JHEP10(2017)080 Springer April 12, 2017 : October 11, 2017 : September 28, 2017 : Received Published Accepted . It generalizes the supercohomology G Published for SISSA by https://doi.org/10.1007/JHEP10(2017)080 [email protected] , b . 3 and Ryan Thorngren 1701.08264 a The Authors. c Anomalies in Field and String Theories, Effective Field Theories, Topological

We discuss bosonization and Fermionic Short-Range-Entangled (FSRE) phases , [email protected] Walter Burke Institute forPasadena, Theoretical Physics, U.S.A. California Institute ofDepartment Technology, of Mathematics, UniversityBerkeley, of U.S.A. Calfornia, E-mail: b a Open Access Article funded by SCOAP Field Theories, Topological States of Matter ArXiv ePrint: dimensions with an arbitrary finiteof point Gu symmetry and Wen. We argue“shadow” that by the condensing most both general such fermionic phase particles can and be obtained strings. from aKeywords: bosonic Abstract: of matter in one,the two, cohomological and parameters three which label spatial suchsymmetries. dimensions, phases emphasizing and We the propose the connection physical a with meaning classification higher-form of scheme for fermionic SPT phases in three spatial Anton Kapustin Fermionic SPT phases inbosonization higher dimensions and JHEP10(2017)080 11 6 3 19 29 41 34 40 15 21 symmetry symmetry – i – 2 -crossed 2-Ising 2 Z G Z 16 36 37 10 1 12 31 23 36 5 8 11 3 6 33 38 22 27 ) 25 B 21 ( ζ symmetry in 3+1d symmetry in 2+1d Q symmetry 2 2 E Z Z -crossed 2-Ising 5.1.1 State5.1.2 sum G B.1 1-form B.2 2-form 5.2 Fermion number 5.3 5.4 Super-cohomology phases from 4.3 The ground4.4 states of the More CYKWW general4.5 model 3d FSRE phases Bosonization of 3d FSRE phases and 3-group symmetry 5.1 2-Ising theory 3.3 The string-net ground state 4.1 Bosonization in4.2 3d Supercohomology phases 2.1 Bosonization in2.2 1d FSRE phases in 1d 3.1 Bosonization in3.2 2d FSRE phases in 2d D ’t Hooft anomalies for aE 1-form ’t Hooft Anomalies for a 2-form B Anomaly descendants C The function 6 Fermionic string phases 7 Concluding remarks A Steenrod squares and Stiefel-Whitney classes 5 Bosonic shadows of 3d FSRE phases 4 Bosonization and FSRE phases in three spatial dimensions 3 Bosonization and FSRE phases in two spatial dimensions Contents 1 Introduction and summary 2 Bosonization and FSRE phases in one spatial dimension JHEP10(2017)080 ]. of 44 1 14 ]. The drawback of 24 ] (the abelian group structure 18 , 4 ]). Gu and Wen also constructed a ]. Z 17 which has an inverse. SRE topological 14 , 13 , × G form an abelian group. 2 11 is an equivalence class of gapped quantum 12 , Z G G 10 = , ]. In terms of ground states, the equivalence 45 42 7 5 – 1 – G , 3 into a commutative unital semigroup, a set with an G . One can study either ground states or Hamiltonians. G symmetry . Topological phases of matter can be “stacked together”, G ]. In the interacting case, there is a fairly complete picture E ] that FSRE phases can be classified using spin-cobordism 2 , 15 1 . This conjecture is supported by a recent mathematical result G is a topological phase with symmetry spatial dimensions with symmetry G d symmetry of the stack to be the diagonal one. This operation makes the set of It was conjectured in [ SRE topological phases are interesting in part because they are more manageable G This is a refinement of the supercohomology proposal of [ 1 well understood. In particular,to given construct a a spin-cobordism lattice classNeither fermionic it is system is it not which clear clear which belongscobordism physical in to classes. properties general the distinguish how corresponding systems corresponding FSRE to phase. different But it is clear by now that this constructionthe does classifying not space produce of allthat possible relates FSRE (spin) phases. cobordisms withthis unitary approach invertible is (spin) that TQFT the [ relation between TQFTs and topological phases of matter is not ie. equivalence classes of quadraticin Hamiltonians all of spatial hopping dimensions fermions,of [ have FSRE been phases classified inon dimensions the 1 set and of 2large 1d [ class FSREs of was recently FSRE studied phases in in [ all dimensions using the “supercohomology” approach [ symmetry phases in than general topologicalFermionic phases SRE but topological phases still (FSRE phases) retain are many particularly rich. interesting Free topological FSRE phases, properties. ground state commuting with by taking tensor productthe of Hilbert spaces, Hamiltonians,topological and phases with ground a states, symmetry associative and and taking commutative binarywith operation an and inverse a for neutral every element, element. but A not short-range-entangled necessarily (SRE) topological phase with A topological phase of matter withlattice a symmetry systems with aFor symmetry classification purposes, itrelations is are the of same two kinds: [ paramagnet tensoring (this with adds a new product degrees state, of eg. freedom), the and ground local state of unitary a transformations trivial of the 1 Introduction and summary G Vanishing of the obstructions for H Fermion parity of the Kitaev string F ’t Hooft anomalies for JHEP10(2017)080 G spatial can get d F 1) − ]. ]: 11 5 symmetries, with , , 3 2 10 symmetry generated Z ) (1.1) 2 2 Z Z G, ( 0 H 1)-form − × d ) 2 symmetry must have a particular ’t ]. The physical significance of each Z 2 Z 21 G, , ( as well as a simple fermionic system. In 1 20 2 H Z symmetry, i.e. by coupling the bosonic system × – 2 – ) 2 Z Z / R gauge field, gives the bosonic shadow of a general 3d G, ( 2 G ] generalizing [ H 10 [ ∈ ) symmetry. This is usually explained using the Jordan-Wigner BG 2 ] it was argued that one can obtain fermionic systems in Z ν, ρ, σ ( 10 -form gauge field valued in d ]), which may have a fermionic ground state depending on how it is embedded 1 The organization of the paper is as follows. In sections 2 and 3 we recall topological In fact, we will argue that in 3d a new phenomenon occurs which makes the bosoniza- The main goal of this paper is to extend some of these results to dimension 3. Our In dimensions 1 and 2 these problems have been solved, at least in the case when propose a classification of 3dwhich FSRE can phases. serve In as sectiona a 5 we bosonic new write shadow class down for a ofcontain 3d 3d 3d bosonic “fermionic FSRE phases model phases. strings”. which Inhigher-dimensional seem In section generalizations. to section 6 be 7 we neither briefly we bosonic discuss summarize nor our fermionic, results although and they discuss possible FSRE phase. This theory istopological interesting order, in analogous its to own the right Ising as anyons a in very simple 2+1d. bosonization non-abelian in 3+1d one and twoWe spatial also dimensions interpret and the classification how in it termsin is of a used properties broken to of symmetry domain classify walls phase. FSRE and phases. In their junctions section 4 we describe our proposal for 3d bosonization and means that the bosonica shadow nontrivial has “interaction” both betweenFSRE 2-form them, phase. and and We propose 1-form both a global needfor generalization of to this the new be Gu-Wen supercohomology phenomenon. gauged whichwhen We accounts in also coupled order write to to down a a get concrete background an 3d lattice bosonic model which, tion approach a bitcontributions more from involved. both particle Namely,written and the in string a fermion states, local parity andchain way. operator the [ Microscopically, string ( these contribution strings cannotinto carry be space. a 1d We call FSRE phase these (the objects Kitaev Kitaev strings. From the mathematical viewpoint, this system can be recovered byto gauging a the dynamical 1d and 2d everynatural FSRE to phase conjecture arises that in this this is way also from true a in suitable higher bosonic dimensions. system, and it is system with a global transformation. In [ dimensions starting from bosonic systems withby a a global ( fermionic quasiparticle.Hooft More anomaly precisely, which the is trivialized when the spin structure is introduced. The fermionic member of thesystem). triple Similar is results understood for (they 2d FSRE describe systems properties have beenapproach of is obtained the based in on [ edge the ideaevery modes of lattice bosonization/fermionization. of fermionic It the is system a in well-known result one that spatial dimension corresponds to a lattice bosonic acts unitarily. For example, 1d FSRE phases are classified by a triple [ To each such triplespin-cobordism one can class assign of a concrete integrable lattice Hamiltonian as well as a JHEP10(2017)080 : 2 P Z . (2.2) (2.1) A 1. We will  symmetry. On 2 are Z F 1) gauge field. To obtain − -untwisted sector and a 2 2 Z Z . are defined only as elements − R ]. This tranformation is not . 8 F − 1 ⊕ F ]. While the massive fermion on 9 + ⊕ B R F + 1 B = = R 1 F . Each of these can be further decomposed B 1 , B , – 3 – − NS − 0 and ⊕ F 0 ⊕ B B + 0 + NS B F = = 0 symmetry, it can be coupled to a B 2 NS Z global symmetry: F 2 gauge fields.” The scare quotes indicate that certain topological Z ] (a discretization of the massive Majorana fermion) to the quantum 2 5 Z , and each of them decomposes into eigenspaces of the fermion parity R F . The eigenvalues of the fermion parity operator ( Z and 2 / Z NS F = 2 One can describe the connection between bosonic and fermionic Hilbert spaces on a A. K. would like to thank Greg Brumfiel, John Morgan and Anibal Medina for com- We will be interested in models where the fermion number is conserved modulo 2. Z -twisted sector, which we denote 2 On the other hand, thedenote fermionic Hilbert space has an NS sector and a R sector, which we terms are important in these sums. circle in complete generality.Z The bosonic Hilbertinto space eigenspaces has of a the a circle requiresother a hand, spin since structure, itthe has the a bosonic corresponding Hilbert bosonicstructures”. space system Conversely, from the does fermionic the not. Hilbertby fermionic space “summing can Hilbert On be over the space obtained from one the has bosonic one to “sum over spin degenerate ground state onthe a other circle hand, thanks the to groundthe spontaneous state fermion breaking of the parity of Majorana cannottransformation chain be as on a “gauging spontaneously circle theconsiders broken. is fermion the always unique, It parity”. bosonization since is transformation This on best becomes a to more circle think obvious [ about when the one JW It is well-known thatsymmetry 1d by means fermionic of systemsan the can equivalence, Jordan-Wigner as be transformation it mapped [ the does to Majorana not chain bosonic preserve [ systems certainIsing physical chain. with properties. Depending on the For values example, of the it parameters, maps the latter model can have a doubly- of this project. 2 Bosonization and FSRE2.1 phases in one Bosonization spatial in dimension 1d municating to him someWilliamson, of Dave their Aasen, unpublished and results.was Ethan supported R. Lake in T. for part would many byEnergy like enlightening the to Physics, discussions. U.S. thank under Department This Dominic Award ofported paper Number Energy, by Office DE-SC0011632. the of Simons Science, The Investigator OfficeK. work Award. of and of R. R. High A. T. T. K. is are supported was grateful by also to an KITP, sup- NSF Santa GRFP Barbara, grant. for hospitality A. during the initial stages Accordingly, the eigenvalues of the fermionof number operator freely use simplicial cochainsproperties and of operations Steenrod on squares them, and including Stiefel-Whitney Steenrod classes are squares. recalled Some in appendix JHEP10(2017)080 , 2 + R Z (2.3) gauge ⊕ F 2 Z + NS -odd one), F 2 Z 1 component − = F . 1) + 1 − B = 1 in both cases. = . , one needs to project to a acts in the sector with the (paramagnet) and a gapped 2 + 1 F -even one and a − R 2 2 2 Z F 1) Z Z Z ∈ ⊕ B − = 1, while for a negative mass it , − 1 α F is not a bosonic SRE phase. Thus − 0 , while the ( B ] and the quantum Ising chain. The 2 1) B − 0 Z 5 − = 0 corresponds to the NS sector and , = 1 s ⊕ B + R + 0 F gauge field (a B 2 2 , Z – 4 – − 1 so that . Note that the weight is a (exp-)linear function ]) and thereby flip the fermion parity of the Ramond-sector (ferromagnet). Consider the limit of an infinite B α 2 1 + 2 Z s gauge field on a circle. = Z 2 1) ∈ . For each value of symmetry and identifies the holonomy of the − gauge field is turned on because in the limit of infinite NS Z − s P 2 F 2 gauge field. On the other hand, in the broken phase, the . Z Z , while the negative-mass Majorana chain is mapped to the 2 2 2 , Z Z Z + 0 B -charge to select either the NS or R sector states: more precisely, = 1 component of the total fermionic Hilbert space, = 2 F Z + NS 1) F − with the weight ( α 1. The ground state in the NS sector has ( − 1) as the fermion parity − = , is the twisted sector of the bosonic theory α F − R 1) 1) − − ⊕ F When considering 1d systems on a circle, it is easy to mistake a spin structure for a The JW transformation maps the positive-mass Majorana chain toNote the that Ising the Majorana chain chain (for either sign of the mass) is an FSRE, but the quantum As an example, consider the Majorana chain [ These relations can be interpreted as follows: to get the fermionic Hilbert space from There is an ambiguity here, since we can tensor an arbitrary fermionic phase with a nontrivial fermionic 2 = 1 corresponds to the R sector, then the generator of − NS between the partition functions of the fermionicSRE theory phase (the and negative-mass Majoranastates its chain [ while bosonic leaving the “shadow”sum NS over on sector spin structures a unaffected. and reversesour This the conventions amounts correspondence so to between that fermionic multiplying higher-dimensional and by generalizations bosonic the phases. are Arf more We invariant choose straightforward. when we bosonization and fermionization do notin map higher SRE dimensions, phases as to we SRE will phases. see. This alsogauge applies field. The distinctiona between curved them space-time becomes with a clearer nontrivial when topology. we consider It systems will on be useful to write down a relation with a spontaneously broken Ising chain with an unbroken Ising chain in a phase with a spontaneously broken chain also has twothe phases, fermion depending mass on in theground sign the state of for continuum either the limit. choicepositive parameter of mass For which the the both corresponds spin Ramond-sector signs to structure groundhas of on state ( has a the ( circle. mass, The there difference is is a that unique for a phase with a spontaneouslyenergy gap. broken Then inthe trivial the and unbroken the phase nontrivial system the has two system ground has states with aand a trivial unique no ground ground states stateenergy when both gap the for the energy of the domain wall between the two vacua is infinite. The Majorana s holonomy ( both of the spin structure and the quantum Ising chain has a gapped phase with an unbroken the bosonic one, one gaugesfield the ( particular value of the if we label the spin structures by In particular, the ( is the untwisted sector ofF the bosonic theory These decompositions are related as follows: JHEP10(2017)080 ) G α ( δ (2.6) (2.4) (2.5) ) we let 2 . One says Z M M, = 1 and a sin- (i.e. an element ( ) = 1) with the 1 σ g α ) tells us whether ( H 2 (i.e. it is the trivial ρ ] using bosonization Z ), one finds that the ∈ . Note that the set of η 5 ]: every spin structure b 0 , ] has G, ). All of these param- 3 . 2.6 ∪ 1 25 ( ) ) is a delta function 1 a α α, α ( 1.1 α η H ( M q b b. R gauge field ∈ Z 1) . ∪ 2 0 7→ times fermion parity, rather than an − ρ a Z α ) , such that )( G 2 ∪ M α Z Z ( α a, b b is as follows [ Z M → ) ) = Z M ) 2 b 2 Z ( η Z q ) + M, 0 ( 1 ], where the same results were obtained using X α − ) = 0) or anti-commutes ( M, ( H ( g ) – 5 – η 1 ∈ ( a q 13 α ρ , ( H η 2 : q 12 / ) ) = η 0 − q M 1 α ( ) ( 1 b b α 2 + + ) of group cohomology classes ( η a q ] (i.e. it is the Kitaev chain). This agrees with what we expect ( commutes ( ) = η . The fermionic partition function depends on a spin structure 15 q η G ( gauge fields is an abelian group. We note for future use that the M σ, ρ, ν f ∈ 2 Z is the number modulo two of Majorana zero modes at each edge Z g 2 Z ) = by triples ( 2 by fermion parity. The generalization to nontrivial extensions is straightforward. )). Z 2 3 G = 0 in the ferromagnetic phase (because of the infinite energy of the domain Z G, ( α is a quadratic refinement of the bilinear form ( 0 M, η ( , while the bosonic partition function depends on a H q 1 In our example of the Ising/Majorana correspondence, Given this relation between spin structures and quadratic refinements, the relation A nice way to think about a spin structure on For simplicity, we are assuming that the total symmetry is M 3 H ∈ gives rise to a quadratic function of the system.gle For example, Majorana the zero negative-mass modea Majorana particular at chain element every [ edge. The parameter extension of and Matrix Product States.fermionic See MPS. also [ The resultis is classified that theeters set can of be FSRE interpretedσ phases in with terms a of unitary properties symmetry of the edge zero modes. The parameter invariant of spin structure [ from the microscopic JW transformation. 2.2 FSRE phases inFSREs in 1d 1d (with arbitrary interactions) have been classified in [ setting wall) and the constantfermionization 1 of in the former the hasphase), paramagnetic a while phase. partition the function Applying fermionization independent of ( of the latter has the partition function which is the Arf between the partition functions can be written as a nonlinear discrete Fourier transform: equivalence classes of latter group naturally acts on the set of spin structures: for all Conversely, every such quadratic function correspondsthat to a spin structurespin on structures is not anon abelian the group set (there is of no quadratic natural refinements way of to a define fixed a bilinear group form). operation On the other hand, the set of on of η general Riemann surface JHEP10(2017)080 ) 2 λ Z Z / = 0. R ) tells σ g via the tells us = 1). ( G, = ( ρ σ there is a σ 2 ρ BG G H ∈ ∈ symmetry is a g ν 2 1-form symmetry. Z ) tells us whether it 2 g , defined up to a ( Z 2 ρ is broken, turning on a Z G ]. The parameter = 0) or fermionic ( 27 σ Σ. There are two spin structures ∂ on the edge zero modes. If any one of ) is a phase attached to a junction of 2 G , g gauge field and consider the ground states on 1 g ( G – 6 – ν ). When the symmetry σ gauge field. Let us imagine that the IR limit of ]. It is helpful to introduce a nontrivial spacetime ]. A parameter of a global 1-form 13 G . 28 2 g (and an arbitrary spin structure). For any describes the “S-matrix” of the domain walls obtained when G and ν ∈ 1 g g when acting on the edge zero modes. The parameter F 1) is spontaneously broken. To be more precise, let us assume that − G around the circle leads to a particle-like domain wall; the paremeter Σ. For a fermionic system, this requires choosing a spin structure on Σ, which g ∂ Together, these parameters define a 2-dimensional spin cobordism class of Finally, the parameter One can also couple the system to a flat It is instructive, although somewhat nontrivial, to interpret these parameters without gauge field, i.e. a 1-cocycle (Cech or simplicial) with values in 2 a 2+1d bosonic systemLet (its us bosonic remind “shadow”) whatZ with this an means anomalous [ gauge transformation (i.e. up topreserve adding the an action, exact but 1-cocycle). cannot This be symmetry gauged. is assumed That to is, one cannot promote the parameter Atiyah-Hirzebruch spectral sequence. 3 Bosonization and FSRE3.1 phases in two Bosonization spatial in dimensions Recently, 2d it has been shown that a 2+1d lattice fermionic system can be obtained from Then all domain wallswalls, are one bosonic, and should sincedomain-wall be the worldlines. able theory to isdomain The trivial compute walls away parameter the labeled from by partition the function domain by summing over possible parity of the ground stateholonomy is shifted by us whether it is bosonic or fermionic. the symmetry whether the ground state in the Ramond sector isa bosonic circle with ( a holonomy unique ground state (again by theis SRE bosonic assumption). a The fermionic parameter for the NS spin structure (for the Ramond spin structure, the fermion as non-bounding and boundingon spin a structures, circle respectively, canRamond since be spin the obtained structure NS cannot by spin bethe restricting structure so ground the obtained. state unique Since on spinone we a structure can are circle on show dealing is that with a non-degenerate in an disk, for the FSRE while either NS phase, the choice sector of it the is always spin parity-even structure, [ and geometry and athe fixed system background isWick-rotate described it by and a place itboundary unitary on continuum an 2d arbitrary Riemann quantumalso surface induces field a Σ, theory, spin perhaps structure then withon on a we a each nonempty circle: boundary can circle periodic in (Ramond) and anti-periodic (Neveu-Schwarz). They are also known controls the projective naturethese of parameters the is non-vanishing, action thetherefore of system the must ground have state nontrivial on edge an zero interval is modes, degenerate and appealing in to the large-volume the limit. edge zero modes [ fermion parity ( JHEP10(2017)080 (3.5) (3.4) (3.1) (3.2) which 2 . Thanks Z by a factor = 1 sector C → F δλ ) symmetry acts 2 1) ) introduced by ) corresponding + Z 2 2 2 are in one-to-one − B Z Z Z Y, Y ) while maintaining ( ) (3.3) Y, → Y, ). 2 ( ( 2 δλ δλ 3 Z 2 , B Z 0 : ∪ C Z + on a triangulated closed B ζ Y, λ = 1 is identified with the ( B ). This action is projective ζ 1 → Q 2 ) 2 + ∪ ) B B Z is needed to cancel the gauge → H 2 ( depends on additional choices: λ B ζ Z ) M ζ B Q ∪ . R Y B M, 0 ( Q ( Y, Z λ . 1) ζ ( B 1 ) + 2 Q − ). By fixing a gauge, we can assume Z λ δλ + 2 are discussed in appendix C 1) )( ) = U ∪ 0 0 Z λ ζ B − B λ × is a closed Riemann surface. There are B + ( but depends on these extra choices: transforms under ∪ Q ) ( ∪ b λ λ 2 M, ζ ∪ ζ Z ( λ Y M Z M B ) ( Q 1 R 2 = 0 is identified with the ( + ]. We get C Y Z Y, − 1) B ( Z – 7 – Y, ) B 2 ∪ ∈ 11 ( − λ 2 B ( C λ M ( X H ) + Y R , ζ , where = ( R ∈ ] λ B which transforms as 0 Q R ( B and [ 1) U λ [ ζ 2 − − U × Z B Q ( ∼ λ ) 0 ]. ] that to every spin structure ) U M B ζ ). One can show that spin structures on 37 ) is a well-defined function on 10 ( ) = is independent of = f + ζ A δλ Z 3.2 Q B Y ( + and [ ζ one can associate a quadratic function ] that one can obtain the fermionic partition function by performing ) says that the fermionic theory is obtained from the bosonic one B Q C ( ζ Y transforms as 11 3.2 1-form symmetry. The factor ( , Q 2 10 δλ Z -valued 1-cochain even at the expense of introducing a 2-form gauge field 2 + . The untwisted sector ) on a closed oriented 3-manifold Z ] (see appendix B B M 1 sector of the fermionic Hilbert space. The gauge 1-form ( b 36 is a certain bilinear operation − → Z 1 = is closed, so that each sector is acted upon by ∪ ), see appendix ), the summand in ( B λ F The equation ( Unlike in 2d, the definition of the quadratic function 3.3 3.3 through 1) (i.e. a 2-cocycle with values in − in each sector by unitarythat operators because of the ’tstandard procedure, Hooft see anomaly. appendix The corresponding 2-cocycle is computed following a a space-time of thetwo form sectors in the gauged bosonicB theory distinguished by theof flux of the the fermionic 2-form gauge Hilbert( field space, while the twisted sector correspondence with quadratic refinements ofto this ( bilinear form which transform according by gauging the anomaly. It is instructive to see how the gauging works on the Hamiltonian level. Consider where Steenrod [ The construction and propertiesto of ( the function a branching structure on theto triangulation. the quadratic The function bilinear form on Here we use anoriented observation 3-manifold [ under It was shown in [ a nonlinear discrete Fourier transform: B gauge invariance. The anomaly offunction a bosonic shadow has a very specific form: the partition to a general JHEP10(2017)080 ) g ( and σ λ (3.8) (3.9) (3.6) (3.7) U ). One ) 2 λ ( , namely Z η λ q U Y, 1) ( − tells us which 1 2 Z = ( = 0. Then the Z ∈ λ ˜ σ U , ]. They are labeled λ → ) , 2 , and the number of 11 G λ Z to each triple junction, gh : + 2 D σ b Z BG, ( ∈ 7→ 1 . ) ). Z 2 Z × λ, b i = 0 g, h ) ( produces M, + , , η 2 ρ ( ) 1 a Z h Ψ h | the domain wall corresponding to the Z ( ) D , δσ λ g σ 7→ ∈ ( η D BG, a λ q ( = 0 ) + 2 and 1) bda g ) can be trivialized by defining Z g ( 4 − Y σ ). D for all Z × 3.5 – 8 – ) λ = ( 1 2 ˜ 2.4 ) is most easily interpreted if U Z ρ, δρ i ) = 2 / excitations of the toric code. The ground states of ) in physical terms. As stated above, a nonzero ∪ Z , η R gh ( ρ m Ψ | σ 1 2 G, λ ( 2 ν, ρ, σ BG, U = ( and Z 2 e δν ]. In particular, the homomorphism C ∈ of the background gauge field carries a Majorana zero mode (or ) thanks to ( 11 ∈ acts as particle-vortex symmetry of the toric code. This implies ) [ g ) g 3.5 symmetry which acts by G g, h 2 ( ρ Z ν, ρ, σ ( . Note that since fusing = 1. As in the 1d case, the sector corresponding to a particular spin struc- g 2 exchange the λ U is obtained by decomposing the Hilbert space into eigenspaces of G M on is a homomorphism. η The parameter One can interpret the data ( σ Alternatively, one can say that the cocycle in ( 4 endpoint of eachassume domain that wall the carriesdomain endpoint no walls, has fermionic we might fermion zero need parity modes, to assign zero. and fermion one parity Butrequiring might physical when as states considering to well be networks invariant under of Majorana zero modes must be preserved modulo 2, we must have i.e. more precisely, an odd numberthis of insertion Majorana becomes zero anwall modes). endpoint carries of In a a the negative-mass domain symmetry-broken Majoranathis wall, phase, chain. property and In a thus what Kitaev the followsgroup string. corresponding we element domain Let will call us a denote 1d by defect with and a global 1-form can check that this 1-form symmetry has the rightmeans ’t that Hooft the anomaly. element that an insertion of a flux antized with respect to elements of the toric code can be described by a topological action The first two of theseThe are bosonic the shadow Gu-Wen of equations all which these describe FSRE supercohomology phases phases. can be taken to be the toric code equivari- 3.2 FSRE phasesLet in us 2d now recall theby triples classification of 2d FSRE phases proposed in [ which satisfy the equations ture This is consistent with ( In particular JHEP10(2017)080 ), ). 2 Z g, h (3.11) (3.10) ( ρ M, ( 1 H ∈ ) ) is a 2-cocycle )) with the con- A ( A ( g, h σ ( σ ρ ( η q . ) gh ( f ) + h ( . Note that it is nonlocal, as expected, ). Then the spin structure induced on f F A ρ. ( ) + σ ∪ ) is a 2-cocycle defined up to a coboundary. g ρ ( 1 2 f g, h – 9 – ( = ρ ) + δν has the same meaning as in the bosonic case, i.e. it has a physical meaning. g, h ( . Assuming that the fermion number depends only on ρ ρ Z / without changing the net fermion parity of the network. M )) = 1. Therefore we can identify R 7→ 2 A ) Z ∈ ( is nontrivial, some domain walls are Kitaev strings and con- σ ) ∈ meet. Requiring that the fermion number of the network does ). They act irreducibly on a fermionic Fock space, and one can ( g, h ) η σ ( 1 ]. The resulting constraint is the Gu-Wen equation q g ρ − 3.9 ( g 11 1 f = 0. g, h, k − ( h ν by δρ D ). Equivalently, one may consider the surface of a tetrahedron, and gauge field on 2 g ) by a coboundary: Z D G whose homology class is dual to and ) is non-vanishing, the situation is not very different. The key point is that g, h , g ( ( h M ρ σ D , is the g on A D γ When the parameter The parameter When Note also that since every domain wall has two ends, we can shift the fermion parity of is Ramond precisely if the homology class ofcurve the string, weγ may assume thattribution the of Kitaev Kitaev string strings to wrapsand the conserved. a fermion This number closed explains why we could ignore it when identifying the fermion number that a homologically trivialcause Kitaev the string spin automatically structure carries inducedNS zero on type. fermion it Therefore by number, the the be- cal contribution spin of and structure the depends in Kitaev on the strings thefirst ambient to that homology space the the class is homology fermion of class of number the ofwhere the is the string nonlo- Kitaev network. strings To is determine the its Poicare-dual of form, note sequently carry fermion number when wrapping cycles with Ramond spin structure. Note into a union ofalong four every tetrahedra, 2-face of passbecause this to such cell the a complex. dual configurationother On cell of hand, the domain complex one walls one and can hand, can insertdomain be the evaluate wall a created amplitude it junctions domain out muct [ taking wall be of into a trivial, account trivial the one. fermionic On statistics the of the triple one of these statesThe the same arguments ground as state. above show The that fermion paritydescribes of the this amplitude ground assignedin state to space-time. is a point-like To derive junction a of constraint four on domain it, wall one worldsheets needs to consider a 3-sphere triangulated Thus only the cohomology class of at the junction ofthanks three to domain the condition walls ( weimagine have turning an on a even local number interaction of at Majorana the zero junction modes, that lifts the degeneracy and makes the endpoint of This shifts not change under Pachner(with moves, one values gets in aregard constraint each saying edge that asthe a interior domain of wall. the Since tetrahedron,gives this the the network fermion condition can parity be of the consistently network continued must into vanish. This again where JHEP10(2017)080 2 2 ), 1, 1. Z Z C − ( ' Z (3.13) (3.12) m ◦ . symmetry ) m 2 2 ' Z Z as follows: . The string- e via the Atiyah- ν ◦ M is a fermion. 1. One can think e BG, ( 1. Such an object BG ψ 1 ' graded vector spaces C ' F 2 is invariant under local , and ∈ ◦ ψ Z i ψ ◦ γ F | ' , f gauge theory, also known as ψ with a chosen triangulation. ) 2 Z m Z / M ◦ R e with the boundary. The toric code . symmetry is a bulk line defect and i ψ 2 BG, γ | ( Z 2 ]) ) of the state C γ . γ ([ is the category of by coboundaries and shifts ( ∈ , such that C C σ C ) BG m 2 , h Z = 1) and thus correspond to non-anomalous = 1. It measures the anomaly of the 1-form ) – 10 – , with the fusion rule ]. This is a warm-up for a similar discussion of 3d f ) with a fusion rule and M, 4 ψ m F ( C and θ 11 θ ∪ 1 ρ X ( e Z ρ Z = ∈ γ e + ∈ θ linearly independent ground states on ). Thus a general ground state is ρ 2 | ψ ∪ ) Z as the result of fusing 2 f : in 2d FSRE phases, the particle and strings contrubutions Z F M, + B ( 1 M, ). A 1-cocycle on a triangulated surface can be thought of more M ( . A generator of a 1-form which satisfies 2 δf R H 1 C Z ∪ ψ are bosons ( H ∈ θ | f ] ( M, m γ ( ] that equivalence classes of such triples are in 1-1 correspondence with 1 2 . This relation shifts 1 37 Z + and BG ∈ e δh γ . Its objects can be thought of as boundary line defects for a particular boundary + C ν must be a fermion. 7→ Let the spatial slice be a closed oriented 2d manifold The simplest 2+1d TQFT with these properties is the As in the 1+1D case, these triples define spin cobordism classes of ψ ν , where i γ states have the propertyrearrangements that the of coefficient the string-netcohomology which class do [ not change its homology class, or dually, the The toric codenet has construction describes these| ground states as particulargeometrically linear as a combinations 1-cycle of on states net”. a dual cell The complex, string-net i.e. a Hamiltonian collection of is closed a curves, a commuting “string projector Hamiltonian whose ground of the boundary line defect has two more irreducible lineThe defects, objects symmetries, but since they are muutually nonlocal, their bound state symmetry. Since we want thei.e. anomaly to be of order 2, thethe topological toric spin must code. be and The has corresponding two category irreducible objects: 1 and category condition. Bulk line defectsthe are Drinfeld described center by of thus objects corresponds in to a modular anhas tensor object topological category spin 3.3 The string-net groundIn this state section we discuss groundof states the of trivial 2d a FSRE simple phaseFSRE lattice following model phases [ which in is later a sections.with bosonic We shadow the need a correct bosonic anomaly. TQFT which A has a bosonic 1-form TQFT can be constructed from a spherical fusion It is shown inthree-dimensional [ spin-cobordism classes of to the fermion number are separately conserved. Hirzebruch spectral sequence. There isensuring a that certain equivalence the relation set oncobordism the of of triples equivalence as classes well, is isomorphic to the three-dimensional spin- with a local expresson JHEP10(2017)080 bulk ). (3.15) (3.17) (3.19) (3.20) m ν, ρ, σ gauge field. and G e ) (3.18) 2 . ) (3.14) ) Z 2 2 Z Z ) (3.16) M, . 2 ( . ). In order to get a nontrivial i gives rise to the unique ground i 1 M, Z . M, i ( γ , η i ( Z | 1 1 00 γ γ , η | M, M, η ∈ ) ∪ H Z 00 Ψ ( is a simultaneous eigenvector of the | γ β 1 ) ( symmetry. The anomaly is again quite Ψ ∈ ∈ i | η α Z M ] ( 2 q ] R η β β Z q , α [ ∈ 1) [ 1) i , 1) − α , α − Ψ ( i | ( , − i + γ ( γ | , α γ γ X – 11 – i γ γ | ) whose variation is a boundary term cancelling X | ] ∪ 2 α γ β = α i 7→ = Z ) acts on these states as follows: ∪ + . Its simultaneous eigenvectors are labeled by spin i α M i ]=[ X , η ] R P, ψ γ γ [ 00 , η M ( β 3.18 R [ 00 1) 3 Ψ , = | 0 Ψ Z − 1) | : ( i Ψ ] i 7→ | − | ∈ ( α β γ symmetry, the state [ | , 2 C i 7→ Ψ Z γ | i 7→ | , one has to couple the toric code to a background γ | ] that one should be able to construct fermionic phases in 3d from G 10 ], this leads to the most general 2d FSRE with parameters ( 11 gauge field 2 Z : η symmetries of the toric code. The most obvious basis 2 Z The most concise way to describe the anomaly is to write down a 5d topological action The “diagonal” 1-form symmetry which acts by There are several natural bases in the space of ground states associated to various It was mentioned in [ bosonic phases with an anomalousspecial: global 2-form it trivializes when a spin structure isfor specified. a 3-form As explained in [ 4 Bosonization and FSRE4.1 phases in three Bosonization spatial in dimensions 3d Upon gauging the 1-form state of the fermionicFSRE with TQFT symmetry on the spin manifold ( corresponds to the bulkstructures line defect The anomalous 1-form symmetry ( These two 1-formline defects. symmetries are non-anomalous and correspond to has simultaneous eigenvectors of the form 1-form symmetry transformations The 1-form symmetry which acts by 1-form can be characterized by the property that JHEP10(2017)080 B . So , the (4.7) (4.4) (4.5) (4.6) (4.1) (4.2) (4.3) β . The X U Y . In the present . ). This transfor- . ) 2 X 2 ) Z Z = δβ X, 1 X, , and neither is ( ( . Thus once we fixed a ∪ ∂P 2 . 1 δλ Y ]: C Z β , C ) Y + symmetry acts projectively 32 ∈ + R ∈ – δb β 2 C, β β 1) 1 , 2 Z 30 ] , − ∪ ∪ 7→ , λ β β i ) Sq [ b has a nonempty boundary β 2 β , ζ + P ∪ Z + + P Z b Ψ 2 b is a spin 4-manifold. This means that | with ( ) 1 2 X, w δβ ∪ β ( 7→ ( X . F 0 X 2 0 b = ζ b Z C β ∪ 1) Q ), and we know from the previous section + C 1 2 1 2 − ∈ 1) C 1 ∪ π Z ( δb − ∪ β X ∪ Y, – 12 – = 0 and C ( Z = 2 Y = ( a 2 Z ( P 1 2 β i Z δβ Z X δλ, f ∪ is closed. When , ζ Z 1 2 + ) = β Ψ 1 2 ) is the Steenrod square. This action is invariant under ) are subject to gauge symmetries | P b 2 2 C β X ( ) = Z Z ) is not invariant under Z U 5 7→ ) = C S 1 2 symmetry on a spin 4-manifold. ( P, 4.3 X, ( : 5 ( 2 − 5 a, b ζ 2 S ) when ( Z ) 2 H C S Z δf, b δβ ∈ → , we can impose a Gauss law constraint selecting the states in the symmetry acts by shifting P, + + ) corresponding to this projective action is computed in appendix ) ( , b 2 Y 2 2 a 2 can be interpreted as an ’t Hooft anomaly for a 3+1d bosonic phase ) C Z 2 Z Z ( C 5 5 Z 7→ on Y, S S P, ∈ ( a ( ζ 2 X, 3 β ( Z , 1 H : C δβ 2 ∈ + a Sq C Note that the 2-cocycle ( As usual, the anomaly implies that the global 2-form 7→ The global 2-form mation shifts the action by where the anomaly is more severe than in the4.2 2d case. Supercohomology phases To obtain the supercohomology phasesthe of simplest Gu Crane-Yetter-Kauffman-Walker-Wang model and [ Wen, we take the bosonic shadow to be fermionic Hilbert space for We also identify the fermion parity operator ( and turns out to be This is a symmetricthat bilinear its form quadratic on refinementsspin correspond structure to spin structures on the variation of which has a global 2-form on the Hilbert space2-cocycle of on the bosonic theory associated to a compact 3-manifold action varies as follows: Note that the variation vanishes when case, this anomaly action is where C the variation of the partition function of the anomalous theory on JHEP10(2017)080 ), is A ( (4.8) X ρ (4.11) (4.10) (4.12) ). It is + ⊂ a ) which is ; then and given C γ 2 R C ˆ Z Σ, this loop 1 7→ πi a, b ∪ symmetry, but is not assumed X, ( C is homologically 2 2 C . X Z ) γ C ] = 0. = δb ∈ β = exp( 2 )) (4.9) C 2 ˆ Σ 2 ∪ γ A β Sq ( such that . Sq W ρ ) C is a generator of a 2-form ). It is also charged under 2 Z is not longer an observable, 2 is a closed spin 4-manifold, ) has loop observables and γ / + is Z ∪ 4.6 Σ R vanishes if X V 4.5 δb C X, . This anomaly can be canceled X γ ( ( ˆ 1 Σ for some 2-chain δ 2 G  ∪ ∂ W ⊂ ) Z b ρ. . Alternatively, if A = γ ) + 1 ∈ β ( + ρ ∪ A γ b ( 2 ρ ρ ∪ ∪ 1 2 2 symmetry, , β b 2 C  Sq = with values in Z is charged under the global 1-form symmetry 1 2 β ρ ) + + γ G 2 Σ C – 13 – ). When Z W C ) + -equivariant model by replacing Sq 2 on + A 1 2 πi G 4.6 ( ) is the Gu-Wen equation for 3d supercohomology ν Sq  ν δb = ( symmetry, the model (  ∪ 4.10 2 exp X δν )) = a Z Z Σ ( A V ( X ρ . To simply notation, let us denote Z )), while for homologically trivial 7→ 2 G 1 2 + under a gauge transformation is independent of Z Σ C = V ( 2 X, ) is the 2nd Stiefel-Whitney class. If ( symmetry: 2 1 Sq ). This does not change the anomaly of the 2-form 2 ) in the action). The first term gives the usual anomaly for the 2-form Z gauged Z 2 gauged Z A S Z ˆ Σ. After gauging the 2-form ( ∈ S ). It is invariant under the gauge symmetry ( X, ρ b ( λ 2 G, 2 Σ ). We modify the action to ( R 2 ∪ 3 H Z for Z C πi ∈ λ ∈ = 0 and the action is invariant for arbitrary 2 X X, ( R ). Thus the theory has the correct ’t Hooft anomaly to be a bosonic shadow of a 2 w ρ + 3 1 2 w a Z Before gauging the 2-form We can promote this theory to a To gauge this 2-form symmetry, we introduce a 3-form gauge field, i.e. a 3-cocycle 4.2 = exp( ∈ 7→ Σ observable generates the 2-form gauge symmetryPoincar´edual to with a parameter because it is notnontrivial gauge-invariant. (this follows The from loop thea fact observable that The loop observable localizedinvariant on under the a gauge 1d symmetry submanifold ( to the 4d action.phases. The equation ( surface observables.V The surface observablethe localized global on 2-form a 2d submanifold Σ Then we can cancel the anomaly by adding the term term symmetry, while the second term leadsif to and an only anomaly for if there exists a 4-cochain where introduces an anomaly for The last term is exact and thus does not lead to anomaly (it can be absorbed into a contact The variation of by ( fermionic theory. then to be spin, the action is invariant only ifC we impose a constraint [ where JHEP10(2017)080 , h cut D c ). If , gauge g , then = D ∂T G G y + x ). In a fixed 2 Z domain walls. We contains 10 zippers . The SPT ground BG, ( G 3 G where four zippers meet Z ∂T ∈ 5 ∈ g g,h,k ρ J through zero. We will see this interpretation where the domain walls c ) defines the fermion parity of the 6 consists of five 3-simplices and is g,h Z g, h ( ∂T and have fermion parity determined by ρ G – 14 – singularity, where four zippers meet. With the x axis 3 A is a domain wall labeled with an element of whose boundary ) can be thought of as a way of assigning fermion parity is a boundary of a 4-simplex, the net fermion number of ∂T T singularity. This is because if we choose a foliation of space ∂T symmetry leads to a theory without any nontrivial observables 3 g, h, k ) meeting at four F-junctions (dual to 3-simplices of ( 2 A ρ Z ∂T is a 3-cocycle follows from the conservation of fermion number (mod 2), the codimension-1 domain wall labeled by ρ g D at a particular instant of time, we see a network of meet. There is also a particle-like fusion junction 1 A − . A picture of the F-junction or ) gh . This suggests that the gauged theory is a fermionic SPT. ( A The fact that Let us discuss the physical significance of the 3-cochain D To establish this, one alsoWe needs assume to here prove that that the a partition zipper function does is nonzero not on carry any a spin 4-manifold. nontrivial 1d FSRE phase, and thus its endpoints do 5 6 . On the other hand, since ρ this configuration of domain walls must vanish mod 2. This is equivalent tonot have the Majorana condition zero modes. We will discuss zippers with Majorana zero modes later. if we assume thatIndeed, the fermion consider number a ishomeomorphic 4-simplex the to sum of a fermion 3-sphere.(dual numbers to of The 2-simplices the dual of F-junctions. the of dual this of triangulation everythe of 1-simplex F-junctions are of labeled by three elements of to F-junctions: someis are natural fermionic, from some thethe Atiyah-Hirzebruch are 2+1D spectral situation bosonic. sequence where later. thetriple analogous junction It 2-cocycle of is domain exactly walls. analogous to and we call the F-junctionby or planes transverse to theor F-junction, associator as where we one scanparticle-like would across objects, apply we and see the a F movie symbol of in the tensor “F move” category theory. These are gauge of will denote by state should be invariantseveral under kinds the of reconnection defects of incocycles. different the For dimensions domain example, corresponding there wall to is network. different a degrees There string-like of “zipper” are group gauge transformation and therefore is 1that when inserted gauging into the any 2-form correlator.except The conclusion the is partition function, whichfield depends on the spin structure as well as the Figure 1 along the blue-grey junction andthrough the this y picture axis to along give the a green-grey movie junction, of the the planes F move as we vary JHEP10(2017)080 - C 2 Z . If lines ρ (4.14) (4.15) (4.13) ψ . , where ρ i B | , it is easy to see that line defects. A 1-form 2 Z ψ ∈ ) . g, h B ( C f B 1 . ∪ i ), one must have δλ 2 B | + is a network of Z B δf. B Y, λdλ C ( + ) as linear combinations of states Y 1 R ρ whose objects represent boundary defect lines B C is shifted by X of the toric code. Each configuration of 4.5 1) – 15 – C . This gauge field is associated to the anomalous 7→ ∈ = − ψ B g,h lines in the toric code, or equivalently as the toric ρ λ i Z corresponds to a local rearrangement of the string , ) can be constructed by categorifying the string-net Ψ = ( 1. It is a fermion and therefore must have topological ψ | ) belongs to this class. The input of the CYKWW δλ ' δλ δλ 4.5 4.5 + + ψ + B ◦ B C B ψ gauge field 7→ 2 ) is unchanged. 7→ 2 Z B Z B BG, ( 3 H ). In the Poincare-dual picture, ], and the model ( can be identified with the ∈ ]. Roughly speaking, instead of a spherical fusion category, one needs to 2 ] Z 32 ψ 30 ρ be a closed oriented 3-manifold with a triangulation. The CYKWW construction – and which satisfies X, Y 30 ( ψ 1. This encodes the fact that the 2-form symmetry has a nontrivial anomaly. If we 2 can be viewed as a network of Z − Let Since every zipper has two ends, we can flip the fermion parity of each end without = 0. Alternatively, we can require the fermion parity of a network of domain walls to Y ∈ The explanation for thisas rule a is subcategory the of following.line the defect The category category of ofin bulk supervector line spaces defects occurs forcode the coupled to toric a code. 2-form Specifically, the The string-net Hamiltonian is a commutingdistinguished projector by Hamiltonian the whose way ground their statesnetwork. components are transform Namely, under under a re-arrangement of the string gauge transformation network. A general state has the form assume that there areis no equivalent other to irreducible the objects category of in super-vector the spaces. braideddescribes fusion the category, ground then statesB of the model ( model [ construction is a braidedfor fusion a category particular boundary condition.symmetry, so In we the present expect case,denote that the there bulk is TQFT an hasspin a invertible 2-form line defect on the boundary which we The ground-states of theapproach model [ ( take a sphericalaccepted semi-simple definition monoidal of this 2-category. object,of and Unfortunately, constructing consequently there there 4d is is TQFTs. no no completely general But generally method there is a well-understood special case, the CYKWW The class [ 4.3 The ground states of the CYKWW model remain unchanged under 3d Pachner moves. This leads tochanging the the same fermion condition number on the of fermion the parity whole of network. thethe But fermion endpoint numbers this of of changes the the F-junctions 3-cocycle change according to δρ JHEP10(2017)080 G G g,h Z (4.16) lines is ψ must be exact. a closed Kitaev g C D , we let ζ which tells us whether (a junction of three domain 2 . Rearranging the Z singularity, which is equivalent ψ g,h 3 creates a pair of string endpoints . The rules of the 3d string-net ∈ Z i A ) f δλ B | ) ). The zipper is a place where three + ). g, h B ( ( B ζ σ g, h Q ( 4.15 gives rise to a unique ground state of the 7→ σ 1) i which is easily seen to be a homomorphism from transform in the same way as the partition − B Z , ζ ) allows strings to end at certain points (they ( 2 B Ψ ∈ | – 16 – B Z C does not vanish). Since the number of endpoints ) X g ( Y, C = ( τ 7 . A natural basis in the space of ground states is means that strings can end anywhere. Furthermore, 3 | i ) ). Z 2 C , ζ ). This gives ( Z ) makes it clear that the number of linearly independent ∈ Ψ | Y, ζ . Namely, given a spin structure Y, 3.1 . C symmetry, ( 4.15 σ Y 2 2 H Z is finite, this parameter vanishes. But this is an interesting possibility if | . Thus 2-form gauge symmetry relates all possible arrangements ) may carry the nontrivial 1d FSRE, i.e. the Kitaev string. The f 1 G − ) gh ( ) by a coboundary. D g, h ( and σ , h symmetry of the toric code whose generator is D 2 , Z g D There is an ambiguity in the definition of One can argue that the ground-state is at most unique after gauging the 2-form sym- The transformation rule ( . Since we assumed that One may ask if the boundary of a domain wall can have gapless modes with a nonzero chiral central 7 Z the boundary of every domainthe wall is 2-cocycle closed and can be contracted to a point), butcharge. it This shifts would leadto to a new parameter is infinite and should lead to a new class of fermionic SPT phases. requires an even number of Majoranato zero the modes 2-cocycle at each condition on domain walls meet. We canstring; attach this to does the not boundary affect of any the observables, domain like degeneracies wall and fermion parities (because ways to see this.walls For example, oneendpoint may ask of if such a aphase zipper zipper would be would characterized have bycarries a an the new odd Kitaev parameter number string or of not. Majorana This zero parameter modes. must be Such a a 2-cocycle. Indeed, consistency unique on any closedthere are 3-manifold no (orientability nontrivial is observables. clearly irrelevant4.4 here). In More particular, generalThe 3d supercohomology FSRE phases do phases not exhaust all possible 3d FSRE phases. There are several Summing over all (exact) valuesa of 2-form gauge transformation supported atat a the 3-simplices 2-simplex sharing of string endpoints. In particular,configuration by of creating strings pairs of to string the endpoints, trivial one can one reduce (no any strings). Hence the ground state is at most metry more simply, usingbackground only 3-form the gauge string-net field pictureare dual of to the those ground-state. 3-simplicesmust on be Coupling which even, to the a corresponding homology 0-cycle must be trivial, hence labeled by spin structures on After we gauge the3d 2-form FSRE on the spin manifold ( equivalent to 1-form gaugeconstruction transformations tell us thatfunction the of the coefficients toric code, i.e. ( ground states is given by 1-form JHEP10(2017)080 ). K by ) is F A = 0 2 ( ζ σ Z (4.21) (4.19) (4.20) (4.17) (4.18) δρ = Y, ( 2 B H ∈ by the ambient )] γ A can be interpreted ( σ 0 ). K F A ( , is modified. To see how is not necessarily trivial, ρ η ρ Y can be reinterpreted as the 0 K R . ) F 0 K A F ( of Kitaev strings to the fermion . σ , ) σ 0 K K ∪ 1 A F F ( ∪ α ρ Hence we expect Y Y 8 δλ Z Z )) + ). . Indeed, shifting the spin structure σ. + A α Y ∪ ( ) + as expected, provided we identify σ + δλ ζ σ )) + ( ( ζ ζ ζ A ∪ ) is not conserved (the conservation law K = ( – 17 – Q λ A F σ 7→ necessarily depends on ( ( δρ ρ )) is that it is not invariant under replacing the 2- ζ + ζ Y ) = A ρ Q ) = ζ ( ( α σ = 7→ ( K + ζ ρ F F ζ Q ( ) depends on K B )) but instead satisfies the twisted conservation law F ( ζ Q 4.20 must transform as well: = 1 (since this is when the spin structure induced on )] is the Poincare-dual of the homology class of the Kitaev strings. ρ α A , ( is flipped by the shift γ R σ δλ ζ should shift by one the fermion number of a Kitaev string wrapping a curve is nonvanishing, the constraint on the 3-cochain + ) with a cohomologous one. In the language of Kitaev strings, this means that α σ σ is independent of the spin structure. Since the fermion number of Kitaev strings A ] for a description of how the spin structure (discretized as a Kastelyn orientation + dimer ( 0 7→ K σ 20 F σ . In fact, [ It follows that the junction parity An important property of We conclude that the total fermion parity is given by When changes when Kitaev strings are deformed and reconnected. For example, when we have See [ Y 8 if and only if . However, by shrinking the link to a microscopic scale, K is not invariant under ( covering) is implemented microscopically on the Kitaev string. with nonzero fermion parity so thatwhen the overall fermion parity is conserved. In other words, so that the total fermion parity is gauge-invariant. F two linked but homologicallyuntil trivial they are loops neither of linked norto Kitaev knotted. the strings, After fermion we this parity. deformation canprocess each This loop try Kitaev does contributes strings deforming zero must not them intersect result and/or in self-intersect. a This paradox can leave because behind during particles the deformation cocycle as the contribution ofsince homologically two trivial homologically Kitaev trivialY strings. closed curves canparticle form contribution to a the nontrivial fermion link parity and in absorbed a into 3-manifold The quadratic function Therefore one must have where wrapped around nontrivial cycles of a 1-cocycle γ spin structure this comes about, letnumber. us It try is to clear guess thatnonzero, the such because a contibution this contribution means can be thaton present there whenever are [ Kitaev stringsmust wrapping depend noncontractible linearly loops on the spin structure on JHEP10(2017)080 ) ]. σ ). 2 Z 38 this 4.22 σ (4.22) (4.23) (4.24) ). The . Thus BG, ν ( 3 4.22 is modified C to be closed ) means that ρ ∈ ρ , so that overall ρ , 4.21 σ ) 1 ), σ ( Z Sq x / 1 2 R ) is given by by 2 + ρ Z ˆ σ BG, 2 δ ( (235) (4.25) ). We thus propose that 3d 4 σ ∪ BG, C . The failure of ( . ˆ ρ σ 5 ρ 4.21 ∈ 1 4 C ν (012) that there is an essentially unique ). This subgroup consists of bosonic  ∈ σ and Z 2 ) F / ν σ σ ( ambiguity in the 1st equation in ( R 2 x (245) ∪ ) and thus can be absorbed into  . Now let us shift σ ρ Z σ signs. To see this, note that flipping the sign / BG, 1 2 1 ( . It is shown in appendix D that for any = 0, in agreement with the physical argument R 4 , and − σ + – 18 – Sq (023) , suggesting that the equation for 1 σ Z ρ 2 δσ σ ∪ ∈ 1 Sq BG, σ Sq ( ∪ ν 1 2 ∪ 5 ρ σ C 1 2 1 2 + ) = ) assuming the relation between 3d FSRE phases and spin- σ 1 )(012345) = 0), where ρ, σ Sq σ 4.21 , ( σ, 2 ( = 0, but have corrections which are encoded in the differentials of 0 ). It is shown in appendix 2  but it must receive higher-order modifications as well, in order to x ∪ . ν, Sq = 0 is modified at leading order as well, to the Gu-Wen equation ˜ δσ σ Sq 1 2 ρ, 4.21 ). These parameters must satisfy constraints which at linearized level 1 ) is an integral lift of = = = 0 = 2 δν ∪ Z Z ρ δρ δρ δν δσ 1 2 = BG, BG, = ( ( 2 δν 2 ρ 2 C C ). ∈ Sq ∈ 1 2 σ σ indicates that there should be three parameters: There are also several nontrivial identifications on the set of solutions of ( Some comment is required regarding the The equation We can also deduce ( 4.21 = flipping the sign can be absorbed into a changeabelian of group structure is alsoSuffice highly it nontrivial, since to the equationssolutions say appear of that nonlinear the [ the form ( space of solutions has an obvious subgroup corresponding to We claim that thissolutions sign of the is equations unimportant, withchanges because the the r.h.s. + there of and is thethe equation a r.h.s. by changes 1-1 by correspondenceexpression between is an exact element of Here ˆ δν be consistent with ( modification of theFSRE Gu-Wen phases equation are consistent classified with by solutions ( of the equations the spectral sequence. The spectralis sequence not immediately modified implies by thatabove, the the but differentials, constraint that on other i.e. constraints areis modified. known The to 1st differential beto in the the ( spectral Steenrod sequence square of Kitaev strings. cobordism. The Atiyah-Hirzebruch spectralBG sequence converging to theand spin-cobordism of are simply paricles can be identifiedmeans with that the the worldlines dual ofthe of fermionic endpoints the particles need of 3-cocycle not thesestrings. be worldlines closed. Put must Eq. lie differently, ( thisstrings. at equation More special means precisely, points as that of explained fermions above, the can fermions worldsheets can be of be created Kitaev created out when of we unlink Kitaev loops The physical meaning of this equation can be explained as follows. Worldlines of fermionic JHEP10(2017)080 , 2 Z , and (4.26) (4.29) (4.27) (4.28) = 2 . G 2 ) 2 symmetry G Z 2 = Z symmetry and 1 BG, 2 ( G Z 2 C ∈ , ) 0 . σ ) , β 0 ) 1 σ 2 ∪ Z + σ ( is still closed, while the 3-form δ , σ 0 BG, + σ ( B 1 0 1 σ C ∪ , its 2-form symmetry group 2-form symmetry provided this 2-form ∪ σ ∈ 1 0 ) provided the l.h.s. does. It is straight- 2 ] that one can construct a 3d fermionic B. σ G ). In the present case + Z 2 0 + ∪ ρ 11 4.21 , ∪ ,G B σ, λ + 1 10 σ 1 ρ = – 19 – ) gives rise to a modified group law for global ∪ BG = 0. The equivalence relation arises from a gauge ( ) = δC 0 δλ 4 ) = ( 4.29 σ 0 H + + , σ σ, δσ 0 ). δλ σ ρ ∪ ( ∪ σ ∪ λ 4.21 ) ) + ( 0 = + σ δρ ρ, σ δβ + ( , so there is only one nontrivial possibility for the Postnikov class. σ + 2 ( ) satisfies the equation ( ρ Z symmetry. Particles will be associated with generators of the 2-form 7→ 4.27 ) = 2 2 Z satisfies Z , ) satisfying 2 C Z δλ, ρ ρ, σ B + ( 4 σ H The modified Bianchi identity ( In general, when a field theory has 0-form, 1-form and 2-form symmetries, the whole 7→ σ gauge field Note the similarity with eq. ( symmetry transformations. To derive the group law, we assume that 2-form symmetry is characterized by itsa 1-form Postnikov symmetry class group takingand values in If the Postnikov classsymmetries. vanishes, If the it 3-group is is nontrivial, the simply 2-form a gauge product field of 1-form and 2-form and a 1-form symmetry, while strings will be associated with generators ofsymmetry the structure 1-form is symmetry. describedobject, by but a 3-group. it simplifies A when general 3-group we is ignore quite 0-form a symmetries. complicated In that case, the 3-group to combine it withthe gauging symmetry some of the other bosonic symmetries systemother is of symmetries, not the simply but a bosonic a productphases system. of more are the general Now, 2-form supposed suppose structure. to containof Specifically, both Kitaev since strings, a we the condensate are of general led fermionic 3d to particles consider FSRE and bosonic a shadows condensate with both a 2-form 4.5 Bosonization ofLet 3d us FSRE phases interpret the andtheir 3-group above bosonic proposal symmetry shadows. forsystem the It from classification was a of argued 3d 3dsymmetry bosonic in FSRE has system [ a phases with suitable in a ’t terms global Hooft of anomaly. A natural generalization of this construction is the r.h.s. of eq.forward ( to verify thethe group full axioms. group We structure expect,the of group e.g. FSRE of by phases bosonic comparison is phases. with a 2d nonsplit phases, extension that of this quotient group by The abelian group structure on these equivalence classes is easily guessed: Indeed, since group of 3d FSRE phases moduloof bosonic pairs SRE ( phases) which consistssymmetry of equivalence classes SRE phases. Taking a quotient by this subgroup leads to a more manageable object (the JHEP10(2017)080 ) σ A ( and ρ (4.35) (4.30) (4.31) (4.32) (4.33) (4.34) λ = C . ) 2 Z ). To get the 3d X, must be closed, ( 1 B by letting 4.21 Z ∈ A to ( , δλ. ) symmetry (this symmetry ρ , λ 1 2 ) 2 ∪ Z 2 ). Since 0 Z Z 2 . ) X, Z δλ 2 ( X, 2 fixed. To ensure gauge-invariance B. Z ( + gauge field 2 C ]. To see this, note that the 2-form 1 A BG, Z X, ∪ ( . ∈ G 10 ( 0 2 λδλ 1 ∈ λ δλ C C and an undetermined 2-form symmetry + ∪ 0 + 0 ∈ : ) (see appendix F for the definition of the ∈ λ λ , β δλ : σ B 0 δλ ) δβ, β 0 ). + C λ ), we must subject λ ) = ∪ C,B + 0 λ ( – 20 – = λ +  C ). The second equation then implies that 4.22 4.29 δλ, λ 0 ) and + λ, λ ]. But this fermionic phase is not an FSRE yet: it 2 ˜ Sq , λ ( + 0 7→ Z C β 10 λ λ, λ ,C ( B ) 0 ∪ β 7→ = 0. We get λ BG, λ 7→ ( C + satisfies ( 3 + B,C ,C B , we conclude that the group law for global 3-group symmetry λ 0 C ( λ C β δ 7→ ∈ = 0 + invariant and shift ρ = B to transform as follows under 1-form gauge symmetry: B and β B B β C gauge transformations, we need to impose further constraints on the ) = ( 0 G , λ 0 ) for some β , like the first equation in ( A σ ) requires ( ) and an ’t Hooft anomaly σ ) + ( E and = 4.29 ρ β, λ B Thus our proposal for 3d bosonization can be formulated as follows: every fermionic Consider now coupling the bosonic theory to a ( symmetry is a proper subgroup of the 3-group symmetry, so we are free to gauge it first on the configuration 2 0 has nontrivial observables chargedis under what the remains global of 1-form theFSRE 3-group we symmetry must after also we gaugeprocedure gauge the cannot this 2-form be 1-form symmetry). reversed, symmetry. since Tosymmetry, the get The and 1-form an cannot order symmetry be of is gauged the not without steps a gauging subgroup in the of this whole the two-step 3-group. 3-group theory has a bosonic shadow3-group with a global 3-group symmetrylatter). as In above particular, (we will we propose denoteconstruction that this is every more 3d general FSRE can thanZ be that constructed proposed in in thisand [ way. This get a 3d fermionic phase as in [ must be a 2-cocycle.FSRE, we Since gauge the 3-groupwith symmetry, while respect keeping to data and transformation with a parameter Specializing to closed transformations is λ The first equation showswith that a this 1-form is symmetry equivalent transformation to a 3-group symmetry transformation Then ( Now consider the effect of two 1-form symmetry transformations with parameters while 1-form symmetry transformations shift transformations leave JHEP10(2017)080 . 2 O Z (5.1) satisfies B fusion rule 2 Z ). The fusion of the fusion of the ψ E,E -graded vector spaces 2 Z Hom( symmetries, it contains a ∈ ) is a braided fusion category, 2 ψ Z . Physically we imagine a gapped symmetry) and a codimension-2 E,E E 2 Z E. ' O and the identity object 1. The – 21 – ⊗ symmetry). Let us denote by ψ -symmetry generators which are charged under the O 2 E Z ) is equivalent to the category of E,E of the monoidal 2-category, and thus ] = 0. It is shown in the appendix that no anomaly is possible, B E [ 2 Sq ]. From the physical viewpoint, this monoidal 2-category describes boundary 33 . This is a natural assumption since gauging both 2-form and 1-form symmetries 1 means that Hom( O ' Next we need to describe morphism categories. Hom( This TQFT is nonabelian, so we will need an algebraic approach to construct it. This Since the 4d TQFT has both 2-form and 1-form The description of the 3-group gauging as a two-step process makes it intuitively clear ψ symmetry, but gauging removes them. and ◦ and by assumption itψ is generated by as a fusion category. Theretensor are product, two and braided structures the on other it: one one to corresponds the to the supertensor usual product. They correspond to two TQFT, and thus every object inE the monoidal 2-category is a(i.e. direct proliferating sum of the several bulk copiesnontrivial of defects observables, i.e. which a generate fermionic SRE them)there phase. should are It bulk lead should defects be to other stressedE a that than before theory the gauging with no the codimension-2 defect withWe the have the boundary fusion gives algebra us another object which we denote We postulate that there are no further indecomposable defects on the boundary of the 4d codimension-3 defect (thedefect generator (the of generator the ofcodimension-3 the 2-form defect with 1-form the boundary.a It line is a defect line on defect thethe on “transparent” the identity surface boundary, object or defect. equivalently Algebraically, the ‘transparent” defect is modes, it will also describe the behavior of theseapproach is charges a and 4d vortex analog strings. ofan the input Turaev-Viro construction [ and takes adefects monoidal for 2-category a as particular topological boundary condition. The goal of thisIsing TQFT section in is 2+1d, and to hassuperconductor construct global with 3-group a fermionic symmetry 3+1d charges and TQFTmodes vortex on which lines the is which boundary. terminate a at Since 3+1d Majorana the analogue Ising zero category of describes the the behavior of Majorana zero and thus the 3-group4-manifold. symmetry with the above anomaly can always be gauged5 on a spin Bosonic shadows of5.1 3d FSRE phases 2-Ising theory that the resulting phase isat a fermionic the phase, first since step. thethe spin second structure But step. is it The introduced is questionsymmetry already boils in not a down clear to fermionic computing that theory,the taking possible a into anomalies constraint account further for that geometric a the 1-form structure 2-form gauge is field not needed at JHEP10(2017)080 κ (5.2) (5.3) (5.4) ), and E,E can terminate -graded fusion 2 ). We postulate gauge field while O Z = Hom( . G i 0 O,E is not invertible. We C , ψ σ 1 h -independent. Physically, ]. This is facilitated by the ) = and has the correct anomaly κ ]. This is a ) is equivalent to the category must be its own dual, we can 33 E 34 ) and Hom( O E,E , because E,E is an Ising braided fusion category E σ, E,O C ' Hom( σ ψ. ' 1. Strictly speaking, the 4d TQFT we are ◦ ⊕ ) − = -action and additional data which generalizes = 1 2 8 O,E is isomorphic to – 22 – σ, ψ κ Z σ ⊗ ◦ ψ ' with a fixed branching structure. A coloring is an -action is trivial. In our case, O σ 2 ) are non-empty and each of them has a single irre- ψ Z X ◦ and is not equivalent to . This means that the surface defect σ σ E,O O Hom( O ]. We will call the 4d TQFT obtained by taking the braided . This is not very important for what follows, since all our such that -crossed braided category [ κ 2 ' action is trivial. Thus 35 -crossed category). All possible braided fusion structures on κ 2 Z ) 2 Z Z ): O,O with a compatible -crossed category one can associate a 4d TQFT using a generalization 2 ) and Hom( 1 Z O,O C Hom( + O,E ), and the 0 . We need the latter option, so that Hom( C C = 2 E,O is invertible. C Sq ψ R To any braided To complete the construction of the monoidal 2-category we need to specify all the Finally, we need to describe the categories Hom( = Hom( 1 keeping the anomalies intact. Below we provide some5.1.1 evidence for this. State sum In this section weover describe colorings the of state a sum triangulation for of the 2-Ising model. The state sum is a sum of the CYKWW construction [ Ising category as an3-group input symmetry for generated by thisto construction be a a 2-Ising bosonic model.obtained by shadow. taking We the will We 2-Ising propose show model that that and the the coupling bosonic it to shadows a of background all 3d FSREs can be labeled by a complex number constructing might depend on arguments only use propertiesencodes of how Ising many categories Majorana whichmod zero are modes 16. exist at the hypothetical end of a Kitaev string, category braiding and reduces to itC when the (and therefore is a braided an Ising category are known, and it turns out there are eight inequivalent ones, naturally because associator morphisms and thefact pentagonator that 2-morphisms the [ monoidalare 2-category equivalent to we those are of constructing a has a very special form: its data We also necessarily have that both Hom( ducible object which weon denote the boundary. Nevertheless, postulate the simplest non-invertible fusion rule: action of supervector space asalso a compute braided Hom( fusion category. Since possible anomalies for the 3-form symmetry: the trivial one and the one with the anomaly JHEP10(2017)080 ] O 35 or (5.5) E are even (01) = 0 1  ψ lines around σ worldsheet. and , in which case the ’s, in which case the σ O ψ O ) as shorthand for the object ij grading where 1 and ( -sheets and 1 2  O ] to evaluate it to a number. This Z . 6 1 d ∪ 1 worldsheet.  on each of four fusion junctions which meet worldvolume and was already defined above. is Poincar´edual to the (01). In Poincar´e-duallanguage, an is a sum of weights over all colorings. Let us 1 ψ ; or an even number of = σ 1  O σ 2 X d – 23 – d , or ψ . In Poincar´e-duallanguage, the morphism describes ] and is the weight of the coloring in the state sum. element ψ 31 2 Z . In particular, we have either a single (02)) where we have used 1 ij ,  ). 1 (12) 1 ∈ C  as the braided Ising category with ) is to Poincar´edual the ) is Poincar´edual to the 2 2 ⊗ C Z Z : lines. Choosing a resolution of the 4-valent vertex into two 3-valent vertices X, X, labeling edge is odd ( and (01) X ( ( σ 1 1 2 2  σ O symmetry C C Z or ∈ ∈ E = , and 2 1  It follows from the rules of the Ising category that  It follows from the fusion rules that and we can usedefines the the rules “15j” of symbol theSee of Ising figure [ category 2. [ ψ defines a basis for thisparticle fusion whose space. worldsheet meets In the Poincar´edual langauge, tetrahedron. theThe 2-morphism coloring is around a a 4-simplexits is boundary, a a collection 3-sphere. of The branching structure defines a framing of this 3-sphere At a tetrahedron, itsix is sheets useful are to meeting, imagine withat a the a 1, dual point picture, inthe shown the in fusion center. figure junctions. At 1,category. this The where That point, rules is, we for we need this can to forget is have the precisely something sheets the gluing and same together just as think of in this the as usual a Ising junction of 1, Hom( E only morphism in the fusionmorphism space may is be either 1a or string whose 2D worldsheet meets that triangle. Edges 01 are labeled3D by worldvolume a intersects this edgeor 1, transversally respectively. depending on whether To a triangle 012, we assign a (simple) morphism in the fusion space (a category) ) and G 0 • • • • • • • • ∈ C It is useful to encode thein state spacetime sum as a sum over cochains. We define the following cochains 5.1.2 morhisms to tetrahedra. Theeach weight 4-simplex. of The each partition coloringspell function is on out a what product thiswith means over the for 15j 2-Ising. symbol This( in should be compared with definition 3.1 of [ assignment of (simple) objects to edges, (simple) morphisms to triangles, and (simple) 2- JHEP10(2017)080 σ O ]), this 6 line. From σ ] gives us a way to connecting them. 6 ψ surfaces making the O ], represents the boundary of line, but these cannot move 35 034 ψ surfaces which evolve according lines have a lines along the intersections of surface intersects the O surfaces defining the into-the-page σ ψ O 023 O 012 lines and ψ – 24 – 123 0123 1234 234 and we see a snapshot of the action. In this snapshot, we 0124 0234 013 σ 0134 124 134 X surface must create ). We stress that we need not impose it as a constraint 024 O 014 5.5 by line objects. Representing triangles, these edges are labeled by triples lines, which will be where the matrices. X ψ , and the rules of the Ising braided fusion category of [ F σ lines in a Hopf-link formation with σ and , and R ψ as a movie of fluctuating . This image, essentially a reproduction of figure 16 from [ X For this rule to be insured by the local dynamics of the 2-Ising Hamiltonian, the term On the boundary of a 4-ball in which creates small discssurfaces. of There are alsothe terms endpoints which of the create smallthis loops follows of the equation ( of the Ising may have two framing as shown in figureconfiguration 3. can According only to be the rules filled of into the the Ising 4-ball category if (see those e.g. [ This last point deserves somesum elaboration. on We can imagineto each the configuration in local the movesworldsheet state always of a the boundary usual and inducing Ising the category, into-the-page except framings for for the the evaluation labeling of triangles in of vertices, and there areway 5 intersections choose to 3 make of the those.with graph trivalent. There the are These 5 some are choose extra labelededges edges 4 by by where tetrads tetrahedra we 1, of of have vertices resolvedevaluate the and 4- this coincide dual picture 4-simplex. to a The number. state This sum defines gives the us 15j a symbol. labeling of these Figure 2 the 4-simplex, considered as apage triangulation using of a the framing 3-sphere.through induced from 4. We have the flattened This branching theedges structure, picture image here which is onto gives representing actually the us triangles Poincar´edual a of in labeling the the of vertex-ordered 3-sphere 4-simplex. vertices 0 to This that is 4-simplex, so the with graph (most) depicts the JHEP10(2017)080 , 2 β Z + (5.7) (5.8) (5.6) ’s are σ 2  7→ surface that 2  curve and the O ) one must also ): σ 5.5 4.35 . . ) 2 α (13)) +  1 − β. , α + 2 1 α (23)  . As explained in section 4.5, with 1 ∪ ’s intersect the orange disc (red stars),  ∪ E 1 σ α α is a spatial 3-manifold. The 1-form + + Y curves with respect to the framing defined 2 2 mod 2 (5.9) (12) +  β 1 σ  1 + 7→ 1 δ 2 ), and to be consistent with ( β – 25 – 2 ∪ (01)( Z 1 1   Y, α,  ) = ( ( 2 Z 1 + ) and acts as follows: anyons (black circles) in a Hopf link formation. The is counting crossings between these two curves, and of Z , α 2 1 2  σ 1 Z ∈ β unchanged, where . A general symmetry transformation is parameterized by a δ α 7→ 1 1 X, 1 )(0123) = ,  (   1 1 2  α ) + ( ∪ 1 δ Z + α by displacing it a small distance into the piece of ∪ , α × 1 ) implies that the 2-Ising model carries an action of the symmetry 1  + 1 ) surface (orange discs). Where the σ  β 2 2 ( (  5.5 Z 7→ O 1 ) has other interesting consequences. To evaluate the right hand side on Y,  7→ ( . One way to understand this is that 2 1 5.5  Z X ) while leaving : 2 anyon (wavy black curve) being born. 2 ∈ . To see this, note that the global 2-form symmetry acts by  Z surfaces. That is, it equals the mod 2 linking number of the ψ ) . A configuration of two E Y, O ( 2 α, β The equation ( Z ∈ computes the mod 2 self-linking numberby of the the curve obtained from bounds it. Thecourse integral where of the crossings are depends on the local framing of space. This quantity depends on the choicestructure of ordering, on which we take to be defined by a branching 5.2 Fermion number The relation ( a tetrahedron, we need to order the vertices 0, 1, 2, 3 and compute Thus the 2-Ising model isthe proper acted anomaly upon such by a thea symmetry 3-group spin is “fermionic” structure. in that we can gauge it by introducing pair ( Performing two consecutive transformation we get the group law eq. ( 3-group β symmetry shifts transform the boundary of the we have a by hand. Configurationscontribution of to defects the which state do sum. not solve this equation will have vanishing Figure 3 JHEP10(2017)080 σ B lines . Let (5.10) ζ induced ψ σ 2-form 2 ), so the two 1 Z line connecting δ symmetry. We ∪ ψ 1 loops. From what  E σ by thinking about the ζ , ) of the loops to be well-defined, B ], one can think of a spin structure -matrix because the framing of ( . It is possible to achieve this by 1 S K 26  mod 2 F by 1 δ λ . All of the Ising category numbers are computed 3 ∪ surface which is a twice-twisted ribbon. See line connecting them. S 1  ψ loops will depends on the spin structure on the spin structure O -surfaces become equivalent to the trivial surface – 26 – σ given a spin structure on spacetime. This is very Z O . . The configurations which appear in the 2-Ising K,ζ K O surface (orange skeleton). With this configuration of the F K,ζ F ) = F 1 ) replacing O anyons (black circles) in a Hopf link formation given by the δ ( σ -loops (its integral over a surface counts the number of 4.20 ) has to satisfy K σ F B ( ) then says that the total fermion number of the state is the K symmetry, the F surfaces so that they induce into-the-page framings and then computing the 5.5 E lines by having an O ψ loops framed by σ loops lose their framings. Their density is measured by a surface has odd self-linking in each component (measured by σ O lines are line defects in the 4d theory whose endpoints represent the fundamental This is a new ingredient for topological order in 3+1D. Quasiparticles can only . A configuration of two ψ surface does not extend to any framing of 9 O We can infer the dependence of When we gauge the The This diagram is not subject to the rules of the Ising category 9 surface, the self-linking of each component is even, so there is no need for a loops without any spin structure induced by a framing. As discussed in [ by the in 2-Ising by choosingwavefunction the overlap with the empty picture. are well-defined, but this is notin very section physical 4.4, and we too can restrictivephysical, for define since our such goals. we an As wishconclude discussed that to the describe fermion fermionic numberus of systems write the the by string gauging fermion the number which is a specialframing case all of of space eq. since ( this frames all curves so that their mod 2 self-linking numbers defect, and the which is Poincar´edual to the strings piercing it). In orderone for needs the some fermion geometric number inputwe that have discussed stands so in far, for the framing of the are attached depends cruciallyσ on the framing. Forfigure example, 4. we can createbe a the Hopf boundary link of of stringa operators quasistring in depends one on way, but how 2-Ising it illustrates is how framed the by statistics its of bounding surface operator. fermion. The equationself-linking ( of the state sum all have even net fermion number, but the number of points where the boundary of a twice-twistedO ribbon of them. This contrasts withinduced by the the into-the-page framedcomponents Hopf are link fermionic we and drew there above, must where be the a framing Figure 4 JHEP10(2017)080 ), ψ 2 Z (5.11) (5.12) -loops G, ψ ( 3 there is a C ) that is governed G ∈ g -surfaces can 5.5 ∈ -surface, as in is shifted by a O ρ O -lines and their . Although the g O ζ g ψ ) to be the func- -surfaces (namely, E and B O . If the symmetry is ( g ⊗ G 1 E K,ζ O . F ' -surfaces, is at most unique, and there (the generator of the 1-form g . O O -surfaces. The mod 2 Y E O O symmetry. We do this by extend- -lines which can be “pair-created” G B . It follows from eq. ( ), a group 3-cochain ψ 2 , the fusion of times time. Its wave-function can be ∪ means that for every C -symmetry, we introduce an invertible Z G α E Y G G, B. Z ( 2 ∪ -lines and -surface intersects another . Obviously, C ψ . This happens because the termination of a g ) + B O 2 O ∈ B Z = -surface is not an invertible defect, one cannot ( – 27 – mod 2 when the spin structure σ O by K,ζ α -symmetry of the 2-Ising model gives a theory with δC F R E G into a 3-cochain 2 -lines is determined by its value on the trivial (empty) of an U(1)), satisfying some conditions we presently derive. ) = ψ δ σ B ( G, . Fusing each of them with -symmetry allows us to move ( g α 4 , this requirement essentially fixes E + E C C K,ζ ∈ -lines can terminate only at special points on F ψ ν . Thus, changing the spin structure is equivalent to twisting certain α symmetry also frees the fundamental fermion, the endpoint of the + -surfaces and ζ E O -surfaces can be deformed away. Similarly, homologically nontrivial -surfaces, and also allow endpoints for to O O α ). B -crossed 2-Ising ( ζ G , considered as a monoidal 2-category with trivial morphisms, by 2-Ising. The data Q Physically, the presence of a global symmetry One can argue that gauging the Gauging the G 1-cocycle 2 unbroken on the boundary, each suchboundary defect which gives we rise to denote ansymmetry), invertible we surface defect get on another the fusion surface of bulk defect domain wallsby obeys the the group group law law of of an extension of Now we want to enlarge 2-Ising toing a theory with a global for this will consist ofand a a group group 4-cochain 2-cochain codimension-1 invertible defect. Their fusion obeys the group law of configuration of configuration. Thus after gaugingare the no ground-state on nontrivial observables. any 5.3 argue that all cannot be deformed away. Butboundary when for we gauge the from vacuum. Theboundaries gauged in an arbitrary way and implies that the value of the wave-function on any the 2-Ising model onrepresented a closed as oriented a 3-manifold sumhave boundaries, over while configurations of points where a boundaryfigure component 3). of Since an the boundary This reflects the non-trivial Postnikov class of the 3-group only trivial observables using the same approach as for the CYKWW model. First consider As discussed intion appendix lines, turning the 3-coboundary the curve is twisted.Z It also flips by framings. This alsothe changes fermion the number mod is 2 linear self-linking in by the the spin structure: same amount, so we find that in 3d as a mod 2 invariant of any framed curve which increments by 1 when the framing of JHEP10(2017)080 - ) 2 ),  G . If A ( g 5.15 , this σ (5.13) (5.14) (5.19) (5.16) (5.17) (5.18) (5.15) O on the A = A and B g E -crossed 2-Ising ) and G A ( ρ -lines are absent, we σ = . Taking into account C . Thus a ∂X ) describes precisely a map lines. Thus the 2-cochain E ) is part of the boundary of = σ, ρ ψ and using the Stockes theorem . . ) g, h ) Y 2 ( labels on objects on edges in the X , ,g A ) 1 Z ( . one can always swap by letting g Z 1 G σ ( / g σ δ E ∪ R O ). Whenever this 3-cochain is nonzero, ) , ∪ 2 ) A ⊗ σ. 1 G, ), the pair ( Z . ( (  A into 2 5 ( ∪ σ g 1 G, σ G C σ g = 0 ( X 5.18 ) + 3 Z E = ∈ = A – 28 – C ( δσ ) 1 ' , assuming that ρ δρ ∈ δ ) = 2 Y surface cannot terminate, we postulate that ( g , we can take a short-cut and require the symmetry = ρ, σ ) and ( A ρ ( ( E should follow from the topological invariance of the A 2 O 2  σ δρ δ ⊗ 5.14 ˜ Sq 1 X ) = 1. Thus, in the gauge where g ) over Z E and satisfying all the constraints. For a trivial gauge field g, h ρ 2 ( 5.16  σ ) are restrictions of cochains on is the classifying space for the 3-group and 1 ). 5.16  ) represents the configurations of BE 5.6 are arbitrary, we must have X,G -lines must satisfy is a 2-cocycle describing this extension, then the fusion rule is ( 1 ψ X 2 Z Z , where ), we get ∈ ∈ and ) singularities where four zippers meet are sources of BE A symmetry ( 2 A 3 5.15 , g → One more constraint on With the constraints in eqs. ( Now let us integrate ( Next we interpret the 3-cochain Geometrically, this means the following: a zipper A E 1 surfaces if and only if to be non-anomalous. Since we embedded g ( is ensured by theformidable properties computation of for the general 2-IsingG 15j symbol. Insteadthis means of that performing the this cocycle rather BG model is an equivariantization of the 2-Ising model. crossed 2-Ising model. In principle,on it a can boundary be of obtained5-ball by a and evaluating arbitrary 5-ball the partition and function requiring it to be 1 for an arbitrary gauge field Since the that all cochains in ( and ( representing The second term is required to ensure that the constraint is invariant under the action of where state sum. Inholds the without bulk, any restrictions. where an the O must have σ Associativity of the fusion algebra is equivalent to bulk defect is not canonically defined, so for a given JHEP10(2017)080 σ σ ∈ to σ 1 . We (5.22) (5.21) (5.20) (01) 2  1 , and Sq  ψ ˜ Sq + ρ 7→ lines satisfying -lines decouples, . ρ ψ σ σ ). σ, ρ, ν ) = 1 is a source for a ) such that -crossed 15j symbol. Z G / lines on a framed 3-sphere as well as elements classified by R σ (02) = 1 mod 2, we have a line coming out of the domain g, h, k G 2 surfaces and 1 ( . Z  G, ρ = 0. The remaining constraints ψ 2 ( O  4 1 by  ∪ C can be obtained by gauging the 3- , and G 2 ψ ∈ with  G . (12) + to obtain the partition function of the 1 2 ) ν 1 .  depends on the braiding structure of the ) + X labels is unconstrained. g,h,k 2  ρ, σ A ρ ( 1 ( J  1 ˜ ρ 2  Sq -crossed 2-Ising (34)) to define the . ∪ -crossed state sum for the 2-Ising model, com- ˜ = (01) + (01) of the group G Sq 2 ψ G – 29 – 1 ,A  2 A  = 1 2 δ (23) + δν ν ,A 10 singularity ]. = (12)) + 3 4 ). We resolve the 4-valent vertex into two 3-valent vertices (23)) = 1, we add an extra 35 -crossed 2-Ising containing (12) A ˆ ,A domain walls, we sum over all networks of α 2 G ,A ,A -crossed 2-Ising model. Changing which braiding structure we G while the cochain (01) G , A (12) (01) ( A σ (02) ,A ( ν A of the elements are required to satisfy a cocycle condition on triangles 012: ). We evaluate the braided Ising category invariant of this diagram and (01) E should be taking to be the extension of our A 2 for more detail on this constraint and an explicit formula for G ( G ρ F -lines cannot end anywhere else. (12) = ψ A = 0, the sector of the . The (01) 2 σ and choose a basis vector in each fusionAround space. a 4-simplex we(see have figure a collection ofmultiply 1, it by label, otherwise we may choose 1 or At a tetrahedron, inlines. the If dual picture, wewall get junction some (see 4-valent figure junction of 1, Edges 01 are labeled withZ elements A At a triangle, if Let us study the 15j symbol of this 2-category. This is the quantity we will multiply Before moving on, we summarize the Shawn Cui’s • • • • 10 -crossed 2-Ising theory. As we see from figure 5, the 15j symbol is the exponential of -line, and 4-simplex-by-4-simplex along a triangulationG of Thus for a fixed networkthe of following condition:ψ each 5.4 Super-cohomology phases from When and we can restrict oursimplify attention to to the networks with propose that the generalgroup 3d symmetry FSRE withuse symmetry permutes how these FSRE phases are associated topare the definitions data 3.1 ( and 3.2 of [ Ising category we use, butswap it the is signs. not Thus really there physical, must since exist we a may 4-cochain redefine See appendix must be exact. Here the choice of the sign in JHEP10(2017)080 , ), ). 2 δβ A Z ( ρ + (5.23) (5.24) (5.25) X, ) as (up = C ( 3 2 )(01234). Z 7→ . 2 δ  X,G  ∈ )(01234). The ( C 2 ρ ∪ 1  , C 1 . The partition 2 Z β  ∪ ∪ C 2 ∈ = 0. The tetrahedra 2 +   σ 2 1 2 A ) +  A = 0. Since ( but 7→ ) + 4 (234) = ( ρ ρ 2 ˆ 2 A α   ( δ ρ (1234) = ( 034 1 ρ lines. In evaluating the diagram ∪ (012) lines go and join “the condensate”, ) with 2 2 background ψ   A ψ ( (014) 1 2 G 2 ρ  023 with non-zero ) + C. ρ. C ) with A 1 ( 012 ν ∪ ) + 1 to power gauge symmetry  5.21 ρ A (0123) + − ( ρ 1 2 2 X ρ Z Z – 30 – 123 0123 = = 1234 234 πi (034) 2 δν 2 lines, indicated by red curves coming out of the resolved 4-  δ 0124 0234 ]. Evaluating the partition function on the boundary 013 0134 124 134 ψ 11 exp 2 ) A ( ρ 024 014 = 2 1 to power δ | ) − 2 Z X X, ( ). Thus we must merely replace 2 2 C Z ∈ singularities) dual to the tetrahedra. The 2  3 X, ( A ' 2 ) C ∈ . We revisit the 15j symbol in the presence of 6= 0 have a non-conservation of X,A β ( ρ Z Consider now coupling this theory to a background 3-form gauge field Now we can write the partition function in a fixed This is achieved by replacing the constraint ( This ensures symmetry underwhere the 2-form to positive multiplicative factors) black with black crossing gives a contribution of This is analogous toof Eq (3.12) a in 5-ball [ shouldthis give is 1, equivalent which to is the equivalent Gu-Wen to equation the condition Figure 5 where way junctions ( represented by a redaccording to ball the which rules of maygive the absorb a Ising category, sign any we get contribution number contributions of of from crossings. The red with black JHEP10(2017)080 ), ). .  6.1 (6.2) (6.1) F.8 (5.26) (5.27) (5.28) δβ . 1  ∪ 2  β ) symmetry, 1 2 ∪ 2 +  ) is twice ( C,B 1 2 ( β ) evaluated at the 2 6.1  ∪ B is any closed oriented ˜ ) + Sq β . Making a change of A 1 2 P extension. We leave the ( B. ). This is easily fixed by δβ ρ 1 + − A 1 ( + ensuring that the model has ∪ Sq ρ δβ . 2 C ) A 2 ∪  + 2 ∪ 1 2 )] Z 7→ C (see appendix )) P and + ( P, A C 2 ( C ( C. C C U(1)), where 2 2 w ρ 2 [ 1 , not Z P, 1 2 ∪ ∪ + Sq ( C ∈ ) 5 2  = C A ( H ( 1 2 symmetry and an anomaly given by ( 1 2 B ρ ) by an additional factor 1 2  X Z ) + Z – 31 – Sq B,B X A 2 1 Z ( ). This is almost the expected transformation law πi symmetry is nonanomalous now) and reduce to the ν Sq A Sq πi 2 X, A, C (  1 2 ( symmetry of this theory has the ρ Z ∪ X Z exp 2 = , this is determined once one knows the anomaly for the + Z E B F P B does not enter the anomaly, it is irrelevant whether the 2- C ) exp 2 πi 1 Z C 1 2 Sq ) is trivialized by the spin structure, and ( ): , the two possible anomalies for a bosonic shadow of a fermionic exp 2 ∪ D ) ) F F.8 2 X, A, C A B Z ( ( 1 2 ρ Z , we find after some work: X, + ) which differ by ( β 2 C X C = + ) = 2 ∈ 2 C,B 2 δ  (  δβ 2  ' + 7→ ˜ ) Sq 2  symmetry is present or not. If it is present, one can gauge it without introducing 2 Z X, A, C Since the anomaly ( X, A, C ( ( part only, though we cannot decide whether to take the + or Z Z this means that themore directly, latter we anomaly use the is following5-manifold identities also (see in trivialized appendix by the spin structure. To see this Since the 3-form gaugeform field the spin structure (since thecase 2-form when it is absent. theory are In this section weconsider would a like bosonic to theory investigate with the a physics of 1-form this term alone. That is, we explicit construction of state sums for more general 3d6 FSREs to future work. Fermionic string phases As discussed in appendix This is a non-minimal contact-term couplingthe between proper anomaly for theIn 2-form fact, symmetry it to also be showssince that a by the bosonic our full shadow results of in aC appendix fermionic phase. value of the 3-form gaugefor field the partition function,multiplying except the that partition we function expect Observe the appearance of the first descendant of Now consider the effectvariables of the gauge transformation function is thus JHEP10(2017)080 ) B 2 Z . It (6.3) F ), but Z, 2 )] = 0. ( 1) 1 when 2 Z P − − ( , hence C 3 2 Z, ( ∈ w w 2 ) to define a 1 H X -field). It seems Sq ( 3 C can be thought of = w structures differ by 3 B 3 1 w w . This means that we -grading on its Hilbert satisfying [ 2 B Sq structure may be called Z 1 P 3 symmetry group and then is a 2-cocycle Γ Sq . This means that in general w 2 Z Y ] and were dubbed fermionic = Z 19 C is assumed to be oriented, Σ need B. structure. This can be easily shown X ∪ -valued charge analogous to ( 3 ] 2 -manifold 3 w Z n w ) and set = [ 4.1 B -structures, we note that 1 . – 32 – structures can be identified with 3 statistics relation unless one considers also string ]. Just like a spin structure can be thought of as a B 3 w Sq 1 )] = 0, so the anomaly is trivial. But there are no -structure but does not have fermions evades the w , we need a trivialization Γ of 19 ⇒ 3 ∪ P Sq X ] ( w 2 2 ). The anomalous nature of the 1-form symmetry means + w w 2 [ -symmetry means proliferating strings. Their worldsheets C Z 2 symmetry into the 2-form ,[ Z P -structure for their definition. Such strings were discussed 2 7→ X, 3 ( Z 2 -structure [ w C 3 Z w vanishes for any oriented 3-manifold -structure on an oriented ), so the set of ∈ 3 which cancels the anomaly. 2 B w Z B 1 B and properties of Steenrod squares. , a symmetry and anomaly ( , defined up to exact 2-cocycles. Clearly, any two Z, ∪ Sq 2 ( 3 because we can flip the sign by a redefinition of the symmetry operators 2 2 2 Γ Y w w w R Z 2  1 H X R ˜ Sq Sq Γ = δ in = 3  w A further insight is obtained by noticing that while the homology 2-cycle dual to A simple way to construct a fermionic string phase is to start with a bosonic model Second, gauging a 1-form First, although not every closed oriented 4-manifold is spin (a counter-example being A model which depends on a In fact, it appears possible to fermionize the theory with something less restrictive ), every closed oriented 4-manifold admits a 2 corresponding to the shift represents the string worldsheet Σ,as the the homology 1-cycle 1-cycle onnormal dual Σ bundle to which of is Σ. dual (in Note the that 2d while sense) the to 4-manifold the 1st Stiefel-Whitney class of the are embedding thegauge 1-form the 1-form symmetry. Thespace, because resulting theory clearly hasfermionic no string phases doalso not illustrates have that ordinary a fermionic conserved phasesto do not the really come in two types corresponding strings. Their normalthe bundle is framing framed, is and twistedfermionic the string by wavefunction phases. one is unit. multiplied by Thus phaseswith requiring a 2-form using are Poincar´e-dualto that these strings needrecently a in a somewhat different context by one of us [ contradiction with the spin-statisticsbosonic phase relation. either. But Inunusual the it phases. remainder does of not this section correspond we to make a aCP few normal remarks about such Hence the anomaly isOn trivial a on 5-manifold a closedcounterterm with orientable a 5-manifold boundary than a spin structure: a trivialization of such that an element of not canonically. To see the relevance of fermionic particles, because thegauging 2-form it merely symmetry, leads even to if athat condensation present, we of have is bosons a not (the violation worldlines anomalous,statistics of of on the and the spin the right = hand side. For a closed spin 5-manifold JHEP10(2017)080 (6.7) (6.4) (6.5) (6.6) gauge 2 Z . odd, both symmetry. λ 2 n -structure on Z 3 w symmetry which with 2 n Z and any Z X . . = ) ) 2 2 . ) G Z Z 2 symmetry into a product of Z X, X, 2 ( ( 1 1 X, Z ( Z C 1 Z ∈ ∈ , we conclude that fermionic string ∈ B is achieved by proliferating fermionic ) satisfying certain rather complicated 1 , a symmetry into the 2-form ) E 2 Sq B 2 λ, λ Z Z ∪ ∪ λ, λ as ν, ρ, σ X, C λ ∪ ( C 2 P λ – 33 – + Z C b ) and consider a global 1-form + 1 2 ∈ b 7→ 4.5 symmetries and a mixed anomaly ) by considering the following action of a global 1-form 7→ 2 6.1 Z δa, b λ, b ) vanishes, we can replace this 4d TQFT with the simplest 2 . Let us give a few examples. If ∪ a, b + Z G b a 7→ X G, -equivariant versions of a certain 4d TQFT which we called the ( Z . That is, we embed the 1-form a 7→ 2 G symmetries in a nonstandard way. As a simple example, consider 1 2 B a with an anomaly. Further, we argued that 3d FSREs with a finite 1 H are classified by triples ( 2 Z ∈ E Sq G σ = C symmetry only) and embed the 1-form gauge theory in 3+1d with an action 2 2 Z Gauging the anomalous 3-group symmetry Our proposed classification of 3d FSRE phases can be made concrete once we pick a Another way to obtain a shadow of a fermionic string phase is to start with a model Finally, let us give a couple of examples of bosonic shadows of fermionic string phases. Z symmetry: to make the result well-defined. 2 particles and Kitaev strings.lattice model. It would be interesting to construct explicitlyparticular the symmetry resulting group 3-group symmetry unitary symmetry equations generalizing the Gu-Wen supercohomology.of We all proposed such that models bosonic2-Ising are model. shadows If Crane-Yetter-Walker-Wang model and recover the supercohomology phases. X 7 Concluding remarks We have argued that every 3d fermionic model has a bosonic shadow which has a certain Z Gauging this symmetry meanstheory, proliferating but the with strings particles confined andparticles to the are the particles not string local of worldsheets with the in respect a to particular the way. strings, Since one the is forced to choose a the One can get the desired anomaly ( and then set 1-form and 2-form acts as follows: It is easy to see that the action iswith invariant both for a any 1-form closed and oriented a 2-form phases have fermion worldlines confined to fermionic string worldsheets. First, as remarked above weform can take a shadow ofFor example, any standard we can fermionic take phase the (with model a ( 2- not be orientable. Since we may think of JHEP10(2017)080 ), G X ]). 2 40 , g (A.1) 4 ). The g )[ 2 ( 2 Z Z Z , n X, ( Z p ( is a product of • ) which squares Z 2 H G Z . ) 2 G, . ( Z 2 G = 0. As usual, a cochain H X, ) has a unique nontrivial ( 2 2 q of degree 1 and an element δ ]. Z C is even, the generator of this 39 ∈ G, ] because we lack an algebraic n x, y ( 2 if 11 , b H ) 2 ] = 0. Similarly, if -cocycles is denoted 2 ] and [ p σ Z Z ) satisfying 36 ' 2 X, ) Z ( 2 p Z C X, ]. , ( a n ], and any element of ∈ +1 Z = 0. Thus – 34 – 40 ( p 2 2 C x , a , we get 3d FSRE phases where the zipper H ) → 2 xy ) Z 2 = U(1)). We show this in appendix Z is denoted [ , X, has solutions only if [ σ ( 2 a q X, Z σ ( + p p ∪ × ) is generated by two elements even, we only get Gu-Wen supercohomology phases, because the C C 2 4 σ : Z Z n ∈ ( = δ 5 b G, . If we set ( δρ H ∪ , we only get supercohomology phases, since the cohomology ring of • with 2 a is a Kitaev string, while all other zippers are “trivial”. Overall, for xy . We assume a local order on vertices of the triangulation. There is a H is called a cocycle, and the space of Z n 2 2 . Z Z δ g 2 we get four supercohomology phases (including the trivial phase), and four vanishes. Indeed, while Z = 2 σ Z × G and vanish, and 3d FSRE phases are classified by the same data as bosonic FSRE 4 ) is zero in coefficients is a polynomial ring [ × 4 -structure on the 4-manifold. It would be very interesting to explore the physics of Z g σ 2 4 3 Z Z w ρ, σ = There is a well-known product operation In this paper we mostly work with simplicial cochains of a triangulated manifold Our results suggest that bosonization in higher dimensions will get progressively more Our discussion was not as systematic as that of [ We also found a new class of phases which are neither bosonic, nor fermionic, in that The simplest example where there are phases which are not supercohomology phases ( does not square to zero (in fact, it generates a polyominal ring inside of degree 2. The only relation is 2 =  G and 2 ˜ annihilated by cohomology class of a cocycle A Steenrod squares and Stiefel-WhitneyWe review classes here some definitions and results from [ with values in coboundary operation would have to dealsymmetries. with Bosonic a shadows symmetry ofsince Gu-Wen 4-group they supercohomology which only phases involves possess would 3-form, 3-form be symmetries. 2-form quite special and 1-form structure of various monoidal 2-categorieswhich enter relevant into to a us, definition we of did these not objects. describe We hope allcomplicated to the as return the data to dimension increases. this issueof This in complexity the the reflects the future. spin-bordism topological complexity spectrum. For example, in four spatial dimensions presumably one they have “fermionic strings” buton no a fermionic particles. Theirthese partition new function phases. depends description of completely general unitary 4d TQFT. In particular, while we outlined the where G non-supercohomology phases. To be precise, weSq also have to check that the final obstruction with to zero must be trivial. is w nilpotent element phases. If parameter Z Hence the equation several copies of ρ JHEP10(2017)080 . , ∈ 2 + → ] = ] a w [ ) b a [ 1 2 is an (A.6) (A.7) (A.8) (A.2) (A.3) (A.4) (A.5) + Z ∪ 2 , . . . , n 2 1 ] Sq a w w X, ( = 0 p = k H 2 v : . For example, ), , n 2 q 1 . Z ) vanishes. ) w 2 2 Sq Z Z X, = ( , k 1 ) X, X, , ( ( b ) v H . , ). In particular, on a Rie- q 2 b ] 2 ) 2 b 1 2 C ∈ [ H ∪ Z 1 ∪ . Z k a ∈ ) ( a 2 X, w Sq ]. In particular, for any [ ( δ X, ( Z a δ ( ) vanishes. Another consequence 1 , b [ ∪ p + 2 ) vanishes, then the class − ] + 2 δb. X, ∪ b n Z a H 1 ( b Z ] 2 ∪ p H w a 1 ∈ ∪ X, a Z ] ( ∪ X, ∈ ] + [ 1 ( a b ∈ [ + δa [ p a δa H b ] = [ C ∪ + a , + ] ∪ [ ] ∈ a p δb a [ [ δb 1 a, a δa 2 – 35 – Sq 1 q ∪ ∪ Sq 1: ∪ − p , a a is a differential, i.e. p ) ) = − a − 2 b ∪ 1 n = Z v = a ∪ ]) = ] = 0 for any ]. a Sq b a a a a [ [ ), then 1 [ = X, ( . There also relations which depend on 1 2 1 ] = ( ∪ ∪ δ ∪ a 1 the square of every element in n a we also have the Wu formula: Z appears, etc. One defines an operation ] [ q b b Sq − Sq p a 2 q X X + + − X, ([ Sq + ∪ ( n has degree 1 p , so any orientable 2-manifold admits a spin structure. b b p ] = we have Stiefel-Whitney classes ) is a certain polynomial in Stiefel-Whitney classes independent 2 1 1 1 C a 2 ∪ Sq for all Sq [ H w ∪ ∪ Z X a ∈ 2 ∪ a = ∈ b ] w We note that X, 2 a 1 ] 1 ( . , by the following formula on the cochain level: ] a p w ∪ q -manifold b Sq − [ the square of every element of n a 1 n ≥ = is an obstruction to orientability. If X H p Sq 3 -manifold product is not commutative either, rather one has 1 -manifold one has ∈ ), n w 2 w n 1 . p ] + ) one has [ 2 Z ∪ a − 2 [ w n 1 Z 1 ]. v X, = 2 we have a ( w . It is known as the Wu class. The lowest Wu classes are [ Sq q The Wu formula has many useful consequences. For example, it implies that on any On a closed On an The X, n ( + X ∪ = 1 p ] 3 ] = b v orientable mann surface is that on a spin 4-manifold where of The class obstruction to having aparticular, spin structure.for These classes satisfy a number of relations. In b Note also that ifH [ H Despite appearances, this operation is linear on the level of cohomology classes, i.e. [ where yet another product where the new product Note that the cup product is commutative on the level of cohomology classes, i.e. [ The cup product is not (super)commutative on the level of chains, rather one has It is bilinear and associative and satisfies the Leibniz rule JHEP10(2017)080 2 is Z K (B.2) (B.3) (B.4) (B.1) and is is a X , denoted 2) = β , X 2 Z of the projec- ( ω K has the anomaly c to a point. From } ω , where → coming from group c 0 to some other gauge β ω } ) (B.5) + . × { 0 CX λ 2. It also determines : λ / X ∪ 2 × { B β . B 7→ U(1)) X ( i . , δ i λ λ | 1 2 λ 2) | ,  2  = ) = δλ Z ( B δλ ( ∪ δλ K δλ ∪ ω ( ∪ 4 δλ ∪ λ β 1] by collapsing λ H X 2 1 , 1 2 CX Z ∈ ). [0 Z 2 2 iπ × ) = Z iπ where we remember the (ungauged) boundary ) = symmetry in 2+1d with the anomaly B  – 36 – X  , λ 1 2 2 λ X, , λ is short-range entangled, since it is produced from ( Z (0 exp 1 = (0 i 1 exp 1 C ω ω | λ B ω evolution of a Hamiltonian defined by 2 X λ ∈ − X π ) λ = Sq β = i 1 2 i ω + | ω | ) = , λ B (0 ( ). We find a variation in the exponent by time 2 1 2 . We consider the path integral on the cone with base ω 1-cochain i ω Z λ 2 X | Z symmetry in 2+1d λ X, symmetry. We can use it to compute the SPT ground state on a closed ( 2 . This is of course the same thing as a gauge transformation on 1 P 2 we obtain the state } Z Z Z 1 X ∈ × { β can be made from the cylinder symmetry anomaly is beyond the scope of this paper. X E CX Now we consider a global symmetry transformation Computing the sum over all these . symmetries relevant for fermionization. Unfortunately the calculation of descendants 2 . In this section, we discuss the descent procedure for anomalies of 1-form and 2-form symmetry transformation. 1-cocycle, a product state This function isthe called variation the of the first partition descendant function of of the 2+1d theory under the 1-form gauge We can rewrite this state as This expression makes it clear that condition on CX this description one sees thatthe a same 2-form thing gauge as a fieldfield homotopy on on from the the cone trivial gaugeparametrized field by on a This cohomology class also definesby an the effective action 1-form for aoriented bosonic 3-manifold SPT in 3+1d protected Z of the B.1 1-form As a warm-up, we consider the 1-form The ’t Hooft anomaliestively reveal themselves by in unitary how operators thecohomology, on symmetry one algebra can the is compute Hilbert realized thetive space. projec- so-called action. descendants For Conversely, to a an find systemω anomaly the with class a projective symmetry in class B Anomaly descendants JHEP10(2017)080 . X C for β (B.9) (B.6) (B.7) (B.8) (B.14) (B.10) (B.11) (B.12) (B.13) Λ + . i 7→ Λ |  Λ , 1 2 ∪ β . β . ∪ i 1 λ | β ∂X . Z 1 2  U(1)) is any different than trans- , λ , i iπ i 2 ∪ 3) Λ Λ . β | , | 0) = β 2  ,  + Λ) Z 1 . It can also be regarded as an Λ Λ) + 1 ( 2 ∂X ∪ δ Λ) β , β ∪ Z , Z K 1 Λ) mod 2 λ. ( (0 ∪ (0 5 iπ 2 ∪ ' ∪ 1 Λ ω H ω + β symmetry. As before, on a 4-manifold δ Λ + Λ 1 2 δ ∈ Λ + Λ − X 2 δ Z 1 dλ ) Z CX C 2 1 Λ + Λ ∪ U(1)) ∪ Z 1 iπ δ ) = ∪ , , β 2 λ ∪ 1 1 – 37 – (Λ iπ 3)  ∪ (Λ , 1 2 X  C , β 2 ). We find Z , λ, β 1 2 X 2 Z (Λ (0 Z ( exp (0 Z δ 2 iπ exp = 2 ω K  iπ Λ) = ω Λ ( X, , C Λ X 3  − ( 2 X Λ = 2 (0 ) H = exp δ 1 2 the SPT ground state is invariant under global 1-form sym- Z = Sq i 1 β ω exp . This is measured by i 1 2 λ symmetry in 3+1d with an anomaly -valued 2-cochain parametrizing a gauge transformation of ∈ ω 2 ∪ 2 | X X ω + 2 β Λ | β Z Λ X 1 Z δ ) =  . As we have seen above, such cocycles are trivialized by quadratic , β i 7→ C 2 2 0 ( is nonempty, it is invariant only up to a boundary term: ω β , Z | ω ) is a 2 (0 X B symmetry in 3+1d 2 Z Z ∂X ω 2 iπ Z followed by X, (  1 2 β C exp ] is the generator of ∈ -valued 2-cocycle C 2 Z i 7→ ω | Now we compute the transformationsome of this state under a global symmetry Λ So we can rewrite using the first descendant where Λ We compute where [ effective action of a 4+1dwe SPT obtain with an a SPT 2-form ground state B.2 2-form Next we consider 2-form so indeed we dosymmetry have 2-group a projectiverefinements symmetry of action the measured form, which by in a this bilinear dimension form we may on obtain the from a spin structure. It also tells us whetherof the the global 2+1d 1-form theory. symmetryforming That acts by is, projectively whether on transforming the by Hilbert space This means that for closed metry, but when which defines the second descendant JHEP10(2017)080 ] ) 2 2 X Z CP (C.2) (C.1) (B.18) (B.15) (B.16) (B.17) = X,Y, X ( • H with boundary 2 X β 1. Such a symmetry 1 ∪ − , we may consider the ) and not cohomology . In other words, the 2 1 X X β Z 1 2 on ∂X, ( ) = X 2 1 . For example, when B β Z . 0 . , β ] . 2 0 0 β ) which measures the obstruction to ∈ ] = 0. Alternatively, we can restrict , Z X 2 β 1 [ . β and β Z 1 (0 ) ∪ ∩ 2 → ∪ ∪ β ) ω X ) δλ β β ∪ 2 1 2 1. Thus we are dealing with an SPT phase − 1 2 Z X,Y, β ) ( − 2 2 ] = [ X Y, ( ) = Z β X, Y, ζ ) = H , β 2 2 ( – 38 – 1 ). Indeed, it has a variation 2 , β ), and that the fundamental homology class [ Z iπ , β ∈ . 0 2 0 Sq w , β ) Z ) is surjective. Given such Y ): , β 2 , β ∪ (0 ] exp( 2 Z B X,Y (0 . Recall also that the cup product makes X, , there exists an oriented 4-manifold (0 ( ( ( 2 ω X 2 ζ • 1 X Y Y, ). ω B ( − X, Y, ζ Q δω [ 2 C H 2 ) extends to a 2-cocycle ( ) 2 Z 2 2 ∈ H Z for which [ w β ) λ ). Thus it makes sense to consider the expression → β Y, + ∂X, ( 2 B ( ) 1 2 ( 2 Z 2 ζ Z , β Z Z 0 Q , ∈ , where X, ( X,Y, (0 ( 2 B 2 δλ 4 is a spin structure on ω H + H ζ β to a spin structure on 7→ ζ β is a closed oriented 3-manifold equipped with a triangulation and a branching represents the unique nonzero degree-2 class, this prefactor is Y β Given an oriented 3-manifold After this is done, we can extract the second descendant which measures to what to be a spin 4-manifold. such that any relative Stiefel-Whitney class extending into a module overtakes the values algebra in Here structure, and Y restriction map C The function We summarize here the definition and properties of the function is only well-defined on which looks like it couldaction be on exact but the unfortunately boundary isclasses. is not. Likewise, only This the defined means bilinear that for form the cocycles symmetry is projective: An interesting feature aboutmations this term is that it is not invariant under 2-gauge transfor- transformation multiplies the ground stateonly by if we restrict toX those extent the action of the global 2-form symmetry on the Hilbert space of the 3+1d theory but also a new ingredient: a prefactor This factor is notand a boundary term for a general This variation involves the boundary variation we expected from the previous calculation JHEP10(2017)080 ) B and C.2 , the com- (C.7) (C.3) (C.4) (C.5) (C.6) (C.8) (C.9) 0 ζ (C.10) ζ implies X restricts Q B ,B T 0 B X ), but not on 2 ) vanishes, and Z ) for all X, ) and B ( ( as well as the choice X ), the quantity ( 2 0 . B , nor on the choice of 2 ζ ) . X, Y, ζ 2 H Z ( ∪ ) X X Q 2 Z ζ ∈ B X,B Y, w is closed, and to ( ] 1 Y, Y ). On the other hand, this ) = 1 ( 2 Z ∪ X 1 Y Y Z B Z B ( Z δλ ζ ∈ ) + Y, ∈ ( Q + α 1 . from α along . Then B, Z X ∀ 0 δλ B X ∪ B ∈ X ∪ X, Y, ζ ζ . ( ∪ is an affine space over the vector space α λ 2 T T Y B, ( X w Z B B B. and Y Y ∪ B X ∪ ∪ + Z ∪ . α Z 0 X . Indeed, given any two spin structures 2 X T α X is shifted by ζ ζ Z Y w ). It is easy to see that + B – 39 – ) + B Y Z 2 ζ extends to T ∼ Z for some T X B Z ∪ Z ) = ( ζ Z ζ B 2 ζ ) + B Y, B ( w ( ∪ Q . Given two such choices, ( B 2 ∪ ζ ( . By Poincar´eduality, X X ζ X is an absolute 2-cocycle, it does depend on the choice of Q H α ) is an affine linear function on it. Z B Q B ) = Y X to ) is linear and depends only on the cohomology class of ∪ R X δλ C.2 ) B B 0 Z Y ) = . But this is the same as ζ + )( 0 0 B ζ − ( X B ) = obtained by gluing α ζ ( Q ( ζ B + from ) depends on the cohomology class [ ( on ζ T Y − Q ζ R 0 Q ζ B Q C.2 X Q B and is exact, which means that X 0 ), and the quantity ( ζ 2 on − Z ζ X Now suppose the spin structure Conversely, for a fixed triangulation and branching structure, the function This implies that the expression The quantity ( Y, ( B within its cohomology class in 1 . On the other hand, since , the difference ( 0 pletely determines the equivalence classζ of and thus must havedifference the is form equal to that therefore we get This property is used in section 4 to argue the conservation of fermion number. It is also easy to see that does not depend eitherX of the choice ofB extension of to of the extension of difference between the corresponding expressions is where the 4-manifold In other words, theH set of spin structures on the concrete representative. But it also depends on the choice of to indicate that whenshifts the by spin structure We write it somewhat schematically as JHEP10(2017)080 ). 2 Z , we 1 (D.2) (D.3) (D.4) (D.1) P, ∪ ( 5 H ... ]. Further, since → 39 , the 4th one is the 2 w 1 4 times the Pontryagin ) can become identical / U(1)) ). Orientability implies 2 2 Sq symmetry in 3+1d, both and certain characteristic Z Z ). By exactness, it follows K, U(1)) using the embedding = ( 2 2 B. P, 5 B 3 P, 3 Z Z ], so we have four candidates: ( P, ( 5 ( H w 5 5 39 K, H H ( H → )[ 5 2 , ) ˜ B, w B, Z ) 2 2). The 1st map is multiplication by Z 1 δ ˜ , Z B 1 2 in P, Sq 1 . Using the defining property of ∪ ( Z 2 K, ∪ ( 4 B ( B ˜ symmetry is a closed oriented 5-manifold. All such B 1 ˜ 5 K H B δ 2 δ and is generated by 1 H 1 8 P Sq ∈ = Z 2 4 + B, w U(1)). x + → Z 1 K – 40 – ˜ Sq U(1)). In fact, we will find they map to the same B ˜ B K, Sq δ ∪ + ( P, 2 5 ( ˜ ∪ 5 B B U(1)) H ( ), where 1 ˜ B H 2 1 4 , namely the Stiefel-Whitney classes and the Pontryagin 1 4 K, Z ( P ), for which a representative may be written 4 BSq ) is = 0 for any 2 B, Sq P, ( 1 Z H x 2 , 1 ) and then embed them into D.3 Z 2 → ) is an integral lift of 2) Sq , BSq Z Z ∈ 2 , It is the Pontryagin-dual of the oriented bordism group Z P, ( 2) ( B , 5 . U(1)) U(1)) is isomorphic to 2 K , ( H U(1)). Z 2 ( K, 2) ). Physically, these classify possible 5d topological actions built out of ( P, , H 4 K ( 2 U(1)) Z ( U(1). We are interested in the image of the Bockstein homomorphism. The 5 , , ∈ 2 Z H ( ] H 2) 2) C → , , B K → 2 2 ( ∈ 4 Z Z ( ( ˜ U(1), keeping in mind that distinct elements of ... B H U(1) ) = 0 and it follows K K ( ( satisfies the Leibniz rule w. r. to the cup product, and Let us write down candidate independent terms in In the bosonic case, we need to compute the oriented cobordism group → P → 1 ( SO SO 5 5 1 2 2 which is a cocyclethat formula this for difference maps to zero in square of [ where find that the Bockstein of ( where we have introduced the2, short-hand and the 2nd mapZ is the Bockstein homomorphism associatedgroup to the short exact sequence same as the 3rdNow one. we must Thus map we these are(nonzero) classes left element. to with To see only this, two one independent needs elements to of use the long exact sequence w The 2nd and the 3rdSq are actually the same thanks to the Wu formula [ Whitney classes, itstruct is elements sufficient in to considerZ the latter.elements of An obvious approach is to con- Ω a 2-form gauge field topological terms will beclasses of integrals the of tangent densities bundleclasses. of made out Actually, of since Pontryagin classes modulo 2 can be expressed through Stiefel- In this section wefor classify bosonic and possible fermionic anomaliesignoring theories, time-reversal for symmetries, assuming a if the 1-form any). space-time symmetry isΩ orientable (i.e. D ’t Hooft anomalies for a 1-form JHEP10(2017)080 3 P w . 2 and ] and ) = 0 (E.1) B w (D.5) (D.6) 2 P B Z ⊕ ( ) in the ) 2 ∪ 2 K w Z ( B 5 U(1)). These K, H 2) are known, , and the non- ( , 2 4 . K, . 2 differential since 2 Z → ( 2 H Z Z 2 ) ( Z . d 5 SO ∈ K K C ( 1 ) = C = 1 SO 5 B. , they all vanish because K Sq 1 U(1)) = ( ) becomes trivial for such which splits off from the P U(1)) is trivial. However, K Sq 5 U(1)) = Sq 3 K, 2 ( D.6 K, K, w ( w ( 2 ,H 5 SO P w 4 ) and ˜ Z Ω U(1)). Note that since 2 5 Spin Z 5 Spin . 1 2 3 Z Ω K, ). This cohomology group is 3), an Eilenberg-Maclane space with w ( 2 = 2 , ) = K, → 2 Z ] = 0, then the corresponding 5d spin- ) which is Poincar´e-dualto [ ( K B 2 3 Z B symmetry ( 5 Spin 1 ( Z K, 4 H ( 2 ∪ K Sq 4 P, C, w Z 2 ∈ ( U(1)) B 2 H = 1 ,H Sq C – 41 – H 2 K, ( K P Z , the bosonic action ( Z . For bosonic systems, the possible anomalies are U(1)) which do not come from Ω term 5 SO 2 C, w P 1 2 2 2 Z ) = K, E = ( K Sq = ( ]. The corresponding spin-topological action is evaluated B 2 3 which realizes it. Then we restrict the spin structure of 1 B π 5 Spin ) and so describes a purely gravitational anomaly. So as 2 γ 2 Z Sq ,H BSq P, Z )). The integral homology groups of P ( ? Z 3 ( ] = [ 1 2 Z ) = B SO q symmetry. The calculations are much the same as the previous section, satisfies the constraint [ K ∈ ∪ 2 Ω ( 0 Z B B C K, H ( 6= 0. The second, however, defines a topological term which measures the p H C 2 Sq = and evaluate the corresponding holonomy. Thus Ω When we consider these terms on a closed spin 5-manifold Note that if In the fermionic case, we need to compute Ω To verify that this exhausts all possible topological actions, one can use the Atiyah- = 0. For new anomalies we look to γ C . That is, the image of the map Ω 2 2 w Atiyah-Hirzebruch spectral sequence.d The first does notholonomy survive of the the spin structure along the curve Poincar´edual to The first two are actuallythe equal thanks gauge to the field Wubefore, formula, while we the find third that does (3-)group not cohomology involve describes all the gauge anomalies. E ’t Hooft AnomaliesIn for this a section we 2-form wishtems to with 2-form discuss possible ’texcept Hooft where anomalies now for the bosoniconly and classifying nonzero fermion space homotopy sys- group pick a closed 1d submanifold to topological action is zero. Thus the fermionic anomaly is necessarily trivial for such topological terms use theacteristic spin structure classes. in a Lookingonly key at such way element and the arises won’t Atiyah-Hirzebruchis from be spectral generated the just sequence, by integrals [ we ofas char- see follows: that we take the the homology class in for a closed orientableP spin manifold there can also be elements of Ω The first termanomaly defines group. Then a the spectral purelyis sequence implies an gravitational that isomorphism. the anomaly map Hence Ω trivial the anomaly group action of can gauge anomalies be written as Hirzebruch spectral sequence computing orientedogy bordism groups groups starting from theand homol- the nonzero ones up to degree 5 are JHEP10(2017)080 ) ]. 2 38 B ( (F.6) (F.3) (F.4) (F.5) (F.1) (F.2) has a 2 U(1)). . This ). Now, Sq E 2 ), a first C as a pair Z , reflecting B BE, ( 2 ( 2 ) such that 5 ) is obviously BE Sq B H Sq BE, ( ( 1 2 x F.4 6 , → 5 , H 0 2 , which is a non-zero P 3 . E ' ) B 2 ∈ 1 2 α  2) by forgetting , see appendix D, and thus , whose elements are pairs + ˆ B , 2 2 Its differential is 1 E δ 2 U(1)) Z . Z ) ∪ , α ( 11 2 B
U(1)), we use the Serre spectral K B BE, α ( 2 . (  ) with 5 ∪ B 2 2 1 δx = 1 2 H B C U(1)) = BE, and the Postnikov class + α ( ∪ ( 1 2 K 2 2 , 2 + ) ∪ 3 K, C 2 Z B. 5 SO ∪ ( 2 B E 5 )). The three possible terms are → C ∪ δ β = C 1 1 2 L ∈ H 1 2 3 1 2 ( ∪ B + ] q π 1 + , BE + C B = 3). This fibration is very useful for computing – 42 – β 2 2 , ∪ C 2 Z B = ( K,H δC 1 Z 2 ( B ( = 2 [ ∪ p ) = ( ∪ 1 2 2 symmetry 2 K B 2 H C π 2 ), we note that it can also be written as , α 1 2 B . The differential in this spectral sequence comes from E = = ∪ 0 2 of the fibration. , 1 2 B β 5 2 2 F.4 ) satisfying ). The group law is not the product, but has a twist L ( L with p,q 2 2 2 E B ◦ . Thus there should exist a mod-2 5-cochain Z Z E ) ∈ B 1 ] 1 P, BE X, ( ( C is a little complicated to deal with. , α 1-form symmetry considered in the previous appendix, . For instance, to compute 3 1 Sq 1 1 C 2 β Z C ∪ ∪ ( Z BE 1 × × C B ∪ [ ) ) 1 2 2 1 2 C Z Z U(1)), so this term does not contribute. The differentials of the third term Sq 1 2 P, X, K, ( ( ( 2 2 6 is an integral lift of Z Z H ). For example, the differential of the second term above is ˆ ∈ B ∈ ) ) F.2 The term We are grateful to Greg Brumfiel and John Morgan for communicating to us some related results [ 11 B,C β, α the Cartan formula for Steenrodis squares the says same that as the mod-2 cohomology class of While it is nonzero,exact, we so now it show can that be it taken is care exact. of by The replacing second term in ( To deal with the first term in ( evaluates to zero on the fiber eq. ( class in are all zero, but itin may the be right a degree, differentialThis so of class the something is third else. the term pull-back However, survives of there to the is generator give no of a candidate nontrivial class in sequence which starts with where From this one seesmap is that a there fibrationthe is with cohomology a fiber of map approximation to the bosonic anomaly group Ω In this section we discusssystems. possible anomalies for Like such a the classifying symmetry space denoted in bosonic andthe fermionic twisted group( law. This means that one can think of a map We have discussed the appearance( of the 3-group symmetry F ’t Hooft anomalies for JHEP10(2017)080 , to G BE (F.7) (F.8) (F.9) (F.10) BG subgroup for certain 2 Z 4 2 5. Thus, ex- Z , and that it is B 4 . × ) . Z as taking values 0 8 2 q < B Z ( Z ˆ B x → 1 2 . . . It also means that to = 5 4 . ! G + ] Z G ˆ , so the difference between B B 2 ˆ anomalies are classified by B 2 − 2 1 δ (235) δ = 0, i.e. it is a extension of for B ˜ ∪ Sq E is exact or not. In appendix ∪ . Thus we can write: ˆ 2 B  BSq B ˆ ˆ 2 [  symmetry are actually classified B . An explicit simplicial expression B is cohomologous to the Bockstein

δ 1 2 Sq 1 4 and (012) 1 2 Sq δ E B 2 B + = F.6 1 4  1 = ]. Note also that ) # ˜  ∪ Sq 2 , all the B ˆ U(1)) is isomorphic to B B 3 2 1 B = B (245) 1 δ ( w . In fact, if we define 2 ∪ 2 2 B = . Note that the extension is not unique, and ∪ B ), we can just check that the ), for either choice of the sign. The ambiguity  BE, ∪ w BSq B ( ˆ [ B For example, given an explicit map from B B 5 when one sets 1 2 C 1 BE δ – 43 – " F.8 1 2 (023) H C,B 1 2 12 1 2 exchanges ( C Sq B  ) does indeed give rise to a cohomology class of 2 2 U(1)). We have shown in the above section that +  B U(1)). There is another useful spectral sequence 1 2 U(1)) is an isomorphism and it is not hard to show 1 ˜ C Sq Sq , F.3 )] = 1 2 1 3 = BE, Sq ]. ∪ ( 2 U(1)) is also an isomorphism for all K BE, ˆ B q ) 2 ( , + ( δ C C,B 3 + B ( p SO 1 2 C SO 5 B 5 SO K 1 2 ) is not that useful unless one has a powerful computer method  ( 1 2 ˜ Ω ∪ )(012345) = 7→ ˜ Sq ) = q SO F.8 is an integral lift of B B → 2[ C ( ( Ω ˆ . B x 1 2 0)] = [ C C,B ) restricts to 2 → . ( 4 B, 2  ˜ U(1)) to the total space U(1)) Z Sq F.8 1 U(1))) to Ω coefficients suffice, so we will be iterating over functions . , , ˜ 8 Sq 3 3 L, Sq Z , where ( ( K BE U(1)) K 5 ˆ ( ( 2 B ,  symmetry has anomaly 5 3 δ ] H → 1 2 ˜ q SO Sq H ) is a mod-2 cochain defined by the equation K E 41 ∈ U(1)) = ( Ω ] B q , ( BG 2 C H x 2 , i.e. Note that the shift The anomalies for 3+1d bosonic systems with An explicit formula ( K BE, , one would like to know whether the pull-back of To this end, ( ( B 5 p Sq 12 1 2 H the anomalies is not really physical,Further, amounting [ to a redefinition of the symmetry operators. for computing this,H the Atiyah-Hirzebruch-Serrethe spectral map sequence,that which goescept from for the purely gravitational anomaly BE we indirectly show themaps exactness of the pull-back of by the cobordism group Ω generated by the cohomology class ofin eq. the ( sign is simplysee if the an ambiguity in choosinghas the the generator anomaly of for showing that a given cocycle is exact. This means that the cohomology group for it is [ The expression ( [ that the two possible extensions differ by We conclude that the first termand in that ( the corresponding cochain-level expression is Here of and 1 only, there is a cochain-level identity On the other hand, we also know that JHEP10(2017)080 . 2 4 ]. ]. ∈ → Y Z Z 26 and C 26 , ) = 0 2 (G.1) (G.2) (G.3) 2 and so is a 15 w xy Sq = 0 and ( 1 2 4 2 2 y U(1)) gauge field w = 0 we can NY Sq 2 ⊕ Z , a 3-manifold U(1)) sends all BE, ( NY X 2 TY 5 SO w ⊂ BE, ( Y , equipped with a pair of . Next we use Thom’s 2 5 Spin y y it suffices to consider the Ω X gauge field and U(1)) come from NY. + = Y 2 2 → 4 Z defines a spin structure on w Z = 0, which implies ). Direct computation is rather BE, TY ( + 2 y X domain wall in the 3+1D system, NY × 1 w 1 is a 4 U(1)) 5 SO 2 w Sq Z domain wall carries a Kitaev chain, 2 = σ, ρ, ν NY Z is a line bundle so x 1 2 is the inclusion map. This tells us that 2 -equivariant 4d spin-cobordisms which ) BE, 4 Z . The construction mirrors the decorated = ( and the restriction of the X 4 Z y NY NY, T Y w NY G 5 SO : 2 Y × defines an orientation of defines one on → 1 1 1 x Y 2 1 2 w w w , where Y and a Z X X 4 : – 44 – + + = y 4 + ( i 2 too, for that matter). σ Z x U(1)). Because the first is proportional to 3 1 2 TY TY , TY 1 2 2 w domain walls. We want to construct a spin structure 2 2 w w = K 4 w w ( . We consider the Poincar´edual Z 5 = = σ 4 = restricted to ] and its generalization: the Smith homomorphism [ H Y Y Z 4 | | Y = 0, where | ∈ ∈ y TX 1 2 TX TX y B TX 1 2 1 cobordism invariant which characterizes a 2+1D phase whose w 2 Sq w w 8 ∗ w i Z and BSq and = 2 1 2 and further that a spin structure on y Z ∗ , these map to zero on closed spin 5-manifolds, which all have representing the . Using this and the fact that i TX 3 ∈ ∗ 1 TY i w 2 X . It will be interesting in future work to investigate how our framework of w x = 0. This implies that the map Ω NY w = . This shows that there exist U(1)). For this we can use naturality of the Atiyah-Hirzebruch-Serre spec- 1 2 xy 2 w = w xy y 1 U(1)) and = Y , = BE, | = 3 Sq ( σ . To see that the spin structure on With a spin structure on the 3-manifold The construction begins with an arbitrary spin 4-manifold Now we want to consider what happens with the map Ω K σ Y = ( TX TY 5 Spin 5 2 1 3 , we can evaluate a constructions. Combined with the fact thatwe this also 3-manifold was see the thati.e. the intersection ofhave a bosonization in various dimensions interact via this decorated domain wall and similar Note that the important formulabut for this is to stronger. work was x domain walls carry Kitaev chains.the Transversality Smith arguments homomorphism carried show that over this from construction the defines proof a 4d of cobordism invariant [ where we havetheorem used the Poincar´edualityw property The first line saysw that an orientationsimplify of the second line to gauge fields immersed in on obstructions difficult, so we will show it incobordism another invariants way, using via the fact the thatthere such Atiyah-Hirzebruch is triples spectral a correspond spin to sequence. cobordism spin invariantdomain with Indeed, wall we constructions of will [ show the gauge anomalies to zero (and G Vanishing of theIn obstructions this for appendix wegauge show field, can that be extended to fermionic SPT data ( Ω tral sequence. We knowH that our nonzero classesthe in second to Ω w JHEP10(2017)080 − ), A ( pin (H.3) (H.4) (H.1) (H.2) σ to set 6= 0, it X 2 structure σ − structure on ). The key is − pin structure on Σ. A ( ρ − pin pin 1, corresponding to the in the presence of such a , − Σ . Therefore, if F , 2 σ ). Equivalently, its Poincar´e N 1) 2 σ structure should be chosen to 2 as the ambient 4D spacetime, − is the particle contribution to , adding to Σ a thin tube along Z w − ρ X δλ Σ = + X, Σ = ( pin 2 + N 3 N 2 σ . 2 Σ) C w 2 w T 7→ ∈ ∗ 1 ): i Σ) σ w χ σ, structure on Σ and no hope therefore of ]. Recall T 1 ∪ 4.21 − ) = 15 w 2 σ Σ + ( pin Σ) = – 45 – T 2 T Σ + ( for some 1 δρ w T w 2 = δχ w Σ = | the inclusion map, we can then use Thom’s theorem: Σ + ( 2 σ T TX describes the worldlines of fermion “particles”, it is a subset 2 X 2 ) only used the partition function of the Kitaev chain and w is Poincar´edual to this collection of points. χ w ( → ∗ 2 i 4.21 Σ) . :Σ 0 = 2 T i 1 σ w = to the fermion parity (recall above we went the opposite direction). Indeed, δρ Σ + ( ) K F T 2 4.21 w Σ is the normal bundle of Σ and we have used the spin structure on as a cocycle then restricts to Σ as = 0. Denoting N This derivation of ( We must assume then that When Σ is immersed in a spin 4-manifold, this local . Assuming all other sorts of fermionic particles are conserved, we derive the crucial ρ TX TX 2 2 of equation ( the conservation of fermionic particles.contribution We can therefore use it to derive the Kitaev string these curves with periodic spinstrings structure appear in as the fermionic small particles, circularstring, since direction. if it we is Such compute small equivalent Tr( to Kitaev the placing partition a function periodic-periodic of the spinunique Kitaev structure chain nontrivial on in Arf the this invariant worldsheet, spinthe on structure and fermion is the parity. torus Because [ seems like we havedefining no a hope Kitaev chain of on defining that a worldsheet. dual is a union ofWe can curves make with a boundary surgery of at Σ, the amounting singular to a points shift of the where w where we have used the Poincar´eduality property be compatible with the ambientw spin structure. Denoting This class can be describedΣ as patch-by-patch, a and 2-cocycle there will in be thecannot certain following be singular way: extended. points choose across These a whichperiodic are the vortices rather in than the antiperiodicThe spin class structure, boundary around conditions which (only fermions have the later extend to a disc). and the fermion parityto of realize the that junctions, proper Poincar´edualstructure definition in on of spacetime Σ. the to Kitaev The chain obstruction along to a such a worldsheet Σ structure requires is a In this appendix we take another look at eq. ( which ties the fermion parity of the Kitaev strings, Poincar´edual in spacetime to H Fermion parity of the Kitaev string JHEP10(2017)080 , ]. ] J. 83 , (H.5) JHEP , spin systems (2006) 2 D Int. J. Mod. Phys. Rev. B 1 , , arXiv:0912.2157 321 AIP Conf. Proc. [ arXiv:1605.01640 , [ . On the other hand, if Y (2017) 075108 Fermionic matrix product . ρ (2017) 096 Annals Phys. interacting spinless Fermi gas ): ]. B 95 (2010) 065010 , σ. D 04 1 12 1 4.20 ∪ SPIRE IN Topological insulators and superconductors: δλ IPAM program Symmetry and Topology in JHEP ][ , + Phys. Rev. , ]. δλ State sum constructions of spin-TFTs and string ∪ – 46 – New J. Phys. λ , SPIRE + IN , we must shift ( Symmetry protected topological phases from decorated ρ ][ (2014) 3507. . Interpreting it as the contribution of Kitaev strings, , talk given at δρ ]. ρ ) up to a redefinition of 5 7→ σ = Y ( ]. ρ ), which permits any use, distribution and reproduction in Classification of gapped symmetric phases in Spin TQFTs and fermionic phases of matter R ζ 2 arXiv:1505.05856 [ σ Q Topological phases of fermions in one dimension SPIRE ) = IN σ ( ][ K CC-BY 4.0 ‘Luttinger liquid theory’ of one-dimensional quantum fluids. I. Properties of (2011) 035107. F , January 26–30, UCLA, U.S.A. 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