素粒子論研究 ･ 電子版 Vol. 42 (2024) No. 3

Soryushiron Kenkyu

Master's Thesis
On Orbifolds in Closed Superstring Field
TheoryDepartment of Physics
Ludwig-Maximilians-Universität München

Xianghang Zhang

Munich, September 12th , 2023

Submitted in partial fulfillment of the requirements for the degree of M. Sc. Physics
Supervised by Prof. Dr. Ivo Sachs

Soryushiron Kenkyu

Soryushiron Kenkyu

Zur Orbifolds in der geschlossenen
Superstring-Feld-Theorie

Soryushiron Kenkyu

Soryushiron Kenkyu

Abstract
In this thesis we study the deformation of orbifold conformal field theories (CFT) in theframework of closed superstring field theory (superSFT). In particular, we consider thetype II theory on ℂ4 /ℤ2 and its possible deformation of massless modes to the nearby
theory on a smoothed-out manifold. This process is described by so-called ``resolving''or ``blowing-up'' in algebraic geometry.
Description of orbifold CFT is based on twist field [1] that changes the periodic
boundary condition to anti-periodic one. This allows us to explore the moduli space ofmarginal deformations corresponding to the twist modes. By analysing the equation of
motion of closed bosonic SFT, we show that the blowing-up is obstructed at the second
order.
Furtherly, we move on to superstrings. Since computation shows the second-orderobstruction vanishes, it is possible to derive the blown-up metric in terms of the twisted
moduli. We note that, although the initial orbifold theory posesses a 𝒩 = 2 superconformal symmetry, the blown-up mode is not necessarily Kähler, indicating a broken
superconformal symmetry. This makes sense because the closed superSFT is defined ona generic 𝒩 = 1 superconformal string background.
We carry on the analysis to the third order, where we find out that due to the chiralstructure of the closed superSFT, the third-order obstruction is simply reduced to zero.
This is unlike the open-string case [2].

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Contents
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Content of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
2

2 Worldsheet Content
2.1 String Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Path Integral Construction of Amplitudes . . . . . . . . . . . . . .
2.1.2 Measures on the (Super)moduli . . . . . . . . . . . . . . . . . . . .
2.1.3 Reducing to the Bosonic Moduli . . . . . . . . . . . . . . . . . . .
2.2 CFT Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Free Boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Ghost Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Free Fermion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Bosonization of the Superghosts . . . . . . . . . . . . . . . . . . . .
2.2.5 Spin Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Orbifold Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 CFT on Z2 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Deformations of the Background . . . . . . . . . . . . . . . . . . . . . . .

4
4
4
5
6
6
7
8
9
10
11
12
12
13

3 Closed Superstring Field Theory
3.1 String Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Geometric BV Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 L∞ Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Closed Superstring Field Theory . . . . . . . . . . . . . . . . . . . . . . .
3.5 Assigning PCO's . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15
15
15
20
21
22

4 Deformation via SFT Action
4.1 Perturbative Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Second-order Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Bosonic String . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Superstring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Third-order Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26
26
27
27
27
30
32

5 Conclusion

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IISoryushiron Kenkyu

On Orbifolds in Closed Superstring Field Theory

A Operadic Structures in Homotopy and BV Algebras

V

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Chapter 1
Introduction
1.1

Motivation

Theoretical physicists seek a mathematical description of the nature, in particular, amathematical model of the fundamental forces and particles. This led to two major
milestones in the last century: quantum mechanics and general relativity. In spite of
their huge experimental success, it is still not known how the nature coorperates theprinciples of quantum mechanics into the theory of gravity.
String theory, although not originally proposed for it, is a theory of quantum gravity.
It is a theoretical framework in which the point-like particles are blown-up into onedimensional objects called strings. While string theory may or may not be the dreamt
theory of everything, the understanding of string theory and its dualities has cast profoundinsights into quantum field theories and mathematics. To name a few, AdS/CFT correspondence, Non-abelian gauge theories, supersymmetry, modular forms, etc. A proper
understanding of string theory is thus still relevant.
A string theory can be broken down into a sum of all different conformal field theories
on Riemann surfaces, called the worldsheet. A sensible theory of fundamental particles
should contain fermions in its spectrum. This is achieved by the NSR formalism of thesuperstring theory replacing the ordinary Riemann surfaces with super Riemann surfaces. There are five consistent superstring theories: type I, type IIA, type IIB, heterotic
𝑆𝑂(32)/ℤ2 and heterotic 𝐸8 × 𝐸8 . However, all those theories require a dimensionality
dramatically larger than four of our observed universe. Therefore, it's been a major taskof string phenomenology to study the compactification of these extra dimentions into
micro scale.
The basic example would be compactifying on a torus, however, it seems diﬀicult togo very far beyond that. One thus needs a systematical method to build new phenomenological models from the old ones. One such way is the orbifold theory, which is a classicalgeometrical method of constructing new spaces that can be implemented directly into
string theory. On the worldsheet it corresponds to the twisted boundary conditions. The
mathematical consequence of allowing twisted strings is that string theory can somehow``probe'' the theory where the singularities in the orbifold theoy are resolved. This raises
an interesting question on the interplay between string theory and the geometry of theresolved orbifold, that is, to study the latter from a stringy point of view.
1Soryushiron Kenkyu

On Orbifolds in Closed Superstring Field Theory

The deformation among different target spaces is realized in string theory by shifting
the string vacuum. This process is most suitable to be described in the second quantizedpicture known as string field theory. The textbook approach to string theory is known tobe perturbative. Conceptually it plays a similar role to the Feynman rules and propagators
in particle physics. One may wonder, however, if there is an action principle governing
the string interaction, and hence comes the string field theory. The field theory approachis especially helpful for our purpose, since different string backgrounds are understood tobe the classical solutions to the string field equation of motion. That is to say, string
field theory provides a unified viewpoint of all the string backgrounds that serves as astandpoint to start our analysis of the deformation.

XμFigure 1.1: String theory studies the map from worldsheet into targe spacetime.

1.2

Content of the Thesis

The main goal of the thesis is to exploit the viewpoint stated above. More specifically,
we try to apply the type II string field theory to understand the spacetime geometry
deformed from an orbifold model. Before really tackling the problem, we will develop
necessary machineries. The outline of the thesis is as follows:
In chapter 2 we review the worldsheet approach of string theory. We will see how pathintegral of Polyakov action gives a measure on the moduli space of Riemann surfaces andamplitudes of a proper worldsheet conformal field theory. We then review common CFT
techniques to treat the latter, including the CFT resulted from an orbifold construction.
In chapter 3 we review the construction of the type II closed string field theory action.
The measure on the moduli space provided by string theory should be extended off-shell.
Geometrically this corresponds to pulling back to an appropriate bundle over the moduli
space. We explain the construction of string field vertices in such a geometric setup,yielding a geometric Batalin-Vilkovisky structure. Passing to the worldsheet CFT we
obtained the algebraic vertices appearing in tha action, satisfying a 𝐿∞ relation in theclassical part. We then move on to the theory of superstrings. We see that integratingout the odd modulus results in dressing the vertices with the picture-changing operators,
which is achieved in the NS-NS sector by a recursive procedure.
Chapter 4 contains all the calculations of the main problem. The problem of whether
we can find an exactly marginal deformation is formulated in a form of perturbativeequations. We show that, in the bosonic case, the deformation is obstructed already at
the second order. However, this obstruction is resolved in superstring theory. We are2

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On Orbifolds in Closed Superstring Field Theory

thus able to compute the second-order deformation of the target spacetime geometry. In
particular we compute the spacetime metric, and find out that it is not Kähler anymore.
Moreover, we are capable of moving further, by showing that the third-order obstruction
is still vanishing.

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Chapter 2
Worldsheet Content
In this chapter we review the basics of string theory and how it is reduced to an integrated worldsheet CFT amplitude. We do it in a manner that treats bosonic strings andsuperstrings similarly, namely from a (super)Riemann surface point of view. As we willsee, this approach is readily generalized to string field theory. The main reference wefollow here is [3]. Moreover, we review the notion of an orbifold CFT introduced in [1],which is of particular interest in this thesis.

2.1

String Theory

2.1.1

Path Integral Construction of Amplitudes

The string worldsheet is a Riemann surface Σℎ of genus ℎ with a Riemannian metric 𝑔.
It is worth noting that the complex structure on such a surface is uniquely determinedby the conformal structure [𝑔] induced by the metric, and vice versa. The time slice of
these Riemann surface is typically depicted as one or a few loops of string, propagating
through the Minkowski spacetime 𝑀 1,𝐷−1 .
We start with the well-known Polyakov action as a functional on the mapping space∞
𝒞 (Σℎ , 𝑀 1,𝐷−1 )𝜕𝑋 𝜇 𝜕𝑋 𝜈
1
2 √ 𝑎𝑏
∫
d
𝜎
𝑔𝑔
𝐺
(𝑋)
,
(2.1)
𝑆𝑃 =
𝜇𝜈
4𝜋𝛼′
𝜕𝜎𝑎 𝜕𝜎𝑏where 𝜎𝑎 = (𝜏 , 𝜎) are local coordinates on the worldsheet and 𝑋 ∈ 𝒞∞ (Σℎ , 𝑀 1,𝐷−1 ). The
action itself is nothing more than a simplest example of 𝜎-model. However, the metric𝑔𝑎𝑏 (𝜎) is taken to be an auxiliary field and needs to be integrated out in the path integral
𝑍ℎ = ∫ d𝑔 ∫ d𝑋e−𝑆𝑃 [𝑔,𝑋] .

(2.2)

The fact that action 𝑆𝑃 possesses a infinite-dimensional gauge symmetry gives divergence. We denote the gauge group as 𝒟𝑖𝑓𝑓 ⋊ 𝒲𝑒𝑦𝑙. The physically meaningful partition
function should be modified as𝑍ℎ = ∫d𝑔∫ d𝑋e−𝑆𝑃 [𝑔,𝑋] ,
𝒱𝑜𝑙
4

(2.3)

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On Orbifolds in Closed Superstring Field Theory

where 𝒱𝑜𝑙 is the volumn of the gauge group.
The path integral is evaluated via the standard Faddeev-Popov procedure, and furtherly replace the Faddeev-Popov determinent with an integral over two anticommuting
ghost fields, the ghost 𝑐𝑎 field and the symmetric traceless anti-ghost 𝑏𝑎𝑏 field. The path
integral can be rewritten as
𝑍ℎ = ∫ d𝑡ℳℎdet(𝜙𝑖 , 𝜇𝑗̂ )𝑔 ̂Ω𝑐𝑘𝑣 [𝑔]̂ −1√det(𝜙𝑖 , 𝜙𝑗 )𝑔 ̂ √det(𝜓𝑖 , 𝜓𝑗 )𝑔 ̂̂̂′′ −𝑆𝑃 [𝑔,𝑋]−𝑆𝑔ℎ [𝑔,𝑏,𝑐]∫ d𝑔 𝑋d,̂𝑔 𝑏d
̂ 𝑔 𝑐ê(2.4)

where d′ denotes integration without the zero modes that give infinite result in Gaussian
integral, 𝜓𝑖 and 𝜙𝑖 are respectively a basis for zero-modes of the CKV operator andits adjoint, and 𝜇𝑖 are Beltrami differentials that correspond to traceless Teichmüller
deformations. One may refer textbooks like [4]. The benefit is that the integrand depends
only on a fixed reference metric we choose, so it is suﬀicient to study a fixed conformal
field theory given by 𝑆𝑃 [𝑔,̂ 𝑋] + 𝑆𝑔ℎ [𝑔,̂ 𝑏, 𝑐], the conformal background, then integrate on
the moduli space.

2.1.2

Measures on the (Super)moduli

It would be beneficial to write the integral (2.4) as an integral of superfunctions. One
needs to choose such a function that the Berezin integral along the odd variables gives
the right measure on the moduli space. Here we merely quote the conclusion and key
properties, details can be found in [3].

Figure 2.1: Supermanifolds can be thought of as spaces with a ``mist'' of odd coordinates
One initiates the re-formulating by defining a superfunction 𝐹𝒱 (𝑔, 𝛿𝑔) on the superspace (𝑔, 𝛿𝑔), where the vertex operators 𝒱1 , ⋯ , 𝒱𝑛 are included in the definition. The

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On Orbifolds in Closed Superstring Field Theory

natural requirements on the function would be Weyl invariance and diffeomorphism invariance, making it into a pullback from the 𝑛-point, genus ℎ moduli space ℳℎ,𝑛 . One
thus defines
1
√
̂
𝑆[𝑋,
𝑏, 𝑐, 𝑔, 𝛿𝑔] = 𝑆𝑃 + 𝑆𝑔ℎ +∫ d2 𝜎 𝑔𝛿𝑔𝑖𝑗 𝑏𝑖𝑗
4𝜋

(2.5)

and
̂

𝐹𝒱 (𝑔, 𝛿𝑔) = ∫ d𝑋d𝑏d𝑐e−𝑆[𝑋,𝑏,𝑐,𝑔,𝛿𝑔] 𝒱1 ⋯ 𝒱𝑛 .

(2.6)

The introducing of 𝑆 ̂ actually gives the correct Jacobian passing from 𝑔 to the modular
parameters {𝑚dim(ℳℎ,𝑛 ) }. Therefore, 𝐹𝒱 (𝑔, 𝛿𝑔) provides a form on the moduli space ℳℎ,𝑛that will be denoted by Ω𝒱;ℎ,𝑛 .
The superstring amplitudes are more or less similarly given bŷ

𝐹𝒱 (𝒥, 𝛿𝒥) = ∫ d𝑋d𝑏d𝑐d𝛽d𝛾e−𝑆[𝑋,𝑏,𝑐,𝛽,𝛾,𝒥,𝛿𝒥] 𝒱1 ⋯ 𝒱𝑛 ,where
𝑆̂= 𝑆 +1̄̄ ̄
− 𝛿 𝒥𝐵)∫ d𝑧d𝑧d𝜃d̄ 𝜃(𝛿𝒥𝐵
2𝜋

(2.7)

(2.8)

with 𝐶 𝑧 = 𝑐𝑧 + 𝜃𝛾 𝜃 and 𝐵𝑧𝜃 = 𝛽𝑧𝜃 + 𝜃𝑏𝑧𝑧 being the complexified ghost superfields
introduced to fix the super-diffeomorphic gauge redundancy and 𝛿𝒥 being the variationof the complex structure on the worldsheet.

2.1.3

Reducing to the Bosonic Moduli

Integrating out all the fermionic degrees of freedom in (2.7) needs a basis chosen for the
odd moduli. By turning on a gravitino field 𝜒𝜃𝑧 ̄ we can deform the super Riemann surface
Σ by𝛿𝒥𝑧𝑧 ̄ = 𝜃𝜒𝜃𝑧 .̄
(2.9)
The construction is most often considered when the gravitino modes are delta functions
supported at some point 𝑝𝑖 ∈ Σ𝑟𝑒𝑑 , and then the integration gives a factor that can be
represented by𝒳(𝑧) = 𝛿(𝛽(𝑧))𝑆𝑧𝜃 (𝑧),(2.10)
the picture-changing operator.

2.2

CFT Systems

In the previous section we have identified the string amplitudes with the amplitudes ofconformally invariant vertex operators on the worldsheet, provided that they have theright ghost and picture number and an integration over the moduli space. This allows us
to use the standard CFT techniques in the evaluation of the string amplitudes. In thissection we review some CFT systems that serve as the building blocks of string theory.
All the CFT systems will be discussed in the radial quantization picture, where the radial
direction represents time.
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On Orbifolds in Closed Superstring Field Theory

Figure 2.2: Radial quantization mapping from a cylinder worldsheet into the complex
plane

2.2.1

Free Boson

Each of the 𝐷 copies of free scalar fields has the action𝑆=1√∫ d2 𝑥 𝑔𝑔𝜇𝜈 𝜕𝜇 𝑋𝜕𝜈 𝑋,′
4𝜋𝛼(2.11)

with the identification 𝑋(𝜎) = 𝑋(𝜎 + 2𝜋). The cylinder can be conformally mapped intoℂ ∖ {0} via radial quantization. In complex coordinates, the action on flat space reads𝑆=1∫ d𝑧d𝑧 ̄ 𝜕𝑧 𝑋𝜕𝑧 𝑋.
̄
2𝜋𝛼′(2.12)The energy-momentum tensor of the theory is
𝑇 (𝑧) =

1∶ 𝑗𝑗 ∶ (𝑧).
𝛼′(2.13)The propagator can be easily derived from the path integral identity
0 = ∫ d𝑋

𝛿e−𝑆[𝑋]𝑋(𝑤)𝛿𝑋(𝑧)(2.14)as𝛼′ln(𝑧 − 𝑤).
2
The operator product expansion can also be read off⟨𝑋(𝑧)𝑋(𝑤)⟩ = −𝑋(𝑧)𝑋(𝑤) = −𝛼′ln(𝑧 − 𝑤) + 𝑂(𝑧 − 𝑤).
2(2.15)(2.16)

To prevent the logarithm we introduce the current
𝑗(𝑧) = 𝑖𝜕𝑋(𝑧) = ∑ 𝑗𝑛 𝑧 −𝑛−1 ,(2.17)

𝑛∈ℤ

where the modes satisfy the commutation relation [𝑗𝑛 , 𝑗𝑚 ] = 𝑛𝛿𝑛+𝑚 encoded in the OPE
𝑗(𝑧)𝑗(𝑤) =𝛼′1+ 𝑂(𝑧 − 2).
2 (𝑧 − 𝑤)2(2.18)
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On Orbifolds in Closed Superstring Field Theory

To construct the Hilbert space of the theory one hopes to identify all the primary fields.
They are exactly the current 𝑗(𝑧) satisfying
𝑗(𝑤)
𝜕𝑗(𝑤)
+
+ 𝑂(𝑧 − 𝑤)
2(𝑧 − 𝑤)𝑧−𝑤(2.19)

𝜕𝑉𝑝 (𝑤)
𝑝2 𝑉𝑝 (𝑤)++ 𝑂(𝑧 − 𝑤).
2
2 (𝑧 − 𝑤)
𝑧−𝑤(2.20)

𝑇 (𝑧)𝑗(𝑤) =and 𝑉𝑝 (𝑧) =∶ e𝑖𝑝𝑋 ∶ (𝑧) satisfying
𝑇 (𝑧)𝑉𝑝 (𝑤) =Then one can construct the Verma module as usual, starting with a unique vacuum |0⟩.
The modes 𝑗𝑛 are used to construct the Fock space:

2.2.2

𝑗𝑛 |0⟩ = 0,𝑛 ≥ 0,(2.21)⟨0|𝑗𝑛 = 0,𝑛 ≤ 0.

(2.22)

Ghost Systems

A ghost system, or a first-order system describes two symmetric traceless fields 𝑏𝜇1 ⋯𝜇𝜆and 𝑐𝜇1 ⋯𝜇𝜆−1 via the action𝑆=1√∫ d2 𝑥 𝑔𝑔𝜇𝜈 𝑏𝜇𝜇1 ⋯𝜇𝜆−1 ∇𝜈 𝑐𝜇1 ⋯𝜇𝜆−1 .
4𝜋(2.23)

In complex coordinates the action writes
𝑆=1
̄ 𝑧 𝑐)∫ d𝑧d𝑧 ̄ (𝑏𝜕𝑧 𝑐 + 𝑏𝜕
̄ ̄ ,
2𝜋

(2.24)

where the parameter 𝜆 can actually be generalized to include half-integers that should be
written as spinors in real coordinates. Ghost fields carry wrong Grassmann parity against
spin-statistic theorem, which we denote by 𝜖. The conformal weights of the fields areℎ(𝑏) = 1 − 𝜆,ℎ(𝑐) = 𝜆.

(2.25)The energy-momentum tensor of the theory is
𝑇 (𝑧) = −𝜆 ∶ 𝑏𝜕𝑐 ∶ (𝑧) + (1 − 𝜆) ∶ 𝜕𝑏𝑐 ∶ (𝑧).

(2.26)

Again, we can obtain the propagator and thus the OPE
𝑐(𝑧)𝑏(𝑤) =1+ 𝑂(𝑧 − 𝑤),
𝑧−𝑤𝑏(𝑧)𝑐(𝑤) =𝜖+ 𝑂(𝑧 − 𝑤).
𝑧−𝑤(2.27)In string theory we will encounter two types of ghost systems. The ususal 𝑏𝑐-ghost
system is parameterized by (𝜖, 𝜆) = (1, 2). The 𝛽𝛾 superghost system is parameterized by(𝜖, 𝜆) = (−1, 32 ).
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On Orbifolds in Closed Superstring Field Theory

2.2.3

Free Fermion

In the super Riemann surface formulation of superstring theory, one derives the worldsheet
action introducing the odd variable 𝜃. This results in an worldsheet fermion 𝜓 with theaction1
̄ + 𝜓𝜕̄ 𝜓)̄ .
∫ d2 𝑧 (𝜓𝜕𝜓
(2.28)
2𝜋
With this action at hand one can derive the equation of motion, which implies 𝜓(𝑧)and 𝜓(̄ 𝑧)̄ are holomorphic and anti-holomorphic field respectively. The OPE can also be
write down straightforwardly,
𝑆[𝜓] =1+ 𝑂(𝑧 − 𝑤).
(2.29)𝑧−𝑤
The subtlety here is that for the worldsheet fermions, there is no canonical choice ofthe spin structure or, in terms of super Riemann surface, the superschemes that reduce
to a Riemann surface are not unique. This results in different sectors for the fermions.
A consistent superstring theory requires a proper summation of these sectors in the path
integral, and the consequence is known as the GSO projection.
For fermions on the complex plane there are two inequivalent spin structures that are
encoded in the boundary conditions
𝜓(𝑧)𝜓(𝑤) =𝜓(e2𝜋𝑖 ) = ±𝜓(𝑧).

(2.30)The boundary condition can be periodic or antiperiodic under a rotation around theorigin, and the fermion sujecting to these conditions are called in the Neveu-Schwarz (NS)sector and the Ramond (R) sector respectively.
Solving the equation of motion with different boundary conditions actually gives different mode expansion. In the NS sector we have∑ 𝜓𝑟 𝑧 −𝑟−1/2 ,𝜓(𝑧) =(2.31)

𝑟∈ℤ+1/2

where the modes 𝜓𝑟 satisfy the anticommutation relation {𝜓𝑟 , 𝜓𝑠 } = 𝛿𝑟+𝑠 .
The energy-momentum tensor is1∶ 𝜓𝜕𝜓 ∶ (𝑧).
2(2.32)

𝜓(𝑧) = ∑ 𝜓𝑟 𝑧 −𝑟−1/2 ,(2.33)

𝑇 (𝑧) =In the Ramond sector we have𝑟∈ℤ

where the modes 𝜓𝑟 again satisfy the relation {𝜓𝑟 , 𝜓𝑠 } = 𝛿𝑟+𝑠 . Note the presence of zeromode here. The Ramond Hilbert space is built from a different (degenerate) vacuum |𝑆⟩.
Some correlation functions are thus different from the NS sector, even though the fields
share the same OPE, for example we have⟨𝜓(𝑧)𝜓(𝑤)⟩𝑆 =𝑧𝑤
1(√ + √ ) ,2(𝑧 − 𝑤)𝑤
𝑧

(2.34)
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with the subscript 𝑆 indicates a different vacuum.
It is possible to connect the two vacuua |0⟩ and |𝑆⟩ by defining that|𝑆⟩ = lim 𝑆(𝑧)|0⟩.
𝑧→0(2.35)

The field 𝑆(𝑧) is called the fermionic twist field and is with conformal dimension 1/16.
We see, in this subsection, that half-integer modes come into play because of theboundary conditions determined by different spin structures, and it results in two distinct
vacua that can be connected by introducting a new field corresponding the operation of``twisting'' the boundary condition. We will show that similar things need to be done
for the bosons too, but for a slightly different reason.

2.2.4

Bosonization of the Superghosts

The 𝛽𝛾-ghost system is obtained by taking 𝜖 = −1 and 𝜆 = 32 . It can be bosonized intoanother fermionic first-order system (𝜉, 𝜂) with (𝜖, 𝜆) = (1, 1) and a scalar boson 𝜙. which
are decoupled. The bosonization is explicitly written as𝛾 = 𝜂e𝜙 ,𝛽 = 𝜕𝜉e−𝜙 .

(2.36)

𝑇 (𝑧) = 𝑇 𝜉𝜂 (𝑧) + 𝑇 𝜙 (𝑧)(2.37)The energy-momentum tensor iswith1
𝑇 𝜙 (𝑧) = − ∶ 𝜕𝜙𝜕𝜙 ∶ (𝑧) − 𝜕 2 𝜙(𝑧).
2The OPE's of the new fields are listed below
𝑇 𝜉𝜂 (𝑧) = − ∶ 𝜂𝜕𝜉 ∶ (𝑧),1
+ 𝑂(𝑧 − 𝑤)
𝑧−𝑤

(2.39)e(𝑞1 +𝑞2 )𝜙(𝑤)+ 𝑂(𝑧 − 𝑤)(𝑧 − 𝑤)𝑞1 +𝑞2(2.40)1+ 𝑂(𝑧 − 𝑤).
(𝑧 − 𝑤)2(2.41)

𝜉(𝑧)𝜂(𝑤) =e𝑞1 𝜙(𝑧) e𝑞2 𝜙(𝑤) =(2.38)

𝜕𝜙(𝑧)𝜕𝜙(𝑤) = −Upon bosonization the PCO in (2.10) becomes𝒳(𝑧) = {𝑄, 𝜉(𝑧)} ,(2.42)and similar for the anti-holomorphic sector.

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On Orbifolds in Closed Superstring Field Theory

2.2.5

Spin Fields

Bosonization can also be applied to the free fermion systems. For a 𝐷-dimensional target
spacetime there are 𝐷 copies of the free fermion systems denoted by 𝜓𝑖 . There exists aequivalence between two free fermion systems 𝜓2𝑖−1 , 𝜓2𝑖 and a free boson 𝜙𝑖 realized by1
Ψ±,𝑖 = √ (𝜓1 (𝑧) ± 𝑖𝜓2 (𝑧)),2𝑖

Ψ±,𝑖 =∶ e±𝑖𝜙 ∶ .

(2.43)(2.44)As indicated by the half-integer modes, for the Ramond vacuum the aforementioned
fermionic twist field (2.35) is bosonized as
1𝑖

𝑆 𝑖 (𝑧) =∶ e 2 𝑖𝜙 ∶ .

(2.45)In order to get spacetime spinor from the worldsheet spinor, one has to constructrepresentations of 𝑆𝑂(𝐷) group (in the Euclidean case). This is done by introducing
indices11(2.46)𝐴 = ( ± ,⋯,± )
⏟⏟⏟⏟⏟
2
2
𝐷/2 times

for the spin fields
𝐷/2𝑖𝑆 = ∏ ∶ e𝑖𝐴𝑖 𝜙 ∶ .
𝐴(2.47)

𝑖=1

All the 2𝐷/2 spin fields split into two irreducible chiral representation: The left handedspinor 𝑆𝛼 with an even number of minus signs and the right handed spinor 𝑆 𝛼̇ with an
odd number of minus signs.
To be precise the bosonization procedure needs to be modified according to [5], inorder to enforce the right anticommutation relations between different fermions.
The 𝜙𝑖 system is a free CFT of bosons, its correlation function obeys𝑖𝑖⟨e𝑖𝑝1 𝜙 1 (𝑧1 ) ⋯ e𝑖𝑝𝑛 𝜙 𝑛 (𝑧𝑛 )⟩𝑛= e∑𝑖<𝑗 𝑝𝑖 𝑝𝑗 𝐺(𝑧𝑖 ,𝑧𝑗 ) 𝛿 (∑ 𝑝𝑖 )𝑖=1(2.48)𝑛= ∏(𝑧𝑖 − 𝑧𝑗 )𝑝𝑖 𝑝𝑗 𝛿 (∑ 𝑝𝑖 ) ,
𝑖<𝑗

𝑖=1

where 𝐺(𝑧𝑖 , 𝑧𝑗 ) is just the Green's function
𝐺(𝑧𝑖 , 𝑧𝑗 ) = − ⟨𝜙(𝑧𝑖 )𝜙(𝑧𝑗 )⟩ = ln(𝑧𝑖 − 𝑧𝑗 ).

(2.49)

Therefore, spin fields also simplify some calculation of the correlation functions, itwill be useful to know(𝜎̄𝜇 )𝛽𝛼̇̇1⟨𝑆 (𝑧1 )𝜓𝜇 (𝑧2 )𝑆 (𝑧3 )⟩ = − √1/2 1/2
2 𝑧12 𝑧23𝛼̇𝛽̇(2.50)
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and

1/2

1
̇
̇ 𝑧
⟨𝑆 𝛼̇ (𝑧1 )𝜓𝜇𝜈 (𝑧2 )𝑆 𝛽 (𝑧3 )⟩ = − (𝜎̄𝜇𝜈 )𝛼̇𝛽 13 ,2𝑧12 𝑧23(2.51)

𝜓𝜇𝜈 =∶ 𝜓𝜇 𝜓𝜈 ∶ .

(2.52)

with

2.3

Orbifold Theory

Orbifold CFT's are introduced in [1] for model-building in string theory. In fact, otherthan string theory, it is a general procedure to obtain new CFT from an old one.
An orbifold CFT is constructed as follow. Consider an automorphism 𝜎 of the original
CFT ℋ such that 𝜎𝑛 = idℋ . We have the twisted sector generated by the fractional powerfield
𝜓(𝑧) = ∑ 𝜓𝑟 𝑧 𝑟 .
(2.53)ℤ
𝑟∈ 𝑛This is nothing more than the notion of a 𝜎-twisted representation of the orinal CFT
operator algebra [6]. For a finite group, rather than a single generator, this notion iseasily generalized by assigning a twisted sector for each conjugacy class.

2.3.1

CFT on ℤ2 Orbifolds

The simplest example for the orbifold construction, which also has a clear geometricpicture, is the CFT with a ℤ2 = {1, 𝜎}-orbifold. One can simply view it as the theory ofa string propagating in the quotient spacetime 𝑀 / ∼ identifying 𝑋 ∼ −𝑋. In this casethere is only one twisted sector, ℋ′ , and it is generated by the half-integer modes 𝑗𝑛 with𝑛 ∈ ℤ + 21 , satisfying the commutation relations [𝑗𝑛 , 𝑗𝑚 ] = 𝑛𝛿𝑛+𝑚 .
Again, different sectors correspond to different vacua. For the twisted vacuum |𝜎⟩ . one
can define, using the state-field correspondence, the bosonic twist field 𝜎(𝑧) of conformal
1such that
dimension 16|𝜎⟩ = lim 𝜎(𝑧) |0⟩ .
(2.54)
𝑧→0

The correlation functions in the twisted sector should be evaluated in the twisted vacuum|𝜎⟩. For example, we have the propagators
1 − √ 𝑤𝑧
⟨𝑋(𝑧)𝑋(𝑤)⟩𝜎 = − ln ()
1 + √ 𝑤𝑧
⟨𝑗(𝑧)𝑗(𝑤)⟩𝜎 = −1𝑧𝑧(√ + √ ) .
22(𝑧 − 𝑤)𝑤
𝑤

(2.55)(2.56)The OPE of the twist field contians a branch cut. For example, we write𝑗(𝑧)𝜎(𝑤) =𝑠(𝑤)1(𝑧 − 𝑤) 2+ 𝑂(𝑧 − 𝑤)

(2.57)

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𝑗(𝑧)𝑠(𝑤) =2𝜕𝜎(𝑤)
1 𝜎(𝑤)
3 +1 + 𝑂(𝑧 − 𝑤)
2 (𝑧 − 𝑤) 2
(𝑧 − 𝑤) 2

𝜎(𝑧)𝜎(𝑤)̃=11(𝑧 − 𝑤) 8+ 𝑂(𝑧 − 𝑤),(2.58)(2.59)

where 𝑠(𝑧) is the excited twist field (see [1] for details) and 𝜎(𝑧)̃is the conjugate fielddefined by BPZ inner product (not to be confused with operator in the anti-holomorphic
sector).

2.4

Deformations of the BackgroundConformal field theories are not isolated. In a given CFT, operators are classified as
relevant, irrelevant and marginal according to their conformal dimensions. Among thosethe marginal operators of conformal dimension one are related to the possible deformationsof a conformal field theory. Such a deformation generated by 𝑉𝑖 (𝑧) is of the form𝛿𝑆 ∝ ∫ d2 𝑧 ∑ 𝑔𝑖 𝑉𝑖 (𝑧),(2.60)𝑖where 𝑔𝑖 's are constants corresponding to coordinates in the moduli space of CFT's (rigorously speaking they're the coordinates in the tangent space). We can easily seen that theoperator 𝑉𝑖 should be of conformal dimension 1. Those operators are called the marginal
operators. However, the marginality of 𝑉 (𝑧) can only preserve the classical conformal
symmetry of the action. At the quantum level, one needs a refined notion of exactly
marginal deformation, meaning that the deformations generated by them are tangent
vectors on the moduli space of conformal field theories, which is also called the conformal
manifold in the literature. Checking exact marginality ususally breaks into an orderby-order procedure to see if the deformation preserves the conformal dimension of itself.
One should compute the exact beta-function to see if it vanishes. Intuitively, we need torequire the vanishing of integrals of the three-point functions⟨𝑉𝑖 (𝑧)𝑉𝑖 (𝑤) ∫ d2 𝑧 ′ 𝑉𝑖 (𝑧′ )⟩ = 0(2.61)

to guarantee the two-point function remains unperturbed. If all those perturbation, including higher order terms, vanish, then the perturbation is exactly marginal. In general,
it is diﬀicult to verify by examination of (𝑛 + 2)-point functions that an operator remains
marginal to all orders.
Studying the conformal manifold of two-dimensional conformal field theory is of particular interest as it describes the landscape of string theory vacua. In such cases, these
conformal manifold is endowed with a spacetime interpretation of the conformal field theory when we view it as a classical solution of string theory [7]. Various works has been
done in this direction. However, it is diﬀicult in general without an extended symmetry. For example, the 𝒩 = 2 superconformal field theories are fairly understood. One
can identify all the exactly marginal fields, and, even better, find out that the conformal
manifold have locally the structure of the product of the complexified Kähler moduli andthe complex moduli.
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The case with 𝒩 = (1, 1) supersymmetry is, however, elusive. The main purpose ofthis thesis is to explore the exactly maringal deformation of 𝒩 = (1, 1) theory around theorbifold point. The string theory picture again plays an important role here, since it is
known that the string field theory provide a somehow unified framework to treat different
backgrounds as the classical solution to the string field equation of motion. This would
be the central topic of the rest of the thesis. Therefore, it is worthwhile to spend the next
chapter in introducing the formulation of closed string field theory.

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Chapter 3
Closed Superstring Field Theory
The goal of this chapter is to present the geometric construction of the Type II superstring
field theory, as outlined by [8], to initiate our calculation in the next chapter. We explain
the geometric setup in the bosonic case first, where the mathematical structures such as
the BV structure and strong homotopy Lie algebras (𝐿∞ algebras) shows up (details canbe found in [9] and references therein). Going to the super case requires detailed studyof super Riemann surfaces and their sewing property. Finally, we explain the explicit
construction of the classical string field action in the NS-NS sector given by [10], which
is needed for our calculations in the next chapter.

3.1

String Field Theory

String field theory (SFT) is originally proposed as a logical completion to the perturbative formulation of worldsheet string theory. One imetates the procedure of second
quantization to obtain an interacting theory of free propagating strings. Like the case ofquantum field theory, the summation over topologically different worldsheet is supposed
to be derived as Feynman rules from the string field theory action.
However, the fact that there is in general no natrual choice of the field variables leads
to a complecated non-polynomial action and various different construction, related byfield redefinition and/or gauge fixing. It thus remains elusive that how the string field
theory plays the fundamental role like QFT. Nevertheless, recent research has shown that
SFT is indispensable in the non-perturbative region such as the infrared behaviour ofstring theory and D-brane backgrounds.
String field theory is assumed to be background independent (and proven so in, e.g.,[11]), in a sense that the landscape of different string backgrounds are organized in theform of classical solutions to the string field theory equation of motion. We shall exploit
this point in Type II string field theory.

3.2

Geometric BV Structure

The construction of string field theory requires full-fledged Batalin-Vilkovisky formalism,
though many intuitions can be gained from the second quantization from point particle
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Quantum Mechanics

Quantum Field Theory
Second Quantization

String Theory

String Field Theory

Figure 3.1: String field theory plays the role of quantum field theory in the second quantized picture
to ordinary quantum field theory. Such a generalization requires off-shell amplitudes instring theory. Vertex operators 𝒱 in the on-shell amplitudes (2.6) are subject to certain
conditions. Primarily, they are taken to be BRST closed
𝑄𝒱𝑖 = 0,(3.1)

which is analogous to the (linearized) equation of motion in QFT, for it reduces to the
Klein-Gordon equation for point particles [9].
Generalizing to an arbitrary operator in the background CFT faces an immediate
diﬀiculty that the conformal invariance is lost because we lose control of the conformal
weights of the operator products. One compensates it by introducing local coordicate
curves near the punctures𝑓𝑖 ∶ 𝐷 → Σℎ,𝑛(3.2)
such that 𝑓𝑖 (0) = 𝑧𝑖 . The off-shell amplitude thus reads∫
𝑚ℎ,𝑛d(𝑔|𝛿𝑔)𝐹𝒱 (𝑔, 𝛿𝑔) = ∫̂d(𝑔|𝛿𝑔) ∫ d𝑋d𝑏d𝑐e−𝑆[𝑋,𝑏,𝑐,𝑔,𝛿𝑔] (𝑓1 ∘ 𝒱1 ) ⋯ (𝑓𝑛 ∘ 𝒱𝑛 ) .

𝑚ℎ,𝑛

(3.3)
All the data for defining the off-shell amplitudes can be collected into a bundle
strucutre 𝒫̂ℎ,𝑛 → 𝑚ℎ,𝑛 with each fiber the local curves around the punctures. A choice ofthe global section 𝒜ℎ,𝑛 of 𝒫̂ℎ,𝑛 fixes the off-shell amplitudes. However, the existence of
global sections is obstructed in general, even for 𝒫̂0,4 . The established way to deal withthis problem is to restrict the class of states to those which have no sense of ``direction''on the unit disk. That is, we will only attempt to define off-shell amplitudes for vertex
operators satisfying
𝑓 ∘ 𝒱(0) = 𝒱(0)(3.4)
with 𝑓(𝑧) = e𝑖𝜃 𝑧 a rotation around the origin. The local coordinate systems up to
rotation are called coordinate curves for that they can be uniquely characterized by ancircle embedded around the punctured point 𝑆 1 → Σ ∶ 𝑧 ↦ 𝑓(𝑧), |𝑧| = 1. We denote theset of all the choices of such coordinate curves by the bundle 𝒫ℎ,𝑛 .
New punctured Riemann surfaces with coordinate curves can be generated from old
ones. This is done by the sewing operation. Given two punctures 𝑝1 and 𝑝2 with local
coordinates 𝑧1 and 𝑧2 , one can glue these two by identifying via the map
𝐼(𝑧1 ) = −1= 𝑧2 .
𝑧1

(3.5)
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On Orbifolds in Closed Superstring Field Theory

...

×

...

...

...

Figure 3.2: The identification creates a ``tube'' connecting two punctures
The sewing operation provides two maps on the bundle 𝒫ℎ,𝑛 , by gluing two pieces ofdifferent Riemann surfacesΦ ∶ 𝒫ℎ1 ,𝑛1 +1 × 𝒫ℎ2 ,𝑛2 +1 → 𝒫ℎ1 +ℎ2 ,𝑛1 +𝑛2 ,(3.6)

and by gluing two punctures on the same Riemann surface𝜁 ∶ 𝒫ℎ,𝑛 → 𝒫ℎ+1,𝑛−2 .

(3.7)

In order to extract the mathematical structures related to the moduli space, one must
introduce the geometric chains 𝐶 ∗ (𝒫ℎ,𝑛 ) on 𝒫ℎ,𝑛 . The definition of this chain complex,
however, requires mathematical technical that we would not like to involve, so we refer[12] for interested readers. Intuitively speaking, we can think of the generators 𝑎ℎ,𝑛 assome dim(𝑎ℎ,𝑛 )-dimensional space mapping into 𝒫ℎ,𝑛 , with the grading
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On Orbifolds in Closed Superstring Field Theory

deg(𝑎ℎ,𝑛 ) = dim(ℳℎ,𝑛 ) − dim(𝑎ℎ,𝑛 ).

(3.8)

Therefore, one can choose some zero-degree generators 𝜈ℎ,𝑛 ∈ 𝐶 0 (𝒫ℎ,𝑛 ) and thinkof them as some spaces with the same dimension as the moduli space ℳℎ,𝑛 mapping
into the bundle 𝒫ℎ,𝑛 , that is, some local sections characterizing continuous choices ofcoordinate curves along the moduli. They will be called the geometric vertices in thefollowing context.
The 𝜈ℎ,𝑛 's contain all the geometric data to construct a string field theory. But wemust consider some compatibility conditions among them. The operations (3.6) and (3.7)
induce operations on the (co)chain complex, that, by abuse of notations, we still denoteas Φ and 𝜁,Φ ∶ 𝐶 𝑘1 (𝒫ℎ1 ,𝑛1 +1 ) × 𝐶 𝑘2 (𝒫ℎ2 ,𝑛2 +1 ) → 𝐶 𝑘1 +𝑘2 +1 (𝒫ℎ1 +ℎ2 ,𝑛1 +𝑛2 ),(3.9)

𝜁 ∶ 𝐶 𝑘 (𝒫ℎ,𝑛 ) → 𝐶 𝑘+1 (𝒫ℎ+1,𝑛−2 ).

(3.10)∗
From now on we restrict to the subchains 𝐶inv
(𝒫ℎ,𝑛 ) that are invariant under the
permutation of punctures, as required by the indistinguishability of particles. Together
with the boundary operator 𝜕 of the chain complex, we have three operations 𝜕, {−, −}and Δ. The latter two are induced by Φ and 𝜁,𝑘𝑘 +𝑘2 +1𝑘1
2
1
(𝒫ℎ2 ,𝑛2 +1 ) → 𝐶inv
(𝒫ℎ1 ,𝑛1 +1 ) × 𝐶inv{−, −} ∶ 𝐶inv

(𝑏ℎ1 ,𝑛1 +1 , 𝑏ℎ2 ,𝑛2 +1 ) ↦(𝒫ℎ1 +ℎ2 ,𝑛1 +𝑛2 )∑𝜎 ⋅ Φ(𝑏ℎ1 ,𝑛1 +1 , 𝑏ℎ2 ,𝑛2 +1 ), (3.11)𝜎∈sh(𝑛1 ,𝑛2 )where sh(𝑛1 , 𝑛2 ) denotes the shuffles, that is, the subset of the 𝑛1 + 𝑛2 -permutation group
under the constraint that 𝜎1 < ⋯ < 𝜎𝑛1 and 𝜎𝑛1 +1 < ⋯ < 𝜎𝑛1 +𝑛2 .
𝑘+1
𝑘(𝒫ℎ+1,𝑛−2 )Δ ∶ 𝐶inv(𝒫ℎ,𝑛 ) → 𝐶inv

𝑏ℎ,𝑛 ↦ 𝜁(𝑏ℎ+1,𝑛−2 ).

(3.12)∗
Together with 𝜕, Δ, and {−, −}, 𝐶inv
(𝒫ℎ,𝑛 ) yields a structure of BV algebra:

𝜕2 = 0Δ2 = 0𝜕Δ + Δ𝜕 = 0
𝜕{−, −} = {𝜕, −} − {−, 𝜕}Δ{−, −} = {Δ, −} − {−, Δ}(3.13){𝑎, 𝑏} = (−1)(deg(𝑎)+1)(deg(𝑏)+1) {𝑏, 𝑎}(−1)(deg(𝑎)+1)(c+1) {{𝑎, 𝑏}, 𝑐} + cycl. = 0.
Constructing a string field theory action is then an decomposition of the moduli spaceℳℎ,𝑛 into pieces corresponding different Feynman diagrams. It can be achieved by sewing
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On Orbifolds in Closed Superstring Field Theory

geometric vertices 𝜈ℎ,𝑛 in a way that reproduces all the entire off-shell amplitudes. For
instance, one can sew two cubic vertices 𝜈0,3 in all possible, nonequivalent ways gives
the boundary of a region the moduli space ℳ0,4 . In order to cover the whole moduli
space one needs to add an elementary four-vertex 𝜈0,4 . This can be cast into the form ofa geometric BV equation1𝜕𝜈0,4 + {𝜈0,3 , 𝜈0,3 } = 0.
(3.14)
2
In general, the moduli space ℳℎ,𝑛 can be covered by the elementary vertex 𝜈ℎ,𝑛 ,sewing two different vertices by {−, −} and self-sewing one vertex by Δ. More thoroughconsideration of the geometric vertices shows that𝜕𝜈ℎ,𝑛 +12

{𝜈ℎ1 ,𝑛1 +1 , 𝜈ℎ2 ,𝑛2 +1 } + Δ𝜈ℎ−1,𝑛+2 = 0.

∑

(3.15)

𝑛1 ≤𝑛2 ; ℎ1 ≤ℎ2
𝑛1 +𝑛2 =𝑛; ℎ1 +ℎ2 =ℎThe geometric vertices are data that are independent of the background. One introduces the background dependence via the worldsheet CFT. More specifically, via theoff-shell amplitude (3.3). The algebraic vertices corresponding to a given background are
now defined by integrating the geometric vertices 𝜈ℎ.𝑛 over the appropriate differential
forms:
𝑓ℎ,𝑛 ≡ ∫ 𝜋∗ Ωℎ,𝑛 ,(3.16)
𝜈ℎ,𝑛where Ωℎ,𝑛 ∈ Ω∗ (ℳℎ,𝑛 , Hominv (ℋ⊗𝑛
𝑆𝐹 𝑇 , ℂ)) is obtained from the form
Ωℎ,𝑛 ∶ (𝒱1 , ⋯ , 𝒱𝑛 ) ↦ Ω𝒱;ℎ,𝑛 .

(3.17)

The corresponding BV master action is written as∞

𝑆[Ψ] = ∑ ℏℎ 𝑓ℎ (Ψ).

(3.18)ℎ=0

where 𝑓ℎ represents the genus ℎ interaction and has the structure of a sum of 𝑛 point
vertices,∞1𝑓(Ψ, ⋯ , Ψ),(3.19)𝑓ℎ (Ψ) = ∑(𝑛 + 1)! (ℎ,𝑛) ⏟𝑛=1
𝑛 timesand

The consequence of passing from 𝒫̂ℎ,𝑛 to 𝒫 is to further restrict the space of string
fields into a subspace satisfying(𝑏0 − 𝑏̄0 )Ψ = 0.
(3.20)
Note that 𝑄 is a valid operator on the resticted state space since {𝑄, 𝑏0 − 𝑏̄0 } = 𝐿0 − 𝐿̄ 0 ,and (𝐿0 − 𝐿̄ 0 ) Ψ = 0 by the level-matching condition.
Simliar to the BV structure introduced in (3.11) and (3.12), we define a BV structureon Hominv (ℋ⊗𝑛
𝑆𝐹 𝑇 , ℂ), which is induced by the odd symplectic structure. Again, we havethe algebraic BV master equation

19

Soryushiron Kenkyu

On Orbifolds in Closed Superstring Field Theory𝜕𝑓ℎ,𝑛 +

1
2∑

{𝑓ℎ1 ,𝑛1 +1 , 𝑓ℎ2 ,𝑛2 +1 } + Δ𝑓ℎ−1,𝑛+2 = 0.

(3.21)

𝑛1 ≤𝑛2 ; ℎ1 ≤ℎ2
𝑛1 +𝑛2 =𝑛; ℎ1 +ℎ2 =ℎConstructing explicitly these vertices is highly non-trivial even in the bosonic case(and more so for closed strings). For closed string field theory, a general construction
procedure can be given by the minimal area metric [13], but writing down the explicit
forms are still a highly tedious task.

3.3

𝐿∞ Algebras

Now we can summerize what we have obtained for the bosonic closed string field theory.
From the background CFT we get a Hilbert space ℋ. The space of string fields is arefinement on ℋ. A state Ψ ∈ ℋ is said to be a string field if it satisfies:
1. gh(Ψ) = 2,
2. (𝑏0 − 𝑏̄0 )Ψ = 0,
3. (𝐿0 − 𝐿̄ 0 )Ψ = 0.
To get meaningful amplitudes one has to keep track of the total grading.
The consequence of (3.15) is the quantum BV master equation (3.21) on the algebraic
vertices, or the string products. It is sometimes also written in a form that fits into thegeneral BV quantization framework
1{𝑆[Ψ], 𝑆[Ψ]} + ℏΔ𝑆[Ψ] = 0.
2(3.22)

proposed as a consistency requirement of the action.
From a physical point of view, introducing the symplectic form, namely a gradedantisymmetric, non-degenerate bilinear form defined via the BPZ inner product𝜔(Ψ1 , Ψ2 ) = (−1)deg(Ψ1 ) ⟨Ψ1 , 𝑐0− Ψ2 ⟩ ,(3.23)

helps us to write down the free field action in terms of the propagator 𝑄.
Considering only the genus zero contribution to the string interaction, that is, ignoring
all higher-genus effects, we obtain the classical master equation{𝑆[Ψ], 𝑆[Ψ]} = 0.

(3.24)

We can define, since 𝜔 is non-degenerate𝜔(−, 𝐿𝑛 ) ≡ 𝑓𝑛+1,0 ,(3.25)

which is compatible with the identification 𝐿1 = 𝑄. The classical action is then of the
form
∞
1
1
𝑆 = 𝜔(Ψ, 𝑄Ψ) + ∑𝜔(Ψ, 𝐿𝑛+1 (Ψ, ⋯ , Ψ)).
(3.26)
2(𝑛 + 2)!
𝑛=1
20

Soryushiron Kenkyu

On Orbifolds in Closed Superstring Field Theory

It appears to be quite astonishing that the classical master equation ranslates exactly into
the 𝐿∞ relations𝑛∑[𝐿𝑝 , 𝐿𝑛−𝑝+1 ] = 0.

(3.27)

𝑝=1

We will show that, however, in the appendix A that it is because BV structures andhomotopy algebras are unified under the theory of operads.
(0,0)
The construction of all closed bosonic products 𝐿𝑛 is another story which can beorganized under a minimal area metric (see [13]).

3.4

Closed Superstring Field Theory

Closed superstring field theory takes the Hilbert space of the closed superstring as it spaceof field configuration. We'll focus on Type II theory in the RNS formalism here. The
procedure is very similar to the bosonic case, except for two points: The moduli spaceis replaced by the super moduli of super Riemann surfaces, and one must distinguishbetween punctures in Neveu-Schwarz and Ramond sectors. The geometric setup is thesame in spirit, except for that now we have to types of punctures, NS and R. And we still
have the result that the vertices of the classical BV master equation of closed superstring
field theory satisfy the axioms of a 𝐿∞ -algebra.
Now let's take a look at the 𝐿∞ algebra structures in the classical closed superstring
field theory action in details. Once more, we begin by the space of states ℋ in superstring
perturbation theory. Now the space ℋ is bi-graded by ghost and picture number, andthe BRST charge 𝑄 together with ghost number grading yields a cochain complex, whose
first cohomology has been as with the physical states.
The space of superstring fields is a refinement on ℋ. A state Ψ ∈ ℋ is said to be astring field if it satisfies:
1. gh(Ψ) = 2,

pic(Ψ) = (−1, −1),

2. (𝑏0 − 𝑏̄0 )Ψ = 0,
3. (𝐿0 − 𝐿̄ 0 )Ψ = 0.
The first condition is usually called the ``natural'' ghost and picture number to computegeneric amplitudes in literature, for example see [14]. To get meaningful amplitudes one
has to keep track of the total grading, since the symplectic form 𝜔 is nonvanishing only onstates whose ghost number adds up to five and whose picture number ads up to (−2, −2).
The string field action is defined by an 𝐿∞ algebra consisting of a series of multilinear
products of ghost degree 1(𝑛−1,𝑛−1)
𝐿𝑛∶ ℋ∧𝑛 → ℋ,
(3.28)
where the 1-product is just 𝑄. The superscript indicates that the picture number of the𝑛-product is (𝑛 − 1, 𝑛 − 1) to ensure the output has the natrual picture (−1, −1) as astring field. The full classical action thus reads
∞
1
1
(𝑛,𝑛)
𝜔(Ψ, 𝐿𝑛+1 (Ψ, ⋯ , Ψ)).
𝑆 = 𝜔(Ψ, 𝑄Ψ) + ∑
2
(𝑛
+
2)!
𝑛=1

(3.29)
21

Soryushiron Kenkyu

On Orbifolds in Closed Superstring Field Theory

The problem of constructing superstring products is then how to place picture changing operators on the bosonic string products.
After construction in the supermoduli one would naturally come to the question thatwhether it is possible to split the integration over the even and odd moduli, integrate out
the latter, and reduce the integral to the one on the bosonic moduli. Unfortunately, thisis shown not to be true in [15]. One can not find a global holomorphic projection from thesupermoduli to the bosonic one. We've seen that the picture-changing operators must be
inserted as a result of integration over the odd moduli. The problem is closely related tothe so-called spurious singularity of those PCO's, where it is not possible to find a global
way to insert the PCO's for all the moduli space. There is a workaround, though, by
the method of vertical integration [16], whose idea is to assgin the PCO's locally, withdiscontinuity and carefully consider the ``vertical'' sections connecting them. Therefore,
one can still construct the theory by properly dressing the bosonic products with PCO's
locally. This will be the topic of the next section.

3.5

Assigning PCO's

A construction of Type II superstring field theory in the NS-NS sector [10] is summarized
briefly here. As explained before, the string field Ψ has ghost number 2 and picture
number −1, and satisfies (𝑏0 − 𝑏̄0 )Ψ = (𝐿0 − 𝐿̄ 0 )Ψ = 0. An on-shell state in Siegel gauge𝑏0 Ψ = 𝑏̄0 Ψ = 0 takes the form̄

Ψ ∼ 𝑐𝑐ē −𝜙 e−𝜙 𝑉 (0, 0),(3.30)

where 𝑉 is a worldsheet matter primary field with conformal weight ( 12 , 12 ). The amplitudes(2.7) is nonvanishing only if the states are of ghost numbers adding up to 5 and picture
numbers adding up to −2.
The picture changing operator has a zero mode
d𝑧 1
𝒳(𝑧),
|𝑧|=1 2𝜋𝑖 𝑧

𝒳=∮(3.31)

which is to be used in the string field theory vertices. Unlike the open strings, the 𝑏d𝑧 1constraint prohibits a more general charge 𝜉 = ∮|𝑧|=1 2𝜋𝑖𝑧 𝜉(𝑧) to construct 𝒳 = {𝑄, 𝜉} .
We start by correctly assigning picture changing operators in the holomorphic sector.
By collecting superstring products into a generating function
∞

(𝑛,0)

𝐿[0] (𝑡) = ∑ 𝑡𝑛 𝐿𝑛+1 ,(3.32)

𝑛=0

where the (𝑛 + 1)-st superstring product is the 𝑛-th coeﬀicient of the formal variable 𝑡.
The upper index on the generating function is to indicate the picture dificit relative towhat is needed in the superstring amplitudes. The products should be independent of the𝜉 zero-mode,[𝜂, 𝐿[0] (𝑡)] = 0.
(3.33)
22

Soryushiron Kenkyu

On Orbifolds in Closed Superstring Field Theory

In our notation the 𝐿∞ relations can be written as[𝐿[0] (𝑡), 𝐿[0] (𝑡)] = 0.

(3.34)

Expanding it in powers of 𝑡 gives the 𝐿∞ relations. Such a series of products can berealized by a differential equation postulate on the generating function
𝜕 [0]𝐿 (𝑡) = [𝐿[0] (𝑡), 𝜆[0] (𝑡)] ,
𝜕𝑡
where

∞
[0]

(𝑛+1,0)𝜆 (𝑡) = ∑ 𝑡𝑛 𝜆𝑛+2,(3.35)(3.36)

𝑛=0

is a generating function for deficit-free auxiliary gauge products.
An non-trivial condition is proposed on the products by requiring are in the small
Hilbert space. Taking 𝜂 of the differential equation (3.35) we see[𝐿[0] (𝑡), 𝐿[1] (𝑡)] = 0,
where(3.37)∞(𝑛,0)

𝐿 (𝑡) = [𝜂, 𝜆 (𝑡)] = ∑ 𝑡𝑛 𝐿𝑛+2
[1]

[0]

(3.38)

𝑛=0

is the single deficit generating function. It can be solved by introducing new auxiliary
products with proper deficit, that is,𝜕 [1]
𝐿 (𝑡) = [𝐿[0] (𝑡), 𝜆[1] (𝑡)] + [𝐿[1] (𝑡), 𝜆[0] (𝑡)] ,
𝜕𝑡
where

∞

(𝑛+1,0)𝜂 (𝑡) = ∑ 𝑡𝑛 𝜂𝑛+3
[1].

(3.39)(3.40)

𝑛=0

This iterative procedure on picture deficit can be carried out as follows. Taking 𝜂 ofeach differential equation implies a constraint on the generating function for productswith higher deficits, which can be solved by yet another differential equation. Introducing
another formal variable 𝑠 for the recursion on deficit, the procedure can be compactly
expressed by the generating functions∞(𝑛)(3.41)(𝑛+1)(3.42)𝐿(𝑠, 𝑡) = ∑ 𝑠𝑚 𝑡𝑛 𝐿𝑚+𝑛+1 ,𝑚,𝑛=0∞𝜆(𝑠, 𝑡) = ∑ 𝑠𝑚 𝑡𝑛 𝜆𝑚+𝑛+2 .
𝑚,𝑛=0Note that at 𝑡 = 0, 𝐿(𝑠, 𝑡) reduces to a generating function of the bosonic string products,
and at 𝑠 = 0 it reduces to a generating function for the desired superstring products.
The recursion is dictated by the differential equations
𝜕𝐿(𝑠, 𝑡) = [𝐿(𝑠, 𝑡), 𝜆(𝑠, 𝑡)] ,𝜕𝑡(3.43)
23

Soryushiron Kenkyu

On Orbifolds in Closed Superstring Field Theory𝜕𝐿(𝑠, 𝑡) = [𝜂, 𝜆(𝑠, 𝑡)] ,(3.44)
𝜕𝑠whose solutions automatically determines a set of superstring products which live in thesmall Hilbert space and satisfy the 𝐿∞ relations.
Expanding (3.43) is 𝑠, 𝑡 and comparing the coeﬀicients give(𝑛+1,0)𝐿𝑚+𝑛+2 =

𝑛 𝑚
1
(𝑘,0)
(𝑛−𝑘+1,0)∑ ∑ [𝐿𝑘+𝑙+1 , 𝜆𝑚+𝑛−𝑘−𝑙+2 ] ,𝑛 + 1 𝑘=0 𝑙=0(3.45)(𝑛+1,0)which determines the explicit form of the product 𝐿𝑚+𝑛+2 provided that we know all the(𝑘,0)gauge products 𝜆𝑙 . The gauge products can be read off from (3.44) by expanding,(𝑛+1,0)(𝑛,0)[𝜂, 𝜆𝑚+𝑛+2 ] = (𝑚 + 1)𝐿𝑚+𝑛+2 .
(𝑛+1,0)(3.46)(𝑚,0)This equation will determine 𝜆𝑚+𝑛+2 in terms of 𝐿𝑚+𝑛+2 and the solution to it is notunique. One naturally suggested solution similar to the resolving procedure in [10] would
be
𝑚 𝑛
1
(𝑘,0)
(𝑚−𝑘+1,0)(𝑚+1,0)∑ ∑ [𝐿𝑘+𝑙+1 , 𝜆𝑚+𝑛−𝑘−𝑙+2 ] .
(3.47)
𝐿𝑚+𝑛+2 =𝑚 + 1 𝑘=0 𝑙=0The construction is recursive and provides a specific way to assgin correct holomorphic picture number to the superstring products. Similar needs to be done to the antiholomorphic sector. In conclusion, we now have a way to construction the (𝑛+1)-st closed
superstring field theory product by climbing a ladder of products and auxiliary products
(0,0)

(1,0)

(1,0)

(𝑛,0)

(𝑛,1)

(𝑛,1)

(𝑛,𝑛)

𝐿𝑛+1 , 𝜆𝑛+1 , 𝐿𝑛+1 , ⋯ , 𝐿𝑛+1 , 𝜆𝑛+1 , 𝐿𝑛+1 , ⋯ , 𝐿𝑛+1 ,(3.48)

where the iteration is given as
𝑚 𝑛
1
(𝑘,0)
(𝑚−𝑘+1,0)∑ ∑ [𝐿𝑘+𝑙+1 , 𝜆𝑚+𝑛−𝑘−𝑙+2 ] ,𝑚 + 1 𝑘=0 𝑙=0(3.49)

𝑛+1
(𝑚,0)
(𝑚,0)
(𝜉𝐿𝑚+𝑛+2 − 𝐿𝑚+𝑛+2 (𝜉 ∧ 𝕀𝑚+𝑛+1 )) ,𝑚+𝑛+3

(3.50)(𝑚+1,0)𝐿𝑚+𝑛+2 =(𝑚+1,0)𝜆𝑚+𝑛+2 =(𝑚+𝑛+1,𝑚+1)
𝐿𝑚+𝑛+2=𝑚 𝑛
1(𝑘+𝑙,𝑘)(𝑚+𝑛−𝑘−𝑙+1,𝑚−𝑘+1)∑ ∑ [𝐿𝑘+𝑙+1 , 𝜆𝑚+𝑛−𝑘−𝑙+2],
𝑚 + 1 𝑘=0 𝑙=0(3.51)𝑛+1
̄ (𝑚+𝑛+1,𝑚) − 𝐿(𝑚+𝑛+1,𝑚) (𝜉 ̄ ∧ 𝕀𝑚+𝑛+1 )) .
(𝜉𝐿
(3.52)
𝑚+𝑛+2
𝑚+𝑛+2
𝑚+𝑛+3
For our purpose it is enough to know these relations up to 3-products, which are listedbelow
1 (1,1) (1,1)
1
(2,2)
(2,2)
(3.53)
𝐿3 = [𝑄, 𝜆3 ] + [𝐿2 , 𝜆2 ] ,2
2
1 ̄ (2,1)
(2,2)
(2,1)
𝜆3 = (𝜉𝐿− 𝐿3 (𝜉 ̄ ∧ 𝕀2 )) ,(3.54)
3
4(𝑚+𝑛+2,𝑚+1)

𝜆𝑚+𝑛+2=(1,1)[𝑄, 𝐿2] = 0,(3.55)
24

Soryushiron Kenkyu

On Orbifolds in Closed Superstring Field Theory

1 (1,1) (1,1)[𝐿, 𝐿2 ] = 0.
(3.56)
2 2
In this chapter, we described the construction of closed string field theory, otherrealizations of string field theory can be described similarly, for example, the open-closed
theory [17]. The most successful realization of string field theory is definitely Witten’s
open string field theory [18]. It is special in the sense that besides the kinetic term it
involves only a cubic vertex – the star product. Witten constructed this theory in acompletely different manner than described above, by seeking for a Chern-Simons likeaction which possesses decent gauge symmetries. Anyhow, it has been realized later onthat Witten’s cubic string field theory arises indeed form the geometrical approach withappropriate minimal area metrics [19].
We conclude this chapter with the fact that the string field theory action is not limited
to the NS-NS sector which we discussed here. The definition of the products should be
extended to include the Ramond sectors, but it is beyond the scope of this thesis. For acomplete theory of closed superstrings see, for example, [20].
(2,2)[𝑄, 𝐿3]+25

Soryushiron Kenkyu

Chapter 4
Deformation via SFT Action
In this chapter we study the possibility of a marginal deformation in the orbifold theory
to be exactly marginal corresponding to the blow-up of the orbifold singularity. The
framework of (bosonic or supersymmetric) SFT is needed and exploited in this problem.
We study the marginality up to the third order, and derive the perturbated metric at thesecond order.

4.1

Perturbative Approach

In this section we formulate the problem of exact marginality corresponding to twist fields
in the framework of string field theory. Suppose we have identified the form of marginal
operators in both bosonic and supersymmetric cases that we denote as Ψ(1) here. The
equation of motion of closed string field theory can be simply derive from the action as𝑄Ψ + 𝐿2 (Ψ, Ψ) + 𝐿3 (Ψ, Ψ, Ψ) + ⋯ = 0.

(4.1)

(1)

Note that the on-shell condition 𝑄Ψ = 0 can be understood as the linearizationof the full equation of motion. The question of exact marginality is then translated into
whether this solution can be lifted to a full solution, written as a perturbative series
Ψ = 𝜆Ψ(1) + 𝜆2 Ψ(2) + 𝜆3 Ψ(3) + ⋯.
This gives a non-homogeneous linear equation at each order of 𝜆. In particular, wehave𝑄Ψ(2) + 𝐿2 (Ψ(1) , Ψ(1) ) = 0,
(4.2)
and
𝑄Ψ(3) + 𝐿2 (Ψ(2) , Ψ(1) ) + 𝐿3 (Ψ(1) , Ψ(1) , Ψ(1) ) = 0,

(4.3)

which will be of primary interest and serve as a starting point of our perturbative analysis.
At any order, to solve the equation we need to check if any obstruction appear, namely if
the BRST charge 𝑄 is invertible on the linear subspace where the non-homogeneous partlies. For example, at the second order, the solution
Ψ(2) = 𝑄−1 𝐿2 (Ψ(1) , Ψ(1) )(4.4)

is well-defined only if 𝒪2 = 𝑃0 [𝐿2 (Ψ(1) , Ψ(1) )] = 0, where 𝑃0 is the projector on thespace on which 𝑄 is not invertible. Whenever the obstruction is non-vanishing, the lifting
procedure is blocked.
26

Soryushiron Kenkyu

On Orbifolds in Closed Superstring Field Theory

4.2

Second-order Deformation

In this section we explicitly evaluate the obstruction appears in the second order equation,in both bosonic and supersymmetric SFT. However, as we'll see immediately, lifting to
the full exactly marginal operator requires us to consider the theory of superstrings.

4.2.1

Bosonic String
25

The massless field in the twist sector is 𝕍 = ∏𝑘=10 |0⟩ ≡ Δ |0⟩ . Together with its rightmoving partner 𝕍̄ it describe a tangent vector in the moduli space of the target spacetimeat the orbifold point.Considering the ghost contribution, the marginal operator reads
Ψ(1) = 𝑐𝕍𝑐𝕍.
̄ ̄
The second order obstruction 𝒪2 is expressed by the projection
𝑃0 𝐿2 (Ψ(1) , Ψ(1) ) = lim 𝑃0 [𝑐𝕍(𝑧)𝑐𝕍(−𝑧)] ⊗ lim 𝑃0 [𝑐𝕍(
̄ ̄ 𝑧)̄ 𝑐𝕍(−̄ ̄ 𝑧)].
̄
𝑧→0

𝑧→0
̄

(4.5)

We now show the holomorphic part is non-zero by noting that the OPE of two ghost
fields selects out in the OPE of matter field the term to 𝑧 −1 . The OPE is given by√√1∶ 𝑒𝑖 2𝑋 ∶ − ∶ 𝑒−𝑖 2𝑋 ∶̄Δ(𝑧)Δ(𝑤)∼−+ ⋯,(𝑧 − 𝑤)2
𝑧−𝑤

(4.6)

and contributes to the projection√√

lim 𝑃0 [𝑐𝕍(𝑧)𝑐𝕍(−𝑧)] = 𝑐𝜕𝑐(∶ 𝑒𝑖 2𝑋 ∶ − ∶ 𝑒−𝑖 2𝑋 ∶)(0)

𝑧→0

(4.7)

as shown in [21]. The anti-holomorphic part of the projection is similar, and in conclusion
we will have a non-vanishing obstruction in the bosonic case.
The presence of this obstruction means that it is impossible to find a bosonic string
background near the orbifold theory, and a geometric picture of blown up sigma model
is absent. As a consequence, we are led to the superstring field theory. This is necessary
anyhow, since the bosonic string field theory is not considered a consistent theory in many
senses, both for the presence of tachyon that indicates a wrong choice of vacuum and forbeing unable to describe fermions. In the next subsection we will see that this obstruction
𝒪2 is not there anymore in the superstring theory, and it will allow us to calculation thegraviton contribution of the perturbative solution, and thus compute the metric of theblown up model.

4.2.2

Superstring

The existence of obstruction leads us to consider the Type II superstring field theory. The
method of constructing such a theory was reviewed in the last chapter. There are various
approaches to a consistent SFT action. We will use the one that was proposed in [22],which we've introduced in 3.5. In the NS-NS sector, the twisted deformation moduli isgiven by the spin fields as
𝑤𝛼 𝑆 𝛼 Δ |0⟩𝑁𝑆 ⊗ 𝑤̄ 𝛽 𝑆 𝛽̄ Δ̄ ∣0⟩̄ 𝑁𝑆 .

(4.8)
27

Soryushiron Kenkyu

On Orbifolds in Closed Superstring Field Theory

In order to test exact marginality, we define the massless field in the (−1, −1) picturēΨ = 𝑐𝑐𝕍̄ 1 𝕍̄ 1 𝑒−𝜙 𝑒−𝜙 , where 𝕍 1 = 𝑤𝛼 𝑆 𝛼 Δ and 𝕍̄ 1 = 𝑤̄ 𝛽 𝑆 𝛽̄ Δ.̄
2
2
22
Taking into account the picture-changing operator arising from integrating over odd
moduli, the two-product in superstring field theory is defined by (see [22])
(1)

(1,1)

𝐿2

(Ψ(1) , Ψ(1) )1̄ (0,0) (Ψ(1) , Ψ(1) ) + 𝐿(0,0) (𝒳Ψ̄ (1) , Ψ(1) ) + 𝐿(0,0) (Ψ(1) , 𝒳Ψ
̄ (1) ))= 𝒳 (𝒳𝐿
2
2
2
9
1
(0,0)
(1)
̄ (0,0) (𝒳Ψ(1) , Ψ(1) ) + 𝐿(0,0) (𝒳𝒳Ψ̄
̄ (1) ))+ (𝒳𝐿
, Ψ(1) ) + 𝐿2 (𝒳Ψ(1) , 𝒳Ψ22
91
̄ (0,0) (Ψ(1) , 𝒳Ψ(1) ) + 𝐿(0,0) (𝒳Ψ̄ (1) , 𝒳Ψ(1) ) + 𝐿(0,0) (Ψ(1) , 𝒳𝒳Ψ̄ (1) )) ,
+ (𝒳𝐿
2
2
2
9

(4.9)

(0,0)

where the bare product or the ``buttom-of-the-ladder'' product 𝐿𝑛
is just thebosonic product 𝐿𝑛 by construction. This serves as a starting point of the recursive
construction of all superstring product decribed in 3.5. Making use of the factorization
property (4.5) we can factorize each term into left- and right-movers
(1,1)

𝑃0 𝐿2

(Ψ(1) , Ψ(1) ) = lim 𝑃0 (𝒳 ⋅ 𝑐𝕍𝑒−𝜙 (𝑧)𝑐𝕍𝑒−𝜙 (−𝑧)𝑧→0+ 𝒳𝑐𝕍𝑒−𝜙 (𝑧)𝑐𝕍𝑒−𝜙 (−𝑧)+ 𝑐𝕍𝑒−𝜙 (𝑧)𝒳𝑐𝕍𝑒−𝜙 (−𝑧))̄̄⊗ lim 𝑃 (𝒳̄ ⋅ 𝑐𝕍𝑒̄ ̄ −𝜙 (𝑧)̄ 𝑐𝕍𝑒̄ ̄ −𝜙 (−𝑧)̄

(4.10)

0

𝑧→0
̄

̄
̄
+ 𝒳̄ 𝑐𝕍𝑒̄ ̄ −𝜙 (𝑧)̄ 𝑐𝕍𝑒̄ ̄ −𝜙 (−𝑧)̄
̄
̄
+ 𝑐𝕍𝑒
̄ ̄ −𝜙 (𝑧)̄ 𝒳̄ 𝑐𝕍𝑒̄ ̄ −𝜙 (−𝑧))̄ ,and deal with them separately.
The holomorphic part vanishes by the following argument, which is essentially thesame as for the open string in [22]. By noting that the picture-changed marginal operator
takes the form
1
(4.11)
𝒳𝑐𝕍 1 𝑒−𝜙 = −𝑐𝕍1 + 𝛾𝕍 1 ,2
4 2
where 𝕍1 is a Grassmann-even superconformal primary of weight one defined by
𝑇𝐹 (𝑧)𝕍 1 (0) =2

𝕍1 (0)+ 𝑂(1),
𝑧(4.12)

we compute the last two terms in the holomorphic part aslim 𝑃0 (𝒳𝑐𝕍 1 𝑒−𝜙 (𝑧)𝑐𝕍 1 𝑒−𝜙 (−𝑧) + 𝑐𝕍 1 𝑒−𝜙 (𝑧)𝒳𝑐𝕍 1 𝑒−𝜙 (−𝑧))

𝑧→0

2

2

2

2

(4.13)
1
1= lim 𝑃0 ((−𝑐𝕍1 + 𝛾𝕍 1 )(𝑧)𝑐𝕍 1 𝑒−𝜙 (−𝑧) + 𝑐𝕍 1 𝑒−𝜙 (𝑧)(−𝑐𝕍1 + 𝛾𝕍 1 )(−𝑧)) .
2
2
𝑧→0
4 2
4 2

28

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On Orbifolds in Closed Superstring Field Theory

Since the OPE of 𝕍 1 (𝑧)𝕍 1 (−𝑧) contains a single pole while that of 𝕍 1 (𝑧)𝕍1 (−𝑧) doesn't,
2
2
2
we conclude thatlim 𝑃0 (𝒳𝑐𝕍 1 𝑒−𝜙 (𝑧)𝑐𝕍 1 𝑒−𝜙 (−𝑧) + 𝑐𝕍 1 𝑒−𝜙 (𝑧)𝒳𝑐𝕍 1 𝑒−𝜙 (−𝑧))

𝑧→0

2

2

2

2

1
1
= lim 𝑃0 ( 𝛾𝕍 1 (𝑧)𝑐𝕍 1 𝑒−𝜙 (−𝑧) + 𝑐𝕍 1 𝑒−𝜙 (𝑧) 𝛾𝕍 1 (−𝑧)) .
2
2
𝑧→0
4 2
4 2

(4.14)

Observe that the two terms have the same operator products in both the matter andghost sector, but commuting 𝑐 with 𝕍 1 results in an extra minus sign. Therefore the two2
terms cancel out and give a vanishing result.
The remaining term can be evaluated as follows. First notice that 𝒳 and 𝑃0 commute
lim 𝑃0 (𝒳 ⋅ 𝑐𝕍 1 𝑒−𝜙 (𝑧)𝑐𝕍 1 𝑒−𝜙 (−𝑧)) = 𝒳 lim 𝑃0 (𝑐𝕍 1 𝑒−𝜙 (𝑧)𝑐𝕍 1 𝑒−𝜙 (−𝑧)) .

𝑧→0

2

𝑧→0

2

2

2

(4.15)The un-picture-changed part is obtained via the OPE
lim 𝑃0 (𝑐𝕍 1 𝑒−𝜙 (𝑧)𝑐𝕍 1 𝑒−𝜙 (−𝑧)) = 𝑃0 [𝜕(𝑐𝜕𝑐𝑒−2𝜙 𝕍0 ) + 𝑐𝜕𝑐𝕍1 𝑒−2𝜙 ].

𝑧→0

2

2

(4.16)

It's not hard to see that the first term 𝜕(𝑐𝜕𝑐𝑒−2𝜙 𝕍0 ) = 𝑄(𝜕𝑐𝑒−2𝜙 𝕍0 ) is 𝑄-exact and thus
projected out, leaving
(0,0)

𝑃0 𝐿2

̄
(Ψ(1) , Ψ(1) ) = 𝑐𝜕𝑐𝕍1 𝑒−2𝜙 ⊗ 𝑐𝜕̄ 𝑐̄ 𝕍̄ ̄ 1 𝑒−2𝜙 .

(4.17)

We proceed using the identity 𝒳 = {𝑄, 𝜉} in the large Hilbert space. The BRST charge
is given by 𝑄 = 𝑄0 + 𝑄1 + 𝑄2 , whered𝑧(𝑐𝑇 + 𝑐𝜕𝑐𝑏)(𝑧),2𝜋𝑖d𝑧
𝛾𝑇 (𝑧),
𝑄1 = − ∮
2𝜋𝑖 𝐹
1
d𝑧 2
𝑄2 = − ∮𝑏𝛾 (𝑧).
42𝜋𝑖
𝑄0 = ∮(0,0)

Since 𝑄𝑃0 = 0, we have 𝒳𝑃0 𝐿2(1,1)

𝑃 0 𝐿2(4.18)(4.19)(4.20)(0,0)(Ψ(1) , Ψ(1) ) = 𝑄𝜉𝑃0 𝐿2. And then̄(Ψ(1) , Ψ(1) ) = (𝑄0 + 𝑄1 + 𝑄2 ) (𝜉𝑐𝜕𝑐𝕍1 𝑒−2𝜙 ⊗ 𝜉 𝑐̄ 𝜕̄ 𝑐̄ 𝕍̄ ̄ 1 𝑒−2𝜙 ) ,(4.21)

where the only non-trivial term is the second one. It is proportional to the double poleof the OPE explicitly evaluated below
1
𝑇𝐹 (𝑧)𝕍1 (0) ∼ (𝜕𝑋 𝜇 𝜓𝜈 − 𝜕𝑋 𝜈 𝜓𝜇 ) + 𝑂(1),𝑧(4.22)

where 𝑇𝐹 is the supercurrent defined by
𝑇𝐹 (𝑧) = √𝑖𝜓𝜇 𝜕𝑋 𝜇 .
2𝛼′(4.23)The double pole is absent in the OPE, therefore, we establish that
(1,1)

𝑃0 𝐿2

(Ψ(1) , Ψ(1) ) = 0.

(4.24)
29

Soryushiron Kenkyu

On Orbifolds in Closed Superstring Field Theory

4.3

MetricWe've seen that in a superstring setup, the perturbative solution(1,1)

Ψ(2) = 𝑄−1 𝐿2(Ψ(1) , Ψ(1) )(4.25)

is well-defined. Using this we can compute the perturbated metric by evaluating thesolution against the vertex operator for the metric(2)(−1,−1)𝐺𝐼 𝐽 ̄ = ⟨𝑉𝐺

𝐼𝐽 ̄(−1,−1)

where the vertex operator 𝑉𝐺

𝐼𝐽 ̄

(1,1)

, 𝑄−1 𝐿2

(Ψ(1) , Ψ(1) )⟩ ,(4.26)

is taken to be in the (−1, −1) picture. However, the(1,1)

effect of assgined picture-changing operator in 𝐿2
can be moved around, resulting in a(0, 0) picture number for the metric operator. Consequently we have
(2)

(0,0)

𝐺𝐼 𝐽 ̄ = ⟨(𝑓∞ ∘ 𝑉 (−1) 𝑉 ̄ (−1) ) (∞) (𝑓1 ∘ 𝑉𝐺

𝐼𝐽 ̄

) (0; −𝑘) (𝑓0 ∘ 𝑉 (−1) 𝑉 ̄ (−1) ) (0)⟩ ,(4.27)

where the operator in natural picture number1̄̄̄ 𝐼 𝜓𝐽̄ ē −𝜙 e−𝜙 e−𝑖𝑘⋅𝑋 e−𝑖𝑘⋅𝑋
𝑐𝑐𝜓
=
𝐼𝐽 ̄
4
needs to be assigned the picture-changing operator 𝒳𝒳̄ and turns into(−1,−1)

𝑉𝐺

(0,0)

𝑉𝐺

𝐼𝐽 ̄

′
1
𝛼′
−𝑖𝑘⋅𝑋 −𝑖𝑘⋅𝑋̄̄ ̄ − 𝛼 (𝑘 ⋅ 𝜓)̄ 𝜓 ̄ )= 𝑐𝑐 ̄ (𝑖𝜕𝑋𝐼 − (𝑘 ⋅ 𝜓)𝜓𝐼 ) (𝑖𝜕𝑋
e
.
𝐽
𝐽̄ e
4
2
2

(4.28)(4.29)

Note that in the following calculation we may drop the terms containing the boson current since they give contributions that are diffeomoephic to zero. So there is only thecontributions due to the non-vanishing momentum 𝑘𝜇 . We note here that the action of
the maps 𝑓𝑧0 reduces for primary operator 𝑂(𝑧) with conformal dimension ℎ to
𝑓𝑧0 ∘ 𝑂(𝑧) = 𝑓𝑧′0 (𝑧)ℎ 𝑂(𝑓𝑧0 (𝑧)).

(4.30)For Witten's cubic action we use three maps 𝑓0 , 𝑓1 and 𝑓∞ , however the only nontrivial action is 𝑓1 on the off-shell operator for the graviton. Therefore the correlation
function reduces to(2)

𝐺𝐼 𝐽 ̄(𝑘) = |𝑓1′ (0)|

𝛼′ 𝑘2

(0,0)

⟨𝑉 (−1) 𝑉 ̄ (−1) (∞)𝑉𝐺

𝐼𝐽 ̄(1; −𝑘)𝑉 (−1) 𝑉 ̄ (−1) (0)⟩ .

(4.31)In the holomorphic sector, it can be split into four sub-amplitudes with generic vertex
operator insertion, which belong to different CFT's and are independent from each other:𝑤𝛼 𝑤𝛽 𝑘𝐽 ⟨𝑐(𝑧1 )𝑐(𝑧2 )𝑐(𝑧3 )⟩ ⟨e−𝜙(𝑧1 ) e−𝜙(𝑧3 ) ⟩̇⟨Δ(𝑧1 )e−𝑖𝑘⋅𝑋 (𝑧2 )Δ(𝑧3 )⟩ ⟨𝑆 𝛼̇ 𝜓𝐾 𝜓𝐼 (𝑧2 )𝑆 𝛽 (𝑧3 )⟩ .

(4.32)All the correlation functions are well know and thus we can calculate′ 2

𝛼 𝑘|𝑓1′ (0)|(2)
𝐺𝐼 𝐽 ̄(𝑘) =16𝑧𝑧̄
( 13 ⋅ 13 )𝑧12 𝑧23 𝑧12
̄ 𝑧23
̄

𝛼′ 𝑘2 /2𝑤𝛼 (𝜎𝐴𝐼 )𝛼𝛽 𝑤𝛽 𝑤̄ 𝛾 (𝜎𝐵𝐽 )𝛾𝛿 𝑤̄ 𝛿 e−𝑖𝑘⋅𝑥0 −𝑖𝑘⋅𝑥̄0 ,(4.33)
30

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On Orbifolds in Closed Superstring Field Theory

where 𝑥0 , 𝑥0̄ are the zero-modes of 𝑋(𝑧), 𝑋(̄ 𝑧)̄ corresponding to the position of the orbifold
singularity and can be taken to zero.
After setting 𝑧1 → ∞, 𝑧2 = 1 and 𝑧3 = 0 we end up with′ 2

𝛼 𝑘|𝑓1′ (0)|(2)
𝐺𝐼 𝐽 ̄ =16

𝑘𝐴 𝑘𝐵 𝑤𝛼 (𝜎𝐴𝐼 )𝛼𝛽 𝑤𝛽 𝑤̄ 𝛾 (𝜎𝐵𝐽 )𝛾𝛿 𝑤̄ 𝛿 .

(4.34)

By construction of Witten's cubic vertex, the map 𝑓1 is given by the composition 𝑓1 (𝑧) =𝐺(𝑔1 (𝑧)) with𝑖 − 𝑧 2/3
𝑔1 (𝑧) = ()(4.35)𝑖+𝑧
and(1 − e2𝜋𝑖/3 )(𝑧 − e−2𝜋𝑖/3 )𝐺(𝑧) =,(4.36)(1 − e−2𝜋𝑖/3 )(𝑧 − e2𝜋𝑖/3 )so its derivative at the origin is4𝑓1′ (0) = √ .
3 3(4.37)

Now let's take a closer look at the metric (4.34) we obtained.
We would like to comment that even though we start with the flat metric on ℂ2 /ℤ2 ,which is Kähler, the full metric solution to the blowing-up equation is not necessarily
Kähler already at the second order. This point can be seen by taking 𝑤𝛼 = 𝑤̄ 𝛼 = ( √12 , √12 ),then we have
2𝑖𝑘1̄ 𝑘1̄⎛⎜ 2𝑘1̄ 𝑘1(2)
𝐺𝐼 𝐽 ̄ ∼ ⎜⎜⎜−2𝑖𝑘 𝑘1̄ 0̄

⎝ −2𝑘1̄ 𝑘0

2𝑘1̄ 𝑘1 −2𝑖𝑘1̄ 𝑘0̄ −2𝑘1̄ 𝑘0−2𝑖𝑘1 𝑘1 −2𝑘0̄ 𝑘1
2𝑖𝑘0 𝑘1 ⎞⎟⎟⎟−2𝑘0̄ 𝑘1 2𝑖𝑘0̄ 𝑘0̄
2𝑘0̄ 𝑘0 ⎟2𝑖𝑘0 𝑘1
2𝑘0̄ 𝑘0 −2𝑖𝑘0 𝑘0 .⎠(4.38)

Consider a diffeomorphism
𝑘1𝑘̄ 1̄

1+ 𝑘0 ,
𝑘02
𝑘0𝑘̄ ̄
1
𝜅 1 = 𝑖 0 + 𝑘1 ,𝑘12
𝜅0 = 𝑖𝑘 𝑘1
𝜅0̄ = −𝑖 1 1 + 𝑘0̄ ,
𝑘0̄
2
𝑘 𝑘
1
𝜅1̄ = −𝑖 0 0 + 𝑘1̄
𝑘1̄
2(4.39)(4.40)

̄𝐼
under which the Kähler form becomes 𝜔 = 𝑖𝐺𝐼 𝐽 d𝑥
∧ d𝑥𝐽 . We then havē𝑘 ̄𝑘 ̄𝑘 ̄𝑘
𝑘1̄ 𝑘1̄ 𝑘1 𝑘0̄
̄+ 𝑖 0 0 0 0 ) d𝑥1 ∧ d𝑥0 ∧ d𝑥0 + ⋯ ,d𝜔 = 𝑖 (2(𝑘𝐼 𝑘̄ 𝐼 )𝑘1 − 𝑖
𝑘0
𝑘1(4.41)

which is non-vanishing away from the origin. This indicates a broken of the 𝒩 = 2
spacetime supersymmetry in the shifted vacuum.

31

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On Orbifolds in Closed Superstring Field Theory

4.4

Third-order Deformation

At the third order in the blow-up parameter of the orbifold, the string field equation of
motion is given by(1,1)

𝑄Ψ(3) − 𝐿2

(2,2)

(Ψ(2) , Ψ(1) ) + 𝐿3

(Ψ(1) , Ψ(1) , Ψ(1) ) = 0.

(4.42)We thus require the projection(1,1)

𝒪3 = 𝑃0 [𝐿2(1,1)(𝑄−1 𝐿2(2,2)(Ψ(1) , Ψ(1) ), Ψ(1) ) − 𝐿3(Ψ(1) , Ψ(1) , Ψ(1) )] ,(4.43)

namely, the third order obstruction to vanish. In the same spirit of [23] we can reduce
this expression to a four-point function in an SCFT which can be evaluated using standard methods, except for the projector 𝑃0 . Effectively this amount to the reverse of theconstruction of string field theory from amplitudes. The fact that closed string theory
contains two holomorphic parts greatly simplifies our evaluation.
To see this, consider the inner product against an generic on-shell state 𝑒(1,1)

𝒪𝑒3 = ⟨𝑒, 𝐿2(1,1)

𝑄−1 𝐿2

(2,2)

− 𝐿3

⟩ (Ψ(1) , Ψ(1) , Ψ(1) ).

(4.44)

It can be easily seen that the vanishing of 𝒪3 = ∑ 𝒪𝑒3 , where the summation is taken overa basis of im(𝑃0 ), is equivalent to the vanishing of component 𝒪𝑒3 . To continue we use the
(1,1)
(1,1)
construction 𝐿2 = [𝑄, 𝜆2 ] to write
1
(1,1)
(1,1)
𝒪𝑒3 = ⟨𝑒, 𝜉 ̄ ( [𝑄, 𝜆2 ]𝑄−1 𝐿2
2
1 (1,1)(1,1)(2,2)+ 𝐿2 𝑄−1 [𝑄, 𝜆2 ] − 𝐿3 )⟩ (Ψ(1) , Ψ(1) , Ψ(1) )2
𝐿(4.45)

Note that we've entered the large Hilbert space in the anti-holomorphic sector by natureof the construction we use. We now use the BPZ inner product with an extra insertion of̄ . In the following whenever the𝜉 ̄ to provide the extra zero-mode 𝜉0̄ , i.e., ⟨⋅, ⋅⟩ = ⟨⋅, 𝜉⋅⟩𝐿
BRST charge 𝑄 hits the ghost 𝜉 ̄ we use
𝒳̄ = {𝑄, 𝜉}̄

(4.46)

to get a picture-changing operator, and whenever it hits the propagator 𝑄−1 we use thecontracting homotopy relation
1 − 𝑃0 = 𝑄𝑄−1 + 𝑄−1 𝑄(4.47)

until it finally hits an on-shell state and vanish. After this process we are left with1
̄ (1,0) 𝑄−1 𝐿(1,1) − 1 𝒳̄ 𝜉𝐿
̄ (1,1) 𝑄−1 𝐿(1,0)
𝒪𝑒3 = ⟨𝑒, 𝒳̄ 𝜉𝐿
2
2
2
2
2
2
1 ̄ (1,1)
1 ̄ (1,1)
(1,1)
(1,1)
𝑃0 𝜆2 − 𝜉𝜆
𝑃0 𝐿2
− 𝜉𝐿
2
2
2
2
1 ̄ (1,1) (1,1) 1 ̄ (1,1) (1,1)̄ (2,2) ⟩ (Ψ(1) , Ψ(1) , Ψ(1) ).
+ 𝜉𝐿2 𝜆2 + 𝜉𝜆2 𝐿2 − 𝜉𝐿
3
2
2
𝐿

(4.48)

32

Soryushiron Kenkyu

On Orbifolds in Closed Superstring Field Theory

L2

Q-1

L2

L3

P0

P0

Figure 4.1: The structure of the third-order obstruction
The propagator-terms in the first line of (4.48) can be furtherly disentangled by the same

33

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On Orbifolds in Closed Superstring Field Theory

procedure,1̄ (1,0) 𝑄−1 𝐿(1,1) − 1 𝒳̄ 𝜉𝐿
̄ (1,1) 𝑄−1 𝐿(1,0) ⟩⟨𝑒, 𝒳̄ 𝜉𝐿
2
2
2
2
2
2
𝐿
1 ̄ 2 (1,0) −1 (1,1) 1 ̄ 2 (1,1) −1 (1,0)= ⟨𝑒, − 𝒳 𝐿2 𝑄 𝜆2 − 𝒳 𝜆2 𝑄 𝐿222
1 ̄ ̄ (1,0)
1 ̄ ̄ (1,1)
(1,1)
(1,0)
− 𝒳𝜉𝐿2 𝑃0 𝜆2 − 𝒳𝜉𝜆2 𝑃0 𝐿2
2
2
1 ̄ ̄ (1,0) (1,1) 1 ̄ ̄ (1,1) (1,0)+ 𝒳𝜉𝐿2 𝜆2 + 𝒳𝜉𝜆2 𝐿2 ⟩ (Ψ(1) , Ψ(1) , Ψ(1) ).
2
2
𝐿(4.49)

Regarding the projector terms in the second line of (4.48) and (4.49), note that
(1,1)

(Ψ(1) , Ψ(1) ) = 0,(4.50)(1,0)(Ψ(1) , Ψ(1) ) = 0,(4.51)

1(1,0)(Ψ(1) , Ψ(1) ) = 𝑃0 {𝜉,̄ 𝐿2 }(Ψ(1) , Ψ(1) ) = 0
3(4.52)

𝑃0 𝐿2
𝑃0 𝐿2
and
(1,1)

𝑃0 𝜆2

as we have shown in the previous section. We are left only with the 𝒳̄ 2 -terms
1
1
(1,0)
(1,1)
(1,1)
(1,0)
⟨𝑒, − 𝒳̄ 2 𝐿2 𝑄−1 𝜆2 − 𝒳̄ 2 𝜆2 𝑄−1 𝐿2 ⟩2
2
𝐿

(4.53)1 ̄ (1,1) (1,1) 1 ̄ (1,1) (1,1)⟨𝑒, 𝜉𝐿𝜆2 + 𝜉𝜆2 𝐿2
2 2
2
1 ̄ ̄ (1,0) (1,1) 1 ̄ ̄ (1,1) (1,0)̄ (2,2) ⟩ .
+ 𝒳𝜉𝐿2 𝜆2 + 𝒳𝜉𝜆2 𝐿2 − 𝜉𝐿
3
2
2
𝐿

(4.54)

andWe have in (4.54)
1 (1,1) (1,1)
1 ̄ (1,0) (1,1)(2,2)⟨𝑒, [𝐿2 , 𝜆2 ] − 𝐿3 + 𝒳[𝐿, 𝜆2 ]⟩ (Ψ(1) , Ψ(1) , Ψ(1) ),2
2
2(4.55)

where we can make use of the relation
(2,2)

𝐿3
and

1
(2,2)
(1,1)
(1,1)
= ([𝑄, 𝜆3 ] + [𝐿2 , 𝜆2 ])
2

(2,1)

𝐿3

(2,1)

= [𝑄, 𝜆3(1,0)] + [𝐿2

(1,1)

, 𝜆2

].

(4.56)(4.57)In the large Hilbert state the zero-mode 𝜉0̄ is present, and thus it will be useful to notethat⟨𝑒, 𝑏𝑛 ⟩ = (−1)deg(𝑏𝑛 ) ⟨𝑒, 𝑏𝑛 (𝕀⊗𝑘 ⊗ 𝜉 ⊗ 𝕀𝑛−𝑘 )⟩𝐿 ,(4.58)

34

Soryushiron Kenkyu

On Orbifolds in Closed Superstring Field Theory

allowing us to move the 𝜉 ghost freely. We get the remaining term in (4.54)
1 ̄ (2,1)
1
̄ (2,1) ⟩ (Ψ(1) , Ψ(1) , Ψ(1) )⟩ (Ψ(1) , Ψ(1) , Ψ(1) ) = ⟨𝑒, 𝒳̄ 𝜉𝐿⟨𝑒, 𝒳𝐿33
22𝐿
1 ̄ (2,2)= ⟨𝑒, 𝒳𝜆3 ⟩ (Ψ(1) , Ψ(1) , Ψ(1) ).
2
𝐿(4.59)

However, the picture-changing operator only gives 𝑄-exact terms in large Hilbert space,
namely1 ̄ (2,2)⟨𝑒, 𝒳𝜆⟩32𝐿
1 ̄ (2,2)
1 ̄ (2,2)= ⟨𝑒, 𝑄𝜉𝜆⟩ + ⟨𝑒, 𝜉𝑄𝜆
⟩
3
3
2
2
𝐿
𝐿
1 ̄ (2,2)
1 (2,2)= ⟨𝑄𝑒, 𝜉𝜆3 ⟩ + ⟨𝑄𝑒, 𝜆3 ⟩
2
2
𝐿
= 0.

(4.60)The remaining 𝒳̄ 2 -terms is what initially gives rise to a constraint on the open moduli(see [24] for a more general discussion). In the following we show that it is absent again
by the doubling of the holomorphic sectors in closed string theory,
1
1
(1,0)
(1,1)
(1,1)
(1,0)
⟨𝑒, 𝒳̄ 2 𝐿2 𝑄−1 𝜆2 + 𝒳̄ 2 𝜆2 𝑄−1 𝐿2 ⟩ (Ψ(1) , Ψ(1) , Ψ(1) )
2
2
𝐿
(1,0)
(1,0)
2 ̄
−1
(1)
(1)
(1)
̄
= ⟨𝑒, 𝒳 𝜉𝐿
𝑄 𝐿
⟩ (Ψ , Ψ , Ψ ),2

2

(4.61)

𝐿

where we've moved the ghost between the inputs. We apply the strategy once more tocommute 𝑄 with 𝑄−1 using the contracting homotopy relation
1
(1,0)
(1,0)
⟨ 𝒳̄ 2 𝐿2 𝑄−1 𝐿2 ⟩ (Ψ(1) , Ψ(1) , Ψ(1) )2
1 ̄ ̄ (1,0)(1,0)= ⟨ 𝜉 𝒳𝐿𝑄𝑄−1 𝐿2 ⟩ (Ψ(1) , Ψ(1) , Ψ(1) )2
2
1 ̄ ̄ (1,0)(1,0)= ⟨ 𝜉 𝒳𝐿
(1 − 𝑃0 )𝐿2 ⟩ (Ψ(1) , Ψ(1) , Ψ(1) ).
2
2(4.62)̄ ̄ (1,0) 𝐿(1,0) ⟩ (Ψ(1) , Ψ(1) , Ψ(1) )The second term vanishes by (4.51), and we have only ⟨ 21 𝜉 𝒳𝐿
2
2
left. Moving the 𝜉 ghost in the holomorphic sector we get
1 ̄ ̄ (1,0) (1,0)1(0,0) (0,0)⟨ 𝜉 𝒳𝐿𝐿2 ⟩ (Ψ(1) , Ψ(1) , Ψ(1) ) = ⟨ 𝒳2 𝒳̄ 2 𝐿2 𝐿2 ⟩ (Ψ(1) , Ψ(1) , Ψ(1) ),
2
2
2
(0,0)

(4.63)(0,0)

where the composition of the bosonic product 𝐿2∘ 𝐿2
appears. In contrast to
Witten's cubic open string field theory [18], the closed string product is not associative.
As a matter of fact, the 𝐿∞ relations give
1 (0,0) (0,0)(0,0)[𝐿2 , 𝐿2 ] = [𝑄, 𝐿3 ],2(4.64)
35

Soryushiron Kenkyu

On Orbifolds in Closed Superstring Field Theory

but obviously the 𝑄-exact term vanishes when inserted into the on-shell amplitude. Consequently, we conclude that the third-order obstruction component 𝒪𝑒3 vanishes on anarbitrary state 𝑒 in the image of 𝑃0 , and thus the whole obstruction 𝒪3 vanishes. The
result is valid without assuming any other constraint analogous to the generalized ADHM
constraint in [24] on the moduli. This means that any one choice of moduli corresponds
to a deformation tangent at the orbifold point, at least so to the third-order correction.
The vanishing of 𝒪3 means that the perturbative solution to the string field theory
equation of motion is well-defined up to the third order. Furthermore, it is in principle
possible to compute all contributions to the metric at order 𝑂(𝜆3 ). The explicit calculation
would be, however, rather cumbersome and not discussed here.

36

Soryushiron Kenkyu

Chapter 5Conclusion
In this thesis we explore exactly marginal deformations near an orbifold conformal field
theory. The analysis is carried out in the string field theory framework. The vertex
operator corresponding to the twist field is considered, both in the bosonic and the supersymmetric case, to generate a deformation. We are able to show that, although thebosonic deforming is blocked, it is possible to generate an exactly marginal deformation
up to the third order. We also compute the metric profile generated by the deformation
at the second order. The metric is not Kähler and hence indicates a breaking of 𝒩 = 2
supersymmetry of the original ℤ2 orbifold model.

37

Soryushiron Kenkyu

Appendix A
Operadic Structures in Homotopy
and BV Algebras
Definition 1. An operad is a collection of objects of a symmetric monoidal category𝒪(𝑛), 𝑛 ≤ 1, with:
The action of the symmetric group 𝑆𝑛 on 𝒪(𝑛).
The composition law:𝛾 ∶ 𝒪(𝑘) × 𝒪(𝑛1 ) × ⋯ × 𝒪(𝑛𝑘 ) → 𝒪(𝑛1 + ⋯ + 𝑛𝑘 ),(𝑓; 𝑓1 , ⋯ , 𝑓𝑘 ) ↦ 𝛾(𝑓; 𝑓1 , ⋯ , 𝑓𝑘 ) ≡ 𝑓(𝑓1 , ⋯ , 𝑓𝑘 ).
The unit 𝑒 ∈ 𝒪(1) satisfying 𝑒(𝑓) = 𝑓 and 𝑓(𝑒, ⋯ , 𝑒) = 𝑓 for any 𝑓 ∈ 𝒪(𝑛).
The composition is equivariant with respect to the symmetric group actions:(𝜎; 𝜎1 , ⋯ , 𝜎𝑘 )𝑓(𝑓1 , ⋯ , 𝑓𝑘 ) = 𝑓(𝜎𝜎(1) 𝑓𝜎(1) , ⋯ , 𝜎𝜎(𝑘) 𝑓𝜎(𝑘) ).
The composition is associative, i.e., 𝛾 ∘ (id × 𝛾 𝑘 ) = 𝛾 ∘ (𝛾 × id).
Theorem 1. The correspondence𝐶 ∗ (ℳ𝑛+1 ) → Hom((𝐻rel )⊗𝑛 , 𝐻rel )𝐶 ↦ ∫ 𝑤𝑛+1

(A.1)

𝐶

defines the structure of an algebra over the operad 𝐶 ∗ (ℳ𝑛+1 ) on the space 𝐻rel .
Theorem 2. 𝐻𝑝 (𝐹𝑝 , 𝐹𝑝−1 )is naturally isomorphic to the 𝑛-th component 𝒮𝑛 of the homotopy Lie operad. Therefore the correspondence
𝐻𝑝 (𝐹𝑝 , 𝐹𝑝−1 ) → Hom((𝐻rel )⊗𝑛 , 𝐻rel )𝑍 ↦ ∫ 𝑤𝑛+1

(A.2)

𝑍

defines the structure of an algebra over the homotopy Lie operad 𝒮(𝑛) on the relative state
space 𝐻rel .
VSoryushiron Kenkyu

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Acknowledgements
First and foremost, I would like to thank my advisor Prof. Ivo Sachs for his patienceand careful guidance during my master's study. I barely knew anything about string
theory when I walked into his oﬀice last year. I also want to thank my parents, for theirconstant support and unconditional love, which could never be expressed in limited
words. I'm grateful for the companion of my friends that I met in Munich. We sharedan unforgetable period of time in my life. In the end I give thank to all those I love andlove me. Thanks for being there.
Special thank to Paulaner Münchner Hell. String theory becomes lovely with your help.

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On Orbifolds in Closed Superstring Field Theory

Declaration
I hereby declare that this thesis is my own work, and that I have not used any sources
and aids other than those stated in the thesis.

Location, date

Name

X