Naively it seems all of the zero modes involving the Yukawa couplings are non-localised in the base manifold $B$. However, note that the extension groups that mentioned above, can kill these zero modes or localise them over curves in $B$. This will be illustrated in the examples. In addition to these couplings there are other candidates for example corresponding the maps in the first row, there are other versions,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
H^0(\mathcal{B}_1)\otimes H^1(\mathcal{L}_1\otimes K_B^{-1})\arrow[r]\& H^1(\mathcal{B}_1\otimes\mathcal{L}_1\otimes K_B^{-1}) \\
H^1(\mathcal{B}_1\otimes K_B^{-1})\otimes H^1(\mathcal{L}_1)\arrow[r]\& H^1(\mathcal{B}_1\otimes\mathcal{L}_1\otimes K_B^{-1})
\end{tikzcd}
\end{eqnarray}

There are similar stories for other vector bundles. Since the vertical components didn't contribute in the $\overline{\textbf{5}}$-$\overline{\textbf{5}}$-10, it is interesting to check to see whether when spectral cover completely degenerates it is possible to have this kind of coupling or not. The analysis is the same as before. Consider only $\bar{V}_5$ part in defVVertical, and it is simply defined iteratively as the extension of line bundles,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
0\arrow[r]\& \mathcal{W}_1\arrow[r]\& \bar{V}_5\arrow[r]\& V_4\arrow[r]\& 0, \\
0\arrow[r]\& \mathcal{U}_1\arrow[r]\& V_4\arrow[r]\& V_3\arrow[r]\& 0, \\
0\arrow[r]\& \mathcal{B}_1\arrow[r]\& V_3\arrow[r]\& V_2\arrow[r]\& 0, \\
0\arrow[r]\& \mathcal{A}_1\arrow[r]\& V_2\arrow[r]\& \mathcal{L}_1\arrow[r]\& 0,
\end{tikzcd}
\end{eqnarray}

and the corresponding sequences for $\Lambda^2\bar{V}$ are

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
0\arrow[r]\& \mathcal{W}_1\otimes V_4\arrow[r]\& \Lambda^2\bar{V}_5 \arrow[r]\& \Lambda^2V_4\arrow[r]\& 0,\\
0\arrow[r]\& \mathcal{U}_1\otimes V_3\arrow[r]\& \Lambda^2 V_4 \arrow[r]\& \Lambda^2V_3\arrow[r]\& 0,\\
0\arrow[r]\& \mathcal{B}_1\otimes V_2\arrow[r]\& \Lambda^2 V_3 \arrow[r]\& \mathcal{A}_1\otimes \mathcal{B}_1 \arrow[r]\& 0.
\end{tikzcd}
\end{eqnarray}

There is also a relation due to the $c_1(V)=0$,

\begin{eqnarray}
\mathcal{W}_1\otimes\mathcal{U}_1\otimes\mathcal{B}_1\otimes\mathcal{A}_1\otimes\mathcal{L}_1 = K_B^5\otimes\mathcal{O}(c_V).
\end{eqnarray}

One can prove there are candidates for $\overline{\textbf{5}}$-$\overline{\textbf{5}}$-10 using the relation above between the line bundles. We are looking for morphisms $H^1(\bar{V})\otimes H^1(\Lambda^2\bar{V})\longrightarrow H^2(\Lambda^3 \bar{V})\simeq H^1(\Lambda^2\bar{V}^* \otimes \mathcal{O}(c_V))$. Where in the last isomorphism we used $\Lambda^5\bar{V}=\mathcal{O}(c_V)$. Putting everything together one can show for example,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
H^1(\bar{V})\otimes H^1(\Lambda^2\bar{V})\arrow[r]\&  H^2(\Lambda^2\bar{V}^* \otimes \mathcal{O}(c_V)) \\
H^1(B,\mathcal{W}_1)\otimes H^1(B,\mathcal{U}_1\otimes \mathcal{B}_1)\arrow[u, hook']\arrow[r]\& H^2(B, \mathcal{A}_1^*\otimes \mathcal{L}_1^*\otimes \mathcal{O}(c_V))\arrow[u, dashed]
\end{tikzcd}
\end{eqnarray}

Where the dashed arrow emphasizes that the relation between these to spaces is given by the Leray spectral sequence, and it is not an injection necessarily. There are 15 Yukawa coupling candidates like this one which, as mentioned in the $E_6$ model, seems to come from the bulk zero modes inside the 7-branes. However one need to check, case by case, whether these zero modes are killed or localised over some curve (by coboundary morphisms or extension maps of the short exact sequences) or not. There are also 30 other candidates. For example corresponding to the diagram above there are two other diagrams,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
H^1(\bar{V})\otimes H^1(\Lambda^2\bar{V})\arrow[r]\&  H^2(\Lambda^2\bar{V}^* \otimes \mathcal{O}(c_V)) \\
H^0(B,\mathcal{W}_1\otimes K_B)\otimes H^1(B,\mathcal{U}_1\otimes \mathcal{B}_1)\arrow[u, dashed]\arrow[r]\& H^1(B, \mathcal{A}_1^*\otimes \mathcal{L}_1^*\otimes \mathcal{O}(c_V)\otimes K_B)\arrow[u, dashed]
\end{tikzcd}
\end{eqnarray}

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
H^1(\bar{V})\otimes H^1(\Lambda^2\bar{V})\arrow[r]\&  H^2(\Lambda^2\bar{V}^* \otimes \mathcal{O}(c_V)) \\
H^1(B,\mathcal{W}_1)\otimes H^0(B,\mathcal{U}_1\otimes \mathcal{B}_1\otimes K_B)\arrow[u, dashed]\arrow[r]\& H^1(B, \mathcal{A}_1^*\otimes \mathcal{L}_1^*\otimes \mathcal{O}(c_V)\otimes K_B)\arrow[u, dashed]
\end{tikzcd}
\end{eqnarray}

§ EXAMPLES

In this section we illustrate the ideas of the previous sections by means of simple examples. In this paper we don't try to find models that have correct phenomenological properties such as correct number of generations, moduli stabilization etc.

§.§ $E_6$ Model

For this subsection lets consider a rank three bundle defined as

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
\& \& \& 0\arrow[d] \& \\
\& \& \& \mathcal{O}(-\sigma+D_2)\arrow[d] \& \\
0\arrow[r]\& \mathcal{O}(D_1)\arrow[r] \& V_3 \arrow[r]\& V_2\arrow[r]\arrow[d]\& 0 \\
\& \& \& \mathcal{O}(\sigma+D_3)\arrow[d] \& \\
\& \& \& 0 \&
\end{tikzcd}
\end{eqnarray}

Where $D_1,D_2,D_3$ are divisors in the base that satisfy

\begin{eqnarray}
\end{eqnarray}

The Fourier transform of this bundle us quite easy to find,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
\& \& \& 0\arrow[d] \& \\
\& \& \& \mathcal{O}(-\sigma+D_3+K_B)\arrow[d,"\mathcal{S}_2"] \& \\
\& \& \& \mathcal{O}(\sigma+D_2)\arrow[d] \& \\
0\arrow[r]\& \mathcal{O}_{\sigma}(D_1+K_B)\arrow[r] \& \Phi^1(V_3) \arrow[r]\& \mathcal{O}_{S_2}(\sigma+D_2)\arrow[r]\arrow[d]\& 0 \\
\& \& \& 0 \&
\end{tikzcd}
\end{eqnarray}

Therefore the spectral cover in this case is

\begin{eqnarray}
S_3=Z \mathcal{S}_2 = Z(a_2 X+a_0 Z^2),\quad [\mathcal{S}_2]=2\sigma-D_3+D_2-K_B
\end{eqnarray}

This is our starting point. The purpose is to compute the Yukawa couplings of this bundle, then consider various deformations of this bundle and see how they affect the Yukawa couplings. The Parameters of this bundle are the polynomials $a_2$ and $a_0$ which determine the double cover, and the extension group of $V_2$ and $\mathcal{O}(D_1)$,

\begin{eqnarray}
Ext^1(\mathcal{O}_{S_2}(\sigma+D_2),\mathcal{O}_{\sigma}(D_1+K_B))&=&H^0(c,\mathcal{O}(D_1-D_2)\otimes \mathcal{O}(c))\nonumber \\
&=& H^0\left(c,\mathcal{O}(K_B+D_1-D_3)\right),
\end{eqnarray}

where $c$ is the “matter curve" $c=\sigma\cap \mathcal{S}_2=K_B+D_2-D_3$.

The first task will be computing the cohomologies and we can just use the results of <ref> with $\mathcal{L}_1=\mathcal{O}(D_1+K_B)$ and $\mathcal{L}_2=\mathcal{O}_{S_2}(\sigma+D_2)$. We have seen that there are possible Yukawa coupling in this case from the map

\begin{eqnarray}
H^1(\mathcal{L}_1\otimes K_B^{-1}) \otimes H^0(i_{c*} \mathcal{L}_2) \longrightarrow H^1(i_{c*} (\mathcal{L}_1\otimes \mathcal{L}_2\otimes K_B^{-1})).\nonumber
\end{eqnarray}

We only need to check whether they actually contribute in the cohmologies $H^1(V)$ and $H^2(\Lambda^2V)$. Let us start with the former,

\begin{eqnarray}
&\begin{tikzcd}[ampersand replacement=\&]
0\arrow[r]\& \mathcal{O}(D_1+K_B)\arrow[r]\& R^1\pi_*V\arrow[r]\& \mathcal{O}_c(D_2+K_B)\arrow[r] \& 0.
\end{tikzcd}&
\end{eqnarray}

Clearly $H^1(\mathcal{O}(D_1))$ injects into $H^1(V)$. But whether any element in  $H^0(\mathcal{O}_c(D_2+K_B)$ can be inside $H^1(V)$ or not, clearly depends on the choices of the divisors. For example when $B=\mathbb{F}_0$ and $D_1=(2,-4)$, $D_2=(2,9)$ ($[c]=(4,12)$) then $h^*(\mathcal{O}_c(D_2+K_B))=(8,12)$ and it is indeed part of the $H^1(V)$ because $H^2(\pi_*V)=0$ in LerayE6Red. In addition the extension of the defining equation is non-zero : $h^0(c,\mathcal{O}(K_B+D_1-D_3))=12$.[It is easy to check that with this choice of divisors there is always a region in Kahler cone such that the slope of $\mathcal{O}(D_1)$ is negative, so the bundle is not semistable.]

We also need to check whether $H^1(i_{c*} (\mathcal{L}_1\otimes \mathcal{L}_2\otimes K_B^{-1}))= H^1(c,\mathcal{O}(K_B+D_1+D_2))$ also maps into $H^1(R^1\pi_*\Lambda^2V)$. It turns out that this is not the case. So there are no 27-27-27 couplings in this case and they are killed by the extension of the defining equation. To see this in more detail remember the the long exact sequence that gave us the derived pushforwards of $\Lambda^2V$ in section <ref>,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, row sep=small]
\& 0\arrow[r] \& \pi_* \Lambda^2V\arrow[r]\arrow[d, phantom, ""{coordinate, name=Z}]\& \mathcal{O}(-D_1)\arrow[dll,
to path={ -- ([xshift=2ex]\tikztostart.east)
|- (Z) [near end]\tikztonodes
-| ([xshift=-2ex]\tikztotarget.west)
-- (\tikztotarget)}]\& \\
\& \mathcal{O}_c(K_B+D_1+D_2) \arrow[r]\& R^1\pi_* \Lambda^2V\arrow[r]\& \mathcal{O}(-D_1+K_B)\arrow[r]\& 0.
\end{tikzcd}\nonumber
\end{eqnarray}

Note that the coboundary map is exactly the same as the extension of the original defining sequence. So the cokernel of this map are torsion sheaves supported over the points where the extension vanishes i.e. the points where a global section (that one chooses as the extension “parameter") of $\mathcal{O}_c(K_B+D_1-D_3)$ vanishes. So we can conclude

\begin{eqnarray}
\pi_* \Lambda^2V = \mathcal{O}(-K_B+2D_3), \quad  R^1\pi_* \Lambda^2V = T\oplus \mathcal{O}(-D_1+K_B),
\end{eqnarray}

where the $T$ is the torsion sheaf mentioned before, and with our choice of the divisors, $T$ is supported over 44 generic points over the curve $c$. So we see that elements of  $H^1(i_{c*} (\mathcal{L}_1\otimes \mathcal{L}_2\otimes K_B^{-1}))$ cannot inject into $H^2(\Lambda^2V)$ and these modes are removed by the coboundary map (or extension). So when the extension is chosen to be zero there can be 27-27-27 couplings from localised over $c$.

For $\overline{\textbf{27}}$-$\overline{\textbf{27}}$-$\overline{\textbf{27}}$ couplings we can use the same techniques as before and we have all information that we need,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
H^1(V^*)\otimes H^1(V^*) \arrow[r]\& H^2(\Lambda^2V^*) \\
H^2(V)\otimes H^2(V) \arrow[r]\& H^2(V)
\end{tikzcd}
\end{eqnarray}

The sequences above are equivalent. So in this case all we need is just the spectral data of $V$. It is not too hard to show that the relevant cohomologies for this particular coupling are

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
H^1(\mathcal{O}(D_1+K_B))\otimes H^1(\mathcal{O}_c(D_2+K_B)) \simeq H^1(\mathcal{O}(-D_1))\otimes H^0(\mathcal{O}_c(K_B-D_3))\arrow[r] \& H^1(\mathcal{O}_c(D_2+K_B),
\end{tikzcd}
\end{eqnarray}

where we have used $D_1+D_2+D_3=0$. The only thing wee need to show is that whether the elements of $H^1(\mathcal{O}(D_1+K_B))$ enter into $H^1(R^1\pi_*V)$,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
\dots\arrow[r]\& H^0(\mathcal{O}_c(D_2+K_B)) \arrow[r]\& H^1(\mathcal{O}(D_1+K_B)) \arrow[r]\& H^1(R^1\pi_* V)\arrow[r]\& \dots,
\end{tikzcd}
\end{eqnarray}

where the second map is simply the extension of the short exact sequence. With the choices of $D_1$, $D_2$ and $D_3$ in this example, one should find the cokernel of $H^0(\mathcal{O}_c(0,7))\longrightarrow H^1(\mathcal{O}(0,-6))$ where the maps is an element of $H^1(\mathcal{O}_c(4,-1))$. The basis of these cohomology groups and the generic form of the homomorphism can be represented by (rational) monomials as follows

\begin{eqnarray}
&X_1^7,\quad X_1^6 X_2,\quad X_1^5 X_2^2,\quad X_1^4 X_2^3,\quad X_1^3 X_2^4,\quad X_1^2 X_2^5,\quad X_1 x_2^6,\quad X_2^7,& \\
&\frac{1}{X_1 X_2 X_1^4},\quad \frac{1}{X_1 X_2 X_1^3 X_2},\quad \frac{1}{X_1 X_2 X_1^2 X_2^2},\quad \frac{1}{X_1 X_2 X_1 X_2^3},\quad \frac{1}{X_1 X_2 X_2^4},& \\
&\frac{1}{X_1 X_2 f_{11}(X_1,X_2)},&
\end{eqnarray}

where the variables $X_1$ and $X_2$ are homogeneous coordinates of the $\mathbb{P}^1$ base of $\mathbb{F}_0$, the monomials in the first row are the bases of $H^0(\mathcal{O}_c(0,7))$, the rational monomials of the second row are the bases of $H^1(\mathcal{O}(0,-6))$, and the third row is the rational monomial that corresponds to the extension of the defining short exact sequence (with $f_{11}$ being a generic degree 11 polynomial). It is now very easy to show that this map is surjective, in other words $H^1(\mathcal{O}(D_1+K_B))$ do not inject into $H^1(R^1\pi_*V)$ and hence no contribution in $H^2(V)$. So there is no  $\overline{\textbf{27}}$-$\overline{\textbf{27}}$-$\overline{\textbf{27}}$ coupling in this model. Here we clearly see one of the main points of this paper. Instead of computing the Yukawa couplings by directly computing the cohomology groups of $V$, one can dissect this problem to smaller pieces using the spectral data, reduce the problem to simple calculations.

Before considering the deformations, it is somewhat important to mention that there is another Yukawa coupling, namely 1-27-$\overline{\textbf{27}}$, which can be non-zero. This is given by the Yoneda pairing,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
Ext^1(\mathcal{O},V)\otimes Ext^1(V,V)\otimes Ext^1(V,\mathcal{O})\arrow[r]\& Ext^3(\mathcal{O},\mathcal{O})=H^3(\mathcal{O})=\mathbb{C}.
\end{tikzcd}
\end{eqnarray}

The middle term in this pairing counts the infinitesimal deformations of the bundle. In the current example they correspond to the deformations of $\mathcal{S}_2$ and the extension group elements. When the bundle moduli are stabilized to some fixed values, these couplings give mass to the 27-$\overline{\textbf{27}}$ pairs but the net number of generations remains unchanged. Anyways, in this example these couplings are non-zero and are localised over the curve $c$ and they are counted by the following map

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
H^0(\mathcal{O}_c(K_B+D_2))\otimes H^0(\mathcal{O}_c(K_B+D_1-D_3))\otimes H^1(\mathcal{O}(-D_1)) \arrow[r]\& H^1(c,K_c) \simeq \mathbb{C}.
\end{tikzcd}
\end{eqnarray}

With the choices we have made all of the cohomologies above are non-zero. But the previous calculations shows that $H^1(\mathcal{O}(-D_1))$ doesn't contribute to $H^2(\mathcal{O}(-D_1))$. Therefore this coupling is also vanishing.

In the next part we deform the spectral data to a completely degenerate one by setting the polynomial $a_2$ to zero. The spectral sheaf can be rewritten in the form of the previous subsection,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
0\arrow[r]\&\mathcal{L}_{c_V}\arrow[r]\& \mathcal{L}_3\arrow[r]\& \bar{\mathcal{L}}_3\arrow[r]\& 0.
\end{tikzcd}
\end{eqnarray}

It is possible to find $\mathcal{L}_V$ and $\bar{\mathcal{L}}_3$ by computing $Li_{c_V}^*\mathcal{L}_3$ and $Li_{\sigma}^*\mathcal{L}_3$ using two different representations of $\mathcal{L}_3$, and compare them. The final result is,

\begin{eqnarray}
&\mathcal{L}_{c_V}= \mathcal{O}_{c_V}(-2\sigma+D_2),& \\
&\begin{tikzcd}[ampersand replacement=\&, row sep=small]
0\arrow[r]\& \mathcal{O}_{\sigma}(-2D_3)\arrow[r]\& \bar{\mathcal{L}}_3\arrow[r]\& \bar{\mathcal{L}}_2 \arrow[r]\& 0, \\
0\arrow[r]\& \mathcal{O}_{\sigma}(D_2)\arrow[r]\& \bar{\mathcal{L}}_2\arrow[r]\& \mathcal{O}_{\sigma}(D_2+K_B) \arrow[r]\& 0.
\end{tikzcd}&
\end{eqnarray}

All of the extension morphisms are non-zero and generic. As usual, instead of working with the vertical components directly, one can perform an inverse Fourier transform,

\begin{eqnarray}
&\begin{tikzcd}[ampersand replacement=\&, row sep=small]
0\arrow[r]\& V_3\arrow[r]\& \bar{V}_3\arrow[r]\& i_{c_V*}\mathcal{F}\arrow[r]\& 0,
\end{tikzcd}&\\
&\begin{tikzcd}[ampersand replacement=\&, row sep=small]
0\arrow[r]\& \mathcal{O}_{c_V}(-4\sigma+D_2+K_B)\arrow[r]\& (K_B\oplus K_B^{-1}\oplus K_B^{-2})\otimes \mathcal{O}_{c_V}(-\sigma+D_2)\arrow[r]\& i_{c_V*}\mathcal{F}\arrow[r]\& 0.
\end{tikzcd}&\nonumber \\
\quad
\end{eqnarray}

To do concrete calculations one needs to choose the divisors $D_1$, $D_2$ and $D_3$.[With the previous choices the bundle becomes unstable as in the limit where the spectral cover is degenerate.] We choose $D_1=(2,-4)$ and $D_2(2,3)$. Then $D_3=(-4,1)$ and $c_V=(8,4)$. In this case one can show the cohomologies of $i_{c_V*}\mathcal{F}=(60,0,0,0)$. Pushforwads are also easy to find,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, row sep=small]
0\arrow[r]\& \mathcal{O}(-2D_3-K_B)\arrow[r]\& \pi_*V\arrow[r]\& \mathcal{O}(D_2-K_B)\arrow[r]\& 0, \\
0\arrow[r]\& \mathcal{O}(-2D_3)\arrow[r]\& \pi_*V\arrow[r]\& \mathcal{O}(D_2+K_B)\arrow[r]\& 0.
\end{tikzcd}
\end{eqnarray}

In principle between the first row and the second raw is non-zero, but in this example it is vanishing. By the analysis of the previous section, especially the relations degE6Coup, there are only three “types" of candidates for the Yukawa couplings from the degenerate part of the spectral cover

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, row sep=small]
H^1(\mathcal{B}_1\otimes K_B^{-1})\otimes H^1(\mathcal{A}_1\otimes K_B^{-1})\arrow[r]\& H^2(\mathcal{B}_1\otimes\mathcal{A}_1\otimes K_B^{-2}), \\
H^0(\mathcal{B}_1)\otimes H^1(\mathcal{A}_1\otimes K_B^{-1})\arrow[r]\& H^1(\mathcal{B}_1\otimes\mathcal{A}_1\otimes K_B^{-1}), \\
H^1(\mathcal{B}_1\otimes K_B^{-1})\otimes H^0(\mathcal{L}_1)\arrow[r]\& H^1(\mathcal{B}_1\otimes\mathcal{A}_1\otimes K_B^{-1}),
\end{tikzcd}
\end{eqnarray}

where $\mathcal{B}_1=\mathcal{O}(-2D_3)$, $\mathcal{A}_1=\mathcal{O}(D_2)$ and $\mathcal{L}_1=\mathcal{O}(D_2+K_B)$. One needs to check whether these cohomologies can contribute into $H^1(V)$ and $H^2(\Lambda^2 V)$. Since the cohomologies of $\mathcal{F}$ are $(60,0,0)$, any element of $H^1(\bar{V})$ can be lifted to an element in $H^1(V)$. However, all of the relevant cohomologies of the line bundles $\mathcal{A}_1$ $\mathcal{B}_1$ and $\mathcal{L}_1$, that can feed non-zero elements into $H^1(\bar{V})$, and at the same time give non-zero Yukawa couplings, are vanishing. Therefore there is no 27-27-27 couplings of this type.

The final part is checking for the couplings induced by the vertical component of the spectral cover. The candidates are given in the relations E6VC1 and E6VC2. The cohomologies of $i_{c_V*}\mathcal{F}^{*\otimes 2}\otimes \mathcal{O}(c_V)$ are $(0,824,0)$. Therefore the couplings corresponding to E6VC2 are zero, consistent with our expectation. For the other option one gets $h^1(\bar{V})=2$ which is lifted from $h^1(\mathcal{O}(D_2))$, and $h^0(i_{c_V*}\mathcal{F})=60$. However one can check easily $h^1(i_{c_V*}\mathcal{F}\otimes \bar{V})=0$. So nothing left and all of the 27-27-27 couplings are zero.

§.§ SU(5) Model

We consider a vector bundle defined by V5S2S2,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
\& \& \&0\arrow[d] \& \\
\& \& \&\mathcal{O}_{S_2}(\sigma+D_2)\arrow[d] \& \\
0\arrow[r]\&i_{\sigma*}\mathcal{O}(D_1)\arrow[r] \& \mathcal{L}_5\arrow[r] \&\mathcal{L}_4\arrow[r]\arrow[d] \& 0 \\
\& \& \&\mathcal{O}_{S_2}(\sigma+D_3)\arrow[d] \& \\
\& \& \&0 \&
\end{tikzcd}
\end{eqnarray}

where $S_2$ is a double cover with divisor class $[S_2]=2\sigma+\eta$. We have showed that there is a 10-10-5 coupling candidate YCV5S2S2, and three candidates for $\overline{\textbf{5}}$-$\overline{\textbf{5}}$-10 YC2V5S2S2,YC3V5S2S2,YC4V5S2S2. We only need to check whether in a certain model the cohomologies in the couplings are non-zero, they contribute to the relevant cohomologies of $V$ and $\Lambda^2V$. We need a bundle $V$ which is stable and degree zero. For the later condition, $Det(\pi_*\mathcal{L}_5)=K_B^5$. This puts a constraint between $\eta$, $D_1$, $D_2$ and $D_3$. To find this one should compute the pushforward of $\pi_*\mathcal{O}_{S_2}(\sigma)$ first,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
0\arrow[r]\& \mathcal{O}\arrow[r]\& \pi_*\mathcal{O}_{S_2}(\sigma)\arrow[r]\& \mathcal{O}(K_B-\eta)\arrow[r]\& 0.
\end{tikzcd}
\end{eqnarray}

\begin{eqnarray}
Det(\pi_* \mathcal{O}_{S_2}(\sigma+D_2)) = \mathcal{O}(K_B-\eta+2D_3), \quad Det(\pi_* \mathcal{O}_{S_2}(\sigma+D_2)) = \mathcal{O}(K_B-\eta+2D_3).
\end{eqnarray}

Deriving the final result is now easy,

\begin{eqnarray}
\end{eqnarray}

On the other hand, the stability requires $h^0(V)=h^3(V)=0$. So combining PiV5S2S2 with the Leray spectral sequence, one gets

\begin{eqnarray}
H^0(V)=H^0(\pi_*V),\quad H^3(V)=H^2(R^1\pi_*V).
\end{eqnarray}

The following choice satisfies the these conditions,

\begin{eqnarray}
B=\mathbb{F}_1,\quad D_1=(-4,3),\quad D_2=(3,3),\quad D_3=(1,2) \Rightarrow \eta=(5,11).
\end{eqnarray}

With this choice the extension of the vertical column can be computed as usual,[$\Gamma$ is the global section functor.]

\begin{eqnarray}
&RHom(i_{S_2*}\mathcal{O}(\sigma+D_3),i_{S_2*}\mathcal{O}(\sigma+D_2)) = R\Gamma(\mathcal{O}_{S_2}(D_2-D_3)\otimes (\mathcal{O}\oplus \mathcal{O}(S_2)[-1]))&\nonumber\\
&\Downarrow& \nonumber\\
&\begin{tikzcd}[ampersand replacement=\&, column sep=small]
0\arrow[r]\& Ext^1_X(\mathcal{O}_{S_2}(\sigma+D_3),\mathcal{O}_{S_2}(\sigma+D_2))\arrow[r]\& H^0(\mathcal{O}_{S_2}(S_2+D_2-D_3)) \arrow[r]\& H^2(\mathcal{O}_{S_2}(D_2-D_3))\arrow[r]\& \dots
\end{tikzcd}&\nonumber\\
\end{eqnarray}

It is not hard to show $h^0(\mathcal{O}_{S_2}(S_2+D_2-D_3))=95$ and $h^2(\mathcal{O}_{S_2}(D_2-D_3))=39$. So there are at least 56 independent extension morphisms that can give a “line bundle" over the degree four cover $S_2^2$ (as opposed to a rank two bundle over $S_2$).

To have non-zero Yukawa couplings $H^1(B,\mathcal{O}(D_1-K_B))$ but be non-empty. The cohmologies of this line bundle are $(0,8,0)$. Together with the Leray spectral sequence,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
0\arrow[r]\& H^1(\mathcal{O}(D_1)\otimes K_B^{-1})\arrow[r]\& H^1(V)\arrow[r]\& H^0(R^1\pi_*V)\arrow[r]\& 0,
\end{tikzcd}
\end{eqnarray}

it is clear that the elements of $H^1(B,\mathcal{O}(D_1-K_B))$ injects into $H^1(V)$. What about $H^0(R^1\pi_*V)$? For Yukawa couplings one needs to check whether at least some elements of $\mathcal{L}_4|_c$ (where $c=S_2\cdot \sigma$) can lift into some non-zero elements in $H^0(R^1\pi_*V)$. Recall the following diagram,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
\& \& \&0\arrow[d] \& \\
\& \& \&\mathcal{O}_c(K_B+D_2)\arrow[d]\arrow[dll,dotted,"extension=zero"description, bend right=35] \& \\
0\arrow[r]\& \mathcal{O}(D_1)\arrow[r]\& R^1\pi_*V\arrow[r]\&\mathcal{L}_4|_c\arrow[r]\arrow[d]\& 0 \\
\& \& \&\mathcal{O}_c(K_B+D_3)\arrow[d]\arrow[ull,dotted,"extension=non-zero"description,bend left=35] \& \\
\& \& \&0 \&
\end{tikzcd}
\end{eqnarray}

The extension of the sequence that defines $R^1\pi_*V$ is non zero, and in principle it comes from two pieces one piece $\mathcal{O}_c(K_B+D_2)$ and one from $\mathcal{O}_c(K_B+D_3)$. With this information we can show that $h^*(R^1\pi_*V)=(5,22,0)$, and all of the five independent elements of $h^0(R^1\pi_*V)$ are coming from $\mathcal{L}_4|_c$. More precisely,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
h^*(\mathcal{O}(D_1)) \& h^*(R^1\pi_*V) \& h^*(\mathcal{L}_4|_c) \\
0 \& 5 \& 5 \leftrightarrow h^0(\mathcal{O}_c(K_B+D_2)) \arrow[dll,dotted,"extension=zero"description] \\
18 \& 22 \& 4 \leftrightarrow h^1(\mathcal{O}_c(K_B+D_3))\\
0 \& 0 \& 0
\end{tikzcd}
\end{eqnarray}

So far we have identified 40 possible 10-10-5 couplings from $H^1(B,\mathcal{O}(D_1-K_B))\otimes H^0(c,\mathcal{O}_c(D_2+K_B))$. To make sure they are actually non-zero the cohomologies of $\Lambda^2V$ is required. The generalities are explained before in previous sections, here we only add some of the details. Remeber we had the following short exact sequence,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
0\arrow[r]\& \mathcal{O}(D_1)\otimes \mathcal{L}_4 \arrow[r]\& \mathcal{L}_5\star_A\mathcal{L}_5\arrow[r]\& \mathcal{L}_4\star_A\mathcal{L}_4\arrow[r]\& 0.
\end{tikzcd}
\end{eqnarray}

The first term on the right is hard to compute. But it is already done before,

\begin{eqnarray}
\mathcal{L}_4\star_A\mathcal{L}_4 &=& i_{\sigma*}Det(\pi_*\mathcal{O}_{S_2}(\sigma)+D_2)\oplus  i_{\sigma*}Det(\pi_*\mathcal{O}_{S_2}(\sigma)+D_3)\oplus \mathcal{L}_2\star\mathcal{L}_2'\nonumber \\
&=& \mathcal{O}_\sigma (K_B-\eta+2D_2)\oplus \mathcal{O}_\sigma (K_B-\eta+2D_3)\oplus \mathcal{L}_2\star\mathcal{L}_2'.
\end{eqnarray}

We only need to compute the last term,

\begin{eqnarray}
\mathcal{L}_2\star\mathcal{L}_2'=\mathcal{O}(D_2+D_3)\otimes \mathcal{O}_{S_2}(\sigma)\star\mathcal{O}_{S_2}(\sigma).
\end{eqnarray}

Here we give a rather heuristic, but direct way, and in the appendix we prove this formula indirectly by using inverse Fourier transform. It has been shown that the antisymmetric part of this product is simply the determinant of the pushforward. For general product we know the product should be supported over a degree four but reducible cover, which one component of that is a non-reduced degree two copy of the zero section. The definition of the Pontrjagin product helps to derive a formula for the restriction of $\mathcal{O}_{S_2}(\sigma)\star\mathcal{O}_{S_2}(\sigma)$ on the zero section,

\begin{eqnarray}
r:\mathcal{O}_{S_2}(\sigma)\star\mathcal{O}_{S_2}(\sigma) \longrightarrow \pi_{*} \left(\mathcal{O}_{S_2}(\sigma)\otimes \tau^*\mathcal{O}_{S_2}(\sigma)\right),
\end{eqnarray}

where $\tau$ is the involution of the elliptic fibers ($y\rightarrow -y$). Since any double cover and also the divisor $\sigma$ are invariant under the involution then $\tau^*\mathcal{O}_{S_2}(\sigma) = \mathcal{O}_{S_2}(\sigma)$. Therefore we can compute the restriction over zero section,

\begin{eqnarray}
r(\mathcal{O}_{S_2}(\sigma)\star\mathcal{O}_{S_2}(\sigma)) = \pi_*\mathcal{O}_{S_2}(2\sigma),
\end{eqnarray}

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
0\arrow[r]\& \mathcal{O}(2K_B+\eta)\arrow[r]\& \pi_*\mathcal{O}_{S_2}(2\sigma) \arrow[r]\& \mathcal{O}(K_B-\eta)\arrow[r]\& 0.
\end{tikzcd}
\end{eqnarray}

Note that the last term on the right is just the antisymmetric part of the Pontrjagin product, i.e. $Det(\pi_*\mathcal{O}_{S_2}(\sigma))=\mathcal{O}_{S_2}(\sigma)\star_A \mathcal{O}_{S_2}(\sigma)$. So the kernel should be the restriction of the symmetric product restricted over the fiber.[This can also be seen by noting that the two terms differ by $\frac{Br}{2}$ where $Br=4\eta+2K_B$ is the branching divisor of $S_2$ [29]] So we expect $\mathcal{O}_{S_2}(\sigma)\star\mathcal{O}_{S_2}(\sigma)$ can be represented by the following diagram,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
\&0\arrow[d] \& \& \& \\
\&\mathcal{T}_1\arrow[d] \& \& \& \\
0\arrow[r]\&\mathcal{O}_{S_2}(\sigma)\star_S\mathcal{O}_{S_2}(\sigma)\arrow[r]\arrow[d] \& \mathcal{O}_{S_2}(\sigma)\star\mathcal{O}_{S_2}(\sigma)\arrow[r] \& \mathcal{O}_{S_2}(\sigma)\star_A\mathcal{O}_{S_2}(\sigma) \& 0, \\
\&\mathcal{F}_2 \arrow[d] \& \& \& \\
\&0 \& \& \& \\
\end{tikzcd}
\end{eqnarray}

where $\mathcal{T}_1$ is a line bundle over the zero section to be determined soon, $\mathcal{F}_2$ is a line bundle over the double cover mentioned in sec ??. As emphasized before, we expect that $i_{\sigma}^*\mathcal{O}_{S_2}(\sigma)\star_S\mathcal{O}_{S_2}(\sigma)=\mathcal{O}(2K_B+\eta)$.[Of course $i_{\sigma}^*\mathcal{O}_{S_2}(\sigma)\star_S\mathcal{O}_{S_2}(\sigma)=\mathcal{O}(2K_B+\eta)=\mathcal{O}(K_B-\eta)$.] To determine $\mathcal{T}_1$ note that the zero section and the support of $\mathcal{F}_2$ intersect on the branching divisor of $S_2$. Therefore we expect that

\begin{eqnarray}
\mathcal{T}_1=\mathcal{O}(2K_B+\eta)\otimes\mathcal{O}(-Br)=\mathcal{O}(-3\eta).
\end{eqnarray}

So the final result about $Li_{\sigma}^*\mathcal{L}_2\star\mathcal{L}_2'$ can be derived easily,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small,row sep=small]
0\arrow[r]\& \mathcal{O}(-3\eta+D_2+D_3-K_B)\arrow[r]\& L^{-1}i_{\sigma}^*\mathcal{L}_2\star\mathcal{L}_2' \arrow[r]\arrow[d, phantom, ""{coordinate, name=Z}]\& \mathcal{O}(-\eta+D_2+D_3)\arrow[dll,to path={ -- ([xshift=1ex]\tikztostart.east)
|- (Z) [near end]\tikztonodes
-| ([xshift=-2ex]\tikztotarget.west)
-- (\tikztotarget)}]\&\&  \\
\& \mathcal{O}(\eta+D_2+D_3+2K_B)\arrow[r]\& i_{\sigma}^*\mathcal{L}_2\star\mathcal{L}_2' \arrow[r]\& \mathcal{O}(-\eta+D_2+D_3+K_B)\arrow[r]\& 0.
\end{tikzcd}
\end{eqnarray}

A comment on the coboundary maps is useful at this point. Note that the coboundary map of the sequence cannot be induced by the Extension groups of the original sequence that defined $\mathcal{L}_4$  for us. Similarly, same thing happens for the maps between $Det\pi_*\mathcal{L}_2$ and $\mathcal{L}_2'$ and other components of $\mathcal{L}_4\star_A\mathcal{L}_4$.

Now everything for computing $Li_{\sigma}^*\mathcal{L}_5\star_A\mathcal{L}_5$ is ready,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
\&0\arrow[r] \& \pi_*\Lambda^2V\arrow[r]\arrow[d, phantom, ""{coordinate, name=Z}] \&\begin{array}{c}
\mathcal{O}(-K_B-\eta+2D_2) \oplus \mathcal{O}(-K_B-\eta+2D_3)  \\
\oplus \mathcal{O}(-2K_B-3\eta+D_2+D_3)\oplus \mathcal{O}(-K_B-\eta+D_2+D_3)
\end{array}\arrow[dll,to path={ -- ([xshift=1ex]\tikztostart.east)
|- (Z) [near end]\tikztonodes
-| ([xshift=-2ex]\tikztotarget.west)
-- (\tikztotarget)}]\& \\
\&  \mathcal{O}(D_1)\otimes \mathcal{L}_4|_c\arrow[r] \& R^1\pi_*\Lambda^2V\arrow[r] \& \begin{array}{c}
\mathcal{O}(-\eta+2D_2) \oplus \mathcal{O}(-\eta+2D_3)  \\
\oplus \mathcal{O}(\eta+D_2+D_3+K_B)\oplus \mathcal{O}(-\eta+D_2+D_3)
\end{array}\arrow[r]\& 0
\end{tikzcd}
\nonumber\\
\quad
\end{eqnarray}

Even though the cokernel of this long exact sequence is non-zero, with the current choices of $D_1$, $D_2$ and $D_3$ all of the cohomologies of $\mathcal{W}_1\otimes \mathcal{L}_4|_c\otimes K_B^{-1} = \mathcal{O}_c(D_1-K_B)\otimes \mathcal{L}_4$ actually inject into the cohomologies of $R^1\pi_*\Lambda^2V$. In particular, for the 10-10-5
coupling only $H^1(R^1\pi_*\Lambda^2V)$ is needed YCV5S2S2, and by Leray sequence all the elements of this cohomology group lift into $H^2(\Lambda^2V)$. However, the relevant cohomologies of $Det(\pi_*\mathcal{L}_2)$ and $Det(\pi_*\mathcal{L}'_2)$ in YC2V5S2S2 and YC3V5S2S2 are zero. So there are no $\overline{\textbf{5}}$-$\overline{\textbf{5}}$-10 couplings in this theory. To check whether the 10-10-5 couplings actually exist in the effective theory, one needs to check whether the elements of $H^1(\mathcal{O}(D_1-K_B))$ and $H^0(\mathcal{L}_4|_c)$ can “contract" with each other and give an element of $H^1(\mathcal{O}_c(D_1-K_B)\otimes \mathcal{L}_4)$. Luckily in this example we can say that without any calculations. Recall that $H^0(\mathcal{L}_4|_c)$ has five independent elements which are injected from $H^0(\mathcal{O}_c(K_B+D_2))$. So to check whether the map in YCV5S2S2 is non-zero, we should examine whether the following map is non-zero

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&,column sep=small]
H^1(\mathcal{O}(D_1-K_B))\otimes H^0(\mathcal{O}_c(D_2+K_B)) \arrow[r, "?" ]\& H^1(\mathcal{O}_c(D_1+D2)) \arrow[r,dashed]\& H^1(\mathcal{O}_c(D_1-K_B)\otimes \mathcal{L}_4).
\end{tikzcd}\nonumber\\
\quad
\end{eqnarray}

It tuns out $h^1(\mathcal{O}_c(D_1+D_2))=0$. So there is no 10-10-5 couplings in this effective theory either.

§ ELLIPTIC FIBRATIONS WITHOUT HOLOMORPHIC SECTION

So far, we assumed the Calabi-Yau threefold is given by the Weierstrass model. As long as the elliptic fibration has a holomorphic (in other words, the section is isomorphic to the base manifold), it is easy to generalize these results. Because to define the Poincare sheaf, the section of the fibration should hit to every fiber at exactly one point. However, most of the elliptically fibered Calabi-Yau threefolds do not have any holomorphic sections. In these cases if someone defines an integral transform with kernel chosen to be the “naive" Poincare sheaf $\mathcal{I}_{\Delta}\otimes \pi_1^*\mathcal{O}(\sigma)\otimes\pi_2^*\mathcal{O}(\sigma)$, then that integral transform is not a Fourier-Mukai transform anymore.

It is very important for phenomenological applications to generalize the approach presented in the previous sections such that they can be used for the non-holomorphic section too. We only need to find the right kernel such that when the transform is restricted to any elliptic fiber (reducible or not)  is reduced to the ordinary Fourier-Mukai transform.

Here we only briefly propose a solution for case the reducible fibers are $I_2$. This approach can be generalized for $I_n$ fibers. However the for other types of reducible elliptic fibers, one needs more sophisticated approaches.

First note that for a single elliptic curve, reducible or not, the Poincare sheaf always defines a Fourier-Mukai transform. The reason is one can always choose the zero section to be a point on the curve. For smooth elliptic curves it is well known, and for $I_2$ fibers it is proved by Caldararu [32]. The problem appears when one consider an elliptic fibration where both types of fibers can appear in one family. When the section of the family is rational, the restriction of $\mathcal{I}_{\Delta}\otimes \pi_1^*\mathcal{O}(\sigma)\otimes\pi_2^*\mathcal{O}(\sigma)$ over smooth fibers is the Poincare sheaf, but its restriction over $I_2$ fibers (if the section wraps around one component, say $c_1$) is not. More precisely, suppose $i_p : E_p \hookrightarrow X$ be the inclusion morphism of the fiber over the point $p\in B$. If $E_p$ is $I_2$ ans $\sigma$ is the rational section that wraps around $c_1$, then the restriction over $E_p$ is given by,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
0\arrow[r]\& i_p^*\mathcal{O}_X(\sigma) \arrow[r]\& \mathcal{O}_{c_1}(-1)\oplus \mathcal{O}_{c_2}(+2)\arrow[r]\& T\arrow[r]\& 0,
\end{tikzcd}
\end{eqnarray}

where $T$ is a torsion sheaf supported over the intersection points of $c_1$ and $c_2$. However, a true section that hits a point in $c_1$ (denoted by $\sigma_0$) can be represented as

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
0\arrow[r]\& \mathcal{O}_{E_p}(\sigma_0) \arrow[r]\& \mathcal{O}_{c_1}(+1)\oplus \mathcal{O}_{c_2}\arrow[r]\& T\arrow[r]\& 0.
\end{tikzcd}
\end{eqnarray}

Ideally, if one can modify $\mathcal{O}(\sigma)$ such that over smooth fibers it is just the ordinary section that hits a single point, i.e. the modification has no effect over smooth fibers, but over the $I_2$ fibers it modifies $i_p^*\mathcal{O}_X(\sigma)$ to $\mathcal{O}_{E_p}(\sigma_0)$ then we reach our purpose. The way we have been able to modify $\mathcal{O}(\sigma)$, turns the restriction on the reducible elliptic fiber into $\mathcal{O}(-\sigma_0)$. The corresponding integral transform is still a Fourier-Mukai.[The kernel of the integral transform is basically the same as the Poincare sheaf twisted by a $\mathcal{O}(+2)$] However, this special transform does not map the semistable degree zero bundles over the elliptic fiber into spectral cover. But it is in principle possible to easily study the Fourier transform of semistable sheaves over such isolated elliptic fibers. On the other hand, over generic irreducible fibers, the Fourier-Mukai is just the same as the usual transform that we used so far. So globally, the Fourier-Mukai transform of a stable degree zero bundle will be a spectral sheaf plus some additional data over the irreducible fibers. We will explain this briefly in the following.

Define the sheaf $\Sigma$ as

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
0\arrow[r]\& \Sigma \arrow[r]\& \mathcal{I}_{c_1}\otimes \mathcal{O}_X(2\sigma)\arrow[r]\& i_{\sigma*} \mathcal{O}_{\sigma} (c_1)\arrow[r]\& 0.
\end{tikzcd}
\end{eqnarray}

Clearly the restriction over smooth fibers is the zero section. Let us compute the restriction over the components of the $I_2$ fiber

\begin{eqnarray}
Li_{c_1}^* \Sigma= \mathcal{O}_{c_1}(-1),\quad  Li_{c_2}^* \Sigma =\mathcal{O}_{c_2}.
\end{eqnarray}

So the restriction of $\Sigma$ on the reducible fiber is $\mathcal{O}(-\sigma_0)$

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&,column sep=small]
0\arrow[r]\& \Sigma\arrow[r]\& \mathcal{O}_{c_1}(-1)\oplus \mathcal{O}_{c_2}\arrow[r]\& T\arrow[r]\& 0.
\end{tikzcd}
\end{eqnarray}

Then define the modified Poincare sheaf as

\begin{eqnarray}
\overline{\mathcal{P}} := \mathcal{I}_{\Delta}\otimes \pi_1^* \Sigma \otimes \pi_2^* \Sigma \otimes K_B^{-1}.
\end{eqnarray}

As mentioned before with this kernel the new or modified Fourier-Mukai is almost the same as the usual one for Weierstrass models, but over the reducible fibers, it maps the degree zero line bundles to rank two bundles. For example

\begin{eqnarray}
R\pi_{2*}(\pi_1^* \mathcal{O}_{E_p}(q-\sigma_0)) = \mathcal{F},
\end{eqnarray}

where $q$ and $\sigma_0$ are points in the same component $c_1$, and $\mathcal{F}$ is

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&,column sep=small]
0\arrow[r]\& \Sigma^{\otimes 2}\otimes \mathcal{O}_{E_p} (p-\sigma_0) \arrow[r]\& \mathcal{F}\arrow[r]\& \Sigma\arrow[r]\& 0.
\end{tikzcd}
\end{eqnarray}

So the usual “spectral data" is modified by adding some exceptional data on non-generic fiber. More clearly, if the vector bundle over the reducible fiber is a direct sum on line bundles $\bigoplus_i \mathcal{O}(q_i-\sigma_0)$ then the Fourier-Mukai transform of that is $\bigoplus_i \mathcal{F}_i$. We ignore the calculations, but it is basically similar to the standard case that can be found in the literature.

It is now possible to continue the classification of the “spectral data" for non-Weierstrass Calabi-Yau threefolds. However this is a very formidable work to do. Because there hundreds of millions of Calabi-Yau threefolds and each one may have many rational zero sections that wrap around many rational curves. In addition the fibers doesn't necessarily degenerate in a simple form like the $I_2$ that we considered here, and for each case maybe another “zero section" $\Sigma$ needs to be defined. On the other hand, the F-theory dual of such heterotic models (it they exist) is not very well known at least to the author. So we leave the complete classification the Yukawa couplings to the future.

§ CONCLUSION

In this paper, we tried to introduce a new approach for computing the Yukawa couplings of the effective $4D$ $\mathcal{N}=1$ GUT theories, derived from heterotic string compactifications over Calabi-Yau threefold manifolds. The key inside was that the Fourier-Mukai transform (with the kernel being the Poincare sheaf) interchanges the tensor product and the Pontrjagin product, at least over a smooth elliptic curve. We argued that for computing the cohomologies of a product of vector bundles, we can still use the Pontjagin product in most cases. In other words, when the spectral cover has no vertical component, it is still possible to use the antisymmetrized Pontrjagin product for computing the cohomologies of $\Lambda^2V$. On the other hand, when there are vertical components, we used indirect computations, even though it seems it is still possible to directly use the spectral data.

Assuming the Calabi-Yau manifold is defined by the Weierstrass equation, one can classify possible spectral data that can appear for a smooth degree zero vector bundle and use the Pontrjagin product to find the relevant “pieces" of the spectral data that can give specific Yukawa couplings. This has two advantages. First, relative to the standard approaches, the calculations become much simpler, or at least the problem dissects into smaller and easy steps. Second, the relation between the intersecting brane models and T-brane becomes very clear. We didn't study the F-theory implications of these calculations thoroughly. However, in principle, these spectral data can be used to define the Higgs bundle one the 7-branes. Hence, we can see whether the specific couplings in the effective heterotic theory result from the triple intersection of branes or the interaction of the zero modes localized inside the 7-brane and curves.

Finally, since most of the elliptically Calabi-Yau manifolds constructed so far has only rational sections, it is interesting to see whether this idea can be extended to the Calabi-Yau manifolds without holomorphic sections. We gave the idea of how this can be done, but the full classification of the spectral data over such manifolds is much harder, plus there are many manifolds that need to be checked. In addition, the F-theory dual of such heterotic models is not very well known. So this generalization is outside of the scope of this paper, and we leave it for the future.

§ ACKNOWLEDGMENTS

MK is grateful for the insightful discussions with Tony Pantev, James Gray, Juntao Wang, Wei Cui, Seung-Joo Lee, Xin Gao, and Paul Oehlmann. Also, special thanks to Elham Mahmoudian for being supportive during the writing and preparing this paper.

§ USEFUL FORMULAS/TERMINOLOGIES

In this appendix we briefly explain the terminologies, and some of the formulas that we used in the bulk of this paper.

* Integral Transform

Integral transform is defined as

\begin{eqnarray}
\Phi: D(X)&\longrightarrow& D(Y)\\
\Phi(\bullet) &:=& R\pi_{2*} (\pi_1^*\bullet \otimes K^{\bullet}),
\end{eqnarray}

where $K^{\bullet}$ is an object of $D(X\times Y)$, and $\pi_1$ and $\pi_2$ are projection morphisms

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
\& X\times Y \arrow[dl,"\pi_1"']\arrow[dr,"\pi_2"] \& \\
X \& \& Y
\end{tikzcd}
\end{eqnarray}

An integral transform is called a Fourier-Mukai when it is an equivalence of the derived categories. For this to be true, the kernel of the integral transform should satisfy an “orthogonality condition", see [28] Theorem 1.27

* WIT-i

As mentioned before $\Phi$ “maps" objects $D(X)$, i.e. complexes of coherent sheaves in $X$, to objects of $D(Y)$. So one can view a coherent sheaf $\mathcal{F}$ as an object in derived category

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
\& \& \& \& 0-th \arrow[d]\& \& \\
\mathcal{F}^{\bullet} \simeq \& \& \dots \arrow[r]\& 0\arrow[r]\& \mathcal{F}\arrow[r]\& 0\arrow[r]\& \dots
\end{tikzcd}
\end{eqnarray}

We call a sheaf $\mathcal{F}$ is WIT-i respect to the functor $\Phi$ if its integral transform $\mathcal{G}^{\bullet}=\Phi(\mathcal{F})$ is a coherent sheaf located on the i-th position

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
\& \& \& \& i-th \arrow[d]\& \&\\
\mathcal{G}^{\bullet} \simeq \& \& \dots \arrow[r]\& 0\arrow[r]\& \mathcal{G}\arrow[r]\& 0\arrow[r]\& \dots
\end{tikzcd}
\end{eqnarray}

In this case we write

\begin{eqnarray}
\Phi(\mathcal{F})=\mathcal{G}[-i].
\end{eqnarray}

Note that $[-i]$ is a shift functor, and it simply shift the complex i places to the right.

* If $V$ is a rank $n$ stable degree zero vector bundle over X (which is an elliptically fibered Calabi-Yau threefold), then $c_1(\pi_*\mathcal{L}_n)=n c_1(K_B)$.

This can be proved easily. First note that the Chern character of $\Phi(V)=i_{S_n*}\mathcal{L}_n[-1]$ can be computed in terms of the Chern classes of $V$ [13]

\begin{eqnarray}
ch(i_{S_n*}\mathcal{L}_n) = [S_n]+[S_n]\cdot (-\frac{c_1(B)}{2}) + \frac{1}{2}c_2(V)f + \dots
\end{eqnarray}

where $[S_n]= n\sigma+\eta$ is the divisor class of the spectral cover, and $f$ is the class of the elliptic fiber. One can use this information, and the Grothendieck-Riemann-Roch theorem to find $ch(\pi_*i_{S_n*}\mathcal{L}_n)$

\begin{eqnarray}
ch(\pi_* i_{S_n*}\mathcal{L}_n) = \pi_*\left(ch(i_{S_n*}\mathcal{L}_n) \frac{td(X)}{td(B)} \right) = \pi_* \left( (n\sigma+\eta) -c_1(B)(n\sigma+\eta)-\frac{c_3(V)}{2}f +\dots\right).
\end{eqnarray}

Then we use the fact that $\pi_*\sigma = 1$, $\pi_*\eta = 0$ (when we right $\eta$ we really mean $\pi^*\eta$) and $\pi_* f=0$. Therefore

\begin{eqnarray}
ch(\pi_* i_{S_n*}\mathcal{L}_n) = n -n c_1(B)+\dots=n+ n c_1(K_B) +\dots.
\end{eqnarray}

* Numeric Rank.
By numeric rank of a sheaf $\mathcal{E}$ which is supported over a non-reduced subscheme we mean the Hilbert polynomial of that sheaf $P(s):\chi(X,\mathcal{E}\otimes\mathcal{O}(s))$, is the same as a the Hilbert polynomial of a coherent sheaf over a smooth subscheme with the same class.

* Proof of defLVVertical

Consider a short exact sequence,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
0\arrow[r]\& A \arrow[r,"f"]\& B \arrow[r]\& C \arrow[r]\& 0,
\end{tikzcd}
\end{eqnarray}

where $A$ and $B$ are locally free, but $C$ can be a general coherent sheaf. In derived category we can rewrite this as an isomorphism

\begin{eqnarray}
&C^{\bullet}& : \dots \longrightarrow 0 \longrightarrow C \longrightarrow 0 \longrightarrow \dots \\
&Cone(f)&: \dots\longrightarrow 0\longrightarrow A \longrightarrow B \longrightarrow 0 \longrightarrow \dots
\end{eqnarray}

\begin{eqnarray}
C^{\bullet} \simeq Cone(f).
\end{eqnarray}

Note that $Cone(f)^0=B$ and $Cone(f)^{-1}=A$ So there is a isomorphism after antisymmetrized product

\begin{eqnarray}\label{higherisom}
\Lambda^2 C^{\bullet} \simeq \Lambda^2 Cone(f).
\end{eqnarray}

When $C$ is a vector bundle

\begin{eqnarray}
&\Lambda^2 C^{\bullet}& = \Lambda^2C, \\
&\Lambda^2 Cone(f)&: 0\longrightarrow S^2A \longrightarrow A\otimes B \longrightarrow \Lambda^2B\longrightarrow 0.
\end{eqnarray}

Again please remember $\Lambda^2 Cone(f)^0=\Lambda^2B$, $\Lambda^2 Cone(f)^{-1}=A\otimes B$ and $\Lambda^2 Cone(f)^{-2}=S^2A$. So higherisom implies the usual long exact sequence

\begin{eqnarray}
0\longrightarrow S^2A \longrightarrow A\otimes B \longrightarrow \Lambda^2B\longrightarrow \Lambda^2C\longrightarrow 0.
\end{eqnarray}

However, if $C$ is a torsion sheaf supported on a smooth divisor $D$

\begin{eqnarray}
C^{\bullet} = i_{D*}\mathcal{M},
\end{eqnarray}

where $\mathcal{M}$ is line bundle on $D$, then the antidymmetrized product will be in the $(-1)$-th position

\begin{eqnarray}
\Lambda^2C^{\bullet} = i_{D*}\mathcal{M} i_{D*}\mathcal{M} &=& i_{D*}\left(\mathcal{M}\otimes_{\Lambda} (\mathcal{M}\oplus\mathcal{M}\otimes \mathcal{N}^*_{D/X} [+1])\right) \nonumber\\
&=& i_{D*}  \mathcal{M}^{\otimes 2}\otimes  \mathcal{N}^*_{D/X} [+1],
\end{eqnarray}

$ \mathcal{N}_{D/X}$ is the normal bundle of $D$. So the isomorphism higherisom implies

\begin{eqnarray}\label{finalcoh}
\mathcal{H}^{-1}(Cone(f)) \simeq i_{D*}  \mathcal{M}^{\otimes 2}\otimes  \mathcal{N}^*_{D/X}.
\end{eqnarray}

Namely, the complex $Cone(f)$ is not exact on the middle term anymore. Using the following short exact sequence for the bundle $A$

\begin{eqnarray}
0\longrightarrow S^2A \longrightarrow A\otimes A \longrightarrow \Lambda^2A \longrightarrow 0
\end{eqnarray}

we can rewrite finalcoh as

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
\& \& 0\arrow[d] \& \& \\
\& \& \Lambda^2 A\arrow[d] \& \& \\
0\arrow[r]\& i_{D*}  \mathcal{M}^{\otimes 2}\otimes  \mathcal{N}^*_{D/X} \arrow[r]\& Q \arrow[d]\arrow[r] \& \Lambda^2B \arrow[r]\& 0 .\\
\& \& B\otimes i_{D*}\mathcal{M} \arrow[d] \& \& \\
\& \& 0 \& \&
\end{tikzcd}
\end{eqnarray}

* $Li_{\sigma}^*\mathcal{O}_{S_2}(\sigma)\star\mathcal{O}_{S_2}(\sigma)$

Here instead of computing the Pontrjagin product we use the converse transform and compute the tensor product of the corresponding bundles and then we take the Fourier-Mukai transform of that. First to fix the notation, assume $[S_2]=2\sigma+\eta$.

It is easy to find the inverse transform of $\mathcal{O}_{S_2}(\sigma)$, which is denoted by $V$

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
0\arrow[r]\& \mathcal{O}_X(-\sigma)\arrow[r]\& V \arrow[r]\& \mathcal{O}_X(\sigma-\eta-K_B)\arrow[r]\& 0.
\end{tikzcd}
\end{eqnarray}

Now all we need is the Fourier transform of $V\otimes V$ and use the identity

\begin{eqnarray}
Li_{\sigma}^*\Phi^1(V\otimes V)=Li_{\sigma}^* \mathcal{O}_{S_2}(\sigma)\star\mathcal{O}_{S_2}(\sigma)\otimes K_B^{-1}[-1].
\end{eqnarray}

We ignore the detailed calculations,[All we need are the Fourier transforms of $\mathcal{O}_X$, $\mathcal{O}_X(2\sigma)$ and $\mathcal{O}_X(-2s)$. They can be computed by using the defining short exact sequence for $\mathcal{I}_{\Delta}$ [28, 13]] and we just write the final result for $\Phi(V\otimes V)$

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
0\arrow[r]\&\mathcal{F}_2\arrow[r]\&\mathcal{F}_1\arrow[r]\&\Phi^1(V\otimes V)\arrow[r]\&0,
\end{tikzcd}
\end{eqnarray}

where $\mathcal{F}_1$ is defined as

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
\&0\arrow[d] \& \& \& \\
\&K_B^{-1}\arrow[d]\& \& \& \\
0\arrow[r]\& \mathcal{B}\arrow[d]\arrow[r] \&\mathcal{F}_1\arrow[r]\& \mathcal{O}_{\sigma}(-\eta)\arrow[r]\&0 \\
\&\mathcal{O}_X(\sigma)\arrow[d] \& \& \& \\
\&0 \& \& \& \\
\end{tikzcd}
\end{eqnarray}

and $\mathcal{F}_2$,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
\& \&0\arrow[d] \& \& \\
0\arrow[r]\&\mathcal{F}_2\arrow[r]\&\mathcal{A}\otimes\mathcal{O}(-2\eta-2K_B)\arrow[r]\arrow[d] \&\mathcal{O}_{\sigma}(-\eta)\arrow[r]\&0\\
\& \&(\mathcal{O}\oplus K_B^2\oplus K_B^3)\otimes \mathcal{O}(\sigma-2\eta-3K_B)\arrow[d] \& \& \\
\& \&\mathcal{O}(4\sigma-2\eta-3K_B)\arrow[d] \& \& \\
\& \&0 \& \&
\end{tikzcd}
\end{eqnarray}

Even though these diagrams look very difficult, the restriction on the zero section is quite easy

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
\mathcal{M}^{\bullet}\simeq\& 0\arrow[r]\&\begin{array}{c}
\mathcal{O}_B(-K_B-\eta)   \\
\oplus \\
\mathcal{O}_B(-3\eta-2K_B)
\end{array} \arrow[r]\& \begin{array}{c}
\mathcal{O}_B(\eta+K_B) \\
\oplus\\
\mathcal{O}_B(-\eta)
\end{array}\arrow[r]\& 0,
\end{tikzcd}\nonumber \\
\quad
\end{eqnarray}

where the sequence $\mathcal{M}^{\bullet}$ on the left is not exact, and it is isomorphic (in derived category $D(B)$) to the pullback
$Li_{\sigma}^*\Phi^1(V\otimes V)$. This mean the zeroth cohomology $\mathcal{H}^0(\mathcal{M}^{\bullet})$ of this sequence gives $i_{\sigma}^*\Phi^1(V\otimes V)$, and the “(-1)-th" cohomology $\mathcal{H}^{-1}(\mathcal{M}^{\bullet})$ gives $L^{-1}i_{\sigma}^*\Phi^1(V\otimes V)$. In addition to this, there is a surjection

\begin{eqnarray}
V\otimes V\longrightarrow \Lambda^2V=\mathcal{O}_X(-\eta-K_B)\longleftrightarrow 0,
\end{eqnarray}

which induces a surjection

\begin{eqnarray}
\mathcal{O}_B(\eta+K_B)\longrightarrow i_{\sigma}^*\Phi^1(V\otimes V)\longrightarrow \mathcal{O}_{\sigma}(-\eta)\longrightarrow 0.
\end{eqnarray}

On the other hand, the map between the line bundles $\mathcal{O}_V(-\eta-K_B)$ and $\mathcal{O}_V(\eta+K_B)$ can be induced by the extension morphism of the original sequence.[The extension group decomposes into two subgroups $Ext^1(\mathcal{O}(\sigma-\eta-K_B),\mathcal{O}(-\sigma)) = H^0(\mathcal{O}(\eta))\oplus H^0(\mathcal{O}(\eta+2K_B))$. The choice of the extension morphism uniquely fixes the algebraic equation of the double cover $S_2= a_2 X+a_0 Z^2$ and the coefficients are identified with the elements of the subgroups mentioned.] They could be $a_0 a_2$. But between the line bundles $\mathcal{O}_V(-3\eta-2K_B)$ and $\mathcal{O}_V(\eta+K_B)$ cannot be induced by the extension morphism. Since the bundle $V\otimes V$ is associated to $V$, we expect that the morphisms between the constituents of $V\otimes V$ is also induced by the morphisms that defined $V$, i.e. the extension morphism. So We conclude that $\mathcal{O}(-3\eta-2K_B)$ should be injected intro $\mathcal{H}^{-1}$. But whether $\mathcal{O}(-\eta-K_B)$ is inside $\mathcal{H}^{-1}$ and $\mathcal{O}(\eta+K_B)$ is inside $\mathcal{H}^0$ depends on the the morphism mentioned above. Therefore the final result is

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&]
0\arrow[r]\& \mathcal{O}(-3\eta-2K_B)\arrow[r]\& \mathcal{H}^{-1}\arrow[r]\arrow[d, phantom, ""{coordinate, name=Z}]\& \mathcal{O}(-\eta-K_B) \arrow[dll,to path={ -- ([xshift=2ex]\tikztostart.east)
|- (Z) [near end]\tikztonodes
-| ([xshift=-2ex]\tikztotarget.west)
-- (\tikztotarget)}] \& \\
\& \mathcal{O}(\eta+K_B)\arrow[r] \& \mathcal{H}^{0}\arrow[r]\& \mathcal{O}(-\eta)\arrow[r]\& 0.
\end{tikzcd}
\end{eqnarray}

Using the relations mentioned above, we can see that this is in complete agreement with our heuristic result in $SU(5)$ example.

* Using Pontrjagin Product For Double Cover With A Vertical Component.

Without loss of generality,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
\& \& \&0\arrow[d] \& \\
\& \& \&\mathcal{L}'_1\arrow[d] \& \\
0\arrow[r] \& \mathcal{A}\arrow[r] \& \mathcal{L}_2\arrow[r] \&\mathcal{L}_1\arrow[r]\arrow[d] \& 0 \\
\& \& \&\mathcal{L}_V \arrow[d] \& \\
\& \& \&0 \& \\
\end{tikzcd}
\end{eqnarray}

where $\mathcal{A}$ and $\mathcal{L}'_1$ are line bundles supported over $\sigma$ and $\mathcal{L}_V$ is a line bundle over $V$. The parameters needed to define $\mathcal{L}$ are given by the extension groups,

\begin{eqnarray}
R^*Hom_{D(X)}(\mathcal{L}_1,\mathcal{A}) = R^* Hom_{D(B)}(Li^*_{\sigma}\mathcal{L}_1, \mathcal{A}) =  R^* Hom_{D(B)} (\mathcal{L}'_1\otimes K_B^{-1} [+1]\oplus \mathcal{L}_1|_{\sigma},\mathcal{A})\\
R^*Hom_{D(X)}(\mathcal{L}_V,\mathcal{L}'_1)= R^*Hom_{D(c)} (\mathcal{L}_V|_c,\mathcal{L}'_1|_c\otimes \mathcal{O}_c(c)[-1]),
\end{eqnarray}

where $c$ in the last line is the intersection of the vertical and horizontal components, and this extension group parameterizes  the gluing. There are three parameters in $Ext^1(\mathcal{L}_1,\mathcal{A})$,

\begin{eqnarray}
&\mathcal{E}xt\in  Hom_{D(B)}(\mathcal{L}'_1\otimes K_B^{-1},\mathcal{A})&,\nonumber \\
&Ext^1_{D(B)}(\mathcal{L}'_1,\mathcal{A})&,\nonumber \\
&Ext^1_{D(B)}(i_{\sigma}^*\mathcal{L}_V,\mathcal{A})=Hom_{D(c)} (\mathcal{L}_V,\mathcal{A}\otimes \mathcal{O}_c(c))&
\end{eqnarray}

The first group parameterizes a line bundle over the non-reduced components. The second one gives a rank two bundle over $\sigma$, and the third group corresponds to gluing on the intersection. We turn the second factor off, so we don't get the a vector bundle over $B$. Also for finding $R\pi_*V$ we need to restricted the spectral data on the zero section,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
0\arrow[r]\& \mathcal{A} \otimes K_B^{-1} \arrow[r]\& L^{-1}i^*_{\sigma}\mathcal{L}_2 \arrow[r] \& \mathcal{L}'_1 \otimes K_B^{-1} \\
\arrow[r,hook]\& \mathcal{A}\arrow[r]\&  L^{0}i^*_{\sigma}\mathcal{L}_2\arrow[r] \& \mathcal{L}_1|_{\sigma} \arrow[r] \& 0.
\end{tikzcd}
\end{eqnarray}

\begin{eqnarray}
R\pi_* V = \mathcal{A} \otimes K_B^{-1} \oplus \mathcal{R}[-1], \\
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
0\arrow[r]\& \mathcal{A}|_{\mathcal{E}xt=0} \arrow[r]\& \mathcal{R}2\arrow[r]\& \mathcal{L}_1|_{\sigma} \arrow[r] \& 0.
\end{tikzcd}
\end{eqnarray}

The Extension for this sequence is induced by $Ext_{D(B)}^1(\mathcal{L}'_1,\mathcal{A})$ restricted on $\mathcal{E}xt=0$. But we chose this to be zero everywhere. Also note that there is a topological constraint on this as before,

\begin{eqnarray}
c_1(\mathcal{A})+c_1(\mathcal{L}'_1)+[V]=2 c_1(K_B).
\end{eqnarray}

There is also a smoothness condition, that is $\pi_*V \sim i_{\sigma}^*\mathcal{L}_2$ must be locally free (smooth coherent sheaf). Also from the reasons mentioned before $i_V^*\mathcal{L}_2$ must have relative +1. Putting these requirements together one gets $\mathcal{L}_V$,

\begin{eqnarray}
\mathcal{L}_V = \mathcal{O}_V(\sigma+K_B-c_1(\mathcal{A})).
\end{eqnarray}

Now we can proceed to find the Fourier transform of $\Lambda^2V$. To compute $\mathcal{L}_2\star_A\mathcal{L}_2$ we should be careful. This is because of the existence of the vertical components, which is already mentioned. To see how to deal with this, first note that,

\begin{eqnarray}
\mathcal{L}_V\star_A\mathcal{L}_V \simeq_{Qis} 0\longrightarrow \mathcal{L}'_1\star_S \mathcal{L}'_1 \longrightarrow \mathcal{L}'_1\star \mathcal{L}_1 \longrightarrow \mathcal{L}_1 \star_A \mathcal{L}_1 \longrightarrow 0.
\end{eqnarray}

Instead of computing $\mathcal{L}_V\star_A\mathcal{L}_V$ directly, we can compute the antisymmetrized product of the inverse Fourier transform. In other words, we use,

\begin{eqnarray}
\mathcal{L}_V\star_A\mathcal{L}_V = i_{V*} \mathcal{L}_V \star \mathcal{L}_V \otimes \mathcal{O}(-V) [+1] = i_{V*}\Phi \left(\mathcal{O}_V \right)(-2\sigma+c_1(\mathcal{L}'_1)-c_1(\mathcal{A}))\otimes K_B[+2].
\end{eqnarray}

It is easy to compute the right hand side,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
0\arrow[r]\& K_B^{-1}\arrow[r]\& \mathcal{G}\arrow[r]\& \mathcal{O}(\sigma) \arrow[r]\& 0
\end{tikzcd}\\
\mathcal{L}_V\star_A\mathcal{L}_V = i_{V*} \mathcal{G}\otimes \mathcal{O}(+K_B+c_1(\mathcal{L}'_1)-c_1(\mathcal{A})) [+1]
\end{eqnarray}

Therefore one gets a diagram as follows,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
\& 0\arrow[d] \& 0\arrow[d] \& \& \\
\& \mathcal{L}'_1\star_S\mathcal{L}'_1\arrow[r, "\simeq"]\arrow[d] \& \mathcal{L}'_1\star\mathcal{L}'_1 \arrow[d] \& \& \\
0\arrow[r] \& Q \arrow[d]\arrow[r] \& \mathcal{L}'_1\star\mathcal{L}_1 \arrow[d]\arrow[r] \& \mathcal{L}_1\star_A\mathcal{L}_1 \arrow[r] \& 0 \\
\& \mathcal{L}_V\star_A \mathcal{L}_V \arrow[d]\arrow[r]\& \mathcal{L}'_1\otimes \mathcal{L}_V \arrow[d]\arrow[r] \& 0 \& \\
\& 0 \& 0 \& \&
\end{tikzcd}
\end{eqnarray}

Putting every thing together we get the result for $\mathcal{L}_1\star_A \mathcal{L}_1$,

\begin{eqnarray}
\mathcal{L}_1\star_A \mathcal{L}_1 = \mathcal{O}_V(c_1(\mathcal{L}'_1)-c_1(\mathcal{A}))[+1].
\end{eqnarray}

The idea that we want to emphasize is that we can continue this approach for finding the “antisymmetric" Pontrjagin product of spectral sheaves of higher degrees. So when we add another sheet, one gets the following quasi-isomorphism,

\begin{eqnarray}
&&\mathcal{L}_1 \star_A \mathcal{L}_1 \simeq_{Qis} \nonumber \\
&&\begin{tikzcd}[ampersand replacement=\&, column sep=small]
0\arrow[r] \& \mathcal{A} \star_S \mathcal{A} \arrow[r]\& \pi^*\pi_*\mathcal{A}\otimes \mathcal{L}_2 \arrow[r]\& \mathcal{L}_2\star_A\mathcal{L}_2 \arrow[r]\& 0.
\end{tikzcd}
\end{eqnarray}

Similar to the previous part, this quasi isomorphism is equivalent to the following diagram,

\begin{eqnarray}
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
\& 0\arrow[d] \& 0\arrow[d] \& \& \\
\& \mathcal{A}\star_S\mathcal{A}\arrow[r, "\simeq"]\arrow[d] \& \mathcal{A}\star\mathcal{A} \arrow[d] \& \& \\
0\arrow[r] \& Q \arrow[d]\arrow[r] \& \mathcal{A}\star\mathcal{L}_2 \arrow[d]\arrow[r] \& \mathcal{L}_2\star_A\mathcal{L}_2 \arrow[r] \& 0 \\
\& \mathcal{L}_1\star_A \mathcal{L}_1 \arrow[d]\arrow[r]\& \mathcal{A}\otimes \mathcal{L}_1 \arrow[d] \& \& \\
\& 0 \& 0 \& \&
\end{tikzcd}
\end{eqnarray}

Again by putting the pieces together and chasing the diagrams, one gets the final result, (if $c_1(\mathcal{A})-c_1(\mathcal{L}'_1+c_1(K_B))=0$, which is induced again by the stability of $V_2$) one gets,

\begin{eqnarray}
\mathcal{L}_2\star_A\mathcal{L}_2 = \mathcal{O}_{\sigma}\otimes K_B^{\otimes 2}.
\end{eqnarray}

This consistent with the expectation of $\Lambda^2V=\mathcal{O}_X$.

[1]
M. B. Green, J. H. Schwarz and E. Witten,
“Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology,”

Cambridge, Uk: Univ. Pr. (1987) 596 P. (Cambridge Monographs On Mathematical Physics)

[2]
V. Braun, Y. H. He, B. A. Ovrut and T. Pantev,
“A Standard model from the E(8) x E(8) heterotic superstring,"
JHEP 06, 039 (2005)
[arXiv:hep-th/0502155 [hep-th]].

[3]
L. B. Anderson,

[arXiv:0808.3621 [hep-th]].

[4]
J. Gray and J. Wang,
“Jumping Spectra and Vanishing Couplings in Heterotic Line Bundle Standard Models,"
JHEP 11, 073 (2019)
[arXiv:1906.09373 [hep-th]].

[5]
L. B. Anderson, J. Gray, A. Lukas and B. Ovrut,
“The Edge Of Supersymmetry: Stability Walls in Heterotic Theory,"
Phys. Lett. B 677, 190 (2009)
[arXiv:0903.5088 [hep-th]].

[6]
L. B. Anderson, J. Gray, A. Lukas and B. Ovrut,
“Stability Walls in Heterotic Theories,"
JHEP 0909, 026 (2009)
[arXiv:0905.1748 [hep-th]].

[7]
L. B. Anderson, J. Gray and B. Ovrut,
“Yukawa Textures From Heterotic Stability Walls,”
JHEP 1005, 086 (2010)
[arXiv:1001.2317 [hep-th]].

[8]
M. F. Atiyah,
“Complex Analytic Connections in Fibre Bundles,”
Trans. AMS, Vol. 85, No. 1 (May, 1957), pp. 181-207.

[9]
L. B. Anderson, X. Gao, J. Gray and S. J. Lee,
“Fibrations in CICY three-folds,"
JHEP 10, 077 (2017)
[arXiv:1708.07907 [hep-th]].

[10]
Y. C. Huang and W. Taylor,
“On the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau three-folds,"
JHEP 03, 014 (2019)
[arXiv:1809.05160 [hep-th]].

[11]
R. Friedman, J. Morgan and E. Witten,
“Vector bundles and F theory,"
Commun. Math. Phys. 187, 679-743 (1997)
[arXiv:hep-th/9701162 [hep-th]].

[12]
R. Friedman, J. W. Morgan, E. Witten,
“Vector bundles over elliptic fibrations,"

[13]
L. B. Anderson, X. Gao and M. Karkheiran,
“Extending the Geometry of Heterotic Spectral Cover Constructions,"
arXiv:1912.00971 [hep-th].

[14]
L. B. Anderson, H. Feng, X. Gao and M. Karkheiran,
“Heterotic/Heterotic and Heterotic/F-theory Duality,”
Phys. Rev. D 100, no.12, 126014 (2019)
[arXiv:1907.04395 [hep-th]].

[15]
R. Donagi and M. Wijnholt,
“Model Building with F-Theory,”
Adv. Theor. Math. Phys. 15, no.5, 1237-1317 (2011)
[arXiv:0802.2969 [hep-th]].

[16]
C. Beasley, J. J. Heckman and C. Vafa,
“GUTs and Exceptional Branes in F-theory - I,”
JHEP 01, 058 (2009)
[arXiv:0802.3391 [hep-th]].

[17]
R. Donagi and M. Wijnholt,
“Higgs Bundles and UV Completion in F-Theory,”
Commun. Math. Phys. 326, 287-327 (2014)
[arXiv:0904.1218 [hep-th]].

[18]
L. B. Anderson, J. J. Heckman and S. Katz,
“T-Branes and Geometry,”
JHEP 05, 080 (2014)
[arXiv:1310.1931 [hep-th]].

[19]
S. Cecotti, C. Cordova, J. J. Heckman and C. Vafa,
“T-Branes and Monodromy,”
JHEP 07, 030 (2011)
[arXiv:1010.5780 [hep-th]].

[20]
R. Donagi and M. Wijnholt,
“Gluing Branes, I,”
JHEP 05, 068 (2013)
[arXiv:1104.2610 [hep-th]].

[21]
R. Donagi and M. Wijnholt,
“Breaking GUT Groups in F-Theory,”
Adv. Theor. Math. Phys. 15, no.6, 1523-1603 (2011)
[arXiv:0808.2223 [hep-th]].

[22]
L. B. Anderson, J. Gray, M. Larfors, M. Magill and R. Schneider,
“Generalized Vanishing Theorems for Yukawa Couplings in Heterotic Compactifications,”
JHEP 05, 085 (2021)
[arXiv:2103.10454 [hep-th]].

[23]
L. B. Anderson, J. Gray, D. Grayson, Y. H. He and A. Lukas,
“Yukawa Couplings in Heterotic Compactification,”
Commun. Math. Phys. 297, 95-127 (2010)
[arXiv:0904.2186 [hep-th]].

[24]
E. I. Buchbinder, A. Constantin, J. Gray and A. Lukas,
“Yukawa Unification in Heterotic String Theory,”
Phys. Rev. D 94, no.4, 046005 (2016)
[arXiv:1606.04032 [hep-th]].

[25]
H. Hayashi, T. Kawano, R. Tatar and T. Watari,
“Codimension-3 Singularities and Yukawa Couplings in F-theory,”
Nucl. Phys. B 823, 47-115 (2009)
[arXiv:0901.4941 [hep-th]].

[26]
H. Hayashi, R. Tatar, Y. Toda, T. Watari and M. Yamazaki,
“New Aspects of Heterotic–F Theory Duality,”
Nucl. Phys. B 806, 224-299 (2009)
[arXiv:0805.1057 [hep-th]].

[27]
L. B. Anderson, J. Gray, N. Raghuram and W. Taylor,
“Matter in transition,”
JHEP 04, 080 (2016)
[arXiv:1512.05791 [hep-th]].

[28]
C. Bartocci, U. Bruzzo and D. Hernández Ruipérez,
“Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics,”
Progress in Mathematics 276 (2009)

[29]
R. Friedman,
“Algebraic Surfaces and Holomorphic Vector Bundles,”
Springer-Verlag New York, Inc. (1998)

[30]
R. Donagi, Y. H. He, B. A. Ovrut and R. Reinbacher,
“The Particle spectrum of heterotic compactifications,”
JHEP 12, 054 (2004)
[arXiv:hep-th/0405014 [hep-th]].

[31]
A. P. Braun and T. Watari,
“On Singular Fibres in F-Theory,”
JHEP 07, 031 (2013)
[arXiv:1301.5814 [hep-th]].

[32]
A. Căldăraru,
“Derived Categories of Twisted Sheaves on Elliptic Threefolds,"
arXiv Mathematics e-prints
arxiv: math/0012083

[33]
R. Donagi, S. Katz and E. Sharpe,

Adv. Theor. Math. Phys. 8, no.5, 813-859 (2004)
[arXiv:hep-th/0309270 [hep-th]].