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Add first part of the non Abelian rotations

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-09-09 21:06:20 +02:00
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\subsection{Motivation}
As seen in the previous sections, the study of viable phenomenological models in string theory involves the analysis of the properties of systems of D-branes.
The inclusion of the physical requirements deeply constrains the possible scenarios.
In particular the chiral spectrum of the \sm acts as a strong restriction on the possible setup.
In this section we study models based on \emph{intersecting branes}, which represent a relevant class of such models with interacting chiral matter.
We focus on the development of technical tools for the computation of Yukawa interactions for D-branes at angles~\cite{Chamoun:2004:FermionMassesMixing,Cremades:2003:YukawaCouplingsIntersecting,Cvetic:2010:BranesInstantonsIntersecting,Abel:2007:RealisticYukawaCouplings,Chen:2008:RealisticWorldIntersecting,Chen:2008:RealisticYukawaTextures,Abel:2005:OneloopYukawasIntersecting}.
The fermion--boson couplings and the study of flavour changing neutral currents~\cite{Abel:2003:FlavourChangingNeutral} are keys to the validity of the models.
These and many other computations require the ability to calculate correlation functions of (excited) twist and (excited) spin fields.
The goal of the section is therefore to address such challenges in specific scenarios.
The computation of the correlation functions of Abelian twist fields can be found in literature and plays a role in many scenarios, such as magnetic branes with commuting magnetic fluxes~\cite{Angelantonj:2000:TypeIStringsMagnetised,Bertolini:2006:BraneWorldEffective,Bianchi:2005:OpenStoryMagnetic,Pesando:2010:OpenClosedString,Forste:2018:YukawaCouplingsMagnetized}, strings in gravitational wave background~\cite{Kiritsis:1994:StringPropagationGravitational,DAppollonio:2003:StringInteractionsGravitational,Berkooz:2004:ClosedStringsMisner,DAppollonio:2005:DbranesBCFTHppwave}, bound states of D-branes~\cite{Gava:1997:BoundStatesBranes,Duo:2007:NewTwistField,David:2000:TachyonCondensationD0} and tachyon condensation in Superstring Field Theory~\cite{David:2000:TachyonCondensationD0,David:2001:TachyonCondensationUsing,David:2002:ClosedStringTachyon,Hashimoto:2003:RecombinationIntersectingDbranes}.
A similar analysis can be extended to excited twist fields even though they are more subtle to treat and hide more delicate aspects~\cite{Burwick:1991:GeneralYukawaCouplings,Stieberger:1992:YukawaCouplingsBosonic,Erler:1993:HigherTwistedSector,Anastasopoulos:2011:ClosedstringTwistfieldCorrelators,Anastasopoulos:2012:LightStringyStates,Anastasopoulos:2013:ThreeFourpointCorrelators}.
Results were however found starting from dual models up to modern interpretations of string theory~\cite{Sciuto:1969:GeneralVertexFunction,DellaSelva:1970:SimpleExpressionSciuto}.
Correlation functions involving arbitrary numbers of plain and excited twist fields were more recently studied~\cite{Pesando:2014:CorrelatorsArbitraryUntwisted,Pesando:2012:GreenFunctionsTwist,Pesando:2011:GeneratingFunctionAmplitudes} blending the CFT techniques with the path integral approach and the canonical quantization~\cite{Pesando:2008:MultibranesBoundaryStates,DiVecchia:2007:WrappedMagnetizedBranes,Pesando:2011:StringsArbitraryConstant,DiVecchia:2011:OpenStringsSystem,Pesando:2013:LightConeQuantization}.
We consider D6-branes intersecting at angles in the case of non Abelian relative rotations presenting non Abelian twist fields at the intersections.
We try to understand subtleties and technical issues arising from a scenario which has been studied only in the formulation of non Abelian orbifolds \cite{Inoue:1987:NonAbelianOrbifolds,Inoue:1990:StringInteractionsNonAbelian,Gato:1990:VertexOperatorsNonabelian,Frampton:2001:ClassificationConformalityModels} and for D-branes relative \SU{2} rotations~\cite{Pesando:2016:FullyStringyComputation}.
In this configuration we study three D6-branes in $10$-dimensional Minkowski space $\ccM^{1,9}$ with an internal space of the form $\R^4 \times \R^2$ before the compactification.
The D-branes are embedded as lines in $\R^2$ and as two-dimensional surfaces inside $\R^4$.
We focus on the relative rotations which characterise each D-brane in $\R^4$ with respect to the others.
In total generality, they are non commuting \SO{4} matrices.
In this paper we study the classical solution of the bosonic string which dominates the behaviour of the correlator of twist fields.
Using the path integral approach we can in fact separate the classical contribution from the quantum fluctuations and write the correlators of $N_B$ twist fields $\sigma_{M_{(t)}}( x_{(t)} )$ as:\footnotemark{}
\footnotetext{%
Ultimately $N_B = 3$ in our case.
}
\begin{equation}
\left\langle
\finiteprod{t}{1}{N_B}
\sigma_{M_{(t)}}(x_{(t)})
\right\rangle
=
\cN
\left(
\left\lbrace x_{(t)},\, M_{(t)} \right\rbrace_{1 \le t \le N_B}
\right)\,
e^{-S_E\left( \left\lbrace x_{(t)}, M_{(t)} \right\rbrace_{1 \le t \le N_B} \right)},
\end{equation}
where $M_{(t)}$ (for $1 \le t \le N_B$) are the monodromies induced by the twist fields, $N_B$ is the number of D-branes and $x_{(t)}$ are the intersection points on the worldsheet.
Even though quantum corrections are crucial to the complete determination of the normalisation of the correlator, the classical contribution of the Euclidean action represents the leading term of the Yukawa couplings.
We focus on its contribution to better address the differences from the usual factorised case and generalise the results to non Abelian rotations of the D-branes.
We do not consider the quantum corrections as they cannot be computed with the actual techniques.
Their calculations requires the correlator of four twist fields which in turn requires knowledge of the connection formula for Heun functions which is not known.
We therefore study the boundary conditions for the open string describing the D-branes embedded in $\R^4$.
In particular we first address the issue connected to the global description of the embedding of the D-branes.
In conformal coordinates we rephrase such problem into the study of the monodromies acquired by the string coordinates.
These additional phase factors can then be specialised to \SO{4}, which can be studied in spinor representation as a double copy of \SU{2}.
We thus recast the issue of finding the solution as $4$-dimensional real vector to a tensor product of two solutions in the fundamental representation of \SU{2}.
We then see that these solutions are well represented by hypergeometric functions, up to integer factors.
Physical requirements finally restrict the number of possible solutions.
\subsection{D-brane Configuration and Boundary Conditions}
We focus on the bosonic string embedded in $\ccM^{1, d + 4}$.
The relevant configuration of the D-branes is seen as two-dimensional Euclidean planes in $\R^4$.
We specifically concentrate on the Euclidean solution for the classical bosonic string in this scenario.
The mathematical analysis is however more general and can be applied to any Dp-brane embedded in a generic Euclidean space $\R^q$.
The classical solution can in principle be defined in this case provided the ability to write the explicit form of the basis of functions with the proper boundary and monodromy conditions.
This is possible in the case of three intersecting D-branes but in general it is an open mathematical issue.
In the case of three D-branes with generic embedding we can in fact connect a local basis around one intersection point to another, the third depending on the first two intersections, by means of Mellin-Barnes integrals.
This way the solution can be explicitly and globally constructed.
With more than three D-branes the situation is by far more difficult since the explicit form of the connection formulas is not known.
\subsubsection{Intersecting D-branes at Angles}
Let $N_B$ be the total number of D-branes and $t = 1,\, 2,\, \dots,\, N_B$ be
an index defined modulo $N_B$ to label them.
We describe one of the D-branes in a well adapted system of coordinates
$X_{(t)}^I$, where $I = 1,\, 2,\, 3,\, 4$, as:
\begin{equation}
X_{(t)}^3 = X_{(t)}^4 = 0.
\label{eq:well-adapt-embed}
\end{equation}
We thus choose $X_{(t)}^1$ and $X_{(t)}^2$ to be the coordinates parallel to the D-brane $D_{(t)}$ while $X_{(t)}^3$ and $X_{(t)}^4$ are the coordinates orthogonal to it.
\begin{figure}[tbp]
\centering
\begin{subfigure}[b]{0.45\linewidth}
\centering
\def\svgwidth{\linewidth}
\import{img/}{branesangles.pdf_tex}
\caption{%
D-branes as lines on $\R^2$.
}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\linewidth}
\centering
\def\svgwidth{\linewidth}
\import{img/}{welladapted.pdf_tex}
\caption{Well adapted system of coordinates.}
\end{subfigure}
\caption{%
Geometry of D-branes at angles.
}
\label{fig:branes_at_angles}
\end{figure}
The well adapted reference coordinates system is connected to the global $\R^{4}$ coordinates $X^I$ by a transformation:
\begin{equation}
\tensor{(X_{(t)})}{^I}
=
\tensor{(R_{(t)})}{^I_J}\, \tensor{X}{^J}
-
\tensor{(g_{(t)})}{^I},
\qquad
I,\, J = 1,\, 2,\, 3,\, 4,
\label{eq:brane_rotation}
\end{equation}
where $R_{(t)}$ represents the rotation of the D-brane $D_{(t)}$ and $g_{(t)} \in \R^4$ its translation with respect to the origin of the global set of coordinates (see \Cref{fig:branes_at_angles} for a two-dimensional example).
While we could naively consider $R_{(t)} \in \mathrm{SO}(4)$, rotating separately the subset of coordinates parallel and orthogonal to the D-brane does not affect the embedding.
In fact it just amounts to a trivial redefinition of the initial well adapted coordinates.
The rotation $R_{(t)}$ is actually defined in the Grassmannian:
\begin{equation}
R_{(t)}
\in
\mathrm{Gr}(2, 4)
=
\frac{\SO{4}}{\rS\left( \OO{2} \times \OO{2} \right)},
\end{equation}
that is we just need to consider the left coset where $R_{(t)}$ is a representative of an equivalence class
\begin{equation}
\left[ R_{(t)} \right]
=
\left\lbrace R_{(t)} \sim \cO_{(t)} R_{(t)} \right\rbrace,
\end{equation}
where $\cO_{(t)} = \rS\left( \OO{2} \times \OO{2} \right)$ is defined as
\begin{equation}
\cO_{(t)}
=
\mqty( \dmat{\cO^{\parallel}_{(t)}, \cO^{\perp}_{(t)}} )
\end{equation}
with $\cO^{\parallel}_{t} \in \OO{2}$, $\cO^{\perp}_{t} \in \OO{2}$ and $\det \cO_{(t)} = 1$.
The superscript $\parallel$ represents any of the coordinates parallel to the D-brane, while $\perp$ any of the orthogonal.
Notice that the additional $\Z_2$ factor in $\rS\left( \OO{2} \times \OO{2} \right)$ with respect to $\SO{2} \times \SO{2}$ can be used to set $g_{(t)}^{\perp} \ge 0$.
% vim ft=tex

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@@ -1,32 +1,32 @@
In this first part we focus on aspects of string theory directly connected with its worldsheet description and symmetries.
The underlying idea is to build technical tools to address the study of viable phenomenological models in this framework.
In fact the construction of realistic string models of particle physics is the key to better understanding the nature of a theory of everything such as string theory.
The construction of realistic string models of particle physics is the key to better understanding the nature of a theory of everything such as string theory.
As a first test of validity, the string theory should properly extend the known Standard Model (\sm) of particle physics, which is arguably one of the most experimentally backed theoretical frameworks in modern physics.
In particular its description in terms of fundamental strings should be able to include a gauge algebra isomorphic to that of
In particular its description in terms of fundamental strings should be able to include a gauge algebra isomorphic to the algebra of the group
\begin{equation}
\SU{3}_{\rC} \otimes \SU{2}_{\rL} \otimes \U{1}_{\rY}
\label{eq:intro:smgroup}
\end{equation}
in order to reproduce known results.
For instance, string theory could provide a unified framework by predicting the existence of a larger gauge group containing the \sm{} as a subset.
In what follows we deal with the definition of mathematical tools to compute amplitudes to be used in phenomenological calculations related to the study of particles in string theory.
For instance, a good string theory could provide a unified framework by predicting the existence of a larger gauge group containing the \sm as a subset.
In what follows we deal with the definition of mathematical tools to compute amplitudes to be used in phenomenological calculations related to the study of particles.
In this introduction we present instruments and preparatory frameworks used throughout the manuscript as many tools are strongly connected and their definitions are interdependent.
In this introduction we present instruments and preparatory frameworks used throughout the manuscript as many other aspects are strongly connected and their definitions are interdependent.
In particular we recall some results on the symmetries of string theory and how to recover a realistic description of physics.
\subsection{Properties of String Theory and Conformal Symmetry}
Strings are extended one-dimensional objects.
They are curves in space parametrized by a coordinate $\sigma \in \left[0, \ell \right]$.
Propagating in $D$-dimensional spacetime they span a two-dimensional surface, the \emph{worldsheet}, described by the position of the string at given values of $\sigma$ at a time $\tau$, i.e.\ $X^{\mu}(\tau, \sigma)$ where $\mu = 0, 1, \dots, D - 1$ indexes the coordinates.
They are curves in spacetime parametrized by a coordinate $\sigma \in \left[0, \ell \right]$.
When propagating they span a two-dimensional surface, the \emph{worldsheet}, described by the position of the string at given values of $\sigma$ at a time $\tau$, i.e.\ $X^{\mu}(\tau, \sigma)$ where $\mu = 0, 1, \dots, D - 1$ indexes the coordinates.
Such surface can have different topologies according to the nature of the object propagating in spacetime: strings can be \emph{closed} if $X^{\mu}(\tau, 0) = X^{\mu}(\tau, \ell)$ or \emph{open} if the endpoints in $\sigma = 0$ and $\sigma = \ell$ do not coincide.
\subsubsection{Action Principle}
As the action of a point particle is proportional to the length of its trajectory (its \emph{worldline}), the same object for string is proportional to the area of the worldsheet in the original formulation by Nambu and Goto.
As the action of a point particle is proportional to the length of its trajectory (its \emph{worldline}), the same object for a string is proportional to the area of the worldsheet in the original formulation by Nambu and Goto.
The solutions of the classical equations of motion (\eom) are therefore strings spanning a worldsheet of extremal area.
While Nambu and Goto's formulation is fairly direct in its definition, it usually best to work with Polyakov's action~\cite{Polyakov:1981:QuantumGeometryBosonic}
\begin{equation}
@@ -89,12 +89,12 @@ implies
=
S_{NG}[X],
\end{equation}
where $S_{NG}[X]$ is the Nambu--Goto action for the classical string.
where $S_{NG}[X]$ is the Nambu--Goto action for the classical string, $\dX = \ipd{\tau} X$ and $\pX = \ipd{\sigma} X$.
The symmetries of $S_P[\gamma, X]$ are the keys to the success of the string theory framework~\cite{Polchinski:1998:StringTheoryIntroduction}.
Specifically~\eqref{eq:conf:polyakov} displays symmetries under:
\begin{itemize}
\item $D$-dimensional Poincaré invariance
\item $D$-dimensional Poincaré transformations
\begin{equation}
\begin{split}
X'^{\mu}(\tau, \sigma)
@@ -108,7 +108,7 @@ Specifically~\eqref{eq:conf:polyakov} displays symmetries under:
\end{equation}
where $\Lambda \in \SO{1, D-1}$ and $c \in \R^D$,
\item 2-dimensional diffeomorphism invariance
\item 2-dimensional diffeomorphisms
\begin{equation}
\begin{split}
X'^{\mu}(\tau', \sigma')
@@ -124,7 +124,7 @@ Specifically~\eqref{eq:conf:polyakov} displays symmetries under:
\end{equation}
where $\sigma^0 = \tau$ and $\sigma^1 = \sigma$,
\item Weyl invariance
\item Weyl transformations
\begin{equation}
\begin{split}
X'^{\mu}(\tau', \sigma')
@@ -169,15 +169,16 @@ While its conservation $\nabla^{\alpha} T_{\alpha\beta} = 0$ is somewhat trivial
\end{equation}
In other words, the $(1 + 1)$-dimensional theory of massless scalars $X^{\mu}$ in~\eqref{eq:conf:polyakov} is \emph{conformally invariant} (for review and details see \cite{Friedan:1986:ConformalInvarianceSupersymmetry,DiFrancesco:1997:ConformalFieldTheory,Ginsparg:1988:AppliedConformalField,Blumenhagen:2009:IntroductionConformalField}).
Finally we can set $\gamma_{\alpha\beta}(\tau, \sigma) = e^{\phi(\tau, \sigma)}\, \eta_{\alpha\beta}$, known as \emph{conformal gauge} where $\eta_{\alpha\beta} = \mathrm{diag}(-1, 1)$, using the invariances of the action.
Using the invariances of the actions we can set $\gamma_{\alpha\beta}(\tau, \sigma) = e^{\phi(\tau, \sigma)}\, \eta_{\alpha\beta}$, known as \emph{conformal gauge} where $\eta_{\alpha\beta} = \mathrm{diag}(-1, 1)$.
This gauge choice is however preserved by the residual \emph{pseudoconformal} transformations
\begin{equation}
\tau \pm \sigma = \sigma_{\pm} \mapsto f_{\pm}(\sigma_{\pm}),
\label{eq:conf:residualgauge}
\end{equation}
where $f_{\pm}(\xi)$ are arbitrary functions.
It is natural to introduce a Wick rotation $\tau_E = i \tau$ and the complex coordinates $\xi = \tau_E + i \sigma$ and $\bxi = \xi^*$.
The transformation maps the Lorentzian worldsheet to a new surface: an infinite Euclidean strip for open strings or a cylinder for closed strings.
In these terms, the tracelessness of the stress-energy tensor translates to
\begin{equation}
T_{\xi \bxi} = 0,
@@ -235,7 +236,7 @@ In fact a transformation $\xi \mapsto \chi(\xi)$ and $\bxi \mapsto \bchi(\bxi)$
\label{fig:conf:complex_plane}
\end{figure}
An additional conformal map
An additional conformal transformation
\begin{equation}
z = e^{\xi} = e^{\tau_e + i \sigma} \in \left\lbrace z \in \C | \Im z \ge 0 \right\rbrace,
\qquad
@@ -295,17 +296,17 @@ we find the short distance singularities of the components of the stress-energy
\begin{split}
T( z )\, \phi_{\omega, \bomega}( w, \bw )
& =
\frac{\omega}{(z - w)^2} \phi_{\omega, \bomega}( w, \bw )
\frac{\omega}{(z - w)^2}\, \phi_{\omega, \bomega}( w, \bw )
+
\frac{1}{z - w} \ipd{w} \phi_{\omega, \bomega}( w, \bw )
\frac{1}{z - w}\, \ipd{w} \phi_{\omega, \bomega}( w, \bw )
+
\order{1},
\\
\bT( \bz )\, \phi_{\omega, \bomega}( w, \bw )
& =
\frac{\bomega}{(\bz - \bw)^2} \phi_{\omega, \bomega}( w, \bw )
\frac{\bomega}{(\bz - \bw)^2}\, \phi_{\omega, \bomega}( w, \bw )
+
\frac{1}{\bz - \bw} \ipd{\bw} \phi_{\omega, \bomega}( w, \bw )
\frac{1}{\bz - \bw}\, \ipd{\bw} \phi_{\omega, \bomega}( w, \bw )
+
\order{1},
\end{split}
@@ -336,7 +337,7 @@ which is an asymptotic expansion containing the full information on the singular
}
The constant coefficients $\cC_{ijk}$ are subject to restrictive constraints given by the properties of the conformal theories to the point that a \cft is completely specified by the spectrum of the weights $(\omega_i, \bomega_i)$ and the coefficients $\cC_{ijk}$ \cite{Friedan:1986:ConformalInvarianceSupersymmetry}.
The \ope can also be compute on the stress-energy tensor itself.
The \ope can also be computed on the stress-energy tensor itself.
Focusing on the holomorphic component we find
\begin{equation}
\begin{split}
@@ -467,14 +468,16 @@ From the commutation relations~\eqref{eq:conf:virasoro} we finally compute its c
=
(\omega + m) \ket{\phi_{\omega}^{\lbrace n_1, n_2, \dots, n_m \rbrace}}.
\end{equation}
States corresponding to primary operators have therefore the lowest energy (intended as eigenvalue of the Hamiltonian $L_0 + \bL_0$) in the entire representation.
States corresponding to primary operators have therefore the lowest energy (intended as eigenvalue of the Hamiltonian) in the entire representation.
They are however called \emph{highest weight} states from the mathematical literature which uses the opposite sign for the Hamiltonian operator.
The particular case of the \cft in \eqref{eq:conf:polyakov} can be cast in this language.
In particular the solutions to the \eom factorise into a holomorphic and an anti-holomorphic contributions:
\begin{equation}
\ipd{z} \ipd{\bz} X( z, \bz ) = 0
\qquad
\Rightarrow
\qquad
X( z, \bz ) = X( z ) + \bX( \bz ),
\end{equation}
and the components of the stress-energy tensor~\eqref{eq:conf:stringT} are
@@ -486,7 +489,7 @@ and the components of the stress-energy tensor~\eqref{eq:conf:stringT} are
\end{split}
\label{eq:conf:bosonicstringT}
\end{equation}
Using the normalisation of the 2-points function $\left\langle X^{\mu}( z, \bz ) X^{\nu}( w, \bw ) \right\rangle = - \frac{1}{2} \eta^{\mu\nu} \ln\left| z - w \right|$ we can show that $c = D$ in~\eqref{eq:conf:TTexpansion}, where $D$ is the dimensions of spacetime (or equivalently the number of scalar fields $X^{\mu}$ in the action).
Using the normalisation of the 2-points function $\left\langle X^{\mu}( z, \bz ) X^{\nu}( w, \bw ) \right\rangle = - \frac{1}{2} \eta^{\mu\nu} \ln\left| z - w \right|$ we can prove that $c = D$ in~\eqref{eq:conf:TTexpansion}, where $D$ is the dimensions of spacetime (or equivalently the number of scalar fields $X^{\mu}$ in the action).
It can be shown that in order to cancel the central charge in bosonic string theory we need to introduce a pair of conformal ghosts $b(z)$ and $c(z)$ with conformal weights $(2, 0)$ and $(-1, 0)$ respectively, together with their anti-holomorphic counterparts $\overline{b}(z)$ and $\overline{c}(z)$.
The non vanishing components of their stress-energy tensor can be computed as:\footnotemark{}
\footnotetext{%
@@ -957,7 +960,7 @@ These results will also be the starting point of~\Cref{part:deeplearning} in whi
\subsection{D-branes and Open Strings}
Dirichlet branes, or \emph{D-branes}, are another key mathematical object in string theory.
They are naturally included as extended object as hypersurfaces supporting strings with open topology and as physical objects with charge and tension~\cite{Polchinski:1995:DirichletBranesRamondRamond,Polchinski:1996:TASILecturesDBranes,DiVecchia:1999:DbranesStringTheory,DiVecchia:2000:BranesStringTheory,DiVecchia:1997:ClassicalPbranesBoundary,Taylor:2003:LecturesDbranesTachyon,Taylor:2004:DBranesTachyonsString,Johnson:2000:DBranePrimer}.
They are naturally included as extended hypersurfaces supporting strings with open topology and as physical objects with charge and tension~\cite{Polchinski:1995:DirichletBranesRamondRamond,Polchinski:1996:TASILecturesDBranes,DiVecchia:1999:DbranesStringTheory,DiVecchia:2000:BranesStringTheory,DiVecchia:1997:ClassicalPbranesBoundary,Taylor:2003:LecturesDbranesTachyon,Taylor:2004:DBranesTachyonsString,Johnson:2000:DBranePrimer}.
They are relevant in the definition of phenomenological models in string theory as they can be arranged to support chiral fermions and bosons in \sm-like scenarios as well as beyond~\cite{Honecker:2012:FieldTheoryStandard,Lust:2009:LHCStringHunter,Zwiebach::FirstCourseString}.
We are ultimately interested in their study to construct Yukawa couplings in string theory.
@@ -1138,7 +1141,7 @@ When $R \to 0$ modes with non vanishing momentum (i.e.\ with $n \neq 0$) become
=
\frac{n}{R} \stackrel{R \to 0}{\longrightarrow} \infty.
\end{equation}
The behaviour resembles field theory: the compactified dimension disappears and open string endpoints live in a $(D-1)$-dimensional hypersurface.
The behaviour is similar to the traditional field theory: the compactified dimension disappears and open string endpoints live in a $(D-1)$-dimensional hypersurface.
This is a consequence of the T-duality transformation applied on the compact direction.
In fact the original Neumann boundary condition~\eqref{eq:tduality:bc} becomes a \emph{Dirichlet condition} for $Y^{D-1}$ defined as in~\eqref{eq:tduality:compactdirection}:
\begin{equation}
@@ -1182,7 +1185,7 @@ The coordinate of the endpoint in the compact direction is therefore fixed and c
\end{equation}
The only difference in the position of the endpoints can only be a multiple of the radius of the compactified dimension.
Otherwise they lie on the same hypersurface.
The procedure can be generalise to $p$ coordinates and constraining the string to live on a $(D - p)$-brane.
The procedure can be generalises to $p$ coordinates and constraining the string to live on a $(D - p - 1)$-brane.
This geometric interpretation of the Dirichlet branes and boundary conditions is the basis for the definition of more complex scenarios in which multiple D-branes are inserted in spacetime.
D-branes are however much more than mathematical entities.
@@ -1203,7 +1206,7 @@ Specifically a Dp-branes breaks the original \SO{1, D-1} symmetry to $\SO{1, p}
Notice that usually $D = 10$, but we keep a generic indication of the spacetime dimensions when possible.
}
The massless spectrum of the theory on the D-brane is easily computed in lightcone gauge~\cite{Goddard:1973:QuantumDynamicsMassless,Polchinski:1998:StringTheoryIntroduction,Green:1988:SuperstringTheoryIntroduction,Angelantonj:2002:OpenStrings}.
Using the residual symmetries of the two-dimensional diffeomorphism (i.e.\ armonic functions of $\tau$ and $\sigma$) we can set
Using the residual symmetries~\eqref{eq:conf:residualgauge} of the two-dimensional diffeomorphism (i.e.\ armonic functions of $\tau$ and $\sigma$) we can set
\begin{equation}
X^+( \tau, \sigma ) = x_0^+ + 2 \ap\, p^+\, \tau,
\end{equation}
@@ -1256,7 +1259,7 @@ Thus the gauge field in the original theory is split into
\end{split}
\end{equation}
In the last expression $\cA^A$ forms a representation of the Little Group \SO{p-2} and as such it is a vector gauge field in $p$ dimensions.
The field $\cA^a$ form a vector representation of the group \SO{D-1-p} and from the point of view of the Lorentz group they are $D - 1 - p$ scalars in the light spectrum.
The field $\cA^a$ forms a vector representation of the group \SO{D-1-p} and from the point of view of the Lorentz group they are $D - 1 - p$ scalars in the light spectrum.
\begin{figure}[tbp]
\centering
@@ -1288,7 +1291,7 @@ Matrices $\tensor{\lambda}{^a_{ij}}$ thus form a basis for expanding wave functi
In general Chan-Paton factors label the D-brane on which the endpoint of the string lives as in the left of~\Cref{fig:dbranes:chanpaton}.
Notice that strings stretching across different D-branes present an additional term in the mass shell condition proportional to the distance between the hypersurfaces: fields built using strings with Chan-Paton factors $\lambda^a_{ij}$ for which $i \neq j$ will therefore be massive.
However when $N$ D-branes coincide in space and form a stack their mass vanishes again: it then possible to organise the $N^2$ resulting massless fields in a representation of the gauge group \U{N}, thus promoting the symmetry $\bigotimes\limits_{a = 1}^N \rU_a( 1 )$ of $N$ separate D-branes.
It is also possible to show that in the field theory limit the resulting gauge theory is a Yang-Mills gauge theory.
It is also possible to show that in the field theory limit the resulting gauge theory is a Super Yang-Mills gauge theory.
Eventually the massless spectrum of $N$ coincident $Dp-branes$ is formed by \U{N} gauge bosons in the adjoint representation, $N^2 \times (D - 1 - p)$ scalars and $N^2$ sets of $(p+1)$-dimensional fermions~\cite{Uranga:2005:TASILecturesString}.
These are the basic building blocks for a consistent string phenomenology involving both gauge bosons and matter.

801
thesis.bib
View File

@@ -1,4 +1,58 @@
@article{Abel:2003:FlavourChangingNeutral,
title = {Flavour {{Changing Neutral Currents}} in {{Intersecting Brane Models}}},
author = {Abel, S. and Masip, M. and Santiago, J.},
date = {2003-04-30},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2003},
pages = {057--057},
issn = {1029-8479},
doi = {10.1088/1126-6708/2003/04/057},
abstract = {Intersecting D-brane models provide an attractive explanation of family replication in the context of string theory. We show, however, that the localization of fermion families at different brane intersections in the extra dimensions introduces flavour changing neutral currents mediated by the Kaluza-Klein excitations of the gauge fields. This is a generic feature in these models, and it implies stringent bounds on the mass of the lightest Kaluza-Klein modes (becoming severe when the compactification radii are larger than the string length). We present the full string calculation of four-fermion interactions in models with intersecting D-branes, recovering the field theory result. This reveals other stringy sources of flavour violation, which give bounds that are complementary to the KK bounds (i.e. they become severe when the compactification radii are comparable to the string length). Taken together these bounds imply that the string scale is larger than \$M\_s\textbackslash gtrsim 10\^2\$ TeV, implying that non-supersymmetric cases are phenomenologically disfavoured.},
archivePrefix = {arXiv},
eprint = {hep-ph/0303087},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/abel_et_al_2003_flavour_changing_neutral_currents_in_intersecting_brane_models.pdf;/home/riccardo/.local/share/zotero/storage/U4DUJK6P/0303087.html},
number = {04}
}
@article{Abel:2005:OneloopYukawasIntersecting,
title = {One-Loop {{Yukawas}} on {{Intersecting Branes}}},
author = {Abel, S. A. and Schofield, B. W.},
date = {2005-06-28},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2005},
pages = {072--072},
issn = {1029-8479},
doi = {10.1088/1126-6708/2005/06/072},
abstract = {We calculate Yukawa interactions at one-loop on intersecting D6 branes. We demonstrate the non-renormalization theorem in supersymmetric configurations, and show how Yukawa beta functions may be extracted. In addition to the usual logarithmic running, we find the power-law dependence on the infra-red cut-off associated with Kaluza-Klein modes. Our results may also be used to evaluate coupling renormalization in non-supersymmetric cases.},
archivePrefix = {arXiv},
eprint = {hep-th/0412206},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/abel_schofield_2005_one-loop_yukawas_on_intersecting_branes5.pdf;/home/riccardo/.local/share/zotero/storage/2LCUN7BE/0412206.html},
number = {06}
}
@article{Abel:2007:RealisticYukawaCouplings,
title = {Realistic {{Yukawa Couplings}} through {{Instantons}} in {{Intersecting Brane Worlds}}},
author = {Abel, Steven A. and Goodsell, Mark D.},
date = {2007-10-05},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2007},
pages = {034--034},
issn = {1029-8479},
doi = {20071006031312},
abstract = {The Yukawa couplings of the simpler models of D-branes on toroidal orientifolds suffer from the so-called ``rank one'' problem -- there is only a single non-zero mass and no mixing. We consider the one-loop contribution of E2-instantons to Yukawa couplings on intersecting D6-branes, and show that they can solve the rank one problem. In addition they have the potential to provide a geometric explanation for the hierarchies observed in the Yukawa coupling. In order to do this we provide the necessary quantities for instanton calculus in this class of background.},
archivePrefix = {arXiv},
eprint = {hep-th/0612110},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/abel_goodsell_2007_realistic_yukawa_couplings_through_instantons_in_intersecting_brane_worlds8.pdf;/home/riccardo/.local/share/zotero/storage/5HFIE6GC/0612110.html},
number = {10}
}
@article{Aldazabal:2000:DBranesSingularitiesBottomUp,
title = {D-{{Branes}} at {{Singularities}} : {{A Bottom}}-{{Up Approach}} to the {{String Embedding}} of the {{Standard Model}}},
shorttitle = {D-{{Branes}} at {{Singularities}}},
@@ -18,6 +72,57 @@
number = {08}
}
@article{Anastasopoulos:2011:ClosedstringTwistfieldCorrelators,
title = {On Closed-String Twist-Field Correlators and Their Open-String Descendants},
author = {Anastasopoulos, Pascal and Bianchi, Massimo and Richter, Robert},
date = {2011-10-24},
url = {http://arxiv.org/abs/1110.5359},
urldate = {2020-09-09},
abstract = {In a recent paper we have proposed the possibility that the lightest massive string states could be identified with open strings living at intersections of D-branes forming small angles. In this note, we reconsider the relevant twist-field correlation functions and perform the analysis of the sub-dominant physical poles in the various channels. Our derivation is new in that it is based on the algebraic procedure for the construction of open string models starting from their closed-string `parents' rather than on the stress-tensor method. We also indicate possible generalizations and diverse applications of our approach.},
archivePrefix = {arXiv},
eprint = {1110.5359},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/anastasopoulos_et_al_2011_on_closed-string_twist-field_correlators_and_their_open-string_descendants5.pdf;/home/riccardo/.local/share/zotero/storage/FNUNE3XD/1110.html},
keywords = {⛔ No DOI found},
primaryClass = {hep-th}
}
@article{Anastasopoulos:2012:LightStringyStates,
title = {Light Stringy States},
author = {Anastasopoulos, Pascal and Bianchi, Massimo and Richter, Robert},
date = {2012-03},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energ. Phys.},
volume = {2012},
pages = {68},
issn = {1029-8479},
doi = {10.1007/JHEP03(2012)068},
abstract = {We carefully study the spectrum of open strings localized at the intersections of D6-branes and identify the lowest massive 'twisted' states and their vertex operators, paying particular attention to the signs of the intersection angles. We argue that the masses of the lightest states scale as M\^2 \textasciitilde{} \textbackslash theta M\^2\_s and can thus be parametrically smaller than the string scale. Relying on previous analyses, we compute scattering amplitudes of massless 'twisted' open strings and study their factorization, confirming the presence of the light massive states as sub-dominant poles in one of the channels.},
archivePrefix = {arXiv},
eprint = {1110.5424},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/anastasopoulos_et_al_2012_light_stringy_states5.pdf;/home/riccardo/.local/share/zotero/storage/V9K88VXH/1110.html},
number = {3}
}
@article{Anastasopoulos:2013:ThreeFourpointCorrelators,
title = {Three- and {{Four}}-Point Correlators of Excited Bosonic Twist Fields},
author = {Anastasopoulos, Pascal and Goodsell, Mark D. and Richter, Robert},
date = {2013-10},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energ. Phys.},
volume = {2013},
pages = {182},
issn = {1029-8479},
doi = {10.1007/JHEP10(2013)182},
abstract = {We compute three- and four-point correlation functions containing excited bosonic twist fields. Our results can be used to determine properties, such as lifetimes and production rates, of massive string excitations localised at D-brane intersections, which could be signatures of a low string scale even if the usual string resonances are inaccessible to the LHC.},
archivePrefix = {arXiv},
eprint = {1305.7166},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/anastasopoulos_et_al_2013_three-_and_four-point_correlators_of_excited_bosonic_twist_fields9.pdf;/home/riccardo/.local/share/zotero/storage/I9E3VKI4/1305.html},
number = {10}
}
@article{Anderson:2018:TASILecturesGeometric,
title = {{{TASI Lectures}} on {{Geometric Tools}} for {{String Compactifications}}},
author = {Anderson, Lara B. and Karkheiran, Mohsen},
@@ -29,6 +134,24 @@
file = {/home/riccardo/.local/share/zotero/files/anderson_karkheiran_2018_tasi_lectures_on_geometric_tools_for_string_compactifications.pdf}
}
@article{Angelantonj:2000:TypeIStringsMagnetised,
title = {Type-{{I}} Strings on Magnetised Orbifolds and Brane Transmutation},
author = {Angelantonj, C. and Antoniadis, I. and Dudas, E. and Sagnotti, A.},
date = {2000-09},
journaltitle = {Physics Letters B},
shortjournal = {Physics Letters B},
volume = {489},
pages = {223--232},
issn = {03702693},
doi = {10.1016/S0370-2693(00)00907-2},
abstract = {In the presence of internal magnetic fields, a D9 brane can acquire a D5 (or anti-D5) R-R charge, and can therefore contribute to the corresponding tadpole. In the resulting vacua, supersymmetry is generically broken and tachyonic instabilities are present. However, suitable choices for the magnetic fields, corresponding to self-dual configurations in the internal space, can yield new chiral supersymmetric vacua with gauge groups of reduced rank, where the magnetic energy saturates, partly or fully, the negative tension of the O5+ planes. These models contain Green-Schwarz couplings to untwisted R-R forms not present in conventional orientifolds.},
archivePrefix = {arXiv},
eprint = {hep-th/0007090},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/angelantonj_et_al_2000_type-i_strings_on_magnetised_orbifolds_and_brane_transmutation.pdf;/home/riccardo/.local/share/zotero/storage/3T3IDPLZ/0007090.html},
number = {1-2}
}
@article{Angelantonj:2002:OpenStrings,
title = {Open {{Strings}}},
author = {Angelantonj, Carlo and Sagnotti, Augusto},
@@ -59,6 +182,61 @@
number = {1-2}
}
@article{Berkooz:2004:ClosedStringsMisner,
title = {Closed {{Strings}} in {{Misner Space}}: {{Stringy Fuzziness}} with a {{Twist}}},
shorttitle = {Closed {{Strings}} in {{Misner Space}}},
author = {Berkooz, M. and Durin, B. and Pioline, B. and Reichmann, D.},
date = {2004-10-02},
journaltitle = {Journal of Cosmology and Astroparticle Physics},
shortjournal = {J. Cosmol. Astropart. Phys.},
volume = {2004},
pages = {002--002},
issn = {1475-7516},
doi = {20041012031301},
abstract = {Misner space, also known as the Lorentzian orbifold \$R\^\{1,1\}/boost\$, is the simplest tree-level solution of string theory with a cosmological singularity. We compute tree-level scattering amplitudes involving twisted states, using operator and current algebra techniques. We find that, due to zero-point quantum fluctuations of the excited modes, twisted strings with a large winding number \$w\$ are fuzzy on a scale \$\textbackslash sqrt\{\textbackslash log w\}\$, which can be much larger than the string scale. Wave functions are smeared by an operator \$\textbackslash exp(\textbackslash Delta(\textbackslash nu) \textbackslash partial\_+ \textbackslash partial\_-)\$ reminiscent of the Moyal-product of non-commutative geometry, which, since \$\textbackslash Delta(\textbackslash nu)\$ is real, modulates the amplitude rather than the phase of the wave function, and is purely gravitational in its origin. We compute the scattering amplitude of two twisted states and one tachyon or graviton, and find a finite result. The scattering amplitude of two twisted and two untwisted states is found to diverge, due to the propagation of intermediate winding strings with vanishing boost momentum. The scattering amplitude of three twisted fields is computed by analytic continuation from three-point amplitudes of states with non-zero \$p\^+\$ in the Nappi-Witten plane wave, and the non-locality of the three-point vertex is found to diverge for certain kinematical configurations. Our results for the three-point amplitudes allow in principle to compute, to leading order, the back-reaction on the metric due to a condensation of coherent winding strings.},
archivePrefix = {arXiv},
eprint = {hep-th/0407216},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/berkooz_et_al_2004_closed_strings_in_misner_space8.pdf;/home/riccardo/.local/share/zotero/storage/FLJKNMJR/0407216.html},
number = {10}
}
@article{Bertolini:2006:BraneWorldEffective,
title = {Brane World Effective Actions for {{D}}-Branes with Fluxes},
author = {Bertolini, Matteo and Billo, Marco and Lerda, Alberto and Morales, Jose F. and Russo, Rodolfo},
date = {2006-05},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {743},
pages = {1--40},
issn = {05503213},
doi = {10.1016/j.nuclphysb.2006.02.044},
abstract = {We develop systematic string techniques to study brane world effective actions for models with magnetized (or equivalently intersecting) D-branes. In particular, we derive the dependence on all NS-NS moduli of the kinetic terms of the chiral matter in a generic non-supersymmetric brane configurations with non-commuting open string fluxes. Near a N=1 supersymmetric point the effective action is consistent with a Fayet-Iliopoulos supersymmetry breaking and the normalization of the scalar kinetic terms is nothing else than the Kahler metric. We also discuss, from a stringy perspective, D and F term breaking mechanisms, and how, in this generic set up, the Kahler metric enters in the physical Yukawa couplings.},
archivePrefix = {arXiv},
eprint = {hep-th/0512067},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/bertolini_et_al_2006_brane_world_effective_actions_for_d-branes_with_fluxes5.pdf;/home/riccardo/.local/share/zotero/storage/DP7XCH8F/0512067.html},
number = {1-2}
}
@article{Bianchi:2005:OpenStoryMagnetic,
title = {The Open Story of the Magnetic Fluxes},
author = {Bianchi, Massimo and Trevigne, Elisa},
date = {2005-08-09},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2005},
pages = {034--034},
issn = {1029-8479},
doi = {10.1088/1126-6708/2005/08/034},
abstract = {We discuss the effects of oblique internal magnetic fields on the spectrum of type I superstrings compactified on tori. In particular we derive general formulae for the magnetic shifts and multiplicities of open strings connecting D9-branes with arbitrary magnetic fluxes. We discuss the flux induced potential and offer an interpretation of the stabilization of R-R moduli associated to deformations of the complex structure of T\^6 in terms of non-derivative mixing with NS-NS moduli. Finally we briefly comment on how to extract other low energy couplings and generalize our results to toroidal orbifolds and other configurations governed by rational conformal field theories on the worldsheet.},
archivePrefix = {arXiv},
eprint = {hep-th/0502147},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/bianchi_trevigne_2005_the_open_story_of_the_magnetic_fluxes5.pdf;/home/riccardo/.local/share/zotero/storage/NH9MPEJT/0502147.html},
number = {08}
}
@article{Blumenhagen:2007:FourdimensionalStringCompactifications,
title = {Four-Dimensional String Compactifications with {{D}}-Branes, Orientifolds and Fluxes},
author = {Blumenhagen, Ralph and Körs, Boris and Lüst, Dieter and Stieberger, Stephan},
@@ -138,6 +316,23 @@
number = {4}
}
@article{Burwick:1991:GeneralYukawaCouplings,
title = {General {{Yukawa}} Couplings of Strings on Orbifolds},
author = {Burwick, T.T. and Kaiser, R.K. and Müller, H.F.},
date = {1991-05},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {355},
pages = {689--711},
issn = {05503213},
doi = {10.1016/0550-3213(91)90491-F},
annotation = {http://web.archive.org/web/20200909161147/https://linkinghub.elsevier.com/retrieve/pii/055032139190491F},
file = {/home/riccardo/.local/share/zotero/files/burwick_et_al_1991_general_yukawa_couplings_of_strings_on_orbifolds.pdf},
keywords = {archived},
langid = {english},
number = {3}
}
@inproceedings{Calabi:1957:KahlerManifoldsVanishing,
title = {On {{Kähler}} Manifolds with Vanishing Canonical Class},
booktitle = {Algebraic Geometry and Topology. {{A}} Symposium in Honor of {{S}}. {{Lefschetz}}},
@@ -164,6 +359,61 @@
langid = {english}
}
@article{Chamoun:2004:FermionMassesMixing,
title = {Fermion Masses and Mixing in Intersecting Brane Scenarios},
author = {Chamoun, N. and Khalil, S. and Lashin, E.},
date = {2004-05-26},
journaltitle = {Physical Review D},
shortjournal = {Phys. Rev. D},
volume = {69},
pages = {095011},
issn = {1550-7998, 1550-2368},
doi = {10.1103/PhysRevD.69.095011},
archivePrefix = {arXiv},
eprint = {hep-ph/0309169},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/chamoun_et_al_2004_fermion_masses_and_mixing_in_intersecting_brane_scenarios.pdf},
keywords = {archived},
langid = {english},
number = {9}
}
@article{Chen:2008:RealisticWorldIntersecting,
title = {A {{Realistic World}} from {{Intersecting D6}}-{{Branes}}},
author = {Chen, Ching-Ming and Li, Tianjun and Mayes, V. E. and Nanopoulos, Dimitri V.},
date = {2008-07},
journaltitle = {Physics Letters B},
shortjournal = {Physics Letters B},
volume = {665},
pages = {267--270},
issn = {03702693},
doi = {10.1016/j.physletb.2008.06.024},
abstract = {We briefly describe a three-family intersecting D6-brane model in Type IIA theory on the T\^6/(Z\_2 x Z\_2) orientifold with a realistic phenomenology. In this model, the gauge symmetry can be broken down to the Standard Model (SM) gauge symmetry close to the string scale, and the gauge coupling unification can be achieved. We calculate the supersymmetry breaking soft terms, and the corresponding low energy supersymmetric particle spectrum, which may be tested at the Large Hadron Collider (LHC). The observed dark matter density may also be generated. Finally, we can explain the SM quark masses and CKM mixings, and the tau lepton mass. The neutrino masses and mixings may be generated via the seesaw mechanism as well.},
archivePrefix = {arXiv},
eprint = {hep-th/0703280},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/chen_et_al_2008_a_realistic_world_from_intersecting_d6-branes.pdf;/home/riccardo/.local/share/zotero/storage/3MKTIA92/0703280.html},
number = {4}
}
@article{Chen:2008:RealisticYukawaTextures,
title = {Realistic {{Yukawa Textures}} and {{SUSY Spectra}} from {{Intersecting Branes}}},
author = {Chen, Ching-Ming and Li, Tianjun and Mayes, V. E. and Nanopoulos, D. V.},
date = {2008-06-20},
journaltitle = {Physical Review D},
shortjournal = {Phys. Rev. D},
volume = {77},
pages = {125023},
issn = {1550-7998, 1550-2368},
doi = {10.1103/PhysRevD.77.125023},
abstract = {We study the possible phenomenology of a three-family Pati-Salam model constructed from intersecting D6-branes in Type IIA string theory on the T\^6/(Z2 x Z2) orientifold with some desirable semi-realistic features. In the model, tree-level gauge coupling unification is achieved automatically at the string scale, and the gauge symmetry may be broken to the Standard Model (SM) close to the string scale. The small number of extra chiral exotic states in the model may be decoupled via the Higgs mechanism and strong dynamics. We calculate the possible supersymmetry breaking soft terms and the corresponding low-energy supersymmetric particle spectra which may potentially be tested at the Large Hadron Collider (LHC). We find that for the viable regions of the parameter space the lightest CP-even Higgs boson mass usually satisfies m\_H {$<$} 120 GeV, and the observed dark matter density may be generated. Finally, we find that it is possible to obtain correct SM quark masses and mixings, and the tau lepton mass at the unification scale. Additionally, neutrino masses and mixings may be generated via the seesaw mechanism. Mechanisms to stabilize the open and closed-string moduli, which are necessary for the model to be truly viable and to make definite predictions are discussed.},
archivePrefix = {arXiv},
eprint = {0711.0396},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/chen_et_al_2008_realistic_yukawa_textures_and_susy_spectra_from_intersecting_branes.pdf;/home/riccardo/.local/share/zotero/storage/3CUNZ86H/0711.html},
number = {12}
}
@article{Cleaver:2007:SearchMinimalSupersymmetric,
title = {In {{Search}} of the ({{Minimal Supersymmetric}}) {{Standard Model String}}},
author = {Cleaver, Gerald B.},
@@ -175,6 +425,147 @@
file = {/home/riccardo/.local/share/zotero/files/cleaver_2007_in_search_of_the_(minimal_supersymmetric)_standard_model_string.pdf}
}
@article{Cremades:2003:YukawaCouplingsIntersecting,
title = {Yukawa Couplings in Intersecting {{D}}-Brane Models},
author = {Cremades, D. and Ibanez, L. E. and Marchesano, F.},
date = {2003-07-16},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2003},
pages = {038--038},
issn = {1029-8479},
doi = {10.1088/1126-6708/2003/07/038},
abstract = {We compute the Yukawa couplings among chiral fields in toroidal Type II compactifications with wrapping D6-branes intersecting at angles. Those models can yield realistic standard model spectrum living at the intersections. The Yukawa couplings depend both on the Kahler and open string moduli but not on the complex structure. They arise from worldsheet instanton corrections and are found to be given by products of complex Jacobi theta functions with characteristics. The Yukawa couplings for a particular intersecting brane configuration yielding the chiral spectrum of the MSSM are computed as an example. We also show how our methods can be extended to compute Yukawa couplings on certain classes of elliptically fibered CY manifolds which are mirror to complex cones over del Pezzo surfaces. We find that the Yukawa couplings in intersecting D6-brane models have a mathematical interpretation in the context of homological mirror symmetry. In particular, the computation of such Yukawa couplings is related to the construction of Fukaya's category in a generic symplectic manifold.},
archivePrefix = {arXiv},
eprint = {hep-th/0302105},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/cremades_et_al_2003_yukawa_couplings_in_intersecting_d-brane_models5.pdf;/home/riccardo/.local/share/zotero/storage/49DFN84N/0302105.html},
number = {07}
}
@article{Cvetic:2010:BranesInstantonsIntersecting,
title = {Branes and Instantons Intersecting at Angles},
author = {Cvetič, Mirjam and García-Etxebarria, Iñaki and Richter, Robert},
date = {2010-01},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energ. Phys.},
volume = {2010},
pages = {5},
issn = {1029-8479},
doi = {10.1007/JHEP01(2010)005},
abstract = {We study in detail the system of D6 branes and euclidean D2-brane instantons intersecting at angles in type IIA string theory. We find that in the absence of orientifolds the system does not contribute to the low energy superpotential, in agreement with expectations based on effective field theory arguments. We also comment on the implications of our results for dual string theory pictures.},
archivePrefix = {arXiv},
eprint = {0905.1694},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/cvetič_et_al_2010_branes_and_instantons_intersecting_at_angles.pdf;/home/riccardo/.local/share/zotero/storage/H2QKB4Y5/0905.html},
number = {1}
}
@article{DAppollonio:2003:StringInteractionsGravitational,
title = {String Interactions in Gravitational Wave Backgrounds},
author = {D'Appollonio, Giuseppe and Kiritsis, Elias},
date = {2003-12},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {674},
pages = {80--170},
issn = {05503213},
doi = {10.1016/j.nuclphysb.2003.09.020},
abstract = {The non-compact CFT of a class of NS-supported pp-wave backgrounds is solved exactly. The associated tree-level covariant string scattering amplitudes are calculated. The S-matrix elements are well-defined, dual but not analytic as a function of \$p\^+\$. They have poles corresponding to physical intermediate states with \$p\^+\textbackslash not =0\$ and logarithmic branch cuts due to on-shell exchange of spectral-flow images of \$p\^+=0\$ states. When \$\textbackslash mu\textbackslash to 0\$ a smooth flat space limit is obtained. The \$\textbackslash mu\textbackslash to\textbackslash infty\$ limit, unlike the case of RR-supported pp-waves, gives again a flat space theory.},
archivePrefix = {arXiv},
eprint = {hep-th/0305081},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/d'appollonio_kiritsis_2003_string_interactions_in_gravitational_wave_backgrounds5.pdf;/home/riccardo/.local/share/zotero/storage/RDEYCNST/0305081.html},
number = {1-2}
}
@article{DAppollonio:2005:DbranesBCFTHppwave,
title = {D-Branes and {{BCFT}} in {{Hpp}}-Wave Backgrounds},
author = {D'Appollonio, G. and Kiritsis, E.},
date = {2005-04},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {712},
pages = {433--512},
issn = {05503213},
doi = {10.1016/j.nuclphysb.2005.01.020},
abstract = {In this paper we study two classes of symmetric D-branes in the Nappi-Witten gravitational wave, namely D2 and \$S 1\$ branes. We solve the sewing constraints and determine the bulk-boundary couplings and the boundary three-point couplings. For the D2 brane our solution gives the first explicit results for the structure constants of the twisted symmetric branes in a WZW model. We also compute the boundary four-point functions, providing examples of open string four-point amplitudes in a curved background. We finally discuss the annulus amplitudes, the relation with branes in \$AdS\_3\$ and in \$S\^3\$ and the analogy between the open string couplings in the \$H\_4\$ model and the couplings for magnetized and intersecting branes.},
archivePrefix = {arXiv},
eprint = {hep-th/0410269},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/d'appollonio_kiritsis_2005_d-branes_and_bcft_in_hpp-wave_backgrounds5.pdf;/home/riccardo/.local/share/zotero/storage/JEE8EF46/0410269.html},
number = {3}
}
@article{David:2000:TachyonCondensationD0,
title = {Tachyon Condensation in the {{D0}}/{{D4}} System},
author = {David, Justin R.},
date = {2000-10-03},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2000},
pages = {004--004},
issn = {1029-8479},
doi = {20050405175528},
abstract = {The D0/D4 system with a Neveu-Schwarz B-field in the spatial directions of the D4-brane has a tachyon in the spectrum of the (0,4) strings. The tachyon signals the instability of the system to form a bound state of the D0-brane with the D4-brane. We use the Wess-Zumino-Witten like open superstring field theory formulated by Berkovits to study the tachyon potential for this system. The tachyon potential lies outside the universality class of the D-brane anti-D-brane system. It is a function of the B-field. We calculate the tachyon potential at the zeroth level approximation. The minimum of the tachyon potential in this case is expected to reproduce the mass defect involved in the formation of the D0/D4 bound state. We compare the minimum of the tachyon potential with the mass defect in three cases. For small values of the B-field we obtain 70\% of the expected mass defect. For large values of the B-field with Pf\$(2\textbackslash pi\textbackslash alpha' B) {$>$}0\$ the potential reduces to that of the D-brane anti-D-brane reproducing 62\% of the expected mass defect. For large values of the B-field with Pf\$(2\textbackslash pi\textbackslash alpha' B) {$<$}0\$ the minimum of the tachyon potential gives 25\% of the expected mass defect. At the tachyon condensate we show that the (0,4) strings decouple from the low energy dynamics.},
archivePrefix = {arXiv},
eprint = {hep-th/0007235},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/david_2000_tachyon_condensation_in_the_d0-d4_system.pdf;/home/riccardo/.local/share/zotero/storage/2U52Z9Q4/0007235.html},
number = {10}
}
@article{David:2001:TachyonCondensationUsing,
title = {Tachyon Condensation Using the Disc Partition Function},
author = {David, Justin R.},
date = {2001-07-10},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2001},
pages = {009--009},
issn = {1029-8479},
doi = {10.1088/1126-6708/2001/07/009},
abstract = {It has been recently proposed that the background independent open superstring field theory action is given by the disc partition function with all possible open string operators inserted at the boundary of the disc. We use this proposal to study tachyon condensation in the D0-D2 system. We evaluate the disc partition function for the D0-D2 system in presence of a large Neveu-Schwarz B-field using perturbation theory. This perturbative expansion of the disc partition function makes sense as the boundary tachyon operator for the large Neveu-Schwarz B-field is almost marginal. We find that the mass defect for the formation of the D0-D2 bound state agrees exactly with the expected result in the large B-field limit.},
archivePrefix = {arXiv},
eprint = {hep-th/0012089},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/david_2001_tachyon_condensation_using_the_disc_partition_function5.pdf;/home/riccardo/.local/share/zotero/storage/VRQZCMVC/0012089.html},
number = {07}
}
@article{David:2002:ClosedStringTachyon,
title = {Closed {{String Tachyon Condensation}} on {{Twisted Circles}}},
author = {David, Justin R. and Gutperle, Michael and Headrick, Matthew and Minwalla, Shiraz},
date = {2002-02-26},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2002},
pages = {041--041},
issn = {1029-8479},
doi = {10.1088/1126-6708/2002/02/041},
abstract = {We study IIA/B string theory compactified on twisted circles. These models possess closed string tachyons and reduce to type 0B/A theory in a special limit. Using methods of gauged linear sigma models and mirror symmetry we construct a conformal field theory which interpolates between these models and flat space via an auxiliary Liouville direction. Interpreting motion in the Liouville direction as renormalization group flow, we argue that the end point of tachyon condensation in all these models (including 0B/A theory) is supersymmetric type II theory. We also find a zero-slope limit of these models which is best described in a T-dual picture as a type II NS-NS fluxbrane. In this limit tachyon condensation is an interesting and well posed problem in supergravity. We explicitly determine the tachyon as a fluctuation of supergravity fields, and perform a rudimentary numerical analysis of the relevant flows.},
archivePrefix = {arXiv},
eprint = {hep-th/0111212},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/david_et_al_2002_closed_string_tachyon_condensation_on_twisted_circles5.pdf;/home/riccardo/.local/share/zotero/storage/LVI9CWIS/0111212.html},
number = {02}
}
@article{DellaSelva:1970:SimpleExpressionSciuto,
title = {A Simple Expression for the {{Sciuto}} Three-Reggeon Vertex-Generating Duality},
author = {Della Selva, A. and Saito, S.},
date = {1970-10},
journaltitle = {Lettere al Nuovo Cimento},
shortjournal = {Lett. Nuovo Cimento},
volume = {4},
pages = {689--692},
issn = {0375-930X, 1827-613X},
doi = {10.1007/BF02755329},
file = {/home/riccardo/.local/share/zotero/files/selva_saito_1970_a_simple_expression_for_the_sciuto_three-reggeon_vertex-generating_duality.pdf},
langid = {english},
number = {15}
}
@book{DiFrancesco:1997:ConformalFieldTheory,
title = {Conformal {{Field Theory}}},
author = {Di Francesco, Philippe and Mathieu, Pierre and Sénéchal, David},
@@ -243,6 +634,81 @@
file = {/home/riccardo/.local/share/zotero/files/di_vecchia_et_al_2006_boundary_state_for_magnetized_d9_branes_and_one-loop_calculation.pdf}
}
@article{DiVecchia:2007:WrappedMagnetizedBranes,
ids = {DiVecchia:2007:WrappedMagnetizedBranesa},
title = {Wrapped Magnetized Branes: Two Alternative Descriptions?},
author = {Di Vecchia, Paolo and Liccardo, Antonella and Marotta, Raffaele and Pezzella, Franco and Pesando, Igor},
date = {2007-12},
journaltitle = {Journal of High Energy Physics},
volume = {2007},
pages = {100--100},
issn = {1029-8479},
doi = {10.1088/1126-6708/2007/11/100},
abstract = {We discuss two inequivalent ways for describing magnetized D-branes wrapped N times on a torus T\^2. The first one is based on a non-abelian gauge bundle U(N), while the second one is obtained by means of a Narain T-duality transformation acting on a theory with non-magnetized branes. We construct in both descriptions the boundary state and the open string vertices and show that they give rise to different string amplitudes. In particular, the description based on the gauge bundle has open string vertex operators with momentum dependent Chan-Paton factors.},
annotation = {ZSCC: 0000022},
archivePrefix = {arXiv},
eprint = {0709.4149},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/di_vecchia_et_al_2007_wrapped_magnetized_branes.pdf},
issue = {11},
number = {DSF-32-2007, NORDITA-2007-28},
primaryClass = {hep-th}
}
@article{DiVecchia:2011:OpenStringsSystem,
title = {Open Strings in the System {{D5}}/{{D9}}},
author = {Di Vecchia, P. and Marotta, R. and Pesando, I. and Pezzella, F.},
date = {2011-06-17},
journaltitle = {Journal of Physics A: Mathematical and Theoretical},
shortjournal = {J. Phys. A: Math. Theor.},
volume = {44},
pages = {245401},
issn = {1751-8113, 1751-8121},
doi = {10.1088/1751-8113/44/24/245401},
abstract = {We construct the six-dimensional Lagrangian for the massless twisted open strings with one end-point ending on a stack of D5 and the other on a stack of D9 branes, interacting with the gauge multiplets living respectively on the D5 and D9 branes. It is first obtained by uplifting to six dimensions the four-dimensional Lagrangian of the N=2 hypermultiplet and manifestly exhibits an SU(2) symmetry. We show by an explicit calculation that it is N=1 supersymmetric in six dimensions and then we check various terms of this Lagrangian by computing string amplitudes on the disk. Finally, starting from this Lagrangian and assuming the presence of non-zero magnetic fluxes along the extra compact dimensions, we determine the spectrum of the Kaluza-Klein states which agrees with the corresponding one obtained from string theory in the field theory limit.},
archivePrefix = {arXiv},
eprint = {1101.0120},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/di_vecchia_et_al_2011_open_strings_in_the_system_d5-d5.pdf;/home/riccardo/.local/share/zotero/storage/5AHE9Z2W/1101.html},
number = {24}
}
@article{Duo:2007:NewTwistField,
title = {New Twist Field Couplings from the Partition Function for Multiply Wrapped {{D}}-Branes},
author = {Duo, Dario and Russo, Rodolfo and Sciuto, Stefano},
date = {2007-12-12},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2007},
pages = {042--042},
issn = {1029-8479},
doi = {10.1088/1126-6708/2007/12/042},
abstract = {We consider toroidal compactifications of bosonic string theory with particular regard to the phases (cocycles) necessary for a consistent definition of the vertex operators, the boundary states and the T-duality rules. We use these ingredients to compute the planar multi-loop partition function describing the interaction among magnetized or intersecting D-branes, also in presence of open string moduli. It turns out that unitarity in the open string channel crucially depends on the presence of the cocycles. We then focus on the 2-loop case and study the degeneration limit where this partition function is directly related to the tree-level 3-point correlators between twist fields. These correlators represent the main ingredient in the computation of Yukawa couplings and other terms in the effective action for D-brane phenomenological models. By factorizing the 2-loop partition function we are able to compute the 3-point couplings for abelian twist fields on generic non-factorized tori, thus generalizing previous expressions valid for the 2-torus.},
archivePrefix = {arXiv},
eprint = {0709.1805},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/duo_et_al_2007_new_twist_field_couplings_from_the_partition_function_for_multiply_wrapped5.pdf;/home/riccardo/.local/share/zotero/storage/V7S6EP95/0709.html},
number = {12}
}
@article{Erler:1993:HigherTwistedSector,
title = {Higher {{Twisted Sector Couplings}} of \${{Z}}\_{{N}}\$ {{Orbifolds}}},
author = {Erler, J. and Jungnickel, D. and Spalinski, M. and Stieberger, S.},
date = {1993-05},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {397},
pages = {379--414},
issn = {05503213},
doi = {10.1016/0550-3213(93)90348-S},
abstract = {We derive the basic correlation functions of twist fields coming from arbitrary twisted sectors in symmetric \$Z\_N\$ orbifold conformal field theories, keeping all the admissible marginal perturbations, in particular those corresponding to the antisymmetric tensor background field. This allows a thorough investigation of modular symmetries in this type of string compactification. Such a study is explicitly carried out for the group generated by duality transformations. Thus, apart from being of phenomenological use, our couplings are also interesting from the mathematical point of view as they represent automorphic functions for a large class of discrete groups.},
archivePrefix = {arXiv},
eprint = {hep-th/9207049},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/erler_et_al_1993_higher_twisted_sector_couplings_of_$z_n$_orbifolds.pdf;/home/riccardo/.local/share/zotero/storage/R797YJWK/9207049.html},
number = {1-2}
}
@article{Finotello:2019:ClassicalSolutionBosonic,
ids = {Finotello:2019:ClassicalSolutionBosonica},
title = {The {{Classical Solution}} for the {{Bosonic String}} in the {{Presence}} of {{Three D}}-Branes {{Rotated}} by {{Arbitrary SO}}(4) {{Elements}}},
@@ -262,6 +728,42 @@
file = {/home/riccardo/.local/share/zotero/files/finotello_pesando_2019_the_classical_solution_for_the_bosonic_string_in_the_presence_of_three_d-branes.pdf;/home/riccardo/.local/share/zotero/files/finotello_pesando_2019_the_classical_solution_for_the_bosonic_string_in_the_presence_of_three_d-branes2.pdf}
}
@article{Forste:2018:YukawaCouplingsMagnetized,
title = {Yukawa Couplings from Magnetized {{D}}-Brane Models on Non-Factorisable Tori},
author = {Forste, Stefan and Liyanage, Christoph},
date = {2018-08},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energ. Phys.},
volume = {2018},
pages = {169},
issn = {1029-8479},
doi = {10.1007/JHEP08(2018)169},
abstract = {We compute Yukawa couplings in type IIB string theory compactified on a non factorisable six-torus in the presence of D9 branes and fluxes. The setting studied in detail, is obtained by T-dualising an intersecting brane configuration of type IIA theory compactified on a torus generated by the SO(12) root lattice. Particular deformations of such torus are taken into account and provide moduli dependent couplings. Agreement with the type IIA result is found in a non trivial way. The classical type IIB calculation gives also information on a factor accessible only by quantum computations on the type IIA side.},
archivePrefix = {arXiv},
eprint = {1802.05136},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/forste_liyanage_2018_yukawa_couplings_from_magnetized_d-brane_models_on_non-factorisable_tori.pdf;/home/riccardo/.local/share/zotero/storage/6I8WLCI5/1802.html},
number = {8}
}
@article{Frampton:2001:ClassificationConformalityModels,
title = {Classification of {{Conformality Models Based}} on {{Nonabelian Orbifolds}}},
author = {Frampton, Paul H. and Kephart, Thomas W.},
date = {2001-09-27},
journaltitle = {Physical Review D},
shortjournal = {Phys. Rev. D},
volume = {64},
pages = {086007},
issn = {0556-2821, 1089-4918},
doi = {10.1103/PhysRevD.64.086007},
abstract = {A systematic analysis is presented of compactifications of the IIB superstring on \$AdS\_5 \textbackslash times S\^5/\textbackslash Gamma\$ where \$\textbackslash Gamma\$ is a non-abelian discrete group. Every possible \$\textbackslash Gamma\$ with order \$g \textbackslash leq 31\$ is considered. There exist 45 such groups but a majority cannot yield chiral fermions due to a certain theorem that is proved. The lowest order to embrace the nonSUSY standard \$SU(3) \textbackslash times SU(2) \textbackslash times U(1)\$ model with three chiral families is \$\textbackslash Gamma = D\_4 \textbackslash times Z\_3\$, with \$g=24\$; this is the only successful model found in the search. The consequent uniqueness of the successful model arises primarily from the scalar sector, prescribed by the construction, being sufficient to allow the correct symmetry breakdown.},
archivePrefix = {arXiv},
eprint = {hep-th/0011186},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/frampton_kephart_2001_classification_of_conformality_models_based_on_nonabelian_orbifolds5.pdf;/home/riccardo/.local/share/zotero/storage/JDK2473W/0011186.html},
number = {8}
}
@article{Friedan:1986:ConformalInvarianceSupersymmetry,
title = {Conformal Invariance, Supersymmetry and String Theory},
author = {Friedan, Daniel and Martinec, Emil and Shenker, Stephen},
@@ -276,6 +778,41 @@
number = {3-4}
}
@article{Gato:1990:VertexOperatorsNonabelian,
title = {Vertex Operators, Non-Abelian Orbifolds and the {{Reimann}}-{{Hilbert}} Problem},
author = {Gato, Beatriz},
date = {1990-04},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {334},
pages = {414--430},
issn = {05503213},
doi = {10.1016/0550-3213(90)90485-V},
annotation = {http://web.archive.org/web/20200909163742/https://linkinghub.elsevier.com/retrieve/pii/055032139090485V},
file = {/home/riccardo/.local/share/zotero/files/gato_1990_vertex_operators,_non-abelian_orbifolds_and_the_reimann-hilbert_problem.pdf},
keywords = {archived},
langid = {english},
number = {2}
}
@article{Gava:1997:BoundStatesBranes,
title = {On the {{Bound States}} of P- and (P+2)-{{Branes}}},
author = {Gava, E. and Narain, K. S. and Sarmadi, M. H.},
date = {1997-10},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {504},
pages = {214--238},
issn = {05503213},
doi = {10.1016/S0550-3213(97)00508-7},
abstract = {We study bound states of D-p-branes and D-(p+2)-branes. By switching on a large magnetic field F on the (p+2) brane, the problem is shown to admit a perturbative analysis in an expansion in inverse powers of F. It is found that, to the leading order in 1/F, the quartic potential of the tachyonic state from the open string stretched between the p- and (p+2)-brane gives a vacuum energy which agrees with the prediction of the BPS mass formula for the bound state. We generalize the discussion to the case of m p-branes plus 1 (p+2)-brane with magnetic field. The T dual picture of this, namely several (p+2)-branes carrying some p-brane charges through magnetic flux is also discussed, where the perturbative treatment is available in the small F limit. We show that once again, in the same approximation, the tachyon condensates give rise to the correct BPS mass formula. The role of 't Hooft's toron configurations in the extension of the above results beyond the quartic approximation as well as the issue of the unbroken gauge symmetries are discussed. We comment on the connection between the present bound state problem and Kondo-like problems in the context of relevant boundary perturbations of boundary conformal field theories.},
archivePrefix = {arXiv},
eprint = {hep-th/9704006},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/gava_et_al_1997_on_the_bound_states_of_p-_and_(p+2)-branes.pdf;/home/riccardo/.local/share/zotero/storage/UP983WKN/9704006.html},
number = {1-2}
}
@article{Ginsparg:1988:AppliedConformalField,
title = {Applied {{Conformal Field Theory}}},
author = {Ginsparg, Paul},
@@ -382,6 +919,24 @@
number = {1-2}
}
@article{Hashimoto:2003:RecombinationIntersectingDbranes,
title = {Recombination of {{Intersecting D}}-Branes by {{Local Tachyon Condensation}}},
author = {Hashimoto, Koji and Nagaoka, Satoshi},
date = {2003-06-18},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2003},
pages = {034--034},
issn = {1029-8479},
doi = {10.1088/1126-6708/2003/06/034},
abstract = {We provide a simple low energy description of recombination of intersecting D-branes using super Yang-Mills theory. The recombination is realized by condensation of an off-diagonal tachyonic fluctuation localized at the intersecting point. The recombination process is equivalent to brane-antibrane annihilation, thus our result confirms Sen's conjecture on tachyon condensation, although we work in the super Yang-Mills theory whose energy scale is much lower than alpha'. We also discuss the decay width of non-parallelly separated D-branes.},
archivePrefix = {arXiv},
eprint = {hep-th/0303204},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/hashimoto_nagaoka_2003_recombination_of_intersecting_d-branes_by_local_tachyon_condensation5.pdf;/home/riccardo/.local/share/zotero/storage/8I2UAHW2/0303204.html},
number = {06}
}
@article{He:2020:CalabiyauSpacesString,
title = {Calabi-Yau Spaces in the String Landscape},
author = {He, Yang-Hui},
@@ -437,6 +992,40 @@
number = {11}
}
@article{Inoue:1987:NonAbelianOrbifolds,
title = {Non-{{Abelian Orbifolds}}},
author = {Inoue, K. and Sakamoto, M. and Takano, H.},
date = {1987-10-01},
journaltitle = {Progress of Theoretical Physics},
shortjournal = {Progress of Theoretical Physics},
volume = {78},
pages = {908--922},
issn = {0033-068X, 1347-4081},
doi = {10.1143/PTP.78.908},
annotation = {http://web.archive.org/web/20200909163620/https://academic.oup.com/ptp/article/78/4/908/1865644},
file = {/home/riccardo/.local/share/zotero/files/inoue_et_al_1987_non-abelian_orbifolds2.pdf},
keywords = {archived},
langid = {english},
number = {4}
}
@article{Inoue:1990:StringInteractionsNonAbelian,
title = {String {{Interactions}} on {{Non}}-{{Abelian Orbifold}}},
author = {Inoue, K. and Nima, S.},
date = {1990-10-01},
journaltitle = {Progress of Theoretical Physics},
shortjournal = {Progress of Theoretical Physics},
volume = {84},
pages = {702--727},
issn = {0033-068X, 1347-4081},
doi = {10.1143/ptp/84.4.702},
annotation = {http://web.archive.org/web/20200909163702/https://academic.oup.com/ptp/article/84/4/702/1892630},
file = {/home/riccardo/.local/share/zotero/files/inoue_nima_1990_string_interactions_on_non-abelian_orbifold.pdf},
keywords = {archived},
langid = {english},
number = {4}
}
@article{Johnson:2000:DBranePrimer,
title = {D-{{Brane Primer}}},
author = {Johnson, Clifford V.},
@@ -490,6 +1079,24 @@
number = {4}
}
@article{Kiritsis:1994:StringPropagationGravitational,
title = {String {{Propagation}} in {{Gravitational Wave Backgrounds}}},
author = {Kiritsis, E. and Kounnas, C.},
date = {1994-01},
journaltitle = {Physics Letters B},
shortjournal = {Physics Letters B},
volume = {320},
pages = {264--272},
issn = {03702693},
doi = {10.1016/0370-2693(94)90655-6},
abstract = {The Conformal Field Theory of the current algebra of the centrally extended 2-d Euclidean group is analyzed. Its representations can be written in terms of four free fields (without background charge) with signature (\$-\$+++). We construct all irreducible representations of the current algebra with unitary base out of the free fields and their orbifolds. This is used to investigate the spectrum and scattering of strings moving in the background of a gravitational wave. We find that all the dynamics happens in the transverse space or the longitunal one but not both.},
archivePrefix = {arXiv},
eprint = {hep-th/9310202},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/kiritsis_kounnas_1994_string_propagation_in_gravitational_wave_backgrounds5.pdf;/home/riccardo/.local/share/zotero/storage/JKGFBSHR/9310202.html},
number = {3-4}
}
@article{Krippendorf:2010:CambridgeLecturesSupersymmetry,
title = {Cambridge {{Lectures}} on {{Supersymmetry}} and {{Extra Dimensions}}},
author = {Krippendorf, Sven and Quevedo, Fernando and Schlotterer, Oliver},
@@ -536,6 +1143,166 @@
number = {3}
}
@article{Pesando:2008:MultibranesBoundaryStates,
title = {Multi-Branes Boundary States with Open String Interactions},
author = {Pesando, Igor},
date = {2008-04},
journaltitle = {Nuclear Physics B},
volume = {793},
pages = {211--245},
issn = {05503213},
doi = {10.1016/j.nuclphysb.2007.10.002},
abstract = {We derive boundary states which describe configurations of multiple parallel branes with arbitrary open string states interactions in bosonic string theory. This is obtained by a careful discussion of the factorization of open/closed string states amplitudes taking care of cycles needed by ensuring vertexes commutativity: in particular the discussion reveals that already at the tree level open string knows of the existence of closed string. We also give a formal expression for computing pure closed string amplitudes using the open string formalism.},
annotation = {ZSCC: 0000011},
file = {/home/riccardo/.local/share/zotero/files/pesando_2008_multi-branes_boundary_states_with_open_string_interactions.pdf},
number = {1-2}
}
@article{Pesando:2010:OpenClosedString,
title = {Open and {{Closed String Vertices}} for Branes with Magnetic Field and {{T}}-Duality},
author = {Pesando, Igor},
date = {2010-02},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energ. Phys.},
volume = {2010},
pages = {64},
issn = {1029-8479},
doi = {10.1007/JHEP02(2010)064},
abstract = {We discuss carefully the vertices which describe the dipole open strings and closed strings on a D-brane with magnetic flux on a torus. Translation invariance along closed cycles forces surprisingly closed string vertices written in open string formalism to acquire Chan-Paton like matrices. Moreover the one loop amplitudes have a single trace for the part of gauge group with the magnetic flux. These peculiarities are also required by consistency of the action of T-duality in the open string sector. In this way we can show to all orders in perturbation theory the equivalence of the T-dual open string theories, gravitational interactions included. We provide also a new and direct derivation of the bosonic boundary state in presence of constant magnetic and Kalb-Ramond background based on Sciuto-Della Selva-Saito vertex formalism.},
archivePrefix = {arXiv},
eprint = {0910.2576},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/pesando_2010_open_and_closed_string_vertices_for_branes_with_magnetic_field_and_t-duality5.pdf;/home/riccardo/.local/share/zotero/storage/7G2ISTM4/0910.html},
number = {2}
}
@article{Pesando:2011:GeneratingFunctionAmplitudes,
title = {The Generating Function of Amplitudes with {{N}} Twisted and {{M}} Untwisted States},
author = {Pesando, Igor},
date = {2011-07-27},
url = {http://arxiv.org/abs/1107.5525},
urldate = {2020-09-09},
abstract = {We show that the generating function of all amplitudes with N twisted and M untwisted states, i.e. the Reggeon vertex for magnetized branes on R\^2 can be computed once the correlator of N non excited twisted states and the corresponding Green function are known and we give an explicit expression as a functional of the these objects},
archivePrefix = {arXiv},
eprint = {1107.5525},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/pesando_2011_the_generating_function_of_amplitudes_with_n_twisted_and_m_untwisted_states.pdf;/home/riccardo/.local/share/zotero/storage/RFT279XA/1107.html},
keywords = {⛔ No DOI found},
primaryClass = {hep-th}
}
@article{Pesando:2011:StringsArbitraryConstant,
title = {Strings in an Arbitrary Constant Magnetic Field with Arbitrary Constant Metric and Stringy Form Factors},
author = {Pesando, Igor},
date = {2011-06},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energ. Phys.},
volume = {2011},
pages = {138},
issn = {1029-8479},
doi = {10.1007/JHEP06(2011)138},
abstract = {We quantize the open string in an arbitrary constant magnetic field with a non factorized metric on a torus. We then discuss carefully the vertexes which describe the emission of dipole open strings and closed strings in the non compact limit. Finally we compute various stringy form factors which in the compact case induces a Kaehler and complex structure dependence and suppression of some amplitudes with KK states.},
archivePrefix = {arXiv},
eprint = {1101.5898},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/pesando_2011_strings_in_an_arbitrary_constant_magnetic_field_with_arbitrary_constant_metric5.pdf;/home/riccardo/.local/share/zotero/storage/5WS8IWA9/1101.html},
number = {6}
}
@article{Pesando:2012:GreenFunctionsTwist,
ids = {Pesando:2013:GreenFunctionsTwist},
title = {Green Functions and Twist Correlators for {{N}} Branes at Angles},
author = {Pesando, Igor},
date = {2012-06},
journaltitle = {Nuclear Physics B},
volume = {866},
pages = {87--123},
issn = {05503213},
doi = {10.1016/j.nuclphysb.2012.08.016},
abstract = {We compute the Green functions and correlator functions for N twist fields for branes at angles on T\^2 and we show that there are N-2 different configurations labeled by an integer M which is roughly associated with the number of obtuse angles of the configuration. In order to perform this computation we use a SL(2,R) invariant formulation and geometric constraints instead of Pochammer contours. In particular the M=1 or M=N-1 amplitude can be expressed without using transcendental functions. We determine the amplitudes normalization from N -\textbackslash textgreater N-1 reduction without using the factorization into the untwisted sector. Both the amplitudes normalization and the OPE of two twist fields are unique (up to one constant) when the \$\textbackslash backslash\$epsilon \textbackslash textless-\textbackslash textgreater 1-\$\textbackslash backslash\$epsilon symmetry is imposed. For consistency we find also an infinite number of relations among Lauricella hypergeometric functions.},
annotation = {ZSCC: 0000000[s0]},
archivePrefix = {arXiv},
eprint = {1206.1431},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/pesando_2012_green_functions_and_twist_correlators_for_$n$_branes_at_angles.pdf},
issue = {2},
number = {DFTT-6-2012},
primaryClass = {hep-th}
}
@article{Pesando:2013:LightConeQuantization,
title = {Light Cone Quantization and Interactions of a New Closed Bosonic String Inspired to {{D1}} String},
author = {Pesando, Igor},
date = {2013-11},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {876},
pages = {1--15},
issn = {05503213},
doi = {10.1016/j.nuclphysb.2013.07.022},
abstract = {We quantize the bosonic part of the D1 string with closed boundary conditions on the light cone and we consider the U(1) worldsheet gauge field a dynamical variable. We compute also 3-Reggeon vertex by the overlapping technique. We find that the Fock space is the sum of sectors characterized by the momentum of the U(1) Wilson line and that these sectors do not interact among them. Each sector has exactly the same spectrum of the usual bosonic string when expressed in properly sector dependent rescaled variables. Rescaling is forced by factorization of the string amplitudes. We are also able to determine the relative string coupling constant of the different sectors. It follows a somewhat unexpected picture in which the effective action is always the same independently on the sector but string amplitudes are only the same when expressed in sector dependent rescaled variables.},
archivePrefix = {arXiv},
eprint = {1305.2710},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/pesando_2013_light_cone_quantization_and_interactions_of_a_new_closed_bosonic_string5.pdf;/home/riccardo/.local/share/zotero/storage/YUJ2G4EN/1305.html},
number = {1}
}
@article{Pesando:2014:CanonicalQuantizationString,
title = {Canonical Quantization of a String Describing \${{N}}\$ Branes at Angles},
author = {Pesando, Igor},
date = {2014-12},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {889},
pages = {120--155},
issn = {05503213},
doi = {10.1016/j.nuclphysb.2014.10.005},
abstract = {We study the canonical quantization of a bosonic string in presence of N twist fields. This generalizes the quantization of the twisted string in two ways: the in and out states are not necessarily twisted and the number of twist fields N can be bigger than 2. In order to quantize the theory we need to find the normal modes. Then we need to define a product between two modes which is conserved. Because of this we need to use the Klein-Gordon product and to separate the string coordinate into the classical and the quantum part. The quantum part has different boundary conditions than the original string coordinates but these boundary conditions are precisely those which make the operator describing the equation of motion self adjoint. The splitting of the string coordinates into a classical and quantum part allows the formulation of an improved overlap principle. Using this approach we then proceed in computing the generating function for the generic correlator with L untwisted operators and N (excited) twist fields for branes at angles. We recover as expected the results previously obtained using the path integral. This construction explains why these correlators},
archivePrefix = {arXiv},
eprint = {1407.4627},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/pesando_2014_canonical_quantization_of_a_string_describing_$n$_branes_at_angles.pdf;/home/riccardo/.local/share/zotero/storage/IXMFPPX9/1407.html}
}
@article{Pesando:2014:CorrelatorsArbitraryUntwisted,
ids = {Pesando:2014:CorrelatorsArbitraryUntwisteda},
title = {Correlators of Arbitrary Untwisted Operators and Excited Twist Operators for {{N}} Branes at Angles},
author = {Pesando, Igor},
date = {2014-01},
journaltitle = {Nuclear Physics B},
volume = {886},
pages = {243--287},
issn = {05503213},
doi = {10.1016/j.nuclphysb.2014.06.010},
abstract = {We compute the generic correlator with L untwisted operators and N (excited) twist fields for branes at angles on T\^2 and show that it is given by a generalization of the Wick theorem. We give also the recipe to compute efficiently the generic OPE between an untwisted operator and an excited twisted state.},
annotation = {ZSCC: 0000012},
archivePrefix = {arXiv},
eprint = {1401.6797},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/pesando_2014_correlators_of_arbitrary_untwisted_operators_and_excited_twist_operators_for_n.pdf;/home/riccardo/.local/share/zotero/files/pesando_2014_correlators_of_arbitrary_untwisted_operators_and_excited_twist_operators_for_n2.pdf},
primaryClass = {hep-th}
}
@article{Pesando:2016:FullyStringyComputation,
ids = {Pesando:2016:FullyStringyComputationa},
title = {Towards a Fully Stringy Computation of {{Yukawa}} Couplings on Non-Factorized Tori and Non-Abelian Twist Correlators ({{I}}): {{The}} Classical Solution and Action},
author = {Pesando, Igor},
date = {2016-09},
journaltitle = {Nuclear Physics B},
volume = {910},
pages = {618--664},
issn = {05503213},
doi = {10.1016/j.nuclphysb.2016.06.013},
abstract = {We consider the simplest possible setting of non-abelian twist fields which corresponds to SU(2) monodromies. We first review the theory of hypergeometric function and of the solutions of the most general Fuchsian second order equation with three singularities. Then we solve the problem of writing the general solution with prescribed U(2) monodromies. We use this result to compute the classical string solution corresponding to three D2 branes in R4. Despite the fact that the configuration is supersymmetric the classical string solution is not holomorphic. Using the equation of motion and not the KLT approach we give a very simple expression for the classical action of the string. We find that the classical action is not proportional to the area of the triangle determined by the branes intersection points since the solution is not holomorphic. Phenomenologically this means that the Yukawa couplings for these supersymmetric configurations on non-factorized tori are suppressed with respect to the factorized case.},
annotation = {ZSCC: 0000003},
archivePrefix = {arXiv},
eprint = {1512.07920},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/pesando_2016_towards_a_fully_stringy_computation_of_yukawa_couplings_on_non_factorized_tori.pdf;/home/riccardo/.local/share/zotero/files/pesando_2016_towards_a_fully_stringy_computation_of_yukawa_couplings_on_non-factorized_tori.pdf},
primaryClass = {hep-th}
}
@article{Polchinski:1995:DirichletBranesRamondRamond,
title = {Dirichlet Branes and {{Ramond}}-{{Ramond}} Charges},
author = {Polchinski, Joseph},
@@ -605,6 +1372,21 @@
number = {3}
}
@article{Sciuto:1969:GeneralVertexFunction,
title = {The General Vertex Function in Dual Resonance Models},
author = {Sciuto, S.},
date = {1969-09},
journaltitle = {Lettere al Nuovo Cimento},
shortjournal = {Lett. Nuovo Cimento},
volume = {2},
pages = {411--418},
issn = {0375-930X, 1827-613X},
doi = {10.1007/BF02755622},
file = {/home/riccardo/.local/share/zotero/files/sciuto_1969_the_general_vertex_function_in_dual_resonance_models5.pdf},
langid = {english},
number = {9}
}
@article{Sheikh-Jabbari:1998:ClassificationDifferentBranes,
title = {Classification of {{Different Branes}} at {{Angles}}},
author = {Sheikh-Jabbari, M. M.},
@@ -623,6 +1405,25 @@
number = {3-4}
}
@article{Stieberger:1992:YukawaCouplingsBosonic,
title = {Yukawa {{Couplings}} for {{Bosonic}} \${{Z}}\_{{N}}\$ {{Orbifolds}}: {{Their Moduli}} and {{Twisted Sector Dependence}}},
shorttitle = {Yukawa {{Couplings}} for {{Bosonic}} \${{Z}}\_{{N}}\$ {{Orbifolds}}},
author = {Stieberger, S. and Jungnickel, D. and Lauer, J. and Spalinski, M.},
date = {1992-10-30},
journaltitle = {Modern Physics Letters A},
shortjournal = {Mod. Phys. Lett. A},
volume = {07},
pages = {3059--3070},
issn = {0217-7323, 1793-6632},
doi = {10.1142/S0217732392002457},
abstract = {The three point correlation functions with twist fields are determined for bosonic \$Z\_N\$ orbifolds. Both the choice of the modular background (compatible with the twist) and of the (higher) twisted sectors involved are fully general. We point out a necessary restriction on the set of instantons contributing to twist field correlation functions not obtained in previous calculations. Our results show that the theory is target space duality invariant.},
archivePrefix = {arXiv},
eprint = {hep-th/9204037},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/stieberger_et_al_1992_yukawa_couplings_for_bosonic_$z_n$_orbifolds.pdf;/home/riccardo/.local/share/zotero/storage/7UDTMVHY/9204037.html},
number = {33}
}
@article{Susskind:2003:AnthropicLandscapeString,
title = {The {{Anthropic Landscape}} of {{String Theory}}},
author = {Susskind, Leonard},

3
thesis.tex
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@@ -16,6 +16,9 @@
\addbibresource{thesis.bib}
\fancyhead[L]{}
\fancyhead[R]{\rightmark}
\hypersetup{%
pdfauthor={Riccardo Finotello}
}