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Algorithmically Solving the Tadpole Problem

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Abstract

The extensive computer-aided search applied in Bena et al. (The tadpole problem, 2020) to find the minimal charge sourced by the fluxes that stabilize all the (flux-stabilizable) moduli of a smooth K3 \(\times \) K3 compactification uses differential evolutionary algorithms supplemented by local searches. We present these algorithms in detail and show that they can also solve our minimization problem for other lattices. Our results support the Tadpole Conjecture: The minimal charge grows linearly with the dimension of the lattice and, for K3 \(\times \) K3, this charge is larger than allowed by tadpole cancellation. Even if we are faced with an NP-hard lattice-reduction problem at every step in the minimization process, we find that differential evolution is a good technique for identifying the regions of the landscape where the fluxes with the lowest tadpole can be found. We then design a “Spider Algorithm,” which is very efficient at exploring these regions and producing large numbers of minimal-tadpole configurations.

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Notes

  1. In Type IIB language, most of this negative contribution comes, in the large-tadpole limit, from D7 branes wrapping highly curved four-cycles on the base manifold.

  2. Remember that K3 has 22 homologically-different two-cycles.

  3. This condition is satisfied if there are no integer eigenvectors of norm minus two orthogonal to the eigenvectors of positive norm (see details in Sect. 2).

  4. An NP-class problem is one in which a candidate solution can be verified in polynomial time, thus our problem appears to be beyond this class. For a more detailed discussion of computational complexity in the context of the string theory landscape see also [43, 56].

  5. A false-positive means that the algorithm failed to find the shortest vector, and when that happens we cannot exclude the corresponding flux matrix during the search (only post-search). The rate at which we can verify matrices post-search and eliminate false-positives depends on the hardware; on our table-top machines we could verify about 1000 matrices per second.

  6. Note that some of these matrices are related to one another by the Weyl group of automorphisms of the K3 lattice. One could reduce the search space by modding out by this discrete group. We have not implemented this here.

  7. When referring to eigenvalues/eigenvectors of \(g {\tilde{g}}\) and \({\tilde{g}} g\), we always implicitly consider eigenvalues and eigenvectors of the induced maps \(g {\tilde{g}}, {\tilde{g}} g :\Lambda \otimes \mathbb {R} \rightarrow \Lambda \otimes \mathbb {R}\). Generically there will be no integer eigenvalues or eigenvectors in \(\Lambda \).

  8. There is a further subtlety if the multiplicity of \(p_r\) is larger than one. It is not hard to see that when this happens the second condition guarantees also that \(N_r\) can only contain an integer vector orthogonal to \(\Sigma \) if it is positive definite.

  9. The package BlackBoxOptim.jl supports only populations consisting of vectors but not of matrices. We therefore encode the \(D \times D\) matrices as vectors of length \(D^2\) by concatenating their columns.

  10. There are two design paths one can take:

    1. The DE can act on real-valued matrix entries in \(\mathbb {R}^{D\times D}\), and the fitness function \(\texttt {fitness}: \mathbb {R}^{D\times D} \mapsto \mathbb {R}^+\) can be used to enforce the fact that the input consists of real numbers

    2. The DE can act on integer-valued matrix entries in \(\mathbb {Z}^{D\times D}\) with fitness function \(\texttt {fitness}: \mathbb {Z}^{D\times D} \mapsto \mathbb {R}^+\).

    The first option, which is the path we chose, contains more freedom for the algorithm to move. Consider the mutation (see Eq. 24) of one entry when the DE acts on \(\mathbb {R}^{D\times D}\): \(0.4 \approx 0 \rightarrow 0.4 + 1*(0.9-0.7) = 0.6 \approx 1\), This mutation managed to switch the effective entry from 0 to 1, while the same DE acting on \(\mathbb {Z}^{D\times D}\) with these numbers rounded, \(0 \rightarrow 0 + 1*(1-1) = 0\), provides no evolution.

  11. This filtering is a Diophantine problem, which oftentimes (though not always) can be solved by Mathematica. This can only be used for post-search verification, as the algorithms Mathematica uses are too slow for running our search.

  12. The algebraic multiplicity denotes the multiplicity of an eigenvalues as the root of the characteristic polynomial. The geometric multiplicity is the dimension of the corresponding eigenspace. If the geometric multiplicity is smaller than the algebraic multiplicity for at least one eigenvalue, a matrix is not diagonalizable.

  13. We will gloss over the details of the defect, which is a central part of the LLL algorithm, since we will not make use of that feature.

  14. The asterisk refers to the corresponding vector after a Gram-Schmidt procedure (no normalization), and the Gram-Schmidt procedure is carried out at every redefinition of \(v_k\).

  15. We use Nemo.jl’s [49] nullspace_right_rational-function that can return the nullspace as integer vectors.

  16. As mentioned above, the LLL algorithm is not guaranteed to always converge to the shortest vector for a given lattice. Therefore, it sometimes misses existing norm-minus-two vectors.

  17. The implementation we have used does this every 10 samplings.

  18. As described in Sect. 3, we use adaptive DE/rand/1/bin with radius limited sampling for our DE searches, from BlackBoxOptim.jl [48]. The same implementation also includes a random sampling method that we use as a benchmark here.

  19. For actual compactifications, such as K3 \(\times \) K3, this matrix corresponds to the intersection matrix of the middle cohomology.

  20. This probability density could roughly be estimated in the following way: The number of flux configurations with a certain charge Q is given by the number of lattice points in a shell of radius Q and thickness 1. In the continuum limit, for \(Q \gg 1\), the number of points is proportional to the volume of the shell. Hence, for a spherical shell this would imply the naïve estimate \(\rho (Q) \sim Q^{D^2}\). This, however, does not take into account that we are dealing with a space with non-definite inner product. Moreover, our constraints on moduli stabilization and absence of singularities further reduce the number of admissible matrices. Finally, this simple counting argument breaks down for small values of Q, when the effects of flux quantization become important. We thank D. Junghans for interesting discussions on this argument.

  21. We thank Cumrun Vafa for suggesting this.

  22. We would like to thank one of the referees for this suggestion.

  23. If we do not use rounding, our algorithm calculates all combinations of rounding to largest/smallest integer, to get the best possible solution. For large matrices with many vectors in the kernel this however becomes too slow.

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Acknowledgements

We would like to thank Andreas Braun, Thomas Grimm, Jim Halverson, Arthur Hebecker, Lena Heijkenskjöld, Daniel Junghans, Daniel Mayerson, Ruben Monten, Jakob Moritz and Cumrun Vafa for interesting discussions. The work of I.B., M.G. and S.L. was supported in part by the ANR grant Black-dS-String ANR-16-CE31-0004-01, by the ERC Grants 772408 “Stringlandscape” and 787320 “QBH Structure”, by the John Templeton Foundation grant 61149 and by the NSF Grant PHY-1915071, PHY-1748958 and PHY-1607611. The work of J.B. was supported by the MIUR-PRIN contract 2015MP2CX4002 “Non-perturbative aspects of gauge theories and strings”. We are grateful to all contributors of the open-source projects that made this work possible, which includes: Jupyter notebooks [70], Julia [14] and packages BlackboxOptim.jl [48], Nemo.jl [49], Polynomials.jl, GenericLinearAlgebra.jl, Plots.jl, Measures.jl, StatsBase.jl, UnicodePlots.jl, BenchmarkTools.jl, ArgParse.jl, ProgressMeter.jl, and all of their dependencies.

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Correspondence to Severin Lüst.

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This article is part of the Topical Collection on Machine-Learning Mathematical Structures edited by Yang-Hui He, Pierre Dechant, Alexander Kasprzyk, and Andre Lukas.

Appendices

Appendix A: Integer Linear Combinations of Eigenvectors

In this Appendix we discuss under which conditions a subset of the eigenvectors of an integer matrix has an integer vector in its span.

In Sect. 2.1 we explained that the K3 \(\times \) K3 compactifications develop an orbifold singularity if one of the integer matrices \(g {\tilde{g}}\) and \({\tilde{g}} g\) with g and \({\tilde{g}}\) defined in (18) has the following property: There is a root of the K3 lattice (11) which is orthogonal to all eigenvectors with positive norm square. A necessary condition for this is that there is a linear combination of the eigenvectors with negative norm square which is integer (see the discussion in Sect. 4 for more details). In this Appendix we will outline the underlying mathematics of this question.

Let \(M \in \mathbb {Z}^{D \times D}\) be a diagonalizable \(D \times D\) matrix with integer entries and \(\lambda _i\) and \(v_i\) (\(i = 1, \dots , D\)) its eigenvalues and the corresponding eigenvectors. In general neither the eigenvalues nor the eigenvectors are integer. We want to know under which circumstances there exists a subset \(\{v_k\, |\, k = 1, \dots , p < D\}\) of the eigenvectors such that there exists a linear combination that has only integer entries:

$$\begin{aligned} \sum _{i=1}^p a^i v_i \in \mathbb {Z^D},\qquad a^i \in \mathbb {C}. \end{aligned}$$
(37)

The two limits of this questions are easy to understand. For \(p = D\) Eq. (37) is always true since \(\mathbb {Z}^D\) is contained in the (complex) span of all eigenvectors, which is \(\mathbb {C}^D\). In the opposite limit, \(p = 1\), Eq. (37) is also true if there exists an eigenvalue \(\lambda \in \mathbb {Z}\); its corresponding eigenvector, v, is given by the solution of

$$\begin{aligned} \left( M - \lambda \mathbb {1}\right) v = 0. \end{aligned}$$
(38)

This is a linear equation with integer coefficients and admits therefore an integer solution. We would now like to understand whether there exist solutions to this problem away from these limits.

The eigenvalues \(\lambda _i\) are the D (complex) roots of the characteristic polynomial

$$\begin{aligned} p(\lambda ) = \det \left( M - \lambda {\mathbb {1}}\right) = \lambda ^D + c_{d-1} \lambda ^{D-1} + \dots + c_0, \end{aligned}$$
(39)

where the coefficients, \(c_i\), are integers. This polynomial is not always irreducible over the integers. When it is not, we can decompose it into a product of smaller polynomials with integer coefficients:

$$\begin{aligned} p(\lambda ) = \prod _r p_r(\lambda ), \end{aligned}$$
(40)

where each of the factors is irreducible over the integers. This means that the polynomials

$$\begin{aligned} p_r(\lambda ) = \lambda ^{d_r} + c^{(r)}_{d_r - 1} \lambda ^{d_r -1} + \dots + c^{(r)}_{0},\qquad c^{(r)}_i \in \mathbb {Z} \end{aligned}$$
(41)

cannot be further decomposed into a product of non-constant integer polynomials of smaller rank. Here we denote the degree of \(p_r(\lambda )\) by \(d_r < D\).

Let us now focus on one of these polynomials, \(p_r(\lambda )\). In principle, this polynomial can occur multiple times in the factorization (40), so we denote its multiplicity by \(m_r\). Furthermore, \(p_r(\lambda )\) has \(d_r\) complex roots \(\lambda ^{(r)}_i\). By construction these roots are eigenvalues of M, so we denote the corresponding eigenvectors by \(v^{(r)}_i\). Notice, that if \(m_r > 1\) there are more eigenvectors than eigenvalues, since for each eigenvalue, \(\lambda ^{(r)}_i\), there are \(m_r\) independent eigenvectors. Next, we consider the matrix

$$\begin{aligned} p_r(M) = M^{d_r} + c^{(r)}_{d_r - 1} M^{d_r -1} + \dots + c^{(r)}_{0} {\mathbb {1}}. \end{aligned}$$
(42)

For every eigenvector, \(v^{(r)}_i\), it satisfies

$$\begin{aligned} p_r(M) v^{(r)}_i = p_r(\lambda ^{(r)}_i) \, v^{(r)}_i = 0. \end{aligned}$$
(43)

Therefore, \(\dim \ker p_r(M) \ge m_r d_r\). This inequality can only hold for every r if it is saturated and we have

$$\begin{aligned} \ker p_r(M) = \mathrm {span} \left\{ v_i^{(r)}\,\bigl |\, i = 1, \dots , m_r d_r\right\} . \end{aligned}$$
(44)

On the other hand, \(p_r(M)\) is an integer matrix and thus its kernel admits an integer basis \(\{b^{(r)}_i,\, i = 1, \dots , m_r d_r\} \in \mathbb {Z}^{D \times m_r d_r}\). Therefore, there exist \(m_r p_r\) independent linear combinations

$$\begin{aligned} \sum _{i=1}^{m_r d_r} a^i v^{(r)}_i \in \mathbb {Z}^D. \end{aligned}$$
(45)

Notice that, if \(m_r > 1\), we can always redefine the eigenvectors in such a way that the sum contains only \(p_r\) non-vanishing terms. To conclude, for each irreducible factor \(p_r(\lambda )\) there exist \(p_r\) independent eigenvectors such that one can form \(d_r\) independent integer vectors from their linear combinations. We can finally express the matrix M with respect to the integer basis \(\bigcup _r \{b^{(r)}_i\}\) to block-diagonalize it over the integers. The characteristic polynomials of the individual blocks will be the irreducible polynomials \(p_r(\lambda )\).

Let us also look at the reverse situation and assume there is a subset of eigenvectors \(\{{\tilde{v}}_i\}\), \(i = 1, \dots , {\tilde{d}} < D\), such that there is a linear combination

$$\begin{aligned} b_1 = \sum _{i = 1}^{{\tilde{d}}} a^i {\tilde{v}}_i \in \mathbb {Z}^D. \end{aligned}$$
(46)

However, denoting the corresponding eigenvalues by \({\tilde{\lambda }}_i\), we can show iteratively that all

$$\begin{aligned} b_n \equiv \sum _{i = 1}^{{\tilde{d}}} a^i {\tilde{\lambda }}_i^n {\tilde{v}}_i = M b_{n-1} \in \mathbb {Z}^D. \end{aligned}$$
(47)

On the other hand, at most \({\tilde{d}}\) of these vectors can be linearly independent. Therefore, there exists a linear relation between the \(b_i\), \(i = 0, \dots , {\tilde{d}}\). This shows that the \({\tilde{\lambda }}_i\) are roots of an integer polynomial of degree \({\tilde{d}}\) and therefore \(p(\lambda )\) must be reducible. Alternatively, one can extend the \(b_i\) (\(i = 0, \dots , {\tilde{d}}\)) to an integer basis of \(\mathbb {C}^D\) and express M with respect to this basis. This brings M into block diagonal form which shows that \(p(\lambda )\) is reducible.

This shows that there exists an integer linear combination of \(p < D\) eigenvectors if and only if the characteristic polynomial of M is reducible and if in its factorization there is an irreducible factor of degree p.

Clearly, the \(p=1\) example mentioned above, where there is a solution if one of the eigenvalues is integer, can be seen as limit of our general result, obtained when the corresponding irreducible factor \(p_r(\lambda )\) has degree \(d_r = 1\).

The roots of any of the irreducible factors \(p_r\) form an algebraic extension of the rationals and we are thus dealing with linear algebra over these algebraic fields. The discussion here therefore seems to indicate that there are potentially very interesting connections of our problem to algebraic number theory, which should be elaborated on elsewhere.

Appendix B: Lattice Reduction

In this Appendix we present some details of the algorithms which we use to determine if there are integer norm-minus-two vectors (roots) in the kernel of an integer matrix. In B.1 we outline a simple algorithm which computes a primitive basis of the kernel of an integer matrix. In B.2 we discuss our adaptation of the LLL algorithm which tries to find the shortest vector in a given lattice.

1.1 B.1: Rational Reduction

Let \(M \in \mathbb {Z}^{D \times D}\) be an integer matrix and \(v_i \in \mathbb {Z}^D\) (\(i = 1, \dots , k\); \(k = \dim \ker M\)) a basis of the kernel of M. The vectors \(\{v_i\}\) are called a primitive basis (of the lattice \(\ker M \cap \mathbb {Z}^D\)) if every integer vector in the kernel of M can be expressed as a linear combination o the \(v_i\) with integer coefficients:

$$\begin{aligned} \mathrm {span}_\mathbb {Z} (v_1, \dots , v_k) = \ker M \cap \mathbb {Z}^D \equiv \ker _\mathbb {Z} M. \end{aligned}$$
(48)

Not every integer basis is necessarily primitive as it can be possible to obtain integer vectors as rational linear combinations of integer vectors.

Here, we give a simple algorithm which generates for a set of integer vectors, \(\{v_i\}\), a new, primitive basis of \(\mathrm {span}_\mathbb {Q} (\{v_i\}) \cap \mathbb {Z}^D\). We start by writing the \(v_i\) as the columns of a matrix V:

$$\begin{aligned} V \equiv \begin{pmatrix} \vert &{} &{} \vert \\ v_1 &{} \cdots &{} v_k \\ \vert &{} &{} \vert \\ \end{pmatrix}. \end{aligned}$$
(49)

The algorithm then works as follows:

  1. 1.

    Bring V into column Hermite normal form and normalize all columns by dividing them by the GCD of their entries. If V was already in this form skip to the next step. The matrix V is now in column echelon form.

  2. 2.

    Iterate over \(i = 1, \dots , k\): Introduce the reduced matrix

    $$\begin{aligned} V_i \equiv \begin{pmatrix} \vert &{} &{} \vert \\ v_i &{} \cdots &{} v_k \\ \vert &{} &{} \vert \\ \end{pmatrix}. \end{aligned}$$
    (50)

    Determine all non-zero elements of \(v_i\) such that all other entries in the same row of \(V_i\) are zero (\(v_{i,j} \ne 0\) and \(v_{m,j} = 0\) for \(m > i\)). Denote the GCD of these elements by Q and decompose Q into prime factors. For each prime factor p of Q do:

    1. (a)

      Compute a basis \(n_j \in \left( \mathbb {Z}_p\right) ^D\) of the kernel of \(V_i\) in \(\mathbb {Z}_p \equiv \mathbb {Z} / p \mathbb {Z}\). (This means each \(n_j\) satisfies \(V_i \, n_j = 0 \mod p\).)

    2. (b)

      Bring the matrix

      $$\begin{aligned} \begin{pmatrix} \vert &{} &{} \vert \\ n_1 &{} \cdots &{} n_l \\ \vert &{} &{} \vert \\ \end{pmatrix} \end{aligned}$$
      (51)

      into column Hermite normal form. Divide each \(n_j\) by its leading element (its first non-zero entry). These operations are done mod p.

    3. (c)

      For each \(n_j\), denote the position of its leading element by \(l_i\) (this means that \(n_{j, l_j} = 1\) and \(n_{j, m} = 0\) for \(m > l_j\)). Replace \(v_{i + l_j -1}\) by \(V_i \, n_j / p\).

1.2 B.2: LLL

Given a set of integer vectors, \(\{ v_k \}\), which span a lattice, the “shortest vector problem” asks what for the length of the shortest non-zero vector in the this lattice. The LLL algorithm [72] (outlined in 3.2) gives an approximate solution to this problem in polynomial time. Here, we use the following adaptation of LLL:

  1. 1.

    Iterating over all vectors \(v_k\) for \(k \in [1,n]\), replace \(v_k\) with a new vector

    $$\begin{aligned} v_k \leftarrow v_k + \lambda ^i v_i. \end{aligned}$$
    (52)

    where \(\lambda ^i\) is the nearest integer to the solution of

    $$\begin{aligned} d_{ij} \lambda ^j + c_i = 0;\quad d_{ij} = (v_i,v_j),\ c_i = (v_k,v_i), \end{aligned}$$
    (53)

    and \(i,j \in [1,n]\) except \(i,j = k\) and any l with vector \(v_l\) that makes \(d_{ij}\) singular. Notice that the left hand side of (53) is the derivative of

    $$\begin{aligned} \Vert v_k + \lambda ^i v_i \Vert ^2 = \Vert v_k\Vert ^2 + 2\lambda ^i c_i + \lambda ^i \lambda ^j d_{ij}, \end{aligned}$$
    (54)

    with respect to \(\lambda ^i\). Hence a solution of (53) extremizes the length of the new vector. Existence and uniqueness of \(\lambda ^i\) is guaranteed if \(d_{ij}\) is non-degenerate. If it is moreover positive (negative) definite, the extremum of (54) will be indeed a minimum (maximum).

  2. 2.

    For all vectors \(v_i\) with \(i \ne k\) that have \(||v_i|| = 0\) but \((v_i,v_k) \ne 0\) we replace \(v_k\) with

    $$\begin{aligned} v_k \leftarrow v_k + v_i \kappa _i;\quad \kappa _i = \frac{2-||v_k||^2}{2(v_k,v_i)}, \end{aligned}$$
    (55)

    for the \(v_i\) that leaves the new \(v_k\) closest to \(||v_k||^2 = 2\).

  3. 3.

    Repeat the above procedure until no vector in the collection has changed.

Both the \(\lambda _i\) and \(\kappa _i\) must be integer (or an appropriate rational) to make the result integer. While we have chosen rounding for most problems we have studied,Footnote 23 that is not always the one that gives the result closest to \(||v_k||^2 = 2\), but it performs better than using any of the floor/ceil functions.

If we compare our Item 1. above to the conventional LLL, we find that the two (disregarding rounding) solve the same analytical problem. Since we opted to solve Eq. (53) in the most intuitive way – by matrix inversions – LLL would probably have a computational complexity advantage to our implementation, even if the LLL performed more iterations finish. The two algorithms also differ in that LLL rounds at every iteration.

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Bena, I., Blåbäck, J., Graña, M. et al. Algorithmically Solving the Tadpole Problem. Adv. Appl. Clifford Algebras 32, 7 (2022). https://doi.org/10.1007/s00006-021-01189-6

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