428216622053327300 16,10 16520552027852727161441204150 25,7 148525 17,9
60438328652210496631088500 17,16 21983863245711864916146007548819920783200
18,8 18428114211388322399550 18,15 7380688961068662640476640993756532397450
19,7 135767820281303350 19,14 793510861046493220781342058077180934050 20,6
303610050 20,13 24681216007215096706670234574240357500 14,13
206807254359453225476163905375300 21,12 186826323241756428747689323102774900
15,12 54117569679383890033269008256800 22,11 260054031226217864291042886267450
16,11 3390671418197231819920137904000 23,10 41701179917420632714678892850
17,10 41701179917420632714678892850 24,9 322658009930052499145200 18,9
69860383953641753591175350 25,8 13948250904141600 19,8 8054854261423242104500
17,17 544441278609756260678075733228688830971900 20,7 13948250904141600 18,16
321496661014907630133282804314901041773300 21,6 250550 19,15
64811223550225163381933657270162177991375 14,14
5291910264323169346514519871886400 20,14
4173727947710375348427686143589666021850 15,13
2787959348321923974606513689960775 21,13
76049417732402225585766755889537693250 16,12 392220183802149653816905782829075
22,12 321974061113618057538097390567433500 17,11
13018317138588633706128782358825 23,11 231063040271162785144114394861625

table 3.6 continued.

$\bm{p}$ $n^{(1)}_{\bm{p}}$ 24,10 16520552027852727161441204150 25,9
42137416928528774899500 26,8 217335663077200 18,17
10364068253575077523538680193754837859801100 19,16
3707131738581515278187095373395684544431750 20,15
455990757375323038689152386845291780997980 21,14
17736215361969271597308350645891203085550 22,13
189541398071564094191656020501685749700 23,12
445795478920394854284300481376874700 24,11 161956900412845480278959039133500
25,10 4956305470261096852879492520 26,9 3733143718641168532250 27,8
1207298050100 18,18 254628881812169884863144737907071363850384275 19,17
154848648177340961815627818425036135998755750 20,16
34212563460249368657994755599372244965731625 21,15
2598107539139344864070481843220029434149400 22,14
61346528057542695552753204264092617938350 23,13
383896440392157077525881068439189823375 24,12
496770523785734165693381637859440175 25,11 89342683428580923415751095264250
26,10 1113322592174963231485326725 27,9 213796802016132371125 28,8 1249719025
19,18 4879913180567429516866248000617525315434124200 20,17
1846594387973758522093973511561276425586720500 21,16
255797150291719302931481295802475608460495500 22,15
12092961029767718697960992221193131180072800 23,14
173738502202568390450523304606890631973000 24,13
634051294393795149147968219347562400500 25,12
445795478920394854284300481376874700 26,11 38633710437048284516888316634750
27,10 184364061855133820169125300 28,9 7360276988409757150 29,8 29350

table 3.6 continued.

###  3.3 Discriminant loci and Yukawa couplings

We do not give genus one numbers for six and seven parameter families (3.12),
(3.13). Performing this computation would be possible if we knew the relevant
discriminants $\Delta$, which we do not. To make progress on this, we compute
a number of discriminants for other models, in the hopes of recognising a
formula that reproduces the known examples. This can be done for a CICY matrix
with identical columns, as in this case $\Delta$ takes a factored form as for
the two families above.

To compute a model’s discriminant, we compute the Yukawa couplings. These are
rational functions (in the coordinates defined in Appendix §A), and the
discriminant can be identified in the denominators. This process can be run
over all two-parameter CICYs, and we also do this for a number of three-
parameter CICYs. However, this has not yet led to any good candidates for the
models (3.12), (3.13).

The Yukawa couplings $C_{ijk}$ can be computed from the formula

$\displaystyle C_{ijk}$
$\displaystyle=\frac{\partial}{\partial\varphi^{i}}\frac{\partial}{\partial\varphi^{j}}\frac{\partial}{\partial\varphi^{k}}\mathcal{F}^{(0)}=-\Pi^{T}\cdot\Sigma\cdot\frac{\partial}{\partial\varphi^{i}}\frac{\partial}{\partial\varphi^{j}}\frac{\partial}{\partial\varphi^{k}}\Pi$
$\displaystyle=-\varpi^{T}\cdot\sigma\cdot\frac{\partial}{\partial\varphi^{i}}\frac{\partial}{\partial\varphi^{j}}\frac{\partial}{\partial\varphi^{k}}\varpi~{}.$
(3.20)

We give again here for ease of reading the matrix $\Sigma$, and also the
matrix $\sigma$:

$\Sigma=\begin{pmatrix}\phantom{-}0&\mathbbm{1}_{m+1}\\\
-\mathbbm{1}_{m+1}&0\end{pmatrix}~{},\qquad\sigma=\begin{pmatrix}0&\phantom{-}0&0&-1\\\
0&\phantom{-}0&\mathbbm{1}_{m}&\phantom{-}0\\\
0&-\mathbbm{1}_{m}&0&\phantom{-}0\\\
1&\phantom{-}0&0&\phantom{-}0\end{pmatrix}~{},\qquad m=h^{2,1}(X)~{}.$ (3.21)

These functions $C_{ijk}$ are rational function of $\varphi^{i}$, and their
denominators all contain a factor of $\Delta$, which equals the polynomial
least common multiple of the denominators of $C_{ijk}$. In practice formula
(3.20) serves to compute $C_{ijk}$ as power series in $\varpi^{i}$, and one
can fit a rational function to this. If the discriminant $\Delta$ is known,
then one can form the power series of
$\varphi^{i}\varphi^{j}\varphi^{k}\cdot\Delta\cdot C_{ijk}$ to find a
polynomial, the numerator of $C_{ijk}$. If $\Delta$ is not known then one must
fit a rational function to $C_{ijk}$, which becomes prohibitively complicated
at large numbers of parameters. A more detailed discussion of the Yukawa
couplings can be found in [63, 64].

We remark that a rule holds for CICYs with identical columns
$(d^{(1)},\,d^{(2)},\,...\,,d^{(k)})^{T}$, so of the form

$\begin{matrix}\mathbbl{P}^{n_{1}}\\\ \vdots\\\
\mathbbl{P}^{n_{k}}\end{matrix}\\!\left[\begin{matrix}d^{(1)}&...&d^{(1)}\\\
\vdots&...&\vdots\\\ d^{(k)}&...&d^{(k)}\end{matrix}\right].$ (3.22)

This rule is

$\Delta=\prod_{j_{1}=1}^{n_{1}+1}\cdot\cdot\cdot\prod_{j_{k}=1}^{n_{k}+1}\left(1-\sum_{i=1}^{k}d^{(i)}\exp\left(\frac{2\pi\text{i}\,j_{i}}{n_{i}+1}\right)\varphi_{i}^{\frac{1}{n_{i}+1}}\right)~{}.$
(3.23)

This does not give the discriminant of the families (3.22), but of their
mirrors. We discuss the parametrisation of the mirror manifolds that we are
employing in Appendix §A. We verify that this holds for all CICY
configurations of the form (3.22), so the following 11 families:

$\begin{gathered}\begin{matrix}\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}5\end{matrix}\right]~{},\qquad\begin{matrix}\mathbbl{P}^{5}\end{matrix}\\!\left[\begin{matrix}3&3\end{matrix}\right]~{},\qquad\begin{matrix}\mathbbl{P}^{7}\end{matrix}\\!\left[\begin{matrix}2&2&2&2\end{matrix}\right]~{},\\\\[20.0pt]
\begin{matrix}\mathbbl{P}^{4}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&1&1&1&1\\\
1&1&1&1&1\end{matrix}\right]~{},\qquad\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{2}\end{matrix}\\!\left[\begin{matrix}3\\\
3\end{matrix}\right]~{},\qquad\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}2\\\
4\end{matrix}\right]~{},\\\\[10.0pt] \begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{2}\\\ \mathbbl{P}^{2}\end{matrix}\\!\left[\begin{matrix}1&1&1\\\
1&1&1\\\ 1&1&1\end{matrix}\right]~{},\qquad\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{2}\end{matrix}\\!\left[\begin{matrix}2\\\ 2\\\
3\end{matrix}\right]~{},\qquad\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}1&1\\\
1&1\\\ 2&2\end{matrix}\right]~{},\qquad\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\end{matrix}\\!\left[\begin{matrix}2\\\ 2\\\ 2\\\
2\end{matrix}\right]~{},\qquad\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\end{matrix}\\!\left[\begin{matrix}1&1\\\ 1&1\\\ 1&1\\\ 1&1\\\
1&1\end{matrix}\right]~{}.\end{gathered}$ (3.24)

For the last model above, with five parameters, this discriminant is

$\Delta=\prod_{\epsilon_{i}\in\\{1,-1\\}}\left(1+\epsilon_{1}\sqrt{\varphi^{1}}+\epsilon_{2}\sqrt{\varphi^{2}}+\epsilon_{3}\sqrt{\varphi^{3}}+\epsilon_{4}\sqrt{\varphi^{4}}+\epsilon_{5}\sqrt{\varphi^{5}}\right)~{}.$
(3.25)

We proceed to tabulate all CICYs with $h^{1,1}=2$ [31], together with
$\Delta$, which we compute by forming the polynomial least common multiple of
the Yukawa coupling denominators [27].

$\tiny{\begin{array}[]{|l|}\hline\cr\vrule
depth=5.0pt,height=10.0pt,width=0.0pt\text{CICYs with }h^{1,1}=2\text{, with
the discriminant polynomial }\Delta\text{ of the mirror.
\cite[cite]{[\@@bibref{Number}{mcgovern2023a}{}{}]}}\\\
\hline\cr\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{5}\end{matrix}\\!\left[\begin{matrix}0&0&2&1\\\
2&2&1&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-4\varphi_{1}\right){}^{2}\left(1-8\varphi_{1}+16\varphi_{1}^{2}-48\varphi_{2}-320\varphi_{1}\varphi_{2}+768\varphi_{2}^{2}+256\varphi_{1}\varphi_{2}^{2}-4096\varphi_{2}^{3}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{4}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}2&0&1&1&1\\\
0&2&1&1&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-4\varphi_{1}\right)\left(1-4\varphi_{2}\right)\left(1-12\varphi_{1}+48\varphi_{1}^{2}-64\varphi_{1}^{3}-12\varphi_{2}-336\varphi_{1}\varphi_{2}-192\varphi_{1}^{2}\varphi_{2}+48\varphi_{2}^{2}-192\varphi_{1}\varphi_{2}^{2}-64\varphi_{2}^{3}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}0&2&1\\\
3&1&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-4\varphi_{1}\right){}^{2}\left(1-8\varphi_{1}+16\varphi_{1}^{2}-81\varphi_{2}-540\varphi_{1}\varphi_{2}+2187\varphi_{2}^{2}+729\varphi_{1}\varphi_{2}^{2}-19683\varphi_{2}^{3}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{6}\end{matrix}\\!\left[\begin{matrix}0&0&1&1&1\\\
2&2&1&1&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-\varphi_{1}\right){}^{2}\left(1-3\varphi_{1}+3\varphi_{1}^{2}-\varphi_{1}^{3}-48\varphi_{2}-336\varphi_{1}\varphi_{2}-48\varphi_{1}^{2}\varphi_{2}+768\varphi_{2}^{2}-768\varphi_{1}\varphi_{2}^{2}-4096\varphi_{2}^{3}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{3}\\\
\mathbbl{P}^{5}\end{matrix}\\!\left[\begin{matrix}0&1&1&1&1\\\
2&1&1&1&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-\varphi_{1}\right)\left(1-4\varphi_{1}+6\varphi_{1}^{2}-4\varphi_{1}^{3}+\varphi_{1}^{4}-16\varphi_{2}-496\varphi_{1}\varphi_{2}-496\varphi_{1}^{2}\varphi_{2}-16\varphi_{1}^{3}\varphi_{2}+96\varphi_{2}^{2}-1984\varphi_{1}\varphi_{2}^{2}+96\varphi_{1}^{2}\varphi_{2}^{2}-256\varphi_{2}^{3}-256\varphi_{1}\varphi_{2}^{3}+256\varphi_{2}^{4}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}0&2&1\\\
2&1&2\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-4\varphi_{1}\right)\left(1-12\varphi_{1}+48\varphi_{1}^{2}-64\varphi_{1}^{3}-48\varphi_{2}-696\varphi_{1}\varphi_{2}+96\varphi_{1}^{2}\varphi_{2}+768\varphi_{2}^{2}+384\varphi_{1}\varphi_{2}^{2}-432\varphi_{1}^{2}\varphi_{2}^{2}-4096\varphi_{2}^{3}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{3}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}0&2&1&1\\\
2&1&1&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-4\varphi_{1}\right)\left(1-12\varphi_{1}+48\varphi_{1}^{2}-64\varphi_{1}^{3}-16\varphi_{2}-664\varphi_{1}\varphi_{2}-544\varphi_{1}^{2}\varphi_{2}+96\varphi_{2}^{2}-1280\varphi_{1}\varphi_{2}^{2}+16\varphi_{1}^{2}\varphi_{2}^{2}-256\varphi_{2}^{3}+128\varphi_{1}\varphi_{2}^{3}+256\varphi_{2}^{4}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{4}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&1&1&1&1\\\
1&1&1&1&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt1-5\varphi_{1}+10\varphi_{1}^{2}-10\varphi_{1}^{3}+5\varphi_{1}^{4}-\varphi_{1}^{5}-5\varphi_{2}-605\varphi_{1}\varphi_{2}-1905\varphi_{1}^{2}\varphi_{2}-605\varphi_{1}^{3}\varphi_{2}-5\varphi_{1}^{4}\varphi_{2}+10\varphi_{2}^{2}-1905\varphi_{1}\varphi_{2}^{2}+1905\varphi_{1}^{2}\varphi_{2}^{2}-10\varphi_{1}^{3}\varphi_{2}^{2}-10\varphi_{2}^{3}-605\varphi_{1}\varphi_{2}^{3}-10\varphi_{1}^{2}\varphi_{2}^{3}+5\varphi_{2}^{4}-5\varphi_{1}\varphi_{2}^{4}-\varphi_{2}^{5}\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{3}\\\
\mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}2&1&1\\\
1&2&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt1-16\varphi_{1}+96\varphi_{1}^{2}-256\varphi_{1}^{3}+256\varphi_{1}^{4}-16\varphi_{2}-1261\varphi_{1}\varphi_{2}-2968\varphi_{1}^{2}\varphi_{2}+304\varphi_{1}^{3}\varphi_{2}+96\varphi_{2}^{2}-2968\varphi_{1}\varphi_{2}^{2}+69\varphi_{1}^{2}\varphi_{2}^{2}-27\varphi_{1}^{3}\varphi_{2}^{2}-256\varphi_{2}^{3}+304\varphi_{1}\varphi_{2}^{3}-27\varphi_{1}^{2}\varphi_{2}^{3}+256\varphi_{2}^{4}\\\
\hline\cr\end{array}}$ $\tiny{\begin{array}[]{|l|}\hline\cr\vrule
depth=5.0pt,height=10.0pt,width=0.0pt\text{CICYs with }h^{1,1}=2\text{, with
the discriminant polynomial }\Delta\text{ of the mirror.
\cite[cite]{[\@@bibref{Number}{mcgovern2023a}{}{}]}}\\\
\hline\cr\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}0&2\\\
3&2\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-4\varphi_{1}\right){}^{2}\left(1-8\varphi_{1}+16\varphi_{1}^{2}-216\varphi_{2}-864\varphi_{1}\varphi_{2}+11664\varphi_{2}^{2}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{5}\end{matrix}\\!\left[\begin{matrix}0&1&1&1\\\
2&2&1&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-\varphi_{1}\right)\left(1-4\varphi_{1}+6\varphi_{1}^{2}-4\varphi_{1}^{3}+\varphi_{1}^{4}-48\varphi_{2}-664\varphi_{1}\varphi_{2}-320\varphi_{1}^{2}\varphi_{2}+8\varphi_{1}^{3}\varphi_{2}+768\varphi_{2}^{2}-2176\varphi_{1}\varphi_{2}^{2}+16\varphi_{1}^{2}\varphi_{2}^{2}-4096\varphi_{2}^{3}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{5}\end{matrix}\\!\left[\begin{matrix}0&1&1&1\\\
3&1&1&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-\varphi_{1}\right){}^{2}\left(1-3\varphi_{1}+3\varphi_{1}^{2}-\varphi_{1}^{3}-81\varphi_{2}-567\varphi_{1}\varphi_{2}-81\varphi_{1}^{2}\varphi_{2}+2187\varphi_{2}^{2}-2187\varphi_{1}\varphi_{2}^{2}-19683\varphi_{2}^{3}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{3}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&1&1&1\\\
2&1&1&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt1-5\varphi_{1}+10\varphi_{1}^{2}-10\varphi_{1}^{3}+5\varphi_{1}^{4}-\varphi_{1}^{5}-16\varphi_{2}-905\varphi_{1}\varphi_{2}-1889\varphi_{1}^{2}\varphi_{2}-318\varphi_{1}^{3}\varphi_{2}+3\varphi_{1}^{4}\varphi_{2}+96\varphi_{2}^{2}-4296\varphi_{1}\varphi_{2}^{2}+814\varphi_{1}^{2}\varphi_{2}^{2}-3\varphi_{1}^{3}\varphi_{2}^{2}-256\varphi_{2}^{3}-784\varphi_{1}\varphi_{2}^{3}+\varphi_{1}^{2}\varphi_{2}^{3}+256\varphi_{2}^{4}\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{5}\end{matrix}\\!\left[\begin{matrix}0&1&1\\\
2&2&2\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-\varphi_{1}\right)\left(1-4\varphi_{1}+6\varphi_{1}^{2}-4\varphi_{1}^{3}+\varphi_{1}^{4}-128\varphi_{2}-768\varphi_{1}\varphi_{2}-128\varphi_{1}^{2}\varphi_{2}+4096\varphi_{2}^{2}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{6}\end{matrix}\\!\left[\begin{matrix}0&0&1&1\\\
2&2&2&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-\varphi_{1}\right){}^{2}\left(1-3\varphi_{1}+3\varphi_{1}^{2}-\varphi_{1}^{3}-128\varphi_{2}-320\varphi_{1}\varphi_{2}+16\varphi_{1}^{2}\varphi_{2}+4096\varphi_{2}^{2}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{7}\end{matrix}\\!\left[\begin{matrix}0&0&0&1&1\\\
2&2&2&1&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-\varphi_{1}\right){}^{3}\left(1-2\varphi_{1}+\varphi_{1}^{2}-128\varphi_{2}-128\varphi_{1}\varphi_{2}+4096\varphi_{2}^{2}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&1&1\\\
2&2&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt1-5\varphi_{1}+10\varphi_{1}^{2}-10\varphi_{1}^{3}+5\varphi_{1}^{4}-\varphi_{1}^{5}-48\varphi_{2}-1280\varphi_{1}\varphi_{2}-1672\varphi_{1}^{2}\varphi_{2}-124\varphi_{1}^{3}\varphi_{2}-\varphi_{1}^{4}\varphi_{2}+768\varphi_{2}^{2}-5888\varphi_{1}\varphi_{2}^{2}-128\varphi_{1}^{2}\varphi_{2}^{2}-4096\varphi_{2}^{3}\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{5}\end{matrix}\\!\left[\begin{matrix}0&0&2\\\
2&2&2\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-4\varphi_{1}\right){}^{2}\left(1-8\varphi_{1}+16\varphi_{1}^{2}-128\varphi_{2}-512\varphi_{1}\varphi_{2}+4096\varphi_{2}^{2}\right)\\\
\hline\cr\end{array}}$ $\tiny{\begin{array}[]{|l|}\hline\cr\vrule
depth=5.0pt,height=10.0pt,width=0.0pt\text{CICYs with }h^{1,1}=2\text{, with
the discriminant polynomial }\Delta\text{ of the mirror.
\cite[cite]{[\@@bibref{Number}{mcgovern2023a}{}{}]}}\\\
\hline\cr\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{6}\end{matrix}\\!\left[\begin{matrix}0&0&0&2\\\
2&2&2&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-4\varphi_{1}\right){}^{3}\left(1-4\varphi_{1}-128\varphi_{2}+4096\varphi_{2}^{2}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}2&1\\\
1&3\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt1-16\varphi_{1}+96\varphi_{1}^{2}-256\varphi_{1}^{3}+256\varphi_{1}^{4}-81\varphi_{2}-1809\varphi_{1}\varphi_{2}+440\varphi_{1}^{2}\varphi_{2}-400\varphi_{1}^{3}\varphi_{2}+2187\varphi_{2}^{2}+1458\varphi_{1}\varphi_{2}^{2}-3375\varphi_{1}^{2}\varphi_{2}^{2}+3125\varphi_{1}^{3}\varphi_{2}^{2}-19683\varphi_{2}^{3}\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{5}\end{matrix}\\!\left[\begin{matrix}0&1&1\\\
3&2&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-\varphi_{1}\right){}^{2}\left(1-3\varphi_{1}+3\varphi_{1}^{2}-\varphi_{1}^{3}-216\varphi_{2}-540\varphi_{1}\varphi_{2}+27\varphi_{1}^{2}\varphi_{2}+11664\varphi_{2}^{2}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}2&1\\\
2&2\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt1-16\varphi_{1}+96\varphi_{1}^{2}-256\varphi_{1}^{3}+256\varphi_{1}^{4}-48\varphi_{2}-2656\varphi_{1}\varphi_{2}-5120\varphi_{1}^{2}\varphi_{2}+512\varphi_{1}^{3}\varphi_{2}+768\varphi_{2}^{2}-8704\varphi_{1}\varphi_{2}^{2}+256\varphi_{1}^{2}\varphi_{2}^{2}-4096\varphi_{2}^{3}\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}0&2&1\\\
2&2&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-4\varphi_{1}\right)\left(1-12\varphi_{1}+48\varphi_{1}^{2}-64\varphi_{1}^{3}-48\varphi_{2}-1344\varphi_{1}\varphi_{2}-768\varphi_{1}^{2}\varphi_{2}+768\varphi_{2}^{2}-3072\varphi_{1}\varphi_{2}^{2}-4096\varphi_{2}^{3}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&1\\\
3&2\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt1-5\varphi_{1}+10\varphi_{1}^{2}-10\varphi_{1}^{3}+5\varphi_{1}^{4}-\varphi_{1}^{5}-216\varphi_{2}-2052\varphi_{1}\varphi_{2}-873\varphi_{1}^{2}\varphi_{2}+16\varphi_{1}^{3}\varphi_{2}+11664\varphi_{2}^{2}\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{3}\\\
\mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}2&1&1\\\
2&1&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt1-16\varphi_{1}+96\varphi_{1}^{2}-256\varphi_{1}^{3}+256\varphi_{1}^{4}-16\varphi_{2}-1984\varphi_{1}\varphi_{2}-7936\varphi_{1}^{2}\varphi_{2}-1024\varphi_{1}^{3}\varphi_{2}+96\varphi_{2}^{2}-7936\varphi_{1}\varphi_{2}^{2}+1536\varphi_{1}^{2}\varphi_{2}^{2}-256\varphi_{2}^{3}-1024\varphi_{1}\varphi_{2}^{3}+256\varphi_{2}^{4}\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{6}\end{matrix}\\!\left[\begin{matrix}0&0&1&1\\\
3&2&1&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-\varphi_{1}\right){}^{3}\left(1-2\varphi_{1}+\varphi_{1}^{2}-216\varphi_{2}-216\varphi_{1}\varphi_{2}+11664\varphi_{2}^{2}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&1&1\\\
3&1&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt1-5\varphi_{1}+10\varphi_{1}^{2}-10\varphi_{1}^{3}+5\varphi_{1}^{4}-\varphi_{1}^{5}-81\varphi_{2}-1674\varphi_{1}\varphi_{2}-1417\varphi_{1}^{2}\varphi_{2}+55\varphi_{1}^{3}\varphi_{2}-8\varphi_{1}^{4}\varphi_{2}+2187\varphi_{2}^{2}-10206\varphi_{1}\varphi_{2}^{2}+108\varphi_{1}^{2}\varphi_{2}^{2}-16\varphi_{1}^{3}\varphi_{2}^{2}-19683\varphi_{2}^{3}\\\
\hline\cr\end{array}}$ $\tiny{\begin{array}[]{|l|}\hline\cr\vrule
depth=5.0pt,height=10.0pt,width=0.0pt\text{CICYs with }h^{1,1}=2\text{, with
the discriminant polynomial }\Delta\text{ of the mirror.
\cite[cite]{[\@@bibref{Number}{mcgovern2023a}{}{}]}}\\\
\hline\cr\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{5}\end{matrix}\\!\left[\begin{matrix}0&0&2\\\
3&2&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-4\varphi_{1}\right){}^{3}\left(1-4\varphi_{1}-216\varphi_{2}+11664\varphi_{2}^{2}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{5}\end{matrix}\\!\left[\begin{matrix}0&1&1\\\
2&3&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-\varphi_{1}\right)\left(1-4\varphi_{1}+6\varphi_{1}^{2}-4\varphi_{1}^{3}+\varphi_{1}^{4}-216\varphi_{2}-864\varphi_{1}\varphi_{2}+72\varphi_{1}^{2}\varphi_{2}-16\varphi_{1}^{3}\varphi_{2}+11664\varphi_{2}^{2}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}0&2\\\
2&3\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-4\varphi_{1}\right)\left(1-12\varphi_{1}+48\varphi_{1}^{2}-64\varphi_{1}^{3}-216\varphi_{2}-2592\varphi_{1}\varphi_{2}+11664\varphi_{2}^{2}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}2&1\\\
3&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt1-16\varphi_{1}+96\varphi_{1}^{2}-256\varphi_{1}^{3}+256\varphi_{1}^{4}-81\varphi_{2}-4968\varphi_{1}\varphi_{2}-11664\varphi_{1}^{2}\varphi_{2}+256\varphi_{1}^{3}\varphi_{2}+2187\varphi_{2}^{2}-31347\varphi_{1}\varphi_{2}^{2}-19683\varphi_{2}^{3}\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{2}\end{matrix}\\!\left[\begin{matrix}3\\\ 3\end{matrix}\right]\\\
\vrule
depth=5.0pt,height=15.0pt,width=0.0pt1-81\varphi_{1}+2187\varphi_{1}^{2}-19683\varphi_{1}^{3}-81\varphi_{2}-15309\varphi_{1}\varphi_{2}-59049\varphi_{1}^{2}\varphi_{2}+2187\varphi_{2}^{2}-59049\varphi_{1}\varphi_{2}^{2}-19683\varphi_{2}^{3}\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&1\\\
4&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt1-5\varphi_{1}+10\varphi_{1}^{2}-10\varphi_{1}^{3}+5\varphi_{1}^{4}-\varphi_{1}^{5}-512\varphi_{2}-2816\varphi_{1}\varphi_{2}+320\varphi_{1}^{2}\varphi_{2}-144\varphi_{1}^{3}\varphi_{2}+27\varphi_{1}^{4}\varphi_{2}+65536\varphi_{2}^{2}\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{5}\end{matrix}\\!\left[\begin{matrix}0&1&1\\\
4&1&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-\varphi_{1}\right){}^{3}\left(1-2\varphi_{1}+\varphi_{1}^{2}-512\varphi_{2}-512\varphi_{1}\varphi_{2}+65536\varphi_{2}^{2}\right)\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}2\\\ 4\end{matrix}\right]\\\
\vrule
depth=5.0pt,height=15.0pt,width=0.0pt1-16\varphi_{1}+96\varphi_{1}^{2}-256\varphi_{1}^{3}+256\varphi_{1}^{4}-512\varphi_{2}-12288\varphi_{1}\varphi_{2}-8192\varphi_{1}^{2}\varphi_{2}+65536\varphi_{2}^{2}\\\
\hline\cr\vrule
depth=5.0pt,height=15.0pt,width=0.0pt\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}0&2\\\
4&1\end{matrix}\right]\\\ \vrule
depth=5.0pt,height=15.0pt,width=0.0pt\left(1-4\varphi_{1}\right){}^{3}\left(1-4\varphi_{1}-512\varphi_{2}+65536\varphi_{2}^{2}\right)\\\
\hline\cr\end{array}}$

###  3.4 Contractions and instanton summations

Upon inspecting the CICY matrices, one can see that the (7,27) model333It is
clearest here for us to refer to these manifolds by their Hodge numbers
$(h^{1,1},h^{2,1})$. The configurations for the families referred to in this
paragraph can be found in formulae (3.13), (3.12), (3.11), (3.14). is a split
of the (6,36) model, which is itself a split of the (5,45) Mirror Hulek
Verrill family. Moreover, this (5,45) family is a split of the Tetraquadric
with Hodge numbers (4,68).

Further still, a sequence of five splits from the (2,52) family444See the
configuration (3.15). yields the (7,27) family. Finally, the (2,52) family is
a split of the Quintic
$\begin{matrix}\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}5\end{matrix}\right]^{(1,101)}_{-200}$.

To pass from a manifold $Y$ to its split $\widetilde{Y}$ involves blowing up
some number of exceptional divisors, which are degree-1 rational curves (so
genus 0). Now consider a rational curve $\widetilde{C}$ on the split manifold.
Corresponding to this curve is a degree vector $\widetilde{\textbf{b}}$,
giving the curve’s homology class. In blowing down, we project out one of the
homology classes on $\widetilde{Y}$. The image of $\widetilde{C}$ in $Y$ is a
rational curve $C$ with degree vector b obtained from $\widetilde{\mathbf{b}}$
by deleting the entries corresponding to the homology class that was projected
out. Let $\rho$ denote the projection operator on degree vectors that deletes
the entry corresponding to the projected-out homology class. We expect to be
able to recover the instanton numbers $n^{(0)}_{\mathbf{b}}$ for $Y$ from the
numbers $n^{(0)}_{\widetilde{\mathbf{b}}}$ for $\widetilde{Y}$ via a sum rule

$n^{(0)}_{\mathbf{b}}=\sum_{\widetilde{\mathbf{b}}|\,\rho\left(\widetilde{\mathbf{b}}\right)=\mathbf{b}}n^{(0)}_{\widetilde{\mathbf{b}}}~{}.$
(3.26)

This can also be seen at the level of the Calabi-Yau genus prepotential $F$,
which is closely related to the topological string free energy $F_{0}$, but
incorporates the full set of topological quantities $Y_{abc}$ as follows:

$F=-\frac{1}{6}Y_{abc}t^{a}t^{b}t^{c}-\frac{1}{(2\pi\text{i})^{3}}\sum_{\mathbf{b}\geq
0}n^{(0)}_{\mathbf{b}}\,\text{Li}_{3}\left(\mathbf{q}^{\mathbf{b}}\right)~{}.$
(3.27)

Reiterating, $a,\,b,\,c$ run from $0$ to $h^{1,1}(Y)$ while $i,\,j,\,k$ run
from $1$ to $h^{1,1}(Y)$. The $t^{i}$ give coefficients in the expansion of
$Y$’s complexified Kähler form in a basis of $H^{2}(Y,\mathbbl{Z})$. We have
said this again in order to set the stage for shrinking two-cycles, a
birational map between threefolds realised by sending one of the $t^{i}$ to
zero.

Let $Y$ have $h^{1,1}=n$. Take $\widetilde{Y}$ with $h^{1,1}=n+1$ to be a
split of $Y$. The coordinates $t^{i}$, $i=1,...,n+1$ give the complexified
Kähler structure parameters on $\widetilde{Y}$, which equal the integral of
the complexified Kähler form555$J$ is the Kähler form and $B$ is the Kalb-
Ramond B-field. $B+\text{i}J$ of two-cycles in a basis of
$H_{2}(\widetilde{Y},\mathbbl{Z})$. Let $t^{n+1}$ be the integral of
$B+\text{i}J$ over the two-cycle wrapped by the collapsing two-spheres. Then
we should have

$F_{Y}(t^{1},...,t^{n})=F_{\widetilde{Y}}(t^{1},...,t^{n},t^{n+1})\bigg{|}_{t^{n+1}=0}~{}.$
(3.28)

Inspecting (3.4), setting $t_{n+1}=0$, and comparing the $q$-expansions of
both sides, we recover the sum rule for all degree vectors $\mathbf{b}$ except
the zero vector. Equating the terms linear, quadratic, and cubic in $t^{i}$ we
get equalities

$\widetilde{Y}_{abc}=Y_{abc}~{},\qquad 0\leq a,b,c\leq n\text{ and
}(a,b,c)\neq(0,0,0)~{},$ (3.29)

with the left hand side giving topological numbers $\widetilde{Y}_{abc}$ on
$\widetilde{Y}$ and the right hand side giving the numbers $Y_{abc}$ for $Y$.

The exception to the sum rule is related to the change in Euler
characteristic. In light of $\text{Li}_{3}(1)=\zeta(3)$, equating constant
terms in (3.28) yields

$\sum_{k}n^{(0)}_{0,...,0,k}=-\frac{1}{2}\bigg{[}\chi\left(\widetilde{Y}\right)-\chi\left(Y\right)\bigg{]}~{}.$
(3.30)

This formula could have been anticipated more geometrically, as the left hand
side counts the curves that are blown down in the birational map
$\widetilde{Y}\mapsto Y$. Since $\chi\left(\mathbbl{P}^{1}\right)=2$, minus
twice the above left hand side gives the change in Euler characteristic.

This can be verified for each pair of CICYs where one splits the other, but
other examples can be found for suitable $\widetilde{Y}$. For instance, the
genus 0 invariants for the families

$\begin{matrix}\mathbbl{P}^{3}\\\
\mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}1&1&2\\\
1&1&2\end{matrix}\right]~{},\qquad\begin{matrix}\mathbbl{P}^{3}\\\
\mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}1&2&1\\\
1&1&2\end{matrix}\right]~{},\qquad\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}2\\\
4\end{matrix}\right]~{},$ (3.31)

can be summed in this manner to recover invariants for the degree 8
hypersurface in the weighted projective space
$\mathbbl{W}\mathbbl{P}^{4}_{(1,1,1,1,4)}$. This is mirror to a manifold with
Picard-Fuchs operator labelled AESZ7 in the database [65], a hypergeometric
model with indices
$\left(\frac{1}{8}~{},\,\frac{3}{8}~{},\,\frac{5}{8}~{},\,\frac{7}{8}\right)$.
This family has also seen recent attention in [66], wherein three-loop “wheel”
Feynman diagrams were evaluated in terms of Calabi-Yau periods.

In [34], a two-parameter non-Abelian gauged linear sigma model (GLSM) [67]
with gauge group

$G=\frac{U(1)\times U(1)\times O(2)}{\\{\pm 1\\}\times\\{\pm
1\\}\times\\{\pm\mathbbm{1}_{2}\\}}$ (3.32)

was studied. This model has six phases, and of relevance to our discussion is
their phase $\text{I}_{+}$ and phase IV. These were both geometric, in the
sense that the GLSM flowed to a nonlinear sigma model on Calabi-Yau spaces
with Hodge numbers $(h^{1,1},h^{2,1})=(2,24)$. The phase $\text{I}_{+}$ model
was a quotient of a complete intersection in a toric variety, while the phase
IV geometry was a determinental hypersurface in another toric variety. We
remark that the phase $\text{I}_{+}$ geometry can be contracted in either of
two ways, and the instanton sums recover the genus-0 invariants and Euler
characteristic of either the Reye congruence [60, 68] or a $\mathbbl{Z}_{2}$
quotient of the quadriconic in $\mathbbl{P}^{7}$:

$\begin{matrix}\mathbbl{P}^{4}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&1&1&1&1\\\
1&1&1&1&1\end{matrix}\right]_{/\mathbbl{Z}_{2}}~{},\qquad\mathbbl{P}^{7}[2,2,2,2]_{/\mathbbl{Z}_{2}}~{}.$
(3.33)

On the other hand, instanton number summations suggest that the phase IV
geometry can be contracted to the intersection of two degree four
hypersurfaces in $\mathbbl{W}\mathbbl{P}^{5}_{(1,1,1,1,2,2)}$. This manifold
is mirror to a geometry with Picard-Fuchs operator AESZ10, a hypergeometric
operator with indices
$\left(\frac{1}{4}~{},\,\frac{1}{4}~{},\,\frac{3}{4}~{},\,\frac{3}{4}\right)$.

We will briefly mention, but say no more about, the question of connecting all
Calabi-Yau threefolds by such transitions (including generalisations, such as
where instead of two-spheres, orbifolds shrink), which in the mathematics
literature relates to a conjecture of Reid [69], see also [70] . The conifold
transition in string theory was studied in [71, 72, 73]. It was shown in [74]
that these transitions could be explained by black hole condensation.

Studying the prepotential $F$ makes clear that the genus 0 instanton numbers
on the split manifold can be summed into the genus 0 numbers of the contracted
manifold, with the topological quantities and change in Euler character
accounted for. At higher genera, setting a $t^{k}$ to zero reproduces the same
summation rules for higher genus instanton numbers. So the rule (3.26) also
holds at higher genus:

$n^{(g)}_{\mathbf{b}}=\sum_{\widetilde{\mathbf{b}}|\,\rho\left(\widetilde{\mathbf{b}}\right)=\mathbf{b}}n^{(g)}_{\widetilde{\mathbf{b}}}~{}.$
(3.34)

Such data can serve as useful boundary data in higher genus free energy
computations, as was noted in [56].

###  3.5 Coxeter groups

When the tables of genus 0 and 1 instanton numbers in [2] were produced, a
number of repetitions of values were noted. A similar phenomenon can be
observed in our tables table 3.3 and table 3.5 for the Tetraquadric. For
example, we have equalities

$\displaystyle n^{(0)}_{(1,1,1,0)}=n^{(0)}_{(6,1,1,0)}=n^{(0)}_{(10,3,1,1)}$
$\displaystyle=2432$ (3.35) $\displaystyle
n^{(0)}_{(3,3,1,1)}=n^{(0)}_{(7,3,1,1)}=n^{(0)}_{(13,3,3,1)}$
$\displaystyle=241754112$ $\displaystyle
n^{(1)}_{(4,2,1,0)}=n^{(1)}_{(7,2,2,0)}=n^{(1)}_{(8,4,1,0)}$
$\displaystyle=-96$ $\displaystyle
n^{(1)}_{(3,2,2,2)}=n^{(1)}_{(9,2,2,2)}=n^{(1)}_{(12,3,2,2)}$
$\displaystyle=1812768672~{}.$

It should be noted that when genus 0 numbers $n^{(0)}_{\bm{p}}$ and
$n^{(0)}_{\bm{q}}$ are equal, then the genus 1 numbers $n^{(1)}_{\bm{p}}$ and
$n^{(1)}_{\bm{q}}$ are equal.

In fact, instanton numbers are equal for degree vectors related by a certain
linear transformation:

$n^{(g)}_{(i,j,k,l)}=n^{(g)}_{-i+2j+2k+2l,j,k,l}~{}.$ (3.36)

Our tables support this claim for genera 0 and 1, but this is expected to hold
at every genus. It is true to the full extent of the tables of genus one
numbers, but at genus 0 there is one exception, which is that this operation
takes $(1,0,0,0)$ to $(-1,0,0,0)$, yet the instanton number
$n^{(0)}_{(1,0,0,0)}=48$ is nonzero. For this particular Tetraquadric family,
the above identity (including the mentioned exception) can be proven at every
genus, either by inheriting an action from the splitting (5,45) family (which
itself has a proven Coxeter symmetry [2]), or by the arguments from flop
transitions that appear in [75]. As we go on to give more examples of such
identities, they will for the purposes of this thesis strictly be conjectures
supported by the instanton number computations. Any example where these
reflections can be traced to a flop transition (such as CICYs that split
another family) would see a rigorous proof along the same lines, and indeed a
number of our examples appear in [75] with proofs.

We will denote this operation by

$g_{1}:(i,j,k,l)\mapsto(2d-3i,j,k,l)~{},\qquad d=i+j+k+l~{}.$ (3.37)

This operation is an involution, with $g_{1}^{2}=1$. Additionally, there are
the obvious permutation operations that leave instanton numbers invariant,
because the mirror Tetraquadric’s complex structure moduli $\varphi_{i}$ can
be exchanged by an $S_{4}$ symmetry. The group $S_{4}$ has three generators,
whose actions on $\mathbbl{Z}^{4}$ are

$s_{1}:(i,j,k,l)\mapsto(j,i,k,l)~{},\quad
s_{2}:(i,j,k,l)\mapsto(i,k,j,l)~{},\quad s_{3}:(i,j,k,l)\mapsto(i,j,l,k)~{}.$
(3.38)

We should be clear that there are a few distinct group orbits with the same
instanton number, a point that demands further study. Collecting these
together, we find the following group presentation

$\langle
g_{1},\,s_{1},\,s_{2},\,s_{3}\,|d_{1}^{2}=s_{1}^{2}=s_{2}^{2}=s_{3}^{2}=1~{},\left(d_{1}s_{2}\right)^{2}=\left(d_{1}s_{3}\right)^{2}=1~{},\,\left(s_{1}s_{2}\right)^{3}=\left(s_{1}s_{3}\right)^{3}=\left(s_{2}s_{3}\right)^{3}=1\rangle~{}.$
(3.39)

Note that the product $d_{1}s_{1}$ is a group element of infinite order. This
is a group generated by reflections $r_{i}$, and a symmetric matrix $m_{ij}$
(with two formally infinite entries) gives the relations that products of
reflections obey:

$\left(r_{i}r_{j}\right)^{m_{ij}}=1~{}.$ (3.40)

Groups with such a presentation are termed _Coxeter groups_. For our group,
the Coxeter matrix $m_{ij}$ reads

$m_{ij}=\begin{pmatrix}1&\infty&2&2\\\ \infty&1&3&2\\\ 2&3&1&3\\\
2&2&3&1\end{pmatrix}~{}.$ (3.41)

The number of generators $r_{i}$ is termed the _rank_ of the Coxeter group.
The data of a Coxeter matrix can be neatly encoded in a Coxeter diagram. This
is a graph with a number of nodes equal to the group’s rank. If $m_{ij}=2$,
then no edges connect the $i,\,j$ nodes. If $m_{ij}=3$ then an unlabelled edge
connects nodes $i,\,j$. If $m_{ij}\geq 4$ then nodes $i,\,j$ are connected by
an edge labelled with the entry $m_{ij}$. The Tetraquadric’s Coxeter group has
the following diagram:

$\infty$

The appearance of Coxeter group actions as symmetries of sets of instanton
numbers was also discussed in [75, 76], and were explained by flop transitions
between manifolds in the same complex deformation family. We incorporate
permutation symmetries in our discussion, so our presented Coxeter groups
differ to those of [75] . The paper [77] explained how an infinite symmetry
group could arise from infinite sequences of birational transformations.

We list some CICY matrices for which a Coxeter group symmetry acts on the
instanton numbers. Each of these geometries possesses an $S_{h^{1,1}}$
symmetry, which is extended by an operation $g$ for which we give the action
on the second homology, to get the group specified by the diagram that we
give. When we give the operation $g$, the symbol $d$ refers to to the sum of a
vector’s elements. For instance in the first example $d=i+j+k+l+m$, while in
the third $d=i+j+k$.

$\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\end{matrix}\\!\left[\begin{matrix}1&1\\\
1&1\\\ 1&1\\\ 1&1\\\ 1&1\end{matrix}\right]~{},\qquad
g:(i,j,k,l,m)\mapsto(d-2i,j,k,l,m)~{},$

$6$

$\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\end{matrix}\\!\left[\begin{matrix}2\\\ 2\\\ 2\\\
2\end{matrix}\right]~{},\qquad g:(i,j,k,l)\mapsto(2d-3i,j,k,l)~{},$

$\infty$

$\begin{matrix}\mathbbl{P}^{2}\\\ \mathbbl{P}^{2}\\\
\mathbbl{P}^{2}\end{matrix}\\!\left[\begin{matrix}1&1&1\\\ 1&1&1\\\
1&1&1\end{matrix}\right]~{},\qquad g:(i,j,k)\mapsto(2d-3i,j,k)~{},$

$\infty$

$\begin{matrix}\mathbbl{P}^{4}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&1&1&1&1\\\
1&1&1&1&1\end{matrix}\right]~{},\qquad g:(i,j)\mapsto(4d-5i,j)~{},$

$\infty$

$\begin{matrix}\mathbbl{P}^{3}\\\
\mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}1&1&2\\\
1&1&2\end{matrix}\right]~{},\qquad g:(i,j)\mapsto(6d-7i,j)~{},$

$\infty$

$\begin{matrix}\mathbbl{P}^{3}\\\
\mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}1&1&2\\\
1&2&1\end{matrix}\right]~{},\qquad g:(i,j)\mapsto(7d-8i,j)~{},$

$\infty$

In the latter three two-parameter examples we see the appearance of the
infinite dihedral group. In §3.7 we will discuss the application of this
infinite symmetry to the computation of higher genus invariants. All of the
above examples possess an $S_{h^{1,1}}$ permutation symmetry, but Coxeter
group symmetries can also be found for manifolds that do not have this
permutation symmetry. Some examples, for which we now give a full set of
generators, are as follows:

$\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}2\\\
4\end{matrix}\right]~{},\qquad g:(i,j)\mapsto(4j-i,j)~{},$

$\displaystyle\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}1&1\\\ 1&1\\\
2&2\end{matrix}\right]~{},$ $\displaystyle g:(i,j,k)\mapsto(2k+j-i,j,k)~{},$
$\displaystyle s:(i,j,k)\mapsto(j,i,k)$

$6$

$\displaystyle\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{2}\end{matrix}\\!\left[\begin{matrix}2\\\ 2\\\
3\end{matrix}\right]~{},$ $\displaystyle g:(i,j,k)\mapsto(3k+2j-i,j,k)~{},$
$\displaystyle s:(i,j,k)\mapsto(j,i,k)$

$\infty$

In the first of the above two examples, we have a $\mathbbl{Z}_{2}$ action but
no permutation symmetry. This should be contrasted with manifolds that possess
a permutation symmetry but have no extra $\mathbbl{Z}_{2}$ symmetry action
that changes the sum $i+j$, for example the $h^{1,1}=2$ bicubic.

The second of the above three examples is interesting insofar as the Coxeter
group is finite, being the dihedral group of order 12. This is the Weyl group
of $G_{2}$. It would be interesting in future work to address the question of
whether a Kac-Moody symmetry can be realised in string compactifications on
manifolds with Coxeter symmetries of the forms that we describe, and were this
the case then this example could be the simplest place to start.

While the instanton numbers are constant across orbits of the Coxeter groups
we have given, it is not the case that each such orbit has a distinct
associated instanton number. There is additional work to be done in explaining
the additional repetitions. For example, the Coxeter group actions that we
have given above for the families

$\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{2}\end{matrix}\\!\left[\begin{matrix}2\\\ 2\\\
3\end{matrix}\right]^{(3,75)}_{-144}~{},\qquad\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{3}\end{matrix}\\!\left[\begin{matrix}1&1\\\
1&1\\\ 2&2\end{matrix}\right]^{(3,55)}_{-104}~{},$ (3.42)

do not change the third component $k$ of the index vector $(i,j,k)$.
Nonetheless, the tables of instanton numbers reveal some repetitions of
instanton numbers for different values of $k$. We produce those tables to a
low order here. Note for instance that for the $\chi=-144$ family

$n^{(0)}_{(1,0,13)}=n^{(0)}_{(2,0,7)}=5221882080\qquad\text{and}\qquad
n^{(0)}_{(1,0,13)}=n^{(0)}_{(5,0,4)}=2115255492,$ (3.43)

while for the $\chi=-104$ family we observe

$n^{(0)}_{(1,0,5)}=n^{(0)}_{(3,0,3)}=125824\qquad\text{and}\qquad
n^{(0)}_{(3,0,9)}=n^{(0)}_{(5,0,7)}=130181768448~{}$ (3.44)

Equalities like this are not explained by the flop operations appearing in
[75], and for the time being we cannot explain them. It may be of note that
the above index vectors have components that are 0.

In another attempt to generalise the results of [75], we can see manifolds
with Coxeter symmetries that are not CICYs. We defer discussion of examples
with groups larger than $\mathbbl{Z}_{2}$ to future work, and here remark that
nontrivial $\mathbbl{Z}_{2}$ symmetries can be found for the following
manifolds, which appeared in the tables of [78]. Details of Calabi-Yau
intersections in weighted projective space that we avail of are found in [79].
Since the instanton numbers that we compute for their $\chi=-112$ families
agree, we conjecture that these two families are identical. We also fix a typo
in the configuration matrix for the $\chi=-98$ family.

$\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{W}\mathbbl{P}^{5}_{(1,1,1,1,2,2)}\end{matrix}\\!\left[\begin{matrix}0&1&1\\\
4&2&2\end{matrix}\right]_{\chi=-112}\;\cong\;\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{W}\mathbbl{P}^{4}_{(1,1,1,1,2)}\end{matrix}\\!\left[\begin{matrix}0&2\\\
4&2\end{matrix}\right]_{\chi=-112}~{},\qquad g:(i,j)\mapsto(2j-i,j)~{}.$
(3.45) $\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{W}\mathbbl{P}^{4}_{(1,1,1,1,2)}\end{matrix}\\!\left[\begin{matrix}1&1&1\\\
2&2&2\end{matrix}\right]_{\chi=-98}~{},\qquad g:(i,j)\mapsto(4j-i,j)~{}.$
(3.46)

Finally, we remark that the instanton numbers for the two-parameter geometry
given in Phase IV of the Knapp-Hori model [34] possess a $\mathbbl{Z}_{2}$
symmetry given by the action $g:(i,j)\mapsto(i,5i-j)$, but this only holds for
vectors with $i>0$ at genus 0. Such $\mathbbl{Z}_{2}$ symmetries and their
utility in higher genus computations were identified in [56].

The content of the following tables is available in electronic form [1].

$\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$
$n^{(0)}_{\bm{p}}$ 0,0,1 128 3,2,3 598081536 2,1,8 1320335902720 1,0,0 32
3,3,2 11713408 2,2,7 22815978525184 0,0,2 120 4,0,4 4264256 3,0,8 21118162688
1,0,1 384 4,1,3 27253248 3,1,7 5536125819904 1,1,0 32 4,2,2 2005824 3,2,6
39458161495040 0,0,3 128 4,3,1 384 3,3,5 26335226884096 1,0,2 2368 5,0,3 10496
4,0,7 28827891712 1,1,1 3072 5,1,2 2368 4,1,6 3144631756416 2,0,1 128 0,0,9
128 4,2,5 8535762597376 0,0,4 104 1,0,8 2692704 4,3,4 1874830230016 1,0,3
10496 1,1,7 1625063424 4,4,3 31657668096 1,1,2 75648 2,0,7 119377920 5,0,6
5865465088 2,0,2 5056 2,1,6 23048243712 5,1,5 224531369984 2,1,1 3072 2,2,5
111830923392 5,2,4 186198155264 0,0,5 128 3,0,6 368140672 5,3,3 8915561216
1,0,4 38624 3,1,5 25668767488 5,4,2 11713408 1,1,3 958464 3,2,4 40106733568
6,0,5 119377920 2,0,3 70656 3,3,3 3737054208 6,1,4 1082082304 2,1,2 293376
4,0,5 119377920 6,2,3 142328576 2,2,1 8000 4,1,4 2487784448 6,3,2 293376 3,0,2
2368 4,2,3 945297408 7,0,4 38624 3,1,1 384 4,3,2 11713408 7,1,3 10496 0,0,6
120 4,4,1 128 0,0,12 104 1,0,5 125824 5,0,4 2692704 1,0,11 36442112 1,1,4
8368448 5,1,3 9744384 1,1,10 113160255872 2,0,4 626432 5,2,2 293376 2,0,10
8125777088 2,1,3 9744384 6,0,3 128 2,1,9 8186269737984 2,2,2 2005824 0,0,10
120 2,2,8 238488938059776 3,0,3 125824 1,0,9 6679296 3,0,9 130181768448 3,1,2
293376 1,1,8 7268894432 3,1,8 58621972428800 3,2,1 3072 2,0,8 529534464 3,2,7
755784523825152 4,0,2 120 2,1,7 188236812288 3,3,6 1028951153011840 0,0,7 128
2,2,6 1803958100480 4,0,8 310514571520 1,0,6 373952 3,0,7 3018823680 4,1,7
63737659456000 1,1,5 57106432 3,1,6 428709670912 4,2,6 351572541694080 2,0,5
4264960 3,2,5 1533012331520 4,3,5 186761410553984 2,1,4 180973568 3,3,4
430848150336 4,4,4 11079782988096 2,2,3 142328576 4,0,6 2151740096 5,0,7
130181768448 3,0,4 2692704 4,1,5 111475558400 5,1,6 10983601432704 3,1,3
27253248 4,2,4 128007090432 5,2,5 22728981286912 3,2,2 3662848 4,3,3
8915561216 5,3,4 3805839345664 3,3,1 3072 4,4,2 20508560 5,4,3 47775759360
4,0,3 70656 5,0,5 174847616 5,5,2 11713408 4,1,2 75648 5,1,4 2487784448 6,0,6
8125777088 4,2,1 128 5,2,3 598081536 6,1,5 224531369984 0,0,8 104 5,3,2
3662848 6,2,4 128007090432 1,0,7 1033472 6,0,4 626432 6,3,3 3737054208 1,1,6
326085888 6,1,3 958464 6,4,2 2005824 2,0,6 24162944 6,2,2 5056 7,0,5 36442112
2,1,5 2324460544 0,0,11 128 7,1,4 180973568 2,2,4 5014078976 1,0,10 15887680
7,2,3 9744384 3,0,5 36442112 1,1,9 29762052096 7,3,2 2368 3,1,4 1082082304
2,0,9 2151748608 8,0,4 104 Table 3.7: The genus 0 instanton numbers of total
degree $\leqslant 12$ for the $\chi=-104$ family (3.42). The numbers not in
this list are either zero, or given by those in the table after permuting the
first two indices.

$\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$
$n^{(0)}_{\bm{p}}$ $\bm{p}$ $n^{(0)}_{\bm{p}}$ 0,0,1 168 5,0,4 2115255492
5,4,3 6055793581127544 6,4,4 19702612873693234944 1,0,0 54 5,1,3 30179989584
5,5,2 29817003490128 6,5,3 400441073987635488 0,0,2 168 5,2,2 6738481008 6,0,6
29850028039080 6,6,2 678024552756840 1,0,1 1080 5,3,1 35294184 6,1,5
2540902631155632 7,0,7 6624537453484920 1,1,0 180 5,4,0 54 6,2,4
5011092898162560 7,1,6 1009832275947370032 0,0,3 144 6,0,3 5686200 6,3,3
801503420918760 7,2,5 3836114182573117632 1,0,2 9504 6,1,2 9589752 6,4,2
9842930030808 7,3,4 1448583132341564928 1,1,1 22968 6,2,1 84240 6,5,1
3363048504 7,4,3 65054184230486808 2,0,1 1080 0,0,10 168 6,6,0 144 7,5,2
246714051981816 2,1,0 54 1,0,9 112746384 7,0,5 685227318336 7,6,1 24516763128
0,0,4 168 1,1,8 408220124400 7,1,4 18624092277168 7,7,0 180 1,0,3 55080 2,0,8
29153182176 7,2,3 9866075528304 8,0,6 170870441516784 1,1,2 801720 2,1,7
26462705388768 7,3,2 275408356176 8,1,5 9942236934310944 2,0,2 55080 2,2,6
565228067371704 7,4,1 179638056 8,2,4 12999327967495584 2,1,1 84240 3,0,7
414019483488 8,0,4 828397800 8,3,3 1333022580438624 2,2,0 144 3,1,6
137077369593336 8,1,3 3897248904 8,4,2 9842930030808 3,0,1 168 3,2,5
1091242103367168 8,2,2 212527800 8,5,1 1672396776 0,0,5 168 3,3,4
771574529680320 8,3,1 84240 9,0,5 414019483488 1,0,4 258876 4,0,6 685227318336
9,0,3 144 9,1,4 6324878723688 1,1,3 14272344 4,1,5 86145995276352 0,0,13 168
9,2,3 1709274209400 2,0,3 1045440 4,2,4 249147504104832 1,0,12 2115255492
9,3,2 19569181320 2,1,2 9589752 4,3,3 56646795125808 1,1,11 42320995599600
9,4,1 2286360 2,2,1 823968 4,4,2 950998199904 2,0,11 2966972060160 10,0,4
12531888 3,0,2 94248 5,0,5 147357745992 2,1,10 13783408869528072 10,1,3
14272344 3,1,1 84240 5,1,4 6324878723688 2,2,9 1570644007964714736 10,2,2
55080 3,2,0 54 5,2,3 5553133901424 3,0,10 211313193184296 0,0,15 144 0,0,6 144
5,3,2 275408356176 3,1,9 391767299135571456 1,0,14 12507646968 1,0,5 1045440
5,4,1 387427104 3,2,8 19261188790077746538 1,1,13 669793850973648 1,1,4
169945416 5,5,0 180 3,3,7 108334675692791766768 2,0,13 46536192247248 2,0,4
12531888 6,0,4 2868991776 4,0,9 2030806663104960 2,1,12 532443125472289380
2,1,3 422121240 6,1,3 30179989584 4,1,8 1732761009324236286 2,2,11
148032484883296635024 2,2,2 212527800 6,2,2 4691149344 4,2,7
39401939588604883920 3,0,12 8041290548966712 3,0,3 5686200 6,3,1 14832456
4,3,6 102365041178451406320 3,1,11 37113307255824419664 3,1,2 37017000 7,0,3
1045440 4,4,5 42886986818729501952 3,2,10 4635692472845342699712 3,2,1 2286360
7,1,2 801720 5,0,8 3683509791835230 3,3,9 71228168955194892241824 3,3,0 180
7,2,1 1080 5,1,7 1475899281351731232 4,0,11 190978119261948528 4,0,2 55080
0,0,11 168 5,2,6 15437538722622496320 4,1,10 428898270234051873648 4,1,1 22968
1,0,10 312318288 5,3,5 17666622173504257920 4,2,9 26635577378628399257880
0,0,7 168 1,1,9 2077856570952 5,4,4 2952225900540710424 4,3,8
208542584860846725573792 1,0,6 3781080 2,0,9 147357745992 5,5,3
65441894934804480 4,4,7 310728018286366928659728 1,1,5 1538714160 2,1,8
241891770932622 6,0,7 1358732492843328 5,0,10 936802041472321344 2,0,5
112746384 2,2,7 9578647470994416 6,1,6 243556290929859120 5,1,9
1084155094548776246256 2,1,4 10651393728 3,0,8 3764269848150 6,2,5
1092271314577190688 5,2,8 34504030323455683365540 2,2,3 18704746728 3,1,7
2356453861300944 6,3,4 485201908448389176 5,3,7 137334444454235316481488 3,0,4
159172380 3,2,6 37164898364815152 6,4,3 25578337459800960 5,4,6
100771753799790046258992 3,1,3 3897248904 3,3,5 58494821385825792 6,5,2
113389478053344 5,5,5 15002509184093317799448 3,2,2 1536760944 4,0,7
12074918985360 6,6,1 12857494104 6,0,9 1114166808793427904 3,3,1 14832456
4,1,6 3130393529188872 7,0,6 89683487215200 6,1,8 662747101613708918166 4,0,3
12531888 4,2,5 20023868970613584 7,1,5 6340790927783952 6,2,7
10648710770871871691424 4,1,2 57195792 4,3,4 12074786392584528 7,2,4
10262891970293004 6,3,6 20684312507372786347752 4,2,1 2286360 4,4,3
815165175453336 7,3,3 1333022580438624 6,4,5 6916476884469525094032 4,3,0 54
5,0,6 6028970554656 7,4,2 13029988164048 6,5,4 411546921208916198364 5,0,2
9504 5,1,5 618734398390992 7,5,1 3363048504 6,6,3 3494470176162937224 5,1,1
1080 5,2,4 1475296893039852 7,6,0 54 7,0,8 326486183204225142 0,0,8 168 5,3,3
283970290298616 8,0,5 685227318336 7,1,7 95346746262039029904 1,0,7 12531888
5,4,2 4180709760048 8,1,4 14260130266464 7,2,6 729068917506240634848 1,1,6
11407448232 5,5,1 1672396776 8,2,3 5553133901424 7,3,5 629003708472330915264
2,0,6 828397800 6,0,5 414019483488 8,3,2 105124396536 7,4,4
82046840116560882948 2,1,5 185136252912 6,1,4 14260130266464 8,4,1 35294184
7,5,3 1485487116716515272 2,2,4 868185209088 6,2,3 9866075528304 9,0,4
159172380 7,6,2 2309426676472032 3,0,5 2868991776 6,3,2 377099230176 9,1,3
422121240 7,7,1 85286277432 3,1,4 196654202136 6,4,1 387427104 9,2,2 9589752
8,0,7 20915462494951344 3,2,3 268467230952 6,5,0 54 9,3,1 168 8,1,6
2715306672487631616 3,3,2 19569181320 7,0,4 2115255492 0,0,14 168 8,2,5
8705702714908296384 4,0,4 828397800 7,1,3 15496835472 1,0,13 5221882080 8,3,4
2759145061379596596 4,1,3 15496835472 7,2,2 1536760944 1,1,12 173059142952312
8,4,3 102967117070019696 4,2,2 4691149344 7,3,1 2286360 2,0,12 12074918985360
8,5,2 318500079686208 4,3,1 35294184 8,0,3 55080 2,1,11 89309343886076376
8,6,1 24516763128 4,4,0 144 8,1,2 9504 2,2,10 16165319885559734832 8,7,0 54
5,0,3 12531888 0,0,12 144 3,0,11 1358732492843328 9,0,6 211313193170328 5,1,2
37017000 1,0,11 828397800 3,1,10 4045017572706202992 9,1,5 9942236934310944
5,2,1 823968 1,1,10 9721605877056 3,2,9 323487707741738306640 9,2,4
10262891970293004 6,0,2 168 2,0,10 685227318336 3,3,8 3085305331283577953136
9,3,3 801503420918760 0,0,9 144 2,1,9 1932024378377232 4,0,10
20915462494951344 9,4,2 4180709760048 1,0,8 38713950 2,2,8 133082696708836560
4,1,9 29595710277925790904 9,5,1 387427104 1,1,7 72542163168 3,0,9
29850028039080 4,2,8 1140647229711563099904 10,0,5 147357745992 2,0,7
5221882080 3,1,8 33021971994940200 4,3,7 5313981987997190937072 10,1,4
1560583187460 2,1,6 2455050545136 3,2,7 949255452430119360 4,4,6
4394846243085819779592 10,2,3 268467230952 2,2,5 26018243190288 3,3,6
2949179390777334672 5,0,9 64083374604252864 10,3,2 1536760944 3,0,6
38437207344 4,0,8 170870441516784 5,1,8 44892419985923962284 10,4,1 22968
3,1,5 6157144423728 4,1,7 83229565047206400 5,2,7 846447272733537148032 11,0,4
258876 3,2,4 22069473542568 4,2,6 1047154408278044472 5,3,6
1887461261292502202904 11,1,3 55080 3,3,3 5853218557032 4,3,5
1392090222866615136 5,4,5 706177757611379483424 4,0,5 29153182176 4,4,4
259340008844756376 5,5,4 45586829807821571112 4,1,4 1560583187460 5,0,7
170870441516784 6,0,8 44262594615526560 4,2,3 1709274209400 5,1,6
36397140905520432 6,1,7 15022136654860343184 4,3,2 105124396536 5,2,5
193464393164553024 6,2,6 133596413514797157576 4,4,1 179638056 5,3,4
100247137259690592 6,3,5 132683253333213798864 Table 3.8: The genus 0
instanton numbers of total degree $\leqslant 15$ for the $\chi=-144$ family
(3.42). The numbers not in this list are either zero, or given by those in the
table after permuting the first two indices.

###  3.6 Higher genus mirror symmetry

The higher genus B-model prepotentials can be computed via a recursive
procedure, originally due to Bershadsky, Cecotti, Ooguri, and Vafa [16, 15].
This was refined in work by Yamaguchi and Yau [80] who demonstrated that the
genus-$g$ prepotential was a polynomial in a set of propagators. This was
further refined in work by Alim, Laenge, and Scheidegger [81, 82], and in [83]
Huang, Katz, and Klemm phrased the recursion as a set of PDEs. This has also
been reviewed by Elmi [84] and in the notes [85]. Work by Klemm, Huang, and
Quackenbush [17] drove the computation to genus 51 for the Quintic, and this
has recently been driven higher still by incorporating the modularity of
D4-D2-D0 bound states [86, 87]. Additional higher genus results and other
developments are found in [83, 88] .

The point of departure for this computation is in realising the complex
structure moduli space of $X$ as a complex, Kähler metric space, with Kähler
potential

$K(\bm{\varphi},\overline{\bm{\varphi}})=-\log\left(-\text{i}\Pi^{\dagger}\Sigma\Pi\right)~{}.$
(3.47)

The metric then is

$G_{i\bar{j}}=\partial_{i}\partial_{\bar{j}}K~{}.$ (3.48)

This metric can be used to raise and lower indices, and doing so twice on the
complex conjugate of the holomorphic Yukawa coupling gives a quantity

$\bar{C}_{\bar{i}}^{\phantom{-}jk}=G^{j\bar{j}}G^{k\bar{k}}\overline{C_{ijk}}~{}.$
(3.49)

From this, we define non-holomorphic propagators $S^{ij},\,S^{i},\,S$ as
potentials:

$\partial_{\bar{i}}S^{ij}=\bar{C}_{\bar{i}}^{\phantom{-}ij}~{},\qquad\partial_{\bar{i}}S^{j}=G_{i\bar{i}}S^{ij}~{},\qquad\partial_{\bar{i}}S=G_{i\bar{i}}S^{i}~{}.$
(3.50)

Note that $S^{ij}$ is a symmetric tensor, $S^{ij}=S^{ji}$. These functions
transform as tensors under coordinate transformations of the moduli space.
Under a Kähler transformation ${K\mapsto K+f+\bar{f}}$ for some holomorphic
$f$, each propagator transforms as ${S\mapsto\text{e}^{2f}S}$,
${S^{i}\mapsto\text{e}^{2f}S^{i}}$, ${S^{ij}\mapsto\text{e}^{2f}S^{ij}}$ (one
says that they have weight (2,0) under Kähler transformations). We display
their covariant, with respect to coordinate and Kähler transformations,
derivatives below, and also do so for $K_{i}=\partial_{\varphi^{i}}K$:

$\displaystyle D_{i}S^{jk}$
$\displaystyle=\partial_{i}S^{jk}+\Gamma^{j}_{il}S^{lk}+\Gamma^{k}_{il}S^{jl}-2K_{i}S^{jk}~{},$
(3.51) $\displaystyle D_{i}S^{j}$
$\displaystyle=\partial_{i}S^{j}+\Gamma^{j}_{il}S^{l}-2K_{i}S^{j}~{},$
$\displaystyle D_{i}S$ $\displaystyle=\partial_{i}S-2K_{i}S~{},$
$\displaystyle D_{i}K_{j}$
$\displaystyle=\partial_{i}K_{j}-\Gamma^{l}_{ij}K_{l}~{}.$

The Christoffel symbols $\Gamma^{i}_{jk}$ give the Levi-Civita connection for
the metric $G$. We also have provided the covariant derivative $D_{i}K_{j}$ to
avoid any confusion in what follows.

For $g\geq 2$, the genus-$g$ B-model free energy is a polynomial of degree
$3g-3$ in the functions $S^{ij}$ (degree 1), $S^{i}$ (degree 2), $S$ (degree
3), and $K_{i}$ (degree 1). The coefficients of this polynomial are rational
functions of the moduli $\bm{\varphi}$, to be determined soon. The Kähler
potential $K$ can be computed from the periods as in (3.47), and we now
explain how to obtain the functions $S^{ij},\,S^{i},\,S$ from the periods. The
key insight is the following special geometry relation, which originates in
the $tt^{*}$ equations as explained in [85] (which we avoid discussing):

$\displaystyle R_{i\bar{i}\,j}^{\phantom{-}l}$
$\displaystyle=\left[\partial_{\bar{i}},D_{i}\right]^{l}_{\phantom{-}j}$
(3.52)
$\displaystyle=\partial_{\bar{i}}\Gamma^{l}_{ij}=\delta^{l}_{i}G_{j\bar{i}}+\delta^{l}_{j}G_{i\bar{i}}-C_{ijk}\bar{C}_{\bar{i}}^{kl}~{}.$

This equation is a total antiholomorphic derivative, with the left and right
hand side explicitly given as $\partial_{\bar{i}}$ of quantities that we can
either already compute, or of $C_{ijk}S^{kl}$. So we integrate this equation,
to get the _integrated special geometry relation_ below that we use to fix the
first propagator $S^{ij}$:

$C_{ijk}S^{kl}=-\Gamma^{l}_{ij}+\delta^{l}_{i}K_{j}+\delta^{l}_{j}K_{i}+s^{l}_{ij}~{}.$
(3.53)

The quantity $s^{l}_{ij}$ is the holomorphic ‘constant of integration’ which
lives in the kernel of $\partial_{\bar{i}}$. It is the first of several
‘propagator ambiguities’ that we will meet. There is some freedom in the
choice of $s^{l}_{ij}$. Note that to solve the above equation for $S^{kl}$ we
must clear the $C_{ijk}$ from the left hand side. We shall make comments on
how to go about this later, and for now press on with reviewing the BCOV
procedure.

We now seek to express covariant derivatives of the propagators in terms of
the propagators themselves. This involves some algebra. Consider for example
$D_{i}S^{jk}$. One hits this with $\partial_{\bar{i}}$, and then expands the
expression to get
$\partial_{\bar{i}}\left(\partial_{i}S^{jk}+\Gamma^{j}_{il}S^{lk}+\Gamma^{k}_{il}S^{jl}-2K_{i}S^{jk}\right)$.
We expand this by the Leibniz rule, with antiholomorphic derivatives of the
Christoffel symbols replaced by the special geometry relation (3.52). After
some manipulation, each term on the right hand side can be written as
$\partial_{\bar{i}}$ of something, and this is integrated to get the
following:

$\displaystyle D_{i}S^{jk}$
$\displaystyle=\delta^{j}_{i}S^{k}+\delta^{k}_{i}S^{j}-C_{imn}S^{mj}S^{nk}+h^{jk}_{i}~{},$
(3.54) $\displaystyle D_{i}S^{j}$
$\displaystyle=2\delta^{j}_{i}S-C_{imn}S^{m}S^{nj}+h^{jk}_{i}K_{k}+h^{j}_{i}~{},$
$\displaystyle D_{i}S$
$\displaystyle=-\frac{1}{2}C_{imn}S^{m}S^{n}+\frac{1}{2}h^{mn}_{i}K_{m}K_{n}+h^{j}_{i}K_{j}+h_{i}~{},$
$\displaystyle D_{i}K_{j}$
$\displaystyle=-K_{i}K_{j}-C_{ijk}S^{k}+C_{ijk}S^{kl}K_{l}+h_{ij}~{}.$

$h^{jk}_{i},\,h^{j}_{i},\,h_{i},\,h_{ij}$ are all further propagator
ambiguities, rational functions of $\bm{\varphi}$ over which there is a degree
of choice. It is the first and second of the above equations that are used in
practice to compute the propagators $S^{i}$ and $S$.

The BCOV recursion relation takes its simplest form after a change of
variables. One writes the tilded propagators

$\widetilde{S}^{ij}=S^{ij}~{},\qquad\widetilde{S}^{i}=S^{i}-S^{ij}K_{j}~{},\qquad\widetilde{S}=S-S^{i}K_{i}+\frac{1}{2}S^{ij}K_{i}K_{j}~{},\qquad\widetilde{K}_{i}=K_{i}~{},$
(3.55)

which, we parenthetically remark, obey

$\partial_{\bar{i}}\widetilde{S}^{ij}=\bar{C}_{\bar{i}}^{ij}~{},\qquad\partial_{\bar{i}}\widetilde{S}^{i}=-\bar{C}_{\bar{i}}^{ij}K_{j}~{},\qquad\partial_{\bar{i}}\widetilde{S}=\frac{1}{2}\bar{C}_{\bar{i}}^{ij}K_{i}K_{j}~{}.$
(3.56)

This change of variables allows for a significant simplification: higher genus
prepotentials $\mathcal{F}^{(g\geq 2)}$ are degree $3g-3$ polynomials in
$\widetilde{S}^{ij},\,\widetilde{S}^{i},$ and $\widetilde{S}$ but _do not
explicitly depend on_ $\widetilde{K}$ .

For $g\geq 2$, the BCOV recursion relation takes the form [83]

$\displaystyle\frac{\partial\mathcal{F}^{(g)}}{\partial\widetilde{S}^{ij}}$
$\displaystyle=\frac{1}{2}\partial_{i}\left(\partial_{j}^{\prime}\mathcal{F}^{(g-1)}\right)+\frac{1}{2}\left(C_{ijl}S^{lk}-s^{k}_{ij}\right)\partial_{k}^{\prime}\mathcal{F}^{(g-1)}+\frac{1}{2}\left(C_{ijk}S^{k}-h_{ij}\right)c_{g-1}$
(3.57)
$\displaystyle\phantom{spookyphantomtext}+\frac{1}{2}\sum_{h=1}^{g-1}\left(\partial_{i}^{\prime}\mathcal{F}^{(h)}\right)\left(\partial_{j}^{\prime}\mathcal{F}^{(g-h)}\right)~{},$
$\displaystyle\frac{\partial\mathcal{F}^{(g)}}{\partial\widetilde{S}^{i}}$
$\displaystyle=(2g-3)\partial_{i}^{\prime}\mathcal{F}^{(g-1)}+\sum_{h=1}^{g-1}c_{h}\partial_{i}^{\prime}\mathcal{F}^{(g-h)}~{},$
$\displaystyle\frac{\partial\mathcal{F}^{(g)}}{\partial\widetilde{S}}$
$\displaystyle=(2g-3)c_{g-1}+\sum_{h=1}^{g-1}c_{h}c_{g-h}~{}.$

In the above equations we have used

$\displaystyle c_{g}$
$\displaystyle=\begin{cases}\frac{\chi(Y)}{24}-1~{},&\qquad\qquad\;\,g=1~{},\\\
(2g-2)\mathcal{F}^{(g)}~{},&\qquad\qquad\;\,g>1~{},\end{cases}$ (3.58)
$\displaystyle\partial_{i}^{\prime}\mathcal{F}^{(g)}$
$\displaystyle=\begin{cases}\frac{1}{2}C_{ijk}S^{jk}+\partial_{i}f^{(1)}~{},&\qquad
g=1~{},\\\ \partial_{i}\mathcal{F}^{(g)}~{},&\qquad g>1~{}.\end{cases}$

The above symbol $f^{(1)}$ denotes the genus-1 holomorphic ambiguity, but is
_not_ the function $f$ of (3.18) that appeared inside the logarithm of the
genus 1 prepotential. Instead,

$f^{(1)}=-\log\left(\Delta^{c}\prod_{i=1}^{h^{2,1}(X)}\left(\varphi^{i}\right)^{(1-Y_{00i})/2}\overline{\Delta^{c}\prod_{i=1}^{h^{2,1}(X)}\left(\varphi^{i}\right)^{(1-Y_{00i})/2}}\phantom{-}\right)~{},$
(3.59)

with $c=1/12$ in the CICY examples that concern us. We stress that $f^{(1)}$
is a function of $\bm{\varphi}$ and $\overline{\bm{\varphi}}$. We labour this
point so as to avoid confusion when we take the topological limit.

The recursive procedure involves working out each $\mathcal{F}^{(g)}$ by first
taking as ansatz the most general polynomial in the tilded propagators of
degree $3g-3$, and then comparing the partial derivatives (with respect to the
propagators) thereof to the right hand side of (3.57) (which only involves
lower-genus prepotentials, guaranteeing a recursion). That right hand side can
be written as a polynomial in the tilded propagators by replacing every
derivative of a propagator produced by (3.58) through the BCOV closure
relations (3.54). For simplicity, we rewrite those closure relations here for
the tilded propagators, and expanding the covariant derivatives:

$\displaystyle\partial_{i}\widetilde{S}^{jk}$
$\displaystyle=C_{imn}\widetilde{S}^{mj}\widetilde{S}^{nk}+\delta^{j}_{i}\widetilde{S}^{k}+\delta^{k}_{i}\widetilde{S}^{j}-s^{j}_{im}\widetilde{S}^{mk}-s^{k}_{im}\widetilde{S}^{mj}+h^{jk}_{i}~{},$
(3.60) $\displaystyle\partial_{i}\widetilde{S}^{j}$
$\displaystyle=C_{imn}\widetilde{S}^{mj}\widetilde{S}^{n}+2\delta^{j}_{i}\widetilde{S}-s^{j}_{im}\widetilde{S}^{m}-h_{ik}\widetilde{S}^{kj}+h^{j}_{i}~{},$
$\displaystyle\partial_{i}\widetilde{S}$
$\displaystyle=\frac{1}{2}C_{imn}\widetilde{S}^{m}\widetilde{S}^{n}-h_{ij}\widetilde{S}^{j}+h_{i}~{},$
$\displaystyle\partial_{i}K_{j}$
$\displaystyle=K_{i}K_{j}-C_{ijn}\widetilde{S}^{mn}K_{m}+s^{m}_{ij}K_{m}-C_{ijk}\widetilde{S}^{k}+h_{ij}~{}.$

This procedure fixes the polynomial form of
$\mathcal{F}^{(g)}\left(\widetilde{S}^{ij},\,\widetilde{S}^{i},\,\widetilde{S}\right)$
up to the ‘constant’ term $f^{(g)}$, which is a rational function of
$\bm{\varphi}$ not fixed by (3.57). That this function is holomorphic is
buried in the derivation of (3.57), which is a rearrangement of
antiholomorphic derivatives that annihilate this _holomorphic ambiguity_
$f^{(g)}$.

Determination of this ambiguity $f^{(g)}$ is the major remaining conceptual
problem in topological string theory. It is a rational function, with poles at
the zeroes of $\Delta$ and any singularities of the propagators. That the
propagators can have singularities outside of $\Delta=0$ is an artefact of how
the equations (3.53) and (3.54) are solved for the propagators.
$\mathcal{F}^{(g)}$ should be nonsingular away from $\Delta=0$, so $f^{(g)}$
must have a residue at these spurious poles so that $\mathcal{F}^{(g)}$ is
regular.

The singular behaviour of $\mathcal{F}^{(g)}$ at the conifold locus $\Delta=0$
is fixed by the conifold gap condition [17], which shows that as the conifold
locus is approached $\mathcal{F}^{(g)}$ should go like

$\mathcal{F}^{(g)}\sim\frac{1}{\Delta^{2g-2}}+\text{regular}$ (3.61)

where “regular” denotes terms that are nonsingular at $\Delta=0$. The degree
of $f^{(g)}$’s numerator should not be too high, so that $\mathcal{F}^{(g)}$
is regular at $\infty$.

In this thesis we shall only make it as high as genus 2. To this end we
display an expression for the genus 2 prepotential that we will make use of,
first derived in [15]. The authors thereof were able to compute higher-still
prepotentials but these do not fit on a page. Note that we now revert to using
untilded propagators.

$\displaystyle\mathcal{F}^{(2)}\;=$
$\displaystyle\frac{1}{2}C^{(1)}_{ij}S^{ij}+\frac{1}{2}C_{i}^{(1)}S^{ij}C_{j}^{(1)}-\frac{1}{8}S^{jk}S^{mn}C_{jkmn}-\frac{1}{2}S^{ij}C_{ijm}S^{mn}C_{n}^{(1)}+\frac{\chi(Y)}{24}S^{i}C_{i}^{(1)}$
(3.62)
$\displaystyle\phantom{-}+\frac{1}{8}S^{ij}C_{ijp}S^{pq}C_{qmn}S^{mn}+\frac{1}{12}S^{ij}S^{pq}S^{mn}C_{ipm}C_{jqn}-\frac{\chi(Y)}{48}S^{i}C_{ijk}S^{jk}$
$\displaystyle\phantom{-}+\frac{\chi(Y)}{24}\left(\frac{\chi(Y)}{24}-1\right)S+f^{(2)}\left(\bm{\varphi}\right)~{}.$

We now fix the above missing pieces of notation:

$\displaystyle C^{(1)}_{i}$
$\displaystyle=D_{i}\mathcal{F}^{(1)}=\partial_{i}\mathcal{F}^{(1)}~{},\qquad
C^{(1)}_{ij}=D_{i}D_{j}\mathcal{F}^{(1)}=\partial_{i}\partial_{j}\mathcal{F}^{(1)}-\Gamma^{l}_{ij}\partial_{l}\mathcal{F}^{(1)}~{},$
(3.63) $\displaystyle C_{jklm}$
$\displaystyle=D_{j}C_{klm}=\partial_{j}C_{klm}-\Gamma^{n}_{jk}C_{nlm}-\Gamma^{n}_{jl}C_{knm}-\Gamma^{n}_{jm}C_{kln}+2K_{j}C_{klm}~{}.$

Note that $C_{jklm}$ is symmetric in its four indices.

In order to obtain curve counts, a change of variables to A-model quantities
must be made. The genus-$g$ A model free energy dependence on higher genus
instanton numbers $n^{(g)}_{\mathbf{b}}$ is given by the Gopakumar-Vafa
formula [18, 19]:

$\displaystyle F^{\text{All Genus}}(\lambda,\mathbf{t})$
$\displaystyle=\sum_{g=0}^{\infty}\lambda^{2g-2}F_{g}(t)$ (3.64)
$\displaystyle=\frac{c(\mathbf{t})}{\lambda}+l(\mathbf{t})+\sum_{g=0}^{\infty}\sum_{\mathbf{b}\geq
0}n^{(g)}_{\mathbf{b}}\sum_{k=1}^{\infty}\frac{1}{k}\left(2\sin\frac{k\lambda}{2}\right)^{2g-2}\exp\left[2\pi\text{i}\,k\,\mathbf{b}\cdot\mathbf{t}\right]~{}.$

$\lambda$ is the string coupling. $c(\mathbf{t})$ and $l(\mathbf{t})$ are
respectively cubic and linear polynomials in $t$, This formula fixes
$F_{g}(\mathbf{t}$ at each genus up to a constant term, which we neglect to
discuss. This was studied at different genera in the series of papers [89, 16,
15, 90] and nicely presented in the thesis [84]. We give the genus 2 expansion
below ahead of making use of it in the next section.

$F_{2}(\mathbf{t})=\frac{\chi(Y)}{5760}+\sum_{\mathbf{b}\geq
0}\left(\frac{n^{(0)}_{\mathbf{b}}}{240}+n^{(2)}_{\mathbf{b}}\right)\text{Li}_{-1}\left(\mathbf{q}^{\mathbf{b}}\right)~{}.$
(3.65)

One can note that genus 1 numbers $n^{(1)}_{\mathbf{b}}$ do not appear in this
formula. Their absence was noted in [15], where it was explained as the lack
of “toroidal bubbling” at genus 2.

##### Quantities in the holomorphic limit

Up until now the propagators have had a dependence on $\mathbf{\varphi}$ and
$\overline{\mathbf{\varphi}}$. To compute enumerative invariants, it suffices
to work in the topological limit $\bar{t}\mapsto-\text{i}\infty$, or
$\overline{\varphi}\mapsto\infty$. The Kähler potential can be massaged,

$K(\bm{\varphi},\overline{\bm{\varphi}})=-\log\left(-\text{i}\Pi^{\dagger}\Sigma\Pi\right)=\log\left(k(\overline{\mathbf{\varphi}})\right)-\log\left(\varpi_{0}(\mathbf{\varphi})+p\left(\mathbf{\varphi},\bar{\mathbf{\varphi}}\right)\right)~{},$
(3.66)

where the function
$p\left(\mathbf{\varphi},\overline{\mathbf{\varphi}}\right)$ vanishes when
$\overline{\mathbf{\varphi}}\mapsto\infty$. Since it is derivatives of the
Kähler potential that play a role in computing the propagators, we can removed
the piece $\log\left(k\left(\overline{\mathbf{\varphi}}\right)\right)$ and
retain in this limit

$K(\mathbf{\varphi})=-\log\left(\varpi_{0}(\mathbf{\varphi})\right)~{}.$
(3.67)

Similarly, the Christoffel symbols become in this limit

$\Gamma^{i}_{jk}=\frac{\partial\varphi^{i}}{\partial
t^{l}}\frac{\partial^{2}t^{l}}{\partial a^{j}\partial a^{k}}~{}.$ (3.68)

###  3.7 A worked example at genus 2

We will set about computing genus 2 numbers on the maximally split quintic
(3.15) . These were already computed to degree 18 in [60], where the
holomorphic ambiguity was fixed with an involved implementation of the
conifold gap condition. This involves expanding the periods about the conifold
locus and imposing the gap condition on the prepotential, expressed in terms
of these conifold-adapted periods.

We instead make the following claim: The genus 2 ambiguity, in this example,
can be fixed from two sources of information. These are

  * •

The known vanishing instanton numbers .

  * •

An infinite Coxeter symmetry .

This example will involve several appearances of the coordinates $\varphi^{i}$
raised to powers. In the name of sanity, we will temporarily write these
coordinates with lower indices so that superscripts can be read as exponents.
For instance, $\varphi_{2}^{4}$ is $\varphi^{2}$ to the power of four. The
only exception to this rule will be when the superscript is an abstract index,
as in $\varphi^{i}$, so that tensor contractions behave as they should.

The three-point functions, or Yukawa couplings, are

$\displaystyle C_{111}$
$\displaystyle=\frac{5\left(1+37\varphi_{1}+73\varphi_{1}^{2}+14\varphi_{1}^{3}-3\varphi_{2}+176\varphi_{1}\varphi_{2}-73\varphi_{1}^{2}\varphi_{2}+3\varphi_{2}^{2}+37\varphi_{1}\varphi_{2}^{2}-\varphi_{2}^{3}\right)}{\varphi_{1}^{3}\Delta}~{},$
(3.69) $\displaystyle C_{112}$
$\displaystyle=\frac{5\left(2+14\varphi_{1}-9\varphi_{1}^{2}-7\varphi_{1}^{3}-\varphi_{2}-88\varphi_{1}\varphi_{2}+64\varphi_{1}^{2}\varphi_{2}-4\varphi_{2}^{2}-51\varphi_{1}\varphi_{2}^{2}+3\varphi_{2}^{3}\right)}{\varphi_{1}^{2}\varphi_{2}\Delta}~{},$
$\displaystyle C_{122}$
$\displaystyle=\frac{5\left(2-\varphi_{1}-4\varphi_{1}^{2}+3\varphi_{1}^{3}+14\varphi_{2}-88\varphi_{1}\varphi_{2}-51\varphi_{1}^{2}\varphi_{2}-9\varphi_{2}^{2}+64\varphi_{1}\varphi_{2}^{2}-7\varphi_{2}^{3}\right)}{\varphi_{1}\varphi_{2}^{2}\Delta}~{},$
$\displaystyle C_{222}$
$\displaystyle=\frac{5\left(1-3\varphi_{1}+3\varphi_{1}^{2}-\varphi_{1}^{3}+37\varphi_{2}+176\varphi_{1}\varphi_{2}+37\varphi_{1}^{2}\varphi_{2}+73\varphi_{2}^{2}-73\varphi_{1}\varphi_{2}^{2}+14\varphi_{2}^{3}\right)}{\varphi_{2}^{3}\Delta}~{}.$

The discriminant $\Delta$ is

$\displaystyle\Delta$
$\displaystyle=1-5\varphi_{1}+10\varphi_{1}^{2}-10\varphi_{1}^{3}+5\varphi_{1}^{4}-\varphi_{1}^{5}-5\varphi_{2}-605\varphi_{1}\varphi_{2}-1905\varphi_{1}^{2}\varphi_{2}-605\varphi_{1}^{3}\varphi_{2}$
(3.70)
$\displaystyle\phantom{-}\phantom{-}\phantom{-}-5\varphi_{1}^{4}\varphi_{2}+10\varphi_{2}^{2}-1905\varphi_{1}\varphi_{2}^{2}+1905\varphi_{1}^{2}\varphi_{2}^{2}-10\varphi_{1}^{3}\varphi_{2}^{2}-10\varphi_{2}^{3}-605\varphi_{1}\varphi_{2}^{3}$
$\displaystyle\phantom{-}\phantom{-}\phantom{-}-10\varphi_{1}^{2}\varphi_{2}^{3}+5\varphi_{2}^{4}-5\varphi_{1}\varphi_{2}^{4}-\varphi_{2}^{5}~{}.$

The undisplayed $C_{ijk}$ can be obtained from those given by symmetry, e.g.
$C_{211}=C_{112}$.

##### A digression on matrix inversion

Note that there is a $\mathbbl{Z}_{2}$ symmetry to the example, so that for
instance $C_{222}$ can be obtained from $C_{111}$ by effecting a swap
$\varphi_{1}\leftrightarrow\varphi_{2}$. This symmetry means that in our final
result, the instanton numbers should possess a symmetry
$n^{(g)}_{(i,j)}=n^{(g)}_{(j,i)}$. The A and B model prepotentials should both
be symmetric functions.

This means that, if we set up the problem correctly, we can have the ambiguity
$f^{(2)}$ be a symmetric function of $\varphi_{1},\,\varphi_{2}$. Let us
explain this point. We begin with the determination of $S^{ij}$ from (3.53) .
In the one-parameter setting, one is free to simply divide both sides of this
equation by the quantity $C_{111}$. In multiparameter examples we must invert
$C_{ijk}$. The traditional way to do this is to pick a direction in moduli
space, say $1$, and then write out the $i=1$ component of (3.53):

$C_{1jk}S^{kl}=-\Gamma^{l}_{1j}+\delta^{l}_{1}K_{j}+\delta^{l}_{j}K_{1}+s^{l}_{1j}~{}.$
(3.71)

Now although covariance has been thrown out the window, $C_{1jk}$ is a
legitimate matrix that generically has full rank, so that we can find an
inverse of it, so that we obtain

$S^{kl}=-\left(C_{1}^{\phantom{-}-1}\right)^{kj}\Gamma^{l}_{1j}+\delta^{l}_{1}\left(C_{1}^{\phantom{-}-1}\right)^{kj}K_{j}+\left(C_{1}^{\phantom{-}-1}\right)^{kl}K_{1}+\left(C_{1}^{\phantom{-}-1}\right)^{kj}s^{l}_{1j}~{}.$
(3.72)

Naively we might say that $s^{l}_{1j}=0$ is the simplest choice, so that we
read off $S^{kl}$ directly from the above. This would be wrong, because the
$S^{kl}$ so defined does not equal $S^{lk}$. The way to proceed is to write

$\left(C_{1}^{\phantom{-}-1}\right)^{kj}s^{l}_{1j}=A^{kl}+E^{kl}~{},$ (3.73)

where $A^{kl}$ and $E^{kl}$ are respectively antisymmetric and symmetric in
their indices. Our freedom is to redefine the symmetric part $E^{kl}$, and we
take this to vanish. Then we get

$S^{kl}=\frac{1}{2}\left[\left(-\left(C_{1}^{\phantom{-}-1}\right)^{kj}\Gamma^{l}_{1j}+\delta^{l}_{1}\left(C_{1}^{\phantom{-}-1}\right)^{kj}K_{j}+\left(C_{1}^{\phantom{-}-1}\right)^{kl}K_{1}\right)+\left(k\leftrightarrow
l\right)\right]~{}.$ (3.74)

This choice would be correct, and one could proceed from here to compute all
of the other propagators and propagator ambiguities using the $i=1$ components
of (3.54) . For completeness, we give the components of the symmetric matrix
$\left(C_{1}^{\phantom{-}-1}\right)^{ij}$:

$\displaystyle\left(C_{1}^{\phantom{-}-1}\right)^{11}$
$\displaystyle=\frac{-\varphi_{1}^{3}\left(2{-}\varphi_{1}{-}4\varphi_{1}^{2}{+}3\varphi_{1}^{3}{+}14\varphi_{2}{-}88\varphi_{1}\varphi_{2}{-}51\varphi_{1}^{2}\varphi_{2}{-}9\varphi_{2}^{2}{+}64\varphi_{1}\varphi_{2}^{2}{-}7\varphi_{2}^{3}\right)}{5\left(2-7\varphi_{1}-2\varphi_{2}\right)}~{},$
(3.75) $\displaystyle\left(C_{1}^{\phantom{-}-1}\right)^{12}$
$\displaystyle=\frac{\varphi_{1}^{2}\varphi_{2}\left(2{+}14\varphi_{1}{-}9\varphi_{1}^{2}{-}7\varphi_{1}^{3}{-}\varphi_{2}{-}88\varphi_{1}\varphi_{2}{+}64\varphi_{1}^{2}\varphi_{2}{-}4\varphi_{2}^{2}{-}51\varphi_{1}\varphi_{2}^{2}{+}3\varphi_{2}^{3}\right)}{5\left(2-7\varphi_{1}-2\varphi_{2}\right)}~{},$
$\displaystyle\left(C_{1}^{\phantom{-}-1}\right)^{22}$
$\displaystyle=\frac{-\varphi_{1}\varphi_{2}^{2}\left(1{+}37\varphi_{1}{+}73\varphi_{1}^{2}{+}14\varphi_{1}^{3}{-}3\varphi_{2}{+}176\varphi_{1}\varphi_{2}{-}73\varphi_{1}^{2}\varphi_{2}{+}3\varphi_{2}^{2}{+}37\varphi_{1}\varphi_{2}^{2}{-}\varphi_{2}^{3}\right)}{5\left(2-7\varphi_{1}-2\varphi_{2}\right)}~{}.$

A problem with this approach is that this breaks the $(1\leftrightarrow 2)$
symmetry. For instance, $S^{11}$ does not give $S^{22}$ upon swapping
$(\varphi_{1}\leftrightarrow\varphi_{2})$. At the end of the process,
$f^{(2)}$ will not be a symmetric function.

This is not such a problem for computations with two-parameter manifolds.
However, we are motivated by the power of symmetry evidenced in the genera 0
and 1 computations for manifolds with $\geq 3$ parameters to seek another way
of determining the propagators, so that we maintain any symmetries that we
begin with. The reader who does not care for maintaining this symmetry can use
the previous expressions of this subsection for $S^{ij}$.

##### Resuming the computation

In what follows, we forget the last five equations and compute $S^{ij}$ anew.
To this end, we contract the integrated special geometry relation (3.53) with
$\varphi^{i}$ to obtain

$\varphi^{i}C_{ijk}S^{kl}=-\varphi^{i}\Gamma^{l}_{ij}+\varphi^{l}K_{j}+\delta^{l}_{j}\varphi^{i}K_{i}+\varphi^{i}s^{l}_{ij}~{}.$
(3.76)

The matrix $\mathfrak{C}_{jk}=\varphi^{i}C_{ijk}$ respects the
$(1\leftrightarrow 2)$ symmetry, as does its inverse. We introduce more
notation,

$\mathfrak{C}_{jk}=\varphi^{i}C_{ijk}~{},\qquad\Upsilon^{l}_{j}=\varphi^{i}\Gamma^{l}_{ij}~{},\qquad\mathfrak{K}=\varphi^{i}K_{i}~{},\qquad\mathfrak{s}^{l}_{j}=\varphi^{i}s^{l}_{ij}~{}.$
(3.77)

We shall also use another abuse of notation, whereby we denote the inverse of
$\mathfrak{C}_{ij}$ with raised components:

$\mathfrak{C}^{ij}\mathfrak{C}_{jk}=\delta^{i}_{k}~{}.$ (3.78)

The components of $\mathfrak{C}_{ij}$ are

$\displaystyle\mathfrak{C}_{11}$
$\displaystyle=\frac{5\left(3+51\varphi_{1}+64\varphi_{1}^{2}+7\varphi_{1}^{3}-4\varphi_{2}+88\varphi_{1}\varphi_{2}-9\varphi_{1}^{2}\varphi_{2}-\varphi_{2}^{2}-14\varphi_{1}\varphi_{2}^{2}+2\varphi_{2}^{3}\right)}{\varphi_{1}^{2}\Delta}~{},$
(3.79) $\displaystyle\mathfrak{C}_{12}$
$\displaystyle=\frac{5\left(4+13\varphi_{1}-13\varphi_{1}^{2}-4\varphi_{1}^{3}+13\varphi_{2}-176\varphi_{1}\varphi_{2}+13\varphi_{1}^{2}\varphi_{2}-13\varphi_{2}^{2}+13\varphi_{1}\varphi_{2}^{2}-4\varphi_{2}^{3}\right)}{\varphi_{1}\varphi_{2}\Delta}~{},$
$\displaystyle\mathfrak{C}_{22}$
$\displaystyle=\frac{5\left(3-4\varphi_{1}-\varphi_{1}^{2}+2\varphi_{1}^{3}+51\varphi_{2}+88\varphi_{1}\varphi_{2}-14\varphi_{1}^{2}\varphi_{2}+64\varphi_{2}^{2}-9\varphi_{1}\varphi_{2}^{2}+7\varphi_{2}^{3}\right)}{\varphi_{2}^{2}\Delta}~{}.$

The inverse has components

$\displaystyle\mathfrak{C}^{11}$
$\displaystyle=\frac{-\varphi_{1}^{2}\left(3-4\varphi_{1}-\varphi_{1}^{2}+2\varphi_{1}^{3}+51\varphi_{2}+88\varphi_{1}\varphi_{2}-14\varphi_{1}^{2}\varphi_{2}+64\varphi_{2}^{2}-9\varphi_{1}\varphi_{2}^{2}+7\varphi_{2}^{3}\right)}{5\left(7-2\varphi_{1}-2\varphi_{2}\right)}~{},$
(3.80) $\displaystyle\mathfrak{C}^{12}$
$\displaystyle=\frac{\varphi_{1}\varphi_{2}\left(4+13\varphi_{1}-13\varphi_{1}^{2}-4\varphi_{1}^{3}+13\varphi_{2}-176\varphi_{1}\varphi_{2}+13\varphi_{1}^{2}\varphi_{2}-13\varphi_{2}^{2}+13\varphi_{1}\varphi_{2}^{2}-4\varphi_{2}^{3}\right)}{5\left(7-2\varphi_{1}-2\varphi_{2}\right)}~{},$
$\displaystyle\mathfrak{C}^{22}$
$\displaystyle=\frac{-\varphi_{2}^{2}\left(3+51\varphi_{1}+64\varphi_{1}^{2}+7\varphi_{1}^{3}-4\varphi_{2}+88\varphi_{1}\varphi_{2}-9\varphi_{1}^{2}\varphi_{2}-\varphi_{2}^{2}-14\varphi_{1}\varphi_{2}^{2}+2\varphi_{2}^{3}\right)}{5\left(7-2\varphi_{1}-2\varphi_{2}\right)}~{}.$

The denominator appearing above will recur, and so we give it a name:

$Z=7-2\varphi^{1}-2\varphi^{2}~{}.$ (3.81)

Analogously to what we did with (3.73), we have a freedom to change the
symmetric part of $\mathfrak{C}^{kj}\mathfrak{s}^{l}_{j}$, and we set this to
zero. Then from (3.76) we get

$S^{kl}=\frac{1}{2}\left[\left(-\mathfrak{C}^{kj}\Upsilon^{l}_{j}+\varphi^{l}\mathfrak{C}^{kj}K_{j}+\mathfrak{K}\mathfrak{C}^{kl}\right)+\left(k\leftrightarrow
l\right)\right]~{}.$ (3.82)

The first few terms in $S^{ij}$’s Taylor expansions are

$\displaystyle S^{11}$
$\displaystyle=-\frac{3\varphi_{1}^{2}}{35}+\frac{64\varphi_{1}^{3}}{245}-\frac{102}{49}\varphi_{1}^{2}\varphi_{2}+\frac{79\varphi_{1}^{4}}{1715}-\frac{7654\varphi_{1}^{3}\varphi_{2}}{1715}-\frac{7488\varphi_{1}^{2}\varphi_{2}^{2}}{1715}+\,...~{},$
(3.83) $\displaystyle S^{12}$
$\displaystyle=\frac{4\varphi_{1}\varphi_{2}}{35}+\frac{24}{49}\varphi_{1}^{2}\varphi_{2}+\frac{24}{49}\varphi_{1}\varphi_{2}^{2}-\frac{1426\varphi_{1}^{3}\varphi_{2}}{1715}-\frac{11182\varphi_{1}^{2}\varphi_{2}^{2}}{1715}-\frac{1426\varphi_{1}\varphi_{2}^{3}}{1715}+\,...~{},$
$\displaystyle S^{22}$
$\displaystyle=-\frac{3\varphi_{2}^{2}}{35}-\frac{102}{49}\varphi_{1}\varphi_{2}^{2}+\frac{64\varphi_{2}^{3}}{245}-\frac{7488\varphi_{1}^{2}\varphi_{2}^{2}}{1715}-\frac{7654\varphi_{1}\varphi_{2}^{3}}{1715}+\frac{79\varphi_{2}^{4}}{1715}+\,...~{}.$

Now we turn to computing $S^{i}$, which in the interests of symmetry we do by
considering $\varphi^{i}D_{i}S^{jk}$ as in (3.54). There is another new
notation,

$\mathfrak{h}^{jk}=\varphi^{i}h^{jk}_{i}~{}.$ (3.84)

We have the freedom to set $\mathfrak{h}^{1,1}=\mathfrak{h}^{2,2}=0$, and then
we can find $S^{1}$ and $S^{2}$ from the $(j,k)=(1,1)$ and $(j,k)=(2,2)$
components of $\varphi^{i}D_{i}S^{jk}$ as below:

$\varphi^{i}D_{i}S^{jk}=\varphi^{j}S^{k}+\varphi^{k}S^{j}-\mathfrak{C}_{mn}S^{mj}S^{nk}+\mathfrak{h}^{jk}~{}.$
(3.85)

The first few terms in the expansions of the $S^{i}$ so obtained are

$\displaystyle S^{1}$
$\displaystyle=-\frac{3\varphi_{1}}{70}+\frac{64\varphi_{1}^{2}}{245}-\frac{929\varphi_{1}\varphi_{2}}{490}+\frac{1389\varphi_{1}^{3}}{1715}-\frac{13539\varphi_{1}^{2}\varphi_{2}}{1715}-\frac{7641\varphi_{1}\varphi_{2}^{2}}{1715}+\,...~{},$
(3.86) $\displaystyle S^{2}$
$\displaystyle=-\frac{3\varphi_{2}}{70}-\frac{929\varphi_{1}\varphi_{2}}{490}+\frac{64\varphi_{2}^{2}}{245}-\frac{7641\varphi_{1}^{2}\varphi_{2}}{1715}-\frac{13539\varphi_{1}\varphi_{2}^{2}}{1715}+\frac{1389\varphi_{2}^{3}}{1715}+\,...~{}.$

We will use the $(j,k)={1,2}$ component of (3.85) in order to compute
$\mathfrak{h}^{12}$, over which we have no freedom having fixed
$\mathfrak{h}^{11}=\mathfrak{h}^{22}=0$. We will need all components of
$\mathfrak{h}^{ij}$ when we consider $\varphi^{i}D_{i}S^{j}$, which we will
use to compute the remaining propagator $S$ . We find

$\displaystyle\mathfrak{h}^{12}=\frac{\varphi_{1}\varphi_{2}}{10Z^{2}\Delta}\bigg{(}98$
$\displaystyle+679\varphi_{1}-5669\varphi_{1}^{2}-6701\varphi_{1}^{3}-29717\varphi_{1}^{4}-29363\varphi_{1}^{5}-4824\varphi_{1}^{6}-2656\varphi_{1}^{7}$
(3.87)
$\displaystyle+21\varphi_{1}^{8}+7\varphi_{1}^{9}+679\varphi_{2}-72532\varphi_{1}\varphi_{2}-848410\varphi_{1}^{2}\varphi_{2}-2858677\varphi_{1}^{3}\varphi_{2}$
$\displaystyle-2218750\varphi_{1}^{4}\varphi_{2}-307684\varphi_{1}^{5}\varphi_{2}+97632\varphi_{1}^{6}\varphi_{2}+4550\varphi_{1}^{7}\varphi_{2}+67\varphi_{1}^{8}\varphi_{2}$
$\displaystyle-5669\varphi_{2}^{2}-848410\varphi_{1}\varphi_{2}^{2}-1272670\varphi_{1}^{2}\varphi_{2}^{2}+8757690\varphi_{1}^{3}\varphi_{2}^{2}+3502930\varphi_{1}^{4}\varphi_{2}^{2}$
$\displaystyle-588156\varphi_{1}^{5}\varphi_{2}^{2}+880\varphi_{1}^{6}\varphi_{2}^{2}+280\varphi_{1}^{7}\varphi_{2}^{2}-6701\varphi_{2}^{3}-2858677\varphi_{1}\varphi_{2}^{3}$
$\displaystyle+8757690\varphi_{1}^{2}\varphi_{2}^{3}-13388420\varphi_{1}^{3}\varphi_{2}^{3}+1228780\varphi_{1}^{4}\varphi_{2}^{3}-37094\varphi_{1}^{5}\varphi_{2}^{3}+672\varphi_{1}^{6}\varphi_{2}^{3}$
$\displaystyle-29717\varphi_{2}^{4}-2218750\varphi_{1}\varphi_{2}^{4}+3502930\varphi_{1}^{2}\varphi_{2}^{4}+1228780\varphi_{1}^{3}\varphi_{2}^{4}-66890\varphi_{1}^{4}\varphi_{2}^{4}$
$\displaystyle+1022\varphi_{1}^{5}\varphi_{2}^{4}-29363\varphi_{2}^{5}-307684\varphi_{1}\varphi_{2}^{5}-588156\varphi_{1}^{2}\varphi_{2}^{5}-37094\varphi_{1}^{3}\varphi_{2}^{5}$
$\displaystyle+1022\varphi_{1}^{4}\varphi_{2}^{5}-4824\varphi_{2}^{6}+97632\varphi_{1}\varphi_{2}^{6}+880\varphi_{1}^{2}\varphi_{2}^{6}+672\varphi_{1}^{3}\varphi_{2}^{6}-2656\varphi_{2}^{7}$
$\displaystyle+4550\varphi_{1}\varphi_{2}^{7}+280\varphi_{1}^{2}\varphi_{2}^{7}+21\varphi_{2}^{8}+67\varphi_{1}\varphi_{2}^{8}+7\varphi_{2}^{9}\bigg{)}~{}.$

We continue to contract propagator ambiguities with $\varphi^{i}$, and write

$\mathfrak{h}^{j}=\varphi^{i}h^{j}_{i}$ (3.88)

so that $\varphi^{i}D_{i}S^{j}$ reads

$\varphi^{i}D_{i}S^{j}=2\varphi^{j}S-\mathfrak{C}_{mn}S^{m}S^{nj}+\mathfrak{h}^{jk}K_{k}+\mathfrak{h}^{j}~{}.$
(3.89)

We now make another choice,
$\frac{1}{\varphi_{1}}\mathfrak{h}^{1}+\frac{1}{\varphi_{2}}\mathfrak{h}^{2}=0$.
Then we can get a symmetric $S$ by taking

$S=\frac{1}{2}\left[\frac{1}{2\varphi_{1}}\left(\varphi^{i}D_{i}S^{1}+\mathfrak{C}_{mn}S^{m}S^{nj}-\mathfrak{h}^{1k}K_{k}\right)+\left(\varphi_{1}\leftrightarrow\varphi_{2}\right)\right]~{}.$
(3.90)

The first few terms of $S$’s Taylor expansion are

$S=-\frac{3}{140}-\frac{188\varphi_{1}}{245}-\frac{188\varphi_{2}}{245}-\frac{5659\varphi_{1}^{2}}{6860}-\frac{34163\varphi_{1}\varphi_{2}}{3430}-\frac{5659\varphi_{2}^{2}}{6860}+\,...~{}.$
(3.91)

Having computed each of the propagators, we can work back through the
relations (3.54) to obtain all of the propagator ambiguities, which would be
necessary to go beyond genus 2. We decline to show those functions here, but
remark that all of their denominators are products of powers of $Z$ and
$\Delta$. The most complicated propagator ambiguity is $h_{i}$, which has as
denominator $40\varphi_{i}Z^{4}\Delta^{3}$, and a numerator of degree 21.

We now plug our propagators into (3.62). It remains to fix the holomorphic
ambiguity, which is of the form

$f^{(2)}=\frac{P_{15}}{Z^{3}\Delta^{2}}~{},$ (3.92)

where $P_{15}$ is a degree-15 symmetric polynomial in
$\varphi_{1},\,\varphi_{2}$.

We consider the combined set of unknowns, coefficients of $P_{15}$ and the
genus 2 instanton numbers $n^{(2)}_{(i,j)}$. We have 73 unknown coefficients
of $P_{15}$. Increasing the order to which we work at provides more unknowns
$n^{(2)}_{(i,j)}$, and as many equations for these. Some of these new unknowns
we can fix in terms of other unknowns by Coxeter symmetry, until with a high
enough order we exhaust the independent constraints that Coxeter symmetry
offers.

Demanding that the $q$-expansion of $\mathcal{F}^{(2)}$ possesses infinite
dihedral Coxeter symmetry reduces the number of unknowns, but unfortunately
these constraints are not all independent and we cannot get by solely by
working to a higher order and imposing Coxeter symmetry. Fortunately, for this
example, we know ahead of time which instanton numbers should vanish from the
tables of Hosono and Takagi [60]. These are the numbers $n^{(2)}_{(i,j)}$ with
$i$ or $j$ less than 4, and also $n^{(2)}_{(4,4)}$. If we include these zeroes
as known values, then we get an integer set of solutions for the instanton
numbers that agrees with Hosono and Takagi’s tables where they overlap, as
well as fixing the ambiguity $f^{(2)}$. Curiously, we need _all_ of these
zeroes. If we leave out $n^{(2)}_{(4,4)}=0$ then we have a solution with one
degree of freedom, eliminated by imposing $n^{(2)}_{(4,4)}=0$.

We have taken a barbaric way to know these zeroes, from Hosono and Takagi’s
computation. Since they used the conifold gap condition, we have not escaped
its necessity. Future investigation will determine whether there is a means of
fixing all zeroes from another principle, like a Castelnuovo bound (vital for
the computation of [17], with a possible generalisation discussed in [87]),
and then we could hope to check how high of a genus Coxeter symmetry will take
us. The genus 2 numbers that we compute are tabulated in table 3.9 .

In [60], Hosono and Takagi display numbers $n^{(g)}_{(i,j)}$ for $0\leq g\leq
2$, $0\leq i\leq 12$, $0\leq j\leq 6$. We have in this thesis displayed
numbers for this model at the same genera, but for degrees $i+j\leq 37$, which
includes numbers not displayed before (even when the Coxeter action is taken
into account). It would be interesting to return to this problem at genus 3.

The contents of the following table is available in electronic form [1].

$\bm{p}$ $n^{(2)}_{\bm{p}}$ $\bm{p}$ $n^{(2)}_{\bm{p}}$ 5,4 2500 12,8
307911899250593057650 5,5 2238300 13,7 1083681563433259200 6,4 36800 14,6
91091698085900 6,5 101188225 15,5 2238300 7,4 132150 11,10
389901597341682586245250 6,6 10885677450 12,9 59381136406580839191450 7,5
1276419800 13,8 1088683595004165702950 8,4 191850 14,7 1328923645662952325 7,6
321011805475 15,6 25512018106050 8,5 6764994000 16,5 2500 9,4 132150 11,11
13181153378953434491429200 7,7 19785981206800 12,10 5475299266824539894041150
8,6 4007017841650 13,9 356988331876350933661250 9,5 17585700200 14,8
2645148270492863092650 10,4 36800 15,7 1083681563433259200 8,7 497495418446900
16,6 4007017841650 9,6 25512018106050 12,11 282712962376014816940856175 10,5
24017901850 13,10 52176320562765050835877250 11,4 2500 14,9
1500298428800313660065325 8,8 23487690756165150 15,8 4481489156958762453000
9,7 6354579702758450 16,7 585085137096464550 10,6 91091698085900 17,6
321011805475 11,5 17585700200 12,12 9047714570978243256779338500 9,8
546653742258204975 13,11 4082629478671789440045910650 10,7 46325408551373425
14,10 349654611337758233664220900 11,6 192086807308450 15,9
4499834280177683754617100 12,5 6764994000 16,8 5337842575936922957650 9,9
22058659953217981800 17,7 206305163005291900 10,8 7207510049560719850 18,6
10885677450 11,7 206305163005291900 13,12 192252132113764705987353896200 12,6
245649059538250 14,11 41394311546286990570767724825 13,5 1276419800 15,10
1690595367815132163307836750 10,9 492220346020866314150 16,9
9770462296518547305273800 11,8 58488168886336533925 17,8
4481489156958762453000 12,7 585085137096464550 18,7 46325408551373425 13,6
192086807308450 19,6 101188225 14,5 101188225 13,13
5896815924361936623044444626800 10,10 17908353146570145820800 14,12
2847327956358643190359156085300 11,9 6703938825093094075300 15,11
303781402592961107944725255700 Table 3.9: The genus 2 instanton numbers of
total degree $\leqslant 37$ for the family (3.15). The numbers not in this
list are either zero, or given by those in the table after permuting indices.

$\bm{p}$ $n^{(2)}_{\bm{p}}$ 16,10 6008644011089373537248656800 17,9
15502680926528679507634750 18,8 2645148270492863092650 19,7 6354579702758450
20,6 36800 14,13 124565089119075700247816681507000 15,12
30450650946759882533741285631525 16,11 1650396599182174240918908567600 17,10
15912299153776396184830406150 18,9 18071193444493417015126625 19,8
1088683595004165702950 20,7 497495418446900 14,14
3693414921590391334562655080548700 15,13 1887312824415684091294137374297850
16,12 241434752646453338262009308998450 17,11 6750850050453875649938033430650
18,10 31701961212371099761950706550 19,9 15502680926528679507634750 20,8
307911899250593057650 21,7 19785981206800 15,14
77734854231576430735155964501483500 16,13 21141578696054339753623844206366900
17,12 1447892872574562822004056745806925 18,11
21056470435268391362146639838000 19,10 47822198309268358643758769375 20,9
9770462296518547305273800 21,8 58488168886336533925 22,7 321011805475 15,15
2242005633475379243138889611530224350 16,14
1202166271275554852096319284514781050 17,13
179193911305392936323515910617613500

table 3.9 continued.

$\bm{p}$ $n^{(2)}_{\bm{p}}$ 18,12 6669236758606676544482977924485150 19,11
50556113264139731816624429986400 20,10 54823814338848321169711111550 21,9
4499834280177683754617100 22,8 7207510049560719850 23,7 1276419800 16,15
47083170110743580006872424368633531875 17,14
14025908356241673430452097295836041700 18,13
1169983477395354290630422427608175550 19,12
23876076766141789732446125447010225 20,11 94078435233076347832828692471750
21,10 47822198309268358643758769375 22,9 1500298428800313660065325 23,8
546653742258204975 24,7 132150 16,16 1327080986654885164312931683366667358200
17,15 741736053327841335761519208560504382350 18,14
126013267019536739983934798259312856100 19,13
5967526679788804873035522660386672150 20,12
67042622214148592803250933902047750 21,11 136315316135643017078981825252200
22,10 31701961212371099761950706550 23,9 356988331876350933661250 24,8
23487690756165150 17,16 27835064011278355479795425089182885779850 18,15
8971331196558149334377988222442191858000 19,14
885994523265940239167051685255212381600 20,13
24041369335716235988574456740045846350 21,12
148673916874105664720736408737803900 22,11 154206354444180862857899859688400
23,10 15912299153776396184830406150 24,9 59381136406580839191450 25,8
497495418446900 17,17 769459996977013484532529807380612024950900 18,16
445912967342824716142914500096902010818850

table 3.9 continued.

$\bm{p}$ $n^{(2)}_{\bm{p}}$ 19,15 84847587251331673881762239411974072983050
20,14 4938016593847719974314383381477983741100 21,13
77169654314981775489114851835598126800 22,12
261702486079440770739117430365216750 23,11 136315316135643017078981825252200
24,10 6008644011089373537248656800 25,9 6703938825093094075300 26,8
4007017841650 18,17 16130063331032815277955386421606900094355875 19,16
5569304833367577335205278477402592908516350 20,15
636726757004598625316020644295171791557250 21,14
22041710721238453533303229084611701286825 22,13
198702763601020024146439712249527136275 23,12
366918571341072517525517535457060450 24,11 94078435233076347832828692471750
25,10 1690595367815132163307836750 26,9 492220346020866314150 27,8 6764994000
18,18 438539109875379528721919654529846561401409750 19,17
262362104844021123045694638864495986589287000 20,16
55131178495812759321176609589875814140037750 21,15
3836414759887507545665007708487359916157800 22,14
79449270275648960296194539460339370694050 23,13
412564879312424855910387890012240484400 24,12
410576929861745683208869036387758800 25,11 50556113264139731816624429986400
26,10 349654611337758233664220900 27,9 22058659953217981800 28,8 191850 19,18
9192063299791665753644261196926491990414773625 20,17
3372577227429580157843234196650003432550205075 21,16
438359755343431485741657365115289687329201500 22,15
18737856667070391183320951837536512233506700 23,14
232781528281126665497995449029490453266300 24,13
693400092277019907121874794244775804900 25,12
366918571341072517525517535457060450 26,11 21056470435268391362146639838000
27,10 52176320562765050835877250 28,9 546653742258204975

table 3.9 continued.

##  Chapter 4 Supergravity Compactifications

> Rosea : I can’t even touch it. Seems to be in a state of flux.
> Donna : What does that mean?
> Rosea : I don’t know. Sort of thing the Doctor would say.
>
> Russell T Davies, Turn Left

Our discussion of background material in this section follows the textbooks
[91, 92]. If massive degrees of freedom are neglected, then any of the five
ten-dimensional superstring theories reduce to ten-dimensional supergravity
theories. Of foremost concern to us are the two type II superstring theories
(type IIA and IIB), whose low energy effective theories are the two maximal
$\mathcal{N}=2$ supergravities (of type IIA and IIB respectively). Either of
the maximal supergravities can be compactified on a Calabi-Yau
threefold111Complex dimension three, so real dimension six. to obtain a four-
dimensional $\mathcal{N}=2$ matter coupled supergravity, and at the level of
supergravity, mirror symmetry gives a duality between each of the supergravity
theories obtained by the compactifications

$\begin{matrix}\text{Type IIB supergravity}\\\ \text{on CY threefold
X}\end{matrix}\qquad\cong\qquad\begin{matrix}\text{Type IIA supergravity}\\\
\text{on mirror Y}\end{matrix}~{}.$ (4.1)

We will review briefly the field content of these 4d theories, in particular
recalling that the scalar fields in the matter content include geometric
moduli of the threefolds. By comparing supersymmetric black hole solutions to
both of the above theories, we can in some cases obtain formulae that relate
number-theoretic quantities computed on $X$ to the enumerative invariants of
$Y$.

Moreover, one approach to realising realistic phenomenological models is _flux
compactifications_ , which we shall always view from the IIB perspective.
Here, in addition to compactifying IIB supergravity (or indeed the full string
theory) on a Calabi-Yau threefold $X$, one gives nonzero vacuum expectation
values to form fields supported on the cohomology of $X$. This breaks
supersymmetry further, and the four-dimensional massless theory has
$\mathcal{N}=1$ supersymmetry. We will study supersymmetric vacuum
configurations, and provide new examples in support of the _flux modularity
conjecture_ [12, 11]. Such IIB setups can be related to F-theory on an
elliptically fibred Calabi-Yau _fourfold_ $\widetilde{X}$, and we will see
that (in line with previous conjectures) the elliptic fibre in a
supersymmetric vacuum configuration has a surprising relation to the
modularity of the threefold $X$.

###  4.1 4d $\mathcal{N}=2$ matter coupled supergravities

The $\mathcal{N}=2$ gravity multiplet consists of the spin-2 spacetime metric
$g_{\mu\nu}$, two spin-$3/2$ gravitini, and a spin-1 vector field dubbed the
graviphoton. The CPT conjugates to all of these must be included in the theory
as well, so as to have CPT symmetry. Additional vector fields can be found in
the matter content, residing in 4d $\mathcal{N}=2$ vector multiplets which
each consist of a spin-1 vector field, two spin-$1/2$ gaugini, and two real
scalars. These scalars are packaged into one complex scalar. Again, this
content is not CPT self-conjugate and so fields with opposite helicities must
also be included. Finally, there can be CPT self-conjugate hypermultiplets,
which contain two spin-$1/2$ fields, their CPT conugate spin-$-1/2$ fields,
and four real scalars.

The attractor mechanism governs the values of the vector multiplet scalars
near a stationary black hole solution, so we pause to comment on the possible
manifolds in which the vector multiplet scalars of a $\mathcal{N}=2$ theory
can live in. With there being $n$ vector multiplets, we have $n+1$
$\text{U}(1)$ vector fields when the graviphoton is incorporated. We will only
be concerned with gauge fields for Abelian gauge groups $\text{U}(1)^{n+1}$,
i.e. extended Maxwell theory. Just as classical electromagnetism (albeit with
electric and magnetic charges) possesses an electromagnetic duality group
$\text{SL}(2,\mathbbl{Z})$ under which the charge vector
$(q_{\text{electric}},q_{\text{magnetic}})^{T}$ transforms as a doublet,
extended Maxwell theory enjoys a symplectic electromagnetic duality group
$\text{Sp}(2n+2,\mathbbl{Z})$ under which the charge vector
$(\mathbf{q}_{\text{electric}},\mathbf{q}_{\text{magnetic}})$ transforms in
the vector representation.

If the 4d theory is to both have these duality transformations and be
supersymmetric, then the vector multiplet scalars must be coordinates on a
Kähler manifold with an
${\text{SP}(2n+2,\mathbbl{R})\otimes\text{GL}(1,\mathbbl{C})}$ bundle that
possesses a holomorphic symplectic section222When we turn to studying IIB
compactifications on a CY threefold $X$, the bundle will simply be
$H^{3}(X,\mathbbl{C})$ and the section will be the holomorphic three-form
$\Omega$. $\mathcal{S}$ [93]. The $n$ vector multiplet scalars can be
projectivised into $n+1$ scalar fields living on a Kähler manifold
$\mathcal{M}$. A symplectic duality transformation on the $2n+2$-component
charge vector is paired with a symplectic transformation of the section

$\mathcal{S}=\begin{pmatrix}X^{I}\\\ \mathcal{F}_{I}\end{pmatrix}$ (4.2)

on the scalar manifold $\mathcal{M}$. We are assuming the existence of such a
section $\mathcal{S}$ in line with [93], and denoting components of
$\mathcal{S}$ by $X_{I}$ and $\mathcal{F}_{I}$. In this equation and the
following, $I,J,K$ run from 0 to $n$. With $\langle\circ,\circ\rangle$ giving
the symplectic form, the Kähler potential of $\mathcal{M}$ is

$K=\text{i}\langle\mathcal{S},\bar{\mathcal{S}}\rangle~{}.$ (4.3)

There is an additional requirement, that

$\langle\partial_{I}\mathcal{S},\partial_{J}\mathcal{S}\rangle=0~{}.$ (4.4)

Kähler manifolds possessing such a section $\mathcal{S}$ are termed _special
Kähler_. So long as the matrix $\partial_{I}X^{J}$ is invertible, then there
must locally exist a holomorphic function $\mathcal{F}(X)$, homogeneous of
degree 2, such that

$\mathcal{F}_{I}=\frac{\partial\mathcal{F}(X)}{\partial X^{I}}.$ (4.5)

In the absence of fluxes, the Kaluza-Klein reduction (as recounted for
instance in the textboook [91]) of either maximal 10d supergravity on a
Calabi-Yau is a 4d $\mathcal{N}=2$ supergravity with some number of the
aforementioned two kinds of matter multiplets. We turn now to describing these
theories, with particular mind to the prepotential for their vector multiplet
scalars.

##### Type IIA compactifications

If type IIA supergravity is compactified on a Calabi-Yau $Y$, the resulting
theory possesses $h^{1,1}$ vector multiplets and $h^{2,1}+1$ hypermultiplets.
The $h^{1,1}$ vector multiplet’s complex scalars parametrise the space of
complexified Kähler classes on $Y$. The hypermultiplet moduli space contains
the space of complex structures on $Y$, in addition to further scalars coming
from reduction of the type IIA form fields on $Y$.

The first half of our symplectic section $\mathcal{S}$, which we denoted
$X^{I}$ in the previous subsection, will be taken to be the projective
coordinates $z^{I}$.

The prepotential is closely related to the genus-0 topological A-model free
energy,

$\displaystyle\mathcal{F}$
$\displaystyle=-\frac{1}{6z^{0}}Y_{IJK}z^{I}z^{J}z^{K}-\frac{\left(z^{0}\right)^{2}}{(2\pi\text{i})^{3}}\sum_{\mathbf{k}\geq
0}n^{(0)}_{\mathbf{k}}\,\text{Li}_{3}\left(\exp\left(2\pi\text{i}\,k_{i}\frac{\bm{z}^{i}}{z^{0}}\right)\right)$
(4.6)
$\displaystyle\equiv-\frac{1}{6z^{0}}Y_{IJK}z^{I}z^{J}z^{K}-\left(z^{0}\right)^{2}\mathcal{I}\left(\frac{\bm{z}}{z^{0}}\right)~{}.$

We have collected the $z^{i}$ into the vector quantity $\bm{z}$. Here $i$ runs
from 1 to $h^{1,1}(Y)$ while $I,\,J,\,K$ run from 0 to $h^{1,1}(Y)$. The
$Y_{ijk}$ are the topological numbers that we encountered in (2.6), (2.10),
(2.12) and (2.11).

The projective coordinates $z^{I}$ relate to the usual complexified Kähler
coordinates $t^{i}$ via

$t^{i}=\frac{z^{i}}{z^{0}}~{}.$ (4.7)

Note that in (4.6) we have given the prepotential $\mathcal{F}$ together with
its nonperturbative instanton corrections
$\left(z^{0}\right)^{2}\mathcal{I}\left(z^{i}/z^{0}\right)$ and perturbative
quantum corrections, those being the terms in
$\frac{1}{6z^{0}}Y_{IJK}z^{I}z^{J}z^{K}$ where any index takes the value 0. We
will understand the purely classical value of the prepotential to be

$\mathcal{F}_{\text{Classical}}=-\frac{\left(z^{0}\right)^{2}}{6}Y_{ijk}t^{i}t^{j}t^{k}~{}.$
(4.8)

##### Type IIB compactifications

Compactifying the type IIB supergravity on a CY threefold $X$ leads to a 4d
$\mathcal{N}=2$ theory with $h^{2,1}(X)$ vector multiplets and $h^{1,1}(X)+1$
hypermultiplets. The $h^{2,1}$ vector multiplet scalars parametrise complex
structures on $X$, and within the hypermultiplet moduli are $h^{1,1}(X)$
complex scalars parametrising the complexified Kähler structures on $X$. When
$X$ is mirror to $Y$, the field content of IIB on $X$ and IIA on $Y$ are
identical. Both theories are actually identical, but this is a more involved
duality than simply identifying dimensions by swapping the Hodge numbers, as
the hypermultiplet moduli space of either compactification must contain the
vector multiplet moduli space of the other. In both compactifications there is
a universal hypermultiplet containing the axiodilaton field.

By using the mirror map, so that the $t^{i}$ parametrising $Y$’s Kähler
structures are coordinates on the complex structure moduli space of $X$, we
can find the same symplectic section as in the previous subsection on IIA
compactifications,

$\mathcal{S}=\begin{pmatrix}z^{I}\\\
\displaystyle\frac{\partial\mathcal{F}}{\partial z^{I}}\end{pmatrix}~{},$
(4.9)

with the same $\mathcal{F}$ as in (4.6). There is an important point to be
made on classical versus nonclassical physics. In the IIA compactification, we
have seen that the prepotential $\mathcal{F}$ is the sum of a classical part
and quantum corrections. However, in the IIB frame this quantity is purely
classical. Only after changing duality frames does the instanton sum
$\mathcal{I}$ concealed in $\mathcal{F}$ above take on a nonperturbative
interpretation.

The symplectic section $\mathcal{S}$ has a natural relation to the holomorphic
three-form $\Omega$ on $X$. After introducing a symplectic basis
$\alpha_{I},\,\beta^{I}$ of $H^{3}(X,\mathbbl{Z})$ we can write

$\Omega=z^{I}\alpha_{I}-\frac{\partial\mathcal{F}}{\partial
z^{I}}\beta^{I}~{}.$ (4.10)

Or in other words, the symplectic section is the integral period vector

$\mathcal{S}=\Pi~{}.$ (4.11)

###  4.2 The attractor mechanism

This subsection serves to review some background material. We follow [92] and
display a derivation of the attractor equations, including the necessary
manipulations to obtain Strominger’s form of the equations [94]. We adopt a
set of notations in this section that make the supergravity analysis
tractable, but this section is mostly self-contained and included to provide
better background to the work on rank-two attractors.

In words, the attractor mechanism fixes the values of the vector multiplet
scalars at the horizon of a supersymmetric black hole configuration in terms
of the charges of said black hole. We shall at the end of this section explain
how in Calabi-Yau compactifications these attractor equations can be
interpreted in terms of the internal geometry’s cohomology.

We begin with a more concrete account of electromagnetic duality, which was
important in arguing for the existence of a holomorphic symplectic section on
the scalar manifold. Then we set about describing dilatation-gauge fixing,
which is necessary in order to fix the form of the action that we will
consider black hole solutions for, and the kinetic terms for the gauge fields.

Subsequently, we will take a spherically symmetric solution ansatz and derive
the equations of motion. The scalar field’s equations of motions will be
recast as a gradient flow equation. Finally we will follow Ferrara and Kallosh
[95] to find algebraic expressions for the fixed points of these gradient
flows. Note that although we will study attractor points, we never work with
the gradient flow equations, so do not find attractors by solving differential
equations. We instead will in later sections deal with solutions to the
algebraic equations given at the end of this section.

##### Electromagnetic duality

We shall write the Maxwell fields as $A_{\mu}^{I}$, with $\mu$ a spacetime
index and $I$ running from $0$ to $n$ (one of these is the graviphoton and the
remaining n are fields in the vector multiplets). The field strengths will
then be

$F_{\mu\nu}^{\phantom{\mu\nu}I}=\partial_{\mu}A^{I}_{\nu}-\partial_{\nu}A^{I}_{\mu}.$
(4.12)

These field strengths form the first $n+1$ components of a symplectic vector
with $2n{+}2$ components. With the action $S$ given below, the quantities

$G_{\mu\nu I}=\kappa^{2}\epsilon_{\mu\nu\rho\sigma}\frac{\delta S}{\delta
F_{\rho\sigma}^{\phantom{\rho\sigma}I}}$ (4.13)

form the remaining $n+1$ components. If $F_{\mu\nu}^{\phantom{\mu\nu}I}$ and
$G_{\mu\nu I}$ solve the gauge field equations of motion then so do
$\widehat{F}_{\mu\nu}^{\phantom{\mu\nu}I}$ and $\widehat{G}_{\mu\nu I}$
defined by

$\left(\begin{matrix}\widehat{F}_{\mu\nu}^{\phantom{\mu\nu}I}\\\\[5.0pt]
\widehat{G}_{\mu\nu
I}\end{matrix}\right)=\mathbf{W}\left(\begin{matrix}F_{\mu\nu}^{\phantom{\mu\nu}I}\\\\[5.0pt]
G_{\mu\nu I}\end{matrix}\right),$ (4.14)

where $\mathbf{W}$ is a $(2n+2)\times(2n+2)$ symplectic matrix. There is a
corresponding transformation law for the gauge kinetic matrix (and hence the
scalars, upon which this matrix depends). This is the statement of
electromagnetic duality for Maxwell fields.

The electric and magnetic charges $q_{I}$ and $p^{I}$ in a volume $V$ are
given by

$q_{I}=-\frac{1}{2}\int_{V}\text{d}^{3}x\,\epsilon^{ijk}\partial_{i}G_{jkI}~{},\qquad
p^{I}=-\frac{1}{2}\int_{V}\text{d}^{3}x\,\epsilon^{ijk}\partial_{i}F_{jk}^{\phantom{jk}I}~{}.$
(4.15)

Under electromagnetic duality the charge vector
$\left(p^{I},q_{I}\right)^{\text{T}}$ is mapped to
$\mathbf{W}\left(p^{I},q_{I}\right)^{\text{T}}$. Quantization forces the
charges to lie on a lattice:

$\left(p^{I},q_{I}\right)\left(\begin{matrix}\phantom{-}\mathbf{0}_{n\times
n}&\mathbf{1}_{n\times n}\\\ -\mathbf{1}_{n\times n}&\mathbf{0}_{n\times
n}\end{matrix}\right)\left(\begin{matrix}p^{I}\\\
q_{I}\end{matrix}\right)=2\pi n,\qquad n\in\mathbb{Z},$ (4.16)

and so $\mathbf{W}$ must have integer entries:
$\mathbf{W}\in\text{Sp}\left(2n+2,\mathbb{Z}\right)$.

##### Dilatation gauge fixing and the gauge field kinetic term

We have already argued that this electromagnetic duality, plus supersymmetry,
leads to the requirement that the vector multiplet scalars take values in a
special Kähler manifold with symplectic section $\mathcal{S}$ and Kähler
potential $K=\text{i}\langle\mathcal{S},\bar{\mathcal{S}}\rangle$.

With the symplectic section $\mathcal{S}$ defined in terms of a prepotential
$\mathcal{F}$ as in (4.2), the gauge field kinetic matrix is

$\kappa^{2}\mathcal{N}_{IJ}\left(z,\bar{z}\right)=\bar{F}_{IJ}+\text{i}\frac{K_{IK}X^{K}\,K_{JL}X^{L}}{K_{MN}X^{M}X^{N}},$
(4.17)

in which all subscripts denote partial derivatives. This couples the scalars
to the gauge fields. We write

$\text{{R}}_{IJ}=\text{Re}\left[\mathcal{N}_{IJ}\right],\qquad\text{{I}}_{IJ}=\text{Im}\left[\mathcal{N}_{IJ}\right].$
(4.18)

The dilatation operator’s action on the scalars is to flow them along a
Killing vector of this manifold. To gauge fix, one identifies all points in
the same orbits of this action. Using the Frobenius theorem one can choose
coordinates $y,z^{\alpha}$ so that $\partial_{y}$ is parallel to this Killing
vector.

We take the quantities $X^{I}$ to be projective by introducing functions
$Z^{I}(z)$, that do not depend on $y$, such that $X^{I}=yZ^{I}(z)$. One can
then write

$\mathcal{S}=\left(\begin{matrix}X^{I}\\\\[5.0pt]
F_{I}(X)\end{matrix}\right)=y\left(\begin{matrix}Z^{I}(z)\\\\[5.0pt]
F_{I}\left(Z(z)\right)\end{matrix}\right).$ (4.19)

The dilatation gauge is fixed by choosing a specific value for $y$. This value
is chosen to be real and to keep the Kähler potential $K$ at a fixed constant
value of $-\kappa^{-2}$.

Performing this fixing we are left with a projective special Kähler manifold.
This has coordinates $z^{\alpha}$, which are the scalars that will appear in
the gauge-fixed action. The Kähler potential of this projective manifold is
$\mathcal{K}\left(z,\bar{z}\right)$, where

$\text{e}^{-\kappa^{2}\mathcal{K}}=-\kappa^{2}\frac{\partial K}{\partial
X^{I}\partial\bar{X}^{\bar{J}}}Z^{I}\bar{Z}^{\bar{J}}=\frac{1}{|y|^{2}}.$
(4.20)

There is some freedom in our choice of scale for $Z^{I}$ and $y$. Had we
instead opted for

$\widehat{y}=y\text{e}^{\kappa^{2}f(z)},\quad\widehat{Z}^{I}=Z^{I}\text{e}^{-\kappa^{2}f(z)}$
(4.21)

with $f$ an arbitrary holomorphic function, then we would have arrived at a
projective potential

$\widehat{\mathcal{K}}=\mathcal{K}+f+\bar{f}.$ (4.22)

A connection is introduced to give derivatives covariant under the
transformations (4.21):

$\displaystyle\nabla_{\alpha}Z^{I}=\partial_{\alpha}Z^{I}+\kappa^{2}\left(\partial_{\alpha}\mathcal{K}\right)Z^{I},$
(4.23)
$\displaystyle\overline{\nabla}_{\bar{\alpha}}\bar{Z}^{\bar{I}}=\partial_{\bar{\alpha}}\bar{Z}^{\bar{I}}+\kappa^{2}\left(\partial_{\bar{\alpha}}\mathcal{K}\right)\bar{Z}^{\bar{I}},$
$\displaystyle\nabla_{\alpha}\bar{Z}^{\bar{I}}=\partial_{\alpha}\bar{Z}^{\bar{I}}=0,$
$\displaystyle\overline{\nabla}_{\bar{\alpha}}Z^{I}=\partial_{\bar{\alpha}}Z^{I}=0.$

The metric on the projective manifold can be written as

$g_{\alpha\bar{\beta}}=\partial_{\alpha}\partial_{\bar{\beta}}\mathcal{K}=\text{i}\langle\nabla_{\alpha}\mathcal{S},\bar{\nabla}_{\bar{\beta}}\bar{\mathcal{S}}\rangle.$
(4.24)

We end this subsection discussing the scalars by stating the bilinear form
$\mathcal{N}_{IJ}$ used in the kinetic term for the gauge fields when a
prepotential does not exist. This is

$\kappa^{2}\bar{\mathcal{N}}_{IJ}=\left(\bar{F}_{I}\;\nabla_{\alpha}F_{I}\right)\left(\bar{X}^{J}\;\nabla_{\alpha}X^{J}\right)^{-1}.$
(4.25)

We will still in this case use the definitions (4.18).

##### Equations of motion

The bosonic part of the action for this theory is

$S\\!=\frac{1}{2\kappa^{2}}\\!\int\\!\text{d}^{4}x\left[\sqrt{-g}\left(R{-}2g^{\mu\nu}g_{\alpha\bar{\beta}}\partial_{\mu}z^{\alpha}\partial_{\nu}\bar{z}^{\bar{\beta}}+\frac{1}{2}\text{{I}}_{IJ}F_{\mu\nu}^{\phantom{\mu\nu}I}F^{\mu\nu
J}\right){-}\frac{1}{4}\text{{R}}_{IJ}\,\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}^{\phantom{\mu\nu}I}F_{\rho\sigma}^{\phantom{\rho\sigma}J}\right].$
(4.26)

One takes the following ansatz for a spherically symmetric, static metric:

$\text{d}s^{2}=-\text{e}^{2U(\tau)}\text{d}t^{2}+\text{e}^{-2U(\tau)}\left[\frac{\text{d}\tau^{2}}{\tau^{4}}+\frac{1}{\tau^{2}}\left(\text{d}\theta^{2}+\sin^{2}\left(\theta\right)\,\text{d}\phi^{2}\right)\right].$
(4.27)

The coordinates $\theta$ and $\phi$ are the familiar 2-sphere coordinates. The
extremal Reissner-Nordstrom coordinate can be brought to this form, and in
this case $\tau$ is $1/r$, where the shifted radial variable $r$ is 0 at the
horizon.

In a static, spherically symmetric solution the three-dimensional electric and
magnetic fields will only have radial components. Thus our field strengths are
of the form

$F^{I}=F^{\phantom{t\tau}I}_{t\tau}\text{d}t\wedge\text{d}\tau+F_{\theta\phi}^{\phantom{\theta\phi}I}\text{d}\theta\wedge\text{d}\phi.$
(4.28)

Staticity and spherical symmetry gives

$\nabla_{t}F_{\mu\nu}^{\phantom{\mu\nu}I}=\nabla_{\theta}F_{\mu\nu}^{\phantom{\mu\nu}I}=\nabla_{\Phi}F_{\mu\nu}^{\phantom{\mu\nu}I}=0.$
(4.29)

The latter two of these enter into a Bianchi identity from which one obtains
the equation

$\nabla_{\tau}F_{\theta\phi}^{\phantom{\theta\phi}I}=0.$ (4.30)

These equations are solved by

$F^{\phantom{t\tau}I}_{t\tau}=f^{I}(\tau),\qquad
F_{\theta\phi}^{\phantom{\theta\phi}I}=-\frac{p^{I}}{4\pi}\sin\left(\theta\right),$
(4.31)

with the constant of integration $-p^{I}/4\pi$ fixed by the magnetic charge
integral in (4.15).

The equation of motion $\delta S/\delta A^{I}_{t}=0$ following from the action
(4.26) is

$\sin\left(\theta\right)\,\partial_{\tau}\left[\text{e}^{-2U}\text{{I}}_{IJ}f^{J}(\tau)-\text{{R}}_{IJ}\frac{p^{I}}{4\pi}\right]=0.$
(4.32)

Integration gives $f^{I}(\tau)$.

$f^{I}(\tau)=\text{e}^{2U}(\text{{I}}^{-1})^{IJ}\left[\text{{R}}_{JK}\frac{p^{K}}{4\pi}-\frac{q_{J}}{4\pi}\right].$
(4.33)

Here the constant of integration is obtained by comparing the expression in
brackets in (4.32) with that of $G_{\mu\nu
I}=\kappa^{2}\epsilon_{\mu\nu\rho\sigma}\frac{\delta S}{\delta
F_{\rho\sigma}^{\phantom{\rho\sigma}I}}$ and then using the electric charge
integral in (4.15).

The stress energy tensor $T_{\mu\nu}$ for the above field content is

$\kappa^{2}T_{\mu\nu}\\!=-\text{{I}}_{IJ}\left(F_{\mu\rho}^{\phantom{\mu\rho}I}F_{\nu}^{\phantom{\nu}\rho
J}-\frac{1}{4}g_{\mu\nu}F_{\rho\sigma}^{\phantom{\rho\sigma}I}F^{\rho\sigma
J}\right)+g_{\alpha\bar{\beta}}\left(\partial_{\mu}z^{\alpha}\partial_{\nu}\bar{z}^{\bar{\beta}}+\partial_{\nu}z^{\alpha}\partial_{\mu}\bar{z}^{\bar{\beta}}-g_{\mu\nu}\partial_{\rho}z^{\alpha}\partial^{\rho}\bar{z}^{\bar{\beta}}\right).$
(4.34)

It is most convenient to work with the following form of the Einstein
equations:

$R_{\mu\nu}=\kappa^{2}\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T_{\rho}^{\phantom{\rho}\rho}\right).$
(4.35)

Expanding the right hand side using (4.34) leads to

$R_{\mu\nu}=-\text{{I}}_{IJ}\left(F_{\mu\rho}^{\phantom{\mu\rho}I}F_{\nu}^{\phantom{\nu}\rho
J}-\frac{1}{4}g_{\mu\nu}F_{\rho\sigma}^{\phantom{\rho\sigma}I}F^{\rho\sigma
J}\right)+g_{\alpha\bar{\beta}}\left(\partial_{\mu}z^{\alpha}\partial_{\nu}\bar{z}^{\bar{\beta}}+\partial_{\nu}z^{\alpha}\partial_{\mu}\bar{z}^{\bar{\beta}}\right).$
(4.36)

A significant simplification is met by introducing the black hole potential
$V_{BH}$.

$V_{BH}=\frac{1}{32\pi^{2}}\left(p^{I}\,\,q_{I}\right)\left(\begin{matrix}-(\text{{I}}+\text{{R}}\text{{I}}^{-1}\text{{R}})_{IJ}&(\text{{R}}\text{{I}}^{-1})_{I}^{\phantom{I}J}\\\\[5.0pt]
(\text{{I}}^{-1}\text{{R}})^{I}_{\phantom{I}J}&-(\text{{I}}^{-1})^{IJ}\end{matrix}\right)\begin{pmatrix}p^{J}\\\\[5.0pt]
q_{J}\end{pmatrix}.$ (4.37)

There are only two independent Einstein equations.

$\displaystyle R_{tt}$
$\displaystyle=\text{e}^{4U}\tau^{4}\ddot{U}=\text{e}^{6U}\tau^{4}V_{BH},$
(4.38) $\displaystyle R_{\tau\tau}$
$\displaystyle=\ddot{U}-2\dot{U}^{2}=-\text{e}^{2U}V_{BH}+2g_{\alpha\bar{\beta}}\dot{z}^{\alpha}\dot{\bar{z}}^{\bar{\beta}}.$

The dot represents differentation with respect to $\tau$. Neatening these
gives

$\displaystyle\ddot{U}$ $\displaystyle=\text{e}^{2U}V_{BH},$ (4.39)
$\displaystyle\dot{U}^{2}$
$\displaystyle=\text{e}^{2U}V_{BH}-g_{\alpha\bar{\beta}}\dot{z}^{\alpha}\dot{\bar{z}}^{\bar{\beta}}.$

The equations of motion of the scalars are

$\frac{\text{d}}{\text{d}\tau}\left(g_{\alpha\bar{\beta}}\dot{\bar{z}}^{\bar{\beta}}\right)-\left(\partial_{\alpha}g_{\gamma\bar{\delta}}\right)\dot{z}^{\gamma}\dot{\bar{z}}^{\bar{\delta}}-\text{e}^{2U}V_{BH}=0,$
(4.40)

plus the complex conjugate of these equations.

##### The black hole potential and the central charge

The central charge $\mathcal{Z}$ is related to the electromagnetic charges and
the scalar Kähler geometry by

$\mathcal{Z}=2\kappa^{-2}\text{e}^{\frac{\kappa^{2}}{2}\mathcal{K}}Z^{I}\left(q_{I}-\mathcal{N}_{IJ}p^{J}\right)=2\kappa^{-2}\left(X^{I}q_{I}-F_{I}p^{I}\right).$
(4.41)

After some unwinding, one can relate the central charge and the black hole
potential.

$(4\pi)^{2}V_{BH}=\frac{\kappa^{4}}{4}|\mathcal{Z}|^{2}+\frac{\kappa^{2}}{4}\left(\nabla_{\alpha}\mathcal{Z}\right)g^{\alpha\bar{\beta}}\left(\nabla_{\bar{\beta}}\bar{\mathcal{Z}}\right).$
(4.42)

Some formulae from special geometry give further simplification.
$\nabla_{\alpha}\bar{\mathcal{Z}}=0$ and
$\nabla_{\alpha}|\mathcal{Z}|^{2}=\partial_{\alpha}|\mathcal{Z}|^{2}$ lead to

$\nabla_{\alpha}\mathcal{Z}=2\sqrt{\frac{\mathcal{Z}}{\bar{\mathcal{Z}}}}\partial_{\alpha}|\mathcal{Z}|.$
(4.43)

This allows for

$\frac{V_{BH}}{G^{2}}=|\mathcal{Z}|^{2}+4g^{\alpha\bar{\beta}}\partial_{\alpha}|\mathcal{Z}|\partial_{\bar{\beta}}|\mathcal{Z}|~{},\qquad
G=\frac{\kappa}{8\pi}\frac{64\pi^{2}}{\kappa^{2}}$ (4.44)

##### The one-dimensional effective action

The large amount of symmetry has greatly reduced the number of equations.
Moreover, the equations (4.40) and the first of (4.39) extremise a single one-
dimensional effective action:

$S_{1D}\left[U,z,\bar{z}\right]=\int_{0}^{\infty}\text{d}\tau\left(\dot{U}^{2}+g_{\alpha\bar{\beta}}\dot{z}^{\alpha}\dot{\bar{z}}^{\bar{\beta}}+\text{e}^{2U}V_{BH}\right).$
(4.45)

Since the above Lagrangian is independent of $\tau$ there is a conserved
quantity:

$\mathcal{E}=\dot{U}^{2}+g_{\alpha\bar{\beta}}\dot{z}^{\alpha}\dot{\bar{z}}^{\bar{\beta}}-\text{e}^{2U}V_{BH}.$
(4.46)

By fixing $\mathcal{E}=0$ the second equation of (4.39) is satisfied. With
this constant of integration fixed, this one-dimensional action captures the
metric and scalar dynamics.

One can go further, by inserting the central charge expression (4.44) into the
action (4.45) the attractor behaviour can be made clear. By completing squares
and recognising a total derivative, one can write

$\displaystyle S_{1D}\left[U,z,\bar{z}\right]=$
$\displaystyle\int_{0}^{\infty}\text{d}\tau\left[\left(\dot{U}+G\text{e}^{U}|\mathcal{Z}|\right)^{2}+g_{\alpha\bar{\beta}}\left(\dot{z}^{\alpha}+2G\text{e}^{U}g^{\alpha\bar{\delta}}\partial_{\bar{\delta}}|\mathcal{Z}|\right)\left(\dot{\bar{z}}^{\bar{\beta}}+2G\text{e}^{U}g^{\gamma\bar{\beta}}\partial_{\gamma}|\mathcal{Z}|\right)\right]$
(4.47)
$\displaystyle-2G\text{e}^{U}|\mathcal{Z}|\bigg{|}_{\tau=0}^{\tau=\infty}.$

##### The attractor equations

The integral in this action has an integrand that is the sum of two squares.
It is thus minimised when both of these squares vanish. This requires that the
following attractor equations are satisfied:

$\dot{U}=-G\text{e}^{U}|\mathcal{Z}|,\qquad\dot{z}^{\alpha}=-2G\text{e}^{U}g^{\alpha\bar{\beta}}\partial_{\bar{\beta}}|\mathcal{Z}|,$
(4.48)

in addition to the complex conjugate of this last equation.

Note that these minimising equations together directly give an expression for
$\dot{U}^{2}+g_{\alpha\bar{\beta}}\dot{z}^{\alpha}\dot{\bar{z}}^{\bar{\beta}}$.
This is

$\dot{U}^{2}+g_{\alpha\bar{\beta}}\dot{z}^{\alpha}\dot{\bar{z}}^{\bar{\beta}}=G^{2}\text{e}^{2U}\left(|\mathcal{Z}|^{2}+4g^{\alpha\bar{\beta}}\partial_{\alpha}|\mathcal{Z}|\partial_{\bar{\beta}}|\mathcal{Z}|\right)=\text{e}^{2U}V_{BH}.$
(4.49)

Equation (4.44) has been used to make the rightmost transition. This implies
that $\mathcal{E}=0$ for any solutions to (4.48) and so the second equation in
(4.39) is satisfied.

The formulae (4.48) describe the evolution of $U$ and motion of the scalars on
their target space as $\tau$ increases. The second formula implies that under
this evolution

$\frac{\text{d}}{\text{d}\tau}|\mathcal{Z}|=-4G\text{e}^{U}g^{\alpha\bar{\beta}}\partial_{\alpha}|\mathcal{Z}|\partial_{\bar{\beta}}|\mathcal{Z}|.$
(4.50)

.

Since $g_{\alpha\bar{\beta}}$ is positive definite this equation informs us
that $|\mathcal{Z}|$ is monotonically decreasing along attractor flows, and
also that the flows end where $|\mathcal{Z}|$ takes a critical value. Since
$|\mathcal{Z}|$ is bounded below (by virtue of being nonnegative), there must
be a limiting value for $|\mathcal{Z}|$ as $\tau\rightarrow\infty$. The
scalars $z^{\alpha}$ thus flow to the values such that $|\mathcal{Z}|$ takes a
critical value. These are the attractor points of the scalar target manifold.

##### Saturation of the BPS bound

One can read off the mass of the black hole from the leading correction to
$g_{tt}$ at large $r$. It will be shown that this mass equals the modulus of
the central charge at infinity, which is the BPS condition. We seek to find
$M$ as per $g_{tt}=-\text{e}^{2U}=-1+2MG\tau+...~{},$ and obtain

$M=\frac{-1}{2G}\frac{\text{d}}{\text{d}\tau}\text{e}^{2U}\bigg{|}_{\tau=0}.$
(4.51)

The flow equations (4.48) facilitate such a computation. Rearranging the
equation involving $\dot{U}$, one obtains

$\frac{\text{d}}{\text{d}\tau}\text{e}^{-U}=G|\mathcal{Z}|.$ (4.52)

One can proceed by

$M=\frac{-1}{2G}\frac{\text{d}}{\text{d}\tau}\text{e}^{2U}\bigg{|}_{\tau=0}=\frac{-1}{2G}\frac{\text{d}}{\text{d}\tau}\left(\text{e}^{-U}\right)^{-2}\bigg{|}_{\tau=0}=\frac{1}{G}\text{e}^{3U}\frac{\text{d}}{\text{d}\tau}\text{e}^{-U}\bigg{|}_{\tau=0}=|\mathcal{Z}|_{\infty}.$
(4.53)

##### Stabilisation equations

We have seen that that the scalars flow to attractor points where
$|\mathcal{Z}|$ is minimised. A set of equations give the locations of these
attractor points as a function of the black hole charges $p^{I}$ and $q_{I}$.
These were initially derived by Strominger in 1996 [94]. Later that year they
were derived using a slightly different approach by Ferrara and Kallosh [95].

Strominger considered solutions to the field equations with constant scalars
(which must necessarily have an attractor point value throughout spacetime).
The spacetime geometry in such setups is Reissner-Nordstrom. As these black
holes are BPS and have vanishing Fermi fields, the bosonic parts of the
fermionic supersymmetry variations necessarily vanish. The gaugino variations
give the sought equations for the scalars in terms of the charges.

Ferrara and Kallosh’s approach, which we detail here, involves directly
extremising the central charge’s modulus. The starting point is formula
(4.41), reproduced here:

$\mathcal{Z}=2\kappa^{-2}\left(X^{I}q_{I}-F_{I}p^{I}\right).$ (4.54)

By formula (4.43), $\partial_{\alpha}|\mathcal{Z}|$ is a nonzero multiple of
$\nabla_{\alpha}\mathcal{Z}$. Thus the scalars that minimise $|\mathcal{Z}|$
satisfy

$\frac{\kappa^{2}}{2}\nabla_{\alpha}\mathcal{Z}=\left(\nabla_{\alpha}X^{I}\right)q_{I}-\left(\nabla_{\alpha}F_{I}\right)p^{I}=0.$
(4.55)

The argument will make use of some formulae from special geometry. There are
the raising and lowering identities

$F_{I}=\mathcal{N}_{IJ}X^{J},\qquad\bar{F}_{I}=\bar{\mathcal{N}}_{IJ}\bar{X}^{J},\qquad\nabla_{\alpha}F_{I}=\bar{\mathcal{N}}_{IJ}\nabla_{\alpha}X^{J},\qquad\bar{\nabla}_{\bar{\alpha}}\bar{F}_{I}=\mathcal{N}_{IJ}\overline{\nabla}_{\bar{\alpha}}\bar{X}^{J}.$
(4.56)

Using these one arrives at the following identities involving the vector
$\mathcal{S}$ of (4.19):

$\displaystyle\langle\mathcal{S},\bar{\mathcal{S}}\rangle$
$\displaystyle=X^{I}\bar{F}_{I}-\bar{X}^{I}F_{I}=X^{I}\bar{X}^{J}\left(\bar{\mathcal{N}}_{IJ}-\mathcal{N}_{IJ}\right)=-2\text{i}X^{I}\bar{X}^{J}\text{{I}}_{IJ},$
(4.57)
$\displaystyle\langle\nabla_{\alpha}\mathcal{S},\overline{\nabla}_{\bar{\beta}}\bar{\mathcal{S}}\rangle$
$\displaystyle=\left(\nabla_{\alpha}X^{I}\right)\left(\overline{\nabla}_{\bar{\beta}}\bar{F}_{I}\right)-\left(\overline{\nabla}_{\bar{\beta}}\bar{X}^{I}\right)\left(\nabla_{\alpha}F_{I}\right)=2\text{i}\left(\nabla_{\alpha}X^{I}\right)\left(\overline{\nabla}_{\bar{\beta}}\bar{X}^{J}\right)\text{{I}}_{IJ},$
$\displaystyle\langle\mathcal{S},\overline{\nabla}_{\bar{\alpha}}\bar{\mathcal{S}}\rangle$
$\displaystyle=X^{I}\overline{\nabla}_{\bar{\alpha}}\bar{F}^{J}-F_{I}\overline{\nabla}_{\bar{\alpha}}\bar{X}^{I}=X^{I}\mathcal{N}_{IJ}\overline{\nabla}_{\bar{\alpha}}\bar{X}^{J}-X^{J}\mathcal{N}_{JI}\overline{\nabla}_{\bar{\alpha}}\bar{X}^{I}=0,$
$\displaystyle\langle\bar{\mathcal{S}},\nabla_{\alpha}\mathcal{S}\rangle$
$\displaystyle=\bar{X}^{I}\nabla_{\alpha}F_{I}-\bar{F}_{I}\nabla_{\alpha}X^{I}=\bar{X}^{I}\bar{\mathcal{N}}_{IJ}\nabla_{\alpha}X^{J}-\bar{X}^{J}\bar{\mathcal{N}}_{IJ}\nabla_{\alpha}X^{I}=0.$

The second of these gives a formula for the metric component
$g_{\alpha\bar{\beta}}=\text{i}\langle\nabla_{\alpha}\mathcal{S},\overline{\nabla}_{\bar{\beta}}\bar{\mathcal{S}}\rangle$.
The first formula above is an equation for the Kähler potential of the rigid
manifold inside which our projective manifold is embedded333This embedding is
one stage in the construction of supergravity actions by gauge-fixing
conformal supergravity, as discussed in the textbook [92].,
$K=\text{i}\langle\mathcal{S},\bar{\mathcal{S}}\rangle$. In choosing a
dilatation gauge we chose values for the coordinate $y$ so that $K$ had a
constant value of $-\kappa^{-2}$, but so as to find a general set of
stabilisation equations not depending on gauge we do not fix this value here.
One should bear in mind that since $\mathcal{S}$ is holomorphic, $K$ is real.

Noting that by construction $\nabla_{\alpha}y=\nabla_{\alpha}\bar{y}$, all of
the equations in (4.57) can be packaged into a single $(n+1)\times(n+1)$
matrix equation: