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

Classification and construction of symmetry protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been shown that the (generalized) group cohomology theory or cobordism theory can give rise to a complete classification of SPT phases in interacting boson/spin systems. Nevertheless, the construction and classification of SPT phases in interacting fermion systems are much more complicated, especially in 3D. In this work, we revisit this problem based on the equivalent class of fermionic symmetric local unitary (FSLU) transformations. We construct very general fixed point SPT wavefunctions for interacting fermion systems. We naturally reproduce the partial classifications given by special group super-cohomology theory, and we show that with an additional $\tilde{B}H^2(G_b, \mathbb Z_2)$ (the so-called obstruction free subgroup of $H^2(G_b, \mathbb Z_2)$) structure, a complete classification of SPT phases for three-dimensional interacting fermion systems with a total symmetry group $G_f=G_b\times \mathbb Z_2^f$ can be obtained for unitary symmetry group $G_b$. We also discuss the procedure of deriving a general group super-cohomology theory in arbitrary dimensions.

In recent years, a new type of topological order -symmetry protected topological (SPT) order [1][2][3] has been proposed and intensively studied in interacting boson and fermion systems. 2D and 3D topological insulators (TI) [4,5] are the simplest examples of SPT phases, which are protected by time reversal and charge conservation symmetries. Although topological insulators were initially proposed and experimentally realized in (almost) non-interacting electron systems, very recent studies have established their existence and stability even in the presence of strong interactions by computing (non-perturbative) quantum anomaly on various manifolds [6]. The first attempt to systematically understand SPT phases in interacting systems was proposed in Ref. 1, which pointed out that the well known Haldane chain [7] is actually an SPT phase! Later, a systematic classification of SPT phases for interacting bosonic systems in arbitrary dimension with an arbitrary global symmetry was achieved by using (generalized) group cohomology theory [2,3,8] or cobordism theory [9], which essentially classifies the quantum anomalies associated with the corresponding global symmetries in interacting bosonic systems. In terms of physical picture, it has been further pointed out that by gauging the global symmetry G, different SPT phases can be characterized by different types of braiding statistics of G-flux/flux lines in 2D/3D [10][11][12][13][14][15][16][17]. Moreover, anomalous surface topological order was also proposed as another very powerful way to identify and characterize different 3D SPT phases in interacting boson and fermion systems [18][19][20][21][22][23][24][25][26][27].
From the quantum information perspective, (intrinsic) topological phases are gapped quantum states that can be defined and classified by inequivalent class of (finite depth) local unitary (LU) transformations [28], which leads to the novel concept of long range entanglement. However, in contrast to those (intrinsic) topological phases, all SPT phases can be adiabatically connected to a trivial disordered phase or an atomic insulator without symmetry protection. Therefore, SPT phases only contain short-range entanglement and they can be constructed by applying local unitary transformations on a trivial product state. In particular, Ref. 2 and 3 introduced a systematic way to construct fixed point ground state wavefunction for bosonic SPT phases on arbitrary triangulations in arbitrary dimensions. It turns out that such a construction is fairly complete for bosonic SPT phases protected by unitary symmetry group(up to 3D). So far the only known example beyond this construction is the so-called ef mf SPT state [18] protected by time reversal symmetry in 3D. Later, it was shown that such an exotic bosonic SPT state can be realized by Walker-Wang model [29,30].
Nevertheless, the classification and systematic understanding of SPT phases in interacting fermion systems are much more complicated. One obvious way with fruitful results is the study on reduction of the free-fermion classifications [31,32] under the effect of interactions [33][34][35][36][37][38][39]. However, these works miss those fermionic SPT (FSPT) phases that can not be realized in free fermion systems. A slightly generalized way to understand FSPT in interacting fermion systems is to stack some additional bosonic SPT states onto a free fermion SPT state [40]. For example, an arguable fairly complete classification of TI in interacting electron systems [41] can be constructed in such a way. However, there is no natural reason to believe that all FSPT phases in interacting systems can be realized by the above stacking constructions and counter examples can be constructed explicitly. Moreover, it has also been shown that certain bosonic SPT phases will become "trivial" (adiabatically connected to a product state) [40,42] when embedding into interacting fermion systems. Apparently, the stacking construction can not explain all these subtle issues. Therefore, a systematic understanding and construction of interacting FSPT phases are very desired.
The first attempt to classify interacting FSPT phases in general dimensions was proposed in Ref. 42, where a class of FSPT phases was constructed systematically by generalizing the usual group cohomology theory into the so-called special group super-cohomology theory. However, it turns out that such a construction can not give rise to all FSPT phases (except in 1D, where the obtained classification of FSPT phases perfectly agrees with previous results [43,44]). On the other hand, quantum anomaly characterized by spin cobordism [45] or invertable spin topological quantum field theory (TQFT) [46][47][48] suggests a very rich world of FSPT phases, though it is not clear how to construct these FSPT states in an explicit and systematic way.
Alternatively, the idea of gauging fermion parity [37,[49][50][51][52] provides us another way to understand FSPT. In 2D, fairly complete classification of FSPT can be obtained in this way, which also agrees with the anomaly classification given by spin cobordism and invertable spin TQFT [45,46,48]. It has been shown that the mathematical objects that classify 2D FSPT phases with a total symmetry G f = G b × Z f 2 can be summarized as three group cohomologies of the symmetry group [48,50]: H 1 (G b , Z 2 ), BH 2 (G b , Z 2 ) and H 3 (G b , U T (1)). H 1 (G b , Z 2 ) which one-to-one corresponds to the Z 2 subgroups of G b , classifies FSPT phases with Majorana edge modes. BH 2 (G b , Z 2 ), the obstruction free subgroup of H 2 (G b , Z 2 ) is formed by elements n 2 ∈ H 2 (G b , Z 2 ) that satisfy Sq 2 (n 2 ) = 0 in H 4 (G b , U T (1)), where Sq 2 is the Steenrod square Sq 2 : (1)) is the wellknown classification of bosonic SPT phases. Physically, the H 1 (G b , Z 2 ) layer can be constructed by decorating a Majorana chain [53] (which is a 1D invertable fermionic TQFT) onto the domain walls of symmetry group G b and the BH 2 (G b , Z 2 ) layer can be constructed by decorating complex fermion (which is a 0D invertable TQFT) onto the intersection points of G b -symmetry domain walls. Nevertheless, the decoration scheme could suffer from obstructions and only a subgroup BH 2 (G b , Z 2 ) classifies valid and inequivalent 2D FSPT phases. In 3D, a few interesting examples of SPT phases are studied based on Walker-Wang model construction [54], e.g., DIII class topological superconductor [24,[55][56][57]. However, it is unclear how to construct all FSPT phases by using Walker-Wang model. In fact, it is even unclear how to reproduce all the special group super-cohomology constructions in this way. Very recently, some new interacting FSPT phases beyond special group super-cohomology are formally proposed by using spin TQFT [58]; however, there is still lacking of general principle and lattice model realizations.
In this paper, we would like to propose a general physical principle to obtain all FSPT phases in 3D with total symmetry group G f = G b × Z f 2 , where G b is the bosonic global symmetry and Z f 2 is the fermion parity conservation symmetry. In a previous work, it has been shown that in the presence of global symmetry, symmetry enriched topological (SET) phases can be defined and classified by equivalent class of symmetric local unitary (SLU) transformations [59,60]. In particular, SPT phases can be realized as a special class of SET phases whose bulk excitations are trivial and can be adiabatically connected to a product state in the absence of global symmetry. In Ref. 61, it has been shown that fermionic local unitary (FLU) transformations can be used to define and classify (intrinsic) topological phases for interacting fermion systems. The Fock space structure and fermion parity conservation symmetry of fermion systems can be naturally encoded into FLU transformations. Thus, it is not quite surprising that in the presence of global symmetry, we can just introduce the notion of fermionic symmetric local unitary (FSLU) transformations to define and classify fermionic SET (FSET) phases in interacting fermion systems. Similar to the bosonic case, FSPT phases are special class of FSET phases that have trivial bulk excitations and can be adiabatically connected to a product state in the absence of global symmetry. Technically, we only need to enforce the support dimension of FSLU transformations equal to one to classify FSPT states.
Essentially, the novel concept of FSLU transformation allows us to construct very general fixed point FSPT states of 2D and 3D FSPT phases. Moreover, all these fixed point wavefucntions admit exact solvable parent Hamiltonians consisting of commuting projectors on an arbitrary triangulation with an arbitrary branching structure. We will begin with the 2D case where the discrete spin structure can be implemented by Kasteleyn orientation [62][63][64] that allows us to decorate Majorana chain onto G b -symmetry domain walls [65,66]. Then we will show how to implement the discrete spin structure on a triangulation of 3D (orientable) spin manifold, which is a nontrivial generalization of Kasteleyn orientation. The discrete spin structure allows us to decorate the Majorana chain onto the intersection lines of G b -symmetry domain walls in a self consistent and topological invariant way. The fundamental mathematical data describing such a decoration scheme belong to H 2 (G b , Z 2 ), subjected to an obstructions on H 4 (G b , Z 2 ). The obstruction can be understood by the following physical picture: Since Kasteleyn orientation is not always possible for a large loop (the 3D discrete spin structure can be used to construct local Kasteleyn orientation of small loops within a tetrahedron), a complex fermion decoration on the intersection points of G b -symmetry domain walls is typically required and this is only possible when the H 4 (G b , Z 2 ) obstruction vanishes. Furthermore, another obstruction on H 5 (G b , U T (1)) will be generated by wavefunction renormalization to finally determine whether the entire decoration scheme of Majoarana chain is valid or not for a fixed point wavefunction in 3D.
The precise mathematical objects that classify 3D FSPT phases with a total symmetry G f = G b × Z f 2 can also be summarized as three group cohomologies of the symmetry group: where O is some unknown cohomology operation (to the best of our knowledge) that mapsñ 2 satisfying The explicit expression of O is very complicated and will be computed in a physical way in section IV B.
/Γ were derived in the special group super-cohomology classification. Recall that H 4 (G b , U T (1)) is the well-known classification of bosonic SPT phases and Γ is a normal subgroup of H 4 (G b , U T (1)) generated by Sq 2 (n 2 ), where n 2 ∈ H 2 (G b , Z 2 ) and Sq 2 (n 2 ) are viewed as elements of H 4 [G b , U T (1)]. Physically, Γ describes those trivialized bosonic SPT phases when embedded into interacting fermion systems.
Together with several previous works, we conjecture that up to spacial dimension d sp = 3, FSPT with symmetry [G f , U T (1)] defined by the exact sequences summarized in table I. We note that for spacial dimension d sp > 1, the general group super-cohomology theory will be defined by two short exact sequences. The first short exact sequence can be understood as decoration of complex fermion onto the intersection points of G b -symmetry domain walls, which was first derived by the special group super-cohomology theory. The second exact sequence can be understood as decoration of Kitaev's Majorana chain onto the intersection lines of G b -symmetry domain walls and our construction give rise to a general scheme to Finally, regarding the completeness of general group super-cohomology classification for 3D FSPT phases, we would like to give some physical arguments here. Although the decoration of complex fermion on the intersection points of G b -symmetry domain walls and the decoration of Majorana chain on G b -symmetry domain walls give rise to a complete classification of 2D FSPT phases, it does not necessarily imply this is also true in 3D. In fact, it has been pointed out [8] that the decoration of invertable TQFT on the G b -symmetry domain walls might also gives rise to new SPT states. For bosonic SPT states, a decoration of the so-called E 8 state on the G b -symmetry domain walls indeed produces the ef mf SPT state beyond group cohomology classification. It was also pointed out that H 1 (G b , Z) classifies these additional bosonic SPT. Since H 1 (G b , Z) is trivial for unitary symmetry group G b and H 1 (Z T 2 , Z) = Z 2 for the anti-unitary time reversal symmetry, we understand the reason why ef mf state is the only non-trivial route state of bosonic SPT states beyond group cohomology classification with time reversal symmetry. For interacting fermion systems, in principle we can decorate a p + ip state (the root state of 2D fermionic invertable TQFT) onto the G b -symmetry domain walls. However, since H 1 (G b , Z) is trivial for unitary symmetry group G b , there will be no new FSPT states with unitary symmetry group G b . For time reversal symmetry, it is possible to generate new FSPT states in this way, and we will discuss this possibility in our future work.
The rest part of the paper is organized as follows. We will begin with the definition of Hilbert space and the basic structure of fixed point wavefunctions for FSPT states with a total symmetry G f = G b × Z f 2 in 2D and 3D. In section III A, we will give a brief review for discrete spin structure and Kasteleyn orientation in 2D. In section III B, we will derive the fixed point conditions for FSLU transformations under wavefunction renormalization and re-derive the classifications of 2D FSPT phases. In section IV A, we will discuss how to generalize the discrete spin structure in 3D. In section IV B, we will use the concept of equivalent class of FSLU transformations and wavefunction renormalization to obtain the classifications and constructions of 3D FSPT phases. Finally, there will be conclusions and discussions for possible future directions.

II. FIXED POINT WAVEFUNCTION OF FSPT STATES
Similar to the wave function renormalization scheme for 2D bosonic SET phases, here we will consider quantum state defined on an arbitrary triangulation for 2D and 3D FSPT phases. The triangulation admits a branching structure that can be labelled by a set of local arrows on all links (edges) with no oriented loop for any triangle. Mathematically, the branching structure can be regarded as a discrete version of spin c structure and can be consistently defined on arbitrary triangulations of orientable manifolds.
Let us begin with the construction of fixed point wavefunction in 2D, and the generalization into 3D will be straightforward. Since any 2D FSPT state can be naturally mapped to a 2D bosonic SET state by gauging the fermion parity symmetry, our construction for fixed point wavefucntions will be greatly inspired by such kind of connections. In particular, an FSPT state with total symmetry G f = G b ×Z f 2 can be mapped to a G b symmetry enriched toric code model. As a simple example, fixed point wavefucntions and commuting projector parent Hamiltonians on arbitrary trivalent graph (due of a triangulation) with G b = Z 2 are constructed in Ref. 59. It has also been shown that all these SET states can be obtained by gauging fermion parity from FSPT states with total symmetry G f = Z 2 × Z f 2 . The building block of bosonic and fermionic degrees of freedom of our 2D FSPT model is shown in Fig. 1. Exactly the same as bosonic SET phase, every vertex i of the space triangulation has a bosonic degrees of freedom labelled by a group element g i ∈ G b (recall that the FSPT phases have a total symmetry G f = G b × Z f 2 ). A spinless complex fermion c lives at the center of each triangle/face (see blue ball in Fig. 1), and the fermion occupation number is either is a subset of all triangles F including the empty set. In addition, each link has two Majorana fermions on its two sides, which is equivalent to a spinless complex fermion a. Similar to the c fermion on each triangle, let |0 be the ground state of no fermions on all links, then a generating set of the Fock space is given by (ij)∈l a † (ij) |0 , where l ⊂ L is a subset of all links L including the empty set. Thus, the full local Hilbert space for our 2D model on a fixed triangulation T is Here |G b | is the order of the bosonic symmetry group G b . As a vector space, the fermionic Hilbert space on the triangles and links are the same as the tensor product F (T ) C 2 ⊗ L(T ) C 2 ; however, the Fock space structure means that a local Hamiltonian for a fermion system is non-local when regarded as one for a boson system. We note that the above structure of total fermionic Hilbert space on arbitrary triangulations is slightly more general than Ref. 61, and it will allow us to construct very general FSET states in 2D. However, the constructions of general FSET states are beyond the scope of this paper.
As mentioned before, for FSPT states, the support space of FSLU transformations must be one dimensional since it can be connected to a product state in the absence of global symmetry. Therefore, the fermionic states of c and a fermions on the triangles and edges are completely fixed by the configurations of group elements {g i } on vertices. In particular, the complex fermion occupation number of c fermion is uniquely determined by the elements in BH 2 (G b , Z 2 ) (the obstruction free subgroup of H 2 (G b , Z 2 )), as first proposed by the special group super-cohomology construction of FSPT phases. Essentially, the complex fermion c can be regarded as a decoration on the intersection points of G b -symmetry domain walls. In Ref. 50, 65, and 66, it was pointed out that a Majorana chain can be decorated onto the G b -symmetry domain walls to generate a complete set of FSPT states in 2D. This layer of decoration is uniquely determined by the elements in H 1 (G b , Z 2 ). It requires a discrete spin structures and Kasteleyn orientations on the dual trivalent lattice (with a proper resolution for the lattice sites, as seen in Fig. 1), such that the total fermion parity of a fermion is always even on any closed loop, and we note that for FSPT state, the Majorana fermions must be paired up (see gray ellipse in Fig. 1) to form Kitaev's Majorana chains on the G b symmetry domain walls (see green strip in Fig. 1). We will review all the details in section III A. An example of triangulation of torus and decoration of Kitaev's Majorana chains is given in Fig. 3. The full details will be discussed in section III B.
Thus, our 2D fixed-point state is a superposition of those basis states on all possible triangulations T .
In section III B, we will derive the rules of wavefucntion renormalization generated by FSLU transformations. We will also obtain the conditions for fixed point wavefunction and show how to construct all FSPT states with total symmetry G f = G b × Z f 2 on arbitrary triangulations in 2D. In the following, we would like to generalize all the above constructions into 3D.
The building block of bosonic and fermionic degrees of freedom is shown in Fig. 2. Again, every vertex i of the 3D space triangulation has a bosonic degree of freedom labelled by a group element g i ∈ G b . However, the spinless complex fermion c introduced by group super-cohomology theory is now living on each tetrahedron (see blue ball in Fig. 2). In addition, each triangle of the space tetrahedron has two Majorana fermions on its two sides, which is again equivalent to a spinless complex fermion a. Similar to 2D case, let |0 and |0 be the ground states of no fermions on all tetrahedron and triangles, then a generating set of the Fock space is given by , where t ⊂ T is a subset of all tetrahedra T including the empty set and f ⊂ F is a subset of all triangles F including the empty set. Thus, the full local Hilbert space of our 3D model on a fixed triangulation T is: Similar to the 2D case, the fermionic states of c and a fermions on the tetrahedra and triangles are also completely fixed by the configurations of group elements {g i } on vertices. The complex fermion occupation number of c fermion is uniquely determined by the elements in BH 3 (G b , Z 2 ) (the obstruction free subgroup of H 3 (G b , Z 2 )), which was also first proposed by the special group super-cohomology construction of FSPT states. It is not a surprise that in 3D the complex fermion c can also be regarded as a decoration (subjected to obstructions) on the intersection points of G b -symmetry domain walls. The most interesting new feature here is that a Majorana chain can also be decorated onto the intersection lines of G b -symmetry domain walls, and such a construction will generate a new set of FSPT states in 3D! As expected, this layer of decoration also requires a discrete spin structure on the dual trivalent lattice (with a proper resolution for the lattice sites as well, as seen in Fig. 2), and the Majorana fermions must be also be paired up to form Kitaev's Majorana chains (see green line in Fig. 2). However, such kind of decorations will be subjected to a fundamental obstruction on H 4 (G b , Z 2 ) due to fermion parity conservation. We will discuss all the details in section IV A. Furthermore, the fixed point condition of wavefunction renormalization will give rise to a secondary obstruction on H 5 (G b , U T (1)). We will explore full details in section IV B.
Finally, our 3D fixed-point state is a superposition of those basis states on all possible triangulations T : In section IV B, we will derive the rules of wavefucntion renormalization generated by FSLU transformations. We will also obtain the conditions for fixed point wavefunction and show how to construct all FSPT states with total symmetry G f = G b × Z f 2 on arbitrary triangulations in 3D.

III. CONSTRUCTIONS AND CLASSIFICATIONS FOR FSPT STATES IN 2D
In Ref. 65, it was first pointed out that discrete spin structure and Kasteleyn orientations play an essential role for constructing FSPT phases with a decoration of Kitaev's Majorana chains on G b -symmetry domain walls. In this section, we will give a brief review of the essential idea and generalize the constructions onto arbitrary triangulations in 2D (see Fig. 3). In particular, we use Poincaré dual to show how to implement discrete spin structure and Kasteleyn orientations in 2D. The Poincaré dual will enable us to define discrete spin structure in arbitrary dimensions and give rise to the notion of local Kasteleyn orientations, which serves as the key step towards decorating Kitaev's Majorana chains onto the intersection lines of G b -symmetry domain walls in higher dimensions. It is well known that an oriented manifold M (with dimension n) admits spin structures if and only if its second Stiefel-Whitney class [w 2 ] ∈ H 2 (M, Z 2 ) vanishes. In the construction of lattice models (on triangulation of M ), we find it is more convenient to use the (n − 2)-th Stiefel-Whitney homology class [w n−2 ] which is the Poincaré dual of [w 2 ].
In this subsection, we only consider the 2D case. For a spatial manifold M (n = 2) with triangulation T , the Stiefel-Whitney homology class [w 0 ] has a representative which is the summation of all vertices v with some (mod 2) coefficients as follows [67] [68]: Here, v ⊆ σ means that v is a sub-simplex of simplex σ. v ⊆ σ is called regular if v and σ have one of the three relative positions shown in Fig. 4. #{σ|v ⊆ σ is regular} · v denotes the formal product of the (mod 2) number of regular pairs v ⊆ σ and the vertex v. We will call a vertex v singular if #{σ|v ⊆ σ is regular} is odd. In this language, w 0 in Eq. (5) is the formal summation of all singular vertices. w 0 is a vector (0-th singular chain) in the vector space (of 0-th singular chains) spanned by formal bases of all vertices with Z 2 coefficients. It is known that all oriented 2D surfaces admit spin structures. In other words, the second Stiefel-Whitney class [w 2 ], or the zeroth Stiefel-Whitney homology class [w 0 ], of any oriented surface is (co)homologically trivial. As a result, the collection of singular vertices w 0 in Eq. (5) can be viewed as a boundary ∂S for some lines S (we will call them singular lines). For a fixed collection of singular vortices (a fixed representative w 0 for [w 0 ]), different inequivalent choices of singular lines S correspond to different spin structures which are isomorphic to H 1 (M, Z 2 ) non-canonically. The singular lines S will be colored blue in the following figures.

Kasteleyn orientations and gauge transformations
To decorate Kitaev chains to domain walls of a 2D spin model, it is useful to find out Kasteleyn orientations [62] for edges of the lattice [65,66]. In this section, we will relate the existence of discrete spin structures (the vanishing of [w 0 ]) of a triangulation T to the existence of Kasteleyn orientations of the resolved dual lattice. Then in the next section, we can use FSLU transformations [28,61] to classify FSPT states and define exactly solvable models on arbitrary triangulation lattices.
Our set up begins with a fixed triangulation T of the surface M . The first step is to construct a polyhedral decomposition P of M which is a trivalent graph dual to T . We add a spinless fermionic degree of freedom to every link of T and split it into two Majorana fermions on the two sides of this link for convenience. Or equivalently, we can resolve the triangulation T by adding a new vertex to each triangle center and obtain a new triangulationT . The Majorana fermions live on the vertices of the resolved dual latticeP, which is a trivalent graph dual toT (see    The second step is adding directions to links in T andP. We order all the vertices in T and use the convention that all links are from vertices of smaller number to vertices of larger number. This is a branching structure of T such that there is no cycle for any triangle. The dual link direction in P is obtained from T using the convention shown in Fig. 6a. The directions of new links inP are also obtained from the triangulation T , by using the conventions in Fig. 5. The essential point of the above link orientation conventions is as follows. When going along the smallest red loop inP around a vertex v ∈ T counterclockwise, we will encounter even number (due to the resolvation) of red links with the direction along or opposite to our direction (for example, the red loop inside the green strip around vertex 4 in Fig. 3). Using the conventions in Fig. 5, the red link direction is opposite to the counterclockwise direction if and only if: (1) The red link is dual to a black link in T such that v is the initial point of this black link. This corresponds to the case in Fig. 4b; (2) The red link is a resolved new link inside a triangle in T such that v is the first point of this triangle, i.e. the 0 point of the triangle 012 . This is the case in Fig. 4c. If the total number of the red links with opposite direction is odd, the vertex v will be called Kasteleyn-oriented. Since the smallest loop inP around v has even number of red links, it does not matter whether we use counterclockwise or clockwise conventions. Under the above construction, we relate the zeroth Stiefel-Whitney homology class w 0 in Eq. (5) and the orientations of links iñ P, i.e. w 0 is the summation of all non-Kasteleyn-oriented vertices: Since the zeroth Stiefel-Whitney homology class [w 0 ] for any oriented surface is trivial, we have w 0 = ∂S for some singular line S. If we further reverse the directions of links inP crossing the singular lines S as shown in Fig. 6b, then all the vertices in T will be Kasteleyn-oriented. This is because this operation only changes the Kasteleyn property of the singular vertices in w 0 while preserving this property for all other vertices, including those in the interval of S. After all the above procedure, we relate the vanishing of the zeroth Stiefel-Whitney homology class [w 0 ] to the property of Kasteleyn orientation of the smallest loop around each vertex.
Note that the above construction of link directions inP depends on the choices of singular lines S. On the one hand, the local shape of S is not important, as long as ∂S are fixed. In fact, if we change the shape of S locally, the changes of link directions inP can be obtained by several "gauge transformations" of Kasteleyn orientations, which relate two different but equivalent Kasteleyn orientations (simultaneously changing the directions of links sharing a common vertex inP) [63]. An example of the basic shape changes of singular lines on T and gauge transformation of Kasteleyn orientations onP are shown in Fig. 7. Note that the Majorana degrees of freedom on vertices ofP are mapped from one lattice to another according to the link directions under the gauge transformation of Kasteleyn orientations (n to n in Fig. 7). In this way, the vacuum state without fermion (without Kitaev chain) on the left lattice is mapped to the vacuum state on the right lattice without fermion parity changing.

Kasteleyn orientations under retriangulations
In the above subsection, we only focus on a fixed triangulation T of M and relate its discrete Stiefel-Whitney homology class [w 0 ] to the Kasteleyn orientations and spin structures. In order to use FSLU transformations to classify FSPT phases, we have to understand the relation of Kasteleyn orientations for different triangulations. In fact, we only have to find out the changes of Kasteleyn orientations under Pachner moves, which are basic moves of retriangulations [69].
Ordinary Pachner moves for two dimensional manifold consist of (2-2) move and (1-3) move. With branching structure, there are 3 types of (2-2) moves and 4 types of (1-3) moves in total (we do not consider the mirror images of these moves, otherwise the number of moves will be doubled). Only 2 types of (2-2) moves and 2 types of (1-3) moves have branching structure that can be induced by a global time ordering [42]. Examples of these moves are shown as follows: Other types of (2-2) and (1-3) moves are shown in Appendix A. For Pachner moves not induced by a global time ordering, the representative w 0 of Stiefel-Whitney class [w 0 ] in Eq. (5) may be changed. For Pachner moves that are induced by a global time ordering, the representative w 0 of Stiefel-Whitney class [w 0 ] is unchanged. This makes 2D case much easier than the 3D case.

B. Fermionic symmetric local unitary transformations and consistent equations
In the above section, we discussed discrete spin structure and Kasteleyn orientation construction on arbitrary 2D triangulation lattice. We can now decorate Kitaev's Majorana chains using these rules and systematically classify 2D FSPT states using FSLU transformations.

Decoration of Kitaev's Majorana chains
As discussed above, our model has two types of fermionic degrees of freedom. The first type is the complex fermion c (ijk) which lives at the center of triangle ijk of space manifold triangulation T . We use n 2 (g i , g j , g k ) = 0, 1 to denote the number of c fermion at the triangle ijk . In fact, the parity conservation constraint for c fermions under retriangulation is dn 2 = 0 (mod 2). Therefore n 2 is an element of The second type of (complex) fermion a (ij) lives on the link ij of T . In order to describe Kitaev's Majorana chain more conveniently, we separate the fermion a (ij) to two Majorana fermions: The Majorana fermions γ ijA and γ ijB live on the two sides of link ij . Dually, they live on the two ends of the link inP dual to link ij . Our convention is that the dual link has direction from vertex ijA to ijB . The fermion parity operator of a fermion or γ fermion at link ij is P γ f = −iγ ijA γ ijB . Now let us decorate Kitaev's Majorana chains to the dual lattice. We use a Z 2 -valued 1-cochainñ 1 (g i , g j ) to indicate whether there is a domain wall between vertices i and j. After fermion decoration, the (2-2) move becomes a fermionic unitary transformation between the fermionic Fock spaces on two different triangulation lattices T and T . An example of this standard F move is (there are Z 2 domain walls on links 01 , 02 , 03 , and no domain wall on other links): where the F operator is defined as Here, ν 3 (g 0 , g 1 , g 2 , g 3 ) is a U T (1)-valued 3-cochain and c † (012) is the creation operator for c fermions at triangle 012 , etc. X[ñ 1 (g i , g j )] is a projection operator changing the Majorana fermion configurations. In the above example, X operator has an explicit form: where P a,b = (1 − iγ a γ b )/2 is the projection operator for Majorana pairs a, b (the direction is from vertex a to vertex b). The first two projection operators in the above equation project the state to the Majorana dimer configuration in the left figure. Note that the Majorana fermions which do not appear explicitly on one lattice are considered to be in the vacuum pairs. For example, the two Majorana fermions γ 13A and γ 13B only appear in the right figure. They are considered to be paired from γ 13A to γ 13B in the left figure. Therefore, we have a third projection operator in Eq. (11) to put the two Majorana fermions γ 13A and γ 13B to vacuum state (a † (13) a (13) = 0) in the left figure. All other Majorana fermions that are not shown in Eq. (11) are unchanged under the above F move.
Since we are constructing FSPT states, the fermionic local unitary transformation F should be G b symmetric in the sense that for all g ∈ G b if G b is a unitary symmetry group. That is why ν 3 (g 0 , g 1 , g 2 , g 3 ), n 2 (g 0 , g 1 , g 2 ) andñ 1 (g 0 , g 1 ) are all cochains which are invariant under unitary g action. (We note that ν 3 (gg 0 , gg 1 , gg 2 , gg 3 ) = ν * 3 (g 0 , g 1 , g 2 , g 3 ) for antiunitary g action.) In general, there are eight kinds of domain wall configurations in the above F move. One can show that for all configurations, the fermion parities of Majorana fermions are the same in the initial and final wave functions (this comes from the Kasteleyn orientation property of retriangulations, see section III A and Appendix A). Therefore, the fermion parities of c fermions and γ fermions should also be conserved separately, and both n 2 ∈ H 2 (G b , Z 2 ) and The phase factor in the front of the F operator can be changed from ν 3 to where µ 2 (g 0 , g 1 , g 2 ) is a U T (1)-valued 2-cochain. If ν 3 (g 0 , g 1 , g 2 , g 3 ) is a consistent solution of fixed-point state (see section III B 2), so is ν 3 (g 0 , g 1 , g 2 , g 3 ). However, we can use a FSLU transformation U µ2,m1 to smoothly connect the these two fixed-point states with ν 3 and ν 3 in F symbols. The FSLU transformation U µ2,m1 decorates fermions to the links of the triangulation and is defined as where s ijk = ±1 is the orientation of the triangle ijk , and f (ij) is the annihilation operator of the new fermions decorated on the link ij . Although we decorate new f fermions with fermion number m 1 to the links, it does not induce additional sign factor [42]. So the equivalence relation of ν 3 and ν 3 in Eq. (13) is the same as the bosonic one. Therefore, elements ν 3 in the group cohomology class H 3 (G b , U T (1)) correspond to the same 2D FSPT phase [42]. Apart from the (2-2) move, there is another (2-0) move which plays the role of unitary condition. An example of domain wall configurations for (2-0) move is where c (012) andc (012) are the annihilation operators of the c fermions at the center of two triangles with opposite orientation in the left figure. X[ñ 1 ] is also the projection operator from the state of Majorana dimer pairs in the right figure to the state of the left figure. Note that there are six Majorana fermions (γ 02A , γ 02B , γ 01A , γ 01B , γ 12A , γ 12B ) that do not appear explicitly in the right figure. Similar to the case of (2-2) move, they should also be considered to be in vacuum pairs in the right figure state, such that −iγ 02A γ 02B = −iγ 01B γ 01A = −iγ 12B γ 12A = 1 when acting on the right figure state. This choice is possible because the dimer loop 01A-01B -01A -01B-01A is Kasteleyn oriented. Therefore one can also use the convention that the two Majorana fermions on the two sides of a link are paired up, by regarding the projection operators −iγ 01A γ 01B , −iγ 01A γ 01B , −iγ 12A γ 12B , −iγ 12A γ 12B as 1 when acting on the vacuum state of the left figure. The X operator then projects the state to the Majorana dimer configuration state in the left figure. The fermion parities of the left and right states are always the same. An explicit expression of X for this particular (2-0) move is Using the (2-0) moves, we can deduce all (3-1) moves and other (2-2) moves form the standard (2-2) F move in Eq. (9).

Fermionic pentagon equations
In the above section, we have discussed the FSLU moves. The most important one is the standard F move in Eq. (9). Similar to the bosonic pentagon equation for the bosonic F move, we have a fermionic pentagon equation as consistent equation for FSLU transformations (see Fig. 10). This fermionic pentagon equation only involves the standard F move. Using the unitary conditions, one can also derive other pentagon equations and they essentially give the same constraint for ν 3 . Let us now calculate the constraint for ν 3 from the pentagon equation in Fig. 10. Since the c fermions and Majorana fermions are decoupled (the c fermion part and Majorana fermion part of X in Eq. (10) commute) in the F move, only the c fermion will twist the cocycle condition for ν 3 . The X operators are merely projection operators that do not introduce any nontrivial phases in two different paths of pentagon equation. The final result of equation for ν 3 is the same as the (special) group super-cohomology theory [42,61], i.e., (dν 3 )(g 0 , g 1 , g 2 , g 3 , g 4 ) = (−1) Sq 2 (n2)(g0,g1,g2,g3,g4) = (−1) n2(g0,g1,g2)n2(g2,g3,g4) .
Now, we see that only BH 2 (G b , Z 2 ), the obstruction free subgroup of H 2 (G b , Z 2 ) formed by elements n 2 ∈ H 2 (G b , Z 2 ) that satisfy Sq 2 (n 2 ) = 0 in H 4 (G b , U T (1)) can give rise to solutions for ν 3 , and inequivalent solutions of ν 3 is still given by H 3 (G b , U T (1)) according to the gauge transformations of ν 3 . In conclusion, the mathematical objects that classify 2D FSPT phases with a total symmetry G f = G b × Z f 2 can be summarized as three group cohomologies of the symmetry group [48,50]: (1)). Finally, by using the method proposed in Ref. 61, one can derive the commuting projector parent Hamiltonian of all these FSPT states on arbitrary 2D triangulations(with a branching structure).

IV. CONSTRUCTIONS AND CLASSIFICATIONS FOR FSPT STATES IN 3D
In this section, we will classify the 3D FSPT states parallel to the discussions of 2D FSPT states. Compared to the 2D case, the most nontrivial part of 3D phases are the fermion parity mixing of the c fermions and Majorana fermions. In section IV A, we find that there are in general no Kasteleyn orientations on a 3D lattice. The existence of spin structure only implies local Kasteleyn orientations. If we decorate Kitaev's Majorana chains to 3D lattice, the shape changing process of the chain may change the fermion parity of the Majorana fermions. In this case, we should use the c fermion to compensate the fermion parity changes. Because of the fermion parity mixing, the cocycle equation for ν 4 is much more complicated than the special group super-cohomology model.
A. Discrete spin structure in 3D and local Kasteleyn orientations In this subsection, we will discuss the first Stiefel-Whitney homology class on discrete lattice and relate it to the local Kasteleyn orientations on the dual lattice. The overall constructions are parallel to the 2D case. The difference is that Kasteleyn orientations are only satisfied for the smallest loops in 3D, not for the general large loops. The fermion parity of Kitaev chain decorated on fluctuating loop is therefore not conserved.

Discrete Stiefel-Whitney homology class w1
Similar to the oriented 2D manifolds, all oriented 3D manifolds admit spin structures. The second Stiefel-Whitney cohomology class [w 2 ] is always trivial. Dually, we can consider the first discrete Stiefel-Whitney homology class for a triangulation T (with branching structure) of 3D spatial manifold M [67]: w 1 is the summation of all links in T with some Z 2 coefficients. Again, l ⊆ σ means that link l is a sub-simplex of simplex σ. And l ⊆ σ is called regular if l and σ have one of the three relative positions shown in Fig. 11. If #{σ|l ⊆ σ is regular} is odd, we will call the link l singular. So w 1 in Eq. (18) is the formal summation of all singular lines. w 1 is a vector (1-th singular chain) in the vector space (of 1-th singular chains) spanned by formal bases of all links with Z 2 coefficients. Since the second Stiefel-Whitney cohomology class [w 2 ] of any oriented 3D manifold is trivial, we can find some surface S such that w 1 = ∂S. For a fixed collection of singular links (a fixed representative w 1 for [w 1 ]), different inequivalent choices of S correspond to different spin structures.

Local Kasteleyn orientations and gauge transformations
In 3D model, we also want to decorate Kitaev chains to some loops. A natural question inherited from 2D is whether there are Kasteleyn properties for all even-link loops in 3D. This is related to the fermion parity of the Kitaev chain. The answer is that the existence of discrete spin structures (the vanishing of [w 1 ]) is related to the existence of local Kasteleyn orientations of the resolved dual lattice. In other words, Kasteleyn properties are satisfied for the smallest loops, but broken for large loops in 3D in general.
Here is the construction which is similar to the 2D case. For a fixed triangulation T of 3D manifold M , the first step is to construct a polyhedral decomposition P of M which is a 4-valent graph dual to T . We now add a spinless fermionic degree of freedom to every face (triangle) of T and split it into two Majorana fermions on the two sides of this face for convenience. Equivalently, we resolve the triangulation T by adding a new vertex to each tetrahedron center and obtain a new resolved triangulationT . The Majorana fermions live on the vertices of the resolved dual latticeP, which is a 4-valent graph dual toT (see Fig. 12).
The second step is again adding directions to links in T andP. The directions of links in T are given by the branching structure. The dual link direction in P is obtained from T using the convention shown in Fig. 13a. The directions of new links inP are obtained from the triangulation T , by using the conventions in Fig. 12.
The above link direction construction has the following properties. Consider a fixed link l ∈ T . When going along the smallest red loop inP around this link l along the right-hand rule direction, we will encounter even number of red links with the direction along or opposite to our direction. Using the conventions in Fig. 12, the red link direction is opposite to our direction if and only if: (1) The red link is dual to a black triangle in T such that the initial and final vertices of l are the first and the last vertices of this black triangle. This corresponds to the case in Fig. 11b; (2) The red link is a resolved new link inside a tetrahedron in T such that the initial and final vertices of l are the first and the last vertices of this tetrahedron. This is the case in Fig. 11c. If the total number of the red links with opposite    direction is odd, we will call the link l Kasteleyn-oriented. Under this construction, we relate the first Stiefel-Whitney homology class w 1 in Eq. (18) to the orientations of links inP, i.e. w 1 is the summation of all non-Kasteleyn-oriented links: = l∈T l (l is non-Kasteleyn-oriented).
As discussed above, the first Stiefel-Whitney homology class [w 1 ] for any oriented 3D manifold is trivial. Therefore we have some singular surface S such that w 1 = ∂S. Now if we reverse the directions of links inP crossing the singular surface S as shown in Fig. 13b, then all the links in T will be Kasteleyn-oriented. After all the above procedures, we relate the vanishing of the zeroth Stiefel-Whitney homology class [w 0 ] to the property of local Kasteleyn orientations of the smallest loops around all link in T . Here, "local" means that only the smallest loops inP around links in T are Kasteleyn-oriented. Larger loops with even number of links do not have Kasteleyn property in general.
The above construction of link directions inP depends on the choices of singular surface S. The shape of S can be also changed with fixed ∂S. If we change the shape of S locally, the changes of link directions inP can be obtained by several "gauge transformations" of Kasteleyn orientations. We define this by simultaneously changing the directions of links sharing a common vertex inP, similar to the 2D case [63]. Different Kasteleyn orientations related by this "gauge transformations" are said to be equivalent. An example of shape changes of singular surfaces on T and gauge transformation of Kasteleyn orientations onP are shown in Fig. 14. The Majorana degrees of freedom on vertices of P are also mapped from one lattice to another according to the link directions (similar to the 2D case in Fig. 7). This makes sure that the vacuum state (without Kitaev chain) on one lattice is mapped to the vacuum state on another lattice, without fermion parity changing (no fermion on both lattice).

Local Kasteleyn orientations under retriangulations
In order to perform FSLU transformations, we now consider that the Kasteleyn orientation changes under retriangulations of M . Pachner moves for 3D manifold consist of (2-3) move and (1-4) move [69]. When introducing branching structure, there are totally 10 types of (2-3) moves and 5 types of (1-4) moves (again we do not consider the mirror images of these moves, otherwise the number will be doubled) [42]. 8 types of (2-3) moves and 3 types of (1-4) moves have branching structure that can be induced by global time ordering [42,70]. The standard (2-3) move is given in Fig. 15

B. Fermionic symmetric local unitary transformations and consistent equations
With the above setup of discrete spin structure and Kasteleyn orientation construction on 3D lattice, we can now use the FSLU transformation to classify 3D FSPT states systematically.

Fermion parity and the obstruction of Kitaev's Majorana chain decoration
Similar to the 2D case, our 3D model has two types of fermionic degrees of freedom. The first type is the complex fermion c (ijkl) which lives at the center of tetrahedron ijkl of triangulation T of space manifold. In the (special) group super-cohomology wave function, the c fermion parity P c f is unchanged under (2)(3) and (1)(2)(3)(4) moves. If we use n 3 (g i , g j , g k , g l ) = 0, 1 to denote the number of c fermion at the tetrahedron ijkl , then the parity conserved condition becomes dn 3 = 0 (mod 2). Therefore n 3 is an element of H 3 (G b , Z 2 ). This will be not true if we introduce the second type of fermions. The second type of (complex) fermion a (ijk) lives on the triangle ijk of T . Similar to the 2D case, we also separate the fermion a (ijk) to two Majorana fermions: The Majorana fermions γ ijkA and γ ijkB live on the two sides of triangle ijk , or dually, on two ends of the link iñ P dual to triangle ijk . Our convention is that the dual link (we will also use ijk to denote the dual link) has direction from vertex ijkA to ijkB . So the fermion parity operator of a fermion or γ fermion at triangle ijk is P γ f = −iγ ijkA γ ijkB . Now let us decorate Kitaev's Majorana chains to the loops in dual lattice P. We introduce a Z 2 cochaiñ n 2 (g i , g j , g k ) = 0, 1 to specify the decoration configuration of Kitaev's Majorana chain. If there is a Kitaev chain goes though link ijk in P [see the green links in Fig. 2 and figures in Eq. (25)], then we setñ 2 (g i , g j , g k ) = 1. On the other hand,ñ 2 (g i , g j , g k ) = 0 means there is no Kitaev chain. The Kitaev chain decorations can be translated to dimer configurations of Majorana pairs in the resolved dual latticeP.ñ 2 (g i , g j , g k ) = 0 indicates the vacuum pairing, i.e., the two Majorana fermions at triangle ijk are paired up from ijkA to ijkB . Ifñ 2 (g i , g j , g k ) = 1, Since we are constructing an SPT state without intrinsic anyonic excitations, the decorated Kitaev chain should form a closed loop without ends. Therefore, similar to the 2D case, we have the equation dñ 2 = 0 (mod 2), which means that the cochainñ 2 is an element of H 2 (G b , Z 2 ). It is possible that all the four faces of a tetrahedron 0123 in T are decorated with Kitaev chains, i.e., dñ 2 (g 0 , g 1 , g 2 , g 3 ) = 4. There are ambiguities of pairing four Majorana fermions inside the tetrahedron. In the total three possible pairings , we use the convention that the Majorana fermion 0 is paired to2, and1 is paired to3 (see Fig. 18). One can also choose other conventions which will essentially produce the same results [71]. Sq 2 (ñ 2 )(g 0 , g 1 , g 2 , g 3 , g 4 ) =ñ 2 (g 0 , g 1 , g 2 )ñ 2 (g 2 , g 3 , g 4 ) = 1.

Fermionic symmetric local unitary transformations
After fermion decoration, the standard (2-3) move Fig. 15 becomes a fermionic unitary transformation between the fermionic Fock spaces on two different triangulation lattices T (left) and T (right). An example of this standard F move which changes the fermion parity of Majorana fermions is (on lattice T , P in Eq. (25) and onP in Eq. (26)): where the F operator is given by F (g 0 , g 1 , g 2 , g 3 , g 4 ) = ν 4 (g 0 , g 1 , g 2 , g 3 , g 4 )c †n3 (0124 Here, c †n3(0124) is the abbreviation of c †n3(g0,g1,g2,g4) (0124) which is the creation operator for c fermion at tetrahedron 0124 , etc. The X[ñ 2 ] operator in the above F move from the resolved dual latticeP toP has the following general expression: When the F move changes the Majorana fermion parity, the last term of X operator is the Majorana fermion operator γ 012A = a (012) + a † (012) . The X operator is now an operator with odd number of a fermion creation or annihilation operators, and indeed changes the fermion parity of the state. We have checked that, for all possible Kitaev's Majorana chain configurations, the loop breaking Kasteleyn orientation in the transition graph of the two Majorana dimer states always contains the vertex 012A. Therefore the X operator with γ 012A will indeed project the state to the desired Majorana configuration state (not 0). In fact, γ 234B is also an allowed choice. We will calculate the consistent equation of ν 4 for both choices later.
Using the unitary conditions, we can deduce all (1-4) moves and other (2-3) moves from the standard F move in Eqs. (25) and (26).  Fig. 25 and Fig. 26). Colored numbers i and j in the subscript of F indicate that the link ij with the same color is added after this F move. All the six F moves do not introduce new singular lines and surfaces. There is a global time direction from left to right such that the vertex with smaller number has earlier time.
Here, (012345) is the abbreviation of (g 0 , g 1 , g 2 , g 3 , g 4 , g 5 ), etc. Note that the obstruction only depends on n 3 and dn 3 (orñ 2 2 through the fermion parity equation dn 3 =ñ 2 2 ). In the following, we will derive the above obstruction equation in detail.
After adding a cochain (−1) d(n3 3dn3) = (−1) dn3 3 dn3+n3 2 dn3+dn3 2n3 to the combination of the phase factor O c and O cγ , we get a simpler expression Now let us turn to the subtlest part O γ coming from decorated Majorana chain. Besides ±1, O γ can also takes value in ±i. If all the six F moves do not change the Majorana fermion parity, then X γ operators in F moves are merely projections with even number of γ Majorana operators. The c fermions and Majorana fermions are decoupled, and both O cγ and O γ are trivial. The obstruction O is the same as the (special) group super-cohomology result. Therefore, we only need to check the case that some of the six F moves in Fig. 19 change the Majorana fermion parity, i.e., some ofñ 2 2 (01234),ñ 2 2 (01245),ñ 2 2 (02345),ñ 2 2 (01345),ñ 2 2 (01235) andñ 2 2 (12345) equal to one. Let us denote the six X operators in F moves as X 01235 = P 1 γñ  |final .
The above equation suggests that O γ depends only on the values ofñ 2 2 (01235),ñ 2 2 (01345),ñ 2 2 (12345),ñ 2 2 (02345), n 2 2 (01245), andñ 2 2 (01234), i.e., the Majorana parity changes of the six F moves. Consider, for example, the case where only F 01345 and F 12345 change the Majorana fermion parity [(0, 1, 1, 0, 0, 0) in the eighth row of Table II. See also Fig. 21 for an example ofñ 2 (g i , g j , g k ) satisfying this condition]. We can expand projection operators P i to Majorana fermion operators. For simplicity, we can consider the term with only contributions −iγ 013A γ 013B and −iγ 123A γ 123B from P 1 . We can also add −iγ 013B γ 123B which equals to 1 when acting on |final in front of this state. The result is then We have assumed that there are only two Kitaev strings meeting at at tetrahedron 0123 (we find this is always true for all possible choicesñ 2 (g i , g j , g k ) that belong to the eighth row of Table II). Similarly, one can calculate other choices ofñ 2 , and at last obtain the results listed in Table II  from all possible Kitaev chain configurations in hexagon equation. The first column has entry ñ 2 2 (01235),ñ 2 2 (01345),ñ 2 2 (12345),ñ 2 2 (02345),ñ 2 2 (01245),ñ 2 2 (01234) , indicating whether the six F moves in the hexagon equation change the Majorana fermion parity or not. The second column shows the γ operators appearing in Eq. (43). The third and fourth columns are lines of Majorana dimer pairs (∅ means there are nõ n 2 (g i , g j , g k ) that has 2 or 4 strings at tetrahedron 0123 ). The last column is the value of O γ forñ 2 (g i , g j , g k ) belongs to this row. Nontrivial results are labelled by red color. The results in this table can be summarized to an expression shown in Eq. (45).

V. CONCLUSIONS AND DISCUSSIONS
In conclusion, we have constructed fixed point wavefucntions for FSPT states in 2D and 3D based on the novel concept of FSLU transformations. All these FSPT states admit parent Hamiltonians consisting of commuting projectors on arbitrary triangulations(with a branching structure). We believe that our construction will give rise to a complete classification for FSPT states with total symmetry G f = G b × Z f 2 when G b is a unitary symmetry group. Mathematically, our constructions naturally define a general group super-cohomology theory that generalizes the so-called special group super-cohomology theory proposed in Ref. 42.
In particular, one can start with a spin manifold in arbitrary spacial dimension d sp and define the corresponding discrete spin structure via Poincare dual, then one can decorate Kitaev's Majorana chain on the intersection lines of G b symmetry domain walls if the first obstruction vanishes for elementsñ dsp−1 ∈ H dsp−1 (G b , Z 2 ). That is, Z 2 ), and the obstruction free elementsñ dsp−1 ∈ H dsp−1 (G b , Z 2 ) will give rise to all inequivalent pattern of Mjorana chain decoration. Finally, by applying the wavefunction renormalization on arbitrary triangulations, one may derive a twisted co-cycle equations where the twisted factors define some unknown cohomolgoy map O that maps elements in , and the second obstruction free condition requires O(ñ dsp−1 ) = 0 in H dsp+2 (G b , U T (1)). Elementsñ dsp−1 ∈ H dsp−1 (G b , Z 2 ) satisfying both the first and second obstruction free conditions might define a subgroupBH dsp−1 (G b , Z 2 ) ∈ H dsp−1 (G b , Z 2 ), which allows us to write down another short → 0 to define a general group super-cohomology theory. We note that here H dsp+1 [G f , U T (1)] is the special group super-cohomology class defined by Ref. 42.
For future work, it would be of great importance to understand the physical properties of 3D FSPT phases classified by general group super-cohomology theory, e.g. understanding the braiding statistics of G b -flux lines. Of course, constructing time reversal symmetry protected FSPT states with both T 2 = 1 and T 2 = P f (where P f is the total fermion parity) is another interesting direction. Moreover, it will also be interesting to investigate the phase transition theory among FSPT phases in arbitrary dimensions, which might define a new type of fractionalized string theory in arbitrary dimensions. Note that the (2-2) move in Eq. (A2) is the standard move in Fig. 8. The other (2-2) move in Eq. (A1) can be derive from a sequence of the standard move and the (2-0) moves in Eq. (15), as shown below: To derive the (2-2) move in Eq. (A1), we first add a vertex g 3 and a triangle labelled by g 0 , g 2 , g 3 on the right of the figure. The first step is a standard (2-2) F move. Then after a (2-0) move and a (0-2) move, we finally obtain the desired triangulation.
When decorating Kitaev's Majorana chains to the dual lattice, there are in total 2 3 = 8 kinds of different decoration configurations (see Table III) Fig. 26 from the standard move and (2-0) moves: We first add a vertex g 4 and a tetrahedron labelled by g 1 , g 2 , g 3 , g 4 on the lower right of the figures on both side of Fig. 26. From the figure on the right hand side, we first perform a (2-0) move to remove the two tetrahedra labelled by g 1 , g 2 , g 3 , g 4 but have opposite orientations. There are now only two tetrahedra left as shown in the second figure above. Now we can use the standard (2-3) F move to add a line connecting g 1 and g 3 . The resulting figure (the third one) is the same as the last figure, after identifying the two vertices labelled by g 4 . Therefore, the inverse of the (2-3) move in Fig. 26              out-leg string # configuration # configuration-1 configuration-2  Fig. 18.        To construction θ[n 3 , n 3 ,ñ 2 ,ñ 2 ] in the Eq. (C17), we only need to find a solution y[ñ 2 ,ñ 2 ] of Eq. (C15). Note that, Eq. (C15) is a (mod 2) equation when evaluating for arbitrary (g 0 , g 1 , g 2 , g 3 , g 4 , g 5 ) ∈ G 6 b . y[ñ 2 ,ñ 2 ](g 0 , g 1 , g 2 , g 3 , g 4 ) is automatically symmetric under G b , for it depends on g i throughñ 2 andñ 2 .
The number of equations in (C15) is the number of all possible cocycleñ 2 andñ 2 , i.e., the number of different cochain patternsñ 2 andñ 2 satisfying dñ 2 (g i , g j , g k , g l ) = dñ 2 (g i , g j , g k , g l ) = 0 (0 ≤ i < j < k < l ≤ 5). One can solve the cocyle equation and find that the number of differentñ 2 (g i , g j , g k ) (0 ≤ i < j < k ≤ 5) pattern is 2 10 . This is the number of Kitaev chain decoration configurations of a 5-simplex, or the hexagon equation (see Fig. 21 for one configuration). So the number of equations is p = 2 20 = 1048576.
The right hand side of Eq. (C15) involves f (ñ 2 ). Because O γ = (−1) f [ñ2] can be ±1 or ±i, f (ñ 2 ) takes value in {0, 1 2 , 1, 3 2 }. To make the right hand side of Eq. (C15) integers, we can multiply Eq. (C15) by 2, and scale y by The final equation for x in the matrix form is This is an overdetermined system of linear equations with p = 2 20 equations and q = 2 12 unknowns. Using the procedure list in the next subsection, we find that the above linear equation indeed has solutions. We can use the solution x to construct θ. And the group structure of the general super-cohomology solutions is given by Eq. (C17).

Solving system of linear equations in R/4Z
In this subsection, we will discuss how to solve the equations of the form Here, A ∈ M p,q (Z) is a p × q (p = 2 20 , q = 2 12 ) matrix with elements in Z (the matrix form of coboundary operator "d" has elements in Z). On the right hand side, b ∈ Z p is a Z-valued vector with length p. However, x ∈ R p is a R-valued vector with length p. All the equations hold modulo 4. Note that this equation is not a system of linear equations in ring Z/4Z, for elements of x can take value in R/4Z, not merely Z/4Z. In fact, we find that the above equation in our case has no solution of x in Z/4Z, but has in R/4Z. To solve the above (mod 4) equation, we introduce an integer variable n such that Eq. (C21) becomes an equation in R without modulo 4: Now the equation holds in R and the unknowns are x ∈ R p and n ∈ Z p . The basic idea to solve Eq. (C22) is Gaussian elimination. Since x takes value in R, we can use usual Gaussian elimination (column reduction) in field R. However, for n takes value in Z, the row reduction of Gaussian elimination can be only performed in ring Z (the invertible scalars are only ±1). Performing Gaussian elimination in Z to solve where U is a invertible matrix in R, and I r is the identity matrix with size r × r. The equation becomes The solution of the new equation Eq. (C28) can be also used to construct a unique solution of Eq. (C24), for U is invertible in R (recall that x has elements in R).
Eq. (C28) has solution (x , n ) (x ∈ R p , n ∈ Z p ) if and only if the last p − r elements of b are all 0 (mod 4), i.e., b p−r ∈ (4Z) p−r , where we have used notation b = b r b q−r . If it is true, one can construct all the solutions of x in Eq. (C28) and use them to obtain all x from Eq. (C29).
In summary, to know whether Eq. (C21) has solution (x, n) (x ∈ R p , n ∈ Z p ) or not, we need to find the Hermite normal H of A (U A = H). Denote the number of nonzero rows of H by r. Then the next step is to check whether the last p − r elements of U b are all 0 (mod 4) or not. In practice, calculating the unimodular matrix U is very time consuming. We only need to find out the Hermite normal form of the augmented matrix (A, b). The original equation (C21) has solution if and only if the element in the (r + 1)-th row and (q + 1)-th column of the Hermite normal form is 0 (mod 4).
We used SageMath [74] to calculate the Hermite normal form of the augmented matrix (A, b) with size p × (q + 1) = 2 20 ×(2 12 +1). The rank of A is found to be r = 4035, and the matrix element at the place (r+1, q+1) = (4036, 4097) of the Hermite normal form is 4 (see Fig. 33). Therefore, the highly overdetermined Eq. (C20) indeed has solutions! We can use the solutions to construct the group structure of the general group super-cohomology solutions by Eq. (C17).
We further checked that the first nonzero element in each row of the Hermite normal form of A is either 1 or 2. So the solution x can be chosen to be in 1 2 Z. And the phase factor e iπθ in ν tot 4 takes values in Z 8 ∼ = {e inπ/4 |n ∈ Z}. This implies that the cohomology operation defined by the obstruction is in fact in Z 8 rather than Z 4 . This is a very interesting hierarchy structure: the obstruction in the special group super-cohomology has a Z 2 phase factor which is the fermion sign, the Majorana fermion decoration gives a Z 4 phase factor obstruction from Kitaev chain (p-wave superconductor), and the Abelian group structure of the general group super-cohomology theory takes value in Z 8 .