A framing is the choice of a normal vector get the topology of the disk. 12 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web

lShort Propagator (UV Region)

Fig. 10. The Wilson loop observable has


















































































































































































































































































































































































































































































































































































































































































































































































































































































































UV divergencies which need to be reg-






















































































































































































































































































































































































































































































































































































































































































































































ulated, coming from points along the

























































































































































































































































































































































































































































































































































































































































knot where the gauge boson propaga-

N




























tor is of zero size (a). In the large limit these map to disks touching at a a. b. point (b), which can be viewed via a con- formal transformation as a long propa-

gator (c). Note that the boundaries of the























































































































































































































































































































































































































































































long propagator are on the Lagrangian




























































































































































































































































































































































































































L


























































submanifold (or more precisely its




























































































































































































































































































































































































































N
























































































large dual) and correspond to an open













































































































































































































































































































































































































































































































































































































































string propagating on it. Thus the UV









































































































































































































































































IR Region













































































































































































































framing choice of CS gets mapped to























































































































































































































































































































































































































































































the choice of large distance (IR) physics




































































































































































































































































































































of modes on the brane. c.

dual channel a long Schwinger time of an open string

L

ending on . This means that the issue of ambiguity

L

is mapped to an IR behaviour of fields living on ,





























and that is exactly where we found the ambiguity in







































the computation of the superpotential in the context





















































































































of mirror symmetry.

Other examples of the large N limit of UV regions



























































































































































for the computation of the Wilson loop observable































































































































































































along the knot get mapped to disks shown in Fig. 11,































































































































































































and look like branched trees of disks.





















































































































































































5.2. Calabi-Yau Geometry, 5-brane Perspective











































































and the Integral Ambiguity













































































As discussed in Sect. 3, we have dual descriptions







































of type IIA geometry with D6 branes wrapped over

Lagrangian submanifolds in terms of M-theory on G2 Fig. 11. Other examples of the large N limit of UV diver- gencies of Wilson loop observables. holonomy manifolds (viewed as circle fibration over CY manifolds) or in terms of type IIB web of 5 branes 6

Now consider where the UV divergencies of the in an ALF-like background in R . To be precise, the

2 1 S gauge theory would arise. They would arise from M-theory on T /type IIB on duality relates the cou- Feynman diagrams where the Schwinger parameter pling constant of type IIB to the complex structure of 2

for the gauge field goes to zero–an example of this is the T . Fibering this duality gives rise to the duality depicted in Figure 10a. Note that the two end points of just mentioned. However, in the type IIB picture we

the short propagator will be mapped to the same point typically fix the type IIB coupling at infinity. This

on the knot in the limit of zero length propagator. means that, by this chain of duality, the G2 holonomy

In the large N limit, where the disk gets filled, these manifold is a circle fibration over the CY where the 2

get mapped to configurations such as that shown in complex structure of the T fibration of CY is fixed Figure 10b. In this dual description by a conformal at infinity. Turning this around, this duality predicts

transformation the worldsheet can be viewed as that the existence of a particular class of CY and G2 holo- depicted in Figure 10c. In other words, we have in the nomy metrics with a particular behaviour of the metric M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web 13

(1,n+1) (0,1) (1,1) (0,1)

(1,0) (1,n)

Fig. 12. In the weak coupling limit, we consider an NS (0; 1) 5-brane with an ALF like space ending on it. If this is ; connected to (1; 0) and (1 1) 5-branes in a junction, the only instantons allowed involve the worldsheet instanton stretched along the interval shown on the left-hand side of the figure above, times the 5-th circle which shrinks at the attachment

point of the ALF space. There is no other instanton allowed in this geometry. However if an NS (0; 1) 5-brane is attached

;n n 6 ; to (1;n) and (1 + 1) 5-branes, the geometry of the intersection dramatically changes (as long as =0 1) and now we

can have many more allowed configurations for the worldsheet (p; q ) instantons. This is depicted on the right hand side of

the above figure.

2 1 1

T n

at infinity (we can also fix the area of at infinity as formation ! + . Said differently, we can take

 

! i1 that gets mapped to the inverse of the radius of type the weak coupling limit  by considering the 1

IIB S ). We now argue that the choice of the complex inverse modular transformation where we consider

2 ;

structure of T , or equivalently the type IIB coupling the 5-brane web given by the NS fivebrane (0 1) and ;n

constant at infinity, changes the number of Euclidean the fivebranes (1;n) and (1 +1). It is easy to see that

! n M2 branes, exactly as is expected by the ambiguity. this geometry of webs has an n ( + 1) symme- Instead of being general we simply illustrate this idea try, when we reverse the direction of the handedness in the context of a simple example, namely the C3 on the D5 brane. Thus, the counting of the worldsheet

geometry discussed before and represented by Fig.6, instantons here will exhibit this symmetry. This is ex- ; consisting of NS 5-brane (0; 1) and D5-brane (1 0) actly the symmetry we will find for the ambiguity of 3

and the (1; 1) 5-brane. We already did show that M2 the mirror to C , as discussed in Section 6. Moreover

; brane instantons get mapped to worldsheet disk in- it is easy to see that except for n =0 1, where

stantons consisting of the (p; q ) string shown in Fig. 7 there is exactly one choice of disk instanton, for all

times an extra circle (the ‘5-th’ circle) which vanishes other n’s we get a large number of disk instantons, at the position of the ALF space. However now we allowed by the geometry of the 5-branes, as shown consider changing the coupling constant. Then the in Figure 12. This is also in line with the result we angles between the 5-branes will be changed. Beyond find from mirror symmetry in Section 6. Thus as the some critical angles we can get new worldsheet in- geometry of Calabi-Yau changes, we get jumps in the stantons. Basically we can consider the web of strings, number of instantons as predicted by this picture.

and the only data that we need to take into account is It is natural to ask what IR aspect of the type IIA CY

p; q p; q that a BPS ( ) string can end only on a ( ) 5-brane geometry with the brane, this choice of  is reflected (and perpendicularly). This web of strings, however, in. One’s natural guess is that the normalizability of can end on the projection of the ALF space on the web the 1-form on the brane represented by the Wilson at any angle, since at that point now the ‘5-th’ circle line vev, which represents the mode corresponding is shrinking (again in a perpendicular fashion). To be to moving the brane, is the relevant issue. If this is

concrete, let us consider the limiting choices of the the case the possible choices of metrics for which

! =n

type IIB coupling constant given by  1 . This a particular 1-form is normalizable will have to be

 ! i1 can be obtained from by the modular trans- classified by the integer ambiguity n, which in turn is 14 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web

3 related to the choice of the CY metric with fixed  at 6.1. Almost C

infinity, labeled by n. Similarly, in the M-theory lift 3

this would be related to the possible choices of nor- We consider the simplest Calabi-Yau X = C .

i X malizable deformations of the G2 holonomy metric. It is described by just three chiral fields and

It would be interesting to study these issues further. no gauge group. The special Lagrangian D-brane L

; ;

we are interested in is given by q1 =(10 1) and

; ; 6. Examples q2 =(0 1 1):

1 2 3 2 2 2 3 2

X j jX j c jX j jX j : In this section we will consider a number of exam- j = 1 =0 ples which illustrate the general discussion we have This geometry can be regarded as a local approxi- presented for the computation of disk amplitudes for mation, in the limit that all radii are large, to a more D-branes wrapping a Lagrangian submanifold of non- involved geometry it is embedded in, for example the compact CY 3-folds. The examples include consid-

3 1 1 2 one discussed in Section 2. The value of c1 on the erations of Lagrangian branes in C , P  P , P brane and the Wilson-line around the one finite circle

and its blow up at a point F1. In these examples we u on L form one complex modulus of the A-brane, consider inequivalent configurations of Lagrangian which measures the classical BPS tension of the D4 branes. For the case of C3 we exhibit the dependence brane domain wall. of the topological string computations on the integer The mirror manifold can be written as

ambiguity we discussed earlier. For all of these cases v

we find the predicted integrality properties of the disk u

P u; v e e ; xz = ( )= + +1 (6.1) amplitudes4. We also consider different Lagrangian

3 1 2

Y u Y v branes (for example ending on the “internal” or “ex- where we have set Y to zero, and = , = . ternal” edges of the toric diagrams). Equations (2.3) fix the classical limit of the brane As discussed in Sect. 4, a non-trivial part of the

1

u ; story involves finding the flat coordinates for the open c = Re( ) string variables. As discussed there, we find the flat coordinates to be given by so the transverse coordinate to the B-brane is u.In

this limit, v is zero on the brane corresponding to the

u
t ; u vanishing of the classical superpotential.

ˆ = + ( i )

ZZ

The SL(2; ) group of reparametrizations acts on

   

0

t

a b u where ( i ) (up to an addition of a constant) is an ˆ

(6.1) by linear transformations 0 =

v d

exponentially suppressed function of Kahler¨ moduli ˆ c

 

t

i

O e

of the Calabi-Yau ( ( )), and depends on the uˆ

that leave the holomorphic (3; 0) form =
choice of the brane. We compute using the methods vˆ

discussed in Sect. 4 for all the examples. dz

v ; ZZ

dud invariant. The subgroup of SL(2 ) that pre-

z

@ W v u

Note that using uˆ = ˆ(ˆ ) and (2.11), we have

  

serves the classical limit above consists of transfor-

p a b X 1

mations = . These transformations

@ W v u kN k u
: d

ˆ c k; uˆ = ˆ(ˆ )= mlog(1 exp[ ˆ m t]) 01

k result in a family of parameterizations of the mirror

geometry

u P u; v

So by solving for vˆ in terms of ˆ (from ( )=0

pv u v

! u; v

and the correction to u; v ˆ ˆ due to the mirror +

xz P u; v e e ;

= p ( )= + +1

map) we find a prediction of an expansion in terms N

of integers k;m. We verify these highly non-trivial and a family of superpotentials obtained by solving

v e integrality predictions in the examples below which the order p polynomial in ,

we now turn to. Z

4

W u v u u: p The integrality requirement is highly non-trivial. Originally, p ( )= ( )d we experimentally found the highly non-trivial flat coordinates only by requiring the integrality of the domain wall degeneracies. Later we showed that they agree with the BPS tension of domain To compute the numbers of disk domain walls we walls, which is the definition used in this section. need to know the flat coordinates.

M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web 15

p v @ W

For = 0 we have computed in Sect. 4 the BPS Solving now for ˆ = uˆ and integrating we get, up

tension of the D-brane in the u-phase and also the to trivial integration constants, the superpotential

v u u i

phase, by trivial relabeling, to be ˆ p=0 = + ,

m

1 1

X Y

v v i :

ˆ p=0 = + The knowledge of these suffices to 1

mu

mp j e : find the BPS tensions for a different choice of fram- W = ( ) (6.4)

mm! j m=1 =1

ing p.

ZZ u ! u pv v ! v

The SL(2; ) transformations + , We can write this in the general form (2.11). The case

P

N

km u

m ˆ e

act on the basis of one-cycles on the Riemann sur- at hand, that is given by W = 2 , yields

m;k

k u

face that are associated to periodicity of Im( ) and N

the following integers for m :

v

v

Im( ). Since u and which provide the quantum

p

N ; v

corrections to u and 1 =( 1)

i h

p

p

1 ( 1) p

;

N = + =

u
; v v
;

u 2 v

ˆ = + u ˆ = + 2 4 2 p

are obtained by evaluating periods of the one-form (1)

p p ;

N3 = ( 1) vdu

 = over these, changing the basis of one-cycles 2

0

ZZ u au bv by an SL(2; ) transformation to = + and

0 1

cu dv

v = + acts in an analogous way on the periods,

p p p; N = (2 1)( 1) i. e. 4 3 5

p 2

0

N p p p p ;

a
b
:

5 = ( 1) ( 1)(5 5 +2) v = u + u 24

Thus, we find that 1 3 2

N p p p p ; = (p 1) (36 54 +31 9)

6 20

u u p i ; v v i : p

ˆ p = +( +1) ˆ = + 7

p

p p N = ( 1) ( 1) 7 720 The equation

4 3 2

p p p p ;

(343 686 + 539 196 + 36)

pv v

uˆ + ˆ ˆ .

P u; v e e

p (ˆ ˆ)=0=1 (6.2) .

vˆ

is solved iteratively, using the ansatz e =

N p

P P P

To show integrality for a given expression m ( ) for

1 1 1

u p k m

k ˆ

a e a x c x ;

k m

k . Using ( ) =

k k m

2 ZZ =0 =0 =0 p

P all , one may factorize the denominator into m

1 Q

n

i

a c c kp m

where = = 1 and m = ( +

p p

0 0 k m

=1 with i prime and check that one can factor

i

i

n n

i i

k a c

mk

) k we get immediately a recursive formula

p p p n k

from the numerator for all = i with

i i

P m 1

1 n

i

a kp m k a a

k ;:::;p m 

k mk

m = ( + +1) for the i

m =1 . That has been checked for 50.

m

1 =1 i

N p m

coefficients. This can be summed using Stirlings co- Note that m is a polynomial in of degree 1.

efficients of the first kind (see e. g. [31]), which As discussed in Sect. 5, the p dependence of the super-

x x
:::
x n

are defined by x( 1)( 2) ( +1)= potential for this particular Lagrangian brane (viewed 

 1

P

t O  O ! m

n

m m

(m) ( ) as a large limit of the ( 1) ( 1) P geom-

S x

. Using the relation S =

n n m=0

r etry) can be mapped to the framing ambiguity of the

 

P

n

nr

mr

(r ) ( ) unknot for the Chern-Simons theory. In Appendix A S

S the result of the sum-

mr nk k k = k we discuss this computation and verify that the large

mation is N Chern-Simons computation leads to a polynomial

m p N

of degree 1in for m . Moreover we have ver-

N i ; ; p

m

2 ified that for i for =12 3 the dependence of

m Y

(1)

mp j : a the framing choice agrees with the above result. We

m = ( ) (6.3)

m!

j =0 also have shown from the computation of the framing 16 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web

v X³=0

C

X¹=0 0 X=0 u u

X4 =0

a. b.

1 1

O K !  Fig. 13. The Fig. a. depicts the base of the toric fibration for X = ( ) P P together with a special Lagrangian 1

brane in X . The minimal disk which ends on the A-brane in the figure wraps partly the P over which the brane is. Figure b.  depicts the Riemann surface  of the mirror geometry, and the mirror B-brane as a choice of a point on . The D4 brane

wrapping the disk in Fig. a. is mirror to a domain-wall which projects to path C in Figure b.

3 2 4 2 0 2

m jX j jX j jX j r :

dependence of the CS theory that for all the coeffi- + 2 = s

m

m 1 2

m =m cient of the leading term p is given by ! in agreement with the above result. The solutions to the D-term equation, projected to the Results of the prediction of [4] for the topologi- 1 1 toric base are given in Figure 13. The F0 = P  P cor- cal string amplitudes for the case of the unknot have 0 responds to the divisor class represented by X =0 recently been verified mathematically using localiza- and is visible in Fig. 13 as the minimal parallelo-

tion methods in [32, 33]. In verifying the predictions K gram in the toric base. The fiber O ( ) corresponds of [4], there were some choices made for the toric to the normal direction. The special Lagrangian D- action in these works. The authors of [32] have re- branes of the A-model for this Calabi-Yau have in- cently checked and verified that the integer choice of equivalent phases, depending on where one puts the ambiguity changes the topological string amplitudes brane, together with a Z-family of choices of fram- 5 exactly as predicted by the above results . Moreover ing in each. The D-brane charge can be taken to be

this ambiguity arises from the boundary of moduli

; ; ; ; q ; ; ; ; q1 =( 1 0 1 0 0), 2 =( 1 0 0 1 0) and so space of Riemann surfaces with holes, as we already

discussed. 2 2 0 2 1 3 2 0 2 2

X j jX j c ; jX j jX j c : j = =

1 1

K ! 

6.2. O ( ) P P c

The constants i are required to be chosen so that

1 1 the Lagrangian D-brane lies on one dimensional toric This manifold involves a compact P  P geom- edges, and different toric legs give rise to generally etry inside a CY 3-fold with a non-compact direc- different phases of the theory.

tion given by the canonical line bundle. It can be i

2 The mirror variables Y satisfy U

described by a linear sigma model with G = (1)

i

i ;:::

and five chiral fields X , for =0 4 with charges 1 2 0 3 4 0

Y Y t; Y Y Y s;

1 2 Y + 2 = + 2 =

; ; ; ; Q ; ; ; ; Q =( 2 1 1 0 0) and =( 2 0 0 1 1) which

leads to the D -terms where the real parts of the complex structure parame-

1 2 2 2 0 2

t; s

X j jX j jX j r ; j ters measure the classical sizes of the two minimal

+ 2 = t

1

P t r s r s ’s in the toric base, Re( )= t , Re( )= . The mir- 5We are grateful to S. Katz and C.-C. Liu for performing these ror manifold is given by computations upon our request. The form we have presented our

answer (6.4) was chosen to simplify the comparison with their

u tu v sv

e e e e ; formula. xz = + + + +1 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web 17

τ u+¹

β 1 u−¹ β ¹ C

C α u u v (u) ² v (u) α1,2 2 α 1 ¹

β 2 β u+² ²

u−²

τ

P u; v u

Fig. 14. The Riemann surface  : ( ) = 0 can be viewed as a branched cover of the -cylinder by two solutions



v u u  v $ v

; ; 1;2( ). The solutions are branched over 4 points , with monodromies 1 2 which exchanges 1 2, and 1 2 which

1;2

v ! v i v ;

take 1;2 1 2 +2 corresponding to the -cylinders which glue the two roots on the figure to the right. Monodromies

u 1;2 arise from the periodicity of itself.

2 3 0

u Y v Y where Y = , = , = 0 and the two holo- In the limit the A-model geometry is classical, it

morphic constrains are solved in terms of these. The follows from the mirror map (2.3) that the mirror

 v

Riemann surface  which is the configuration space B-brane is located in the region on , where is

u; v v  u

of the mirror B-brane is given by 0 = P ( )= constant, 0, and is a parameter which is large,

u tu v sv

e e e u  t=

e + + + + 1. When viewed in terms 2. Away from the large radius limit, this de-

u v

e  P u; v

of the single valued variables e and , is re- forms to a root of the equation ( ) = 0. This has u lated to the A-model geometry by “thickening” of the two solutions for v at every value of , one-dimensional edges of the toric diagram (or more

0 

u tu

precisely, their projection onto the X = 0 plane). In e

1+e +

v v u

= ; ( ) = log t; s the large radius limit 0  the A-model geometry 1 2 2

becomes classical and the legs of the Riemann surface p

of the B-model become long and thin, so the A- and 

tu s

u 2

e e

(1 + e + ) 4 i the B-model can be related already at the classical  + level. 2

For example, the A-brane on Fig. 13 is on the inter-

c ;r c and the mirror B-brane propagates along a region of

nal leg which is given by 2 = 0, and 1 in the [0 t ]

v v u

r the root = 1( ). As we discussed above, the super- interval. The large radius limit is, in addition to t and

r potential on this leg is given by

s being large, the limit of large disk size. For exam-

c

ple in the regime 1 t 2 the basic disk is the one Z

on Fig.13a, and the classical limit corresponds to c1

W u u u:

( )= v1( )d

r = c

being of order of t 2. The size parameter 1, together

R

c i A

with the Wilson line u = 1 + , is the classical

@ W

tension of the D4 brane domain wall wrapping this This vanishes in the classical limit, since u =

u ! disk and ending on the D6 brane. v1( ) 0. 18 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web

The flat coordinate computes the tension uˆ of the D4 brane domain wall wrapped on the basic disk as in Fig. 13a, and we will compute this by computing the tension of the mirror B-brane domain wall. As explained in Sect. 4. the mirror domain wall projects β1

to the loop C on the Riemann surface which starts at αs

the location of the B-brane at a fixed u, winds around

! v i v +2 before “attaching” again (see Fig.13b). In this configuration, the tension is given by a classical integral on the Riemann surface α1 α 2

1 Z

v u u:

uˆ = 1( )d i

2 C β 2

1 R βt

u u

The period integral v1( )d is, as explained C 2i in Sect. 4, the sum of two pieces. There is the classical

contribution to the domain wall tension which is equal

s

1 Fig. 15. The closed string periods t , are linear com-

v u u

i ;

to ( )= , as the initial and final end-point i 1 binations of small periods i , for =12 which come

2i

u v

v ! v i of C differ by +2 . The quantum correction from the periodicity of and variables. to the BPS mass comes from the “small period”, the

contour integral around the 1 cycle (see Fig. 14) of

1 R II III v

ud .

2i 1 The small periods can be found as follows. Notice

v^

v s i

that, since v1 + 2 = +2 , the sum of two periods

! u i around u +2 is I

u^

Z Z

1 1 rt

u v u s i:

v1d + 2d = +2

i i

2 1 2 2

On the other hand, the closed string period sˆ, that rs

measures the mass of the D4 brane wrapping a P1 of r

size s can also be expressed in terms of small periods

1 1

A O K ! 

Fig. 16. Three phases of the brane on ( ) P P . as it is computed along the contour s = 1 2, (orientations are fixed up to an over-all sign by the The phases I and II are related by the exchange of the two

P1’s in the base, and this is reflected in the disc domain wall v requiring that both u and have trivial monodromy

numbers.

around s ),

For the B-brane at hand we need the small period

Z Z v around the cycle on the -leg (Fig. 15), and by a

1 1 1

u v u s:

v1d 2d = ˆ computation analogous to the one we just did,

i i

2 1 2 2

t

From this we can find for example the small period t ˆ

R

ss

u i:

6 ˆ 1 uˆ = + +

i v u

along 1 as + = 1d .

2 2i 1 2

6This method of calculating the domain-wall tension gives the

In fact the small period of 1 is the correction for the result up to factors of i . The direct evaluation of the period around B-brane on the other leg of the Riemann surface – C can be done, and it determines the answer to be the one we

presented above. the leg parameterized by v which corresponds to the M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web 19 phase II (see Fig. 16). In this phase we have therefore To compute disk numbers in these various phases, we

need to write the superpotential in each case in terms

s

s ˆ of the open and closed string flat coordinates.

v i : vˆ = + + 2 The closed string flat coordinates have a com- plicated dependence on the classical, linear-sigma According to the discussion in the C3 case, which

model, coordinates t. Quite analogously to what we carries over here as well, the correction terms we found for the open string, the A-model closed string computed amplitudes have integrality properties when expanded

in terms of flat coordinates which measure the BPS

t s s

t ˆ ˆ

i ;
i ;

X v u = + = + mass of D2-branes wrapped on rational curves in . 2 2

The corrected quantities are related transcendentally

t t

tˆ ˆ

; ZZ

i j

form a doublet under the SL(2 ) transformations, by the mirror map i = ( ) to the complex structure

t

i

z e t

; ZZ u; v ˆ

just as do. Consider now an SL(2 ) group ele- variable i = . In fact, as discussed above, and  ment which takes sˆ are among the periods of the Riemann surface .

Alternatively, they can be obtained from the solutions

0 0

u ! u u v; v ! v v: f = = L

to Picard-Fuch’s equations i =0,

Y Y

Q Q

i i

@ @ ;

The new equation of the curve becomes L

i i

i = ( ) ( ) (6.5)

Q > Q < i

i 0 0

0 0 0 0 0 0

u v tu v sv

+v

e e e : e + + + +1=0

i 1 1

@ @=@z O K ! P  P

where i = . For the ( ) the L

This change of coordinates does not change the clas- linear differential operators t;s are sical value of the superpotential in phase I (the phase

2

0

L  z    

t t t s s t

t = 2 ( + )(2 +2 +1)

v v  originally parameterized by u) since = 0, (6.6)

but corresponds to a different choice of framing with 2

L  z     ;

s s t s s t

s = 2 ( + )(2 +2 +1)

n = 1, which changes the fluctuating field on the

0

u u u v

z @=@z

brane from to = . The quantum corrected 

t;s t;s where t;s = . There is a constant solution

7

z z

domain wall tension is now f s

0 = 1, and near t = = 0 there are two logarithmic

;f t; s

solutions f1 2. The flat coordinates ˆ ˆ are given by

0

u v

u =

f =f ;i

linear combinations of ratios of the periods i 0 =

1; 2 picked by the correct classical behavior For the brane in phase II it is no longer true that the

0 2 2 3

u u v

t z z z z z z O z ;

classical superpotential is zero, since = is tˆ

s t s

= (2 t +2 +3 +12 +3 + ( ))

t s

ZZ not zero after the SL(2; ) transformation.

With this choice of parameterization, consider the 2 2 3

s s z z z z z z O z :

s t s

ˆ = (2 t +2 +3 +12 +3 + ( )) s superpotential t

(6.7)

Z

0 0 0

z

W v v u

= ( )d Note that the solutions are symmetric in t;s except for

the logarithmic term. To expand the disk amplitude 0

0 in terms of the flat coordinates we need the inverse

v P u ;v

with u ( ) obtained by solving ( ) = 0. This

s

tˆ ˆ

z q q e q e

0

t;s t s

relations t;s ( ) for = and = . The first

! 1

superpotential has no classical piece, as v , 0

0 few terms of the expansion are

 u  u 0 is on the curve. In fact, 0 is the equation

of the outer leg of  (see Fig. 16), so this computes the 2 3 2

z q q q q q q q

s s s

s = 2 2 +3 +3

s t superpotential of the brane in phase III. The domain s 2

4 3 2 2 3 5

q q q q q q q O q ; s

wall tension in the phase III (the outer leg) is given by 4 4 t 4 4 + ( )

s s t t s (6.8)

2 2 3

z q q q q q q q

t t s t t

s

s = 2 2 +3 +3

s t

ˆ t

0

i : v v = = +

2 3 2 2 3 4 5

q q q q q q q O q ; s

4 t 4 4 4 + ( )

s s t t t

7As we will see when we discuss the closed string flat coordi-

t s

ˆ ˆ

q e q e

0 s

where t = and = . nates in more detail below, u turns out to be zero.

20 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web

K ! Table 1. The disk domain wall degeneracies for brane I in Table 2. Disk degeneracies for brane I in the O ( )

1 1 1 1

K !  m> s  m the O ( ) P P geometry for 0. Exchanging P P geometry for = 1.

with t yields the result for brane II.

k k t

s — —

k k t s — — 012345 01 2 3 4 5 00–10000 m =1: 10–1 –2 –3 –4 –5 01 0 0 0 0 020–1 –10 –45 –140 –350 11 2 3 4 5 630–1 –30 –300 –1776 –7650 2 1 10 45 140 350 756 40–1 –70 –1332 –13400 –91070 3 1 30 300 1776 7650 26532 50–1 –140 –4590 –72856 736270 4 1 70 1332 13400 91070 472368 5 1 140 4590 72856 736270 5437530 In phase I we get the values for numbers of primitive

m =2:

N

;k ;m

disks k given in Table 1.

s t 00 0 0 0 0 0 There is a symmetry which relates the numbers of 11 2 3 4 5 6

2 2 16 62 180 428 896 disk instantons with negative m to those above. We 3 4 70 552 2856 11280 36828 have

4 6 224 3130 26336 159078 759200 

k m< ;

5 9 588 13420 171720 1503135 10016490 0ift 0

N

;k ;m

k = (6.9)

s t

m N k m 

;k m;m t

=3: k if 0.

s t 00 0 0 0 0 0 11 2 3 4 5 6Examples are given in Table 2. 2 4 24 85 230 525 1064 For the numbers of primitive disks for the brane III 3 11 146 977 4542 16644 51420

4 25 618 6975 50912 278134 1231230 see Table 3.

N N

; ; ;k ;m

Up to the invariant = 2 we have k =0 t

5 49 2070 36637 395818 3068331 18655290 0 0 1 s

k

t t m =4: for and for there is a symmetry

00 0 0 0 0 0

N N ;

;k ;m k m;k m;m

k = (6.10)

t t s 11 2 3 4 5 6 s +

2 6 34 112 290 638 1260 t 3 25 276 1645 7040 24246 71400 which reflects the exchange symmetry of s and for 4 76 1498 14496 94830 476900 1979098 the brane in phase III. The instantons with negative m

5 196 6248 91935 870220 6103867 34309080

k jmj

are absent for s and their numbers are related

m =5:

to those with positive m by 00 0 0 0 0 0

11 2 3 4 5 6

: N N

m;k m;m ;k ;m k

k = (6.11)

t t s 2 9 46 145 360 770 1484 s + 3 49 482 2640 10592 34674 98028 4 196 3270 28240 169402 795998 3126928 Examples are given in Table 4. 5 635 16642 213083 1816038 11729677 61675880

m =6: 2

! 6.3. O ( 3) P 00 0 0 0 0 0 11 2 3 4 5 6 The linear sigma model for this geometry has

2 12 60 182 440 918 1736

U Q

3 87 790 4060 15478 48600 132732 G = (1) and four matter fields with charges =

; ; ; 4 440 6560 51906 290600 1290870 4840248 (3 1 1 1) 5 1764 40050 460355 3604656 21737688 107979508

1 2 2 2 3 2 0 2

X j jX j jX j jX j r: j + + 3 = (6.12)

To get the disk numbers the only remaining task is u to expand W ( ) in terms of open and closed string The base of the toric fibration is shown in the Fig- flat coordinates. The integrality of the disk amplitude ure 17.

The D-brane charge can be taken to be q =

@ u v W 1

(2.11) implies that we write uˆ (ˆ)=ˆ as

; ; ;q ; ; ;

(1; 0 1 0) 2 =(0 1 0 1) (the choice is unique

1

  X

k in all local models), and so

k mu t

s ˆ

@ u mN q q e : W

k ;k ;m

u (ˆ)= log 1

t

t s

ˆ s

k ;k

s =0 =0

t 1 2 0 2 1 2 2 0 2 2

jX j jX j c ; jX j jX j c :

k

m= = = t

M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web 21

K ! O K ! Table 3. The disk degeneracies for brane III in the O ( ) Table 4. The disk degeneracies for brane III in the ( )

1 1 1 1

m>  m

P  P geometry for 0. P P geometry for = 1.

k k k k

t s t s — — — — 01 2 3 4 5 01 2 3 4 5

m =1: 00 0 0 0 0 0 02 2 2 2 2 2 1 –2 –2 –2 –2 –2 –2 1 0 2 12 40 100 210 2 –2 –12 –40 –100 –210 –392 2 0 2 40 310 1520 5628 3 –2 –40 –310 –1520 –5628 –17184 3 0 2 100 1520 12908 76488 4 –2 –100 –1520 –12908 –76488 –353316 4 0 2 210 5628 91070 680940 5 –2 –210 –5628 –76488 –680940 –4515558 5 0 2 392 17184 353316 4515558

m =2: III 00 0 –2 –4 –8 –12 II 10 0 –4 –32 –140 –448 20 0 –8 –140 –1188 –6580 30 0 –12 –448 –6580 –58240 I 40 0 –28 –1176 –27840 –370428 50 0 –24 –2688 –97020 –1859648

m =3: 00 0 0 2 10 28 1 0 0 0 10 100 540

2 0 0 0 28 540 5012 2

O ! 3 0 0 0 62 2100 317072 Fig. 17. Three phases of the A brane on ( 3) P . 2 4 0 0 0 120 6600 147420 Phases I and II are related by Z3 symmetry of the P . 5 0 0 0 210 17820 576212

1 2

c ;r >c > ;

Phase II : =0 t 0

k k t s — —

0 ::: 45 6 7

1 2 1

c ;

m =4:

00::: 0 –4 –28 –104 As the branes in phases I and II are related by the

10::: 0 –28 –336 –2156 2 Z3 symmetry of P , the special Lagrangian D-branes

20::: 0 –104 –2156 –21888 of the A-model for this Calabi-Yau have two inequiv- 30::: 0 –300 –9856 –149940

40::: 0 –720 –36036 –787640 alent phases, together with a Z-family of choices of

50::: 0 –1540 –112112 – 3406480 framing in each. m =5: The mirror B-model geometry (see e. g. [34 - 36])

00::: 0 10 84 396 can be written as

10::: 0 84 1176 8736

u v tuv

20::: 0 396 8736 96660

e e e ; xz = + + +1 (6.14)

30::: 0 1386 45864 724800

40::: 0 4004 191100 4273840 1 2 3 0

Y Y t Y where we have “solved” Y + + = +3 50::: 0 10090 672672 –

0 1 2

Y u Y v Y t u v by Y =0, = , = and = . m =6: 3 In terms of these coordinates, the brane in phase I 00::: 0 –26 –264 –1504

10::: 0 –264 –4224 –35640 propagates on the internal leg of the Riemann surface

:::

 u t= 20 0 –1504 –35640 –427540 where v 0 and large of order 2, the brane in

30::: 0 –6228 –211200 –3484800

 v  t= phase II is on u 0, and 2, and the brane on

40::: 0 –21028 –987360 –

 v external leg has u , both being large.

50::: 0 –61152 ––

In the variables of (6.14) the classical superpoten-

R

v u u u tial vanishes in phase I and W ( )= ( )d com- Consider three phases of the A-brane that are visible putes the disk instanton generated superpotential. classically (see Fig. 17). To compute disk numbers we need the flat coordi- nate. According to the discussion in Sect. 5, this is

1 2

r >c > ; c ;

Phase I : t 0 =0 given by the difference of superpotentials on the two

22 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web

K ! Table 5. Disk degeneracies for brane I or II in the O ( )

P2 geometry. k m — — 01234567 8

–5000 0 0 5 –84 1200 –16854 –4000 0 –228–344 4360 –57760 –3000 1–10 102 –1160 14274 –185988 –200–14–32 326 –3708 45722 –598088 –101–212–104 1085 –12660 159208 –2112456 11–15–40 399 –4524 55771 –729256 9961800 20–17–61 648 –7661 97024 –1293185 17921632 30–19–93 1070 –13257 173601 –2371088 33470172 40–112–140 1750 –22955 312704 –4396779 63460184

50–115–206 2821 –39315 559787 –8136830 120497011 u; v Fig. 18. The curve P ( ) = 0 associated to the mirror of

2

!

O ( 3) P . 2

K ! Table 6. Disk degeneracies for brane III in the O ( ) P geometry.

sides of the domain wall as computed along the con-

m k tour C on Figure 18. The non-trivial contribution to — — 012345678 uˆ is the exponentially suppressed shift which comes from the small period on Fig. 18: 1 –12 –532–286 3038 –35870 454880 –6073311

201–421–180 1885 –21952 275481 –3650196 Z

301–318–153 1560 –17910 222588 –2926959

u v u u u
: ˆ = ( )d = + u 401–420–160 1595 –17976 220371 –2869120

C 501–526–196 1875 –20644 249120 –3205528 601–736–260 2403 –25812 306095 –3889116 We find, using the same kind of methods discussed 701–952–365 3254 –34089 397194 –4981102 1 1

for the P  P example, that 801–12 76 –528 4578 –46812 535639 –6627840

t

t ˆ For the phase III, to compute the integer invariants

i ;

u = +

3 we need to change the parameterization of the curve.

ZZ

Consider the SL(2; ) transformation where tˆ is the closed string period on the contour 1

in Figure 18. 0

! u u v; u =

The closed string period can be computed either 0

directly by integration on  , or from the Picard-Fuchs

! v v; v =

3

 z  

equation with L = +3 (3 + 1)(3 + 2), where

t

 z@ z e = z and = . The solutions to the Picard- which will allow us to compute the superpotential in Fuchs equation can be expressed, e. g., as Mejer G- phase III since in these variables the equation of the

functions [37] and 0  phase III leg is u = 0. The equation for written in

the new variables becomes

1

n

X n

(1) (3 )!

n

t z :

tˆ

0 0 0

= 0

v t v u

3 u +3

n n

e e ; ( !) e + +1+ =0 (6.16)

n=1

0 v

upon trivial multiplication by e . The good flat coor- t

The inverse z (ˆ) relation is

tt

0 0

0 ˆ

v v
v dinate for this phase is ˆ = + v = + 3 .For

2 3 4 5 6

d

z q q q q q q :::;

= +6 +9 +56 300 +3942 + (6.15) the integer invariants in phase III we get that k;m =0 m

for m<0 and, for positive , we obtain the values t

ˆ e

where q = . of Table 6. u; t

After rewriting W ( ) in terms of the flat coordi- From (6.16) we obtain another description of phase

t I nates u;ˆ ˆ, and expanding as in (2.11) we obtain the , which is at the classical level equivalent to the one integer invariants of Table 5. already given, but differes in the quantum theory by M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web 23

2

K !

Table 7. Disk degeneracies for brane I in the O ( ) P Integrality of the Bulk Mirror Map

2 Z geometry for various choice of the ambiguity n .

The integrality of mirror map in the bulk, even k m — — though it has been proven in some cases, has not been 012 3 4 5 6 7 8

physically explained. Here we will connect this to the

n N

= 1: integrality of the number of domain walls k;m.Inthe

I –5000 0 0 40–1274 27885 –528934 case n= 1 we can explicitly show that all coefficients

–4000 0 10–253 4604 –76068 1214324 N

of the form k;1 are directly related to the coefficients –3000 3–54 783 –11058 157347 –2274642 of the bulk mirror map and using this relation the –2001–13 142 –1657 20785 –274473 3769424

–101–429–274 3002 –36144 464522 –6262370 integrality of one follows from the other. a

11–14–29 274 –3002 36144 –464522 6262370 Define the numbers i by

200–113–142 1657 –20785 274473 –3769424

1

X

t 3000–354–783 11058 –157347 2274642 tˆ

2 i

e a q : 4000 0–10 253 –4604 76068 –1214324 =1+ i

5000 0 0 –40 1274 –27885 528934 i=1 k

m — — a

One can show the integrality of i using the integrality

01234of the mirror map (6.15). Furthermore one can show,

v u; q n =1: by explicitly investigating the Taylor series of (ˆ ), –400000that

–30000 –1

N P a ;:::;a a ;

k k k –2000 –17 k;1 = ( 1 1)

–10 1 –15–40 P

11 –212–104 1085 where k is a polynomial with integer coefficients in 2 –14–32 326 –3708 a

the i .E.g.weget 31 –10 102 –1160 14274

4 –228–344 4360 –57760

; P1 = 3

n =2:

–300000 2

P a ; 2 =30+12a1 + 1 –2000 –113

–11 –14–29 274 2 3

P a a a a a a ; 11 –429–274 3002 3 = 420 210 1 30 1 1 +12 2 +2 1 2 21 –13 142 –1657 20785

33 –54 758 –11058 157347 2 3 4

a a a a a P4 = 6930 + 4200 1 + 840 +60 + 210 2 410 –253 4608 –76068 1214324 1 1 1 540–1274 27885 –528934 9380474

2 2

a a a a a a a a ;

60 1 2 3 1 2 + 2 +12 3 +2 1 3

the relative framing n = 1. The flat coordinate in P

etc.. The proof that all k are integer polynomials is

I

the phase n= 1 is tedious and relies on some formulas for the derivatives

v u; q 0

0 of (ˆ ) for this special example. From this one sees

u ;

uˆ =

N a k

that the integrality of k;1 and are equivalent, and

a

; ZZ

the integrality of k follows from the integrality of v since under the SL(2 ) transformation u and

cancel off. We have also considered other values of the mirror map in the bulk (6.15), and vice-versa. The

0

u ! u u nv n, i.e the D-branes with shifted = + as integrality of the mirror map in the bulk had not been

I physically explained before. Here, by relating it to the

the dynamical field on the brane. First, note that n

N

I I I

k;

n

and (n+1) (where denotes the brane in phase integrality of numbers of domain walls 1 we have

v !v t

and with framing n) are related by + , with found a physical explanation for it. The integrality of

N m 6 a

u i fixed. Thus, we expect k;m = 1 requires special properties of the and

seems much more involved.

n

n ( +1)

N ;

N =

k;m

m;m

k +

K ! 6.4. O ( ) F1

where m denotes the is boundary class of the disk. The

; ; integer invariants for n = 1 1 2, given in Table 7, Consider a CY geometry containing the blowup 2 clearly respect this. of P at one point, which we denote by F1. The

24 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web

K ! III Table 8. Disk degeneracies for brane I and II in the O ( )

F1 geometry. k

II k f b — — r

t 0123 45 m = 5: rb I Phase II: 501 –614–14 5

602 –30 140 –280 252

K ! Fig. 19. Toric base of O ( ) F and the A-brane in

1 Phase I:

K ! three phases. O ( ) F1 can be viewed as a blowup of

2 1 500 0 0 0 0

O ! r

( 3) P at a point by a P of size t .

600 0 0 0 42 m = 4: charges describing the gauged linear  -model on the

Phase II:

Q ; ; ; ;

non-compact Calabi-Yau are b =( 2 0 0 1 1)

401 –45–20

Q ; ; ; ;

and f =( 1 1 1 1 0), so the corresponding 502 –20 60 –70 28

D -terms are 603 –68 400 –936 –344

3 2 4 2 0 2 Phase I:

jX j jX j jX j r ;

+ 2 = b (6.17) 400 0 0 –20

1 2 2 2 0 2 3 2

jX j jX j jX j jX j r ;

+ = f 500 0 0 –14 28

600 0 0 –52 308

r r f

where b and are the areas of the base and the m = 3: fiber of the Hirzebruch surface F1. The equations are Phase II: “solved” in Figure 19. The threefold has two classes of 301 –21 00

2

PI divisors: H coming from itself and the exceptional 402 –12 20 –10 0

2 2 503 –45 170 –230 102

H E EH divisor E with =1, = 1 and =0. The charge vectors (or generators of the mori-cone) 604–130 958 –2612 2940

Phase I:

H E correspond to the fiber F = and the base,

300 0 1 0 0

Q Q Q Q

E 2 2

b f P which is . Note that + = P with is

1 400 0 5 –10 0

! P

the charge for the O ( 3) , and correspondingly 500 0 14 –90 102

F H

E + = . 600 0 31 –450 1428 The A-brane charges are again such that the brane m = 2:

1

; ; ; ; is special Lagrangian, e.g. q =( 1 1 0 0 0) and Phase II:

2

; ; ; ; c ;c q =( 1 0 0 0 1) which defines the moduli 1 2: 201 –10 00 302 –64 00

1 2 0 2

X j jX j c ; j = 1 403 –28 57 –32 0 504 –90 424 –664 326

4 2 0 2 605–237 2172 –6872 8640

jX jX j c : j = 2 Phase I: We will consider the following 3 phases (out of a total 200 –10 00 of 8 possibilities)8 300 –24 00 400 –428–32 0

500 –6 112 –390 326

r r = >c > c ; f Phase I : ( b + ) 2 1 0 2 =0

600 –9 336 –2500 5638

t t

b f

c r = >c > ;

z e z e

Phase II : =0 b 2 0 (6.18) f 1 2 where b = , = , and we have solved

1 4

u; Y v the mirror relations in terms of Y = = .

0

r = >c > c r :

Y b

Phase III : f 2 1 0 2 = with = 0. The mirror of the brane in phase I has

u  t t = v  II f

( b + ) 2 as a variable with 0, has

K ! F

The mirror of O ( ) 1 is given by

v  t = u 

f 2 as a variable with 0 on the relevant

v u v v

u leg of the toric diagram. One finds using the methods

xz e e z z e z e ;

b b =1+ + f + + (6.19) 1 1

discussed for the P  P example, that

8We have checked that all other phases also lead to integral

t t i :

ˆ

f f expansions for disk amplitudes. u;v =( )+ M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web 25

Table 8 (continued). Table 8 (continued).

k k k k

f b f b — — — —

012 3 4 5 012 34 5

m m = 1: =3: Phase II: Phase II: 1 –110 0 0 0000000 0 20 2 –2 00010 –10 00 0 30 3–15 12 0 0 20 –29 00 0 40 4–60 160 –104 0 30 –348–93 0 0 50 5–175 1080 –1995 1085 40 –4 155 –884 1070 0 Phase I: 50 –5 390 –4682 15134 –13257 1010 0 0 0Phase I: 20 1 –2 000000000 0 30 1–10 12 0 0 11 –10 00 0 40 1–30 120 –104 0 24–139000 50 1–70 648 –1596 1085 311 –85 167 –93 0 0 425–382 1555 –2268 1070 0 m =1: Phase II: 549–1344 9813 –27584 32323 –13257

0100 0 0 0m =4: 10 –10000Phase II: 20 –25000000000 0 30 –330 –400010 –10 00 0 40 –4 105 –432 399 0 20 –212000 50 –5 280 –2520 6370 –4524 30 –360–140 0 0 Phase I: 40 –4 188 –1260 1750 0 50 –5 460 –6397 23482 –22955 0100 0 0 0 11 –10000Phase I: 21 –65000000000 0 31–20 59 –400011 –10 00 0 41–50 356 –706 399 0 26–18 12 0 0 0 51–105 1500 –6244 9372 –4524 325–155 270 –140 0 0 476–887 3056 –3995 1750 0 m =2: Phase II: 5 196 –3873 23040 –56429 60021 –22955

0000 0 0 0m =5: 10 –10000Phase II: 20 –27000000000 0 30 –338 –610010 –10 00 0 40 –4 128 –616 648 0 20 –215000 50 –5 330 –3420 9744 –7661 30 –374–206 0 0 Phase I: 40 –4 225 –1772 2821 0 0000 0 0 0Phase I: 11 –10000000000 0 22 –9700011 –10 00 0 34–43 100 –610029–24 15 0 0 0 46–147 756 –1263 648 0 349–264 421 –206 0 0 59–406 3920 –13122 17260 –7661 4 196 –1876 5700 –6841 2821 0

The numbers of primitive disks in these two phases coordinate on the brane. The correction to the flat are listed in Tabel 8. coordinate is again found by requiring integrality of

We consider now the phase III, for which we must the amplitude, and we find that

0

! u

change the parameterization of the curve by u =

v
v t t :

v ˆ

f f

ˆ = + v = +

t u

b , if we are to have the superpotential which

is zero classically. In this phase, v is the transverse The disk numbers follow (Table 9).

26 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web

K ! F

Table 9. Disk degeneracies for brane III in the O ( ) 1. Table 9 (continued).

k k

k k f

b — — f b — —

01 2 3 4 5 01 2 3 4 5 m m =1: =4: 01–10 0 0 000–14 –52 0 10–22 0 0 010–220 –60 70 –28 20–315 –12 0 0 20–368–400 936 –945 30–460–160 104 0 30–4 180 –1912 7910 –15030 40–5 175 –1080 1995 –1085 40–5 412 –7290 49096 –155035 50–6 420 –5040 19110 –27144 50–6 840 –23520 243558 –1185830

60–7 882 –18480 124033 –337617 60–7 1576 –66660 1015960 –7246659

m m = 1: = 4: 000 0 0 0 000 0 0 0 0 0 1 –10 0 0 0 0 10 0 0 0 0 0 201 0 0 0 020 0 0 0 0 0 302 –500 030 0 0 0 0 0 403–30 40 0 0 40 0 0 0 0 0 504–105 432 –399 0 50 1 0 0 0 0

605–280 2520 –6370 4524 60 2 –12 0 0 0 m m =2: =5: 00–11 0 0 000–16–14 14 –5 10–26 –40 010–230–140 280 –252 20–328 –57 32 0 20–395–810 2870 –4858 30–490–424 664 –326 30–4 240 –3472 20150 –56728 40–5 237 –2172 6872 –8640 40–5 525 –12156 109167 –475047 50–6 532 –8640 48208 –114774 50–6 1036 –36648 487382 –3116370

60–7 1072 –28578 258516 –1023679 60–7 1890 –98340 1869595 –16909871

m m = 2: = 5: 000 0 0 0 000 0 0 0 0 0 100 0 0 0 010 0 0 0 0 0 200 0 0 0 020 0 0 0 0 0 301 0 0 0 030 0 0 0 0 0 402 –700 040 0 0 0 0 0 503–38 61 0 0 50 0 0 0 0 0 604–128 616 –648 0 60 1 0 0 0 0

m =3: 00–12 –10 0Sheldon Katz, Chiu-Chu Liu, Marcos Marino, Cliff 10–212 –20 10 0 Taubes, Shing-Tung Yau and Eric Zaslow for very 20–345–170 230 –102 valuable discussions. 30–4 130 –958 2612 –2940 40–5 315 –4116 19750 –41996 The research of M.A. and C.V. is supported in part 50–6 672 –14520 112970 –398970 by NSF grants PHY-9802709and DMS 9709694. The

60–7 1302 –44073 525031 –2854610 research of A.K. is supported in part by the GIF grant m = 3: I-645-130.14/1999. 000 0 0 0 0 100 0 0 0 0Appendix. Large N Limit of Chern-Simons and 200 0 0 0 0 300 0 0 0 0 Framing of the Knot 401 0 0 0 0 502 –900 0As discussed in Sect. 5, Wilson loop observables of

603–48 93 0 0 Chern-Simons theory at large N are naturally encoded in terms of the expectation values

Acknowledgements

D E

X 1 n

We are grateful to Volker Braun, Stefan Freden- n

U V F V ; t; g ;

exp tr tr = exp( ( s ))

hagen, Kentaro Hori, Amer Iqbal, Dominic Joyce, n n

M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web 27

n

U U where V is viewed as a source and is the holonomy representation of .Tofind the change in this cor- of the connection along the loop. If we are interested relation function due to change in framing, we will

in extracting the amplitudes with a single hole, we first have to write it in terms of trace of U in different n

consider terms on the right hand side of the form tr V representations, compute each one, and multiply each

g ph R

in the exponent. This will correspond to contributions one with exp( s ). n where the large N worldsheet wraps times around There is an identity which is useful for this purpose:

the loop. From the left-hand side this is equivalent to n 1

computation of X

n s

U U;

tr = ( 1) Tr R

n;s

E D

1 s=0

n :

tr U

n

R N

where n;s denotes representations of SU( ) with For a special choice of framing of the unknot this was n boxes for the Young Tableau which looks like ‘ ’ computed in [4] and it was shown that and which consists of only one non-trivial row and

one non-trivial column. s + 1 denotes the number of E

D 1 1 elements in the first column.

n n

U ;

tr U = tr 0 Combining all this, we find that the coefficient of

n n

n p

tr V for the unknot and with framing number is

SU N

where U0 is a particular element of ( ) given by given by

N r i ( +1 2 )

a diagonal matrix with entries exp( ), as r

N k

+ n 1 X

N 1 1 s

ranges from 1 to . This leads, in the leading order n

h U i pg h U

R R

tr = ( 1) exp( s )Tr n;s

n;s 0

g g

n n in s , corresponding to disk amplitude ( = 0 and one hole), to s=0

We are interested in extracting the disk amplitude E D 1 1

n which is the leading term for this expansion as

nt= nt= ;

tr U = 2 [exp( 2) exp( 2)]

g !

n n g

s 0. Moreover we will be interested in the limit

s

! 1 V where t , but where we rescale by a fac-

t=2

t V where in the large limit, and by redefining the tor of e so that we would be computing the brane 3 (and absorbing a factor of exp(t=2) in it ) we obtain in the C geometry. The most complicated aspect of 3

the C answer for the disk amplitude U

this computation is finding Tr R . We will reduce n;s 0 this computation to a group theory computation as 1 X 1 n follows: Let

2 tr V

g n

s

n

n

X Y

i R n

n;s i

U U :

Tr R = [tr ] n;s 0 n

i 0

u

P e (where we have identified in this paper V = ).

i=1

in n

i =

Now we ask what happens if we choose a differ- R

ent framing. For the expectation value of the Wilson n;s

n

for some group theoretic factor i . Using this and

loops with holonomies in a given representation R of the leading computation at large N for the unknot

SU(N ) the answer is relatively simple: with the standard framing, we can reduce the above

computation to

h

R

h U i h U i ip ;

p R

Tr R = Tr exp(2 )

0 n 1

X N

k + 1 1

n s

h U i pg h R

tr = ( 1) exp( s )

n;s

n n p where denotes the change in the framing from a s=0

given one denoted by 0 (characterized by an element

n Y

1 1 X S

S 1 n

of topological class of winding of an over an ), R

n;s i

( )

n

i

h ig

and R denotes the Casimir of the representation. In

s

i

Σ in n

other words, the result of the change in framing is a i = =1

g ph R

multiplication by exp( s ). Using this and the fact that up to a further redefi-

V g

However we need to compute the correlation func- nition of and dropping a subleading term in s ,

n 1

h U i h n n s

tion for tr which is the trace in the fundamental R = ( 1 2 ) we have computed the n;s 2

28 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web

h

; ;

above expression for n =12 3 and found perfect Furthermore we have verified using the identity

i

P

n

n

match with the result of the computation done for n 2 1 2

n

n r

n n r +1 ( 2 +1)

3

n 1 ( 1) = that the

r u ! u pv

nr r n

C when we shift + . Moreover we have n!2 =1 ( )!( 1)! ! p verified that we obtain for all n a polynomial of de- leading power in agrees between both computations

3

n

p gree n 1in in agreement with the results of C . for all .

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