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Blow-up equations encode the 1-form anomalies and 2-group structure of five-dimensional SCFTs.

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2026-07-10 23:50 UTC pith:6G4OFKKP

load-bearing objection Solid computational shortcut: fractional exponents of the known blow-up prefactor give 1-form anomalies and 2-group data, with new prepotentials and coefficients for non-Lagrangian families. the 1 major comments →

arxiv 2607.06663 v1 pith:6G4OFKKP submitted 2026-07-07 hep-th

Generalised global symmetries in 5d mathcal{N}=1 theories from the blow-up equations

William Harding , Noppadol Mekareeya This is my paper
classification hep-th
keywords 5d SCFTsblow-up equations1-form symmetries2-group symmetries't Hooft anomaliessuperconformal indexinstanton partition functionsnon-Lagrangian theories
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Five-dimensional N=1 superconformal theories carry higher-form symmetries and 2-group structures whose 't Hooft anomalies are hard to extract without geometry or a Lagrangian. This paper shows that the classical prefactor weighting magnetic fluxes in the blow-up equations already contains that information. Evaluated on a genuine electric 1-form background, the fractional parts of its exponents give the cubic self-anomaly of the 1-form symmetry and its mixed anomalies with instanton, flavour, gravitational and R-symmetry currents. Combined with the faithful continuous UV symmetry read from the superconformal index, the same fractional data decide whether the theory realises a non-split 2-group or only a mixed anomaly. The method works for ordinary gauge theories and for non-Lagrangian families, yielding new prepotentials and previously unknown anomaly coefficients.

Core claim

The classical blow-up prefactor exp(-V_n), when evaluated on a genuine electric 1-form background, has fractional exponents that encode the cubic self-anomaly of the 1-form symmetry together with its mixed anomalies with U(1)_I, flavour, gravity and SU(2)_R; the same forced fractional fluxes, interpreted inside the faithful continuous UV symmetry obtained from the superconformal index, distinguish non-split 2-groups from mixed 't Hooft anomalies.

What carries the argument

The blow-up function exp(-V_n) that multiplies every magnetic-flux summand in the blow-up equations: its fractional exponents on genuine 1-form cosets supply the anomaly coefficients, while integrality of its gauge-fugacity exponents fixes the forced partner fluxes that diagnose 2-groups.

Load-bearing premise

That the integrality condition on the gauge-fugacity exponents of the blow-up prefactor cleanly isolates the unscreened 1-form backgrounds, and that any half-integer spin^c offset can be subtracted without contaminating the physical anomaly.

What would settle it

Recompute the one-instanton partition function and superconformal index for SU(4)_0 + 2 antisymmetrics (or Spin(7)+3F) by an independent ADHM or geometric method and check whether the forced half-unit flux extracted from exp(-V_n) still matches the 2-group or mixed-anomaly diagnosis claimed in the tables.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper shows that the classical prefactor exp(−V_n) of the 5d blow-up equations, evaluated on a genuine electric 1-form background, encodes the cubic self-anomaly of the 1-form symmetry and its mixed anomalies with U(1)_I, flavour, gravity and SU(2)_R via the fractional parts of its exponents. Combined with the faithful continuous UV symmetry extracted from the superconformal index, the same data decide whether a theory realises a non-split 2-group or only a mixed ’t Hooft anomaly. The method is developed for pure SU(N)_kCS, for SU(4) and USp(4) with antisymmetric matter, for Spin(7)/Spin(8) with vectors, and for the non-Lagrangian B_N, B_N^(1,2,3) and P^{2}∪F_{3,6} families, yielding new closed-form prepotentials and several previously unreported anomalies.

Significance. If correct, the work supplies a purely field-theoretic diagnostic for 1-form anomalies and 2-groups that does not require the full Calabi–Yau geometry or the symmetry TFT. The closed-form expression (3.17) for general gauge group plus matter, the two-step quantisation recipe of §3.2, and the flux criterion of §6.1 are concrete and reusable. New results for the B_N and B_N^(1,2,3) prepotentials, the cubic anomalies of P^{2}∪F_3 and P^{2}∪F_6, and the mixed 1-form–flavour anomalies of B_N^(2,3) enlarge the catalogue of known 5d generalised symmetries. The systematic use of the superconformal index to fix the faithful continuous group (Tables 2–3 and §5) cleanly separates 2-groups from mixed anomalies, a distinction that is otherwise subtle.

major comments (1)
  1. The central claim is sound and the derivations are carried through explicitly. No load-bearing technical error that would reverse a 2-group versus mixed-anomaly diagnosis was found. The only soft spot is the clean isolation of bare 1-form backgrounds for non-Lagrangian theories (spin^c offset δ of (A.12)–(A.13) and evaluation of Ω only on n = jk/m). The paper recovers every previously known geometric coefficient and produces new ones that are internally consistent with the same intersection data that fix the prepotential, so the assumption is well-supported rather than circular. No major revision is required on this point.
minor comments (4)
  1. §3.7 and Table 1: the distinction between dynamical flux n = n + δ and bare 1-form background n is stated clearly, but a one-sentence reminder at the opening of §4.3 would help readers who skip the appendices.
  2. Appendix C: the parallel treatment of SU(3)_6 and USp(4)_0 + 2Λ^{2} is valuable; a short remark that the free-hypermultiplet factor (C.12) is required precisely because the one-instanton partition function starts at order x would make the logic more self-contained.
  3. Notation: the interchangeable use of B_h, B_m and B_ma for flavour fluxes is harmless once defined, but a single consistent choice in the main text (with the multi-index reserved for appendices) would reduce minor friction.
  4. References: the recent literature on 5d 2-groups from 6d reduction (e.g. [30]) is cited; a brief cross-check that the flux criterion of §6.1 reproduces the known E_1 2-group would strengthen the pure-gauge subsection.

Circularity Check

2 steps flagged

Mild definitional reading of prepotential CS terms as anomalies; 2-group criterion and new non-Lagrangian data remain independent.

specific steps
  1. self definitional [§4.2, eqs. (4.6)–(4.8) and surrounding text; cf. (2.7), (3.3b)]
    "We therefore isolate the term proportional to k_CS that is cubic in the gauge fluxes. This is precisely the fractional part of the (p1p2)-exponent P(n) evaluated with the instanton background switched off, B_m0=0: P(n)|_{B_m0=0} ≡ k_CS/6 ∑_a ẽn_a^{3} (mod 1) … so that P(n)|_{B_m0=0} ≡ -N(N-1)(N-2)k_CS/(6m^{3}) (mod 1), in agreement with the cubic self 't Hooft anomaly of the 1-form symmetry"

    The cubic self-anomaly is defined as the Chern–Simons term k_CS CS_5(A) evaluated on the 1-form background. The prepotential F in (2.7) already contains exactly that term (k_CS/6 ∑ ẽφ_i^{3}), and V_n/P(n) is constructed from E which includes F. Isolating the fractional part of the CS contribution on the flux therefore recovers the anomaly coefficient by construction of the blow-up function, not by an independent derivation.

  2. self definitional [Appendix A.1, eqs. (A.2)–(A.4) and the paragraph bridging prepotential and anomaly]
    "The same intersection data fix the Coulomb branch prepotential. … The gravitational level C^G_i entering (A.2) is thus the very same coefficient that multiplies ϵ_{1}^{2}+ϵ_{2}^{2} in (A.3); this is the bridge between the prepotential read off from the blow-up and the topological anomaly."

    For the non-Lagrangian families the paper first computes the intersection numbers c_ijk, C^G_i from the toric geometry, assembles the prepotential E (and hence P(n)), then shows that the fractional parts of P(n) reproduce the geometric anomaly formula (A.2) that is built from the identical intersections. The match is therefore tautological once the geometric data are fixed; the 'extraction from the blow-up' does not supply independent information beyond the geometry already used to write E.

full rationale

The paper's core diagnostic evaluates the classical blow-up prefactor exp(-V_n), which is built directly from the effective prepotential E (itself containing the cubic CS/prepotential F of the 5d theory or the geometric intersections c_ijk, C^G_i). The fractional parts of the resulting exponents Q(n) and P(n) on 1-form cosets are therefore the CS/instanton numbers by construction of V_n; the paper isolates them and matches known geometric/SymTFT coefficients. This is a mild self-definitional step for the anomaly extraction itself (especially pure SU(N) and the non-Lagrangian appendices, where the same intersection data fix both E and the geometric formula for Ω). It is not load-bearing for the paper's main claims: the practical field-theoretic recipe, the new closed-form prepotentials, the new cubic/mixed anomalies for P^{2}∪F_{3,6} and B_N^{(2,3)}, and especially the flux criterion for 2-groups (forced μ from Z_i integrality + independent faithful F/Z_F from the superconformal index). No fitted parameters are re-labelled as predictions, no uniqueness theorems are imported from overlapping authors to forbid alternatives, and the blow-up equations themselves are external literature. Score 2 reflects one minor definitional reduction that does not force the central results.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The work rests on the established existence and form of the blow-up equations for 5d N=1 theories, on the geometric formula for the cubic anomaly of a 1-form symmetry, and on the standard dictionary between superconformal indices and faithful continuous global symmetries. No free parameters are fitted; the only inputs are the classical prepotential (or its geometric intersection data) and the representation content of the theory.

axioms (4)
  • domain assumption The blow-up equations of Nakajima–Yoshioka and their 5d N=1 extensions correctly determine the instanton partition function on the Omega-background, with classical prefactor exp(-V_n) fixed by the effective prepotential.
    Invoked throughout §§2–3; taken from the literature without re-derivation.
  • domain assumption The cubic self-anomaly of a discrete 1-form symmetry Z_m^(1) is given by the fractional part of the Chern–Simons and gravitational terms evaluated on the central divisor (geometric formula of Apruzzi et al.).
    Used in §4 and appendices A–B to identify the fractional part of P(n) with the physical anomaly.
  • domain assumption The superconformal index of a 5d N=1 theory, computed from the IR gauge theory (or from the blow-up equations), correctly captures the protected current multiplets of the UV SCFT and therefore determines the faithful continuous global symmetry.
    Central to §5 and the 2-group criterion of §6.
  • standard math Standard representation theory of classical Lie algebras and the definition of the plethystic exponential and Haar measures.
    Used for all character expansions and index integrals.

pith-pipeline@v1.1.0-grok45 · 86830 in / 2864 out tokens · 38160 ms · 2026-07-10T23:50:16.100911+00:00 · methodology

0 comments
read the original abstract

Five-dimensional $\mathcal{N}=1$ superconformal field theories admit a rich variety of generalised global symmetries, including higher-form symmetries, 2-group symmetries, and their 't$~$Hooft anomalies. We show that this data can be extracted directly from the blow-up equations that govern the instanton partition functions of such theories on the $\Omega$-background. The central object is the classical prefactor $\exp(-V_n)$ that weights each magnetic flux on the blown-up geometry: evaluated on a background for the electric 1-form symmetry, the fractional parts of its exponents encode the cubic self-anomaly of the 1-form symmetry, as well as its mixed anomalies with the instanton, flavour, gravitational, and $\mathrm{SU}(2)_R$ symmetries. Combined with the faithful continuous global symmetry of the ultraviolet fixed point, which we determine from the superconformal index, the same data decides whether the theory possesses a 2-group symmetry or a mixed 't$~$Hooft anomaly. We illustrate the method in gauge theories, including $\mathrm{SU}(4)$ and $\mathrm{USp}(4)$ with antisymmetric hypermultiplets, and $\mathrm{Spin}(7)$ and $\mathrm{Spin}(8)$ with vector hypermultiplets, as well as in several families of non-Lagrangian theories, for which we obtain new results: notably, the effective prepotentials of the $B_N$ and $B_N^{(1,2,3)}$ families, the cubic 1-form anomalies of the rank-two theories $\mathbb{P}^2\cup\mathbb{F}_3$ and $\mathbb{P}^2\cup\mathbb{F}_6$, and several mixed anomalies involving the 1-form and flavour symmetries.

discussion (0)

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