Notes on gauging noneffective group actions

E. Sharpe; T. Pantev

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Notes on gauging noneffective group actions

T. Pantev, E. Sharpe

High Energy Physics - Theory · Submitted 2 Feb 2005 · 17 references

Pith's verdict: A short, honest founding note: the explicit partition functions are solid and the root-of-unity twist-field picture is genuinely new, but the spectrum argument rests on a prefactor-counts-states step that is defended by analogy rather than direct count.

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arXiv IDhep-th/0502027 (version 2)

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CitationE. Sharpe, T. Pantev. "Notes on gauging noneffective group actions." arXiv:hep-th/0502027 (2005). https://pith.science/paper/TWQ2QQ46

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Editor's note

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Worth your time, with one caveat about where the weight sits. The real content is in §2: for D4, D8, the binary dihedral case, and the non-banded quaternion and A4 examples, the authors write down the one-loop partition function as a sum over commuting pairs and show it is not the partition function of the effective quotient. The D4 case is the cleanest — four copies of most Z2×Z2 twisted sectors, none of the (a|b)-type sectors, and the omitted set is exactly an SL(2,Z) orbit, so modular invariance survives. That is a checkable computation and it holds up. The second contribution, in §5.2, is that twist fields for trivially-acting elements can be rewritten as fields valued in roots of unity, with correlation functions vanishing unless the exponent is a multiple of k. That is a real reformulation, not a relabeling, and it is what the sequels build on. §4.3's genus-g computation (k^g times the partition function of C) is a nice independent corroboration that the reader's report undersells slightly — it is not just a one-loop statement. On the stress-test note: the concern about §4.1 is fair but a bit overstated. The prefactor-counts-states step is supported by the flat-space pole expansion, yes, but also by the disjoint-union comparison in §5.3 and by the g-loop count in §4.3, which together make the k-copy reading hard to avoid. I would call it a soft spot in the exposition, not in the conclusion. The genuinely weak pieces are elsewhere: the tensor-product claim C[X/G] ≅ C_X ⊗ C_G in §5.1 is asserted, and the all-genus version is only checked for the specific Z_kn-of-[C/Z_n] setup; the unitarity argument in §4.2.3 is heuristic and the authors say so; and the deformation-theory resolution is deferred to [1,2], which were unpublished at the time. None of that undermines the headline claim. This is a preliminary note and reads like one, but the preliminary parts are flagged. Audience: anyone working on orbifolds, gerbes, or gauged linear sigma models with nonminimal charges. Send it out; the referees will have useful things to say about §4.1 and §5.1, and the paper deserves that attention.

Referee report

Significance

The explicit one-loop partition functions in section 2 are directly checkable and the modular-invariance checks are carried out, which gives the central distinction between noneffective and effective gaugings a solid computational basis. The identification of trivially-acting twist fields with root-of-unity-valued fields is checked against explicit correlation functions in section 5.2 and realized concretely in the Landau-Ginzburg construction of section 6. The mirror-symmetry prediction of section 8 (a trivial G-gerbe on X is mirror to a trivial G-gerbe on Y) is a falsifiable statement. The paper is exploratory in character and explicitly defers several derivations to companion papers, but the results it does present are well motivated and likely correct.

Major objections (1)

[Section 4.1] The inference from the multiplicative prefactor in Z[X/Z_k] = (1/|Z_k|) sum_{g,h} Z_{g,h} = k Z_X to k copies of the Hilbert space is the load-bearing step for the spectrum claim and for the root-of-unity identification, but the support offered is indirect. The analogy to Polchinski's flat-space pole expansion (eq. 7.3.15) is drawn from a free bosonic string on R^24, not from a sigma model with a trivially-acting Z_k, and the two-loop factorization argument of section 4.2.3 is explicitly heuristic ('we appear to violate the optical theorem ... unless all relevant scattering amplitudes have no imaginary part, which seems extremely unlikely'). A direct demonstration would put the claim on firmer footing: for example, a tau_2 -> infinity expansion of Z[X/Z_k] exhibiting the k-fold degeneracy of the light states, or a computation of a specific correlator that requires k distinct dimension-zero operators. The correlation-function matching for [point/Z_k] in section 5.2 is the cleanest direct evidence and should be highlighted as such.

Minor objections (10)

[Section 2.0.1] The sentence 'and H has five conjugacy classes' uses H without defining it; the group under discussion is D4 (or G). This also clashes with the later use of H for the quaternion group in section 2.0.4, so the notation should be made consistent.
[Section 2.0.3] The relations 'aba = b' and 'a'b'a' = b'' are the relations in the chosen matrix representations, not the abstract group presentations (where the dihedral relation is aba = b^{-1}). The paper should say explicitly that these are representation-specific relations to avoid confusing readers.
[Section 3] The claims about anomaly breaking (U(1)_A to Z_{2N} versus Z_{2kN}) and about the quantum cohomology rings C[x]/(x^N - q) versus C[x]/(x^{kN} - q) are stated without derivation and deferred to reference [2]. A brief derivation or a more specific pointer to the relevant section of [2] would help the reader assess these claims.
[Section 4.2.3] The unitarity argument is explicitly heuristic, as the quoted sentence acknowledges. Since it argues against an alternative spectrum calculation rather than for the main claim, this is acceptable, but the paper should state clearly that it is a plausibility argument and not a proof.
[Section 5.1] The claim C[X/G] is isomorphic to C_X tensor C_G is introduced as 'we claim' and supported by the massless spectrum, correlation functions, and the one-loop partition function (with genus-g matching in section 5.3). This is strong evidence, but partition-function matching at all genus plus spectrum matching is not by itself a proof of CFT equivalence. The paper should state explicitly what level of evidence supports the isomorphism and whether it is expected to hold as an equivalence of full operator algebras or only as a matching of genus-g partition functions and spectra.
[Section 5.2] The notation '<phi^n> = sum_k phi_k' is loose; it should be a sum over the kth roots of unity of phi^n, and the normalization of the measure should be stated.
[Section 6] The unobstructedness argument ('there is no way to get nonvanishing correlation functions involving these operators unless there are some nonvanishing correlation functions among the original untwisted sector operators') should be made precise: which selection rule is being used, and why the twist-field correlator is nonzero when the net twist is trivial. The selection rule established in section 5.2 supplies the ingredients, but section 6 should cite it explicitly.
[Section 7] The claim that B-branes in such orbifolds are twisted sheaves on the underlying space is stated without derivation and deferred to reference [1]. A brief explanation or a more specific pointer would help the reader.
[Section 1] There are several typographical errors, including 'oppotunity' and 'spectru' in the introduction, and 'multplicative' and 'as sume' in section 4.2. These should be corrected.
[References] Reference [3] is listed as 'to appear'; its status should be clarified, since section 5.3 attributes the identification with disjoint unions to it while also giving an independent partition-function argument in the present paper.

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The paper

The paper claims

Gauging a group that acts trivially is not the same as gauging the group it acts through.

The reading

A sigma model in which a group $G$ is gauged, but where some nontrivial elements of $G$ act as the identity on the target space, is usually assumed to be physically the same as gauging the smaller group $G/K$ that acts faithfully. The paper argues this assumption is false. Working through explicit examples ($D_4$ with a central $Z_2$, $D_8$, a binary dihedral group, the quaternions, and $A_4$), the authors compute one-loop partition functions and find that the noneffective gauging either deletes whole orbits of twisted sectors or reweights them with unequal multiplicities, while remaining modular invariant. The massless spectrum is then computed in the same formal way as for an effectively-acting group, giving one sector per conjugacy class even when the corresponding group element does nothing. The consequence the authors draw is that twist fields for trivially-acting elements can be rewritten as fields valued in roots of unity, so deforming along those twisted moduli produces a new, purely algebraic presentation of a conformal field theory.

What carries the argument

The load-bearing device is the one-loop orbifold partition function $Z(G) = frac{1}{|G|} sum_{g,h in G, gh=hg} Z_{g,h}$, summed over commuting pairs, together with the observation that in a theory coupled to worldsheet gravity a multiplicative constant in front of the partition function counts states rather than being physically inert. That identification is what converts the truncated or reweighted twisted-sector sum into a statement about the Hilbert space: $k$ copies of the untwisted sector for a trivially-acting $Z_k$. The second device is the inertia stack $I[X/G] = prod_{[g]} [X^g/Z(g)]$, which gives the same sector count for noneffective and effective actions and is used to cross-check the spectrum. The third is the rewriting of a trivially-acting twist field as a field $Upsilon$ valued in roots of unity, which makes the deformation story algebraic: a Landau-Ginzburg superpotential can be deformed by terms such as $Upsilon,psi,x_1 x_2 x_3 x_4 x_5$, with the path integral summing over the root-of-unity values of $Upsilon$.

Extended reading notes

Core claim

The central claim is that gauging a finite group $G$ that acts noneffectively is a genuinely different physical operation from gauging the effective quotient $G/K$, where $K$ is the normal subgroup acting trivially, and that the difference is nonperturbative. The evidence is a set of worked examples. In the $D_4$ case, with $1 to Z_2 to D_4 to Z_2 times Z_2 to 1$ a nontrivial central extension, the one-loop partition function takes the form $Z(D_4) = frac{|Z_2 times Z_2|}{|D_4|},|Z_2|^2,[Z(Z_2 times Z_2) - text{some twisted sectors}]$: the omitted sectors are exactly those whose defining pairs of group elements cannot be lifted to commuting pairs in $D_4$, and they fill a complete $SL(2,mathbb{Z})$ orbit, so modular invariance survives the truncation. In non-banded cases, such as the quaternion group over $Z_2$ or $A_4$ over $Z_3$, no sectors are omitted but they appear with unequal multiplicities, and again the multiplicities are constant on $SL(2,mathbb{Z})$ orbits. The authors then argue that the massless spectrum must be computed with one sector per conjugacy class of $G$, so that a completely trivial $Z_k$ action on $X$ yields $k$ copies of $H^*(X;mathbb{C})$ and a one-loop partition function $k,Z(X)$. From this they conclude that the twist field of a trivially-acting element does not change the moding of worldsheet fields, and can therefore be described algebraically as a field taking values in $k$-th roots of unity, with correlation functions that vanish unless the exponent is a multiple of $k$.

Load-bearing premise

The argument rests on the premise that quantizing one branch of the semiclassical moduli space for each principal $G$-bundle on the circle still gives a genuine Hilbert-space sector when the group element acts trivially, and that a constant prefactor in a gravity-coupled partition function really does count states; if either fails, the $k$-copy spectrum and the root-of-unity identification lose their support.

Editorial extensions

If this is right

Two-dimensional gauge theories that differ only by nonminimal charges are perturbatively identical but nonperturbatively distinct, with different correlation functions, different $R$-symmetry anomalies, and different quantum cohomology rings: $mathbb{C}[x]/(x^N - q)$ for minimal charges versus $mathbb{C}[x]/(x^{kN} - q)$ for charges multiplied by $k$.
The massless spectrum of a noneffective orbifold has as many sectors as conjugacy classes of the group, so a trivial $Z_k$ quotient of $X$ carries $k$ dimension-zero operators and $k$ copies of the cohomology of $X$, and the CFT factorizes as $C_X otimes C_{[text{point}/G]}$.
Because the spectrum of a trivial $G$-gerbe over $X$ is the tensor product of the CFT of $X$ with that of $[text{point}/G]$, a mirror pair $X$, $Y$ implies that the trivial $G$-gerbe on $X$ is mirror to the trivial $G$-gerbe on $Y$, with Hodge numbers satisfying $h^{i,j}([X/G]) = h^{n-i,j}([Y/G])$.
Chan-Paton factors can carry a nontrivial action of a group that acts trivially on the target, consistently with the Cardy condition, so B-branes in such orbifolds are twisted sheaves on the underlying space.
Deformations along twisted-sector moduli are unobstructed whenever the untwisted moduli are, producing families of CFTs that have no counterpart in the mathematical deformation theory of the corresponding stack.
The one-loop partition functions of these models equal those of sigma models on $k$ disjoint copies of the target, which reproduces the multiple dimension-zero operators and makes the failure of cluster decomposition mild.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the identification with disjoint unions is taken literally at the level of the full CFT, then a noneffective gauging is a tractable stand-in for a disconnected target, which suggests using it as a computational probe of string theory on spaces with several components without giving up a connected worldsheet description.
The root-of-unity description turns a geometric deformation question into a finite sum over a discrete variable, which suggests that the resulting family of CFTs might be classified by simple data (the order $k$ and the exponent of $Upsilon$) rather than by continuous moduli.
The argument that omitted twisted sectors reappear inside higher-genus diagrams that fail to factorize is a general diagnostic; it could be applied to any proposed truncation of an orbifold partition function to test whether the truncation corresponds to a consistent theory.
The mismatch between physical moduli and the moduli of the corresponding stack suggests that two stacks with the same coarse moduli but different banding data should be distinguishable by counting exactly marginal operators, which is in principle a computable invariant.

Original abstract

In this paper we study sigma models in which a noneffective group action has been gauged. Such gauged sigma models turn out to be different from gauged sigma models in which an effectively-acting group is gauged, because of nonperturbative effects on the worldsheet. We concentrate on finite noneffectively-acting groups, though we also outline how analogous phenomena also happen in nonfinite noneffectively-acting groups. We find that understanding deformations along twisted sector moduli in these theories leads one to new presentations of CFT's, defined by fields valued in roots of unity.

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In this paper we study sigma models in which a noneffective group action has been gauged. Such gauged sigma models turn out to be different from gauged sigma models in which an effectively-acting group is gauged, because of nonperturbative effects on the worldsheet. We concentrate on finite noneffectively-acting groups, though we also outline how analogous phenomena also happen in nonfinite noneffectively-acting groups. We find that understanding deformations along twisted sector moduli in these theories leads one to new presentations of CFT's, defined by fields valued in roots of unity.

background contribution method result interpretation main claim

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s0	In this paper we study sigma models in which a noneffective group action has been gauged.	contribution 0.92	0.66	0.05	asserted 0.95
s1	Such gauged sigma models turn out to be different from gauged sigma models in which an effectively-acting group is gauged, because of nonperturbative effects on the worldsheet.	result 0.92	0.60	0.44	asserted 0.99
s2	We concentrate on finite noneffectively-acting groups, though we also outline how analogous phenomena also happen in nonfinite noneffectively-acting groups.	limitation 0.74	0.50	0.04	asserted 0.90
s3	We find that understanding deformations along twisted sector moduli in these theories leads one to new presentations of CFT's, defined by fields valued in roots of unity.	result 0.96	0.94	0.53	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.78.

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not examined

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Figures

**Figure 1.** The four two-torsion points on an elliptic curve.

**Figure 2.** An annulus diagram, interpreted in two ways

**Figure 3.** An annulus diagram in an orbifold trivially on X, then even though G acts trivially on X, we still must distinguish boundary states in each of |Zn| = n twisted sectors, even though many of these boundary states appear otherwise identical, just as the closed string massless spectrum is n copies of the untwisted sector. Mathematically, the combination of a trivial action on the underlying space and a nontrivial action on the Chan-Paton factors means that B-branes in such orbifolds are twisted sheaves on the underlying space. In fact, there is a general statement that sheaves on…

Figures reproduced from arXiv:hep-th/0502027 by the authors.

References

Works cited (17)

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Global 3-group symmetry and 't Hooft anomalies in axion electrodynamics
hep-th 2020-09 conditional novelty 5.0 of 10

Massless axion electrodynamics has a global 3-group symmetry whose 't Hooft anomalies include a cubic mixed anomaly between 0-form and 1-form symmetries.
Sums over topological sectors and quantization of Fayet-Iliopoulos parameters
hep-th 2010-12 conditional novelty 5.0 of 10

Fractional Fayet-Iliopoulos parameters are argued to be consistent in supergravities whose moduli space is a gerbe, with new examples of gerby moduli spaces in heterotic and F-theory compactifications.
Notes on generalized global symmetries in QFT
hep-th 2015-08 conditional novelty 4.0 of 10

Higher-form symmetries of quantum field theories are recast as special cases of 2-groups and higher groups, with examples and a conjecture that anomalies transmute ordinary symmetries into higher-group symmetries.
Generalized Symmetries in Condensed Matter
cond-mat.str-el 2022-04 conditional novelty 2.0 of 10

Generalized symmetries and their anomalies may extend the Landau classification to cover topological order, fracton phases, symmetry-protected topological phases and Fermi liquids, according to this review.
Lectures on Generalized Symmetries
hep-th 2023-07 unverdicted novelty 1.0 of 10

Lecture notes that systematically introduce higher-form symmetries, SymTFTs, higher-group symmetries, and related concepts in QFT using gauge theory examples.
An Introduction to Higher-Form Symmetries
hep-th 2023-03 accept novelty 1.0 of 10

A pedagogical set of lecture notes that develops higher-form symmetries from ordinary Noether symmetries and applies them to Maxwell, Chern-Simons, and SU(N) gauge theories, including spontaneous breaking.

How the literature uses it

80 arXiv papers cite this in 144 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 79 sentences without a confident label

Builds on it 1 paper

This operation also extends to the $(-1)$-form case, where it corresponds to summing over only field configurations with topological charge a multiple of $k$ [ref] .

Spontaneously Broken $(-1)$-Form U(1) Symmetries · Defining a $(-1)$-form $$ global symmetry · 0.80

Compares with it 4 papers

We end this section with a brief comparison with [ref] .

McKay quivers and decomposition · Extensions and decompositions · 1.00
As a consistency check, write $X = exp(-2 πR σ)$, $q = R^ℓn q_2d$, then the relation ([ref]) reduces to [equation] which is the same quantum cohomology ring relation for $ℙ^n-1_[ℓ,⋯,ℓ]$ discussed in e.g. [ref] .

Decomposition squared · Special case: Weighted projective spaces · 0.84
We have contrasted the construction focusing on missing charges in the club sandwich approach [ref] with a symmetry-generator-centric perspective that has been employed in previous works on the subject [ref] .

Decomposition and (Non-Invertible) (-1)-Form Symmetries from the Symmetry Topological Field Theory · Conclusions · 0.99

Cites it as background 53 papers

Our analysis was motivated by the consideration of the relationship between the higher symmetry [ref] and the emergent dynamical decomposition in field theory, which was first observed in two-dimensions [ref] , and then in other dimensions [ref] .

Vacuum Structures Revisited · Introduction · 0.83
In these cases, the QFT can be seen as equivalent to the disjoint union of other theories often called `universes' [ref] .

McKay quivers and decomposition · Introduction · 0.86
Recently, the notion of symmetries has been generalized to ones for extended objects, called higher-form symmetries [ref] (see also related topics [ref] ).

Global 3-group symmetry and 't Hooft anomalies in axion electrodynamics · Introduction · 0.94

Discussion

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