Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms

We formulate a family of spin Topological Quantum Field Theories (spin-TQFTs) as fermionic generalization of bosonic Dijkgraaf–Witten TQFTs. They are obtained by gauging G-equivariant invertible spin-TQFTs, or, in physics language, gauging the interacting fermionic Symmetry Protected Topological states (SPTs) with a finite group G symmetry. We use the fact that the torsion part of the classification is given by Pontryagin duals to spin-bordism groups of the classifying space BG. We also consider unoriented analogues, that is G-equivariant invertible \(\hbox {pin}^\pm \)-TQFTs (fermionic time-reversal-SPTs) and their gauging. We compute these groups for various examples of abelian G using Adams spectral sequence and describe all corresponding TQFTs via certain bordism invariants in 3, 4 and other dimensions. This gives explicit formulas for the partition functions of spin-TQFTs on closed manifolds with possible extended operators inserted. The results also provide explicit classification of ’t Hooft anomalies of fermionic QFTs with finite abelian group symmetries in one dimension lower. We construct new anomalous boundary spin-TQFTs (surface fermionic topological orders). We explore SPT and symmetry enriched topologically (SET) ordered states, and crystalline SPTs protected by space-group (e.g. translation \({\mathbb {Z}}\)) or point-group (e.g. reflection, inversion or rotation \(C_m\)) symmetries, via the layer-stacking construction.

1. There are generalizations without theses conditions.

2. In physics literature such spaces are often denoted by a different symbol, e.g. \({\mathcal {H}}(M^{n-1})\), while Z is only used for partition function, that is the value of the functor Z on a closed n-manifold.

3. The morphisms are grading preserving linear maps.

4. Which can be defined as a connected topological space (unique up to homotopy) satisfying \(\pi _1(BG)=1\), \(\pi _{i}(BG)=0,\;i>1\).

5. The bordism data contains a choice of the isomorphism between \(M^{n-1}_2\) and the boundary, however we will usually not indicate it explicitly.

6. In the sense that one can couple the theory to a background G gauge field. There is no condition that G acts faithfully on the operators of the theory.

7. Note that the free part of the classification, conjecturally given by

   $$\begin{aligned} {\text {Hom}}(\Omega _{n+1}^{Spin}(BG),{\mathbb {Z}}), \end{aligned}$$

   (2.15)

   contains Chern–Simons-like terms, which are not strictly topological.

8. Meaning that there are continuous maps from a path connecting points in \(\text {Hom}(\Omega _n^{Spin}(BG),U(1))\) to all values of the TQFT functor.

9. Of course, it is known that Dijkgraaf–Witten theories corresponding to different elements of group cohomology can become equivalent after dynamical gauging (e.g. [63] and References therein). However, here we mean a stronger statement: There are identical TQFTs even before gauging (i.e. SPTs). After gauging, there might be additional identifications corresponding to field redefinitions. In dimension \(n>6\) Dijkgraaf–Witten theories labeled by different elements of \(H^n(BG,U(1))\) can become equivalent even as non-spin TQFTs, cf. [33]. This is because starting from dimension \(n=7\) the map

   $$\begin{aligned} H^n(BG,U(1)) \rightarrow \mathrm {Hom}(\mathrm {Tor}(\Omega ^{SO}_n(BG)),U(1) ) \end{aligned}$$

   is not injective in general. This is because there are examples in degree 7 when a homology class of BG cannot be represented by an image of a smooth manifold continuously mapped to BG. Since SPTs/invertible TQFTs are classified by r.h.s. of the above equation, the TQFTs labeled by the elements of l.h.s. that map to the same element realize equivalent TQFTs (more concretely, the actions constructed via two different classes of \(H^n(BG,U(1))\) will have the same value on any smooth n-manifold).

10. See a recent discussion in Ref. [70] and Ref. [47]’s Section 5.4 along this statement, and References therein. For example, the 2d Arf invariant or equivalently the 1+1D Kitaev fermionic chain [40], obtained from the generator of \({\text {Hom}({\text {Tor}}\,\Omega _2^{Spin}(pt),U(1))}\cong {\mathbb {Z}}_2\), is actually not a short-range entangled fSPTs, but instead a long-range entangled invertible fermionic topological order in 1+1D (in 2d).

11. Physically the value of q(a) corresponds to periodicity condition on spinors along the Poincaré dual 1-cycle: 1 for periodic and 0 for anti-periodic.

12. There is an analogous definition of linking pairing on \(\mathrm {Tor}\,H_m(M^{2m+1},{\mathbb {Z}})\) for any odd dimensional manifold.

13. This follows from the fact that an element \(a\in H^1(M^3,{\mathbb {Z}}_2)\) can be represented by a smooth map \(M^3\rightarrow \mathbb {RP}^N\), for sufficiently large N, so that a is the pullback of the generator \(w_1\) of \(H^*(\mathbb {RP}^N,{\mathbb {Z}}_2)={\mathbb {Z}}_2[w_1]/w_1^{N+1}\) and \(\text {PD}(a)\) is the preimage of \(\mathbb {RP}^{N-1}\subset \mathbb {RP}^{N}\).

14. We use the usual normalization of connection 1-form/curvature such that the first Chern class is \(c_1=F/(2\pi )=dA/(2\pi )\).

15. Alternatively, one can also mathematically define spin-Chern–Simons TQFT via spin-generalization (see e.g. [5, 8, 37]) of Reshetikhin-Turaev [50] construction where the input data is the spin modular tensor category of representations of lattice vertex operator algebra associated to K.

16. Such that \(Z_\text {spin-CS}^K(S^2\times S^1)=1\).

17. Note that in the case of spin structure Thom spectrum is equivalent to Madsen-Tillmann spectrum: \(MSpin \cong MTSpin\).

18. In general \(A_p^\wedge \), for a prime p, denotes a p-completion of the abelian group A, that is the following inverse limit of abelian groups:

    $$\begin{aligned} A_p^\wedge :=\varprojlim A/p^mA \end{aligned}$$

    (4.3)

    taken with respect to quotient homomorphisms \(A/p^{m+1}A\rightarrow A/p^{m}A\). For example: \(({\mathbb {Z}}/p^k{\mathbb {Z}})^\wedge _p={\mathbb {Z}}/p^k{\mathbb {Z}}\), \(({\mathbb {Z}}/q{\mathbb {Z}})^\wedge _p=0\) when q and p are coprime, and \({\mathbb {Z}}_p^\wedge \) is the group of p-adic integers. In general, if A is finitely generated, \(A_p^\wedge =A\otimes _{\mathbb {Z}}{\mathbb {Z}}_p^\wedge \).

19. Note that via Yoneda lemma there is one-to-one correspondence between cohomology operations

    $$\begin{aligned} H^1(\,\cdot \,,G)\longrightarrow H^n(\,\cdot \,,U(1)) \end{aligned}$$

    (4.11)

    and elements of \(H^n(BG,U(1))\).

20. There is natural action

    $$\begin{aligned} {\text {Ext}}^{s,t}_{{\mathcal {A}}(1)}({\mathbb {Z}}_2,{\mathbb {Z}}_2)\otimes {\text {Ext}}^{s',t'}_{{\mathcal {A}}(1)}(M,{\mathbb {Z}}_2) \rightarrow {\text {Ext}}^{s+s',t+t'}_{{\mathcal {A}}(1)}(M,{\mathbb {Z}}_2) \end{aligned}$$

    (4.17)

    realized by.

21. In fact, for the purposes of inducing spin structure from the ambient space, as described below, it is enough to have an immersion.

22. Note that if \(f^*(x)\equiv f\;\;\mathrm {mod}\;2\ne 0\) and a smooth representative \(N^{n-1}\) of the Poincaré dual of f exists, it will be necessarily orientable, because \([\text {PD}(f)]\in H_{n-1}(M^n,{\mathbb {Z}}_4)\) can be obtained by the pushforward of \([N^{n-1}]\) under the embedding. On the other hand, if \(f\;\;\mathrm {mod}\;2=0\) this implies that \(f\in H^1(M^n,{\mathbb {Z}}_4)\) is the image of some \(g'\in H^1(M^n,{\mathbb {Z}}_2)\) under the canonical map induced by the non-trivial homomorphism \({\mathbb {Z}}_2\rightarrow {\mathbb {Z}}_4\). Such homomorphism also induces a homomorphism \(\Omega _n^{Spin}(B{\mathbb {Z}}_4\times B{\mathbb {Z}}_2)\rightarrow \Omega _n^{Spin}(B{\mathbb {Z}}_2\times B{\mathbb {Z}}_2)\) and the corresponding bordism invariant for \(G={\mathbb {Z}}_2\times {\mathbb {Z}}_4\) should reduce (i.e. pulled back) to the one of the invariants in Sect. 4.1.2. A naive argument shows that it should be \(\int _{M_4}g'g^3\).

23. This follows from the fact that an element \(h\in H^1(M^n,{\mathbb {Z}})\) can be represented by a smooth map \(M^n\rightarrow S^1\), so that h is the pullback of the generator of \(H^1(S^1,{\mathbb {Z}})={\mathbb {Z}}\) and \(M^{n-1}\) is the preimage of a regular value.

24. In the cases when \(H=Spin\) and \(H=Spin\times _{{\mathbb {Z}}_2}{\mathbb {Z}}_{2m}\) the stable and unstable structures are equivalent, see [39].

25. Note that the free part of the classification contains Chern–Simons-like theories, which are not strictly topological.

26. Unlike in the \(H=Spin\) case it is not the same as Thom spectrum MH. However, \(MTPin^\pm \cong MPin^\mp \).

27. As \({\text {Hom}}({\mathbb {C}},{\mathbb {C}})\cong {\mathbb {C}}\).

28. As \({\text {Hom}}({\mathbb {C}},Z(Y^k\times S^{n-k-1}))\cong Z(Y^k\times S^{n-k-1})\).

29. The case of the multi-component link is analogous.

30. A possible way to argue it by realising \(\Sigma \) to be a smooth representative of the Poincaré-Lefschetz dual to the generator of \(H^1(S^3\setminus {\mathcal {N}}(L),{\mathbb {Z}})\cong {\mathbb {Z}}\). It exists by the same argument as in footnote 23. Alternatively, there is a known algorithm of constructing \(\Sigma \) from a knot diagram of L.

31. Meridian cycle is a small cycle surrounding the knot and longitude is the cycle given by the push-off \(L'\) towards the framing vector.

32. The identification of the map between one-dimensional complex spaces with \({\mathbb {C}}\) itself requires a choice of basis for each \(Z^\mu (M^2,f)\), that is a linear map \({\mathbb {C}}\rightarrow Z^\mu (M^2,f)\). Due to the monoidal property \(Z^\mu (M_1^2\sqcup M^2_2,f_1\sqcup f_2)={\mathbb {Z}}^\mu (M_1^2,f_1)\otimes Z^\mu (M_2^2,f_2)\) and existence of a canonical bordism between disjoint union and connected sum, it is sufficient to consider only for genus one Riemann surfaces. For the case when \(M^2=S^1_+\times S^1_+\) and f is such that \((M^2,f)\) represents a zero class in \(\Omega _2^{Spin}(B{\mathbb {Z}}_2)\cong {\mathbb {Z}}_2^2\) the basis has been fixed above. All other null-bordant pairs \((M^2,f)\) can be related by the mapping class group action on \(T^2\). For pairs \((M^2,f)\) which represent non-trivial elements in \(\Omega _2^{Spin}(B{\mathbb {Z}}_2)\) one can first obtain a map \({\mathbb {C}}\rightarrow Z^\mu (M^2,f)\otimes Z^\mu (M^2,f)\) by cutting a 3-torus \(T^3\) in half. This fixes a basis in \(Z^\mu (M^2,f)\) up to a sign. Since such (M, f) only appear in pairs in the boundary of the bordism \((M^3,g)\), different choices do not affect the choice of the isomorphism \(Z^\mu (M^3,g)\cong {\mathbb {C}}\).

33. The more usual notation is \(\beta \), however we are already using this symbol for a different invariant.

34. For completeness, let us remind it. Let \(N^{n-m}\) be a codimension n submanifold in \({\mathbb {R}}^n\) with a framing on the normal bundle. The corresponding continuous map \(S^n\rightarrow S^{m}\) then can be explicitly constructed as follows. First let us identify the source \(S^n\) with the ambient \({\mathbb {R}}^n\) with added point at infinity, and the target \(S^n\) as \(D^n/\partial D^n\) where \(D^n\) is a unit ball. Pick a tabular neighborhood \({\mathcal {T}}(N^{n-m})\subset {\mathbb {R}}^n\) of \(N^{n-m}\). The framing on the normal bundle then provides an explicit isomorphism \({\mathcal {T}}(N^{n-m})\cong N^{n-m}\times D^{m}\) where \(D^{m}\) is the m-dimensional unit ball. The map \(S^n\rightarrow S^m\) is then given by taking all the points outside of the tabular neighborhood to \(\partial D^m\) (collapsed to a single point in \(S^m\)) and the points inside to be projected on \(D^m\).

35. The crucial property of the direct sum operation on n-dimensional TQFTs is

    

    The direct sum operation is then naturally extended to the TQFT values on disjoint \((n-1)\)-dimensional manifolds and bordisms between them so that functoriality and symmetric monoidal property hold.

36. Here we are not trying to define a full-fledged extended TQFT, which also encodes the set of possible boundary conditions, but we only describe a single boundary condition for a given fSPT.

37. That is, \(Z_0(M^n_1)\) and \(Z_0(M^n_2)\) are the same if \(\partial M^n_1\) is spin-diffeomorphic to \(\partial M^n_2\). This is because we can drill out \(M^n\setminus N_{\partial }\) where \(N_{\partial }\) is the tubular neighborhood of the boundary, and then fill there with the \(n\)-dimensional ball without changing the value of \(Z_0\), since the bulk theory is trivial.

38. This might seem contradicting since, while the \({{\,\mathrm{ABK}\,}}\) invariant is \({\mathbb {Z}}_8\) valued, the \({\mathbb {Z}}_2\) defect should vanish when two of them are stacked. This is actually not the case because when \({{\,\mathrm{PD}\,}}(g)\) is oriented \({{\,\mathrm{ABK}\,}}\) has order 2, and when \({{\,\mathrm{PD}\,}}(g)\) is unorientable it has non-vanishing self-intersection and therefore two of symmetry defect occupying the same unorientable homology class cannot be stacked in a parallel way.

39. The one-dimensional \({\mathbb {Z}}_2\)-graded vector space \(Z^{{{\,\mathrm{ABK}\,}}}(S^{1}_-)\) has the odd \({\mathbb {Z}}_2\) degree, which can be understood from the partition function of \(Z^{{{\,\mathrm{ABK}\,}}}\) on a torus.

40. Note that because \(\partial '(c)=\alpha \in H^1(T(L),{\mathbb {Z}}_2)\), in principle a smooth representative of c ends on a cycle in T(L) which represents an element in integral homology \(H_1(T(L),{\mathbb {Z}})\) that can be different from the one given by the map \(H_1(L,{\mathbb {Z}})\rightarrow H_1(T(L),{\mathbb {Z}})\) via pushoff towards a framing vector. However, by gluing the appropriate number of Möbius strips to the boundary components of c one can always fix this mismatch. This can be seen from the fact that framings at each component of L are (non-canonically) in one-to-one correspondence with integers and gluing a single Möbius strip changes the integer by 2 and that the values of framings mod 2 is fixed by \(\alpha \in H^1(T(L),{\mathbb {Z}}_2)\). See [38].

41. This means that the \(Z_\text {gauged}^{0\beta }\) can be a boundary theory of the invertible TQFT defined on orientable manifolds with structure maps \((a_1,a_2):M^4 \rightarrow K({\mathbb {Z}}_2,2)\times K({\mathbb {Z}}_2,2)\) that have the partition function \((-1)^{\int _{M^4}a_1\cup a_2}\).

42. Note that the classifying space \(B{\mathbb {Z}}=S^1\). The formula (9.3) can be understood as a generalization of the particular case of the Künneth formula for integral homology: \(H_n(B({\mathbb {Z}}\times G_o))=H_n(BG_o) \times H_{n-1}(BG_o)\) to the spin-bordism generalized homology.

43. Thanks to the local unitary transformation, this 2+1D crystalline fSPTs can be deformed to a trivial tensor product state once we break the \({\mathbb {Z}}\)-translational symmetry.

44. For example, it is not possible to obtain the \({\mathbb {Z}}_8\) classes generated by \({{\hat{\gamma }}_{\text {PD}(h)}(f)}\) in \(\Omega ^4_{Spin,\mathrm {Tor}}(B({\mathbb {Z}} \times {\mathbb {Z}}_4))\) from the \({\mathbb {Z}}_4\) classes generated by \({\delta _{\text {PD}(f)}(g,g)} =f\cup (g\cup \text {ABK})\) in \(\Omega _4^{Spin}(B({\mathbb {Z}}_2\times {\mathbb {Z}}_4))\), via obtaining a new the \({\mathbb {Z}}_2\) gauge field g from the \({\mathbb {Z}}\) gauge field h by mod 2 reduction. We can prove it is impossible by contradiction. Suppose it is possible, then

    $$\begin{aligned} \delta _{\text {PD}(f)}(h \;\;\mathrm {mod}\;2, h \;\;\mathrm {mod}\;2) \overset{?}{=} {\hat{\gamma }}_{\text {PD}(h)}(f) \;\;\mathrm {mod}\;4 \end{aligned}$$

    (9.13)

    for any \(f \in H^1(M^4,{\mathbb {Z}}_4)\), \(h \in H^1(M^4,{\mathbb {Z}})\) and any spin \(M^4\). But then one can take \(M^4=S^1 \times M^3\), h to be the generator of \(H^1(S^1,{\mathbb {Z}})\), and f to be in \(H^1(M^3,{\mathbb {Z}}_4)\). Then the right hand side of the above formula (9.13) becomes \(\int _{M^3} f {\mathcal {B}} f\) (where \(B: H^1(M^4,{\mathbb {Z}}_4) \rightarrow H^2(M^4,{\mathbb {Z}}_4)\) is Bockstein morphism) and in general is not zero, while the left hand side is identically zero in this setup (because self-intersection of PD(\(h\;\;\mathrm {mod}\;2\)) inside PD(f) is trivial), so we have a contradiction. The formula (9.13) is false.

45. See for example, Ref. [63] on the computation of modular \({\mathcal {S}}^{xyz}\)-matrix (a generator of modular SL(\(3,{\mathbb {Z}}\)) data) that shows this non-abelian property with \(({\mathcal {S}}_{0 \alpha }/{\mathcal {S}}_{0 0}) >1\) for certain non-abelian TQFTs and a certain anyonic string excitation from a surface operator labeled by \(\alpha \). More examples of non-abelian TQFTs are given in Ref. [62] and in Tables 3 and 4.

46. As discussed in Sect. 4, in such expression we assume that there exists a representation of Poincaré dual of f by an immersed manifold \(\text {PD}(f)\).

47. See a field theory derivation of decorated domain walls proliferation construction of SPTs and topological terms related to bosonic TQFTs and Dijkgraaf–Witten gauge theory in Ref. [28]. Similar construction for fermionic SPTs is discussed in Ref. [25]

48. Hereby the chiral and anti-chiral-p-wave superconductors, or the \(p_x\pm ip_y\) superconductors, we mean the Cooper pairing of two fermions are in the \(p_x\pm ip_y\)-orbital of p-orbital pairing states (p in terms of the angular momentum \(\ell =1\) in the spherical harmonics, or the p in the s, p, d, f, etc. of atomic orbitals). The pairing function results in the superconductor order parameter \(\Delta ({\mathbf {k}})\propto k_x\pm ik_y\), where \(k_x,k_y\) are spatial momentum in the x, y-directions.

49. Which is generated by \(\int _{M^4} (g \cup \text {ABK}) \cup (h \;\;\mathrm {mod}\;8)\) with the \({\mathbb {Z}}_2\) gauge field g and the \({\mathbb {Z}}\) gauge field of h along the z-direction.

50. This means we can also re-interpret the role of crystalline-\({\mathbb {Z}}\) lattice translation as a new internal symmetry instead.

51. For even d, the internal symmetry corresponds to extensions

    $$\begin{aligned} 1 \rightarrow {\mathbb {Z}}_2^f \rightarrow G \rightarrow {\mathbb {Z}}_2^T \rightarrow 1, \end{aligned}$$

    where \(G= {\mathbb {Z}}_4^{Tf}\) (\(Pin^+\)) for \(I^2=+1\), while \(G={\mathbb {Z}}_2^T\times {\mathbb {Z}}_2^f\) (\(Pin^-\)) for \(I^2=(-1)^F\). For odd d, the internal symmetry corresponds to extension

    $$\begin{aligned} 1 \rightarrow {\mathbb {Z}}_2^f \rightarrow G \rightarrow {\mathbb {Z}}_2 \rightarrow 1, \end{aligned}$$

    where \(G= {\mathbb {Z}}_4^{f}\) (\({(Spin \times {\mathbb {Z}}_4)/{\mathbb {Z}}_2}\)-structure in the cobordism approach) for \(I^2=+1\), while \(G={\mathbb {Z}}_2 \times {\mathbb {Z}}_2^f\) (\({(Spin \times {\mathbb {Z}}_2)}\)-structure in the cobordism approach) for \(I^2=(-1)^F\).

52. For all d-dimensions where \(d>2\), the internal symmetry corresponds to the extension

    $$\begin{aligned} 1 \rightarrow {\mathbb {Z}}_2^f \rightarrow G \rightarrow {\mathbb {Z}}_{2m} \rightarrow 1, \end{aligned}$$

    where \(G={\mathbb {Z}}_{2m}^f\) (in terms of bordism of \({(Spin \times {\mathbb {Z}}_{2m})/{\mathbb {Z}}_2}\)-structure) for \(R^m=+1\), while \(G={\mathbb {Z}}_{m} \times {\mathbb {Z}}_2^f\) (in terms of bordism of \({(Spin \times {\mathbb {Z}}_m)}\)-structure) for \(R^m=(-1)^F\). This is a natural generalization from \(m=2\) of \(C_2\)-symmetry in 2+1D to generic m in other dimensions. Note that R is the generator of crystalline rotation symmetry group, not the internal symmetry G. The difference between two (factor \((-1)^F\) in the m-th power of the generator) has been discussed e.g. in [16].

53. \(U_T(1)\) is the notation indicating that the coefficient U(1) of the cohomology group is non-trivially acted by the antiunitary time-reversal symmetry if exists. Here we do not pay attention to the antiunitary symmetry, and assume that all the (considered) symmetries are unitary.

The authorship is listed in the alphabetical order. PP would like to thank Anna Beliakova, Ryan Thorngren for useful discussions and the hospitality of SCGP, Weizmann Institute and KITP where parts of the work were done. JW warmly thanks the participants and organizers of the PCTS workshop on Fracton Phases of Matter and Topological Crystalline Order (December 3–5, 2018) for many inspiring conversations near the completion of this work. JW also thanks Meng Cheng, Dominic Else, Sheng-Jie Huang, and Hao Song, for clarifying their own works. MG thanks the support from U.S.-Israel Binational Science Foundation. KO gratefully acknowledges the support from NSF Grant PHY-1606531 and Paul Dirac fund. PP gratefully acknowledges the support from Marvin L. Goldberger Fellowship and the DOE Grant 51 DE-SC0009988 during his appointment at IAS. ZW is supported by the Shuimu Tsinghua Scholar Program. ZW gratefully acknowledges support from NSFC Grants 11431010, 11571329. JW gratefully acknowledges the Corning Glass Works Foundation Fellowship and NSF Grant PHY-1606531. This work was also supported in part by NSF Grant DMS-1607871 “Analysis, Geometry and Mathematical Physics”, Center for Mathematical Sciences and Applications at Harvard University, and the National Science Foundation under Grant No. NSF PHY-1748958.

Authors and Affiliations

1. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

   Meng Guo

2. Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada

   Meng Guo

3. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

   Meng Guo

4. School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, 08540, USA

   Kantaro Ohmori, Pavel Putrov & Juven Wang

5. ICTP, 34151, Trieste, Italy

   Pavel Putrov

6. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

   Zheyan Wan

7. School of Mathematical Sciences, USTC, Hefei, 230026, China

   Zheyan Wan

8. Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA, USA

   Juven Wang

Corresponding author

Correspondence to Pavel Putrov.

Communicated by H. T. Yau.

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