Databases of quantum periods for Fano manifolds

Abstract

Fano manifolds are basic building blocks in geometry – they are, in a precise sense, atomic pieces of shapes. The classification of Fano manifolds is therefore an important problem in geometry, which has been open since the 1930s. One can think of this as building a Periodic Table for shapes. A recent breakthrough in Fano classification involves a technique from theoretical physics called Mirror Symmetry. From this perspective, a Fano manifold is encoded by a sequence of integers: the coefficients of a power series called the regularized quantum period. Progress to date has been hindered by the fact that quantum periods require specialist expertise to compute, and descriptions of known Fano manifolds and their regularized quantum periods are incomplete and scattered in the literature. We describe databases of regularized quantum periods for Fano manifolds in dimensions up to four. The databases in dimensions one, two, and three are complete; the database in dimension four will be updated as new four-dimensional Fano manifolds are discovered and new regularized quantum periods computed.

Measurement(s)	Regularized quantum period
Technology Type(s)	Computational algebra system • Mathematics

Background & Summary

Algebraic geometry describes geometric shapes as the solution sets of systems of polynomial equations, and lets us exchange geometric understanding of a shape X with structural understanding of the equations that define it. This geometric analysis has become vital in mathematics and physics, with applications as diverse as motion planning for robots^1,2, optimisation^3,4, algebraic statistics^5,6, coding^7,8, gauge theory^9,10, and string theory^11,12,13. The Minimal Model Program decomposes shapes defined by polynomial equations into basic pieces^14,15. After birational transformations, which are modifications on subsets with zero volume (and codimension at least one), any such shape decomposes into ‘atomic pieces’ of three types: positively curved, flat, and negatively curved. Fano manifolds are the positively curved smooth pieces. These are arguably the most important of the basic pieces, because analysis of basic pieces that are flat (Calabi–Yau) or negatively curved (general type) typically starts by realising them inside a Fano manifold by imposing additional equations. For example, four-dimensional Fano manifolds naturally contain three-dimensional Calabi–Yau manifolds, as their ‘anticanonical sections’, which are of particular importance in constructing models of spacetime in Type II string theory^16,17.

There are finitely many Fano manifolds, up to deformation, in each dimension¹⁸. There is exactly 1 one-dimensional Fano manifold, the Riemann sphere ${{\mathbb{P}}}^{1}$; this follows from the Klein–Poincaré Uniformization Theorem. The two-dimensional case is also classical¹⁹; there are 10 deformation families of two-dimensional Fano manifolds. The classification of three-dimensional Fano manifolds, due to Fano²⁰, Iskovskikh^21,22,23, and Mori–Mukai^24,25 is one of the triumphs of 20th-century algebraic geometry; there are 105 deformation families. The classification of Fano manifolds in dimensions four and higher is still far from understood.

Recently a new approach to the classification of Fano manifolds has been proposed, which relies on a conjectural link between Fano manifolds and Laurent polynomials provided by Mirror Symmetry²⁶. This has already led to new classification results²⁷ and the discovery of many new four-dimensional Fano manifolds^28,29,30. The key invariant to be analysed is the regularized quantum period of a Fano manifold X. This is a power series

$${\widehat{G}}_{X}(t)=1+\mathop{\sum }\limits_{d=2}^{\infty }{c}_{d}{t}^{d}$$

(1)

where c_d = r_d d! and r_d is a certain genus-zero Gromov–Witten invariant of X. Intuitively speaking, r_d is the number of degree-d rational curves in X that pass through a fixed generic point of X and have a certain constraint on their complex structure. In general r_d is a rational number, because curves with a symmetry group of order n are counted with weight 1/n, but in all known cases the coefficients of ${\widehat{G}}_{X}$ are integers. Gromov–Witten invariants remain constant under deformation of X, so ${\widehat{G}}_{X}$ is a deformation invariant of X.

This paper describes databases of regularized quantum periods of Fano manifolds in dimensions one, two, three, and four^31,32,33,34. The databases were prepared by aggregating and standardising existing descriptions of regularized quantum periods in the literature, and computing the regularized quantum periods of various four-dimensional Fano manifolds where geometric constructions were known. The databases in dimensions up to three are complete; the database in dimension four will be updated as further four-dimensional Fano manifolds are discovered and new regularized quantum periods computed. This data will be useful to mathematicians and physicists who wish to identify Fano manifolds, to construct Calabi–Yau manifolds, or to investigate further the connection between Fano manifolds and Mirror Symmetry.

Methods

Let X be a Fano manifold. General theory implies that there is a differential operator

$$L=\mathop{\sum }\limits_{k=1}^{N}{l}_{k}{t}^{{n}_{k}}{D}^{{m}_{k}}$$

(2)

where the l_k are non-zero rational numbers and $D=t\frac{d}{dt}$, such that $L{\widehat{G}}_{X}\equiv 0$; see e.g. [²⁶, Theorem 4.3]. We normalise the operator L in (2) by taking the l_k to be integers with no common factor and insisting that l_M >0, where M is the index k such that the pair (m_k, n_k) is lexicographically maximal. The identity $L{\widehat{G}}_{X}\equiv 0$ translates into a recurrence relation for the coefficients c_d in (1) which determines all of the c_d from the first few of them, and typically just from c₀ and c₁; here and henceforth we write c₀ = 1 and c₁ = 0.

Each entry in our databases describes the regularized quantum period of a Fano manifold. Two regularized quantum periods are assumed to be distinct unless it has been proven that they agree (i.e. that all the infinitely many coefficients c_d in (1) agree). For example, two period sequences are the same if they have been shown to have the same differential operator, or if the Fano manifolds they are derived from have been shown to be deformation equivalent. More precisely, an entry in one of the databases corresponds to a sequence c₀, c₁,… of coefficients of the regularized quantum period (1), where in some cases only finitely many terms c_d may be known, along with information about the geometric origins of this period sequence. In dimensions one, two, and three the classification of Fano manifolds is known, and each deformation class of Fano manifolds has a distinct regularized quantum period. Thus in dimensions one, two, and three we can also interpret each entry in our databases as corresponding to a deformation class of Fano manifolds. In dimension four the classification of Fano manifolds is unknown, and it is possible that there exist Fano manifolds X₁ and X₂ that have the same regularized quantum period but that are not deformation equivalent. (No such examples are known, in any dimension, and we expect that no such examples exist.) One should therefore take care to think of each entry in the four-dimensional database as representing a regularized quantum period sequence of a Fano manifold rather than a deformation class of Fano manifolds. It may happen that two period sequences in the four-dimensional database are later proven to be equal. When this happens, we will update the database as described in the ‘Data records’ section.

The following methods were used to compute regularized quantum periods and the differential operators that annihilate them.

Mirror symmetry and the Lairez algorithm

For X a smooth Fano toric variety or toric complete intersection, mirror constructions by Givental³⁵ and Hori–Vafa³⁶ give a Laurent polynomial f that corresponds to X under Mirror Symmetry – see e.g. ²⁸, for a summary of this. Given such a Laurent polynomial, one can compute a differential operator L such that $L{\widehat{G}}_{X}\equiv 0$ using Lairez’s generalised Griffiths–Dwork algorithm³⁷. These steps are implemented in the Fanosearch software library; see the ‘Code availability’ section.

Products

If the regularized quantum periods of Fano manifolds X₁ and X₂ are known then the regularized quantum period of the product X₁ × X₂ is determined by [³⁸, Corollary E.4]. Furthermore if X₁ and X₂ correspond under Mirror Symmetry to, respectively, the Laurent polynomials ${f}_{1}\in {\mathbb{C}}[{x}_{1}^{\pm 1},\ldots ,{x}_{k}^{\pm 1}]$  and ${f}_{2}\in {\mathbb{C}}[{y}_{1}^{\pm 1},\ldots ,{y}_{\ell }^{\pm 1}]$, then the product X₁ × X₂ corresponds under Mirror Symmetry to ${f}_{1}+{f}_{2}\in {\mathbb{C}}[{x}_{1}^{\pm 1},\ldots ,{x}_{k}^{\pm 1},\,{y}_{1}^{\pm 1},\ldots ,{y}_{\ell }^{\pm 1}]$. One can then use the Lairez algorithm to compute a differential operator L such that $L{\widehat{G}}_{{X}_{1}\times {X}_{2}}\equiv 0$.

Numerical linear algebra

Given sufficiently many coefficients c₀, …, c_M in (1), one can find a recurrence relation satisfied by this sequence using linear algebra. This determines a candidate for the differential operator L in (2). We applied this method to the Strangeway fourfolds, where closed formulas for the c_i are known³⁹. When the number M of coefficients involved is large, the linear system that determines the recurrence relation becomes highly overdetermined and so we can be confident that the operator L is correct.

A note on rigour

When computing differential operators (2), neither the approach based on the Lairez algorithm nor the approach based on numerical linear algebra provides a proof that $L{\widehat{G}}_{X}\equiv 0$. In the latter case this is because the method cannot do so; in the former case this is because of an implementation detail in Lairez’s algorithm: certain calculations over ${\mathbb{Q}}(t)$ are made using reductions to ${{\mathbb{F}}}_{p}(t)$ for randomly-chosen primes p followed by a reconstruction step, and there is a (very small) probability of erroneous reconstruction. In each case, however, the probability of error is tiny.

In more detail: the Lairez algorithm computes the differential operator L by first computing a certain connection matrix M with entries in ${\mathbb{Q}}(t)$. This matrix determines L uniquely. In the implementation of the algorithm that we used, the matrix M is reconstructed from its reductions M_i to ${{\mathbb{F}}}_{{p}_{i}}(t)$ for a sequence p₁, …, p_k of randomly-chosen 32-bit primes p_i. The entries of M are recovered from the entries of M_i using rational reconstruction⁴⁰, where we increase the number k of primes until the reconstruction stabilises. Let us consider a heuristic estimate of the probability of a single coefficient $q\in {\mathbb{Q}}$ in a single entry of the matrix M being erroneously reconstructed from its mod-p_i reductions. Set m = p₁ p₂⋯p_k−1. We have

$$q\equiv A\quad {\rm{mod}}\;m\quad \quad q\equiv B\quad {\rm{mod}}\;m{p}_{k}\quad \quad {\rm{for}}\;{\rm{some}}\;A,\;B$$

and since the reconstruction stabilises it follows that B ≡ A + rm mod mp_k for some integer r. There are p_k possible choices for B given A and (since p_k was chosen uniformly at random among 32-bit primes) it seems reasonable that, if the reconstruction at step k − 1 was erroneous, then all possibilities for B are equally likely. Thus the probability of erroneous stabilisation is 1/p_k. This is just one coefficient among many. Assuming that erroneous reconstructions of individual coefficients are independent, and using worst-case sizes for M and degrees of entries in M gives a probability of erroneous reconstruction on the order of 10⁻⁶. Furthermore the operators that we found satisfy a number of stringent checks described in the ‘Technical validation’ section. It is reasonable to conclude that they are correct.

Dimension one

The Fano manifold ${{\mathbb{P}}}^{1}$ is toric, and corresponds under Mirror Symmetry to the Laurent polynomial x + x⁻¹. The database of regularized quantum periods for one-dimensional Fano manifolds³¹, which contains one record, was constructed from this Laurent polynomial using Lairez’s algorithm. To cross-check, one can use Givental’s mirror theorem⁴¹ to compute the regularized quantum period, finding

$${\widehat{G}}_{{{\mathbb{P}}}^{1}}=\mathop{\sum }\limits_{d=0}^{\infty }{t}^{2d}\frac{(2d)!}{d!d!}$$

It is then elementary to check that $L{\widehat{G}}_{{{\mathbb{P}}}^{1}}\equiv 0$, where $L=(4{t}^{2}-1)D+4{t}^{2}$.

Dimension two

The database in dimension two³² was constructed by applying the Givental/Hori–Vafa mirror construction and the Lairez algorithm to the models of two-dimensional Fano manifolds as toric complete intersections given in [³⁸, §G].

Dimension three

Regularized quantum periods for three-dimensional Fano manifolds are known, as are Laurent polynomials that correspond to each three-dimensional Fano manifold under Mirror Symmetry³⁸. For 89 of the 105 deformation families, this correspondence follows from the Givental/Hori–Vafa construction. In the remaining cases, which are those in Table 1 of [³⁸, Appendix A] where ‘Method’ is equal to ‘Abelian/non-Abelian correspondence’ or ‘Quantum Lefschetz with mirror map’, the correspondence is conjectural but is supported by strong numerical evidence, including the computation of the first several hundred terms of the expansion (1). The database in dimension three³³ was constructed by applying Lairez’s algorithm to these Laurent polynomials.

Dimension four

The database of regularized quantum periods for four-dimensional Fano manifolds³⁴ was constructed as follows.

Four-dimensional Fano toric complete intersections

Four-dimensional Fano manifolds that are complete intersections in smooth Fano toric varieties of dimension up to 8 have been classified²⁸. The electronic supplementary material for that paper also provides the regularized quantum periods and differential operators (2) in machine-readable form. These differential operators were normalised as discussed after Eq. (2) and added to the database.

Four-dimensional Fano manifolds with classical constructions

Ref. ³⁹ computes regularized quantum period sequences and differential operators (2) for many four-dimensional Fano manifolds with classical constructions, including all four-dimensional Fano manifolds with Fano index greater than one. These differential operators were normalised according to our conventions and added to the database. That paper also gives regularized quantum period sequences, but not differential operators, for a number of other four-dimensional Fano manifolds:

• four-dimensional Fano toric varieties;

• products of lower-dimensional Fano manifolds;

• the Strangeway fourfolds.

Toric varieties are toric complete intersections, so four-dimensional Fano toric varieties were already included in the database. Differential operators for products of lower dimensional Fano manifolds were computed using the methods discussed in the section ‘Products’ above. Differential operators for the Strangeway fourfolds were computed as discussed in the section ‘Numerical linear algebra’ above. These differential operators were then normalised as discussed after Eq. (2) and added to the database.

Four-dimensional quiver flag zero loci

Four-dimensional Fano manifolds that are quiver flag zero loci in Fano quiver flag varieties of dimension up to 8 were classified by Kalashnikov²⁹, who also computed the coefficients c_d in (1) for these for d ≤ 15. Kalashnikov partitions the four-dimensional Fano quiver flag zero loci into equivalence classes depending on the values of the coefficients (c₀, …, c₁₅) and finds 749 equivalence classes. It is not known, however, whether quiver flag zero loci with the same (c₀, …, c₁₅) are deformation equivalent, so we regard the regularized quantum period sequence represented by (c₀, …, c₁₅) as coming from the four-dimensional Fano manifold specified as the representative of the equivalence class in [²⁹, Appendix B, Table 1]. These representatives were added to the database.

Table 1 The keys and values in each of the databases.

Data Records

We provide four key-value databases, containing regularized quantum periods for Fano manifolds in dimensions one, two, three, and four. These databases have been committed to the public domain using a CC0 license. They are available from the open access data repository Zenodo^31,32,33,34. Zenodo provides versioned DOIs, and new versions of the databases will be produced if the data needs to be updated – for example because new four-dimensional quantum periods are computed. Each database is presented as an ASCII text file, called ‘smooth_fano_N.txt’ where N is the appropriate dimension. Each line of a record in that file contains a key, followed by ‘:’, followed by a value. Records are separated by a single blank line.

Table 1 describes the keys provided by the records in each database and their corresponding values, where we think of each database as a single key-value table. Keys are strings of lower-case characters. Each entry in each database has an ID, which is a positive integer. IDs are sequential and start from 1, but carry no meaning or information other than to identify that particular entry in that particular database. Each entry in each database also specifies a non-empty sequence of names. These names identify Fano manifolds with that regularized quantum period in various published (partial) classifications, and hence determine constructions of these Fano manifolds. Names of the form ‘S1 x S2’, where S1 and S2 are names of Fano manifolds X₁ and X₂, indicate that the corresponding Fano manifold is a product X₁ × X₂. Fano manifolds can have many different names. The databases of regularized quantum period sequences for one-, two-, and three-dimensional Fano manifolds use names as in Table 2; the database of regularized quantum period sequences for four-dimensional Fano manifolds uses names as in Table 3, and as specified in the file ‘README.txt’ in the Zenodo dataset³⁴.

Table 2 Names of entries in the databases for dimensions one, two, and three.

Table 3 Names of entries in the database for dimension four.

The differential operator L in (2) is recorded as parallel sequences of coefficients l_k and exponents (m_k, n_k); the corresponding keys in the database are named ‘pf_coefficients’ and ‘pf_exponents’ to reflect the fact that under Mirror Symmetry L corresponds to a Picard–Fuchs operator. The databases in dimensions one, two, and three contain values for all keys except ‘duplicate’; in particular this determines the differential operator L in (2), and thus all coefficients c_d in the expansion (1) of ${\widehat{G}}_{X}$. For a number of the entries in the four-dimensional database, the differential operator (2) is unknown; in this case the key-value pairs related to the differential operator are omitted from the database entry.

The databases contain a key ‘pf_proven’, with boolean values, that has the value false for each entry in each database such that (2) is known. This reflects the fact that, as discussed above, at the time of writing it is not proven that the differential operators (2) recorded in the database actually annihilate the corresponding regularized quantum periods. If this situation changes in the future – for example, if the certificated version of the Lairez algorithm described in [³⁷, §7.3] is implemented – then we will make new versions of the databases available.

The presence of the key ‘duplicate’ in an entry indicates that this entry E is a duplicate of another entry in the same database, with the indicated ID, and that the entry E will not change further as the database is updated. The key ‘duplicate’ is not (and will never be) present in any of the databases in dimensions one, two, or three; at the time of writing it is also not present in the database in dimension four, but it will be added in future updates to indicate when two entries in that database have been proven to coincide.

Technical Validation

The records in our databases satisfy a number of consistency checks. These can be verified, for example, using the computational algebra system Magma⁴² and the Fanosearch software library; see Table 4 for the relevant function names, and the software documentation for these functions for the arguments and parameters required. Firstly, the period sequences in each database are annihilated by the corresponding differential operators (2) whenever they are known. Secondly. the differential operators L from (2) are expected to be of Fuchsian type, that is, to have only regular singular points. This is an extremely delicate condition on the coefficients l_k, and can be checked by exact computation; in particular calls to RamificationData(L) or RamificationDefect(L) in the Fanosearch Magma library will raise an error if L is not Fuchsian. All the differential operators in the databases in dimensions one, two, and three are Fuchsian. Checking this for some of the entries in the four-dimensional database is prohibitively expensive, because it involves linear algebra over the splitting field of the symbol of the differential operator and this symbol can be of very high degree, but all of the four-dimensional operators for which the calculation was possible are Fuchsian, and all of them have regular singularities at those singular points defined over number fields of low degree.

Table 4 Useful Magma functions (also known as intrinsics) in the Fanosearch library.

As a further check, the differential operator (2) is expected to be of low ramification in the following sense²⁶. Let $S\subset {{\mathbb{P}}}^{1}$ be a finite set and ${\mathbb{V}}\to {{\mathbb{P}}}^{1}\backslash S$ a local system. Fix a basepoint $x\in {{\mathbb{P}}}^{1}\backslash S$. For s ∈ S, choose a small loop that winds once anticlockwise around s and connect it to x via a path, thereby making a loop γ_s about s based at x. Let ${T}_{s}:{{\mathbb{V}}}_{x}\to {{\mathbb{V}}}_{x}$ denote the monodromy of ${\mathbb{V}}$ along γ_s. The ramification of ${\mathbb{V}}$ is:

$${\rm{ramif}}({\mathbb{V}})=\sum _{s\in S}{\rm{\dim }}({{\mathbb{V}}}_{x}/{{\mathbb{V}}}_{x}^{{T}_{s}})\quad \quad \quad {\rm{where}}\;{{\mathbb{V}}}_{x}^{{T}_{s}}\,\,{\rm{is}}\;{\rm{the}}\;{T}_{s} \mbox{-} {\rm{invariant}}\;{\rm{part}}\;{\rm{of}}\,{{\mathbb{V}}}_{x}.$$

The ramification defect of ${\mathbb{V}}$ is the quantity ${\rm{r}}{\rm{a}}{\rm{m}}{\rm{i}}{\rm{f}}({\mathbb{V}})-2{\rm{r}}{\rm{a}}{\rm{n}}{\rm{k}}({\mathbb{V}})$. Non-trivial irreducible local systems ${\mathbb{V}}\to {{\mathbb{P}}}^{1}\backslash S$ have $\mathrm{ramif}({\mathbb{V}})\ge 2\mathrm{rank}({\mathbb{V}})$, and hence have non-negative ramification defect. A local system of ramification defect zero is called extremal. The ramification (and respectively ramification defect) of a differential operator L is the ramification (and respectively ramification defect) of the local system of solutions Lf ≡ 0. All the differential operators in the databases in dimensions one, two, and three are of low ramification; indeed with the exception of the two-dimensional Fano manifolds with names ‘dP(7)’ and ‘dP(8)’, which have ramification defect 1, all of these operators are extremal. Computing the ramification for some of the operators in the four-dimensional database is prohibitively expensive, for the same reason as before, but all of the four-dimensional operators for which the calculation was possible are of low ramification, and many of them extremal or of ramification defect 1.

Usage Notes

The key-value data files provided in the Zenodo datasets are easy to parse in any computational algebra system, and in particular can be parsed using the function KeyValueFileProcess provided by the Fanosearch Magma library. The four-dimensional database is available for interactive searching via the Graded Ring Database (http://grdb.co.uk) and programmatically via the Graded Ring Database API. For example, a request to the URL http://grdb.co.uk/xml/search.xml?agent=curl&dataid=smoothfano4&c4=72&c5=360&printlevel=1 will return XML-formatted data as follows:

<?xml version=“1.0”?> <!-- Graded Ring Database --> <results numrows=“1”> <result row=“1” printlevel=“1”> <id>32</id> <names>CKP(31), Obro(4,31)</names> <c2>0</c2> <c3>18</c3> <c4>72</c4> <c5>360</c5> <c6>2430</c6> </result> </results>

Increasing the value of printlevel in the request, to a maximum of 3, will return more information. Users are encouraged to change the value of agent to something more appropriate for their application. For example, a request to the URL http://grdb.co.uk/xml/search.xml?agent=my_app&dataid=smoothfano4&id=340&printlevel=2 will return

<?xml version=“1.0”?> <!-- Graded Ring Database --> <results numrows=“1”> <result row=“1” printlevel=“2”> <id>340</id> <names>CKK(262), CKP(332)</names> <c2>6</c2> <c3>6</c3> <c4>114</c4> <c5>360</c5> <c6>3390</c6> <period>[1,0,6,6,…]</period> <notes>This period sequence is realised by…</notes> </result> </results>

Here ‘…’ indicates that some output has been omitted, for readability. The XML elements c2, …, c6 of the result element record the coefficients c₂, …, c₆ in (1); there are also XML elements of the result element with the same names as the corresponding keys in Table 1. Elements pf_coefficients, pf_exponents, and pf_proven are included only when printlevel = 3.

Calabi–Yau differential operators

When X is a four-dimensional Fano manifold of Picard rank one, and in certain other sporadic cases, the differential operator L in (2) is a Calabi–Yau differential operator⁴³ of order 4. Thus some of the regularized quantum periods that we describe also appear in the AESZ table⁴⁴ of Calabi–Yau differential operators (https://cydb.mathematik.uni-mainz.de).