Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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SFT Diagrams and Minimal Area Metrics

Imposing the Siegel gauge condition b

=0, one

finds the gauge-fixed action

¼



hjc

jiþ

hj  i

þhjb

ji



½32

where  is a Lagrangian mul tiplier. The propagator

reads

¼ b

dT e

TL

½33

Since L

is the first-quantized open-string Hamilto-

nian, e

TL

is the operator that evolves the open-

string wave functions [X()] by Euclidean world

sheet time T. It can be visualized as a flat

rectangular strip of ‘‘horizontal’’ width  and

‘‘vertical’’ height T. Each pro pagator comes with

an antighost insertion



bðÞ½34

integrated on a horizontal trajectory.

The only elementary interaction vertex is the mid-

point three-string overlap, visualized in Figure 3.We

are instructed to draw all possible diagrams with

given external legs (represented as semi-infinite

strips), and to integrate over all Schwinger para-

meters T

2 [0, 1) associated with the internal

propagators. The claim is that this prescription

reproduce precisely the first-quantized result [1].

This follows if we can show that (1) the OSFT

Feynman rules give a unique cover of the moduli

space of open Riemann surfaces; (2) the integration

measure agrees with the measure [d



]in[1]. The

latter property holds because the antighost insertion

[34] is precisely the one prescribed by the Polyakov

formalism for integrating over the moduli T

.To

show point (1), we introduce the concept of

minimal-area metrics, which has proved very

fruitful. (Here and below, our discussion of

minimal-area metrics will summarize ideas devel-

oped mainly by Zwiebach.) Quite generally, the

Feynman rules of an SFT provide us with a cell

decomposition of the appropriate moduli space of

Riemann surfaces, a way to construct surfaces in

terms of vertices and propagators. Given a Riemann

surface (for fixed values of its complex moduli), the

SFT must associate with it one and only one string

diagram. The diagram has more structure than the

Riemann surface: it defines a metric on it. In all

known covariant SFTs, this is the met ric of minimal

area obeying suitable length conditions. Consider

the following:

Minimal-area problem for open SFT Let R

be a

Riemann surface with at least one boundary

component and possibly punctures on the boundary.

Find the (conformal) metric of minimal area on R

such that all nontrivial Jordan open curves have

length greater than or equal to . (A curve is said to

be nontrivial if it cannot be continuously shrunk to a

point without crossing a puncture.)

An OSFT diagram (for fixed values of its T

defines a Riemann surface R

endowed with a

metric solving this minimal-area problem. This is the

metric implicit in its picture: fla t everywhere except

at the conical singularities of defect angle (n  2)

when n propagators meet symmetrically. (For n = 3,

these are the elementary cubic vertices; for n > 3,

they are effective vertices, obtained when propaga-

tors joining cubic vertices collapse to zero length.) It

is not difficult to see both that the length conditions

are obeyed, and that the metric cannot be made

smaller without violating a length condition. Con-

versely, any surface R

endowed with a minimal-

area metric, corresponds to an OSFT diagr am. The

idea is that the minimal-are a metric must have open

geodesics (‘‘horizontal trajectories’’) of length 

foliating the surface. The geodesics intersect on a

set of measure zero – the ‘‘critical graph’’ where the

propagators are glued. Bands of open geodesics of

infinite height are the external legs of the diagram,

while bands of finite height are the internal

propagators.

The single cover of moduli space is then ensured

by an existence and uniqueness theorem for metrics

solving the minimal-area problem for OSFT. These

metrics are seen to arise from Jenkins–Strebel

quadratic differentials. Existence shows that the

Feynman rules of OSFT generate each Riemann

surface R

at least once. Uniqueness shows that

there is no overcounting: since different diagrams

correspond to different metrics (by inspection of

their picture), no Riemann surface can be generated

twice.

Figure 3 The cubic vertex represented as the mid-point gluing

of three strips.

String Field Theory 99

Closed Strings in OSFT

As is familiar, the open-string S-ma trix con tains

poles due to the exchange of on-shell open and

closed strings. The closed-string poles are present in

nonplanar loop amplitudes. We have seen that

OSFT reproduces the standard S-matrix. Factoriza-

tion over the open-string poles is manifest, it

corresponds to propagator lengths T

going to

infinity. Su rprisingly, the closed-string poles are

also correctly reproduced, despite the fact that

OSFT treats only the open strings as fundamental

dynamical variables. In some sense, closed strings

must be considered as derived objects in OSFT.

Factorizing the amplitudes over the closed-string

poles, one finds that on-shell closed-string states can

be represented, at least formally, as certain singular

open-string fields with G = þ2, closely related to the

(formal) identity string field. The picture is that of a

folded open string, whose left and right halves

precisely overlap, with an extra closed-string vertex

operator inserted at the mid-point. The correspond-

ing open/closed vertex is given by

h

phys

ji

h

phys

ð0ÞI  ð0Þi

I¼

1 þ iz

1  iz



½35

and describes the coupling to the open-string field of

a nondynamical, on-shell closed string j

phys

i.Itis

possible to add this open/closed vertex to the OSFT

action. Remarkably, the resulting Feynman rules

give a single cover of the moduli space of Riemann

surfaces with at least one boundary, with open and

closed punctures. This is shown using the same

minimal-area problem as above, but now allowing

for surfaces with closed punctures as well.

We should finally mention that the structure of

OSFT emerges frequently in topological string

theory, in contexts where open/closed duality plays

a central role. Two examples are the interpretation

of Chern–Simons theory as the OSFT for the

A-model on the conifold, and the intepretation of

the Kontsevich matrix integral for topological

gravity as the OSFT on FZZT branes in (2, 1)

minimal string theory.

Closed Bosonic SFT

The generalization to covariant closed SFT is

nontrivial, essentially because the requisite closed-

string decomposition of moduli space is much more

complicated.

The free theory parallels the open case, with a

minor complication in the treatment of the CFT zero

modes. The closed-string field is taken to live in a

subspace of the matter þ ghost state space, j i2

CFT

, where the tilde means that we impose the

subsidiary conditions



ji¼L



ji¼0; b



 b





;



 L





½36

In the classical theory, the string field carries ghost

number G = þ2, since it is the off-shell extension of

the familiar closed-string physical states, and the

quadratic action reads

S h; Q

i½37

Here Q

is the usual closed BRST operator. The inner

product h, i is defined in terms of the BPZ inner

product, with an extra insertion of c



 c





hA; BihAjc



jBi½38

In [37] G

top

= þ6, as it should be. Without the

extra ghost insertion and the subsidiary conditions

[36] it would not be possible to write a quadratic

action. The linearized equations of motion and

gauge invariance,

ji¼0; jijiþQ

ji; ji2

ð1Þ

CFT

½39

give the expected cohomological problem. The fact

that the cohomology is computed in the semirelative

complex, b



ji= b



ji= 0, well known from the

operator formalism of the first-quantized theory, is

recovered naturally in the second-quantized treatment.

The interacting action is constructed iteratively,

by demanding that the resulting Feynman rules give

a (unique) cover of moduli space. This requires the

introduction of infinitely many elementary string

vertices V

g, n

, where n is the number of closed-string

punctures and g the genus. This decomposition of

moduli space is more intricate than the decomposi-

tion that arises in OSFT, but is in fact analogous to

it, when characterized in terms of the following.

Minimal-area problem for closed SFT Let R

be a

closed Riemann surface, possibly with punctures.

Find the (conformal) metric of minimal area on R

such that all nontrivial Jordan closed curves have

length greater than or equal to 2.

The minimal-area metric induces a foliation of

by closed geodesics of length 2. In the classical

theory (g = 0), the minimal-area metrics arise from

Jenkins–Strebel quadratic differentials (as in the open

case), and geodesics intersect on a measure-zero set.

For g > 0, however, there can be foliation bands of

geodesics that cross. By staring at the foliation, we can

break up the surface into vertices and propagators. In

correspondence with each puncture, there is a band of

100 String Field Theory

infinite height, a flat semi-infinite cylinder of circum-

ference 2, which we identify as an external leg of the

diagram. We mark a closed geodesic on each semi-

infinite cylinder, at a distance  from its boundary.

Bands of finite height (internal bands not associated to

punctures) correspond to propagators if their height is

greater than 2, otherwise they are considered part of

an elementary vertex. Along any internal cylinder of

height greater than 2, we mark two closed geodesics,

at a distance  from the boundary of the cylinder. If we

now cut open all the marked curves, the surface

decomposes into a number of semi-infinite cylinders

(external legs), finite cylinders (internal propagators)

and surfaces with boundaries (elementary interac-

tions). Each elementary interaction of genus g and

with n boundaries is an element of V

g, n

.Acrucialpoint

of this construction is that we took care of leaving a

‘‘stub’’ of length  attached to each boundary. Stubs

ensure that sewing of surfaces preserves the length

condition on the metric (no closed curve shorter

than 2).

These geometric data can be translated into an

iterative algebraic construction of the full quantum

action S[]. The V

g, n

satisfy geometric recursion

relations whose algebraic counterpart is the quan-

tum BV master equation for S[]. Remarkably, the

singularities of the  operator encountered in OSFT

are absent here, precisely because of the prese nce of

the stubs. We refer to Zwiebach (1993) for a

complete discussion of closed SFT.

Open/Closed SFT

There is also a covariant SFT that includes both open

and closed strings as fundamental variables. The

Feynman rules arise from the following problem.

Minimal-area problem for open/closed SFT Let

be a Riemann surface, with or without

boundaries, possibly with open and closed punctu-

res. Find the (conformal) metric of minimal area on

such that all nontrivial Jordan open curves have

length greater than or equal to l

= , and all

nontrivial Jordan closed curves have length greater

than or equal to l

= 2.

The surface R

is decomposed in terms of

elementary vertices V

g, n

b, m

(of genus g, b boundary

components, n closed-string punctures and m open-

string punctures) joined by open and closed propa-

gators. Degenerations of the surface correspond

always to propagators becoming of infinite lengt h –

factorization is manifest both in the open and in the

closed channel.

The S FT descr ibed in the section ‘‘Clos ed strings

in OSFT’’ (Witten OSFT au gmented with the single

open/closed vertex [35]) corresponds to taking l

= 

and l

= 0. Varying l

2 [0, 2], we find a whole

family of interpolating SFTs. This construction

clarifies the special status of the Witten theory:

moduli space is covered by a single cubic open

overlap vertex, with no need to introduce dynamical

closed strings, but at the price of a somewhat

singular formulation.

Classical Solutions in Open SFT

In the present formulation of SFT, a background (a

classical solution of string theory) must be chosen from

the outset. The very definition of the string field

requires to specify a (B)CFT

. Intuitively, the string

field lives in the ‘‘tangent’’ to the ‘‘theory space’’ at a

specific point – where ‘‘theory space’’ is some notion of

a ‘‘space of 2D (boundary) quantum field theories,’’

not necessarily conformal. In the early 1990s indepen-

dence from the choice of background was demon-

strated for infinitesimal deformations: the SFT actions

written using neighboring (B)CFTs are indeed related

by a field redefinition. In recent years, it has become

apparent that at least the open-string field reaches out

to open-string backgrounds a finite distance away –

possibly covering the whole of theory space. (Classical

solutions of closed SFT are beginning to be investi-

gated at the time of this writing (2005)).

The OSFT action written using BCFT

data is just

the full world volume action of the D-brane with

BCFT

boundary conditions. Which classical solu-

tions should we expect in this OSFT? In the bosonic

string, Dp branes carry no conserved charge and are

unstable. This instability is reflected in the presence

of a mode with m

= 1=

, the open-string

tachyon T(x



),  = 0, ..., p. From this physical pic-

ture, Sen argued that:

1. the tachyon potential, obtained by eliminating

the higher modes of the string field by their

equations of motion, must admit a local mini-

mum corresponding to the vacuum with no

D-brane at all (henceforth, the tachyon vacuum,

T(x



) = T

);

2. the value of the potential at T

(measured with

respect to the BCFT

point T = 0) must be

exactly equal to minus the tension of the brane

with BCFT

boundary conditions;

3. there must be no perturbative open-string excita-

tions around the tachyon vacuum; and

4. there must be space-dependent ‘‘lump’’ solutions

corresponding to lower- dimensional branes. For

example, a lump localized along one world

volume direction, say x

, such that T(x

) !T

as x

!1, is identified with a D(p  1) brane.

String Field Theory 101

Sen’s conjectures have all been verified in OSFT.

(See Sen (2004) and Taylor and Zwiebach (2003)

for reviews). The deceptively simple-looking equa-

tions of motion (in Siegel gauge)

jiþb

ðjijiÞ ¼ 0 ½40

are really an infinite system of coupled equations,

and no analytic solutions are known. Turning on a

vacuum expectation value (VEV) for the tachyon

drives into condensation an infinite tower of modes.

Fortunately, the approximation technique of ‘‘level

truncation’’ is surprisingly effective. The string field

is restricted to modes with an L

eigenvalue smaller

than a prescribed maximal level L. For any finite L,

the truncated OSFT contains a finite number of

fields and numerical computations are possible.

Numerical results for various classical solutions

converge quite rapidly as the level L is increased.

The most important solution is the string field jT i

that corresponds to the tachyon vacuum. A remark-

able feature of jT i is universality: it can be written

as a linear combi nation of modes obtained by acting

on the tachyon c

j0i with ghost oscillators and

matter Virasoro operators,

jT i ¼ T

j0iþuL

2

j0iþvc

1

j0iþ

This implies that the properties of jT i are indepen-

dent of any detail of BCFT

, since all computations

involving jT i can be reduced to purely combinator-

ial manipulations involving the ghosts and the

Virasoro algebra. The numerical results strongly

confirm Sen’s conjectures, and indicate that the

tachyon vacuum is located at a non-singular point in

configuration space. Numerical solutions describing

lower-dimensional branes and exactly marginal

deformations are also available. For example, the

full family of solutions interpolating between a

D1 and a D0 brane at the self-dual radius has

been found. There is increasing evidence that the

open-string field provides a faithful map of the

open-string landscape.

Vacuum SFT: D-branes as Projectors

In the absence of a closed-form expression for jT i,

we are led to guesswork. When expanded around

jT i, the OSFT is still cubic, only with a different

kinetic term Q,

S ¼

hjQjiþ

hj  i



½41

The operator Q must obey all the formal properties

[17], must be universal (constructed from gh osts and

matter Virasoro operators), and must have trivial

cohomology at G = þ1. Another constraint comes

from requiring that [41] admits classical solutions in

Siegel gauge. The choice

Q¼

ðcðiÞ



cðiÞÞ

¼ c

ðc

þ c

2

Þþðc

þ c

4

Þ ½42

satisfies all these requirements. The conjecture

(Rastelli et al. 2001) is that, by a field redefinition,

the kinetic term around the tachyon vacuum can be

cast into this form. This ‘‘purely ghost’’ Q is

somewhat singular (it acts at the delicate string

mid-point), an d presumably should be regarded as

the leading term of a more complicated operator

that includes matter pieces as well. The normal-

ization constant 

is formally infinite. Nevertheless,

a regulator (e.g., level truncation) can be introduced,

and physical observables are finite and independent

of the regulator. The vacuum SFT ([41]–[42] )

appears to capture the correct physics, at least at

the classical leve l. Taking a matter/ghost factorized

ansatz

j

ij

i½43

and assuming that the ghost part is universal for all

D-branes solutions, the equations of motion reduce

to following equations for the matter part:

j

ij

i¼j

i½44

A solution j

i can be regarded as a projector

acting in ‘‘half-string space.’’ Recall that the

-product looks formally like a matrix multiplica-

tion [19]: the matrices are the string fields, whose

‘‘indices’’ run over the half-string curves. These

projector equations have been exactly solved by

many different techniques (see Rastelli (2004) for a

review). In particular, there is a general BCFT

construction that shows that one can obtain solu-

tions corresponding to any D-b rane configuration,

including multiple branes – the rank of the projector

is the number of branes. A rank-one projector

corresponds to an open-string functional which is

left/right split, [X( )] = F

). There is

also clear analogy between these solutions and the

soliton solutions of noncommutative field theory.

The analogy can be made sharper using a formalism

that rewri tes the open-string -product as the tensor

product of infinitely many Moyal products. (See

Bars (2002) and references therein).

It is unclear whether or not multiple-brane

solutions (should) exist in the or iginal OSFT – they

are yet to be found in level truncation. Under-

standing this and other issues, like the precise role of

closed strings in the quantum theory seems to

require a precise characterization of the allowed

102 String Field Theory

space of open-string functionals. In principle, the

path integral over such functionals would define the

theory at the full nonperturbative level. This remains

a challenge for the future.

Note Added in Proof Very recently, M Schnabl,

building on previous work on star algebra projectors

and related surface states (Rastelli L (2004) and

references therein) was able to find the exact

solution for the universal tachyon condensate in

OSFT. This breakthrough is likely to lead to rapid

new developments in SFT.

See also: Boundary Conformal Field Theory; BRST

Quantization; Chern–Simons Models: Rigorous Results;

Fedosov Quantization; The Jones Polynomial; Large-N

and Topological Strings; Large-N Dualities;

Noncommutative Geometry from Strings;

Noncommutative Tori, Yang–Mills, and String Theory;

Operads; Superstring Theories; Topological Quantum

Field Theory: Overview; Two-Dimensional Conformal

Field Theory and Vertex Operator Algebras.

Further Reading

Bars I (2002) MSFT: Moyal Star Formulation of String Field Theory.

Proceedings of 3rd International Sakharov Conference on

Physics, Moscow, Russia, 24–29 Jun, arXiv:hep-th/0211238.

Berkovits N (2001) Review of open superstring field theory,

arXiv:hep-th/0105230.

Berkovits N, Okawa Y, and Zwiebach B (2004) WZW-like action

for heterotic string field theory. Journal of High Energy

Physics 0411: 038 (arXiv:hep-th/0409018).

Bordes J, Chan HM, Nellen L, and Tsou ST (1991) Half string

oscillator approach to string field theory. Nuclear Physics B

351: 441.

Erler TG and Gross DJ (2004) Locality, causality, and an initial

value formulation for open string field theory, arXiv:hep-th/

0406199.

Ohmori K (2001) A Review on Tachyon Condensation in Open

String Field Theories. Master’s thesis, University of Tokyo

(arXiv:hep-th/0102085).

Okawa Y (2002) Open string states and D-brane tension from

vacuum string field theory. Journal of High Energy Physics

0207: 003 (arXiv:hep-th/0204012).

Rastelli L (2004) Open string fields and D-branes. Fortschritte der

Physics. 52: 302.

Rastelli L, Sen A, and Zwiebach B Vacuum String Field Theory,

Proceedings of the Strings 2001 Conference, TIFR, Mumbai,

India, arXiv:hep-th/0106010.

Schnabl M (2005) Analytic solution for tachyon condensation in

open string theory. arXiv:hep-th/0511286.

Sen A (2004) Tachyon dynamics in open string theory, arXiv:hep-

th/0410103.

Shatashvili SL On Field Theory of Open Strings, Tachyon

Condensation and Closed Strings, Proceedings of the Strings

2001 Conference, TIFR, Mumbai, India, arXiv:hep-th/

0105076.

Siegel W (1988) Introduction To String Field Theory. Advanced

Series in Mathematical Physics 8: 1–244 (arXiv:hep-th/

0107094).

Taylor W and Zwiebach B (2001) D-branes, tachyons, and string

field theory. In: Gubser SS and Lykken JD (eds.) Boulder

2001, Strings, Branes and Extra Dimensions, TASI 2001

Lectures, pp. 641–759. Singapore: World Scientific.

Thorn CB (1989) String field theory. Physics Report 175: 1.

Witten E (1986) Noncommutative geometry and string field

theory. Nuclear Physics B 268: 253.

Zwiebach B (1993) Closed string field theory: quantum action

and the B-V master equation. Nuclear Physics B 390: 33

(arXiv:hep-th/9206084).

Zwiebach B (1998) Oriented open–closed string theory revisited.

Annals of Physics 267: 193 (arXiv:hep-th/9705241).

String Theory: Phenomenology

A M Uranga, Consejo Superior de Investigaciones

Cientificas, Madrid, Spain

String Theory and Compactification

The string theory provides a setup in which gauge

and gravitational interactions can be described in a

unified framework consistently at the quantum level.

As such, it provides a candidate theory in which to

describe the standard model of particle physics

(describing quarks and leptons and their strong and

electroweak interactions) and gravity within the

same quantum theory.

The string theory has a unique fundamental scale

, fixed by the string tension, often encoded in the

parameter 

of dimension (length)

2

. All other

scales are derived from this one and are background

dependent.

Most of the string theory phenomenological

model building has centered on the critical super-

strings, which are ten dimensional (10D) and

involve spacetime (as well as world-sheet) super-

symmetry. There are five such different 10D

theories: type IIA, type IIB, type I, and the E

 E

and SO(32) heterotic theories. The heterotic theories

include nonabelian gauge fields and charged fer-

mions in ten dimensions; hence, they constitute a

promising setup to embed the standard model. On

the other hand, the possibility of including D-branes

String Theory: Phenomenology 103

(which carry nonabelian gauge symmetries and

charged matter) in compactifications of type II

theories (and orientifolds thereof, like the type I

theory itself) makes the latter reasonable alternative

setups to embed the standard model as a brane

world. The different 10D theories (as well as the

11D M-theory) are related by diverse dualities, also

upon compactification. This suggests that they are

just different limits of a unique underlying theory.

For 4D models, this implies that the different classes

of constructions are ultimately related by dualities,

and that often a given model may be realized using

different string theory constructions as starting

points.

In order to recover 4D physics at low energies,

compactification of the theory is required. In

geometrical terms, the theory is required to propa-

gate on a spacetime with geometry M

 X

, where

is a 4D Minkowski space, and X

is a compact

manifold. This description is valid in the regime of a

large compactification volume, 

 1 (where R

is the overall scale of the compact manifold), where



string theory corrections are negligible. Other 4D

string models may be constructed using abstract

conformal field theories. They may often be

regarded as extrapolations of geometric compactifi-

cations to the regime of sizes comparable with the

string length, where string theory corrections are

relevant and the classical geometric picture does not

hold.

In the simplest situation of geometrical compacti-

fication, not including additional backgrounds

beyond the metric, the requirement of 4D spacetime

supersymmetry (useful for the stability of the model,

as well as of phenomenological interest) implies that

the space X

is endowed with an SU(3) holonomy

metric. Existence of such metrics is guaranteed for

Calabi–Yau spaces, namely Ka¨ hler manifolds with

vanishing first Chern class.

There are a very large number of 4D super-

symmetric string models that can be constructed

using different starting string theories and different

compactification manifolds. They lead to different

4D spectra, often including nonabelian gauge sym-

metries and charged chiral fermions (but only rarely

resembling the actual standard model). In addition,

for each given model, there exist, in general, a large

number of massless 4D scalars, known as moduli,

whose vacuum expectation values are not fixed.

They parametrize different choices of the compacti-

fication data in a given topological sector (e.g.,

Ka¨ hler and complex structure moduli of the internal

Calabi–Yau space). All physical parameters of the

4D theory vary continuously with the vacuum

expectation values of these scalars.

All such models are on equal footing from the

point of view of the theory. Hence, 4D string models

suffer from a large arbitrariness. Although the

breaking of supersymmetry clearly changes the

picture qualitatively (e.g., flat directions associated

to moduli are lifted by radiative corrections), it is

difficult to evaluate this impact.

In this situation, most of the research in string

theory phenomenology has centered on the study of

generic properties of certain classes of compactifica-

tions, with the potential to lead to realistic struc-

tures (such as N = 1 or no supersymmetry,

nonabelian gauge symmetries with replicated sets

of charged chiral fermions). Within each class,

explicit models (as close as possible to the standard

model) have also been constructed. Generic predic-

tions or expectations for phenomenology can be

obtained within each setup, but quantitative results,

even for explicit models, are always functions of

undetermined moduli vacuum expectation values.

Tractable mechanisms for moduli stabilization are

under active research, although only preliminary

results are available presently.

The better-studied classes of models are compac-

tifications of heterotic theories on Calabi–Yau

spaces, and compactifications of type II theories (or

orientifolds thereof) with D-branes. Other possibi-

lities include the heterotic M-theory, the M-theory

on G

holonomy varieties, the F-theory on Calabi–

Yau 4-folds, etc. As already mentioned, different

classes (or even explicit models) are often related by

string duality.

Heterotic String Phenomenology

A large class of phenomenologically interesting

string vacua, which has been explored in depth, is

provided by 4D compactifications of (any of the

two) perturbative heterotic string theories. Compac-

tification on large volume manifolds can be

described in the supergravity approximation. As

described by Candelas, Horowitz, Strominger, and

Witten, the requirement of 4D N = 1 supersymmetry

requires the internal manifold to be of SU(3)

holonomy, a condition which is satisfied by

Calabi–Yau manifolds. In the presence of a curva-

ture, the Bianchi identity for the Kalb–Ramond

2-form B is modified, so that, in general, it reads

dH ¼ tr R



tr F

½1

where H is the field strength 3-form, R is the Ricci

2-form, and F is the field strength, in the adjoint

representation, of the 10D gauge fields. Regarding

the above equation in cohomology leads to a

104 String Theory: Phenomenology

consistency condition, forcing the background gauge

bundle V to be topologically nontrivial, with

ðVÞ¼c

ðTX

Þ½2

where c

denotes the second Chern class, and TX

the compactification tangent space.

The condition of supersymmetry implies that the

gauge fields must be solutions of the Donaldson–

Uhlenbeck–Yau equations. Existence of such a solu-

tion is guaranteed for holomorphic and stable gauge

bundles. The simplest solution to these conditions is

the so-called standard embedding, where the gauge

connection is locally identical to the spin connection,

but more general solutions exist and have been

characterized for particular classes of Calabi–Yau

manifolds (e.g., when they are elliptically fibered).

The gauge background bundle V,withstructure

group H, breaks the 10D gauge symmetry G to its

commutant subgroup G

. The latter corresponds to

the 4D gauge symmetry. Moreover, the background

bundle modifies the Kaluza–Klein reduction of the

10D charged fermions, leading to a nonzero number

of replicated 4D chiral fermions. Decomposing the

adjoint representation of G (in which 10D fermions

transform) with respect to G

 H,

Adj G ¼



ðR

; R

H;i

Þ½3

the net number of 4D chiral fermions in the

representation R

is given by the index of the

Dirac operator coupled to V in the representation

H, i

. Condition [1] implies proper cancellation of

chiral anomalies in the resulting theory. A simple

and well-studied class is provided by standard

embedding compactifications of the E

 E

hetero-

tic string theory, whose unbroken 4D gauge group is

 E

. The number of families (i.e., chiral multi-

plets in the representation 27 of E

) and conjugate

families (in the

27) are given by the Hodge numbers

¼ h

1;1

ðX

Þ; n

¼ h

2;1

ðX

Þ½4

More specifically, the harmonic representatives in

each cohomology class represent the internal profile

of the corresponding 4D fields. The net number of

families is thus determined by the Euler character-

istic (X

)

fam

¼jh

1;1

 h

2;1

j¼

jðX

Þj ½5

Recently, much progress in heterotic model building

has been achieved in nonstandard embedding com-

pactifications by the detailed construction of holo-

morphic stable bundles and the computation of the

diverse indexes. In particular, explicit models with

just the minimal supersymmetric standard model

spectrum have been constructed.

The above geometric approach has several limita-

tions. On the technical side, the construction of

explicit holomorphic and stable gauge bundles is

nontrivial from the mathematical viewpoint. On the

more fundamental side, it allows one to explore only

the large volume limit of heterotic compactifications.

Further insight into the latter aspect can be

obtained via constructions based on exactly solvable

conformal field theories (CFTs), which describe the

world-sheet string dynamics in compactifications,

including all 

corrections, and, therefore, allowing

one to enter the small volume regime. The simplest

such compactifications are provided by toroidal

orbifolds, which describe string propagation in

quotients of toroidal compactifications by a discrete

group . From the world-sheet viewpoint, they are

described by 2D free CFT, but which include sectors

of closed strings with boundary conditions twisted

by elements of . The resulting 4D theory contains

chiral fermions, arising from the untwisted and

twisted sectors. In the former, the nonchiral spec-

trum of toroidal compactification suffers a projec-

tion onto the -invariant states and leads to

chirality. Twisted sectors are localized at the fixed

points of the orbifold action, where the local

supersymmetry is reduced, leading naturally to

chiral fermions.

Many of these models can be regarded as limits of

compactifications on Calabi–Yau spaces in the limit

in which they become locally flat and develop

conical singularities (and similarly, their gauge

bundles become locally flat and with curvature

localized near the singular points). Indeed, flat

directions involving moduli fields in the twisted

sector often exist, which correspond to geometric

blow-ups of the singular point that resolve the

conical singularities to yield a smooth Calabi–Yau.

The theories remain simple and solvable for any

value of the untwisted moduli (namely moduli of the

underlying toroidal compactification). This allows

the discussion of their low-energy effective action

including the explicit dependence on the untwisted

moduli, while only partial results for the dependence

on twisted moduli are known.

Other approaches, such as free fermion construc-

tions or Gepner models, also provide exact descrip-

tions of compactifications, although only at a point

of the moduli space, deep inside the small volume

regime.

Exact CFT constructions provide a small volume

description of Calabi–Yau compactifications, at

least for particular models. Moreover, their consis-

tency conditions (modular invariance of the parti-

tion function) provide a stringy version of the large

volume geometric condition implied by eqn [2]. The

String Theory: Phenomenology 105

constructions also show the existence of full-fledged

string theory constructions with properties similar to

geometric compactifications, but incorporating all 

corrections.

Within the general class of perturbative heterotic

string models, a certain number of phenomenologi-

cally interesting statements are quite generic.



The 4D Planck scale M

and gauge couplings g

(at the string scale) are related to the fundamental

string scale by

¼ M

½6

This implies that the string scale is close to the 4D

Planck scale. In this situation, supersymmetry can

stabilize the electroweak scale against radiative

corrections.



4D heterotic models contain certain U(1) symme-

tries, whose gauge bosons actually get Stuckelberg

masses due to B ^ F couplings to components of

the 2-form. Such U(1)’s would correspond to

global symmetries, but are violated at tree level by



nonperturbative effects, namely world-sheet

instantons. Hence, no continuous global symme-

tries exist, even perturbatively, in these models.

Proton decay might, however, be avoided by

discrete global symmetries. In any event, even

without such symmetries, the large fundamental

scale suppresses the processes mediating proton

decay. Thus, the proton lifetime is naturally larger

than present experimental bounds.



Gauge coupling constants for the different gauge

factors in the standard model unify at the string

scale. This agrees with extrapolation from their

electroweak values, assuming the minimal super-

symmetric standard model content between the

electroweak and string scale, up to a mismatch of

scales (by a factor of 20). The latter may be

addressed in diverse ways, such as threshold

corrections, intermediate scales, or in the heterotic

M-theory.



Yukawa couplings are, in principle, computable.

Explicit computations have been carried out in

standard embedding geometric compactifications

(where they amount to the overlap integral of the

internal profiles of the 4D fields, namely a

topological intersection number), and in orbifold

models. They are in general moduli dependent, so

their quantitative analysis is involved. Qualita-

tively, however, interesting patterns, such as

hierarchical structures, are possible, for example,

in specific orbifold models.

Heterotic models have been studied beyond the

perturbative regime. For instance, the construction

of compactifications including nonperturbative

objects, namely 5-branes, has been pursued; so has

been the strong coupling limit of the E

 E

heterotic, described by compactifications of the

M-theory on an interval (the so-called heterotic

M-theory or Horava–Witten theory). The strong

coupling phenomena of the SO(32) heterotic theory

can be addressed using dual type I (or other type II

orientifold) constructions.

D-Brane Phenomenology

A different setup for realistic string theory compac-

tifications, within the so-called brane-world con-

structions, is provided by compactifications of type II

string theories containing D-branes, or quotients

thereof. A particularly relevant class of quotients

involves quotienting out by world-sheet parity,

accompanied by some Z

geometric action. The

resulting theories are denoted type II orientifolds, and

contain orientifold planes, subspaces fixed under the

geometric action, corresponding to regions where the

orientation of a string can flip. Type II compactifica-

tions with D-branes filling the noncompact dimen-

sions must satisfy a set of consistency conditions,

known as RR tadpole cancellation. This is the

condition that, in the compact space, the charge of

D-branes and orientifold planes under the different

RR forms must cancel. For the Z-valued charges, the

conditions read

þ Q

¼ 0 ½7

where N

denotes the multiplicity of D-branes with

charge vector and Q

under the RR fields, Q

is the

charge vector of the orientifold planes. Additional

discrete conditions may be present if the relevant

K-theory group (classifying D-brane charges in the

corresponding background) contains torsion pieces.

The most familiar example of these constructions

is provided by the type I string theory, which is an

orientifold quotient of the type IIB theory by world-

sheet parity (with no geometric action). The model

can be regarded as containing one orientifold

9-plane and 32 D9 branes (all filling out 10D

spacetime), such that their RR charges with respect

to the (nondynamical) RR 10-form cancel.

Supersymmetric geometric compactifications of

type II theories and orientifolds must correspond to

compactification on Calabi–Yau spaces in order to

have a preserved spinor. Models with D-branes

filling the noncompact dimensions may be broadly

classified into two classes: type IIB compactifications

with D(3 þ 2p)-branes, wrapped on holomorphic

2p-cycles, and carrying holomorphic and stable

106 String Theory: Phenomenology

world-volume gauge bundles, and type IIA compac-

tifications with D6 branes wrapped on special

Lagrangian 3-cycles (in general, models with D4

and D8 branes are not allowed since Calabi–Yau

spaces do not have nontrivial 1- or 5-cycles on

which to wrap the branes). This classification is a

large volume realization of the general classification

of supersymmetric configurations of D-branes into

two classes, denoted A and B.

Intersecting Brane Worlds

Type IIA compactifications with A-branes corre-

spond to compactifications of type IIA theory (or

orientifolds thereof) with D6 branes wrapped on

3-cycles of the internal Calabi–Yau space. In these

models, each stack of N D6 branes generically leads

to a U(N) gauge factor. Chirality arises from open

strings stretched between pairs of branes at the

corresponding intersections. The chiral fermions

from an open string stretched between branes a

and b transform in the bifundamental representation

, &

) of the gauge factors U(N

)  U(N

) of the

intersecting D6 brane stacks. In general, two

3-cycles in a 6D manifold intersect at points of the

internal space. Hence, such fermions arise in several

families, whose (net) number is given by the (net)

number of intersections of the corresponding

3-cycles 

, 

, namely the topological invariant

intersection number of their homology classes

¼½

½

½8

Simple modifications of the above rules arise in

some sectors in the presence of orientifold planes

(e.g., the reduction of the gauge symmetry from

unitary to orthogonal or symplectic factors for

branes on top of orientifold planes).

The RR tadpole cancellation conditions specify

that the total homological charge carried by the D6

branes (and the orientifold 6-planes) cancel. They

imply automatic cancellation of cubic nonabelian

anomalies, and the cancellation of mixed U(1)

anomalies by a Green–Schwarz mechanism mediated

by 4D scalars from the RR closed-string sector.

Explicit models with SM spectrum have been

constructed in orientifolds of toroidal compactifica-

tions in the nonsupersymmetric case, and in orbi-

folds thereof in supersymmetric cases. The

generalization of the above construction beyond

toroidal situations is, in principle, possible, but

difficult, due to the mathematically challenging

task of constructing special Lagrangian submani-

folds for general Calabi–Yau manifolds.

Certain phenomenologically interesting quantities,

such as gauge couplings and their threshold

corrections, Yukawa couplings, and other diverse

correlation functions have been computed in toroi-

dal cases, where the corresponding correlators are

computable exactly in 

. Particularly interesting is

the computation of Yukawa couplings, or, in

general, of couplings involving only fields at inter-

sections. These couplings arise from open-string

world-sheet instantons, namely disks with bound-

aries on the D-branes corresponding to those

intersections.

Type IIB Orientifolds

Type IIB compactifications with B-type branes

contain several familiar classes of 4D models, for

instance, compactifications of type I string theory on

smooth Calabi–Yau spaces (whose description may

be carried out using the effective supergravity

action, in close analogy with the heterotic compac-

tifications). Compactifications of type I string theory

on orbifolds can be regarded as a particular

realization of this, easily described using exact

CFTs (although from the viewpoint of the general

description as B-branes, the appearance of lower-

dimensional branes requires their mathematical

description to involve coherent sheaves). Since

open strings at orbifolds do not have twisted

boundary conditions, chirality arises from the orbi-

fold projection of the toroidally compactified theory

on the spectrum.

Another example within this kind is provided by

the so-called magnetized D-brane models. These

correspond to toroidal compactifications of type I

theory, with D9 branes carrying constant magnetic

backgrounds for the internal components of the

world-volume gauge fields. In this kind of model,

although the closed-string sector is highly super-

symmetric, the open-string spectrum has reduced

supersymmetry, or no supersymmetry (if the bundle

stability condition is relaxed). Chirality arises from

the nontrivial index of the Dirac operator for open

strings ending on D-branes with different world-

volume magnetic fields. Explicit models have mainly

centered on nonsupersymmetric models from orien-

tifolds of T

, and on supersymmetric models from

orientifolds of the T

=(Z

 Z

) orbifold. In both

contexts, models with semirealistic spectra have

been obtained: concretely nonsupersymmetric mod-

els with just the standard model spectrum, or

supersymmetric models with the minimal super-

symmetric standard model spectrum, plus nonchiral

matter. Further, properties of the gauge coupling

constants and the computation of the Yukawa

couplings have been studied as functions of unde-

termined moduli.

String Theory: Phenomenology 107

Finally, a second large class of models constructed

using B-type branes are given by lower-dimensional

D-branes, for example, D3 branes, located at singular

points in the internal compactification space. Since the

massless sector of open strings is determined only in

terms of the local structure of the singularity, these

models have been mostly studied in noncompact

setups. Resulting spectra can be encoded in quiver

diagrams, related to those in the mathematical litera-

ture on the McKay correspondence. Semirealistic three-

family models have been constructed based on systems

ofD3andD7branesattheC

orbifold singularity.

Type IIB orientifold compactifications are also

intimately related to F-theory compactifications on

Calabi–Yau 4-folds, which provide a nonperturba-

tive completion for such models.

Mirror symmetry exchanges type IIB and IIA

compactifications with B- and A-type branes. Hence,

it provides a map between the above two kinds of

compactifications. This shows that type IIB orienti-

fold models lead to spectra with structure similar to

that of intersecting-branes worlds, and that they

share many of their general properties.

As a particular example, toroidal models of

intersecting D6 branes are mapped under mirror

symmetry to models of magnetized D9 branes. This

mirror map has been exploited to construct the same

theories from both starting points and to recover

certain quantities, such as the 

-exact Yukawa

couplings in the IIA picture from a purely classical

(no 

corrections) computation in the mirror IIB

model. This is a particular application of the general

proposal of homological mirror symmetry in com-

pactifications with branes.

Type II orientifold compactifications with

D-branes have also been explored beyond the

geometric regime, using exact CFTs to describe the

(analog of the) internal space, and crosscap and

boundary states to describe (the analogs of) orienti-

fold planes and D-branes. Formal developments in

the construction of the latter in Gepner models have

been successfully applied to obtain large classes of

semirealistic 4D string models in this setup.

As compared with heterotic compactifications, the

setup of D-brane models leads to several generic

features:



Since gauge sectors are localized on D-branes, and

have a dilaton dependence different from gravita-

tional interactions, the relation between the

fundamental string scale and the 4D Planck scale

and gauge coupling reads

11p

½9

where V

is a measure of the volume in the

directions transverse to the brane, and g

is the

10D string coupling. The above relation shows that

it is possible to achieve large 4D Planck mass with

a lower fundamental string scale by adjusting the

transverse volume and the string coupling. This has

been proposed by Antoniadis, Arkani-Hamed,

Dimopoulos, and Dvali as an alternative to explain

the Planck/weak hierarchy without supersymmetry.



The compactifications contain several U(1) gauge

symmetries. For some of the corresponding gauge

bosons, the 4D effective theory contains Stuckel-

berg masses of order M

, due to B ^ F couplings

to fields in the RR sector. These couplings make

the U(1) gauge bosons massive; hence, they are

absent from the low-energy physics. Nevertheless,

the U(1)’s remain as global symmetries exact in 

and to all orders in the perturbation theory in g

They are violated by D-brane instantons, which

are nonperturbative in g

. In many realistic

models, the baryon number is one such global

symmetry, and it prevents proton decay, even if

the string scale is not large.



In general, each gauge factor in the standard

model arises from a different brane stack, and

their gauge couplings at the string scale are

controlled by different moduli. This implies that,

generically, it is not natural to have gauge

coupling unification in D-brane models. Particular

models may enjoy enhanced discrete global

symmetries at special points in moduli space

where unification is achieved, thus making uni-

fication appear more natural in such examples.

Similar statements apply for constructions which

realize complete or partial unification of gauge

groups at large scales (like string models of grand

unification or of Pati–Salam type).



As already mentioned, important quantities such

as Yukawa couplings are, in principle, computa-

ble, although quantitative expressions have been

derived only in a few examples, mostly in toroidal

compactifications or quotients thereof. The results

are moduli dependent, making it difficult to

derive model-independent patterns.

M-Theory Phenomenology

Most of the phenomenological models from the

M-theory have been constructed using the Horava–

Witten theory (compactification of M-theory on

) as starting point. This theory provides a

description of the strong coupling regime of the

 E

heterotic theory, and many of its basic

features are similar to those in the perturbative

regime. In particular, the techniques used in model

108 String Theory: Phenomenology