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nontrivial checks of this duality comes from a

comparison of the free energies. In eqns [8] and

[9], we already have carried out the sum over the

holes in the Chern–Simons theory and organized it

as a closed-string genus expansion. Note that these

expressions are already of the form [11] expected of

a closed topological string. One simply has to check

that it is indeed that on the S

resolved conifold.

In the language of the integer invariants n



, the S

resolved conifold is particularly simple. The only

nonzero invariant is n

= 1. Physically, this corre-

sponds to a single brane wrap ped on the genus-zero

. Putting this into eqn [13], and making the

expansion in powers of g

, we find exactly eqn [9]

for the genus-g contribution to the free energy. This

is quite a remarkable agreement and represents a

triumph for the ideas of large-N duality.

Geometric Transitions and Large-N

Duality

To understand the reason for this duality a bit

better, we utilize an old observation of Witten that

Chern–Simons theory is an open topological string

theory. As mentioned earlier, the expansion [2] (or

[6]) is suggestive of an open-string expansion in

terms of handles and holes. Witten observed that

open topological strings on the noncompact 3-fold



M (with Dirichlet boundary conditions on M for

the end points of the string) is Chern–Simons theory

on M. In fact, in the modern language of D-b ranes,

we would say that U(N) Chern–Simons theory is the

world-volume theory of N D-branes wrapped on M,

for the topological A-model on T



In particular, Chern–Simons theory on S

is the

theory of branes wrapped on S

inside T



. The

latter is the conifold geometry but now deformed by

a nonzero size S

. It is described by the equation

xy  zw ¼  ½16

where  is the deformation which parametrizes the

size of the S

The above large-N duality can be considered as an

open–closed string duality. Namely , that the theory

of open A-model topological strings on the S

resolved conifold (with N D-branes) is dual to closed

A-model topological strings on the S

resolved

conifold. Cast in this way, we see that the duality

involves a transition in the background g eometry in

going from the open-string to the closed-string

description. The sum over the holes changes the

background. The S

, as it were, shrinks to zero size

and a transverse S

opens up. This geometric

transition makes the connection between the

Chern–Simons theory and the closed topological string

somewhat less mysterious. Maldacena’s conjecture for

super Yang–Mills involves a similar passage from

D-branes in flat space to a closed-string theory on

anti-de Sitter space. In fact, it appears as if the best way

to understand ’t Hooft’s idea in generality is to think of

it as an open–closed string duality.

Further Checks and Consequences

The free energy is not the only gauge-invariant

observable in Chern–Simons theory. One important

class of observables, which played an important role

in the connection with knot invariants, are the

Wilson loop expectation values. Given a knot K in

, we can defin e, in terms of an arbitrary

representation R of U(N), the trace of the holonomy

around the knot averaged with respect to the Chern–

Simons path-integral measure:

ðKÞ ¼< tr

P exp i



> ½17

P denotes path ordering. Similarly, we can also

define the expectation values of links: products of

traces of holonomies around various interlinked

paths. The nonpe rturbative solution of Chern–

Simons theory gives exact answers for the expecta-

tion values of these Wilson loops. The discussion

below is, however, confined to knots.

Since the trace of holonomies is being considered

in different representations, it makes sense to study

the generating functional

ZðU; VÞ¼

ðUÞ tr

ðVÞ

¼ exp

n¼1

tr U

trV

½18

The source V here is a U(M) matrix, unrelated to the

U(N) holonomy U around K. The second equality in

[18] follows from use of the Frobenius formula. It

was shown by Ooguri and Vafa that this generating

functional is the natural objec t from the point of

view of the open–closed string duality.

We have already mentioned that the U(N) Cher n–

Simons theory can be thought of as the theory of N

topological D-branes wrapped on the Lagrangian S

cycle inside T



. For a knot K in the S

, we consider

another Lagrangian 3-cycle

in T



which

intersects the S

exactly in K. A canonical construc-

tion for

ðqðsÞ; p Þ2T



¼ 0

½19

Large-N and Topological Strings 267

where the knot K is parametrized by the closed curve

q(s). By construction,

intersects the S

in K.

Now consider M D-branes wrapped on

. One now

has to consider the fields coming from the strings

stretching between the two sets of branes. One can

show that integrating out these fields (which are in the

bifundamental of the product group U(N)  U(M))

modifies the original Chern–Simons action to

eff

ðAÞ¼S

ðAÞþ

n¼1

trU

trV

½20

Here V is the holonomy around K of the U(M)

gauge field

A. Thus, this configuration of M probe

branes gives rise exactly to the generating function

eqn [18] for Wilson loops of K.

The geometric transition which relates the Chern–

Simons theory to the closed-string theory now

suggests what one needs to do to compute this

generating function on the closed-string side. We

have to follow the configuration of the M probe

branes on

through the conifold transition in

which the S

shrinks and one blows up the S

.Itis

not easy in general to figure out the Lagrangian

cycle C

which results from following

through

the transition. It has only been done in a class of

knots including the simple unknot. But assuming we

know C

, the generating function for Wilson loops

is given by the free energy on the S

resolved

conifold in the presence of M probe branes on C

This requires one to know more than the closed-

string partition function computed earlier. We now

also need to compute amplitudes for world sheets

with boundary on C

. These are called open-string

Gromov–Witten invariants and the study of this

subject is in its infancy. For simple knots such as the

unknot, for which C

is known, these can be

computed. One finds again a remarkable agreement

with the nonperturbative answers of Chern–Simons

theory. Thus, the computation of knot invariants

gets related to open-string Gromov–Witten invar-

iants. There have been a number of other tests

involving more general knots and links. One also

has to be careful of subtleties such as in the choice

of framing. The reader is referred to the articles

by Marino (2002, 2004) for these topics.

Conclusions

The large-N duality of ’t Hooft is realized in Chern–

Simons theory in a very explicit way. Thanks to the

analytic control we have over both Chern–Simons

theory as well as clos ed topological strings, the

conjecture pa sses very nontrivial checks that extend

to all-genus case. This is more than we can do in the

AdS/CFT conjecture where most computations are

at tree level in the supergravity limit. In contrast,

here we see the essential stringiness of the closed-

string dual to Chern–Simons theory.

Also, by viewing it as an open –closed string

duality, many aspects of the correspondence were

clarified. It, therefore, provides a useful toy model

for a general understanding of open–closed string

duality. Indeed, a proof of this duality using world

sheet techniques has been proposed by Ooguri and

Vafa. One would like to carry over some of the

intuition that operates in this duality to the case of

other physically interesting gauge theories.

From the mathematical point of view, as already

indicated, this duality leads to previously unsuspected

relations between Gromov–Witten invariants and

invariants of 3-manifolds, including those of knots.

In fact, by considering more general geometric

transitions and using this duality locally, one can

learn about all-genus topological string amplitudes

for a wide class of noncompact toric geometries. This

line of development culminated in the formulation of

the topological vertex by Aganagic, Klemm, Marino,

and Vafa, which captures the essence of the

topological closed-string amplitudes for noncompact

toric geometries. As in the case of the general

correspondence between the gauge theory and grav-

ity, this duality sheds new light on both sides of the

equation. We learn to see new integrality properties

in knot and 3-manifold invariants which have an

interpretation in terms of enumerative problems in

3-folds. The surprises that such a deep connection

presages have not yet been exhausted.

See also: AdS/CFT Correspondence; Chern–Simons

Models: Rigorous Results; Duality in Topological

Quantum Field Theory; Free Probability Theory; The

Jones Polynomial; Knot Theory and Physics; Large-N

Dualities; Quantum 3-Manifold Invariants; Schwarz-Type

Topological Quantum Field Theory; String Field Theory;

Topological Gravity, Two-Dimensional; Topological

Quantum Field Theory: Overview.

Further Reading

Brezin E and Wadia SR (eds.) (1993) The Large N Expansion in

Quantum Field Theory and Statistical Physics. Singapore:

World Scientific.

Gopakumar R and Vafa C (1999) On the gauge theory/geometry

correspondence. Advances in Theoretical and Mathematical

Physics 3: 1415 (arXiv:hep-th/9811131).

Hori K, Katz S, Klemm A, Pandharipande R, and Thomas R

(2003) Mirror Symmetry. New York: AMS Publishing.

Marino M (2002) Enumerative geometry and knot invariants

(arXiv:hep-th/0210145).

Marino M (2004) Chern–Simons theory and topological strings

(arXiv:hep-th/0406005).

268 Large-N and Topological Strings

Large-N Dualities

A Grassi, University of Pennsylvania, Philadelphia,

PA, USA

Introduction

Gopakumar and Vafa (1999) conjectured that U(N)

Chern–Simons gauge theory on S

is dual, for large

values of N, to a closed topological string theory

on a suitable Calabi–Yau 3-fol d X. They suggested

that this duality is realized by a geometric ‘‘transi-

tion,’’ a topological surgery which can be realized by

birational contractions follo wed by the complex

deformations of Calabi–Yau varieties. Here we will

give some general comments on the history of this

conjecture and then present some of its mathema-

tical implications; we will focus on the geometric

transition and the novel mathematics that it has

generated.

A duality relating gauge theories and string

theories (with gravity) was first conjectured by

’t Hooft (1974). In 1998 Maldacena conjectured a

duality between Yang–Mills gauge theory with

N = 4 SUSY on a four-dimensional manifold M

and IIB type closed string on the anti-de Sitter space

AdS

 S

. Chern–Simons string theory is a three-

dimensional theory and purely topological, hence it

is in principle simpler than four-dimensional Yang–

Mills theory, which also involves a metric.

In this survey, we discuss the IIA open/closed

dualities: we will mostly be concerned with the partition

function, that is we will be working in the context of

‘ ‘top ological strings.’’ The duality has been extended to

a duality of strings, adding fluxes on the closed sector

and branes on the open sector. There is much

mathematical evidence supporting the conjecture.

Overview

The conjecture says that U(N) Chern–Simons gauge

theory on S

is dual, for large values of N, to type

IIA closed topological string theory on a suitable

Calabi–Yau manifold X. A starting point for the

geometry, and its mathematical implications, is that

can be thought of as a vanishing cycle in a local

Calabi–Yau manifold Y = T



, which deforms to a

singular Calabi–Yau Y

; X is a Calabi–Yau bira-

tional resolution of Y

. X are Y are related by a

geometric transition. In fact, Witten showed that

quantum Chern–Simons theory on S

can be thought

of as open IIA (with U(N) branes) on Y = T



; thus,

a more general conjecture says, loosely speaking,

that open IIA theory on a Calabi–Yau manifold Y is

dual, for large N, to closed IIA on a Calabi–Yau X

which is related to Y via a geometric transition. A

consequence of a physics ‘‘duality’’ is a matching of

the free energies of the dual theories. In this

particular case, if the conjecture is true, the Chern–

Simons free energy Z(S

,U(N)) should determine,

and be determined by, the closed prepotential

(X, t). Note that Z(S

,U(N)) is purely topologi-

cal, and that F

(X, t) includes all genera, as we will

discuss later. A mathematical application is comput-

ing Gromov–Witten invar iants for higher genus via

large-N dualities (Marin˜ o 2004). Another con se-

quence involves the matching of the observable in S

and X.

This conjecture is now supported by a vast

amount of evidence. Vafa, Gopakumar and Ooguri

noted, via a string-theory analysis, that topological

and knot invariants of S

(computed through U(N)

Chern–Simons theory on S

) determine and are

determined by , for large N, the Gromov–Witten

invariants of X in a neighborhood of the exceptional

locus of the birational contraction X !Y

The extension to the full string theory would say

that open string of type IIA compacti fied on a

Calabi–Yau manifold Y with branes is conjectured

to be dual to closed string of type IIA compactified

on a Calabi–Yau manifold X with fluxes, if X and Y

are related by a geometric transition.

A mathematical consequence of this stat ement

is that the closed Gromov–Witten invariants of X

agree, with a suitable identification of the para-

meters, with combinations of open Gromov–Witten

invariants and knot invariants of Y. This has been

shown to hold for some classes of examples .

This circle of ideas has stimulated much work in

physics and mathematics on the nature of the

mathematical correspondence behind this duality, as

well as the property of the enumerative and topo-

logical invariants involved. The ‘‘mirrors’’ of the above

transitions have been studied in a series of papers,

starting with the work of Dijkgraaf and Vafa (2002).

The mathematics behind the open/closed dualities

is still not understood: it is reasonable to speculate

that the natural setup is a framework of symplectic

field theory.

We shall start by discussing the principal topics

of this large-N duality: Chern–Simons quantum field

theory, IIA closed prepotential (and Gromov–Witten

invariants), and Chern–Simons as open string (and

IIA open prepotential). Next we shall study the

geometric transitions and conclude with some

mathematical predictions of the duality.

Large-N Dualities 269

We shall not discuss some other interesting

implications of this duality. For example, we shall

not discuss its mirror IIB duality: it is known that

the part of the closed prepotential in IIA correspond-

ing to rational curves can be expressed as its IIB

mirror dual with periods over certain suitable cycles;

the IIA open contribution corresponding to open

discs is expressed in terms of integrals over chains

and the Abel–Jacobi map. We only remark that this

large-N duality has also been interpreted as a duality

between seven-dimensional manifolds with G

holonomy.

History

The chronology of various important contributions

in the field of large-N duality is as follows:



1976: ’t Hooft’s conjecture



1988: Clemens introduces transitions



1988: Witten introduces quantum Chern–Simons

theory on 3-manifolds



1992: Witten discusses Chern–Simons theory as

open string



1998: Gopakumar–Vafa–Ooguri



2001: Verification for unknot, Katz–Liu, Li, and

Song



2001: Lift to manifolds with G

holonomy



2002: The conjecture verified for many examples

of conifold transitions, including compact case;

the topological vertex is introduced



2003: Relations with Donaldson–Thomas invariants

Background

The varieties of interest in the physical theory

must satisfy certain ‘‘supersymmetry’’ conditions; in

particular, a complex algebraic manifold is required

to be Calabi–Yau, a real seven-dimensional Rieman-

nian manifold is required to have G

holonomy

group. Also of particular interest are the Lagrangian

real submanifolds of the Cal abi–Yau 3-folds. By a

Calabi–Yau manifold X we mean a manifold with

(X) = 0, h

(

) = 0, where 

is the sheaf of

holomorphic k-forms, and 0 < k < dim (X). If

dim X  2, we also assume that X is simply

connected, but not necessarily compact. For exam-

ple, if dim (X) = 1, X is a torus, if dim (X) = 2, X is

a K 3 surface, if dim (X)  3, X is simply called a

Calabi–Yau manifold. A compact Ka¨ hler manifold

(M, g, J) of complex dimension m  3 is a Calabi–

Yau variety if and only if its holonomy is SU(m). A

subvariety L of a symplectic manifold (X, !)is

Lagrangian if !

= 0 and dim L = (1=2) dim X.

Sometimes we consider noncompact manifolds,

thought of as neighborhoods of a compact projective

Calabi–Yau manifold. Typically, our symplectic

manifold is a Calabi–Yau 3-fold (X, !) together

with its Ka¨hler form !. If there exists an antiholo-

morphic involuti on, then the fixed locus is a

Lagrangian submanifold.

The Dualities

We will take the point of view that dualities in

physics imply relations between geometric invari-

ants, without dwelling on the physics of the dualities

themselves. A consequence of a physics ‘‘duality’’ is

the matching of the prepotential of two dual string

theories.

A Few Comments on Chern–Simons Theory: Free

Energy (Partition Function)

Let L be a closed oriented manifold together with a

principal G-bundle. The classical Chern–Simons

action is defined as S(L, A) =

(A), where  is a

3-form on L which dep ends on a connection A and a

suitable bilinear invariant form on the Lie algebra g.

It is well defined under gauge transformations

modulo the integers; e

2iS(L, A)

is well defined. In

the large-N dualities consi dered here, the groups of

interests are SU(N) and U(N). The first check of the

duality was found with G = SU(N) and M = S

; later

it was discovered that the correct group for the

matching of the observables must be U(N), while

both can be used for the free energies. We shall

consider G = SU(N)andM = S

. Without loss of

generality, the bundle can be taken to be the product

U(N)  S

; any bilinear invariant form on the Lie

algebra su(N) is necessarily an integer multiple k of

the Cartan–Killing form on the Lie algebra. Then

S = S(k, A) and

Sðk; AÞ¼

8

tr A ^ dA þ

A ^ A ^ A



where k is the ‘‘level’’ of the theory. Witten defines

the quantum Chern–Simons theory by taking the

integral of the Chern–Simo ns action over all possible

connections A modulo gauge equ ivalence G:

ZðS

;SUðNÞÞ

A=G

ðDAÞe

2iSðAÞ

A=G

ðDAÞexp 

4

tr A^dA þ

A^A^A





Witten shows how to calculate the free energy

Z(S

,SU(N)) through topological surgery, assuming

Z(S

S

)=1. Witten also defines the partition

270 Large-N Dualities

function of knots and links in L (the ‘‘expectation

values’’), which are knot and link invariants. The

expectation values are computed by evaluating the

trace of the holonomy transformation of a U(N)

connection around the knot, and then taking a

suitable average of the U(N) connections. These

invariants depend on a choice of the framing of the

knot (or link).

The explicit computations involve physics, repre-

sentation theory, and topology. If L = S

, then:

; SUðNÞ



¼ðk þ NÞ

N=2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

k þ N



N 1

j¼1

2 sin

j

k þ N



Nj

Reshetikin and Turaev, among others, described

mathematically the Chern–Simons free energy and

the expectation values.

A Few Comments on Closed-String Theory: Free

Energy (Prepotential)

In IIA closed-string theory on X, a Calabi–Yau

manifold, one considers holomorphic stable maps

of closed Riemann surfaces of genus g,  : 

!X,

with 



(

) = [] 2H

(X, Z), for all genera g and

homology classes  2H

(X, Z).

Then one forms the closed prepotential F

(X, t),

which encodes the enumerative invariants of

such maps to X, and which depends on the

Ka¨ hler parameters t of X. Sometimes the prepoten-

tial is also called ‘‘free energy’’ in the physics

literature or Gromov–Witten prepotential, as it

contains the Gromov–Witten invariants of X.

Setting F

(q) =

 2H

(X, Z)

g, 



, the closed pre-

potential is defined as

ðX; qÞ¼

g>0

2g2

ðqÞ

Here q is a formal variable such that q



þ

= q



q



(for 

, 

(X, Z)) and g

is the string coupling

constant. C

g, 

are the genus g Gromov–Witten

invariants of X, corresp onding to the class  and

they have been defined as

g;

½M

g;0

ðX;Þ

virt

It is difficult to explicitly compute the invariants

g, 

; in particular, there is no known general

method for calculating these invariants. They are

computed most ly via ‘‘localization’’ methods, in the

presence of a suitable torus action. In the case of

g = 0 the invariants are often computed via IIA–IIB

duality, calculating certain periods in the mirror

manifold W.

Example (Faber–Pandharipande). Let X ﬃO

(1)

O

(1); X is a neighborhood of a rigid

rational curve, which can be thought of as a local

Calabi–Yau manifold; then all the effective curves

 2H

(X,Z) must be of the form  = d[P

], 8d2N.

Faber and Pandharipande showe d that

ðX; qÞ¼

d¼1

2 sinðdg

=2Þ

½1

This formula was proved with localization methods

after it was conjectured by Gopakumar and Vafa using

large-N dualities. In fact, a consequence of a duality

between two theories is the matching of the free energies

of two dual string theories. In this particular case, the

conjectures imply that Chern–Simons free energy

determines, and is determined by, the all-genus closed

prepotential of a suitable Calabi–Yau manifold X:

ZðS

; UðNÞÞ ¸ F

ðX; tÞ

Note that the left-hand side is purely topological,

as we saw in the previous section, while the right-

hand side is holomorphic.

The trait d’union between the two prepotentials is

given by the interpretation of Chern–Simons theory

on S

as open-string theory on T



and the

geometric transition.

A Few Comments on Open-String Theory

with Branes: Open Prepotential

Let Y be a Calabi–Yau manifold together with {[L

Lagrangian submani folds; to each submanifold

is assig ned a gauge group G

: L

is wrapped

with G

-branes. Here we shall focus on the case

= U( N

) and we will write (Y; L

,U(N

)).

Witten shows that the open prepotential

(Y, , t

, g

) depends on ’t Hooft’s coupling con-

stants 

associated to Chern–Simons theory on the

Lagrangian submanifolds (L

,U(N

)), together with

the open Ka¨ hler parameters t

(X; [ L

, Z), and

the string coupling constant g

. To describe the open

prepotential, Witten argues, we consider all maps

of Riemann surfaces with boundary to Y,with

the condition that the boundaries are mapped to the

Lagrangian submanifolds L

; one should also include

all the ‘‘highly degenerate holomorphic maps,’’ in

particular those which contract 

g, h

to a ‘‘ribbon

graph’ ’ on the Lagrangian [L

. The contribution of

these highly degenerate maps is captured by the

quantum Chern–Simons theory of the Lagrangians

,U(N

)}.

Large-N Dualities 271

Application 1 (Chern–Simons free energy as open

prepotential). Let us consider open IIA on

Y = T



with U(N)-branes wrapped on L = S

L is a Lagrangian submanifold with the standard

symplectic structure ; note that in T



L there are no

nontrivial homology curves. Then, according to

Witten, the corresponding open prepotential

(Y, [ L

) must only depend on the ‘‘highly

degenerate’’ maps and must consist of the Chern–

Simons term F

on L = S

. In particular,

¼ log ZðS

Þ¼F

ðY;;g

where  = 2N=(k þ N) is the ’t Hooft coupling

constant. Periwal (1993) showed that, for large N,

log Z(S

) could be expanded as a closed-string

expansion:

ðÞ¼

g0

ðÞg

22g

where g

=:2=(k þ N) is the Chern–Simons cou-

pling constant. In 1998 Gopakumar and Vafa, using

physics arguments, deduced that the expansion

would have the closed form [1], which was later

proved by Faber and Pandharipande.

The explicit description of the open prepotential

in the presence of homology classes is not known;

one woul d need to combine the enumerative

invariants of open maps toget her with the quantum

Chern–Simons factor. We shall discuss an approach

at the end of this note, but consider first the

geometric transition.

The Transition

The conjecture says that U(N) Chern–Simons gauge

theory on S

is dua l, for large values of N, to IIA

closed topological string theory on a suitable

Calabi–Yau manifold X. A starting point to find

such X is that S

is a Lagrangian 3-cycle in the

manifold Y = T



; performing a topological surgery

by replacing S

with S

one obtains a (local) Calabi–

Yau manifold X, on whi ch the dual IIA theory is

compactified. The key observation is that Y can be

identified with the algebraic variety of equation

{xy  zw = t}  C

and that this is a complex

smoothing (in fact the Milnor fiber) of Y

with

equation {xy  zw = 0}  C

. On the other hand, X

is a small resolution of this singularity, where P

the exceptional locus of the birational contraction.

The origin is an ‘‘ordinary double point’’ singularity

and the nontrivial sphere S

 Y is the vanishing

cycle of the degeneration. The manifolds involved

are noncompact: the exceptional curve [P

] = t is

the only nontrivial homology class in X, and the

enumerative invariants in X can be thought as the

contribution of the exceptional curve in a neighbor-

hood of a Calabi–Yau manifold. We shall present

the steps leading to this construction and the

evidence for the conjecture.

The Local Construction of X

Let Y



= f(w

, ..., w

) 2C

such that

j = 1

= g.

Proposition 1 Let  be a nonzero real positive

parameter; then:



L = S

 T



is a Lagrangian submanifold of



with its standard symplectic structure;





ﬃ Y



and L ﬃ L



def

fRe(

j = 1

= ) g .

In fact, we can embed T



in R

j¼1

¼ 1;

j¼1

¼ 0

where S

= {p

= 0}; consider then the morphism

defined by setting

Reðw

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

 þ

; p

¼ Imðw

which induces the diffeomorphism Y



ﬃ T



of the

statement.

Remark 1 Let Y

= f

j = 1

= 0gC

; then:



is singular at the origin,





is a complex deformation of Y

, and





is called a ‘‘vanishing cycle.’’

With a change of coordinates we can write the

equation of Y



as {xy zw = 0}; the singularity is

still at the origin. This singularity is an ordinary

double point, which is often referred in physics

literature as ‘‘the conifold singularity.’’ Let

X C

P

be defined:

z þ y ¼ 0;x þ w ¼ 0

[, ] 2P

Remark 2 X is smooth and the morphism

 : X ! Y

; ððx; y; z; wÞ; ½; Þ7!ðx; y; z; wÞ

is an isomorphism 

XnP

:(XnP

) ’ (Y

n{0}) and

7!(0, 0, 0, 0, ) C

.  is a small (nondivisorial)

birational resolution of the singularity at the origin.



is a deformation (smoothing) of Y

. Note that

topologically S

ﬃ L



 Y



has been replaced by

ﬃ S

 X. The algebraic properties of the topo-

logical surgery between Y



and X were first studied

by Clemens in 1988.

272 Large-N Dualities

Transitions in Geometry

A transition between X and Y is a birational

contraction from a smooth Calabi–Yau X to a

singular variety Y

followed by a complex deforma-

tion to another smooth Calabi–Yau manifold Y:

The vanishing cycles of the complex deformation

are always Lagrangian submanifolds of Y. The

transition makes sense if dim (X) = dim (Y)  2 and

it is nontrivial if dim (X) = dim (Y)  3, when the

topology of X is different from the topology of Y.

The possible transitions among Calabi–Yau 3-folds

have been classified.

Conjecture 1 Let X and Y be Calabi–Yau mani-

folds related by a geometric transition: then IIA

open theory with U(U) branes compactified on

(Y, [L

) is dual to IIA closed theory compactified

on X (with fluxes).

As a consequence:

Conjecture 2 Let X and Y be Calabi–Yau mani-

folds related by a geometric transition: then

(Y, , g

, t

) = F

(X, q, g

) for a suitable identi-

fication of the parameters.

The results stated in the previous section can be

summarized in the the following statement, which is

the proof of the above con jecture for the special case

of a local conifold transition:

Theorem 1 Let X ﬃO

(1) O

(1) and

Y = T



with U(N) branes wrapped on L = S

Then X and Y are related by a conifold transition

and log F

) = F

(Y, ) = F

(X, q), with the

identification

 ¼

2N

k þ N

¼ q; g

2

k þ N

This matching of the free energies is supporting

evidence for the large-N conjecture. At this moment,

we still do not know if Conjectures 1 and 2 hold for

more general transitions.

A Few Comments on Knots and Links

Later, Ooguri and Vafa extended the conjecture to

the observables, that is, by adding knots and links in

; the guiding principle is that a knot (or link) CS

should determine a noncompact Lagrangian sub-

manifold L

 X; it is conjectured that the knot

(and link) invariants, expressed as expectation

values, should determine and be determined by the

enumerative invariants of morphisms of bounded

Riemann surfaces, with boundaries mapped onto

. We refer to these invariants as open Gromov–

Witten invariants. While both statements have been

verified with mathematical techniques only when C

is the unknot, there is much supporting evidence for

the conjecture in general. We will not describe these

aspects here but only make a few remarks.

The expectation values of a knot C are computed

by taking first the trace of a holonomy matrix of a

U(N) connection A along C and then integrating over

all connections (modulo gauge equivalence). As for

the case of the Chern–Simons free energy, the

definition of expectation values has been worked

out both in the realm of physics and of mathematics.

The expectation values are knot and link invariants,

and depend on a choice of the framing of the knot (or

link). The open Gromov–Witten invariants have not

yet been constructed, as we shall discuss in the

following section; however, starting with the work of

Katz and Liu, Li and Song open invariants have been

successfully calculated in the presence of a torus

action. The resulting invariants do depend on the

choice of the torus action, which has been shown to

match the choice of the framing of the knot (or link).

Further Reading

Clemens CH (1983) Double solids. Advances in Mathematics 47:

107–230.

Diaconescu D-E, Florea B (2003) Large N duality of compact

Calabi–Yau three-folds, hep-th/0302076.

Dijkgraaf R and Vafa C (2002) Matrix models, topological strings

and supersymmetric gauge theories. Nuclear Physics B 644(3):

3–20 (hep-th/0206255).

Faber C and Pandharipande R (2000) Hodge integrals and

Gromov–Witten theory. Inventiones Mathematicae 139:

173–199 (math.AG/9810173).

Freed DS (1995) Classical Chern–Simons theory, Part 1. Advances

in Mathematics 113: 237–303.

Gopakumar R and Vafa C (1999) On the gauge theory/geometry

correspondence. Advances in Theoretical and Mathematical

Physics 3: 1415–1443.

Katz S and Liu CCM (2002) Enumerative geometry of stable

maps with Lagrangian boundary conditions and multiple

covers of the disk. Advances in Theoretical and Mathematical

Physics 5: 1–49.

Li J and Song YS (2001) Open string instantons and relative

stable morphisms, hep-th/0103100.

Marin˜ o M (2004) Chern–Simons theory and topological strings,

hep-th/0406005.

Ooguri HH and Vafa C (2000) Knot invariants and topological

strings. Nuclear Physics B 577: 419–438.

Periwal V (1993) Topological closed-string interpretation of

Chern–Simons theory. Physical Review Letters 71:

1295–1298.

’t Hooft G (1974) A planar diagram theory for strong interac-

tions. Nuclear Physics B 72: 461–473.

Witten E (1989) Quantum field theory and the Jones polynomial.

Communications in Mathematical Physics 121: 351–399.

Witten E (1995) Chern–Simons gauge theory as a string

theory. In: The Floer Memorial Volume,Progressin

Mathematics, vol. 133, pp. 637–678. Basel: B irkha¨user

(hep-th/9207094).

274 Large-N Dualities

Lattice Gauge Theory

A Di Giacomo, Universita

di Pisa, Pisa, Italy

Introduction

As a prototype of lat tice gauge theory, quantum

chromodynamics (QCD) will be considered in this

article. All statements about QCD can easily be

extended to other theories, with different gauge

group and different content of particles.

QCD is a gauge theory with gauge group SU(3)

(color group), coupled to spin-1/2 particles (quarks)

belonging to the fundamental representation of the

color group. There exist in Nature six different

species (flavors) of quarks, with masses ranging

from m

 5 MeV to m

top

 180 GeV: the values of

these masses are determined by other interactions

and can be treated as input parameters of the theory

as well as the number of quark flavors. In standard

notation, the Lagrangian reads

L ¼

trðG



Þþ



ði6D  m

½1

The sum runs over the six quark flavors f.



= @





 @



þ ig[A



, A



] is the field strength

tensor, A



the (gluon) gauge field,

(a = 1,...,8) are the eight generators of the

gauge group in the fundamental representation,

normalized as tr(T

) = (1=2)

is a color

triplet of fields. Under a gauge transformation U(x),

ðxÞ!

ðxÞ¼UðxÞ

ðxÞ½2



ðxÞ!A



ðxÞ

¼ UðxÞA



ðxÞþiUð xÞ@



ðxÞ½3



is the covariant derivative of



¼ð@



 igA



½4

and transforms like

by construction.

L is invariant under the gauge transformation

equations [2] and [3]. As a consequence of gauge

invariance, the theory has one single coupling

constant g.

To make connection with the observations, one

has to solve the theory, that is, one has to construct

a H ilbert space on which the fields act as operators

obeying the equations of motion and the canonical

commutation relations. In textbook field theory,

this is done by splitting the Lagrangian L into two

parts:

L ¼ L

þ L

½5

with L

the part of L which is bilinear in the fields

and L

the rest. L

can be solved exactly since it

describes free particles and the corresponding

equations of motion are linear. The resulting Hilbert

space is the Fock space of free particles. L

is treated

as a perturbation producing scattering between the

fundamental particles. This approach works well

in quantum electrodynamics, where the observed

particles (electrons and photons) coincide with the

excitations of the fundamental fields of the

Lagrangian.

In QCD, the fundamental excitations (the quarks

and the gluons) are observed as particles neither in

Nature nor as a product of high-energy collisions

between elementary particles. This feature is known

as confinement of color. The conjecture is that

excitations with nontrivial color are forbi dden to

propagate as free particles. However, if hadrons are

probed at short distances by photons or by leptons,

everything works as if they were composite states of

quarks. The accepted explanation relies on asymp-

totic freedom: the effective coupling constant

becomes small at short distances (high momentum

transfers) and the constituents behave as free

particles.

At large distances, the fundamental excitations are

not observed, the interaction is strong and the

perturbative picture describing scattering between

quarks and gluons is not adequate for the real

world.

An alternative quantization procedure is needed

which does not rely on perturbation theory. A

formally exact quantization procedure is the Feynman

path integral. The solution of the theory is given in

terms of a functional integral Z[J], which generates

the correlators of the fields in the ground state

(vacuum). Indicating symbolically the Lagrangian

coordinates, namely the fields, by a single symbol ,

one has

Z½J ¼

dðxÞexp  S ½

JðxÞðxÞdx



½6

The connected Euclidean vacuum correlators are

given in terms of functional derivati ves of Z[J]

< 0jTððx

Þðx

Þðx

ÞÞj0 >

conn

Z½0



Z½J 

Jðx

ÞJðx

ÞJðx



JðxÞ¼0

½7

Lattice Gauge Theory 275

‘‘Euclidean’’ means that they are analytic con-

tinuations to imaginary times. Going to Euclidean

system is necessary to isolate the vacuum state. The

amplitudes can be analytically continued back to

Minkowski space. The Hilbert space and all the

physical observables can be constructed in terms of

the corre lators, a property known as reconstruction

theorem. Formally (i.e., assuming that everything

makes sense only if the functional integral exists),

< 0jTððx

Þðx

Þðx

ÞÞj0 >

conn

dðxÞ

 expðS½Þðx

Þðx

Þðx

Þ½8

The continuation to imaginary time changes sign to

the kinetic energy, and Z formally becomes the partition

function of a four-dimensionalstatisticalmodelwith

Hamiltonian S

[], a general fact in Feynman integrals.

By definition of functional integral, Z is defined

by discretizing a finite volume V of spacetime to a

finite set of points and then sending their number to

infinity, making a set dense in V. If the limit exists, a

is obtained. The volume V is then sent to infinity,

to cover the whole spacetime (thermodynamical

limit) and Z

eventually converges to Z. A rigorous

proof of the existence of these limits does not exist

for QCD, but there are qualitative arguments that

this is the case, which will be presented below.

In the lattice formulation of field theory, a regular

lattice, usually cubic, is taken as a discretization of

spacetime.

From the very definition of Feynman integral, it

follows that the formulation of field theory on the

lattice is nothing but an approximation to the limit

which defines Z. It will provide a good approxima-

tion if the lattice spacing is small enough with

respect to the physical lengths involved and if the

lattice is large compared to them.

Perturbation theory amounts to split the action

into a bilinear term S

and an interaction term S

containing the higher powers of the fields. The Z

integral is then computed by exp anding the weight

in a power series of S

dðxÞexpðS

 S

dðxÞexp ðS

ðS

½9

The Feynman integral thus becomes Gaussian, can

be computed, and gives the usual perturbative

expansion. The two limits (integral and series

expansion) do not commute in general. For QCD,

there are indeed arguments that the renormalized

perturbative expansion does not converg e and is

plagued by singularities known as renormalons.

Wilson’s Formulation

For field theories of scalar particles, the lattice

discretization is performed by assigning a value of

the field to each site of the lattice. The Wilson

formulation for gauge theories is not made in terms of

the fields A



, which are defined in the Lie algebra of

the gauge group, but in terms of parallel transports,

which are elements of the group itself. The building

blocks are parallel transports along links parallel to

spacetime axes connecting neighboring sites



ðxÞ

 P exp



xþ







 expðigaA



ðxÞÞ ½10

where

 indicates the vector of length a in the 

direction and P the ordered product. The last

approximate equality is valid in the limit of small

lattice spacing a. g is the coupling constant.

Under a gauge transformation V(x);



ðxÞ!VðxÞU



ðxÞV

ðx þ

Þ½11

It follows from eqn [11] that the parallel transport

along a closed path is gauge invariant. The density

of action can be written in terms of the parallel

transport along the elementary square of links in the

hyperplanes  



, known as plaquette:



¼ tr½U



ðxÞU



ðx þ

ÞU



ðx þ

ÞU



ðxÞ ½12

By expanding in powers of a, one easily finds



¼ N



tr½G



þOða

Þ½13

with N

the number of colors, 3 for QCD. The

lattice action can be defined as

S ¼

x

 1 







½14

with  = 2N

, and tends to the continuum action

as a ! 0, O(a

). An infinite number of higher-orde r

terms in a exist, which come from the expansion of

the links, but they are expected to be irrelevant in

the continuum limit a ! 0.

The measure of the Feynman integral is assumed

to be the Haar measure of the gauge group for each

link, which again can be shown to tend to the

continuum measure in the continuum limit.

276 Lattice Gauge Theory