classS-schur-db — exact Schur indices of four-dimensional N=2 class S theories (release 2)

A database of exact finite expansions of the Schur index of four-dimensional N=2 superconformal theories of class S. Release 2 holds 115,806 records, one per construction: untwisted theories of type A₁–A₄, D₄, E₆, E₇ and E₈, ℤ₂-twisted theories of type A₃, D₄ and E₆, and Argyres–Douglas theories with one irregular puncture. Every series is computed from a published index formula with exact integer arithmetic and stored through x²⁰ (x = q^(1/2)), and every record names its sources: the index formula, the catalogue location of every piece of its construction, the sources of its annotations, and earlier computations of the same index in the literature.

The release covers a stated finite range (below) and is complete within it: every candidate inside the range is present as a record, admitted or excluded. It is complete within its twelve types, not within class S.

Release 2 is the revision tagged v2 of this dataset repository, https://huggingface.co/datasets/cms1308/classS-schur-db. A search site that reads these files, with a page per record, is at https://huggingface.co/spaces/cms1308/classS-schur-db. Release 1 stays at the tag v1 (section "Versions").

Contents

type	category	what	records	admitted	excluded	finite window	series through x^40
A1	A-untwisted	untwisted A₁, genus g with s punctures	526	526	0	386	0
A2	A-untwisted	untwisted A₂	456	453	3	27	164
A3	A-untwisted	untwisted A₃	2,620	2,593	27	231	2,077
A4	A-untwisted	untwisted A₄	22,673	22,571	102	7,897	19,976
D4	D-untwisted	untwisted D₄	22,704	22,524	180	5,888	8,314
E6	E-untwisted	untwisted E₆, genus 0	2,133	954	1,179	0	0
E7	E-untwisted	untwisted E₇, genus 0	8,518	1,134	7,384	0	0
E8	E-untwisted	untwisted E₈, genus 0	10,299	68	10,231	0	0
A3tw	A-twisted-Z2	ℤ₂-twisted A₃, nontrivial twist sector	18,678	18,607	71	233	5,864
D4tw	D-twisted-Z2	ℤ₂-twisted D₄, nontrivial twist sector	22,972	21,991	981	0	0
E6tw	E-twisted-Z2	ℤ₂-twisted E₆, genus 0, nontrivial twist sector	2,922	1,550	1,372	0	0
AD	AD	Argyres–Douglas theories (J^h[k], Y)	1,305	1,305	0	453	0

In all 115,806 records: 94,276 admitted with a series, 21,530 excluded without one, none unresolved. The type is the prefix of the identifier id: <type>-g<genus>-s<punctures>-<sorted puncture labels> for class S constructions, AD-<J>-b<b>-k<k>-<Y> for the Argyres–Douglas labels. The last column counts the records with series_K40; every Argyres–Douglas record stores instead its series through x^K_deep with K_deep ≥ 40 (series_deep).

A construction is a genus g, a multiset of s regular punctures with 2g − 2 + s ≥ 1 and, for the twisted types, the twist sector, together with a witness: a sewing of fixtures and cylinders taken from the published catalogues (or, for a three-punctured sphere, one catalogue fixture). It is admitted when its graded Coulomb-branch dimensions are nonnegative, its cutoff weights are positive and a sourced witness exists, and excluded otherwise. Admission does not certify that a construction is a superconformal field theory, and distinct constructions are not proven to be inequivalent theories. An Argyres–Douglas record is a label (J^h[k], Y) of arXiv:1706.01607 with b = h, an irreducible singularity and c ≤ 40: J of type A with a regular puncture Y or none, or J of type D or E with the full puncture or none. Most E-type candidates are excluded: a sphere with three punctures of an E type is a fixture only when every graded Coulomb dimension is nonnegative, and most multisets of the range fail that.

Range of release 2

Release 2 contains every candidate of the twelve types above that satisfies at least one of two conditions:

1. it lies in the (genus, s) box of its type, where s is the number of punctures (n_punctures);
2. the formula value of its central charge, virtual_annotations.c, satisfies c ≤ 40.

types	genus 0	genus 1	genus 2	genus 3
A2, A3, A4	s = 3–12	s = 1–8	s = 0–6	s = 0–4
D4, A3tw	s = 3–8	s = 1–6	s = 0–4	s = 0–2

A1, D4tw, AD, E6, E7, E8 and E6tw have no box and enter by the bound on c alone; every E-type record has genus 0, since a surface of higher genus with these punctures has c > 40. For a class S construction c = (2n_v + n_h)/12, with n_v and n_h computed from the puncture data; it is evaluated for every candidate, excluded ones included. For an Argyres–Douglas label c is taken from the sources named in its provenance_annotations.

Not in release 2: class S theories of type A_{N−1} with N > 5 and D_N with N > 4; twists other than the ℤ₂ twists of A₃, D₄ and E₆; Argyres–Douglas theories with b ≠ h, with J of type D or E and a general puncture or gcd(k, h) > 1, with J of type A, k < 0 and a puncture other than the full one and [N−2, 1, 1], and with reducible singularities.

The stored series

The index is I(q) = Tr (−1)^F q^(E−R) with R = 1/2 on the hypermultiplet scalars, unrefined (every flavor fugacity set to one), with the vacuum-energy factor of the source stripped. A record stores I(x²) = Σ b_n x^n for n = 0, …, K with K = 20 inclusive and b₀ = 1, as exact signed integers written as decimal strings. Zeros are explicit; the tail beyond x^K is uncomputed, not zero.

With each series the record stores:

• pl: the plethystic logarithm PL[I] through x^K;
• quotient: S = I/P, where P = 𝒜^(2g−2+s) Π_p K_p is the prefactor of the index formula of the type, 𝒜 independent of the punctures and K_p the factor of puncture p (for an Argyres–Douglas record, P is the factor K_Y of its regular puncture); with quotient_pl = PL[S] and prefactor_pl = PL[P]. S belongs to the construction, not to the theory: two constructions of one theory with different prefactors have different quotients.
• finite_window: true when the smallest cutoff weight exceeds K, so that the stored series equals the prefactor P alone through x²⁰;
• series_K40, quotient_K40, quotient_pl_K40: the same through x⁴⁰ where stored (36,395 records);
• series_deep, pl_deep (Argyres–Douglas only): the series through x^K_deep with K_deep = max(40, 2(h+k+j_max)), at most x⁶⁴⁶. For the 453 Argyres–Douglas records in the finite window, the dependence on the irregular singularity is in these fields only.

Index formulas

formula	types	records with a series	status	sources	restatements
A	A2, A3, A4	25,617	conjectural	1104.3850, 1110.3740	1408.6522
A1	A1	526	proven	1104.3850, 1110.3740, 0910.2225	1707.07679, 2112.10715
D4	D4	22,524	conjectural	1212.1271, 1212.0545	—
E6	E6	954	conjectural	1212.0545	—
E7	E7	1,134	conjectural	1212.0545	—
E8	E8	68	conjectural	1212.0545	—
A3tw	A3tw	18,607	conjectural	1212.1271, 1212.0545, 1501.00357	1711.04727
D4tw	D4tw	21,991	conjectural	1212.1271, 1212.0545, 1501.00357	1711.04727
E6tw	E6tw	1,550	conjectural	1501.00357, 1212.0545	1711.04727
AD	AD	1,305	conjectural	1706.01607	—

The status is that of the formula in its sources. The sources column names the papers that are the original source of some part of the formula; the restatements column names papers that print the formula in the form used here but credit it to earlier work (SOURCES.json gives the originality of every entry). The class S formulas are the index TQFT sums over representations with the puncture factors of the sources: for E₆, E₇ and E₈ the general simply-laced sum of 1212.0545 Section 4.1 at t = q; for twisted E₆ the fixture formula of 1501.00357 Section 4.2 with the K-factor of 1711.04727 Section 3, sewn to spheres with any number of punctures as 2201.13435 Section 2.3 prints it for a general ℤ₂ twist (an additional formula entry of these records). The Argyres–Douglas formula is the closed plethystic form of 1706.01607, which rests on a conjectured irregular wave function. For A₁ the formula is an identity for the Lagrangian descriptions, whose proof its source outlines (1110.3740, Appendix E). A stored series is compared with a second implementation or route where one exists, through the order that the caveats of its archive state, and every stored unflavored series was recomputed by the independent reviews (below); nothing is claimed beyond the stored order. For 495 Argyres–Douglas records no second route applies and the series rests on the formula alone (caveat ad-formula-alone, attached to them: its field records lists 628 records, of which its scope excepts the 133 that a review compared with a second route). The Argyres–Douglas formula of 1706.01607 has earlier sources for part of its labels: the type-A irregular wave function was conjectured in 1509.06730, the Schur index with the full puncture or without puncture was proposed as a vacuum character in 1604.02155; and the Schur indices of the (A₁, G) theories, among them (A₁, D_{2n+1}) with the full puncture and the type-A labels without puncture, were computed earlier in 1506.00265 and 1505.05884. SOURCES.json records the originality of each (the entries of 1706.01607, 1506.00265 and 1505.05884), and a record lists the earlier computations of its theory in provenance_literature.

Files

file	rows	content
records.parquet	115,806	the records, one row per construction; nested fields as JSON strings
records.jsonl.gz	115,806	the same records as gzip JSON Lines, nested fields as objects
equal_windows.parquet	12,742	groups of records with identical stored series
equal_windows.jsonl.gz	12,742	the same, JSON Lines
known_equivalences.parquet	11,083	sets of records that a source states to be one theory
known_equivalences.jsonl.gz	11,083	the same, JSON Lines
caveats.json	129	the caveats that records refer to by caveat_ids, with their scope and release notes
schema.json	51	the record fields with their types and descriptions
SOURCES.json	163	the registry of cited locations: one entry per location in a paper, with its verification
RELEASE-CAVEATS.json	4	the release layer over the archive caveats: the scopes and release notes written into caveats.json, and the 4 caveats of the release
RELEASE-STATEMENTS.json	79	the release layer of printed statements: statements about fixtures, read in the sources at the reviews of release 2, that the records carry besides those of their archive files
BUILD.json	—	the build record: input hashes, counts and the provenance report
manifest.json	—	the manifest of the database: release files, range, formulas, relations, and the archive files the release is built from
README.md	—	this card
SHA256SUMS	—	SHA-256 of every other file

Records

The 51 fields are described in schema.json. The main ones:

• identity: id, category, algebra, twist, twist_sector, genus, n_punctures, punctures (label, partition as SU(2)-block dimensions, twisted flag, color of a D₄ puncture, the label of the source), irregular (Argyres–Douglas: J, b, k);
• status: status (admitted, excluded, unresolved) and status_reasons;
• series: series_K, series, pl, quotient, quotient_pl, prefactor_pl, finite_window, cutoff_weights, the x⁴⁰ and deep fields above;
• annotations: a, c (exact rationals), d_k, n_h, n_v, coulomb_rank, coulomb_dimensions, manifest_flavor_dimension, manifest_flavor_rank, known_free_hypers, named_theory, enhanced_flavor, source_discrepancy, and virtual_annotations (the same quantities computed formally for every candidate);
• construction: witnesses;
• provenance: provenance_formula, provenance_construction, provenance_annotations, provenance_literature, provenance_missing;
• caveat_ids and archive_records.

Puncture labels. A label is the partition joined by dashes, with t in front for a twisted puncture. Untwisted D₄ punctures of the two triality families carry a color in the label and in color: green, red, blue for [3,1⁵], [2⁴], [2⁴] (3-1-1-1-1-1g, 2-2-2-2r, 2-2-2-2b) and for [5,1³], [4²], [4²] (5-1-1-1g, 4-4r, 4-4b). Triality permutes the three colors on all punctures at once and leaves the index unchanged, and a construction is stored once, under the image whose identifier is lexicographically smallest; a search by partition must take every image (caveat release-d4-triality). Twisted A₃ punctures are stored in the D₃ frame of the engine: untwisted ones as D₃ partitions of 6, twisted ones as C₂ partitions of 4, with the label of 1212.3952 in source_label ([3,1²]_t is t2-2, [2²,1]_t is t2-1-1; caveat release-a3tw-puncture-labels, which also says where the stored dictionary departs from the one printed in 1212.3952). Punctures of E₆, E₇ and E₈ are nilpotent orbits with their Bala–Carter labels: the label is the ASCII spelling (p for + and, for E₇, for a prime: A3pA1p is (A₃+A₁)′, E6a1 is E₆(a₁)), partition is null and source_label holds the label printed in the source. The twisted punctures of twisted E₆ are F₄ orbits, labelled t and the F₄ Bala–Carter label, with s after the rank of a short-root component (tA2s is Ã₂, tF4a1 is F₄(a₁)).

A null annotation means unrecorded, not zero or absent. known_free_hypers, named_theory and enhanced_flavor are filled only where a source prints the statement (a catalogue row, a labelled figure or a sentence); enhanced_flavor of an E-type three-punctured sphere lists every such statement with its location in statements, where the source's own data contradict it the values of the series in formula_values. A record is named where a printed row or a statement that names its sector as an SCFT is read by its archive or by the release layer, by its printed flavor symmetry: for E₇, E₈ and twisted E₆ the caveats of the archive (e7-statement-reading, e8-section3-reading, e6tw-statement-reading) list these statements; for E₆ they are the rows of 1403.4604 that print an SCFT, the preamble of its Section 3.5 and its Section 5, 1501.00357 Section 7, 2203.05040 Section 3.1 and 1704.07890 Sections 4.3.2–4.3.3 and 4.4.1, each cited in the record's provenance, the earliest print first; the names of the records of v1 are those of their catalogue rows; a product of sectors is written with x, and a sector whose name has several factors in brackets, for example E_7 x [(E_7)_16 x SU(2)_9]. In Parquet the coefficients are lists of decimal strings and the fields irregular, refined_probes, virtual_annotations, enhanced_flavor, source_discrepancy, the provenance objects of the formula and the annotations, and the data of a witness are JSON strings; the JSON Lines file keeps them as objects.

Relations between records

• Two records are never merged; the identifier is the construction.
• known_equivalences (11,083 rows) is the only relation that groups records as one theory. Each row gives its statement and the registry entries of its sources, and every row has one stored series. A row is declared when a source states the identification, or when two catalogue rows print the same global symmetry with levels, graded dimensions and (n_h, n_v) and a source uses that symmetry as the name of an SCFT, provided the stored series agree and no source calls the identification a conjecture (174 rows, each with a name; 166 of them are the k ↔ N pairs of Argyres–Douglas labels). The pairs of E₆ fixtures that 1403.4604 Section 6 conjectures to be isomorphic are not declared, nor the D₄ fixture 120 of 1106.5410, which prints the data of the first pair. The other 10,909 rows, without a name, come from a rule: a record with an atypical twisted puncture (t4-2, t3-3 of twisted D₄; t2-2 of twisted A₃; tF4a1 of twisted E₆) and the records obtained by resolving such punctures into their pairs are one theory, by 1309.2299 Section 2.5, 1212.3952 Section 4.1 (stated there for the fixture and the four-punctured sphere; the formula of 1309.2299 at N = 3 gives the same resolution for twisted A₃) and 1501.00357 Section 3.6 with 1711.04727 Section 5 (5,857 rows with 16,801 twisted D₄ records, 4,761 rows with 13,681 twisted A₃ records, 291 rows with 621 twisted E₆ records; caveat release-atypical-punctures).
• Fixtures that a source conjectures to be isomorphic, or whose interacting parts a source identifies while their free hypermultiplets differ, are not known equivalences; the caveats of their archives name them, and for E₇ and E₈ records enhanced_flavor lists the partner records (e6_sector_records, e7_sector_records).
• equal_windows (12,742 rows, 37,508 members) lists the groups of records with identical stored series through x²⁰. It serves a reader who wants one record per distinct series. Equal stored series are not an identification. compared_members lists the members whose series were compared beyond x²⁰, through x^compared_through; persists_through says that they agree through that order, first_differing_order where they first differ (null when no member was compared beyond x²⁰). Imported records were compared through x⁴⁰; a group of Argyres–Douglas records is compared through the smallest K_deep of its members. relation is the first class that fits: cross_type (members of more than one type), e12 (twisted A₃ records that agree after every t2-2 is resolved into 3-3 and t4, the atypical puncture above; the label is that of the imported relation file), color_only (untwisted D₄ records that differ only in the colors of their punctures), prefactor_only (every member in the finite window), other.

Provenance

Each record carries its sources at four levels. Every source identifier is an entry of SOURCES.json, which gives the arXiv identifier, the location in the paper (section, equation label, table or figure), an excerpt of at most 25 words, the status of the statement, whether the location supports the use made of it, and whether the paper is the original source of the statement or restates an earlier one.

1. Formula (provenance_formula): the source identifiers and status of the index formula, and the files of the code that computed the series, by path and SHA-256.
2. Construction (provenance_construction): the catalogue location of every fixture and cylinder of the witness; a fixture that its catalogue counts without printing it is cited by the count statement with the rule that recovers it.
3. Annotations (provenance_annotations): the sources of every annotation, and whether a value is quoted from a source or computed from its formulas.
4. Literature (provenance_literature): earlier computations of the same index; same_index is false for a related quantity, such as a Macdonald or full index whose Schur limit the source does not write out, or a series printed for the interacting part or a factor of the record, which the release check compares after removing the free hypermultiplets or dividing by the release records of the other factors.

provenance_missing lists the levels not filled. The formula, construction and annotation levels are filled on every record to which they apply; the literature level is filled on 1,267 admitted records and missing on 93,009, for which no earlier computation of the same Schur index was found (computations of a related index are listed with same_index false and do not fill the level). Some entries rest on the full text of a paper as read on alphaXiv, not on a copy of its source; their registry entry says so (read_via).

archive_records links every record to the lines of the archive files it was converted from, with their SHA-256. The archive files and the code that computed the series are not part of this release; they are named here by path and hash. Paths under imported/ are archive files imported byte for byte from an earlier project of the author.

Caveats

caveats.json holds 129 caveats: 125 quoted from the documentation of the archive files and 4 of the release itself (archives: release). A record lists in caveat_ids the caveats of the archive files whose content it carries, each restricted by its scope: the field applies of a caveat states the records it concerns (null: every record of its archives), for example only the records in the finite window, or only the records listed in its field records less those that its scope excepts. The quotes are verbatim; part of them name results, open questions or files of the earlier project in which an archive was first produced, or files of the code repository, none of which is part of this release, and the field release_note of such a caveat says what the reference means for a reader of the release. The main caveats:

• the index formulas are conjectural in their sources except for A₁, and a series is certified through its stored order only;
• admission does not certify a superconformal field theory, and distinct constructions are not proven to be inequivalent theories;
• in a finite-window record the stored series is the prefactor alone through x²⁰;
• one witness is stored per construction, and the witness search has no second implementation; the E-type fixtures that a catalogue counts without printing are recovered by the rule that every graded Coulomb dimension be nonnegative;
• free-hypermultiplet counts, theory names and enhanced flavor symmetries are recorded only where an archive or the release reads them from a printed statement of a source (for A₁, only on the three-punctured sphere), and null means unrecorded;
• printed rows that the stored series contradict are kept as printed and name the conflict: one twisted A₃ fixture (1212.3952, A3twistedinteracting7; caveats oq30-open, oq30-constraints on every twisted A₃ record), six E₆ rows (e6-flavor-conflicts), nine E₇ rows (e7-printed-conflicts), one E₈ statement (e8-printed-conflict) and three twisted E₆ records (e6tw-printed-conflicts);
• the punctures of twisted A₃ and the colors of untwisted D₄ follow conventions stated in release-a3tw-puncture-labels and release-d4-triality.

Independent review

Each archive was reviewed independently before it was published, by a Claude subagent (Claude Opus 5.5) in a fresh context that wrote no file of the database: the eight archives of release 1 on 2026-10-02, the four added in release 2 (e6, e7, e8, e6tw) on 2026-10-05. The reviewers are the same model family as the author and read the author's notes, so the reviews are not blind comparisons. With code of their own, which shares no code with the engines, the reviewers recomputed every stored unflavored series of every archive (through x²⁰, through x⁴⁰ where stored, and through K_deep for the Argyres–Douglas records) and 600 of the 9,994 stored flavored probe series (refined_probes), enumerated the candidate sets of the range, checked the witnesses and compared the release with the archive files; they found no differing coefficient, status, witness or annotation value. For E₆ and twisted E₆ the reviewers used character methods other than those of the author's engines (the Weyl character formula at a regular torus element, a Brauer–Klimyk recursion); for E₇ and E₈ they used the method of the engines, the Weyl character formula restricted to a Levi subalgebra, in code of their own, which differs from the method of the second implementation. For twisted E₆ they also used routes without class S: the E-string series of 1609.00310, which give the rank-one factors, and a modular differential equation of 1907.08629, which gives the rank-two E₆ Minahan–Nemeschansky series of the records through x¹⁸. Lagrangian routes (Haar integrals) and character formulas that differ from the index formula cover part of the records; elsewhere a recomputation of the same formula supports the coefficients, not the formula.

The corrections the reviews of release 2 required concerned printed statements that the archive transcriptions lacked (79 statements on 64 records, from tables, labelled figures and sentences of 1704.07890, 1802.09626, 1501.00357, 1403.4604, 2203.05040, 2212.11983, 1803.02425, 1503.06217 and 1504.08348, now in RELEASE-STATEMENTS.json; a further review of the corrections found the statements of 1403.4604 Section 5 and 1803.02425, which name 24 more records), names, literature entries, originality verdicts and the text of caveats; no stored coefficient, status or witness changed. The outcome for each archive is its caveat release-review-imported, c40-review, a1-review, twisted-d4-review, ad-review, e6-review, e7-review, e8-review or e6tw-review.

Versions

• v1 (2026-10-03): release 1, 91,934 records of the types A₁–A₄, D₄, ℤ₂-twisted A₃ and D₄ and the Argyres–Douglas labels, at commit 22dc4f0 of this repository. It is unchanged.
• v2 (this release): release 1 with the 23,872 records of untwisted E₆, E₇, E₈ and ℤ₂-twisted E₆ added. Every record of v1 is a record of v2 with the same identity, status, series and views, cutoff data, annotations and witnesses; of the records of v1 only two D₄ records differ, in their literature level: those of the rank-two E₆ Minahan–Nemeschansky theory and of the S-fold theory 𝒯^(2)_{E₆,2} gained the series of 1905.00864 and 2110.14897, and of 2007.00647 Section 5. The relations, caveats, registry and the release layers grow with the new records.

Reading the data

The Parquet files can be read with pyarrow, pandas or the Hugging Face datasets library; the coefficients are decimal strings, to be converted to arbitrary-precision integers.

import json
import pyarrow.parquet as pq

columns = ['id', 'category', 'genus', 'n_punctures', 'status', 'c', 'named_theory', 'series', 'provenance_formula']
records = pq.read_table('records.parquet', columns=columns).to_pylist()
admitted = [r for r in records if r['status'] == 'admitted']
print(len(records), 'records,', len(admitted), 'admitted')

r = next(r for r in records if r['named_theory'] == 'E_6')
print(r['id'], 'c =', r['c'])
print([int(b) for b in r['series'][:9]])                 # b_0, ..., b_8 of I(x^2)
print(json.loads(r['provenance_formula'])['source_ids'])  # entries of SOURCES.json

The JSON Lines files are read one line at a time with gzip.open(path, 'rt') and json.loads; loading every record at once with its nested fields takes several GiB.

Verifying the files

SHA256SUMS gives the SHA-256 of every file. manifest.json lists the release files with their hashes, row counts and the hash of each gzip payload (compressed bytes can differ between zlib builds; the payload hash does not), states the range and the counts by type, and lists the formulas with their source identifiers.

License

The database (records, relations, caveats, schema, registry, release layers, manifest, build record) is released under the Creative Commons Attribution 4.0 International license (CC BY 4.0). The excerpts in SOURCES.json and the quotes in RELEASE-STATEMENTS.json are short quotations from the cited papers, each with its arXiv identifier and location; they remain under the terms of those papers.

Citation

If you use the database, please cite it and the sources of the formulas for the records you use (the sources column of the formula table above, which names the original sources, and for Argyres–Douglas records the earlier sources named below the table that apply to them; each record lists every entry, restatements included, in provenance_formula).

@misc{classS-schur-db,
  author       = {Cho, Minseok},
  title        = {{classS-schur-db}: exact Schur indices of four-dimensional {$\mathcal N=2$} class {S} theories, release 2},
  year         = {2026},
  version      = {v2},
  howpublished = {Dataset},
  url          = {https://huggingface.co/datasets/cms1308/classS-schur-db}
}