Topological Transition on the Conformal Manifold
TTopological Transition on the Conformal Manifold
Wenjie Ji a , Shu-Heng Shao b , and Xiao-Gang Wen a
1, 2 a Department of Physics, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139, USA b School of Natural Sciences, Institute for Advanced Study,Princeton, New Jersey 08540, USA
Despite great successes in the study of gapped phases, a comprehensive understanding of thegapless phases and their transitions is still under developments. In this paper, we study a generalphenomenon in the space of (1+1) d critical phases with fermionic degrees of freedom describedby a continuous family of conformal field theories (CFT), a.k.a. the conformal manifold. Alonga one-dimensional locus on the conformal manifold, there can be a transition point, across whichthe fermionic CFTs on the two sides differ by stacking an invertible fermionic topological order(IFTO), point-by-point along the locus. At every point on the conformal manifold, the order anddisorder operators have power-law two-point functions, but their critical exponents cross over witheach other at the transition point, where stacking the IFTO leaves the fermionic CFT unchanged.We call this continuous transition on the fermionic conformal manifold a topological transition .By gauging the fermion parity, the IFTO stacking becomes a Kramers-Wannier duality betweenthe corresponding bosonic CFTs. Both the IFTO stacking and the Kramers-Wannier duality areinduced by the electromagnetic duality of the (2+1) d Z topological order. We provide severalexamples of topological transitions, including the familiar Luttinger model of spinless fermions (i.e.the c = 1 massless Dirac fermion with the Thirring interaction), and a new class of c = 2 examplesdescribing U (1) × SU (2)-protected gapless phases. I. INTRODUCTION
The study of gapped topological ordered phases hasbeen systematically developed over the last thirty years.The field has moved to the direction of studying (1) thecritical theories between topological phases, or betweena topological phase and a trivial phase. (2) the gaplessphases without quasiparticles. In recent years, there areseveral progresses in studying transitions between topo-logical phases. One is to consider the continuous transi-tion between gapped trivial phase and gapped topologi-cal ordered phases that do not involve any spontaneouslysymmetry breaking[1–5] (but may be viewed as the spon-taneously breaking of emergent higher-symmetry[6] inthe sense of [7, 8]). The critical points between sym-metry protected topological phases are also examplesof continuous transitions that do not involve any spon-taneously symmetry breaking[9–13]. With or withoutthe spontaneously symmetry breaking, the critical pointsfor continuous phase transition are often gapless stateswithout well defined quasiparticles [5, 14, 15]. More-over, some gapless phases can be strongly correlated andhave no well-defined quasiparticles down to zero energy,such as the large N QED in (2+1) d , certain U (1) spinliquid[14, 15], and QED in (3+1) d .To characterize gapless phases without quasiparticles,inspired by the success in gapped phases, we start withthe question whether they can be topologically nontriv-ial. A far from exhaustive list of reference are [16–27].To give an example, one construction is to impose sym-metries and to decorate domain walls in gapless phasesby symmetry charges, in analogy of construction of sym-metry protected topological phases [28, 29]. Overall, thisline of questions is hard and a universal understanding is in demand.More generally, we would like to understand how todistinguish the topological nature in a gapless phase. Itis proposed recently to use the topological edge modein gapless phases[30–32]. A numerical success has beenmade in identifying ground state degeneracy in (1+1) d gapless models with open boundary condition. The en-ergy splitting scales with the system size either expo-nentially or with power law with a large exponent. Thelow energy theories are conformal field theories (CFT)without symmetry-preserving relevant operator. It is alsoproposed in the above literature that there can be topo-logical invariant defined in the gapless bulk, based on dif-ferent ways to assign symmetry charges to the non-localoperators in the low energy effective field theory.In this paper, we introduce a concrete field-theoreticsetup where the topological nature of a gapless statechanges as we dial the parameters of the model. Weconsider a one-parameter family of CFTs in (1+1) d with fermionic degrees of freedom, labeled by an ex-actly marginal coupling g . The latter parametrizes aone-dimensional locus of the conformal manifold whereevery point defines a fermionic (spin) CFT with thesame central charge. We will be interested in the sce-nario where the CFT F [ − g ] with negative coupling dif-fers from that F [ g ] with positive coupling by an invert-ible fermionic topological order (IFTO) ( a.k.a. the Arfinvariant), point-by-point for all couplings g . In otherwords, the gapless state F [ g ] can be viewed as the otherstate F [ − g ] stacked with a (1+1) d p -wave superconduct-ing state [33].The CFT F [0] at the transition point g = 0 enjoys anenhanced Z IFTO2 global symmetry: it is invariant understacking an IFTO. Away from the transition point, the a r X i v : . [ c ond - m a t . s t r- e l ] A ug Z IFTO2 action is not a symmetry, but maps F [ g ] to F [ − g ]by stacking an IFTO. Since the CFTs on the two sidesof F [0] differ by a (1+1) d topological order, we will callthe transition F [ − g ] → F [0] → F [ g ] a topological tran-sition on the conformal manifold. The simplest exampleis a c = 1 massless Dirac fermion with a quartic fermioninteraction ( i.e. the Thirring coupling) [34], which canbe equivalently described by the Luttinger model of spin-less fermions. We will also discuss a c = 2 example with U (1) × SU (2) global symmetry.The topological transition is similar to the standardsecond-order phase transition, and yet it is different inmany aspects. It is similar in that the gapless states onboth sides differ by a topological order, much as a (1+1) d massive Majorana fermions with m > m < e.g. the spectrum and thequantum numbers of the local operators) change con-tinuously as we vary the exactly marginal coupling g .Another resemblance is that while the order and the dis-order operators both have power-law two-point functionsalong the topological transition, their critical exponentscross over with each other at the transition point F [0].This is to be contrasted with the standard order-disorderphase transition where in one phase the order operatorshave asymptotic constant correlations while the disorderoperators have exponentially decaying correlations, andvice versa in the other phase.The topological transition F [ − g ] → F [0] → F [ g ] hasa parallel story in the bosonized picture. The bosoniza-tion and fermionization maps in (1+1) d quantum fieldtheory have been recently revisited from a more modernpoint of view [34–39], which we will review in SectionIII. Let B [ g ] be the bosonization of F [ g ] by gauging thefermion parity, then B [ g ] and B [ − g ] differ by a Z B orb-ifold [7, 34]. In particular, the bosonic transition point B [0] is self-dual under the Z B orbifold, which general-izes the notion of the Kramers-Wannier duality [40–42]to more general bosonic (non-spin) CFT than the IsingCFT. By exploiting our knowledge on the bosonic confor-mal manifold, we produce several examples of topologicaltransitions for fermionic CFTs.The line of CFTs under consideration can also be real-ized by (1+1) d lattice models where Z IFTO2 is a symmetryof the models. But such a Z IFTO2 symmetry has a ’t Hooftanomaly and is not on-site in the lattice models. We canalso realize the line of CFTs by the boundaries of (2+1) d Z topological order ( i.e. Z gauge theory), where Z IFTO2 is the on-site symmetry of the 2+1 models that exchangethe Z -charge e and the Z -vortex m . In the above twofamilies of CFTs with Z IFTO2 symmetry, F [ g ] and F [ − g ]represent the two degenerate ground states from sponta- On the lattice model, (In the bosonic model, the Z orbifold isrealized by translation by half a site, mapping site degrees offreedom to link degrees of freedom. The simplest example is theIsing model.) neous Z IFTO2 -symmetry breaking.Both the Kramers-Wannier duality of (1+1) d bosonictheories and the IFTO stacking Z IFTO2 action of (1+1) d fermionic theories are intimately related to electromag-netic duality in the (2+1) d bosonic untwisted Z gaugetheory [43]. Indeed, the electromagnetic duality of the Z gauge theory extends to the boundary with local bosons(with an anomaly-free Z B twisting) as the Kramers-Wannier duality [44, 45]. On the other hand, whenthe (1+1) d boundary has local fermions, the electromag-netic duality extends to stacking an IFTO [46]. Purelyfrom the (1+1) d boundary point of view, the Kramers-Wannier duality is related to the Z IFTO2 stacking viabosonization/fermionization (as we will discuss in SectionIII), which provides a direct translation between the twoextensions. This is analogous to the relation between the(3+1) d Maxwell theory and the (2+1) d particle-vortexdualities [47, 48]. See Figure 1. The generalized Kramers-Wannier duality of a bosonicCFT B can be implemented by the non-invertible dual-ity defect line in the (1+1) d spacetime [49–55]. We givea detailed analysis of the duality defect in several ex-amples, and discuss its relation to the Z IFTO2 symmetrydefect of the corresponding fermionic CFT F .This paper is organized as follows. In Section II, wereview the IFTO and discuss general features of topo-logical transitions on fermionic conformal manifolds. Wealso discuss the interpretation of stacking an IFTO fromthe (2+1) d Z gauge theory point of view. In SectionIII, we review the bosonization and the fermionizationprocedures in (1+1) d . In particular, we show that twofermionic theories differ by an IFTO if and only if theirbosonized theories are related by a Z orbifold. In Sec-tion IV, we discuss several examples of bosonic confor-mal manifolds where CFTs are related by a Z orbifold,which includes the c = 1 compact boson S , the c = 1orbifold theory S / Z , as well as a c = 2 T CFT ex-ample. These bosonic examples pave the way for thetopological transition of their fermionizations, includingthe c = 1 massless Thirring model ( a.k.a. the Luttingerliquid), which we discuss in Section V. We will also dis-cuss a c = 2 fermionic CFT describing a spin − gaplessphase beyond the Luttinger liquid type. In Section VI,we discuss how a symmetry defect in a fermionic theorybecomes a duality defect under bosonization. We endwith several future directions in Section VII. II. TOPOLOGICAL TRANSITION ON THEFERMIONIC CONFORMAL MANIFOLD
In this paper we consider gapless states described byCFTs with fermionic degrees of freedom. In particular, Throughout the paper, we will assume that the gravitationalanomaly of the (1+1) d fermionic theory is c L − c R = 0 mod 8,and the ( − F fermion parity has no ’t Hooft anomaly. (2+1)d bosonic Z gauge theoryelectromagnetic duality(1+1)d bosonic theory with Z B Kramers-Wannier duality (1+1)d fermionic theoryIFTO-stacking ( Z IFTO2 ) fermionize Z B gauge ( − F FIG. 1. We can couple the (2+1) d bosonic Z gauge theory to a (1+1) d bosonic theory with a non-anomalous Z B symmetry, orto a (1+1) d fermionic theory. The (2+1) d electromagnetic duality implements either the Kramer-Wannier duality ( Z B orbifold)when the (1+1) d boundary is bosonic, or the IFTO stacking ( Z IFTO2 ) when the boundary is fermionic. we focus on CFTs with exactly marginal deformations.By turning on the exactly marginal deformations, onegenerates a continuous family of CFTs parametrized bythe exactly marginal couplings. The space of this familyof CFTs is called a conformal manifold . In this section,we describe a general phenomenon where along a one-dimensional slice of the conformal manifold, there is atransition point across which the CFTs on the two sidesdiffer by an IFTO stacking.
A. Invertible Fermionic Topological Order
We start by reviewing the IFTO, which is the Kitaevchain of p -wave superconductor [33] (see [7, 34, 56–59]for further discussions).Given a Riemann surface Σ with a spin structure ρ ,the Arf invariant Arf[ ρ ] is defined as [60]Arf[ ρ ] = (cid:40) , if ρ is odd , , if ρ is even . (II.1)On a genus- g Riemann surface, there are 2 g − (2 g −
1) oddspin structures and 2 g − (2 g + 1) even spin structures.A Z gauge connection s ∈ H (Σ , Z ) is specified bythe holonomy (cid:72) γ s ∈ { , } around each cycle γ of Σ.Given a spin structure ρ and a Z connection, we canconstruct a new spin structure modified by the Z twist.We will denote this new spin structure by s + ρ .Using the Arf invariant, we can define a (1+1)-dimensional IFTO. The partition function of this IFTOon a Riemann surface with spin structure ρ is simplygiven by Z IFTO [ ρ ] = e i π Arf[ ρ ] . (II.2)This IFTO is protected by the ( − F symmetry. The signof Z IFTO [ ρ ], compared to the trivially fermionic gappedphase Z [ ρ ] = 1, measures the parity of the number ofMajorana zero modes[33], or the parity change of thenumber of negative energy eigenstates. If we stack two IFTOs together, then it becomes a trivial phase. Let the Z background gauge field of ( − F be S . The partitionfunction of the IFTO coupled to a background ( − F gauge field S is Z IFTO [ S + ρ ] = e i π Arf[ S + ρ ] . B. Topological Transition of the Fermionic GaplessStates
We start with a general discussion on fermionic statesin (1+1) dimensions described by a fermionic quantumfield theory (QFT) F . From F we can construct anotherfermionic theory F (cid:48) by stacking with an IFTO Z IFTO : Z F (cid:48) [ ρ ] = Z IFTO [ ρ ] Z F [ ρ ] . (II.3)The partition functions for F and F (cid:48) are identical onRiemann surfaces with even spin structure, but differ bya sign for odd spin structure.On a spacetime torus with complex structure moduli q = exp(2 π i τ ), a fermion system has four partition func-tions Z F [ AE ] = Tr H A (cid:20) − F q H + K ¯ q H − K (cid:21) ,Z F [ AO ] = Tr H A (cid:20) − ( − F q H + K ¯ q H − K (cid:21) ,Z F [ P E ] = Tr H P (cid:20) − F q H + K ¯ q H − K (cid:21) ,Z F [ P O ] = Tr H P (cid:20) − ( − F q H + K ¯ q H − K (cid:21) , (II.4) A QFT whose partition function requires a choice of the spinstructure is called a spin or a fermionic QFT, such as the Majo-rana fermion. By contrast, a QFT whose partition function doesnot require a choice of the spin structure is called a non-spin ora bosonic QFT, such as the Ising CFT. where H P ( H A ) is the Hilbert space with periodic (an-tiperiodic) boundary condition for the fermions. H and K are the eigenvalues of the Hamiltonian and momen-tum in the corresponding Hilbert space. In a CFT withcentral charges c = c L = c R , H and K are related to theconformal weights as h − c = H + K and ¯ h − c = H − K .Here E and O stand for the ( − F -even and ( − F -oddsectors, respectively.Alternatively, we may define the torus partition func-tions for fermion systems through the space-time path in-tegral, which also include four types, Z F [ AA ], Z F [ AP ], Z F [ P A ], and Z F [ P P ]. Here the first and second sub-scription P or A corresponds the periodic or anti-periodicboundary condition for fermions in x and t direction, re-spectively. The two sets of partition functions are related Z F [ AE ] = 12 ( Z F [ AP ] + Z F [ AA ]) ,Z F [ AO ] = −
12 ( Z F [ AP ] − Z F [ AA ]) ,Z F [ P E ] = 12 ( Z F [ P P ] + Z F [ P A ]) ,Z F [ P O ] = −
12 ( Z F [ P P ] − Z F [ P A ]) . (II.5)The boundary conditions AP, P A, AA correspond to theeven spin structures and
P P corresponds to the odd spinstructure. Thus, on a torus, (II.3) implies • IFTO Stacking: Z F [ AA ] = Z F (cid:48) [ AA ] , Z F [ AP ] = Z F (cid:48) [ AP ] ,Z F [ P A ] = Z F (cid:48) [ P A ] , Z F [ P P ] = − Z F (cid:48) [ P P ] . (II.6)This implies that F and F (cid:48) share the same Hilbert spacein the anti-periodic sector, while the fermion parity dif-fers by a sign in the periodic sector.If a fermionic CFT F satisfies Z F [ ρ ] = Z IFTO [ ρ ] Z F [ ρ ] . (II.7)then F remains unchanged after stacking with an IFTO. This means that the fermionic CFT F is at the phasetransition boundary between a trivial order and an IFTO.In other words, the fermionic CFT F describes a con-tinuous phase transition between two fermionic CFTs F and F (cid:48) that differ by an IFTO. For such a fermionic CFT,it satisfies Z F [ ρ ] = 0 , ρ : odd . (II.8)On the torus, its partition function with periodic condi-tions on both cycles ( P P ) vanishes, i.e. Z F [ P P ] = 0.The classic example is a single Majorana fermion the-ory, governed by the action S = 12 π (cid:90) d z (cid:0) ψ ¯ ∂ψ + ¯ ψ∂ ¯ ψ + i m ¯ ψψ (cid:1) , (II.9) In the high energy terminology, H P is the Ramond sector while H A is the Neveu-Schwarz sector. where ψ ( z ) and ¯ ψ (¯ z ) are right and left-moving Majo-rana fermion field. It is the low energy description ofKitaev’s fermionic chain model [33] near the transitionfrom the trivial insulator to the p -wave superconductor.Here F [ m ] is the Majorana fermion with mass m . Thedifference of a Majorana fermion with positive mass m and that with a negative mass − m is precisely the IFTO: Z Maj [ ρ, m ] = Z IFTO [ ρ ] Z Maj [ ρ, − m ] . (II.10)This implies at the critical point m = 0, a single masslessMajorana fermion CFT F = F [0], is invariant under thestacking with the IFTO. a. Exactly Marginal Deformation In the example ofa single Majorana fermion, the mass term, i.e. the de-formation operator O that drives the transition through F , is relevant. When O is exactly marginal, it movesthe CFT F along two different directions onto the con-formal manifold (leaving the central charge unchanged).Let g be the exactly marginal coupling, we will denotethe fermionic CFT on the conformal manifold as F [ g ]with F [ g = 0] = F . The main point of this paper isto give examples of the transition on a one-dimensionallocus of the conformal manifold, where CFTs on the twosides F [ g ] and F [ − g ] differ by an IFTO, much as thepositive mass and the negative mass Majorana fermionsdo. More precisely, Z F [ ρ, g ] = Z IFTO [ ρ ] Z F [ ρ, − g ] , (II.11)point-by-point for every g . We will call such a transition F [ − g ] → F → F [ g ] on the conformal manifold a topo-logical transition . This is to be distinguished from thestandard second order phase transition where the gap isclosed at the critical point but then opens again. In thetopological transition, every point is a CFT and the gapis always zero.More generally, starting from a fermionic CFT F , wecan turn on two different exactly marginal deformations O and O (cid:48) with couplings g ≥ g (cid:48) ≥
0. There canalso be topological transitions from F (cid:48) [ g (cid:48) ] → F → F [ g ]such that F [ g ] differs from F (cid:48) [ f ( g )] by an IFTO for somefunction f ( g ).In the order phase of a standard second order phasetransition, the two-point function of an order operatorapproaches a constant at large separation, while that ofthe disorder operator decays exponentially. The situationis reversed in the disorder phase. By contrast, along thetopological transition F (cid:48) → F → F on the conformalmanifold, the two-point functions of the order and disor-der operators both fall off by power laws. We illustratethe two-point functions of order and disorder operatorsin both cases in Figure 2. The critical exponents ( i.e. the This difference in the IFTO can also be thought of as an anomalyinvolving the coupling g and the spin structure ρ in the sense of[61, 62]. scaling dimensions) of the order operator and the disor-der operator cross over with each other at the transitionpoint F . We will demonstrate this in explicit examplesin Section V.The mapping from F to F (cid:48) , i.e. the stacking withthe IFTO, resembles a Z transformation. Indeed, in theMajorana fermion example, such a Z transformation isthe chiral fermion parity ( − F L , which flips the sign ofthe left-moving fermion but not that of the right. This Z transformation maps F [ m ] to F [ − m ], and it is a globalsymmetry of the theory at the transition point F . Moregenerally in the topological transition, we will denote this Z action as Z IFTO2 , since its action is to stack an IFTO. b. (2+1) d Z Topological Order
The ’t Hooftanomaly of a Z internal, unitary global symmetry in(1+1) d has a Z classification [56, 63–67]. The Z IFTO2 has odd units of the mod 8 anomaly (see Section VI).Consequently, there is no (1+1) d lattice UV completionof F such that Z IFTO2 is realized as an on-site symme-try. Instead, there is a (2+1) d lattice UV completion of F as a boundary theory, such that Z IFTO2 is realized asan on-site symmetry on the (2+1) d lattice [68, 69]. Tosee this, we note that the four-component partition func-tions for a fermionic CFT F is given by the partitionfunctions on the four sectors of boundary of the (2+1) d Z topological order ( i.e. the untwisted Z gauge theory[70]): Z F [ AE ] = Z , Z F [ P O ] = Z e ,Z F [ P E ] = Z m , Z F [ AO ] = Z f , (II.12)that are labeled by four types of anyons: 1 , e, m, f [71].Physically, the operators in each of the four sectors H E/OA/P become the operators on the (1+1) d boundary that liveat the end of the corresponding anyon in the coupledsystem. The Z IFTO2 transformation maps Z F [ P P ] to − Z F [ P P ] for the boundary theory on a torus. Thus,in the (2+1) d system, the IFTO-stacking Z IFTO2 symme-try is the electromagnetic duality (which is a 0-form Z symmetry of the Z gauge theory) that exchanges e and m but leaves 1 and f unchanged. III. INVERTIBLE FERMIONIC TOPOLOGICALORDER AND Z ORBIFOLD
In this section we provide an equivalent bosonic de-scription of the topological transition for fermionic CFTdiscussed in Section II B. Despite the notation might have suggested, Z IFTO2 is not a globalsymmetry of the IFTO (II.2) (but of the theory F ). The only Z symmetry of the IFTO is the fermion parity ( − F . A. Bosonization and Fermionization
We start by reviewing the procedure of bosonizationand fermionization in (1+1) dimensions that has beendeveloped in [34, 35, 37–39]. See [72] for related discus-sions on the lattice from a modern perspective. a. Fermion → Boson
Our starting point is a general(1+1)-dimensional fermionic QFT F with partition func-tion Z F [ ρ ]. A universal symmetry for any fermionic QFTis the fermion parity ( − F . The partition function witha nontrivial background field S for the ( − F is Z F [ S + ρ ].Next, we would like to gauge the ( − F to obtain abosonic theory B which is independent of the choice of thespin structure. We will promote the ( − F backgroundgauge field S to a dynamical gauge field s , and sum overit with an overall normalization factor g . The resultingpartition function of the bosonic theory is Z B = 12 g (cid:88) s Z F [ s + ρ ] . (III.1)Note that the RHS is independent of the choice of ρ .In (1+1) dimensions, gauging a Z symmetry (in thiscase, the fermion parity ( − F ) gives rise to a dual Z B symmetry [73] in the gauged theory. The partition func-tion of the bosonic theory B with a nontrivial dual Z B background field T is Z B [ T ] = 12 g (cid:88) s Z F [ s + ρ ] · exp (cid:20) i π (cid:18)(cid:90) s ∪ T + Arf[ T + ρ ] + Arf[ ρ ] (cid:19)(cid:21) . Indeed, one can check that the RHS is independent of thechoice of the spin structure ρ . We will call the bosonictheory B the bosonization of the fermionic theory F .Note that in this terminology, bosonization is a map froma fermionic theory to a bosonic theory, not an equivalenceof the two theories. In string theory, this is known as theGliozzi-Scherk-Olive projection [74]. b. Boson → Fermion
Suppose instead we start witha bosonic theory B with a non-anomalous Z B symmetry.How do we obtain a fermionic theory via gauging? Wefirst couple B to the Z B background field T in a way thatdepends on the choice of the spin structure: Z B [ T ] e i π Arf[ T + ρ ]+i π Arf[ ρ ] . (III.2)This can be interpreted as coupling the bosonic CFT B to the IFTO via the term Arf[ T + ρ ]. Next, we promotethe background field T to a dynamical field t , and obtain x h O ( x ) O (0) i Order operator (constant)Disorder operator x h O ( x ) O (0) i Disorder operator (constant)Order operator x h O ( x ) O (0) i | x | | x | x h O ( x ) O (0) i | x | | x | phase transitiontopological transition FIG. 2. Top: The Landau phase transition between the symmetry breaking phase (the order phase) and the symmetric phase(the disorder phase). Bottom: The topological transition between two families of gapless phases with power-law decayingcorrelation functions for both the order and the disorder operators. In one family, the scaling dimension of the order operatoris smaller than that of the disorder operator, i.e. ∆ ord < ∆ dis , while in the other family we have ∆ ord > ∆ dis . a fermionic theory that depends on the spin structure, Z F [ S + ρ ] = 12 g (cid:88) t Z B [ t ] · exp (cid:20) i π (cid:18) Arf[ t + ρ ] + Arf[ ρ ] + (cid:90) t ∪ S (cid:19)(cid:21) . In the resulting fermionic theory, the dual symmetry isidentified as the fermion parity ( − F , and S is its back-ground field. We will call the fermionic theory F the fermionization of the bosonic theory B with respect to Z B . This can be thought of as the continuum versionof the Jordan-Wigner transformation on the lattice [77].Using (A.2), we see that the fermionization (III.3) withrespect to the Z B symmetry is the inverse of bosonization(i.e. gauging ( − F ) (III.2). Importantly, the fermioniza-tion depends on a choice of a non-anomalous Z B globalsymmetry. Generally, a bosonic theory might have morethan one non-anomalous Z symmetries, and its fermion-ization might not be unique. For those not familiar about cup products, The condensed-matter oriented reference are [75] In the context of vertex operator algebra, the fermionization F is called a non-local Z B cover of B [76]. The fermionization described above does not hold when the Z B is anomalous, in which case the partition function Z [ A ] dependsnot just on the cohomology class of the background field A , butalso on the choice of the representative. c. Torus Partition Functions Let us apply thebosonization and fermionization procedures to torus par-tition functions of CFTs. In a bosonic CFT with a globalsymmetry Z B , we define Z B [ α x α t ] with α x , α t = 0 , α = 1) or without( α = 0) the Z B twists in the x and t directions. Thesetorus partition functions have the following trace inter-pretations: Z B [00] = Tr H (cid:104) q H + K ¯ q H − K (cid:105) ,Z B [01] = Tr H (cid:104) η q H + K ¯ q H − K (cid:105) ,Z B [10] = Tr (cid:101) H (cid:104) q H + K ¯ q H − K (cid:105) ,Z B [11] = Tr (cid:101) H (cid:104) η q H + K ¯ q H − K (cid:105) , (III.3)where H is the Hilbert space where all operators havethe periodic boundary condition, while (cid:101) H is the Hilbertspace where the Z B -even ( Z B -odd) operators have theperiodic (antiperiodic) boundary condition. Here η is the Z B charge operator. We will call H the untwisted sector and (cid:101) H the twisted sector with respect to Z B . Via theoperator-state correspondence, states in H are in one-to-one correspondence with the local operators, while statesin (cid:101) H are in one-to-one correspondence with the non-localoperators living at the end of the Z B line defect [54].Alternatively, we can consider the Z -even/odd sub-sectors H E/O of the untwisted sector H , and similarlythe Z B -even/odd subsectors (cid:101) H E/O of the twisted sector (cid:101) H . The associated torus partition functions are Z B [0 E ] = 12 ( Z B [00] + Z B [01]) ,Z B [0 O ] = 12 ( Z B [00] − Z B [01]) ,Z B [1 E ] = 12 ( Z B [10] + Z B [11]) ,Z B [1 O ] = 12 ( Z B [10] − Z B [11]) . (III.4)Following the fermionization procedure (III.3), we canrelate the four bosonic torus partition functions Z B [0 E ], Z B [0 O ], Z B [1 E ], Z B [1 O ] to the four fermionic torus par-tition functions Z F [ AE ] , Z F [ AO ] , Z F [ P E ] , Z F [ P O ] (de-fined in (II.4) and (II.5)): • Fermionization/Bosonization : Z F [ AE ] = Z B [0 E ] , Z F [ AO ] = Z B [1 O ] ,Z F [ P E ] = Z B [0 O ] , Z F [ P O ] = Z B [1 E ] . (III.5)As a special example, consider the case when F is themassless Majorana fermion and B is the Ising CFT. TheVirasoro primaries and their conformal weights ( h, ¯ h ) inthe four Hilbert spaces of the Majorana fermion CFT, orequivalently via (III.5), the four sectors of the Ising CFTHilbert space, are H EA [Maj] = H E [Ising] 1 : (0 , , ε : ( 12 ,
12 ) , H OA [Maj] = (cid:101) H O [Ising] ψ : ( 12 , , ¯ ψ : (0 ,
12 ) , H EP [Maj] = H O [Ising] σ : ( 116 ,
116 ) , H OP [Maj] = (cid:101) H E [Ising] µ : ( 116 ,
116 ) . (III.6)Here ψ, ¯ ψ are the left- and the right-moving Majoranafermions, ε = ψ ¯ ψ is the energy operator, σ is the spin (ororder) operator, and µ is the disorder operator.This bosonization/fermionization relation generalizesthe familiar relation between the Ising CFT and the Majorana fermion to any bosonic CFT with a non-anomalous Z B global symmetry and any fermionic CFT.In going from F to B , we gauge the ( − F of F to ob-tain a bosonic theory B with Z B symmetry via (III.2).Conversely, using (III.3), we gauge Z B (with a nontriv-ial coupling to the IFTO) of B to retrieve the fermionictheory F we start with. B. Gauging with the Invertible FermionicTopological Order
Starting with a fermionic theory F , consider anotherfermionic theory F (cid:48) defined as multiplying F by theIFTO (II.2): Z F (cid:48) [ S (cid:48) + ρ ] = Z F [ S (cid:48) + ρ ] e i π Arf[ S (cid:48) + ρ ] . (III.7)Here S (cid:48) is the background field for the ( − F symmetryin F (cid:48) . The partition functions for F and F (cid:48) are identicalon Riemann surfaces with even spin structures, but differby a sign on manifolds with odd spin structures. Nextwe gauge the ( − F of F (cid:48) to obtain a bosonic theory B (cid:48) following the recipe (III.2): Z B (cid:48) [ T (cid:48) ]= 12 g (cid:88) s (cid:48) Z F [ s (cid:48) + ρ ] · exp (cid:20) i π (cid:18) Arf[ s (cid:48) + ρ ] + (cid:90) s (cid:48) ∪ T (cid:48) + Arf[ T (cid:48) + ρ ] + Arf[ ρ ] (cid:19)(cid:21) (III.8)We can use (A.1) to rewrite the partition function of B (cid:48) as Z B (cid:48) [ T (cid:48) ] = 12 g (cid:88) s (cid:48) Z F [ s (cid:48) + ρ ] exp [i π Arf[ s (cid:48) + T (cid:48) + ρ ]](III.9)How is B (cid:48) related to B ? Let us consider the Z B orb-ifold of B , which has a dual Z symmetry. Its partitionfunction for B (cid:48) , with the background field T (cid:48) for the dual Z symmetry is g (cid:80) t Z B [ t ] e i π (cid:82) t ∪ T (cid:48) , where t is the dy-namical gauge field for Z B . It follows from (III.2) thatwe can write Z B (cid:48) in terms of the fermionic theory Z F :12 g (cid:88) t Z B [ t ] e i π (cid:82) t ∪ T (cid:48) = 12 g (cid:88) t g (cid:88) s Z F [ s + ρ ] exp (cid:20) i π (cid:18)(cid:90) t ∪ T (cid:48) + (cid:90) s ∪ t + Arf[ t + ρ ] + Arf[ ρ ] (cid:19)(cid:21) = 12 g (cid:88) s (cid:48) Z F [ s (cid:48) + ρ ] exp [i π Arf[ s (cid:48) + T (cid:48) + ρ ]] (III.10)where in the second line we have used (A.1) and renamed s as s (cid:48) . Matching (III.9) with (III.10), we have shownthat B (cid:48) is the Z B orbifold of B , i.e. B (cid:48) = B / Z B . a. (2+1) d Z Topological Order and the Electromag-netic Duality
The torus partition functions of B and B (cid:48) are related as follows: • Z B orbifold : Z B [0 E ] = Z B (cid:48) [0 E ] , Z B [0 O ] = Z B (cid:48) [1 E ] ,Z B [1 E ] = Z B (cid:48) [0 O ] , Z B [1 O ] = Z B (cid:48) [1 O ] . (III.11)The Z B -odd, untwisted sector H O and the Z B -even,twisted sector (cid:101) H E are exchanged under the Z B orb-ifold. The Z B orbifold in (1+1) d has a natural interpreta-tion from the (2+1) d Z gauge theory, which we explainbelow. In Section II B, we realize the four-componentfermionic partition functions in terms of the four bound-ary partition functions Z ,e,m,f of the (2+1) d Z topo-logical order, as in (II.12). The same four boundarypartition functions Z ,e,m,f also give rise to the four-component partition functions for a bosonic CFT with Z B symmetry [71, 78]: Z B [0 E ] = Z , Z B [0 O ] = Z e ,Z B [1 E ] = Z m , Z B [1 O ] = Z f . (III.12)The operators in each of the four sectors H E/O , (cid:101) H E/O become the operators on the (1+1) d boundary that liveat the end of the corresponding anyon in the coupled sys-tem. From the point of view of the (2+1) d Z topologicalorder, the Z B orbifold exchanges Z e and Z m , which isthe electromagnetic duality [43–45]. Therefore, the elec-tromagnetic duality of the (2+1) d Z topological orderinduces the Kramers-Wannier duality of the boundary(1+1) d bosonic CFTs.We summarize the above discussion in the commuta-tive diagram [7, 39] shown in Figure 3. Given any bosonic F F B B× IFTO Z B orbifold gauge ( − F fermionize Z B gauge ( − F fermionize Z B FIG. 3. The commutative diagram of fermionic CFTs F , F (cid:48) and their bosonizations B and B (cid:48) . CFT B with a non-anomalous Z B symmetry, we obtaintwo fermionic CFTs F and F (cid:48) that differ by an IFTO.Conversely, given any fermionic CFT F and its IFTO-stacked theory F (cid:48) , we obtain two bosonic CFTs B and B (cid:48) related by a Z B orbifold. Under the IFTO-stacking,the bosonization/fermionization, and the Z B orbifold, theHilbert spaces are permuted as in (II.6), (III.5), and(III.11), respectively. Both the fermionic CFT F andthe bosonic CFT B can be realized as the boundary ofthe (2+1) d untwisted Z gauge theory. For the fermionicboundary, the electromagnetic duality in the bulk ex-changing the electric e and the magnetic m anyons in- duces the IFTO stacking, while for the bosonic bound-ary, it induces the Z B orbifold ( i.e. the Kramers-Wannierduality). IV. TRANSITION BETWEEN BOSONIC CFTSAND THEIR Z ORBIFOLDS
In Section III, we see that the IFTO stacking offermionic CFTs is mapped to the Z B orbifold of bosonicCFTs via bosonization. Therefore, the topological tran-sition introduced in Section II B on the fermionic confor-mal manifold is equivalent to a one-dimensional bosonicconformal manifold where the CFTs are related by a Z B orbifold point-by-point across the transition point B . Inthis section we discuss several such bosonic examples,paving the way for their corresponding fermionic modelsin Section V. A. Kramers-Wannier Duality Defect a. Kramers-Wannier Duality
Let B (cid:48) → B → B bethe bosonization ( i.e. gauging ( − F ) of the topologi-cal transition F (cid:48) → F → F , and let Z B be the emergentsymmetry from gauging ( − F . The exactly marginal de-formation that interpolates between F and F (cid:48) survivesthe bosonization, and gives an exactly marginal deforma-tion interpolating between B and B (cid:48) . From Section III Band (II.3), we learn that B (cid:48) = B / Z B . (IV.1)In particular, at the origin of the deformation, thebosonic CFT B is self-dual under the Z B orbifold: B = B / Z B . (IV.2)In the example when F is a single Majorana fermion, itsbosonization B is the Ising CFT. The self-duality (IV.2)is then nothing but the Kramers-Wannier duality [40, 41].More generally, (IV.2) generalizes the familiar Kramers-Wannier duality to any bosonic CFT that is self-dualunder Z B orbifold. b. Topological Defect Line Our description of thebosonic model will rely on the topological defect lines ,which are one-dimensional extended objects in space-time. We will give a brief review on this subject, whilethe readers are referred to [53, 54] for a more complete in-troduction. Any global symmetry in (1+1) dimensions isassociated with a topological defect line that implementsthe symmetry action on local operators [7, 8]. However,not all topological defect line is associated with a globalsymmetry. Such topological defect line is called non-invertible , or non-symmetry [54]. One feature of the non-invertible topological defect line is that its action, whenrestricted to the local operators, is not invertible, andtherefore not group-like. More precisely, as we bring a N σ = N µ η (a) N ση = N µη (b) N ε = N− ε (c) ησ = η − σ (d) ηε = ηε (e) FIG. 4. The action of topological defect lines on local and non-local operators in the Ising CFT [49]. The Z B defect line η acts on operators with ± N exchanges the local, order operator σ ( z, ¯ z ) with the non-local, disorderoperator µ ( z, ¯ z ), which lives at the end of the Z B defect line η . non-invertible topological defect line past a local opera-tor, we might create a non-local operator.The Kramers-Wannier duality (IV.2) of the bosonicCFT B is implemented by such a non-invertible defect N , sometimes also called the duality defect [49–51, 53,54]. The duality defect N , together with the Z B defect η , form a fusion category known as the Z Tambara-Yamagami (TY) category [79] with the Ising fusion rules: η = I , N = 1 + η , N η = η N = N , (IV.3)where I is the trivial topological line. Let us discuss the duality defect in the Ising CFT[49] in details as an example. In the Ising sector, wewill denote the energy operator as ε ( z, ¯ z ) with confor-mal weights ( h, ¯ h ) = ( , ), and the spin operator (alsoknown as the order operator) as σ ( z, ¯ z ) with conformalweights ( h, ¯ h ) = ( , ). As we sweep the duality defect N past the energy operator ε , the latter obtains a minussign. On the other hand, as we sweep the duality de-fect N past the spin order σ ( z, ¯ z ), a Z B line η is createdwith the disorder operator µ ( z, ¯ z ) sitting at the endpoint.Therefore the duality defect N exchanges the order oper-ator σ ( z, ¯ z ) (which is a local operator) with the disorderoperator µ ( z, ¯ z ) (which is a non-local operator attachedto a line). See Figure 4. c. Duality Interface Now consider the bosonic CFT B with a duality defect N inserted along the time di-rection (see Figure 5). Let O be an exactly marginal Given the above fusion rules, there are two solutions to the pen-tagon identities for the F -moves. One of them is realized in theIsing CFT, and other is realized in the su (2) WZW model. See[79] for their respective F -symbols. deformation, and O (cid:48) be the local operator obtained bysweeping the duality defect N past O . We now turnon the exactly marginal deformation O to the left of N ,and O (cid:48) to the right of N . The deformation drives the sys-tems on the two sides to two different CFTs, B and B (cid:48) ,and the duality defect N becomes a topological dualityinterface between the two CFTs. The duality interfaceimplements the duality B (cid:48) = B / Z B , but it is not a topo-logical defect line in either B or B (cid:48) .Under fermionization, the duality defect N of thebosonic CFT B becomes the Z IFTO2 symmetry in thefermionic CFT F . See Figure 6 and Section VI for morediscussions. By turning on the deformation O and O (cid:48) ,the Z IFTO2 symmetry defect becomes an interface sepa-rating two fermionic theories F and F (cid:48) that differ by anIFTO. See Figure 5.To summarize, the topological transition of fermionicCFT can be equivalently recast into the following bosonicdata (see Figure 7): • A bosonic CFT B that is self-dual under gauginga non-anomalous Z B global symmetry, i.e. B = B / Z B . The self-duality is implemented by a dual-ity defect N . • An exactly marginal deformation O . Here we assume O (cid:48) (cid:54) = O . If O (cid:48) = O , i.e. if the duality defect N commutes with O , then B = B (cid:48) and F = F (cid:48) , and there is nointeresting topological transition to discuss. B B e i g ∫ d z O ′ e i g ∫ d z O N (cid:212)⇒ N BB ′ F F e i g ∫ d z O ′ e i g ∫ d z O Z IFTO2 (cid:212)⇒ Z IFTO2 FF ′ FIG. 5. The duality defect N in the self-dual bosonic CFT B becomes a duality interface between B and B (cid:48) after renormalizationgroup flow. The duality defect N in B turns into the Z IFTO2 symmetry defect of F under fermionization.Fermionic CFT F Bosonic CFT B fermion parity dual ←−−−−−→ non-anomalous( − F Z B IFTO-stacking non-symmetry extension −−−−−−−−−−−−−−−−→
Duality defect Z IFTO2 N FIG. 6. Under bosonization/fermionization, the fermion parity ( − F is the dual symmetry of Z B , while the IFTO stacking Z IFTO2 symmetry of F is extended to a non-invertible duality defect N in B . BB ′ FF ′ F OO ′ OO ′ B Z B orb. × IFTO Fermionize Z B Gauging (− ) F FIG. 7. The topological transition on the fermionic conformalmanifold can be equivalently bosonized to a family of bosonicCFTs that are related by the Z B orbifold. B. Free Compact Boson S Our first example is the c = 1 free compact boson the-ory (see, for example, [80] for a review). The conformalmanifold of c = 1 CFTs consists of two branches, the S branch and the S / Z branch, together with three iso-lated points. See Figure 8. In this section we start withthe S branch, which has the description of the free com-pact boson X ( z, ¯ z ) = X L ( z ) + X R (¯ z ) with identification X ( z, ¯ z ) ∼ X ( z, ¯ z ) + 2 πR with R ≥
1. Our convention forthe radius R is such that T-duality acts as T-duality : S [ R ] = S (cid:20) R (cid:21) . (IV.4)The free boson field is normalized such that X ( z, ¯ z ) X (0 , ∼ − log | z | . On the S branch, Our convention for the radius is related to that in [80] as R Ginsparg (cid:48) s = R Ours / √ u (1) × u (1) chiral algebra generatedby the currents ∂X ( z ) and ¯ ∂X (¯ z ). At a generic radius,there is one exactly marginal operator generating theconformal manifold: O = ∂X ¯ ∂X . (IV.5) T O I R S R S / Z √ √ Z orb: R S → / R S Z orb: R S / Z → / R S / Z FIG. 8. The conformal manifold of the bosonic c = 1 CFTs.The conformal manifold has an S branch labeled by R S ≥ S / Z labeled by R S / Z ≥
1. The endpoint of S branch at R S = 1 is the su (2) WZW model. Thetwo branches meet at the Kosterlitz-Thouless point, whichis described by R S = 2 or equivalently by R S / Z = 1.The bosonized Dirac fermion and the Ising theories are theself-dual points of the Z B symmetries defined in (IV.11) and(IV.16), respectively. The su (2) / Γ orbifold models for Γ =T,O, I are isolated points in the moduli space. T, O, I each rep-resents the tetrahedral, octahedral and icosahedral groups. a. Primary Operators
The local primary operatorswith respect to the u (1) × u (1) chiral algebra are V n,w ( z, ¯ z )= exp (cid:104) i (cid:16) nR + wR (cid:17) X L ( z ) + i (cid:16) nR − wR (cid:17) X R (¯ z ) (cid:105) , (IV.6)which are labeled by two integers, the momentum num-ber n ∈ Z and the winding number w ∈ Z . The conformalweights of V n,w are h = 14 (cid:16) nR + wR (cid:17) , ¯ h = 14 (cid:16) nR − wR (cid:17) . (IV.7) The torus partition function Z S ( R ) is therefore: Z S ( R ) = 1 | η ( q ) | (cid:88) n,w ∈ Z q ( nR + wR ) ¯ q ( nR − wR ) . (IV.8)The global symmetry at a generic radius contains( U (1) n × U (1) w ) (cid:111) Z , where the Z acts as X → − X .The U (1) n and U (1) w correspond to momentum andwinding, which act by phases e i nθ and e i wθ on the pri-mary operator (IV.6), respectively. They act on X L ( z )and X R (¯ z ) by shifts: U (1) n : X L ( z ) → X L ( z ) + R θ n ,X R (¯ z ) → X R (¯ z ) + R θ n ,U (1) w : X L ( z ) → X L ( z ) + 12 R θ w ,X R (¯ z ) → X R (¯ z ) − R θ w , (IV.9)with θ n,w ∼ θ n,w + 2 π . Both U (1) n and U (1) w are non-anomalous for all R . In particular, this implies that U (1) n is neither holomorphic nor anti-holomorphic atany radius R . The same is true for U (1) w . When R is ra-tional, certain integral combination of U (1) n and U (1) w becomes holomorphic or anti-holomorphic, and the CFTenjoys an enhanced chiral algebra.Let Z (1 , and Z (0 , be the Z subgroups of U (1) n and U (1) w , which act on the primary operators by signs e i πn and e i πw , respectively. There is no ’t Hooft anomaly forthe momentum Z (1 , , nor for the winding Z (0 , alone,but there is a mixed anomaly between the momentum Z (1 , and the winding Z (0 , . b. Twisted Sector (cid:101) H Let us discuss the non-local op-erators that live in the twisted sector with respect to a Z global symmetry. The twisted sector operators of Z ( m ,m )2 ( m i = 0 ,
1) are given by the same form as(IV.6), but generally with fractional momentum ˜ n andwinding number ˜ w : V ˜ n, ˜ w ( z, ¯ z ) : ˜ n ∈ m Z , ˜ w ∈ m Z . (IV.10)In other words, the momentum Z (1 , twist makes thewinding number fractional due to the mixed anomaly,and vice versa. c. Z B Orbifold of S We will choose the Z B symme-try to be the momentum Z (1 , : Z B : V n,w → ( − n V n,w . (IV.11)The Z B orbifold of the c = 1 compact boson theory atradius R is another compact boson at radius R/
2, which Here, η ( q ) is the Dedekind eta function defined as η ( q ) = q / (cid:81) ∞ i =1 (1 − q i ). R : S [ R ] Z B = S (cid:20) R (cid:21) . (IV.12)We compute the twisted torus partition functions of the S [ R ] theory and give the proof of this relation in Ap-pendix B 1.The fixed point of this orbifold, i.e. the Kramers-Wannier self-dual point, is at R = √
2, which is describedby the bosonization of a Dirac fermion. This is our firstexample of a family of bosonic CFTs related by the Z B orbifold.The Kramers-Wannier duality is implemented by the(0+1) d duality defect line N satisfying (IV.11). As webring a local operator V n,w past through N , it is mappedto (see Section V A for derivation) V n,w → V − w, − n . (IV.13)The righthand side is only a local operator if n ∈ Z , i.e. when V n,w is Z B even. On the other hand, when V n,w is Z B odd, i.e. n ∈ Z + 1, it is mapped to a non-local operator living at the end of the Z B line η . This isindeed the characteristic way how a duality defect acts onoperators. For example, the duality defect N in the IsingCFT maps the local, Z B -odd, order operator σ to thenon-local, disorder operator µ , while it maps the local, Z B -even, energy operator ε to itself with a sign. See [81]for a thorough discussion on the topological defect linesin c = 1 CFTs. C. Bosonic S / Z Orbifold
The next example is the c = 1 theory S / Z [ R ] definedas the Z orbifold of the c = 1 compact free boson theoryat radius R , where the Z acts as X → − X . The exactlymarginal operator is again (IV.5).The torus partition function of the c = 1 S / Z orb-ifold theory at radius R is Z S / Z ( R ) = 12 Z S ( R ) + (cid:12)(cid:12)(cid:12)(cid:12) η ( q ) θ ( q ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) η ( q ) θ ( q ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) η ( q ) θ ( q ) (cid:12)(cid:12)(cid:12)(cid:12) , (IV.14)where the first two terms come from the untwisted sectorand the last two terms come from the twisted sectors ofthe S theory. The latter comes from the two twist fields σ , ( z, ¯ z ), corresponding to the two fixed points of S / Z and their descendants. Both σ , have h = ¯ h = .At R = √
2, the S / Z theory is equivalent to twocopies of the Ising CFT. At this point, the two twist Here θ ( q ) = 2 q / (cid:81) ∞ i =1 (1 − q i )(1 + q i ) , θ ( q ) = (cid:81) ∞ i =1 (1 − q i )(1 + q i − / ) , θ ( q ) = (cid:81) ∞ i =1 (1 − q i )(1 − q i − / ) , and η ( q ) isthe Dedekind eta function as given in footnote 13. fields σ , are the spin operators of the two Ising CFTs.The exactly marginal operator O in (IV.5) becomes O ( z, ¯ z ) = ε ( z, ¯ z ) ε ( z, ¯ z ) , (IV.15)where ε i ( z, ¯ z ) is the energy operator of weight ( h, ¯ h ) =( , ) of the i -th Ising CFT. a. Z B Orbifold of S / Z At a generic radius of the S / Z theory with radius R ≥
1, the theory has a Z B symmetry [82]: Z B : σ → − σ , σ → σ , V n,w → ( − n V n,w . (IV.16)At the Ising point, Z B is just the Z symmetry of one ofthe Ising CFT. The theory enjoys the Kramers-Wannierduality for each copy of the Ising CFT: S / Z (cid:2) R = √ (cid:3) Z B = S / Z [ R = √ . (IV.17)The Kramers-Wannier duality is implemented by a du-ality defect N , which flips the sign of ε and maps theorder operator σ to the disorder operator µ .The exactly marginal deformation O = ε ε is odd theduality defect N . This implies that starting from the B ≡ Ising point, the theory B deformed by + O andthe theory B (cid:48) deformed by −O are related to each otherby the Z B orbifold. The radii of the theories on two sidescan be worked out to be S / Z [ R ] Z B = S / Z (cid:20) R (cid:21) ( R ≥ . (IV.18)We show this equality explicitly at the level of the toruspartition function in Appendix B 2. D. Bosonic T CFT
The third example is the c = 2 CFT whose targetspace is a torus T . The conformal manifold is four-dimensional, and we will identify a particular locus alongwhich the family of CFTs are related to each other by Z B orbifold. Our exposition follows [83] (see also [84] for theclassification of rational points on the conformal manifoldof the T CFT).We will normalize the two scalar fields to have period-icities X ( z, ¯ z ) ∼ X ( z, ¯ z ) + 2 πR , X ( z, ¯ z ) ∼ X ( z, ¯ z ) +2 πR . The metric and the B field of the T CFT will be If we choose to describe the same theory in the T-dual frame withradius R T = 1 /R ≤
1, then the Z B symmetry acts on V n,w inthe T-dual frame by a phase ( − w because T-duality exchanges n with w . Using T-duality we can rewrite (IV.18) as S / Z [ R ] Z B = S / Z [ R/ Z B in (IV.16) is notT-duality invariant, so S / Z [ R ] Z B (cid:54) = S / Z [1 /R ] Z B . G ij and B ij with i, j = 1 ,
2, parametrizingthe conformal manifold of the T CFT. Since we onlyhave two scalars, there is only one B field, b ≡ B . The B field modulus is periodic, b ∼ b + 1.The metric moduli includes the K¨ahler modulus R andthe complex structure moduli τ . The latter is encoded in G ij as G ij = (cid:18) τ τ | τ | (cid:19) , (IV.19)where τ = τ + i τ and | τ | = τ + τ . The complexstructure moduli τ are subject to the P SL (2 , Z ) identifi-cation. Let us summarize the exactly marginal deforma-tions of the T CFT:
R > , τ ∼ a τ + bc τ + d , b ∼ b + 1 , (IV.20)where a , b , c , d ∈ Z and ad − bc = 1. There are also T-duality identifications but we will not discuss them here. a. Primary Operators The local primary operators V n ,w ,n ,w ( z, ¯ z ) of the u (1) L × u (1) R current algebra arelabeled by four integers, two momentum numbers n , n and two winding numbers w , w . Its conformal weightsare given as follows. Let v i ≡ n i R − B ij w j R , (IV.21)Next we define v iL = v i + w i R , v iR = v i − w i R , wherethe indices are raised and lowered by G ij and G ij . Theconformal weights of V n ,w ,n ,w are h = 14 G ij v iL v jL , ¯ h = 14 G ij v iR v jR . (IV.22)The Lorentz spin of the operator is s = h − ¯ h = n i w i . Theprimary operator V n ,w ,n ,w can be written in terms ofthe left- and right-moving compact bosons as V n ,w ,n ,w ( z, ¯ z ) = exp (cid:34) (cid:88) i =1 i v iL X iL ( z ) + i v iR X iR (¯ z ) (cid:35) (IV.23)There are special points on the conformal manifoldwhere the CFT is described by the WZW model: • su (2) × su (2) = so (4) : R = 1 , ( τ , τ ) =(0 , , b = 0 . • su (3) : R = 1 , ( τ , τ ) = ( , − √ ) , b = 1 / . • su (2) × u (1) : R = 1 , ( τ = 0 , τ = √ , b = 0 . To distinguish the target space torus from the spacetime torus,we use τ for the complex structure of the former, while τ for thelatter. For the spacetime torus, we also use q = e π i τ . b. Z B Self-Dual Locus
We will be interested in aparticular non-anomalous Z B global symmetry that ex-ists at any point on the conformal manifold of the T CFT. Its action on the local operator is Z B : V n ,w ,n ,w → ( − n + w + n + w V n ,w ,n ,w . (IV.24)At the special point of su (2) × su (2) , this Z B symmetryis the diagonal subgroup of the center Z ’s of the two left-moving su (2)’s. We discuss the twisted torus partitionfunctions of the T CFT with respect to Z B in AppendixB 3.Now consider two one-dimensional loci B and B (cid:48) on theconformal manifold, joining at the su (2) × su (2) point( R = 1 , τ = 0 , τ = 1 , b = 0): B [ τ ] : R = 1 , τ = 0 , τ ≥ , b = 0 , B (cid:48) [ b ] : R = 1 , τ = b , τ = (cid:112) − b , > b ≥ . (IV.25)The two families of theories B and B (cid:48) are obtained from su (2) × su (2) by the following two exactly marginaldeformations O and O (cid:48) , respectively: O ( z, ¯ z ) = ∂X ( z ) ¯ ∂X (¯ z ) , O (cid:48) ( z, ¯ z ) = ∂X ( z ) ¯ ∂X (¯ z ) . (IV.26)Note that each O and O (cid:48) preserves a copy of the su (2) × u (1) × su (2) × u (1) current algebra, but they preservedifferent subalgebras of su (2) × su (2) × su (2) × su (2). Moreexplicitly, the su (2) × su (2) currents along the B [ τ ] pathare: B [ τ ] : ( h, ¯ h ) = (1 ,
0) : J x L ± i J y L ∼ V ( ± , ± , , ,J z L ∼ i ∂X ( z ) , ( h, ¯ h ) = (0 ,
1) : J x R ± i J y R ∼ V ( ± , ∓ , , ,J z R ∼ i ¯ ∂X (¯ z ) . (IV.27)On the other hand, those on the B (cid:48) [ b ] path are B (cid:48) [ b ] : ( h, ¯ h ) = (1 ,
0) : J x L ± i J y L ∼ V ( ± , ± , , ,J z L ∼ i ∂X ( z ) , ( h, ¯ h ) = (0 ,
1) : J x R ± i J y R ∼ V (0 , , ± , ∓ ,J z R ∼ i ¯ ∂X (¯ z ) . (IV.28)Below we will show that B (cid:48) [ b ] = B [ τ ] Z B , τ = (cid:114) b − b . (IV.29)The structure of this one-dimensional locus on the con-formal manifold is shown in Figure 9. To show (IV.29), The theory B [ τ ] with τ ≤ τ ≥ B (cid:48) [ b ] with b < [ su ( ) ] τ b = τ su ( ) su ( ) × u ( ) B[ τ ] B ′ [ b ] Z B orbifold FIG. 9. A one-dimensional locus on the conformal manifoldof the T CFT. The two families of CFTs B [ τ ] and B (cid:48) [ b ] arerelated by gauging Z B point-by-point with τ = (cid:113) b − b . we will write down a one-to-one map between the op-erators V (cid:48) n (cid:48) ,w (cid:48) ,n (cid:48) ,w (cid:48) of B (cid:48) [ b ] and the untwisted andthe twisted sectors of the orbifold theory B [ τ ] / Z B forall τ ≥
1. Let us start with the untwisted sector,which consists of Z B even operators V n ,w ,n ,w satis-fying n + w + n + w ∈ Z . The untwisted sectoroperators are mapped to V (cid:48) n (cid:48) ,w (cid:48) ,n (cid:48) ,w (cid:48) of B (cid:48) [ b ] as n (cid:48) = 12 ( n + w − n + w ) ,w (cid:48) = 12 ( n + w + n − w ) ,n (cid:48) = 12 ( n − w − n − w ) ,w (cid:48) = 12 ( − n + w − n − w ) . (IV.30)Note that since the untwisted sector operators satisfy n + w + n + w ∈ Z , the resulting n (cid:48) i , w (cid:48) i are integers.Furthermore, n (cid:48) + w (cid:48) + n (cid:48) + w (cid:48) ∈ Z , so these operatorsare even under the Z B (cid:48) symmetry in B (cid:48) [ b ].The rest of the V (cid:48) n (cid:48) ,w (cid:48) ,n (cid:48) ,w (cid:48) operators come from the Z B -even, twisted sector of B / Z B , which consists of oper-ators V ˜ n , ˜ w , ˜ n , ˜ w with ˜ n i , ˜ w i ∈ + Z and ˜ n ˜ w + ˜ n ˜ w ∈ Z . The latter two conditions imply that ˜ n , ˜ w , ˜ n , ˜ w ∈ Z + 1. The twisted sector operator are mapped to V (cid:48) n (cid:48) ,w (cid:48) ,n (cid:48) ,w (cid:48) of B (cid:48) [ b ] by the same map (IV.30) but withtilde on the righthand side. These V (cid:48) n (cid:48) ,w (cid:48) ,n (cid:48) ,w (cid:48) ’s have n (cid:48) + w (cid:48) + n (cid:48) + w (cid:48) ∈ Z + 1, so they are odd under the Z B (cid:48) symmetry in B (cid:48) [ b ].We have therefore shown that the spectrum of local op-erators of B [ τ ] / Z B is isomorphic to B (cid:48) [ b ] with the moduliidentified as τ = (cid:113) b − b .The duality between B and B (cid:48) is implemented by a (0+1)-dimensional duality interface. At the su (2) × su (2) point, this duality interface becomes a duality de-fect N . From (IV.26), we see that O and O (cid:48) are relatedby exchanging the X R with X R (but leaving X , L as theywere). Naively, one might think this exchange action isa Z global symmetry of the full (bosonic) theory. Thisis, however, not true, because we cannot consistently ex-tend such an exchange action to an invertible map fromlocal operators to local operators. For example, this ex-change action would have mapped the primary operator V n ,w ,n ,w to V n w n − w , n w − n w , n w n − w , n w − n w . (IV.31)Similar to the compact boson CFT, the righthand sideis only a local operator if n + w + n + w ∈ Z , i.e. when V n ,w ,n ,w is Z B even. On the other hand, when V n ,w ,n ,w is Z B odd, it is mapped to a non-local oper-ator in the twisted sector of Z B . Therefore, we concludethat N acts on local operators in a non-invertible way,which is a characteristic feature of a duality defect. V. FERMIONIC MODELS
In this section, we discuss the fermionic dual of theKramers-Wannier transitions on the bosonic conformalmanifold. See [34, 85] for discussions on the c = 1fermionic (spin) CFTs. A. Dirac Fermion
The simplest example of a topological transition onthe conformal manifold is to take F to be the c = 1 freemassless Dirac fermion, which is equivalent to two left-moving Majorana fermions ψ iL ( z ) and two right-movingMajorana fermions ψ iR (¯ z ), with i = 1 , Let Ψ
L,R ≡ ψ L,R + i ψ L,R , Ψ † L,R ≡ ψ L,R − i ψ L,R . The theory has aleft and a right u (1) × u (1) current algebras generated byΨ L ( z )Ψ † L ( z ) and Ψ R (¯ z )Ψ † R (¯ z ), respectively. The Diracfermion theory is the fermionization of the c = 1 compactboson discussed in Section IV B at R = √ Z B = Z (1 , symmetry defined in (IV.11).There is one exactly marginal deformation, theThirring deformation: O ( z, ¯ z ) = Ψ L ( z )Ψ † L ( z )Ψ R (¯ z )Ψ † R (¯ z ) . (V.1) As a fermionic theory, there is no distinction between a Diracfermion and two Majorana fermions. However, there are differentways to sum over the spin structures when trying to obtain abosonic theory. In [86], the authors use “Dirac fermion” and “twoMajorana fermions” to refer to two different ways of summingover the spin structures, corresponding to the R = √ R = √ S / Z (discussed in Section IV C), respectively. R of the compact boson theory was derived in [87].We will therefore use R to denote exactly marginal cou-pling for (V.1) and denote the deformed Dirac fermiontheory as Dirac[ R ]. In particular, Dirac[ √
2] is the free,massless Dirac fermion. a. Topology of the Dirac Branch
Let us commenton the global topology of the conformal manifold for theDirac fermion branch parametrized by R . We start bynoting that, due to the T-duality S [ R ] = S [1 /R ], the S branch of the compact boson theory is a half-line, withthe endpoint located at R = 1, i.e. the su (2) WZWmodel. On the fermion side, Dirac[ R ] is the fermion-ization of S [ R ] with respect to Z B = Z (1 , . Since themomentum Z B = Z (1 , is exchanged with the winding Z (0 , under the T-duality, the fermionization does not commutes with the T-duality of the bosonic theories.Therefore Dirac[ R ] (cid:54) = Dirac[1 /R ]. Consequently, thetopology of the Dirac branch of the c = 1 fermionic CFTis R instead of a half line [34]. See Figure 10. b. Z IFTO2
Symmetry
The Z IFTO2 symmetry is definedas Z IFTO2 : Ψ L ( z ) → Ψ † L ( z ) , Ψ † L ( z ) → Ψ L ( z ) , Ψ R (¯ z ) → Ψ R (¯ z ) , Ψ † R (¯ z ) → Ψ † R (¯ z ) , (V.2)which is the particle-hole transformation Z C L on the left-moving fermion operators. Given a fixed spin structure,if we treat the Dirac fermion as two Majorana fermions,then Z IFTO2 is the chiral fermion parity ( − F L for onecopy of the Majorana fermion. Note that the exactlymarginal operator O is Z IFTO2 -odd.In Section IV B we showed that the bosonic theory S [ R ] is related to S [2 /R ] by the Z B orbifold. It followsfrom the commutative diagram in Figure 3 that: Z Dirac [ ρ, R ] = Z IFTO [ ρ ] Z Dirac [ ρ, /R ] , (V.3)Hence as we move along the one-dimensional conformalmanifold generated by O form R < √ R > √
2, themodels on the two sides F and F (cid:48) of a free massless Diracfermion differ by an IFTO [34]. This is our first exam-ple of a topological transition on the fermionic conformalmanifold. Table I shows the operator spectrum of c = 1fermionic CFTs.Let us describe the local operators of the fermionicmodel in terms of V n,w . In the anti-periodic ( A ) sectorof the Dirac fermion, the local operators come from (1) Put differently, we can define another family of fermionic theo-ries, denoted as (cid:94)
Dirac[ R ], by fermionizing S [ R ] with respect to Z (0 , . Then (cid:94) Dirac[ R ] = Dirac[1 /R ]. One can alternatively regard the IFTO as a (1+1) d local coun-terterm, and identify Dirac[ R ] with Dirac[2 /R ]. From this per-spective the Dirac branch of the fermionic conformal manifold isagain a half line, but the origin is now located at the Dirac point R = √ su (2) point R = 1. the Z B -even, local operators plus (2) the Z B -odd non-local operators from the Z B twisted sector (see (III.5)).The former are operators V n,w with n ∈ Z , w ∈ Z , whilethe latter are V ˜ n, ˜ w with ˜ n ∈ Z + 1, ˜ w ∈ + Z . We haveused the fact that the twisted sector operator V ˜ n, ˜ w is Z B -even if the spin s = ˜ n ˜ w is an integer, while it is Z B -oddif s is a half-integer [54, 78, 88]. We identify the fermionoperators asΨ L ( z ) = V , , Ψ † L ( z ) = V − , − , ( h, ¯ h ) = (cid:18) , (cid:19) , Ψ R (¯ z ) = V , − , Ψ † R (¯ z ) = V − , , ( h, ¯ h ) = (cid:18) , (cid:19) . (V.4)It is then straightforward to see that Z IFTO2 acts as Z IFTO2 : V n,w ( z, ¯ z ) → V − w, − n ( z, ¯ z ) . (V.5)For example, Z IFTO2 exchanges V , / = Ψ L ( z ) with V − , − / = Ψ † L ( z ), but leaves V , − / = Ψ R (¯ z ) with V − , / = Ψ † R (¯ z ) invariant. Note that Z IFTO2 anticom-mutes with the left-moving current algebra generatorΨ L ( z )Ψ † L ( z ) but commutes with the right-moving oneΨ R (¯ z )Ψ † R (¯ z ).What happens when we extend the Z IFTO2 symmetryin the fermionic theory to its bosonization? In the lat-ter, the local primary operators are labeled by integral n and w . Hence the Z IFTO2 symmetry in the fermionictheory does not extend to an action that maps a localoperator to another local operator. In fact, it maps the Z B -odd operators ( i.e. those with odd n ) to a non-localoperator in the Z B -twisted sector. This is indeed how thenon-invertible duality defect N acts as discussed in Sec-tion IV B. We have therefore demonstrated how a sym-metry action in a fermionic theory is extended to a non-invertible defect under bosonization (see Section VI). c. Order and Disorder Operators From the bosonic S [ R ] point of view, it is natural to identify the lightest Z B -odd, local operator V , in H O as the order operator,while the lightest Z B -even, non-local operator V , − / in (cid:101) H E as the disorder operator:Order: O ord = V , , ∆ ord = 12 R Disorder: O dis = V , − , ∆ dis = R . (V.6)That the order and disorder fields are mutually non-localin the field theory with U (1) symmetry has been studiedfor long [89–92] in one dimension, and more recently inhigher dimension [93, 94]. Here we identify the orderfield as through the study of (non-anomalous) Z globalsymmetry. It does not require U (1) symmetries. Notethat both O ord and O dis are scalar operators, i.e. s = h − ¯ h = 0. From the fermionic CFT Dirac[ R ] point of view,both O ord and O dis are in the P sector, with opposite( − F quantum numbers. At the Dirac point R = √ RR = √ R = Z B orb: R → / RS BosonDirac RR = √ R → / R FIG. 10. The conformal manifold of the c = 1 compact boson theory S [ R ] (top) and that of the Dirac fermion perturbed bythe Thirring coupling Dirac[ R ] (bottom). The former is a half-line, while the latter is a full line.bosonic sector fermionic sector range of n range of w example operators H E H EA Z Z V , = Ψ L Ψ R , V , − = Ψ † L Ψ R H O H EP Z + 1 Z O ord = V , (cid:101) H E H OP Z + Z O dis = V , − (cid:101) H O H OA Z + 1 + Z Ψ L , Ψ † L , Ψ R , Ψ † R TABLE I. Operators V n,w in the c = 1 fermionic CFT Dirac[ R ]. This is analogous to (III.6) for the Ising CFT and the Majoranafermion. Together with chiral fermion operator Ψ L and by operator product expansion, they generate other sectors from H EA , O e : H EA → H EP , O m : H EA → H OP , Ψ L : H EA → H OA . Z IFTO2 is a global symmetry that exchanges the orderwith the disorder operators: Z IFTO2 : O ord (cid:55)→ O dis . (V.7)Indeed, the Z IFTO2 becomes the Kramers-Wannier dualitydefect under bosonization.In the topological transition from Dirac[
R < √
2] toDirac[
R > √ (cid:104) O ord ( z, ¯ z ) O ord (0) (cid:105) = 1 | z | ord , (cid:104) O dis ( z, ¯ z ) O dis (0) (cid:105) = 1 | z | dis . (V.8)The exponents of the power law fall-off obey∆ ord > ∆ dis , R < √ , ∆ ord < ∆ dis , R > √ . (V.9)When R < √
2, the two-point function of the order op-erator O ord approaches zero asymptotically faster thanthat of the disorder operator, and vice versa for R > √ d. Ultraviolet Realization on the Lattice The com-plex fermion CFT arises as a the infrared (IR) descrip-tion of the Luttinger liquid [95–98], a (1+1) d spinlesselectron system appearing in condensed matters. In theLuttinger liquid at a generic filling, the UV symmetry is G UV = U (1) C × U (1) trn , where U (1) C is the total chargeconservation, U (1) trn is the translation symmetry. Theoperator V n,w ( z, ¯ z ) has charge q C = n, q trn = − wk F ,where k F is the Fermi momentum. This is determinedfrom the fact that Ψ L ( z ) carries q C = 1 , q trn = − k F ,and Ψ R (¯ z ) carries q C = 1 , q trn = k F . In particular,the fermion bilinear Ψ L ( z )Ψ R (¯ z ) has charge q C = 2and Ψ † L ( z )Ψ R (¯ z ) has charge q trn = 2 k F . There is nosymmetric relevant operator in the A sector satisfying∆ < , q C = q trn = 0. The Thirring deformation O in (V.1) is the symmetric perturbation with the lowestweight. Tuning its coupling from negative to positivethrough the free Dirac fermion, drives a topological phasetransition differing by an IFTO.In the lower energy theory of the Luttinger liquid, V , = Ψ L Ψ R carries q C = 2 , q trn = 0, representingthe superconducting (SC) order operator, while V , − =7Ψ † L Ψ R carries q C = 0 , q trn = 2 k F , representing the chargedensity wave (CDW) order operator. At the Dirac point R = √
2, the Z IFTO2 acts as Z IFTO2 : V , = Ψ L Ψ R (cid:55)→ V , − = Ψ † L Ψ R (V.10)This is the well-known duality between the SC order andthe CDW order in the Luttinger liquid. The duality ex-changes the Luttinger parameter K ↔ /K , which mapsthe K <
K > K is related to thecompact boson radius R as R = √ K .) Meanwhile, the Z IFTO2 exchanges the order and the disorder operatorsas in (V.7). We have therefore demonstrated that theattractive/repulsive duality in the bosonized Luttingerliquid is parallel to the Kramers-Wannier duality in theIsing CFT.
B. Majorana × Ising
Next we consider a fermionic c = 1 CFT that is thedirect product of a single Majorana fermion ψ L ( z ) , ψ R (¯ z )and the bosonic Ising CFT, i.e. F = Maj × Ising.The Z IFTO2 symmetry is taken to be the left-movingfermion parity ( − F L that flips the sign of ψ L ( z ) butnot that of ψ R (¯ z ): Z IFTO2 = ( − F L : ψ L ( z ) → − ψ L ( z ) , ψ R (¯ z ) → ψ R (¯ z ) . (V.11)The theory F has one exactly marginal operator O ( z, ¯ z ) = ψ L ( z ) ψ R (¯ z ) ε ( z, ¯ z ) , (V.12)which is odd under Z IFTO2 . We immediately learn that Z Maj × Ising [ ρ, R ] = Z IFTO [ ρ ] Z Maj × Ising [ ρ, /R ] , (V.13)where we have used the radius R of S / Z to representthe coupling of O . Again, we have shown in SectionIV C that the corresponding bosonic theories B and B (cid:48) are related by the Z B orbifold.There is no continuous internal global symmetry in F . The discrete symmetry is D [82], whose action onthe twist fields is generated by σ → − σ , σ → σ and σ ↔ σ . There is, however, a D -invariant relevant de-formation, ε + ε . C. Four Majorana Fermions
Let us take F to be a free theory of four Majo-rana fermions, which is the fermionization of the c = 2model in Section IV D. We will denote the four left-moving (right-moving) Weyl fermions as Ψ L, ± , Ψ † L, ± (Ψ R, ± , Ψ † R, ± ). (The meaning of the subscripts ± willbe explained momentarily from the lattice realization.) τ b = τ Maj F[ τ ] F ′ [ b ] × IFTO
FIG. 11. A one-dimensional locus on the conformal manifoldof four Majorana fermions. The two families of CFTs F [ τ ]and F (cid:48) [ b ] differ by an invertible fermionic topological order. These fermion operators can be written in terms of theexponential operators V n ,w ,n ,w in (IV.23) asΨ L, ± = V ± , ± , , = e i ( ± X L + X L ) , Ψ † L, ± = V ± , ± , − , − = e i ( ∓ X L − X L ) , Ψ R, ± = V ± , ∓ , , − = e i ( ± X R + X R ) , Ψ † R, ± = V ∓ , ± , − , = e i ( ∓ X R − X R ) (V.14)The theory F has a so (4) × so (4) = su (2) × su (2) × su (2) × su (2) current algebra at level 1. The Z IFTO2 is given in by the duality defect action (IV.31) in thebosonic model, which exchanges the currents su (2) with su (2) . It acts on the fermion by Z IFTO2 : Ψ R, − ↔ Ψ † R, − (V.15)leaving the other fermions Ψ invariant.The theory F has two exactly marginal operators O ( z, ¯ z ) ∼ (Ψ L, Ψ † L, − + Ψ L, − Ψ † L, ) · (Ψ R, Ψ † R, + Ψ R, − Ψ † R, − ) , O (cid:48) ( z, ¯ z ) ∼ (Ψ L, Ψ † L, − + Ψ L, − Ψ † L, ) · (Ψ R, Ψ † R, − Ψ R, − Ψ † R, − ) , (V.16)that are mapped to each other under Z IFTO2 . Note that O and O (cid:48) are not proportional to each other as in theprevious examples, but different operators. O and O (cid:48) drive the transitions from F to F and F (cid:48) , respectively.Following the discussions in Section II B and in SectionIV D, we conclude that there is a topological transitionfrom F (cid:48) [ b ] → F → F [ τ ]. In particular, the bosonic8relation (IV.29) is translated into Z F (cid:48) [ ρ, b ] = Z IFTO [ ρ ] Z F [ ρ, τ ] , τ = (cid:114) b − b . (V.17)That is, the two families of theories F and F (cid:48) differ byan IFTO point-by-point. a. Ultraviolet Realization on the Lattice Thefermionic CFT can arise as the infrared descrip-tion of a lattice theory in the ultraviolet. Letus suppose the UV symmetry on the lattice is G UV ≡ U (1) C × U (1) trn × SU (2), where U (1) C isthe total charge conservation, SU (2) is the spin rotationsymmetry in a spin- fermion lattice model, and U (1) trn is the translation symmetry for fermions at incommen-surate filling. The UV symmetry is embedded in theemergent IR symmetries for the theory F and F (cid:48) in twodifferent ways. The symmetry generators, actions andthe U (1) C , U (1) trn , SU (2) L , SU (2) R quantum numbersfor V n ,w ,n ,w are as follows. • F [ τ ] J C = i( ∂X + ¯ ∂X ) ,U (1) C : V n ,w ,n ,w → e i2 n γ C V n ,w ,n ,w J trn = i( ∂X − ¯ ∂X ) ,U (1) trn : V n ,w ,n ,w → e i2 w γ trn V n ,w ,n ,w . (V.18) q C = 2 n , q trn = 2 w ,j zL = 12 ( n + w ) , j zR = 12 ( n − w ) . (V.19) • F (cid:48) [ b ] J C = i( ∂X + ¯ ∂X ) ,U (1) C : V n ,w ,n ,w → e i( n − w + n + w ) γ C V n ,w ,n ,w J trn = i( ∂X − ¯ ∂X ) ,U (1) trn : V n ,w ,n ,w → e i( − n + w + n + w ) γ trn V n ,w ,n ,w . (V.20) q C = n − w + n + w , q trn = − n + w + n + w ,j zL = 12 ( n + w ) , j zR = 12 ( n − w ) . (V.21) b. Absence of Symmetry-Preserving Relevant Op-erators To ensure the perturbative stability of thefermionic theories, we need to exclude G UV -invariantrelevant operators in the anti-periodic ( A ) sector. Re-call that the theory F and F (cid:48) share the same A -sector, i.e. Z F [ AA ] = Z F (cid:48) [ AA ] and Z F [ AP ] = Z F (cid:48) [ AP ]. From (III.5) and Section IV D, we learn that the A -sector pri-mary operators of F are of the form V n ,w ,n ,w satisfy-ing( i ) n i , w i ∈ Z , n + w + n + w ∈ Z , ( ii ) n i , w i ∈
12 + Z , n w + n w ∈ Z (V.22)The G UV -invariant operators further satisfy q C =0 , q trn = 0 , j zL + j zR = 0, which implies F : n = n = w = 0 , F (cid:48) : n = w = − n = w . (V.23)The lightest ( i.e. the smallest scaling dimension) scalaroperators satisfying the above constraints are F : V , , , , ( h, ¯ h ) = (1 , , F (cid:48) : V , , − , , ( h, ¯ h ) = (1 , , (V.24)which are marginal but not relevant. Therefore, we con-clude that in either F or F (cid:48) , there is no relevant op-erator neutral under the microscopic symmetry G UV .Both theories describe (1+1) d symmetry-protected gap-less phases. c. Topological Transition and Symmetry Embedding A given embedding of the UV symmetry G UV into theIR emergent symmetry is consistent with either thefermionic CFT F or F (cid:48) , but not both. Hence a fermioniclattice model can realize either F or F (cid:48) , but not thetopological transition from F to F (cid:48) . However, if there isno symmetry constraint, both theories can describe themulti-critical theories of the same lattice model. In thiscase, by tuning the exactly marginal perturbations O , O (cid:48) along the path described by Figure 11, one can realizethe topological transition from the lattice model. VI. DUALITY DEFECT AND THE CHIRALFERMION PARITY
The duality defect N arises from the Z IFTO2 line in F before gauging ( − F . Due to the mixed anomalybetween Z IFTO2 and ( − F , the former becomes a non-invertible topological defect N in the gauged theory B [36, 38]. This is a generalization to the symmetry ex-tension in gauging a bosonic global symmetry in (1+1)dimensions with mixed anomaly, which we will brieflydiscuss below. We start with an example of the usual symmetry ex-tension from the mixed anomaly. Consider the c = 1compact boson CFT and let Z (1 , and Z (0 , be the π rotations of the momentum and winding symmetries, re-spectively. While Z (1 , and Z (0 , by themselves are free We thank Kantaro Ohmori and Yuji Tachikawa for discussionson this point. Z (1 , , then we throw away op-erators with odd momentum number n ∈ Z + 1, butintroduce operators with half-integral winding numbers w ∈ + Z from the twisted sector. It follows that thewinding Z (0 , acts by a factor of ± i on these new twistedsector states, and hence it is extended to Z in the gaugedtheory.In the case of gauging the ( − F of F , we might in-troduce twisted sector states that might not have a well-defined Z IFTO2 action. In the example of a single mass-less Majorana fermion, the Z IFTO2 symmetry is the chiralfermion parity ( − F L . In the corresponding bosonic the-ory, i.e. the Ising CFT, the would-be Z IFTO2 acts by a signon the energy operator ε ( z, ¯ z ) = ψ L ( z ) ψ R (¯ z ). However,this is incompatible with the fusion rule of the primaryoperators, σ × σ = 1 + ε . The obstruction is preciselythat there is no invertible action of the would-be Z IFTO2 on the spin operator σ , which comes from the twistedsector when gauging F . Hence there is no extension of Z IFTO2 in F to any bigger group that can be consistentwith the above fusion rule. Instead, the Z IFTO2 of F isextended to a non-invertible defect N of B , satisfyinga non-group-like fusion rules (IV.3) (see Figure 6). Onecan show that the duality defect N is compatible withthe fusion rule of local operators σ × σ = 1 + ε [49, 54].More generally, consider N Majorana fermions. Let( − F L be the chiral fermion parity that flips the signsof all the left-moving fermions. The (1+1) d fermionic ’tHooft anomaly of ( − F L is N mod 8 [56, 63–67]. Thebosonization is the bosonic Spin ( N ) WZW model, inwhich the chiral fermion parity ( − F L has become [78,99]: • N = 1 , N obeying the Z Tambara-Yamagami category TY + [79]. • N = 3 , N obeying the other Z Tamabara-Yamagami category TY − . • N = 0 mod 8: a Z line that is non-anomalous. • N = 4 mod 8: a Z line that is anoma-lous, corresponding to the nontrivial element of H ( Z , U (1)) = Z . • N = 2 , Z line. The Z is anoma-lous, corresponding to the square of the generatorof H ( Z , U (1)) = Z .Both TY ± share the same fusion rules (IV.3), but differ-ent F symbols. TY + is realized by the Ising CFT, whileTY − is realized by the su (2) = Spin (3) WZW model. The fusion category of the
Spin ( N ) WZW model can also beread off from the modular tensor category for the
Spin ( N ) Chern-Simons theory (which is a non-spin TQFT) by forgettingthe braiding. The latter can be found, for example, in Table 1-3of [100]. Note that the periodicity of N for the former is 8, whilethat for the latter is 16. Note that in our discussion of the three fermionic mod-els in Section V, our choice of the Z IFTO2 (see (V.2),(V.11), (V.15)) always flips the sign of a single
Majorana-Weyl fermion. Hence our Z IFTO2 has ’t Hooft anomaly 1mod 8 and turns into a duality defect TY + as in the N = 1 mod 8 case above. VII. OUTLOOK
A natural configuration for the topological transitionis a (1+1) d system with a (0+1) d interface separatinga fermionic CFT F with F (cid:48) = F ⊗
IFTO as in Figure5. Since the two fermionic CFTs F and F (cid:48) differ by anIFTO, one may wonder if the domain wall between F and F (cid:48) contains a Majorana zero mode or not. This questioncan be approached by DMRG [101], tensor network [102],MERA [103] or by the conformal interface [81].We can also study a (1+1) d gas of spinless fermions onan open chain with attractive or repulsive interactions.We know that a single IFTO on an open chain has atwo-fold topological degeneracy, which comes from twoMajorana zero modes from the two ends of the chain.Here the topological degeneracy has an energy splittingof order e − L/ξ , where L is the length of the chain and ξ is a length scale. Since the fermion system with at-tractive interaction and that with repulsive interactiondiffer by an IFTO, one may wonder if one of the abovetwo systems might have topological degeneracy ( i.e. theMajorana zero modes) localized at the chain ends. Moregenerally, an energy splitting of order O (cid:0) L α (cid:1) with α > L → ∞ , is a splitting less than that of the many-bodyenergy levels, which is of order 1 /L , and can indeed beviewed as a topological degeneracy even for gapless CFT.The above question can be addressed via bosonizationwhich maps an interacting fermion gas on a 1 d chain toa free compact boson system on a 1 d chain, at low ener-gies. The free compact boson system can be solved ex-actly, and we find that the degeneracy of the ground stateis always independent of the interaction. More specifi-cally, the Hilbert space of a compact boson with Neu-mann boundary condition on an open chain correspondsto that of a Dirac fermion with the same left and rightboundary condition ψ iL = + ψ iR ( i = 1 ,
2) on an openchain. The spectrum does not have nearly degenerateground state with splitting of order O (cid:0) L α (cid:1) with α > L → ∞ , regardless the fermion interaction is repulsiveor attractive. We confirm this with the exact computa-tion of the low energy spectrum of the following spinless More generally, there are two natural boundary conditions fora single Majorana fermion: ψ L = ηψ R , η = ±
1. On an openchain, we will choose the same boundary condition on the left endas that of the right end. Therefore, a Dirac fermion composedof two flavors of Majorana fermions can have four choices ofthe boundary conditions. Here, we choose η = +1 for bothMajorana fermions and for both boundaries, which correspondsto the Neumann boundary condition after bosonization. L , H = L − (cid:88) i =1 ( − c † i c i +1 + h.c. ) + V (cid:18) c † i c i − (cid:19) (cid:18) c † i +1 c i +1 − (cid:19) . (VII.1)After shifting the ground state energy to zero, we findthe spectrum E n / (cid:16) πL +1 (cid:17) as follows,V L n = 0 1 2 3-1 20 0.0 0.439154 0.439154 1.3235-1 10 0.0 0.444899 0.444899 1.358150 20 0.0 0.999068 0.999068 1.998140 10 0.0 0.996605 0.996605 1.993211 20 0.0 1.65908 1.65908 2.449061 10 0.0 1.61755 1.61755 2.35127We see that, indeed, regardless the fermion interaction isrepulsive or attractive, the ground state is always unique,and the energy to reach the first excited state is alwaysof order O ( L ).The topological transition in the c = 2 fermionicCFT involves two different interactions O and O (cid:48) . Thetransition is forbidden if we impose the UV symmetry U (1) c × U (1) trn × SU (2). This exotic phenomenon mayappear in the doped spin- fermionic models[26].In this paper, we explored an interesting phenomenonthat a Z IFTO2 symmetry of a (1+1) d system is generatedby stacking an (1+1) d invertible topological order. Sucha symmetry turns out to be anomalous as discussed inSection VI.This phenomenon can be quite general. For bosonicsystems, we have invertible topological orders in (2+1) d classified by Z , and invertible topological orders in(4+1) d classified by Z . For fermionic systems, we haveinvertible topological orders in (0+1) d and (1+1) d clas-sified by Z , and invertible topological orders in (2+1) d classified by Z . Therefore, we may have (0+1) d fermionicsystems with a Z symmetry generated by adding afermion. Such a symmetry is analogous to supersym-metry. In fact, such a (0+1) d fermionic system does ex-ist, which can be formed by two Majorana zero modesat the two ends of p -wave superconducting chain. Simi-larly, we may have (1+1) d fermionic systems with a Z symmetry generated by adding a (1+1) d IFTO, which iswhat we studied in this paper. We may also have (4+1) d bosonic systems with a Z symmetry generated by addinga (4+1) d invertible bosonic topological order. Acknowledgments
We would like to thank N. Benjamin, C. Cordova, L.Fu, T. Hsieh, P.-S. Hsin, T. Johnson-Freyd, Z. Komar-godski, J. Kulp, H.T. Lam, M. Metlitski, K. Ohmori,N. Seiberg, T. Senthil, C. Wang, J. Wang, Y. Wang foruseful conversations. We also thank Y. Tachikawa for comments on a draft. The work of S.H.S. is supportedby the National Science Foundation grant PHY-1606531,the Roger Dashen Membership, and a grant from the Si-mons Foundation/SFARI (651444, NS). X.G.W. is sup-ported by NSF Grant No. DMS-1664412. W.J. thanksthe Yukawa Institute for Theoretical Physics at KyotoUniversity, where part of this work was completed duringthe workshop YITP-T-19-03 “Quantum Information andString Theory 2019.” This work benefited from the 2019Pollica summer workshop, which was supported in partby the Simons Foundation (Simons Collaboration on theNon-Perturbative Bootstrap) and in part by the INFN.S.H.S. is grateful for the hospitality of the Physics De-partment of National Taiwan University during the com-pletion of this work. This research was supported in partby Perimeter Institute for Theoretical Physics. Researchat Perimeter Institute is supported by the Government ofCanada through the Department of Innovation, Scienceand Economic Development and by the Province of On-tario through the Ministry of Research and Innovation.This work was performed in part at Aspen Center forPhysics, which is supported by National Science Founda-tion grant PHY-1607611.
Appendix A: Identities for the Arf Invariants
Here we record some important identities for the Arfinvariant and Z connections (see, for example, [34]):Arf[ s + t + ρ ] =Arf[ s + ρ ] + Arf[ t + ρ ]+ Arf[ ρ ] + (cid:90) s ∪ t , g (cid:88) s e i π (Arf[ s + ρ ]+Arf[ ρ ]+ (cid:82) s ∪ t ) = e i π Arf[ t + ρ ] , (A.1)and 12 g (cid:88) s e i π (cid:82) s ∪ t = (cid:40) g , if t = 0 , , otherwise . (A.2) Appendix B: Twisted Torus Partition Functions1. Compact Boson
In this Appendix we compute the torus partition func-tion of the S CFT with non-trivial Z B = Z (1 , twist(defined in (IV.11)), and prove (IV.12). We will use 0(1) to denote the trivial (nontrivial) Z B twist in eitherthe space or time direction. In particular, the torus par-tition function Z S ( R ) (IV.8) with trivial Z B backgroundwill be denoted as Z S [00].1The torus partition function of S with a Z B twist inthe time direction is Z S [01] = 1 | η ( q ) | (cid:88) n,w ∈ Z ( − n q ( nR + wR ) ¯ q ( nR − wR ) . (B.1)Next, we compute the torus partition function Z S [10] forthe twisted sector of Z B , corresponding to a non-trivialtwist along the spatial direction. From the discussion inSection IV B, we see that the twisted sector of Z B = Z (1 , are operators V ˜ n, ˜ w with half-integral winding number w .Hence Z S [10] = 1 | η ( q ) | (cid:88) ˜ n ∈ Z , ˜ w ∈ + Z q ( ˜ nR + ˜ wR ) ¯ q ( ˜ nR − ˜ wR ) . (B.2)Finally, the torus partition function with non-trivial Z B twists in both cycles is the modular T transform τ → τ +1of Z S [10]: Z S [11] = 1 | η ( q ) | (cid:88) ˜ n ∈ Z , ˜ w ∈ + Z ( − ˜ n q ( ˜ nR + ˜ wR ) ¯ q ( ˜ nR − ˜ wR ) . (B.3)Adding the above four twisted torus partition functions Z S ’s together and dividing by 2, we obtain the toruspartition function of the S [ R ] / Z B theory: Z S / Z B ( R )= 12 (cid:16) Z S [00]( R ) + Z S [01]( R ) + Z S [10]( R ) + Z S [11]( R ) (cid:17) = Z S ( R/ . (B.4)Finally by T-duality, Z S ( R/
2) = Z S (2 /R ). Hence, wehave shown (IV.12) at the level of the torus partitionfunction. S / Z In this Appendix we compute the twisted torus parti-tion functions of S / Z [ R ] and show (IV.18). The toruspartition function of S / Z with a non-trivial Z B twistin the time direction is Z S / Z [01] = 1 | η ( q ) | (cid:88) n,w ∈ Z ( − n q ( nR + wR ) ¯ q ( nR − wR ) + (cid:12)(cid:12)(cid:12)(cid:12) η ( q ) θ ( q ) (cid:12)(cid:12)(cid:12)(cid:12) . (B.5)Note that the twisted sector of σ contributes oppositelycompared to that of σ , so their contribution cancel witheach other. The other two can be immediately obtainedfrom the modular S : τ → − /τ and T : τ → τ + 1 transformations: Z S / Z [10] = 1 | η ( q ) | (cid:88) n ∈ Z ,w ∈ + Z q ( nR + wR ) ¯ q ( nR − wR ) + (cid:12)(cid:12)(cid:12)(cid:12) η ( q ) θ ( q ) (cid:12)(cid:12)(cid:12)(cid:12) Z S / Z [11] = 1 | η ( q ) | (cid:88) n ∈ Z ,w ∈ + Z ( − n q ( nR + wR ) ¯ q ( nR − wR ) + (cid:12)(cid:12)(cid:12)(cid:12) η ( q ) θ ( q ) (cid:12)(cid:12)(cid:12)(cid:12) . (B.6)Adding the four Z S / Z ’s together and dividing by 2, weobtain the partition function for S / Z Z B : Z S / Z Z B ( R )= 12 1 | η ( q ) | (cid:88) n ∈ Z ,w ∈ Z + (cid:88) n ∈ Z ,w ∈ + Z q ( nR + wR ) ¯ q ( nR − wR ) + (cid:12)(cid:12)(cid:12)(cid:12) η ( q ) θ ( q ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) η ( q ) θ ( q ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) η ( q ) θ ( q ) (cid:12)(cid:12)(cid:12)(cid:12) . (B.7)Finally, we note that (cid:88) n ∈ Z ,w ∈ Z + (cid:88) n ∈ Z ,w ∈ + Z q ( nR + wR ) ¯ q ( nR − wR ) = (cid:88) n,w ∈ Z q ( nR (cid:48) + wR (cid:48) ) ¯ q ( nR (cid:48) − wR (cid:48) ) (B.8)with R (cid:48) = R . Comparing the above with (IV.14), wehave shown (IV.18). T CFT
The torus partition function of the T CFT at a genericpoint on the conformal manifold is given by Z B [00] = Tr H [ q L − c ¯ q ¯ L − c ] = 1 | η ( q ) | (cid:88) n i ,w i ∈ Z q h − ¯ q ¯ h − (B.9)where h, ¯ h are given as in (IV.22).The torus partition function with a Z B twist along thetime direction is Z B [01] =Tr H [ η q L − c ¯ q ¯ L − c ]= 1 | η ( q ) | (cid:88) n i ,w i ∈ Z ( − n + w + n + w q h − ¯ q ¯ h − . Next, we consider the torus partition function witha Z B twist in the spatial direction, which counts non-local operators living in the twisted sector (cid:101) H . These non-local operators are of the form (IV.23) but with fractional2momentum and winding numbers ˜ n i , ˜ w i : V ˜ n , ˜ w , ˜ n , ˜ w : ˜ n i , ˜ w i ∈
12 + Z . (B.10)Note that the spin s = ˜ n i ˜ w i of such operator is either aninteger or a half integer, consistent with the spin selectionrule for a non-anomalous Z B [54, 78, 88]. The Z B chargeof V ˜ n , ˜ w , ˜ n , ˜ w is given by ( − s . The partition functionwith a Z B twist in the time direction is therefore: Z B [10] =Tr (cid:101) H [ q L − c ¯ q ¯ L − c ]= 1 | η ( q ) | (cid:88) ˜ n i , ˜ w i ∈ + Z q h − ¯ q ¯ h − . Finally, the torus partition function with Z B twists bothin the spatial and time direction is Z B [11] =Tr (cid:101) H [ η q L − c ¯ q ¯ L − c ]= 1 | η ( q ) | (cid:88) ˜ n i , ˜ w i ∈ + Z ( − n i ˜ w i q h − ¯ q ¯ h − . At the special point of su (2) × su (2) , the torus par-tition functions are Z B [00] = (cid:16) | χ su | + | χ su | (cid:17) ,Z B [01] = (cid:16) | χ su | − | χ su | (cid:17) ,Z B [10] = (cid:16) χ su ¯ χ su + χ su ¯ χ su (cid:17) ,Z B [11] = − (cid:16) χ su ¯ χ su − χ su ¯ χ su (cid:17) , (B.11)where χ su j ( q ) is the su (2) current algebra characterswith su (2) spin j . At level 1, there are only two allowedspins, j = 0 and j = 1 /
2, whose conformal weights h are0 and 1 /
4, respectively. Their characters are χ su ( q ) = θ ( q ) η ( q ) , χ su ( q ) = θ ( q ) η ( q ) . The torus partition functionof the fermionized theory, i.e. four Majorana fermions,with respect to the four spin structures are Z F [ AA ] = (cid:104) ( χ su ) + ( χ su ) (cid:105) (cid:104) ( ¯ χ su ) + ( ¯ χ su ) (cid:105) ,Z F [ AP ] = (cid:104) ( χ su ) − ( χ su ) (cid:105) (cid:104) ( ¯ χ su ) − ( ¯ χ su ) (cid:105) ,Z F [ P A ] = 4 χ su χ su ¯ χ su ¯ χ su ,Z F [ P P ] = 0 . (B.12) Appendix C: The Tomonaga-Luttinger LiquidTheory
In this appendix we distinguish two different dualitymaps in the Tomonago-Luttinger (TL) model[97, 104, 105]. The TL liquid is described by the following ac-tion, in the imaginary time with coordinate ( x , x ) =( x, − i t ), S TL = v F π (cid:90) d x (cid:20) K ( ∂ φ ) + K ( ∂ θ ) (cid:21) − i π (cid:90) d x∂ θ∂ φ (C.1)The canonical commutation relation is (cid:20) ∂ φ ( x ) , π ∂ θ ( x (cid:48) ) (cid:21) = i δ ( x − x (cid:48) ) (C.2)which is independent of K . It follows that when K = 1,the theory is equivalent to the bosonized theory of thefree Dirac fermion.From this action, one can integrate out θ to obtain anaction for the φ field, S TL [ φ ] = 12 πKv F (cid:90) d x (cid:2) ( ∂ φ ) + v F ( ∂ φ ) (cid:3) (C.3)where φ ( x + 2 π, x ) = φ ( x ) + 2 πR φ Q φ , Q ∈ Z (C.4)with R φ = √ K , normalized such that the free Diractheory with K = 1 has R φ = √
2. This is nothing butthe action of free boson CFT S [ R ] with compactificationradius R = R φ . The Luttinger liquid theory of the inter-acting spinless fermion is, more precisely, the fermionizedtheory Dirac[ R ].Alternatively, one can integrate out φ to obtain an ac-tion for the θ field, S TL [ θ ] = K πv F (cid:90) d x (cid:2) ( ∂ θ ) + v F ( ∂ θ ) (cid:3) (C.5)where θ ( x + 2 π, x ) = θ ( x ) + 2 πR θ Q θ , Q θ ∈ Z (C.6)with R θ = √ K . The normalization is to be consistentwith (C.2). The field φ and dual field θ can be equiv-alently represented by chiral fields φ ( z, ¯ z ) = X L ( z ) + X R (¯ z ) and θ ( z, ¯ z ) = X L ( z ) − X R (¯ z ).There are two distinct maps along the moduli space of c = 1 CFT parametrized by the radius R = R φ . • T-duality: R → R , X L → X L , X R → − X R , (C.7)under which the bosonic operator V n,w ( R ) → V w,n (cid:0) R (cid:1) . Namely, the T-duality exchanges themomentum n and the winding w numbers of lo-cal operators V n,w in the bosonic CFT. In termsof the fields in the Luttinger model, the T-dualitymaps φ → θ, R φ → R φ (cid:54) = R θ . Under T-duality, thefree Dirac point with R φ = √ not the self-dualpoint. Instead, the su (2) CFT with R = 1 is theself-dual point.3 • IFTO-stacking: R → R , X L → X L , X R → − X R , (C.8) under which the fermionic operator V n,w ( R ) → V − w, − n (cid:0) R (cid:1) . The corresponding map in the Lut-tinger model is K → K , φ → θ, R φ → R θ . Underthis map, the Dirac theory is the self-dual point[34]. [1] X.-G. Wen and Y.-S. Wu, Phys. Rev. Lett. , 1501(1993).[2] W. Chen, M. P. A. Fisher, and Y.-S. Wu, Phys. Rev.B , 13749 (1993).[3] X.-G. Wen, Phys. Rev. Lett. , 3950 (2000), cond-mat/9908394.[4] X.-G. Wen, Phys. Rev. B , 165113 (2002), cond-mat/0107071.[5] T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, andM. P. A. Fisher, Phys. Rev. B , 144407 (2004).[6] X.-G. Wen, (2018), arXiv:1812.02517.[7] A. Kapustin and N. Seiberg, JHEP , 001 (2014),arXiv:1401.0740 [hep-th].[8] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett,JHEP , 172 (2015), arXiv:1412.5148 [hep-th].[9] X. Chen, F. Wang, Y.-M. Lu, and D.-H. Lee,Nucl. Phys. B , 248 (2013), arXiv:1302.3121 [cond-mat.str-el].[10] L. Tsui, H.-C. Jiang, Y.-M. Lu, and D.-H. Lee,Nucl. Phys. B , 330 (2015), arXiv:1503.06794 [cond-mat.str-el].[11] L. Tsui, F. Wang, and D.-H. Lee, (2015),arXiv:1511.07460 [cond-mat.str-el].[12] N. Bultinck, Phys. Rev. B , 165132 (2019),arXiv:1905.05790 [cond-mat.str-el].[13] Z. Bi and T. Senthil, Physical Review X , 021034(2019).[14] W. Rantner and X.-G. Wen, Phys. Rev. Lett. , 3871(2001), cond-mat/0010378.[15] M. Hermele, T. Senthil, M. P. A. Fisher, P. A. Lee,N. Nagaosa, and X.-G. Wen, Phys. Rev. B , 214437(2004), cond-mat/0404751.[16] W. Ji and X.-G. Wen, Phys. Rev. Lett. , 035301(2019).[17] J. P. Kestner, B. Wang, J. D. Sau, and S. Das Sarma,Phys. Rev. B , 174409 (2011).[18] M. Cheng and H.-H. Tu, Physical Review B , 094503(2011).[19] L. Fidkowski, R. M. Lutchyn, C. Nayak, and M. P.Fisher, Physical Review B , 195436 (2011).[20] J. D. Sau, B. Halperin, K. Flensberg, and S. D. Sarma,Physical Review B , 144509 (2011).[21] C. V. Kraus, M. Dalmonte, M. A. Baranov, A. M.L¨auchli, and P. Zoller, Physical review letters ,173004 (2013).[22] A. Keselman and E. Berg, Physical Review B , 235309(2015).[23] F. Iemini, L. Mazza, D. Rossini, R. Fazio, and S. Diehl,Physical review letters , 156402 (2015).[24] A. Montorsi, F. Dolcini, R. C. Iotti, and F. Rossi, Phys-ical Review B , 245108 (2017).[25] J. Ruhman and E. Altman, Physical Review B ,085133 (2017). [26] H.-C. Jiang, Z.-X. Li, A. Seidel, and D.-H. Lee, Sciencebulletin , 753 (2018).[27] A. Keselman, E. Berg, and P. Azaria, Phys. Rev. B ,214501 (2018).[28] T. Scaffidi, D. E. Parker, and R. Vasseur, Physical Re-view X , 041048 (2017).[29] D. E. Parker, T. Scaffidi, and R. Vasseur, Physical Re-view B , 165114 (2018).[30] R. Verresen, R. Moessner, and F. Pollmann, PhysicalReview B , 165124 (2017).[31] R. Verresen, N. G. Jones, and F. Pollmann, Physicalreview letters , 057001 (2018).[32] R. Verresen, R. Thorngren, N. G. Jones, and F. Poll-mann, (2019), arXiv:1905.06969 [cond-mat.str-el].[33] A. Y. Kitaev, Phys. Usp. , 131 (2001), arXiv:cond-mat/0010440 [cond-mat.mes-hall].[34] A. Karch, D. Tong, and C. Turner, SciPost Phys. ,007 (2019), arXiv:1902.05550 [hep-th].[35] D. Gaiotto and A. Kapustin, Proceedings, Gribov-85Memorial Workshop on Theoretical Physics of XXICentury: Chernogolovka, Russia, June 7-20, 2015 , Int.J. Mod. Phys.
A31 , 1645044 (2016), arXiv:1505.05856[cond-mat.str-el].[36] L. Bhardwaj, D. Gaiotto, and A. Kapustin, JHEP ,096 (2017), arXiv:1605.01640 [cond-mat.str-el].[37] A. Kapustin and R. Thorngren, JHEP , 080 (2017),arXiv:1701.08264 [cond-mat.str-el].[38] R. Thorngren, (2018), arXiv:1810.04414 [cond-mat.str-el].[39] Y. Tachikawa, “TASI 2019 Lectures,” https://member.ipmu.jp/yuji.tachikawa/lectures/2019-top-anom .[40] H. A. Kramers and G. H. Wannier, Phys. Rev. , 252(1941).[41] M. I. Monastyrsky and A. B. Zamolodchikov, (1978).[42] R. Savit, Reviews of Modern Physics , 453 (1980).[43] S.-J. Rey and A. Zee, Nuclear Physics B , 897(1991).[44] P. Severa, JHEP , 049 (2002), arXiv:hep-th/0206162[hep-th].[45] D. S. Freed and C. Teleman, (2018), arXiv:1806.00008[math.AT].[46] W. Ji and X.-G. Wen, (2019), arXiv:1905.13279 [cond-mat.str-el].[47] E. Witten, , 1173 (2003), arXiv:hep-th/0307041 [hep-th].[48] N. Seiberg, T. Senthil, C. Wang, and E. Witten, AnnalsPhys. , 395 (2016), arXiv:1606.01989 [hep-th].[49] J. Frohlich, J. Fuchs, I. Runkel, and C. Schweigert,Phys. Rev. Lett. , 070601 (2004), arXiv:cond-mat/0404051 [cond-mat].[50] J. Frohlich, J. Fuchs, I. Runkel, and C. Schweigert,Nucl. Phys. B763 , 354 (2007), arXiv:hep-th/0607247[hep-th]. [51] J. Frohlich, J. Fuchs, I. Runkel, and C. Schweigert, in Proceedings, 16th International Congress on Mathemat-ical Physics (ICMP09): Prague, Czech Republic, August3-8, 2009 (2009) arXiv:0909.5013 [math-ph].[52] D. Aasen, R. S. K. Mong, and P. Fendley, J. Phys. A ,354001 (2016), arXiv:1601.07185 [cond-mat.stat-mech].[53] L. Bhardwaj and Y. Tachikawa, JHEP , 189 (2018),arXiv:1704.02330 [hep-th].[54] C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, andX. Yin, JHEP , 026 (2019), arXiv:1802.04445 [hep-th].[55] M. Buican and A. Gromov, Commun. Math. Phys. ,1017 (2017), arXiv:1701.02800 [hep-th].[56] A. Kapustin, R. Thorngren, A. Turzillo, andZ. Wang, JHEP , 052 (2015), [JHEP12,052(2015)],arXiv:1406.7329 [cond-mat.str-el].[57] K. Shiozaki, H. Shapourian, and S. Ryu, Phys. Rev. B95 , 205139 (2017), arXiv:1609.05970 [cond-mat.str-el].[58] R. Dijkgraaf and E. Witten, Int. J. Mod. Phys.
A33 ,1830029 (2018), arXiv:1804.03275 [hep-th].[59] T. Senthil, D. T. Son, C. Wang, and C. Xu, (2018),arXiv:1810.05174 [cond-mat.str-el].[60] M. F. Atiyah, Annales scientifiques de l’´Ecole NormaleSup´erieure
Ser. 4, 4 , 47 (1971).[61] C. Cordova, D. S. Freed, H. T. Lam, and N. Seiberg,(2019), arXiv:1905.09315 [hep-th].[62] C. Cordova, D. S. Freed, H. T. Lam, and N. Seiberg,(2019), arXiv:1905.13361 [hep-th].[63] S. Ryu and S.-C. Zhang, Phys. Rev. B , 245132(2012).[64] X.-L. Qi, New Journal of Physics , 065002 (2013).[65] H. Yao and S. Ryu, Phys. Rev. B , 064507 (2013).[66] Z.-C. Gu and M. Levin, Phys. Rev. B , 201113 (2014).[67] D. S. Freed and M. J. Hopkins, (2016),arXiv:1604.06527.[68] C. Heinrich, F. Burnell, L. Fidkowski, and M. Levin,Phys. Rev. B , 235136 (2016), arXiv:1606.07816.[69] M. Cheng, Z.-C. Gu, S. Jiang, and Y. Qi, PhysicalReview B , 115107 (2017).[70] R. Dijkgraaf and E. Witten, Commun. Math. Phys. ,393 (1990).[71] W. Ji and X.-G. Wen, (2019), arXiv:1905.13279.[72] D. Radicevic, (2018), arXiv:1809.07757 [hep-th].[73] C. Vafa, Mod. Phys. Lett. A4 , 1615 (1989).[74] F. Gliozzi, J. Scherk, and D. I. Olive, Nucl. Phys. B122 ,253 (1977).[75] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen,Phys. Rev. B , 155114 (2013), arXiv:1106.4772 [cond-mat.str-el].[76] L. J. Dixon, P. H. Ginsparg, and J. A. Harvey, Com-mun. Math. Phys. , 221 (1988).[77] P. Jordan and E. P. Wigner, Z. Phys. , 631 (1928). [78] Y.-H. Lin and S.-H. Shao, Phys. Rev. D100 , 025013(2019), arXiv:1904.04833 [hep-th].[79] D. Tambara and S. Yamagami, Journal of Algebra ,692 (1998).[80] P. H. Ginsparg, in
Les Houches Summer School in The-oretical Physics: Fields, Strings, Critical PhenomenaLes Houches, France, June 28-August 5, 1988 (1988)pp. 1–168, arXiv:hep-th/9108028 [hep-th].[81] J. Fuchs, M. R. Gaberdiel, I. Runkel, andC. Schweigert, Journal of Physics A: Mathematical andTheoretical , 11403 (2007).[82] R. Dijkgraaf, E. P. Verlinde, and H. L. Verlinde, Com-mun. Math. Phys. , 649 (1988).[83] J. Polchinski, String theory. Vol. 1: An introduction tothe bosonic string , Cambridge Monographs on Mathe-matical Physics (Cambridge University Press, 2007).[84] S. Gukov and C. Vafa, Commun. Math. Phys. , 181(2004), arXiv:hep-th/0203213 [hep-th].[85] D. Gaiotto and T. Johnson-Freyd, (2019),arXiv:1904.05788 [hep-th].[86] S. Elitzur, E. Gross, E. Rabinovici, and N. Seiberg,Nucl. Phys.
B283 , 413 (1987).[87] S. R. Coleman, Phys. Rev.
D11 , 2088 (1975),[,128(1974)].[88] L.-Y. Hung and X.-G. Wen, Phys. Rev.
B89 , 075121(2014), arXiv:1311.5539 [cond-mat.str-el].[89] E. Marino and J. Swieca, Nucl. Phys. B , 175 (1980).[90] R. Koberle and E. Marino, Phys. Lett. B , 475(1983).[91] E. Marino, Nuclear Physics B , 413 (1983).[92] E. Marino, Nuclear Physics B , 149 (1984).[93] E. Marino, Phys. Lett. B , 63 (1991).[94] E. Marino, J. Stat. Mech. , 033103 (2017).[95] A. Luther and V. Emery, Physical Review Letters ,589 (1974).[96] A. Theumann, Journal of Mathematical Physics , 2460(1967).[97] F. D. M. Haldane, J. Phys. C14 , 2585 (1981).[98] J. Voit, Reports on Progress in Physics , 977 (1995).[99] T. Numasawa and S. Yamaguch, JHEP , 202 (2018),arXiv:1712.09361 [hep-th].[100] A. Kitaev, Ann. Phys. , 2 (2006), cond-mat/0506438.[101] Y. He, B. Tian, D. Pekker, and R. S. K. Mong, (2018),arXiv:1811.06066 [cond-mat.str-el].[102] R. Vanhove, M. Bal, D. J. Williamson, N. Bultinck,J. Haegeman, and F. Verstraete, Phys. Rev. Lett. ,177203 (2018), arXiv:1801.05959 [quant-ph].[103] J. C. Bridgeman and D. J. Williamson, Phys. Rev. B , 125104 (2017), arXiv:1703.07782 [quant-ph].[104] S. Tomonaga, Prog. Theor. Phys. , 544 (1950).[105] J. M. Luttinger, J. Math. Phys.4