Pith. sign in

REVIEW 4 major objections 6 minor 92 references

Statistics of invertible mixed-dimensional excitations conserved by a higher group G are classified by a single cohomology class in H^{d+2}(BG; R/Z).

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-10 04:49 UTC pith:7JHXQKGI

load-bearing objection Solid Abelian holographic construction of hopping algebras from WZW data; the full higher-group classification is only an embedding plus conjectures, not a proved isomorphism. the 4 major comments →

arxiv 2607.08583 v1 pith:7JHXQKGI submitted 2026-07-09 cond-mat.str-el hep-thquant-ph

Holographic Theory of Mixed-Dimensional Statistics and Conservation-Encoding Hopping-Operator Algebras

Hanyu Xue , Xiao-Gang Wen This is my paper
classification cond-mat.str-el hep-thquant-ph
keywords mixed-dimensional statisticshopping-operator algebrahigher groupsfusion categoriesWZW termsholographic realizationinvertible excitationsconservation laws
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.

The paper defines quantum statistics of point, string, and higher excitations through the algebra of local hopping operators that move them while preserving conservation laws. For invertible excitations whose fusion rules come from a higher group G, that algebra—and therefore the statistics—is completely classified by a cohomology class [ω] in H^{d+2}(BG; R/Z). Changing the class by a coboundary only rephases the operators, so the physical statistics are an invariant of the class. The same class supplies a holographic picture: the excitations live on the boundary of a G higher-group gauge theory twisted by ω in one higher dimension. More generally, non-invertible mixed-dimensional statistics are classified by fusion d-categories, recovering the higher-group story as the fully pointed special case.

Core claim

For G-conserved invertible excitations in d-dimensional space the corresponding hopping-operator algebra (and hence the statistics it defines) is classified by a cohomology class [ω] ∈ H^{d+2}(BG; R/Z); rephasing of local operators corresponds only to coboundaries. The same class realizes the excitations holographically as boundary degrees of freedom of an ω-twisted G higher-group gauge theory.

What carries the argument

The hopping-operator algebra (local operator subalgebra, LOsA) generated by weakly local operators that move or deform excitations while preserving the conservation law; its discrete classification relative to a fixed excitation complex is the statistics.

Load-bearing premise

The map from cohomology classes to statistics is assumed to be an isomorphism for generic combinatorial spheres, and the higher-group and non-invertible extensions rest on the same unproved identification.

What would settle it

Compute the discrete statistics group T* of the hopping algebra on the boundary of a (d+1)-simplex for a concrete Abelian higher group and check whether it equals the predicted cohomology group H^{d+2}(BG; R/Z).

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

If this is right

  • Fermi statistics, Abelian anyon statistics, and Abelian string statistics in any dimension all arise as special cases of the same cohomology class.
  • Nontrivial Z2 string statistics in 3+1D require spacetime to admit a w3-structure, the precise analogue of a spin structure for fermions.
  • A mixed Z2-particle–string system with twisted conservation law has statistics forming a Z4 group for d>2 (Z2 for d=2), linked to p-wave superconducting strings and fermionic bosonization.
  • Any statistics classified by the cohomology can be detected by a local statistical process supported near a single top-dimensional simplex.
  • Non-pointed conservation laws and their statistics are classified by fusion d-categories, exactly as generalized symmetries are.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same cohomology class that classifies the anomaly of a higher-group symmetry also classifies the statistics of its defects, giving a concrete dictionary between anomaly indicators and measurable braiding phases.
  • Lattice models whose hopping operators realize a nontrivial class should exhibit protected ground-state degeneracy or anomalous edge modes even without an explicit bulk topological order.
  • The framework suggests a systematic way to engineer mixed-dimensional anyons by choosing Postnikov data of G and a cocycle twist, then reading off the allowed statistical processes.

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

4 major / 6 minor

Summary. The paper defines statistics of mixed-dimensional excitations via hopping-operator algebras (LOsAs) that encode conservation laws. For Abelian higher-form conservation (G-valued q-cocycles), it constructs an explicit WZW-boundary realization of hopping operators and configuration states from a cocycle ω ∈ Z^{d+2}(K(G,q),R/Z), proving a canonical embedding H^{d+2}(K(G,q),R/Z) ↦ T^*(m_q(X,G)) (Theorem V.1) with dual surjection on statistical processes; cohomologous cocycles differ only by rephasing. The same data supply a holographic bulk: the excitations live on the boundary of an ω-twisted higher-form gauge theory. The authors conjecture that the same cohomology class classifies LOsAs for invertible (fully pointed) excitations of a general higher group G, and that non-invertible mixed-dimensional statistics are classified by fusion d-categories. Supporting material includes prism-integral descendants, transfer of statistical processes, an anomaly/statistics dictionary, and worked examples (fermions, Abelian anyons/strings, mixed Z_2 particle-string systems with Serre spectral-sequence computations).

Significance. If the classification claims hold, the work unifies mixed-dimensional statistics, conservation laws, and holographic bulk theories under a single cohomological/categorical language, extending Fermi statistics and anyon braiding to intertwined particle-string systems and higher groups. The Abelian construction is concrete and usable: Theorem V.1, the descendant calculus (Sec. V A, App. C–E), and the prism integration give an algorithmic route from cocycles to hopping operators, while the examples (Secs. VIII A–E, App. I) produce falsifiable spacetime constraints (e.g., w_3-structure for Z_2 strings) and order-4 statistics for twisted particle-string systems. The proved embedding and holographic existence results are already of independent value for lattice models and bosonization, even before the reverse inclusion and higher-group conjectures are settled.

major comments (4)
  1. Abstract and opening claim: the abstract states that for G-conserved invertible excitations the LOsA “is classified by” [ω] ∈ H^{d+2}(BG;R/Z) and that “we show” this. In the body, Theorem V.1 only constructs a canonical embedding H^{d+2}(K(G,q),R/Z) ↦ T^*(m_q(X,G)) for Abelian higher-form conservation, with surjectivity left as Conjecture IV.1 of prior work (explicitly unproved for generic combinatorial spheres). Higher-group statements are Conjectures VI.1–VI.2. The abstract and strongest claim therefore assert a full classification where only injectivity/existence is proved. Please rephrase the abstract and introduction to match the proved embedding plus conjectural reverse inclusion, or prove the reverse inclusion.
  2. Theorem V.1 and Conjecture IV.1: the dual map T(m_q) ↠ H^{d+2}(K(G,q),Z) is a surjection on statistical processes, but the identification T^* ≅ H^{d+2} remains one-sided for generic X. Because the paper’s central classification slogan rests on this isomorphism, either (i) restrict all classification statements to the image of the embedding, or (ii) supply a proof (or a sharp reduction) of surjectivity for combinatorial spheres. Leaving the reverse inclusion as an external conjecture undercuts the claim that every LOsA of this type arises from a cohomology class.
  3. Section VI (Conjectures VI.1–VI.2) and Section VII: the higher-group and fully-pointed fusion-category classification are presented as the natural extension of the Abelian theorem, yet no axiomatic realization of the excitation complex or locality axiom is given for non-Abelian higher groups (only a sketch in App. B). The mathematical equivalence “fully-pointed fusion n-category ↔ (G,[ω])” supports the conjecture but does not prove that every LOsA realizing a higher-group conservation law is captured by H^{d+2}(BG;R/Z). Either demote these statements clearly to conjectures throughout (including the abstract’s “we show” language) or add a precise reduction to the Abelian case / a definition of T^* for higher groups.
  4. Section V D and Table I: the claim that “a symmetry anomaly manifests itself as nontrivial statistics of symmetry defects” is physically appealing and consistent with the constructions, but the table equates transformation-patch operators with hopping operators and anomaly indicators with statistical processes while noting “we are not very sure about it.” If this dictionary is load-bearing for the holographic interpretation, it needs a theorem (or a counter-example free statement of scope); otherwise mark it as heuristic so that the proved embedding is not read as depending on it.
minor comments (6)
  1. Notation: the paper mixes multiplicative U(1) and additive R/Z conventions; a short global convention paragraph (beyond Sec. II) would reduce ambiguity in phases e^{2πi∫} vs (-1)^{∫}.
  2. Eq. (41) vs Theorem V.1: the prior-work isomorphism for X=∂Δ^{d+1} is cited as proved, while the generic-sphere case is conjectural; cross-references should flag this distinction whenever T^* ≅ H^{d+2} is invoked.
  3. Section VIII E: the belief that H^{d+2}(K_1,R/Z)=Z_4 for d>3 is stated without a full spectral-sequence argument (only d=2,3 are treated carefully). Label it as a conjecture or sketch the missing pages.
  4. Appendix A: the Majorana realization correctly relaxes orthogonality of configuration states; a one-sentence pointer in Def. IV.2 would help readers who know only the stricter definition of Ref. [16].
  5. Typos/style: occasional missing articles and inconsistent hyphenation (“higher group” / “higher-group”); “prop ersubalgebra” line break in the introduction; “cohain” in Sec. II.
  6. References: the link between LOsAs and braided fusion d-categories in the trivial Witt class cites [18,34,35]; a brief comparison with the Doplicher–Haag–Roberts reconstruction already listed as [43,44] would orient algebraic readers.

Circularity Check

1 steps flagged

Mild self-citation for the unproved reverse inclusion; the proved embedding and holographic construction are independent and non-circular.

specific steps
  1. self citation load bearing [Theorem V.1 (and Remark after it); abstract claim of classification]
    "Thus we have a canonical embedding H^{d+2}(K(G,q),R/Z)↪T^*(m_q(X,G)). … We believe (Conjecture IV.1 of Ref. [16]) this map is an isomorphism, but we do not have a proof yet."

    The abstract and strongest claim assert that LOsAs (hence statistics) of G-conserved invertible excitations are classified by the cohomology class [ω]. The body only constructs and proves injectivity of the map; the reverse inclusion that would make it a classification is imported verbatim as an unproved conjecture from the same author's prior paper [16]. Without that self-citation the proved result is only a lower bound (every class yields a distinct LOsA).

full rationale

The paper's core technical contribution (Theorem V.1 and Appendix C) constructs explicit hopping operators U(s) from a cocycle ω via the second descendant L of a WZW term, verifies the configuration and locality axioms directly from the cochain identities (C4)–(C6), and proves injectivity of the resulting map H^{d+2}(K(G,q),R/Z)↪T^*(m_q(X,G)) by transferring statistical processes from the simplex (where nontrivial phases are known) to a generic combinatorial sphere. Different cohomology classes produce observably different statistical phases Φ_ω(P_σ) by construction of the transfer map r^*; this is not a tautology or a fit. The reverse inclusion (surjectivity, hence full classification) is explicitly left as Conjecture IV.1 of the authors' prior work [16] and is never claimed as proved here. Higher-group and fusion-category statements are likewise labeled Conjectures VI.1–VI.2 or consequences of the known LOsA–symmetry correspondence. No parameters are fitted to data and then re-predicted; no uniqueness theorem is imported to forbid alternatives; the continuous redundancies quotiented to obtain T^* are the same linear algebra already present in the definition of statistics. The abstract's stronger wording (“classified by”) slightly overstates the proved content, but the body is careful and the derivation chain itself does not reduce to its inputs. Score 2 reflects only the load-bearing self-citation of the unproved reverse map.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 2 invented entities

The central claims rest on standard algebraic topology, the authors’ prior axiomatization of statistics via excitation complexes, and a small number of domain assumptions about weak locality and configuration states. No free parameters are fitted; the only invented entities are definitional packaging of already-used notions.

axioms (4)
  • domain assumption Weak locality: hopping operators with disjoint supports commute (and higher nested commutators vanish when the common intersection is empty).
    Stated in Definition IV.2 and used throughout to define a LOsA; not derived from a tensor-product Hilbert space.
  • domain assumption Configuration axiom: U(s)|a angle = e^{i heta(s,a)} |a+∂s angle (or its non-Abelian analogue).
    Definition IV.2; encodes conservation while allowing projective phases that become the statistics.
  • standard math Standard facts of group cohomology, higher groups, Steenrod squares and Serre spectral sequences.
    Used to compute H^{d+2}(BG;R/Z) and to construct descendants; taken as background.
  • domain assumption The discrete part T^*(m) of the solution space of phases is a topological invariant independent of continuous rephasings.
    Inherited from Ref. [16] and used to define the classification group of statistics.
invented entities (2)
  • Hopping-operator algebra / LOsA no independent evidence
    purpose: Microscopic definition of statistics as the isomorphism class of the local operator subalgebra generated by conservation-preserving hops.
    Packaging of earlier operator-algebra ideas into a single object that can be classified by cohomology or fusion categories; no independent experimental handle beyond the statistics themselves.
  • Excitation complex no independent evidence
    purpose: Geometric data (configuration group, elementary hops, supports) that encode a conservation law abstractly, independent of a Hilbert-space realization.
    Introduced in prior work of the authors and used here as the fixed background against which statistics are classified; again definitional rather than a new physical object.

pith-pipeline@v1.1.0-grok45 · 62643 in / 2761 out tokens · 35870 ms · 2026-07-10T04:49:16.594480+00:00 · methodology

0 comments
read the original abstract

We develop a general framework for the statistics of mixed-dimensional excitations subject to intertwined conservation laws, extending the familiar Fermi statistics with conserved particle number. We define statistics microscopically through a \emph{hopping-operator algebra}: a local operator subalgebra (LOsA) generated by operators that locally move or deform excitations while preserving the conservation law. Nontrivial statistics arise when this subalgebra is nontrivial. We first focus on LOsAs that encode \emph{pointed} conservation laws. These give rise to invertible excitations, whose fusion rules are exactly those of the symmetry defects of a higher group $\cG$. For such $\cG$-conserved excitations in $d$-dimensional space, we show that the corresponding LOsA -- and hence the statistics it defines -- is classified by a cohomology class $[\omega] \in H^{d+2}(B\cG;\R/\Z)$, where changing $[\omega]$ by a coboundary corresponds merely to a rephasing of the local operators. We further provide a holographic realization: excitations with this prescribed conservation law and statistics live on the boundary of a $\cG$ higher-group gauge theory in $(d+1)$-dimensional space, twisted by $[\omega]$. More generally, non-pointed conservation laws and the associated statistics of non-invertible excitations are defined by a pair: a LOsA together with its excitation-complex representation. This is equivalent to the pair consisting of a LOsA and its Hilbert-space representation, which is the data defining a generalized symmetry. Consequently, non-pointed conservation laws and their statistics in $d$-dimensional space are classified by fusion $d$-categories, just as generalized symmetries are. The higher-group results above are the fully-pointed special cases of this more general classification.

Figures

Figures reproduced from arXiv: 2607.08583 by Hanyu Xue, Xiao-Gang Wen.

Figure 1
Figure 1. Figure 1: FIG. 1. A particle on site [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. An isomorphic holographic decomposition [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The dots represent the sites that form the space [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The statistical phase of the T-junction process detects an anyon’s self-statistics. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) Two branched simplices with opposite [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. A triangle, or 2-simplex, in [ [PITH_FULL_IMAGE:figures/full_fig_p035_6.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

92 extracted references · 92 canonical work pages · 38 internal anchors

  1. [1]

    The statistics are classified byH d+2(K0,R/Z)

    The untwisted conservation law (ρ= 0) Whenρ= 0, particles and strings are independently conserved (df d−1 2 = 0 and df d 2 = 0). The statistics are classified byH d+2(K0,R/Z). We find two universal self-statistics terms ford >3: string self-statistics term 1 2Sq2Sq1fd−1 and particle self- statistics term 1 2Sq2fd, which generateH d+2(K0,R/Z) ∼= (Z2)2. For...

    work page

  2. [2]

    Physically, specific self-intersections of the string worldsheets act as sources for the particle current

    The twisted conservation law (ρ= 1) Whenρ= 1, the particle current is no longer indepen- dently conserved: df d 2 = Sq 2fd−1. Physically, specific self-intersections of the string worldsheets act as sources for the particle current. The statistics of the intertwined Z2-conserved particle-string system are classified by H d+2(K1,R/Z).(174) One of the gener...

    work page

  3. [3]

    Ford= 2, 1 8 Sq3f1 = 0

    Therefore,ω d+2(fd−1, fd) is indeed a cocycle. Ford= 2, 1 8 Sq3f1 = 0. The above calculation and the fact that ∆4(f1) = 0 implies that 1 2 Sq2f2 is aR/Z-valued cocycle, despitef 2 is not a cocycle. Our calculation indi- cates that the condition df 2 =Sq 2f1 is enough to make 1 2 Sq2f2 to be a cocycle. In this caseω 4(f1, f2) = 1 2 Sq2f2 is of order 2, and...

    work page

  4. [4]

    We believe that H d+2(K1,R/Z) =Z 4,(184) since if the particlef d is a fermion, it must cou- ple to stringf d−1 in the way described by (175)

    ThusZ 4 ⊂H d+2(K1,R/Z). We believe that H d+2(K1,R/Z) =Z 4,(184) since if the particlef d is a fermion, it must cou- ple to stringf d−1 in the way described by (175). ButH d+2(K1,R/Z) could be larger, since string could have its own independent statistics described by ωd+2(fd−1, fd) that depend only onf d−1. Such a ωd+2(fd−1) is in H d+2(K(Z2, d−1),R/Z) =...

    work page

  5. [5]

    This is achieved by replacing lattice higher gauge theory with a more general state-sum theory

    Holographic description from state-sum theories The holographic framework of using higher gauge the- ory to describe statistics in one lower dimension nat- urally extends to non-invertible excitations. This is achieved by replacing lattice higher gauge theory with a more general state-sum theory. Specifically, a (d+ 2)- dimensional state-sum theory provid...

    work page

  6. [6]

    [16] applies only to Abelian cases,i.e., the config- urations form an Abelian groupAand hopping opera- torsSact additively

    Hopping algebra description: a potential generalization The axiomatic framework of statistics established in Ref. [16] applies only to Abelian cases,i.e., the config- urations form an Abelian groupAand hopping opera- torsSact additively. The corresponding statistics are classified by cohomology classesH d+2(BG,R/Z), where G= Qd q=1 K(π q, q−1) is a higher...

    work page

  7. [7]

    a finite setAcalled configurations

    work page

  8. [8]

    a finite setSand a map·:S×A→Asuch that for everys∈S,a7→s·ais a bijection

    work page

  9. [9]

    •Configuration axiom:for anys∈Sanda∈A, U(s)|a⟩=e iθ(s,a)|s·a⟩(B4) for someθ(s, a)∈R/2πZ

    a topological spaceXand a subspace supp(s)⊂ X for eachs∈S, such that supp(P) =∅=⇒P·a=a,∀a∈A.(B3) Definition B.2.A realization of the (invertible) exci- tation complex (A, S,·,supp) consists of a Hilbert space H, a collection of normalizedconfiguration states{|a⟩ | a∈A}inH, and a collection ofhopping operators {U(s)|s∈S}, satisfying the following two axiom...

    work page

  10. [10]

    (C10) is indepen- dent ofea∈B q(M, G)

    Independence of bulk extensions We first show that the state in Eq. (C10) is indepen- dent ofea∈B q(M, G). Consider an alternative choice ea′ =ea+ deµ.(C12) Because the configurationa∈B q(X, G) should be the same, the restriction ofeµtoXsatisfies µ∈Z q−1(X, G).(C13) We compare|a⟩ conf ea with |a⟩conf ea+deµ= X ev′∈Cq−1(M,G) d ev′=ea+deµ e2πi R M β( ev′)|v...

    work page

  11. [11]

    (C11) to Eq

    The configuration axiom Lets∈C q−1(X, G) and choose an extension es∈Cq−1(M, G),es| X =s.(C20) Applying Eq. (C11) to Eq. (C10) gives U(s)|a⟩ conf ea = X dev=ea e2πi R M β(ev)e−2πi R X L(s,v)|v+s⟩. (C21) By Stokes’ theorem and Eq. (C6), Z X L(s, v) = Z M dL(es,ev) = Z M Θ(es,dev) + Z M β(ev)− Z M β(ev+es). (C22) Since dev=ea, the summand in Eq. (C21) become...

    work page

  12. [12]

    The locality axiom Write U(s) =M sTs, T s|c⟩=|c+s⟩,(C25) whereM s is the diagonal phase in Eq. (C11). The trans- lationsT s commute. Hence all commutators come from finite differences of the local phaseL(s, c). For example, with the convention [A, B] = A−1B−1AB, one obtains [U(s2), U(s1)]|c⟩=e −2πi R X Γ2(s1,s2;c)|c⟩,(C26) where Γ2(s1, s2;c) =L(s 1, c) +L...

    work page

  13. [13]

    Changing cocycle representatives and descendants The construction uses a cocycle representativeωto- gether with a normalized full descent datum (Θ, β, L). We now show that the statistics depend only on the co- homology class [ω]∈H d+2 K(G, q),R/Z .(C29) Let (ω ′,Θ ′, β′, L′) be another normalized descent datum with ω′ =ω+ dµ, µ∈C d+1 K(G, q),R/Z ,(C30) wh...

    work page

  14. [14]

    Therefore, when one extends a boundary cochain such ass∈C q−1(∂∆d+1, G) to the bulk, no additional choice is required

    The special caseX=∂∆ d+1 Now take X=∂∆ d+1, M= ∆ d+1.(C44) This case is special becauseC n(∂∆d+1, G)≃ C n(∆d+1, G). Therefore, when one extends a boundary cochain such ass∈C q−1(∂∆d+1, G) to the bulk, no additional choice is required. The general configuration-state formula reduces to |a⟩conf = X v∈C q−1(∆d+1,G) dv=a e2πi R ∆d+1 β(v) |v⟩. (C45) The hoppin...

    work page

  15. [15]

    Transfer of statistical processes In the previous subsections we have constructed the map in Eq. (C42). Now we prove that this map is in- jective,i.e., different cohomology classes give different statistics. Because the map is evidently an Abelian-group homomorphism, it is enough to prove that for any non- trivial cohomology class [ω]∈H d+2(K(G, q),R/Z), ...

    work page

  16. [16]

    The complex is constructed from simplices: vertices, edges, triangles, tetrahedra, and so on

    Cochains and cocycles In this paper, we model theD-dimensional spacetime MD using a triangulated spacetime complex, denoted by M D, withD=d+ 1. The complex is constructed from simplices: vertices, edges, triangles, tetrahedra, and so on. We label vertices with indicesi, j, k, . . ., edges by pairs (i, j), and triangles by triples (i, j, k). To unambiguous...

    work page

  17. [17]

    , m+n)⟩ =⟨f m,(0,1,

    Cup products and higher cup products Forf m ∈C m(M;M 1) andh n ∈C n(M;M 2), together with a bilinear pairingM 1 ×M 2 →M 3, the Alexander- Whitney cup product is ⟨fm ⌣ h n,(0,1, . . . , m+n)⟩ =⟨f m,(0,1, . . . , m)⟩⟨h n,(m, m+ 1, . . . , m+n)⟩.(E4) We often writef mhn forf m ⌣ h n. Let us check the coboundary sign explicitly. On the simplex (0,1, . . . , m...

    work page

  18. [18]

    , q)⟩ = X i• (−1)ϵ(i•)⟨fm, F(i •)⟩⟨hn, H(i•)⟩,(E8) where (−1)ϵ(i•) is the orientation sign of the correspond- ing Steenrod shuffle

    (Section 2) ⟨fm ⌣ k hn,(0,1, . . . , q)⟩ = X i• (−1)ϵ(i•)⟨fm, F(i •)⟩⟨hn, H(i•)⟩,(E8) where (−1)ϵ(i•) is the orientation sign of the correspond- ing Steenrod shuffle. It is the sign of permutation to connect two ordered array’s (0, . . . , i0, i 1, . . . , i2, i 3, . . . , i4, . . .) (i0, . . . , i1, i 2, . . . , i3, i 4, . . . , i5, . . .) →(0, . . . , i...

    work page

  19. [19]

    This is aZ 2-valued cocycle, and its cohomology class depends only on the cohomology class ofz n

    Steenrod squares For aZ 2-valuedn-cocyclez n, the Steenrod square is represented on cochains by Sqk(zn) =z n ⌣ n−k zn. This is aZ 2-valued cocycle, and its cohomology class depends only on the cohomology class ofz n. The Steenrod squares also obey the Adem relations. For 0< a <2b, SqaSqb = ⌊a/2⌋X j=0 b−j−1 a−2j Sqa+b−jSqj,(E15) where the binomial coeffici...

    work page

  20. [20]

    Ifc n is closed, this reduces to Sqkcn =c n ⌣ n−k cn

    Generalized Steenrod squares For an arbitraryn-cochainc n, we define Sqkcn ≡c n ⌣ n−k cn +c n ⌣ n−k+1 dcn.(E25) The sign of the second term is chosen to be compatible with (E10). Ifc n is closed, this reduces to Sqkcn =c n ⌣ n−k cn. For aZ 2-valued cocycle,Sq k therefore reduces to the usual Steenrod square Sq k. We now compute its coboundary. In (E14), s...

    work page

  21. [21]

    , An defining the ho- motopy groups of the target: π1(K•) =G, π j(K•) =A j,2≤j≤n

    A sequence of groupsG, A 2, . . . , An defining the ho- motopy groups of the target: π1(K•) =G, π j(K•) =A j,2≤j≤n

    work page

  22. [22]

    , n, a group homomorphism αj :G→Aut(A j) 35 x01 1 x12 1x02 1 pt pt pt x012 2 FIG

    For eachj= 2, . . . , n, a group homomorphism αj :G→Aut(A j) 35 x01 1 x12 1x02 1 pt pt pt x012 2 FIG. 6. A triangle, or 2-simplex, in [K •]2. The labels satisfy xpq 1 ∈G, withx 12 1 x01 1 =x 02 1 , andx 012 2 ∈A 2. All vertices pt0, pt1, pt2 are identified with the single vertex pt. specifying the action of the fundamental group G=π 1(K•) on the higher ho...

    work page

  23. [23]

    A sequence of Postnikov cocyclesk 3, k4, . . . , kn+1. Inductively, suppose the (j−1)-stage K(j−1) =K α2,k3;···;α j−1 ,kj(G, A2, . . . , Aj−1) has been constructed. The next Postnikov class is represented by a cocycle kj+1 ∈Z j+1 K(j−1), Aαj j , or equivalently, by a cohomology class [k j+1]∈ H j+1 K(j−1), Aαj j . This cocycle determines the next fibratio...

    work page

  24. [24]

    Components with base degree larger thanpare invisible onE p,q 0

    TheE 0-page TheE 0-page is the associated graded object of the fil- tered cochain complex: Ep,q 0 = F pC p+q F p+1C p+q .(H1) ThusE p,q 0 keeps only the component of a total degree- (p+q) cochain with base degree exactlyp. Components with base degree larger thanpare invisible onE p,q 0 . For example, if a cochain has the form Ω =b pfq +b p+1fq−1 +b p+2fq−...

    work page

  25. [25]

    Cycles and boundaries on ther-th page Forr≥0, define Z p,q r = Ω∈ F pC p+q dΩ∈ F p+rC p+q+1 .(H2) Thus Ω∈Z p,q r means that dΩ has no terms of base degree p, p+ 1, . . . , p+r−1. Equivalently, Ω is closed up to filtration degreep+r−1. The subgroup of terms that are already trivial on the r-th page is Bp,q r =Z p+1,q−1 r−1 + dZ p−r+1,q+r−2 r−1 . The first ...

    work page

  26. [26]

    By definition, dΩ∈ F p+rC p+q+1

    The spectral-sequence differential Let Ω∈Z p,q r . By definition, dΩ∈ F p+rC p+q+1. Furthermore, d2Ω = 0, so dΩ automatically represents anr-cycle in bidegree (p+r, q−r+ 1). Therefore dΩ defines a class in Ep+r,q−r+1 r . We define dr[Ω] = [dΩ].(H4) Thus dr :E p,q r − →Ep+r,q−r+1 r .(H5) The differential dr raises the base degree byr, lowers the fiber degr...

    work page

  27. [27]

    it keeps only the classes whose outgoing d r- differential vanishes

    work page

  28. [28]

    it quotients out classes that are incoming d r- differentials. In physics language: nonzero outgoing differential⇐ ⇒the candidate topological term has an obstruction, incoming differential⇐ ⇒the candidate term is equivalent to a trivial term.(H7) 39 This is analogous to ordinary cohomology: cocycles are killed by d, while coboundaries are hit by d. A spec...

    work page

  29. [29]

    The first few pages We now spell out the first few pages in a way that is useful for computations. a.E 0-page.Since Z p,q 0 = Ω∈ F pC p+q dΩ∈ F pC p+q+1 =F pC p+q, and Bp,q 0 =F p+1C p+q, we obtain Ep,q 0 = F pC p+q F p+1C p+q .(H8) ThereforeE p,q 0 consists of cochains of bidegree exactly (p, q). b.E 1-page.The condition Ω∈Z p,q 1 means dΩ∈ F p+1C p+q+1....

    work page

  30. [30]

    However,E ∞ is not usually the cohomology group itself

    TheE ∞-page and the extension problem After all differentials are taken into account, the spec- tral sequence stabilizes to theE ∞-page. However,E ∞ is not usually the cohomology group itself. Instead, it gives the associated graded pieces of a filtration of the desired cohomology group. For fixed total degreen, there is a filtration 0 =F n+1H n(E;A)⊂F nH...

    work page

  31. [31]

    The relevantE 2-page We only need classes that can affect total degree 5, together with their nearby differential targets. On the baseK(Z 2,2), define Q4 := 1 4 P(x2)∈H 4(K(Z2,2);R/Z),(I3) S5 := 1 2 x2Sq1x2 ∈H 5(K(Z2,2);R/Z),(I4) R6 := 1 2 x3 2 ∈H 6(K(Z2,2);R/Z).(I5) Here H4(K(Z2,2);R/Z) ∼= Z4 is generated byQ 4. The classS 5 is the string self- statistic...

    work page

  32. [32]

    Hence we assign bidegrees x2 : (2,0), x 3 : (0,3)

    The Postnikov differentiald 4 In this fibration,x 2 is a base cochain andx 3 is a fiber cochain. Hence we assign bidegrees x2 : (2,0), x 3 : (0,3). The Postnikov relation dx3 2 =x 2 2 takes a cochain of bidegree (0,3) to a cochain of bidegree (4,0). Therefore it raises base degree by 4 and lowers fiber degree by 3. This is precisely the bidegree of a d 4-...

    work page

  33. [33]

    Tracking the total-degree-5diagonal We now examine the three total-degree-5 candidates: P5, M 5, S 5. a. The classS 5.The base class S5 = 1 2 x2Sq1x2 lies in E5,0 4 . It has no possible outgoing differential, because a differ- ential dr :E 5,0 r →E 5+r,1−r r would land in negative fiber degree forr≥2. It also has no incoming differential relevant to this ...

    work page

  34. [34]

    Becausex 3 is not closed onK 1, one must use the gener- alized Steenrod square Sq2x3 =x 3 ⌣ 1 x3 +x 3 ⌣ 2 dx3

    Solving the extension problem by an explicit cocycle The naive particle self-statistics term is 1 2 Sq2x3. Becausex 3 is not closed onK 1, one must use the gener- alized Steenrod square Sq2x3 =x 3 ⌣ 1 x3 +x 3 ⌣ 2 dx3. This expression reduces to the ordinary representative Sq2x3 =x 3 ⌣ 1 x3 on the fiber, where dx 3 = 0. Hence 1 2 Sq2x3 is the natural lift ...

    work page

  35. [35]

    Since Sq2x3 and ∆ 5(x2) are integer-valued cochains, multiplying their coefficients by 2 makes those terms vanish inR/Z

    The order ofΩ 5 We now compute twice this class. Since Sq2x3 and ∆ 5(x2) are integer-valued cochains, multiplying their coefficients by 2 makes those terms vanish inR/Z. Therefore 2Ω5(x2, x3) 1 = 1 4 Sq3x2.(I25) Using Sq3x2 =x 2 dx2 = 2x2B2x2, we obtain 2Ω5(x2, x3) 1 = 1 4 ·2x 2B2x2 = 1 2 x2B2x2.(I26) Again using B2x2 2,d = Sq 1x2, we find 2Ω5(x2, x3) 1,d...

    work page

  36. [36]

    J. M. Leinaas and J. Myrheim, On the theory of identical particles, Nuovo Cim B37, 1 (1977)

    work page 1977

  37. [37]

    Wilczek, Quantum mechanics of fractional-spin parti- cles, Physical Review Letter49, 957 (1982)

    F. Wilczek, Quantum mechanics of fractional-spin parti- cles, Physical Review Letter49, 957 (1982)

    work page 1982

  38. [38]

    B. I. Halperin, Statistics of quasiparticles and the hierar- chy of fractional quantized Hall states, Physical Review Letter52, 1583 (1984)

    work page 1984

  39. [39]

    Arovas, J

    D. Arovas, J. R. Schrieffer, and F. Wilczek, Fractional statistics and the quantum Hall effect, Physical Review Letter53, 722 (1984)

    work page 1984

  40. [40]

    Wu, Multiparticle quantum mechanics obeying fractional statistics, Physical Review Letter53, 111 (1984)

    Y.-S. Wu, Multiparticle quantum mechanics obeying fractional statistics, Physical Review Letter53, 111 (1984)

    work page 1984

  41. [41]

    Witten, Quantum field theory and the Jones polyno- mial, Commun

    E. Witten, Quantum field theory and the Jones polyno- mial, Commun. Math. Phys.121, 351 (1989)

    work page 1989

  42. [42]

    Moore and N

    G. Moore and N. Read, Nonabelions in the fractional quantum hall effect, Nucl. Phys. B360, 362 (1991)

    work page 1991

  43. [43]

    Wen, Non-Abelian statistics in the FQH states, Phys

    X.-G. Wen, Non-Abelian statistics in the FQH states, Phys. Rev. Lett.66, 802 (1991)

    work page 1991

  44. [44]

    Braiding statistics of loop excitations in three dimensions

    C. Wang and M. Levin, Braiding statistics of loop exci- tations in three dimensions, Physical Review Letter113, 080403 (2014), arXiv:1403.7437. 45

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  45. [45]

    Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding

    S. Jiang, A. Mesaros, and Y. Ran, Generalized Modu- lar Transformations in (3 + 1)D Topologically Ordered Phases and Triple Linking Invariant of Loop Braiding, Phys. Rev. X4, 031048 (2014), arXiv:1404.1062

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  46. [46]

    J. C. Wang and X.-G. Wen, Non-Abelian string and par- ticle braiding in topological order: Modular SL(3,Z) rep- resentation and(3+1)-dimensional twisted gauge theory, Phys. Rev. B91, 035134 (2015), arXiv:1404.7854

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  47. [47]

    Framed Wilson Operators, Fermionic Strings, and Gravitational Anomaly in 4d

    R. Thorngren, Framed Wilson operators, fermionic strings, and gravitational anomaly in 4d, J. High Energ. Phys.2015(2), 152, arXiv:1404.4385

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  48. [48]

    Gravitational anomaly of 3+1 dimensional Z_2 toric code with fermionic charges and fermionic loop self-statistics

    L. Fidkowski, J. Haah, and M. B. Hastings, Gravita- tional anomaly of 3+1 dimensionalz 2 toric code with fermionic charges and fermionic loop self-statistics, Phys- ical Review B106, 10.1103/physrevb.106.165135 (2022), arXiv:2110.14654

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevb.106.165135 2022

  49. [49]

    Z. Wan, J. Wang, and X.-G. Wen, 3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems, (2021), arXiv:2112.12148

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  50. [50]

    Kobayashi, Y

    R. Kobayashi, Y. Li, H. Xue, P.-S. Hsin, and Y.-A. Chen, Generalized statistics on lattices, Physical Review X16, 10.1103/6k88-w52n (2026), arXiv:2412.01886

    work page doi:10.1103/6k88-w52n 2026

  51. [51]

    Xue, Statistics of abelian topological excitations, Physical Review B 10.1103/g3nc-fwqg (2026)

    H. Xue, Statistics of abelian topological excitations, Physical Review B 10.1103/g3nc-fwqg (2026)

    work page doi:10.1103/g3nc-fwqg 2026

  52. [52]

    Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions

    W. Ji and X.-G. Wen, Categorical symmetry and non- invertible anomaly in symmetry-breaking and topologi- cal phase transitions, Phys. Rev. Res.2, 033417 (2020), arXiv:1912.13492

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  53. [53]

    L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang, and H. Zheng, Algebraic higher symmetry and categorical symmetry: A holographic and entanglement view of symmetry, Phys. Rev. Res.2, 043086 (2020), arXiv:2005.14178

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  54. [54]

    Fermions, strings, and gauge fields in lattice spin models

    M. Levin and X.-G. Wen, Fermions, strings, and gauge fields in lattice spin models, Phys. Rev. B67, 245316 (2003), arXiv:cond-mat/0302460

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  55. [56]

    Higher symmetry and gapped phases of gauge theories

    A. Kapustin and R. Thorngren, Higher Symmetry and Gapped Phases of Gauge Theories, inAlgebra, Geometry, and Physics in the 21st Century: Kontsevich Festschrift, edited by D. Auroux, L. Katzarkov, T. Pantev, Y. Soibel- man, and Y. Tschinkel (Birkh¨ auser, Cham, 2017) pp. 177–202, arXiv:1309.4721

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  56. [57]

    Braiding statistics approach to symmetry-protected topological phases

    M. Levin and Z.-C. Gu, Braiding statistics approach to symmetry-protected topological phases, Physical Review B86, 115109 (2012), arXiv:1202.3120

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  57. [58]

    Symmetry Fractionalization, Defects, and Gauging of Topological Phases

    M. Barkeshli, P. Bonderson, M. Cheng, and Z. Wang, Symmetry fractionalization, defects, and gauging of topological phases, Physical Review B100, 115147 (2019), arXiv:1410.4540

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  58. [59]

    Emergent (anomalous) higher symmetries from topological orders and from dynamical electromagnetic field in condensed matter systems

    X.-G. Wen, Emergent (anomalous) higher symmetries from topological order and from dynamical electromag- netic field in condensed matter systems, Phys. Rev. B 99, 205139 (2019), arXiv:1812.02517

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  59. [60]

    Dijkgraaf and E

    R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Commun. Math. Phys.129, 393 (1990)

    work page 1990

  60. [61]

    D. S. Freed and F. Quinn, Chern–simons theory with finite gauge group, Communications in Mathematical Physics156, 435 (1993)

    work page 1993

  61. [62]

    Joyal and R

    A. Joyal and R. Street, Braided tensor categories, Ad- vances in Mathematics102, 20 (1993)

    work page 1993

  62. [64]

    Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory

    Z.-C. Gu and X.-G. Wen, Symmetry-protected topologi- cal orders for interacting fermions: Fermionic topological nonlinear sigma models and a special group supercoho- mology theory, Physical Review B90, 115141 (2014), arXiv:1201.2648 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  63. [65]

    Fermionic Symmetry Protected Topological Phases and Cobordisms

    A. Kapustin, R. Thorngren, A. Turzillo, and Z. Wang, Fermionic symmetry protected topological phases and cobordisms, J. High Energ. Phys.2015(12), 1, arXiv:1406.7329

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  64. [66]

    Exactly soluble local bosonic cocycle models, statistical transmutation, and simplest time-reversal symmetric topological orders in 3+1D

    X.-G. Wen, Exactly soluble local bosonic cocycle mod- els, statistical transmutation, and simplest time-reversal symmetric topological orders in 3+1 dimensions, Phys. Rev. B95, 205142 (2017), arXiv:1612.01418

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  65. [67]

    Fermionic SPT phases in higher dimensions and bosonization

    A. Kapustin and R. Thorngren, Fermionic SPT phases in higher dimensions and bosonization, J. High Energ. Phys.2017(10), 80, arXiv:1701.08264

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  66. [68]

    Towards a complete classification of fermionic symmetry protected topological phases in 3D and a general group supercohomology theory

    Q.-R. Wang and Z.-C. Gu, Towards a complete classifica- tion of symmetry-protected topological phases for inter- acting fermions in three dimensions and a general group supercohomology theory, Physical Review X8, 011055 (2018), arXiv:1703.10937 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  67. [69]

    Symmetry as a shadow of topological order and a derivation of topological holographic principle

    A. Chatterjee and X.-G. Wen, Symmetry as a shadow of topological order and a derivation of topological holo- graphic principle, Phys. Rev. B107, 155136 (2023), arXiv:2203.03596

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  68. [70]

    D. S. Freed, G. W. Moore, and C. Teleman, Topo- logical symmetry in quantum field theory (2022), arXiv:2209.07471

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  69. [71]

    Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers

    L. Kong, X.-G. Wen, and H. Zheng, Boundary-bulk rela- tion for topological orders as the functor mapping higher categories to their centers (2015), arXiv:1502.01690

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  70. [72]

    Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases

    R. Thorngren and Y. Wang, Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases, J. High Energ. Phys.2024, 132, arXiv:1912.02817

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  71. [73]

    Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions

    L. Kong and X.-G. Wen, Braided fusion categories, gravitational anomalies, and the mathematical frame- work for topological orders in any dimensions, (2014), arXiv:1405.5858

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  72. [74]

    Bulk Anyons as Edge Symmetries: Boundary Phase Diagrams of Topologically Ordered States

    T. Lichtman, R. Thorngren, N. H. Lindner, A. Stern, and E. Berg, Bulk anyons as edge symmetries: Boundary phase diagrams of topologically ordered states, Physical Review B104, 075141 (2021), arXiv:2003.04328

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  73. [75]

    Orbifold groupoids

    D. Gaiotto and J. Kulp, Orbifold groupoids, J. High En- erg. Phys.2021(2), 132, arXiv:2008.05960

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  74. [76]

    Symmetry TFTs from String Theory

    F. Apruzzi, F. Bonetti, I. G. Etxebarria, S. S. Hosseini, and S. Sch¨ afer-Nameki, Symmetry TFTs from string the- ory, Communications in Mathematical Physics402, 895 (2023), arXiv:2112.02092 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  75. [77]

    Emergent generalized symmetry and maximal symmetry-topological-order

    A. Chatterjee, W. Ji, and X.-G. Wen, Emergent general- ized symmetry and maximal symmetry-topological-order (2022), arXiv:2212.14432

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  76. [78]

    Doplicher, R

    S. Doplicher, R. Haag, and J. E. Roberts, Fields, observ- ables and gauge transformations. I, Communications in Mathematical Physics13, 1 (1969)

    work page 1969

  77. [79]

    Doplicher, R

    S. Doplicher, R. Haag, and J. E. Roberts, Fields, observ- ables and gauge transformations. II, Communications in Mathematical Physics15, 173 (1969)

    work page 1969

  78. [80]

    T. Lan, C. Zhu, and X.-G. Wen, Fermion decoration con- struction of symmetry protected trivial orders for fermion systems with any symmetriesg f and in any dimensions, 46 Phys. Rev. B100, 235141 (2019), arXiv:1809.01112

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  79. [81]

    Spin TQFTs and fermionic phases of matter

    D. Gaiotto and A. Kapustin, Spin TQFTs and fermionic phases of matter, Int. J. Mod. Phys. A31, 1645044 (2016), arXiv:1505.05856

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  80. [82]

    Symmetry and Topological Order

    Z. Nussinov and G. Ortiz, Sufficient symmetry condi- tions for topological quantum order, Proc. Natl. Acad. Sci. U.S.A.106, 16944 (2009), arXiv:cond-mat/0605316

    work page internal anchor Pith review Pith/arXiv arXiv 2009

Showing first 80 references.