[style=statementstyle]statementThe Swampland Program
[style=statementstyle]statement1Higher-form gauge symmetries
[style=statementstyle]statement2Boundary symmetries in non-compact spaces
[style=statementstyle]statement3Net gauge charge in compact spaces
[style=statementstyle]statement4Baby universe hypothesis
[style=statementstyle]statement6Distance conjecture
[style=statementstyle]statement5Weak Gravity Conjecture (basic version)
[style=statementstyle]statement7Particles vs instantons
[style=statementstyle]statement8de Sitter conjecture
[style=statementstyle]statement9Trans-Planckian censorship conjecture (TCC)

CALT-TH-2022-042

Lectures on the string landscape
and the Swampland
Nathan Benjamin Agmon1, Alek Bedroya1, Monica Jinwoo Kang2, and Cumrun Vafa1
1 Physics Department, Harvard University, Cambridge, MA 02138, U.S.A.

2 Walter Burke Institute for Theoretical Physics, California Institute of
Technology

Pasadena, CA 91125, U.S.A.

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

We provide an overview of the string landscape and the Swampland program. Our
review of the string landscape covers the worldsheet and spacetime
perspectives, including vacua and string dualities. We then review and
motivate the Swampland program from the lessons learned from the string
landscape. These lecture notes are aimed to be self-contained and thus can
serve as a starting point for researchers interested in exploring these ideas.

These notes are an expanded version of two courses _The String Landscape and
the Swampland_ taught by C. Vafa at Harvard University in 2018 with a focus on
the landscape, written by M. J. Kang with additional material from N. B.
Agmon, and in 2022 with a focus on the Swampland, written by A. Bedroya.

###### Contents

  1. Introduction
  2. I The string landscape
    1. 1 Bosonic string theory
      1. 1.1 Conventions
      2. 1.2 Freely propagating strings
      3. 1.3 Basics of conformal field theory
      4. 1.4 Worldsheet gauge anomalies and the critical string
      5. 1.5 BRST formalism
      6. 1.6 Quantum string
      7. 1.7 String perturbation theory
      8. 1.8 Tree-level scattering
      9. 1.9 One-loop scattering
    2. 2 Bosonic string compactifications
      1. 2.1 Geometric CFTs
      2. 2.2 WZW models and current algebras
      3. 2.3 Orbifolds
      4. 2.4 Noncritical string theory
    3. 3 Superstring theory
      1. 3.1 Basics of the NSR formalism
      2. 3.2 Modular invariance and the GSO projection
      3. 3.3 Green-Schwarz superstring
      4. 3.4 Examples of superstring compactifications
      5. 3.5 The type I string
      6. 3.6 The heterotic string
      7. 3.7 Superstring compactifications
      8. 3.8 Model building
    4. 4 String dualities
      1. 4.1 Supergravities in $d\geq 10$
      2. 4.2 T-duality for superstring theories
      3. 4.3 Branes
      4. 4.4 M-theory
      5. 4.5 Completing the web of dualities for $N_{SUSY}=16$
      6. 4.6 F-theory
      7. 4.7 More dualities in lower dimensions
    5. 5 Complex geometry
      1. 5.1 Preliminary definitions
      2. 5.2 Examples of Calabai–Yau manifolds
      3. 5.3 Calabi–Yau manifolds from complex projective spaces
      4. 5.4 Singularities of K3 surfaces
      5. 5.5 Singularities of Calabi–Yau threefolds
      6. 5.6 Toric geometry
    6. 6 Sigma models
      1. 6.1 Supersymmetric sigma models and mirror symmetry
      2. 6.2 Supersymmetric minimal models
      3. 6.3 Mirror symmetry in minimal models
      4. 6.4 Calabi–Yau SCFT from minimal models
      5. 6.5 Minimal models and Calabi–Yau $\sigma$-models
    7. 7 Black Holes and holography
      1. 7.1 Black holes in string theory
      2. 7.2 Holography
  3. II The Swampland program
    1. 1 Introduction to Swampland program
      1. 1.1 Basic features of quantum field theories
      2. 1.2 Quantum gravity vs quantum field theory
      3. 1.3 Why is gravity special?
      4. 1.4 Problems of treating gravitational theories as EFTs
      5. 1.5 The Swampland program
    2. 2 Swampland I: No global symmetry conjecture
      1. 2.1 No global symmetry: black hole argument
      2. 2.2 What is a global symmetry?
      3. 2.3 Non-invertible symmetries
      4. 2.4 What is a gauge symmetry?
      5. 2.5 Non-compact spaces and boundary symmetries
      6. 2.6 No global symmetry: holographic argument
      7. 2.7 Symmetries in string theory
      8. 2.8 Cobordism conjecture
      9. 2.9 Baby universe hypothesis
    3. 3 Swampland II: Completeness of spectrum
      1. 3.1 Completeness hypothesis
      2. 3.2 Evidence in string theory
      3. 3.3 Completeness of spectrum for discrete symmetries
      4. 3.4 Completeness of spectrum: arguments
    4. 4 Swampland III: Weak gravity conjecture
      1. 4.1 Evidence from string theory
      2. 4.2 Weak gravity conjecture: formulation
      3. 4.3 Motivation
      4. 4.4 Festina lente
      5. 4.5 Applications
    5. 5 Swampland IV: Distance conjectures
      1. 5.1 Introduction
      2. 5.2 Moduli space
      3. 5.3 Supergravities
      4. 5.4 Dualities and infinite distances limits
      5. 5.5 Universal properties of infinite distance limits
      6. 5.6 AdS and CFT distance conjectures
    6. 6 Swampland V: de Sitter conjectures
      1. 6.1 (Non)-supersymmetry and (in)stability
      2. 6.2 de Sitter and tree-level string theory
      3. 6.3 de Sitter conjectures
    7. 7 Swampland VI: Finiteness and string lamppost
      1. 7.1 String lamppost and finiteness principles
      2. 7.2 Finiteness principle in AdS
      3. 7.3 String lamppost principle from brane probes
      4. 7.4 Finiteness of light species

### Introduction

After more than five decades of research, string theory has emerged as the
most promising candidate for describing the connection between the observed
universe and a theory of quantum gravity. It offers our deepest understanding
of how quantum theory of gravity works. We have learned how to construct large
classes of vacua in various string theories, including solutions which contain
standard model matter content, such as non-abelian gauge fields, chiral
fermions, and multiple generations of matter fields. We have also discovered
that string dualities lead not only to connections between various string
theories, but also to the discovery of new quantum systems decoupled from
gravity, in up to six dimensions.

The remarkable success of string theory may have led to the misunderstanding
that any quantum field theory that appears consistent without gravity can be
coupled to quantum gravity with string theory serving as the bridge. This
expectation cannot be further from the truth. As we currently understand, only
a few special quantum field theories emerge as the low energy limits of string
theory. Essentially no generic quantum field theories can emerge, and only
very special ones do! That may sound unnatural from the viewpoint of effective
field theory but that is the lesson string theory is teaching us.

On the other hand, particle phenomenology and cosmology face a crisis of
naturalness. Parameters and choices of theories needed to explain our universe
seem highly fine-tuned and unnatural. It is not difficult to imagine that the
reason they look fine-tuned is because the consistency of quantum gravity, as
string theory solutions offer, is not incorporated into the notion of
naturalness. With a correct prior, namely being able to couple the QFT to
gravity, the naturalness criteria changes dramatically enough to make QFTs
that describe our universe not as fine-tuned as they appear.

The Swampland program aims to delineate conditions on effective field theories
which distinguish the “good ones” (those that can couple to gravity
consistently) from the “bad ones”. The aim of the courses serving as the basis
for these lecture notes was to introduce this topic to students interested in
doing research in this direction.

These lecture notes contain two basic parts. The first part includes several
chapters dealing with an overview of string theory, with a focus on the
landscape from both the worldsheet and spacetime perspectives. It is rather
brief, but we try to be self-contained. The topics we cover serve as useful
background for ideas in the Swampland program. The second part of these notes
provides motivation for the program and explains its guiding principles.

##  Part I The string landscape

###  1 Bosonic string theory

We begin with a lightning review of perturbative bosonic string theory. String
theory is a theory of $1+1$ dimensional relativistic fundamental objects that
propagate in some target space with an action defined on the worldsheet traced
out by the strings.

####  1.1 Conventions

Before diving into the details, it is useful to set some of the conventions
first. Our discussion closely follows the modern textbook route [1, 2, 3, 4]
with many of the conventions of [1, 2]. We use $(\sigma^{1},\sigma^{2})$ to
parameterize the the string worldsheet $\Sigma$ in Euclidean signature, where
$\sigma^{2}$ plays the conventional role of Euclidean time. We often extend
the domain of physical operators on the worldsheet by analytically extending
them to complex values of $\sigma^{1}$ and $\sigma^{2}$. A particularly useful
coordinate system for the resulting $\mathds{C}\times\mathds{C}$ space is

$\displaystyle z=\sigma^{1}+i\sigma^{2},\quad\bar{z}=\sigma^{1}-i\sigma^{2}.$
(I.1.1)

Note that $z$ and $\bar{z}$ are not necessarily complex conjugates111We denote
the complex conjugate by $z^{*}$.. We can set the signature of the theory by
restricting to the appropriate 2d subspace of $\mathds{C}\times\mathds{C}$.
For example, the $z^{*}=\bar{z}$ subspace corresponds to the Euclidean
parametrization of the worldsheet, while the $z,\bar{z}\in\mathds{R}$ subspace
corresponds to Lorentzian signature.

Since the functions are meromorphic in $z$ and $\bar{z}$, complex derivatives
$\partial_{z}$ and $\partial_{\bar{z}}$ are well defined. We refer to them
respectively as holomorphic and anti-holomorphic derivatives. When there is no
possibility for confusion, we drop the subscripts and denote the associated
derivatives by

$\displaystyle\partial\equiv\frac{\partial}{\partial
z}\quad\text{and}\quad{\bar{\partial}}\equiv\frac{\partial}{\partial{\bar{z}}}.$
(I.1.2)

They satisfy $\partial z=\bar{\partial}\bar{z}=1$ and
$\partial\bar{z}=\bar{\partial}z=0$. Holomorphic functions
(${\bar{\partial}}f(z,\bar{z})=0$) are written as $f(z)$ whereas anti-
holomorphic functions (${\partial}f(z,\bar{z})=0$) are denotes by
$f(\bar{z})$.

The measure $d^{2}z=dz\wedge d{\bar{z}}$ satisfies
$d^{2}z=2d\sigma^{1}d\sigma^{2}$, with the Jacobian factor included. The
integral over a closed contour in the complex plane is taken to satisfy

$\displaystyle\oint\frac{1}{z}=2\pi i.$ (I.1.3)

For convenience, we work in units where the string length $\ell_{s}=1$, _i.e._
where the string tension is

$\displaystyle T=\frac{1}{2\pi}.$ (I.1.4)

####  1.2 Freely propagating strings

At its core, perturbative string theory is a theory of fundamental
$1+1$-dimensional objects moving in a target space. If we view the amplitude
as a function of the worldsheet of the string, string theory is a 2d field
theory where the amplitude of a given worldsheet configuration is given by
$e^{iS}$ where $S$ is the action of the corresponding two dimensional field
theory.

$\mathcal{M}\sim\exp(iS_{worldsheet})$ Figure I.1.1: A 2 $\rightarrow$ 2
scattering event of strings represented by a single string worldsheet. The
action on the worldsheet determines the scattering amplitude $\mathcal{M}$.

Recall that the action for a relativistic particle of mass $m$ is defined to
be proportional to the proper length of its worldline $\gamma$, _i.e._

$\displaystyle
S=-m\int_{\gamma}d\tau\sqrt{\partial_{\tau}X^{\mu}\cdot\partial_{\tau}X_{\mu}}.$
(I.1.5)

The same line of reasoning suggests a natural candidate for strings, namely
introducing an action proportional to the area of string worldsheet. This
action is known as the Nambu–Goto action. For a string that traces $X^{\mu}$
in a Minkowski background, the Nambu–Goto action reads [1]

$\displaystyle
S_{NG}=-T\int_{\Sigma}d^{2}\sigma\sqrt{\det\left(\partial_{a}X^{\mu}\cdot\partial_{b}X_{\mu}\right)}$
(I.1.6)

where $T=1/(2\pi)$ is the string tension, _i.e._ mass per unit length, and
$\Sigma$ is the worldsheet of the string. A priori, this action is difficult
to quantize due to the presence of the square root. By introducing a dynamical
worldsheet metric $g_{ab}$, we can instead consider the simpler Polyakov
action [5, 6, 7],

$\displaystyle
S_{P}=-\frac{T}{2}\int_{\Sigma}d^{2}\sigma\sqrt{g}g_{ab}\partial^{a}X^{\mu}\cdot\partial^{b}X_{\mu}.$
(I.1.7)

Note that the metric $g_{ab}$ is an auxiliary variable and we can solve for it
up to an overall scaling from the equation of motion. It is straightforward to
verify that the two theories lead to the same equations of motion for
$X^{\mu}$ and are therefore classically equivalent. The classical theory of
(I.1.7) enjoys several internal symmetries. The worldsheet theory is invariant
under global target space translations and Lorentz transformations,

$\displaystyle
X^{\prime\mu}(\sigma)=\Lambda^{\mu}_{\>\>\>\nu}X^{\nu}(\sigma)+v^{\mu},\quad\Lambda\in
SO(d-1,1).$ (I.1.8)

Locally, this two-dimensional field theory is also invariant under several
gauge symmetries, including _diffeomorphisms_ (_i.e._ reparametrizations),

$\displaystyle X^{\prime\mu}(\sigma^{\prime})=X^{\mu}(\sigma),\quad
g^{\prime}_{ab}(\sigma^{\prime})=\frac{\partial\sigma^{c}}{\partial\sigma^{\prime
a}}\frac{\partial\sigma^{d}}{\partial\sigma^{\prime b}}g_{cd}(\sigma),$
(I.1.9)

with $\sigma^{\prime a}=f^{a}(\sigma^{b})$, and Weyl transformations (_i.e._
local rescalings),

$\displaystyle g_{ab}^{\prime}(\sigma)=e^{2\phi(\sigma)}g_{ab}(\sigma).$
(I.1.10)

The reason the above symmetries must be gauge symmetries and not ordinary
symmetries is to ensure that the metric $g$ is a fictitious field and the
Nambu–Gotu and Polyakov theories are equivalent.

We perform a Wick rotation of the worldsheet theory to work in the Euclidean
signature. The Wick rotation will remove the minus sign in front of the
Lorentzian action. The quantized 2d Polyakov theory is then defined in terms
of a path integral over all field configurations as

$\displaystyle
Z=\int\frac{DXDg}{V_{\text{Diff}\times\text{Weyl}}}e^{-S_{P}[X,g]},$ (I.1.11)

where we have divided by the volume of the gauge group to render the
expression finite. The expression in (I.1.11) includes an implicit sum over
inequivalent topologies (_i.e._ manifolds which are not connected through the
action of the gauge group). To make progress in the quantum theory, we utilize
the standard Fadeev-Popov gauge-fixing procedure [8], which localizes the path
integral to a single gauge slice at the cost of introducing a set of anti-
commuting $b$,$c$ ghost fields. There is a single $c$ ghost for every gauge
parameter, and one $b$ ghost for every gauge-fixing condition. A particularly
convenient choice of gauge-fixing condition is the conformal gauge222Although
it is always possible to choose $g_{ab}=\delta_{ab}$ locally in each
coordinate patch of $\Sigma$, there can be global obstructions to this gauge
choice. We postpone the study of these global issues to the discussion of
moduli spaces of Riemann surfaces.

$\displaystyle g_{ab}=\delta_{ab}.$ (I.1.12)

An infinitesimal gauge transformations is specified by three worldsheet
functions: infinitesimal coordinate transformations
$\delta\sigma^{1}(\sigma)$, $\delta\sigma^{2}(\sigma)$ as well as the Weyl
rescaling function $\phi(\sigma)$. Therefore, we expect to have three ghost
$c$ fields in total. Moreover, every pair of worldsheet indices ${a,b}$
corresponds to a gauge-fixing condition $g_{ab}=\delta_{ab}$ and should lead
to a ghost field $b_{ab}$. As the gauge-fixing conditions are symmetric in $a$
and $b$, $b_{ab}$ must also be symmetric.

It turns out the ghost field associated with Weyl transformations is easy to
integrate out since it only appears as a quadratic term in the action. Doing
so imposes a tracelessness condition on $b_{ab}$. Thus, we end up having two
ghost fields $c^{a}$ corresponding to the coordinates reparameterizations
$\delta\sigma^{a}$, a traceless symmetric tensor ghost field $b_{ab}$, and $d$
free massless scalars. The action for the massless scalars is

$\displaystyle
S_{X}=\frac{T}{2}\int_{\Sigma}d^{2}\sigma\partial_{a}X^{\mu}\partial_{a}X_{\mu},$
(I.1.13)

while the ghosts $c^{a}$, $b_{ab}$ are governed by the action

$\displaystyle S_{gh}=\frac{1}{2\pi}\int_{\Sigma}d^{2}\sigma
b_{ab}\partial^{a}c^{b}.$ (I.1.14)

The path integral of the gauge-fixed theory thus reduces to

$\displaystyle Z=\int DXDbDc\>e^{-S_{X}-S_{gh}}.$ (I.1.15)

####  1.3 Basics of conformal field theory

It turns out that our choice of conformal gauge does not completely eliminate
all of the gauge redundancy. There are still residual gauge transformations
corresponding to combinations of diffeomorphisms and Weyl transformations that
leave the metric invariant, referred to collectively as the group of conformal
transformations. In the conventional definition of conformal theories in flat
space, a diffeomorphism is accompanied by a rescaling of other fields rather
than the metric. It is straightforward to see these two definitions are
equivalent and the rescaling of metric could be absorbed in the rescaling of
the fields. For example, in the bosonic string action (I.1.13)$+$(I.1.14), a
rescaling of the metric under the residual conformal transformation

$\displaystyle g_{ab}\rightarrow e^{2\omega(\sigma)}g_{ab}$ (I.1.16)

could be replaced with a transformation of the fields given by

$\displaystyle(X,b_{ab},c^{a})\rightarrow(X,b_{ab}e^{2\omega(\sigma)},c^{a}e^{-\omega(\sigma)})$
(I.1.17)

As a result, we can view the worldsheet theory as a theory in 2d flat space
which has conformal symmetry [9, 10]. However, it is important to remember
that the conformal symmetry is not a global symmetry, but a gauge symmetry of
the theory! This is an important distinction. There is no a priori reason why
the conformal symmetry of a classical theory must be preserved at the quantum
level. However, in the case of string worldsheet, this is a requirement for
the mathematical consistency of the theory.

Since the dynamics of the strings are captured by a conformal theory, it is
useful to briefly review the basic properties of CFTs in two
dimensions.333Most of the essential CFT features in string theory are covered
or at least mentioned in [1, 2]. For a more complete reference, see the yellow
book [10].. Every local QFT by definition has a conserved operator known as
the stress-energy tensor $T_{ab}$. For theories which admit a Lagrangian
description, the stress tensor can be defined unambiguously by minimally
coupling the theory to gravity444Minimal coupling refers to the gauging of the
local Lorentz symmetry $SO(1,d-1)$.

$\displaystyle T_{ab}=-4\pi g^{-1/2}\frac{\delta S}{\delta g^{ab}},$ (I.1.18)

where the overall coefficient is a matter of convention. In scale invariant
theories the trace of the energy momentum tensor vanishes on shell [11]. In
conformal field theories, one can show that the energy momentum tensor can be
improved by adding a total derivative $\partial_{c}\partial_{d}N^{abcd}$, such
that the trace vanishes off shell as well. To ensure that $T^{ab}$ remains
symmetric and conserved, $N$ needs to be symmetric under $a\leftrightarrow b$
and $c\leftrightarrow d$ and antisymmetric under $\\{a\leavevmode\nobreak\
\text{or}\leavevmode\nobreak\ b\\}\leftrightarrow\\{c\leavevmode\nobreak\
\text{or}\leavevmode\nobreak\ d\\}$. In fact, such an improvement of the
energy momentum tensor can be achieved by adding a boundary term to the action
[11]. The boundary term ensures the action is Weyl invariant. If we work with
this action, then $T^{a}_{a}$ must vanish identically, which in complex
coordinates reads

$\displaystyle T_{z\bar{z}}=0.$ (I.1.19)

Similarly, the conservation equation can be written as

$\displaystyle\bar{\partial}T_{zz}=\partial T_{\bar{z}\bar{z}}=0,$ (I.1.20)

which implies that the diagonal components $T\equiv T_{zz}(z)$ and
$\bar{T}=T_{\bar{z}\bar{z}}(\bar{z})$ are holomorphic and anti-holomorphic,
respectively. It follows that both admit Laurent expansions around the origin

$\displaystyle
T(z)=\sum_{n}L_{n}z^{-n-2},\quad\bar{T}(\bar{z})=\sum_{n}\bar{L}_{n}\bar{z}^{-n-2},$
(I.1.21)

where the overall factor $z^{-2}$ ($\bar{z}^{-2}$) is chosen to agree with its
scaling dimension $\Delta=2$. We know that the energy momentum tensor is the
generator of coordinate transformations, and conformal transformations are a
special type of coordinate transformation in flat space. Thus, we should
expect the generators of conformal transformations can be expressed in terms
of $T(z)$ and $\bar{T}(\bar{z})$. In fact, $L_{n}$ is the generator of
$(z,\bar{z})\rightarrow(z+\epsilon z^{n},\bar{z})$ and $\bar{L}_{n}$ is the
generator of
$(z,\bar{z})\rightarrow(z,\bar{z}+\epsilon\bar{z}^{n})$.555Conformal
transformations in two-dimensions generically take the form of holomorphic
functions $f(z)$.  These generators satisfy the following classical
commutation relations,

$\displaystyle[L_{m},L_{n}]$ $\displaystyle=(m-n)L_{m+n},$ (I.1.22)

and are known as the Witt algebra. After quantizing the theory, the algebra
acquires a central extension $c$. The centrally extended algebra, known as the
Virasoro algebra $\text{Vir}_{c}$, takes the following form [12]

$\displaystyle[L_{m},L_{n}]$
$\displaystyle=(m-n)L_{m+n}+\frac{c}{12}(m^{3}-m)\delta_{m+n,0}.$ (I.1.23)

Similarly, the anti-holomorphic modes $\bar{L}_{n}$ generate an independent
copy $\text{Vir}_{\bar{c}}$ with central charge $\bar{c}$ not necessarily
equal to $c$. The central charges $c$ and $\bar{c}$ represent the breaking of
the conformal symmetry at the quantum level.

All CFT states on the cylinder, which are dual to local operators at the
origin of $\mathds{C}$ via the _state-operator correspondence_ , can written
in terms of states with definite weights $(h,\bar{h})$ under
$L_{0},\bar{L}_{0}$. For unitary theories, these can be further decomposed
into linear combinations of primary and descendant states. A primary
$\ket{\psi}$ satisfies

$\displaystyle L_{n}\ket{\psi}=\bar{L}_{n}\ket{\psi}=0,\quad n>0,$ (I.1.24)

while a descendant takes the general form

$\displaystyle\ket{\psi}^{\prime}=L_{-n_{1}}\cdots L_{-n_{k}}\ket{\psi},\quad
n_{1},\dots n_{k}>0$ (I.1.25)

Analogously, a primary operator ${\cal O}$ of weight $(h,\bar{h})$ obeys the
operator product expansion (OPE)

$\displaystyle T(z){\cal O}(0)=\frac{h}{z^{2}}{\cal
O}(0)+\frac{1}{z}\partial{\cal O}(0)+\text{non-singular},$ (I.1.26)

and similarly for $\bar{T}{\cal O}$. In particular, higher-order singularities
are absent. This implies that under a conformal transformation, $\cal{O}$
transforms as

$\displaystyle\mathcal{O}(z^{\prime},\bar{z}^{\prime})=(\partial
z^{\prime})^{-h}(\bar{\partial}\bar{z^{\prime}})^{-\bar{h}}\mathcal{O}(z,\bar{z}).$
(I.1.27)

where $z^{\prime}=f(z)$ and $\bar{z}^{\prime}=\bar{f}(\bar{z})$ are the
associated conformal maps.

It is easy to verify this definition is equivalent to (I.1.24) via (I.1.23).
The stress tensor is not a primary, but nonetheless satisfies the OPE:

$\displaystyle
T(z)T(0)=\frac{c}{2z^{4}}+\frac{2}{z^{2}}T(0)+\frac{1}{z}\partial
T(0)+\text{non-singular},$ (I.1.28)

which through (I.1.24) leads to the aforementioned Virasoro algebra (I.1.23).

####  1.4 Worldsheet gauge anomalies and the critical string

We now return to the gauge-fixed worldsheet theory, whose action in complex
coordinates takes the form

$\displaystyle S=\frac{1}{2\pi}\int d^{2}z\left(\partial
X^{\mu}\bar{\partial}X_{\mu}+b{\bar{\partial}}c+{\bar{b}}\partial{\bar{c}}\right),$
(I.1.29)

where $c=c^{z}(z)$, $b=b_{zz}(z)$, $\bar{c}=c^{\bar{z}}(\bar{z})$,
$\bar{b}=b_{\bar{z}\bar{z}}(\bar{z})$. The energy-momentum tensor for the
matter fields $X^{\mu}$ is666All operator products are assumed to be normal
ordered with respect to some procedure, such as creation-annihilation ordering
(with creation operators on the left and annihilation operators on the right)
or conformal normal ordering.

$\displaystyle T_{m}=-\partial X^{\mu}\partial
X_{\mu},\quad\bar{T}_{m}=-\bar{\partial}X^{\mu}\bar{\partial}X_{\mu},$
(I.1.30)

with central charge $c=\bar{c}=d$, where $\mu=1,\ldots,d$. This agrees with
the notion that the central charge is a rough measure of the number of degrees
of freedom in a CFT. The ghost theory admits an independent stress tensor of
the form

$\displaystyle T_{gh}=2(\partial c)b+c\partial
b,\quad\bar{T}_{gh}=2(\bar{\partial}\bar{c})\bar{b}+\bar{c}\bar{\partial}\bar{b}$
(I.1.31)

with central charge $c=\bar{c}=-26$. Therefore, the central charge of the
combined matter+ghost CFT is

$\displaystyle c_{\text{tot}}=d-26.$ (I.1.32)

Recall that before gauge-fixing, the classical worldsheet theory possessed a
$[\text{Diff}\times\text{Weyl}]$ gauge symmetry. The quantum theory, as
defined by the Polyakov path integral, can suffer from both global and gauge
anomalies. Indeed, on a general curved worldsheet, bosonic string theory
suffers from a Weyl anomaly, which manifests itself through the non-vanishing
of the trace of the stress tensor777Note that any CFT with $c\neq\bar{c}$
necessarily has a gravitational (diffeomorphism) anomaly, and so we will
usually assume $c=\bar{c}$ unless explicitly stated.

$\displaystyle T_{z\bar{z}}=-\frac{c}{24}\mathcal{R}.$ (I.1.33)

Here, $\mathcal{R}$ is the 2d Ricci scalar on the worldsheet, which completely
captures the geometry of a 2d manifold. To preserve the full
$[\text{Diff}\times\text{Weyl}]$ gauge symmetry and consistently quantize the
theory, we must therefore take the number of scalar fields to be equal to 26.
This in turn implies that the dimensionality of spacetime is [13]

$\displaystyle d=26.$ (I.1.34)

The resulting Weyl-invariant theory is known as critical bosonic string
theory.

To summarize, the only dimension $d$ where the $d$-dimensional Euclidean space
could be the target space of a non anomalous 2d CFT is $d=26$.

####  1.5 BRST formalism

As we discussed in subsection 1.3, the conformal gauge choice
$g_{ab}=\delta_{ab}$ does not completely eliminate the gauge redundancy. As a
consequence, the naive Hilbert space of the theory over-counts the physical
states. The BRST formalism is a systematic method that allows us to restrict
to a faithful subspace of the Hilbert space where every physical state is
represented exactly once. It is based on introducing a new global fermionic
symmetry known as the BRST symmetry [14, 15].888BRST refers to Becchi, Rouet,
Stora, and Tyutin. For the combined matter+ghost CFT, the current $j_{B}^{a}$
associated with this symmetry is [16]

$\displaystyle j_{B}=c\cdot T_{m}+\frac{1}{2}c\cdot
T_{gh},\quad\bar{j}_{B}=\bar{c}\cdot\bar{T}_{m}+\frac{1}{2}\bar{c}\cdot\bar{T}_{gh},$
(I.1.35)

where $j_{B}\equiv j_{B}^{z}$ and $\bar{j}_{B}\equiv j_{B}^{\bar{z}}$. Its
components are are (anti-)holomorphic like the stress tensor, which implies
$j_{B}^{a}$ is conserved

$\displaystyle\partial_{a}j_{B}^{a}=\bar{\partial}j_{B}+\partial\bar{j}_{B}=0.$
(I.1.36)

The associated topological charge operator acting on a local operator ${\cal
O}(0)$ is given by

$\displaystyle
Q_{B}=\oint_{0}dzj_{B}(z)+\oint_{0}\bar{dz}\bar{j}_{B}(\bar{z}),$ (I.1.37)

where the contour necessarily surrounds the origin. The action of the BRST
symmetry can be thought of as a gauge transformation with the gauge parameters
replaced with $c$ ghosts. Note that the BRST symmetry acts on the $b$ ghost as

$\displaystyle Q_{B}\cdot b(z)=T(z).$ (I.1.38)

A key property of the BRST formalism is that $Q_{B}$ is nilpotent, _i.e._

$\displaystyle Q_{B}^{2}=0,$ (I.1.39)

and so admits a cohomology of states. By gauge-fixing the worldsheet theory,
we have enlarged the original space of states for the matter CFT to include
additional states generated by free field oscillators built from the ghost
fields. Naively, one might think that all such states contribute to on-shell
scattering processes in spacetime; as it turns out, many of these states lead
to equivalent spacetime physics. To account for this redundancy, we must
restrict the set of physical states to be cohomology classes of $Q_{B}$. That
is, a physical state is the coset of $Q_{B}$-closed states modulo
$Q_{B}$-exact states:999To obtain a sensible unitary S-matrix, we must also
supply an extra constraint on physical states:
$b_{0}\ket{\Psi}=\bar{b}_{0}\ket{\Psi}=0$, where $b_{0}=\oint dzzb(z)$ and
$\bar{b}_{0}=\oint d\bar{z}\bar{z}\bar{b}(\bar{z})$. In the string field
theory terminology, this constraint is known as Siegel gauge.

$\displaystyle
Q_{B}\ket{\Psi}=0,\quad\ket{\Psi}\simeq\ket{\Psi}+Q_{B}\ket{\Lambda},$
(I.1.40)

where $\ket{\Lambda}$ is an arbitrary state. The statement that $Q_{B}=0$ on
physical states can be seen as reformulation of the fact that all physical
observables must be gauge-invariant.

####  1.6 Quantum string

We now turn to analyzing the spectrum of physical states of the (free) closed
bosonic string. In conformal gauge, we found that such states are in one-to-
one correspondence with BRST cohomology classes. Using (I.1.38), it is clear
that all physical states satisfy $L_{0}=\bar{L}_{0}=0$. In principle, we could
use this to derive the masses of closed string excitations by choosing a
particular BRST representative for each cohomology class (for a detailed
analysis, see [1]) and extracting the oscillators and their algebras from the
matter and ghost fields. A faster approach to obtain the spectrum, which
easily generalizes to more complicated string theories, is to abandon
conformal gauge and completely fix the gauge. A convenient gauge choice that
has this property is the so-called light-cone gauge.

##### Light-cone quantization

Light-cone (LC) gauge quantization is a particular approach to the
quantization of sigma models whose target space admits a light-like Killing
vector. For string theory, the key is to gauge-fix Diff$\times$Weyl such that
there is a map from a worldsheet light-like Killing vector to one in the
target spacetime. This removes the longitudinal degrees of freedom. As a
bonus, the associated $bc$ ghosts completely decouple and can be neglected in
the quantum theory. The price we must pay for these simplifications is the
loss of manifest Lorentz invariance at the level of the classical worldsheet
theory. Furthermore, we must also verify that it remains a symmetry in the
quantum theory.

Working with a worldsheet in Lorentzian signature, we define null spacetime
and worldsheet coordinates as,

$X^{\pm}:=\frac{X^{0}\pm
X^{1}}{\sqrt{2}},\quad\sigma^{\pm}:=\frac{\sigma^{0}\pm\sigma^{1}}{\sqrt{2}}.$
(I.1.41)

Recall that taking the worldsheet metric flat, $g_{ab}=\eta_{ab}$, does not
completely fix the gauge redundancy. The remaining gauge degrees of freedom
can be eliminated by choosing the light-cone gauge,

$X^{+}=x^{+}+\alpha^{\prime}p^{+}\sigma^{+},\quad\sigma^{\pm}=\sigma^{0}\pm\sigma^{1},$
(I.1.42)

for some constants $x^{+}$ and $p^{+}$ which ties $\sigma^{+}$ to $X^{+}$. We
can set $x^{+}$ to zero by reparametrizing $X^{+}\rightarrow X^{+}-x^{+}$.
Note that $X^{+}$ is no longer a dynamical field. Naively this gauge-fixing
condition would appear to leave a free field theory with $d-1=25$ free bosons.
However, this is not the case since the equation for the metric forces us to
set

$T_{ab}=0$ (I.1.43)

as a constraint which must be imposed by hand. This set of constraints are
collectively referred to as Virasoro constraints. In light-cone gauge, they
take the form

$\displaystyle T_{++}$
$\displaystyle=\partial_{+}X^{+}\cdot\partial_{+}X^{-}-\partial_{+}X^{i}\cdot\partial_{+}X^{i}=0,\quad
i=1,\cdots,24,$ (I.1.44)

with a similar expression for $T_{--}$. Together with (I.1.42) we then have

$\displaystyle\partial X^{-}=\frac{1}{p^{+}}\partial X^{i}\cdot\partial
X^{i},$ (I.1.45)

and so the nonzero oscillatory modes of $X^{-}$ also decouple. Thus, by
choosing light-cone gauge we have explicitly removed the longitudinal degrees
of freedom; only the 24 transverse fields $X^{i}$ remain. The quantum theory
of the free string in lightcone gauge can be defined by canonical
quantization. We can expand the transverse fields in terms of their Fourier
modes on the cylinder, _i.e._

$\displaystyle
X^{i}(\sigma)=x^{i}+\frac{p^{i}}{p^{+}}\sigma^{0}+\frac{i}{\sqrt{2}}\sum_{n\neq
0}\frac{1}{n}\left(\alpha_{n}^{i}e^{in\sigma^{-}}+\bar{\alpha}_{n}^{i}e^{-in\sigma^{+}}\right).$
(I.1.46)

The Hamiltonian $H$, which can now be identified with $p^{-}$, is given by

$H=\frac{p^{i}p^{i}}{2p^{+}}+\frac{1}{2p^{+}}\left(\sum_{n=1}^{\infty}\alpha_{-n}^{i}\alpha_{n}^{i}+\bar{\alpha}^{i}_{-n}\bar{\alpha}^{i}_{n}+A+\bar{A}\right),$
(I.1.47)

where $A$ and $\bar{A}$ are constants to be computed from the usual ordering
ambiguity once we quantize the theory. We can proceed by imposing equal-time
commutation relations on the $X^{i}$ and their conjugate momenta, or
equivalently the following commutation relations on the oscillators:

$\displaystyle[\alpha^{i}_{n},\alpha^{j}_{m}]=[\bar{\alpha}^{i}_{n},\bar{\alpha}^{j}_{m}]=n\delta^{ij}\delta_{n,-m},\quad[x^{i},p^{j}]=i\delta^{ij}.$
(I.1.48)

Up to an overall normalization, the $\alpha_{n}^{i}$ and
$\bar{\alpha}_{n}^{i}$ satisfy the bosonic creation-annihilation commutation
relations, with $n<0$ corresponding to creation operators and $n>0$ to
annihilation operators. There are still two unconstrained modes, the zero mode
$x^{-}$ of $X^{-}$ and $p^{+}$, which satisfy

$\displaystyle[x^{-},p^{+}]=-i.$ (I.1.49)

The minus sign reflects the fact that the flat metric $\eta_{ab}$ in the
conformal coordinates $\sigma^{\pm}$ takes the form $\eta^{-+}=-1$. Thus, the
$x^{\mu}$ describe the center-of-mass coordinates of the string, while the
oscillators describe its vibrational modes. We can decompose the Hilbert space
into representations of the Lorentz group $SO(1,d-1)$. In particular, the
vacua of the harmonic oscillators furnish an irreducible representation with
states $\ket{p}$ labeled by a center-of-mass momentum vector
$p\equiv(p^{+},p^{i})$. They satisfy

$\alpha^{j}_{n}\ket{p}=\bar{\alpha}^{j}_{n}\ket{p}=0,\quad n>0.$ (I.1.50)

The Fock space ${\cal H}$ of free string states can be constructed by repeated
applications of the creation operators to the boosted ground states above,

$\ket{\mathbf{N},\mathbf{\bar{N}};p}=\left(\prod_{j=1}^{24}\prod_{n_{j}=1}^{\infty}(\bar{\alpha}_{-\bar{n}}^{j})^{N_{j\bar{n}}}\right)\left(\prod_{i=1}^{24}\prod_{n=1}^{\infty}(\alpha_{-n}^{i})^{N_{in}}\right)\ket{p}$
(I.1.51)

Here, $\textbf{N}=(N_{in})$ and $\mathbf{\bar{N}}\leavevmode\nobreak\
=(N_{i\bar{n}})$ are ordered collections of non-negative integer values that
determine the excitation number of each mode. Occasionally, we will also take
$N$ and $\bar{N}$ to be the sums $\sum_{i=1}^{24}\sum_{n}nN_{in}$ and
$\sum_{i=1}^{24}\sum_{\bar{n}}\bar{n}N_{i\bar{n}}$ respectively. They are
referred to as the levels of the string state
$\ket{\mathbf{N},\mathbf{\bar{N}};p}$.

Note that we have allowed ourselves to be slightly imprecise in describing the
gauge-fixing procedure. For the closed string, the light-cone gauge-fixing
conditions described above leave some residual gauge freedom – namely,
translations in the $\sigma$ direction. The actual space of physical states
${\cal H}_{\text{phys}}$ consists of gauge-invariant states in ${\cal H}$.
These are the states uncharged under translations in $\sigma$, _i.e._ those
that obey the level-matching condition $N=\bar{N}$ [17]. The closed string
Hilbert space is therefore

${\cal H}_{\text{phys}}=\text{Span}\\{\ket{\mathbf{N},\mathbf{\bar{N}};p}\\}.$
(I.1.52)

where $N=\bar{N}$. From now on we will simply refer to $N$ as the level of the
state.

To determine the masses of the closed string states in ${\cal
H}_{\text{phys}}$, notice that there is a spacetime Lorentz invariant

$m^{2}\equiv 2p^{-}p^{+}-p^{i}p^{i},$ (I.1.53)

which is just the usual mass squared invariant of relativistic particles.
Using the fact that $p^{-}=H$, we can work systematically work out the closed
string masses. Recall that the Hamiltonian $H$ includes two unfixed constants,
$A$ and $\bar{A}$. It can be shown that these quantities necessarily take the
values

$A=\bar{A}=-1$ (I.1.54)

for spacetime Lorentz invariance to be preserved. It follows that the closed
string masses are

$m^{2}\ell_{s}^{2}=4(N-1),\quad N=0,1,2,\ldots$ (I.1.55)

where we have temporarily restored the string length $\ell_{s}=1$.

##### Low-lying spectrum and NLSMs

Let us now analyze the states at each level (_i.e._ for each mass squared
value) in detail and discuss their spacetime interpretation. At level $N=0$ we
have a single state

$\displaystyle\ket{T}=\ket{p},\quad m^{2}\ell_{s}^{2}=-4,$ (I.1.56)

which behaves as a spacetime scalar with negative mass squared, _i.e._ a
tachyon. Soon, we will introduce string interactions through a perturbative
expansion. Although the free theory is fully consistent, we will see that the
presence of the tachyon introduces an incurable IR divergence in loop
diagrams, thus spoiling the theory’s consistency at the perturbative level.
This is not unlike what happens in QFT whenever the potential $V(T)$ of a
scalar field $T$ develops a local maximum at $T=0$ instead of a local minimum.
The quantized theory yields a scalar particle with negative mass squared that
inevitably introduces divergences in Feynman diagrams with loops. The solution
in this case is to expand around a stable vacuum (global minimum) where
$T=T_{0}$ and instead consider the quanta of its fluctuations around this
point. The phenomenon of tachyon condensation and whether there exists such a
vacuum remains an open problem for bosonic closed string theory.

The states at level $N=1$ are given by

$\zeta_{ij}\alpha^{i}_{-1}\bar{\alpha}^{j}_{-1}\ket{p},\quad m^{2}=0,$
(I.1.57)

where $\zeta_{ij}$ is a rank-2 tensor with no constraints. According to
Wigner’s classification, massless particles in spacetime are associated with
finite-dimensional irreducible representations (irreps) of the little group
$SO(d-2)$. We therefore decompose $\zeta_{ij}$ into $SO(24)$ irreps,

$\displaystyle\zeta_{ij}=(\zeta_{(ij)}-\zeta\delta_{ij})+\zeta_{[ij]}+\zeta\delta_{ij},$
(I.1.58)

which correspond to a symmetric-traceless irrep, an anti-symmetric irrep, and
a (trace) singlet, respectively. The symmetric-traceless states are simply the
propagating modes of a massless spin 2 particle, _i.e._ the graviton. The
others corresponding to an antisymmetric rank 2 tensor and a scalar are less
familiar in Einstein gravity, but ubiquitous in theories of gravity arising
from stings. The tensor $\zeta_{ij}$ represents the collective polarization of
these three states.

To each massless particle of the string we can associate a spacetime field in
26 dimensions,

$\displaystyle G_{\mu\nu},\quad B_{\mu\nu},\quad\phi,$ (I.1.59)

which are, respectively, the spacetime metric, a 2-form gauge potential (known
as the Kalb-Ramond field or B-field, for short), and a scalar field (the
dilaton). As we will explain below, one can think of these fields as coherent
states of the massless string states. Since coherent states satisfy the
classical equations of motion, we expect these fields to be governed by an
effective action that captures the low energy physics of these particles [18,
19]. The states beyond level $N=1$ have masses on the order of the string
scale $\ell_{p}^{-1}$, and so their dynamics are irrelevant for low energies
$E\ll\ell_{p}^{-1}$, and can be safely neglected in the low energy effective
action.

Recall that the photon is the quantum of the electromagnetic field, and that a
nonzero classical background can be generated via coherent states of photons.
Charged particles couple minimally to this field via a worldline action
$i\int_{\gamma}A_{\mu}(X)dX^{\mu}$. Analogously, coherent states of the
massless particles in string theory generate nonzero backgrounds for (I.1.59).
These fields couple minimally to the string worldsheet, and so it is
permissible to generalize the Polyakov formalism to include the effects of
these backgrounds. The usual Polyakov action in flat Minkowski space
generalizes to

$S_{G}=\frac{1}{4\pi}\int_{\Sigma}G_{\mu\nu}(X)dX^{\mu}\otimes dX^{\nu}.$
(I.1.60)

The other massless fields introduce new terms that take the form

$\displaystyle S_{B}=\frac{i}{4\pi}\int_{\Sigma}B_{\mu\nu}(X)dX^{\mu}\wedge
dX^{\nu},\quad S_{\phi}=\frac{1}{4\pi}\int_{\Sigma}R(g)\Phi(X),$ (I.1.61)

where $R(g)$ is the scalar curvature of the worldsheet. The first term is
similar to the worldline action of a charged particle. Indeed, we say that the
string is (electrically) charged under a one-form gauge symmetry associated
with the B-field. For $\Phi$ constant, the second term is a topological
invariant of the worldsheet, and is proportional to the Euler characteristic
$\chi$. For a compact, oriented worldsheet, it is given by

$\displaystyle\chi=2-2g,$ (I.1.62)

where $g$ is its genus.

By pulling back the spacetime fields to the worldsheet, we find that the
generalized Polyakov action takes the form of a nonlinear sigma model (NLSM)
on the worldsheet [20, 18],

$S_{NLSM}=\frac{1}{4\pi}\int_{\Sigma}d^{2}\sigma\sqrt{g}\>\left[\left(G_{\mu\nu}(X)g^{ab}+iB_{\mu\nu}\epsilon^{ab}\right)\partial_{a}X^{\mu}\partial_{b}X^{\nu}+R(g)\Phi(X)\right].$
(I.1.63)

Some brief comments are in order. The background fields are functions of the
target space coordinates $X^{\mu}$, and so (I.1.63) is generically a strongly
coupled field theory, and with infinitely many terms in the polynomial
expansion! This makes analyzing anything but the simplest of closed string
backgrounds a daunting task. A more fundamental problem is the lack of a
guarantee that this theory is Weyl invariant in the quantum level. Weyl
invariance plays a crucial role in the consistency of BRST quantization and
therefore in the very fabric of the critical bosonic string. Using standard
field theory techniques, it is possible to calculate the beta functions of the
nonlinear sigma model order by order in $\ell_{s}^{2}$. Weyl invariance
requires that the beta functions vanish, and so give constraints on the
allowed string backgrounds. Miraculously, the lowest order constraints
reproduce Einstein’s field equations for $G_{\mu\nu}$ (as well as equations of
motion for $B$ and $\phi$). Of course, there are also higher ordering
corrections in $\ell_{s}^{2}$ that affect the high energy physics (after all,
Einstein gravity is not a UV complete theory, so this better not be the end of
the story). Remarkably, the effective action obtained by imposing Weyl
symmetry can also be directly calculated from string theory amplitudes – and
the two expressions perfectly match [21].

####  1.7 String perturbation theory

Given the quantum closed string, we now want to introduce string interactions
via splitting and joining of strings. For strings in a general background
(with an asymptotic dilaton value $\Phi_{0}$) with action given by (I.1.63),
this is described by the Polyakov path integral

$Z=\sum_{g=0}^{\infty}\lambda_{0}^{2g-2}\int
DXDg\exp\left(-S_{G}[X;g]-S_{B}[X;g]\right)\,,$ (I.1.64)

where we have now explicitly included the sum over worldsheets of different
topology, which for 2d geometries reduces to a sum over discrete topologies
labeled by the genus $g\in\mathbb{N}$ (_i.e._ the number of holes). We are
only considering interactions among closed strings, for which the number of
worldsheet boundaries $b=0$. Note the distinction between the genus $g$ in the
sum and the worldsheet metric $g_{ab}$. From the Gauss-Bonnet theorem we see
that the background dilaton $\exp(\Phi_{0})$ and the bare coupling constant
$\lambda_{0}$ both contribute in the same fashion, _i.e._ as
$\exp(\Phi_{0})^{2g-2}$ and $\lambda_{0}^{2g-2}$ respectively. Thus, we can
combine them into a single renormalized coupling constant $\lambda$ and
redefine the dilaton field such that its VEV reads

$\lambda=\langle e^{\Phi}\rangle.$ (I.1.65)

Notice that (I.1.64) takes the form of a perturbative series expansion, where
$\lambda$ plays the role of the coupling “constant” of the theory. This is a
slight misnomer, since one key property of $\lambda$ is that it is not a free
parameter, bur rather is fixed dynamically. As we will see, this is a
universal feature of string theory backgrounds, whose parameters are generated
dynamically via the VEVs of scalar fields.

This is a remarkable feature of string theory! Even when we start with free
string theory ($\lambda_{0}=0$), the spectrum includes coherent backgrounds
where the perturbative description is given by strings with non zero coupling
$\lambda=\exp{\Phi_{0}}$. In other words, the free theory prescribes the
structure of the interacting theory!

##### Scattering states in conformal gauge

For asymptotically flat backgrounds, e.g. $G_{\mu\nu}=\eta_{\mu\nu}$, we can
define an S-matrix describing the scattering of closed string states. First,
let us revisit the BRST formalism for the closed string in conformal gauge.
Recall that an on-shell physical string state in this background is a weight
(0,0) $Q_{B}$ cohomology class that obeys the Siegel constraints. All states
in the class satisfy the level-matching conditions, and so the complete set of
constraints on any state $\ket{\psi}$ in the class is given by

$\displaystyle b_{0}\ket{\psi}=\overline{b}_{0}\ket{\psi}=0,$ (I.1.66)
$\displaystyle p^{2}=-m^{2}=4(1-N),\quad N\in\mathbb{N}\cup\\{0\\},$ (I.1.67)

where the mass-shell constraint now arises from BRST invariance. We can always
choose their representatives to be states of the form
$c_{1}\bar{c}_{1}\ket{V}$, where $c_{1}$ and $\bar{c}_{1}$ are free field
oscillators of the ghost fields $c(z)$ and $\bar{c}(\bar{z})$, and $\ket{V}$
is a weight (1,1) state built purely from the fields in the matter CFT (_i.e._
the oscillators of $X^{\mu}$). Using the state-operator correspondence, these
are dual to vertex operators of the form

$c(z)\bar{c}(\bar{z})V(z,\bar{z}),\quad
V(z,\bar{z})=\partial^{n_{1}}X(z)\cdots\bar{\partial}^{m_{1}}X(\bar{z})\cdots
e^{ip\cdot X(z,\bar{z})},$ (I.1.68)

where all operator products are assumed to be normal ordered. Note that BRST
invariance implies additional constraints on the tensor coefficients of these
operators.

It is straightforward to determine the vertex operators (states) at each level
in conformal gauge. For instance, the tachyon is represented by the vertex
operator

$V^{(0)}=e^{ip\cdot X},\quad p^{2}=4,$ (I.1.69)

Similarly, the massless states at level 1 are associated with the vertex
operator

$\displaystyle V^{(1)}=\epsilon_{\mu\nu}\partial
X^{\mu}{\bar{\partial}}X^{\nu}e^{ip\cdot X},\quad
p^{2}=0,\quad\epsilon_{\mu\nu}p^{\mu}=\epsilon_{\mu\nu}p^{\nu}=0,$ (I.1.70)

where $\epsilon_{\mu\nu}$ is a polarization tensor. Its transversality
property arises from BRST-invariance, and ensures that the longitudinal
degrees of freedom decouple – this is the expected result for massless quanta
of gauge fields in spacetime. Note that $V^{(1)}$ packages together the
graviton, dilaton, and B-field.

##### The string S-matrix

For a scattering process of $n$ strings, the worldsheet takes the form of some
compact manifold $\Sigma_{g}$ glued to a set of $n$ infinitely long cylinders
carrying the asymptotic closed string states. The Diff$\times$Weyl symmetry
can be used to map this surface to the same $\Sigma_{g}$, but with $n$
punctures (_i.e._ marked points). This transformation effectively carries out
the state-operator mapping (I.1.68), with the ghosts removed. Each such
operator has a natural pairing with $dz\wedge d\bar{z}$ which should be
inserted in the gauge-fixed version of (I.1.64). These objects have conformal
weight $(0,0)$, and so preserve our choice of conformal gauge. It turns out
that while $V$ itself is not BRST-invariant, its transformation is a total
derivative on the moduli space of inequivalent Riemann surfaces, and so its
integrated version is BRST-invariant. We therefore have that S-matrix elements
take the schematic form [22]

${\cal
A}_{n}\sim\sum_{g=0}^{\infty}\lambda^{2g-2}\left\langle\prod_{i=1}^{n}\int_{\Sigma_{g}}V_{i}(z_{i},\bar{z}_{i})dz_{i}\wedge
d\bar{z}_{i}\right\rangle.$ (I.1.71)

There are two primary issues with formula (I.1.71).

First, the summation over Riemann surfaces is incomplete. For example, for
$n=0$ and $g\neq 1$, it is not possible to take the metric to be flat
globally. However, we can make it flat in local patches with coordinates
$(\sigma^{1}_{i},\sigma^{2}_{i})$ that cover the whole worldsheet. Then the
transition functions between two overlapping coordinates
$(\sigma^{1}_{i},\sigma^{2}_{i})$ are given by conformal transformations. If
we consider the complex coordinates $(z_{i},\bar{z}_{i})$ associated with each
coordinate patch $(\sigma^{1}_{i},\sigma^{2}_{i})$, the conformal transition
maps between $\sigma$s take the form of holomorphic maps between the complex
coordinates of each patch. Therefore, we can think of the worldsheet as a 2
dimensional surface with a global complex structure, _i.e._ a Riemann surface.
However, not all Riemann surfaces of the same genus are equivalent. We can
label different Riemann surfaces by continuous parameters, called moduli,
whose values do not change under infinitesimal Diff$\times$Weyl
transformations. To account for this in (I.1.71), we must integrate over the
moduli space ${\cal M}_{g}$ of genus $g$ surfaces in addition to summing over
$g$. The real dimension of ${\cal M}_{g}$ is

$\text{dim}_{\mathbb{R}}\>{\cal M}_{g}=\begin{cases}\ 0\quad&\text{if }g=0,\\\
\ 2\quad&\text{if }g=1,\\\ \ 6g-6\quad&\text{if }g>1.\end{cases}$ (I.1.72)

For nonzero $n$, the moduli space includes the locations of the integrated
vertex operators giving $2n$ additional parameters – this is the moduli space
of genus $g$ Riemann surfaces with $n$ punctures.

Second, the Polyakov path integral in conformal gauge possesses some residual
gauge symmetry associated with global conformal transformations of
$\Sigma_{g}$. These transformations are generated by conformal Killing vectors
(CKVs), which are global vector fields $\xi^{a}$ that satisfy the conformal
Killing equation

$\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}=g_{ab}\nabla_{c}\xi^{c}.$ (I.1.73)

CKVs correspond to conformal transformations that do not change the geometry
of the surface. We can use (and eliminate) these extra gauge redundancies to
fix the position of several vertex operators.

The way in which the moduli and CKVs enter into the path integral relates
directly to the ghost path integral. The ghost action admits a $bc$ ghost
number symmetry which is anomalous on a general Riemann surface. The anomaly
requires that all non-vanishing correlation functions have ghost charge
$6g-6$, where $b$ has charge $-1$ and $c$ has charge $+1$. The result of
gauge-fixing then implies that every modulus comes with a $b$ insertion, while
every CKV comes with a $c$ insertion. Thus, for each complex CKV we insert
$c\bar{c}V(z,\bar{z})$ instead of $\int Vdz\wedge d\bar{z}$. Overall, this
result of gauge-fixing the string theory path integral can be viewed as the
string theory derivation of the Riemann-Roch theorem, which states that the
number of real moduli minus the number of CKVs of any Riemann surface is equal
to $6g-6$.

The precise nature of the $b$ ghost insertions traces back to the gauge-fixing
procedure. Recall that the moduli of $\Sigma_{g}$ are encoded in the gauge-
fixed metric $g_{ab}(t)$. For each modulus $t^{i}$, it can be shown that this
gauge-fixing procedure results in the insertion

$\displaystyle{\cal B}(t)=\frac{1}{2\pi}\int
d^{2}z\left(b_{zz}(\mu_{t})_{\bar{z}}^{\,\,z}+{b}_{\bar{z}\bar{z}}(\mu_{t})_{z}^{\,\,\bar{z}}\right)\,,$
(I.1.74)

where $\mu$ is the Beltrami differential; its components can be computed
directly in terms of the metric and its derivatives,

$\displaystyle(\mu_{t})_{a}^{\,\,\,b}=\frac{1}{2}g^{bc}(t)\frac{\partial}{\partial
t}g_{ac}(t),$ (I.1.75)

and can be viewed as deforming the complex structure of the worldsheet
($\bar{\partial}\rightarrow\bar{\partial}+\mu_{\bar{z}}^{z}\partial$) leading
to an insertion of $\int b_{zz}\mu_{\bar{z}}^{z}+c.c.$.

In summary, the $n$-point scattering amplitude of bosonic strings is given
by101010Moduli space integrals for multi-loop amplitudes are rigorously
derived in [23]. where $t_{i}$ are the moduli of genus $g$ surfaces and the
overall coefficient can be fixed by demanding unitarity. For genus $g$, there
are $3g-3$ complex moduli, or $6g-6$ real moduli.

####  1.8 Tree-level scattering

The discussion in the previous discussion is somewhat formal, so let us
consider some concrete examples of string scattering amplitudes. The simplest
string diagrams arise at tree level ($g=0$). The unique genus $0$ Riemann
surface is the Riemann sphere $\mathds{C}^{*}=\mathds{C}\cup\\{\infty\\}$
which has no moduli. The CKVs can easily be found by searching for global
vector fields $\xi(z)\partial_{z}$ and $\bar{\xi}(\bar{z})\partial_{\bar{z}}$
that are sufficiently regular at $z=\infty$. The solution $\xi(z)=a+bz+cz^{2}$
corresponds to 3 independent complex CKVs, which together generate the Mobius
group PSL(2,$\mathds{C}$) of fractional linear transformations

$z\to\frac{az+b}{cz+d},\quad ad-bc=1,\quad a,b,c,d\in\mathds{C}.$ (I.1.76)

We must therefore fix the locations of three vertex operators. The $n$-point
scattering amplitude thus takes the form [9]111111By treating the vertex
operator positions as moduli, the $n$-point amplitude can be written in a more
symmetric form solely in terms of integrated vertex operators. See [1] for
details.

$\displaystyle{\cal
A}^{(0)}_{n}=\left\langle\prod_{i=1}^{3}c(z_{i})\bar{c}(\bar{z}_{i})V_{i}(z_{i},\bar{z}_{i})\prod_{j=4}^{n}\int
d^{2}z_{j}V_{j}(z_{j},\bar{z}_{j})\right\rangle_{S^{2}},$ (I.1.77)

where $\langle\cdots\rangle_{S^{2}}$ indicates the correlation function on the
Riemann sphere in the full matter and ghost CFTs. For instance, the
$n=5$-point scattering amplitude is depicted in Figure I.1.2.

$\int V$$\int V$$c\bar{c}V$$c\bar{c}V$$c\bar{c}V$

Figure I.1.2: A five-point tree level amplitude requires three fixed vertex
operators $c\bar{c}V$ and two integrated vertex operators $\int d^{2}zV$ on
the Riemann sphere. The value of the amplitude is unaffected by the choice of
location of the fixed vertex operators.

As a warm up exercise, we consider the scattering of three tachyons. The
amplitude can be computed using any free field theory techniques of choice. Up
to an overall constant fixed by unitarity, the result is

$\displaystyle{\cal
A}^{(0)}_{T^{3}}=g_{s}\left\langle\prod_{i=1}^{3}c(z_{i})\bar{c}(\bar{z}_{i})e^{ip_{i}\cdot
X(z_{i},\bar{z}_{i})}\right\rangle_{S^{2}}\simeq
g_{s}\delta^{26}(p_{1}+p_{2}+p_{3}),$ (I.1.78)

Here, $g_{s}$ is the string coupling constant, which is proportional to the
dilaton VEV $\lambda$. More interesting is the four-tachyon amplitude,

$\displaystyle{\cal
A}^{(0)}_{T^{4}}=\left\langle\prod_{i=1}^{3}c(z_{i})\bar{c}(\bar{z}_{i})e^{ip_{i}\cdot
X(z_{i},\bar{z}_{i})}\int d^{2}ze^{ip_{4}\cdot
X(z,\bar{z})}\right\rangle_{S^{2}}\simeq
g_{s}^{2}\delta^{26}(p_{1}+p_{2}+p_{3}+p_{4})A(s,t,u).$ (I.1.79)

where $A$ is the famous Virasoro-Shapiro amplitude [24, 25]

$\displaystyle A(s,t,u)=\int
d^{2}z|z|^{-2u-4}|1-z|^{-2t-4}=2\pi\frac{\Gamma\left(-1-\frac{s}{4}\right)\Gamma\left(-1-\frac{t}{4}\right)\Gamma\left(-1-\frac{u}{4}\right)}{\Gamma\left(2+\frac{s}{4}\right)\Gamma\left(2+\frac{t}{4}\right)\Gamma\left(2+\frac{u}{4}\right)}.$
(I.1.80)

Here, $s$, $t$, and $u$ are the usual Mandelstam invariants

$\displaystyle s=-(p_{1}+p_{2})^{2},\quad t=-(p_{1}+p_{3})^{2},\quad
u=-(p_{1}+p_{4})^{2},$ (I.1.81)

which satisfy $s+t+u=4m_{T}^{2}$ for the tachyon mass squared $m_{T}^{2}=-4$.
The amplitude in (I.1.80) has many interesting properties, such as duality,
which we do not have time to properly cover. Most important is that the
4-point amplitude obeys the unitarity properties expected of an S-matrix
element [24]. In particular, we expect the 4-point diagram to factorize into
two 3-point diagrams along with a simple pole via a unitarity cut in the
$s$-channel (_i.e._ separating $p_{1},p_{2}$ from $p_{3},p_{4}$). Indeed, we
see that as an intermediate state goes on-shell with $s=m_{N}^{2}=4(N-1)$, the
4-point amplitude reduces to

$\displaystyle A(s,t,u)\xrightarrow[]{s=4(N-1)}\frac{P_{2N}(t)}{s-4(N-1)},$
(I.1.82)

where $P_{2N}(t)$ is a degree $2N$ polynomial in $t$ that accounts for which
spins up to $2N$ are exchanged, consistent with maximum spin at level $N$.

####  1.9 One-loop scattering

Now let us consider 1-loop amplitudes, which correspond to genus $g=1$. Such
worldsheets have the topology of a torus $T^{2}$. It is worthwhile to spend
some time discussing the geometry of the torus and the structure of its moduli
space. We can construct a torus as a quotient $\mathds{C}/\Lambda$, where
$\Lambda$ is a lattice generated from two basis vectors $e_{1}$ and $e_{2}$.
Its fundamental domain consists of the parallelogram with two of its sides
given by $e_{i}$, with opposite sides identified. As Riemann surfaces, two
tori are equivalent if they are related by a conformal transformation. For us,
this means a particular torus can be described by a single complex modulus
$\tau$, known as a Techmüller parameter, which corresponds to $e_{1}/e_{2}$.
Without loss of generality, such a torus can be constructed from the
associated parallelogram with sides given by $e_{1}=1$ and $e_{2}=\tau$, as
shown in Figure I.1.3.

$0$$1$$\tau$$\mathbb{R}$$i\mathbb{R}$

Figure I.1.3: The torus $T^{2}$ with complex structure $\tau$ as described by
the quotient $\mathds{C}/\Lambda$, where $\Lambda$ is the lattice generated by
$\\{1,\tau\\}$. The fundamental domain is shaded in green, with opposite sides
identified.

$i\mathbb{R}$$\tau$$\mathbb{R}$$1$$\tau+1$

Figure I.1.4: Different parameterizations of the same torus are related by the
action of the modular group $PSL(2.\mathbb{Z})$. A torus with modulus $\tau$
can also be described with modulus $\tau+1$. $0$$-1/2$$1/2$ Figure I.1.5: A
fundamental domain ${\cal F}$ of the modulus space of $T^{2}$. The two
vertical sides are identified, as are the two circular arcs. The image of
${\cal F}$ under $SL(2,\mathbb{Z})$ is the entirety of $\mathbb{H}$.

Naively, each torus can be parametrized by a modulus $\tau$ taking value in
the upper-half plane $\mathbb{H}$. However, there is in fact an equivalence
class of $\tau$ values which correspond to tori that can be mapped to one
another by different choice of basis for $\Lambda$. The group which relates
these different parameterizations is known as the modular group, which for
$T^{2}$ is isomorphic to $PSL(2,\mathbb{Z})$, and is generated by
$\tau\rightarrow\tau+1$ and $\tau\rightarrow-1/\tau$. The modular group acts
on a given modulus as121212The correct modular group is $PSL(2,\mathbb{Z})$
since a given element $A\in SL(2,\mathbb{Z})$ and its additive inverse $-A$
both yield the same transformed value $\tau^{\prime}$.

$\displaystyle\tau\longrightarrow\tau^{\prime}=\frac{a\tau+b}{c\tau+d},\quad\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in SL(2,\mathbb{Z}).$ (I.1.83)

To avoid over-counting in the path integral, we should really be integrating
over values of $\tau$ in a single fundamental domain of
$\mathbb{H}/PSL(2,\mathbb{Z})$ such as

$\displaystyle{\cal
F}=\left\\{\tau\in\mathbb{H}\,\bigg{|}-\frac{1}{2}\leq\text{Re}(\tau)\leq\frac{1}{2},|\tau|\geq
1\right\\},$ (I.1.84)

shaded in blue in Figure I.1.5.

We are now ready to talk about the genus 1 contribution to worldsheet
amplitudes. Recall that the nature of the ghost insertion is determined by the
number of conformal Killing group (CKV) and moduli. First, we find the number
of $c$-ghosts required. The conformal Killing group (CKG) of the torus,
$T^{2}$, is easy to describe: it consists of translations. There is thus a
single (complex) CKV, which leads to the insertion of a single pair $c\bar{c}$
in all scattering amplitudes. Intuitively, their location can be fixed using
the translation symmetry. Second, we work out the number of $b$-ghost
insertions and their form. Earlier, we found that the torus had a single
complex modulus $\tau=\tau_{1}+i\tau_{2}$. For convenience, we encode the
moduli dependence of $T^{2}$ in the metric,

$\displaystyle ds^{2}=|d\sigma^{1}+\tau d\sigma^{2}|^{2}$ (I.1.85)

such that the coordinates maintain the usual periodicity properties

$(\sigma^{1},\sigma^{2})\sim(\sigma^{1},\sigma^{2})+(2\pi,2\pi).$ (I.1.86)

The associated $b$-ghost insertions are then given by

$\frac{1}{2}{\cal B}_{\tau}{\cal B}_{\bar{\tau}}=\frac{i}{4\pi\tau_{2}}\int
d^{2}zb(z)\frac{i}{4\pi\tau_{2}}\int d^{2}\bar{w}\bar{b}(\bar{w}),$ (I.1.87)

where the factor of 1/2 accounts for the fact that $d\tau
d\bar{\tau}=2d\tau_{1}d\tau_{2}$. Note that
$\text{Vol}(T^{2})=(2\pi)^{2}\tau_{2}$. From our previous discussion, the
integration range of $\tau$ should be restricted to the chosen fundamental
domain ${\cal F}$; we must also include an extra factor of 1/2 in the measure
to account for the $\mathbb{Z}_{2}$ in the CKG. Putting everything together,
the 1-loop scattering amplitude takes the form

$\displaystyle{\cal A}_{n}^{(1)}=\int_{\cal
F}\frac{d^{2}\tau}{2(4\pi\tau_{2})^{2}}\int d^{2}zd^{2}w\left\langle
b(z)\bar{b}(\bar{w})c\bar{c}V_{1}(z_{1},\bar{z}_{1})\prod_{j=2}^{n}\int_{T^{2}}d^{2}z_{j}V_{j}(z_{j},\bar{z}_{j})\right\rangle_{T^{2}},$
(I.1.88)

where $\langle\cdots\rangle_{T^{2}}$ indicates the path integral over the
fields on $T^{2}$ with implicit modulus $\tau$.

The 1-point amplitude can be recast in a more symmetric form by recognizing
that the path integral associated to the correlation function in (I.1.88) is
independent of the ghost positions: this permits the substitution $\int
d^{2}zb(z)=2\text{Vol}(T^{2})b(0)$ for both $b$-ghost insertions. Since the
full CKG $T^{2}\rtimes\mathbb{Z}_{2}$ is finite (with volume
$2\text{Vol}(T^{2})$), it is equivalent to inserting $c$-ghosts at arbitrary
positions, using only integrated vertex operators, and then dividing by the
volume of the CKG to account for the gauge redundancy. The result is

$\displaystyle{\cal A}_{n}^{(1)}=\int_{\cal
F}\frac{d^{2}\tau}{4\tau_{2}}\left\langle
b(0)\bar{b}(0)c(0)\bar{c}(0)\prod_{j=1}^{n}\int_{T^{2}}d^{2}z_{j}V_{j}(z_{j},\bar{z}_{j})\right\rangle_{T^{2}}.$
(I.1.89)

##### The torus partition function

Consider the vacuum amplitude ${\cal A}^{(1)}_{0}$, which for the torus is
well-defined and physically meaningful. We will temporarily ignore the moduli
space integration; the integrand

$Z_{T^{2}}(\tau)=\langle b(0)\bar{b}(0)c(0)\bar{c}(0)\rangle_{T^{2}}$ (I.1.90)

is simply the path integral over the matter and ghost fields with some extra
ghost insertions to saturate the fermionic zero modes. In the QFT language,
this is often referred to as the torus partition function. It is most readily
calculated using the path integral formalism for states in the Hilbert space.
In this formalism, the path integral on a (Euclidean) cylinder of length $T$
simply computes the overlap $\langle\psi_{f}|e^{-HT}|\psi_{i}\rangle$, where
$H$ is the Hamiltonian and $\ket{\psi_{i,f}}$ are the final and initial
states, respectively, which appear through the boundary conditions implemented
on each end of the cylinder. This line of reasoning extends to the torus,
which can be thought of as a cylinder of length $2\pi\tau_{2}$ whose
boundaries are glued after a $2\pi\tau_{1}$ twist. This twist is accounted for
by inserting $e^{2\pi i\tau_{1}P}$ in the overlap, where $P$ is the generator
of translations along $\sigma^{1}$, _i.e._ the momentum. The two operators $H$
and $P$ generate the isometries of the cylinder, and can be expressed in terms
of the Virasoro generators as

$H=L_{0}+\overline{L}_{0}\,,\qquad\text{and}\qquad
P=L_{0}-\overline{L}_{0}\,.$ (I.1.91)

Finally, the operation of gluing becomes the trace over all states in the
Hilbert space $\mathcal{H}$ with periodic boundary condition for ghosts. The
path integral can thus be rewritten as

$Z_{T^{2}}(\tau)=\text{Tr}_{\mathcal{H}}\left((-1)^{F}b_{0}\bar{b}_{0}c_{0}\bar{c}_{0}q^{L_{0}}\overline{q}^{\overline{L}_{0}}\right)\,,\quad
q=e^{2\pi i\tau}\,,\quad\overline{q}=e^{-2\pi i\overline{\tau}}\,,$ (I.1.92)

where $(-1)^{F}$, by definition, anti-commutes with the ghost fields and
commutes with the matter fields (this determines its action on all of the
states via the state-operator mapping). The full vacuum amplitude is given by
reintroducing the moduli space integration:

$\displaystyle{\cal A}_{0}^{(1)}=\int_{\cal
F}\frac{d^{2}\tau}{2\tau_{2}}\text{Tr}_{\mathcal{H}}\left((-1)^{F}b_{0}\bar{b}_{0}c_{0}\bar{c}_{0}q^{L_{0}}\overline{q}^{\overline{L}_{0}}\right).$
(I.1.93)

While we now have all the ingredients to compute the vacuum amplitude for the
bosonic string, we pause briefly to flesh out some of details of the CFT
partition function; this will come in handy later when more general matter
CFTs arise, such as in string compactifications. The full state space of the
bosonic string splits into the direct sum
$\mathcal{H}_{m}\oplus\mathcal{H}_{gh}$, where the subscripts denote the
matter and ghost spaces, respectively. Consequently, the partition function
factorizes into separate ghost and matter contributions. The matter CFT
contributes

$Z_{m}(\tau)=\text{Tr}_{\mathcal{H}_{m}}\left(q^{L_{0}-c/24}\overline{q}^{\overline{L}_{0}-\bar{c}/24}\right)\,,$
(I.1.94)

where the central charges $c=\bar{c}=26$ now contribute due to the conformal
anomaly on the cylinder. This equation is often the more familiar one in the
CFT context, and is equal to $\langle 1\rangle_{T^{2}}$ when there is no
possibility for fermionic zero modes. Due to the trace, the partition function
above clearly gives a sum over the matter states, weighted by the conformal
weights assigned to each state. One might initially expect that such a sum is
unconstrained, though thanks to string theory we know this to be untrue. Given
a parametrization $\tau$ of the torus, there is a whole family of equivalent
moduli generated by the action of $PSL(2,\mathbb{Z})$. Since the measure
$\frac{d^{2}\tau}{\tau_{2}}$ is invariant under this action, we therefore
expect that $Z(\tau)$ should generically be modular invariant as well in any
consistent string theory.131313Modular invariance at the level of the CFT
translates to invariance of the worldsheet string theory under large
Diff$\times$Weyl gauge transformations.

Modular invariance has dramatic consequences at the level of the states in the
CFT (and by extension the string spectrum). Using the trace form of $Z(\tau)$
reveals constraints on the operator spectrum. For instance, invariance under
$\tau\to\tau+1$ implies that all local operators must have integer spins,
_i.e._ $h-\bar{h}\in\mathbb{Z}$. Invariance under $\tau\to-1/\tau$ constrains
the high energy spectrum in terms of the low energy data, which leads to
various universal features such as Cardy’s formula for the density of states.

We now return to the calculation of the closed string vacuum amplitude in
(I.1.89), focusing first on the CFT partition function. The matter oscillators
contribute

$\text{Tr}^{\prime}_{\mathcal{H}_{m}}q^{L_{0}-26/24}\bar{q}^{\bar{L}_{0}-26/24}=\left(|q|^{-2/24}\prod_{n=1}^{\infty}|1-q^{n}|^{-2}\right)^{26}=\eta^{-26}\bar{\eta}^{-26},$
(I.1.95)

where $\eta=q^{1/24}\prod_{n}(1-q^{n})$ is the Dedekind eta function. The
trace over zero modes (momentum eigenstates) gives

$iV_{26}\int\frac{d^{26}p}{(2\pi)^{26}}(q\bar{q})^{p^{2}/4}=iV_{26}\int\frac{d^{26}p}{(2\pi)^{26}}e^{-\pi\tau_{2}p^{2}}=iV_{d}(4\pi^{2}\tau_{2})^{-13},$
(I.1.96)

where $V_{26}$ is the 26-dimensional spacetime volume. The factor of $i$ comes
from Wick rotating the $X^{0}$ field. The trace over ghost oscillators gives
the additional factor

$\text{Tr}_{\mathcal{H}_{gh}}(-1)^{F}b_{0}\bar{b}_{0}c_{0}\bar{c}_{0}q^{L_{0}+26/24}\bar{q}^{\bar{L}_{0}+26/24}=\eta^{2}\bar{\eta}^{2},$
(I.1.97)

which serves to cancel the two longitudinal degrees of freedom. This is
consistent with light-cone quantization, where only the transverse degrees of
freedom are dynamical. In addition, it is important to note that the
cancellation between longitudinal and ghost degrees of freedom is only
possible for a single timelike direction in the target space [26]. In a
unitary quantum theory, the Hilbert space of states by definition should have
a positive-definite inner product. Consider the usual worldsheet string theory
with 26 bosons and the $bc$ ghost system, but now suppose $r$ of the bosons
are timelike. The space of states for this theory includes negative norm
states due to the timelike and ghost oscillators that should drop out when we
project to the BRST cohomology. Therefore, in a unitary string theory the
partition function should be unaffected by the insertion of $(-1)^{\sigma}$
into the trace, where $(-1)^{\sigma}=+1$ for spacelike oscillators and
$(-1)^{\sigma}=-1$ for timelike. The ghost oscilators have signature $(1,1)$.
Thus, nonzero modes with this factor contribute

$\text{Tr}^{\prime}_{\mathcal{H}}\left((-1)^{\sigma}(-1)^{F}b_{0}\bar{b}_{0}c_{0}\bar{c}_{0}q^{L_{0}}\bar{q}^{\bar{L}_{0}}\right)=\left|\frac{\prod_{n}(1-q^{n})(1+q^{n})}{\prod_{n}(1-q^{n})^{26-r}(1+q^{n})^{r}}\right|^{2}=|(1-q^{n})^{-24}|^{2},$
(I.1.98)

where the last equality follows assuming unitarity. Therefore, we must take
$r=1$ and so there is only a single timelike direction compatible with
positive definite inner product on BRST cohomology states.

Combining everything yields the vacuum amplitude [27]

${\cal A}_{0}^{(1)}=iV_{26}\int_{\cal
F}\frac{d^{2}\tau}{2\tau_{2}}(4\pi^{2}\tau_{2})^{-13}|\eta|^{-48}.$ (I.1.99)

Given that ${d^{2}\tau}/{\tau_{2}}$ and $\tau_{2}|\eta|^{4}$ are themselves
modular invariant, clearly the full amplitude is as well. Note that the region
$\tau_{2}\to 0$ is absent from our domain of integration. This regime
describes ultraviolet (UV), or high energy, processes. Without restricting the
$\tau$ integration to ${\cal F}$, the integration would produce a UV
divergence due to the $\tau_{2}\to 0$ region. Unlike in QFT, this type of
divergence is not present for the string vacuum amplitude. The modular
invariance of the torus effectively acts as a UV cutoff that renders the
theory UV finite at one loop. This behavior turns out to be true for general
string amplitudes, and thus perturbation of the closed bosonic string is
expected to be UV finite. That being said, if we expand the integrand as a
power series in $q$, we find a term that behaves like $q^{-1}$. This is due to
the tachyon, and ultimately gives rise to an IR divergence. This is the
failure of perturbation theory and so bosonic string amplitudes are only
formally defined. Our efforts have not truly ended in failure, however, since
these IR divergences will turn out to be absent for certain superstring
theories.

###  2 Bosonic string compactifications

Up to now we have considered closed strings propagating in flat Minkowski
spacetime. However, with the NLSM worldsheet theory introduced in Section 2.2,
it is natural to consider more general target spaces for the worldsheet
fields. For example, it could be that the target space of the worldsheet
theory has a component of $\mathds{R}^{k-1,1}$ where the $SO(1,k-1)$ isometry
of $\mathds{R}^{k-1,1}$ extends to an exact global symmetry of the worldsheet
theory. Such a theory could potentially describe a string theory in a $k$
dimensional Minkowski spacetime. We can go one step further and replace
$\mathds{R}^{k-1,1}$ with a space with non-zero curvatures associated with
$G_{ij}$ and $B_{ij}$. Such a theory can potentially describe a theory in a
curved spacetime with torsion! In general we can think of any modular
invariant conformal field theory with zero total central charge as a string
theory. Depending on the structure of the target space of the theory and its
global symmetries, the theory may or may not have a spacetime interpretation.
In the following we review some important examples of 2d CFTs and methods to
construct them.

####  2.1 Geometric CFTs

Circling back to the Minkowski spacetime, an important class of string theory
backgrounds arise from worldsheet theories with the form of a product CFT

$\mathbb{R}^{k-1,1}\times\mathscr{C}_{c=26-k},$ (I.2.1)

where $\mathbb{R}^{k-1,1}$ refers to the $X$ CFT with $k$ free noncompact
scalars, and $\mathscr{C}_{c=26-k}$ is some compact CFT with central charge
$c=26-k$. The condition $c=26-k$ ensures that after taking the ghost theory
into account, the total central charge vanishes. Often times this auxiliary
CFT can be expressed as a NLSM with a compact target space (though in general
the compact CFT need not have a geometric interpretation). Sometimes these
backgrounds can be thought of as $k$ large spacetime dimensions with $26-k$
curled up dimensions where the spectrum of $\mathscr{C}$ is discrete. In this
subsection we study some examples where this geometric interpretation is
explicit in the sense that the $\mathscr{C}_{c=26-k}$ theory is a $26-k$
dimensional NLSM. These theories are often called compactifications of the
$26$ dimensional theory and the target space of the NLSM is called the
internal geometry. Typically, the geometric parametries such as the size of
the cycles become dynamical fields. Thus, the case where the $26$ dimensions
are flat corresponds to a particular limit of the field space where the cycles
have infinite radii.

The simplest string compactification that we can consider is one with a one-
dimensional internal geometry, a circle. In what follows, we will explain some
aspects regarding such theories.

##### Kaluza-Klein mechanism

First recall field theories coupled to classical gravity compactified on a
circle of radius $R$. Suppose the spacetime has $d+1$ dimensions where
$x^{\mu}$ for $\mu\in\\{0,...,d-1\\}$ are non-compact while $x^{d}$ is the
compact coordinate with the periodicity condition $x^{d}\sim x^{d}+2\pi R$.
Since the spacetime momentum $p_{d}$ is generator of translation and a
translation of $2\pi R$ along the circle is identity.

$\displaystyle e^{i2\pi R\leavevmode\nobreak\ \hat{\bf
p}_{d}}\equiv\mathds{1}.$ (I.2.2)

Therefore, the eigenvalues of momentum along the circle $p_{d}=n/R$ are
quantized with $n\in\mathbb{Z}$. Moreover, we can decompose a massless $d+1$
dimensional field $\phi$ into an infinite set of Fourier modes along the
compact dimension.

$\displaystyle\phi(x^{0},...,x^{d})=\sum_{n\in\mathbb{Z}}\phi_{n}(x^{0},...,x^{d-1})e^{in(x^{d}/R)}.$
(I.2.3)

The coefficients $\phi_{n}$ only depend on the non-compact coordinates and
hence are $d$ dimensional fields. The spacetime excitation of $\phi_{n}$ has
$p_{d}=n/R$. And the $(d+1)$-dimensional mass formula

$\displaystyle E^{2}=\sum_{\mu\leq d}p_{\mu}p^{\mu}$ (I.2.4)

becomes

$\displaystyle E^{2}=n^{2}/R^{2}+\sum_{\mu\leq d-1}p_{\mu}p^{\mu}.$ (I.2.5)

In other words, the fields develop nonzero masses via the compact momentum,
_i.e._

$\displaystyle m^{2}=n^{2}/R^{2}.$ (I.2.6)

Such modes are ubiquitous in general compactifications, where they are
referred to as Kaluza-Klein (KK) modes. In string theory, circle
compactifications are more interesting because the string can wind multiple
times around the circle.

##### The compactified worldsheet theory

The worldsheet theory consists of 25 noncompact scalars $X^{\mu}$ as well as a
periodic scalar $X^{25}:\Sigma\to S^{1}_{R}$. For a string which winds $w$
times around the circle we have

$\displaystyle X^{25}(\sigma+2\pi,\tau)=X(\sigma,\tau)+2\pi Rw.$ (I.2.7)

These winding modes are topologically distinct and can be thought of as
solitons of the worldsheet theory. The spacetime momentum is quantized
according to

$\displaystyle
p^{25}=\frac{1}{2\pi}\int_{0}^{2\pi}d\sigma\partial_{\tau}X^{25}=\frac{n}{R}.$
(I.2.8)

We must include both the KK and winding modes to ensure modular invariance on
the torus. The momentum and winding constraints imply that the mode expansion
of $X^{25}$ on the cylinder takes the form [28, 29]

$\displaystyle
X^{25}(\sigma,\tau)=wR\sigma+\frac{n}{R}\tau+\frac{i}{\sqrt{2}}\sum_{n\neq
0}\left(\frac{\alpha_{n}^{25}}{n}e^{-in(\sigma+\tau)}+\frac{\overline{\alpha}^{25}_{n}}{n}e^{in(\sigma-\tau)}\right),$
(I.2.9)

where $\alpha^{25}_{n},\overline{\alpha}^{25}_{n}$ are oscillator modes that
obey the usual free oscillator algebra. The quantized theory is very similar
to that of the noncompact free boson, except for one crucial difference.
Previously, the ground state of the worldsheet theory had a continious
degenerecy labeled by $26$ dimensional momentum of the Tachyon. One can think
of the degenrate vacua as a family of states that transform to each other
under the spacetime Lorentz transformations which are global symmetries of the
worldsheet. Now that one of the components of the momentum is quantized, the
worldsheet ground states are labeled by $\ket{p^{\mu};n,w}$. The integer $n$
is associated with the quantization of momentum which reflects the symmetry of
the worldsheet theory generated by discrete translation along $X^{25}$. But
how about $w$? Could it be that the presence of another integer signals
additional features?

The answer is yes, and global symmetries of the worldsheet lead to gauge
symmetry in the spacetime! The pair $(n,w)$ are in fact charges of a
$U(1)_{n}\times U(1)_{w}$ global symmetry on the worldsheet. In addition to
the massless string states that give the 25-dimensional graviton, B-field, and
dilaton, we now have additional massless vectors generated by

$\displaystyle\bar{\alpha}_{-1}^{[25}\alpha_{-1}^{\mu]}\ket{p^{\mu};0,0}\,,\quad\bar{\alpha}_{-1}^{(25}\alpha_{-1}^{\mu)}\ket{p^{\mu};0,0}\,.$
(I.2.10)

These states are the “photons” of the two $U(1)$ factors. From the 26d point
of view, the gauge field for $U(1)_{n}$ arises from the metric $G_{25\mu}$,
while that of $U(1)_{w}$ arise from the B-field $B_{25\mu}$. Clearly the
$U(1)_{w}$ symmetry is a purely stringy effect, since strings not particles
are charged under the B-field. Note that there is another massless state
$\bar{\alpha}_{-1}^{25}\alpha_{-1}^{25}\ket{p}$ that leads to a massless
scalar in spacetime known as a moduli field. It arises from $G_{2525}$, and so
changes the size of the circle, which is dynamical.

Another difference in the worldsheet theory of the compact boson is that the
oscillators $\alpha_{0}^{25}$ and $\overline{\alpha}_{0}^{25}$ are distinct.
We can see this by writing the zero modes as

$X^{25}(\sigma,\tau)=\frac{1}{2}[p_{L}(\tau+\sigma)+p_{R}(\tau-\sigma)],$
(I.2.11)

where we have introduced the left/right-moving momenta

$\alpha_{0}^{25}p_{L}=\frac{n}{R}+wR,\quad\bar{\alpha}_{0}^{25}=p_{R}=\frac{n}{R}-wR.$
(I.2.12)

These quantities naturally form a vector

$\displaystyle l=\frac{1}{\sqrt{2}}(p_{L},p_{R})$ (I.2.13)

which generates a 2d lattice $p(1,0)+q(0,1)$ for integers $p,q$. The lattice
has some special properties in terms of the following inner product.

$\displaystyle l^{1}\cdot l^{2}=l_{L}^{1}l_{L}^{2}-l_{R}^{1}l_{R}^{2}.$
(I.2.14)

With this norm we can check that $l\cdot l^{\prime}\in\mathbb{Z}$. Such
lattices are called integral. Moreover, the norm

$\displaystyle l\cdot l=2nw\in 2\mathbb{Z}.$ (I.2.15)

is an even integer. Lattices with this property are referred to as _even_.
Moreover, if $l^{\prime}\cdot l\in\mathbb{Z}$ for all $l$ in the lattice, then
$l^{\prime}$ is also in the lattice. This implies that the lattice is _self-
dual_. Such lattices are called Narain lattices [30].

The inner product (I.2.14) is preserved under $SO(1,1)$. Therefore, we can
find a Narain lattice by applying elements of $SO(1,1)$ on another one. We can
generalize this to higher dimensional lattices. Suppose we have a Narain
lattice where $p_{L}$ takes values in a $p$ dimensional lattice and $p_{R}$ in
a $q$ dimensional lattice such that the overall lattice is even self-dual. We
denote such a lattice by $\Gamma^{p,q}$. Applying every element of $SO(p,q)$
to a Narain lattice gives another Narain lattice. However, not all of these
transformations lead to distinct lattices. For example, if we act on $p_{L}$
and $p_{R}$ separately with elements of $SO(p)$ and $SO(q)$ respectively, the
lattices will not change, and they simply reflect a change of coordinates. In
fact, up to a discrete quotient, the moduli space of $(p,q)$ Narain lattices
is $\frac{SO(p,q)}{SO(p)\times SO(q)}$. The dimension of this space is $pq$.
In the special case of $p=q=1$, this space is can be parametrized by one
variable which is the radius $R$. The radius $R$ serves as the $SO(1,1)$ boost
parameter.

##### T-duality

The self-dual property of the lattice has important physical implications.
Consider the $\mathbb{Z}_{2}$ self-dual transformation

$R\rightarrow\frac{1}{R},\quad n\leftrightarrow w,$ (I.2.16)

which can be recast as the map

$p_{L}\rightarrow p_{L},\quad p_{R}\rightarrow-p_{R}.$ (I.2.17)

The invariance of string theory under the duality transformation (I.2.16) is
referred to as T-duality [31].

Instead of $R$, we can consider a parameter $\lambda$ more closely related to
the moduli field,

$R=e^{\lambda}.$ (I.2.18)

T-duality now takes the form $\lambda\to-\lambda$. Under T-duality, the radius
of the circle is no longer an invariant notion. More surprisingly, very small
radii are mapped to large radii, which implies there is no absolute notion of
distance. To resolve these issues, we must consider what it means to have a
circle of small radius. From the quantum viewpoint, it is unclear how to
resolve distances which are smaller than the size of a typical wavepacket. It
is natural then to define the physics of small radii in terms of the dual
large radii theory.

In other words, as seen in Figure I.2.1, the space of inequivalent theories is
the half-line $R\geq 1$. Equivalently, we could take the set of inequivalent
theories to be $0\leq R\leq 1$, but it is more natural to think in terms of
the larger of the two equivalent radii: momenta continua are more familiar
than winding number continua. In particular, questions of locality are clearer
in the larger radius picture. For this parametrization, there is no radius
smaller than the self-dual radius:

$R_{self-dual}=R_{\text{SU}(2)\times\text{SU}(2)}=\sqrt{\alpha^{\prime}}.$
(I.2.19)

Clearly $\lambda=0$ is invariant under this transformation and hence labels
the self-dual point. This allows us to simply the diagram of inequivalent
theories in terms of $\lambda$ as in Figure I.2.1.

SU(2)$\times$SU(2) self dual point$\lambda=0$ Figure I.2.1: The space of
compactified bosonic string theories on $S^{1}$. Here, $\lambda=0$ is the
SU(2)$\times$SU(2) self-dual point, so only theories with $\lambda\geq 0$ are
considered inequivalent.

The self-dual point is related to the emergence of new symmetries. As it turns
out, at the self-dual point $R=\alpha^{\prime}/R$ the duality transformation
becomes a bonafide symmetry! In particular, there is an emergent
SU(2)$\times$SU(2) gauge symmetry in spacetime, as shown in Figure I.2.2.

SU(2)$\times$SU(2) gauge symmetry$R^{2}=1$ Figure I.2.2: At the self-dual
point $R=\sqrt{\alpha^{\prime}}$, there is an SU(2)$\times$SU(2) gauge
symmetry.

It is fairly straightforward to derive the emergent symmetry from the
worldsheet theory, where we expect the appearance of new conserved currents.
The mass-shell conditions for the compactified theory at arbitrary radius are
[1]

$\displaystyle\frac{1}{2}p_{L}^{2}$
$\displaystyle-\frac{1}{2}p_{R}^{2}+\frac{4}{\alpha^{\prime}}(N_{L}-N_{R})=0$
(I.2.20) $\displaystyle\frac{1}{2}m^{2}$
$\displaystyle=\frac{1}{2}p_{L}^{2}+N_{L}-1=\frac{1}{2}p_{R}^{2}+N_{R}-1.$
(I.2.21)

For the self-dual radius $R=\sqrt{\alpha^{\prime}}$ and $m^{2}=0$, these take
the simplified form

$\displaystyle 0=n^{2}+w^{2}+2(N_{L}+N_{R}-2)$ (I.2.22)

Let us focus on a single sector, say the left-moving one with $N_{L}>0$ and
$N_{R}=0$. Then the massless states are given by

$\displaystyle(n,w)=(1,1),\quad N_{L}=1,N_{R}=0\,,$ (I.2.23)
$\displaystyle(n,w)=(-1,-1),\quad N_{L}=1,N_{R}=0.$ (I.2.24)

These two states correspond to the left-moving $W^{\pm}$ bosons! The right-
moving states are recovered from $N_{R}=1$ and $N_{L}=0$. The $Z$ bosons arise
from states corresponding to the aforementioned $U(1)$ fields that emerge due
to the KK compactification at any radius. Altogether, these particles combine
into two independent SU(2) triplets.

There is a nice connection between the duality group $\mathbb{Z}_{2}$ and the
gauge group $SU(2)$. At the self-dual point, the duality group becomes a
symmetry. Therefore, the $\mathbb{Z}_{2}$ must map a representation of $SU(2)$
to another representation of $SU(2)$ with same dimensions. Since every
representation is specified by its weights, the symmetry group
$\mathbb{Z}_{2}$ must map a weight lattice of $SU(2)$ to another weight
lattice. In fact, the $\mathbb{Z}_{2}$ is the Weyl group of reflection
symmetries of the root lattice of $SU(2)$. This connection extends to many
other examples. At the self-dual point, a subgroup of the duality group is the
symmetry group of the root lattice of the emergent gauge group.

###### Optional exercise

: Consider a compactification on the torus $T^{n}(G,B)$. Show that the
partition function takes the form

$Z=\frac{\sum
q^{\frac{1}{2}p_{L}^{2}}\bar{q}^{\frac{1}{2}p_{R}^{2}}}{\eta^{n}\bar{\eta}^{n}},\quad\text{for}\quad\eta=q^{1/24}\prod_{n}(1-q^{n}).$
(I.2.25)

Check that it is invariant under modular transformations, _i.e._
$\tau\to\tau+1$ and $\tau\to-1/\tau$. Show that invariance under
T-transformations implies that the lattice of momenta is even, and invariance
under S-transformations implies that the lattice is self-dual.

Exercise 1:  Suppose we want to compactify $d$ dimensions where $d$ is the
rank of an $ADE$ group $G$. Show that you can choose the background fields
$B_{ij}$ and $G_{ij}$ such that the resulting lattice of $(P_{L},P_{R})$ of
the momenta in the compact directions is a weight lattice $(w,w^{\prime})$ of
$G$ where the difference $w-w^{\prime}$ is in the root lattice of $G$. Using
this, show that this lattice is even and self dual. Now show that we can
choose the size of the compact dimensions such that the compactified theory
has a $G\times G$ gauge symmetry.

####  2.2 WZW models and current algebras

More generally, we can talk about geometric string compactifications via NLSMs
with an arbitrary spacetime metric and H-flux where $H=dB$. For instance, we
can take the target space to be some group manifold $G$. The NLSM action can
be rewritten in Wess-Zumino-Witten (WZW) form [32]

$S=-\frac{k}{8\pi}\int_{\Sigma=\partial
M^{3}}\text{Tr}\left(g^{-1}dg\right)^{2}+ikS_{WZ},$ (I.2.26)

where the target space coordinates are embedded in the group element
$g=\exp(iT^{a}X^{a})$ with $T^{a}$ the generators of the Lie algebra $Lie(G)$.
The Wess-Zumino term [33]

$S_{WZ}=\frac{1}{24\pi}\int_{M^{3}}\text{Tr}(g^{-1}dg)^{3}$ (I.2.27)

arises from the B-field contribution, rewritten in terms of the H-flux via
Stoke’s theorem. Both terms in the WZW action are necessary to preserve
conformal invariance to $O(\alpha^{\prime 2})$ [34].

###### Optional exercise

: Show that the Wess-Zumino term arises from the coupling of the B-field to
the worldsheet.

##### Current algebras

The WZW action is invariant under the the transformation

$g\to LgR^{-1},$ (I.2.28)

where $L$ and $R$ lie in separate copies $G_{L}$ and $G_{R}$ of the group
manifold. The theory thus admits a $G_{L}\times G_{R}$ global symmetry with
conserved currents

$J^{i}(z)=\text{Tr}\left(\partial
gg^{-1}T^{a}\right),\quad\bar{J}^{j}(\bar{z})=\text{Tr}\left(\bar{\partial}gg^{-1}T^{a}\right).$
(I.2.29)

In the quantum theory, the WZ term leads to an infinite-dimensional
enhancement of this global symmetry. The currents satisfy a particular type of
OPE [35]

$J^{a}(z)J^{b}(w)\sim\frac{k\delta^{ab}}{(z-w)^{2}}+\frac{if^{abc}J^{c}}{z-w},$
(I.2.30)

which leads to an extension of the Virasoro algebra known as a current
algebra. Here, $f^{abc}$ are the structure constants for $Lie(G)$ and the
value $k$ is quantized due to the Dirac quantization condition for $B$. As it
is an extension of Virasoro, the currents completely specify the stress tensor
and its associated central charge

$c=\frac{k\ \text{dim }G}{k+\hat{h}_{G}},$ (I.2.31)

where $\hat{h}_{G}$ is the dual Coxeter number for $Lie(G)$. The classical
limit $k\to\infty$ corresponds to the case where the group manifold becomes
flat. In this limit, $c\sim\text{dim}\>G$, coinciding with the fact that the
theory describes $\text{dim}\>G$ free bosons.

Exercise 2: Show that $rank(G)\leq c\leq dim(G)$. For simply laced the left
hand side saturates.

As a simple example, we consider a $\hat{\mathfrak{u}}(N)$ current algebra at
level $k=1$ constructed from $N$ complex Weyl fermions $\psi^{a}$. The
worldsheet action on the complex plane takes the form

$\displaystyle S=\int d^{2}z\bar{\psi}^{\bar{a}}\bar{\partial}\psi^{a},$
(I.2.32)

which have a singular OPE

$\displaystyle\Psi^{a}(z)\Psi^{\bar{b}}(0)\sim\frac{\delta^{a\bar{b}}}{z}.$
(I.2.33)

Despite notation, both $\Psi$ and $\bar{\Psi}$ are holomorphic fields. Their
stress tensor is given by

$\displaystyle
T=-\frac{1}{2}\delta^{a\bar{b}}\Psi^{a}\partial\bar{\Psi}^{\bar{b}}$ (I.2.34)

which fixes the central charge to be $c=N$, _i.e._ each complex fermion
contributes $c=1$. Note that $\bar{T}=0$ and so $\bar{c}=0$ as well. The
theory is naturally invariant under $U(N)$ rotations, under which $\Psi$
($\bar{\Psi}$) transforms as the fundamental representation (anti-
fundamental). The associated conserved currents take the form

$\displaystyle J^{a\bar{b}}=\Psi^{a}\Psi^{\bar{b}}.$ (I.2.35)

It is a straightforward exercise to verify they satisfy the
$\hat{\mathfrak{u}}(N)$ current algebra with $k=1$.

##### Coset models

Another collection of models we can consider are the so-called _coset models_
(gauged WZW models) [36]. Consider a WZW model with group $G$, stress tensor
$T_{G}$, and central charge $c_{G}$. Supposing that $G$ contains some subgroup
$H\subset G$, there is an additional set $(T_{H},c_{H})$ for the WZW model for
$H$. We can then “gauge out” $H$ by constructing a new theory with a stress
tensor and central charge given by

$\displaystyle T_{G/H}=T_{G}-T_{H}\,,\quad c_{G/H}=c_{G}-c_{H}.$ (I.2.36)

This corresponds to taking the original WZW action and gauging the subgroup
$H$.

####  2.3 Orbifolds

Thus far we have studied toroidal compactifications of string theory, with the
key stringy ingredient being the emergence of winding modes, _i.e._ the string
wrapping various one-cycles. We also saw that in general it may be necessary
to turn on H-flux (in addition to the nontrivial metric) to preserve conformal
invariance on the worldsheet. In this section, we consider another example of
worldsheet CFT, orbifolds, which are manifolds with singular curvature [37,
38, 39].

##### Orbifold CFTs

Our first experience with orbifolds will be through _orbifold CFTs_ , which
arises from gauging discrete worldsheet symmetries. As we will see, the
singular geometry of the target space will be emergent from this point of
view.

Consider a unitary CFT invariant under some discrete symmetry group $G$, which
we label by CFTG. By definition, the group is represented unitarily, _i.e._
the theory admits a $G$-action on states of the form

$\displaystyle g:\ket{\psi}\to U(g)\ket{\psi},\quad g\in G,$ (I.2.37)

where $U(g)$ is a unitary operator. When there is no room for ambiguity, we
write $g$ as a shorthand for $U(g)$. A natural attempt at gauging the symmetry
is to project the Hilbert space to a $G$-invariant subspace:

$\mathcal{H}\longrightarrow\mathcal{H}^{G}:=P_{G}\mathcal{H},\quad
P_{G}=\frac{1}{|G|}\sum_{g\in G}g.$ (I.2.38)

Indeed, the map $P_{G}$ is a Hermitian projection operator

$\ln P_{G}=P_{G},\quad P_{G}^{2}=P_{G}.$ (I.2.39)

The evaulation of partition function of this theory simply amounts to
inserting $P_{G}$ in the trace

$Z_{G}=\frac{1}{|G|}\sum_{g\in G}\mathrm{Tr}_{\mathcal{H}}g\
q^{L_{0}-c/24}\bar{q}^{\bar{L}_{0}-\bar{c}/24}.$ (I.2.40)

An immediate problem is that $Z_{G}$ is not modular invariant, and so our
naive gauging procedure is incompatible with string perturbation theory.
Consider an $S$-transformation that interchanges the temporal and spatial
circles. Before, we could think of $g$ as acting on states every time we wind
around the temporal circle. Now, it must act when we wind around the spatial
circle. We can no longer think about $g$ as acting on states but rather as
implementing different boundary conditions for local operators

$\displaystyle{\cal O}(\sigma+2\pi,\tau)=U_{g}{\cal
O}(\sigma,\tau)U_{g}^{\dagger}:=g\cdot{\cal O}(\sigma,\tau).$ (I.2.41)

Under the state-operator mapping, these operators correspond correspond to
states in some new space ${\mathscr{H}}_{g}$ known as a _twisted sector_. Note
that the twisted sector coincides with neither ${\mathscr{H}}$ nor
$P_{G}{\mathscr{H}}$, where all of the (bosonic) operators are periodic.

We should therefore modify the gauged theory CFTG/$G$ to include all
$G$-invariant states, including the twisted sectors

${\mathscr{H}}_{\text{CFT}_{G}/G}=\bigoplus_{g\in G}P_{G}{\mathscr{H}}_{g}.$
(I.2.42)

We are then left with a well-defined, modular invariant partition function

$\displaystyle Z_{\text{CFT}_{G}/G}=\frac{1}{|G|}\sum_{g,h\in
G|[g,h]=1}\mathrm{Tr}_{\mathcal{H}_{h}}g\
q^{L_{0}-c/24}\bar{q}^{\bar{L}_{0}-\bar{c}/24},$ (I.2.43)

where in the h-twisted sector we have restricted to symmetries $g$ that do not
change the sector ($gh=hg$).

$Z=\frac{1}{|G|}\sum_{gh=hg}$$h$$g$ Figure I.2.3: Partition function of an
orbifold theory including the twisted sectors.

We can assign different phases to each sector which corresponds to turning on
the B field. However, theses phases, which are also called discrete torsion,
cannot be assigned arbitrarily. The phase assignment must not break modular
invariance [40]. Of course just as in any gauging, there cannot be anomalies
which renders it inconsistent. In string theory context, this amounts to the
left-right level matching condition.

##### Geometric orbifolds

With the understanding of how to construct orbifold theories on the
worldsheet, we now investigate the consequences on the spacetime geometry.

As a simple example, we begin with toroidal orbifolds. Take a torus $T^{d}$
equipped with a group action for some discrete group $G$. We can consider the
quotient space $T^{d}/G$, which generically has a non-trivial topology. Any
fixed points of $G$ are necessarily singular. For instance, consider the unit
circle $S^{1}$ whose center coincides with the origin of the $(x,y)$ plane. It
admits a $\mathbb{Z}_{2}$ symmetry that sends $y\to-y$. The space
$S^{1}/\mathbb{Z}_{2}$ is clearly an orbifold, since the symmetry leaves the
two points at $y=0$ and $y=\frac{1}{2}$ fixed. We can visualize
$S^{1}/\mathbb{Z}_{2}$ as a line segment stretching from $y-0$ to
$y=\frac{1}{2}$, with two singularities at its endpoints.

We now seek the CFT construction of geometric orbifolds. A NLSM with some
target space manifold $M$ will generically be invariant under a discrete
subgroup $G$ of the full isometry group of $M$. As we saw in the previous
section, from the CFT perspective, discrete gauging requires that orbifold
states are singlets under $G$. Moreover, to ensure modular invariance we must
include all of the twisted sectors. For the closed string, it is easy to see
that the twisted sectors correspond to identifying spacetime points $X^{i}$
with the same $G$-orbit. Now, an open string stretched between these two
points is, in fact, a closed string. This leads to new string states which can
be precisely identified with the twisted sector states.

A particularly illuminating example is the bosonic string on the torus
$T^{2}$. The theory has a $\mathbb{Z}_{2}$ symmetry that maps $x\mapsto-x$ for
$x\in T^{2}$. On the worldsheet, the $\mathbb{Z}_{2}$-action is given by

$\displaystyle\partial X^{1,2}\mapsto-\partial X^{1,2},\quad\partial
X^{\mu}\to\partial X^{i},\quad\mu\neq 1,2\,,$ (I.2.44)

where the $X^{\mu}$ correspond to the transverse spacetime directions (which
we henceforth drop). Note that $(p_{L},p_{R})\mapsto(-p_{L},-p_{R})$ and
$\alpha_{n}\mapsto-\alpha_{n}$ under this map. In the untwisted sector, the
$\mathbb{Z}_{2}$-invariant states are thus given by acting with an even/odd
number of oscillators, respectively, on

$\displaystyle\ket{\pm}=\frac{1}{\sqrt{2}}\left(\ket{p_{L},p_{R}}\pm\ket{-p_{L},-p_{R}}\right).$
(I.2.45)

Recall that the partition function of the ungauged theory is

$\displaystyle
Z_{T^{2}}=\sum_{(p_{L},p_{R})\in\Lambda}\frac{q^{\frac{1}{2}p_{L}^{2}}\bar{q}^{\frac{1}{2}p_{R}^{2}}}{\left|q^{2/24}\prod_{n}(1-q^{n})^{2}\right|^{2}}\,.$
(I.2.46)

Once we insert the action of $g$ into the trace, the ground state now
contributes

$\bra{p_{L},p_{R}}\mathrm{Tr}\
q^{L_{0}}\bar{q}^{\bar{L}_{0}}\ket{-p_{L},-p_{R}},$ (I.2.47)

which is non-zero only for $p_{L}=p_{R}=0$. We thus only keep the states with
$p_{L}=p_{R}=0$. Given that $g$ acts as $q\to-q$, we easily find the partition
function for the untwisted sector

$Z=\frac{1}{2}Z_{T^{2}}+\frac{1}{2}\frac{1}{\left|q^{2/24}\prod_{n}(1+q^{n})^{2}\right|^{2}},$
(I.2.48)

where the factor of $1/2$ arises from the $1/2$ in $P_{g}=(1+g)/2$.

The twisted sectors correspond to the choice of boundary conditions

$X^{1,2}(\sigma+2\pi,\tau)=\pm X^{1,2}(\sigma,\tau).$ (I.2.49)

These periodicity conditions lead to a new ground state as well as a different
set of oscillators with half-integer grading. There are four fixed points of
the $\mathbb{Z}_{2}$ action on $T^{2}$, which simply produces a multiplicative
factor in the twisted sector partition function. The ground state energy in
this sector can be calculated from the regularized sum

$\frac{1}{2}\sum_{n=0}^{\infty}(n+\eta)=-\frac{1}{24}+\frac{1}{4}\left(\eta(1-\eta)\right)\,.$
(I.2.50)

The modes here are half-integers, so $\eta=1/2$ and $E=+1/48$. This gives the
trace in the twisted sector,

$\text{tr}_{\mathcal{H}_{\text{twisted}}}q^{L_{0}-c/24}\bar{q}^{\bar{L}_{0}-\bar{c}/24}=\frac{4}{\left|q^{-2/48}\prod_{n}(1-q^{n+1/2})^{2}\right|^{2}}$
(I.2.51)

Of course, this trace only gives the partition function _before_ gauging,
_i.e._ we’re missing $\text{tr}_{\mathcal{H}_{\text{twisted}}}(g(\cdots))$.
This is given by

$\text{tr}_{\mathcal{H}_{\text{twisted}}}gq^{L_{0}-c/24}\bar{q}^{\bar{L}_{0}-\bar{c}/24}=\frac{4}{\left|q^{-2/48}\prod_{n}(1+q^{n+1/2})^{2}\right|^{2}}.$
(I.2.52)

The complete $T^{2}/\mathbb{Z}_{2}$ partition function is thus

$Z_{T^{2}/\mathbb{Z}_{2}}=\frac{1}{2}Z_{T^{2}}+\frac{1}{2}\frac{1}{\left|q^{2/24}\prod_{n}(1+q^{n})^{2}\right|^{2}}+\frac{2}{\left|q^{-2/48}\prod_{n}(1-q^{n+1/2})^{2}\right|^{2}}+\frac{2}{\left|q^{-2/48}\prod_{n}(1+q^{n+1/2})^{2}\right|^{2}}.$
(I.2.53)

Exercise 3: Verify that the last term in (I.2.53) can be obtained by an
appropriate modular transformation on (I.2.51). Also verify (I.2.53) is
invariant under $\tau\rightarrow\tau+1/2$.

Optional exercise: Consider the orbifold $T^{2}/\mathbb{Z}_{3}$. The
$\mathbb{Z}_{3}$ action on $T^{2}$ can be written as

$z=x^{1}+ix^{2},\quad z\rightarrow\omega z,\quad\omega^{3}=1\,.$ (I.2.54)

Construct the partition function of this theory; it is helpful to note that
the twisted sectors are complex conjugates of one another ($\omega$ and
$\omega^{2}=\omega^{-1}$).

There are also perfectly sensible string theory vacua which are not bonafide
geometries, known as _non-geometric compactifications_ [41, 42, 43]. These can
arise when considering orbifold theories which cannot be interpreted as target
space geometries of the form $M/G$. As an example, we can compactify on a
torus $T^{d}$ to get states whose momenta $(p_{L},p_{R})$ lie in a
$2d$-dimensional lattice $\Lambda$. For certain choices of metric and B-flux,
it is possible to arrange things such that the resulting theory has a larger
symmetry group than that of the original torus. This is due to the fact that
the dimension of the lattice is what constrains the possible global symmetries
of the CFT (_i.e._ group actions). Continuing with the two-dimensional
example, such a lattice can have $\mathbb{Z}_{2}$, $\mathbb{Z}_{3}$,
$\mathbb{Z}_{4}$, or $\mathbb{Z}_{6}$ symmetry. However, for $T^{2}$, the
Narain lattice is four-dimensional; it thus could potentially have a
$\mathbb{Z}_{12}$ symmetry, which is _not_ a symmetry of $T^{2}$. If we
quotient by $\mathbb{Z}_{12}$, the resulting CFT evidently does not possess
the interpretation of a NL$\sigma$M with target space $T^{2}/G$, so it is non-
geometric. These constructions have been studied in [42].

##### Monstrous moonshine

Even before phycisists had encountered orbifolds, mathematicians had been
independently studying asymmetric orbifold constructions with the aim of
understanding conjectures about the so-called monster group - the largest
sporadic finite simple group (e.g. [44]). Recall that all simple Lie groups
belong to either an infinite series, $SU(n)$, $SO(n)$, $Sp(n)$, or one of the
exceptional groups $E_{6}$, $E_{7}$, $E_{8}$, $F_{4}$, or $G_{2}$. This is
similar to the classification of finite simple groups, which can be split into
a set of infinite families together with a finite number of additional groups
(known as the sporadic groups). The monster group happens to be the sporadic
group with the maximum order, namely

$|G|\sim 8\times 10^{53}\,.$ (I.2.55)

For comparison, there are around $10^{80}$ atoms in the universe.

The monster group makes an unexpected appearance in the study of modular
functions which are complex functions with specific transformation properties
under $SL(2,\mathbb{Z})$ transformations. An important modular function is
called the j-function. The j-function is a meromorphic invariant under
$SL(2,\mathbb{Z})$ with a single simple pole in the fundamental region of
$SL(2,\mathbb{Z})$ at $\text{Im}(\tau)\rightarrow\infty$.

The expansion of $j$ reads

$\displaystyle j(q)=q^{-1}+744+196884q+21493760q^{2}+\ldots,$ (I.2.56)

where the first term represents the simple pole at
$\text{Im}(\tau)\rightarrow\infty$. It turns out that the expansion
coefficients have a mysterious connection with the Monster group: Every
coefficient except the constant term $744$, could be written as elementary
sums of the dimensions of some representations of the Monster group! For
example, the smallest two irreducible representations of the Monster group
have dimensions 1 and 196883 which add up to 196884.

${\bf 196884}={\bf 196883}\oplus{\bf 1}\,,$ (I.2.57)

In the 1970s, mathematicians noticed this connection [45] which motivated them
to construct the Monstrous Moonshine representation of the Monster group.

Now, let us go back to physics and see how string theory helps us to
understand this unexpected connection. Modular functions have a very natural
place in string theory. Because of modular invariance, they show up as
partition functions of the worldsheet theory. If j-invariant were to be the
partition function of a worldsheet theory, the Monstrous Moonshine suggests
that states of the theory fall into representations of the Monster group. In
other words, the theory probably has a Monster symmetry. Moreover, the
$q^{-1}$ term signals the existance of a Tachyon, just like the Bosonic string
theory. In fact, it turns out there is an orbifold construction that
reproduces the j-function as the worldsheet partition function!

This construction involves compactifying the 26d target space of the bosonic
string to 2d [46]. The resulting 24d internal space is chosen such that the
left- and right-moving momenta are elements of a particular 24d lattice,
_i.e._

$\ket{p_{L},p_{R}}\in\Gamma_{L}^{24}\otimes\Gamma_{R}^{24}\,.$ (I.2.58)

The relevant conditions are that the lattice is even and self-dual, as
required by modular invariance, as well as that the shortest non-trivial
vector has length $p_{L}^{2}=4$. Recall that the root lattices associated with
the simple Lie algebras have vectors with a minimum length of $2$. There is
exactly one even self-dual lattice in 24 dimensions whose vectors all have
length $p_{L}^{2}>2$, known as the _Leech lattice_ $\Gamma_{L}^{24}$ [47].
Given the string compactification, we are free to gauge the $\mathbb{Z}_{2}$
symmetry (the same symmetry present for $T^{2}$) and study the orbifold
theory:

$\Gamma_{L}^{24}/\mathbb{Z}_{2},\quad p_{L}\longrightarrow-p_{L}\,.$ (I.2.59)

This is an asymmetric action which satisfies the level matching conditions in
the twisted sector. As before, the orbifold theory has a partition function of
the form

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{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.70659pt}{0.0pt}\pgfsys@lineto{27.74617pt}{0.0pt}\pgfsys@moveto{28.45276pt}{0.70659pt}\pgfsys@lineto{28.45276pt}{27.74617pt}\pgfsys@moveto{27.74617pt}{28.45276pt}\pgfsys@lineto{0.70659pt}{28.45276pt}\pgfsys@moveto{0.0pt}{27.74617pt}\pgfsys@lineto{0.0pt}{0.70659pt}\pgfsys@stroke\pgfsys@invoke{
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(I.2.60)

In fact, it can be computed explicitly:

$Z=q^{-1}+A_{1}q^{1}+\cdots\,,\quad A_{1}=196884\,.$ (I.2.61)

There is a single tachyon in the spectrum, hence the $q^{-1}$ term with unit
coefficient. The fact that there is no $O(q^{0})$ term indicates that there
are no massless particles in the spectrum.

The uniqueness properties of the j-function makes it easy to relate the
partition function to the $j$-function,

$Z(q)=j(q)-744\,.$ (I.2.62)

At the level of the string theory, this strongly suggest that the particle
content of the theory organizes into irreducible representations of the
monster group – that is, the monster group is the symmetry group of this
system. It was later proven by Richard Borcherds that the full vertex operator
algebra (VOA) of the CFT is a generalized Kac–Moody algebra with the Monster
group acting on it [48].

####  2.4 Noncritical string theory

A key problem of the critical bosonic string is the existence of the tachyon,
which spoils the validity of perturbation theory. We seek a new background
which admits a stable string vacuum. Thus far, we have considered backgrounds
with nontrivial curvature and B-flux, but with a constant dilaton profile.
Let’s now consider a $d$-dimensional flat Minkowski background
($G_{\mu\nu}=\eta_{\mu\nu}$ and $B_{\mu\nu}=0$) but with a linear dilaton
profile along a particular spatial direction $X^{i}$,

$\phi(X)=QX^{i}.$ (I.2.63)

The nonlinear sigma model that corresponds to this choice of background is
given by the action

$S=\frac{1}{4\pi}\int_{\Sigma}d^{2}\sigma\sqrt{g}\left(g^{ab}\eta_{\mu\nu}\partial_{a}X^{\mu}\partial_{b}X_{\nu}+QR(g)X^{i}\right).$
(I.2.64)

In conformal gauge, the action reduces to that of $d$ free bosons with the
caveat that the stress tensor is modified to

$T_{m}=-(\partial X^{\mu}\partial X_{\mu})+Q\partial^{2}X^{i}.$ (I.2.65)

The central charge changes accordingly:

$c_{m}=d+6Q^{2}.$ (I.2.66)

To cancel the ghost contribution, we set $c_{m}=26$ as usual. This fixes the
value of $Q$ to

$Q=\sqrt{\frac{26-d}{6}}.$ (I.2.67)

We can now use our favorite quantization method (e.g. BRST or light-cone) to
determine the string spectrum in this background. Crucially, we now find that
the mass squared of the tachyon (_i.e._ lowest mode) is

$m_{T}^{2}=-\frac{d-2}{24},$ (I.2.68)

which vanishes for $d=2$. We have thus found a string background where the
tachyon is stable!141414More precisely, the lowest mode of the bosonic string
transforms as a massless scalar, which is erroneously but typically referred
to as the tachyon. Such string theories where the initial background is not of
critical dimension $d=26$ are known as the noncritical string theory
[7].151515An equivalent approach to the noncritical string is to sacrifice
Weyl invariance and promote the Weyl mode on the worldsheet to a dynamical
field.

In two dimensions, the moduli cannot be frozen because the boundary conditions
of scalar fields can be dynamically changed with finite energy. Thus, even
though the theory has a stable vacuum, we must now contend with a variable
coupling without a convergent boundary condition. This will create a problem
since in regions where the effective coupling $\lambda(x)=e^{\phi(X)}$ grows
large, we lose perturbative control of the theory. By imposing Weyl invariance
on the worldsheet of the theory we can find the tachyon profile of the
background. A profile that preserves Weyl invariance is

$\delta S=\frac{1}{4\pi}\int d^{2}z\sqrt{g}\mu e^{\alpha X^{i}}\,,$ (I.2.69)

where $\alpha=Q^{1/2}$ is fixed by Weyl invariance (but $\mu$ is not). In this
background, the region of large coupling is now cutoff due to the tachyon
background. The upshot is we can now do perturbation theory again, although
the theory now depends on an additional parameter $\mu$ set by the VEV of the
tachyon field. Moreover, the worldsheet theory is now described by an
interacting CFT known as Liouville field theory [49, 50].

###  3 Superstring theory

The quantum bosonic string exhibits many interesting physical phenomena, but
cannot be probed beyond tree level due the presence of the tachyon. In this
section, we will consider a set of alternative models, collectively known as
the superstring, with a supersymmetric worldsheet theory. The superstring
admits the requisite features of any quantum string theory: a matter CFT, a
ghost system, and BRST symmetry. Crucially, certain superstring theories admit
consistent truncations which render them free of anomalies and tachyons. There
are also certain models which exhibit spacetime supersymmetry.

####  3.1 Basics of the NSR formalism

The action of the superstring, in Polyakov form, describes ${\cal N}=(1,1)$
supergravity on the worldsheet [51, 52]. It is quite complicated, so we only
focus on the action after gauge-fixing. Similar to the bosonic string, we have
$d$ free scalars $X^{\mu}$ and a pair of $bc$ ghosts with spins $(2,-1)$ that
arise from worldsheet diffeomorphisms and Weyl rescalings. We additionally
have $d$ pairs of Majorana–Weyl fermions $\psi^{\mu}$ and $\bar{\psi}^{\mu}$
which transform as spacetime vectors. There are also new commuting ghosts,
namely the $\beta\gamma$ ghosts with spins $(\frac{3}{2},-\frac{1}{2})$, which
arise from gauge-fixing super-diffeomorphisms and super-Weyl transformations.
The fields and their data are described in Table I.3.1.

Altogether, they sit inside a free worldsheet action given by

$S_{\text{NSR}}=\frac{1}{2\pi}\int d^{2}z\left(\partial
X^{\mu}{\bar{\partial}}X_{\mu}+\frac{1}{2}\psi^{\mu}{\bar{\partial}}\psi_{\mu}+\frac{1}{2}\bar{\psi}^{\mu}\partial\bar{\psi}_{\mu}+b{\bar{\partial}}c+\bar{b}\partial
c+\beta{\bar{\partial}}\gamma+\bar{\beta}\partial\bar{\gamma}\right).$ (I.3.1)

In total, the matter fields contribute $c_{m}={3d}/{2}$ and the ghosts
$c_{gh}=-15$. To avoid the appearance of a Weyl anomaly, we must take
$c_{m}=15$ and so the superstring naturally lives in $d=10$ spacetime
dimensions.

Field content | Conformal weight $h$ | Central charge $c$
---|---|---
$X^{\mu}$ | $0$ | $d$
$\psi^{\mu}$ | $\dfrac{1}{2}$ | $\dfrac{d}{2}$
$(b,c)$ | $(2,-1)$ | $-26$