Higher Derivative Corrections to String Inflation

In the standard approach of moduli stabilisation in 4D type IIB effective supergravity models, one follows a so-called two-step strategy. In the first step, the axio-dilaton $S$ and the complex structure moduli $U^{\alpha}$ are stabilised by the superpotential $W_{\rm flux}$ induced by background 3-form fluxes $(F_{3},H_{3})$. This flux-dependent superpotential can fix all complex structure moduli and the axio-dilaton supersymmetrically at leading order by enforcing:

$$D_{U^{\alpha}}W_{\rm flux}=D_{S}W_{\rm flux}\,=0\,.$$

(2.1)

After fixing $S$ and the $U$-moduli, the flux superpotential can effectively be considered as constant: $W_{0}=\langle W_{\rm flux}\rangle$. At this leading order, the Kähler moduli $T_{i}$ remain flat due to the no-scale cancellation. Using non-perturbative effects is one of the possibilities to fix these moduli. In this context, if we assume $n$ non-perturbative contributions to $W$ which can be generated by using rigid divisors, such as shrinkable dP 4-cycles, or by rigidifying non-rigid divisors using magnetic fluxes Bianchi:2011qh ; Bianchi:2012pn ; Louis:2012nb , the superpotential takes the following form:

$$W=W_{0}+\sum_{i=1}^{n}\,A_{i}\,e^{-a_{i}\,T_{i}}\,,$$

(2.2)

where:

$$T_{i}=\tau_{i}+i\theta_{i}\qquad\text{with}\qquad\tau_{i}=\frac{1}{2}\int_{D_{i}}J\wedge J\quad\text{and}\quad\theta_{i}=\int_{D_{i}}\,C_{4}\,.$$

(2.3)

For the current work we consider CY orientifolds with trivial odd sector in the $(1,1)$-cohomology and subsequently orientifold-odd moduli are absent in our analysis (interested readers may refer to Gao:2014uha ; Cicoli:2021tzt ; Carta:2022web ). Note that in (2.2) there is no sum in the exponents $(a_{i}\,T_{i})$, and summations are to be understood only when upper indices are contracted with lower indices; otherwise we will write an explicit sum as in (2.2). We will suppose that, out of $h^{1,1}_{+}=h^{1,1}$ Kähler moduli, only the first $n$ appear in $W$, i.e. $i=1,...,n\leq h^{1,1}_{+}$.

The Kähler potential including $\alpha^{\prime 3}$ corrections takes the form Becker:2002nn :

$$K=-\ln\left[-i\int\Omega\,(U^{\alpha})\wedge\bar{\Omega}\,(\bar{U^{\alpha}})\right]-\ln\left(S+\bar{S}\right)-2\ln\left[{\cal V}\,(T_{i}+\bar{T_{i}})+\frac{\xi}{2}\left(\frac{S+\bar{S}}{2}\right)^{3/2}\right],$$

where $\Omega$ denotes the nowhere vanishing holomorphic 3-form which depends on the complex-structure moduli, while ${\cal V}$ denotes the CY volume which receives $\alpha^{\prime 3}$ corrections through $\xi=-\frac{\chi(X)\,\zeta(3)}{2\,(2\pi)^{3}}$ where $\chi(X)$ is the CY Euler characteristic and $\zeta(3)\simeq 1.202$.

Assuming that $S$ and the $U$-moduli are stabilised as in (2.1), considering a superpotential given by (2.2) and an $\alpha^{\prime 3}$-corrected Kähler potential given by (2.1), one arrives at the following master formula for the scalar potential AbdusSalam:2020ywo :

$$V=V_{\alpha^{\prime 3}}+V_{\rm np1}+V_{\rm np2}\,,$$

(2.4)

where (defining $\hat{\xi}\equiv\xi g_{s}^{-3/2}$ with $g_{s}=\langle{\rm Re}(S)\rangle^{-1}$):

	$\displaystyle V_{\alpha^{\prime 3}}$	$\displaystyle=$	$\displaystyle e^{K}\,\frac{3\,\hat{\xi}(\mathcal{V}^{2}+7\,\mathcal{V}\,\hat{\xi}+\hat{\xi}^{2})}{({\cal V}-\hat{\xi})(2\mathcal{V}+\hat{\xi})^{2}}\,\,|W_{0}|^{2}\,,$		(2.5)
	$\displaystyle V_{\rm np1}$	$\displaystyle=$	$\displaystyle e^{K}\,\sum_{i=1}^{n}\,2\,|W_{0}|\,|A_{i}|\,e^{-a_{i}\tau_{i}}\,\cos(a_{i}\,\theta_{i}+\phi_{0}-\phi_{i})$
			$\displaystyle\times~{}\biggl{[}\frac{(4\mathcal{V}^{2}+\mathcal{V}\,\hat{\xi}+4\,\hat{\xi}^{2})}{(\mathcal{V}-\hat{\xi})(2\mathcal{V}+\hat{\xi})}\,(a_{i}\,\tau_{i})+\frac{3\,\hat{\xi}(\mathcal{V}^{2}+7\,\mathcal{V}\,\hat{\xi}+\hat{\xi}^{2})}{(\mathcal{V}-\hat{\xi})(2\mathcal{V}+\hat{\xi})^{2}}\biggr{]}\,,$
	$\displaystyle V_{\rm np2}$	$\displaystyle=$	$\displaystyle e^{K}\,\sum_{i=1}^{n}\,\sum_{j=1}^{n}\,|A_{i}|\,|A_{j}|\,e^{-\,(a_{i}\tau_{i}+a_{j}\tau_{j})}\,\cos(a_{i}\,\theta_{i}-a_{j}\,\theta_{j}-\phi_{i}+\phi_{i})\,$
			$\displaystyle\times\biggl{[}-4\left({\cal V}+\frac{\hat{\xi}}{2}\right)\,(k_{ijk}\,t^{k})\,a_{i}\,a_{j}\,+\frac{4{\cal V}-\hat{\xi}}{(\mathcal{V}-\hat{\xi})}\left(a_{i}\,\tau_{i})\,(a_{j}\,\tau_{j}\right)$
			$\displaystyle+\,\frac{(4\mathcal{V}^{2}+\mathcal{V}\,\hat{\xi}+4\,\hat{\xi}^{2})}{(\mathcal{V}-\hat{\xi})(2\mathcal{V}+\hat{\xi})}\,(a_{i}\,\tau_{i}+a_{j}\,\tau_{j})+\frac{3\,\hat{\xi}(\mathcal{V}^{2}+7\,\mathcal{V}\,\hat{\xi}+\hat{\xi}^{2})}{({\cal V}-\hat{\xi})(2\mathcal{V}+\hat{\xi})^{2}}\biggr{]}~{},$

where we have introduced phases into the parameters as $W_{0}=|W_{0}|\,e^{{\rm i}\,\phi_{0}}$ and $A_{i}=|A_{i}|\,e^{{\rm i}\,\phi_{i}}$. The good thing about the master formula (2.5) is the fact that it determines the complete form of $V$ simply by specifying topological quantities such as the intersection numbers $k_{ijk}$, the CY Euler number and the number $n$ of non-perturbative contributions to $W$.

Note that $V_{\alpha^{\prime 3}}$ vanishes for $\hat{\xi}=0$ and reproduces the standard no-scale structure in the absence of a $T$-dependent non-perturbative $W$. On the other hand, for very large volume $\mathcal{V}\gg\hat{\xi}$, this term takes the standard form which plays a crucial rôle in LVS models Balasubramanian:2005zx :

$$V_{\alpha^{\prime 3}}\simeq\left(\frac{g_{s}\,e^{K_{\rm cs}}}{2\,\mathcal{V}^{2}}\,\right)\frac{3\,\hat{\xi}\,|W_{0}|^{2}}{4\,{\cal V}}\,.$$

(2.6)

Let us also stress that $V_{\alpha^{\prime 3}}$ depends only on the overall volume $\mathcal{V}$, while $V_{\rm np1}$ depends on $\mathcal{V}$ and the 4-cycle moduli $\tau_{i}$ (with the additional dependence on the axions $\theta_{i}$). Hence these two contributions to $V$ could be minimised by taking derivatives with respect to $\mathcal{V}$ and $(h^{1,1}-1)$ 4-cycle moduli. However $V_{\rm np2}$ depends on the quantity $k_{ijk}\,t^{k}$ which in general cannot be inverted to be expressed as an explicit function of the $\tau_{i}$’s. It has been observed that using the master formula (2.5) one can efficiently perform moduli stabilisation in terms of the 2-cycle moduli $t^{i}$ as shown in AbdusSalam:2020ywo ; AbdusSalam:2022krp .

For example, considering $h^{1,1}=2$, $n=1$ and $\hat{\xi}>0$ in the master formula (2.5) along with using the large volume limit, one can immediately read-off the following three terms:

	$\displaystyle V$	$\displaystyle\simeq$	$\displaystyle\frac{g_{s}\,e^{K_{\rm cs}}}{2}\biggl{[}\frac{3\,\hat{\xi}\,|W_{0}|^{2}}{4\mathcal{V}^{3}}+\frac{4a_{1}\tau_{1}|W_{0}||A_{1}|}{\mathcal{V}^{2}}\,e^{-a_{1}\tau_{1}}\cos\left(a_{1}\theta_{1}+\phi_{0}-\phi_{1}\right)$
		$\displaystyle-$	$\displaystyle\frac{4a_{1}^{2}|A_{1}|^{2}k_{111}t_{1}}{\mathcal{V}}\,e^{-2a_{1}\tau_{1}}\biggr{]}.$

If the CY $X$ has a Swiss-cheese form, one can find a basis of divisors such that the only non-zero intersection numbers are $k_{111}$ and $k_{222}$. This leads to the relation $t^{1}=-\sqrt{2\tau_{1}/k_{111}}$, where the minus sign is dictated from the Kähler cone conditions as the divisor $D_{1}$ in this Swiss-cheese CY is an exceptional 4-cycle. Using this in (2.1) one gets Balasubramanian:2005zx :¹¹1Ref. AbdusSalam:2020ywo has shown that LVS moduli fixing can be realised also for generic cases where the CY threefold does not have a Swiss-cheese structure.

$$V\simeq\frac{g_{s}\,e^{K_{\rm cs}}}{2}\left(\frac{\beta_{\alpha^{\prime}}}{\mathcal{V}^{3}}+\beta_{\rm np1}\,\frac{\tau_{1}}{\mathcal{V}^{2}}\,e^{-a_{1}\tau_{1}}\cos\left(a_{1}\theta_{1}+\phi_{0}-\phi_{1}\right)\\ +\beta_{\rm np2}\,\frac{\sqrt{\tau_{1}}}{\mathcal{V}}\,e^{-2a_{1}\tau_{1}}\right),$$

(2.8)

with:

$$\beta_{\alpha^{\prime}}=\frac{3\hat{\xi}|W_{0}|^{2}}{4}\,,\qquad\beta_{\rm np1}=4a_{1}|W_{0}||A_{1}|\,,\qquad\beta_{\rm np2}=4a_{1}^{2}|A_{1}|^{2}\sqrt{2k_{111}}\,.$$

(2.9)

Let us start by briefly reviewing the generic methodology for analysing the divisor topologies which is widely adopted for scanning useful CY geometries suitable for phenomenology, e.g. see Cicoli:2021dhg ; Shukla:2022dhz . Subsequently we will continue following the same in our current analysis. The main idea is to consider the CY threefolds arising from the four-dimensional reflexive polytopes listed in the Kreuzer-Skarke (KS) database Kreuzer:2000xy , and classify the divisors based on their relevance for phenomenological model building aiming at explicit orientifold constructions. For that purpose, we rather have a very nice collection of the various topological data of CY threefolds available in the database of Altman:2014bfa which can be directly used for further analysis. In this regard, Tab. 1 presents the number of (favorable) polytopes along with the corresponding (favorable) triangulations and (favorable) geometries for a given $h^{1,1}(X)$ in the range $1\leq h^{1,1}(X)\leq 5$.

For a given CY geometry, the main focus is limited to:

• •

  looking at the topology of the so-called ‘coordinate divisors’ $D_{i}$ which are defined through setting the toric coordinates to zero, i.e. $x_{i}=0$. This means that there is a possibility of missing a huge number of divisors, e.g. those which could arise via considering some linear combinations of the coordinate divisors, and some of such may have interesting properties. However, it is hard to make an exhaustive analysis including all the effective divisors of a given CY threefold.

• •

  focusing on scans using ‘favourable’ triangulations (Triang^∗) and ‘favourable’ geometries (Geom^∗) for a given polytope. This could be justified in the sense that for non-favourable CY threefolds, the number of toric divisors in the basis is less than $h^{1,1}(X)$, and subsequently there is always at least one coordinate divisor which is non-smooth, and one usually excludes such spaces from the scan. However, the number of such CY geometries is almost negligible in the sense that there are just 1, 14 and 141 for $h^{1,1}(X)$ being 3, 4 and 5 respectively.

The role of divisor topologies in the LVS context can be appreciated by noting that the Swiss-cheese structure of the CY volume can be correlated with the presence of del Pezzo (dP_n) divisors $D_{s}$. These dP_n divisors are defined for $0\leq n\leq 8$ having degree $d=9-n$ and $h^{1,1}=1+n$, such that dP₀ is a ${\mathbb{P}}^{2}$ and the remaining 8 del Pezzo’s are obtained by blowing up eight generic points inside ${\mathbb{P}}^{2}$. It turns out that they satisfy the following two conditions Cicoli:2011it :

$$\int_{X}D_{s}^{3}=k_{sss}>0\,,\qquad\int_{X}D_{s}^{2}\,D_{i}\leq 0\qquad\forall\,i\neq s\,.$$

(2.10)

Here the self-triple-intersection number $k_{sss}$ corresponds to the degree of the del Pezzo 4-cycle dP_n where $k_{sss}=9-n>0$, which is always positive as $n\leq 8$ for del Pezzo surfaces. In addition, one imposes the so-called ‘diagonality’ condition on such a del Pezzo divisor $D_{s}$ via the following relation satisfied by the triple intersection numbers Cicoli:2011it ; Cicoli:2018tcq :

$$k_{sss}\,\,k_{sij}=k_{ssi}\,\,k_{ssj}\,\qquad\qquad\forall\,\,\,i,j.$$

(2.11)

It turns out that whenever this diagonality condition is satisfied, there exists a basis of coordinates divisors such that the volume of each of the 4-cycles $D_{s}$ becomes a complete-square quantity as illustrated from the following relations:

$$\tau_{s}=\frac{1}{2}\,k_{sij}t^{i}\,t^{j}=\frac{1}{2\,k_{sss}}\,k_{ssi}\,k_{ssj}t^{i}\,t^{j}=\frac{1}{2\,k_{sss}}\,\left(k_{ssi}\,t^{i}\,\right)^{2}\,.$$

(2.12)

Subsequently what happens is that one can always shrink such a ‘diagonal’ del Pezzo ddP_n to a point-like singularity by squeezing it along a single direction. A systematic analysis on counting the CY geometries which could support (standard) LVS models, in the sense of having at least one diagonal del Pezzo divisor, has been performed in Cicoli:2021dhg and the results are summarised in Tab. 2. Moreover, it is worth mentioning that the scanning result presented in Tab. 2 is quite peculiar in the sense that for all the CY threefolds with $h^{1,1}\leq 5$, one does not have any example having a ‘diagonal’ dP_n divisor for $1\leq n\leq 5$, which has been subsequently conjectured to be true for all the CY geometries arising from the KS database.

Let us mention that the classification of CY geometries relevant for LVS as presented in Tab. 2 corresponds to having a ‘standard’ LVS in the sense of having at least one ‘diagonal’ del Pezzo divisor in a Swiss-cheese like model. However, it has been found in some cases that one can still have alternative moduli stabilisation schemes realising an exponentially large CY volume, e.g. using the underlying symmetries of the CY threefold in the presence of a non-diagonal del Pezzo AbdusSalam:2020ywo , and in the framework of the so-called perturbative LVS Antoniadis:2018hqy ; Antoniadis:2019rkh ; Leontaris:2022rzj ; Leontaris:2023obe .

The higher derivative $F^{4}$ contributions to the scalar potential for a generic CY orientifold compactification take the following simple form Ciupke:2015msa :

$$V_{F^{4}}=-\left(\frac{e^{K_{cs}}\,g_{s}}{8\pi}\right)^{2}\frac{\lambda\,|W_{0}|^{4}}{g_{s}^{3/2}{\cal V}^{4}}\sum_{i=1}^{h^{1,1}}\Pi_{i}\,t^{i}\,\equiv\frac{\gamma}{{\cal V}^{4}}\,\sum_{i=1}^{h^{1,1}}\ \Pi_{i}\,t^{i},$$

(3.1)

where the topological quantities $\Pi_{i}$ are given by:

$$\Pi_{i}=\int_{X}c_{2}(X)\wedge\hat{D}_{i}\,,$$

(3.2)

and $\lambda$ is an unknown combinatorial factor which in the single modulus case is rather small in absolute value Grimm:2017okk :

$$\lambda=-\frac{11}{24}\,\frac{\zeta(3)}{(2\pi)^{4}}=-3.5\cdot 10^{-4}\,.$$

(3.3)

Its value is not known for $h^{1,1}>1$ but we expect it to remain small, in analogy with the $h^{1,1}=1$ case. In fact, one can argue that the factor $\zeta(3)/(2\pi)^{4}$ in $\lambda$ is expected to be always present for generic models with several Kähler moduli as well. This is because the coupling tensor ${\cal T}_{\overline{i}\,\overline{j}\,kl}$ appearing in this correction through the following higher derivative piece Ciupke:2015msa :

$$V_{F^{4}}=-e^{2K}\,{\cal T}^{\overline{i}\,\overline{j}kl}\overline{D}_{\overline{i}}\overline{W}\,\overline{D}_{\overline{j}}\overline{W}\,D_{k}{W}\,{D}_{l}{W}\,,$$

(3.4)

can be schematically written as:

$${\cal T}_{\overline{i}\,\overline{j}\,kl}=\frac{c}{\mathcal{V}^{8/3}}\,\frac{\zeta(3)\,{\cal Z}}{g_{s}^{3/2}}\,,$$

(3.5)

where $c$ can be considered as some combinatorial factor, which for example, in the single modulus case turns out to be $11/384$ Grimm:2017okk , and:

$${\cal Z}=(2\pi)^{2}\,\int_{X}c_{2}(X)\wedge J\,,$$

(3.6)

where we stress that we are working with the convention $\ell_{s}=(2\pi)\sqrt{\alpha^{\prime}}=1$. Subsequently, we have

$${\cal T}_{\overline{i}\,\overline{j}\,kl}=c\,\frac{\zeta(3)}{(2\pi)^{4}\,\mathcal{V}^{8/3}\,g_{s}^{3/2}}\int_{X}c_{2}(X)\wedge J\,.$$

(3.7)

Note that the $\mathcal{V}^{-8/3}$ factor in the above expression cancels off with a $\mathcal{V}^{8/3}$ contribution coming from 4 inverse Kähler metric factors needed to raise the 4 indices of the coupling tensor ${\cal T}_{\overline{i}\,\overline{j}\,kl}$ to go to (3.4).

Here, let us mention that the higher derivative $F^{4}$ correction under consideration appears at $\alpha^{\prime 3}$ order, like the BBHL-correction Becker:2002nn , and both are induced at string tree-level, resulting in a factor of $g_{s}^{-3/2}$. For explicitness, let us also note that the leading order BBHL correction Becker:2002nn appearing at the two-derivative level takes the following form:²²2In this regard, it may be worth noticing that the original result Becker:2002nn has been obtained with the convention $(2\pi\alpha^{\prime})=1$ which removes the $(2\pi)^{-3}$ factor from the denominator of the $\hat{\xi}$ parameter.

$$V_{\alpha^{\prime 3}}=\left(\frac{e^{K_{cs}}\,g_{s}}{8\pi}\right)\,\frac{3\,\xi\,|W_{0}|^{2}}{4\,g_{s}^{3/2}\,{\cal V}^{3}},\qquad\xi=-\frac{\zeta(3)\chi(X)}{2\,(2\pi)^{3}}\,.$$

(3.8)

Now, comparing these two $\alpha^{\prime}$ corrections one finds that:

$$\frac{V_{F^{4}}}{V_{\alpha^{\prime 3}}}=\tilde{c}\,\left(\frac{g_{s}}{8\pi}\right)\,e^{K_{cs}}\,|W_{0}|^{2}\,\left(\frac{\Pi_{i}\,t^{i}}{\chi(X){\cal V}}\right),$$

(3.9)

where $\tilde{c}$ is some combinatorial factor, which for the case of a single Kähler modulus is

$$\tilde{c}=\frac{11}{9(2\pi)}\simeq 0.2\,.$$

(3.10)

One can observe that each factors in (3.9) can be of a magnitude less than one in typical models. For example, demanding large complex-structure limit in order to ignore instanton effects can typically result in having $e^{K_{cs}}\sim 0.01$ Louis:2012nb , the string coupling $g_{s}$ needs to be small and the CY volume large to trust the low-energy EFT, and the ratios between $\Pi_{i}$’s and $\chi(X)$ are typically of ${\cal O}(1)$ Shukla:2022dhz . Having these aspects in mind, it is very natural to anticipate that higher-derivative $F^{4}$ effects are subdominant as compared to the two-derivative corrections. Note that (3.1) can also be rewritten as:

$$V_{F^{4}}=-V_{\alpha^{\prime 3}}\,\frac{\sqrt{g_{s}}}{3\pi}\left(\frac{\lambda}{\xi}\right)\sum_{i=1}^{h^{1,1}}\Pi_{i}\left(\frac{m_{3/2}}{M_{\rm KK}^{(i)}}\right)^{2}$$

(3.11)

where the gravitino mass is:

$$m_{3/2}^{2}=\left(\frac{g_{s}}{8\pi}\right)\frac{|W_{0}|^{2}}{\mathcal{V}^{2}}\,M_{p}^{2}\,,$$

(3.12)

and $M_{\rm KK}^{(i)}$ is the Kaluza-Klein scale associated to the $i$-th divisor:

$$\left(M_{\rm KK}^{(i)}\right)^{2}=\frac{M_{s}^{2}}{t_{i}}=\frac{\sqrt{g_{s}}}{4\pi}\frac{M_{p}^{2}}{t_{i}\mathcal{V}}\,.$$

(3.13)

In the above equation we have used the relation between the string scale and the Planck mass in the convention where $\mathcal{V}_{s}=\mathcal{V}\,g_{s}^{3/2}$ (with $\mathcal{V}_{s}$ the volume in string frame and $\mathcal{V}$ the volume in Einstein frame):

$$M_{s}^{2}=\frac{1}{(2\pi)^{2}\alpha^{\prime}}=\sqrt{g_{s}}\frac{M_{p}^{2}}{4\pi\mathcal{V}}\,.$$

(3.14)

Note that (3.11) makes clear that $V_{F^{4}}$ is an $\mathcal{O}(F^{4})$ correction since $V_{\alpha^{\prime 3}}$ is an $\mathcal{O}(F^{2})$ effect and Cicoli:2013swa :

$$\left(\frac{m_{3/2}}{M_{\rm KK}}\right)^{2}\sim g\frac{|F|^{2}}{M_{\rm KK}^{2}}\ll 1\,,$$

(3.15)

where $g\sim M_{\rm KK}/M_{p}\sim\mathcal{V}^{-2/3}\ll 1$ is the coupling of heavy KK modes to light states.

Two important quantities characterising the topology of a divisor $D$ are the Euler characteristic $\chi(D)$ and the holomorphic Euler characteristic (also known as arithmetic genus) $\chi_{h}(D)$ which are given by the following useful relations Blumenhagen:2008zz ; Collinucci:2008sq ; Cicoli:2016xae :

	$\displaystyle\chi(D)$	$\displaystyle\equiv$	$\displaystyle\sum_{i=0}^{4}{(-1)}^{i}\,b_{i}(D)=\int_{X}\,\hat{D}\wedge\left(\hat{D}\wedge\hat{D}+c_{2}(X)\right)\,,$		(3.16)
	$\displaystyle\chi_{h}({D})$	$\displaystyle\equiv$	$\displaystyle\sum_{i=0}^{2}{(-1)}^{i}\,h^{i,0}(D)=\frac{1}{12}\int_{X}\,\hat{D}\wedge\left(2\,\hat{D}\wedge\hat{D}+c_{2}(X)\right)\,,$		(3.17)

where $b_{i}(D)$ and $h^{i,0}(D)$ are respectively the Betti and Hodge numbers of the divisor. Using these two relations we find that $\Pi(D)$ is related with the Euler characteristics and the holomorphic Euler characteristic as follows:

$$\Pi(D)=\chi(D)-\int_{X}\,\hat{D}\wedge\hat{D}\wedge\hat{D},\qquad\Pi(D)=12\,\chi_{h}(D)-2\,\int_{X}\,\hat{D}\wedge\hat{D}\wedge\hat{D}\,,$$

(3.18)

which also give another useful relation:

$$\Pi(D)=2\,\chi(D)-12\,\chi_{h}(D)\,.$$

(3.19)

Therefore, the topological quantity $\Pi(D)$ vanishes for a generic smooth divisor $D$ if the following simple relation holds,

$$\Pi(D)=0\quad\Longleftrightarrow\quad\chi(D)=6\,\chi_{h}(D)\,.$$

(3.20)

Now, using the relations $\chi(D)=2\,h^{0,0}-4\,h^{1,0}+2\,h^{2,0}+h^{1,1}$ and $\chi_{h}(D)=h^{0,0}-h^{1,0}+h^{2,0}$, we find another equivalent relation for vanishing $\Pi(D)$:

$$h^{1,1}(D)=4\,h^{0,0}(D)-2\,h^{1,0}(D)+4\,h^{2,0}(D)\,.$$

(3.21)

Any divisor satisfying the vanishing $\Pi$ relation (3.21) will be denoted as $D_{\Pi}$. After knowing the topology of a generic divisor $D$, it is easy to check if $h^{1,1}$ satisfies this condition or equivalently $\chi=6\,\chi_{h}$. To demonstrate it, let us quickly consider the following two examples:

$\displaystyle{\mathbb{T}}^{4}\equiv\begin{tabular}[]{ccccc}&&1&&\\ &2&&2&\\ 1&&4&&1\\ &2&&2&\\ &&1&&\\ \end{tabular}\qquad\qquad\text{and}\qquad\qquad{\rm K3}\equiv\begin{tabular}[]{ccccc}&&1&&\\ &0&&0&\\ 1&&20&&1\\ &0&&0&\\ &&1&&\\ \end{tabular}\,.$

(3.32)

Now it is obvious that ${\mathbb{T}}^{4}$ has $\Pi({\mathbb{T}}^{4})=0$ as it satisfies $\chi=0=6\,\chi_{h}$. However, K3 has $\Pi({\rm K3})=24$ and $6\chi_{h}=12=\chi/2$. Alternatively, it can be also checked that the Hodge number condition in (3.21) is satisfied for ${\mathbb{T}}^{4}$ but not for K3.

Therefore, we can generically formulate that a divisor $D$ of a Calabi-Yau threefold having the following Hodge Diamond results in a vanishing $\Pi(D)$:

$\displaystyle D_{\Pi}\equiv\begin{tabular}[]{ccccc}&&$h^{0,0}$&&\\ &$h^{1,0}$&&$h^{1,0}$&\\ $h^{2,0}$&&$\left(4h^{0,0}-2h^{1,0}+4h^{2,0}\right)$&&\quad$h^{2,0}$\\ &$h^{1,0}$&&$h^{1,0}$&\\ &&$h^{0,0}$&&\\ \end{tabular}\,,$

(3.38)

and if we consider that the $D_{\Pi}$ divisor is smooth and connected, then we have $h^{0,0}(D_{\Pi})=1$. Subsequently we can identify three different classes of vanishing $\Pi$ divisors:

1. 1.

   dP₃ divisors: For connected rigid 4-cycles with no Wilson lines we have $h^{1,0}(D)=h^{2,0}(D)=0$, and hence a vanishing $\Pi(D)$ results in the following Hodge diamond:

   $\displaystyle D_{\Pi}\equiv\begin{tabular}[]{ccccc}&&1&&\\ &0&&0&\\ 0&&4&&0\\ &0&&0&\\ &&1&&\\ \end{tabular}\equiv{\rm dP}_{\Pi}\,.$

   (3.44)

   This topology corresponds to the dP surface of degree six, i.e. a dP₃. Moreover, this class of $D_{\Pi}$ which singles out a dP₃ surface, also includes the possibility of the ‘rigid but not del Pezzo’ 4-cycle denoted as NdP_n for $n\geq 9$ Cicoli:2011it . These surfaces are blow-up of line-like singularities and have similar Hodge diamonds as those of the usual dP surfaces dP_n defined for $0\leq n\leq 8$.

2. 2.

   Wilson divisors: For connected rigid 4-cycles with Wilson lines we have $h^{2,0}(D)=0$ but $h^{1,0}(D)>0$, resulting in the following Hodge diamond for $D_{\Pi}$:

   $\displaystyle D_{\Pi}\equiv\begin{tabular}[]{ccccc}&&1&&\\ &$h^{1,0}$&&$h^{1,0}$&\\ 0&&$\left(4-2h^{1,0}\right)$&&\quad 0\\ &$h^{1,0}$&&$h^{1,0}$&\\ &&1&&\\ \end{tabular}\equiv W_{\Pi}\,.$

   (3.50)

   Given that all Hodge numbers are non-negative integers, the only possibility compatible with $h^{1,1}\geq 1$ (to be able to a have a proper definition of the divisor volume) is $h^{1,0}=1$ which, in turn, corresponds to $h^{1,1}=2$. This is a so-called ‘Wilson’ divisor with vanishing $\Pi(W)$ which we denote as $W_{\Pi}$. This $W_{\Pi}$ divisor corresponds to a subclass of ‘Wilson’ divisors, characterised by the Hodge numbers $h^{0,0}=h^{1,0}=1$ and arbitrary $h^{1,1}$, that have been introduced in Blumenhagen:2012kz to support poly-instanton corrections.

3. 3.

   Non-rigid divisors: Now let us consider the third special class which can have deformation divisors, i.e. $h^{2,0}(D)>0$. When the divisor does not admit any Wilson line, i.e. $h^{1,0}(D)=0$, the Hodge diamond for $D_{\Pi}$ simplifies to:

   $\displaystyle D_{\Pi}\equiv\begin{tabular}[]{ccccc}&&1&&\\ &0&&0&\\ $h^{2,0}$&&$\left(4+4\,h^{2,0}\right)$&&\qquad$h^{2,0}$\\ &0&&0\\ &&1&&\\ \end{tabular}\,.$

   (3.56)

   To our knowledge, so far there are no known examples in the literature which have such a topology. The simplest of its kind will have $h^{2,0}(D)=1$ and $h^{1,1}(D)=8$. In this regard, it is worth mentioning that the topology of the so-called ‘Wilson’ divisors which are ${\mathbb{P}}^{1}$ fibrations over ${\mathbb{T}}^{2}$s, has been argued to be useful in Lust:2006zg and some years later it was found to be the case while studying the generation of poly-instanton effects Blumenhagen:2012kz . So it would be interesting to know if such non-rigid divisor topologies of vanishing $\Pi$ exist in explicit CY constructions, and further if they could be useful for some phenomenological applications.

   The last possibility is to consider the most general situation with deformations and Wilson lines, i.e. $h^{2,0}(D)>0$ and $h^{1,0}(D)>0$. As already mentioned, the simplest case is $\mathbb{T}^{4}$ with $h^{2,0}(\mathbb{T}^{4})=1$, $h^{1,0}(\mathbb{T}^{4})=2$ and $h^{1,0}(\mathbb{T}^{4})=4$ which however never shows up in our search through the KS list, as well as more general divisors with both deformations and Wilson lines.

Before coming to the scan of such divisor topologies of vanishing $\Pi$, let us mention a theorem of Oguiso1993 ; Schulz:2004tt which states that if the CY intersection polynomial is linear in the homology class $\hat{D}_{f}$ corresponding to a divisor $D_{f}$, then the CY threefold has the structure of a K3 or a ${\mathbb{T}}^{4}$ fibration over a ${\mathbb{P}}^{1}$ base. Noting the following relation for the self-triple-intersection number of a generic smooth divisor $D$:

$$\int_{X}\hat{D}\wedge\hat{D}\wedge\hat{D}=12\,\chi_{h}(D)-\chi(D)\,,$$

(3.57)

and subsequently demanding the absence of such cubics for $D_{f}$ in the CY intersection polynomial, results in $\chi(D)=12\chi_{h}(D)$ or the following equivalent relation:

$$\quad h^{1,1}(D_{f})=10\,h^{0,0}(D_{f})-8\,h^{1,0}(D_{f})+10\,h^{2,0}(D_{f})\,.$$

(3.58)

This relation is clearly satisfied for K3 and ${\mathbb{T}}^{4}$ divisors, and can be satisfied for some other possible topologies as well. For example, another non-rigid divisor for which the self-cubic-intersection is zero is given by the following Hodge diamond:

$\displaystyle{\rm SD}\equiv\begin{tabular}[]{ccccc}&&1&&\\ &0&&0&\\ 2&&30&&2\\ &0&&0&\\ &&1&&\\ \end{tabular},\qquad\chi(SD)=36,\qquad\chi_{h}(SD)=3\,.$

(3.64)

This is also a very well known surface frequently appearing in CY threefolds, e.g. it appears in the famous Swiss-cheese CY threefold defined as a degree-18 hypersurface in WCP${}^{4}[1,1,1,6,9]$ where the divisors corresponding to the first three coordinates with charge 1 are such surfaces.

Moreover, interestingly one can see that for the ‘Wilson’ type divisor the relation in (3.58) is indeed satisfied for $h^{1,1}(D)=2$ which is exactly something needed for the generation of poly-instanton effects on top of having vanishing $\Pi(D)$ as we have discussed before. In this regard, let us also add that the simultaneous vanishing of $\Pi(D)$ and $D^{3}_{|_{X}}$ results in the vanishing of $\chi(D)$ and $\chi_{h}(D)$ and vice-versa, and so, besides a particular type of ‘Wilson’ divisor, there can be more such divisor topologies satisfying the following if and only if condition:

$$\Pi(D)=0=\int_{X}\,\hat{D}\wedge\hat{D}\wedge\hat{D}\qquad\Longleftrightarrow\qquad\chi(D)=0=\chi_{h}(D)\,.$$

(3.65)

Thus, if a divisor $D$ is connected and has $\Pi(D)=0=D^{3}_{|_{X}}$, then its Hodge diamond is:

$\displaystyle D_{\Pi}\equiv\begin{tabular}[]{ccccc}&&1&&\\ &$1+n$&&$1+n$&\\ $n$&&$2+2n$&&\quad$n$\\ &$1+n$&&$1+n$&\\ &&1&&\\ \end{tabular}\equiv D_{\Pi}^{\rm cubic}\,,$

(3.71)

where $n$ is the number of possible deformations for the divisor $D$. For $n=0$ this corresponds to a $W_{\Pi}$ divisor, and for $n=1$ this corresponds to a ${\mathbb{T}}^{4}$. Although we are not aware of any such examples with $n\geq 2$, it would be interesting to know what topology they would correspond to.