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Up to NS fermions

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
This commit is contained in:
Riccardo
2020-09-25 17:28:21 +02:00
parent e363da4b90
commit 5f7a0f734f
9 changed files with 984 additions and 240 deletions
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20
sec/app/isomorphism.tex
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@@ -33,7 +33,7 @@ The parametrisation is such that:
U(\widehat{\vb{n}})
\label{eq:U_props}
\end{eqnarray}
where $\sigma^2$ is the second Pauli matrix, $\widetilde{\vb{n}} = \left( -n^1, n^2, -n^3 \right)$ and $\widehat{\vb{n}} = - \left(\frac{1}{2} -n \right)\, \frac{\vb{n}}{n}$.
where $\sigma^2$ is the second Pauli matrix, $\widetilde{\vb{n}} = \qty( -n^1, n^2, -n^3 )$ and $\widehat{\vb{n}} = - \qty(\frac{1}{2} -n )\, \frac{\vb{n}}{n}$.
The group product of two elements $U(\vb{n} \circ \vb{m} ) = U(\vb{n})\, U(\vb{m})$ has an explicit realisation as:
\begin{equation}
@@ -58,17 +58,17 @@ The group product of two elements $U(\vb{n} \circ \vb{m} ) = U(\vb{n})\, U(\vb{
Let $I = 1,\, 2,\, 3,\, 4$ and define:
\begin{equation}
\tau_I = \left( i\, \1_2,\, \vb{\sigma} \right),
\tau_I = \qty( i\, \1_2,\, \vb{\sigma} ),
\end{equation}
where $\vb{\sigma} = \left( \sigma^1,\, \sigma^2,\, \sigma^3 \right)$ are the Pauli matrices.
where $\vb{\sigma} = \qty( \sigma^1,\, \sigma^2,\, \sigma^3 )$ are the Pauli matrices.
It is possible to show that:
\begin{equation}
\begin{split}
\left( \tau_I \right)^{\dagger}
\qty( \tau_I )^{\dagger}
& =
\eta_{IJ}\, {\tau}^I,
\\
\left( \tau^I \right)^*
\qty( \tau^I )^*
& =
-\sigma_2\, \tau_I\, \sigma_2,
\end{split}
@@ -85,7 +85,7 @@ The following relations are then a natural consequence:
& = &
2\, \eta_{IJ},
\\
\tr(\tau_I \left( \tau_J \right)^{\dagger})
\tr(\tau_I \qty( \tau_J )^{\dagger})
& = &
2\, \delta_{IJ}.
\end{eqnarray}
@@ -99,7 +99,7 @@ We can recover the components using the previous properties:
X^I
=
\frac{1}{2}\, \delta^{IJ}\,
\tr(X_{(s)} \left( \tau_J \right)^{\dagger})
\tr(X_{(s)} \qty( \tau_J )^{\dagger})
=
\frac{1}{2}\, \eta^{IJ}\, \tr(X_{(s)} \tau_J),
\end{equation}
@@ -126,7 +126,7 @@ A rotation in spinor representation is defined as:
\end{equation}
and it is equivalent to:
\begin{equation}
\left( X' \right)^I
\qty( X' )^I
=
\tensor{R}{^I_J}\,
X^J
@@ -137,7 +137,7 @@ through
=
\frac{1}{2}
\tr(
\left( \tau_I \right)^{\dagger}\,
\qty( \tau_I )^{\dagger}\,
U_{L}(\vb{n})\,
\tau_J\,
U_{R}^{\dagger}(\vb{m})
@@ -167,7 +167,7 @@ From the second equation in \eqref{eq:tau_props} and the first equation in \eqre
\end{equation}
Furthermore the direct computation of the determinant of $R$ using the parametrisation~\eqref{eq:su2parametrisation} shows that $\det R = 1$.
Finally the explicit choice of the basis $\tau$ ensures $R$ to be a real matrix which ensures $R \in \SO{4}$.
Since $\left\lbrace U_{L},\, U_{R} \right\rbrace$ and $\left\lbrace -U_{L},\, -U_{R} \right\rbrace$ generate the same \SO{4} matrix then the correct isomorphism takes the form:
Since $\qty{ U_{L},\, U_{R} }$ and $\qty{ -U_{L},\, -U_{R} }$ generate the same \SO{4} matrix then the correct isomorphism takes the form:
\begin{equation}
\SO{4}
\cong

28
sec/app/parameters.tex
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@@ -15,9 +15,9 @@ In the main text we set
where $\cL(\vb{n}_{\vb{\infty}}) \in \SU{2}$.
The previous equation implies
\begin{equation}
\left( D\, \rM_{\vb{\infty}}\, D^{-1} \right)^\dagger
\qty( D\, \rM_{\vb{\infty}}\, D^{-1} )^\dagger
=
\left( D\, \rM_{\vb{\infty}}\, D^{-1} \right)^{-1},
\qty( D\, \rM_{\vb{\infty}}\, D^{-1} )^{-1},
\end{equation}
which can be rewritten as
\begin{equation}
@@ -147,7 +147,7 @@ which is satisfied by:
k'_{ab} \in \Z,
\end{split}
\end{equation}
where $p^{(L)},\, q^{(L)} \in \left\lbrace 0, 1 \right\rbrace$.
where $p^{(L)},\, q^{(L)} \in \qty{ 0, 1 }$.
Notice that changing the value of $p^{(L)}$ corresponds to swapping $a$ and $b$: since the hypergeometric function is symmetric in those parameters we can fix $p^{(L)}=0$.
Redefining $k'$ we can always set $q^{(L)}=0$.
We therefore have:
@@ -170,16 +170,16 @@ We cannot thus choose a vanishing $k_{\delta^{(L)}_{\vb{\infty}}}$ but we have t
We find a third relation by considering the entry
\begin{equation}
\Im\left(
\Im\qty(
e^{+2\pi i \delta_{\vb{\infty}}^{(L)}}\,
D^{(L)}\,
\rM_{\vb{\infty}}^{(L)}\,
\left( D^{(L)} \right)^{-1}
\right)_{11}
\qty( D^{(L)} )^{-1}
)_{11}
=
\Im\left(
\Im\qty(
\cL(n_{\vb{\infty}})
\right)_{11}.
)_{11}.
\end{equation}
Using
\begin{equation}
@@ -221,7 +221,7 @@ We then write
\qquad
k_{abc}\in \Z,
\end{equation}
with $f^{(L)} \in \left\lbrace 0, 1 \right\rbrace$.
with $f^{(L)} \in \qty{ 0, 1 }$.
The request
\begin{equation}
A
@@ -284,10 +284,10 @@ So far we can summarise the results in
$K^{(L)}$ is finally determined from
\begin{equation}
\left( D^{(L)}\, \rM_{\vb{\infty}}\, \left( D^{(L)} \right)^{-1} \right)_{21}
\qty( D^{(L)}\, \rM_{\vb{\infty}}\, \qty( D^{(L)} )^{-1} )_{21}
=
e^{-2\pi i \delta_{\vb{\infty}}^{(L)}}\,
\left( \cL(n_{\vb{\infty}}) \right)_{21},
\qty( \cL(n_{\vb{\infty}}) )_{21},
\label{eq:fixing_K_21}
\end{equation}
and get:
@@ -310,7 +310,7 @@ We check the consistency condition \eqref{eq:K_consistency_condition} using~\eqr
The result is
\begin{equation}
\begin{split}
\left( K^{(L)} \right)^{-1}
\qty( K^{(L)} )^{-1}
& =
\frac{(-1)^{m_a + m_b + m_c}}{2 \pi^2}\,
\cG(1 - a^{(L)},\, 1 - b^{(L)},\, 2 - c^{(L)})\,
@@ -341,10 +341,10 @@ We can then rewrite~\eqref{eq:cos_n1} as
It is then possible to verify that the sum of the left and right hand sides of~\eqref{eq:n12+n22} and the last equation are equal to $1$.
The same consistency check can also be performed by computing $K^{(L)}$ from
\begin{equation}
\left( D^{(L)}\, \rM_{\vb{\infty}}\, \left( D^{(L)} \right)^{-1} \right)_{12}
\qty( D^{(L)}\, \rM_{\vb{\infty}}\, \qty( D^{(L)} )^{-1} )_{12}
=
e^{-2\pi i \delta_{\vb{\infty}}^{(L)}}\,
\left( \cL(n_{\vb{\infty}}) \right)_{12},
\qty( \cL(n_{\vb{\infty}}) )_{12},
\end{equation}
instead of \eqref{eq:fixing_K_21}.

220
sec/part1/dbranes.tex
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@@ -32,10 +32,10 @@ Using the path integral approach we can in fact separate the classical contribut
\right\rangle
=
\cN
\left(
\left\lbrace x_{(t)},\, \rM_{(t)} \right\rbrace_{1 \le t \le N_B}
\right)\,
e^{-S_E\left( \left\lbrace x_{(t)}, \rM_{(t)} \right\rbrace_{1 \le t \le N_B} \right)},
\qty(
\qty{ x_{(t)},\, \rM_{(t)} }_{1 \le t \le N_B}
)\,
e^{-S_E\qty( \qty{ x_{(t)}, \rM_{(t)} }_{1 \le t \le N_B} )},
\end{equation}
where $\rM_{(t)}$ (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.
@@ -119,15 +119,15 @@ The rotation $R_{(t)}$ is actually defined in the Grassmannian:
\in
\mathrm{Gr}(2, 4)
=
\frac{\SO{4}}{\rS\left( \OO{2} \times \OO{2} \right)},
\frac{\SO{4}}{\rS\qty( \OO{2} \times \OO{2} )},
\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]
\qty[ R_{(t)} ]
=
\left\lbrace R_{(t)} \sim \cO_{(t)} R_{(t)} \right\rbrace,
\qty{ R_{(t)} \sim \cO_{(t)} R_{(t)} },
\end{equation}
where $\cO_{(t)} = \rS\left( \OO{2} \times \OO{2} \right)$ is defined as
where $\cO_{(t)} = \rS\qty( \OO{2} \times \OO{2} )$ is defined as
\begin{equation}
\cO_{(t)}
=
@@ -135,7 +135,7 @@ where $\cO_{(t)} = \rS\left( \OO{2} \times \OO{2} \right)$ is defined as
\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 $g_{(t)}^{\perp} \ge 0$.
Notice that the additional $\Z_2$ factor in $\rS\qty( \OO{2} \times \OO{2} )$ with respect to $\SO{2} \times \SO{2}$ can be used to set $g_{(t)}^{\perp} \ge 0$.
\subsubsection{Boundary Conditions for Branes at Angles}
@@ -150,7 +150,7 @@ We define the usual upper plane coordinates:
=
e^{\tau_E + i \sigma}
& \in &
\ccH \cup \left\lbrace z \in \C \mid \Im z = 0 \right\rbrace,
\ccH \cup \qty{ z \in \C \mid \Im z = 0 },
\\
\bu
=
@@ -158,20 +158,20 @@ We define the usual upper plane coordinates:
=
e^{\tau_E - i \sigma}
& \in &
\overline{\ccH} \cup \left\lbrace z \in \C \mid \Im z = 0 \right\rbrace,
\overline{\ccH} \cup \qty{ z \in \C \mid \Im z = 0 },
\end{eqnarray}
where $\ccH = \left\lbrace z \in \C \mid \Im z > 0 \right\rbrace$ is the upper complex plane and $\overline{\ccH} = \left\lbrace z \in \C \mid \Im z < 0 \right\rbrace$ is the lower complex plane.
where $\ccH = \qty{ z \in \C \mid \Im z > 0 }$ is the upper complex plane and $\overline{\ccH} = \qty{ z \in \C \mid \Im z < 0 }$ is the lower complex plane.
In conformal coordinates $u$ and $\bu$, D-branes at $\sigma = 0$ and $\sigma = \pi$ are mapped onto the real axis $\Im z = 0$.
We use the symbol $D_{(t)}$ to label both the brane and the interval representing it on the real axis of the upper half plane:
\begin{equation}
D_{(t)} = \left[ x_{(t)}, x_{(t-1)} \right],
D_{(t)} = \qty[ x_{(t)}, x_{(t-1)} ],
\qquad
t = 2,\, 3,\, \dots,\, N_B,
\qquad
x_{(t)} < x_{(t-1)}.
\end{equation}
The points $x_{(t)}$ and $x_{(t-1)}$ represent the worldsheet intersection points of the brane $D_{(t)}$ with the branes $D_{(t+1)}$ and $D_{(t-1)}$ respectively.
The choice of the intervals must be carefully considered: since the D-branes are defined modulo $N_B$, the shorthand for the interval $D_{(1)} = \left[ x_{(1)}, x_{(N_B)} \right]$ should actually be:
The choice of the intervals must be carefully considered: since the D-branes are defined modulo $N_B$, the shorthand for the interval $D_{(1)} = \qty[ x_{(1)}, x_{(N_B)} ]$ should actually be:
\begin{equation}
D_{(1)}
=
@@ -193,11 +193,11 @@ In the global coordinates system associated to the subspace $\R^4 \subset \ccM^{
\frac{1}{4 \pi \ap}
\iint\limits_{\R \times \R^+}
\dd{x}\dd{y}\,
\left(
\qty(
\ipd{x} X^I\, \ipd{x} X^J
+
\ipd{y} X^I\, \ipd{y} X^J
\right)\,
)\,
\eta_{IJ},
\end{split}
\label{eq:string_action}
@@ -208,7 +208,7 @@ The \eom in these coordinates are:
\ipd{u} \ipd{\bu} X^I( u, \bu )
=
\frac{1}{4}
\left( \ipd{x}^2 + \ipd{y}^2 \right) X^I( x+iy, x-iy )
\qty( \ipd{x}^2 + \ipd{y}^2 ) X^I( x+iy, x-iy )
=
0.
\label{eq:string_equation_of_motion}
@@ -235,29 +235,29 @@ In the well adapted frame~\eqref{eq:well-adapt-embed} we describe an open string
m = 3,\, 4,
\label{eq:dirichlet_bc}
\end{eqnarray}
where $x \in D_{(t)} = \left[ x_t, x_{t-1} \right]$ and the index $i$ labels the Neumann boundary conditions while $m$ labels the Dirichlet coordinates associated to the direction orthogonal to the D-branes.
where $x \in D_{(t)} = \qty[ x_t, x_{t-1} ]$ and the index $i$ labels the Neumann boundary conditions while $m$ labels the Dirichlet coordinates associated to the direction orthogonal to the D-branes.
As the presence of $g_{(t)}^m$ in \eqref{eq:brane_rotation} and \eqref{eq:dirichlet_bc} may complicate the analysis, we consider the derivative along the boundary direction of \eqref{eq:dirichlet_bc} to remove the dependence on the translation vector.
This procedure produces simpler boundary conditions which are nevertheless not equivalent to the original~\eqref{eq:neumann_bc} and \eqref{eq:dirichlet_bc}: they will be recovered later by adding further constraints.
The simpler boundary conditions we consider in the global coordinates are:
\begin{eqnarray}
\tensor{\left( R_{(t)} \right)}{^i_J}
\tensor{\qty( R_{(t)} )}{^i_J}
\eval{\ipd{\sigma} X^J( \tau, \sigma )}_{\sigma = 0}
& = &
i\, \tensor{\left( R_{(t)} \right)}{^i_J}
\left(
i\, \tensor{\qty( R_{(t)} )}{^i_J}
\qty(
\ipd{u} X^J( x + i\, 0^+ ) - \ipd{\bu} \bX^J( x - i\, 0^+ )
\right)
)
=
0,
\\
\tensor{\left( R_{(t)} \right)}{^m_J}
\tensor{\qty( R_{(t)} )}{^m_J}
\eval{\ipd{\tau} X^J( \tau, \sigma )}_{\sigma = 0}
& = &
i\, \tensor{\left( R_{(t)} \right)}{^m_J}
\left(
i\, \tensor{\qty( R_{(t)} )}{^m_J}
\qty(
\ipd{u} X^J( x + i\, 0^+ ) + \ipd{\bu} \bX^J( x - i\, 0^+ )
\right)
)
=
0,
\end{eqnarray}
@@ -268,7 +268,7 @@ With the introduction of the target space embedding of the worldsheet interactio
\begin{cases}
\ipd{u} X^I( x + i\, 0^+ )
& =
\tensor{\left( U_{(t)} \right)}{^I_J}
\tensor{\qty( U_{(t)} )}{^I_J}
\ipd{\bu} \bX^J( x - i\, 0^+ ),
\qquad
x \in D_{(t)}
@@ -283,11 +283,11 @@ In the last expression we introduced the matrix
\begin{equation}
U_{(t)}
=
\left( R_{(t)} \right)^{-1}\,
\qty( R_{(t)} )^{-1}\,
\cS\,
R_{(t)}
\in
\frac{\SO{4}}{\rS\left( \OO{2} \times \OO{2} \right)},
\frac{\SO{4}}{\rS\qty( \OO{2} \times \OO{2} )},
\label{eq:Umatrices}
\end{equation}
where
@@ -298,7 +298,7 @@ where
\label{eq:reflection_S}
\end{equation}
embeds the difference between Neumann and Dirichlet conditions.
Given its definition $U_{(t)}$ is such that $U_{(t)} = \left( U_{(t)} \right)^{-1} = \left( U_{(t)} \right)^T$.
Given its definition $U_{(t)}$ is such that $U_{(t)} = \qty( U_{(t)} )^{-1} = \qty( U_{(t)} )^T$.
The target space vector $f_{(t)}$ recovers the apparent loss of information suffered when losing $g_{(t)}$.
Consider for instance the embedding equations~\eqref{eq:dirichlet_bc} for any two intersecting D-branes $D_{(t)}$ and $D_{(t+1)}$.
@@ -324,7 +324,7 @@ we can compute the intersection point as:
\begin{equation}
f_{(t)}
=
\left( \cR_{(t,\, t+1)} \right)^{-1}\,
\qty( \cR_{(t,\, t+1)} )^{-1}\,
\cG_{(t,\, t+1)}.
\end{equation}
Information on $g_{(t)}$ is thus recovered through the global boundary conditions in the second equation in \eqref{eq:discontinuity_bc}.
@@ -363,7 +363,7 @@ The boundary conditions in terms of the doubling field are:
\ipd{z} \cX( x_t + \eta - i\, 0^+ ),
\label{eq:bottom_monodromy}
\end{eqnarray}
for $0 < \eta < \min\left( \abs{x_{(t-1)} - x_{(t)}}, \abs{x_{(t)} - x_{(t+1)}} \right)$ in order to consider only the two adjacent D-branes $D_{(t)}$ and $D_{(t+1)}$.
for $0 < \eta < \min\qty( \abs{x_{(t-1)} - x_{(t)}}, \abs{x_{(t)} - x_{(t+1)}} )$ in order to consider only the two adjacent D-branes $D_{(t)}$ and $D_{(t+1)}$.
Matrices $\cU_{(t,\, t+1)}$ and $\widetilde{\cU}_{(t,\, t+1)}$ are the non trivial monodromies arising from the rotation of the D-branes.
Since the relative rotations between consecutive D-branes are non Abelian, for each interaction point there are two monodromies \cU and $\widetilde{\cU}$ depending on the location of the base point of the closed loop: one for paths starting in the upper plane \ccH and one for paths starting in $\overline{\ccH}$.
@@ -451,7 +451,7 @@ We define the spinor representation of $X$ as:
\begin{equation}
X_{(s)}( u, \bu ) = X^I( u, \bu )\, \tau_I,
\end{equation}
where $\tau = \left( i\, \1_2,\, \vb{\sigma} \right)$ and $\vb{\sigma}$ is the vector of the Pauli matrices.
where $\tau = \qty( i\, \1_2,\, \vb{\sigma} )$ and $\vb{\sigma}$ is the vector of the Pauli matrices.
Consider then:
\begin{equation}
\ipd{z} \cX_{(s)}( z )
@@ -470,7 +470,7 @@ Consider then:
\end{equation}
As in the real representation the discontinuities on the D-branes can be cast into monodromy factors with respect to the D-brane $D_{(\bt)}$. Branch cut structure and considerations on the homotopy group are left unchanged as long as we consider both left and right sectors of $\SU{2}_L \times \SU{2}_R$ at the same time.
Let $0 < \eta < \min\left( \abs{x_{(t)} - x_{(t-1)}}, \abs{x_{(t+1)} - x_{(t)}} \right)$.
Let $0 < \eta < \min\qty( \abs{x_{(t)} - x_{(t-1)}}, \abs{x_{(t+1)} - x_{(t)}} )$.
We find:
\begin{eqnarray}
\ipd{z} \cX_{(s)}( x_t + e^{2 \pi i}( \eta + i\, 0^+) )
@@ -749,7 +749,7 @@ We write any possible solution in a factorised form as
(-\omega_z)^{A_{lr}}\,
(1-\omega_z)^{B_{lr}}\,
\cB_{\vb{0},\, l}^{(L)}(\omega_z)
\left( \cB_{\vb{0},\, r}^{(R)}(\omega_z) \right)^T,
\qty( \cB_{\vb{0},\, r}^{(R)}(\omega_z) )^T,
\label{eq:formal_solution_lr}
\end{equation}
where $l$ and $r$ label the parameters associates with the left and right sectors of the hypergeometric function.
@@ -790,14 +790,14 @@ We impose:
&&\begin{cases}
D^{(L)}\,
\rM_{\vb{0}}^{(L)}\,
\left( D^{(L)} \right)^{-1}
\qty( D^{(L)} )^{-1}
=
e^{-2\pi i \delta_{\vb{0}}^{(L)}}\,
\cL(\vb{n}_{\vb{0}})
\\
D^{(R)}\,
\rM_{\vb{0}}^{(R)}\,
\left( D^{(R)} \right)^{-1}
\qty( D^{(R)} )^{-1}
=
e^{-2\pi i \delta_{\vb{0}}^{(R)}}\,
\cR^*(\vb{m}_{\vb{0}})
@@ -815,14 +815,14 @@ We impose:
&&\begin{cases}
D^{(L)},
\rM_{\vb{\infty}}^{(L)}\,
\left( D^{(L)} \right)^{-1}
\qty( D^{(L)} )^{-1}
=
e^{-2\pi i \delta_{\vb{\infty}}^{(L)}}\,
\cL(\vb{n}_{\vb{\infty}})
\\
D^{(R)}\,
\rM_{\vb{\infty}}^{(R)}\,
\left( D^{(R)} \right)^{-1}
\qty( D^{(R)} )^{-1}
=
e^{-2\pi i \delta_{\vb{\infty}}^{(R)}}\,
\cR^*(\vb{m}_{\vb{\infty}})
@@ -964,7 +964,7 @@ We find:
\frac{n^1_{\vb{\infty}}+ i\, n^2_{\vb{\infty}}}{n_{\vb{\infty}}},
\label{eq:K_factor_value}
\end{eqnarray}
where $f^{(L)} \in \left\lbrace 0,\, 1 \right\rbrace$.
where $f^{(L)} \in \qty{ 0,\, 1 }$.
For the sake of brevity we defined two auxiliary functions, namely $\cG(a,\, b,\, c) = \gfun{1-a}\, \gfun{1-b}\, \gfun{a+1-c}\, \gfun{b+1-c}$ and $\cF(a,\, b,\, c) = \sin(\pi c)\, \sin(\pi(a-b))$.
We also introduced the norm $n_{\vb{1}} = \norm{\vb{n}_{\vb{1}}}$ of the rotation vector around $\omega_{\bt+1} = 1$.
Its dependence on the other parameters is encoded in~\eqref{eq:monodromy_relations}, where $\rM^+_{\vb{1}} = \rM^{-1}_{\vb{0}}\, \rM^{-1}_{\vb{\infty}}$, and the composition rule~\eqref{eq:product_in_SU2}:
@@ -1004,28 +1004,28 @@ We can use properties of the hypergeometric functions to show that any choice do
Specifically we can start with certain values but we can recover the others through:
\begin{equation}
\rP
\left\lbrace
\qty{
\mqty{
0 & 1 & \infty & \\
0 & 0 & a & z \\
1-c & c-a-b & b &
}
\right\rbrace
}
=
(1-z)^{c-a-b}\,
\rP
\left\lbrace
\qty{
\mqty{
0 & 1 & \infty & \\
0 & 0 & c-b & z \\
1-c & a+b-c & c-a &
}
\right\rbrace,
},
\end{equation}
where \rP is the Papperitz-Riemann symbol for the hypergeometric functions.
We can then assign any admissible value to $f^{(L)}$ and $f^{(R)}$ and then recover the other through the identification:
\begin{eqnarray}
f^{(L)}{}' & = & \left( 1 + f^{(L)} \right)~\text{mod}~2,
f^{(L)}{}' & = & \qty( 1 + f^{(L)} )~\text{mod}~2,
\\
\ffa_l' & = & \ffc_l - \ffb_l,
\\
@@ -1073,7 +1073,7 @@ Using the \rP symbol the solutions can be symbolically written as
\\
&
\times
\rP \left\lbrace
\rP \qty{
\mqty{
0
&
@@ -1097,11 +1097,11 @@ Using the \rP symbol the solutions can be symbolically written as
-n_{\vb{\infty}} + \ffb^{(L)}
&
}
\right\rbrace
}
\\
&
\times
\rP \left\lbrace
\rP \qty{
\mqty{
0
&
@@ -1125,7 +1125,7 @@ Using the \rP symbol the solutions can be symbolically written as
-m_{\vb{\infty}} + \ffb^{(R)}
&
}
\right\rbrace.
}.
\end{split}
\label{eq:symbolic_solutions_using_P}
\end{equation}
@@ -1226,9 +1226,9 @@ In fact we require the finiteness of the Euclidean action~\eqref{eq:action_doubl
In principle it could appear obvious to use~\eqref{eq:contiguous_functions} to restrict the possible arbitrary integers to:
\begin{eqnarray}
\ffa^{(L)} \in \left\lbrace -1,\, 0 \right\rbrace,
\ffa^{(L)} \in \qty{ -1,\, 0 },
& \qquad &
\ffa^{(R)} \in \left\lbrace -1,\, 0 \right\rbrace,
\ffa^{(R)} \in \qty{ -1,\, 0 },
\\
\ffb^{(L)} = 0,
& \qquad &
@@ -1248,15 +1248,15 @@ We could then use~\eqref{eq:reduction_F_F+} to write the possible solution as
(1-\omega_z)^{n_{\vb{1}} + m_{\vb{1}}}
\\
& \times
\sum\limits_{\ffa^{(L,\,R)} \in \left\lbrace -1, 0 \right\rbrace}
\sum\limits_{\ffa^{(L,\,R)} \in \qty{ -1, 0 }}
h(\omega_z,\, \ffa^{(L,R)})
\times
\\
& \times
\cB_{\vb{0}}^{(L)}(a^{(L)} + \ffa^{(L)},\, b,\, c;\, \omega_z)
\left(
\qty(
\cB_{\vb{0}}^{(R)}(a^{(R)} + \ffa^{(R)},\, b,\, c;\, \omega_z)
\right)^T.
)^T.
\end{split}
\label{eq:doubling_field_expansion}
\end{equation}
@@ -1285,10 +1285,10 @@ It can be verified that the convergence of the action both at finite and infinit
\ipd{u'} \cX_{(s)}(u')
+
U_L^{\dagger}(\vb{n}_{{\bt}})
\left[
\qty[
\finiteint{\bu'}{x_{(\bt-1)}}{\bu}
\ipd{\bu'} \cX_{(s)}(\bu')
\right]
]
U_R(\vb{m}_{{\bt}}),
\label{eq:classical_solution}
\end{equation}
@@ -1319,7 +1319,7 @@ In this case~\eqref{eq:symbolic_solutions_using_P} becomes
\begin{equation}
(-\omega)^\ffA\,
(1-\omega)^\ffB\,
\rP\left\lbrace
\rP\qty{
\mqty{
0
&
@@ -1343,11 +1343,11 @@ In this case~\eqref{eq:symbolic_solutions_using_P} becomes
-n_{\vb{\infty}} + \ffb^{(L)}
&
}
\right\rbrace.
}.
\end{equation}
The only possible solution compatible with~\eqref{eq:constraints_finite_X} is
\begin{equation}
\rP\left\lbrace
\rP\qty{
\mqty{
0
&
@@ -1371,7 +1371,7 @@ The only possible solution compatible with~\eqref{eq:constraints_finite_X} is
-n_{\vb{\infty}} + 2
&
}
\right\rbrace,
},
\label{eq:X_solution_pure_L}
\end{equation}
that is $\ffa^{(L)} = -1$, $\ffb^{(L)} = 0$, $\ffc^{(L)} = 0$, $\ffA = -1$ and $\ffB = -1$.
@@ -1381,7 +1381,7 @@ For each possible case the solution is however unique and it is given by
\begin{enumerate}
\item $n_{\vb{0}} > m_{\vb{0}}$ and $n_{\vb{1}} > m_{\vb{1}}$:
\begin{equation}
\rP\left\lbrace
\rP\qty{
\mqty{
0
&
@@ -1405,8 +1405,8 @@ For each possible case the solution is however unique and it is given by
-n_{\vb{\infty}} + 2
&
}
\right\rbrace
\rP\left\lbrace
}
\rP\qty{
\mqty{
0
&
@@ -1430,13 +1430,13 @@ For each possible case the solution is however unique and it is given by
-m_{\vb{\infty}} + 1
&
}
\right\rbrace,
},
\label{eq:X_solution>>}
\end{equation}
\item $n_{\vb{0}} > m_{\vb{0}}$, $n_{\vb{1}} < m_{\vb{1}}$ and $n_{\vb{\infty}} > m_{\vb{\infty}}$:
\begin{equation}
\rP\left\lbrace
\rP\qty{
\mqty{
0
&
@@ -1460,8 +1460,8 @@ For each possible case the solution is however unique and it is given by
-n_{\vb{\infty}} + 2
&
}
\right\rbrace
\rP\left\lbrace
}
\rP\qty{
\mqty{
0
&
@@ -1485,13 +1485,13 @@ For each possible case the solution is however unique and it is given by
-m_{\vb{\infty}} + 1
&
}
\right\rbrace,
},
\label{eq:X_solution><>}
\end{equation}
\item $n_{\vb{0}} > m_{\vb{0}}$, $n_{\vb{1}} < m_{\vb{1}}$ and $n_{\vb{\infty}} < m_{\vb{\infty}}$:
\begin{equation}
\rP\left\lbrace
\rP\qty{
\mqty{
0
&
@@ -1515,8 +1515,8 @@ For each possible case the solution is however unique and it is given by
-n_{\vb{\infty}} + 1
&
}
\right\rbrace
\rP\left\lbrace
}
\rP\qty{
\mqty{
0
&
@@ -1540,13 +1540,13 @@ For each possible case the solution is however unique and it is given by
-m_{\vb{\infty}} + 2
&
}
\right\rbrace,
},
\label{eq:X_solution><<}
\end{equation}
\item $n_{\vb{0}} < m_{\vb{0}}$, $n_{\vb{1}} > m_{\vb{1}}$ and $n_{\vb{\infty}} > m_{\vb{\infty}}$:
\begin{equation}
\rP\left\lbrace
\rP\qty{
\mqty{
0
&
@@ -1570,8 +1570,8 @@ For each possible case the solution is however unique and it is given by
-n_{\vb{\infty}} + 2
&
}
\right\rbrace
\rP\left\lbrace
}
\rP\qty{
\mqty{
0
&
@@ -1595,13 +1595,13 @@ For each possible case the solution is however unique and it is given by
-m_{\vb{\infty}} + 1
&
}
\right\rbrace,
},
\label{eq:X_solution<>>}
\end{equation}
\item $n_{\vb{0}} < m_{\vb{0}}$, $n_{\vb{1}} > m_{\vb{1}}$ and $n_{\vb{\infty}} < m_{\vb{\infty}}$:
\begin{equation}
\rP\left\lbrace
\rP\qty{
\mqty{
0
&
@@ -1625,8 +1625,8 @@ For each possible case the solution is however unique and it is given by
-n_{\vb{\infty}} + 1
&
}
\right\rbrace
\rP\left\lbrace
}
\rP\qty{
\mqty{
0
&
@@ -1650,13 +1650,13 @@ For each possible case the solution is however unique and it is given by
-m_{\vb{\infty}} + 2
&
}
\right\rbrace,
},
\label{eq:X_solution<><}
\end{equation}
\item $n_{\vb{0}} < m_{\vb{0}}$, $n_{\vb{1}} < m_{\vb{1}}$:
\begin{equation}
\rP\left\lbrace
\rP\qty{
\mqty{
0
&
@@ -1680,8 +1680,8 @@ For each possible case the solution is however unique and it is given by
-n_{\vb{\infty}} + 1
&
}
\right\rbrace
\rP\left\lbrace
}
\rP\qty{
\mqty{
0
&
@@ -1705,7 +1705,7 @@ For each possible case the solution is however unique and it is given by
-m_{\vb{\infty}} + 2
&
}
\right\rbrace.
}.
\label{eq:X_solution<<}
\end{equation}
\end{enumerate}
@@ -1737,7 +1737,7 @@ In the previous section we produced one solution for each ordering of the $n_{\o
There are however other solutions connected to the $\Z_2$ equivalence class in the isomorphism between \SO{4} its double cover.
Given a solution $(\vb{n}_{\vb{0}},\, \vb{n}_{\vb{1}},\, \vb{n}_{\vb{\infty}}) \oplus (\vb{m}_{\vb{0}},\, \vb{m}_{\vb{1}},\, \vb{m}_{\vb{\infty}})$, we can in fact replace any couple of $\vb{n}$ and $\vb{m}$ by $\widehat{\vb{n}}$ and $\widehat{\vb{m}}$ and get an apparently new solution.\footnotemark{}
\footnotetext{%
We need to change two rotation vectors because the monodromies are constrained by \eqref{eq:monodromy_relations}.
We need to change two rotation vectors because the monodromies are constrained by~\eqref{eq:monodromy_relations}.
}
For instance we could consider $(\widehat{\vb{n}}_{\vb{0}},\, \widehat{\vb{n}}_{\vb{1}},\, \vb{n}_{\vb{\infty}}) \oplus (\vb{m}_{\vb{0}},\, \widehat{\vb{m}}_{\vb{1}},\, \widehat{\vb{m}}_{\vb{\infty}})$.
On the other hand the previous substitution would change the \SO{4} in both $\omega = 0$ and $\omega = \infty$: it does not represent a new solution.
@@ -2041,9 +2041,9 @@ Explicitly we impose the four real equations in spinorial formalism
\ipd{\omega} \cX(\omega)
+
U_L^{\dagger}(\vb{n}_{{\bt}})
\left[
\qty[
\finiteint{\bomega}{0}{1} \ipd{\bomega} \cX(\bomega)
\right]
]
U_R(\vb{m}_{{\bt}})
=
f_{{\bt+1}\, (s)} - f_{{\bt-1}\, (s)},
@@ -2181,9 +2181,9 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h
\centering
\begin{tabular}{@{}ccc@{}}
\toprule
$\left( \ffa^{(L)},\, \ffb^{(L)},\, \ffc^{(L)} \right)$ &
$\qty( \ffa^{(L)},\, \ffb^{(L)},\, \ffc^{(L)} )$ &
$n_{\vb{1}}$ &
$\left( \cB^{(L)}( z ) \right)^T$
$\qty( \cB^{(L)}( z ) )^T$
\\
\midrule
\multirow{4}{*}{$(-1,\, 0,\, 0)$} &
@@ -2191,7 +2191,7 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h
$\mqty( (1 - z)^{-2\, n_{\vb{\infty}} - 2\, n_{\vb{0}} + 1} & 0 )$
\\
&
$1 - \left( n_{\vb{0}} + n_{\vb{\infty}} \right)$ &
$1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ &
$\mqty( 1 & 0 )$
\\
&
@@ -2212,7 +2212,7 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h
& 0 )$
\\
&
$1 - \left( n_{\vb{0}} + n_{\vb{\infty}} \right)$ &
$1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ &
$\mqty( 1 & 0 )$
\\
&
@@ -2229,7 +2229,7 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h
$\mqty( (1-z)^{-2\, n_{\vb{\infty}} - 2\, n_{\vb{0}}} & 0 )$
\\
&
$1 - \left( n_{\vb{0}} + n_{\vb{\infty}} \right)$ &
$1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ &
$\mqty( 0 & (1-z)^{2\, n_{\vb{\infty}} + 2\, n_{\vb{0}} - 2}\, (-z)^{1 - 2\, n_{\vb{0}}} )$
\\
&
@@ -2246,7 +2246,7 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h
$\mqty( (1-z)^{-2\, n_{\vb{\infty}} - 2\, n_{\vb{0}}} + 1 & 0 )$
\\
&
$1 - \left( n_{\vb{0}} + n_{\vb{\infty}} \right)$ &
$1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ &
$\mqty( 1 & 0 )$
\\
&
@@ -2263,7 +2263,7 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h
$\mqty( 0 & (-z)^{-2\, n_{\vb{0}}} )$
\\
&
$1 - \left( n_{\vb{0}} + n_{\vb{\infty}} \right)$ &
$1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ &
$\mqty( 0 & (1-z)^{2\, n_{\vb{\infty}} + 2\, n_{\vb{0}} - 1}\, (-z)^{-2\, n_{\vb{0}}} )$
\\
&
@@ -2280,7 +2280,7 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h
$\mqty( (1-z)^{-2\, n_{\vb{\infty}} -2\, n_{\vb{0}}} & 0 )$
\\
&
$1 - \left( n_{\vb{0}} + n_{\vb{\infty}} \right)$ &
$1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ &
$\mqty( 0 & (1-z)^{2\, n_{\vb{\infty}} + 2\, n_{\vb{0}} - 2}\, (-z)^{-2\, n_{\vb{0}}} )$
\\
&
@@ -2305,7 +2305,7 @@ This is the first specific case shown in~\Cref{sec:true_basis}.
In this scenario the left solution $\cB^{(L)}$ is always the same and matches the previous computation, however the right sector seems to give different solutions when different Abelian limits are taken.
Studying all possible solutions we find that all of them give the same answer in the limit $m_{\vb{t}} \to 0$, i.e.\ both $\cB^{(R)} = \mqty(1 & 0)^T$ and $\cB^{(R)} = \mqty(0 & 1)^T$.\footnotemark{}
\footnotetext{%
We write ``possible solutions'' because $m_{\vb{1}} = 1 - \left( m_{\vb{0}} + m_{\vb{\infty}} \right)$ is not.
We write ``possible solutions'' because $m_{\vb{1}} = 1 - \qty( m_{\vb{0}} + m_{\vb{\infty}} )$ is not.
}
The difference is the solution obtained from $n_{\vb{0}} > m_{\vb{0}}$, $n_{\vb{1}} > m_{\vb{1}}$ and $n_{\vb{\infty}} > m_{\vb{\infty}}$ or $n_{\vb{0}} > m_{\vb{0}}$, $\hat{n}_{\vb{1}} < \hat{m}_{\vb{1}}$ and $\hat{n}_{\vb{\infty}} < \hat{m}_{\vb{\infty}}$.
In any case the solution is factorised in the form $\cB^{(L)}(z) \otimes \mqty(C & C')^T$ which is expected since the right sector plays no role.
@@ -2337,7 +2337,7 @@ Comparing with the case $m = 0$ given in~\eqref{eq:Abelian_vs_n_simple_case}, we
\subsubsection{Recovering the Abelian Result: an Example}
To show that the construction of the Abelian limit is indeed working, we consider the first case in~\Cref{sec:true_basis} with $n_{\vb{1}} = 1 - \left( n_{\vb{0}} + n_{\vb{\infty}} \right)$ and $m_{\vb{1}} = -m_{\vb{0}} + m_{\vb{\infty}}$.
To show that the construction of the Abelian limit is indeed working, we consider the first case in~\Cref{sec:true_basis} with $n_{\vb{1}} = 1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ and $m_{\vb{1}} = -m_{\vb{0}} + m_{\vb{\infty}}$.
This leads to two independent rational functions of $\omega_z$:
\begin{equation}
\begin{split}
@@ -2393,7 +2393,7 @@ such that $\sum\limits_{t} \varepsilon_{\vb{t}} = 1$, and
\label{eq:Abelian_rotation_second}
\end{equation}
where $\sum\limits_{t} \varphi_{\vb{t}} = 2$, in order to approach the usual notation in the literature.
As usual we have $\ipd{\omega_z} \cZ^1( \omega_z ) \neq \left[ \ipd{\omega_z} \overline{\cZ}^1( \omega_z ) \right]^*$.
As usual we have $\ipd{\omega_z} \cZ^1( \omega_z ) \neq \qty[ \ipd{\omega_z} \overline{\cZ}^1( \omega_z ) ]^*$.
We can now build the Abelian solution to show the analytical structure of the limit.
We have
@@ -2415,7 +2415,7 @@ We have
where we chose $R_{(\bt)} = \1_4$ so that $U_{(\bt)}$ in~\eqref{eq:Umatrices} is mapped to $(i \sigma_1, i \sigma_1) \in \SU{2} \times \SU{2}$.
Notice however that $\vb{n}_{\vb{t}} = n_{\vb{t}}^3\, \vb{k}$ implies that $v^3_{(t)} = 0$ in~\eqref{eq:special_UL_brane_t}.
Hence $U_L$ and $U_R$ are always off diagonal and their action on~\eqref{eq:Abelian_sol_example} is to fill the first column.
From the previous relations we can also recover the usual holomorphicity $\overline{Z}^1(\bu) = \left[ Z^1(u) \right]^*$ of the sector with $\sum\limits_t \varepsilon_{\vb{t}} = 1$ and $\overline{Z}^2(\bu) = \left[ Z^2(u) \right]^*$ of the sector with $\sum\limits_t \varphi_{\vb{t}} = 2$.
From the previous relations we can also recover the usual holomorphicity $\overline{Z}^1(\bu) = \qty[ Z^1(u) ]^*$ of the sector with $\sum\limits_t \varepsilon_{\vb{t}} = 1$ and $\overline{Z}^2(\bu) = \qty[ Z^2(u) ]^*$ of the sector with $\sum\limits_t \varphi_{\vb{t}} = 2$.
\subsubsection{Abelian Limits}
@@ -2445,14 +2445,14 @@ We get:
\sum\limits_{m \in \{3, 4\}}
g_{(t)\, m}\,
\finiteint{x}{x_{(t)}}{x_{(t-1)}}
\tensor{\left( R_{(t)} \right)}{_{mI}}
\eval{\left( X_L'(x) - X_R'(x) \right)^I}_{y=0^+},
\tensor{\qty( R_{(t)} )}{_{mI}}
\eval{\qty( X_L'(x) - X_R'(x) )^I}_{y=0^+},
\label{eq:area_tmp}
\end{equation}
where indices $I = 1,\, 2,\, 3,\, 4$ are summed over and $m = 3,\, 4$ are the transverse directions in the well adapted frame with respect to the D-brane.
As the number of D-branes is defined modulo $N_B = 3$, $D_{(1)}$ is split on two separate intervals:
\begin{equation}
\left[ x_{(1)},\, x_{(3)} \right]
\qty[ x_{(1)},\, x_{(3)} ]
=
\left[ x_{(1)},\, +\infty \right)
\cup
@@ -2488,7 +2488,7 @@ Now~\eqref{eq:area_tmp} becomes:
\sum\limits_{m \in \{3, 4\}}
\eval{%
g_{(t)\, m}\,
\tensor{\Im \left( R_{(t)} \right)}{_{mI}}
\tensor{\Im \qty( R_{(t)} )}{_{mI}}
X_L^I(x+i0^+)
}^{x = x_{(t-1)}}_{x = x_{(t)}}
\\
@@ -2501,9 +2501,9 @@ Now~\eqref{eq:area_tmp} becomes:
\end{split}
\label{eq:action_with_imaginary_part}
\end{equation}
where $g^{(\perp)}_{(t)\, I} = \sum_{m \in \{3, 4\}} \tensor{\left( R_{(t)}^{-1} \right)}{_{mI}}\, g_{(t)\, m}$ is the transverse shift of $D_{(t)}$ in the global coordinates of the target space:
where $g^{(\perp)}_{(t)\, I} = \sum_{m \in \{3, 4\}} \tensor{\qty( R_{(t)}^{-1} )}{_{mI}}\, g_{(t)\, m}$ is the transverse shift of $D_{(t)}$ in the global coordinates of the target space:
\begin{equation}
g^{(\perp)}_{(t)\, I}\, (f_{(t-1)} - f_{(t)})^I = 0.
g^{(\perp)}_{(t)\, I}\, \qty(f_{(t-1)} - f_{(t)})^I = 0.
\label{eq:g_perp_Delta_f}
\end{equation}
@@ -2514,20 +2514,20 @@ In this case there are global complex coordinates for which the string solution
\begin{equation}
Z^i(u, \bu) = Z^i_L(u),
\qquad
\overline{Z}^i(u, \bu) = \bar{Z}^i(\bu) = \left( Z^i_L(u) \right)^*,
\overline{Z}^i(u, \bu) = \bar{Z}^i(\bu) = \qty( Z^i_L(u) )^*,
\end{equation}
where $i = 1$ in the Abelian case and $i=1,\, 2$ in the \SU{2} case.
We also have $f^i_{(t)} = Z^i_L(x_{(t)} + i\, 0^+)$.
Equations~\eqref{eq:g_perp_Delta_f} and~\eqref{eq:action_with_imaginary_part} then become
\begin{eqnarray}
\Re( g_{(t)\, i}^{(\perp)}\, \left( f_{(t-1)} - f_{(t)} \right)^i )
\Re( g_{(t)\, i}^{(\perp)}\, \qty( f_{(t-1)} - f_{(t)} )^i )
& = &
0,
\\
4 \pi \ap \eval{S_{\R^4}}_{\text{on-shell}}
& = &
-2 \finitesum{t}{1}{3}\,
\Im( g_{(t)\, i}^{(\perp)}\, \left( f_{(t-1)} - f_{(t)} \right)^i ),
\Im( g_{(t)\, i}^{(\perp)}\, \qty( f_{(t-1)} - f_{(t)} )^i ),
\end{eqnarray}
where the last equation shows that the action can be expressed using just the global data.
@@ -2540,10 +2540,10 @@ Since the action is positive then we can write
=
\frac{1}{2 \pi \ap}
\finitesum{t}{1}{3}\,
\left(
\qty(
\frac{1}{2} \abs{g^{(\perp)}_{(t)}} \,
\abs{f_{(t-1)} - f_{(t)}}
\right),
),
\end{equation}
where a factor $\frac{1}{2}$ comes from raising the complex index in $g^{(\perp)}_{(t)\, 1}$.
The right hand side of the previous expression is the sum of the areas of the triangles having the interval between two intersection points on a given D-brane $D_{(t)}$ as base and the distance between the D-brane and the origin as height.

829
sec/part1/fermions.tex
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File diff suppressed because it is too large Load Diff

116
sec/part1/introduction.tex
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@@ -19,7 +19,7 @@ In particular we recall some results on the symmetries of string theory and how
\subsection{Properties of String Theory and Conformal Symmetry}
Strings are extended one-dimensional objects.
They are curves in spacetime parametrized by a coordinate $\sigma \in \left[0, \ell \right]$.
They are curves in spacetime parametrized by a coordinate $\sigma \in \qty[0, \ell ]$.
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.
@@ -46,11 +46,11 @@ The \eom for the string $X^{\mu}(\tau, \sigma)$ is therefore
\begin{equation}
\frac{1}{\sqrt{- \det \gamma}}\,
\ipd{\alpha}
\left(
\qty(
\sqrt{- \det \gamma}\,
\gamma^{\alpha\beta}\,
\ipd{\beta} X^{\mu}
\right)
)
=
0,
\qquad
@@ -66,14 +66,14 @@ In fact
=
- \frac{1}{4 \pi \ap}
\sqrt{- \det \gamma}\,
\left(
\qty(
\ipd{\alpha} X \cdot \ipd{\beta} X
-
\frac{1}{2}
\gamma_{\alpha\beta}\,
\gamma^{\lambda\rho}\,
\ipd{\lambda} X \cdot \ipd{\rho} X
\right)
)
=
0
\label{eq:conf:worldsheetmetric}
@@ -152,13 +152,13 @@ In fact the classical constraint on the tensor is simply
\fdv{S_P[\gamma, X]}{\gamma^{\alpha\beta}}
=
-\frac{1}{\ap}
\left(
\qty(
\ipd{\alpha} X \cdot \ipd{\beta} X
-
\frac{1}{2} \eta_{\alpha\beta}\,
\eta^{\lambda\rho}\,
\ipd{\lambda} X \cdot \ipd{\rho} X
\right)
)
=
0.
\label{eq:conf:stringT}
@@ -188,33 +188,33 @@ while its conservation $\partial^{\alpha} T_{\alpha\beta} = 0$ becomes:\footnote
Since we fix $\gamma_{\alpha\beta}(\tau, \sigma) \propto \eta_{\alpha\beta}$ we do not need to account for the components of the connection and we can replace the covariant derivative $\nabla^{\alpha}$ with a standard derivative $\partial^{\alpha}$.
}
\begin{equation}
\bpd T_{\xi\xi}( \xi, \bxi ) = \pd \bT_{\bxi\bxi}( \xi, \bxi ) = 0.
\bpd T_{\xi\xi}( \xi,\, \bxi ) = \pd \bT_{\bxi\bxi}( \xi,\, \bxi ) = 0.
\end{equation}
The last equation finally implies
\begin{equation}
T_{\xi\xi}( \xi, \bxi ) = T_{\xi\xi}( \xi ) = T( \xi ),
T_{\xi\xi}( \xi,\, \bxi ) = T_{\xi\xi}( \xi ) = T( \xi ),
\qquad
\bT_{\bxi\bxi}( \xi, \bxi ) = \bT_{\bxi\bxi}( \bxi ) = \bT( \bxi ),
\bT_{\bxi\bxi}( \xi,\, \bxi ) = \bT_{\bxi\bxi}( \bxi ) = \bT( \bxi ),
\end{equation}
which are respectively the holomorphic and the anti-holomorphic components of the 2-dimensional stress energy tensor.
which are respectively the holomorphic and the anti-holomorphic components of the bidimensional stress energy tensor.
The previous properties define what is known as a 2-dimensional \emph{conformal field theory} (\cft).
The previous properties define what is known as a bidimensional \emph{conformal field theory} (\cft).
Ordinary tensor fields
\begin{equation}
\phi_{\omega, \bomega}( \xi, \bxi )
\phi_{\omega, \bomega}( \xi,\, \bxi )
=
\phi_{\underbrace{\xi \dots \xi}\limits_{\omega~\text{times}}\, \underbrace{\bxi \dots \bxi}\limits_{\bomega~\text{times}}}( \xi, \bxi )
\left( \dd{\xi} \right)^{\omega}
\left( \dd{\bxi} \right)^{\bomega}
\phi_{\underbrace{\xi \dots \xi}\limits_{\omega~\text{times}}\, \underbrace{\bxi \dots \bxi}\limits_{\bomega~\text{times}}}( \xi,\, \bxi )
\qty( \dd{\xi} )^{\omega}
\qty( \dd{\bxi} )^{\bomega}
\end{equation}
can be classified according to their weight $\left( \omega, \bomega \right)$ referring to the holomorphic and anti-holomorphic parts respectively.
are classified according to their weight $\qty( \omega,\, \bomega )$ referring to the holomorphic and anti-holomorphic parts respectively.
In fact a transformation $\xi \mapsto \chi(\xi)$ and $\bxi \mapsto \bchi(\bxi)$ maps the conformal fields to
\begin{equation}
\phi_{\omega, \bomega}( \chi, \bchi )
=
\left( \dv{\chi}{\xi} \right)^{\omega}\,
\left( \dv{\bchi}{\bxi} \right)^{\bomega}\,
\phi_{\omega, \bomega}( \xi, \bxi ).
\qty( \dv{\chi}{\xi} )^{\omega}\,
\qty( \dv{\bchi}{\bxi} )^{\bomega}\,
\phi_{\omega, \bomega}( \xi,\, \bxi ).
\end{equation}
\begin{figure}[tbp]
@@ -238,9 +238,9 @@ In fact a transformation $\xi \mapsto \chi(\xi)$ and $\bxi \mapsto \bchi(\bxi)$
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,
z = e^{\xi} = e^{\tau_e + i \sigma} \in \qty{ z \in \C | \Im z \ge 0 },
\qquad
\bz = e^{\bxi} = e^{\tau_e - i \sigma} \in \left\lbrace z \in \C | \Im z \le 0 \right\rbrace
\bz = e^{\bxi} = e^{\tau_e - i \sigma} \in \qty{ z \in \C | \Im z \le 0 }
\end{equation}
maps the worldsheet of the string to the complex plane.
On this Riemann surface the usual time ordering becomes a \emph{radial ordering} as constant time surfaces are circles around the origin (see the contours $\ccC_{(0)}$ and $\ccC_{(1)}$ in \Cref{fig:conf:complex_plane}).
@@ -265,14 +265,14 @@ The transformation on a field $\phi_{\omega, \bomega}$ of weight $(\omega, \bome
\liebraket{Q_{\epsilon, \bepsilon}}{\phi_{\omega, \bomega}( w, \bw )}
\\
& =
\cint{0} \ddz \epsilon(z) \left[ T(z), \phi_{\omega, \bomega}( w, \bw ) \right]
\cint{0} \ddz \epsilon(z) \qty[ T(z), \phi_{\omega, \bomega}( w, \bw ) ]
+
\cint{0} \ddbz \bepsilon(\bz) \left[ \bT(\bz), \phi_{\omega, \bomega}( w, \bw ) \right]
\cint{0} \ddbz \bepsilon(\bz) \qty[ \bT(\bz), \phi_{\omega, \bomega}( w, \bw ) ]
\\
& =
\cint{w} \ddz \epsilon(z)\, \rR\!\left( T(z)\, \phi_{\omega, \bomega}( w, \bw ) \right)
\cint{w} \ddz \epsilon(z)\, \rR\!\qty( T(z)\, \phi_{\omega, \bomega}( w, \bw ) )
+
\cint{\bw} \ddbz \bepsilon(\bz)\, \rR\!\left( \bT(\bz)\, \phi_{\omega, \bomega}( w, \bw ) \right),
\cint{\bw} \ddbz \bepsilon(\bz)\, \rR\!\qty( \bT(\bz)\, \phi_{\omega, \bomega}( w, \bw ) ),
\end{split}
\end{equation}
where in the last passage we used the fact that the difference of ordered integrals becomes the contour integral of the radially ordered product computed surrounding $w$.
@@ -393,7 +393,7 @@ This ultimately leads to the quantum algebra
known as Virasoro algebra, unique central extension of the classical de Witt algebra, with central charge $c$.
Operators $L_n$ and $\bL_n$ are called Virasoro operators.\footnotemark{}
\footnotetext{%
Notice that the subset of Virasoro operators $\left\lbrace L_{-1}, L_0, L_1 \right\rbrace$ forms a closed subalgebra generating the group $\SL{2}{\R}$.
Notice that the subset of Virasoro operators $\qty{ L_{-1}, L_0, L_1 }$ forms a closed subalgebra generating the group $\SL{2}{\R}$.
}
Notice that $L_0 + \bL_0$ is the generator of the dilations on the complex plane.
In terms of radial quantization this translates to time translations and $L_0 + \bL_0$ can be considered to be the Hamiltonian of the theory.
@@ -592,13 +592,13 @@ In complex coordinates on the plane it is:
=
- \frac{1}{4 \pi}
\iint \dd{z} \dd{\bz}
\left(
\qty(
\frac{2}{\ap}\, \ipd{\bz} X^{\mu}\, \ipd{z} X^{\nu}
+
\psi^{\mu}\, \ipd{\bz} \psi^{\nu}
+
\bpsi^{\mu}\, \ipd{z} \bpsi^{\nu}
\right)
)
\eta_{\mu\nu}.
\label{eq:super:action}
\end{equation}
@@ -641,7 +641,7 @@ The action~\eqref{eq:super:action} is also invariant under the \emph{supersymmet
- \bepsilon( \bz )\, \ipd{\bz} \bX^{\mu}( \bz )
\end{split}
\end{equation}
generated by the currents $J( z ) = \epsilon( z )\, T_F( z )$ and $\overline{J}( \bz ) = \bepsilon( \bz )\, \bT_F( \bz )$, where $\epsilon( z )$ and $\bepsilon( \bz ) = \left( \epsilon( z ) \right)^*$ are anti-commuting fermions and
generated by the currents $J( z ) = \epsilon( z )\, T_F( z )$ and $\overline{J}( \bz ) = \bepsilon( \bz )\, \bT_F( \bz )$, where $\epsilon( z )$ and $\bepsilon( \bz ) = \qty( \epsilon( z ) )^*$ are anti-commuting fermions and
\begin{equation}
\begin{split}
T_F( z )
@@ -681,7 +681,7 @@ The central charge associated to the Virasoro algebra is in this case given by b
The central charge is therefore $c = \frac{3}{2} D$ for the \cft defined in~\eqref{eq:super:action}.
As in the case of the bosonic string, in order to cancel the central charge we introduce the reparametrisation anti-commuting ghosts $b( z )$ and $c( z )$ and their anti-holomorphic components as well as the commuting \emph{superghosts} $\beta( z )$ and $\gamma( z )$ and their anti-holomorphic counterparts.
These are conformal fields with conformal weights $\left( \frac{3}{2}, 0 \right)$ and $\left( -\frac{1}{2}, 0 \right)$.
These are conformal fields with conformal weights $\qty( \frac{3}{2}, 0 )$ and $\qty( -\frac{1}{2}, 0 )$.
Their central charge becomes $c_{\text{ghost}} = c_{bc} + c_{\beta\gamma} = -26 + 11 = -15$ (see \cref{note:conf:ghosts} for the general computation).
When considering the full theory $T_{\text{full}} = T + T_{\text{ghost}}$ and $\bT_{\text{full}} = \bT + \bT_{\text{ghost}}$ the central charge vanishes only in 10-dimensional spacetime:
\begin{equation}
@@ -738,14 +738,14 @@ The tensor $J$ is called \emph{complex structure} if there exist a tensor $N$ su
\begin{equation}
\tensor{N}{^a_{bc}}\, v_p^b\, w_p^c
=
\left(
\qty(
\liebraket{v_p}{w_p}
+
J
\left( \liebraket{J\, v_p}{w_p} + \liebraket{v_p}{J\, w_p} \right)
\qty( \liebraket{J\, v_p}{w_p} + \liebraket{v_p}{J\, w_p} )
-
\liebraket{J\, v_p}{J\, w_p}
\right)^a
)^a
=
0
\end{equation}
@@ -807,7 +807,7 @@ $(M, J, g)$ is a \emph{Kähler} manifold if:
\begin{equation}
\dd{\omega}
=
\left( \pd + \bpd \right)
\qty( \pd + \bpd )
\omega(z, \bz)
=
0,
@@ -833,7 +833,7 @@ In local coordinates a Hermitian metric is such that
thus the Kähler form becomes $\omega = i g_{a\overline{b}}\, \dd{z}^a \wedge \dd{\bz}^{\overline{b}}$.
The relation~\eqref{eq:cy:kaehler} then translates into:
\begin{equation}
\dd{\omega} = i\, \left( \pd + \bpd \right)\, g_{a\overline{b}}\, \dd{z}^a \wedge \dd{\bz}^{\overline{b}}
\dd{\omega} = i\, \qty( \pd + \bpd )\, g_{a\overline{b}}\, \dd{z}^a \wedge \dd{\bz}^{\overline{b}}
=
0
\quad
@@ -995,20 +995,20 @@ The usual mode expansion in conformal coordinates $X^{\mu}( z, \bz ) = X( z ) +
x_0^{\mu}
+
i\, \sqrt{\frac{\ap}{2}}\,
\left(
\qty(
- \alpha_0^{\mu}\, \ln{z}
+ \sum\limits_{n \in \Z \setminus \{0\}} \frac{\alpha_n^{\mu}}{n} z^{-n}
\right),
),
\\
\bX^{\mu}( \bz )
& =
\overline{x}_0^{\mu}
+
i\, \sqrt{\frac{\ap}{2}}\,
\left(
\qty(
- \balpha_0^{\mu}\, \ln{\bz}
+ \sum\limits_{n \in \Z \setminus \{0\}} \frac{\balpha_n^{\mu}}{n} \bz^{-n}
\right),
),
\end{split}
\label{eq:tduality:modes}
\end{equation}
@@ -1045,9 +1045,9 @@ respectively encoding the quantisation of the momentum for a compact coordinate
We finally have
\begin{equation}
\begin{split}
\alpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \left( n\, \frac{\ap}{R} + m\, R \right),
\alpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \qty( n\, \frac{\ap}{R} + m\, R ),
\\
\balpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \left( n\, \frac{\ap}{R} - m\, R \right),
\balpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \qty( n\, \frac{\ap}{R} - m\, R ),
\end{split}
\end{equation}
@@ -1058,24 +1058,24 @@ From~\eqref{eq:conf:Texpansion} and \eqref{eq:conf:bosonicstringT} we find
L_0
&=
\frac{\ap}{2}\,
\left(
\left( \alpha_0^{D-1} \right)^2
\qty(
\qty( \alpha_0^{D-1} )^2
+
\sum\limits_{i = 0}^{D-2}\, \left( \alpha_0^i \right)^2
\sum\limits_{i = 0}^{D-2}\, \qty( \alpha_0^i )^2
+
\sum\limits_{n = 1}^{+\infty}\, \left( 2 \alpha_{-n}^{\mu} \alpha_n^{\nu}\, \eta_{\mu\nu} + a \right)
\right),
\sum\limits_{n = 1}^{+\infty}\, \qty( 2 \alpha_{-n}^{\mu} \alpha_n^{\nu}\, \eta_{\mu\nu} + a )
),
\\
\bL_0
&=
\frac{\ap}{2}\,
\left(
\left( \balpha_0^{D-1} \right)^2
\qty(
\qty( \balpha_0^{D-1} )^2
+
\sum\limits_{i = 0}^{D-2}\, \left( \balpha_0^i \right)^2
\sum\limits_{i = 0}^{D-2}\, \qty( \balpha_0^i )^2
+
\sum\limits_{n = 1}^{+\infty}\, \left( 2 \balpha_{-n}^{\mu} \balpha_n^{\nu}\, \eta_{\mu\nu} + a \right)
\right),
\sum\limits_{n = 1}^{+\infty}\, \qty( 2 \balpha_{-n}^{\mu} \balpha_n^{\nu}\, \eta_{\mu\nu} + a )
),
\end{split}
\end{equation}
where $a$ is constant given by normal ordering, representing the zero point energy of the theory.
@@ -1084,14 +1084,14 @@ Imposing physical conditions~\eqref{eq:conf:physical} and the \emph{level matchi
\begin{split}
M^2
& =
\frac{1}{\ap^2}\, \left( n\, \frac{\ap}{R} + m\, R \right)^2
\frac{1}{\ap^2}\, \qty( n\, \frac{\ap}{R} + m\, R )^2
+
\frac{4}{\ap}\, \left( \rN + a \right)
\frac{4}{\ap}\, \qty( \rN + a )
\\
& =
\frac{1}{\ap^2}\, \left( n\, \frac{\ap}{R} - m\, R \right)^2
\frac{1}{\ap^2}\, \qty( n\, \frac{\ap}{R} - m\, R )^2
+
\frac{4}{\ap}\, \left( \overline{\rN} + a \right),
\frac{4}{\ap}\, \qty( \overline{\rN} + a ),
\end{split}
\label{eq:dbranes:closedspectrum}
\end{equation}
@@ -1129,7 +1129,7 @@ The usual mode expansion~\eqref{eq:tduality:modes} here leads to
+
i\, \sqrt{\frac{\ap}{2}}\,
\sum\limits_{n \in \Z \setminus \{0\}}
\frac{\alpha_n^{\mu}}{n} \left( z^{-n} + \bz^{-n} \right)
\frac{\alpha_n^{\mu}}{n} \qty( z^{-n} + \bz^{-n} )
\end{equation}
and $\ell = \pi$.
@@ -1217,7 +1217,7 @@ The mass shell condition for open strings then becomes:\footnotemark{}
The constant $a$ in~\eqref{eq:dbranes:closedspectrum} takes here the value $-1$ from the imposition of the canonical commutation relations and a $\zeta$-regularisation.
}
\begin{equation}
M^2 = \frac{1}{\ap} \left( N - 1 \right).
M^2 = \frac{1}{\ap} \qty( N - 1 ).
\end{equation}
Consider for a moment bosonic string theory and define the usual vacuum as

6
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@@ -195,7 +195,7 @@
\thispagestyle{plain}
\noindent {\Large \sc Abstract} \\
\rule{0.99\textwidth}{\sepwidth}
\rule{0.99\linewidth}{\sepwidth}
}
{%
\vfill
@@ -208,11 +208,11 @@
\thispagestyle{plain}
\noindent {\Large \sc Acknowledgements} \\
\rule{0.99\textwidth}{\sepwidth}
\rule{0.99\linewidth}{\sepwidth}
}
{%
\vspace{\parskip}
\raggedleft\theauthor
\theauthor
\vfill
}

5
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@@ -74,7 +74,12 @@
\newcommand{\bomega}{\ensuremath{\overline{\omega}}}
\newcommand{\bepsilon}{\ensuremath{\overline{\epsilon}}}
\newcommand{\balpha}{\ensuremath{\overline{\alpha}}}
\newcommand{\bzeta}{\ensuremath{\overline{\zeta}}}
\newcommand{\halpha}{\ensuremath{\widehat{\alpha}}}
\newcommand{\hbeta}{\ensuremath{\widehat{\beta}}}
\newcommand{\htau}{\ensuremath{\widehat{\tau}}}
\newcommand{\hpsi}{\ensuremath{\widehat{\psi}}}
\newcommand{\Hpsi}{\ensuremath{\widehat{\Psi}}}
%---- BEGIN DOCUMENT