Stabilizing massless fields with fluxes in Landau-Ginzburg models

Because the $1^{9}$ model is non-geometric, it is not possible to study Ramond and Neveu-Schwarz fluxes in the usual fashion in the supergravity approximation. However, the vertex operators creating the corresponding spacetime fields still exist in the worldsheet theory. Their interactions with the moduli induce a superpotential completely analogous to the geometric formulation. The fluxes are also subject to the same quantization and tadpole cancellation conditions. We refer to Becker:2006ks for a rigorous justification of these statements. Here, we only broach some ideas, and summarize the results. Crucially, to describe the wrapped fluxes, we require an integral homology basis, and to understand the space-time superpotential and tadpole cancellation, the pairing with cohomology. Physically, one can think of integral homology in terms of supersymmetric cycles wrapped by D-branes. In type IIB, the cycles that can be threaded by fluxes are represented by A-branes. The cycles that support the orientifold planes and carry (the analogues of) the D3/D7-brane tadpole are represented by B-branes.

The undeformed $1^{9}$ model is an orbifolded tensor product of $\mathcal{N}=2$ minimal models with smallest possible central charge $\hat{c}=\frac{1}{3}$. This has a Landau-Ginzburg representation with a single chiral field, and superpotential

$$\mathcal{W}=x^{3}\,.$$

(11)

The A-branes of this model are represented by contours in the $x$-plane that asymptote to regions in which $\mathop{\rm Im}\nolimits(\mathcal{W})=0$ HoriIqbalVafa . There are three such contours, $\left(V_{0},V_{1},V_{2}\right)$, shown in Fig. 1 below. These are not independent cycles, but satisfy the one relation,

$$V_{0}+V_{1}+V_{2}=0\,.$$

(12)

Under the $\mathbb{Z}_{3}$ action (7), they transform as

$$g:V_{n}\mapsto V_{n+1\bmod 3}\,.$$

(13)

Somewhat fancily, one can think of the charge lattice $\Lambda$ of A-branes in the minimal model as fitting into the exact sequence,

$$0\to\mathbb{Z}\to\mathbb{Z}^{3}\to\Lambda\to 0$$

(14)

where the middle $\mathbb{Z}^{3}$ is generated by the $V_{0}$, $V_{1}$, $V_{2}$, and $\mathbb{Z}$ represents the relation (12).

The chiral ring of the minimal model is spanned by the elements $1,x\in\mathcal{R}=\mathbb{C}[x]/x^{2}$. These correspond by spectral flow to Ramond-Ramond ground states traditionally labelled as $\ket{l}$ with $l=1,2$

$$x^{k=0,1}\ext@arrow 0055{\arrowfill@\leftarrow\relbar\rightarrow}{}{\text{ spectral flow }}\ket{l=1,2}\,.$$

(15)

The overlap between these Ramond ground states and the boundary states represented by the $V_{n}$ (the disk one-point function) can be calculated (after supersymmetric localization) as a contour integral HoriIqbalVafa . Up to normalization, we have

$$\braket{V_{n}|l}=\int_{V_{n}}x^{l-1}e^{-\mathcal{W}}dx=\frac{1}{3}\omega^{nl}(1-\omega^{l})\Gamma\Bigl{(}\frac{l}{3}\Bigr{)}\,,$$

(16)

where $n\in\{0,1,2\}$ and $l\in\{1,2\}$. The same integral also calculates the variation of the overlaps under the deformation $\mathcal{W}\to x^{3}-tx$.

$$\Bigl{(}\frac{\partial}{\partial t}\Bigr{)}^{r}\Bigr{|}_{t=0}\braket{V_{n}|l}=\int_{V_{n}}x^{r+l-1}e^{-x^{3}}dx=\frac{1}{3}\omega^{n(r+l)}(1-\omega^{r+l})\Gamma\Bigl{(}\frac{r+l}{3}\Bigr{)}\,.$$

(17)

This vanishes when $r+l=0\bmod 3$ because the integrand is exact in this case. The fact that it does not vanish when $r+l>2$ (but not $0\bmod 3$), when formally $x^{r+l-1}=0\in\mathcal{R}$ is zero by the equations of motion, is physically a result of “contact terms” in the operator product expansion. Mathematically, this amounts to integration by parts. The formula (17) will be the basis for the calculation of the higher-order terms in the superpotential in section 4.

To determine the contribution of the fluxes to the D3-brane tadpole, we require the intersection form on the charge lattice. Physically, the intersection of $V_{n^{\prime}}$ and $V_{n}$ can be defined as the open string Witten index between the respective branes. Mathematically, it is the geometric intersection between a small counter-clockwise rotation of $V_{n^{\prime}}$ and $V_{n}$ HoriIqbalVafa . In matrix form Brunner:1999jq ,

$$\bigl{(}\braket{V_{n^{\prime}}|V_{n}}\bigr{)}_{n^{\prime},n=0,1,2}=\begin{pmatrix}1&-1&0\\ 0&1&-1\\ -1&0&1\end{pmatrix}=1-g$$

(18)

where $g$ is the matrix representation of (13). The fact that (18) is neither symmetric nor anti-symmetric reflects that a single minimal model is not yet Calabi-Yau.

The calculations are expedited if one uses the Poincaré duals of the Ramond ground states as basis for the charge lattice. This was emphasized in Becker:2006ks ; Becker:2023rqi . Defining for $l=1,2$

$$\Omega_{l}:=\frac{1}{3}\sum_{n}\omega^{nl}V_{n}\,,$$

(19)

with inverse relation

$$V_{n}=\sum_{l}\omega^{-nl}\Omega_{l}\,,$$

(20)

we find from (18)

$$\braket{\Omega_{l^{\prime}}|\Omega_{l}}=\delta_{l^{\prime}+l,3}\,\frac{1}{3}(1-\omega^{l})$$

(21)

and

$$\braket{V_{n}|\Omega_{l}}=\frac{1}{3}\omega^{nl}(1-\omega^{l})\,.$$

(22)

Thus, by comparison with (16),

$$\ket{l}=\Gamma\Bigl{(}\frac{l}{3}\Bigr{)}\ket{\Omega_{l}}\,.$$

(23)

All these relations are compatible with (12) and $\ket{l}=0$ when $l=3$. Eqs. (19) and the reality of the $V_{n}$ also imply that complex conjugation acts on the $\Omega_{l}$ via

$$\overline{\Omega_{l}}=\Omega_{3-l}$$

(24)

In combination with (21), this produces the $tt^{*}$-metric on the RR ground states Cecotti:1991me .

The full orbifoldized $1^{9}$ model can now be worked out straightforwardly. The Ramond ground states are tensor products labelled as $\ket{{\mathbf{l}}}$ with ${\mathbf{l}}=\left(l_{1},l_{2},\ldots,l_{9}\right)$, $l_{i}\in\{1,2\}$, and $\sum l_{i}$ divisible by $3$ in order to satisfy the orbifold projection. These correspond to the basis of the chiral ring (8) by spectral flow and can be classified by Hodge type as shown in table 1.

An (over-complete) integral basis of cycles is obtained by taking tensor products of the $V_{n}$ to $V_{{\mathbf{n}}}=V_{n_{1}}\times\cdots\times V_{n_{9}}$ for ${\mathbf{n}}=\left(n_{1},n_{2},\ldots,n_{9}\right)$, $n_{i}\in\{0,1,2\}$, and summing over $\mathbb{Z}_{3}$ images.

$$\gamma_{{\mathbf{n}}}:=V_{\mathbf{n}}+V_{\mathbf{n+1}}+V_{\mathbf{n+2}}$$

(25)

where $\mathbf{1}=(1,1,1,1,1,1,1,1,1)$ and $\mathbf{2}=2\cdot\mathbf{1}=(2,2,2,2,2,2,2,2,2)$. On the tensor product of (14),

$$0\to\mathbb{Z}\to 9\mathbb{Z}^{3}\to 36(\mathbb{Z}^{3})^{2}\to\cdots\to(\mathbb{Z}^{3})^{9}\to\Lambda\to 0\,,$$

(26)

the $\mathbb{Z}_{3}$ action is free except on the very first term, where it is trivial. This shows that the rank of the lattice $\Lambda$ spanned by the $\gamma_{{\mathbf{n}}}$ is $((3-1)^{9}+1)/3-1=170$. This is equal to the dimension of the chiral ring (8). The overlap integrals (16) become

$$\braket{\gamma_{{\mathbf{n}}}|{\mathbf{l}}}=\frac{1}{3^{8}}\omega^{{\mathbf{n}}.{\mathbf{l}}}\prod_{i=1}^{9}(1-\omega^{l_{i}})\Gamma\Bigl{(}\frac{l_{i}}{3}\Bigr{)}$$

(27)

where $\mathbf{n}.\mathbf{l}=\sum_{i=1}^{9}n_{i}\,l_{i}$, and one factor of $3$ is owed to (25). The intersection form is obtained by orbifolding the tensor product (18).

$$\braket{\gamma_{{\mathbf{n}}^{\prime}}|\gamma_{{\mathbf{n}}}}=\braket{V_{{\mathbf{n}}^{\prime}}|V_{{\mathbf{n}}}}+\braket{V_{{\mathbf{n}}^{\prime}+{\mathbf{1}}}|V_{{\mathbf{n}}}}+\braket{V_{{\mathbf{n}}^{\prime}+{\mathbf{2}}}|V_{{\mathbf{n}}}}$$

(28)

In the Poincaré dual basis

$$\ket{\Omega_{{\mathbf{l}}}}=\frac{1}{3^{9}}\sum_{[{\mathbf{n}}]}\omega^{{\mathbf{n}}.{\mathbf{l}}}\gamma_{{\mathbf{n}}}=\frac{1}{3^{9}}\sum_{{\mathbf{n}}}\omega^{{\mathbf{n}}.{\mathbf{l}}}V_{{\mathbf{n}}}\,,$$

(29)

the intersection form becomes

$$\braket{\Omega_{{\mathbf{l}}^{\prime}}|\Omega_{{\mathbf{l}}}}=\delta_{{\mathbf{l}}^{\prime}+{\mathbf{l}},{\mathbf{3}}}\,\frac{1}{3^{8}}\prod_{i}(1-\omega^{l_{i}})\,.$$

(30)

We will refer to this as the “$\Omega$-basis”. Complex conjugation acts on it by

$$\overline{\Omega_{{\mathbf{l}}}}=\Omega_{\bar{\mathbf{l}}}$$

(31)

where $\bar{\mathbf{l}}={\mathbf{3}}-{\mathbf{l}}$, and ${\mathbf{3}}=3\cdot{\mathbf{1}}=(3,3,3,3,3,3,3,3,3)$. The form (30) is anti-symmetric following the orbifold projection.

We are now in a position to describe supersymmetric 3-form fluxes in the $1^{9}$ model. There are two ways to do this. The first is to expand the standard combination of Ramond and Neveu-Schwarz fluxes $G_{3}=F_{3}-\tau H_{3}$ in terms of the integral cohomology basis given by the $\gamma_{\mathbf{n}}$. Writing

$$G_{3}=\sum_{\mathbf{n}}\left(N^{\mathbf{n}}-\tau M^{\mathbf{n}}\right)\gamma_{\mathbf{n}}\,,$$

(32)

the $N^{\mathbf{n}},M^{\mathbf{n}}$ should be integer. They are not uniquely determined because the $\gamma_{{\mathbf{n}}}$ are not linearly independent. The spacetime superpotential induced by this flux is given by the Landau-Ginzburg version of the standard GVW formula Gukov:1999ya

$$W_{{\rm GVW}}=\int\left(F_{3}-\tau H_{3}\right)\wedge\Omega=\braket{G_{3}|{\mathbf{1}}}$$

(33)

Here, we have used table 1 to identify the holomorphic three-form $\Omega$ with the ground state $\ket{{\mathbf{1}}}$. The overlap should be evaluated with the help of (27). The first (and higher) derivatives of the superpotential with respect to the moduli (including the axio-dilation $\tau$) can be evaluated with the help of (17), see subsection 2.3. Setting them to zero will constrain $G_{3}$ to be of a certain Hodge type as usual. This gives a set of linear equations on the $N^{\mathbf{n}}$, $M^{\mathbf{n}}$, which have to be solved over the integers. The precise formula also depends on the spacetime Kähler potential, see section 3.

The alternative approach is to expand $G_{3}$ in the $\Omega$-basis

$$G_{3}=\sum_{{\mathbf{l}}}A^{\mathbf{l}}\Omega_{{\mathbf{l}}}$$

(34)

This allows to directly constrain its Hodge type by simply setting the undesired $A^{\mathbf{l}}$ to $0$. Flux quantization is equivalent to the condition that in

$$\int_{\gamma_{\mathbf{n}}}G_{3}=\braket{\gamma_{\mathbf{n}}|G_{3}}=N_{\mathbf{n}}-\tau M_{\mathbf{n}}\,,$$

(35)

which is again to be evaluated with (27), the $N_{{\mathbf{n}}}$ and $M_{{\mathbf{n}}}$ have to be integer. They are related to the integers in (32) by lowering indices with the help of the symplectic intersection form (28).

The two formulations (32) and (34) are of course equivalent as far as the parametrization of the supersymmetric fluxes is concerned. However, the calculation of the higher-order terms in the superpotential is considerably more efficient in the $\Omega$-basis. We therefore prefer it.

The final ingredients are the orientifold projection and the comparison between the O-plane charge and flux tadpole. These were determined in Becker:2006ks using the general formulas provided in Hori_2008 . We will restrict to the orientifold of the $1^{9}$ model that is generated by dressing worldsheet parity with the exchange of the first two coordinates. This has to be accompanied by a phase rotation in order to guarantee invariance of the superpotential term in (5). Namely, we are orientifolding by

$$\sigma:\left(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8},x_{9}\right)\mapsto-\left(x_{2},x_{1},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8},x_{9}\right)\,.$$

(36)

There are $63$ invariant monomials under this orientifold. Including the axio-dilaton, this gives a total of $64$ moduli that we wish to stabilize. The above orientifold projection breaks the initial permutation group $S_{9}$ of the $1^{9}$ model to a $\mathbb{Z}_{2}\times S_{7}$ subgroup. We will later use this group to connect different flux configurations. The O-plane associated with the orientifold projection in equation (36) is of “O3-plane type”. Its charge is equal to $12$ in natural units. This induces a RR tadpole that must be cancelled by the fluxes that we turn on, as well as possibly adding $N_{\rm D3}$ background D3-branes. The precise condition is that

$$N_{\rm flux}=\frac{1}{\tau-\bar{\tau}}\int G_{3}\wedge\bar{G}_{3}=\int F_{3}\wedge H_{3}\overset{!}{=}12-N_{\rm D3}\,.$$

(37)

The overlap of fluxes is to be evaluated with the help of (30) or (31), if working with the $\Omega$-basis.

By combining (17) with (25), we obtain the following explicit formula for an arbitrary multi-derivative of the space-time superpotential (33) in the $\gamma$-basis

$$W=\sum(N^{\mathbf{n}}-\tau M^{\mathbf{n}}\bigl{)}\braket{\gamma_{\mathbf{n}}|{\mathbf{1}}}$$

(38)

with respect to the deformation parameters in (10) labelled by $t^{{\mathbf{k}}}$, with ${\mathbf{k}}$ having nine entries, six of which are $0$ and three of which are $1$.

$$\frac{\partial}{\partial t^{\mathbf{k}_{1}}}\frac{\partial}{\partial t^{\mathbf{k}_{2}}}\ldots\frac{\partial}{\partial t^{\mathbf{k}_{r}}}\braket{\gamma_{\mathbf{n}}|{\mathbf{1}}}\bigg{|}_{t^{\mathbf{k}}=0}=\frac{1}{3^{8}}\omega^{{\mathbf{n}}.{\mathbf{L}}}\prod_{i=1}^{9}(1-\omega^{L_{i}})\Gamma\Bigl{(}\frac{L_{i}}{3}\Bigr{)}$$

(39)

Here, as always, $\omega\equiv e^{\frac{2\pi{\rm i}}{3}}$, and we have abbreviated ${\mathbf{L}}=(L_{1},\ldots,L_{9})$ with

$${\mathbf{L}}=\sum_{\alpha=1}^{r}\mathbf{k}_{\alpha}+{\mathbf{1}}\,.$$

(40)

The normalization in (39) is the same as in (27). Transforming to the $\Omega$-basis

$$W=\sum_{\mathbf{l}}A^{\mathbf{l}}\braket{\Omega_{\mathbf{l}}|{\mathbf{1}}}$$

(41)

with the help of (29), we find Becker:2022hse

$$\frac{\partial}{\partial t^{\mathbf{k}_{1}}}\frac{\partial}{\partial t^{\mathbf{k}_{2}}}\ldots\frac{\partial}{\partial t^{\mathbf{k}_{r}}}\int\Omega_{\bf{l}}\wedge\Omega\bigg{|}_{t^{\mathbf{k}}=0}=\delta_{{\mathbf{l}}+{\mathbf{L}}}\,\frac{1}{3^{9}}\prod_{i=1}^{9}\left(1-\omega^{L_{i}}\right)\Gamma\Bigl{(}\frac{L_{i}}{3}\Bigr{)}\,.$$

(42)

Here, the Kronecker-$\delta$ is understood $\bmod 3$ in all 9 components. Taking account of the product of $(1-\omega^{L_{i}})$’s, we find that the derivative in equation (42) vanishes whenever $L_{i}=0\bmod 3$ or $l_{i}+L_{i}\neq 0$ mod 3 for any $i\in\{1,2,\ldots,9\}$. Since all $l_{i}$ and $(L_{i}\bmod 3)$ are either $1$ or $2$, the second condition is equivalent to $\bar{\mathbf{l}}={\mathbf{L}}\bmod 3$, where $\mathbf{\bar{l}}={\mathbf{3}}-\bf{l}$, and ${\mathbf{3}}=(3,3,3,3,3,3,3,3,3)$. Because ${\mathbf{l}}$ has six entries equal to $1$ and $3$ entries equal to $2$, we can simplify

$$\frac{\partial}{\partial t^{\mathbf{k}_{1}}}\frac{\partial}{\partial t^{\mathbf{k}_{2}}}\ldots\frac{\partial}{\partial t^{\mathbf{k}_{r}}}\int\Omega_{\bf{l}}\wedge\Omega\bigg{|}_{t^{\mathbf{k}}=0}=\begin{cases}-\bigl{(}\sqrt{-3}\bigr{)}^{-9}\prod_{i=1}^{9}\Gamma\bigl{(}\frac{L_{i}}{3}\bigr{)}&\text{for $\bf{\bar{l}}=\bf{L}$ mod 3}\,,\\ 0&\text{ otherwise}\,.\end{cases}$$

(43)

Moreover, by the functional equation of the Gamma-function, the product is always a rational multiple of $\Gamma\bigl{(}\frac{2}{3}\bigr{)}^{6}\Gamma\bigl{(}\frac{1}{3}\bigr{)}^{3}$. The importance of the result (43) is computational. It means that before calculating the derivative explicitly, we can check whether $\mathbf{\bar{l}}=\sum_{\alpha}\mathbf{k}_{\alpha}+{\bf 1}\bmod 3$. This substantially speeds up the calculation of higher order terms. We also note that the derivative does not depend on the individual $\mathbf{k}_{\alpha}$ but rather only on their sum.

We now turn to mixed multi-derivatives involving both complex structure moduli and the axio-dilaton. Since by (33), $W$ is linear in $\tau$, we only need to worry about first partial derivatives with respect to $\tau$. The derivative with respect to $\tau$ can be calculated from (32) and the reality of $F_{3}$, $H_{3}$ as usual

$$\partial_{\tau}W=\frac{1}{\tau-\bar{\tau}}\int\bigl{(}G_{3}-\overline{G}_{3}\bigr{)}\wedge\Omega$$

(44)

In the $\gamma$-basis, this reduces a multi-derivative of the type

$$\frac{\partial}{\partial\tau}\frac{\partial}{\partial t^{\mathbf{k}_{1}}}\frac{\partial}{\partial t^{\mathbf{k}_{2}}}\ldots\frac{\partial}{\partial t^{\mathbf{k}_{r}}}W$$

(45)

to (39) with the same ${\mathbf{k}}_{\alpha}$’s, but summed only against $M^{\mathbf{n}}$’s. In the $\Omega$-basis, we can use (31) to similarly reduce to (42). However, we have to be careful to take into account that in general the coefficients $A^{\mathbf{l}}$ will be complex numbers and also have to be complex conjugated along the way. For a single complex $A^{\mathbf{l}}$ with $G_{3}=A^{\mathbf{l}}\Omega_{\mathbf{l}}$ we have

$$\frac{\partial}{\partial\tau}\frac{\partial}{\partial t^{\mathbf{k}_{1}}}\frac{\partial}{\partial t^{\mathbf{k}_{2}}}\ldots\frac{\partial}{\partial t^{\mathbf{k}_{r}}}W=\frac{1}{\tau-\bar{\tau}}\int(A^{\mathbf{l}}\Omega_{\bf{l}}-\bar{A}^{{\mathbf{l}}}\Omega_{\bf{\bar{l}}})\wedge\frac{\partial}{\partial t^{\mathbf{k}_{1}}}\frac{\partial}{\partial t^{\mathbf{k}_{2}}}\ldots\frac{\partial}{\partial t^{\mathbf{k}_{r}}}\Omega\,.$$

(46)

Using the result (43), this becomes

$$\frac{\partial}{\partial\tau}\frac{\partial}{\partial t^{\mathbf{k}_{1}}}\frac{\partial}{\partial t^{\mathbf{k}_{2}}}\ldots\frac{\partial}{\partial t^{\mathbf{k}_{r}}}W\bigg{|}_{t^{\mathbf{k}}=0,\tau=\tau_{0}}=\begin{cases}{\rm i}A^{\mathbf{l}}\frac{(\sqrt{-3})^{-9}}{2\mathop{\rm Im}\nolimits(\tau_{0})}\,\prod_{i=1}^{9}\Gamma\bigl{(}\frac{L_{i}}{3}\bigr{)}&\text{for $\bar{\mathbf{l}}={\mathbf{L}}\bmod 3$}\\ {\rm i}\bar{A}^{{\mathbf{l}}}\frac{(\sqrt{-3})^{-9}}{2\mathop{\rm Im}\nolimits(\tau_{0})}\,\prod_{i=1}^{9}\Gamma\bigl{(}\frac{L_{i}}{3}\bigr{)}&\text{for ${\mathbf{l}}={\mathbf{L}}\bmod 3$}\\ 0&\text{ otherwise}\end{cases}\,,$$

(47)

where we defined $\tau_{0}$ to be the vacuum expectation value of the axio-dilaton. Again, this can be evaluated quite speedily on a computer using only modular arithmetic. Note however that the contributions from ${\mathbf{l}}={\mathbf{L}}$ are proportional to $\Gamma\bigl{(}\frac{1}{3}\bigr{)}^{6}\Gamma\bigl{(}\frac{2}{3}\bigr{)}^{3}$ and are not rationally related to those from $\bar{\mathbf{l}}={\mathbf{L}}$.

By combining all of the above, the exact superpotential for a generic flux $G_{3}=\sum_{{\mathbf{l}}}A^{{\mathbf{l}}}\Omega_{\bf{l}}$ with complex prefactors $A^{{\mathbf{l}}}$ becomes

	$\displaystyle W=-\bigl{(}\sqrt{-3}\bigr{)}^{-9}\sum_{\bf{l}}\sum_{r=1}^{\infty}$	$\displaystyle\frac{1}{r!}\left(\sum_{\{t^{\mathbf{k}_{\alpha}}\}\text{ with }\mathbf{L}=\mathbf{\bar{l}}}\prod_{i=1}^{9}\Gamma\Bigl{(}\frac{L_{i}}{3}\Bigr{)}t^{\mathbf{k}_{1}}t^{\mathbf{k}_{2}}\ldots t^{\mathbf{k}_{r}}\,A^{\bf{l}}\left(1-{\rm i}\frac{\tau-\tau_{0}}{2\mathop{\rm Im}\nolimits(\tau_{0})}\right)\right.$		(48)
		$\displaystyle\left.-{\rm i}\sum_{\{t^{\mathbf{k}_{\alpha}}\}\text{ with }\mathbf{L}=\mathbf{l}}\prod_{i=1}^{9}\Gamma\Bigl{(}\frac{L_{i}}{3}\Bigr{)}t^{\mathbf{k}_{1}}t^{\mathbf{k}_{2}}\ldots t^{\mathbf{k}_{r}}\,\bar{A^{\bf{l}}}\,\frac{\tau-\tau_{0}}{{2\mathop{\rm Im}\nolimits(\tau_{0})}}\right)\,.$		(49)

For the purposes of moduli stabilization, this function has to be restricted to the orientifold fixed locus $t^{\mathbf{k}}=t^{\sigma({\mathbf{k}})}$. Following Becker:2023rqi , we do this in practice by ordering the ${\mathbf{k}}$’s alphabetically and dropping orientifold repetitions. We identify

$$t^{I}=t^{{\mathbf{k}}_{I}}=t^{\sigma({\mathbf{k}}_{I})}$$

(50)

with $I\in\{1,\ldots,63\}$ and include the axio-dilaton via

$$t^{0}=\tau-\tau_{0}\,.$$

(51)

This gives us finally a flux-dependent and highly transcendental function of $64$ variables whose critical behaviour at the origin is the subject of the following sections.

We will now write out these conditions in terms of the cohomology basis reviewed in section 2. In order to satisfy flux quantization, the coefficients with respect to the integral basis $\gamma_{\mathbf{n}}$ must be integral periods of the torus with complex structure $\tau$. In order to be supersymmetric, $G_{3}$ should be purely of Hodge type $(2,1)$. In the dual expansions

$$G_{3}=\sum_{\mathbf{n}}\left(N^{\mathbf{n}}-\tau M^{\mathbf{n}}\right)\gamma_{\mathbf{n}}=\sum_{\mathbf{l}}A^{\mathbf{l}}\Omega_{\mathbf{l}}\,.$$

(54)

the $N^{\mathbf{n}}$, $M^{\mathbf{n}}$ are integer, and the $A^{\mathbf{l}}$ are zero except when $\sum l_{i}=12$. The $\gamma_{\mathbf{n}}$ and $\Omega_{\mathbf{l}}$ are related by (29), (25). If this relation (the “period matrix” of the Landau-Ginzburg model) were completely generic, these conditions would have no non-trivial solution at all. In the situation at hand, in which all period coefficients are integral linear combinations of $\omega=e^{\frac{2\pi{\rm i}}{3}}$ and $\omega^{2}$, there are very many. More precisely, as observed in Becker:2006ks , there are still no solutions unless the axio-dilaton is of the form

$$\tau=\frac{a\omega+b}{c\omega+d}\,,$$

(55)

with integer $a$, $b$, $c$, $d$. This is easiest to see by writing the second condition in (54) as $\braket{G_{3}|\Omega_{\mathbf{l}}}=0$ unless $\sum l_{i}=15$. For simplicity, we will restrict to $\tau=\omega$. For this choice, it follows from (27) that one may set all but one $A^{\mathbf{l}}$ in (54) to zero. Namely, for any ${\mathbf{l}}$ with $\sum l_{i}=12$,

$$G_{({\mathbf{l}})}=27(\omega-\omega^{2})\Omega_{{\mathbf{l}}}$$

(56)

(but no smaller multiple of $\Omega_{{\mathbf{l}}}$) is an integral flux of type $(2,1)$. Here, and from now on, we will replace the subscript ‘3’ on $G$ with labels for various explicit solutions. We will indicate their physical characteristics by a superscript as they become available. We observe that $G_{({\mathbf{l}})}$ and $\omega G_{({\mathbf{l}})}$ are linearly independent over the integers (in fact, the reals). When $l_{1}=l_{2}=1$ or $l_{1}=l_{2}=2$, the flux $G_{({\mathbf{l}})}$ is invariant under the orientifold (36). Its contribution to the D3-brane tadpole is given by (37) in terms of its length (30),

$$\frac{1}{\omega-\omega^{2}}\braket{\overline{G}_{({\mathbf{l}})}|G_{({\mathbf{l}})}}=27\,.$$

(57)

When $l_{1}\neq l_{2}$, we need to add the respective orientifold image. The tadpole contribution doubles. In total, we obtain $126$ linearly independent primitive integral flux vectors

$$\begin{split}G_{({\mathbf{l}},1)}^{[1,27]}=27(\omega-\omega^{2})\Omega_{{\mathbf{l}}}&\qquad G_{({\mathbf{l}},2)}^{[1,27]}=27(\omega^{2}-1)\Omega_{{\mathbf{l}}}\quad\bigl{(}l_{1}=l_{2}\bigr{)}\\ G_{({\mathbf{l}},1)}^{[2,54]}=27(\omega-\omega^{2})\bigl{(}\Omega_{{\mathbf{l}}}+\Omega_{\sigma({\mathbf{l}})}\bigr{)}&\qquad G_{({\mathbf{l}},2)}^{[2,54]}=27(\omega^{2}-1)\bigl{(}\Omega_{{\mathbf{l}}}+\Omega_{\sigma({\mathbf{l}})}\bigr{)}\quad\bigl{(}l_{1}\neq l_{2}\bigr{)}\end{split}$$

(58)

The first entry in the superscript square brackets gives the number of non-zero components in the $\Omega$-basis, and the second, the tadpole contribution. We also use an additional subscript to label different flux choices with the same square bracket superscripts. As a result, the supersymmetric flux lattice in fact has full maximal rank.

All the fluxes in (58) have a tadpole in excess of the orientifold charge (equal to $12$, see (37)). Fluxes with smaller tadpole can be constructed by taking suitable linear (but non-integral!) combinations of (58). For example, one may verify that the flux

$$\begin{split}G^{[2,18]}_{(1)}&=27\bigl{(}\Omega_{1,1,1,1,1,1,2,2,2}-\Omega_{2,2,1,1,1,1,1,1,2}\bigr{)}\\ &=-\frac{1}{3}\bigl{(}G^{[1,27]}_{({\mathbf{l}}_{1},1)}+2G^{[1,27]}_{({\mathbf{l}}_{1},2)}+G^{[1,27]}_{({\mathbf{l}}_{2},1)}+2G^{[1,27]}_{({\mathbf{l}}_{2},2)}\bigr{)}\end{split}$$

(59)

where ${\mathbf{l}}_{1}=(1,1,1,1,1,1,2,2,2)$ and ${\mathbf{l}}_{1}=(2,2,1,1,1,1,1,1,2)$, is integral for $\tau=\omega$ and has tadpole $18$ as indicated. The flux

$$G^{[4,12]}_{(1)}=9(\omega-\omega^{2})\bigl{(}-\Omega_{{1,1,1,1,1,1,2,2,2}}+\Omega_{1,1,1,1,2,1,2,2,1}+\Omega_{1,1,2,2,1,1,1,1,2}-\Omega_{1,1,2,2,2,1,1,1,1}\bigr{)}$$

(60)

has tadpole $12$, and

$$\begin{split}G^{[8,8]}_{(1)}&=9\bigl{(}-\Omega_{1,1,1,2,1,2,1,2,1}+\Omega_{1,1,1,2,1,2,1,1,2}+\Omega_{1,1,1,2,1,1,2,2,1}-\Omega_{1,1,1,2,1,1,2,1,2}\\ &\qquad+\Omega_{1,1,1,1,2,2,1,2,1}-\Omega_{1,1,1,1,2,2,1,1,2}-\Omega_{1,1,1,1,2,1,2,2,1}+\Omega_{1,1,1,1,2,1,2,1,2}\bigr{)}\end{split}$$

(61)

which was first found in Becker:2006ks , has tadpole $8$. The latter two fluxes can hence be used to construct $\mathcal{N}=1$ supersymmetric Minkowski vacua.

To describe the full set of physical flux configurations (in particular, to enumerate integral flux vectors of tadpole $\leq 12$), it is important to first find an integral basis of the supersymmetric flux lattice consisting of vectors of smallest length possible.⁵⁵5An integral basis of a lattice $\Lambda$ is a basis of the vector space $\Lambda\otimes\mathbb{Q}$ with respect to which any lattice vector has integral coefficients. Eq. (58) are not an integral basis, because (for example) it does not contain (59) in its $\mathbb{Z}$-span. Finding lattice vectors of small(est) length in high-dimensional lattices is famously a very hard computational problem. This problem was tackled in Becker:2023rqi , and solved in a two-step process. First, a lucky coincidence that we describe momentarily yields an integral basis containing individual vectors of possibly rather large length. Second, by some judicious computational efforts, one transforms this into another integral basis with smaller lengths. We do not know whether the result is optimal.

The number of $\Omega_{\mathbf{l}}$’s of Hodge-type $(2,1)$ is $63$, and the corresponding complex coefficients $A^{\mathbf{l}}$ in equation (54) parameterize the flux. This amounts to $126$ real parameters, which we assemble in a vector of $\mathbb{R}^{126}$. The flux quantization conditions (35), explicitly

$$\braket{\gamma_{\mathbf{n}}|G_{3}}=\frac{1}{3^{8}}\sum_{\mathbf{l}}A^{\mathbf{l}}\omega^{{\mathbf{n}}.{\mathbf{l}}}\prod_{i=1}^{9}(1-\omega^{l_{i}})=N_{\mathbf{n}}-\tau M_{\mathbf{n}}~,$$

(62)

are linear, complex constraints between the $A^{\mathbf{l}}$ and the $340$ integers $N_{\mathbf{n}}$, $M_{\mathbf{n}}$. (Actually, only $2\times 128=256$ of these are independent because of the orientifold.) Separating real and imaginary parts, and viewing the set $\cup_{\mathbf{n}}\{N_{\mathbf{n}},M_{\mathbf{n}}\}=\{N_{n}:n=1(1)340\}$ as coordinatizing integral points $\mathbb{Z}^{340}\subset\mathbb{R}^{340}$, we can recast (62) in terms of a real linear map⁶⁶6It is to be noted that equation (63) are not (real and imaginary parts of) equation (35) on the nose, but a linear transform of it by an invertible $340\times 340$ matrix. $\mathbf{B}\in\mathbb{R}^{340\times 126}$ from $\mathbb{R}^{126}$ to $\mathbb{R}^{340}$ as

$$\sum_{{\mathbf{l}}}\mathbf{B}_{n{\mathbf{l}}}A^{\mathbf{l}}=N_{n}\,.$$

(63)

According to (58), this map hits a lattice of rank $126$ inside $\mathbb{Z}^{340}\subset\mathbb{R}^{340}$. In particular, the matrix $\mathbf{B}$ has full rank $126$. (This is true on general grounds.) We can pick $126$ $\mathbb{R}$-linearly independent rows from this system. We term the $N_{n}$’s in the corresponding rows independent flux quantum numbers, and denote them $\{y_{i}:i=1(1)126\}$. One can then solve the system

$$\sum_{{\mathbf{l}}}\mathbf{B}_{i{\mathbf{l}}}A^{\mathbf{l}}=y_{i}~$$

(64)

to obtain the $A^{\mathbf{l}}$ as linear functions of $y_{i}$: $A^{\mathbf{l}}=A^{\mathbf{l}}(y_{1},\ldots,y_{126})$. Having done this, it is still a non-trivial demand that the remaining $340-126=214$ flux numbers are integral. Luckily, this in fact is true, as the linearly dependent equations in (63) are $\mathbb{Z}$-linear combinations of the independent ones. This means that the columns of $[\mathbf{B}_{i{\mathbf{l}}}]^{-1}$ are an integral basis of the supersymmetric flux lattice Becker:2023rqi . Many of the elements in this basis have large tadpole values. One would like to swap them for fluxes of smaller length, such as (60), (61) and those presented below. In Becker:2023rqi , it was shown that this can be done via a convenient ${\it SL}(126,\mathbb{Z})$ transformation. This guarantees that the result is still an integral basis. See appendix B of Becker:2023rqi for the explicit list.

Having described the rank and an integral basis, we now turn to the problem of finding the finite set of vectors satisfying the tadpole cancellation condition within the infinite lattice. For a generic flux $G_{3}=\sum_{\mathbf{l}}A^{\mathbf{l}}\Omega_{\mathbf{l}}$, the contribution made by each summand to the flux-tadpole is determined completely by its coefficient $A^{\mathbf{l}}$ because of equations (30), (31). Doing this in practice, one finds Becker:2022hse ; Becker:2023rqi that the contribution to the flux tadpole from each turned-on $\Omega_{\mathbf{l}}$ is a homogeneous quadratic in the $y_{i}$ with positive integer coefficients and hence positive integer-valued. Therefore, to catalogue all physical solutions with $N_{\rm flux}\leq 12$, we only need to turn on at most $12$ of the $\Omega_{\mathbf{l}}$’s. This process has been initiated in Becker:2023rqi where the search for physical solutions was organized by the number of $\Omega_{\mathbf{l}}$’s turned on. In the remainder of this section, we summarize some of these results of Becker:2023rqi to get a sense of the flux vectors satisfying the tadpole constraint, and their simplest physical characteristics. We also provide some additional details on the classification of solutions in this model.

By construction, the superpotential (48) computed in subsection 2.3

$$W=\int G_{3}\wedge\Omega=W(t^{I})$$

(65)

as a function of the $64$ moduli remaining after the orientifold and its first derivatives $\partial_{I}W$ vanish at the origin $t^{I}=0$, for any $G_{3}$ in the supersymmetric flux lattice described in the previous subsection. The simplest non-trivial physical invariant is the Hessian,

$$M_{IJ}=\partial_{I}\partial_{J}W\,.$$

(66)

We think of it as the “holomorphic mass matrix”. As shown in Bardzell:2022jfh ; Becker:2022hse , its rank gives the number of moduli that are rendered massive by turning on the flux. This is $n_{\rm stab}$ appearing in the stronger version (2) of the tadpole conjecture. The result is interesting already for the simplest fluxes listed in (58) (which mind you are non-physical because their tadpole is too large). For the “$1$-$\Omega$” fluxes with tadpole $27$, it turns out that when ${\mathbf{l}}$ has $l_{1}=l_{2}=1$, the rank of $M_{IJ}$ is $16$. When $l_{1}=l_{2}=2$, it is $22$. For the “$2$-$\Omega$” fluxes, i.e., $l_{1}\neq l_{2}$ (tadpole $54$), it is also $22$. For the record, under the $S_{7}$ symmetry group, the $1$-$\Omega$ solutions $G^{[1,27]}_{({\mathbf{l}},1)}$, $G^{[1,27]}_{({\mathbf{l}},2)}$ organize into $6$ distinct orbits, and the $2$-$\Omega$ solutions, in three.

We then proceed by increasing the number of non-zero coefficients in $G_{3}=\sum_{\mathbf{l}}A^{\mathbf{l}}\Omega_{\mathbf{l}}$. It was found in Becker:2023rqi not to be possible to satisfy the tadpole constraint (37) with $2$- or $3$-$\Omega$ fluxes, so we skip the details such as minimum-$N_{\rm flux}$ solutions of these types and the ranks of the corresponding mass matrices. The interested reader may consult Becker:2023rqi . With four $\Omega_{\mathbf{l}}$’s one can produce physical fluxes satisfying (37). The smallest value of $N_{\rm flux}$ in this class is $12$, and is attained by precisely $54$ distinct $S_{7}$ orbits of solutions. Representatives from these orbits are given in equations (67), (68), (69).

	$\displaystyle G_{3}$	$\displaystyle=\left(a_{1}\Omega_{1,1,1,1,1,1,2,2,2}+a_{2}\Omega_{1,1,1,1,1,2,1,2,2}+a_{3}\Omega_{1,1,1,1,2,1,2,1,2}+a_{4}\Omega_{1,1,1,1,2,2,1,1,2}\right)$		(67a)
		$\displaystyle{\rm with}~~~(a_{1},\ldots,a_{4})=9(\omega-\omega^{2})\omega^{p}\left\{\begin{array}[]{ll}(-1,1,1,-1)~,\\ (1,-\omega,-1,\omega)~,\\ (1,-\omega,-\omega,\omega^{2})~,\\ (-1,\omega,\omega,-\omega^{2})~,\end{array}\right.~~~p=0,1,2$		(67f)

all of which have $16$ massive moduli. The solution $G^{[4,12]}_{(1)}$ given in equations (90) and (106) in the next section belongs to the $S_{7}$ orbit of the first of these with $p=0$.

	$\displaystyle G_{3}$	$\displaystyle=\left(a_{1}\Omega_{1,1,1,1,1,1,2,2,2}+a_{2}\Omega_{1,1,1,1,1,2,1,2,2}+a_{3}\Omega_{1,1,1,2,2,1,2,1,1}+a_{4}\Omega_{1,1,1,2,2,2,1,1,1}\right)$		(68a)
		$\displaystyle{\rm with}~~~(a_{1},\ldots,a_{4})=9(\omega-\omega^{2})\omega^{p}\left\{\begin{array}[]{ll}(-1,1,1,-1)~,\\ (-1,1,\omega,-\omega)~,\\ (-1,\omega,1,-\omega)~,\\ (-1,\omega,\omega,-\omega^{2})~,\\ (1,-\omega,-\omega,\omega^{2})~,\end{array}\right.~~~p=0,1,2$		(68g)

all of which have $22$ massive moduli. The solution $G^{[4,12]}_{(2)}$ given in (104) belongs to the $S_{7}$ orbit of the first of these with $p=0$.

	$\displaystyle G_{3}$	$\displaystyle=\left(a_{1}\Omega_{1,1,1,1,1,1,2,2,2}+a_{2}\Omega_{1,1,1,1,1,2,1,2,2}+a_{3}\Omega_{2,2,1,1,1,1,2,1,1}+a_{4}\Omega_{2,2,1,1,1,2,1,1,1}\right)$		(69a)
		$\displaystyle{\rm with}~~~(a_{1},\ldots,a_{4})=9(\omega-\omega^{2})\omega^{p}\left\{\begin{array}[]{ll}(-1,1,1,-1)~,\\ (1,-1,-\omega^{2},\omega^{2})~,\\ (-1,1,\omega,-\omega)~,\\ (1,-\omega,-1,\omega)~,\\ (-1,\omega,1,-\omega)~,\\ (1,-\omega,-\omega^{2},1)~,\\ (-1,\omega,\omega^{2},-1)~,\\ (-1,\omega,\omega,-\omega^{2})~,\\ (1,-\omega,-\omega,\omega^{2})~,\end{array}\right.~~~p=0,1,2~,$		(69k)

all of which have $26$ massive moduli. The solution $G^{[4,12]}_{(3)}$ given in (105) belongs to the $S_{7}$ orbit of the first of these with $p=0$.