aa r X i v : . [ m a t h - ph ] J un Knots in
S U ( M | N ) Chern-Simons Field Theory Xin LIUSchool of Mathematics and Statistics, University of SydneyNSW 2006, [email protected]
Abstract
Knots in the Chern-Simons field theory with Lie super gauge group SU ( M | N )are studied, and the S L ( α, β, z ) polynomial invariant with skein relations are ob-tained under the fundamental representation of su ( M | N ).PACS Numbers: 11.15.-q, 02.10.KnKeywords: Chern-Simons Field Theory; Lie supergroup SU ( M | N ); Link Invariants. Chern-Simons (CS) theories are Schwarz-type topological field theories — a CS actionis both gauge invariant and generally covariant, and a quantum CS theory has generalvariance in the BRST formalism under the Landau gauge although a metric enters thegauge-fixing term [1]. CS theories were first introduced into physics in the study ofquantum anomaly of gauge symmetries by Jackiw et al. [2]. Witten pointed out [3] that CStheories provide a field theoretical origin for polynomial invariants of links in knot theory.Different Lie gauge groups of the CS theories and different algebraic representations ofthe gauge groups lead to different link invariants [3, 4, 5]. Perturbative expansions ofcorrelation functions of Wilson loops in CS theories present Vassiliev invariants [6, 7, 8].Recent developments include the applications of CS theories in topological string theory[9] and the (2 + 1)-dimensional quantum gravity [10].Super symmetries have found realizations in various physical systems [11]. Represen-tation theories for Lie superalgebras have been developed by many authors [12, 13]. Linkinvariants have been obtained from quantum super group invariants by Gould, Bracken,Zhang, Links, Kauffman, et al. from the algebraic point of view [14], including the HOM-FLY polynomial from the U q ( su ( M | N )) invariants ( M = N ), the Kauffman polynomial1rom the U q ( osp ( M | N )) invariants, and the Alexander-Conway polynomial from the U q ( gl ( N | N )) invariants.In this paper we will use the field theoretical point of view to study knots in the CSfield theory with super gauge group SU ( M | N ) , M = N [15, 16]. Under the fundamen-tal representation of the superalgebra su ( M | N ), a correlation function of Wilson loopoperators will be studied and the S L ( α, β, z ) link polynomial be obtained [4]. One willdiscuss the relationships between the S L ( α, β, z ) polynomial and the HOMFLY and Jonespolynomials, and show that the CS theory with super group SU ( N + 2 | N ) has the Jonespolynomial invariant. This is different from the situation of the CS theory with normalLie group SU ( N ) — under the fundamental representation, only the SU (2) CS theoryhas the Jones polynomial.This paper is arranged as follows. In Section 2, the notation of Lie superalgebra su ( M | N ) under the fundamental representation is given. In Section 3, path variationwithin correlation functions of Wilson loops in the CS theory is rigorously studied. InSection 4, the variation of correlation functions obtained in Section 3 is formally discussedwith respect to different link configurations, without integrating out the path integrals.From the formal analysis the S L ( α, β, z ) polynomial with skein relations is obtained, andits relationships to other knot polynomials are discussed. The paper is summarized inSection 5. Let us fix the notation of the superalgebra su ( M | N ) first. Consider the elements { ˆ e ab | a, b = 1 , · · · , M + N, M = N } satisfying the following super commutation relations[17, 18, 19, 20]: [ˆ e ab , ˆ e cd ] = ˆ e ad δ bc − ( − ([ a ]+[ b ])([ c ]+[ d ]) ˆ e cb δ da . (1)Here the Z -grading is given by [ˆ e ab ] = [ a ] + [ b ] with [1] = · · · = [ M ] = 0 and[ M + 1] = · · · = [ M + N ] = 1. In the fundamental representation ˆ e ab is realized byˆ e ab = e ab − δ ab ( − [ a ] M − N I, (2)where e ab is the ( M + N ) × ( M + N ) matrix unit with entry 1 at the position ( a, b ) and0 elsewhere. ˆ e ab satisfies the traceless requirement Str (ˆ e ab ) = 0 , where Str ( X ) is thesupertrace of the representation matrix of X ∈ g , Str ( X ) = P i ( − [ i ] X ii , i denoting theentry indices. The ˆ e ab ’s have the identity P M + Na =1 ˆ e aa = 0. The ( M + N ) − SU ( M | N ), denoted by n ˆ E ab , ˆ F ab , ˆ H cc o , can be constructed in terms of2 e ab : ˆ E ab = i (ˆ e ab − ˆ e ba ) , ˆ F ab = (ˆ e ab + ˆ e ba ) , a, b = 1 , · · · , M + N, a = b ;ˆ H cc = P cl =1 l (ˆ e ll − ˆ e l +1 ,l +1 ) , c = 1 , · · · , M + N − , (3)where no summation for repeating c, l . The ˆ E ab , ˆ F ab and ˆ H cc satisfy the properties oftracelessness and unitarity: Str (cid:16) ˆ E ab (cid:17) = Str (cid:16) ˆ F ab (cid:17) = Str (cid:16) ˆ H cc (cid:17) = 0; (cid:16) ˆ E ab (cid:17) † = ˆ E ab , (cid:16) ˆ F ab (cid:17) † = ˆ F ab and (cid:16) ˆ H cc (cid:17) † = ˆ H cc . The ˆ E ab and ˆ F ab play the role of the raising/loweringgenerators, and ˆ H cc the elements of the Cartan subalgebra of su ( M | N ). Hereinafter forconvenience one uses the basis { ˆ e ab , a = b ; ˆ e cc , c = 1 , · · · , M + N − } .We begin the study of the knots in a CS field theory by considering the correlationfunction of Wilson loops under the fundamental representation of su ( M | N ) [3, 4, 5] h W ( L ) i = D StrP e i H L A µ ( x ) dx µ E = Z − StrP Z D Ae iS e i H L A µ ( x ) dx µ , (4)where Z = R D Ae iS the normalization factor. L denotes the integration loop and P theproper product. S is the non-Abelian CS action, S = k π Z R d xǫ µνρ Str (cid:18) A µ ∂ ν A ρ + 23 A µ A ν A ρ (cid:19) , (5) k being an integer valued constant. A µ is the SU ( M | N ) gauge potential, A µ = A abµ ˆ e ab .The gauge field tensor F µν is induced by A µ : F µν = F abµν ˆ e ab , F abµν = ∂ µ A abν − ∂ ν A abµ − ( − ([ a ]+[ c ])([ c ]+[ b ]) (cid:0) A acµ A cbν − A acν A cbµ (cid:1) . (6)The grading [ A µ ] = [ F µν ] = [ S ] = even.The gauge invariance of the phase of the action, e iS , needs more discussion. The gaugetransformations of A µ and F µν are A µ −→ Ω A µ Ω − + ∂ µ ΩΩ − and F µν −→ Ω F µν Ω − , withΩ denoting a group G transformation. It is known that if G is a normal Lie group theaction S transforms as S −→ S + k π Z R d x∂ µ j µ + 2 πk π Z R d xǫ µνρ Str [ a µ a ν a ρ ] , (7)where a µ = Ω − ∂ µ Ω and j µ = ǫ µνρ Str ( A ν a ρ ). The second term in (7) is a total divergencewhich has no contribution to the action as j µ vanishes at infinity. The third term, markedas S WZW , is a Wess-Zumino-Witten (WZW) term. Jackiw, Cronstr¨om, Mickelsson, etal. [2, 21] examined this term for an arbitrary non-Abelian Lie group G . They pointedout that when Ω satisfies the regular condition — Ω tends to a definite limit at infinity,lim x →∞ Ω ( x ) = I — the WZW term is a total differential S WZW = 2 πk π Z R dx µ ∂ µ [Θ νρ dx ν ∧ dx ρ ] = 2 πk π Z R d Θ , (8)3here Θ is a 2-form constructed by Ω, and d Θ serves as a volume element [21].Since the regular condition implies the compactification R −→ S , Eq.(8) becomes S WZW = 2 πk π R S d Θ, which gives the degree of the homotopy mapping Ω : S → G when G is compact. Hence for a compact group G one has S WZW = 2 πkw (Ω), and theaction transforms as S → S + 2 πkw (Ω), where w (Ω) is the so-called winding number, w (Ω) ∈ π [ SU ( M | N )] = Z . In this paper, the gauge group is the super group SU ( M | N );a point needs clarification is whether the WZW term is able to be written as a total dif-ferential. This problem is being studied by us at present and will be discussed in ourfurther papers.Under the fundamental representation (2) the ˆ e ab has the following supertraces Str (ˆ e ab ˆ e cd ) = ( − [ a ] δ ad δ bc − ( − [ a ]+[ c ] δ ab δ cd M − N , (9)
Str (ˆ e ab ˆ e cd ˆ e ef ) = ( − [ a ] δ af δ bc δ de − ( − [ a ]+[ c ] δ ab δ cf δ de M − N − ( − [ c ]+[ f ] δ cd δ af δ be M − N − ( − [ f ]+[ a ] δ ef δ ad δ bc M − N + 2 ( − [ a ]+[ c ]+[ e ] δ ab δ cd δ ef ( M − N ) . (10)In terms of (9) and (10) the component form of the SU ( M | N ) CS action reads S = k π Z d xǫ µνρ ( − [ b ] " A abµ ∂ ν A baρ + 23 ( − [ c ]+[ a ][ b ]+[ b ][ c ]+[ c ][ a ] A abµ A bcν A caρ − ( − [ a ] A aaµ ∂ ν A bbρ M − N −
23 ( − [ a ] A aaµ A cbν A bcρ M − N + 43 ( − [ a ]+[ c ] A aaµ A bbν A ccρ ( M − N ) . (11)It can be proved that S has an important property [5, 4, 1]2 πk ǫ µνρ ( − [ b ] ∂S∂A abρ ( x ) ˆ e ba = F baµν ( x ) ˆ e ba . (12)This gives the equation of motion of a pure gauge: πk δSδA = F = 0, which is the sameas the commonly known equation of motion in the CS theories with normal Lie gaugegroups. Eq.(12) will be crucial in following sections for derivation of the skein relationsof knots in the CS theory with SU ( M | N ) gauge group. In this section correlation functions of Wilson loops will be studied, with emphasis placedon variation of integration paths and the induced changes of the correlation functions.Consider two knots which are almost the same except at one double-point x , asillustrated by Figure 1. 4
1 2 3 4 x (a)
1 2 3 4 x (b) x
4 3 2 1 (c)
Figure 1: Overcrossing, Undercrossing and Non-Crossing: (a) L + ; (b) L − ; (c) L .Here 1 , , , x , x , x , x . Denote the knot in Figure1(a) as L + and that in Figure 1(b) as L − . Figure 1(c) shows the non-crossing situation.Let U (1 ,
2) [resp. U (3 , →
2) [resp.(3 → U (1 ,
2) in Figure 1(a) as U + (1 , U − (1 , →
2) is prior to(3 →
4) in the sense of proper order. In following we will discuss the difference betweenthe overcrossing L + and undercrossing L − , by fixing the segment (3 →
4) and moving thesegment (1 →
2) from back to front.Let h W ( L + ) i and h W ( L − ) i be the respective correlation functions of L + and L − .Each of them can be written as a series of propagation processes in proper order: h W ( L ± ) i = h Str [ · · · U ± (1 , · · · U (3 , · · · ] i , (13)where the propagators are realized by U ± (1 ,
2) = e i R A µ ( x ) dx µ (cid:12)(cid:12)(cid:12) L ± , U (3 ,
4) = e i R A µ ( x ) dx µ , (14)the grading of U ± (1 ,
2) and U (3 ,
4) being even. The difference between the correlationfunctions of L + and L − is h W ( L + ) i − h W ( L − ) i = h Str ( · · · [ U + (1 , − U − (1 , · · · U (3 , · · · ) i . (15)The path variation L − → L + , given by [ U + (1 , − U − (1 , →
2) in L − corresponds to the path 1ACDB2,and that in L + to 1AEFB2. Then U + (1 , − U − (1 ,
2) = U (1 , A ) (cid:18) i Z AEFB A µ ( x ) dx µ − i Z ACDB A µ ( x ) dx µ (cid:19) U ( B, , (16)where the exponential expansion e i R A µ ( x ) dx µ = 1 + i R A µ ( x ) dx µ applies. In the light ofthe Stokes’ law one has U + (1 , − U − (1 ,
2) = U (1 , A ) i Z ∂ AEFBDC A µ ( x ) dx µ ! U ( B, U (1 , A ) i Z AEFBDC F µν ( x ) dx µ ∧ dx ν ! U ( B, , (17)5 H A C B DE F1 2 x G 3 4 d Figure 2: 3-Dimensional Geometric Illustration of Path Variationwhere ∂ AEFBDC is the boundary of the tiny area
AEFBDC at x . In (17) the curvature F µν ( x ) is the SU ( M | N ) gauge field tensor which has the expansion F µν ( x ) = F abµν ( x ) ˆ e ab .Thus the difference between the path integrals h W ( L − ) i and h W ( L + ) i is h W ( L + ) i − h W ( L − ) i = Z − Z AEFBDC dx µ ∧ dx ν Z D Ae iS Str (cid:2) · · · U (1 , A ) iF abµν ( x ) ˆ e ab U ( B, · · · U (3 , · · · (cid:3) . (18)Using the property of the Chern-Simons action (12), one has h W ( L + ) i − h W ( L − ) i = 2 πk Z − Z AEFBDC d Σ ρ Z D AStr (cid:20) · · · U (1 , A ) ( − [ a ] ˆ e ba ∂e iS ∂A abρ ( x ) U ( B, · · · U (3 , · · · (cid:21) = − πk Z − Z AEFBDC d Σ ρ Z D Ae iS Str (cid:20) · · · U (1 , A ) ( − [ a ] ˆ e ba U ( B, ∂∂A abρ ( x ) [ · · · U (3 , · · · ] (cid:21) , (19)where d Σ ρ = ǫ ρµν dx µ ∧ dx ν is the surface element of AEFBDC , and the technique ofintegration by parts has been used. In (19) the propagators [ · · · U (1 , A ) ˆ e ba U ( B, ∂∂A abρ ( x ) because they are not impacted by the move of Figure2. In the remaining propagation processes [ · · · U (3 , · · · ], only (3 →
4) passes the point x , hence only U (3 ,
4) is impacted by the move. Therefore, h W ( L + ) i − h W ( L − ) i = − πk Z − Z AEFBDC d Σ ρ Z D Ae iS · Str (cid:20) · · · U (1 , A ) ( − [ a ] ˆ e ba U ( B, · · · (cid:18) ∂∂A abρ ( x ) U (3 , (cid:19) · · · (cid:21) . (20)Let us examine the (cid:16) ∂∂A abρ ( x ) U (3 , (cid:17) in (20). It is shown in Figure 2 that U (3 ,
4) = e i R A λ ( y ) dy λ = U (3 , G ) e R HG iA klλ ( y )ˆ e kl dy λ U ( H, , (21)6here GH is a short segment passing x . Thus ∂∂A abρ ( x ) U (3 ,
4) = U (3 , G ) (cid:20)Z HG iδ ( x − x ) dx ρ ˆ e ab e R HG iA klλ ( y )ˆ e kl dy λ (cid:21) U ( H, , (22)and (20) becomes h W ( L + ) i − h W ( L − ) i = − i πk Z − Z AEFBDC Z HG δ ( x − x ) d Σ ρ ⊗ dx ρ Z D Ae iS · Str h · · · U (1 , A ) ( − [ a ] ˆ e ba U ( B, · · · U (3 , x ) ˆ e ab U ( x , · · · i , (23)where the dx ρ is along the direction of the segment GH. In (23) a volume integral isrecognized: [vol] x = Z AEFBDC Z HG δ ( x − x ) d Σ ρ ⊗ dx ρ , (24)which has the evaluation [vol] x ( = 0 , trivial;= ± , non-trivial. (25)In detail, • [vol] x = 0 describes the trivial case that in Figure 2 the dx ρ is parallel to the planeof AEFBDC ; namely, the move from ACDB to AEFB is done by sliding along 3GH4.Therefore d Σ ρ ⊗ dx ρ = 0. • [vol] x = 1 describes the non-trivial move L − → L + , where dx ρ is perpendicular to AEFBDC and d Σ ρ ⊗ dx ρ = 1; otherwise, [vol] x = − L + → L − , where dx ρ isperpendicular to AEFBDC but d Σ ρ ⊗ dx ρ = −
1. The case we come across in Figure2 is the former, so [vol] x = 1.Therefore, (23) becomes h W ( L + ) i − h W ( L − ) i = − i πk Z − Z D Ae iS Str h · · · U (1 , A ) ( − [ b ] ˆ e ab U ( B, · · · U (3 , x ) ˆ e ba U ( x , · · · i . (26) In this section the S L ( α, β, z ) polynomial invariant for knots in the SU ( M | N ) CS fieldtheory will be derived from (26), and its relationship to the HOMFLY and Jones polyno-mials will be discussed. 7nder the fundamental representation the entries of the matrices ˆ e ab satisfy the Fierzidentity [22] ( − [ b ] (ˆ e ab ) ij (ˆ e ba ) kl = ( − [ j ] δ il δ jk − M − N δ ij δ kl . (27)Hence (26) leads to h W ( L + ) i − h W ( L − ) i = − i πk Z − Z D Ae iS · Str [ · · · U (1 , A ) U ( x , · · · ] Str [ U ( B, · · · U (3 , x )]+ i πk M − N Z − Z D Ae iS · Str [ · · · U (1 , A ) U ( B, · · · U (3 , x ) U ( x , · · · ] . (28)When the points A and B approaching x , the first term of (28) corresponds to the non-crossing case L in Figure 1(c). For the second term, however, one has two ways to connect A and B — the undercrossing and the overcrossing — in order to form a propagationprocess (1 → (cid:18) − i πk M − N ) (cid:19) h W ( L + ) i − (cid:18) i πk M − N ) (cid:19) h W ( L − ) i = − i πk h W ( L ) i . (29)Then, considering the weak coupling limit of large k [3], we define β = 1 − i πk M − N ) + O (cid:18) k (cid:19) , z = − i πk + O (cid:18) k (cid:19) , (30)and obtain an important skein relation β h W ( L + ) i − β − h W ( L − ) i = z h W ( L ) i . (31)For the purpose of examining knot writhing, let us consider the special case that thepoint x is identical to x in Figure 1. Then in (26) one haslim B → x ; x = x U ( B, · · · U (3 , x ) = I, (32)and D W (cid:16) ˆ L + (cid:17)E − D W (cid:16) ˆ L − (cid:17)E = − i πk Z − Z D Ae iS Str h · · · U (1 , A ) ( − [ b ] ˆ e ab ˆ e ba U ( x , · · · i , (33)where ˆ L + and ˆ L − are two writhing situations shown in Figure 3(a) and 3(b). Figure 3(c)shows the non-writhing situation ˆ L .In the above the factor ( − [ b ] ˆ e ab ˆ e ba is the Casimir operator( − [ b ] ˆ e ab ˆ e ba = 2 C I, C = ( M − N ) −
12 ( M − N ) , M = N. (34)8 (a) (b) (d) (e) (c) Figure 3: Typical Configurations: (a) writhing ˆ L + ; (b) writhing ˆ L − ; (c) non-writhing ˆ L ;(d) trivial circle ˆ L c ; (e) non-intersecting union ˆ L i .When A approaches x one has D W (cid:16) ˆ L + (cid:17)E − D W (cid:16) ˆ L − (cid:17)E = − i πk C D W (cid:16) ˆ L (cid:17)E , (35)where D W (cid:16) ˆ L (cid:17)E = Z − R D Ae iS Str [ · · · U (1 , x ) U ( x , · · · ]. The move ˆ L − → ˆ L + is achange of the writhe of the path segment. In this regard an intermediate stage ˆ L can beinserted and the move becomes ˆ L − → ˆ L → ˆ L + . Then the correlation function becomes D W (cid:16) ˆ L + (cid:17)E − D W (cid:16) ˆ L − (cid:17)E = hD W (cid:16) ˆ L + (cid:17)E − D W (cid:16) ˆ L (cid:17)Ei + hD W (cid:16) ˆ L (cid:17)E − D W (cid:16) ˆ L − (cid:17)Ei . The two subprocesses ˆ L − → ˆ L and ˆ L → ˆ L + should be equivalent, hence D W (cid:16) ˆ L + (cid:17)E − D W (cid:16) ˆ L (cid:17)E = D W (cid:16) ˆ L (cid:17)E − D W (cid:16) ˆ L − (cid:17)E = − i πk C D W (cid:16) ˆ L (cid:17)E , and we arriveat another skein relation D W (cid:16) ˆ L + (cid:17)E = α D W (cid:16) ˆ L (cid:17)E , D W (cid:16) ˆ L − (cid:17)E = α − D W (cid:16) ˆ L (cid:17)E , α = 1 − i πk C + O (cid:18) k (cid:19) . (36)Besides (31) and (36), one needs the correlation function for the trivial circle ˆ L c shownin Figure 3(d): D W (cid:16) ˆ L c (cid:17)E = Z − Z D Ae iS Str h ˆ L c i = Z − Z D Ae iS Str [ I ] = ( M − N ) . (37)Thus, in summary, we have acquired the following skein relations for knots in the SU ( M | N ) CS field theory: D W (cid:16) ˆ L c (cid:17)E = M − N ( M = N ) , (38) D W (cid:16) ˆ L + (cid:17)E = α D W (cid:16) ˆ L (cid:17)E , D W (cid:16) ˆ L − (cid:17)E = α − D W (cid:16) ˆ L (cid:17)E , (39) β h W ( L + ) i − β − h W ( L − ) i = z h W ( L ) i , (40)with α = 1 − i πk C + O (cid:18) k (cid:19) , β = 1 − i πk M − N ) + O (cid:18) k (cid:19) , z = − i πk + O (cid:18) k (cid:19) . (41)These relations present a polynomial invariant h W ( L ) i for the knots, known as the S L ( α, β, z ) polynomial proposed by Guadagnini et al. [4, 1].9t is checked that Eq.(39) is consistent with (40). Considering the special case x = x for (31) there is β D W (cid:16) ˆ L + (cid:17)E − β − D W (cid:16) ˆ L − (cid:17)E = z D W (cid:16) ˆ L i (cid:17)E , (42)where ˆ L i is the non-intersecting union of a trivial circle and a line segment shown in Figure3(e). The LHS of (42) gives β D W (cid:16) ˆ L + (cid:17)E − β − D W (cid:16) ˆ L − (cid:17)E = ( βα − β − α − ) D W (cid:16) ˆ L (cid:17)E with respect to (39). The RHS of (42) is z D W (cid:16) ˆ L i (cid:17)E = zZ − Z D Ae iS Str [ · · · U (1 , · · · ] Str h ˆ L c i = z ( M − N ) D W (cid:16) ˆ L (cid:17)E . (43)Hence βα − β − α − = z ( M − N ), which is consistent with the definitions of α, β and z .The S L ( α, β, z ) polynomial is regular-isotopic, but not ambient-isotopic. Namely, h W ( L ) i is invariant under the type-II and -III Reidemeister moves (shown in Figure 4),but is not invariant under the type-I move. Indeed, • in a type-II move, path variation of Figure 2 takes place at both the points x a and x b . Then there are volumes of variation given in (24) at both x a and x b , whichare marked as [vol] x a and [vol] x b respectively. It can be checked that [vol] x a and[vol] x b take opposite sign: [vol] x a = 1 , [vol] x b = −
1. Hence totally the type-IImove causes no variation in the correlation function; • in a type-III move, there are neither “undercrossing to overcrossing”nor “overcross-ing to undercrossing ”moves taking place, so the volume of variation is zero, andthe type-III move causes no variation in the correlation function; • in a type-I move, the variation of the correlation function is given by (39). x b x a x b x a (c) (b) (a) Figure 4: Reidemeister Moves: (a) Type-I; (b) Type-II; (c) Type-III.10n following the relationships between the S L ( α, β, z ) polynomial and other knot poly-nomial invariants will be studied. h W ( L ) i will be modified to be an ambient-isotopicinvariant, and a difference between the normal and super Lie gauge groups, SU ( N ) and SU ( M | N ), will arise from the Jones polynomial.Firstly, the ambient-isotopic HOMFLY knot polynomial invariant can be constructedfrom h W ( L ) i by introducing a factor describing knot writhing: h P ( L ) i = α − ω ( L ) h W ( L ) i . (44)Here ω ( L ) is the writhe number of a knot L , defined as ω ( L ± ) = ω ( L ) + ǫ ( L ± ; x ) = ω ( L ) ± , (45)where ǫ ( L ± ; x ) is the sign of the crossing point x on L ± : ǫ ( L ± ; x ) = ±
1. For ˆ L + , ˆ L − and ˆ L , (45) reads ω (cid:16) ˆ L + (cid:17) = ω (cid:16) ˆ L (cid:17) + 1 , ω (cid:16) ˆ L − (cid:17) = ω (cid:16) ˆ L (cid:17) − . (46)(46) means that ˆ L + contributes a 1 to the writhe number, while ˆ L − contributes a ( − D P (cid:16) ˆ L + (cid:17)E = D P (cid:16) ˆ L (cid:17)E , D P (cid:16) ˆ L − (cid:17)E = D P (cid:16) ˆ L (cid:17)E , (47)meaning h P ( L ) i is invariant under the type-I Reidemeister move. Furthermore h P ( L ) i satisfies ( αβ ) h P ( L + ) i − ( αβ ) − h P ( L − ) i = zP ( L ) . (48)Hence one arrives at the skein relations for h P ( L ) i : D P (cid:16) ˆ L c (cid:17)E = M − N, (49) t h P ( L + ) i − t − h P ( L − ) i = z h P ( L ) i , (50)where t ≡ αβ = 1 − i πk ( M − N )2 + O (cid:18) k (cid:19) , z = − i πk + O (cid:18) k (cid:19) . (51)(49) can be obtained from (50) by considering t D P (cid:16) ˜ L + (cid:17)E − t − D P (cid:16) ˜ L − (cid:17)E = z D P D ˜ L c EE ,where ˜ L + , ˜ L − and ˜ L c denote unknots shown in Figure 5.Eqs.(49) and (50) show h P ( L ) i is an ambient-isotopic HOMFLY polynomial invariant.Secondly, if specially M − N = 2 in (49) to (51), the z is related to t as z = t − t − ,up to the first order. This means that in the SU ( N + 2 | N ) CS field theory, under thefundamental representation there is a knot polynomial h V ( L ) i ≡ h P ( L ) i which satisfiesthe skein relation t h V ( L + ) i − t − h V ( L − ) i = (cid:16) t − t − (cid:17) h V ( L ) i . (52)11 x (b)
1 2 3 4 x (a) x
4 321 (c)
Figure 5: Unknots: (a) ˜ L + ; (b) ˜ L − ; (c) ˜ L c .This h V ( L ) i is known as the Jones polynomial . Therefore there are a series of CStheories with Lie super gauge group SU ( N + 2 | N ) , N ∈ Z + , which have the Jonespolynomial. This is different from the situation of the CS theory with normal Lie group SU ( N ) — it is known that under the fundamental representation, only the SU (2) theoryhas the Jones polynomial invariant among all SU ( N ) CS theories, N = 2 , , · · · [3, 1, 8].Different choices of gauge groups with different algebraic representations lead to differ-ent knot polynomials in CS field theories [8]. In our further work the relationship betweenthe S L ( α, β, z ) and the Kauffman polynomials in the OSp (1 |
2) CS field theory will bestudied.Finally, the α, β and z in the S L ( α, β, z ) polynomial and the t in the HOMFLYpolynomial can be expressed in a unified way. Introducing a variable q = e − i πk , (53) α, β , z and t can be regarded as the lower order expansions of the q exponentials [4, 1, 3]: α = q C = q ( M − N )2 − M − N ) , β = q M − N ) , z = q − q − and t = q M − N . Then the S L ( α, β, z )shown in (38)–(40) and HOMFLY polynomial in (49)–(50) can be written more elegantlyas D W (cid:16) ˆ L c (cid:17)E = M − N ( M = N ) , (54) D W (cid:16) ˆ L + (cid:17)E = q ( M − N )2 − M − N ) D W (cid:16) ˆ L (cid:17)E , (55) D W (cid:16) ˆ L − (cid:17)E = q − ( M − N )2 − M − N ) D W (cid:16) ˆ L (cid:17)E , (56) q M − N ) h W ( L + ) i − q − M − N ) h W ( L − ) i = (cid:16) q − q − (cid:17) h W ( L ) i , (57)and D P (cid:16) ˆ L c (cid:17)E = q M − N − q − M − N q − q − , (58) q M − N h P ( L + ) i − q − M − N h P ( L − ) i = (cid:16) q − q − (cid:17) h P ( L ) i . (59) Compared to the standard conventions adopted in mathematics, there is a sign differ-ence in the skein relation (52) of the Jones polynomial. See [1] for this discussion.12
Conclusion
In this paper we have studied knots in the CS field theory with gauge group SU ( M | N ).In Section 2, the notation for the fundamental representation of the Lie superalgebra su ( M | N ) is fixed, and an important property of the CS action, Eq.(12), is presented.In Section 3, variation of the correlation function of Wilson loops is rigorously studied.In Section 4, the variation of correlation functions (26) is discussed for different linkconfigurations. It is addressed that the path integrals have been formally expressed aspropagators instead of being integrated out. A rigorous development of techniques forpath integrals awaits future advances in the mathematical theory of functional integrals.From the formal analysis the S L ( α, β, z ) knot polynomial and its skein relations, (38) to(40), are obtained. In terms of the S L ( α, β, z ) polynomial the HOMFLY and Jones knotpolynomials as well as their skein relations (49) to (52) have been derived by consideringthe knot writhing. The author is indebted to Prof. R.B. Zhang and Dr. W.L. Yang for instructive advicesand warmhearted help. This work was financially supported by the USYD PostdoctoralFellowship of the University of Sydney, Australia.
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