1 Introduction and Summary

Braiding Statistics and Link Invariants of

[2.75mm] Bosonic/Fermionic Topological Quantum Matter

[3.75mm] in 2+1 and 3+1 dimensions

Pavel Putrov, Juven Wang, and Shing-Tung Yau
School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA

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

Department of Physics, Harvard University, Cambridge, MA 02138, USA

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

Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+1 and 3+1 dimensions are explored. Many of our TQFTs are highly-interacting without free quadratic analogs. Some of our bosonic TQFTs can be regarded as the continuum field theory formulation of Dijkgraaf-Witten twisted discrete gauge theories. Other bosonic TQFTs beyond the Dijkgraaf-Witten description and all fermionic TQFTs (namely the spin TQFTs) are either higher-form gauge theories where particles must have strings attached, or fermionic discrete gauge theories obtained by gauging the fermionic Symmetry-Protected Topological states (SPTs). We analytically calculate both the Abelian and non-Abelian braiding statistics data of anyonic particle and string excitations in these theories, where the statistics data can one-to-one characterize the underlying topological orders of TQFTs. Namely, we derive path integral expectation values of links formed by line and surface operators in these TQFTs. The acquired link invariants include not only the familiar Aharonov-Bohm linking number, but also Milnor triple linking number in 3 dimensions, triple and quadruple linking numbers of surfaces, and intersection number of surfaces in 4 dimensions. We also construct new spin TQFTs with the corresponding knot/link invariants of Arf(-Brown-Kervaire), Sato-Levine and others. We propose a new relation between the fermionic SPT partition function and the Rokhlin invariant. As an example, we can use these invariants and other physical observables, including ground state degeneracy, reduced modular and matrices, and the partition function on manifold, to identify all classes of 2+1 dimensional gauged -Ising-symmetric -fermionic Topological Superconductors (realized by stacking layers of a pair of chiral and anti-chiral -wave superconductors [ and ], where boundary supports non-chiral Majorana-Weyl modes) with continuum spin-TQFTs.

## 1 Introduction and Summary

In condensed matter physics, we aim to formulate a systematic framework within unified principles to understand many-body quantum systems and their underlying universal phenomena. Two strategies are often being used: classification and characterization. The classification aims to organize the distinct macroscopic states / phases / orders of quantum matter in terms of distinct classes, give these classes some proper mathematical labels, and find the mathematical relations between distinct classes. The characterization aims to distinguish different classes of matter in terms of some universal physics probes as incontrovertible experimental evidence of their existences. Ginzburg-Landau theory [1, 2, 3] provides a framework to understand the global-symmetry breaking states and their phase transitions. Ginzburg-Landau theory uses the group theory in mathematics to classify the states of matter through their global symmetry groups. Following Ginzburg-Landau theory and its refinement to the Wilson’s renormalization-group theory [4], it is now well-known that we can characterize symmetry breaking states through their gapless Nambu-Goldstone modes, the long-range order (see References therein [5]), and their behaviors through the critical exponents. In this classic paradigm, physicists focus on looking into the long-range correlation function of local operators at a spacetime point , or into a generic -point correlation function:

 ⟨O(x1)O(x2)⟩,⟨O1(x1)O2(x2)⋯On(xn)⟩,etc. (1)

through its long-distance behavior.

However, a new paradigm beyond-Ginzburg-Landau-Wilson’s have emerged since the last three decades [6, 7]. One important theme is the emergent conformal symmetries and emergent gauge fields at the quantum critical points of the phase transitions. This concerns the critical behavior of gapless phases of matter where the energy gap closes to zero at the infinite system size limit. Another important theme is the intrinsic topological order [8]. The topological order cannot be detected through the local operator , nor the Ginzburg-Landau symmetry breaking order parameter, nor the long-range order. Topological order is famous for harboring fractionalized anyon excitations that have the fractionalized statistical Berry phase[9]. Topological order should be characterized and detected through the extended or non-local operators. It should be classified through the quantum pattern of the long-range entanglement (See [10] for a recent review). Topological order can occur in both gapless or gapped phases of matter. In many cases, when topological orders occurr in the gapped phases of condensed matter system, they may have low energy effective field theory descriptions by Topological Quantum Field Theories (TQFTs) [11]. Our work mainly concerns gapped phases of matter with intrinsic topological order that have TQFT descriptions.

One simplest example of topological order in 2+1 dimensions (denoted as 2+1D1) is called the topological order[12], equivalently the spin liquid[13], or the toric code [14], or the discrete gauge theory [15]. Indeed the topological order exists in any dimension, say in D with . Discrete gauge theory can be described by an integer level- field theory with an action , where and are locally 1-form and -form gauge fields. The case of and is our example of 2+1D topological order. Since -point correlation function of local operators cannot detect the nontrivial or topological order, we shall instead use extended operators to detect its nontrivial order. The extended operators pf are Wilson and ’t Hooft operators: carrying the electric charge along a closed curve , and carrying the magnetic charge along a closed surface . The path integral (details examined in the warm up exercise done in Section 2) for the correlator of the extended operators results in

 ⟨WA,en(C1n)WB,qm(Sd−2m)⟩=exp[−i2πNenqmLk(C1n,Sd−2m)]. (2)

With some suitable and values, its expectation value is nontrivial (i.e. equal to 1), if and only if the linking number of the line and surface operator is nonzero. The closed line operator can be viewed as creating and then annihilating a pair of particle-antiparticle 0D anyon excitations along a 1D trajectory in the spacetime. The closed surface operator can be viewed as creating and then annihilating a pair of fractionalized flux-anti-flux D excitations along some trajectory in the spacetime (Note that the flux excitation is a 0D anyon particle in 2+1D, while it is a 1D anyonic string excitation in 3+1D). A nontrivial linking implies that there is a nontrivial braiding process between charge and flux excitation in the spacetime2. The link confugurations shown in terms of spacetime braiding process are listed in Table 1. Physically, we can characterize the topological order through the statistical Berry phase between anyonic excitations, say , via the nontrivial link invariant. Mathematically, the viewpoint is the opposite, the topological order, or TQFT, or here the theory detects the nontrivial link invariant. It shall be profound to utilize both viewpoints to explore the topological order in condensed matter, TQFT in field theories, and link invariants in mathematics. This thinking outlines the deep relations between quantum statistics and spacetime topology[16, 17].

The goals of our paper are: (1) Provide concrete examples of topological orders and TQFTs that occur in emergent low energy phenomena in some well-defined fully-regularized many-body quantum systems. (2) Explicit exact analytic calculation of the braiding statistics and link invariants for our topological orders and TQFTs. For the sake of our convenience and for the universality of low energy physics, we shall approach our goal through TQFT, without worrying about a particular lattice-regularization or the lattice Hamiltonian. However, we emphasize again that all our TQFTs are low energy physics of some well-motivated lattice quantum Hamiltonian systems, and we certainly shall either provide or refer to the examples of such lattice models and condensed matter systems, cases by cases. To summarize, our TQFTs / topological orders shall satisfy the following physics properties:

1. The system is unitary.

2. Anomaly-free in its own dimensions. Emergent as the infrared low energy physics from fully-regularized microscopic many-body quantum Hamiltonian systems with a ultraviolet high-energy lattice cutoff. This motivates a practical purpose for condensed matter.

3. The energy spectrum has a finite energy gap in a closed manifold for the microscopic many-body quantum Hamiltonian systems. We shall take the large energy gap limit to obtain a valid TQFT description. The system can have degenerate ground states (or called the zero modes) on a closed spatial manifold . This can be evaluated as the path integral on the manifold , namely as the dimension of Hilbert space, which counts the ground state degeneracy (GSD). On an open manifold, the system has the lower dimensional boundary theory with anomalies. The anomalous boundary theory could be gapless.

4. The microscopic Hamiltonian contains the short-ranged local interactions between the spatial sites or links. The Hamiltonian operator is Hermitian. Both the TQFT and the Hamiltonian system are defined within the local Hilbert space.

5. The system has the long-range entanglement, and contains fractionalized anyonic particles, anyonic strings, or other extended object as excitations.

As said, the topological order / gauge theory has both TQFT and lattice Hamiltonian descriptions [12, 13, 14, 15]. There are further large classes of topological orders, including the toric code [14], that can be described by a local short-range interacting Hamiltonian:

 ^H=−∑v^Av−∑f^Bf, (3)

where and are mutually commuting bosonic lattice operators acting on the vertex and the face of a triangulated/regularized space. With certain appropriate choices of and , we can write down an exact solvable spatial-lattice model (e.g. see a systematic analysis in [18, 19], and also similar models in [20, 21, 22]) whose low energy physics yields the Dijkgraaf-Witten topological gauge theories [23]. Dijkgraaf-Witten topological gauge theories in -dimensions are defined in terms of path integral on a spacetime lattice (-dimensional manifold triangulated with -simplices). The edges of each simplex are assigned with quantum degrees of freedom of a gauge group with group elements . Each simplex then is associated to a complex phase of -cocycle of the cohomology group up to a sign of orientation related to the ordering of vertices (called the branching structure). How do we convert the spacetime lattice path integral as the ground state solution of the Hamiltonian given in Eq. (3)? We design the term as the zero flux constraint on each face / plaquette. We design that the term acts on the wavefunction of a spatial slice through each vertex by lifting the initial state through an imaginary time evolution to a new state with a vertex via . Here the edge along the imaginary time is assigned with and all are summed over. The precise value of is related to fill the imaginary spacetime simplices with cocycles . The whole term can be viewed as the near neighbor interactions that capture the statistical Berry phases and the statistical interactions. Such models are also named the twisted quantum double model [24, 18], or the twisted gauge theories [22, 19], due to the fact that Dijkgraaf-Witten’s group cohomology description requires twisted cocycles.

With a well-motivated lattice Hamiltonian, we can ask what is its low energy continuum TQFT. The Dijkgraaf-Witten model should be described by bosonic TQFT, because its definition does not restrict to a spin manifold. Another way to understand this bosonic TQFT is the following. Since and are bosonic operators in Eq.3, we shall term such a Hamiltonian as a bosonic system and bosonic quantum matter. TQFTs for bosonic Hamiltonians are bosonic TQFTs that require no spin structure. We emphasize that bosonic quantum matter and bosonic TQFTs have only fundamental bosons (without any fundamental fermions), although these bosonic systems can allow excitations of emergent anyons, including emergent fermions. It has been noticed by [24, 25, 26, 27, 22, 28] that the cocycle in the cohomology group reveals the continuum field theory action (See, in particular, the Tables in [28]). A series of work develop along this direction by formulating a continuum field theory description for Dijkgraaf-Witten topological gauge theories of discrete gauge groups, their topological invariants and physical properties [27, 28, 29, 30, 31, 32, 33, 34, 35, 17, 36]. We will follow closely to the set-up of [28, 17]. Continuum TQFTs with level-quantizations are formulated in various dimensions in Tables of [28]. Dynamical TQFTs with well-defined exact gauge transformations to all orders and their physical observables are organized in terms of path integrals of with linked line and surface operators in Tables of [17]. For example, we can start by considering the Dijkgraaf-Witten topological gauge theories given by the cohomology group , say of a generic finite Abelian gauge group . Schematically, leaving the details of level-quantizations into our main text, in 2+1D, we have field theory actions of , , and , etc. In 3+1D, we have , . Here and fields are locally 2-form and 1-form gauge fields respectively. For simplicity, we omit the wedge product () in the action. (For example, is a shorthand notation for .) The indices of and are associated to the choice of subgroup in . The fields are 1-form U(1) gauge fields, but the fields can have modified gauge transformations when we turn on the cubic and quartic interactions in the actions. We should warn the readers not to be confused by the notations: the TQFT gauge fields and , and the microscopic Hamiltonian operator and are totally different subjects. Although they are mathematically related by the group cohomology cocycles, the precise physical definitions are different. How do we go beyond the twisted gauge theory description of Dijkgraaf-Witten model? Other TQFTs that are beyond Dijkgraaf-Witten model, such as [37, 29] and other higher form TQFTs[38], may still be captured by the analogous lattice Hamiltonian model in Eq. (3) by modifying the decorated cocycle in to more general cocycles. Another possible formulation for beyond-Dijkgraaf-Witten model can be the Walker-Wang model[39, 40]. The lattice Hamiltonian can still be written in terms of certain version of Eq. (3). All together, we organize the list of aforementioned TQFTs, braiding statistics and link invariants that we compute, and some representative realizable condensed matter/lattice Hamiltonians, in Table 1.

Most TQFTs in the Table 1 are bosonic TQFTs that require no spin manifold/structure. However, in 2+1D, and in 3+1D, 3 are two examples of fermionic TQFTs (or the so-called spin TQFTs) when is an odd integer. A fermionic TQFTs can emerge only from a fermionic Hamiltonian that contains fundamental fermionic operators satisfying the anti-commuting relations (see e.g. [41, 42]). We emphasize that the fermionic quantum matter have fundamental fermions (also can have fundamental bosons), although these fermionic systems can allow excitations of other emergent anyons. Mathematically, TQFTs describing fermionic quantum matter should be tightened to spin TQFTs that require a spin structure [43, 44] (see the prior observation of the spin TQFT in [23]).

We shall clarify how we go beyond the approach of [28, 17]. Ref.[28] mostly focuses on formulating the probe-field action and path integral, so that the field variables that are non-dynamical and do not appear in the path integral measure. Thus Ref.[28] is suitable for the context of probing the global-symmetry protected states, so-called Symmetry Protected Topological states [45] (SPTs, see [10, 46, 47] for recent reviews). Ref.[17] includes dynamical gauge fields into the path integral, that is the field variables which are dynamical and do appear in the path integral measure. This is suitable for the context for Ref.[17] observes the relations between the links of submanifolds (e.g. worldlines and worldsheets whose operators create anyon excitations of particles and strings) based on the properties of 3-manifolds and 4-manifolds, and then relates the links to the braiding statistics data computed in Dijkgraaf-Witten model [26, 22, 30] and in the path integral of TQFTs. In this article, we explore from the opposite direction reversing our target. We start from the TQFTs as an input (the first sub-block in the first column in Table 1), and determine the associated mathematical link invariants independently (the second sub-block in the first column in Table 1). We give examples of nontrivial links in 3-sphere and 4-sphere , and their path integral expectation value as statistical Berry phases (the second column in Table 1), and finally associate the related condensed matter models (the third column in Table 1).

In Table 1, we systematically survey various link invariants together with relevant braiding processes (for which the invariant is a nontrivial number as ) that either are new to or had occurred in the literature in a unified manner. The most familiar braiding is the Hopf link with two linked worldlines of anyons in 2+1D spacetime[11, 9] such that . The more general Aharonov-Bohm braiding [48] or the charge-flux braiding has a worldline of an electric-charged particle linked with a -worldsheet of a magnetic flux linked with the linking number in +1D spacetime. The Borromean rings braiding is useful for detecting certain non-Abelian anyon systems[30]. The link of two pairs of surfaces as the loop-loop braiding (or two string braiding) process is mentioned in [49, 50, 51]. The link of three surfaces as the three-loop braiding (or three string braiding) process is discovered in [26, 21] and explored in [22]. The link of four 2-surfaces as the four-loop braiding (or four string braiding) process is explored in [30, 17, 35].

More broadly, below we should make further remarks on the related work [27, 28, 37, 29, 30, 31, 32, 33, 34, 35, 36, 52, 53]. This shall connect our work to other condensed matter and field theory literature in a more general context. While Ref. [27] is motivated by the discrete anomalies (the ’t Hooft anomalies for discrete global symmetries), Ref. [28] is motivated by utilizing locally flat bulk gauge fields as physical probes to detect Symmetry Protected Topological states (SPTs). As an aside note, the SPTs are very different from the intrinsic topological orders and the TQFTs that we mentioned earlier:

• The SPTs are short-range entangled states protected by nontrivial global symmetries of symmetry group . The SPTs have its path integral on any closed manifold. The famous examples of SPTs include the topological insulators[54, 55] protected by time-reversal and charge conjugation symmetries. The gapless boundaries of SPTs are gappable by breaking the symmetry or introducing strong interactions. Consequently, take the 1+1D boundary of 2+1D SPTs as an example, the 1+1D chiral central charge is necessarily (but not sufficiently) .

• The intrinsic topological orders are long-range entangled states robust against local perturbations, even without any global symmetry protection. However, some of topological orders that have a gauge theory description of a gauge group may be obtained by dynamically gauging the global symmetry of SPTs [56, 57]. The boundary theory for topological orders/TQFTs obtained from gauging SPTs must be gappable as well.

In relation to the lattice Hamiltonian, the SPTs has its Hilbert space and group elements associated to the vertices on a spatial lattice [45], whereas the corresponding group cohomology implementing the homogeneous cocycle and the holonomies are trivial for all cycles of closed manifold thus . In contrast, the Eq. (3) is suitable for topological order that has its Hilbert space and group elements associated to the links on a spatial lattice [18, 22, 19], whereas its group cohomology implementing the inhomogeneous cocycle and its holonomies are non-trivial for cycles of closed manifold thus sums over different holonomies.

In relation to the field theory, we expect that the SPTs are described by invertible TQFTs (such as the level in theory), a nearly trivial theory, but implemented with nontrivial global symmetries. (See [58] for the discussions for invertible TQFTs, and see the general treatment of global symmetries on TQFTs in [29].) In contrast, we expect that the intrinsic topological orders are described by generic non-invertible TQFTs (e.g. level theory). Since Ref.[28] implements the nearly flat probed gauge fields, the formalism there could not be the complete story for the intrinsic topological orders and TQFTs of our current interests. It is later found that one can view the topological actions in terms of dynamical gauge fields instead of the probed fields, by modifying the gauge transformations [31, 32]. Up until now, there is good evidence that we can view the discrete spacetime Dijkgraaf-Witten model in terms of some continuum TQFTs (See Tables in [17, 28] and our Table 1). One of the most important issues for understanding the dynamical TQFT is to compute precisely the path integral and to find explicitly the physical observables. To this end, one partial goal for this article, is to explicitly compute the path integral and the braiding statistics / link invariants for these TQFTs in various dimensions. We focus mainly on 2+1D and 3+1D for the sake of realistic dimensions in condensed matter physics, but our formalism can be easily applied to any dimension.

Other than TQFTs and discrete gauge theories in Table 1, we can obtain even more fermionic spin TQFTs by gauging the global symmetries of fermionic SPTs (fSPTs). An interesting example is gauging the fSPTs with symmetry in various dimensions. We are able to address one interesting puzzle concerning the fSPTs as Topological Superconductors with 8 distinct classes labeled by (realized by stacking layers of a pair of chiral and anti-chiral -wave superconductors). Although it is known that gauged fSPTs are bosonic Abelian Chern-Simons (CS) theories for bosonic gauge and twisted gauge theory (toric code and double-semion models), and gauged fSPTs are fermionic Abelian spin-CS theory for fermionic gauge and twisted gauge theory, the field theories description for the odd- classes () are somewhat mysterious. In some sense, the odd- class are fermionic “ gauge spin-TQFTs,” but the statistics is somehow non-Abelian. We solve the puzzle by deriving explicit non-Abelian spin TQFTs obtained from gauging fSPTs, and compute physical observables to distinguish class in Sec. 8.

### 1.1 The plan of the article and the convention of notation

The plan of our article is organized as follows. In Sec. 2, we derive the link invariant of theory in any dimension as the Aharonov-Bohm’s linking number that detects a charge particle and a flux loop braiding process through the Aharonov-Bohm phase. In Sec. 3, we study and in 2+1D and show that its path integral calculates the linking number. In Sec. 4, we study in 2+1D and obtain Milnor’s triple linking number from its path integral. In Sec. 5, we study in 3+1D and obtain triple-linking number of surfaces. In Sec. 6, we study in 3+1D and obtain quadruple-linking number of surfaces. In Sec. 7, we study in 3+1D and obtain intersection number of open surfaces. In Sec. 8, we construct the explicit fermionic SPT path integrals with symmetry, and their gauged versions: fermionic spin TQFTs. We derive the experimentally measurable physics observables, including the ground state degeneracy (GSD), the braiding statistics (the modular matrices and ), etc. In addition, we discuss their relation to various invariants including Arf(-Brown-Kervaire), Rokhlin, Sato-Levine invariants and more. In Sec. 9, we conclude with additional remarks.

We should emphasize that the link invariants we derive are powerful and important in various aspects. (1) A link invariant can detect various possible links in spacetime, or various possible braiding processes (regardless if the braiding process is known or unknown to the literature). While in the literature, few specific braiding processes have been investigated (such as the three or four string braiding processes), we can use our link invariants to identify other braiding processes that produce nontrivial values of topological invariants and thus have nontrivial statistical Berry phases. (2) Our method to derive topological invariants is based on field theory description of TQFTs. In particular, our approach is systematic, using Poincaré duality and intersection theory. Our approach is universal, and our result is more general than what appeared in the literature.

Note: To denote the cyclic group of order , we use and , which are equivalent mathematically, but have different meanings physically. We use to denote a symmetry group and a gauge group. We use the slight different notation to denote the distinct classes in the classification of SPTs/TQFTs or in the cohomology/bordism group. Notation stands for the fermion parity symmetry. We denote and . As usual, notation means the disjoint union between two sets or two manifolds and . The means relative complement of in . We use to denote cup-product in cohomology ring. GSD stands for ground state degeneracy. In Table.1 and elsewhere, the repeated indices is normally assumed to have Einstein summation, except that the term where the prime indices here are fixed instead of summed over.

## 2 ∫BdA in any dimension and Aharonov-Bohm’s linking number

Below we warm up by considering the level- BF theory with an action in any dimension, where is quantized to be an integer. The study of BF theory in physics dates back to the early work of [66, 67]. Consider the following action on any closed -manifold :

 S[A,B]=∫MdN2πB∧dA (4)

where is a 1-form gauge field on and is a -form gauge field on . The partition function or path integral without any additional operator insertion is

Locally the gauge transformation is given by:

 A → A+dg, (6) B → B+dη. (7)

If has non-trivial topology, globally and may have discontinuities such that and are continuous forms representing a cohomology class in and respectively.

Now for a path integral with insertions, let be a gauge invariant functional of the fields and . The path integral with insertion can be formally defined as

Let us note that in the case when has non-trivial topology, the field only locally can be understood as a form. Globally, it can be realized as where is a globally defined form and is a discontinuous -form such that is a continuous form representing a class in , the flux of the gauge field . So the path integral over actually means the following

 ∫DB…≡∑[dβ]∈2πHd−1(Md,Z)∫D~B…. (9)

Below we evaluate the in various scenarios starting from the simplest, almost trivial case and gradually increasing complexity.

1. If is independent of the field, then the integration over gives the equation of motion as constraint of , which localizes to be flat connection. Namely, the curvature is zero . Furthermore, from Poincaré duality , it follows that the sum over fluxes imposes the following constrains on :

 exp(iN∫γA)=1,∀γ (10)

that is, modulo gauge transformations, connection belongs to subset of flat connections:

 [A]∈Hom(H1(Md),ZN)⊂Hom(H1(Md),U(1)). (11)

Note that from the universal coefficient theorem and the fact that is a free group, it follows that . The path integral then reduces to the following finite sum:

 ⟨Φ⟩=1Z∑[A]∈Hom(H1(Md),ZN)Φ(A). (12)

The standard normalization for the partition function is as follows:

 Z=1N∑[A]∈Hom(H1(Md),ZN)1 (13)

so that for .

2. If depends on field as follows

 Φ(A,B)=∏mexp[iqm∫Sd−2mB]⋅Φ0(A). (14)

Where is a family of -dimensional hypersurfaces inside the spacetime manifold and is the insertion that depends only on . Gauge invariance requires . One can also rewrite (21) as follows:

 Φ(A,B)=∏mexp[iqm∫M4B∧δ⊥(Sd−2m)]⋅Φ0(A) (15)

where is the 2-form valued delta function distribution supported on . That is,

 ∫Mdωd−2∧δ⊥(Sd−2m)=∫Sd−2mωd−2. (16)

for any form . After integrating out the path integral Eq. (8) localizes to the solutions of the equations of motion with source:

 FA=dA=−2πN∑mqmδ⊥(Sd−2m). (17)

This equation implies that is a differential form which represents the class in Poincaré dual to the class in homology . Here and below denotes the homology class of the surface . Since represents the first Chern class of the gauge bundle, must represent an integral homology class. This gives the constraint on the allowed charge (the magnetic charge), if some of the classes are nontrivial.

3. If , then there is a unique solution to Eq. (17), modulo the gauge redundancy. The cohomology is then generated by 1-forms such that

 ∫C1nμi=δn,i, (18)

where is a small circle linking . Here we denote means the relative complement of in . The solution of Eq. (17) then becomes:

 A=−2πN∑mqmμm. (19)

One possible choice of forms is using 1-form valued delta functions supported on , Seifert hypersurfaces bounded by (i.e. such that and therefore ):

 A=−2πN∑mqmδ⊥(Vd−1m). (20)
4. If in Eq. (21) is a product of the Wilson loops around the one-dimensional loops separate and disjoint from , such that

 Φ0(A)=∏nexp[ien∫γ1nA] (21)

with the electric charge associated to each loop, then the path integral with insertion can be evaluated as follows:

where the is the linking integer number between the loop and the -dimensional submanifold , which by definition is given by counting intersection points in with signs corresponding to orientation.

## 3 ∫KIJAIdAJ and ∫BdA+AdA in 2+1D and the linking number

In the 2+1D spacetime, as another warp up exercise, consider the action of Chern-Simons theory with level matrix :

 S[A]=∫M3s∑I,J=1KIJ4πAI∧dAJ. (23)

where is a symmetric integral valued matrix. The above most general Abelian Chern-Simons theory includes a particular case:

 S[A,B]=∫M3∑INI2πBI∧dAI+∑I,JpIJ4πAI∧dAJ (24)

where is a symmetric integral valued matrix. When is an odd integer, we have the Abelian spin-Chern-Simons theory (considered in detail in [43]). When is an even integer, we have the Abelian Chern-Simons theory that are within the cohomology group for the Dijkgraaf-Witten theory [28], , and . Here we denote .

Note that when is odd for some , the theory becomes fermionic spin-TQFT that depends on the choice of spin structure. A generic collection of line operators supported on closed disjoint curves embedded in can be realized as follows:

 Wq[{γI}sI=1]=exp(is∑I=1eI∫γIAI)≡exp(is∑I=1eI∫M3AI∧δ⊥(γI)) (25)

for some integer numbers . As we will see the result, up to a sign, only depends on the class of -vector in the cokernel of the level matrix , that is effectively . Suppose . The expectation value of is then given by a Gaussian integral which localizes on the following equations of motion:

 ∑JKIJdAJ=−2πδ⊥(γI) (26)

which, up to a gauge transformation, can be solved as follows:

 AI=−2π∑J(K−1)IJeJδ⊥(ΣJ) (27)

where is a Seifert surface bounded by and we used that . Plugging the solution back into the integrand gives us

 ⟨We[{γI}]⟩=exp{−πi∑I,J(K−1)IJeIeJ∫S3δ⊥(ΣI)∧dδ⊥(ΣJ)}=exp{−πi∑I,J(K−1)IJeIeJLk(γI,γJ)} (28)

where is the linking number between and , which is by definition equal to the intersection number . The physics literature on this invariant dates back to [68, 69].

## 4 ∫BdA+A3 in 2+1D, non-Abelian anyons and Milnor’s triple linking number

In the 2+1D spacetime, we can consider the following action on a 3-manifold :

 S[A,B]=∫M33∑I=1NI2πBI∧dAI+¯p(2π)2A1∧A2∧A3 (29)

where and are 1-form fields. Here with . We have the TQFT that are within the class in the cohomology group for the Dijkgraaf-Witten theory [28].

The gauge transformation is:

 AI→AI+dgIBI→BI+dηI+¯p2πNI