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Adjustments to intros and conclusions

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-10-13 17:48:21 +02:00
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sec/part1/dbranes.tex
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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.
In this section we study \emph{intersecting D-branes}, which represent a relevant class of 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.
Furthermore 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}.
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}.
Results were however found starting from dual models~\cite{Sciuto:1969:GeneralVertexFunction,DellaSelva:1970:SimpleExpressionSciuto} up to modern interpretations of string theory.
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}.
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 relative \SU{2} rotations of the D-branes ~\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.
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 $\sigma_{\rM_{(t)}}( x_{(t)} )$ as:\footnotemark{}
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 $\sigma_{\rM_{(t)}}\qty( x_{(t)} )$ as:\footnotemark{}
\footnotetext{%
Ultimately $N_B = 3$ in our case.
}
@@ -46,7 +46,7 @@ Their calculations requires the correlator of four twist fields which in turn re
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}.
These additional phase factors can then be specialised to \SO{4} and be studied in spinor representation as a tensor product of \SU{2} elements.
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.
@@ -2568,7 +2568,7 @@ Each term of the action can be interpreted again as an area of a triangle where
In the general case there does not seem to be any possible way of computing the action~\eqref{eq:action_with_imaginary_part} in term of the global data.
Most probably the value of the action is larger than in the holomorphic case since the string is no longer confined to a plane.
Given the nature of the rotation its worldsheet has to bend in order to be attached to the D-brane as pictorially shown in~\Cref{fig:brane3d} in the case of a $3$-dimensional space.
The general case we considered then differs from the known factorized case by an additional contribution in the on-shell action which can be intuitively understood as a small ``bump'' of the string worldsheet in proximity of the boundary.
The general case we considered then differs from the known factorised case by an additional contribution in the on-shell action which can be intuitively understood as a small ``bump'' of the string worldsheet in proximity of the boundary.
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