Unoriented 3d TFTs

This paper generalizes two facts about oriented 3d TFTs to the unoriented case. On one hand, it is known that oriented 3d TFTs having a topological boundary condition admit a state-sum construction known as the Turaev-Viro construction. This is related to the string-net construction of fermionic phases of matter. We show how Turaev-Viro construction can be generalized to unoriented 3d TFTs. On the other hand, it is known that the “fermionic” versions of oriented TFTs, known as Spin-TFTs, can be constructed in terms of “shadow” TFTs which are ordinary oriented TFTs with an anomalous ℤ2 1-form symmetry. We generalize this correspondence to Pin+-TFTs by showing that they can be constructed in terms of ordinary unoriented TFTs with anomalous ℤ2 1-form symmetry having a mixed anomaly with time-reversal symmetry. The corresponding Pin+-TFT does not have any anomaly for time-reversal symmetry however and hence it can be unambiguously defined on a non-orientable manifold. In case a Pin+-TFT admits a topological boundary condition, one can combine the above two statements to obtain a Turaev-Viro-like construction of Pin+-TFTs. As an application of these ideas, we construct a large class of Pin+-SPT phases.


Introduction
It is an age-old problem to provide a complete definition of quantum field theories. A part of the problem is to understand on what kinds of manifolds can we put a quantum field theory. For instance, we can ask the following question: given a theory that can be defined on orientable manifolds, what sort of extra data do we need in order to extend the definition of the theory to non-orientable manifolds? First of all, such an extension may not be possible. For instance, if the theory has a framing anomaly, then it will not be well-defined on non-orientable manifolds. This was recently explained in a footnote of [1]. Second, if such an extension is possible, then it need not be unique. That is, there can be different unoriented theories which reduce to the same oriented theory on orientable manifolds. We will see plenty of examples like this in this paper. On a non-orientable manifold, we can choose a consistent orientation everywhere if we remove a locus homologous to the Poincare dual of first Stiefel-Whitney class w 1 . The

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induced local orientation flips as we cross this locus. In order to be able to define an unoriented theory in terms of the data of the oriented theory, we need the existence of orientation reversing codimension one defects which we place along this locus. These orientation reversing defects implement orientation reversing symmetries akin to the orientation preserving codimension one defects which implement a global symmetry transformation [2]. These defects can be placed on top of each other forming the structure of a group G with a homomorphism ρ : G → Z 2 whose kernel G 0 is the global symmetry group of the theory. The set G 1 = G − G 0 parametrizes the orientation reversing symmetries.
In this paper, we explore the consequences of the existence of such orientation reversing defects in the context of 3d TFTs which admit a topological boundary condition. We restrict ourselves to the case in which the structure group of the TFT can be decomposed as O(3)×G for a finite global symmetry group G. In such cases, the properties of orientation reversing defects allow us to propose a generalization of Turaev-Viro state-sum construction of 3d TFTs [3,4] to the unoriented case. 1 We check that this proposal indeed defines a 3d unoriented TFT. From now on, whenever we say "unoriented TFT", we mean this particular structure group.
For an oriented 3d TFT T with global symmetry G, the input data for the construction is a G-graded spherical fusion category C. We will find that an unoriented 3d TFT T extending T is constructed in terms of a G-graded "twisted" spherical fusion category C where C is embedded as a subcategory of C. In terms of the data of C, we give a prescription to construct the partition function of T on any (possibly non-orientable) 3-manifold.
We also apply these ideas to 3d Pin + -TFTs (i.e. TFTs with structure group Pin + (3) × G) which are a generalization of 3d Spin-TFTs. Spin-TFTs are "fermionic" analogs of ordinary oriented TFTs as they are sensitive to the spin structure of the underlying orientable manifold. To define fermions on an unorientable manifold, we need to choose either a Pin +structure or a Pin − -structure on the manifold. Correspondingly, the natural unoriented generalizations of Spin-TFTs are Pin + -TFTs and Pin − -TFTs.
In [5], a recipe was given to construct a 3d Spin-TFT from an ordinary 3d TFT with an anomalous Z 2 1-form symmetry. This ordinary TFT T f was called the shadow of the corresponding Spin-TFT T s . The idea was to use a kernel TFT T k to connect the shadow theories with their spin counterparts. T k is a Spin-TFT with an anomalous Z 2 1-form symmetry. The diagonal Z 2 1-form symmetry in the product theory T f × T k is non-anomalous. This non-anomalous 1-form symmetry is then gauged to obtain the spin TFT T s . We extend their recipe by constructing shadows for Pin + -TFTs. The Pin + -shadows correspond to theories with anomalous Z 2 1-form symmetry and a certain time-reversal 1 Historically, the original construction due to Turaev and Viro was based on a modular tensor category treated as a spherical fusion category. This construction produced theories which could be defined on an unoriented manifold. This construction was generalized to arbitrary spherical fusion categories but such theories could only be defined on oriented manifolds. It is this latter construction that we call "oriented Turaev-Viro construction" in this paper. This paper presents a further generalization of this setup which we call "unoriented Turaev-Viro construction". Our construction can be used to construct any unoriented 3d TFT with a topological boundary condition.

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anomaly in the presence of a background 2-connection for the Z 2 1-form symmetry. The Pin + -kernel TFT T + k has a corresponding time-reversal anomaly which cancels the anomaly of the shadow. Hence, the resulting Pin + -TFTs are time-reversal invariant and can be put on non-orientable manifolds without any ambiguity [6].
As an application, we construct a large class of Pin + -SPT phases with global symmetry G. SPT phases are TFTs which are invertible under the product operation on TFTs. In the condensed matter literature, these are referred to as fermionic SPT phases protected by G × Z T 2 with T 2 = (−1) F . In the case when G is trivial, cobordism hypothesis predicts two Pin + -SPT phases forming a Z 2 group structure [7]. Our construction reproduces both of these SPT phases along with the Z 2 structure. This paper is organized as follows. In section 2, we propose a Turaev-Viro construction for unoriented 3d TFTs. In section 3, we provide a construction of Pin + -TFTs in terms of ordinary unoriented TFTs with a Z 2 1-form symmetry which is anomalous and has a mixed anomaly with time-reversal symmetry. In section 4, we construct a large class of Pin + -SPT phases with global symmetry G and reproduce the Z 2 group of Pin + -SPT phases in the case of trivial G. In section 5, we present our conclusions and comment on future directions which include a strategy to classify all Pin + -SPT phases with global symmetry G.

Turaev-Viro construction
For an exhaustive review and physical understanding of the Turaev-Viro state-sum construction of oriented 3d TFTs, the reader is referred to [5]. In this section, we first review relevant aspects of this construction. Then, we propose a generalization of the construction to the unoriented case. We also provide a physical understanding of our proposal in terms of orientation reversing defects. We close the section by discussing invertible unoriented TFTs with global symmetry G, or in other words bosonic SPT phases protected by G×Z T 2 . A reader only interested in the Turaev-Viro construction of unoriented 3d TFTs is referred to subsections 2.3 and 2.4.

Boundary line defects and spherical fusion category
In general, a boundary condition B allows one to define a TFT T on a manifold M with boundary B by placing B on the boundary. The boundary condition is called topological if topological deformations of M (including topological deformations of B) leave the partition function of T on M invariant.
Turaev-Viro procedure constructs an oriented 3d (unitary) TFT T from the knowledge of a topological boundary condition B of T [8]. T can be recovered from any one of its topological boundary conditions. For simplicity, we will assume that T has a onedimensional Hilbert space on S 2 . The Turaev-Viro construction for such a TFT T is phrased in terms of a (unitary) spherical fusion category C.
The objects of C are line defects living on B. Such line defects are specified by a label L and an orientation along the line corresponding to L. If a line defect with a certain orientation is denoted as an object L in C, the same line defect with opposite choice of orientation is denoted as the dual object L * . See The dual line L * is defined by reversing the orientation of L. Figure 2. A morphism m between outgoing lines A 1 , A 2 and A 3 corresponds to a state m in the Hilbert space on a disk with boundary punctures A 1 , A 2 and A 3 . Consider on a hemisphere geometry with a boundary on the spherical part and the disk shown in (b) being the cross-section. The state shown in (b) is produced on the cross-section if the boundary has the graph shown on (a) inserted on it such that A i end on their respective punctures.
The morphisms m AB from A to B in C are local operators living between two boundary lines. Thus, m AB form a vector space. This vector space can also be identified with the Hilbert space of states on the disk with boundary punctures corresponding to A * and B. Similarly, the local operators living at the junction of multiple outgoing lines A i is the space of states on disk with boundary punctures corresponding to A i . The space of states can be generated by placing a hemispherical cap on which the lines A i emanate from a point on the boundary of the cap and go to their respective punctures. See figure 2.
The composition of morphisms corresponds to fusion of local operators along the line. There is a tensor product corresponding to fusion of lines as they are brought together. There are also canonical associator, evaluation and coevaluation maps which physically correspond to placing the lines in a certain fashion and fusing them. See figures 3 and 4. Using these canonical morphisms, we can assign a morphism m from ⊗ i A i to ⊗ j B j to any planar graph Γ of boundary line defects (with local operators at their junctions) such that Γ has incoming lines A i and outgoing lines B j . The canonical morphisms satisfy certain Figure 3. (a) Composition of morphisms. The box is our alternative notation for a morphism.
(b,c) Tensor product of objects and morphisms. identities which guarantee that a topologically equivalent graph Γ evaluates to the same morphism m. Consider vacua i of B which can be characterized by the expectation value of a line L i . Such lines are called simple lines. Morphism space from L i to L j is empty for i = j and is one-dimensional for i = j. The space of local operators living on L i can be identified as C because there is a canonical identity operator living on L i . Every line L can be written as a sum of simple lines L = ⊕n i L i where n i denotes the multiplicity of the simple line L i in the sum. The identity line 1 can be treated as a special simple line which can be inserted anywhere without changing any answers. The duals of simple lines are simple as well.
Turaev-Viro construction uses C as an input and produces the partition of T on any oriented manifold M as the output. We will describe the construction in a very hands-on fashion in the next subsection. We will see that the basic object in the construction is a graph Γ in C drawn on the sphere. Γ can be projected down to a closed graph Γ p drawn on the plane. Γ p constructs a morphism from identity line to itself which evaluates to a definite number. This number is the partition function Z Γ of T on a 3-ball along with a network of boundary lines (and local operators at their junctions) Γ inserted at the boundary 2-  sphere. The word "spherical" in spherical fusion category corresponds to certain axioms which guarantee that different projections to the plane evaluate to the same number.
This construction can be easily generalized to TFTs with a global symmetry group G. The symmetry manifests itself in the existence of codimension one topological defects U g labeled by g ∈ G. Going across the locus of U g implements a symmetry transformation on the system by g. These defects fuse according to the group law and can end on B giving rise to new lines at the junction. Thus the category of boundary lines living on B becomes graded by G, i.e. C = ⊕ g C g .

Oriented Turaev-Viro
Let's look at the decomposition of the tensor product of two simple lines L i ⊗ L j = ⊕n k ij L k . This means that there is a n k ij dimensional space of morphisms from L k to L i ⊗ L j . We pick a basis of this space labeled by α. Similarly we pick a dual basis for the space of morphisms from L i ⊗ L j to L k which we also label by α.  the basis can be written graphically as in figure 6(b). 2 We can transform to a basis labeled by α . We denote the unitary matrix corresponding to the transformation as (U ij k ) α α . See figure 7.
The associator induces an isomorphism between the morphism space from L l to (L i ⊗ L j ) ⊗ L k and the morphism space from L l to L i ⊗ (L j ⊗ L k ). In terms of our chosen basis, this isomorphism can be captured in terms of F -symbols (F ijk l ) (p,α,β)(q,γ,δ) which are defined in figure 8. Under a change of basis, F -symbols transform as We are now ready to describe Turaev-Viro prescription for the partition function of   to a morphism as shown in the figure 9(b). Thus we have a labeling of edges and faces of a branched triangulation. Call one such labeling asl.
Pick a tetrahedron t inl. To each t we assign a planar graph Γ t in C and we call such a graph as a tetrahedron graph. Notice that t can have two chiralities -positive and negative JHEP05(2017)048 as shown in figure 10. Γ t for a positive chirality t and a negative chirality t are shown in figure 11. The first one evaluates to (F ijk l ) (p,α,β)(q,γ,δ) and the second one evaluates to (F ijk l ) * (p,α,β)(q,γ,δ) . Let's call this number as n t (l) and define N (l) = t n t (l). To each edge e of T , we can associate a number d e (l) which is the quantum dimension of the simple line assigned to e inl. Define d(l) = e d e (l). The partition function Z(M ) is then given by The invariance of Z(M ) under Pachner moves is guaranteed by the pentagon equation satisfied by the associators in C. The pentagon equation says that the following morphism made by composing associators and the following morphism made by composing associators are equal. In terms of F -symbols, this means that One can check that (2.5) is invariant under an arbitrary gauge transformation (2.1).

Twisted spherical fusion category and orientation reversing defects
Consider an oriented theory defined by C. We propose that an unoriented parent theory can be constructed in terms of a larger "twisted" spherical fusion categoryC. This larger category is assembled from four piecesC =C 0,0 ⊕C 0,1 ⊕C 1,0 ⊕C 1,1 such that each of the subcategoriesC , is G-graded.C 0,1 is a bimodule on whichC 0,0 acts from left andC 1,1 acts from right. Similarly,C 1,0 is a (non-empty) bimodule withC 1,1 acting from the left and C 0,0 acting from the right. An object inC 0,1 fuses with an object inC 1,0 to give an object inC 0,0 . Similarly, an object inC 1,0 fuses with an object inC 0,1 to give an object inC 1,1 .C can also be thought of as a 2-category made out of two objects '0' and '1'. C 0,0 is the same as the spherical fusion category C.C 1,1 has same objects as that of C. Similarly,C 1,0 is a copy ofC 0,1 at the level of objects. Graphically, we describe simple objects ofC , as lines with the left plaquette labeled by and right plaquette labeled by . In general, we draw graphs Γ inC in which we label the each plaquette by some . See figure 12.
In our notation, the labels i, j, k etc. tell us that we have a line ofC , with a specific value of + but do not determine the individual , . The data of individual is Figure 13. The action of anti-linear isomorphism I.
captured in the labeling of plaquettes by , , etc. Thus the labeling of plaquettes is slightly redundant. We need only specify the label of a single plaquette and the labels for the other plaquettes can be determined from the labels i, j, k etc. In what follows, we will often just specify the label of the left-most plaquette.
C comes equipped with the data of an anti-linear isomorphism I between various morphism spaces. This map is easy to describe in terms of simple objects. It takes the morphism space (V ij k ) from L k to L i ⊗ L j with some labeling of plaquettes (such that the left-most plaquette is ) to the morphism space (V ij k ) +1 from L k to L i ⊗ L j but with the labeling on plaquettes flipped. Thus, we just have to pick a basis α of morphism spaces (V ij k ) 0 . The basis (V ij k ) 1 is determined by the action of I and we label the resulting basis by the same labels α. See figure 13. Thus, under a change of basis (U ij k ) α α of (V ij k ) 0 , the basis of (V ij k ) 1 transforms by (U ij k ) * α α . The associators are compatible with I. Let's denote F -symbol associated to a graph having the left-most plaquette 0 as (F ijk l ) (p,α,β)(q,γ,δ) as shown in figure 14. Then, compatibility of associator and I implies that the F -symbol associated to the same graph but with the labels of all plaquettes flipped is (F ijk l ) * (p,α,β)(q,γ,δ) . Thus, the pentagon equation for the associator becomes where s(i) = * if i labels a simple object ofC 0,1 orC 1,0 and s(i) = 1 otherwise. As the equation in terms of F -symbols looks different from (2.5), we refer to this equation as the twisted pentagon equation even though it still descends from the pentagon equation on the associators. This equation also appeared in [9]. Notice that the gauge transformations on F -symbols now take the following form and one can check that (2.6) is invariant under this gauge transformation. The identity line of C =C 0,0 can be inserted anywhere in plaquettes labeled by 0 without changing any answers. Similarly, the identity line ofC 1,1 can be inserted anywhere in plaquettes labeled by 1 without changing any answers. For each object inC 0,1 , we define a dual object inC 1,0 and vice-versa. These duals are slightly different from the usual duals in a spherical fusion category C. That is, given a line L inC 0,1 , the evaluation maps take L ⊗ L * to identity inC 0,0 and take L * ⊗ L to identity inC 1,1 . Similar statements hold true if we flip 0 and 1 or replace evaluation with co-evaluation.
Given a graph Γ inC drawn on the sphere, different projections to planar graphs Γ p must be equivalent. In other words, we demand that there are conditions onC similar to that of a spherical structure on a spherical fusion category C.
We now turn our attention to the physical interpretation of the structure we have just described. An unoriented 3d TFTT has an orientation reversing defect U R implementing a reflection transformation. This defect can fuse with other orientation preserving defects U g to form more orientation reversing defects U Rg . The fusion of these defects froms a group G = G × Z 2 and there is a canonical homomorphism ρ 1 fromG to Z 2 whose kernel is G.
U R can be fused with the boundary B to give a new boundary B . Under such a fusion, the orientation of the boundary flips as well. There is a spherical fusion category Figure 15. The action of U R sends a graph (a) on B to a graph (b) on B but in the "wrong" orientation. This means that the tensor product in graph (b) is taken from right to left. In order to get back to our convention of tensor product from left to right, we take a mirror of graph (b) and obtain graph (c). The resulting graph (c) is read in C .  Figure 16. The symmetry of the theory under a reflection guarantees that the above two graphs evaluate to the same number.
C associated to B which is identified asC 1,1 . If there is a line L on B, then fusion of B with U R flips its orientation and we obtain the line L * on B . Consider a morphism from L k to L i ⊗ L j on B. Slapping U R on top of it, we send each line to its dual and B to B . However, since this process flips the orientation of the boundary, we have to take a mirror of the resulting configuration of lines to read it in terms of C . See figure 15.
This is the origin of the anti-linear isomorphism I inC described above.
U Rg can end on the boundary giving an interface between B and B and an interface between B and B. The lines living on these interfaces give rise toC 0,1 andC 1,0 respectively. Together they form a "twisted" spherical fusion categoryC described above.
The label 0 and 1 of plaquettes in a graph inC corresponds respectively to the boundary B and B in the physical setting. A graph Γ inC drawn on a sphere computes the partition function ofT on a 3-ball with a network of boundary lines specified by Γ. The bulk of the JHEP05(2017)048 3-ball contains orientation reversing defects which end on the boundary at the location of lines living inC 0,1 orC 1,0 .
Given such a 3-ball with Γ on the boundary, we can bubble a U R in the bulk of the 3-ball and bring it to the boundary. This sends Γ to Γ (after taking the mirror) and both of these graphs must evaluate to the same number. This is the origin of the compatibility between associators and I. See figure 16.

Unoriented Turaev-Viro
In this subsection, we generalize the Turaev-Viro prescription to compute the partition function of an unoriented theoryT on an unoriented 3-manifold M . We will assume that the reader has read subsection 2.2 before reading this subsection and so we will sometimes cut corners in what follows.
Fix an orientation O on R 3 . An unoriented 3-manifold M can be constructed as follows. We pick open sets of R 3 and glue them along codimension one loci using piecewise-linear maps. This gives us a locus L in M which is defined by the property that the transition functions are orientation reversing. The Poincare dual of this locus is a representative of first Stiefel-Whitney class and we call it w 1 . We assign a local orientation O t to any small tetrahedron t in M − L by first using the local chart to map it to a tetrahedron in R 3 where we have already picked an orientation Pick a branched triangulation T of M . w 1 assigns a number p e valued in {0, 1} to every edge e. And the G-connection α 1 on M assigns an element g e of the group G to each directed edge e.
Let's extract a set of labels S 0,0 such that each label in the set corresponds to a simple object ofC 0,0 . Similarly, extract a set of labels S 0,1 such that each label in the set corresponds to a simple object ofC 0,1 . Define an involution * on S 0,0 induced by taking the dual of simple objects. Similarly, define an involution * on S 0,1 under which i is sent to j if the * operation ofC sends the object L i inC 0,1 to the object L j inC 1,0 . There is also a G-grading on both of these sets descending from the G-grading of simple objects.
We now label each directed edge e by a label in (S 0,pe ) ge . Pick a face f of T . We can label f by some label α just as in the oriented case. Thus we obtain a labeling of edges and faces of a branched triangulation. Call one such labeling asl. Now pick a tetrahedron t in M − L in the labelingl. To each such t we will assign a planar graph Γ t inC. If the chirality of t matches the local orientation O t , we assign the graph shown in figure 17(a) with = = = = 0 and if the chirality doesn't match the local orientation we assign the graph shown in figure 17(b) with = = = = 0. To define Γ t for a t intersecting L, we choose a small neighborhood U t of t such that L looks like a wall cutting U t into two parts. On one side of the wall, we assign 0 to every vertex and on the other side we assign 1. We assign a global orientation O Ut to U t given by local orientation O t of any tetrahedron t lying completely on one side of the wall where vertices are labeled by 0. We now assign the graph shown in figure Figure 17. The possible graphs that we attach to a tetrahedron.
Notice that if we flip the choice of 0 and 1 that we assigned to the sides of the wall and apply the above presciption, then Γ t is flipped to the "reflected" graph Γ t which is the graph obtained by acting U R on Γ t . See figure 16. Γ t and Γ t evaluate to the same number.
Also notice that if we take a tetrahedron t in U t whose vertices are all labeled by 1 and assign to it a new graph Γ t by matching its chirality with O Ut instead, then Γ t will be the "reflected" version of the old graph Γ t that we assigned in the starting of last paragraph by matching its chirality with the local orientation, and hence Γ t will evaluate to the same number as Γ t . Thus, we see that we could have also given the prescription to compute the partition function in various patches U i by using the local orientations and assigning {0, 1} to the two sides produced by an intersection of U i with the w 1 wall. The tetrahedra lying the intersection of U i and U j would give the same contribution in each patch. Thus, we would just have to make sure that we "glue" the tetrahedra in various intersections properly.
Returning back to our original prescription, we just repeat what we already said for the oriented case. Let's call the evaluation of Γ t as n t (l) and define N (l) = t n t (l). To each edge e of T , we can associate a number d e (l) which is the quantum dimension of the simple line assigned to e inl. Define d(l) = e d e (l). The partition function Z(M ) is then given by is what we dub as the total quantum dimension ofC (where d i is the quantum dimension of simple line L i ) and v is the number of vertices in T . We would like to emphasize that we are picking labels i only in "half" ofC i.e.C 0,0 andC 0,1 . Hence, the total quantum dimension only involves square of quantum dimensions of half of the simple lines.
The invariance of Z(M ) under Pachner moves and under change of representative of w 1 is guaranteed by the twisted pentagon equation (2.6) satisfied by the F -symbols inC.
In the rest of the paper, by "twisted spherical fusion category" we will mean the data JHEP05(2017)048 ofC 0,0 ⊕C 0,1 and we will often repackage this data as C = ⊕gC g = ⊕ g C g ⊕ g C Rg wherẽ g ∈G = G × Z 2 and R is the generator of Z 2 inG. C g = (C 0,0 ) g and C Rg = (C 0,1 ) g . We also define a homomorphism ρ (also called ρ 1 ) fromG to Z 2 which sends G × {e} to 0 and G × {R} to 1.

Example: bosonic SPT phases
Bosonic SPT phases protected by G = G 0 × Z T 2 are invertible unoriented TFTs with global symmetry G 0 . Such a phase is constructed by a twisted spherical fusion category C having a single simple object L g in each subcategory C g . The fusion rules are L g ⊗ L g L gg . F-matrices define a U(1) valued function of three group elements α 3 (g, g , g ) = F g,g ,g ;gg g . The twisted pentagon equation (2.6) translates to α s(g) 3 (g , g , g )α 3 (g, g g , g )α 3 (g, g , g ) α 3 (gg , g , g )α 3 (g, g , g g ) = 1 (2.9) where s(g) = (−1) ρ(g) . This means that α 3 is a ρ-twisted group cocycle. On the other hand, gauge transformations (2.7) become which corresponds to adding an exact ρ-twisted cocycle δβ 2 to α 3 . This means that the bosonic SPT phases are classified by the ρ-twisted group cohomology H 3 (BG, U(1) ρ ) [10]. A background connection α 1 for G 0 on M combines with w 1 to give a background connection for G which is represented as a map from M to BG. An element of H 3 (BG, U(1) ρ ) is then pulled back to a density on M which can be integerated on M to produce the partition function Z(M, α 1 ).

Pin + -TFTs
We start this section by reviewing the construction of Spin-TFTs from their shadows [5]. We will argue that the Pin + -shadows must have an additional kind of anomaly which was not present in the case of Spin-shadows. Incorporating this addtional anomaly will allow us to generalize the shadow construction to Pin + -TFTs. We finish the section by showing how to take a product of Pin + -TFTs at the level of shadows.

Review of spin case
Ref. [5] provided a recipe to construct a 3d Spin-TFT T s from its shadow T f . The shadow is an ordinary TFT with an anomalous Z 2 1-form symmetry. This manifests itself in the existence of a bulk line Π which fuses with itself to the identity and has certain properties. See figure 18.
We want to couple T f to a background 2-connection β 2 for the 1-form symmetry. We can do so by inserting Π lines inside a triangulated manifold such that an even number of Π lines cross a face having β 2 = 0 and an odd number of Π lines cross a face having JHEP05(2017)048 Figure 18. β 2 = 1. Since Π has a non-trivial crossing with itself, topologically different ways of gluing Π lines inside the tetrahedron will differ by signs. Hence, we need to pick a convention of how we will glue the Π lines crossing these faces inside each tetrahedron when we say that T f is coupled to a background 2-connection β 2 . Once we have picked this convention, the partition function will not be invariant under gauge transformations of β 2 . After fixing the convention, the change in the partition function under gauge transformations is independent of the theory, however. To see this, consider the product T = T 1 ×T 2 of two shadow theories T 1 and T 2 , and couple it to a background 2-connection for the diagonal Z 2 1-form symmetry. The partition function would then be the product and a gauge transformation would leave Z invariant. This is because resolving a crossing of the product line Π 1 Π 2 gives no minus sign as the signs from crossing of Π 1 and crossing of Π 2 cancel each other. The strategy of [5] was to compute this anomaly for a simple shadow theory, namely the shadow of Gu-Wen fermionic SPT phases. The anomaly under β 2 → β 2 + δλ 1 turns out to be This transformation is the same as the transformation of a spin-structure η 1 dependent sign z(M, η 1 , β 2 ). This sign can be written as [11] z(M, η 1 , where N is a 4-manifold whose boundary is M and w 2 is a representative of second Stiefel-Whitney class. For oriented manifolds, this sign is independent of N because β 2 ∪β 2 +w 2 ∪β 2 is exact if β 2 is a cocycle. It is easy to see from this expression that  Here we have used a representation of spin structure as an equivalence class of 1-cochains η 1 satisfying δη 1 = w 2 under the equivalence relation given by addition of exact 1-cochains to η 1 [11].
Thus combining the shadow theory with this sign gives a theory with a non-anomalous Z 2 1-form symmetry. The spin theory T s is obtained from this by gauging this 1-form symmetry We would like to have a Turaev-Viro construction for Z f (M, β 2 ). To this end, we should understand how to encode the Π line in terms of the spherical fusion category C.

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Notice that Π is mapped to a boundary line P by bringing it to the boundary. If we bring Π to the boundary such that it crosses a boundary line X, we obtain a canonical isomorphism β X : X ⊗ P → P ⊗ X. Bringing Π to the boundary in topologically equivalent ways should lead to same answers. Hence, (P, β) can be moved across other morphisms. See figure 19.
Mathematically, this means that Π is an element (P, β) of Drinfeld center of C. This element fuses with itself to identity and β P = −1. The Turaev-Viro construction for Z f (M, β 2 ) is achieved by inserting a Π line emanating from every vertex whose dual face has β 2 = 1. See figure 20.

Fermion in Pin + -theories
Pin + -TFTs are a generalization of Spin-TFTs to the unoriented case. Spinors can be defined on an n-dimensional non-orientable manifold by using transition functions valued in Pin + (n) group or Pin − (n) group, both of which are double covers of O(n). They are distinguished by the value of R 2 acting on spinors where R is a spatial reflection. R 2 = +1 for Pin + -group and R 2 = −1 for Pin − -group. In terms of time reversal symmetry T , the action on spinors is T 2 = −1 for the Pin + case and T 2 = +1 for the Pin − case. In the Pin + case, there is a choice in defining the action of reflection in i-th spatial direction on spinors. We can either multiply the spinor by the gamma matrix γ i or by −γ i . This suggests that in a Pin + -shadow there are two canonical choices m R and n R = −m R of local operators at the junction of a Π line and R-defect. These operators square to 1. The orientation preserving defects always have a single canonical local operator at the junction. Now we will argue that, in general, there must be a locus L embedded inside the locus M of orientation reversing defects which implements the transformation m ↔ n. Moreover, the homology class of L must be the Poincare dual of [w 2 1 ]. Choose a locus L embedded inside M whose homology class is the dual of [w 2 1 ]. Now bubble a fermion line near M and move it such that it intersects M in two junctions. See figure 21(a). The local opeartors at the two junctions must be inverses of each other. Take one of these junctions around a cycle C in M. If the cycle intersects L , then fusing the fermion line with itself at the end of this process gives a crossing of fermion line which provides a factor of −1. See figure 21(b). Topological invariance demands that C must intersect L as well so that the fusion of the local operators at the end of the process provides a factor of −1 which cancels the sign from the crossing. Similarly, if C doesn't intersect L, it doesn't intersect L either. Hence, L and L are in the same homology class. Thus, we can choose to identify L with the representative w 2 1 . We will see in the next section that this flip m ↔ n as Π crosses L is responsible for the presence of mixed anomaly between time reversal symmetry and Z 2 1-form symmetry in Pin + -shadows.

Shadows of Pin + -TFTs
Just as in the spin case, to define what we mean by a Pin + -shadow T f coupled to a background β 2 , we need to pick a convention for configuring Π lines. In addition to this, JHEP05(2017)048 we also need to choose whether we will put m or n on the junctions when Π crosses R-defect. The Pin + -TFT T + is obtained as where the sign which cancels the anomaly for Z 2 1-form symmetry of T f can be defined as where ∂N = M and η 1 parametrizes Pin + -structures. The expression is independent of N as β 2 ∪ β 2 + (w 2 1 + w 2 ) ∪ β 2 is exact if β 2 is a cocycle. Fliping the choice of local operator changes the partition function as Z f (M, β 2 ) → (−1) M w 1 ∪β 2 Z f (M, β 2 ). This can be absorbed into a permutation of Pin + -structures η 1 → η 1 + w 1 . Thus, as in the spin case, we have a one-to-one correspondence [T + ] ↔ (T f , Π) where [T + ] denotes the equivalence class of Pin + -TFTs under permutations of Pin + -structures. Now, notice that Pin + -shadows have a time reversal anomaly in the presence of a background 2-connection β 2 . As we add δv 0 to w 1 , we add δu 1 to w 2 This corresponds to moving M and L . But during such movements, L will cross some Π lines encoding β 2 and the partition function will change as Under this transformation, the sign also transforms in the same way The signs z + written above implies the following anomaly under β 2 → β 2 + δλ 1 where w 1 is a representative of first Stiefel-Whitney class. As the anomaly is universal, we will verify that this is the correct anomaly by computing the anomaly directly for shadows of Pin + generalization of Gu-Wen fermionic SPT phases in the next section.
To obtain the Turaev-Viro construction for Z f (M, β 2 ), we need to know how to encode the Π line in terms of the data of C. As in the spin case, Π is mapped to some boundary line P with canonical isomorphisms β X : X ⊗ P → P ⊗ X. However, unlike the spin case, Π is not an element of Drinfeld center of C. Rather, we need to insert extra signs whenever we move Π across L . This descends to the statement that (P, β) is an element of a twisted Drinfeld center which is defined in figure 22.

Product of Pin + -TFTs
In this subsection, we want to figure out the shadow of the product of two Pin + -TFTs. This will lead to the definition of a product on the shadow theories which we will call the shadow product.
First, notice that 3 Now consider two Pin + -TFTs T + and T + with their corresponding shadows T f and T f . Using the above, we can write the partition function of the product theory as (3.12) which can be massaged as Figure 23. The properties of a bulk line b generating a non-anomalous Z 2 1-form symmetry are very similar to that of Π. The only difference is that crossing b lines doesn't lead to a sign. with being the partition function of the shadow corresponding to the product theory. We denote this shadow theory as the shadow product T f × f T f . Physically, we are constructing the shadow of the product by gauging the diagonal Z 2 1-form symmetry in the product of the shadow theories. Notice that this 1-form symmetry is non-anomalous and hence gauging it makes sense.
To implement the shadow product in the Turaev-Viro description, we first take a graded product of C × G C of C and C . This means that (C × G C ) g = C g × C g . Now we need a notion of gauging the line b = ΠΠ in the Drinfeld center of C × G C . In general, we can consider the following problem. Take a theory T b specified by a twisted spherical fusion category C b having a non-anomalous Z 2 1-form symmetry. This means that there exists a line b in the Drinfeld center of C b which fuses with itself to identity and has the properties shown in figure 23. We want to construct the twisted spherical fusion category for the theory T Z 2 obtained after gauging the 1-form symmetry generated by b.
b is invisble in the gauge theory. This means that a morphism from A to b ⊗ B in C b has to be regarded as a morphism from A to B in C Z 2 . And the morphisms from A to B in C b are also morphisms from A to B in C Z 2 . The composition and tensor product of new morphisms are defined as shown in the figure 24.
Let's try to understand what happens to the simple objects under this operation. If L is a simple object in C b , M = b ⊗ L is simple as well. If M is not isomorphic to L, then the morphism from L to b ⊗ M in C b provides an isomorphism from L to M in C Z 2 combining them into a single simple object in C Z 2 . If M is isomorphic to L, then the morphism from L to b ⊗ M in C b provides an additional endomorphism ξ L of L in C Z 2 . Since there are two independent morphisms from L to itself in C Z 2 , it must split into two simple objects L + and L − which can be constructed by inserting a projector on L.

Fermionic SPT phases
In this section, we discuss Pin + -SPT phases. We also explicitly compute the partition function on an arbitrary manifold M of a certain Pin + -shadow which gives rise to the Pin + Gu-Wen phases. We can read the anomaly of Pin + -shadows from the expression for the partition function. The anomaly matches the expectation of the previous section. We finish the section by reproducing Z 2 group of Pin + -SPT phases without any global symmetry.

Gu-Wen phases
In this subsection, we will discuss Pin + Gu-Wen SPT (f-SPT) phases with global symmetry G. Gu-Wen fermionic SPT phases were first described in [12] and explored further in [11].

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(a) Figure 26. Intermediate computational steps relatingα 3 and ν 3 . The sign arises from dragging a Π line over a vertex. We can further resolve the crossing on the right hand side to make contact with ν 3 which is defined in figure 27. The twisted spherical fusion category for these phases is such that C g has two simple objects L g,0 and L g,1 for any g in G × Z R 2 . The fusion rule is L g, ⊗ L g , L gg , + +n 2 (g,g ) (4.1) where n 2 is a Z 2 -valued group cocycle, i.e. it is an element of is also the group of central extensions of the form Thus, we can view C as descending fromĈ which is aĜ-graded category with a single simple object in each grade. One obtains C by forgetting the sub-grading corresponding to the Z 2 subgroup appearing in the above central extension. More physically,Ĉ can be viewed as generalizing the notion of unoriented bosonic SPT phases to bosonic SPT phases with more complicated structure group. Forgetting the Z 2 grading corresponds to gauging the Z 2 symmetry. The associator of elements in C can be read from the associator inĈ JHEP05(2017)048 which we denote asα 3 . It is an element of H 3 (BĜ, U(1) ρ ). As a note, we will denote an arbitrary element of G × Z R 2 by g in what follows. We demand the existence a fermionic line Π in the twisted Drinfeld center of C which fuses with itself to the identity. For the 1-form symmetry generated by this line to be compatible with G, the line must be of the form (L e,0 , β) or (L e,1 , β). The former case cannot lead to a fermionic line. Hence, Π must be of the form (L e,1 , β). The existence of such a line will put some constraints on the form of C which we now explore. First, we choose our basis of morphisms as shown in the figure 25. Consider the basic graph dual to the tetrahedron. Using our basis, it can be written as in figure 26(a). This, in turn, can be manipulated to the final graph shown in figure 27 which we define to be ν 3 (g, g , g ). During this manipulation we obtain a sign from resolving a crossing and another sign from moving a Π line across a vertex. See figure 26(b). Thus, we see that where 1 is a Z 2 -valued co-chain which sends (g, ) to .
We find that a Pin + Gu-Wen phase is specified by a double (ν 3 , n 2 ) where ν 3 satisfies However, there is a redundancy in such a description. We will see in subsection 4.3 that the phase defined by ν 3 = 1 and n 2 = ρ 2 1 is the same as the trivial phase specified by ν 3 = 1 and n 2 = 0.

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Since flipping the sign of all the β in the orientation reversing sector doesn't change the resulting Pin + -TFT, we can choose L e,1 equipped with β(g, ) = (−1) (4.10) as the fermion.
For the rest of this subsection, we note that we can write H 2 (B(G × Z R 2 ), Z 2 ) in terms of group cohomology of G. Let's denote an arbitrary element of G × Z R 2 as g 1 , g 2 etc. We also denote an arbitrary element of G as g and R as the generator of Z R 2 . We have the gauge transofrmations n 2 (g 1 , g 2 ) → n 2 (g 1 , g 2 ) + n 1 (g 1 ) + n 1 (g 1 g 2 ) + n 1 (g 2 ) (4.11) Pick n 1 such that n 1 (g) = 0 for all g and n 1 (R) + n 1 (gR) = n 2 (g, R). Thus we have fixed a gauge such that n 2 (g, R) = 0 for all g. Then using the cocycle condition n 2 (g 2 , g 3 ) + n 2 (g 1 g 2 , g 3 ) + n 2 (g 1 , g 2 g 3 ) + n 2 (g 1 , g 2 ) = 0 (4.12) we find that we can express n 2 as where m 2 parametrizes an element of H 2 (BG, Z 2 ), m 1 parametrizes an element of H 1 (BG, Z 2 ) andm 1,2 denotes the pullback of m 1,2 from G to G × Z R 2 . This analysis establishes that

Anomaly for Pin + -shadows
In this subsection we will compute the partition function Z f (M, β 2 ) for a Pin + Gu-Wen phase. The explicit expression will allow us to compute the anomaly under a gauge transformation β 2 → β 2 +δλ 1 . As the anomaly is universal, this will justify our prescription (3.6) for constructing Pin + -TFTs in terms of their shadows.
In the presence of a background β 2 , the basic tetrahedron graph is as shown in figure 29(a). This can be gauge fixed as shown in figure 28. After the gauge fixing, we can move the Π lines to the position shown in figure 29(b). This implies that the partition function can be written as This expression is non-zero only when the G-connection is such that n 2 = β 2 + δα 1 . By shifting 1 → 1 + α 1 , the above expression can be re-written as The sign inside the sum is exact and hence we obtain Shifting β 2 → β 2 + δλ 1 is the same as shifting α 1 → α 1 + λ 1 under which the partition function changes as which matches the expectation in (3.10) exactly.

Group structure of Gu-Wen phases
Now we would like to compute the product of two Gu-Wen phases labeled by (ν 3 , n 2 ) and (ν 3 , n 2 ). The G-graded product of corresponding categories has 4 simple objects in each grade L g, , which fuse according to the cocycle (n 2 , n 2 ) and have associatorsα 3α 3 . The non-anomalous Z 2 1-form symmetry is generated by L e,1,1 which has crossing (−1) + .

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Gauging the symmetry identifies L g, , with L g, +1, +1 . We pick representative objects L g, ,0 in each grade and compute the associator of L g, ,0 , L g , ,0 and L g , ,0 via the tetrahedron graph. Multiplying two representative objects L g, ,0 and L g , ,0 , we obtain L gg , + +n 2 (g,g ),n 2 (g,g ) which can be mapped back to the representative object L gg , + +n 2 (g,g )+n 2 (g,g ),0 by inserting n 2 (g, g ) number of ΠΠ lines emanating from the corresponding vertex. The representative objects thus fuse according to the cocycle n 2 +n 2 . Now, we gauge fix as in the previous subsection. Then, doing same manipulations as in the previous subsection, we find that the tetrahedron graph evaluates to Upto a gauge redefintion, it can be written as Thus the product is a Gu-Wen phase withν 3 = ν 3 ν 3 (−1) n 2 ∪ 1 n 2 andñ 2 = n 2 + n 2 .
However, notice that substituting ν 3 = 1, n 2 = 0 in (4.17) and writing w 2 1 = δσ 1 gives and substituting ν 3 = 1, n 2 = ρ 2 1 gives which are the same expressions! Thus, the Gu-Wen phase labeled by (ν 3 = 1, n 2 = ρ 2 1 ) is the trivial phase. The reader might complain that (4.21) does not seem to describe a trivial phase. We would like to stress that this is the partition function of the shadow theory describing the trivial Pin + -TFT. The trivial Pin + -TFT is obtained by combining a non-trivial shadow with a non-trivial sign.

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4.4 Z R 2 version of Ising As an application of our formalism, we would like to construct all Pin + -SPT phases wth global symmetry group G being the trivial group {id}. There is only one Gu-Wen phase in this class, which is the trivial phase. There is a non-trivial phase in this class which is given by the Z R 2 analogue of a Z 2 graded spherical fusion category I which is known as the Ising fusion category. Below we recall the construction of I and its Z R 2 cousin. It turns out that the analysis for both the cases is similar and we treat both of them together.
We are looking for a Z 2 graded (twisted) spherical fusion category such that C 0 = {I, P } and C 1 = {S} are the simple objects. The fusion rules are P ⊗ P I The F -symbols can be bootstrapped from these fusion rules by using (twisted) pentagon equation and taking advantage of the gauge freedom. When the Z 2 grading corresponds to a Z 2 global symmetry, the non-trivial F -symbols are determined to be That is, the choice of sign becomes a gauge freedom in the Z R 2 case. The fermion line is given by an element of (twisted) Drinfeld center of the form (P, β). Solving the Drinfeld center equations for the Z 2 case, we obtain β P = −1 (4.37) On the other hand, solving the twisted Drinfeld center equations for the Z R 2 case, we obtain However, as we know from before, flipping the sign of all the β in the orientation reversing sector doesn't change the resulting Pin + -TFT and we can fix β S = +1. Hence, in the Z R 2 case, there are no choices and we obtain only one Pin + -TFT which we call I + .

Pin + -SPT phases with no global symmetry
Cobordism hypothesis predicts a Z 2 group of Pin + -SPT phases [7]. We have already found the trivial phase as a Gu-Wen phase. We claim that the non-trivial phase corresponds to the Pin + -TFT I + that we encountered in last subsection. To justify this, we will show that the square of I + is the trivial Gu-Wen phase. This will prove that I + is indeed an SPT phase and provide an explicit construction of Pin + -SPT phases without global symmetry. The existence of this non-trivial phase was also discussed in [13]. The graded product of I + with itself has simple objects II, P I, IP , P P in the trivial grade and a simple object SS in the non-trivial grade. Gauging the 1-form symmetry generated by ΠΠ, we obtain a category C with C 0 having simple objects II, P I and C 1 having simple objects SS + , SS − . SS + and SS − are constructed by using projectors obtained by using the non-trivial endomorphism of SS. See figure 30.
SS + ⊗P I involves the F -symbol F P SP which flips the sign of ξ S and hence SS + ⊗P I SS − . On the other hand, P I ⊗ SS + involves β P and hence P I ⊗ SS + SS − . The computation of SS + ⊗ SS + can be done in a similar but more involved manner which we explain in figure 31. We find that SS + ⊗ SS + II. All the statements above hold true if we replace SS + with SS − . Thus, C has the fusion rules of the Gu-Wen phase which is trivial.
For a general G, we can consider the pullback of I + along ρ 1 which we denote as I + (G). I + (G) g has two simple elements I g , P g if ρ 1 (g) = 0 and has a single simple object  Figure 31. Computation of SS + ⊗SS + is by definition a sum of four terms which involve associators and crossings. II inside SS ⊗ SS is mapped to II by first and fourth terms and to P P (which is isomorphic to II in the new category) by the second and third terms. Similarly, P I is mapped to the zero object as the four terms cancel in pairs. Hence, SS + ⊗ SS + II.
S g if ρ 1 (g) = 1. The fusion rules and associators are just pulled back from I + . Clearly, I + (G) will also square to 0 as our argument above is independent of G-grading. This allows us to construct GW(G)×Z 2 worth of Pin + -SPT phases with global symmetry G. We suspect that this is not the full classification and comment on how to complete the classification in the next section.

Conclusion and future directions
In this paper we discussed the generalization of Turaev-Viro construction of oriented 3d TFTs to unoriented 3d TFTs. We proposed that the input data of this construction in the unoriented case should be a "twisted" spherical fusion category in which the pentagon equation for the F -symbols is modified.
As a generalization of the construction of [5], we also proposed a construction for Pin + -TFTs in terms of their shadows. The shadows are ordinary unoriented TFTs with a Z 2 1-form symmetry which is anomalous and has a mixed anomaly with time-reversal symmetry.
Combining the above two ingredients, we were able to give explicit constructions of a large class of invertible Pin + -TFTs with global symmetry G. Such theories are known as Pin + -SPT phases. We also reproduced the Z 2 group of Pin + -SPT phases without any global symmetry.
There are plenty of interesting directions in which this work can be extended in the future and we make some very speculative comments about them in what follows. Perhaps the most immediate future direction is to use the machinery developed in this paper to provide a classification of Pin + -SPT phases for an arbitrary group G which admit a topological boundary condition. The author suggests to look at a spherical fusion category graded by Z 2 × Z R 2 with simple elements I, P in the (0, 0) grade, I 1 , P 1 in the (0, 1) grade, S in the (1, 0) grade and S 1 in the (1, 1) grade. The fusion rules mimic the Ising category. Is it possible to find a consistent set of F -symbols? If yes, then the class of Pin + -SPT phases JHEP05(2017)048 we presented in this paper is not the full answer. It should then be possible to finish the classification, in a spirit similar to the one in [5], by pulling back this Z 2 × Z R 2 phase and combining it with the class of phases presented in this paper.
It would be very interesting to provide a construction (Turaev-Viro-like or some other construction) for TFTs with more general struture groups. For instance, one could mix O(n) and G or mix Pin + (n) and G in the fermionic case. It seems that a proper treatment of these generalizations should involve a rich interplay of symmetry defects along with higher codimension defects living in the worldvolume of symmetry defects.
Let us comment about the Pin − (n) × G case. It seems natural that the kernel for Pin − -TFTs would be the sign z − (M, η 1 , β 2 ) = (−1) M η 1 ∪β 2 + N β 2 ∪β 2 +(w 2 1 +w 2 )∪β 2 (5.1) which seems to be the same expression as (3.7) but this time we take η 1 to parametrize Pin − -structures. This would suggest that the corresponding shadow theory has no mixed anomaly between time reversal and Z 2 1-form symmetry. Also, the anomaly for the 1-form symmetry should now be Z f (M, β 2 ) → (−1) M λ 1 ∪β 2 +β 2 ∪λ 1 +λ 1 ∪δλ 1 Z f (M, β 2 ) (5.2) However, for Pin + case, we saw in figure 21 that moving the fermion Π across the locus dual to w 2 1 should change the operator at the junction of Π line and the orientation reversing defects. The argument given there was that this sign was needed to cancel the sign coming from the crossing of Π lines. This lead to different anomalies than the ones we want for the Pin − case. So, in the Pin − case, we do not want such a change in the sign of the corresponding local operator. The author suspects that in this case the sign coming from crossing of Π lines will be canceled by factors coming from patching of Π with RΠ where RΠ is Π line with a reflected framing. This would make sure that Π is an element of Drinfeld center rather than a twisted Drinfeld center, which would in turn imply the anomalies given above. It would be interesting to work out the details and provide a Turaev-Viro construction for Pin − shadows.
Of course, this means that one will have to first understand how to compute (in terms of the twisted spherical fusion category) the extra data attached to a bulk line which corresponds to patching the line with itself but with reflected framing. In other words, this corresponds to a generalization of Moore-Seiberg data [14,15] to the unoriented case. A step towards this was recently taken in [16].
A puzzle here is that there should be no non-trivial Pin − -SPT phase according to [7]. So, somehow the Pin − -TFTs produced by the potential Pin − -shadows having Z R 2 version of Ising as their twisted spherical fusion category should be trivial.
Another interesting direction to pursue would be to see if it is possible to find a generalization of Turaev-Viro construction which could construct anomalous 3d TFTs. Such TFTs live at the boundary of a 4d SPT phase. Hence, such TFTs should not admit topological boundaries of their own but they can admit interfaces to other 3d TFTs with the same anomaly. Perhaps it is possible to choose a simple TFT in each anomaly class JHEP05(2017)048 and build a Turaev-Viro construction using a topological interface between the TFT we want to construct and the simple TFT. See [1,6,17,18] for recent interesting work on anomalous unoriented 3d TFTs.
Finally, it would be interesting to concretely construct a time-reversal invariant commuting projector Hamiltonian using the data of twisted spherical fusion category. This Hamiltonian goes into the string-net construction of fermionic phases of matter. See [5,19] for more details.