Large-$N$ Torus Knots in Lens Spaces and Their Quiver Structure

Prep ared for submission to JHEP La rge- N T o rus Knots in Lens Spaces and Their Quiver Structure Ritab rata Bhattachary a, Suvanka r Dutta, Naman P asari, Nitin V erma a Dep artment of Physics, Indian Institute of Scienc e Educ ation and R ese ar ch Bhop al, Bhop al Byp ass R o ad, Bhop al - 462066, India E-mail: ritabratab3@gmail.com, suvankar@iiserb.ac.in, pasari21@iiserb.ac.in, inittinv@gmail.com , Abstract: W e study torus knot inv arian ts in the lens space S 3 / Z p within Chern–Simons theory . Using the surgery and mo dular description of lens spaces, we deriv e a general expression for the inv arian t of an ( α, β ) torus knot in this background. In the large- N limit these inv arian ts simplify and acquire a universal form: the inv arian t of an ( α, β ) torus knot in S 3 / Z p can b e expressed in terms of the inv arian t of the ( α, α + pβ ) torus knot in S 3 . After an appropriate redefinition of knot v ariables, the generating functions of these in v ariants exhibit a structure analogous to quiv er partition functions. Since the associated quiv er is indep enden t of the rank N and level k of Chern–Simons theory , the large- N result pro vides a direct wa y to iden tify the underlying quiver, allowing us to determine the quiver structure asso ciated with torus knots in S 3 / Z p . Con ten ts 1 In tro duction 1 2 Knot in v arian ts in S 3 3 2.1 (2 , 2 n + 1) torus knot inv arian ts in symmetric representation 7 2.2 T orus knot inv arian ts in sheared S 3 8 3 Knot in v arian ts in lens space 10 3.1 ( α, β ) torus knots in lens space 10 3.2 Matrix mo del 12 3.3 Reduced inv arian ts 16 4 Quiv ers for torus knots in lens space 16 4.1 Quiv ers 17 5 Conclusion 19 A Construction of S 3 / Z p 20 B T o c hec k h s ρ ( U ) i L ( p, 1) = ( q κ ρ dim q ( ρ ))   q → q 1 /p in the large N limit 22 1 In tro duction Knot in v ariants in three–manifolds provide a rich framework for exploring the in terpla y b et w een top ology , quantum field theory , and algebraic structures underlying quan tum knot in v ariants. The sub ject has a long history , b eginning with classical p olynomial in v arian ts suc h as the Alexander and Jones p olynomials [ 1 – 3 ]. A ma jor conceptual adv ance came with Witten’s realization that these inv arian ts arise naturally as Wilson lo op observ ables in three–dimensional Chern–Simons gauge theory [ 4 ], leading to a deep connection b etw een knot theory , quantum groups, and represen tation theory [ 5 – 9 ]. While the case of S 3 has b een extensively studied and is by no w well understo o d, considerably less is kno wn ab out knot in v ariants in more general three–manifolds. Among such bac kgrounds, lens spaces S 3 / Z p pro vide a natural and tractable extension of the three–sphere, whose non trivial top ology leads to new structures while still allowing explicit computations through the surgery and mo dular formulation of Chern–Simons theory [ 5 , 6 , 10 – 13 ]. Understanding knot inv arian ts in these spaces is therefore an important step to ward extending large– N dualities, in tegrality structures, and the knot–quiver corresp ondence beyond the simplest S 3 setting [ 14 – 18 ]. In this work w e study torus knots in the lens space S 3 / Z p from the p ersp ective of Chern– Simons theory . Using the surgery description of lens spaces and their mo dular prop erties, – 1 – w e deriv e a generic expression for the in v arian t of an ( α , β ) torus knot in this background. W e show that in the double scaling limit N → ∞ , k → ∞ , N k + N = λ (fixed) . (1.1) these in v ariants admit a particularly simple univ ersal form: the in v ariant of an ( α, β ) torus knot in S 3 / Z p can b e expressed directly in terms of the inv arian t of an ( α , α + pβ ) torus knot in S 3 . This relation pro vides a transparen t mapping b et w een different torus knot sectors and isolates the effect of the Z p quotien t in the large– N regime 1 . The large– N analysis is p erformed using matrix mo del techniques that hav e prov en effective in the study of Chern–Simons knot in v ariants and torus links [ 19 , 20 ]. An imp ortant consequence of this result is that, in the large N limit, the generating func- tions of torus knot in v ariants in S 3 / Z p acquire a structure closely analogous to quiver partition functions after an appropriate redefinition of the knot v ariables. This structure allo ws us to identify the corresp onding quiv er description for torus knots in lens spaces, thereb y extending the knot–quiver correspondence [ 17 , 18 , 21 , 22 ] to the background S 3 / Z p . A crucial ingredien t in this construction is the independence of the quiv er structure from the rank and level of the underlying Chern–Simons gauge theory [ 17 , 21 ]. T o implement this program, we employ matrix mo del techniques to analyze HOMFL Y– PT in v ariants of torus knots in S 3 / Z p in the double-scaling limit ( 1.1 ). Such matrix mo del descriptions of Chern–Simons theory on lens spaces and related Seifert manifolds pro vide an efficient framew ork for large- N computations of correlators and knot observ ables [ 14 , 23 – 26 ]. In particular, matrix mo del metho ds hav e b een successfully applied to torus- knot/torus-link HOMFL Y-t yp e inv arian ts in the large- N regime [ 12 , 19 , 20 ]. The resulting large- N relation then allo ws us to identify the quiver asso ciated with an ( α, β ) torus knot in S 3 / Z p in terms of the quiv er corresp onding to the ( α, α + pβ ) torus knot in S 3 . Since the quiv er structure is indep endent of the rank N and lev el k of Chern–Simons theory , this construction determines the quiver partition function in the same double-scaling limit after an appropriate redefinition of the knot v ariables [ 17 , 18 , 21 ]. Consequently , HOMFL Y–PT in v ariants for torus knots in S 3 / Z p can b e obtained directly from the corresp onding quiver data in this limit. The pap er is organized as follows. Section 2 reviews torus knot inv arian ts in S 3 in Chern– Simons theory and their formulation in terms of mo dular transformations on the torus Hilb ert space. Section 3 extends this construction to lens spaces and derives a general expression for torus knot in v ariants in S 3 / Z p . Section 3.2 studies their double-scaling limit using matrix mo del tec hniques, and section 3.3 introduces reduced inv arian ts. Section 4 analyzes the generating functions of these inv arian ts and determines the asso ciated quiver structure. Finally , section 5 summarizes our results and discusses future directions. 1 T orus knots in lens spaces were studied in [ 12 ], where the corresp onding knot inv ariant w as decomposed in to several comp onents lab eled by partitions. It was shown there that the low est comp onent of any torus knot inv ariant in the lens space can be expressed as the inv arian t of a torus knot with shifted parameters in S 3 . In contrast, w e show that such a relation holds for the full knot inv ariant in the double scaling limit. – 2 – 2 Knot in v arian ts in S 3 Canonical quan tisation of CS theory on a three–manifold X with b oundary Σ asso ciates a ph ysical Hilb ert space H (Σ) to the b oundary . As shown by Witten [ 4 ], this Hilb ert space can b e identified with the space of conformal blo cks of a W ess–Zumino–Witten (WZW) mo del on Σ , with gauge group G at level k . The structure of H (Σ) dep ends strongly on the top ology of Σ . F or instance, when Σ = S 2 the space of conformal blocks is one–dimensional, whereas for Σ = T 2 the Hilb ert space is nontrivial and admits a canonical basis. F or Σ = T 2 , the conformal blocks are in one–to–one corresp ondence with the in tegrable represen tations of the affine algebra g k . A con v enien t basis of H ( T 2 ) is giv en b y v ectors | µ i , eac h lab eled b y an in tegrable represen tation µ . These states arise as the path in tegral of CS theory on a solid torus with a Wilson lo op in represen tation µ wrapping the non– con tractible cycle. F or G = S U ( N ) or U ( N ) , the allow ed representations corresp ond to Y oung diagrams with less than N ro ws and at most k columns (for S U ( N ) ) or Y oung diagrams with negativ e num ber of b o xes and with a maxim um width k for ( U ( N ) ), ensuring that H ( T 2 ) is finite dimensional [ 27 ]. Consider a closed three–manifold M decomp osed in to tw o pieces X L and X R sharing a common b oundary Σ . The path integral on X R defines a state | ϕ i ∈ H (Σ) , while the path integral on X L , whose b oundary has the opp osite orientation, gives a dual state h ψ | ∈ H ∗ (Σ) . Gluing the tw o pieces along Σ yields the partition function Z ( M ) = h ψ | ϕ i . (2.1) Let us consider M = S 3 . Removing a solid torus from S 3 lea v es a complemen tary man- ifold that is again a solid torus. These tw o solid tori are related b y an inv ersion: the in terior of the excised torus b ecomes the exterior of the complementary one, and vice versa (see fig. 1 ). Equiv alently , the contractible cycle J m K of one torus is mapped to the non– con tractible cycle J l K of the other. Thus S 3 can b e reconstructed b y gluing t w o solid tori along their b oundaries after inv erting one of them. This in v ersion exchanges the meridian Figure 1 . Heegaard Splitting of S 3 . (Credit : Mathematic a ) J m K and longitude J l K of the b oundary T 2 and is implemented on the Hilb ert space H ( T 2 ) – 3 – b y the operator S . F or further details, see app endix A . Gluing the tw o solid tori after this op eration pro duces the three–sphere S 3 . In contrast, gluing tw o solid tori without this in v ersion yields the manifold S 2 × S 1 . Applying this construction to Chern–Simons theory , consider the path in tegral on a solid torus without an y Wilson line insertion. The corresp onding state in the Hilb ert space H ( T 2 ) is the v acuum state | 0 i . If w e no w glue t wo such solid tori after p erforming an S mo dular transformation on the boundary of one of them, the resulting partition function is given b y Z ( S 3 ) = h 0 | S | 0 i = S 00 . (2.2) This provides the standard expression for the Chern–Simons partition function on the three–sphere. Next, consider a Wilson lo op in representation µ supp orted on the non–contractible cycle (the longitude) of a solid torus. The path in tegral on the torus then prepares the state | µ i ∈ H ( T 2 ) . W e no w glue this torus to another solid torus without any Wilson line insertion. As discussed earlier, the gluing required to pro duce S 3 in v olv es a mo dular S transformation acting on the b oundary of one torus. The resulting partition function computes the Chern–Simons inv arian t of an unknot in representation µ : W ( unknot ; µ ) = h 0 | S | µ i = S 0 µ . (2.3) Geometrically , the unknot ma y b e characterised b y its winding num b ers around the cycles of the b oundary torus. With resp ect to the torus on which it is initially defined, the unknot wraps once around the longitude and do es not wind along the meridian. It may therefore b e represented as (1 , 0) ≡ 1 J l K + 0 J m K (2.4) Ho w ev er, the mo dular S transformation exchanges the meridian and longitude cycles. Con- sequen tly , after gluing, the same knot can equiv alen tly b e view ed as (0 , 1) ≡ 0 J l K + 1 J m K (2.5) with resp ect to the complementary solid torus. Since knot inv arian ts are indep endent of this description, w e obtain W (1 , 0) µ ( S 3 ) = W (0 , 1) µ ( S 3 ) = S 0 µ . (2.6) W e no w consider a general torus knot embedded in a solid torus. An ( α, β ) - torus knot, T ( α,β ) is sp ecified up to isotopy b y the homology class ( α, β ) = α J l K + β J m K . (2.7) suc h that gcd( α, β ) = 1 . By the abov e Definition, a torus knot T ( α,β ) can b e intuitiv ely – 4 – though t of as a closed lo op that wraps α times along the longitudinal direction and β times around the meridonial direction. An example has b een illustrated in the fig. 2 b elow, Figure 2 . T refoil T (2 , 3) , wrapp ed around the torus. (Credit- Mathematic a ) The path integral on the solid torus in the presence of a Wilson lo op in representation µ wrapping the ( α, β ) cycle prepares a state in the Hilb ert space H ( T 2 ) . This state, often referred to as a knot state , may b e expanded in the standard represen tation basis as [ 6 , 7 , 12 , 13 , 15 , 28 , 29 ]. | T ( α,β ) ; µ, f i = q ( f − αβ ) κ µ X ρ | ρ | = α | µ | c ρ µ,α q β α κ ρ | ρ i , α 6 = 0 . (2.8) Here w e define the knot states with an o verall phase factor q ( f − αβ ) κ µ . Later we see that this extra factor plays the role of framing. c ρ µ,α denotes the Adams co efficients. In Chern– Simons theory these co efficients determine the decomp osition of Wilson lo op op erators, T r µ ( U α ) = X ρ c ρ µ,α T r ρ ( U ) . (2.9) The parameter q is giv en by q = exp  2 π i k + N  . (2.10) W e also use another v ariable v to express the HOMFL Y-PT p olynomials giv en by , v = q N = exp  2 π iN k + N  . (2.11) The quan tit y κ ρ is defined in terms of the quadratic Casimir asso ciated with the represen- tation ρ κ ρ = 1 2 N X i =1 l i ( l i − 2 i + 1) + N | ρ | 2 . (2.12) – 5 – l i is the num b er of b oxes in the i-th row in the Y oung diagram asso ciated with the repre- sen tation ρ and | ρ | is the num b er of b oxes in the Y oung diagram. The notation X ρ | ρ | = α | µ | denotes the sum ov er all represen tations ρ whose total num b er of b oxes equals α | µ | . The states | ρ i app earing on the right–hand side of the knot state ( 2.8 ) corresp ond to Y oung diagram states asso ciated with partitions of α | µ | , and are not, strictly sp eaking, elemen ts of the Hilb ert space H ( T 2 ) . How ev er, in the com bined large N and large k limit, this distinction b ecomes immaterial, and one may consistently regard the states | ρ i as b elonging to H ( T 2 ) . With this understanding, the knot state construction sho ws that a general ( α, β ) torus knot state admits a representation as a linear sup erp osition of unknot states lab elled by different representations. In other words, arbitrary torus knot states can b e decomposed in to sums ov er Wilson lo op states asso ciated with unknots in v arious represen tations. T o compute the torus knot in v arian t in S 3 , we glue this solid torus to an empty solid torus. As discussed earlier, the gluing that pro duces S 3 requires the insertion of the modular S transformation. The resulting amplitude is h 0 | S | T ( α,β ) ; µ, f i = q ( f − αβ ) κ µ X ρ | ρ | = α | µ | c ρ µ,α q β α κ ρ h 0 | S | ρ i . (2.13) Using the definition of the mo dular matrix elemen ts, h 0 | S | ρ i = S 0 ρ , (2.14) w e obtain h 0 | S | T ( α,β ) ; µ, f i = q ( f − αβ ) κ µ X ρ | ρ | = α | µ | c ρ µ,α q β α κ ρ S 0 ρ . (2.15) The normalized torus knot inv arian t is obtained by dividing b y the v acuum partition func- tion on S 3 , Z ( S 3 ) = h 0 | S | 0 i = S 00 . (2.16) Th us, W ( α,β ) µ ( S 3 , f ; q , v ) = h 0 | S | T ( α,β ) ; µ, f i h 0 | S | 0 i = q ( f − αβ ) κ µ X ρ | ρ | = α | µ | c ρ µ,α q β α κ ρ S 0 ρ S 00 . (2.17) Using the standard identit y S 0 ρ S 00 = dim q ( ρ ) , (2.18) – 6 – w e arrive at the final expression W ( α,β ) µ ( S 3 , f ; q , v ) = q ( f − αβ ) κ µ X ρ | ρ | = α | µ | c ρ µ,α q β α κ ρ dim q ( ρ ) . (2.19) This expression w as earlier derived in [ 8 , 20 , 30 ]. F or unknot (1 , 0) , the Adams co efficien ts c ρ µ, 1 are given b y c ρ µ, 1 = δ ρ µ , (2.20) whic h follows directly from their defining relation ( 2.9 ). As a consistency chec k, for the unknot (1 , 0) one recov ers the familiar result (with framing f = 0 ) W ( unknot ) µ ( S 3 , f = 0; q , v ) = S 0 µ S 00 = dim q ( µ ) , (2.21) whic h expresses the unknot inv arian t in terms of the quantum dimension of the represen- tation µ . Consequen tly , the ( α, β ) torus knot inv arian t in S 3 ma y b e expressed as a sum ov er (1 , 0) unknot in v ariants in S 3 . In other words, the general torus knot in v arian t admits a decom- p osition in terms of contributions associated with unknots in v arious representations. As discussed earlier, the mo dular S transformation exchanges the meridian and longitude cycles of the b oundary torus. Consequently , a torus knot c haracterised b y winding num b ers ( α, β ) with resp ect to one solid torus is equiv alently describ ed as a ( β , α ) torus knot with resp ect to the complementary solid torus. Since the resulting three–manifold is unchanged and knot inv arian ts are indep enden t of this choice of description, w e obtain the symmetry W ( α,β ) µ ( S 3 , f ; q , v ) = W ( β ,α ) µ ( S 3 , f ; q , v ) . (2.22) 2.1 (2 , 2 n + 1) torus knot inv ariants in symmetric representation W e use ( 2.19 ) to write down the inv arian ts for any (2 , 2 n + 1) torus knot inv arian ts in symmetric r representation, as a sp ecial case, that will b e used for later purp ose. The A dam’s co efficients for symmetric representations µ = r and α = 2 are non-zero only when the partitions ρ has maximum t w o rows with total n um b er of b oxes 2 r . Such partitions can b e represen ted b y (2 r − k , k ) with 0 ≤ k ≤ r . The v alue of Adam’s co efficien ts for suc h partitions are giv en by c (2 r − k ,k ) r, 2 = ( − 1) k . (2.23) – 7 – Therefore, the (2 , 2 n + 1) torus knot inv arian ts are given by , W (2 , 2 n +1) r ( S 3 , f ; q , v ) = q ( f − 2(2 n +1)) κ r r X k =0 ( − 1) k q 2 n +1 2 κ (2 r − k ,k ) dim q (2 r − k , k ) . (2.24) 2.2 T orus knot in v ariants in sheared S 3 Instead of p erforming a pure S –mo dular surgery , one ma y consider a more general gluing obtained b y com bining in v ersion with a Dehn twist. The Dehn twist along one fundamental cycle of the torus is implemen ted on H ( T 2 ) by the mo dular T -matrix T . A particularly im- p ortan t example is provided by the T S T mo dular transformation. When tw o empt y solid tori are glued using T S T , the resulting three–manifold is again S 3 , although the identifi- cation of cycles differs from that of the simple S –transformation. Recall that the mo dular transformation S exchanges the longitude and meridian cycles, while T implemen ts a Dehn t wist (app endix A ). Under the action of T S T , the cycles transform as J l K − → J l K , J m K − → J m K + J l K . (2.25) Th us, the longitude remains unchanged, while the meridian is shifted by the longitude. In contrast to pure S –surgery , the cycles are therefore not simply interc hanged. In this description the resulting three–manifold remains S 3 , ho w ev er, the S 3 is realised as the union of an ordinary solid torus and a complementary solid torus that is not only in verted but also twisted, and may therefore be viewed as a sheared presen tation of S 3 . Applying this construction to Chern–Simons theory , the path in tegral on an empt y solid torus prepares the v acuum state | 0 i . Gluing t wo such tori using the T S T transformation yields the partition function of Chern–Simons theory in a sheared S 3 [ 31 , 32 ] Z TST ( S 3 ) = h 0 | T S T | 0 i = T 00 S 2 00 . (2.26) Compared to the standard result obtained from pure S –gluing ( 2.2 ), the partition function differs by the additional factor T 2 00 . Since T 00 is a pure phase, this difference reflects a c hange of framing rather than a change of manifold. W e can also apply the T S T mo dular surgery to a torus knot state. A subtlet y in this construction arises in iden tifying the torus knot after the gluing. Because the complemen- tary torus is t wisted b y the mo dular transformation, its meridian and longitude cycles are mixed, making a direct geometric interpretation of the resulting knot less transparen t. T o analyse this situation, we pro ceed as follo ws. W e consider Chern–Simons theory on tw o solid tori, lab elled as the left and right tori, and insert an ( α, β ) torus knot in the righ t torus. The path integral on the right torus prepares the corresponding torus knot state, giv en in ( 2.8 ), while the left torus without Wilson line insertions prepares the v acuum bra h 0 | . W e then act with the T S T mo dular transformation on the left torus and glue it to the right torus. – 8 – With this prescription, the torus on which the knot is initially defined remains un t wisted, so the knot retains its natural interpretation as an ( α, β ) torus knot with respect to that torus. The complementary torus, how ev er, b ecomes b oth inv erted and twisted. As a consequence, its meridian and longitude cycles are no longer simply exchanged but are mixed, and the same em b edded knot may therefore admit a differen t description when expressed in terms of the transformed cycles. F or this reason, when we refer to an ( α, β ) torus knot in the sheared S 3 , w e mean that the knot is defined with resp ect to the un t wisted solid torus inside S 3 . Thus, starting from an ( α, β ) torus knot state, the knot obtained after T S T surgery is naturally in terpreted as an ( α, β ) torus knot in a sheared presentation of S 3 . Starting from the torus–knot state ( 2.8 ) the (unnormalised) amplitude obtained b y T S T – gluing is h 0 | T S T | T ( α,β ) ; µ, f i = q ( f − αβ ) κ µ X ρ | ρ | = α | µ | c ρ µ,α q β α κ ρ h 0 | T S T | ρ i . (2.27) Since T is diagonal, one has h 0 | T S T | ρ i = ( T S T ) 0 ρ = T 00 S 0 ρ T ρρ . (2.28) The corresp onding partition function for this gluing is given by ( 2.26 ). Therefore the normalised knot in v ariant is given b y W ( α,β ) µ ( S 3 , f ; q , v )   TST = 1 Z TST ( S 3 ) h 0 | T S T | T ( α,β ) ; µ i = q ( f − αβ ) κ µ X ρ | ρ | = α | µ | c ρ µ,α q β α κ ρ S 0 ρ S 00 T ρρ T 00 . (2.29) Using ( 2.18 ) and T ρρ T 00 = q κ ρ , (2.30) this can b e written as W ( α,β ) µ ( S 3 , f ; q , v )   TST = q ( f − αβ ) κ µ X ρ | ρ | = α | µ | c ρ µ,α q β + α α κ ρ dim q ( ρ ) . (2.31) The in v ariant obtained from T S T -surgery should not b e interpreted as a mere framing mo dification. Although the resulting three–manifold is still S 3 , the gluing now in v olv es an additional Dehn t wist, which alters the iden tification of cycles on the b oundary torus. Consequen tly , the am bien t geometry is naturally viewed as a sheared presentation of S 3 . – 9 – In this sheared description, a torus knot lab eled b y winding num b ers ( α, β ) retains its original in terpretation with resp ect to the torus on whic h it is defined. How ev er, when the same configuration is expressed in the standard (unsheared) presen tation of S 3 , the mixing of the longitude and meridian cycles induced by the Dehn twists modifies the slop e of the knot. As a consequence, the knot in v ariant computed using T S T -surgery can b e naturally related to the inv arian t obtained in the usual S -surgery description. In particular, one finds W ( α,β ) µ ( S 3 , f ; q , v )    TST = q α 2 κ µ W ( α,β + α ) µ ( S 3 , f ; q , v ) . (2.32) Th us, an ( α, β ) torus knot defined in the sheared presentation of S 3 is equiv alen t to an ( α, α + β ) torus knot in the standard presen tation of S 3 . 3 Knot in v arian ts in lens space A Lens space L ( p, 1) ≡ S 3 / Z p is obtained in the surgery construction b y gluing tw o solid tori with the mo dular transformation (see app endix A ) [ 31 , 32 ] K = ( T S T ) p . (3.1) Under the mo dular transformation ( T S T ) p , the cycles of the b oundary torus transform as ( T S T ) p : J l K − → J l K , J m K − → J m K + p J l K . (3.2) Using the mo dular group relation T S T = S † T − 1 S , (3.3) K may equiv alen tly b e written as K = S † T − p S . (3.4) Th us, the Lens space S 3 / Z p is constructed by gluing tw o solid tori via the transformation ( T S T ) p . In the Chern–Simons formulation, the corresp onding partition function is giv en b y Z  S 3 / Z p  = h 0 | K | 0 i = X ρ S 2 0 ρ T − p ρρ . (3.5) Geometrically , the rep eated T insertions implement Dehn twists, while S exc hanges the meridian and longitude cycles, thereby generating the nontrivial top ology of the Lens space. 3.1 ( α, β ) torus knots in lens space T o compute the torus knot in v arian t in S 3 / Z p , w e b egin with tw o solid tori. The left solid torus is taken to b e empty and therefore prepares the v acuum state h 0 | . On the righ t solid torus we insert a ( α, β ) torus knot, and the corresp onding state is given b y ( 2.8 ). In order to generate the torus knot in v ariants in Lens space, w e apply the mo dular trans- formation K = ( T S T ) p (3.6) – 10 – to one of the solid tori b efore gluing. F or definiteness, let us p erform this operation on the left solid torus. Under this transformation, the meridian and longitude cycles are mo dified. Gluing the transformed left torus with the right torus then pro duces the manifold S 3 / Z p in the presence of a torus knot. In this construction, the right solid torus remains unt wisted and naturally defines an em- b edded torus inside the resulting manifold. Consequen tly , the Wilson lo op retains its in terpretation as a ( α, β ) torus knot with resp ect to this unt wisted torus, while the non- trivial top ology of the am bien t space is en tirely encoded in the mo dular transformation acting on the complementary torus. Starting from the torus knot state ( 2.8 ) the normalized torus knot in v ariant in the Lens space is defined by W ( α,β ) µ ( S 3 / Z p , f ; q , v ) = 1 Z ( S 3 / Z p ) h 0 | S † T − p S | T ( α,β ) ; µ i . (3.7) Substituting the knot state, we obtain W ( α,β ) µ ( S 3 / Z p , f ; q , v ) = q ( f − αβ ) κ µ X ρ | ρ | = α | µ | c ρ µ,α q β α κ ρ ( S † T − p S ) 0 ρ ( S † T − p S ) 00 . (3.8) Using the unitarit y and symmetry of the mo dular S matrix, ( S † ) 0 ρ = S 0 ρ , (3.9) the matrix elemen t may b e written as ( S † T − p S ) 0 ρ = X ν S 0 ν T − p ν ν S ν ρ = X ν S 2 0 ν T − p ν ν S ν ρ S 0 ν . (3.10) Therefore, the normalized torus knot inv arian t takes the form W ( α,β ) µ ( S 3 / Z p , f ; q , v ) = q ( f − αβ ) κ µ X ρ | ρ | = α | µ | c ρ µ,α q β α κ ρ Ξ ( p ) ρ . (3.11) where, Ξ ( p ) ρ = 1 Z ( S 3 / Z p ) X ν S 2 0 ν T − p ν ν S ν ρ S 0 ν . (3.12) This expression is exact for arbitrary rank N and level k of Chern–Simons theory , but its explicit ev aluation is cumbersome since the sum o v er ν runs ov er all in tegrable representa- tions of the affine algebra. In the double-scaling limit ( 1.1 ), it simplifies significan tly and can b e analyzed using matrix mo del techniques. – 11 – 3.2 Matrix model In the large- N limit, Chern–Simons theory admits a matrix mo del description in whic h knot inv arian ts arise as op erator exp ectation v alues [ 20 , 25 , 26 ]. This framework provides an efficient w a y to analyze the double-scaling b eha vior of torus knot in v ariants in lens spaces and unco ver their quiver structure. W e b egin b y recalling the mo dular transformation matrices of the u ( N ) k WZW theory . The mo dular S –matrix is giv en by [ 27 , 33 , 34 ] S µρ = ( − i ) N ( N − 1) / 2 ( k + N ) N/ 2 exp " − 2 π i l ( µ ) l ( ρ ) N ( N + k ) # det M ( µ, ρ ) , (3.13) where M ij ( µ, ρ ) = exp  2 π i k + N ϕ i ( µ ) ϕ j ( ρ )  , i, j = 1 , . . . , N , (3.14) and ϕ i ( µ ) = l ( µ ) i − l ( µ ) N − i + N + 1 2 . (3.15) Here l ( µ ) denotes the total num b er of b oxes in the Y oung diagram asso ciated with the represen tation µ , while l ( µ ) i denotes the n umber of b oxes in the i-th row. The mo dular T –matrix is diagonal and takes the form [ 27 , 33 , 34 ] T µρ = exp h 2 π i  Q µ − c 24 i δ µρ , (3.16) where Q µ = κ µ k + N , c = N ( N k + 1) N + k . (3.17) F ollowing [ 20 , 26 ] we parametrise the representation ρ in terms of angular v ariables θ i , defined by θ i = 2 π N + k  l i − i + N + 1 2  . (3.18) Since ρ is an integrable representation, l i are b ounded: − k 2 ≤ l i ≤ k 2 . In the double scaling limit, the v ariables θ i range con tin uously o v er the in terv al [ − π , π ] , and may therefore be in terpreted as eigenv alues of an N × N unitary matrix, U = diag ( e iθ 1 , . . . , e iθ N ) . (3.19) With this parametrisation, the mo dular matrix elements admit a natural group–theoretic in terpretation. In particular, consider the ratio S ρµ S 0 ρ . (3.20) Substituting the determinant representation of the S –matrix and expressing the result – 12 – in terms of ho ok num b ers, one finds that all o v erall normalisation factors cancel. The dep endence on the represen tation ρ then enters solely through the v ariables θ i . After straigh tforw ard algebra, this ratio reduces to S ρµ S 0 ρ = s µ ( U ) , (3.21) where s µ ( U ) denotes the Sch ur p olynomial, s µ ( U ) = det  e iθ i h ( µ ) j  det  e iθ i ( N − j )  (3.22) where h ( µ ) j = l ( µ ) i + N − i are the ho ok num b ers asso ciated with the representation µ . Th us, the mo dular matrix ratio is identified with the c haracter of the unitary matrix U in represen tation µ , while the eigen v alues of U are determined by the num b er of b o xes in represen tation ρ ( 3.18 ). The summation o ver representations may be naturally in terpreted as a summation ov er unitary matrices. In the ho ok-n um b er parametrisation, the factor S 2 0 ρ tak es a particularly simple form. When expressed in terms of the v ariables θ i , it reduces to the familiar V an- dermonde measure asso ciated with unitary matrix integrals. Consequently , in the large N limit, the discrete sum ov er represen tations can b e replaced by a con tin uous in tegration o v er eigenv alues, X ρ S 2 0 ρ − → Z Y i dθ i Y i 1 case: F rom the structure of the matrix mo del, w e observ e that the expectation v alue h s ρ ( U ) i L (1 , 1) dep ends on the parameter q = e 2 πiλ N . Examining the effectiv e action ( 3.27 ), the corresp onding saddle p oint equation ( 3.28 ), and its solution ( 3.29 ), w e find that in the double-scaling limit all correlation functions ev aluated on the saddle p oin t depend only on the combination λ/p . Consequen tly , exp ectation v alues of correlation functions of any gauge-inv arian t op erators for p > 1 can b e obtained from the corresponding p = 1 results by replacing λ with λ/p 3 . Since q = e 2 πiλ N , this replacement amounts to the transformation q → q 1 /p . Therefore, in the large- N limit w e obtain h s ρ ( U ) i L ( p, 1) = h s ρ ( U ) i L (1 , 1)    q → q 1 /p = q κ µ p dim q 1 /p µ. (3.33) This relation can b e v erified explicitly in the double-scaling limit by ev aluating b oth sides of the equation. In particular, one computes the Sch ur p olynomial on the leading saddle p oin t solution ( 3.29 ) for small represen tations ρ [ 20 ], and compares the result with the corresp onding q -dimension expanded order by order in the large- N limit. See app. B for some explicit calculations. Th us, in the large- N limit we can write W ( α,β ) µ  S 3 / Z p , f ; q , v  = q ( f − αβ ) κ µ X ρ | ρ | = α | µ | c ρ µ,α  q 1 p  α + pβ α κ ρ dim q 1 p ρ. (3.34) Using the definition ( 2.19 ), this expression can b e rewritten in terms of torus knot inv arian ts in S 3 . In particular, w e obtain W ( α,β ) µ  S 3 / Z p , f ; q , v  = q f ( p − 1) κ µ p q α 2 κ µ p W ( α,α + pβ ) µ  S 3 ; q 1 p , v 1 p  . (3.35) In the large- N limit the torus knot inv arian ts in the lens space S 3 / Z p tak e a remarkably simple form. In particular, the in v ariant of an ( α, β ) torus k not in S 3 / Z p can b e expressed directly in terms of the inv arian t of the ( α, α + pβ ) torus knot in S 3 . This relation establishes a direct corresp ondence b et ween torus knot sectors in the lens space and those in the three– 3 The last term in S eff in ( 3.27 ) does not en ter the saddle point equation and therefore do es not contribute to an y normalized correlation functions. – 15 – sphere, with the effect of the Z p quotien t appearing as a shift in the slope of the knot in the large- N regime. 3.3 Reduced in v ariants F or the purp ose of form ulating the knot–quiver correspondence, it is conv enient to w ork with reduced knot inv arian ts [ 17 , 18 , 35 ]. W e therefore define the reduced inv arian t f W ( α,β ) µ ( S 3 / Z p ; q , v ) b y normalizing the torus knot in v ariant W ( α,β ) µ ( S 3 / Z p , f ; q , v ) with re- sp ect to the unknot inv arian t, f W ( α,β ) µ  S 3 / Z p ; q , v  = W ( α,β ) µ  S 3 / Z p , f ; q , v  W (1 , 0) µ ( S 3 / Z p , f ; q , v ) . (3.36) Using the relation ( 3.35 ), the reduced inv arian t in the lens space can b e expressed in terms of torus knot inv arian ts in S 3 , f W ( α,β ) µ  S 3 / Z p ; q , v  = q α 2 − 1 p κ µ W ( α,α + pβ ) µ  S 3 , f ; q 1 /p , v 1 /p  W (1 , 1) µ  S 3 , f ; q 1 /p , v 1 /p  . (3.37) Since W (1 , 1) µ  S 3 , f ; q 1 /p , v 1 /p  = W (1 , 0) µ  S 3 , f ; q 1 /p , v 1 /p  , the expression simplifies to f W ( α,β ) µ  S 3 / Z p ; q , v  = q α 2 − 1 p κ µ f W ( α,α + pβ ) µ  S 3 ; q 1 /p , v 1 /p  . (3.38) The reduced inv arian t f W is indep enden t of the framing f . Rescaling the v ariables q → q p and v → v p , we obtain the k ey identit y f W ( α,β ) µ  S 3 / Z p ; q p , v p  = q ( α 2 − 1) κ µ f W ( α,α + pβ ) µ  S 3 ; q , v  (3.39) whic h relates torus knot inv arian ts in the lens space to those in S 3 . 4 Quiv ers for torus knots in lens space The sup erp olynomial pro vides a refined extension of the HOMFL Y–PT p olynomial by enco ding the full triply graded homological structure asso ciated with a knot K . It is defined as the generating function P K ( q , v , t ) = X i,j,k v i q j t k dim H i,j,k ( K ) , where H i,j,k ( K ) denotes the triply graded knot homology . In this framew ork, the HOMFL Y– PT p olynomial arises as a decategorified limit of the superp olynomial obtained b y taking – 16 – the graded Euler c haracteristic. Equiv alently , one recov ers the classical in v arian t via the sp ecialization [ 36 – 39 ] P K ( q , v ) = P K ( q , v , − 1) . Th us, the HOMFL Y–PT p olynomial captures only the alternating sum of homology dimen- sions, while the sup erp olynomial retains the richer graded information of the underlying homological theory . The sup erp olynomial for (2 , 2 n + 1) torus knot in S 3 is given b y P (2 , 2 n +1) r ( q , v , t ) = a 2 nr q 2 nr X 0 ≤ k n ≤···≤ k 1 ≤ r  r k 1  q 2  k 1 k 2  q 2 · · ·  k n − 1 k n  q 2 q A t B k 1 Y i =1 (1 + a 2 q 2( i − 2) t ) (4.1) where A = 2 n X i =1 ((2 r + 1) k i − k i − 1 k i ) , B = 2 n X i =1 k i , h r k i q 2 = ( q 2 ; q 2 ) r ( q 2 ; q 2 ) k ( q 2 ; q 2 ) r − k . (4.2) The knot in v arian ts obtained from the superp olynomial after setting t = − 1 are the HOMFL Y-PT inv arian ts. They are related to the inv arian ts obtained from the knot op er- ator formalism ( 2.24 ) with f = 0 by a change of v ariables: q − → q − 1 2 and v − → v − 1 2 . (4.3) More explicitly , P (2 , 2 n +1) r ( q − 1 / 2 , v − 1 / 2 , − 1) = f W (2 , 2 n +1) r ( S 3 , q , v ) . (4.4) Before pro ceed further we p oint out an imp ortan t issue here. Let us lo ok at ( 2.24 ) and ( 4.1 ) for n = 1 . Both represen t trefoil in symmetric r representation. Up to some o v erall factors b oth the in v arian ts are given b y sum ov er an integer from 0 to r . Although the final results matc h the individual summands do not match. Similar observ ation p ersists for high v alues of n . 4.1 Quiv ers T o determine the quiver structure, we b egin by in tro ducing the generating function for torus knot inv arian ts in the lens space S 3 / Z p , obtained by summing the reduced in v ariants o v er all symmetric representations, P ( x ; q , v ) = ∞ X r =0 f W ( α,β ) r  S 3 / Z p ; q p , v p  ( q − 1 , q − 1 ) r x r (4.5) where, ( x, y ) r = r − 1 Y i =0  1 − xy i  . (4.6) – 17 – Using the relation ( 3.39 ), the generating function can b e rewritten as 4 P ( x ; q , v ) = ∞ X r =0 q ( α 2 − 1) κ r f W ( α,α + pβ ) r  S 3 ; q , v  ( q − 1 , q − 1 ) r x r . (4.7) The reduced inv arian ts in S 3 can b e expressed in terms of HOMFL Y–PT p olynomials obtained from the sup erp olynomial ( 4.4 ), f W ( α,α + pβ ) r  S 3 ; q , v  = P ( α,α + pβ ) r  q − 1 / 2 , v − 1 / 2 , − 1  . (4.8) Substituting this relation into the generating function yields P ( x ; q , v ) = ∞ X r =0 q ( α 2 − 1) κ r P ( α,α + pβ ) r  q − 1 / 2 , v − 1 / 2 , − 1  ( q − 1 , q − 1 ) r x r . (4.9) Using the knot–quiv er corresp ondence together with the change of v ariables ( 4.3 ), the righ t-hand side can b e written in the standard quiver form P ( x ; q − 2 , v − 2 ) = X d 1 ,...,d m ≥ 0 x ∑ m i =1 d i q ∑ m i,j =1 ( Q ij − α 2 +1) d i d j Q m i =1 q ( l i + α 2 +1) d i v ( a i − α 2 +1) d i ( − 1) t i d i Q m i =1 ( q 2 , q 2 ) d i , (4.10) where Q ij is the quiv er matrix asso ciated with the ( α, α + pβ ) torus knot in S 3 . This relation implies that the quiver structure of an ( α , β ) torus knot in S 3 / Z p is directly related to the quiv er structure of the ( α , α + pβ ) torus knot in S 3 . In particular, the corresp onding quiver matrices are related by Q ( α,β ) S 3 / Z p = Q ( α,α + pβ ) S 3 − ( α 2 − 1)       1 1 · · · 1 1 1 · · · 1 . . . . . . . . . . . . 1 1 · · · 1       . (4.11) The second term corresp onds to a universal shift of the quiver matrix and can b e remov ed b y an appropriate choice of framing [ 17 ]. As a simple example, consider the case α = 2 with β o dd. F or o dd v alues of p , the corresp onding torus knot inv arian t in S 3 / Z p is equiv alen t to the (2 , 2 + pβ ) torus knot in v ariant in S 3 . In S 3 , such knots are con ven tionally written as (2 , 2 n + 1) , with n = 1 2 (1 + pβ ) . (4.12) 4 One can still use the relation ( 3.39 ) for large v alues of symmetric representations in the double scaling limit [ 20 ]. – 18 – The dimension of the corresp onding quiv er matrix for these knots is 2 n + 1 = 2 + pβ . In particular, for the trefoil knot in S 3 / Z p (corresp onding to β = 3 ), the dimension of the asso ciated quiver matrix is 2 + 3 p for o dd v alues of p . 5 Conclusion In this w ork w e ha v e in v estigated torus knot in v arian ts in the lens space S 3 / Z p within the framew ork of Chern–Simons theory . Using the surgery description of lens spaces and the mo dular properties of the torus Hilb ert space, w e first derived a general expression for the in v ariant of an ( α, β ) torus knot in this background. This formula is exact and v alid for arbitrary v alues of the rank N and level k of the Chern–Simons theory , although its direct ev aluation inv olv es summations ov er integrable represen tations of the affine algebra. A substantial simplification arises in the double-scaling limit ( 1.1 ). In this regime the mo dular sums app earing in the lens space inv arian ts admit a natural description in terms of a unitary matrix mo del. Using this form ulation, we show ed that the correlation func- tions en tering the knot inv arian ts dep end only on the com bination λ/p . Consequently , the exp ectation v alues relev ant for torus knot in v ariants in S 3 / Z p can b e obtained from the corresp onding p = 1 results through the simple replacement q → q 1 /p . This observ ation leads to a remarkably simple large- N relation b etw een torus knot in- v ariants in the lens space and those in the three–sphere. In particular, w e find that the in v ariant of an ( α, β ) torus knot in S 3 / Z p can b e expressed directly in terms of the inv ari- an t of the ( α, α + pβ ) torus knot in S 3 . Thus, in the large- N limit the effect of the Z p quotien t manifests itself as a simple shift in the slop e of the torus knot, providing a direct map b etw een torus knot sectors in the lens space and those in S 3 . Motiv ated by the knot–quiver corresp ondence, we then introduced reduced inv arian ts ob- tained b y normalizing the torus knot in v ariant b y the corresp onding unknot inv arian t. In terms of these reduced inv arian ts we derived a compact identit y relating torus knot in v ari- an ts in S 3 / Z p to those in S 3 . This relation allows the generating function of lens space torus knot inv arian ts to b e expressed in the standard form of a quiv er partition function after an appropriate redefinition of v ariables. Using this structure, we determine the quiv er asso ciated with torus knots in S 3 / Z p . In particular, w e show that the quiver matrix for an ( α , β ) torus knot in the lens space is directly related to the quiver matrix for the ( α, α + pβ ) torus knot in S 3 , up to a univ ersal shift that can b e absorb ed into a framing redefinition. Since the quiver structure is indep endent of the rank N and lev el k of Chern–Simons theory , this relation allo ws us to determine the quiver data for torus knots in lens spaces from the double-scaling limit. Our results therefore provide a natural extension of the knot–quiver corresp ondence to lens space backgrounds. More broadly , they reveal a simple and univ ersal relation b etw een torus knot inv arian ts in S 3 / Z p and those in S 3 , suggesting that many structural prop erties – 19 – of knot inv arian ts in nontrivial three–manifolds may ultimately b e understo o d in terms of their counterparts in the three–sphere. Sev eral directions merit further inv estigation. It would b e in teresting to extend the presen t analysis to more general knots and links in lens spaces and other Seifert manifolds and the corresp onding knot complemen ts [ 40 – 42 ]. Another natural problem is to understand the role of the large- N phase structure of the Chern–Simons matrix mo del and its implications for knot in v ariants and their quiver descriptions. Finally , it w ould b e w orth while to explore whether the relation uncov ered here admits a deeper in terpretation in the context of large- N dualities and top ological string theory . A c kno wledgmen t: SD thanks Piotr Sułk owski for fruitful discussions on the knot– quiv er corresp ondence during his visit to W arsaw in 2025 and ackno wledges the hospitalit y of the Universit y of W arsa w during this visit. The work of RB is supp orted by the SERB pro ject SERB/PHY/2023-2024/65. W e thank Dheera j Kulkarni for useful discussions. W e also ackno wledge the use of AI to ols (ChatGPT and Grammarly) solely for impro ving the grammar and clarity of the manuscript. Finally , we are grateful to the p eople of India for their contin ued supp ort of research in basic science. A Construction of S 3 / Z p Let D 2 × S 1 denote a solid torus whose b oundary is the tw o–torus ∂ ( D 2 × S 1 ) = T 2 . Lo ops on the torus are classified by the first homology group H 1 ( T 2 ) = Z × Z . A basis for this homology group is given b y the meridian m and the longitude l . F or a solid torus, the meridian is the con tractible cycle bounding a disk in D 2 × S 1 , while the longitude is the non–contractible cycle that b ounds a punctured disk in the solid torus. An ( α, β ) torus knot is defined as the isotopy class of an embedded simple closed curv e γ in the solid torus whose pro jection to the b oundary torus represents the homology class J γ K = α J l K + β J m K , with gcd( α, β ) = 1 . Such a knot winds α times along the longitudinal direction and β times around the meridional direction of the torus. In his seminal work [ 4 ], Witten exploited the fact that the three–sphere S 3 admits a gen us–one Heegaard splitting. In this description, S 3 is obtained b y gluing t w o solid tori ( D 2 × S 1 ) L and ( D 2 × S 1 ) R along their common b oundary torus T 2 through an orientation– rev ersing diffeomorphism. This p ersp ective plays a cen tral role in the op erator formulation – 20 – of Chern–Simons theory , where the gluing map is interpreted as a mo dular transformation acting on the Hilb ert space H ( T 2 ) asso ciated with the b oundary torus. Let us c ho ose meridian–longitude bases ( J m L K , J l L K ) , ( J m R K , J l R K ) for the homology cycles of the b oundary tori of the solid tori ( D 2 × S 1 ) L and ( D 2 × S 1 ) R , resp ectiv ely . Lens space: The lens space L ( p, 1) is the three–manifold obtained by gluing the tw o solid tori suc h that the meridian of the left torus is mapp ed to the cycle [ 43 ] J m L K 7→ J m R K + p J l R K . With resp ect to the chosen bases, this condition sp ecifies that the meridian of the left torus wraps once around the meridian and p times around the longitude of the right torus. This distinction is essential: the meridian bounds a disk inside the solid torus, while the longitude represents the non–contractible cycle. The ab ov e prescription determines the first column of the corresp onding mapping class group element in S L (2 , Z ) . A con v enien t representativ e of this element is 1 p 0 1 ! ∈ S L (2 , Z ) , whic h acts on the homology cycles J m K J l K ! 7− → 1 p 0 1 ! J m K J l K ! , so that J m K 7→ J m K + p J l K . In tro ducing the standard generators of S L (2 , Z ) , S = 0 1 − 1 0 ! , T = 1 0 1 1 ! , the matrix S exchanges the (1 , 0) and (0 , 1) cycles of the torus, while T acts as a Dehn t wist that shifts the meridian by the longitude. One can readily verify that ( T S T ) p = 1 p 0 1 ! . Th us, up to isotopy , the lens space L ( p, 1) ∼ = S 3 / Z p – 21 – can b e obtained b y gluing the left and right solid tori using the abov e mapping class group elemen t. In Chern–Simons theory , diffeomorphisms of the b oundary torus are represented b y op er- ators acting on the Hilb ert space H ( T 2 ) . In particular, S ≡ U S , T ≡ U T , denote the op erators implemen ting the mo dular transformations S and T on the Hilbert space and is giv en b y ( 3.13 ) and ( 3.16 ) resp ectively . These op erators realize the geometric actions of in v ersion and Dehn t wist on states in the torus Hilb ert space and play a central role in the op erator formulation of knot inv arian ts. B T o chec k h s ρ ( U ) i L ( p, 1) = ( q κ ρ dim q ( ρ ))   q → q 1 /p in the large N limit W e hav e deriv ed the result h s ρ ( U ) i L ( p, 1) = ( q κ ρ dim q ( ρ ))   q → q 1 /p (B.1) The LHS can b e computed upto leading and sub-leading orders in N using saddle point analysis and using the following c haracter expansion of sch ur p olynomials s ρ ( U ) = X  k χ ρ ( c (  k )) Q i k i ! i k i Y j (T r( U j ) k j where the summation vector  k is such that P r r k r = | ρ | . Using this w e find that s ( U ) = T r( U ) s ( U ) = T r( U ) 2 2 + T r( U 2 ) 2 s ( U ) = T r( U ) 2 2 − T r( U 2 ) 2 (B.2) In large- N limit of our matrix mo del admits the follo wing expansion for any observ able O , h O i = h O i 0 + h O i 1 N 2 + h O i 2 N 4 + · · · F urther in order to remov e N dependece from the exp ectation of traces we define a mo dified trace tr( U m ) = 1 N T r( U m ) Th us we ha v e 1 N h s ( U ) i = h tr( U ) i ≈ h tr( U ) i 0 + h tr( U ) i 1 N 2 + · · · (B.3) – 22 – 2 N 2 h s ( U ) i = h tr( U ) 2 i + 1 N h tr( U 2 ) i = h tr( U ) i 2 + h tr U tr U i c N 2 + 1 N  tr( U 2 )  ≈  h tr U i 2 0 + 2 N 2 h tr U i 0 h tr U i 1 + · · ·  + h tr U tr U i c, 0 N 2 + 1 N  h tr( U 2 ) i 0 + h tr( U 2 ) i 1 N 2 + · · ·  = h tr U i 2 0 + h tr( U 2 ) i 0 N + [2 h tr U i 0 h tr U i 1 + h tr U tr U i c, 0 ] N 2 + h tr( U 2 ) i 1 N 3 + O ( 1 N 4 ) (B.4) 2 N 2 h s ( U ) i = h tr( U ) 2 i − 1 N h tr( U 2 ) i = h tr U i 2 + h tr U tr U i c N 2 − h tr( U 2 ) i N ≈  h tr U i 2 0 + 2 N 2 h tr U i 0 h tr U i 1 + · · ·  + h tr U tr U i c, 0 N 2 − 1 N  h tr( U 2 ) i 0 + h tr( U 2 ) i 1 N 2 + · · ·  = h tr U i 2 0 − h tr( U 2 ) i 0 N + [2 h tr U i 0 h tr U i 1 + h tr U tr U i c, 0 ] N 2 − h tr( U 2 ) i 1 N 3 + O ( 1 N 4 ) (B.5) It has b een shown in [ 20 ] that h tr U i 0 = i  1 − e 2 πiλ p  2 π λ p h tr( U 2 ) i 0 = i  4 e 2 πiλ p − 3 e 4 πiλ p − 1  4 π λ p h tr U i 1 = i  1 − e 2 πiλ p  π λ 12 p h tr U tr U i c, 0 = e 2 πiλ p  e 2 πiλ p − 1  (B.6) Plugging these bac k in ( B.3 ), ( B.4 ) and ( B.5 ), we ha v e h s ( U ) i L ( p, 1) = − π iλ  − 1 + e 2 πiλ p  12 N p − iN p  − 1 + e 2 πiλ p  2 π λ (B.7) – 23 – h s ( U ) i L ( p, 1) = − N 2 p 2  − 1 + e 2 πiλ p  2 8 π 2 λ 2 − iN p  − 4 e 2 πiλ p + 3 e 4 πiλ p + 1  8 π λ + π iλ  − 2 e 2 πiλ p + 3 e 4 πiλ p − 1  12 N p − π 2 λ 2  − 1 + e 2 πiλ p  2 288 N 2 p 2 + 1 24  − 10 e 2 πiλ p + 11 e 4 πiλ p − 1  (B.8) h s ( U ) i L ( p, 1) = − N 2 p 2  − 1 + e 2 πiλ p  2 8 π 2 λ 2 + iN p  − 4 e 2 πiλ p + 3 e 4 πiλ p + 1  8 π λ − π iλ  − 2 e 2 πiλ p + 3 e 4 πiλ p − 1  12 N p − π 2 λ 2  − 1 + e 2 πiλ p  2 288 N 2 p 2 + 1 24  − 10 e 2 πiλ p + 11 e 4 πiλ p − 1  (B.9) Using ( 2.12 ) and ( 2.18 ) one can calculate ( q κ ρ dim q ( ρ ))   q → q 1 /p order by order in N for ρ = , , and c hec k that the expressions match exactly . 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