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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-10-20 19:29:13 +02:00
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@@ -9,13 +9,13 @@ The fermion--boson couplings and the study of flavour changing neutral currents~
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}.
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~\cite{Sciuto:1969:GeneralVertexFunction,DellaSelva:1970:SimpleExpressionSciuto} up to modern interpretations of string theory.
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,Bianchi:2005:OpenStoryMagnetic,Pesando:2010:OpenClosedString,Forste:2018:YukawaCouplingsMagnetized}, strings in gravitational wave background~\cite{Kiritsis:1994:StringPropagationGravitational,DAppollonio:2003:StringInteractionsGravitational}, bound states of D-branes~\cite{Gava:1997:BoundStatesBranes,Duo:2007:NewTwistField} and tachyon condensation in Superstring Field Theory~\cite{David:2000:TachyonCondensationD0,David:2001:TachyonCondensationUsing,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,Anastasopoulos:2012:LightStringyStates,Anastasopoulos:2013:ThreeFourpointCorrelators}.
Results were however found starting from dual models~\cite{Sciuto:1969:GeneralVertexFunction} 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 relative \SU{2} rotations of the D-branes ~\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} 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.
@@ -1128,7 +1128,7 @@ Using the \rP symbol the solutions can be symbolically written as
The normalisation parameters $K$ cannot however be guessed from the \rP symbol.
As we are interested in finding the truly independent solutions to the original problem, we can use properties of the hypergeometric functions to reduce the number of possible choices of the integer factors in the definition of the parameters.
It is possible to show that any hypergeometric function $\hyp{a + \ffa}{b + \ffb}{c + \ffc}{z}$ can be written as a combination of \hyp{a}{b}{c}{z} and any of its contiguous functions~\cite{::NISTDigitalLibrary}.
It is possible to show that any hypergeometric function $\hyp{a + \ffa}{b + \ffb}{c + \ffc}{z}$ can be written as a combination of \hyp{a}{b}{c}{z} and any of its contiguous functions~\cite{Olver:2020:NISTDigitalLibrary}.
For instance we can choose:
\begin{equation}
\hyp{a + \ffa}{b + \ffb}{c + \ffc}{z}