Exact bosonization in arbitrary dimensions

We extend the previous results of exact bosonization, mapping from fermionic operators to Pauli matrices, in 2d and 3d to arbitrary dimensions. This bosonization map gives a duality between any fermionic system in arbitrary $n$ spatial dimensions and a new class of $(n-1)$-form $\mathbb{Z}_2$ gauge theories in $n$ dimensions with a modified Gauss's law. This map preserves locality and has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the manifold. A new formula for Stiefel-Whitney homology classes on lattices is derived. In the Euclidean path integral, this exact bosonization map is equivalent to introducing a topological"Steenrod square"term to the spacetime action.


Introduction and Summary
It is well known that every fermionic lattice system in 1d is dual to a lattice system of spins with a Z 2 global symmetry (and vice versa). The duality is kinematic (independent of a particular Hamiltonian) and arises from the Jordan-Wigner transformation. Recently it has been shown that any fermionic lattice system in 2d is dual to a Z 2 gauge theory with an unusual Gauss law [1]. The fermion can be identified with the flux excitation of the gauge theory, which is described by the Chern-Simon term iπ A∪δA in the spacetime action. The 2d duality is also kinematic. This approach has been generalized to 3d [2]. Every fermionic lattice system in 3d is dual to a Z 2 2-form gauge theory with an unusual Gauss law. Here "2-form gauge theory" means that the Z 2 variables live on faces (2-simplices), while the parameters of the gauge symmetry live on edges (1-simplices). 2-form gauge theories in 3+1D have local flux excitations, and the unusual Gauss law ensures that these excitations are fermions. This Gauss's law can be described by the "Steenrod square" topological action iπ B ∪ B + B ∪ 1 δB. The form of the modified Gauss law was first observed in [3]: a bosonization of fermionic systems in n dimensions must have a global (n − 1)-form Z 2 symmetry with a particular 't Hooft anomaly. The standard Gauss law leads to a trivial 't Hooft anomaly, so bosonization requires us to modify it in a particular way.
In this paper, we extend these results to arbitrary n dimensions. We show that every fermionic lattice system in n-dimension is dual to a Z 2 (n−1)-form gauge theory with a modified Gauss law. Our bosonization map is kinematic and local in the same sense as the Jordan-Wigner map: every local observable on the fermionic side, including the Hamiltonian density, is mapped to a local gauge-invariant observable on the Z 2 gauge theory side. In the Euclidean picture, we show explicitly that our bosonization map is equivalent to introducing the topological term in the action: where A n−1 is (n − 1)-form gauge fields, a (n − 1)-cochain A n−1 ∈ C n−1 (Y, Z 2 ), and Y is (n + 1)-dimensional spacetime manifold. When A n−1 is closed (a cocycle), this term reduces to the Steenrod square operator [4]. This "Steenrod square" term appears in the construction of fermionic symmetry-protected-topological phases [5]. There are already several proposals for an analog of the Jordan-Wigner map in arbitrary dimensions [6,7,8,9]. Our construction is most similar to that of Bravyi and Kitaev [6]. One advantage of our construction is that we can clearly identify the kind of n-dimensional bosonic systems that are dual to fermionic systems: they possess global (n−1)-form Z 2 symmetry with a specific 't Hooft anomaly, as proposed in [3]. It is also manifest in our approach that the bosonization map depends on a choice of spin structure.

Notations and Coventions
We will always work with an arbitrary triangulation of a simplyconnected n-dimensional manifold M n equipped with a branching structure (orientations on edges without forming a loop in any triangle). The vertices, edges, faces, and tetrahedra are denoted v, e, f, t, respectively. The general d-simplex is denoted as ∆ d . We can label the vertices of ∆ d as 0, 1, 2, . . . , d such that the directions of edges are from the small number to the larger number. We denote this d-simplex as ∆ d = 01 . . . d . Its boundaries are (d−1)-simplices 0, . . . ,î, . . . , d for i = 0, 1, . . . , d, whereî means i is omitted. A formal sum of d-simplices modulo 2 forms an element of the chain C d (M n , Z 2 ).
For every v, we define its dual 0-cochain v, which takes value 1 on v, and 0 otherwise, i.e. v(v ) = δ v,v . Similarly, e is an 1-cochain e(e ) = δ e,e , and so forth, i.e. ∆ d being a d-cochain All dual cochains will be denoted in bold. An evaluation of a cochain c on a chain c will sometimes be denoted c c. When the integration range is not written, c is assumed to be the top dimension and c ≡ The cup product ∪ of a p-cochain α p and a q-cochain β q is a (p + q)-cochain defined as: The definition of the higher cup product [3,4] is where i ∼ j represents the integers from i to j, i.e. i, i + 1, . . . , j, and {i 0 , i 1 , . . . , i a } are chosen such that the arguments of α p and β q contain p + 1 and q + 1 numbers separately. The boundary operator is denoted by ∂. For an n-simplex ∆ n , ∂∆ n consists of all boundary (n − 1)-simplices of ∆ n . The coboundary operator is denoted by δ (not to be confused with the Kronecker delta previously). On a 0-cochain v, δv is an 1-cochain acting on edges, and is 1 if ∂e contains v and 0 otherwise: It is similar for simplices in any dimension.
Finally, ∆ 1 n ⊃ ∆ 2 n or ∆ 2 n ⊂ ∆ 1 n means that the simplex ∆ 1 n contains ∆ 2 n as a subsimplex. A general rule of thumb is that the subset symbol always points to one higher dimension.

Review of Boson-Fermion Duality in (2+1)D and (3+1)D
We begin by reviewing the duality between fermions and Z 2 lattice gauge theory in both two spatial dimensions [1] and three spatial dimensions [2]. On each face f of the 2-manifold M 2 , we place a single pair of fermionic creation-annihilation operators c f , c † f , or equivalently a pair of Majorana fermions γ f , γ f . The algebra of Majorana fermions is where {A, B} = AB −BA is the anti-commutator. The even fermionic algebra consists of local observables with a trivial fermionic parity (i.e. P F = 1). It is generated by the on-site fermion parity, and the fermionic hopping operator on every edge e, where L(e) and R(e) are faces to the left and right of e, with respect to the branching structure of e. The commutation relation of hopping operators can be expressed as: i.e. the sign from the commutation occurs only when the arrows on the two edges follow head to tail and are on the same triangle. In general, for any 1-cochains λ and λ , In other words, S λ is the product of S e over {e|λ(e) = 1} and the sign in front is consistent with the commutation relations. If we consider the product of fermionic hopping operators on edges around a vertex v, the Majorana operators cancel out up to some P f terms. The two generators P f and S e satisfy the following constraint at each vertex v [1]: is the chain which is PoincarÃľ dual to the second StiefelâĂŞWhitney cohomology class w 2 (M 2 ). The explicit expression of w 2 is given in Appendix A. The second Stiefel-Whitney class is the obstruction to a spin structure. The fermion can only be define on a manifold which admits spin structure The bosonic dual of this system involves Z 2 -valued spins on the edges of the triangulation. The bosonic algebra are generated by Pauli matrix on edges: For every face f , we define the flux operator: and for every edge e we define a unitary operator U e which squares to 1: where X e , Z e are Pauli matrices acting on a spin at the edge e. It has been shown in [1] that the sets {U e , W f } and {S e , P f } satisfy the same commutation relations. The boson-fermion duality map defined on the manifold M 2 is: where w 2 ∈ C 0 (M 2 , Z 2 ) is the chain representation of 2nd Stiefel-Whitney class and E ∈ C 1 (M 2 , Z 2 ) denotes a choice of spin structure (∂E = w 2 ). For the consistency of this duality map, we need to impose the gauge constraints on bosonic side e⊃v X e ( e Z δv∪e e ) = 1. The gauge invariant subspace in the bosonic Hilbert space is dual to the fermionic system with total fermion parity f P f = 1.
The 3d boson-fermion duality can be done in a similar way [2]. The only difference is that the fermions γ t , γ t are at the center of tetrahedra t and Pauli operators X f , Z f live on faces f . In three spaitial dimensions, any fermionic system can be mapped to a 2-form Z 2 gauge theory on the 3-dimensional lattice. The duality dictionary becomes: is the chain representative of the second StiefelâĂŞWhitney class, and E ∈ C 2 (M 3 , Z 2 ) denotes a choice of spin structure (∂E = w 2 ).

Exact bosonization in n dimensions
From the 2d and 3d formulae (11) and (12), it is very natural to conjecture the n-dimensional boson-fermion duality. The fermions live at the center n-simplices, i.e. γ ∆n , γ ∆n for each ∆ n . The Pauli matrices live on (n − 1)-simplices, i.e. X ∆ n−1 and Z ∆ n−1 for each ∆ n−1 . The n-dimensional boson-fermion duality should be: is the chain representative of the second StiefelâĂŞWhitney class, E ∈ C n−1 (M n , Z 2 ) denotes a choice of spin structure (∂E = w 2 ), and for general (n − 1)-cochain λ n−1 and λ n−1 , the product of S operators is defined as This n-dimensional boson-fermion duality (13) is the main theorem of this paper, which will be proved by the end of this section.
We are going to prove that this is equivalent to (15) From the definition of the higher cup product (3), we have The ∪ n−2 only contains the product of boundaries ∆ i n−1 with the same orientation (inward or outward) and each pair of ∆ i n−1 , ∆ i n−1 with the same orientation appears exactly once. Therefore, the ∪ n−2 expression in (15) captures the commutation relations of fermionic hopping operators S ∆ n−1 . It is easy to check that bosonic operators U ∆ n−1 satisfy the same commutation relations: (17) Therefore, {S ∆ n−1 , P ∆n } and {U ∆ n−1 , W ∆n } in (13) have the same commutation relations.

Gauge constraints
In this section, we will derive the constraints on fermionic operators: This follows directly from the following two lemmas. . Lemma 2. The sign difference of S δ∆ n−2 and the product of P ∆n is Fig. 2. This sign is a chain representative of 2nd Stiefel-Whitney class: Proof of Lemma 1. Let us denote ∆ n = 01 . . . n formed by ∆ n−2 and two (n − 1)-simplex ∆ L n−1 and ∆ R n−1 , shown in Fig. 1(a). We know that S δ∆ n−2 contains γ ∆n γ ∆n if and only if ∆ L n−1 , ∆ R n−1 are one inward boundary and one outward boundary of n-simplex ∆ n , as indicated in Fig. 1(b) and (c).
For the product of P ∆n , we simplify the integral as The contribution of ∆ n = 01 . . . n to (20) is which is 1 if and only ∆ L n−1 , ∆ R n−1 are one inward boundary and one outward boundary of the n-simplex ∆ n . This shows that product of P ∆n contain P ∆n ∼ γ ∆n γ ∆n if and only if ∆ L n−1 , ∆ R n−1 are one inward boundary and one outward boundary of the n-simplex ∆ n . This cancels out with S δ∆ n−2 exactly.

(28)
We can modify the sign of S ∆ n−1 as where E ∈ C n−1 (M n , Z 2 ) is a choice of spin structure satisfying ∂E = w 2 . In these modified operators, the constraint on the fermionic operator becomes which is mapped to ).
We also need to impose the even total parity constraint for fermions ∆n P ∆n = 1 (32) since it is mapped to the bosonic operator ∆n W ∆n = 1. After imposing the gauge constraints, the n-dimensional boson-fermion duality (13) is completed.

Gauss's law as boundary anomaly
First, we consider the standard Z 2 lattice gauge theory on the ndimensional manifold M n : with the gauge constraint (Gauss's law) It is well-kwown that its Euclidean theory is (n + 1)-dimensional Ising model (with some choice of A and B) [10]: where A ∈ C n−1 (Y, Z 2 ) is a (n − 1)-cochain on the spacetime manifold Y . In this case, S Ising is invariant under the gauge transformation A n−1 → A n−1 + δΛ n−2 for arbitrary (n − 2)-cochain Λ n−2 ∈ C n−2 (Y, Z 2 ). Therefore, S Ising has no boundary anomaly under the standard Gauss's law. Now, we propose a new class of Z 2 lattice gauge theory: with the modified Gauss's law (gauge constraints) at (n − 2)-simplices This model describes a free fermion system, since it is dual to The modified Gauss's law (37) on a (n − 2)-simplex ∆ n−2 , or equivalently on the dual (n − 2)-cochain ∆ n−2 , can be generalized to an arbitrary (n − 2)-cochain λ n−2 = i ∆ i n−2 , the Gauss's law is (39) where the sign comes from anti-commutation of X and Z on the same simplex. The derivation uses the following property of higher cup products: Consider now the following (n − 1)-form gauge theory defined on a general triangulated (n + 1)-dimensional manifold Y : (41) where A n−1 ∈ C n−1 (Y, Z 2 ), and the gauge symmetry acts by A n−1 → A n−1 +δΛ n−2 for Λ n−2 ∈ C n−2 (Y, Z 2 ). The second term is the generalized Steenrod square term defined in [5]. The action is gauge-invariant up to a boundary term: where we have omited the subscript of A n−1 and Λ n−2 for simplicity. This boundary term determines the Gauss law for the wave-function Ψ(A) on the spatial slice M = ∂Y : where ω(Λ, A) = M (Λ ∪ n−4 Λ + Λ ∪ n−3 δΛ + δΛ ∪ n−2 A). The Gauss law is the same as the gauge constraint (39) if we identify Z ∆ n−1 as (−1) A n−1 (∆ n−1 ) and X ∆ n−1 acts as the transformation A n−1 → A n−1 + ∆ n−1 . The modified Gauss's law (37) represents the boundary anomaly of topological action (41) as we claimed.
In the following subsection, we derive the Euclidean action of the modified Z 2 lattice gauge theory (36) explicitly, which is analogous to (41).

Euclidean path integral of lattice gauge theories
Start with the Hamiltonian of modified Z 2 lattice gauge theory: The partition function is: where we use Trotter-Suzuki decomposition in imaginary time direction and T is the transfer matrix defined as The first factor arises from the gauge constraints on the Hilbert space.
The spacetime manifold consists of many time slices labelled by layers {i}. In the ith layer, we insert a complete basis (in Pauli matrix ). The transfer matrix T between the ith layer and the (i + 1)th layer contains gauge constraints on every spatial (n − 2)-simplex ∆ n−2 : where we introduce the Lagrangian multiplier a i+1/2 n−2 ∈ C n−2 (M n , Z 2 ) (Z 2 fields on each ∆ n−2 of the spatial manifold M n ). Notice that a i+1/2 n−2 defined between two time slices lives on the spatial (n − 2)simplex ∆ n−2 , which can be interpreted as the spacetime (n − 1)simplex between the two layers. From the same calculation in [2], we have and Here J s , J τ are constants depending on A, B, δτ in the original Hamiltonian and we assume J s = J τ = J for simplicity. | · · · | gives the argument's parity 0 or 1. The gauge transformations act as where λ i are arbitrary (n − 2)-cochains and µ i are arbitrary (n − 3)cochains.
If we interpret a i+1/2 n−2 as spacetime (n − 1)-cochains, we can rewrite which is Z 2 fields on (n − 1)-simplices in spacetime manifold Y . It is natural to write S Ising in (50) as The spacetime manifold Y = M n × [−∞, 0] (spatial and temporal parts) is not a triangulation, since we only triangularize the spatial manifold M n under the discretized time. The (higher) cup products are not well-defined in Y . However, we can still write an expression in (n + 1)-dimensional triangulation Y such that Y is a refinement of Y . We can check that (51) and (55) produce the same boundary term under gauge transformations.

Conclusions
We have extended the the exact bosonization (11) in 2d and (12) in 3d to arbitrary dimensions. The dictionary for n-dimensional bosonfermion duality is given in (13). This bosonization is a duality between any fermionic system in arbitrary n spatial dimensions and (n−1)-form Z 2 gauge theories in n dimensions with gauge constraints (modified Gauss's law). This map preserves locality: every local even fermionic observable is mapped to a local gauge-invariant bosonic operator. The formula has an explicit dependence on the second StiefelâĂŞWhitney class of the manifold, and a choice of spin structure is needed. As a side product, we discover a new formula (19) for Stiefel-Whitney homology classes on lattices. In the Euclidean picture, we have shown that the Euclidean path integral of the n-dimensional Z 2 gauge theory with modified Gauss's law is the (n + 1)-dimensional Ising model with an additional topological Steenrod square (41) term. We thank Anton Kapustin and Po-Shen Hsin for many very helpful discussions.

A A formula for Stiefel-Whitney homology classes
In this section, we prove Lemma 3, (28). First, let us recall the theorem proved in [11]. Let s be a p-simplex, say s = v 0 , v 1 , . . . , v p . Let k be another simplex which has s as a face; i.e., s ⊂ k (s may be equal to We say that s is regular in k, if #(B m ) = 0 for every odd m. Let ∂ p (k) denote the mod 2 chain which consists of all p-dimensional simplices s in k so that s is regular in k. For example, 012 and 023 are regular in 0123 and therefore ∂ 2 ( 0123 ) = 012 + 023 . The theorem is [11]: k| dim k≥(n−2) ∂ n−2 (k) is a (n − 2)-chain which represents w 2 .
We now use this theorem to prove lemma 3.
Proof of Lemma 3. For every (n − 2)-simplex ∆ n−2 , it is regular in itself. This contributes the 1 in the coefficient of c(∆ n−2 ) in (28). For every (n − 1)-simplex ∆ n−1 , it is a boundary of two n-simplices ∆ L n and ∆ R n , with ∆ n−1 being an outward boundary of ∆ L n and an inward boundary of ∆ R n . We define that ∆ n−1 belongs to ∆ R n and the summation of dim k = n − 1, n in theorem 1 can be written as: If ∆ n = 0 . . . n is " + "-oriented, the terms in the summation is where we have used the definition of regular simplex defined above. Similarly, we can derive that if ∆ n = 0 . . . n is " − "-oriented, the term is i,j|i<j, i∈even, j∈even 0 . . .î . . .ĵ . . . n .