Extended Equivalence of $U(1)$ Chern-Simons and Reshetikhin-Turaev TQFTs

EXTENDED EQUIV ALENCE OF U (1) CHERN–SIMONS AND RESHETIKHIN–TURAEV TQFTS D ANIEL GAL VIZ Abstra ct. W e establish the equiv alence betw een U (1) Chern–Simons and Reshetikhin– T uraev TQFT s asso ciated with ﬁnite quadratic mo dules. F or gauge group U (1) and even lev el k , we pro v e that the corresp onding Chern–Simons TQFT is naturally isomorphic to the Reshetikhin–T uraev TQFT determined by the pointed mo dular category C ( Z k , q k ) . The equiv alence holds b oth for closed 3 -manifolds and for b ordisms with b oundary , so that the t w o constructions deﬁne naturally isomorphic extended (2 + 1) -dimensional TQFT s. In particular, the ﬁnite quadratic mo dule ( Z k , q k ) completely determines the ev en-lev el U (1) Chern–Simons theory . Main Theorem. Let k ∈ Z b e a nonzero ev en integer, let ( Z k , q k ) b e the asso ciated ﬁnite quadratic mo dule, and let C ( Z k , q k ) denote the corresp onding p ointed mo dular category . Then the Reshetikhin–T uraev theory asso ciated with C ( Z k , q k ) is naturally isomorphic, as an extended (2 + 1) -dimensional TQFT, to the Ab elian Chern–Simons theory with gauge group U (1) and level k . Equiv alently , the t w o theories deﬁne naturally isomorphic symmetric monoidal functors Cob ext 2+1 − → V ect C . Contents 1. In tro duction 2 2. Geometric Quantization of U (1) Chern–Simons Theory 3 2.1. The Ab elian Chern–Simons F unctional 3 2.2. Geometric Quantization of the Boundary Phase Space 4 2.3. Ab elian Chern–Simons TQFT 6 3. Surgery Inv ariants from Abelian Gauge Groups 6 3.1. The Ab elian R T inv ariant for G = Z k 7 3.2. Recipro cit y formulas and the relation with U (1) Chern–Simons theory 9 3.3. The obstruction for G = U (1) 10 4. Ab elian Reshetikhin–T uraev Theory with Boundary 11 4.1. T uraev’s Extended R T F ormalism 12 4.2. Ab elian R T Boundary State Spaces 14 5. Extended Equiv alence of U (1) Chern–Simons and R T Theories 18 5.1. Closed Comparison of Ab elian R T and U (1) Chern–Simons Theory 18 5.2. Boundary Op erator Equiv alence 22 5.3. Extended Equiv alence of Ab elian R T and U (1) Chern–Simons TQFT s 30 5.4. Finite Quadratic Mo dules as Universal Data 33 References 34 1 2 D ANIEL GAL VIZ 1. In tro duction Since the construction of Chern-Simons theory b y Witten and its mathematical re- alization provided b y Reshetikhin and T uraev [ Wit89 , R T90 ], the non–Ab elian theory has historically b een the ﬁrst to attract atten tion; how ever the Ab elian case is by no means trivial. On the con trary , it provides a setting in whic h sev eral constructions can b e made completely explicit and compared in detail. Non–Ab elian Chern–Simons theory with ﬁnite gauge group, dev elop ed b y F reed and Quinn [ F re95 , F Q93 ], already sho ws that top ological gauge theories can often be understo o d by concrete cutting-and-gluing constructions. Ab elian Chern–Simons theory ma y b e view ed as a contin uous coun terpart of this picture, but one in which zero mo des, torsion, and quadratic reﬁnemen ts pla y a more delicate role. T wo approac hes to rank-one Abelian Chern–Simons theory are esp ecially relev ant for this pap er. The ﬁrst is geometric. In a series of pap ers, Manoliu [ Man97 , Man98a , Man98b ] studied the quantization of symplectic tori in real p olarizations and then applied this mac hinery to U (1) Chern–Simons theory on closed orien ted 3 -manifolds and on mani- folds with b oundary . In her formulation, the partition function is expressed in terms of Ra y–Singer/Reidemeister torsion, whic h makes b oth its top ological inv ariance and its dep endence on the torsion subgroup of H 1 ( M ; Z ) transparent. Just as imp ortantly , Manoliu’s theory provides a gen uine b oundary-v alue formalism: the state spaces are obtained b y geometric quantization of symplectic boundary tori, and b ordisms act b y op- erators constructed from Chern–Simons sections and Blattner–K ostant–Stern b erg pairings [ Man97 , Man98a ]. The second approac h is com binatorial. Mattes, Poly ak, and Reshetikhin (MPR) [ MPR93 ], and indep enden tly Murak ami, Oh tsuki, and Ok ada [ MOO92 ], constructed Ab elian 3 -manifold in v ariants in terms of explicit quadratic Gauss sums asso ciated with surgery presen tations. In this framework, the dep endence on linking num b ers and quadratic forms on ﬁnite Ab elian groups is completely explicit. This surgery description is closer in spirit to the Reshetikhin–T uraev formalism. Later work of Deloup and T uraev [ Del99 , DT07 ] further clariﬁed the role of linking forms, recipro city formulas, and ﬁnite quadratic data in these in v ariants. The equiv alence b et w een the Ab elian Reshetikhin–T uraev TQFT and the rank-one U (1) Chern–Simons theory has long b een exp ected, but to our knowledge it had not previously b een established in this extended form. This equiv alence is not immediate, b ecause the t wo theories are form ulated in v ery diﬀerent languages. On the geometric side one encounters symplectic b oundary phase spaces, real p olarizations, and half-densities. On the com binatorial side one works with surgery presen tations, Gauss sums, determinan t corrections, and Kirb y calculus. Man y comparisons in the literature stop at the lev el of closed partition functions and do not address b oundary state spaces or bordism op erators [ MOO92 , Jef92 , MPR93 , Sti08 , GT13 , GT14 , TT25 ]. F rom the TQFT p oint of view, ho w ever, these are precisely the essential structures. Moreo v er, normalization issues are subtle. Geometric form ulas contain Gaussian factors arising from the free part of cohomology together with torsion factors coming from Reidemeister or Ray–Singer torsion, whereas com binatorial form ulas in v olve determinant factors and signature phases coming from quadratic recipro city and surgery normalization. Any satisfactory comparison m ust therefore matc h these contributions term b y term. A further diﬃculty is that the degenerate surgery case det L = 0 , equiv alently b 1 ( M ) > 0 , cannot be ignored. F rom the p ersp ective of TQFT functorialit y , cutting and gluing naturally pro duce in termediate b ordisms whose linking matrices are singular. Th us the correct U (1) CHERN–SIMONS AND RESHETIKHIN–TURAEV TQFTS 3 statemen t is not simply an equalit y of partition functions for rational homology spheres, but an equiv alence v alid in the presence of b oth torsion sectors and zero-mo de sectors. This is one reason wh y the natural framework for the problem is that of extended TQFT. In A tiy ah’s axiomatic language, a 3 -dimensional TQFT is a symmetric monoidal functor from a b ordism category to vector spaces [ A ti88 ]. In the present setting, how ever, one m ust w ork with an extended b ordism category in whic h b oundary surfaces carry additional Lagrangian data and gluing is corrected by Maslo v indices [ W al91 , T ur94 ]. The correct form ulation of the problem is therefore not merely to compare scalar in v ariants of closed 3 -manifolds, but to compare the extended (2 + 1) -dimensional theories. In concrete terms, one seeks canonically isomorphic state spaces for eac h extended surface (Σ , λ ) , and iden tical linear maps for eac h extended b ordism X : (Σ in , λ in ) − → (Σ out , λ out ) , after the Maslo v anomaly has b een normalized in a compatible manner. The pap er pro ceeds as follo ws. First, we review Manoliu’s geometric quan tization construction of rank-one U (1) Chern–Simons theory , including the boundary state spaces, b ordism op erators, and extended gluing formalism. Second, we reformulate the corre- sp onding Ab elian Reshetikhin–T uraev theory in the p oin ted mo dular category C ( Z k , q k ) , iden tifying its surgery expressions and boundary state spaces in a form adapted to com- parison. Third, w e prov e that these tw o constructions agree on closed 3 -manifolds, on b oundary state spaces, and on the op erators assigned to b ordisms, yielding a natural isomorphism of extended (2 + 1) -dimensional TQFT s. In particular, the ﬁnite quadratic mo dule ( Z k , q k ) is shown to enco de exactly the discrete data gov erning the rank-one U (1) Chern–Simons theory . A ckno wledgemen ts. I would lik e to thank Nicolai Reshetikhin for man y helpful con v ersations and suggestions, for in tro ducing me to Abelian Chern–Simons theory , and for his supp ort throughout the dev elopment of this pro ject. 2. Geometric Quantization of U (1) Chern–Simons Theory In this section w e summarize the main results of Manoliu concerning the construction of Ab elian U (1) Chern–Simons theory as a unitary extended (2 + 1) -dimensional top o- logical quan tum ﬁeld theory . W e restrict ourselves to the ingredien ts needed later: the Chern–Simons functional, the geometric quantization of the b oundary phase space, the con- struction of the v ectors assigned to 3 -manifolds, and the extended gluing law. F ull proofs of the U (1) Chern-Simons theory are referred to the original sources [ Man97 , Man98a ]. 2.1. The Ab elian Chern–Simons F unctional. Let X b e a closed orien ted 3 -manifold, let P → X b e a principal U (1) -bundle, and let Θ b e a connection on P . F ollowing [ Man98a ], the Chern–Simons functional is deﬁned b y passing to the asso ciated S U (2) -bundle ˆ P = P × U (1) S U (2) − → X , where U (1)  → S U (2) is the diagonal em b edding e iθ 7− →  e iθ 0 0 e − iθ  . The connection Θ induces an S U (2) -connection ˆ Θ on ˆ P , and one sets (2.1) S X,P (Θ) = Z X ˆ s ∗ α ( ˆ Θ) (mo d 1) , 4 D ANIEL GAL VIZ where ˆ s : X → ˆ P is an y section and (2.2) α ( ˆ Θ) = D ˆ Θ ∧ F ˆ Θ E ♭ − 1 6 D ˆ Θ ∧ [ ˆ Θ , ˆ Θ] E ♭ . Here α ( ˆ Θ ) is the ordinary Chern–Simons 3 -form of the S U (2) -connection ˆ Θ . The v alue of S X,P (Θ) ∈ R / Z is indep enden t of the choice of ˆ s , since diﬀerent c hoices diﬀer b y an in teger winding-num b er term. The functional S X,P : A P → R / Z is gauge in v ariant and therefore descends to the quotien t A P / G P . It satisﬁes the exp ected formal prop erties: functoriality under orien tation- preserving bundle morphisms, S X ′ ,P ′ ( φ ∗ Θ) = S X,P (Θ) , orien tation reversal, S − X,P (Θ) = − S X,P (Θ) , and additivit y under disjoint union, S X 1 ⊔ X 2 , P 1 ⊔ P 2 (Θ 1 ⊔ Θ 2 ) = S X 1 ,P 1 (Θ 1 ) + S X 2 ,P 2 (Θ 2 ) . Moreo v er, its critical p oints are precisely the ﬂat connections: (2.3) dS X,P (Θ) = 0 ⇐ ⇒ F Θ = 0 . Since the gauge group is Ab elian, this condition is simply d Θ = 0 in a lo cal trivialization. When X has b oundary , the Chern–Simons functional is no longer a gauge-inv arian t scalar, but instead giv es rise to a section of the b oundary prequantum line bundle. This is the basic geometric input in the construction of the Ab elian Chern–Simons state Z CS X . 2.2. Geometric Quan tization of the Boundary Phase Space. Throughout, the gauge group is U (1) = { z ∈ C : | z | = 1 } , and the lev el k ∈ Z is assumed to b e ev en. This ensures that the Chern–Simons functional is w ell deﬁned mo dulo Z without additional spin data and that the corresp onding prequan tum line bundle exists. Since U (1) is Ab elian, ev ery ﬂat connection on a closed oriented surface Σ is determined by its holonomy , so the moduli space of ﬂat U (1) -connections modulo gauge equiv alence iden tiﬁes canonically with the torus M Σ ∼ = H 1 (Σ; R ) /H 1 (Σ; Z ) . If g is the genus of Σ , then M Σ is a 2 g -dimensional symplectic torus, with symplectic form ω Σ induced by the cup-product pairing on H 1 (Σ; R ) , equiv alently b y integration ov er Σ . F or ev en k , Manoliu constructs a Hermitian line bundle L Σ → M Σ with unitary connection of curv ature curv ( ∇ L Σ ) = − 2 π i k ω Σ ; this is the Chern–Simons prequan tum line bundle. Although prequan tum line bundles on ( M Σ , k ω Σ ) form a torsor under H 1 ( M Σ ; U (1)) , the Chern–Simons line bundle construction canonically selects L Σ [ F Q93 ]. T o quan tize M Σ , one chooses a rational Lagrangian subspace L ⊂ H 1 (Σ; R ) , that is, a Lagrangian subspace deﬁned o v er Q . Suc h an L determines an inv arian t real p olarization P L of M Σ . Quan tization is p erformed with half-densities: if B S P L denotes the Bohr– Sommerfeld set of the polarization and | det ( P ∗ L ) | 1 / 2 the corresp onding half-densit y bundle, then the Hilb ert space assigned to the extended surface (Σ , L ) is (2.4) H (Σ , L ) = M Λ ⊂BS P L Γ ﬂat  Λ; L Σ ⊗ | det( P ∗ L ) | 1 / 2  . Equiv alently , one may view each summand as the one-dimensional space of cov ariantly constan t sections ov er a Bohr–Sommerfeld leaf. The resulting Hilb ert space is ﬁnite- dimensional and satisﬁes dim H (Σ , L ) = | k | g . If L 1 and L 2 are tw o rational Lagrangians, U (1) CHERN–SIMONS AND RESHETIKHIN–TURAEV TQFTS 5 geometric quantization pro duces a canonical unitary Blattner–K ostan t–Stern b erg operator F L 2 L 1 : H (Σ , L 1 ) → H (Σ , L 2 ) . These operators do not comp ose strictly: their composition is t wisted b y the Maslo v–Kashiw ara index of triples of Lagrangian subspaces, and this defect is precisely what the extended formalism corrects. No w let X b e a compact orien ted 3 -manifold with b oundary . The restriction of ﬂat connections induces a map r X : M X → M ∂ X , and on cohomology the corresp onding map ˙ r X : H 1 ( X ; R ) → H 1 ( ∂ X ; R ) has image L X := Im ( ˙ r X ) , which is a rational Lagrangian subspace of H 1 ( ∂ X ; R ) . Th us X canonically determines a b oundary p olarization P L X and hence a Hilb ert space H ( ∂ X, L X ) . The mo duli space of ﬂat U (1) -connections on X decomp oses according to the torsion top ological type of the underlying bundle as M X = G p ∈ T ors H 2 ( X ; Z ) M X,p . F or eac h torsion class p , the comp onent M X,p carries a canonically deﬁned co v ariantly con- stan t Chern–Simons section σ X,p : M X,p → r ∗ X L ∂ X | M X,p , whose phase is exp (2 π ik C S ( p )) , where C S ( p ) is a quadratic reﬁnemen t of the torsion linking pairing. Summing o ver all comp onen ts giv es σ X = X p ∈ T ors H 2 ( X ; Z ) σ X,p . The second ingredient in the construction of the state asso ciated with X is the canonical half-densit y µ X on the Lagrangian subtorus Λ X := Im ( r X ) ⊂ M ∂ X , obtained from Reidemeister torsion, equiv alently Ra y–Singer torsion. More precisely , the torsion T X , together with Poincaré duality and the long exact sequence of the pair ( X , ∂ X ) , determines an in v arian t half-densit y on Λ X denoted b y µ X = Z H 1 ( X,∂ X ; U (1)) ( T X ) 1 / 2 . Com bining the Chern–Simons section and the torsion half-densit y yields the state (2.5) Z CS X = k m X # T ors H 2 ( X ; Z ) X p ∈ T ors H 2 ( X ; Z ) σ X,p ⊗ µ X ∈ H ( ∂ X, L X ) , where m X = 1 4  dim H 1 ( X ; R ) + dim H 1 ( X , ∂ X ; R ) − dim H 0 ( X ; R ) − dim H 0 ( X , ∂ X ; R )  . In the closed case ∂ X = ∅ , the Hilbert space reduces to C , the half-densit y b ecomes a densit y on M X = H 1 ( X ; U (1)) , and Z CS X b ecomes a scalar in v arian t, equiv alen tly (2.6) Z CS X = k m X Z M X σ X ( T X ) 1 / 2 , with m X = 1 2 (dim H 1 ( X ; R ) − dim H 0 ( X ; R )) . T o obtain strict functoriality under gluing, one passes to W alk er’s category of extended manifolds [ W al91 ]. An extended 2 -manifold is a pair (Σ , L ) , where Σ is a closed orien ted surface and L ⊂ H 1 (Σ; R ) is a rational Lagrangian subspace. An extended 3 -manifold is a triple ( X , L, n ) , where X is a compact orien ted 3 -manifold, L ⊂ H 1 ( ∂ X ; R ) is a rational Lagrangian subspace, and n ∈ Z / 8 Z records the Maslo v correction required for strict gluing. The extended Ab elian Chern–Simons TQFT then assigns the Hilb ert space H (Σ , L ) to (Σ , L ) , and to an extended 3 -manifold ( X , L, n ) it assigns the vector (2.7) Z CS ( X,L,n ) = e πi 4 n F LL X  Z CS X  ∈ H ( ∂ X, L ) , 6 D ANIEL GAL VIZ where F LL X : H ( ∂ X, L X ) → H ( ∂ X, L ) is the BKS op erator asso ciated with the c hange of p olarization. With this correction, the resulting assignmen ts satisfy functoriality , m ultiplicativit y under disjoint union, compatibility with orientation reversal, and the gluing la w; see [ Man98a ] for the full proof. 2.3. Ab elian Chern–Simons TQFT. The Ab elian Chern–Simons TQFT construction can be summarized as follo ws: Theorem 2.1 ([ Man98a ], Theorem VI.11) . The assignments (Σ , L ) 7− → H (Σ , L ) , ( X , L, n ) 7− → Z CS ( X,L,n ) deﬁne a unitary extended (2 + 1) -dimensional TQFT. More precisely: (1) F unctorialit y: extende d diﬀe omorphisms induc e unitary isomorphisms b etwe en the c orr esp onding Hilb ert sp ac es, and the ve ctors Z CS ( X,L,n ) ar e natur al with r esp e ct to these maps. (2) Orien tation: ther e is a natur al antiline ar identiﬁc ation H ( − Σ , L ) ∼ = H (Σ , L ) , and Z CS ( − X,L,n ) = Z CS ( X,L,n ) . (3) Disjoin t union: one has c anonic al unitary identiﬁc ations H (Σ 1 ⊔ Σ 2 , L 1 ⊕ L 2 ) ∼ = H (Σ 1 , L 1 ) ⊗ H (Σ 2 , L 2 ) , and the c orr esp onding states ar e multiplic ative under disjoint union. (4) Cylinder axiom: the cylinder Σ × I deﬁnes the identity op er ator on H (Σ , L ) . (5) Gluing: if X is cut along a close d oriente d surfac e Σ , then the state of the glue d manifold is obtaine d by c ontr acting the state of the cut manifold using the Hermitian p airing on H (Σ , L Σ ) , with the r e quir e d Maslov c orr e ction built into the extende d structur e. Th us Ab elian U (1) Chern–Simons theory at even level k giv es a unitary extended (2 + 1) -dimensional TQFT. Its b oundary state spaces arise by geometric quan tization of symplectic moduli tori, and its b ordism op erators are constructed from the Chern–Simons functional, BKS pairings, and torsion half-densities. This geometric framew ork will be the geometric side of the comparison with Abelian Reshetikhin–T uraev theory carried out in the next sections. 3. Surgery Inv ariants from Ab elian Gauge Groups This section reviews the Ab elian surgery inv ariants introduced by Mattes, Poly ak, and Reshetikhin [ MPR93 ], and also Murak ami, Ohtsuki, and Ok ada [ MOO92 ], with emphasis on the case relev ant for the comparison with rank-one U (1) Chern–Simons theory . W e fo cus on three points. First, w e describ e the p ointed Reshetikhin–T uraev inv ariant asso ciated with the ﬁnite quadratic data ( Z k , q k ) , where k ∈ 2 Z > 0 . Second, we recall the Gauss reciprocity formulas that relate this surgery inv arian t to the decomp osition of the Ab elian Chern–Simons partition function into top ological sectors. Third, we explain wh y an analogous construction based directly on the compact group U (1) do es not yield a non trivial Ab elian Reshetikhin–T uraev theory under natural con tinuit y assumptions. The relev ance of this discussion is that the com binatorial in v ariant attached to ( Z k , q k ) is the correct discrete coun terpart of rank-one U (1) Chern–Simons theory at lev el k . In U (1) CHERN–SIMONS AND RESHETIKHIN–TURAEV TQFTS 7 later sections w e will compare this in v ariant, ﬁrst on closed 3 -manifolds and then in the presence of b oundary , with the geometric theory reviewed ab o v e. 3.1. The Ab elian R T inv ariant for G = Z k . Let k ∈ 2 Z > 0 , and write G = Z k = Z /k Z additiv ely . T o match the rank-one U (1) Chern–Simons theory at lev el k , we ﬁx the p oin ted ribb on data (3.1) F ≡ 1 , q k ( x ) = exp  π i k x 2  , Ω k ( x, y ) = q k ( x + y ) q k ( x ) q k ( y ) = exp  2 π i k xy  . Since k is ev en, q k is w ell deﬁned on Z k , and Ω k is a nondegenerate bicharacter. These data determine a p ointed mo dular category C k := C ( Z k , q k ) , whose simple ob jects are indexed b y elemen ts of Z k , with tensor pro duct induced by addition in Z k , trivial asso ciator, braiding c x,y = Ω k ( x, y ) id x + y , and ribbon t wist θ x = q k ( x ) id x . Let J = J 1 ∪ · · · ∪ J s ⊂ S 3 b e an oriented framed link, and let x = ( x 1 , . . . , x s ) ∈ Z s k b e a coloring of its components. If L J denotes the symmetric linking matrix of J , with framings on the diagonal, then the corresp onding p oin ted Reshetikhin–T uraev ev aluation is (3.2) R T C k ( J ; x ) = exp  π i k ⟨ x, L J x ⟩  , where for vectors u, v and a matrix A we write ⟨ u, v ⟩ = u ⊤ v , ⟨ u, Av ⟩ = u ⊤ Av . Indeed, each framing con tributes the twist q k ( x i ) , each crossing con tributes the braiding Ω k ( x i , x j ) ± 1 , and multiplying all lo cal con tributions yields the quadratic exp onen tial ( 3.2 ) . No w let L = L 1 ∪ · · · ∪ L m b e a surgery link presen ting the closed orien ted 3 -manifold M L , and let L ′ = L ′ 1 ∪ · · · ∪ L ′ r b e a colored link in the complement of L . W e write the linking matrix of L ⊔ L ′ in block form as L L ⊔ L ′ = L L L LL ′ L ⊤ LL ′ L L ′ ! . If g ∈ Z m k colors the surgery components and h ∈ Z r k colors L ′ , then (3.3) R T C k ( L ⊔ L ′ ; g , h ) = exp π i k  ⟨ g , L L g ⟩ + 2 ⟨ g , L LL ′ h ⟩ + ⟨ h, L L ′ h ⟩  ! . The Abelian surgery in v arian t is obtained b y a v eraging ov er the lab els on the surgery link and inserting the usual Kirb y normalization. Using the normalized coun ting measure on Z k , the inv ariant takes the form (3.4) Z Z k ( L ′ h ⊂ M L ) = N k ( L ) 1 k m X g ∈ Z m k R T C k ( L ⊔ L ′ ; g , h ) , where N k ( L ) is c hosen so as to ensure in v ariance under Kirby mov es. The required normalization is expressed in terms of the quadratic Gauss sums (3.5) A + ( k ) := X s ∈ Z k exp  π i k s 2  , A − ( k ) := X s ∈ Z k exp  − π i k s 2  = A + ( k ) . F or ev en k , the classical ev aluation giv es 8 D ANIEL GAL VIZ (3.6) A + ( k ) = k 1 / 2 e π i/ 4 , A − ( k ) = k 1 / 2 e − π i/ 4 . Equiv alently , if a ± ( k ) denote the normalized Gauss sums used in the Mattes–P oly ak– Reshetikhin presen tation [ MPR93 ], then A ± ( k ) = k a ± ( k ) . If σ ( L L ) denotes the signature of the surgery matrix, then the Kirb y normalization is (3.7) N raw k ( L ) = k − 1 / 2 A + ( k ) − m − σ ( L L ) 2 A − ( k ) − m + σ ( L L ) 2 . Substituting ( 3.3 ) in to ( 3.4 ) yields (3.8) Z RT , ra w Z k ( L ′ h ⊂ M L ) = A + ( k ) − m − σ ( L L ) 2 A − ( k ) − m + σ ( L L ) 2 × X g ∈ Z m k exp π i k  ⟨ g , L L g ⟩ + 2 ⟨ g , L LL ′ h ⟩ + ⟨ h, L L ′ h ⟩  ! . When L ′ = ∅ , this reduces to the closed in v ariant (3.9) Z RT , ra w Z k ( M L ) = k − 1 / 2 A + ( k ) − m − σ ( L L ) 2 A − ( k ) − m + σ ( L L ) 2 X g ∈ Z m k exp  π i k ⟨ g , L L g ⟩  . Let us brieﬂy indicate why this is inv arian t under Kirby mov es. In v ariance under isotop y is immediate. Inv ariance under K 2 follo ws from the fact that a handle slide replaces L L b y a congruen t matrix L L 7− → A ⊤ L L A, A ∈ GL( m, Z ) , and the induced c hange of v ariables g 7→ A − 1 g is bijectiv e on ( Z k ) m . F or K 1 , adjoining an unknot of framing ± 1 pro duces an additional factor X s ∈ Z k exp  ± π i k s 2  = A ± ( k ) , whic h is canceled precisely b y the normalization ( 3.7 ) . Therefore ( 3.8 ) deﬁnes a top ological in v ariant of the surgery presentation. This inv arian t is the rank-one p oin ted Reshetikhin– T uraev theory asso ciated with the ﬁnite quadratic mo dule ( Z k , q k ) . It is the combinatorial theory that will later b e compared with U (1) Chern–Simons theory . The case G = R . Mattes, P oly ak, and Reshetikhin also considered the Ab elian surgery construction for the noncompact group G = R , see [ MPR93 ]. In this case one takes the standard con tinuous quadratic data q R ( x ) = e π ix 2 , Ω R ( x, y ) = q R ( x + y ) q R ( x ) q R ( y ) = e 2 π ixy , so that the link ev aluation is again given b y the same quadratic expression as in the ﬁnite case. If L = L 1 ∪ · · · ∪ L m is a surgery link with linking matrix L L , and L ′ is an auxiliary colored link with color h ∈ R r , the corresp onding in v ariant is formally obtained b y replacing the ﬁnite av erage o ver Z m k with an oscillatory Gaussian integral ov er R m : Z R ( L ′ h ⊂ M L ) = b − m − σ ( L L ) 2 + b − m + σ ( L L ) 2 − Z R m exp π i  ⟨ x, L L x ⟩ + 2 ⟨ x, L LL ′ h ⟩ + ⟨ h, L L ′ h ⟩  ! dx, where the normalization constan ts are the F resnel in tegrals b + = Z R e π is 2 ds = e π i/ 4 , b − = Z R e − π is 2 ds = e − π i/ 4 , U (1) CHERN–SIMONS AND RESHETIKHIN–TURAEV TQFTS 9 understo o d in the oscillatory sense. When L L is nondegenerate, this in tegral ev aluates to the familiar Gaussian factor e πi 4 σ ( L L ) | det L L | − 1 / 2 times the phase obtained b y completing the square, while in the degenerate case one obtains a distribution supp orted along the n ull directions. Th us the R -theory should b e view ed as the con tinuous Gaussian analogue of the ﬁnite Z k -theory . Although it is useful conceptually , it is not the com binatorial mo del relev an t for the comparison with rank-one U (1) Chern–Simons theory at lev el k , which is go verned instead by the ﬁnite quadratic data ( Z k , q k ) . 3.2. Reciprocity form ulas and the relation with U (1) Chern–Simons theory. Mattes, Poly ak, and Reshetikhin, and indep endently Murak ami, Oh tsuki, and Ok ada observ ed that the raw R T Gauss sums appearing in ( 3.9 ) are closely related to the decomp osition of the Ab elian U (1) Chern–Simons partition function in to top ological sectors [ MOO92 , MPR93 ]. In the Ab elian case, unlik e the simply connected non–Ab elian case, one must sum o ver isomorphism classes of principal U (1) -bundles: Z CS X = X | P |∈ H 2 ( X ; Z ) Z A P / G P exp  2 π ik S X,P (Θ)  D Θ , F or a rational homology sphere, eac h bundle con tains at most one gauge-equiv alence class of ﬂat connections, so this partition function reduces to a discrete sum ov er topological sectors. The surgery Gauss sums in ( 3.9 ) are therefore exp ected to matc h these ﬂat-sector con tributions term b y term. The bridge b etw een the combinatorial and geometric pictures is provided by quadratic recipro cit y . Let L b e a symmetric, nondegenerate integer m × m matrix, and let r ≥ 1 . Then one has the recipro city form ula (3.10) X n ∈ Z m r exp  π i r ⟨ n, L n ⟩  = r m/ 2  det( L /i )  − 1 / 2 X l ∈ Z m / L Z m exp  − π ir ⟨ l, L − 1 l ⟩  . This iden tity rewrites the ﬁnite Gauss sum on the left as a sum ov er the ﬁnite Ab elian group Z m / L Z m , which in surgery presentations is canonically identiﬁed with the torsion subgroup T = T ors H 1 ( M ; Z ) ∼ = H 2 ( M ; Z ) . Th us the righ t-hand side has exactly the form one expects from a sum o ver top ological sectors in Ab elian Chern–Simons theory . When L is degenerate, an analogous form ula still holds after separating the nondegen- erate and null directions. Let L reg denote the restriction of L to (k er L ) ⊥ , and let d = 1 2 dim k er( L ⊗ R ) . Then one has the generalized recipro cit y form ula (3.11) X n ∈ Z m r exp  π i r ⟨ n, L n ⟩  = r d  det( L reg /ir )  − 1 / 2 X l ∈ ( Z m ∩ (Im L ) # ) / Im L exp  − π ir ⟨ l, L − 1 l ⟩  , where ( Im L ) # denotes the dual lattice with resp ect to the standard pairing. The degenerate form ula is essen tial from the TQFT p oint of view, since in termediate bordisms pro duced b y cutting and gluing naturally lead to singular surgery matrices. A useful reﬁnement of ( 3.10 ) w as later given by Deloup and T uraev [ DT07 ], who made the signature phase 10 D ANIEL GAL VIZ explicit: (3.12) X n ∈ Z m r exp  π i r ⟨ n, L n ⟩  = r m/ 2 e πi 4 σ ( L ) | det L| − 1 / 2 X l ∈ Z m / L Z m exp  − π ir ⟨ l, L − 1 l ⟩  . This is equiv alen t to ( 3.10 ) once one ﬁxes the square-ro ot branc h b y the signature of L . Applied to the in v ariant ( 3.9 ) , these recipro cit y form ulas sho w that the surgery Gauss sums reduce to in v ariants determined b y the torsion subgroup of H 1 ( M ; Z ) together with the induced quadratic reﬁnement of the linking pairing. This is precisely the form that arises in Abelian U (1) Chern–Simons theory . In particular, the com binatorial inv arian t attac hed to ( Z k , q k ) already contains the same ﬁnite quadratic data that con trol the top ological sectors of the geometric theory . Historically , Mattes, P oly ak, and Reshetikhin form ulated this corresp ondence as a comp elling exp ectation rather than as a complete theorem. The missing ingredient was a full comparison b etw een the surgery Gauss sums and a geometric quantization mo del for U (1) Chern–Simons theory . This is exactly the comparison carried out later in the present w ork. 3.3. The obstruction for G = U (1) . It is natural to ask whether one can deﬁne an Ab elian Reshetikhin–T uraev in v ariant directly from the compact group G = U (1) , in exact analogy with the cases G = Z k and G = R . W e now explain why this fails under the natural con tinuit y assumptions on the ribb on data. Let G b e an Ab elian top ological group equipp ed with a normalized Haar probability measure dg . In the Ab elian MPR formalism, one starts with a con tin uous bicharacter Ω : G × G → U (1) and a contin uous quadratic reﬁnement θ : G → U (1) , satisfying (3.13) θ ( x + y ) = Ω( x, y ) θ ( x ) θ ( y ) , θ (0) = 1 . Giv en such data, the ev aluation of a colored framed link J = J 1 ∪ · · · ∪ J r ⊂ S 3 , x = ( x 1 , . . . , x r ) ∈ G r , is (3.14) ⟨ J x ⟩ = r Y i =1 θ ( x i ) fr( J i ) Y 1 ≤ i 0 , the r esulting close d invariant c oincides with the cyclic p ointe d R T invariant use d in Se ction 3 and henc e with the Mattes–Polyak–R eshetikhin Gauss sum after quadr atic r e cipr o city [ MPR93 , MOO92 , DT07 ] . Pr o of. The category C ( G, q ) is C –linear and semisimple with simple ob jects indexed by G and one–dimensional morphism spaces. Duals are giv en b y R ∗ x ∼ = R − x . The braiding and t wist satisfy the rib bon identities b ecause q is a quadratic reﬁnemen t and Ω is its asso ciated bic haracter. The Hopf–link ev aluation giv es the bic haracter pairing S Hopf x,y = Ω( x, y ) , with the present normalization con v en tion for the p oin ted category . Nondegeneracy of Ω implies this pairing is in vertible, so C ( G, q ) is modular. This pro ves (i). Since C ( G, q ) is mo dular, Theorem 4.1 applies. Hence, T uraev’s construction deﬁnes state spaces V G,q (Σ , λ ) and b ordism op erators e Z G,q ( M ) satisfying the gluing axioms of a (2 + 1) –dimensional TQFT. This prov es (ii). U (1) CHERN–SIMONS AND RESHETIKHIN–TURAEV TQFTS 15 The op erator e Z G,q ( M ) is deﬁned through the pairing rule ⟨ w , e Z G,q ( M )( v ) ⟩ = e Z G,q ( M v ,w ) . The closed in v ariant on the righ t is computed by the Reshetikhin–T uraev surgery con- struction. F or a surgery presen tation M v ,w = M L , the ev aluation of the colored surgery link factorizes in the p ointed case as ⟨ L g ⟩ = Y i q ( g i ) L ii Y i 0 , Z ( M ) x = Z RT , ra w Z k ( M x ) = A − m − σ 2 + A − m + σ 2 − 1 k m X g ∈ ( Z k ) m exp  π i k Q L,x ( g )  , wher e Q L,x ( g ) is the quadr atic form obtaine d by adjoining the b oundary ﬁl ling to the linking matrix. Pr o of. Presen t M b y surgery on L ⊂ S 3 . Gluing the solid torus H x corresp onds to p erforming Dehn ﬁlling along the b oundary torus with meridian colored by x . The resulting manifold M x is closed and admits a surgery presentation obtained from L b y adjoining one additional comp onent corresponding to the b oundary ﬁlling. The R T in v ariant of M x is therefore computed b y the standard surgery form ula. In the pointed Ab elian category G = Z k with q ( y ) = exp  π i k y 2  , the R T ev aluation reduces to a quadratic Gauss sum. By the cyclic ﬁnite-group form ula of Section 3 , this equals A − m − σ 2 + A − m + σ 2 − 1 k m X g ∈ ( Z k ) m exp  π i k Q L,x ( g )  . Th us Z ( M ) x = Z RT , ra w Z k ( M x ) . □ Since V (Σ , λ ) ∼ = C [ G ] for a torus, we obtain Z ( M ) = X x ∈ G Z RT , ra w Z k ( M x ) e x . Th us the boundary state is completely determined b y the family of closed-manifold ev aluations { Z Z k ( M x ) } x ∈ G . The MPR Gauss sum do es not directly describ e the inv arian t of a manifold with b oundary . Rather, the b oundary TQFT assigns a v ector or linear map, its matrix elements are deﬁned by closing the manifold, and eac h suc h closed ev aluation is computed by the Ab elian surgery formula. Therefore the b oundary theory is consisten t with MPR in the sense that all of its closed pairings repro duce the Ab elian surgery in v ariant. Let L b e the linking matrix of the surgery presentation of M x . As in the closed case, the torsion Gauss sum obtained from the ab o v e formula equals X [ y ] ∈ Z m / L Z m exp(2 π ik q L ([ y ])) , where , q L ([ y ]) = 1 2 y ⊤ L − 1 y (mo d 1) . Hence the b oundary co eﬃcients are gov erned by the same quadratic reﬁnemen t of the torsion linking form that app ears in the closed theory . In particular, for G = Z k , the Ab elian Reshetikhin–T uraev theory with b oundary reconstructs the U (1) Chern–Simons theory at lev el k , and its closed ev aluations coincide exactly with the p ointed cyclic R T in v ariant after the recipro city comparison of Section 3.2 . The torsion linking form λ ([ x ] , [ y ]) induces the quadratic reﬁnemen t q L ([ x ]) . In the Ab elian R T theory , the matrix elements of a bordism op erator are computed b y closing with handleb o dies and ev aluating the resulting closed surgery presentation. After separating the null directions and applying quadratic recipro city to the nondegenerate blo c k, one reco vers the same ﬁnite torsion Gauss sum X [ x ] ∈ Z ρ / L reg Z ρ exp  2 π ik q L reg ([ x ])  18 D ANIEL GAL VIZ that appears in the closed comparison theorem. This observ ation is compatible with the closed formulas prov ed earlier, but it is not needed in the pro of of the extended equiv alence theorem. 5. Extended Equiv alence of U (1) Chern–Simons and R T Theories 5.1. Closed Comparison of Ab elian R T and U (1) Chern–Simons Theory. In this section, w e pro vide a detailed deriv ation sho wing that the tw o partition functions for the U (1) Chern–Simons and C ( Z k , q k ) Ab elian R T theory are, in fact, equiv alent. On one side, w e hav e Manoliu’s formula [ Man98a ], obtained from geometric quantization ov er U (1) ﬂat connections and expressed in terms of Ra y–Singer torsion. On the other side, we ha v e the Mattes–P olyak–Reshetikhin inv arian t [ MPR93 ], and Murak ami–Ohtsuki–Ok ada indep enden tly [ MOO92 ], constructed in the Reshetikhin–T uraev spirit [ R T91 ] from the p oin ted mo dular category C ( Z k , q k ) with q k ( x ) = exp  π i k x 2  , k ∈ 2 Z > 0 . W e sho w that, once torsion classes, linking pairings, quadratic reﬁnemen ts, and Gauss recipro cit y are consistently matched, the t w o expressions agree with a ﬁxe d universal signature normalization. Let M b e a closed, connected, orien ted 3 –manifold presen ted by in tegral surgery along an m –comp onen t orien ted framed link L = L 1 ∪ · · · ∪ L m ⊂ S 3 , with in teger symmetric linking matrix L = L L = ( L ij ) ∈ M m ( Z ) , L ii = fr( L i ) , L ij = lk( L i , L j ) ( i  = j ) . W rite σ = σ ( L ) for the signature and set ρ = rank( L ) , ν = n ull( L ) = m − ρ. It is standard that ν = b 1 ( M ) and that (5.1) H 1 ( M ; Z ) ∼ = Z ν ⊕ T ors H 1 ( M ; Z ) , # T ors H 1 ( M ; Z ) =   det( L reg )   , where L reg denotes an y nondegenerate ρ × ρ blo c k obtained from L b y an in tegral c hange of basis, equiv alen tly , the induced form on Z m / k er L . If L ′ ⊂ S 3 \ L is an additional framed link (Wilson lines in ph ysics), w e denote its c harge v ector b y h = ( h 1 , . . . , h ℓ ) ∈ ( Z k ) ℓ , and write L LL ′ ∈ M m × ℓ ( Z ) for the mixed linking matrix and L L ′ ∈ M ℓ ( Z ) for the self-linking matrix of L ′ in S 3 , so that after surgery , it b ecomes a framed link in M with the same in teger self-linking matrix. Throughout, the Chern–Simons lev el is denoted k ∈ 2 Z > 0 . In the R T normalization asso ciated with the p oin ted mo dular category C ( Z k , q k ) , it is con v enient to use the unnormalize d Gauss sums (5.2) A + ( k ) := X s ∈ Z k exp  π i k s 2  , A − ( k ) := X s ∈ Z k exp  − π i k s 2  . F or ev en k one has A + ( k ) = √ k e π i/ 4 , A − ( k ) = √ k e − π i/ 4 . F or a surgery link L together with Wilson line insertions L ′ h , the closed R T surgery expression is U (1) CHERN–SIMONS AND RESHETIKHIN–TURAEV TQFTS 19 (5.3) Z RT , ra w Z k ( L ′ h ⊂ M L ) = k − 1 / 2 A + ( k ) − m − σ 2 A − ( k ) − m + σ 2 × X g ∈ ( Z k ) m exp π i k  ⟨ g , L g ⟩ + 2 ⟨ g , L LL ′ h ⟩ + ⟨ h, L L ′ h ⟩  ! , where ⟨ x, Ax ⟩ = x ⊤ Ax ∈ Z . The additional factor k − 1 / 2 is the standard closed-manifold R T normalization, and it is exactly the factor needed to matc h Manoliu’s exp onen t m M b elo w in both the det L  = 0 and det L = 0 cases. In the closed case L ′ = ∅ , one ma y apply quadratic Gauss recipro cit y in the form used b y Murak ami–Ohtsuki–Ok ada and Deloup–T uraev [ MOO92 , DT07 ] to rewrite ( 5.3 ) as a Gauss sum ov er the cokernel lattice; for det L  = 0 this yields the familiar formula (5.4) Z RT , ra w Z k ( M L ) = e − πi 4 σ ( L ) k − 1 / 2 | det L| − 1 / 2 X [ x ] ∈ Z m / L Z m exp  π ik x ⊤ L − 1 x  , where L − 1 is tak en o ver Q and the exp onent is w ell deﬁned on the quotien t. Manoliu’s construction [ Man98a ], see our ﬁrst chapter, assigns to a closed orien ted 3 –manifold M a ﬁnite expression (5.5) Z CS U (1) ,k ( M ) = k m M # T ors H 2 ( M ; Z ) X p ∈ T ors H 2 ( M ; Z ) σ M ,p Z M M ( T M ) 1 / 2 = k m M Z M M σ M ( T M ) 1 / 2 , where M M = H 1 ( M ; U (1)) is the moduli space of ﬂat U (1) –connections, ( T M ) 1 / 2 is the Ra y–Singer half-density , and σ M ,p is the Chern–Simons phase on the comp onen t lab eled b y the torsion class p . F or connected closed M , Manoliu’s exp onent is (5.6) m M = 1 2  dim H 1 ( M ; R ) − dim H 0 ( M ; R )  = 1 2 ( b 1 ( M ) − 1) = 1 2 ( ν − 1) . In particular, when det L  = 0 one has ν = b 1 ( M ) = 0 and thus m M = − 1 2 , whereas for det L = 0 one has ν = b 1 ( M ) > 0 and m M = ( ν − 1) / 2 . The ﬁrst p oint of con tact b etw een the t wo sides is the p o wer of k . Prop osition 5.1. L et M = M L b e obtaine d by sur gery on L with linking matrix L of r ank ρ and nul lity ν . Assume k ∈ 2 Z > 0 . Then the k –dep endenc e c oming fr om the pr efactor in ( 5.3 ) and fr om Gauss r e cipr o city yields the net p ower k m M = k ( ν − 1) / 2 of Manoliu’s formula ( 5.5 ) . Pr o of. The magnitude of A + ( k ) − m − σ 2 A − ( k ) − m + σ 2 con tributes k − m/ 2 to the k –p o wer. If det L  = 0 then ρ = m and the recipro city step pro duces the determinant factor | det L| − 1 / 2 and a Gauss sum o ver Z m / L Z m . The original ( Z k ) m sum is traded for this cok ernel sum with a comp ensating factor k m/ 2 , so the o v erall prefactor in ( 5.3 ) con tributes k − 1 / 2 · k − m/ 2 · k m/ 2 = k − 1 / 2 . This is exactly k m M since m M = − 1 / 2 . If det L = 0 , c ho ose U ∈ GL m ( Z ) with U ⊤ L U =  L reg 0 0 0  , L reg ∈ M ρ ( Z ) in vertible o v er Q . W riting g = ( g reg , g 0 ) ∈ ( Z k ) ρ × ( Z k ) ν , the quadratic form depends only on g reg , so the raw sum factors as 20 D ANIEL GAL VIZ X g ∈ ( Z k ) m exp  π i k ⟨ g , L g ⟩  = X g reg ∈ ( Z k ) ρ exp  π i k ⟨ g reg , L reg g reg ⟩  ! · X g 0 ∈ ( Z k ) ν 1 ! = k ν X g reg ∈ ( Z k ) ρ exp  π i k ⟨ g reg , L reg g reg ⟩  . Applying reciprocity to the nondegenerate blo c k pro duces a comp ensating factor k ρ/ 2 and the determinant factor | det L reg | − 1 / 2 . Collecting the k –p o w ers giv es k − 1 / 2 · k − m/ 2 · k ν · k ρ/ 2 = k − 1 / 2 · k ν − ( ρ + ν ) / 2+ ρ/ 2 = k − 1 / 2 · k ν / 2 = k ( ν − 1) / 2 = k m M , whic h is exactly Manoliu’s exp onent ( 5.6 ). □ This shows that, when b 1 ( M ) > 0 , there are con tinuous families of ﬂat U (1) -connections, whic h are precisely the zero mo des of the quadratic action. These zero mo des modify the normalization of the path integral. The factor k m M = k ( b 1 ( M ) − 1) / 2 is exactly the correction arising from those ﬂat directions. On the R T side, one starts with a sum o ver ( Z k ) m lab elings of the surgery link and then uses Gauss recipro city to rewrite it as a sum o ver the cok ernel of the nondegenerate blo c k L reg . When L is degenerate, the n ull directions contribute an additional factor k ν . After com bining this with the remaining normalization factors, the resulting p o wer is exactly k ( ν − 1) / 2 . Therefore, Proposition 5.1 sho ws that the R T side captures the same zero-mode con tribution. Theorem 5.2. L et M b e a close d, c onne cte d, oriente d 3 –manifold. L et ( T M ) 1 / 2 denote the half-density on M M = H 1 ( M ; U (1)) arising fr om R ay–Singer analytic torsion in Manoliu’s normalization [ Man98a ] . Then (5.7) Z M M ( T M ) 1 / 2 =   T ors H 1 ( M ; Z )   1 / 2 . If mor e over M = M L is given by sur gery on L with linking matrix L and r ank ρ , then (5.8)   T ors H 1 ( M ; Z )   =   det( L reg )   , Z M M ( T M ) 1 / 2 =   det( L reg )   1 / 2 , and in p articular when det L  = 0 one has det( L reg ) = det( L ) . Pr o of. By the Ra y–Singer conjecture prov ed b y Cheeger and Müller [ RS71 , Che79 , Mül78 ], analytic torsion equals Reidemeister torsion. F or a closed orien ted 3 –manifold, the Reidemeister torsion on a torsion character equals | T ors H 1 ( M ; Z ) | , and it is constan t on connected comp onents of M M ; in tegrating the corresp onding half-densit y yields ( 5.7 ) (see Manoliu’s normalization discussion in [ Man98b ]). F or a surgery presen tation, H 1 ( M ; Z ) ∼ = Z m / L Z m and the torsion part has order | det( L reg ) | , giving ( 5.8 ). □ Theorem 5.2 has a clear ph ysical meaning. After expanding Ab elian Chern–Simons theory around ﬂat connections, the quadratic ﬂuctuation determinant is given by the Ra y–Singer analytic torsion, so that ( T M ) 1 / 2 ma y b e view ed as the residual contribution of quan tum ﬂuctuations around the ﬂat sector. The theorem shows that, once this quantit y is in tegrated o ver the mo duli space of ﬂat U (1) -connections, the result is not a gen uinely metric-dep enden t analytic ob ject, but instead reduces to the purely top ological quan tity p | T ors H 1 ( M ; Z ) | . This is also consistent with the Cheeger–Müller theorem, which identiﬁes analytic torsion with Reidemeister torsion and hence shows that the ﬂuctuation determinant is U (1) CHERN–SIMONS AND RESHETIKHIN–TURAEV TQFTS 21 top ological rather than metric-dep enden t. In the surgery presentation, the corresp onding ﬁnite inv arian t is explicit, since   T ors H 1 ( M ; Z )   =   det L reg   . Th us Theorem 5.2 shows that the seemingly contin uous analytic measure is in fact detecting precisely the ﬁnite top ology enco ded in the torsion subgroup of H 1 ( M ; Z ) . Let T := T ors H 2 ( M ; Z ) ∼ = T ors H 1 ( M ; Z ) , and identify T ∼ = Z ρ / L reg Z ρ via surgery . The torsion linking form λ : T × T → Q / Z is represen ted b y L − 1 reg : (5.9) λ ([ x ] , [ y ]) ≡ x ⊤ L − 1 reg y (mo d 1) . A standard surgery quadratic reﬁnement is (5.10) q L ([ x ]) ≡ 1 2 x ⊤ L − 1 reg x (mo d 1) , so that q L ( u + v ) − q L ( u ) − q L ( v ) = λ ( u, v ) . Theorem 5.3. Fix the Chern–Simons c onvention so that for p ∈ T the Manoliu factor is (5.11) σ M ,p = exp  2 π i k q L ( p )  . Then the torsion sum in ( 5.5 ) is the Gauss sum of q L : (5.12) X p ∈ T ors H 2 ( M ; Z ) σ M ,p = X [ x ] ∈ Z ρ / L reg Z ρ exp  π ik x ⊤ L − 1 reg x  . Pr o of. The reﬁnemen t iden tit y follows b y direct expansion of ( 5.10 ) . The iden tiﬁcation of the Abelian Chern–Simons action on torsion classes with a quadratic reﬁnemen t of the torsion linking pairing is classical (see e.g. F reed–Quinn [ F Q93 ] and the surgery discussion surrounding [ DT07 ]); ﬁxing the sign con v ention as in ( 5.11 ) yields ( 5.12 ). □ This tells us that eac h discrete torsion comp onen t on p ∈ T con tributes a phase giv en b y the classical Chern–Simons action, and this action is naturally a quadratic function on the ﬁnite group of torsion classes. Consequen tly , the closed-manifold partition function reduces to a sum of phases e 2 π i q ( u ) o v er all discrete ﬂat bundles. In other words, the Ab elian Chern–Simons path integral b ecomes a ﬁnite-dimensional Gaussian sum. In other words, the quadratic form q L is a reﬁnement of the torsion linking pairing λ : T × T → Q / Z . Hence, the action is completely determined b y the linking behavior of torsion cycles in the manifold. F rom the surgical p oint of view, this is exactly the exp ected algebraic framework. The previous theorem therefore shows that the R T state-sum expression is not simply analogous to the Chern–Simons path integral, but coincides with it after iden tifying the torsion group and its quadratic data. Hence, the closed-manifold conten t of Ab elian Chern–Simons theory is enco ded by a ﬁnite quadratic module, namely a ﬁnite Ab elian group equipp ed with a quadratic reﬁnemen t of its linking pairing, as w e will sho w in more detail later. After putting together Proposition 5.1 , Theorem 5.2 , and Theorem 5.3 , w e can under- stand the Chern-Simons partition function as follo ws: (5.13) Z CS U (1) ,k ( M L ) = k ( b 1 ( M ) − 1) / 2 | {z } F ree part / Contin uous ﬂat directions | T ors H 1 ( M ; Z ) | − 1 / 2 | {z } T orsion normalization factor X x ∈ T ors H 1 ( M ; Z ) e 2 π ik q M ( x ) | {z } Finite quadratic Gauss sum from the torsion linking form Therefore, the closed U (1) Chern–Simons partition function decomp oses into a contribution from the free part of H 1 ( M ; Z ) , the torsion half-densit y factor, and a ﬁnite sum o ver the torsion comp onents, whose phases are determined b y the quadratic reﬁnement of the 22 D ANIEL GAL VIZ torsion linking pairing. W e are now ready to sho w the Comp arison The or em b etw een Ab elian R T in v ariant and U (1) Chern–Simons theory . Theorem 5.4. L et M = M L b e obtaine d by sur gery on a fr ame d link L with linking matrix L of r ank ρ and signatur e σ ( L ) . Fix k ∈ 2 Z > 0 and use the normalizations ( 5.3 ) and ( 5.5 ) . Then, for L ′ = ∅ , (5.14) Z RT , ra w Z k ( M L ) = e − πi 4 σ ( L reg ) Z CS U (1) ,k ( M L ) , wher e L reg is any nonde gener ate ρ × ρ blo ck r epr esenting the induc e d form on Z m / k er L . In p articular: • if det L  = 0 then ρ = m and ( 5.14 ) b e c omes the r ational homolo gy spher e c ase with L reg = L ; • if det L = 0 then ρ < m and the right-hand side uses the torsion data determine d by L reg , while the Manoliu factor k m M uses m M = ( ν − 1) / 2 with ν = b 1 ( M ) . Pr o of. Insert Theorem 5.2 and Theorem 5.3 in to Manoliu’s expression ( 5.5 ) . Using ( 5.8 ) and ( 5.12 ) gives Z CS U (1) ,k ( M L ) = k m M | det( L reg ) | − 1 / 2 X [ x ] ∈ Z ρ / L reg Z ρ exp  π ik x ⊤ L − 1 reg x  . On the R T side, apply Deloup–T uraev recipro cit y to the nondegenerate blo c k L reg , and use the trivial factorization o ver the null directions when det L = 0 to obtain the corresp onding expression with the same Gauss sum and the determinan t factor | det ( L reg ) | − 1 / 2 together with the universal signature correction e − π iσ ( L reg ) / 4 . This is the standard signature term in quadratic recipro city , matc hing [ MOO92 , DT07 ]. Prop osition 5.1 shows that the remaining k –p o wer is exactly k m M in b oth regimes det L  = 0 and det L = 0 . Combining these iden tiﬁcations yields ( 5.14 ). □ Theorem 5.4 compares the r aw closed inv arian ts and sho ws that they diﬀer by the univ ersal quadratic recipro cit y factor e − πi 4 σ ( L reg ) . Consequen tly , this theorem do es not b y itself imply equalit y of the boundary op erators: in the handleb o dy pairing construction, the relev an t closed manifolds M v ,w := H out ( w ) ∪ X ∪ H in ( v ) dep end on the b oundary states v , w , and the residual phase therefore v aries with the c hosen closure. T o obtain exact equality of matrix elements one must pass to the extende d theories, where the Reshetikhin–T uraev functor is corrected by the W alker–T uraev w eight and the Chern–Simons functor is corrected by the Maslov–Kashiw ara/W alker phase. The b oundary equiv alence theorem is therefore reduced to proving that, for the canonical handleb o dy closures, the extended weigh t satisﬁes n ( M v ,w ) ≡ σ ( L v ,w, reg ) (mo d 8) , so that the W alk er correction absorbs the signature defect and the closed extended scalars agree exactly as w e will sho w in the next section. 5.2. Boundary Op erator Equiv alence. In the previous sections we compared the closed partition functions arising from tw o constructions of Ab elian Chern–Simons theory at level k : the geometric quantization approach b y Manoliu [ Man98a , Man98b ] and the surgery approach by Mattes–P oly ak–Reshetikhin (MPR) [ MPR93 ], which is the Ab elian Reshetikhin–T uraev surgery TQFT [ R T91 ]. W e no w put this comparison in a conceptually U (1) CHERN–SIMONS AND RESHETIKHIN–TURAEV TQFTS 23 cleaner form: after passing to the extended formalism and ﬁxing the common W alk er– Maslo v normalization, the tw o constructions deﬁne the same extende d (2 + 1) -dimensional TQFT , not merely the same closed 3 –manifold inv ariants. Let M 1 : (Σ 0 , λ 0 ) − → (Σ 1 , λ 1 ) , M 2 : (Σ 1 , λ 1 ) − → (Σ 2 , λ 2 ) b e comp osable extended bordisms. W rite ∂ M 1 = − Σ 0 ⊔ Σ 1 , ∂ M 2 = − Σ 1 ⊔ Σ 2 . T o form the comp osite M 2 ◦ M 1 , one glues along Σ 1 . The relev ant symplectic v ector space is V Σ 1 := H 1 (Σ 1 ; R ) , equipp ed with the intersection pairing. The gluing determines the standard triple of Lagrangian subspaces in V Σ 1 : L M 1 := Im  H 1 ( M 1 ; R ) → H 1 (Σ 1 ; R )  , λ 1 ⊂ H 1 (Σ 1 ; R ) , L M 2 := Im  H 1 ( M 2 ; R ) → H 1 (Σ 1 ; R )  , where the incoming boundary of M 2 is identiﬁed with Σ 1 . The Maslo v index app earing in the gluing law is therefore µ ( M 1 , M 2 ) := µ  L M 1 , λ 1 , L M 2  ∈ Z . Prop osition 5.5. L et Z RT , ra w Z k denote the unc orr e cte d R eshetikhin–T ur aev op er ator deﬁne d by the hand leb o dy p airing c onstruction. Then (5.15) Z RT , ra w Z k ( M 2 ◦ M 1 ) = κ µ ( M 1 ,M 2 ) Z RT , ra w Z k ( M 2 ) ◦ Z RT , ra w Z k ( M 1 ) , wher e µ ( M 1 , M 2 ) = µ ( L M 1 , λ 1 , L M 2 ) is the Maslov index of the standar d gluing triple and κ ∈ C × is the anomaly c onstant of the mo dular c ate gory. F or the p ointe d A b elian c ate gory C ( Z k , q k ) , with q k ( x ) = exp  π i k x 2  and k ∈ 2 Z > 0 , the anomaly c onstant is the phase of the c orr esp onding quadr atic Gauss sum; with the normalization ﬁxe d in The or em 4.1 , this phase is κ = e − π i/ 4 . Pr o of. Equation ( 5.15 ) is the standard pro jectiv e gluing law in T uraev’s extended R T theory; see [ T ur02 , Sec. IV.3]. The gluing anomaly is measured by the Maslo v index of the standard triple of Lagrangian subspaces attached to the cut boundary . P assing from ra w b ordisms to w eighted extended bordisms restores strict functoriality . In the p oin ted Ab elian case, the anomaly constant is determined by the quadratic Gauss sum of the twists of the simple ob jects. F or the chosen normalization of C ( Z k , q k ) , this phase is e − π i/ 4 . □ Remark. F r om this p oint onwar d, Z RT Z k : Cob ext 2+1 → V ect C denotes the W alker–normalize d Maslov–c orr e cte d extende d R eshetikhin–T ur aev functor alr e ady c onstructe d in The or em 4.1 . Thus, for a close d extende d 3 -manifold ( M , n ) , (5.16) Z RT Z k ( M , n ) := κ − n Z RT , ra w Z k ( M ) , κ = e − π i/ 4 , wher e n ∈ Z / 8 Z is the W alker–T ur aev weight. The symb ol Z RT , ra w Z k ( M ) wil l b e use d only for the r aw close d sur gery sc alar app e aring in Se ction 5.1 . The preceding theorems already provide a rigorous extended R T TQFT in T uraev’s sense. The next theorem do es not redeﬁne that TQFT. Its purp ose is to iden tify , for the sp ecial closed extended 3 -manifolds arising as handleb o dy closures in the pairing construction, the W alk er w eigh t on the R T side with the signature defect of the corresponding surgery presen tation. This is the extra input needed to upgrade the closed comparison theorem of Section 5.1 to an exact b oundary equiv alence statemen t. Theorem 5.6. Fix k ∈ 2 Z > 0 . L et X : (Σ in , λ in ) → (Σ out , λ out ) b e an extende d b or dism, and let v ∈ V RT Z k (Σ in , λ in ) , w ∈ V RT Z k (Σ out , λ out ) ∗ . 24 D ANIEL GAL VIZ Cho ose standar d hand leb o dy r epr esentatives H in ( v ) : ∅ → (Σ in , λ in ) , H out ( w ) : (Σ out , λ out ) → ∅ , and form the close d extende d 3 -manifold M v ,w := H out ( w ) ∪ Σ out X ∪ Σ in H in ( v ) . L et n ( M v ,w ) ∈ Z / 8 Z b e its W alker weight, and let L v ,w b e any inte gr al sur gery matrix for the underlying close d manifold M v ,w . L et L v ,w, reg denote the induc e d nonde gener ate form on Z m / k er L v ,w . Then: (i) The W alker weight of the c anonic al hand leb o dy closur e satisﬁes (5.17) n ( M v ,w ) ≡ σ ( L v ,w, reg ) (mo d 8) . (ii) The W alker-normalize d extende d R T sc alar satisﬁes (5.18) Z RT Z k ( M v ,w ) = e πi 4 σ ( L v,w , reg ) Z RT , ra w Z k ( M v ,w ) . (iii) The right-hand side of ( 5.18 ) is indep endent of the auxiliary sur gery pr esentation and of the chosen standar d hand leb o dy r epr esentatives of v and w . (iv) One has the exact close d e quality (5.19) Z RT Z k ( M v ,w ) = Z C S U (1) ,k ( M v ,w ) . Pr o of. Let W v ,w b e the surgery trace 4–manifold obtained from B 4 b y attaching 2 -handles along a framed link presen ting the closed manifold M v ,w , and let L v ,w denote the corre- sp onding linking matrix. F or the canonical handlebo dy closures, w e use W alker’s surgery framew ork of the extended weigh t, according to whic h the framing defect is measured b y the signature of the surgery data; see [ W al91 , p. 103] (see also [ W al91 , p. 35]). Thus n ( M v ,w ) ≡ σ ( W v ,w ) (mo d 8) . Recall that the intersection form of an oriented compact 4 -manifold W is the symmetric bilinear pairing Q W : H 2 ( W ; Z ) × H 2 ( W ; Z ) − → Z , Q W ([ S ] , [ S ′ ]) = S · S ′ , deﬁned b y the algebraic in tersection num b er of represented surfaces. In the present case, the form Q W v,w is represented, in the natural basis given by the cores of the attached 2 -handles, b y the surgery linking matrix L v ,w ; see [ GS99 , Prop. 4.5.11]. If Q W v,w is degenerate, its radical is ker Q W v,w , and the induced nondegenerate form on H 2 ( W v ,w ; Z ) / k er Q W v,w is represen ted by the regular part L v ,w, reg . Since the signature of a p ossibly degenerate symmetric form is, b y deﬁnition, the signature of its induced nondegenerate quotien t form, it follo ws that σ ( W v ,w ) = σ ( L v ,w, reg ) . Therefore n ( M v ,w ) ≡ σ ( L v ,w, reg ) (mo d 8) , whic h is exactly ( 5.17 ) . F or later use, note that b oth sides satisfy the same Maslo v gluing la w. Indeed, by the deﬁnition of the W alk er w eigh t in the extended b ordism category , (5.20) n ( M 2 ◦ M 1 ) ≡ n ( M 2 ) + n ( M 1 ) + µ ( M 1 , M 2 ) (mo d 8) . U (1) CHERN–SIMONS AND RESHETIKHIN–TURAEV TQFTS 25 Lik ewise, W all’s non-additivit y theorem applied to the corresp onding surgery trace 4 – manifolds yields (5.21) σ ( L 21 , reg ) ≡ σ ( L 2 , reg ) + σ ( L 1 , reg ) + µ ( M 1 , M 2 ) (mo d 8) . See [ W al69 , Ran97 ]. By deﬁnition of the W alk er-normalized closed R T scalar, Z RT Z k ( M v ,w ) = κ − n ( M v,w ) Z RT , ra w Z k ( M v ,w ) . Using κ = e − π i/ 4 from Proposition 5.5 together with ( 5.17 ), we obtain Z RT Z k ( M v ,w ) =  e − π i/ 4  − n ( M v,w ) Z RT , ra w Z k ( M v ,w ) = e πi 4 σ ( L v,w , reg ) Z RT , ra w Z k ( M v ,w ) , (5.22) whic h pro v es (ii). Since the left-hand side dep ends only on the closed extended manifold ( M v ,w , n ( M v ,w )) , the right-hand side is indep enden t of b oth the chosen surgery presen tation and the chosen standard handlebo dy represen tatives of v and w , pro ving (iii). Finally , the closed comparison theorem of Section 5.1 giv es Z C S U (1) ,k ( M v ,w ) = e πi 4 σ ( L v,w , reg ) Z RT , ra w Z k ( M v ,w ) , and comparison with ( 5.18 ) prov es ( 5.19 ). □ Fix an even lev el k ∈ 2 Z > 0 . Let G k = Z k and deﬁne the quadratic function (5.23) q k : G k → U (1) , q k ( x ) = exp  π i k x 2  , with associated bic haracter (5.24) Ω k ( x, y ) = q k ( x + y ) q k ( x ) q k ( y ) = exp  2 π i k xy  . Let C k := C ( G k , q k ) b e the p oin ted mo dular category determined b y ( G k , Ω k , q k ) ; it is mo dular b ecause Ω k is nondegenerate. (A) The R T side. Let Cob ext 2+1 denote T uraev’s category of extende d bordisms [ T ur02 ]: ob jects are extended surfaces (Σ , λ ) , where λ ⊂ H 1 (Σ; R ) is a Lagrangian subspace; morphisms are extended 3 –dimensional b ordisms equipp ed with the appropriate additional data that controls the framing anomaly; comp osition is gluing with Maslov correction. T uraev’s construction [ T ur02 ] yields a symmetric monoidal functor (5.25) Z R T k : Cob ext 2+1 − → V ect C , (Σ , λ ) 7− → V R T k (Σ , λ ) , M 7− → Z R T k ( M ) , whose v alue on a closed 3 –manifold presented by surgery on a framed link L ⊂ S 3 is computed b y the Ab elian surgery sum obtained from the R T ev aluation [ R T91 , MPR93 ]. Here and b elo w, Z R T k denotes the W alk er–normalized Maslov–corrected extended R T functor, in the notation of Section 5.1 . (B) The Chern–Simons side. Manoliu constructs an extended TQFT for U (1) Chern– Simons theory b y geometric quan tization with analytic torsion [ Man97 , Man98a ]. In particular, to each extended surface (Σ , L ) , where L is the p olarization, one assigns a ﬁnite-dimensional Hilbert space H k (Σ , L ) , and to eac h extended b ordism X one assigns a linear map (5.26) Z CS U (1) ,k : Cob ext 2+1 − → V ect C , (Σ , L ) 7− → H k (Σ , L ) , X 7− → Z CS U (1) ,k ( X ) , c haracterized b y the canonical vector ( 2.7 ) and the BKS pairing (gluing formalism), with the standard Maslov–Kashiw ara correction built in to the extended structure. 26 D ANIEL GAL VIZ T o compare ( 5.25 ) and ( 5.26 ) , we ﬁrst identify the extended-surface data via the standard P oincaré-dualit y corresp ondence betw een T uraev’s homological Lagrangian λ ⊂ H 1 (Σ; R ) and Manoliu’s cohomological polarization L λ ⊂ H 1 (Σ; R ) . Th us w e write (5.27) (Σ , λ ) ← → (Σ , L λ ) , where L λ denotes the rational Lagrangian determined b y λ under Poincaré duality . This is the common p olarization data for the classical b oundary phase space, namely the symplectic torus of ﬂat U (1) -connections on Σ at lev el k . Under this iden tiﬁcation, one pro ves that the R T and Chern–Simons state spaces are canonically isomorphic. In particular, for a connected surface of genus g , b oth theories yield a v ector space of dimension k g , and the canonical isomorphism identiﬁes the R T handleb o dy basis with Manoliu’s Bohr–Sommerfeld basis. Lemma 5.7. L et λ Z := λ ∩ H 1 (Σ; Z ) , Π λ := Hom( λ Z , Z /k Z ) ∼ = λ ∨ Z /k λ ∨ Z . Then: (i) F or every adapte d inte gr al symple ctic b asis ( a 1 , . . . , a g , b 1 , . . . , b g ) , λ = ⟨ a 1 , . . . , a g ⟩ R , evaluation on ( a 1 , . . . , a g ) identiﬁes Π λ with ( Z /k Z ) g . (ii) The R T hand leb o dy b asis of V RT k (Σ , λ ) is c anonic al ly indexe d by Π λ : if α ∈ Π λ , then in any adapte d b asis the c orr esp onding b asis ve ctor is the hand leb o dy skein obtaine d by c oloring the i -th c or e curve by α ( a i ) ∈ Z /k Z . (iii) Manoliu’s Bohr–Sommerfeld b asis of H k (Σ , L λ ) is c anonic al ly indexe d by Π λ : if α ∈ Π λ , then in any adapte d b asis the c orr esp onding b asis ve ctor is the unique c ovariantly c onstant se ction over the Bohr–Sommerfeld le af lab ele d by the c o or dinate ve ctor ( α ( a 1 ) , . . . , α ( a g )) ∈ ( Z /k Z ) g . (iv) These two indexings ar e indep endent of the chosen adapte d b asis. (v) If f : Σ → Σ pr eserves the extende d structur e (Σ , λ ) , then f ∗ ( λ Z ) = λ Z , henc e f induc es a p ermutation f # : Π λ → Π λ , f # ( α ) = α ◦ ( f ∗ | λ Z ) − 1 , and b oth the R T b asis and the Bohr–Sommerfeld b asis ar e e quivariant for this action. Pr o of. P art (i) is immediate: an adapted basis ( a 1 , . . . , a g ) is a Z -basis of λ Z , so ev aluation on that basis iden tiﬁes Hom( λ Z , Z /k Z ) with ( Z /k Z ) g . F or part (ii), choose a handleb o dy H with ∂ H = Σ and k er ( H 1 (Σ; R ) → H 1 ( H ; R )) = λ , as in T uraev’s handleb o dy mo del. If ( a i , b i ) is adapted, then the a i b ound compressing disks in H , and the dual core curves γ i form a basis of H 1 ( H ; Z ) dual to a 1 , . . . , a g . F or α ∈ Π λ , deﬁne e α to b e the R T basis vector represen ted b y the skein whose i -th core curv e γ i is colored b y α ( a i ) ∈ Z /k Z . Since in the pointed category ev ery simple ob ject is inv ertible and fusion is addition in Z /k Z , this giv es exactly the standard handleb o dy basis. F or part (iii), in Manoliu’s real-p olarization quantization the state space H k (Σ , L λ ) is the direct sum o ver Bohr–Sommerfeld lea ves of one-dimensional spaces of co v ariantly constan t sections. F or the p olarization determined by L λ , the Bohr–Sommerfeld set is canonically indexed b y λ ∨ Z /k λ ∨ Z ∼ = Π λ . Thus for α ∈ Π λ there is a distinguished basis vector v α supp orted on the Bohr–Sommerfeld leaf lab eled b y α . In coordinates coming from an y adapted basis, this is exactly Manoliu’s basis v ector v (Σ , L λ ) q with q = ( α ( a 1 ) , . . . , α ( a g )) . It remains to pro ve the indep endence asserted in part (iv). Let ( a 1 , . . . , a g , b 1 , . . . , b g ) and ( a ′ 1 , . . . , a ′ g , b ′ 1 , . . . , b ′ g ) b e t w o adapted integral symplectic bases. Since b oth are adapted to the same Lagrangian λ , the c hange-of-basis matrix has U (1) CHERN–SIMONS AND RESHETIKHIN–TURAEV TQFTS 27 the form  A B 0 ( A − 1 ) T  ∈ S p (2 g , Z ) , A ∈ GL ( g , Z ) , with B symmetric, so a ′ = Aa, and b ′ = B a + ( A − 1 ) T b. Hence the dual basis of λ ∨ Z transforms by ( a ′∗ ) = ( A − 1 ) T ( a ∗ ) . Therefore the co ordinate vector of a ﬁxed elemen t α ∈ Π λ c hanges by the same con tragredien t rule: q ′ = ( A − 1 ) T q . On the R T side, the core classes in H 1 ( H ; Z ) dual to a i transform b y the same rule, b ecause the a -part v anishes in H 1 ( H ; Z ) . Th us the skein lab eled b y q in the ﬁrst adapted basis is exactly the sk ein labeled by q ′ = ( A − 1 ) T q in the second. So the abstract vector e α is independent of the adapted basis. On the Chern–Simons side, the Bohr–Sommerfeld leaf lab els are coordinates in λ ∨ Z /k λ ∨ Z , so changing adapted basis changes those coordinates b y the same matrix ( A − 1 ) T . Hence the abstract vector v α is also indep enden t of the adapted basis. This pro ves part (iv). F or part (v), if f : Σ → Σ preserves the extended structure, then f ∗ λ = λ and hence f ∗ ( λ Z ) = λ Z . Therefore f acts on Π λ b y precomp osition. By functoriality of T uraev’s handleb o dy construction, the R T mapping-cylinder action sends e α to e f # α . By functorialit y of Manoliu’s quan tization and the BKS construction, the Chern–Simons action sends v α to v f # α . Th us b oth canonical bases are equiv ariant for the same p erm utation action of f on Π λ . □ Theorem 5.8. L et (Σ , λ ) b e a c onne cte d extende d surfac e of genus g , and let L λ ⊂ H 1 (Σ; R ) b e the r ational L agr angian c orr esp onding to λ ⊂ H 1 (Σ; R ) under Poinc ar é duality. Then ther e exists a c anonic al ve ctor sp ac e isomorphism (5.28) Φ Σ : V R T k (Σ , λ ) ∼ = − − → H k (Σ , L λ ) , which is natur al with r esp e ct to diﬀe omorphisms pr eserving the extende d structur e and satisﬁes dim V R T k (Σ , λ ) = dim H k (Σ , L λ ) = k g . Mor e over, after cho osing any inte gr al symple ctic b asis ( a 1 , . . . , a g , b 1 , . . . , b g ) of H 1 (Σ; Z ) adapte d to λ , i.e. λ = ⟨ a 1 , . . . , a g ⟩ R , the isomorphism Φ Σ is char acterize d by (5.29) Φ Σ ( e q ) = v (Σ , L λ ) q , q ∈ ( Z /k Z ) g , L et Φ ∨ Σ := ((Φ − 1 Σ ) ∗ ) :  V RT k (Σ , λ )  ∗ ∼ − − → H k (Σ , L λ ) ∗ denote the induc e d dual isomorphism, wher e { e q } is the standar d R T hand leb o dy b asis and { v (Σ , L λ ) q } is Manoliu’s Bohr–Sommerfeld b asis. Final l y, Φ Σ identiﬁes the c anonic al cylinder kernels, with the se c ond tensor factor tr ansp orte d by the induc e d dual map: (5.30) (Φ Σ ⊗ (Φ − 1 Σ ) ∗ )  Z RT k (Σ × I , λ ⊕ λ, 0)  = Z C S U (1) ,k (Σ × I , L λ ⊕ L λ , 0) . Equivalently, Φ Σ identiﬁes the c anonic al b oundary p airings on the R T and Chern–Simons sides. Pr o of. Let λ Z := λ ∩ H 1 (Σ; Z ) , Π λ := Hom( λ Z , Z /k Z ) . By Lemma 5.7 (i), b oth the R T state space V RT k (Σ , λ ) and the Chern–Simons state space H k (Σ , L λ ) carry canonical bases indexed b y the same ﬁnite set Π λ : { e α } α ∈ Π λ ⊂ V RT k (Σ , λ ) , { v α } α ∈ Π λ ⊂ H k (Σ , L λ ) . 28 D ANIEL GAL VIZ Moreo v er, these bases are indep endent of ev ery auxiliary adapted basis. W e therefore deﬁne Φ Σ ( e α ) := v α , for α ∈ Π λ . Since | Π λ | = k g , this is a well-deﬁned vector space isomorphism Φ Σ : V RT k (Σ , λ ) ∼ − − → H k (Σ , L λ ) , and dim V RT k (Σ , λ ) = dim H k (Σ , L λ ) = k g . If one no w chooses an y adapted in tegral symplectic basis ( a 1 , . . . , a g , b 1 , . . . , b g ) , then Π λ ∼ = ( Z /k Z ) g b y α 7→ ( α ( a 1 ) , . . . , α ( a g )) , and the deﬁnition ab ov e becomes exactly Φ Σ ( e q ) = v (Σ , L λ ) q , q ∈ ( Z /k Z ) g , whic h is ( 5.29 ) . W e next c heck compatibility with the cylinder kernels. On the R T side, the cylinder kernel is Z RT k (Σ × I , λ ⊕ λ, 0) = X α ∈ Π λ e α ⊗ e ∨ α , where e ∨ α denotes the canonical cov ector dual to e α under the R T boundary pairing. On the Chern–Simons side, Manoliu’s cylinder k ernel is Z C S U (1) ,k (Σ × I , L λ ⊕ L λ , 0) = X α ∈ Π λ v α ⊗ v ∗ α , where v ∗ α is the canonical co vector dual to v α under the BKS pairing. Therefore (Φ Σ ⊗ (Φ − 1 Σ ) ∗ )  Z RT k (Σ × I , λ ⊕ λ, 0)  = Z C S U (1) ,k (Σ × I , L λ ⊕ L λ , 0) . Equiv alently , Φ Σ iden tiﬁes the canonical R T pairing with Manoliu’s canonical pairing, whic h is the con tent of ( 5.30 ) . Finally , let f : Σ → Σ be a diﬀeomorphism preserving the extended structure. By Lemma 5.7 (v), f acts on Π λ , and b oth functorial theories act on their canonical bases b y the same p erm utation: Z RT k ( f )( e α ) = e f # α , Z C S U (1) ,k ( f )( v α ) = v f # α . Hence Φ Σ  Z RT k ( f ) e α  = Φ Σ ( e f # α ) = v f # α = Z C S U (1) ,k ( f )( v α ) = Z C S U (1) ,k ( f )Φ Σ ( e α ) for ev ery α ∈ Π λ . Since the e α form a basis, w e conclude that Φ Σ ◦ Z RT k ( f ) = Z C S U (1) ,k ( f ) ◦ Φ Σ . Th us Φ Σ is canonical and natural with resp ect to diﬀeomorphisms preserving the extended structure. This pro ves ( 5.28 ) , ( 5.29 ) , and the pairing compatibilit y asserted in ( 5.30 ) . □ F or a disconnected extended surface (Σ , λ ) = F r j =1 (Σ j , λ j ) , w e extend ( 5.28 ) b y sym- metric monoidalit y: Φ Σ := r O j =1 Φ Σ j , Φ ∅ = id C . Th us Φ Σ is deﬁned for all ob jects of Cob ext 2+1 . Lemma 5.9. F or e ach c onne cte d extende d surfac e (Σ , λ ) , ﬁx onc e and for al l a standar d hand leb o dy H λ with ∂ H λ = Σ and k er  H 1 (Σ; R ) → H 1 ( H λ ; R )  = λ. L et Π λ b e the c anonic al p olarization lab el set of L emma 5.7 . L et { e α } α ∈ Π λ ⊂ V RT k (Σ , λ ) b e the c anonic al R T b asis, and let { v α } α ∈ Π λ ⊂ H k (Σ , L λ ) b e the c anonic al Chern–Simons b asis deﬁne d by v α = Φ Σ ( e α ) . L et e ∨ α and v ∗ α denote the c orr esp onding c anonic al dual c ove ctors under the R T and Chern–Simons b oundary p airings. U (1) CHERN–SIMONS AND RESHETIKHIN–TURAEV TQFTS 29 Now let X : (Σ in , λ in ) → (Σ out , λ out ) b e an extende d b or dism. F or α ∈ Π λ in and β ∈ Π λ out , let M α,β := H out β ∪ Σ out X ∪ Σ in H in α denote the close d extende d 3-manifold obtaine d by gluing to X the ﬁxe d standar d inc oming and outgoing hand leb o dies c arrying the lab els α and β . Then: ⟨ e ∨ β , Z RT k ( X )( e α ) ⟩ = Z RT k ( M α,β ) , and ⟨ v ∗ β , Z C S U (1) ,k ( X )( v α ) ⟩ = Z C S U (1) ,k ( M α,β ) . In p articular, after identifying the c anonic al b ases by Φ Σ , the matrix c o eﬃcients of the R T and Chern–Simons op er ators ar e c ompute d fr om the same c anonic al ly indexe d family of close d extende d manifolds { M α,β } . Pr o of. The ﬁrst identit y is exactly T uraev’s pairing construction for extended R T theory: for a b ordism X : (Σ in , λ in ) → (Σ out , λ out ) , the matrix co eﬃcien t of Z RT k ( X ) against a c hosen incoming state and outgoing cov ector is, b y deﬁnition, the R T inv ariant of the closed extended 3-manifold obtained b y gluing the corresp onding handlebo dies to X ; see Theorem 4.1 and Theorem 4.2 (iii). Applied to the canonical basis vector e α and canonical dual co vector e ∨ β , this gives ⟨ e ∨ β , Z RT k ( X )( e α ) ⟩ = Z RT k ( M α,β ) . F or the Chern–Simons side, Theorem 5.8 identiﬁes the canonical R T basis and canonical Chern–Simons basis b y the same lab el set Π λ , and also identiﬁes the canonical R T and Chern–Simons b oundary pairings. Therefore the basis v ector v α and dual co v ector v ∗ β are the Chern–Simons b oundary data corresp onding to the same canonical incoming and outgoing labels α, β . Manoliu’s gluing axiom then implies that the corresp onding matrix co eﬃcien t of Z C S U (1) ,k ( X ) is the closed Chern–Simons inv ariant of the extended 3-manifold obtained b y gluing to X the same ﬁxed standard incoming and outgoing handleb o dies with those lab els. By deﬁnition, that closed extended manifold is M α,β . Hence ⟨ v ∗ β , Z C S U (1) ,k ( X )( v α ) ⟩ = Z C S U (1) ,k ( M α,β ) . The ﬁnal statement follo ws immediately . □ Theorem 5.10. Fix k ∈ 2 Z > 0 , and let C k = C ( Z k , q k ) , q k ( x ) = e π ix 2 /k , Ω k ( x, y ) = e 2 π ixy/k . L et X : (Σ in , λ in ) → (Σ out , λ out ) b e an extende d b or dism in T ur aev’s sense, and let Z RT Z k ( X ) ∈ Hom  V RT Z k (Σ in , λ in ) , V RT Z k (Σ out , λ out )  denote the W alker–normalize d extende d R T op er ator. Interpr et the same extende d data as an e - 3 -manifold ( X , L, n ) in Manoliu’s sense, with asso ciate d op er ator Z C S U (1) ,k ( X ) ∈ Hom  H k (Σ in , L in ) , H k (Σ out , L out )  . R ememb er the c anonic al identiﬁc ation of b oundary state sp ac es establishe d e arlier in The or em 5.8 : Φ Σ : V RT Z k (Σ , λ ) ∼ = − → H k (Σ , L λ ) . Then Φ Σ out ◦ Z RT Z k ( X ) = Z C S U (1) ,k ( X ) ◦ Φ Σ in . 30 D ANIEL GAL VIZ Equivalently, after identifying the b oundary state sp ac es via Φ Σ , one has Z RT Z k ( X ) = Z C S U (1) ,k ( X ) . Pr o of. By linearit y , it suﬃces to compare matrix coeﬃcients on the canonical basis of the incoming state space and the canonical dual basis of the outgoing state space. Let α ∈ Π λ in , and β ∈ Π λ out , write e α ∈ V RT k (Σ in , λ in ) , e ∨ β ∈ V RT k (Σ out , λ out ) ∗ for the canonical R T basis vector and canonical dual co v ector. Deﬁne v α := Φ Σ in ( e α ) ∈ H k (Σ in , L in ) , v ∗ β := (Φ − 1 Σ out ) ∗ ( e ∨ β ) ∈ H k (Σ out , L out ) ∗ . By Theorem 5.8 , v α and v ∗ β are precisely the corresp onding canonical Chern–Simons basis v ector and canonical dual co vector. Now let M α,β b e the canonical closed extended 3-manifold obtained by gluing to X the standard incoming and outgoing handleb o dies carrying the lab els α and β , as in Lemma 5.9 . That lemma giv es ⟨ e ∨ β , Z RT k ( X )( e α ) ⟩ = Z RT k ( M α,β ) , and ⟨ v ∗ β , Z C S U (1) ,k ( X )( v α ) ⟩ = Z C S U (1) ,k ( M α,β ) . By Theorem 5.6 , the t wo closed extended in v arian ts agree on ev ery canonical handleb ody closure. Hence Z RT k ( M α,β ) = Z C S U (1) ,k ( M α,β ) , so ⟨ e ∨ β , Z RT k ( X )( e α ) ⟩ = ⟨ v ∗ β , Z C S U (1) ,k ( X )( v α ) ⟩ . Using the deﬁnitions of v α and v ∗ β , this b ecomes ⟨ e ∨ β , Z RT k ( X )( e α ) ⟩ = D (Φ − 1 Σ out ) ∗ ( e ∨ β ) , Z C S U (1) ,k ( X )  Φ Σ in ( e α )  E , or equiv alen tly , ⟨ e ∨ β , Z RT k ( X )( e α ) ⟩ = ⟨ e ∨ β , Φ − 1 Σ out Z C S U (1) ,k ( X )Φ Σ in ( e α ) ⟩ . Since the co v ectors e ∨ β separate p oints and the v ectors e α form a basis, equality of all these matrix co eﬃcien ts implies Z RT k ( X ) = Φ − 1 Σ out Z C S U (1) ,k ( X )Φ Σ in . Equiv alently , Φ Σ out ◦ Z RT k ( X ) = Z C S U (1) ,k ( X ) ◦ Φ Σ in . This pro ves the theorem. □ 5.3. Extended Equiv alence of Ab elian R T and U (1) Chern–Simons TQFT s. W e no w state the main result. The tw o extended TQFT functors agree under the boundary iden tiﬁcation ( 5.28 ) . The only subtlety is that Theorem 5.4 compares the r aw closed in v ariants and lea ves a universal signature phase; in the extended formalism this phase is absorb ed by the W alker–Maslo v correction, as established in Theorem 5.6 . With that understo o d, th e pro of follo ws T uraev’s deﬁnition of b ordism op erators via closed pairings together with Manoliu’s gluing form ula, using the established closed and b oundary comparison theorems as input. U (1) CHERN–SIMONS AND RESHETIKHIN–TURAEV TQFTS 31 Theorem 5.11. Fix k ∈ 2 Z > 0 , and let C k = C ( Z k , q k ) , q k ( x ) = exp  π i k x 2  . L et Z R T k , Z CS U (1) ,k : Cob ext 2+1 − → V ect C denote, r esp e ctively, T ur aev’s W alker–normalize d R eshetikhin–T ur aev TQFT asso ciate d with C k [ T ur02 , §II.2, §IV.1–IV.3] and Manoliu’s extende d U (1) Chern–Simons TQFT at level k [ Man98a ] , with the extende d structur es identiﬁe d via ( 5.27 ) . Assume the normalization c onventions ar e ﬁxe d so that the closed extended invariants ar e c omp ar e d in the same Kirby/Maslov c onvention. Then: (i) (Close d extende d e quality on c anonic al closur es.) F or every close d extende d 3 –manifold X arising as a c anonic al hand leb o dy closur e in the p airing c onstruction, one has Z R T k ( X ) = Z CS U (1) ,k ( X ) . This is exactly the W alker–c orr e cte d close d e quality obtaine d in The or em 5.6 . (ii) (Boundary op er ator e quivalenc e.) F or every extende d b or dism X : (Σ in , λ in ) − → (Σ out , λ out ) one has e quality of b or dism op er ators after identifying b oundary state sp ac es: Φ Σ out ◦ Z R T k ( X ) = Z CS U (1) ,k ( X ) ◦ Φ Σ in , wher e Φ Σ is the c anonic al identiﬁc ation fr om The or em 5.8 . This is pr e cisely The or em 5.10 . (iii) (Natur al monoidal isomorphism of functors.) The family of isomorphisms Φ Σ fr om ( 5.28 ) , extende d to disc onne cte d surfac es by tensor pr o duct over c onne cte d c omp onents, deﬁnes a monoidal natural isomorphism of symmetric monoidal functors Φ : Z R T k = ⇒ Z CS U (1) ,k . Equivalently, Z R T k and Z CS U (1) ,k ar e isomorphic as symmetric monoidal functors Cob ext 2+1 − → V ect C . Pr o of. P arts (i) and (ii) are exactly the previously established closed and b oundary comparison theorems in the ﬁxed common normalization, namely Theorem 5.6 together with Theorem 5.10 . W e therefore pro ve (iii). Let (Σ , λ ) be an ob ject of Cob ext 2+1 and write it as a disjoin t union of connected comp o- nen ts (Σ , λ ) = r G j =1 (Σ j , λ j ) . Both TQFT s are symmetric monoidal, hence pro vide canonical tensor product iden tiﬁca- tions V R T k (Σ , λ ) ∼ = r O j =1 V R T k (Σ j , λ j ) , H k (Σ , L λ ) ∼ = r O j =1 H k (Σ j , L λ j ) , 32 D ANIEL GAL VIZ where L λ = L j L λ j is the induced p olarization on the Chern–Simons side. On eac h connected component, Theorem 5.8 provides the canonical isomorphism ( 5.28 ). Deﬁne (5.31) Φ Σ := r O j =1 Φ Σ j , transp orted through the ab o ve canonical monoidal identiﬁcations. This giv es a w ell-deﬁned v ector space isomorphism for every ob ject (Σ , λ ) . Naturalit y . Let X : (Σ in , λ in ) → (Σ out , λ out ) b e any morphism in Cob ext 2+1 . W e m ust sho w that the square V R T k (Σ in , λ in ) Z R T k ( X ) − − − − → V R T k (Σ out , λ out ) Φ Σ in   y   y Φ Σ out H k (Σ in , L λ in ) − − − − − − → Z CS U (1) ,k ( X ) H k (Σ out , L λ out ) comm utes. But this is precisely the statemen t of part (ii), that is, Theorem 5.10 . Thus { Φ Σ } is a natural transformation Z R T k ⇒ Z CS U (1) ,k . Φ is a natural isomorphism. Eac h Φ Σ is an isomorphism by construction: on connected comp onen ts this is Theorem 5.8 , and on disconnected surfaces it is the tensor pro duct of the isomorphisms ( 5.31 ). Hence Φ is a natural isomorphism. Monoidalit y . Let (Σ 1 , λ 1 ) and (Σ 2 , λ 2 ) be extended surfaces. Symmetric monoidality of b oth TQFT s giv es canonical iden tiﬁcations V R T k (Σ 1 ⊔ Σ 2 , λ 1 ⊕ λ 2 ) ∼ = V R T k (Σ 1 , λ 1 ) ⊗ V R T k (Σ 2 , λ 2 ) , H k (Σ 1 ⊔ Σ 2 , L λ 1 ⊕ L λ 2 ) ∼ = H k (Σ 1 , L λ 1 ) ⊗ H k (Σ 2 , L λ 2 ) . By the deﬁnition ( 5.31 ) of Φ on disjoint unions, the identit y Φ Σ 1 ⊔ Σ 2 = Φ Σ 1 ⊗ Φ Σ 2 holds under the ab o ve identiﬁcations. On the monoidal unit ∅ , both theories assign C , and w e tak e Φ ∅ = id C . Therefore Φ is a monoidal natural isomorphism. Coherence with comp osition and iden tities. Coherence is automatic from naturalit y together with functoriality of eac h TQFT. F or composable b ordisms X 1 , X 2 , naturalit y giv es compatibility with Z R T k ( X 2 ◦ X 1 ) = Z R T k ( X 2 ) ◦ Z R T k ( X 1 ) and Z CS U (1) ,k ( X 2 ◦ X 1 ) = Z CS U (1) ,k ( X 2 ) ◦ Z CS U (1) ,k ( X 1 ) , and similarly for identit y b ordisms. This uses strict functoriality of the extended R T theory [ T ur02 , §I I.2, §IV.3] and of Manoliu’s extended theory [ Man98a ]. This completes the proof. □ The relation b etw een Ab elian Chern–Simons theory and the Reshetikhin–T uraev TQFT asso ciated with the p oin ted mo dular category C k = C ( Z k , q k ) is well known at the lev el of closed partition functions and Hilb ert space dimensions; see [ MOO92 , Jef92 , MPR93 , Sti08 , U (1) CHERN–SIMONS AND RESHETIKHIN–TURAEV TQFTS 33 GT13 , GT14 , TT25 ]. The theorem ab ov e upgrades this corresp ondence to an explicit equiv alence of extended TQFT functors. 5.4. Finite Quadratic Mo dules as Universal Data. The preceding sections show that the closed and extended rank-one Ab elian theories are go v erned b y ﬁnite quadratic data. F or a closed orien ted 3 –manifold M , this data is the ﬁnite quadratic mo dule  T , q M  , T := T ors H 1 ( M ; Z ) , where q M is the quadratic reﬁnemen t of the torsion linking pairing determined b y the Chern–Simons functional. F or the rank-one theory at even lev el k , the corresp onding discrete datum is the ﬁnite quadratic module ( Z k , q k ) , q k ( x ) = exp  π i k x 2  . W e no w summarize the comparison results prov ed ab o ve and record the resulting classiﬁ- cation statemen t. Theorem 5.12. L et M b e a close d, c onne cte d, oriente d 3 –manifold, and let T := T ors H 1 ( M ; Z ) . L et λ M : T × T → Q / Z b e the torsion linking p airing, and let q M : T → Q / Z b e its quadr atic r eﬁnement determine d by the Chern–Simons functional. Fix k ∈ 2 Z > 0 and deﬁne (5.32) G k ( M ) := | T | − 1 / 2 X x ∈ T exp  2 π ik q M ( x )  . Then: (i) (5.33) Z CS U (1) ,k ( M ) = k m M G k ( M ) , m M = 1 2  b 1 ( M ) − 1  . (ii) If M is pr esente d by sur gery on a fr ame d link with linking matrix L , then (5.34) Z RT , ra w k ( M ) = e − πi 4 σ ( L reg ) Z CS U (1) ,k ( M ) . (iii) L et C k := C ( Z k , q k ) , q k ( x ) = exp  π i k x 2  . Then ther e is a symmetric monoidal natur al isomorphism Z RT C k ∼ = = = ⇒ Z C S U (1) ,k . Equivalently, for every extende d surfac e (Σ , λ ) ther e is a c anonic al isomorphism Φ Σ : V RT k (Σ , λ ) ∼ = − − → H U (1) ,k (Σ , L λ ) , c omp atible with the op er ators assigne d to extende d b or disms. Pr o of. P art (i) is exactly ( 5.13 ) , rewritten using ( 5.32 ) . Part (ii) is Theorem 5.4 . Part (iii) is Theorem 5.11 ; the b oundary iden tiﬁcation is Theorem 5.8 , and compatibilit y with b ordism op erators is Theorem 5.10 . □ Theorem 5.13. L et k ,  ∈ 2 Z > 0 . F or m ∈ { k ,  } , set q m ( x ) = exp  π i m x 2  , C m := C ( Z m , q m ) , and let Z C S U (1) ,m : Cob ext 2+1 − → V ect C 34 D ANIEL GAL VIZ denote the extende d U (1) Chern–Simons TQFT at l evel m . Then the fol lowing ar e e quivalent: (i) k =  ; (ii) ( Z k , q k ) ∼ = ( Z ℓ , q ℓ ) as ﬁnite quadr atic mo dules; (iii) Z RT C k ∼ = Z RT C ℓ as symmetric monoidal functors; (iv) Z C S U (1) ,k ∼ = Z C S U (1) ,ℓ as symmetric monoidal functors. In p articular, within the even-level r ank-one family, the extende d A b elian Chern–Simons the ory is classiﬁe d by the ﬁnite quadr atic mo dule ( Z k , q k ) . Pr o of. The equiv alence of (i) and (ii) is immediate, since an isomorphism ( Z k , q k ) ∼ = ( Z ℓ , q ℓ ) iden tiﬁes the underlying ﬁnite groups, hence k = | Z k | = | Z ℓ | = . The implications (i) ⇒ (iii) and (i) ⇒ (iv) are immediate. Assume (iii). Ev aluate the natural isomorphism Z RT C k ∼ = Z RT C ℓ on an y connected extended surface (Σ , λ ) of gen us g ≥ 1 . Then V RT k (Σ , λ ) ∼ = V RT ℓ (Σ , λ ) , so Proposition 5.8 gives k g = dim V RT k (Σ , λ ) = dim V RT ℓ (Σ , λ ) =  g . Hence k =  . Assume (iv). By Theorem 5.12 (iii), Z C S U (1) ,k ∼ = Z RT C k , Z C S U (1) ,ℓ ∼ = Z RT C ℓ . Therefore (iv) implies Z RT C k ∼ = Z RT C ℓ , so (iii) holds, and hence k =  b y the previous paragraph. Thus (i)–(iv) are equiv alen t. □ Th us, at ev en level, the ﬁnite quadratic mo dule ( Z k , q k ) completely determines the U (1) Ab elian Chern–Simons theory . References [A ti88] M. A tiy ah. T op ological quantum ﬁeld theories. Inst. Hautes Etudes Sci. Publ. Math. , 68:175–186, 1988. [Che79] Jeﬀ Cheeger. Analytic torsion and the heat equation. Annals of Mathematics , 109(2):259–322, 1979. [Del99] Florian Deloup. Linking forms, recipro cit y for Gauss sums and in v ariants of 3-manifolds. T r ansactions of the A meric an Mathematic al So ciety , 351(5):1895–1918, 1999. [DT07] Florian Deloup and Vladimir T uraev. On recipro city . Journal of Pur e and Applie d Algebr a , 208(1):153–158, 2007. [F Q93] Daniel S. F reed and F rank Quinn. Chern-Simons theory with ﬁnite gauge group. Commun. Math. Phys. , 156:435–472, 1993. [F re95] Daniel S. F reed. Classical Chern-Simons theory . P art 1. A dv. Math. , 113:237–303, 1995. [GS99] Rob ert E. Gompf and András I. Stipsicz. 4-Manifolds and Kirby Calculus , volume 20 of Gr aduate Studies in Mathematics . American Mathematical So ciety , Providence, RI, 1999. [GT13] Enore Guadagnini and F rank Thuillier. Three-manifold inv ariant from functional integration. J. Math. Phys. , 54:082302, 2013. [GT14] Enore Guadagnini and F rank Thuillier. P ath-integral in v ariants in Abelian Chern-Simons theory. Nucl. Phys. B , 882:450–484, 2014. [Jef92] L. C. Jeﬀrey . Chern-Simons-Witten inv ariants of lens spaces and torus bundles, and the semiclassical appro ximation. Commun. Math. Phys. , 147:563–604, 1992. U (1) CHERN–SIMONS AND RESHETIKHIN–TURAEV TQFTS 35 [Man97] Mihaela Manoliu. Quan tization of symplectic tori in a real p olarization. Journal of Mathematic al Physics , 38(5):2219–2254, 05 1997. [Man98a] Mihaela Manoliu. Ab elian Chern–Simons theory . I. a top ological quantum ﬁeld theory . Journal of Mathematic al Physics , 39(1):170–206, 01 1998. [Man98b] Mihaela Manoliu. Ab elian Chern–Simons theory . I I. a functional integral approach. Journal of Mathematic al Physics , 39(1):207–217, 01 1998. [MOO92] Hitoshi Murak ami, T omotada Ohtsuki, and Masae Ok ada. Inv ariants of three-manifolds derived from linking matrices of framed links. Osaka Journal of Mathematics , 29(3):545–572, September 1992. [MPR93] Josef Mattes, Michael Poly ak, and Nikolai Reshetikhin. On inv ariants of 3 -manifolds deriv ed from Ab elian groups. In Louis H. Kauﬀman and Randy A. Baadhio, editors, Quantum T op olo gy , v olume 3 of Series on Knots and Everything , pages 324–338. W orld Scien tiﬁc Publi shing Co. Inc., Riv er Edge, NJ, 1993. [Mül78] W olfgang Müller. Analytic torsion and R–torsion of Riemannian manifolds. A dvanc es in Mathe- matics , 28:233–305, 1978. [Ran97] Andrew Ranicki. The Maslov index and the Wall signature non-additivity in v ariant. Unpublished man uscript, 1997. [RS71] D. B. Ray and I. M. Singer. R T orsion and the Laplacian on Riemannian manifolds. A dv. Math. , 7:145–210, 1971. [R T90] N. Y u. Reshetikhin and V. G. T uraev. Ribb on graphs and their inv ariants deriv ed from quantum groups. Communic ations in Mathematic al Physics , 127(1):1–26, 1990. [R T91] N. Reshetikhin and V. G. T uraev. Inv ariants of three manifolds via link p olynomials and quan tum groups. Invent. Math. , 103:547–597, 1991. [Sti08] Sp encer D. Stirling. Ab elian Chern-Simons the ory with tor al gauge gr oup, mo dular tensor c ate gories, and gr oup c ate gories . PhD thesis, T exas U., Math Dept., 2008. [TT25] Mic hail T agaris and F rank Thuillier. Reshetikhin-Turaev construction and U(1) n Chern-Simons partition function. arXiv:2507.08587, 2025. [T ur94] Vladimir G. T uraev. Quantum Invariants of Knots and 3-Manifolds , volume 18 of De Gruyter Studies in Mathematics . W alter de Gruyter, 1994. [T ur02] Vladimir G. T uraev. T orsions of 3–dimensional manifolds , v olume 208 of Pr o gr ess in Mathe- matics . Birkhäuser, Basel, 2002. [W al69] C. T. C. W all. Non-additivit y of the signature. Inventiones Mathematic ae , 7:269–274, 1969. [W al91] Kevin W alk er. On Witten’s 3-manifold in v arian ts. Preprin t, 1991. [Wit89] Edw ard Witten. Quantum Field Theory and the Jones Polynomial. Commun. Math. Phys. , 121:351–399, 1989. Y AU MA THEMA TICAL SCIENCES CENTER AND DEP AR TMENT OF MA THEMA TICS, TSINGHUA UNIVERSITY, BEIJING, CHINA.

Extended Equivalence of $U(1)$ Chern-Simons and Reshetikhin-Turaev TQFTs

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