Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond
FField theory representation of gauge-gravity symmetry-protectedtopological invariants, group cohomology and beyond
Juven C. Wang,
1, 2, ∗ Zheng-Cheng Gu, † and Xiao-Gang Wen
2, 1, ‡ Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada
The challenge of identifying symmetry-protected topological states (SPTs) is due to their lackof symmetry-breaking order parameters and intrinsic topological orders. For this reason, it is im-possible to formulate SPTs under Ginzburg-Landau theory or probe SPTs via fractionalized bulkexcitations and topology-dependent ground state degeneracy. However, the partition functions frompath integrals with various symmetry twists are the universal SPT invariants defining topologicalprobe responses, fully characterizing SPTs. In this work, we use gauge fields to represent thosesymmetry twists in closed spacetimes of any dimensionality and arbitrary topology. This allows usto express the SPT invariants in terms of continuum field theory. We show that SPT invariants ofpure gauge actions describe the SPTs predicted by group cohomology, while the mixed gauge-gravityactions describe the beyond-group-cohomology SPTs. We find new examples of mixed gauge-gravityactions for U(1) SPTs in 4+1D via mixing the gauge first Chern class with a gravitational Chern-Simons term, or viewed as a 5+1D Wess-Zumino-Witten term with a Pontryagin class. We ruleout U(1) SPTs in 3+1D mixed with a Stiefel-Whitney class. We also apply our approach to thebosonic/fermionic topological insulators protected by U(1) charge and Z T time-reversal symmetrieswhose pure gauge action is the axion θ -term. Field theory representations of SPT invariants notonly serve as tools for classifying SPTs, but also guide us in designing physical probes for them. Inaddition, our field theory representations are independently powerful for studying group cohomologywithin the mathematical context. Introduction – Gapped systems without symmetrybreaking can have intrinsic topological order.
How-ever, even without symmetry breaking and without topo-logical order, gapped systems can still be nontrivial ifthere is certain global symmetry protection, known asSymmetry-Protected Topological states (SPTs).
Theirnon-trivialness can be found in the gapless/topologicalboundary modes protected by a global symmetry, whichshows gauge or gravitational anomalies.
More pre-cisely, they are short-range entangled states which canbe deformed to a trivial product state by local unitarytransformation if the deformation breaks the globalsymmetry. Examples of SPTs are Haldane spin-1 chainprotected by spin rotational symmetry and the topo-logical insulators protected by fermion number con-servation and time reversal symmetry.While some classes of topological orders canbe described by topological quantum field theories(TQFT), it is less clear how to systematically con-struct field theory with a global symmetry to classify orcharacterize SPTs for any dimension . This challengeoriginates from the fact that SPTs is naturally defined ona discretized spatial lattice or on a discretized spacetimepath integral by a group cohomology construction in-stead of continuous fields. Group cohomology construc-tion of SPTs also reveals a duality between some SPTsand the Dijkgraaf-Witten topological gauge theory.
Some important progresses have been recently madeto tackle the above question. For example, thereare 2+1D Chern-Simons theory, non-linear sigmamodels, and an orbifolding approach implementingmodular invariance on 1D edge modes.
The aboveapproaches have their own benefits, but they may be ei- ther limited to certain dimensions, or be limited to somespecial cases. Thus, the previous works may not fulfill allSPTs predicted from group cohomology classifications.In this work, we will provide a more systematic way totackle this problem, by constructing topological responsefield theory and topological invariants for SPTs (SPTinvariants) in any dimension protected by a symmetrygroup G . The new ingredient of our work suggests aone-to-one correspondence between the continuous semi-classical probe-field partition function and the discretizedcocycle of cohomology group, H d +1 ( G, R / Z ), predictedto classify d + 1D SPTs with a symmetry group G . Moreover, our formalism can even attain SPTs beyondgroup cohomology classifications.
Partition function and SPT invariants – For sys-tems that realize topological orders, we can adiabaticallydeform the ground state | Ψ g.s. ( g ) (cid:105) of parameters g via: (cid:104) Ψ g.s. ( g + δg ) | Ψ g.s. ( g ) (cid:105) (cid:39) . . . Z . . . (1)to detect the volume-independent universal piece of par-tition function, Z , which reveals non-Abelian geometricphase of ground states. For systems that real-ize SPTs, however, their fixed-point partition functions Z always equal to 1 due to its unique ground state onany closed topology. We cannot distinguish SPTs via Z . However, due to the existence of a global sym-metry, we can use Z with the symmetry twist toprobe the SPTs. To define the symmetry twist, we notethat the Hamiltonian H = (cid:80) x H x is invariant underthe global symmetry transformation U = (cid:81) all sites U x ,namely H = U HU − . If we perform the symmetrytransformation U (cid:48) = (cid:81) x ∈ ∂R U x only near the bound-ary of a region R (say on one side of ∂R ), the local a r X i v : . [ c ond - m a t . s t r- e l ] M a y !" !"
2D 3D (a) (b) (c) (d) (e) !" (f) !" !" !" FIG. 1. On a spacetime manifold, the 1-form probe-field A can be implemented on a codimension-1 symmetry-twist (with flat d A = 0) modifying the Hamiltonian H , but theglobal symmetry G is preserved as a whole. The symmetry-twist is analogous to a branch cut, going along the arrow- - - (cid:66) would obtain an Aharonov-Bohm phase e ig with g ∈ G by crossing the branch cut (Fig.(a) for 2D, Fig.(d)for 3D). However if the symmetry twist ends, its ends are monodromy defects with d A (cid:54) = 0, effectively with a gaugeflux insertion. Monodromy defects in Fig.(b) of 2D act like0D point particles carrying flux, in Fig.(e) of 3Dact like 1D line strings carrying flux. The non-flat mon-odromy defects with d A (cid:54) = 0 are essential to realize (cid:82) A u d A v and (cid:82) A u A v d A w for 2D and 3D, while the flat connections(d A = 0) are enough to realize the top Type (cid:82) A A . . . A d +1 whose partition function on a spacetime T d +1 torus with( d +1) codimension-1 sheets intersection (shown in Fig.(c),(f)in 2+1D, 3+1D) renders a nontrivial element for Eq.(2). term H x of H will be modified: H x → H (cid:48) x | x near ∂R .Such a change along a codimension-1 surface is calleda symmetry twist, see Fig.1(a)(d), which modifies Z to Z (sym.twist). Just like the geometric phases of the de-generate ground states characterize topological orders, we believe that Z (sym.twist), on different spacetimemanifolds and for different symmetry twists, fully char-acterizes SPTs. The symmetry twist is similar to gauging the on-sitesymmetry except that the symmetry twist is non-dynamical. We can use the gauge connection 1-form A todescribe the corresponding symmetry twists, with probe-fields A coupling to the matter fields of the system. Sowe can write Z (sym.twist) = e i S (sym.twist) = e i S ( A ) . (2)Here S ( A ) is the SPT invariant that we search for.Eq.(2) is a partition function of classical probe fields, or atopological response theory, obtained by integrating outthe matter fields of SPTs path integral. Below we wouldlike to construct possible forms of S ( A ) based on thefollowing principles: (1) S ( A ) is independent of space-time metrics ( i.e. topological), (2) S ( A ) is gauge invari-ant (for both large and small gauge transformations), and(3) “Almost flat” connection for probe fields. U(1) SPTs – Let us start with a simple example ofa single global U(1) symmetry. We can probe thesystem by coupling the charge fields to an externalprobe 1-form field A (with a U(1) gauge symmetry),and integrate out the matter fields. In 1+1D, we canwrite down a partition function by dimensional count-ing: Z (sym.twist) = exp[ i θ π (cid:82) F ] with F ≡ d A ,this is the only term allowed by U(1) gauge symmetry U † ( A − id) U (cid:39) A + d f with U = e i f . More gener-ally, for an even ( d + 1)D spacetime, Z (sym.twist) =exp[ i θ ( d +12 )!(2 π ) d +12 (cid:82) F ∧ F ∧ . . . ]. Note that θ in such anaction has no level-quantization ( θ can be an arbitraryreal number). Thus this theory does not really corre-spond to any nontrivial class, because any θ is smoothlyconnected to θ = 0 which represents a trivial SPTs.In an odd dimensional spacetime, such as 2+1D,we have Chern-Simons coupling for the probe fieldaction Z (sym.twist) = exp[ i k π (cid:82) A ∧ d A ]. Moregenerally, for an odd ( d + 1)D, Z (sym.twist) =exp[ i πk ( d +22 )!(2 π ) ( d +2) / (cid:82) A ∧ F ∧ . . . ] , which is known tohave level-quantization k = 2 p with p ∈ Z for bosons,since U(1) is compact. We see that only quantized topo-logical terms correspond to non-trivial SPTs, the allowedresponses S ( A ) reproduces the group cohomology de-scription of the U(1) SPTs: an even dimensional space-time has no nontrivial class, while an odd dimension hasa Z class. (cid:81) u Z N u SPTs – Previously the evaluation of U(1) fieldon a closed loop (Wilson-loop) (cid:72) A u can be arbitraryvalues, whether the loop is contractible or not, since U(1)has continuous value. For finite Abelian group symmetry G = (cid:81) u Z N u SPTs, (1) the large gauge transformation δA u is identified by 2 π (this also applies to U(1) SPTs).(2) probe fields have discrete Z N gauge symmetry, (cid:73) δA u = 0 (mod 2 π ) , (cid:73) A u = 2 πn u N u (mod 2 π ) . (3)For a non-contractible loop (such as a S circle of atorus), n u can be a quantized integer which thus allowslarge gauge transformation. For a contractible loop, dueto the fact that small loop has small (cid:72) A u but n u is dis-crete, (cid:72) A u = 0 and n u = 0, which imply the curvatured A = 0, thus A is flat connection locally. (i). For , the only quantized topological termis: Z (sym.twist) = exp[ i k II (cid:82) A A ] . Here and belowwe omit the wedge product ∧ between gauge fields asa conventional notation. Such a term is gauge invari-ant under transformation if we impose flat connectiond A = d A = 0, since δ ( A A ) = ( δA ) A + A ( δA ) =(d f ) A + A (d f ) = − f (d A ) − (d A ) f = 0. Here wehave abandoned the surface term by considering a 1+1Dclosed bulk spacetime M without boundaries. • Large gauge transformation : The invariance of Z under the allowed large gauge transformation viaEq.(3) implies that the volume-integration of (cid:82) δ ( A A )must be invariant mod 2 π , namely (2 π ) k II N = (2 π ) k II N =0 (mod 2 π ). This rule implies the level-quantization . • Flux identification : On the other hand, when the Z N flux from A , Z N flux from A are inserted as n , n multiple units of 2 π/N , 2 π/N , we have k II (cid:82) A A = k II (2 π ) N N n n . We see that k II and k (cid:48) II = k II + N N π giverise to the same partition function Z . Thus they mustbe identified (2 π ) k II (cid:39) (2 π ) k II + N N , as the rule of fluxidentification. These two rules impose Z (sym.twist) = exp[ i p II N N (2 π ) N (cid:90) M A A ] , (4)with k II = p II N N (2 π ) N , p II ∈ Z N . We abbre-viate the greatest common divisor (gcd) N ...u ≡ gcd( N , N , . . . , N u ). Amazingly we have independentlyrecovered the formal group cohomology classification pre-dicted as H ( (cid:81) u Z N u , R / Z ) = (cid:81) u Type IV partitionfunction that is independent of spacetime metrics: Z (sym.twist) = exp[i p IV N N N N (2 π ) N (cid:90) M A A A A ] , (8)where d A i = 0 to ensure gauge invariance. The largegauge transformation δA i of Eq.(3), and flux identifica-tion recover p IV ∈ Z N ⊂ H ( (cid:81) i =1 Z N i , R / Z ). Herethe 3D SPT invariant is analogous to 2D, when thefour codimension-1 sheets ( yzt , xzt , yzt , xyz -branes inFig.1(f)) with flat A j of nontrivial element g j ∈ Z N j in-tersect at a single point on spacetime T torus, it rendersa nontrivial partition function for the Type IV SPTs.Another response is for Type III 3+1D SPTs: Z (sym.twist) = exp[i (cid:90) M p III N N (2 π ) N A A d A ] , (9)which is gauge invariant only if d A = d A = 0. Basedon Eq.(3),(7), the invariance under the large gauge trans-formations requires p III ∈ Z N . Eq.(9) describes TypeIII SPTs: p III ∈ Z N ⊂ H ( (cid:81) i =1 Z N i , R / Z ). Yet another response is for Type II 3+1D SPTs: Z (sym.twist) = exp[i (cid:90) M p II N N (2 π ) N A A d A ] . (10)The above is gauge invariant only if we choose A and A such that d A = d A d A = 0. We denote A =¯ A + A F where ¯ A d ¯ A = 0, d A F = 0, (cid:72) ¯ A = 0 mod2 π/N , and (cid:72) A F = 0 mod 2 π/N . Note that in generald ¯ A (cid:54) = 0, and Eq.(10) becomes e i (cid:82) M p II N N π )2 N A A F d ¯ A .The invariance under the large gauge transformations of A and A F and flux identification requires p II ∈ Z N = H ( (cid:81) i =1 Z N i , R / Z ) of Type II SPTs. For Eq.(9),(10),we have assumed the monodromy line defect at d A (cid:54) = 0 is gapped ; for gapless defects, one will need to introduceextra anomalous gapless boundary theories. SPT invariants and physical probes – Top types: The SPT invariants can help us to designphysical probes for their SPTs, as observables of numer-ical simulations or real experiments. Let us consider: Z (sym.twist)= exp[i p (cid:81) d +1 j =1 N j (2 π ) d N ... ( d +1) (cid:82) A A . . . A d +1 ], ageneric top type (cid:81) d +1 j =1 Z N j SPT invariant in ( d + 1)D,and its observables. • (1). Induced charges : If we design the space to have atopology ( S ) d , and add the unit symmetry twist of the Z N , Z N , . . . , Z N d to the S in d directions respectively: (cid:72) S A j = 2 π/N j . The SPT invariant implies that such aconfiguration will carry a Z N d +1 charge p N d +1 N ... ( d +1) . • (2). Degenerate zero energy modes : We can also ap-ply dimensional reduction to probe SPTs. We can de-sign the d D space as ( S ) d − × I , and add the unit Z N j symmetry twists along the j -th S circles for j =3 , . . . , d + 1. This induces a 1+1D Z N × Z N SPT in-variant exp[ i p N N ... ( d +1) N N πN (cid:82) A A ] on the 1D spa-tial interval I . The 0D boundary of the reduced 1+1DSPTs has degenerate zero modes that form a projectiverepresentation of Z N × Z N symmetry. For example,dimensionally reducing 3+1D SPTs Eq.(8) to this 1+1DSPTs, if we break the Z N symmetry on the Z N mon-odromy defect line, gapless excitations on the defect linewill be gapped. A Z N symmetry-breaking domain wallon the gapped monodromy defect line will carry degen-erate zero modes that form a projective representation of Z N × Z N symmetry. • (3). Gapless boundary excitations : For Eq.(8), we de-sign the 3D space as S × M , and add the unit Z N symmetry twists along the S circle. Then Eq.(8) re-duces to the 2+1D Z N × Z N × Z N SPT invariantexp[ i p IV N N N N N πN (cid:82) A A A ] labeled by p IV N N ∈ Z N ⊂ H ( Z N × Z N × Z N , R / Z ). Namely, the Z N monodromy line defect carries gapless excitations iden-tical to the edge modes of the 2+1D Z N × Z N × Z N SPTs if the symmetry is not broken. Lower types: Take 3+1D SPTs of Eq.(9) as an exam-ple, there are at least two ways to design physical probes.First, we can design the 3D space as M × I , where M is punctured with N identical monodromy defects eachcarrying n unit Z N flux, namely (cid:90) (cid:7)(cid:6) (cid:4)(cid:5) (cid:90) d A = 2 πn ofEq.(7). Eq.(9) reduces to exp[ i p III n N N (2 π ) N (cid:82) A A ],which again describes a 1+1D Z N × Z N SPTs, labeledby p III n of Eq.(4) in H ( Z N × Z N , R / Z ) = Z N . Thisagain has 0D boundary-degenerate-zero-modes.Second, we can design the 3D space as S × M andadd a symmetry twist of Z N along the S : (cid:72) S A =2 πn /N , then the SPT invariant Eq.(9) reduces toexp[ i p III n N (2 π ) N (cid:82) A d A ], a 2+1D Z N × Z N SPTs la-beled by p III n N N of Eq.(6). • (4). Defect braiding statistics and fractional charges :These (cid:82) A d A types in Eq.(6), can be detected by thenontrivial braiding statistics of monodromy defects, suchas the particle/string defects in 2D/3D. More-over, a Z N monodromy defect line carries gapless excita-tions identical to the edge of the 2+1D Z N × Z N SPTs.If the gapless excitations are gapped by Z N -symmetry-breaking, its domain wall will induce fractional quantumnumbers of Z N charge, similar to Jackiw-Rebbi orGoldstone-Wilczek effect. U (1) m SPTs – It is straightforward to apply theabove results to U(1) m symmetry. Again, we find only trivial classes for even ( d + 1)D. For odd( d + 1)D, we can define the lower type action: Z (sym.twist) = exp[ i πk ( d +22 )!(2 π ) ( d +2) / (cid:82) A u ∧ F v ∧ . . . ] . Meanwhile we emphasize that the top type action with k (cid:82) A A . . . A d +1 form will be trivial for U(1) m case sinceits coefficient k is no longer well-defined, at N → ∞ of ( Z N ) m SPTs states. For physically relevant 2 + 1D, k ∈ Z for bosonic SPTs. Thus, we will have a Z m × Z m ( m − / classification for U(1) m symmetry. Beyond Group Cohomology and mixed gauge-gravity actions – We have discussed the allowed ac-tion S (sym.twist) that is described by pure gauge fields A j . We find that its allowed SPTs coincide with groupcohomology results. For a curved spacetime, we havemore general topological responses that contain bothgauge fields for symmetry twists and gravitational con-nections Γ for spacetime geometry. Such mixed gauge-gravity topological responses will attain SPTs beyondgroup cohomology. The possibility was recently discussedin Ref.17 and 18. Here we will propose some additionalnew examples for SPTs with U(1) symmetry.In , the following SPT response exists, Z (sym.twist) = exp[i k (cid:90) M F CS (Γ)]= exp[i k (cid:90) N F p ] , k ∈ Z (11)where CS (Γ) is the gravitations Chern-Simons 3-formand d(CS ) = p is the first Pontryagin class. This SPTresponse is a Wess-Zumino-Witten form with a surface ∂ N = M . This renders an extra Z -class of 4+1D U(1)SPTs beyond group cohomology. They have the follow-ing physical property: If we choose the 4D space to be S × M and put a U(1) monopole at the center of S : (cid:82) S F = 2 π , in the large M limit, the effective 2+1D the-ory on M space is k copies of E bosonic quantum Hallstates. A U(1) monopole in 4D space is a 1D loop. Bycutting M into two separated manifolds, each with a 1D-loop boundary, we see U(1) monopole and anti-monopoleas these two 1D-loops, each loop carries k copies of E bosonic quantum Hall edge modes. Their gravitationalresponse can be detected by thermal transport with athermal Hall conductance, κ xy = 8 k π k B h T .In , the following topological response exists Z (sym.twist) = exp[ i2 (cid:90) M F w ] , (12)where w j is the j th Stiefel-Whitney (SW) class. Let usdesign M as a complex manifold, thus w j = c j mod 2.The first Chern class c of the tangent bundle of M isalso the first Chern class of the determinant line bundleof the tangent bundle of M . So if we choose the U(1)symmetry twist as the determinate line bundle of M ,we can write the above as ( F = 2 πc ): Z (sym.twist) =exp[i π (cid:82) M c c ]. On a 4-dimensional complex manifold,we have p = c − c . Since the 4-manifold CP is nota spin manifold, thus w (cid:54) = 0. From (cid:82) CP p = 3, wesee that (cid:82) CP c c = 1 mod 2. So the above topologi-cal response is non-trivial, and it suggests a Z -class of3+1D U(1) SPTs beyond group cohomology. Althoughthis topological response is non-trivial, however, we do not gain extra 3+1D U(1) SPTs beyond group cohomol-ogy, since exp[ i2 (cid:82) N F w ] = exp[ i4 π (cid:82) N F ∧ F ] on anymanifold N , and since the level of (cid:82) F ∧ F of U(1)-symmetry is not quantized on any manifold. Fermionic/Bosonic topological insulators withU(1) charge and Z T time-reversal symmetries –In 3+1D, the fermionic topological insulator as SPTsprotected by U(1) charge and Z T time-reversal symme-tries is known to have an axionic θ -term response. Wecan verify the claim by our approach. In 3+1D, althoughwe do not have a Chern-Simons form available, we canuse the probeexp[ i k π (cid:90) M F ∧ F ] ≡ exp[ i4 π θ π (cid:90) M F ∧ F ] . (13)The time reversal symmetry Z T on F ∧ F is odd, sothe θ must be odd as θ → − θ under Z T symmetry.On a spin manifold, the π (cid:82) M F ∧ F corresponds toan integer of instanton number, together with our largegauge transformation and flux identification, it dictates θ (cid:39) θ + 2 π . More explicitly, we recover the familiar formexp[ i4 π θ π (cid:82) M (cid:15) µνρσ F µν F ρσ d x ]. If the trivial vacuumhas θ = 0, then the 3+1D fermionic topological insulatorcan be probed by the θ = π response.The 3+1D bosonic topological insulator has the sim-ilar θ -term topological response. Except that the spin structure is not required for bosonic systems, and theearlier quantization becomes doubled as an even integer,thus θ (cid:39) θ + 4 π . If the trivial vacuum has θ = 0, thenthe 3+1D bosonic topological insulator can be probedby the θ = 2 π response. More topological responsesof fermionic/bosonic topological insulators within or be-yond group cohomology are recently discussed in Refs.17, 18, and 79. Conclusion – The recently-found SPTs, described bygroup cohomology, have SPT invariants in terms of puregauge actions (whose boundaries have pure gauge anoma-lies ). We have derived the formal group coho-mology results from an easily-accessible field theory set-up. For beyond-group-cohomology SPT invariants, whileours of bulk-onsite-unitary symmetry are mixed gauge-gravity actions , those of other symmetries (e.g. anti-unitary-symmetry time-reversal Z T ) may be pure grav-ity actions . SPT invariants can also be obtained viacobordism theory, or via gauge-gravity actions whoseboundaries realizing gauge-gravitational anomalies . Wehave incorporated this idea into a field theoretic frame-work, which should be applicable for both bosonic andfermionic SPTs and for more exotic states awaiting fu-ture explorations. Acknowledgements – JW wishes to thank EdwardWitten for thoughtful comments during PCTS workshopat Princeton in March. This research is supported byNSF Grant No. DMR-1005541, NSFC 11074140, andNSFC 11274192. Research at Perimeter Institute is sup-ported by the Government of Canada through IndustryCanada and by the Province of Ontario through the Min-istry of Economic Development & Innovation. Supplemental Material Appendix A: “Partition functions of Fields” - LargeGauge Transformation and Level Quantization In this section, we will work out the details oflarge gauge transformations and level-quantizations forbosonic SPTs with a finite Abelian symmetry group G = (cid:81) u Z N u for 1+1D, 2+2D and 3+1D. We will briefly com-ment about the level modification for fermionic SPTs,and give another example for G = U(1) m (a product of m copies of U(1) symmetry) SPTs. This can be straight-forwardly extended to any dimension.In the main text, our formulation has been focused onthe 1-form field A µ with an effective probed-field par-tition function Z (sym.twist) = e i S ( A ) . Below we willalso mention 2-form field B µν , 3-form field C µνρ , etc. Wehave known that for SPTs, a lattice formulation can eas-ily couple 1-form field to the matter via A µ J µ coupling.The main concern of relegating B , C higher forms to the Appendix without discussing them in the main text isprecisely due to that it is so far unknown how to findthe string (Σ µν ) or membrane (Σ µνρ ) -like excitations inthe bulk SPT lattice and further coupling via the B µν Σ µν , C µνρ Σ µνρ terms. However, such a challenge may be ad-dressed in the future, and a field theoretic framework hasno difficulty to formulate them together. Therefore herewe will discuss all plausible higher forms altogether.For G = (cid:81) u Z N u , due to a discrete Z N gauge symme-try, and the gauge transformation ( δA , δB , etc) must beidentified by 2 π , we have the general rules: (cid:73) A u = 2 πn u N u (mod 2 π ) (A1) (cid:73) δA u = 0 (mod 2 π ) (A2) (cid:90) (cid:7)(cid:6) (cid:4)(cid:5) (cid:90) B u = 2 πn u N u (mod 2 π ) (A3) (cid:90) (cid:7)(cid:6) (cid:4)(cid:5) (cid:90) δB u = 0 (mod 2 π ) (A4) (cid:90) (cid:7)(cid:6) (cid:4)(cid:5) (cid:90)(cid:90) C u = 2 πn u N u (mod 2 π ) (A5) (cid:90) (cid:7)(cid:6) (cid:4)(cid:5) (cid:90)(cid:90) δC u = 0 (mod 2 π ) (A6) . . . Here A is integrated over a closed loop, B is integratedover a closed 2-surface, C is integrated over a closed3-volume, etc. The loop integral of A is performedon the normal direction of a codimension-1 sheet (seeFig.1(a)(d)). Similarly, the 2-surface integral of B is per-formed on the normal directions of a codimension-2 sheet,and the 3-volume integral of C is performed on the nor-mal directions of a codimension-3 sheet, etc. The aboverules are sufficient for the actions with flat connections(d A = d B = d C = 0 everywhere).Without losing generality, we consider a spacetimewith a volume size L d +1 where L is the length of one di-mension (such as a T d +1 torus). The allowed large gaugetransformation implies the A , B , C locally can be: A u,µ = 2 πn u d x µ N u L , δA u = 2 πm u d x µ L , (A7) B u,µν = 2 πn u d x µ d x ν N u L , δB u,µν = 2 πm u d x µ d x ν L , (A8) C u,µνρ = 2 πn u d x µ d x ν d x ρ N u L , δC u,µνρ = 2 πm u d x µ d x ν d x ρ L . (A9) . . . As we discussed in the main text, for some cases, if thecodimension- n sheet (as a branch cut) ends, then its endpoints are monodromy defects with non-flat connections(d A (cid:54) = 0, etc). Those monodromy defects can be viewedas external flux insertions (see Fig.1(b)(e)). In this Ap-pendix we only need non-flat 1-form: d A (cid:54) = 0. We canimagine several monodromy defects created on the space-time manifold, but certain constraints must be imposed, (cid:90) (cid:7)(cid:6) (cid:4)(cid:5) (cid:90) d A v = 0 (mod 2 π ) , (A10) (cid:90) (cid:7)(cid:6) (cid:4)(cid:5) (cid:90) δ d A v = 0 . (A11)This means that the sum of inserted fluxes at monodromydefects must be a multiple of 2 π fluxes. A fractional fluxis allowed on some individual monodromy defects, butoverall the net sum must be nonfractional units of 2 π (see Fig.2).For mixed gauge-gravity SPTs, we have also discussedits probed field partition function in terms of the spinconnection ω , it is simply related to the usual Christof-fel symbol Γ via a choice of local frame (via vielbein),which occurs in gravitational effective probed-field parti-tion function Z (sym.twist) = e i S ( A, Γ ,... ) .We will apply the above rules to the explicit examplesbelow. FIG. 2. The net sum of fluxes at monodromy defects (aspunctures or holes of the spatial manifold) must be 2 πn unitsof fluxes, with n ∈ Z . e.g. (cid:80) j Φ B ( x j ) = (cid:82)(cid:82) d A v = 2 πn . 1. Top Types: (cid:82) A A . . . A d +1 with G = (cid:81) u Z N u a. 1+1D (cid:82) A A For 1+1D bosonic SPTs with a symmetry group G = (cid:81) u Z N u , by dimensional counting, one can thinkof (cid:82) d A = (cid:82) F , but we know that due to F = d A is a total derivative, so it is not a bulk topologicalterm but only a surface integral. The only possibleterm is exp[ i k II (cid:82) A ∧ A ], (here A and A comefrom different symmetry group Z N , Z N , otherwise A ∧ A = 0 due to anti-symmetrized wedge product).Below we will omit the wedge product ∧ as conventionaland convenient notational purposes, so A A ≡ A ∧ A .Such a term A A is invariant under transformationif we impose flat connection d A = d A = 0, since δ ( A A ) = ( δA ) A + A ( δA ) = (d f ) A + A (d f ) = − f (d A ) − (d A ) f = 0. Here we have abandonedthe surface term if we consider a closed bulk spacetimewithout boundaries. • Large gauge transformation : The partition func-tion Z (sym.twist) invariant under the allowed largegauge transformation via Eq.(A7) implies k II (cid:90) δ ( A A ) = k II (cid:90) ( δA ) A + A ( δA )= k II (cid:90) πm d x L πn d x N L + 2 πn d x N L πm d x L = k II (2 π ) ( m n N + n m N ) , which action must be invariant mod 2 π for any largegauge transformation parameter (e.g. n , n ), namely(2 π ) k II N = (2 π ) k II N = 0 (mod 2 π ) ⇒ (2 π ) k II N = (2 π ) k II N = 0 (mod 1) (A12)This rule of large gauge transformation implies the level-quantization . • Flux identification : On the other hand, when the Z N flux from A and Z N flux from A are inserted as n , n multiple units of 2 π/N , 2 π/N , we have k II (cid:90) A A = k II (cid:90) πn d xN L πn d tN L = k II (2 π ) N N n n . No matter what value n n is, whenever k II (2 π ) N N shifts by2 π , the symmetry-twist partition function Z (sym.twist)is invariant. The coupling k II must be identified, via(2 π ) k II (cid:39) (2 π ) k II + N N . (A13)( (cid:39) means the level identification.) We call this rule asthe flux identification. These two rules above imposesthat k II = p II N N (2 π ) N with p II defined by p II (mod N )so p II ∈ Z N , where N is the greatest common divi-sor(gcd) defined by N ...u ≡ gcd( N , N , . . . , N u ). N is the largest number can divide N and N from Chineseremainder theorem . We thus derive Z (sym.twist) = exp[ i p II N N (2 π ) N (cid:90) M A A ] . (A14) b. 2+1D (cid:82) A A A In 2+1D, we have exp[ i k III (cid:82) A A A ] allowed by flatconnections. We have the two rules, large gauge trans-formation k III (cid:90) δ ( A A A )= k III (cid:90) ( δA ) A A + A ( δA ) A + A A ( δA )= k III (2 π ) ( m n n N N + n m n N N + n n m N N ) , which action must be invariant mod 2 π for anylarge gauge transformation parameter (e.g. n , n , . . . )and flux identification with k III (cid:82) A A A = k III (cid:82) πn d xN L πn d yN L πn d tN L = k III (2 π ) N N N n n n . Bothlarge gauge transformation and flux identification respec-tively impose(2 π ) k III N u N v = 0 (mod 1) , (A15)(2 π ) k III (cid:39) (2 π ) k III + N N N , (A16)with u, v ∈ { , , } and u (cid:54) = v . We thus derive k III = p III N N N (2 π ) N and Z (sym.twist) = exp[ i p III N N N (2 π ) N (cid:90) M A A A ] , (A17)with p III defined by p III (mod N ), so p III ∈ Z N . c. ( d + 1) D (cid:82) A A . . . A d +1 In ( d +1)D, similarly, we have exp[ i k (cid:82) A A . . . A d +1 ]allowed by flat connections, where the large gaugetransformation and flux identification respectivelyconstrain (2 π ) d k N u (cid:81) d +1 j =1 N j = 0 (mod 1) , (A18)(2 π ) d k (cid:39) (2 π ) d k + d +1 (cid:89) j =1 N j , (A19) with u ∈ { , , . . . , d + 1 } . We thus derive Z (sym.twist) = exp[ i p (cid:81) d +1 j =1 N j (2 π ) d N ... ( d +1) (cid:90) A A . . . A d +1 ] , (A20)with p defined by p (mod N ... ( d +1) ). We name thisform (cid:82) A A . . . A d +1 as the Top Types , which can berealized for all flat connection of A . Its path integral in-terpretation is a direct generalization of Fig.1(c)(f), whenthe ( d +1) number of codimension-1 sheets with flat A on T d +1 spacetime torus with nontrivial elements g j ∈ Z N j intersect at a single point, it renders a nontrivial parti-tion function of Eq.(2) with Z (sym.twist) (cid:54) = 1. 2. Lower Types in 2+1D with G = (cid:81) u Z N u a. (cid:82) A u d A v Apart from the top Type, we also have Z (sym.twist) = exp[ i k (cid:82) A u d A v ] assuming that A is almost flat but d A (cid:54) = 0 at monodromy defects. Notethat d A is the flux of the monodromy defect, whichis an external input and does not have any dynamicalvariation, δ (d A v ) = 0 as Eq.(A11). For the large gaugetransformation , we have k (cid:82) δ ( A u d A v ) as k (cid:90) (cid:0) ( δA u )d A v + A u δ (d A v ) (cid:1) = 0 (mod 2 π ) ⇒ k π (cid:90) (cid:0) πm u d xL πn v d y d tL + 0 (cid:1) = 0 (mod 1) , for any m u , n v . We thus have(2 π ) k = 0 (mod 1) . (A21)The above include both Type I and Type II SPTs in2+1D: Z (sym.twist) = exp[ i p I (2 π ) (cid:90) M A d A ] , (A22) Z (sym.twist) = exp[ i p II (2 π ) (cid:90) M A d A ] , (A23)where p I , p II ∈ Z integers. Configuration : In order for Eq.(A23), e i p II2 π (cid:82) M A d A to be invariant under the large gauge transformation thatchanges (cid:72) A by 2 π , p II must be integer. In order forEq.(A22) to be well defined, we denote A = ¯ A + A F where ¯ A d ¯ A = 0, d A F = 0, (cid:72) ¯ A = 0 mod 2 π/N ,and (cid:72) A F = 0 mod 2 π/N . In this case Eq.(A22) be-comes e i p I2 π (cid:82) M A F d ¯ A . The invariance under the largegauge transformation of A F requires p I to be quantizedas integers.For the flux identification , we compute k (cid:82) A u d A v = k (cid:82) πn u dxN u L πn v dydtL = k (2 π ) N u n u n v , where k is identifiedby (2 π ) k (cid:39) (2 π ) k + N u . (A24)On the other hand, the integration by parts in the case ona closed (compact without boundaries) manifold impliesanother condition,(2 π ) k (cid:39) (2 π ) k + N v , (A25) Flux identification : If we view k (cid:39) k + N u / (2 π ) and k (cid:39) k + N v / (2 π ) as the identification of level k , then weshould search for the smallest period from their linearcombination. From Chinese remainder theorem , overallthe linear combination N u and N v provides the smallestunit as their greatest common divisor(gcd) N uv :(2 π ) k (cid:39) (2 π ) k + N uv (A26)Hence p I , p II are defined as p I (mod N ) and p II (mod N ), so it suggests that p I ∈ Z N and p II ∈ Z N .Alternatively, using the fully-gauged braiding statisticsapproach among particles, it also renders p I ∈ Z N and p II ∈ Z N . b. (cid:82) A B For A u d A v action, we have to introduce non-flatd A (cid:54) = 0 at some monodromy defect. There is anotherway instead to formulate it by introducing flat 2-form B with d B = 0. The partition function Z (sym.twist) =exp[ i k II (cid:82) A B ]. The large gauge transformation and the flux identification constrain respectively(2 π ) k II N u = 0 (mod 1) , (A27)(2 π ) k II (cid:39) (2 π ) k II + N N , (A28)with u ∈ { , } . We thus derive Z (sym.twist) = exp[ i p II N N (2 π ) N (cid:90) M A B ] , (A29)with p II defined by p II (mod N ) and p II ∈ Z N . 3. Lower Types in 3+1D with G = (cid:81) u Z N u a. (cid:82) A u A v d A w To derive (cid:82) A u ∧ A v ∧ d A w topological term, we firstknow that the (cid:82) F u ∧ F v = (cid:82) d A u ∧ d A v term is only atrivial surface term for the symmetry group G = (cid:81) j Z N j and for G = U(1) m . First, the flat connection d A = 0imposes that F u ∧ F v = 0. Second, for a nearly flat con-nection d A (cid:54) = 0, we have k π d A u ∧ d A v (cid:54) = 0 but the levelquantization imposes k ∈ Z , and the flux identificationensures that k (cid:39) k + 1. So all k ∈ Z is identical to thetrivial class k = 0. Hence, for G = (cid:81) j Z N j , the onlylower type of SPTs we have is that (cid:82) A u A v d A w . Suchterm vanishes for a single cycle group ( A A d A = 0 for G = Z N , since A ∧ A = 0) thus it must come from twoor three cyclic products ( Z N × Z N or Z N × Z N × Z N ). : However, weshould remind the reader that if one consider a differentsymmetry group, such as G = U(1) (cid:111) Z T of a bosonictopological insulator, the extra time reversal symmetry Z T can distinguish two distinct classes of θ = 0 and θ = 2 π for the probe-field partition functionexp[ i4 π θ π (cid:90) M F ∧ F ] . (A30)The time reversal symmetry Z T on F ∧ F is odd,so the θ must be odd as θ → − θ under Z T sym-metry. The π (cid:82) M F ∧ F corresponds to an inte-ger of instanton number, together with our large gaugetransformation and flux identification, it dictates θ (cid:39) θ + 4 π . More explicitly, we recover the familiar formexp[ i4 π θ π (cid:82) M (cid:15) µνρσ F µν F ρσ d x ]. If the trivial vacuumhas θ = 0, then the 3+1D bosonic topological insulatorcan be probed by θ = 2 π response.Similar to Sec.A 2 a, the almost flat connection butwith d A (cid:54) = 0 at the monodromy defect introduces a pathintegral, Z (sym.twist) = exp[ i k (cid:90) M A u A v d A w ] . (A31)For the large gauge transformation , wethus have k (cid:82) δ ( A u A v d A w ) = k (cid:82) ( δA u ) A v d A w + A u ( δA v )d A w + A u A v δ (d A w ) = 0 (mod 2 π ) ⇒ k π (cid:82) πn u d xL πn v d yN v L πn w d z d tL + πn u d xN u L πn v d yL πn w d z d tL =0 (mod 1) . This constrains that(2 π ) kN u = (2 π ) kN v = 0 (mod 1) . (A32)Thus, the large gauge transformation again implies that k has a level quantization .For the flux identification , k (cid:82) A u A v d A w = k (cid:82) πn u dxN u L πn v dyN v L πn w dzdtL = k (2 π ) N u N v n u n v n w . The wholeaction is identified by 2 π under the shift of quantizedlevel k : (2 π ) k (cid:39) (2 π ) k + N u N v . (A33)For the case of a Z N × Z N symmetry, we have TypeII SPTs. We obtain a partition function: Z (sym.twist) = exp[ i p II N N (2 π ) N (cid:90) M A A d A ] , (A34)The flux identification Eq.(A33) implies that the identi-fication of p II (cid:39) p II + N . Thus, it suggests that a cyclicperiod of p II is N , and we have p II ∈ Z N .Similarly, there are also distinct classesof Type II SPTs with a partition functionexp[ i p II N N (2 π ) N (cid:82) M A A d A ] with p II ∈ Z N .We notice that A A d A and A A d A are differenttypes of SPTs, because they are not identified even bydoing integration by parts.For the case of Z N × Z N × Z N symmetry, we haveextra Type III SPTs partition functions (other than theabove Type II SPTs), for example: Z (sym.twist) = exp[ i p III N N (2 π ) N (cid:90) M A A d A ] . (A35)Again, the flux identification Eq.(A33) implies that theidentification of p III (cid:39) p III + N . (A36)Thus, it suggests that a cyclic period of p III is N , and p III ∈ Z N .However, there is an extra constraint on the level iden-tification. Now consider (cid:82) A A d A = (cid:82) − d( A A ) A up to a surface integral (cid:82) d ( A A A ). No-tice that (cid:82) − d( A A )d A = − (cid:82) A A d A − (cid:82) A A d A . If we reconsider the flux identifi-cation of Eq.(A35) in terms of Z (sym.twist) =exp[ − i p III N N (2 π ) N (cid:82) M ( A A d A + A A d A )], wefind the spacetime volume integration yields a phase Z (sym.twist)=exp[ − i p III N N (2 π ) N (cid:0) (2 π ) n n N N + (2 π ) n n N N (cid:1) ]. Z (sym.twist) = exp[ − π i p III n N n N + n N N (cid:1) ] . (A37)We can arbitrarily choose n , n , n to determine the levelidentification of p III from the flux identification. Thefinest level identification is determined from choosing thesmallest n and the smallest n N + n N . We choose n = 1. By Chinese remainder theorem, we can choose n N + n N = gcd( N , N ) ≡ N . Thus Eq.(A37)yields Z (sym.twist) = exp[ − π i p III N ]. It is apparent thatthe flux identification implies the level identification p III (cid:39) p III + N . (A38)Eq.(A36),(A38) and their linear combination togetherimply the finest level p III identification p III (cid:39) p III + gcd( N , N ) (cid:39) p III + N . (A39)Overall, our derivation suggests that Eq.(A35) has p III ∈ Z N . b. (cid:82) A C Similar to Sec.A 2 b, we can introduce a flat 3-form C field with d C = 0 such that Z (sym.twist) =exp[ i k II (cid:82) A C ] can capture a similar physics of (cid:82) A A d A . The large gauge transformation and fluxidentification constrain respectively,(2 π ) k II N u = 0 (mod 1) , (A40)(2 π ) k II (cid:39) (2 π ) k II + N N . (A41)with u ∈ { , } . We derive Z (sym.twist) = exp[ i p II N N (2 π ) N (cid:90) M A C ] , (A42)with p II defined by p II (mod N ), thus p II ∈ Z N . c. (cid:82) A A B Similar to Sec.A 2 b, A 3 b, in 3+1D, by dimen-sional counting, we can also introduce Z (sym.twist) =exp[ i k (cid:82) A A B ]. The large gauge transformation andthe flux identification yield(2 π ) kN u N v = 0 (mod 1) , (A43)(2 π ) k (cid:39) (2 π ) k + N N N . (A44)We thus derive Z (sym.twist) = exp[ i p III N N N (2 π ) N (cid:90) M A A B ] , (A45)with p III defined by p III (mod N ) with p III ∈ Z N . 4. Cases for Fermionic SPTs Throughout the main text, we have been focusing onthe bosonic SPTs, which elementary particle contents areall bosons. Here we comment how the rules of fermionicSPTs can be modified from bosonic SPTs. Due to thatthe fermionic particle is allowed, by exchanging two iden-tical fermions will gain a fermionic statistics e i π = − • Large gauge transformation : The Z invarianceunder the allowed large gauge transformation implies thevolume-integration must be invariant mod π (instead ofbosonic case with mod 2 π ), because inserting a fermioninto the system does not change the SPT class of system.Generally, there are no obstacles to go through the anal-ysis and level-quantization for fermions, except that weneed to be careful about the flux identification. Belowwe give an example of U(1) symmetry bosonic/fermioncSPTs, and we will leave the details of other cases forfuture studies. 5. U (1) m symmetry bosonic and fermionic SPTs For U(1) m symmetry, one can naively generalize theabove results from a viewpoint of G = Π m Z N = ( Z N ) m with N → ∞ . This way of thinking is intuitive (thoughnot mathematically rigorous), but guiding us to obtainU(1) m symmetry classification. We find the classifica-tion is trivial for even ( d + 1)D, due to F u ∧ F v ∧ . . . (where F = d A is the field strength, here u, v canbe either the same or different U(1) gauge fields) isonly a surface term, not a bulk topological term. Forodd ( d + 1)D, we can define the lower type action: Z (sym.twist) = exp[ i πk ( d +22 )!(2 π ) ( d +2) / (cid:82) A u ∧ F v ∧ . . . ] . Meanwhile we emphasize that other type of actions,such as the top type, k (cid:82) A A . . . A d +1 form, orany other terms involve with more than one A (e.g. k (cid:82) A u A u . . . d A u. ) will be trivial SPT class for U(1) m case - since its coefficient k no longer stays finite for0 N → ∞ of ( Z N ) m symmetry SPTs, so the level k isnot well-defined. For physically relevant 2 + 1D, k ∈ Z for bosonic SPTs, k ∈ Z for fermionic SPTs via Sec.A 4.Thus, we will have a Z m × Z m ( m − / classification forU(1) m symmetry boson, and the fermionic classification increases at least by shifting the bosonic Z → Z . Theremay have even more extra classes by including Majoranaboundary modes, which we will leave for future investi-gations. Appendix B: From “Partition Functions of Fields” to “Cocycles of Group Cohomology” and K¨unneth formula In Appendix A, we have formulated the spacetime partition functions of probe fields (e.g. Z ( A ( x )), etc),which fields A ( x ) take values at any coordinates x on a continuous spacetime manifold M with no dynamics. Onthe other hand, it is known that, ( d + 1)D bosonic SPTs of symmetry group G can be classified by the ( d + 1)-thcohomology group H d +1 ( G, R / Z ) (predicted to be complete at least for finite symmetry group G without time reversalsymmetry). From this prediction that bosonic SPTs can be classified by group cohomology, our path integral on thediscretized space lattice (or spacetime complex) shall be mapped to the partition functions of the cohomologygroup - the cocycles . In this section, we ask “whether we can attain this correspondence from “partition functionsof fields” to “cocycles of group cohomology?” Our answer is “yes,” we will bridge this beautiful correspondencebetween continuum field theoretic partition functions and discrete cocycles for any ( d + 1)D spacetime dimension forfinite Abelian G = (cid:81) u Z N u . (d+1)dim partition function Z ( d + 1)-cocycle ω d +1 p I (cid:82) A ) exp (cid:16) π i p I N a (cid:17) p II N N (2 π ) N (cid:82) A A ) exp (cid:16) π i p II N a b (cid:17) p I (2 π ) (cid:82) A d A ) exp (cid:16) π i p I N a ( b + c − [ b + c ]) (cid:17) exp(i p I (cid:82) C ) (even/odd effect) exp (cid:16) π i p I N a b c (cid:17) p II (2 π ) (cid:82) A d A ) exp (cid:16) π i p II N N a ( b + c − [ b + c ]) (cid:17) exp(i p II N N (2 π ) N (cid:82) A B ) (even/odd effect) exp (cid:16) π i p II N a b c (cid:17) p III N N N (2 π ) N (cid:82) A A A ) exp (cid:16) π i p III N a b c (cid:17) (cid:82) p (1 st )II(12) N N (2 π ) N A A d A ) exp (cid:0) π i p (1 st )II(12) ( N · N ) ( a b )( c + d − [ c + d ]) (cid:1) exp(i p II N N (2 π ) N (cid:82) A C ) (even/odd effect) exp (cid:0) π i p II N a b c d (cid:1) (cid:82) p (2 nd )II(12) N N (2 π ) N A A d A ) exp (cid:0) π i p (2 nd )II(12) ( N · N ) ( a b )( c + d − [ c + d ]) (cid:1) exp(i p II N N (2 π ) N (cid:82) A C ) (even/odd effect) exp (cid:0) π i p II N a b c d (cid:1) p (1 st )III(123) N N (2 π ) N (cid:82) ( A A )d A ) exp (cid:0) π i p (1 st )III(123) ( N · N ) ( a b )( c + d − [ c + d ]) (cid:1) exp(i p III N N N (2 π ) N (cid:82) A A B ) (even/odd effect) exp (cid:0) π i p III N a b c d (cid:1) p (2 nd )III(123) N N (2 π ) N (cid:82) ( A A )d A ) exp (cid:0) π i p (2 nd )III(123) ( N · N ) ( a b )( c + d − [ c + d ]) (cid:1) exp(i p III N N N (2 π ) N (cid:82) A A B ) (even/odd effect) exp (cid:0) π i p III N a b c d (cid:1) p IV N N N N (2 π ) N (cid:82) A A A A )] exp (cid:0) π i p IV N a b c d (cid:1) p I (2 π ) (cid:82) A d A d A ) exp (cid:16) π i p I ( N ) a ( b + c − [ b + c ])( d + e − [ d + e ]) (cid:17) . . . . . . p V N N N N N (2 π ) N (cid:82) A A A A A ) exp (cid:0) π i p V N a b c d e (cid:1) TABLE I. Some derived results on the correspondence between the spacetime partition function of probe fields (thesecond column) and the cocycles of the cohomology group (the third column) for any finite Abelian group G = (cid:81) u Z N u .The even/odd effect means that whether their corresponding cocycles are nontrivial or trivial(as coboundary) depends on thelevel p and N (of the symmetry group Z N ) is even/odd. Details are explained in Sec B 2. 1. Correspondence The partition functions in Appendix A have beentreated with careful proper level-quantizations via largegauge transformations and flux identifications. For G = (cid:81) u Z N u , the field A u , B u , C u , etc, take values in Z N u variables, thus we can express them as A u ∼ πg u N u , B u ∼ πg u h u N u , C u ∼ πg u h u l u N u (B1)with g u , h u , l u ∈ Z N u . Here 1-form A u takes g u value onone link of a ( d + 1)-simplex, 2-form B u takes g u , h u val-ues on two different links and 3-form C u takes g u , h u , l u values on three different links of a ( d + 1)-simplex. Thesecorrespondence suffices for the flat probe fields.In other cases, we also need to interpret the non-flat d A (cid:54) = 0 at the monodromy defect as the external insertedfluxes, thus we identifyd A u ∼ π ( g u + h u − [ g u + h u ]) N u , (B2)here [ g u + h u ] ≡ g u + h u (mod N u ). Such identificationensures d A u is a multiple of 2 π flux, therefore it is con-sistent with the constraint Eq.(A10) at the continuumlimit. Based on the Eq.(B1)(B2), we derive the corre-spondence in Table I, from the continuum path integral Z (sym.twist) of fields to a U(1) function as the discretepartition function. In the next subsection, we will verifythe U(1) functions in the last column in Table I indeedare the cocycles ω d +1 of cohomology group. Such a corre-spondence has been explicitly pointed out in our previouswork Ref.68 and applied to derive the cocycles. (d+1)dim Partition function Z of “fields” p ∈ H d +1 ( G, R / Z ) K¨unneth formula in H d +1 ( G, R / Z )0+1D exp(i p.. (cid:82) A ) Z N H ( Z N , R / Z )1+1D exp(i p.. (cid:82) A A ) Z N H ( Z N , R / Z ) (cid:2) Z H ( Z N , R / Z )2+1D exp(i p.. (cid:82) A d A ) Z N H ( Z N , R / Z )2+1D exp(i p.. (cid:82) A d A ) Z N H ( Z N , R / Z ) ⊗ Z H ( Z N , R / Z )2+1D exp(i p.. (cid:82) A A A ) Z N [ H ( Z N , R / Z ) (cid:2) Z H ( Z N , R / Z )] (cid:2) Z H ( Z N , R / Z )3+1D exp(i p.. (cid:82) A A d A ) Z N H ( Z N , R / Z ) (cid:2) Z H ( Z N , R / Z )3+1D exp(i p.. (cid:82) A A d A ) Z N H ( Z N , R / Z ) (cid:2) Z H ( Z N , R / Z )3+1D exp(i p.. (cid:82) ( A A )d A ) Z N [ H ( Z N , R / Z ) (cid:2) Z H ( Z N , R / Z )] ⊗ Z H ( Z N , R / Z )3+1D exp(i p.. (cid:82) ( A d A ) A ) Z N [ H ( Z N , R / Z ) ⊗ Z H ( Z N , R / Z )] (cid:2) Z H ( Z N , R / Z ))3+1D exp(i p.. (cid:82) A A A A ) Z N (cid:2) [ H ( Z N , R / Z ) (cid:2) Z H ( Z N , R / Z )] (cid:2) Z H ( Z N , R / Z ) (cid:3) (cid:2) Z H ( Z N , R / Z )4+1D exp(i p.. (cid:82) A d A d A ) Z N H ( Z N , R / Z )4+1D exp(i p.. (cid:82) A d A d A ) Z N H ( Z N R / Z ) ⊗ Z H ( Z N , R / Z )4+1D exp(i p.. (cid:82) A d A d A ) Z N H ( Z N , R / Z ) ⊗ Z H ( Z N , R / Z )4+1D exp(i p.. (cid:82) A d A A A ) Z N (cid:2) [ H ( Z N , R / Z ) (cid:2) Z H ( Z N , R / Z )] (cid:2) Z H ( Z N , R / Z ) (cid:3) p.. (cid:82) A d A A A ) Z N (cid:2) [ H ( Z N , R / Z ) (cid:2) Z H ( Z N , R / Z )] (cid:2) Z H ( Z N , R / Z ) (cid:3) p.. (cid:82) A d A d A ) Z N [ H ( Z N , R / Z ) ⊗ Z H ( Z N , R / Z )] ⊗ Z H ( Z N , R / Z )4+1D exp(i p.. (cid:82) A A A d A ) Z N (cid:2) [ H ( Z N , R / Z ) (cid:2) Z H ( Z N , R / Z )] (cid:2) Z H ( Z N , R / Z ) (cid:3) p.. (cid:82) A d A A A ) Z N (cid:104)(cid:2) [ H ( Z N , R / Z ) ⊗ Z H ( Z N , R / Z )] (cid:2) Z H ( Z N , R / Z ) (cid:3) (cid:2) Z H ( Z N , R / Z ) (cid:105) p.. (cid:82) A A d A A ) Z N (cid:104)(cid:2) [ H ( Z N , R / Z ) (cid:2) Z H ( Z N , R / Z )] ⊗ Z H ( Z N , R / Z ) (cid:3) (cid:2) Z H ( Z N , R / Z ) (cid:105) p.. (cid:82) A A A d A ) Z N (cid:104)(cid:2) [ H ( Z N , R / Z ) (cid:2) Z H ( Z N , R / Z )] (cid:2) Z H ( Z N , R / Z ) (cid:3) ⊗ Z H ( Z N , R / Z ) (cid:105) p.. (cid:82) A A A A A ) Z N (cid:104)(cid:2) [ H ( Z N ) (cid:2) Z H ( Z N )] (cid:2) Z H ( Z N ) (cid:3) (cid:2) Z H ( Z N ) (cid:105) (cid:2) Z H ( Z N )TABLE II. From partition functions of fields to K¨unneth formula . Here we consider a finite Abelian group G = (cid:81) u Z N u .The field theory result can map to the derived facts about the cohomology group and its cocycles. Here the level-quantizationis shown in a shorthand way with only p.. written, the explicit coefficients can be found in Table II. In some row, we abbreviate H ( Z n j , R / Z ) ≡ H ( Z n j ). The torsion product Tor Z ≡ (cid:2) Z evokes a wedge product ∧ structure in the corresponding fieldtheory, while the tensor product ⊗ Z evokes appending an extra exterior derivative ∧ d structure in the corresponding fieldtheory. This simple observation maps the field theoretic path integral to its correspondence in K¨unneth formula. We remark that the field theoretic path integral’s level p quantization and its mod relation also provide an indepen-dent way (apart from group cohomology) to count the number of types of partition functions for a given symmetrygroup G and a given spacetime dimension. Such the modular p is organized in (the third column of) Table II. Inaddition, one can further deduce the K¨unneth formula (the last column of Table II) from a field theoretic partition2 Type I Type II Type III Type IV Type V Type VI . . . . . . Z N i Z N ij Z N ijl Z N ijlm Z gcd ⊗ i ( N ( i ) ) Z gcd ⊗ i ( N i ) Z gcd ⊗ mi ( N i ) Z gcd ⊗ d − i N i Z gcd ⊗ di N ( i ) H ( G, R / Z ) 1 H ( G, R / Z ) 0 1 H ( G, R / Z ) 1 1 1 H ( G, R / Z ) 0 2 2 1 H ( G, R / Z ) 1 2 4 3 1 H ( G, R / Z ) 0 3 6 7 4 1 H d ( G, R / Z ) (1 − ( − d )2 d − (1 − ( − d )4 . . . . . . . . . . . . . . . d − Z gcd ⊗ mi ( N i ) class in the cohomology group H d ( G, R / Z ) for a finite Abeliangroup G = (cid:81) ku =1 Z N u . Here we define a shorthand of Z gcd( N i ,N j ) ≡ Z N ij ≡ Z gcd ⊗ i ( N i ) , etc also for other higher gcd. Ourdefinition of the Type m is from its number ( m ) of cyclic gauge groups in the gcd class Z gcd ⊗ mi ( N i ) . The number of exponentscan be systematically obtained by adding all the numbers of the previous column from the top row to a row before the wish-to-determine number. This table in principle can be independently derived by gathering the data of Table II from field theoryapproach. For example, we can derive H ( G, R / Z ) = (cid:81) ≤ i 2. Cohomology group and cocycle conditions To verify that the last column of Table I (bridged from the field theoretic partition function) are indeed cocyclesof a cohomology group, here we briefly review the cohomology group H d +1 ( G, R / Z ) (equivalently as H d +1 ( G, U(1))by R / Z = U(1)), which is the ( d + 1)th-cohomology group of G over G module U(1). Each class in H d +1 ( G, R / Z )corresponds to a distinct ( d + 1)-cocycles. The n -cocycles is a n -cochain, in addition they satisfy the n -cocycle-conditions δω = 1. The n -cochain is a mapping of ω ( a , a , . . . , a n ): G n → U(1) (which inputs a i ∈ G , i = 1 , . . . , n ,and outputs a U(1) value). The n -cochain satisfies the group multiplication rule:( ω · ω )( a , . . . , a n ) = ω ( a , . . . , a n ) · ω ( a , . . . , a n ) , (B3)thus form a group. The coboundary operator δδ c ( g , g , . . . , g n +1 ) ≡ c ( g , . . . , g n +1 ) c ( g , . . . , g n ) ( − n +1 · n (cid:89) j =1 c ( g , . . . , g j g j +1 , . . . , g n +1 ) ( − j , (B4)which defines the n -cocycle-condition δω = 1. The n -cochain forms a group C n , while the n -cocycle forms itssubgroup Z n . The distinct n -cocycles are not equivalentvia n -coboundaries, where Eq.(B4) also defines the n -coboundary relation: if n-cocycle ω n can be written as ω n = δ Ω n − , for any ( n − n +1 , then we saythis ω n is a n -coboundary. Due to δ = 1, thus we knowthat the n -coboundary further forms a subgroup B n .In short, B n ⊂ Z n ⊂ C n The n -cohomology group isprecisely a kernel Z n (the group of n -cocycles) mod outimage B n (the group of n -coboundary) relation: H n ( G, R / Z ) = Z n / B n . (B5)For other details about group cohomology (especiallyBorel group cohomology here), we suggest to read Ref.6,68, and 70 and Reference therein.To be more specific cocycle conditions, for finiteAbelian group G , the 3-cocycle condition for 2+1D is (a pentagon relation), δω ( a, b, c, d ) = ω ( b, c, d ) ω ( a, bc, d ) ω ( a, b, c ) ω ( ab, c, d ) ω ( a, b, cd ) = 1 (B6)The 4-cocycle condition for 3+1D is δω ( a, b, c, d, e ) = ω ( b, c, d, e ) ω ( a, bc, d, e ) ω ( a, b, c, de ) ω ( ab, c, d, e ) ω ( a, b, cd, e ) ω ( a, b, c, d ) = 1(B7)The 5-cocycle condition for 4+1D is δω ( a, b, c, d, e, f ) = ω ( b, c, d, e, f ) ω ( a, bc, d, e, f ) ω ( ab, c, d, e, f ) · ω ( a, b, c, de, f ) ω ( a, b, c, d, e ) ω ( a, b, cd, e, f ) ω ( a, b, c, d, ef ) = 1 (B8)We verify that the U(1) functions (mapped from a fieldtheory derivation) in the last column of Table I indeed3satisfy cocycle conditions. Moreover, those partitionfunctions purely involve with 1-form A or its field-strength (curvature) d A are strictly cocycles butnot coboundaries . These imply that those terms withonly A or d A are the precisely nontrivial cocycles in thecohomology group for classification.However, we find that partition functions in-volve with 2-form B , 3-form C or higherforms, although are cocycles but sometimes mayalso be coboundaries at certain quantized level p value. For instance, for those cocycles correspondto the partition functions of p (cid:82) C , p N N (2 π ) N (cid:82) A B , p N N (2 π ) N (cid:82) A C , p N N (2 π ) N (cid:82) A C , p N N N (2 π ) N (cid:82) A A B , p N N N (2 π ) N (cid:82) A A B , etc (which involve with higherforms B , C ), we find that for G = ( Z ) n symmetry, p = 1 are in the nontrivial class (namely not a cobound-ary), G = ( Z ) n symmetry, p = 1 , G = ( Z ) n symmetry of all p and G = ( Z ) n symme-try at p = 2, are in the trivial class (namely a cobound-ary), etc. This indicates an even-odd effect , sometimesthese cocycles are nontrivial, but sometimes are trivial ascoboundary, depending on the level p is even/odd and thesymmetry group ( Z N ) n whether N is even/odd. Suchan even/odd effect also bring complication intothe validity of nontrivial cocycles, thus this is an-other reason that we study only field theory in-volves with only 1-form A or its field strength d A .The cocycles composed from A and d A in Table Iare always nontrivial and are not coboundaries. We finally point out that the concept of boundaryterm in field theory (the surface or total derivativeterm) is connected to the concept of coboundary inthe cohomology group . For example, (cid:82) (d A ) A A are identified as the coboundary of the linear combina-tion of (cid:82) A A (d A ) and (cid:82) A (d A ) A . Thus, by count-ing the number of distinct field theoretic actions (notidentified by boundary term ) is precisely counting thenumber of distinct field theoretic actions (not identified by coboundary ). Such an observation matches the fieldtheory classification to the group cohomology classifica-tion shown in Table III. Furthermore, we can map thefield theory result to the K¨unneth formula listed in Ta-ble II, via the correspondence: (cid:90) A ∼ H ( Z N , R / Z ) (B9) (cid:90) A d A ∼ H ( Z N , R / Z ) (B10) (cid:90) A d A d A ∼ H ( Z N , R / Z ) (B11)Tor Z ≡ (cid:2) Z ∼ ∧ (B12) ⊗ Z ∼ ∧ d (B13) (cid:90) A ∧ A ∼ H ( Z N , R / Z ) (cid:2) Z H ( Z N , R / Z ) (B14) (cid:90) A ∧ d A ∼ H ( Z N , R / Z ) ⊗ Z H ( Z N , R / Z ) (B15) . . . To summarize, in this section, we show that, at leasefor finite Abelian symmetry group G = (cid:81) ki =1 Z N i , fieldtheory can be systematically formulated, via the level-quantization developed in Appendix A, we can count thenumber of classes of SPTs. Explicit examples are orga-nized in Table I, II, III, where we show that our fieldtheory approach can exhaust all bosonic SPT classes (atleast as complete as) in group cohomology: H ( G, R / Z ) = (cid:89) ≤ i In this section, we comment more about the SPT invariants from probe field partition functions, and the derivationof SPT Invariants from dimensional reduction, using both a continuous field theory approach and a discrete cocycleapproach. We focus on finite Abelian G = (cid:81) u Z N u bosonic SPTs.First, recall from the main text using a continuous field theory approach, we can summarize the dimensionalreduction as a diagram below:1 + 1D 2 + 1D 3 + 1D · · · d + 1D A A A A A (cid:111) (cid:111) A A A A (cid:111) (cid:111) · · · (cid:111) (cid:111) A A . . . A d +1 (cid:111) (cid:111) A v d A w A u A v d A w (cid:107) (cid:107) (cid:111) (cid:111) . . . (cid:111) (cid:111) (C1)There are basically (at least) two ways for dimensional reduction procedure: • ( i ) One way is the left arrow ← procedure, which compactifies one spatial direction x u as a S circle while a gauge4field A u along that x u direction takes Z N u value by (cid:72) S A u = 2 πn u /N u . • ( ii ) Another way of dimensional reduction is the up-left arrow (cid:45) , where the space is designed as M × M d − ,where a 2-dimensional surface M is drilled with holes or punctures of monodromy defects with d A w flux, via (cid:90) (cid:7)(cid:6) (cid:4)(cid:5) (cid:90) (cid:88) d A w = 2 πn w under the condition Eq.(A10). As long as the net flux through all the holes is not zero ( n w (cid:54) = 0),the dimensionally reduced partition functions can be nontrivial SPTs at lower dimensions. We summarize theirphysical probes in Table IV and in its caption. Physical Observables Dimensional reduction of SPT invariants and probe-feild actions • degenerate zero energy modes of 1+1D SPT A A ← A A A ← A A A A ← · · · (projective representation of Z N × Z N symmetry) A A ← A u A v d A w ← · · ·• edge modes on monodromy defects of 2+1D SPT - gapless, A v d A w ← A u A v d A w ← · · · or gapped with induced fractional quantum numbers • braiding statistics of monodromy defects TABLE IV. We discuss two kinds of dimensional-reducing outcomes and their physical observables. The first kind reduces to (cid:82) A A type action of 1+1D SPTs, where its 0D boundary modes carries a projective representation of the remained symmetry Z N × Z N , due to its action is a nontrivial element of H ( Z N × Z N , R / Z ). This projective representation also implies thedegenerate zero energy modes near the 0D boundary. The second kind reduces to (cid:82) A v d A w type action of 2+1D SPTs, whereits physical observables are either gapless edge modes at the monodromy defects, or gapped edge by symmetry-breaking domainwall which induces fractional quantum numbers. One can also detect this SPTs by its nontrivial braiding statistics of gappedmonodromy defects (particles/strings in 2D/3D for (cid:82) A d A / (cid:82) AA d A type actions). Second, we can also apply a discrete cocycle approach (to verify the above field theory result). We only need to usethe slant product, which sends a n -cochain c to a ( n − i g c : i g c ( g , g , . . . , g n − ) ≡ c ( g, g , g , . . . , g n − ) ( − n − · n − (cid:89) j =1 c ( g , . . . , g j , ( g . . . g j ) − · g · ( g . . . g j ) , . . . , g n − ) ( − n − j , (C2)with g i ∈ G . Let us consider Abelian group G , in 2+1D,where we dimensionally reduce by sending a 3-cocycle toa 2-cocycle: C a ( b, c ) ≡ i a ω ( b, c ) = ω ( a, b, c ) ω ( b, c, a ) ω ( b, a, c ) . 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