On $\alpha^{\prime}$-effects from $D$-branes in $4d\;\mathcal{N}=1$

Since the discovery of D-branes Polchinski_1995 ; PolchinskiVol2 they have played a crucial role in model building as they contribute non-Abelian gauge groups. In particular, type IIB flux compactification to four-dimensional space-time generically rely on the presence of space-time filling D3/D7-branes Giddings_2002 ; Choi_2004 . In order to cancel the positive charges one is required to introduce O3/O7 orientifold planes in the setup Brunner_2004 ; Blumenhagen_2007 . Due to their back-reaction on the geometry one refers to these backgrounds as Calabi-Yau orientifolds.

In the recent years F-theory vacua gained a lot of attention Vafa_1996 ; weig2018tasi ; donagi2008model ; Grimm_2011 which in the weak string coupling limit naturally incorporate the type IIB D7-branes and O7-planes.

Two main challenges remain in the landscape of type IIB flux vacua. Firstly, the problem of Kähler moduli stabilization. While the Gukov-Vafa-Witten super-potential Gukov_2000 in the presence of non-vanishing background fluxes generically allows to stabilize the complex structure moduli Giddings_2002 ; Choi_2004 , the leading order terms in $\alpha^{\prime}$ and $g_{s}$, do not generate a potential for the geometric Kähler deformations which is referred to as the no-scale property. Secondly, the de Sitter up-lift a commonly used phrase to describe the generation of a potential in a low energy theory of string theory which admits a global or local de Sitter minimum, see e.g. Cicoli_2018 for a review.

Historically, the first issue was addressed by incorporating non-perturbative effects such as instantons to generate a potential for the Kähler moduli, while the de Sitter uplift mainly relied on exotic objects such as anti $D3$ branes Kachru_2003 . Recently, the swampland de Sitter conjecture obied2018sitter has cast doubt on the consistency of those mechanisms. This however, is an ongoing debate see e.g. Palti:2019pca for a review.

Instanton effects are exponentially suppressed at large volumes which make them in general sub-leading to the leading order $\alpha^{\prime}$-correction to the scalar potential. The large volume scenario Balasubramanian_2005 ; Conlon_2005 carefully balances an $\alpha^{\prime 3}g_{s}^{2}$-correction to the Kähler potential Antoniadis_1997 ; Becker_2002 ; Bonetti:2016dqh against instanton effects to the superpotential to achieve moduli stabilization. However, this relies on having small cycles in the Calabi-Yau orientifold while maintaining an overall large volume. Lastly, Kähler moduli stabilization may be achieved by the leading order perturbative corrections Ciupke:2015msa ; Weissenbacher:2019bfb and may potentially induce a natural de Sitter uplift Balasubramanian_2004 ; Westphal_2007 ; Ciupke:2015msa ; Weissenbacher:2019bfb . Latter allows for the cycles in the internal space to be of comparable size.

This provides a strong phenomenological motivation for the study of $\alpha^{\prime}$-corrections to the Kähler potential and coordinates of four-dimensional $\mathcal{N}=1$ supergravity theories. A series of our previous works Grimm:2013gma ; Grimm:2013bha ; Weissenbacher:2019mef addressed the study of the leading order $\alpha^{\prime}$-corrections in F-theory and thus weakly coupled IIB vacua. The foundation of those studies is laid by an extensive analysis of dimensional reduction of higher-derivative terms from eleven to three dimensions Grimm:2014xva ; Grimm:2014efa ; Grimm:2017pid . The correction whose potential existence had been elusive ever since Grimm:2013gma is of order $\alpha^{\prime 2}g_{s}$ and thus leading to the well known Euler characteristic correction Antoniadis_1997 ; Becker_2002 .²²2The dimensional reduction of the Heterotic string at $\alpha^{\prime 2}$-order including Kähler deformations was discussed in Candelas_2017 . Recent developments Weissenbacher:2019mef suggest the existence of another $\alpha^{\prime 2}g_{s}$-correction to the Kähler coordinates proportional to the logarithm of the internal volume which if present breaks the no-scale structure.
This work provides further evidence to the existence of both the correction to the Kähler coordinates as well as the Kähler potential by the taking the type IIB approach. We study Calabi-Yau orientifold compactifications with space-time filling D7-branes and O7 planes Jockers:2004yj ; Grimm:2004uq , in particular we focus on the gravitational four-derivative $\alpha^{\prime 2}g_{s}$-sector of the DBI effective actions Bachas:1999um , respectively. Let us emphasize that our starting point is identical to the one in Junghans:2014zla . However, the approach discussed in Junghans:2014zla lacks certain conceptually required steps to allow conclusions on the Kähler metric and thus the Kähler potential. Namely, the discussion of the perturbation of the internal metric with respect to Kähler deformations³³3Note that Junghans:2014zla uses the absence of an $\alpha^{\prime 2}g_{s}$-correction to the four-dimensional Einstein-Hilbert term to conclude on the absence of an $\alpha^{\prime 2}g_{s}$-correction to the Kähler metric. Not only is this implication flawed, but by discussing the E.O.M’s one indeed concludes that an $\alpha^{\prime 2}g_{s}$-correction to the four-dimensional Einstein-Hilbert is present upon dimensional reduction. , as well as the discussion of the resulting equations of motions which are modified when $\alpha^{\prime 2}g_{s}$-corrections to the DBI actions are present. Thus Junghans:2014zla fails to identify any $\alpha^{\prime 2}g_{s}$-correction in the resulting four-dimensional theory.

In this work we mainly study the gravitational $R^{2}$-terms in the DBI action of D7-branes and O7-planes Bachas:1999um . But we also briefly discuss $\alpha^{\prime}$-corrections from other D$p$-branes, in particular D5-branes and O5-planes and moreover D6-branes and O6-planes in type IIA. A comprehensive study would as well require the discussion of the $F^{2}_{5}R$-sector. However, latter to the best of our knowledge has not been discussed in the literature.
This article is structured as follows. In section 2 we set the stage by introducing the notion of $\alpha^{\prime 2}g_{s}$-corrections to the four-dimensional Kähler potential and coordinates. We continue in section 3 by reviewing the starting point of our computation, namely $\alpha^{\prime 2}g_{s}$-effects to D7-branes and O7-planes effective actions. The dimensional reduction is performed for a single Kähler modulus, i.e  the overall volume in section 5.⁴⁴4To perform the computations in this work we employ computational techniques, in particular we heavily rely on the analytic abstract tensor algebra libraries xAct and xTensor Mart_n_Garc_a_2002 ; Mart_n_Garc_a_2003 ; Mart_n_Garc_a_2008 . Finally, we conclude in section 6.1 on the corrections to the Kähler potential and coordinates by comparison to the F-theory side. Lastly, in section 6.2 we briefly turn to the discussion of $\alpha^{\prime}$-corrections to the Kähler potential originating from other D$p$-branes, in particular we initiate the study in type IIA. Higher-derivative terms in the DBI action of D5-branes potentially give rise to a novel $\alpha^{\prime 3}g_{s}$-correction to the Kähler potential, as well as a potential novel $\alpha^{\prime}{}^{5/2}g_{s}$-correction from D6-branes in type IIA.

The main objective is to provide further evidence for the presence of the conjectured $\alpha^{\prime 2}g_{s}$-correction to the four-dimensional $\mathcal{N}=1$ Kähler coordinates Weissenbacher:2019mef . The latter is proportional to logarithm of the volume of the Calabi-Yau orientifold

$$\;\;\sim\;\;\mathcal{Z}\;\cdot\;\text{log}\hat{\mathcal{V}}\;\;.$$

(1)

with $\hat{\mathcal{V}}$ the Einstein frame volume i.e. the volume of the internal manifold $\mathcal{V}$ equipped with an additional dilaton dependence as

$$\hat{\mathcal{V}}=e^{-\tfrac{3\phi}{2}}\mathcal{V}\;\;,\;\;\;\text{and}\;\;\;\hat{v}^{i}=e^{-\tfrac{\phi}{2}}v^{i}\;\;\;,$$

(2)

and where $\mathcal{Z}$ is a topological correction which will be introduced in detail in section 3.3. Moreover, $v^{i},\,i=1,\dots,h^{1,1}_{+}$ are the Kähler moduli fields in dimensionless units of $2\pi\alpha^{\prime}$, and $\phi\equiv\phi(x)$ is the dilaton.⁵⁵5We denote the external and internal space coordinates with $x^{\mu}$ and $y^{i}$, respectively. Note that this is an abuse of notation of the indices $i,j$, which we also use to denote the abstract index on the Kähler moduli space, see e.g. eq. (2). We present a systematic study of the one-modulus reduction of the $R^{2}$-terms in the DBI and ODBI action of D7-branes and O7’s in sections 4 and 5.

To set the stage let us begin by reviewing the Kähler potential and coordinates which are to be corrected at sub-leading order. The volume $\hat{\mathcal{V}}$ is dimensionless in units of $(2\pi\alpha^{\prime})^{3}$. The Kähler potential and coordinates are given by

$$K=\phi-2\,\text{log}\Big{(}\hat{\mathcal{V}}+\gamma_{1}\mathcal{Z}_{i}\hat{v}^{i}\Big{)}\;\;,$$

(3)

and

$$T_{i}=\rho_{i}+i\,\Big{(}\hat{\mathcal{K}}_{i}+\gamma_{2}\mathcal{Z}_{i}\text{log}\hat{\mathcal{V}}\Big{)}\;\;,$$

(4)

respectively. The axio-dilaton is

$$\tau=C_{0}+ie^{-\phi}\;\;,$$

(5)

where $\rho_{i}$ are the real scalars arising from the reduction of the Ramond-Ramond type IIB four-form fields $C_{4}$, $C_{0}$ the type IIB axion and $\gamma_{1},\gamma_{2}$ are real parameters. The quantity $\hat{\mathcal{K}}_{i}=\hat{\mathcal{K}}_{ijk}\hat{v}^{j}\hat{v}^{k}$ is defined using the intersection numbers $\hat{\mathcal{K}}_{ijk}$.⁶⁶6The intersection numbers are given by $$\hat{\mathcal{K}}_{ijk}=\frac{1}{6!}\int_{Y_{3}}\omega_{i}\wedge\omega_{j}\wedge\omega_{k}\;\;.$$ (6) where $Y_{3}$ is the internal Calabi-Yau manifold, and $\omega_{i},\,i=1,\dots,h^{1,1}$ are the harmonic $(1,1)$-forms. The sub-leading correction $\mathcal{Z}_{i}$ are numbers and thus do not depend on the moduli fields, i.e. those are topological quantities of the internal space. Let us stress that throughout this work we use dimensionless units

$$\mathcal{V}=\mathcal{V}/(2\pi\alpha^{\prime})^{3}\;,\;\;\text{and}\;\;\;v^{i}=v^{i}/2\pi\alpha^{\prime}\;\;,$$

(7)

unless specified else-wise. One thus easily infers that the corrections $\mathcal{Z}_{i}$ in eq.’s (3) and (4) are of order $\alpha^{\prime 2}\,g_{s}$ compared to the leading order term. Moreover let us note that the Kähler coordinates eq. (4) in principle may be corrected by other terms at this order Weissenbacher:2019mef all of which however become constant shifts in the one-modulus case. We thus omit them for simplicity as we focus on the one-modulus case $h^{1,1}=1$ in the following. From (3) and (4) one infers that the Kähler potential becomes

$$K=\phi-2\,\text{log}\Big{(}\hat{\mathcal{V}}+\gamma_{1}\mathcal{Z}\,\hat{\mathcal{V}}^{\tfrac{1}{3}}\Big{)}\;\;,$$

(8)

and the Kähler coordinates result in

	$\displaystyle T$	$\displaystyle=\rho+i\,\Big{(}3\,\hat{\mathcal{V}}^{\tfrac{2}{3}}+\gamma_{2}\mathcal{Z}\,\text{log}\hat{\mathcal{V}}\Big{)}\;\;,$		(9)
	$\displaystyle\tau$	$\displaystyle=C_{0}+i\,e^{-\phi}\;\;.$		(10)

The kinetic terms of the four-dimensional $\mathcal{N}=1$ supergravity theory are given by

$$S=\frac{1}{2\kappa_{4}^{2}}\int R\ast 1-2\,G^{I{\bar{J}}{}}dT_{I}\wedge\ast d{\bar{T}}{}_{J}\;\;\,\text{with}\;\;G^{I{\bar{J}}{}}=\frac{\partial^{2}K(T,{\bar{T}}{})}{\partial T_{I}\,\partial{\bar{T}}{}_{J}}\;\;,$$

(11)

where $T_{I}=(\tau,T_{i})$ and $G^{I{\bar{J}}{}}$ is the Kähler metric. It is convenient to express the Kähler potential (8) in terms of the Kähler coordinates using that $\mathcal{V}\gg 1$, i.e. as an expansion to linear order in $\mathcal{Z}$. One finds that (8) and (9) become

$$K=-\log\big{(}-\tfrac{i}{2}\big{(}\tau-{\bar{\tau}}{}\big{)}\big{)}-\Big{(}3-\frac{9i\,\gamma_{2}\,\mathcal{Z}}{T-{\bar{T}}{}}\Big{)}\cdot\log\big{(}-\tfrac{i}{6}\big{(}T-{\bar{T}}{}\big{)}\big{)}-\frac{12i\,\gamma_{1}\,\mathcal{Z}}{T-{\bar{T}}{}}\;\;.$$

(12)

Thus in the one-modulus case one infers from (8), (9), (10) and (12) that

	$\displaystyle S=\frac{1}{2\kappa_{4}^{2}}\int R\ast 1\;\;-$	$\displaystyle\frac{1}{2}d\phi\wedge\ast d\phi\;-\;\frac{1}{2}\,e^{2\phi}dC_{0}\wedge\ast dC_{0}$		(13)
	$\displaystyle-$	$\displaystyle\left(\frac{2}{3\,\hat{\mathcal{V}}^{2}}-\frac{8\gamma_{1}+3\gamma_{2}}{9\,\hat{\mathcal{V}}^{8/3}}\mathcal{Z}\right)d\hat{\mathcal{V}}\wedge\ast d\hat{\mathcal{V}}\;\;$
	$\displaystyle-$	$\displaystyle\left(\frac{1}{6\,\hat{\mathcal{V}}^{4/3}}-\frac{8\gamma_{1}+9\gamma_{2}}{36\,\hat{\mathcal{V}}^{2}}\mathcal{Z}\right)d\rho\wedge\ast d\rho\;\;.$

Where we have again used the fact that $\mathcal{Z}\ll\mathcal{V}$ for geometries where the internal volume is large in units of $(2\pi\alpha^{\prime})^{3}$. The primary goal is to argue that the obtained correction $\mathcal{Z}$ is indeed of topological nature and moreover agrees with our previous F-theory analysis. For a definition of $\mathcal{Z}$ see eq. (33). Secondly, from (13) one infers that by fixing the kinetic terms obtained by dimensional reduction on the internal Calabi-Yau orientifolds in the presence of D7-branes and O7-planes one can fix $\gamma_{1}$ and $\gamma_{2}$. In particular, we will derive the correction for the case of a Calabi-Yau orientifold with a single Kähler modulus and with eight coinciding D7’s and one O7^-. Note that this setup is different compared to the one studied in Weissenbacher:2019mef where only a single D7-brane is present with the class of the divisor wrapped being a multiple of the one wrapped by the O7-plane, such that tadpole cancellation is guaranteed. Lastly, note that in eq.(13) the correction to the mixed kinetic terms of the dilaton and the Einstein frame volume as well as the correction to the kinetic term of the dilaton is absent.
Scalar Potential. Let us emphasize that the Ansatz for the Kähler potential and Kähler coordinates breaks the no-scale structure and thus generates a scalar potential for the Kähler moduli fields for non-vanishing vacuum expectation value of the super-potential $W$, given by e.g. the flux-superpotential after stabilizing the complex structure moduli denoted by $W_{0}$. The F-term scalar potential is given by

$$V_{F}=e^{K}\big{(}G_{I{\bar{J}}{}}D^{I}W{\bar{D}}{}^{J}{\bar{W}}{}-3|W|^{2}\big{)}\;\;,$$

(14)

where $D^{I}W=\partial_{T_{I}}W+W\partial_{T_{I}}K$. By using (8) and (9) one infers in the one-modulus case that

$$V_{F}=\frac{3\,\gamma_{2}\,e^{\phi}\,\mathcal{Z}}{2\,\hat{\mathcal{V}}^{8/3}}\,|W_{0}|^{2}\;\;.$$

(15)

Comments. Let us close this section with some concluding remarks on the Ansatz for the Kähler potential and coordinates (8) and (9). In particular let us emphasize that it is the complete Ansatz at order $\alpha^{\prime 2}g_{s}$ consistent with the functional form of Kähler metric which will be derived later in this work. Firstly, note that the real part of the Kähler coordinates (9) are protected by shift symmetry of $\rho$ against $\alpha^{\prime}$-corrections.⁷⁷7Note that one may want to write the general Ansatz for an $\alpha^{\prime 2}$-correction up to constant shifts as $$\text{Re}T=\rho\cdot\Big{(}1+d_{1}\,\mathcal{Z}\,\frac{1}{\hat{\mathcal{V}}^{2/3}}\Big{)}+d_{2}\,\mathcal{Z}\,\text{log}\hat{\mathcal{V}}\;\;,$$ (16) with $d_{1},d_{2}$ parameters. However, the first correction breaks shift-symmetry and the term proportional to the logarithm does not constitute a sub-leading term in the limit $\hat{\mathcal{V}}\to\infty$. Thus one concludes that the real part of $T$ is not to be corrected. Analogous conclusions hold for the generic modulus case (3) and (4). Moreover, the imaginary part of the axio-dilaton eq. (10) is not expected to be corrected by $\alpha^{\prime}$-corrections. We proceed in this work without a general proof of this assumption. However, note that in Bonetti:2016dqh we have explicitly confirmed that in the case of the $\alpha^{\prime 3}g_{s}^{2}$-correction to the Kähler-potential there is no correction to the axio-dilaton.

Lastly, let us emphasize that the possibility of a correction to the Kähler coordinates of a $4d,\,\mathcal{N}=1$ theory proportional to the logarithm of the internal volume has already been discussed in the literature Conlon_2009 ; Conlon_2010 . To establish a correction of the latter to our topological coefficient in eq. (1) is of great interest.⁸⁸8Note that Conlon_2010 concludes that the correction is of order $g_{s}$ compared to the leading term. Furthermore, it is worth noting that an analog correction to the Kähler coordinates appears in $3d,\,\mathcal{N}=2$ originating from higher-derivative terms to eleven-dimensional supergravity i.e. the low energy limit of M-theory compactified on a Calabi-Yau fourfold Grimm:2017pid . Moreover, there is no symmetry forbidding such a correction (1) and thus one would generically expect it to be present, see e.g. Palti:2020qlc . Concludingly, the presence of a correction to the Kähler coordinates as in eq.’s (1), (4) and (9) is thus likely. We support this conclusion by providing more explicit evidence in this work.

In this section we discuss the relevant D7-brane and O7-plane actions. Those are are in general composed of the Dirac-Born-Infeld (DBI) as well as the Wess-Zumino⁹⁹9 Also referred to as Chern-Simons action. action as

$$S_{DBI}+S_{WZ}\;\;.$$

(17)

Let us first review the classical contribution to (17). One finds the DBI action Polchinski_1995 ; Witten_1996 ; PolchinskiVol2 for a D7-brane to be

$$S^{\tiny(0)}_{DBI}=-\mu_{7}\int d^{7}Y\,e^{-\Phi}\text{Tr}\sqrt{-\text{det}\big{(}i^{\ast}(g+B)_{ij}+2\pi\alpha^{\prime}F_{ij}\big{)}}$$

(18)

with brane charge $\mu_{7}=((2\pi)^{3}\cdot(2\pi\alpha^{\prime})^{4})^{-1}$, $i$ the embedding map of the D7 into the ten-dimensional space-time and $i^{\ast}$ its pullback. Moreover, $F$ is the gauge field strength on the world-volume of the brane denef2008les , $B$ the NS-NS two-form field and $\Phi$ the dilaton. The leading order Wess-Zumino contribution is given by

$$S^{\tiny(0)}_{WZ}=\mu_{7}\int\text{Tr}\,\big{(}i^{\ast}\big{(}C\wedge e^{B}\big{)}\wedge e^{2\pi\alpha^{\prime}F}\big{)}\;\;,$$

(19)

where $C=\sum_{n}C_{n}$ the sum over the various Ramond-Ramond fields. The $O7^{-}$-plane contributions are

$$S^{\tiny(0)}_{O-DBI}=8\,\mu_{7}\int d^{7}Y\,e^{-\Phi}\text{Tr}\sqrt{-\text{det}\big{(}i^{\ast}g_{MN}\big{)}}\;\;\text{and}\;\;S^{\tiny(0)}_{O-WZ}=-8\,\mu_{7}\int i^{\ast}C_{8}\;\;.$$

(20)

In the following we refer to the $O7^{-}$-plane simply as $O7$. At the relevant $\alpha^{\prime 2}g_{s}$-order there are $R^{2}$-terms in the DBI action of the D7 brane and O7 planes, which we review in section 3.1.

In this section we analyize the E.O.M’s resulting from the leading order ten-dimensional type IIB supergravity and the $\alpha^{\prime 2}g_{s}$-corrected DBI actions of D7-branes and O7-planes. The relevant part of the ten-dimensional type IIB leading order supergravity action in the string frame takes the form¹¹¹¹11Where we use the notation $|H_{3}|^{2}=\tfrac{1}{p!}H_{MNO}H^{MNO}$.

$$S^{0}_{IIB-S}=\frac{1}{2\kappa_{10}}\int e^{-2\Phi}\Big{(}\,R\ast_{10}1+4\,d\Phi\wedge\ast d\Phi-\frac{1}{2}|H_{3}|^{2}\Big{)}-\frac{1}{2}|F_{3}|^{2}-\frac{1}{4}|F_{5}|^{2}\;.$$

(34)

where

$$H_{3}=dB_{2}\;\;,\;\;F_{3}=dC_{2}-C_{0}dB_{2}\;\;\text{and}\;\;F_{5}=dC_{4}-\tfrac{1}{2}C_{2}\wedge dB_{2}+\tfrac{1}{2}B_{2}\wedge dC_{2}\;\;,$$

(35)

are the usual NSNS and RR three-form field strengths and with

$$F_{5}=\ast_{10}F_{5}\;\;,$$

(36)

being self-dual. A Weyl rescaling by

$$g^{S}_{MN}=e^{\Phi/2}g^{E}_{MN}\;\;,$$

(37)

leads to the ten-dimesnioal Einstein frame action ¹²¹²12 The Einstein frame action results in $$S^{0}_{IIB-E}\supset\frac{1}{2\kappa_{10}}\int R\ast_{10}1\dots\;\;.$$ (38) where we have used (130) .

Moreover, we have used $g^{S},\,g^{E}$ to denote the string and Einstein-frame metric, respectively and $2\kappa_{10}=(2\pi)^{7}\alpha^{\prime 4}=\mu_{7}^{-1}$. The basic framework of our discussion are supersymmetric flux compactifications of type IIB super-string theory on a Calabi-Yau threefold Giddings_2002 ; Choi_2004 ; Blumenhagen_2007 ; Jockers:2004yj ; Grimm:2004uq . Setting aside higher-derivative corrections to the ten-dimensional supergravity action Giddings:2001yu the background metric is given by

	$\displaystyle\Phi$	$\displaystyle=\phi^{\tiny(0)}\;\;\;,\;\;\;\;\phi^{\tiny(0)}=\text{const.}\;\;,$		(39)
	$\displaystyle ds^{2}_{S}$	$\displaystyle=e^{\,2A}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+e^{-2A}g_{ij}dy^{i}dy^{j}\;\;$

where $g_{ij}$ denotes the unmodified Calabi-Yau threefold metric. The warp factor $A=A(y)$ determines the background $F_{5}$-flux via

$$F_{5}=\big{(}1+\ast_{10}\big{)}de^{4A}\wedge dx^{0}\wedge dx^{1}\wedge dx^{2}\wedge dx^{4}\;\;.$$

(40)

Moreover, the flux obeys the Bianchi identity

$$dF_{5}=H_{3}\wedge F_{3}+\rho_{6}\;\;,$$

(41)

where $\rho_{6}$ encodes the D3-brane charge density associated with potential localized sources. Integrating (41) one yields the D3-brane tadpole cancellation condition

$$\frac{1}{(2\pi\alpha^{\prime})^{2}}\int_{Y_{3}}H_{3}\wedge F_{3}+N_{D_{3}}\;\;=\;\;0\;\;,$$

(42)

where $N_{D_{3}}$ is the total number of D3 branes. In the presence of D7-branes and O7-planes the D3 tadpole cancellation condition (42) gets modified Plauschinn_2009 ; Blumenhagen_2009 as

$$\frac{1}{(2\pi\alpha^{\prime})^{2}}\int_{Y_{3}}H_{3}\wedge F_{3}+N_{D_{3}}\;\;=\;\;\frac{N_{O_{3}}}{4}+\sum_{i}^{\#D7^{\prime}s}N^{i}_{D7}\frac{\chi(D^{\tiny D7}_{i})}{24}+\sum_{i}^{\#O7^{\prime}s}\frac{\chi(D^{\tiny O7}_{i})}{12}$$

(43)

where the sum runs over the stacks of D7-branes containing the number of $N_{D7}$ each. The $D_{i}$’s are the four-cycles in the Calabi-Yau orientifold $oY_{3}$ wrapped by the D7’s and O7’s, respectively. Moreover $\chi$ is the Euler-characteristic of the 4-cycles and $N_{O_{3}}$ the number of O3-planes. We have set the background gauge flux of the D7-branes to zero throughout this work. We do not discuss localized D3 branes and O3 planes in this work. From (43) one infers that non-vanishing $H_{3}$ and $F_{3}$ flux are consistent with the absence of D3 branes as long as the Euler-characteristic of the divisor wrapped by the D7-branes and O7 planes do not vanish. Moreover, let us note that the D3 and O3 higher-derivative corrections won’t affect our results as the latter cannot affect the Kähler metric of the moduli space with the same functional dependence as D7-branes. However, the presence of D3-branes gives rise to other effects DeWolfe:2002nn . Lastly, the tadpole cancellation condition of D7-branes Plauschinn_2009 is given by

$$\sum_{i}^{\#D7^{\prime}s}N^{i}_{D7}\cdot\Big{(}\,[D^{\tiny D7}_{i}]+[D^{\tiny D7}{}^{\prime}_{i}]\,\Big{)}+8\sum_{i}^{\#O7^{\prime}s}[D^{\tiny O7}_{i}]\;\;=\;\;0\;\;,$$

(44)

where $[\cdot]$ denotes the class of the four-cycle wrapped by the D7’s and O7’s and the prime denotes the orientifold image i.e. the action on the background geometry in the presence of the O7-plane. To accommodate for (115) we choose a setup of eight D7-branes and one O7-plane which wrap the same four-cycle inside the Calabi-Yau fourfold and vanishing gauge flux. Moreover, the Calabi-Yau orientifold admits only one Kähler modulus $h^{1,1}=1$, i.e. the overall volume and moreover one finds that $h^{1,1}_{+}=1$. Although the latter requirement may seem restrictive it is sufficient to deduce the Kähler-potential and coordinates as those may be generalized to the generic Kähler moduli case, see e.g. the original derivation of the Euler characteristic correction to the $\mathcal{N}=1$ Kähler potential Becker_2002 . The generic moduli case derivation was done rather recently by Bonetti:2016dqh .

To combine (21) ,(26) and (34) one equips the DBI and ODBI action with $\delta_{D7}$, which is non-vanishing on the world-volume of the D7-branes. The definition of the scalar quantity $\delta_{D7}$ is simply given by¹³¹³13Note that (45) may be alternatively expressed via a 2-form $\hat{\omega}$ which is Poincare dual to the cycle wrapped by the D7-brane. In other words one finds that $\int_{oY_{3}}J\wedge J\wedge\hat{\omega}=\int_{D7}\ast_{8}1$ and thus $\delta_{D7}\propto\hat{\omega}_{ij}g^{\tiny(0)}{}^{ij}$.

$$\int\delta_{D7}\ast_{10}1=\int_{D_{7}}\ast_{8}1\;,\;\;\text{and in particular}\;\;\int_{oY_{3}}\delta_{D7}\ast_{6}1=\int_{D_{m}}\ast_{4}1\;\;.$$

(45)

Dilaton E.O.M. We proceed by deriving the string frame equation of motion for the dilaton. One infers from (34) and eight-times the DBI action (21) plus the ODBI action (26) for eight coincident D7’s on top of a single O7 to be

	$\displaystyle R^{\tiny(1)}+4\nabla^{\tiny(0)}_{i}\nabla^{\tiny(0)}{}^{i}\Phi^{\tiny(1)}-\tfrac{1}{2}|H_{3}|^{2}$	$\displaystyle+2\alpha\,\delta_{D7}\,e^{\Phi}\cdot\nabla^{\tiny(0)}{}^{\alpha}\nabla^{\tiny(0)}{}^{\beta}R_{T}{}_{\alpha\beta}|_{oY_{3}}$		(46)
		$\displaystyle-2\alpha\,\delta_{D7}\,e^{\Phi}\cdot\nabla^{\tiny(0)}{}^{a}\nabla^{\tiny(0)}{}^{b}{\bar{R}}{}_{ab}|_{oY_{3}}+\dots=0\;\;,$

where we have used the fact that the leading order solution eq. (39) denoted by the superscript ⁽⁰⁾ is Minkowski times internal Calabi-Yau, and moreover that the classical dilaton solution is constant. For notational simplicity we have defined

$$\alpha=12\cdot\frac{(2\pi\alpha^{\prime})^{2}}{192}\;\;.$$

(47)

The ellipsis in (46) denote $\mathcal{O}(\alpha^{2})$ contributions as well as terms quadratic in the Riemann-tensor of the internal space.
External Einstein equation. Let us next turn to derive the Einstein equations from (34) , (21) and (26).¹⁴¹⁴14Note that there is no contribution from the classical DBI and WZ actions due to tadpole cancellation. The external Einstein equation is given by

	$\displaystyle R^{\tiny(1)}+4\nabla^{\tiny(0)}_{i}\nabla^{\tiny(0)}{}^{i}\Phi^{\tiny(1)}-\tfrac{1}{2}|H_{3}|^{2}$	$\displaystyle-\tfrac{1}{2}e^{2\Phi}|F_{3}|^{2}$		(48)
	$\displaystyle+\,4\alpha\,\delta_{D7}\,e^{\Phi}\cdot\nabla^{\tiny(0)}{}^{\alpha}\nabla^{\tiny(0)}{}^{\beta}R_{T}^{\tiny(0)}{}_{\alpha\beta}|_{oY_{3}}$	$\displaystyle-4\alpha\,\delta_{D7}\,e^{\Phi}\cdot\nabla^{\tiny(0)}{}^{a}\nabla^{\tiny(0)}{}^{b}{\bar{R}}{}^{\tiny(0)}_{ab}|_{oY_{3}}+\dots=0\;\;.$

Internal Einstein equations. The internal Einstein equations split in tangent and normal directions. The internal normal components are given by

	$\displaystyle\delta_{ab}\Big{(}$	$\displaystyle\tfrac{1}{2}R^{\tiny(1)}+2\nabla^{\tiny(0)}_{i}\nabla^{\tiny(0)}{}^{i}\Phi^{\tiny(1)}-\tfrac{1}{4}|H_{3}|^{2}-\tfrac{1}{4}e^{2\Phi}|F_{3}|^{2}\Big{)}$		(49)
		$\displaystyle-R^{\tiny(1)}_{ab}+\tfrac{3}{2}|H_{3}|^{2}_{ab}+\tfrac{3}{2}e^{2\Phi}|F_{3}|^{2}_{ab}-2\alpha\,\delta_{D7}\,e^{\Phi}\cdot\nabla^{\tiny(0)}_{\alpha}\nabla^{\tiny(0)}{}^{\alpha}{\bar{R}}{}^{\tiny(0)}_{ab}|_{oY_{3}}+\dots=0\;\;,$

The tangent components are given by ¹⁵¹⁵15Where we have used the notation $$|H_{3}|^{2}=\frac{1}{3!}H_{3}{}_{ijk}H_{3}{}^{ijk}\;\;,\;\;\;|H_{3}|^{2}_{ab}=\frac{1}{3!}H_{3}{}_{aij}H_{3}{}_{b}{}^{ij}\;\;,\;\;\;|H_{3}|^{2}_{\alpha\beta}=\frac{1}{3!}H_{3}{}_{\alpha ij}H_{3}{}_{\beta}{}^{ij}\;\;,$$ (50) and analogously for $|F_{3}|^{2}_{ab}$ and $|F_{3}|^{2}_{\alpha\beta}$ .

		$\displaystyle g^{\tiny(0)}_{\alpha\beta}\Big{(}\tfrac{1}{2}R^{\tiny(1)}+2\nabla^{\tiny(0)}_{i}\nabla^{\tiny(0)}{}^{i}\Phi^{\tiny(1)}-\tfrac{1}{4}|H_{3}|^{2}-\tfrac{1}{4}e^{2\Phi}|F_{3}|^{2}+2\alpha\,\delta_{D7}\,e^{\Phi}\cdot\nabla^{\tiny(0)}{}^{\gamma}\nabla^{\tiny(0)}{}^{\delta}R_{T}^{\tiny(0)}{}_{\gamma\delta}|_{oY_{3}}$
		$\displaystyle-2\alpha\,\delta_{D7}\,e^{\Phi}\cdot\nabla^{\tiny(0)}{}^{a}\nabla^{\tiny(0)}{}^{b}{\bar{R}}{}^{\tiny(0)}_{ab}|_{oY_{3}}\Big{)}$
		$\displaystyle+\alpha\,\delta_{D7}\,e^{\Phi}\Big{(}-4\nabla^{\tiny(0)}_{\gamma}\nabla^{\tiny(0)}_{(\alpha}R_{T}{}^{\tiny(0)}_{\beta)\gamma}|_{oY_{3}}-4\nabla^{\tiny(0)}{}^{\gamma}\nabla^{\tiny(0)}{}^{\delta}R_{T}{}^{\tiny(0)}_{\alpha\gamma\beta\delta}|_{oY_{3}}+2\nabla^{\tiny(0)}_{\gamma}\nabla^{\tiny(0)}{}^{\gamma}R_{T}{}^{\tiny(0)}_{\alpha\beta}|_{oY_{3}}\Big{)}$
		$\displaystyle-R^{\tiny(1)}_{\alpha\beta}+\tfrac{3}{2}|H_{3}|^{2}_{\alpha\beta}+\tfrac{3}{2}e^{2\Phi}|F_{3}|^{2}_{\alpha\beta}+\dots\;\;=\;\;0\;\;,$		(51)

Background solution. We next show that the following ansatz for the type IIB background metric and dilaton

	$\displaystyle\Phi$	$\displaystyle=\phi^{\tiny(0)}+\alpha\,\phi^{\tiny(1)}\;\;\;,\;\;\;\;\phi^{\tiny(0)}=\text{const.}\;\;,$		(52)
	$\displaystyle ds^{2}_{S}$	$\displaystyle=e^{2\alpha\,A^{\tiny(1)}}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+e^{-2\alpha A^{\tiny(1)}}\Big{(}g^{\tiny(0)}_{ij}+\alpha\,\,g^{\tiny(1)}_{ij}\Big{)}dy^{i}dy^{j}\;\;,$		(53)

is a solution to (46)-(4), where $g^{\tiny(0)}_{ij}$ denotes the unmodified Calabi-Yau metric which constitutes a solution to the classical E.O.M.’s. Moreover $g^{\tiny(1)}_{ij}$ can be expressed in terms of tangent and normal indices $i\to(\alpha,a)$ according to (133) and (134). Firstly, one infers from (52) and (53) that (46) may be re-expressed as¹⁶¹⁶16Let us emphasize that in this work for simplicity i.e. the one-modulus $h^{1,1}=1$ Calabi-Yau (52) and (53) are given for the case where the world-volume of 8 D7’s and O7 coincides. However, note that the (52) and (53) can be generalized to account for non-coinciding D7’s and O7’s by simple summing over the respective contributions.

		$\displaystyle\nabla^{\tiny(0)}{}^{\alpha}\nabla^{\tiny(0)}{}^{\beta}\Big{(}g^{\tiny(1)}_{\alpha\beta}+2\,\delta_{D7}\,e^{\phi^{\tiny(0)}}\cdot R_{T}{}_{\alpha\beta}|_{oY_{3}}\Big{)}+\nabla^{\tiny(0)}{}^{a}\nabla^{\tiny(0)}{}^{b}\Big{(}g^{\tiny(1)}_{ab}-2\,\delta_{D7}\,e^{\phi^{\tiny(0)}}\cdot{\bar{R}}{}_{ab}|_{oY_{3}}\Big{)}$		(54)
		$\displaystyle+\nabla^{\tiny(0)}_{i}\nabla^{\tiny(0)}{}^{i}\Big{(}4\,\phi^{\tiny(1)}-g^{\tiny(1)}{}_{i}{}^{i}+10\,A^{\tiny(1)}\big{)}-\tfrac{1}{2}|H_{3}|^{2}\;\;+\dots\;\;=\;\;0\;\;.$

We proceed analogously for the Einstein equation (48)-(4). For details see appendix B.3. A few comments in order. From comparison of eq.’s  (46) - (48) one infers that there exists no solution for vanishing background fluxes $H_{3}$ and $F_{3}$. To solve for $H_{3}$ and $F_{3}$ explicitly is beyond the scope of this work as it additionally requires to check consistency with the E.O.M’s of $H_{3},F_{3}$ and $C_{4}$. For our purpose it is sufficient to limit ourselves to making an Ansatz for the squared contributions

$$|F_{3}|^{2}_{\alpha\beta},\;|H_{3}|^{2}_{\alpha\beta}\;\;\sim\;\;\mathcal{O}(\alpha)\;\;\;\;\text{and}\;\;\;\;|F_{3}|^{2}_{ab},\;|H_{3}|^{2}_{ab}\;\;\sim\;\;\mathcal{O}(\alpha)\;\;,$$

(55)

rather than $H_{3}$ and $F_{3}$ itself. We use that

$$|H_{3}|^{2}=g^{\tiny(0)}{}^{\alpha\beta}|H_{3}|^{2}_{\alpha\beta}+\delta^{ab}|H_{3}|^{2}_{ab}\;\;,\;\text{and}\;\;\;|F_{3}|^{2}=g^{\tiny(0)}{}^{\alpha\beta}|F_{3}|^{2}_{\alpha\beta}+\delta^{ab}|F_{3}|^{2}_{ab}\;\;\;.$$

(56)

Details of the flux-background ansatz can be found in the appendix B.3. Let us take a step back to discuss the factorization of eq.s (46)-(4) in a total derivative contribution and a curvature square density. One finds that the E.O.M’s are of the schematic form¹⁷¹⁷17Note that by commuting two covariant derivatives the two sector in eq.(57) can communicate in principle. However, this does not affect our analysis at hand.

$$\underbrace{\nabla\nabla{}\Big{(}\sum_{n}R_{n}}_{=\;0}\Big{)}\;\;+\;\;\underbrace{\sum_{m}R^{2}_{m}}_{=\;0}\;\;=\;\;0\;\;.$$

(57)

where $R_{n}$ is placeholder for objects in the list $\left\{R_{T}|_{oY_{3}},{\bar{R}}{}|_{oY_{3}},g^{\tiny(1)},\phi^{\tiny(1)},A^{\tiny(1)}\right\}$ and $R^{2}_{m}$ is our notation for curvature objects in the list $\left\{R_{T}^{2}|_{oY_{3}},R^{2}|_{oY_{3}},\hat{R}\Omega^{2}|_{oY_{3}},\Omega^{4}|_{oY_{3}}\right\}$. Moreover, the formal sum in eq. (57) allows for various different index contractions as well as different pre-factors. For simplicity, in this work we only provide a solution to the total-derivative contribution which, however suffices to fix (52) and (53). Note that due to this we can restrict ourselves to making an Ansatz for (55) which only contains total derivative pieces.
Let us now turn to the discussion of the solution of (46)-(4) by the Ansatz (52) and (53) which is fixed to take the form

	$\displaystyle g^{\tiny(1)}_{\alpha\beta}\;\;$	$\displaystyle=-2\,\delta_{D7}\,e^{\phi^{\tiny(0)}}\,R_{T}{}_{\alpha\gamma\beta}{}^{\gamma}|_{oY_{3}}+\hat{\gamma}_{3}\,\,\delta_{D7}\,g^{\tiny(0)}_{\alpha\beta}\cdot e^{\phi^{\tiny(0)}}R_{T}|_{oY_{3}}\;\;,\;\;$		(58)
	$\displaystyle g^{\tiny(1)}_{ab}\;\;$	$\displaystyle=\;\;\;4\,\delta_{D7}\,e^{\phi^{\tiny(0)}}\,{\bar{R}}{}_{a\gamma b}{}^{\gamma}|_{oY_{3}}\;\;+\;\hat{\gamma}_{3}\,\,\delta_{D7}\,g^{\tiny(0)}_{ab}\cdot e^{\phi^{\tiny(0)}}\,R_{T}|_{oY_{3}}\;\;,$

and moreover the background dilaton and warp factor to be

$$\phi^{\tiny(1)}\;\;=\big{(}-\tfrac{6}{5}+\gamma_{3}\big{)}\,\delta_{D7}\,e^{\phi^{\tiny(0)}}\,R_{T}|_{oY_{3}}\;\;,\;\;\;A^{\tiny(1)}\;\;={\bar{\gamma}}{}_{3}\,e^{\phi^{\tiny(0)}}\,\delta_{D7}\,R_{T}|_{oY_{3}}\;,$$

(59)

where

$$\gamma_{3}=6\,\hat{\gamma}_{3}+2\,{\bar{\gamma}}{}_{3}\;\;.$$

(60)

Thus note that the Einstein equations and dilaton equation of motion alone do not completely fix the Ansatz (52) and (53). More precisely, there remains an ambiguity in between the warp factor and the correction to the Calabi-Yau metric parametrized by the $\gamma_{3}{}^{\prime}s$. We expect that a complete treatment of the other E.O.M.’s involving the fluxes will fix the remaining freedom.

Ramond Ramond $C_{4}$. It is interesting to discuss the relationship of the warp-factor (59) and the dynamic equations for the NS-NS and R-R fields given by (40) and (41). Counting derivatives one infers that

$$dF_{5}=d\ast_{10}de^{4A^{\tiny(1)}}dx^{1}\dots dx^{4}\sim\mathcal{O}\big{(}\alpha\big{)}\;\;,\;\;H_{3}\sim\mathcal{O}\big{(}\alpha^{\tfrac{1}{2}}\big{)}\;\;,\;\;F_{3}\sim\mathcal{O}\big{(}\alpha^{\tfrac{1}{2}}\big{)}\;\;.$$

(61)

and thus in particular

$$F_{5}\sim\mathcal{O}\big{(}\alpha\big{)}\;\;,\text{thus}\;\;|F_{5}|^{2}\sim\mathcal{O}\big{(}\alpha^{2}\big{)}\sim\mathcal{O}{\bf\big{(}\alpha^{\prime}{}^{4}\big{)}}\;\;.$$

(62)

This is self-consistent with the E.O.M’s (46)-(4) in which we have omitted $|F_{5}|^{2}$ due to the fact of being of higher order in $\alpha$. Moreover, it implies that the Wess-Zumino contribution (27) can be safely neglected in the next section 5 as it is of higher order as well. Note that (61) is consistent with the well known flux quantization condition Giddings:2001yu given by

$$\frac{1}{2\pi\alpha^{\prime}}\int H_{3}\;\;\in\;\;2\pi\mathbb{Z}\;\;,\;\;\frac{1}{2\pi\alpha^{\prime}}\int F_{3}\;\;\in\;\;2\pi\mathbb{Z}\;\;.$$

(63)

Let us close this section by providing an outlook on the next section. Note that a solution to the equations of motion is a necessary but not a sufficient condition for the background to preserve the required amount of supersymmetry. As we are not aware of a discussion of the $\alpha^{\prime}$-corrected supersymmetry conditions we limited ourselves to the discussion of the E.O.M.’s. To gain confidence in the background solution we will provide a check employing four dimensional $\mathcal{N}=1$ supersymmetry.¹⁸¹⁸18This procedure of fixing higher-derivative terms or for our matter at hand a parameter in the higher-derivative background solution was employed in e.g. our previous work Grimm:2017okk . One may first compactify to lower dimensions and verify consistency with $4d,\,\mathcal{N}=1$ or $\mathcal{N}=2$ supersymmetry. The latter led us to find novel higher-derivative terms in type IIA Grimm:2017okk which were recently confirmed by scattering amplitudes methods Liu:2019ses . We use the simple fact that the $\alpha^{\prime 2}$-correction to the kinetic terms must take the form (13). In particular, it implies the vanishing of the $\alpha^{\prime 2}$-correction to the dilaton kinetic terms as well as the mix terms with the Einstein frame volume $\hat{\mathcal{V}}$. Employing this technique in section 5 will lead us to fix $\hat{\gamma}_{3}$ and ${\bar{\gamma}}{}_{3}$.