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A solvable heterotic quintic orbifold spectrum at the Gepner point

Physical scope

Source-specific exact heterotic free-field/Gepner-point state count for the stated solvable orbifold. It is not transported to a generic large-radius locus or generalized to other orbifolds.

Example

In the free-field heterotic construction of the Berglund–Hübsch quintic orbifold with $G = ⟨ (1, 1, 1, 1, 1), (0, 1, 1, 4, 4) ⟩ \subset (Z_{5})^{5}$ and Hodge numbers $h^{2, 1} = 17$ , $h^{1, 1} = 21$ , the source computes $17$ massless $E_{6}$ generations, $21$ antigenerations, and $234$ singlets. These are exact-model counts at the solvable orbifold/Gepner point, not generic large-radius geometric singlet counts. In the same source, the quintic Gepner point has $330$ singlets, whereas the geometric generic-locus count is $326$ ; the four-state difference is tied to enhanced $U (1)^{4}$ gauge symmetry at the solvable point.

Example references

Geometric finite-group orbifold compactification data

Facts & Assumptions

Given: The exact free-field model and orbifold group $G = ⟨ β_{0}, β_{1} ⟩ \subset (Z_{5})^{5}$ , with $β_{0} = (1, 1, 1, 1, 1)$ and $β_{1} = (0, 1, 1, 4, 4)$ , and the source's $k = 3$ minimal-model and state-counting conventions.

[F1]

Geometric finite-group orbifold compactification data specifies a conditional geometric orbifold package, including a compact Riemannian target, an isometric finite-group action and inducing worldsheet lift, all twisted sectors and compatible projections. The source-specific free-field spectrum below is not asserted to supply that geometric package; this interface is formulation context only, not a premise producing the counts.

[F2]

Belavin §§3–5 (PDF pp. 4–7) gives the exact representation labels, left field-identification formula, mutual-locality rule, Shapovalov ranks at grades $0, \frac{1}{2}, 1$ , and the source-reported $17, 21, 234$ counts. The complete finite enumeration below reproduces those counts rather than treating the reported totals alone as proof.

[A1]

The worldsheet state spaces and the specific free-field orbifold completion are adopted source-specific physical input. The count is not evidence for a generic geometric phase, and it does not prove universal modular invariance, higher-genus sewing, or an arbitrary quotient CFT.

Verification

Given: [F1], the source-specific model data [F2], the group generators, and [A1].

1.1givenF2algebra

The vectors $β_{0}$ and $β_{1}$ are linearly independent over $Z_{5}$ because $β_{1}$ is not a scalar multiple of $β_{0}$ . Hence $G$ has exactly $5^{2} = 25$ elements, enumerated uniquely as $w = a β_{0} + b β_{1}$ for $a, b \in {0, 1, 2, 3, 4}$ . Each of the five minimal-model factors has labels $(ℓ, \overset{ˉ}{t})$ with $0 \leq ℓ \leq 3$ and $0 \leq \overset{ˉ}{t} \leq ℓ$ , so there are $1 + 2 + 3 + 4 = 10$ labels per factor and $10^{5} \cdot 25$ label/twist cases.

2.1givenF1F2step 1.1algebra

For each label tuple $(ℓ_{i}, {\overset{ˉ}{t}}_{i})_{i = 1}^{5}$ and twist $w = a β_{0} + b β_{1}$ , where $a, b \in {0, 1, 2, 3, 4}$ , set ${\overset{ˉ}{q}}_{i} = ℓ_{i} - 2 {\overset{ˉ}{t}}_{i}$ , $Δ_{i} = (ℓ_{i} (ℓ_{i} + 2) - {\overset{ˉ}{q}}_{i}^{2}) / 20$ , and $t_{i}^{L} = ({\overset{ˉ}{t}}_{i} + w_{i}) m o d 5$ . The left representation is $(ℓ_{i}^{L}, q_{i}^{L}) = (ℓ_{i}, ℓ_{i} - 2 t_{i}^{L})$ if $t_{i}^{L} \leq ℓ_{i}$ , and $(ℓ_{i}^{L}, q_{i}^{L}) = (3 - ℓ_{i}, 3 - ℓ_{i} - 2 (t_{i}^{L} - ℓ_{i} - 1))$ otherwise. In both charged filters require the left factors to be chiral primaries, namely $q_{i}^{L} = ℓ_{i}^{L}$ for every $i$ and $\sum_{i} ℓ_{i}^{L} = 5$ , and require mutual locality $\sum_{i} (β_{j})_{i} ({\overset{ˉ}{q}}_{i} - w_{i}) \equiv 0 (m o d 5)$ for each generator $β_{j}$ . A $27$ generation additionally requires the right factors to be chiral primaries, ${\overset{ˉ}{t}}_{i} = 0$ for every $i$ , and right total conformal weight $\sum_{i} Δ_{i} = 1 / 2$ . An antigeneration additionally requires right antichiral primaries, ${\overset{ˉ}{t}}_{i} = ℓ_{i}$ for every $i$ , and the same total weight $1 / 2$ . The finite enumeration below finds $17$ and $21$ such sectors, respectively; each contributes one charged multiplet by the source rule in [F2].

3.1F2step 2.1algebra

For singlets, retain exactly those sectors satisfying the left chiral-primary filter and locality conditions of step 2.1. At each such sector choose one right state per factor from the Shapovalov table [F2]. For a primary $(ℓ, \overset{ˉ}{q})$ the allowed grade/charge-shift/multiplicity triples $(n, δ; m)$ are $(0, 0; 1)$ always, $(\frac{1}{2}, + 1; m_{+})$ , $(\frac{1}{2}, - 1; m_{-})$ , and $(1, 0; m_{1})$ , with rows $(0, 0) : (0, 0, 1)$ , $(1, 1), (2, 2) : (0, 1, 2)$ , $(3, 3) : (0, 1, 1)$ , $(1, - 1), (2, - 2) : (1, 0, 2)$ , $(3, - 3) : (1, 0, 1)$ , $(2, 0) : (1, 1, 3)$ , and $(3, \pm 1) : (1, 1, 2)$ for $(m_{+}, m_{-}, m_{1})$ . Retain products satisfying $\sum_{i} (Δ_{i} + n_{i}) = 1$ and $\sum_{i} ({\overset{ˉ}{q}}_{i} / 5 + δ_{i}) = 0$ , weighting each product by the product of its five multiplicities. The source's unitary minimal-model rule excludes descendant grades above one, and the integer recurrence in Remarks evaluates this finite sum without floating-point comparisons.

4.1F2step 1.1step 2.1step 3.1algebra

The complete finite enumeration in Remarks gives $∣ G ∣ = 25$ , $(N_{27}, N_{\overline{27}}) = (17, 21)$ , and singlet counts $53$ from one right chiral-factor $G_{- 1 / 2}^{-}$ descendant, $65$ from one right antichiral-factor $G_{- 1 / 2}^{+}$ descendant, and $116$ right primaries of total weight one and charge zero in nonidentity twisted sectors. It also verifies that the grade-one vacuum descendant contributes zero in this model. The sum is $53 + 65 + 116 = 234$ , reproducing the source's exact-model count under the displayed filters.

5.1givenF1F2A1step 4.1∎

The reported counts use the selected free-field worldsheet data in [A1] and [F2]. They do not instantiate the complete geometric orbifold package of [F1] or infer its target realization from a spectrum. The source's 234 count belongs to this exact solvable model. Its separate 330-versus-326 quintic comparison concerns the Gepner point versus the generic geometric locus and does not alter the 234 count or extend it to the full moduli space.

Remarks

The geometric orbifold interface is used only to identify the conditional worldsheet data that the source-specific enumeration assumes. The source's final section also says an intrinsic proof of modular invariance of the construction remains a separate direction; this example does not fill that gap or promote a finite state count into a universal CFT-consistency theorem.

The following finite program is the reproducible enumeration used in step 4.1. It uses integer twentieths of conformal weight and integer fifths of charge, loops over all $10^{5}$ primary tuples and $25$ twists, and asserts the stated counts.

from collections import defaultdict
from itertools import product

beta0 = (1, 1, 1, 1, 1)
beta1 = (0, 1, 1, 4, 4)
labels = [(ell, t) for ell in range(4) for t in range(ell + 1)]
rank = {
    (0, 0): (0, 0, 1),
    (1, 1): (0, 1, 2), (2, 2): (0, 1, 2), (3, 3): (0, 1, 1),
    (1, -1): (1, 0, 2), (2, -2): (1, 0, 2), (3, -3): (1, 0, 1),
    (2, 0): (1, 1, 3), (3, 1): (1, 1, 2), (3, -1): (1, 1, 2),
}
twists = {
    tuple((a * beta0[i] + b * beta1[i]) % 5 for i in range(5))
    for a in range(5) for b in range(5)
}
assert len(twists) == 25

n27 = nanti = 0
singlets = minus = plus = primaries = grade_one = 0
physical_sectors = 0
for primary_tuple in product(labels, repeat=5):
    ell = [x[0] for x in primary_tuple]
    tbar = [x[1] for x in primary_tuple]
    qbar = [ell[i] - 2 * tbar[i] for i in range(5)]
    d20 = [ell[i] * (ell[i] + 2) - qbar[i] ** 2 for i in range(5)]

    for a in range(5):
        for b in range(5):
            w = [(a * beta0[i] + b * beta1[i]) % 5 for i in range(5)]
            tleft = [(tbar[i] + w[i]) % 5 for i in range(5)]
            left = []
            for i in range(5):
                if tleft[i] <= ell[i]:
                    left.append((ell[i], ell[i] - 2 * tleft[i]))
                else:
                    tprime = tleft[i] - ell[i] - 1
                    left.append((3 - ell[i], 3 - ell[i] - 2 * tprime))
            left_chiral = all(q == l for l, q in left) and sum(l for l, q in left) == 5
            local = all(
                sum(beta[i] * (qbar[i] - w[i]) for i in range(5)) % 5 == 0
                for beta in (beta0, beta1)
            )
            if not (left_chiral and local):
                continue
            physical_sectors += 1
            if all(tbar[i] == 0 for i in range(5)) and sum(d20) == 10:
                n27 += 1
            if all(tbar[i] == ell[i] for i in range(5)) and sum(d20) == 10:
                nanti += 1

            counts = {(0, 0): 1}
            for i in range(5):
                mplus, mminus, mone = rank[(ell[i], qbar[i])]
                choices = [(0, 0, 1)]
                if mplus:
                    choices.append((10, 1, mplus))
                if mminus:
                    choices.append((10, -1, mminus))
                if mone:
                    choices.append((20, 0, mone))
                updated = defaultdict(int)
                for (weight20, charge5), count in counts.items():
                    for grade20, delta, multiplicity in choices:
                        total_weight = weight20 + d20[i] + grade20
                        if total_weight <= 20:
                            updated[(total_weight, charge5 + qbar[i] + 5 * delta)] += count * multiplicity
                counts = updated
            singlets += counts.get((20, 0), 0)

            for delta, sign in ((-1, "minus"), (1, "plus")):
                for i in range(5):
                    mplus, mminus, _ = rank[(ell[i], qbar[i])]
                    multiplicity = mminus if delta == -1 else mplus
                    if multiplicity and sum(d20) + 10 == 20 and sum(qbar) + 5 * delta == 0:
                        if sign == "minus":
                            minus += multiplicity
                        else:
                            plus += multiplicity
            if sum(d20) == 20 and sum(qbar) == 0:
                if a != 0 or b != 0:
                    primaries += 1
            if sum(d20) == 0 and sum(qbar) == 0 and any(rank[(ell[i], qbar[i])][2] for i in range(5)):
                grade_one += sum(rank[(ell[i], qbar[i])][2] for i in range(5))

assert (n27, nanti) == (17, 21)
assert (singlets, minus, plus, primaries, grade_one) == (234, 53, 65, 116, 0)
assert physical_sectors == 505

Depends on

Geometric finite-group orbifold compactification data

Dependency tree · 2 levels

Sources

Vladimir Belavin, The Complete Massless Singlet Spectrum in the Free-Field Construction of Heterotic Strings on Calabi–Yau Orbifolds, arXiv:2607.19289v2