A New Phase in Chern-Simons Theory on S^3 in Seifert Framing
AA New Phase in Chern-Simons Theory on S in Seifert Framing Kushal Chakraborty ∗ and Suvankar Dutta † Indian Institute of Science Education & Research Bhopal,Bhopal Bypass, Bhopal 462066, India
Abstract
We consider U ( N ) k Chern-Simons theory on S in Seifert framing and write down the partitionfunction as a unitary matrix model. In the large k and large N limit the eigenvalue density satisfiesan upper bound πλ where λ = N/ ( k + N ). We study the partition function under saddle pointapproximation and find that the saddle point equation admits a gapped solution for the eigenvaluedensity. The on-shell partition function on this solution matches with the partition function inthe canonical framing up to a phase. However the eigenvalue density saturates the upper cap ata critical value of λ and ceases to exist beyond that. We find a new phase (called cap-gap phase)in this theory for λ beyond the critical value and see that the on-shell free energy for the cap-gapphase is less than that of the gapped phase. We also check the level-rank duality in the theory andobserve that the level-rank dual of the gapped phase is a capped phase whereas the cap-gap phaseis level-rank dual to itself. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] F e b . INTRODUCTION Being a topological theory the partition function (PF) of Chern-Simons (CS) theory in three dimensions is a topological invariant. At the quantum level topological invariance is alsopreserved but at the expense of a choice of framing [1, 2]. The PF of CS theory with gaugegroup G , rank k on Seifert manifold M ( g,p ) can be obtained by surgery from the expectationvalue of Wilson loop in S × S [3, 4]. Different choices of surgery gives different framingsin M ( g,p ) . The PF on M ( g,p ) is given by Z ( M ( g,p ) , G, k ) = (cid:88) R K ( p )0 R (cid:88) V S − g , V S RV . (1)Here K ( p ) is a surgery/framing dependent matrix, S RR (cid:48) is modular transform matrix asso-ciated with highest weight representations of affine Lie algebra g k of G k under inversion ofmodular parameter. The sum in (1) runs over integrable representations of g k and R = 0corresponds to trivial representation. For K ( p ) = ST − p S , where T RR (cid:48) is the second modu-lar transform matrix associated with translation of modular parameter, the PF is given by[3] Z SF ( M ( g,p ) , G, k ) = (cid:88) R S − g , R T − p RR . (2)We are interested in CS theory on S which is a Seifert manifold with genus g = 0 andthe first Chern class p = 1. On S , there exists a canonical framing K ( p ) = S [3] inwhich the PF is given by Z Can ( S , G, k ) = S . Using the properties of S and T matrices( S = ( ST ) = I ) one can show that PF in canonical and Seifert framings are related by : Z SF ( S , G, k ) = T S . In this paper we focus on the affine gauge group G = U ( N ) k . Onecan compute S for U ( N ) k and it turns out to be a smooth function of λ = N/ ( k + N ).On the other hand, in the large k and large N limit keeping λ fixed, one can computethe PF in Seifert framing (2) under the saddle point approximation and find the dominantrepresentation for 0 ≤ λ ≤ λ . It was first pointed out in [9]. In this paper we address this issue indetail. Using the fact that the sum in (2) runs over integrable representations, we write thePF (for g = 0) as a unitary matrix model. In the large k, N limit the eigenvalue densityis constrained to have a maximum value πλ . We derive the saddle point equation for the A Seifert manifold M ( g,p ) is a circle bundle over genus g Riemann surface Σ g with the first Chern class p . λ >
0. We compute the PF on the gapped solution and see that it is equal to T S for allvalues of 0 ≤ λ ≤
1. Therefore the gapped phase is equivalent to the dominant representationobtained in [5–8]. However, the eigenvalue density of the gapped phase saturates the upperbound at λ = 1 /π log cosh π ≡ λ ∗ and seizes to exists beyond λ ∗ [9]. In this paper wediscover that there exists another phase (we call this phase cap-gap phase) for λ > λ ∗ . Wecompute the free energy of the cap-gap phase and see that it is less than that of the gappedphase for λ > λ ∗ . Therefore our calculation shows that the PF of CS theory in Seifertframing (2) admits a phase transition at λ = λ ∗ in the large k, N limit when we consider theintegrability condition properly. The advantage of converting the PF to a unitary matrixmodel is that the dominant representations for both the gapped and cap-gap phases areintegrable by construction.CS on S also enjoys the level-rank duality [10]. We find the Young diagram (YD) distri-bution for a large k, N phase and its dual and show that they are related by transpositionfollowed by a shift. We also check that the PFs of dual theories are the same in the large k, N limit. The dual of a gapped phase has an upper cap in the eigenvalue distribution. Onthe other hand, the cap-gap phase is dual to itself. This is similar to the matter CS theorieson S × S studied by [11, 12]. II. U ( N ) CHERN-SIMONS THEORY ON SEIFERT MANIFOLD
The affine Lie algebra u ( N ) k is the quotient of su ( N ) k × u (1) N ( k + N ) by Z N . Hence u ( N )representation can be written in terms of su ( N ) representations and eigenvalues of u (1)generator : R = ( R, Q ). We use the notation R for su ( N ) representations and Q is eigenvalueof u (1) generator, given by Q = r ( R ) mod N , where r ( R ) is the number of boxes in R . Trivialrepresentation R = 0 corresponds to R = 0 and Q = 0. The modular transform matrix S RR (cid:48) for u ( N ) k can be written in terms of representations of su ( N ) and the u (1) charges[10, 13, 14] S RR (cid:48) = ( − i ) N ( N − ( k + N ) N e − πiQQ (cid:48) N ( N + k ) det M ( R, R (cid:48) ) (3)3here, M ( R, R (cid:48) ) is a N × N matrix with elements, M ij ( R, R (cid:48) ) = exp (cid:20) πik + N φ i ( R ) φ j ( R (cid:48) ) (cid:21) , φ i ( R ) = l i − r ( R ) N − i −
12 ( N + 1) (4)and l i ’s are the number of boxes in i th row in R . The other modular transformation matrix T RR (cid:48) is given by T RR (cid:48) = exp (cid:2) πi ( h R − c ) (cid:3) δ RR (cid:48) , h R = C ( R ) k + N , c = N ( Nk +1) k + N (5)where C ( R ) is the quadratic Casimir of u ( N ) k . Since Q = r ( R ) + N s for s ∈ Z , u ( N )representations R can be characterised by extended YDs by re-defining number of boxesin i th row ¯ l i = l i + s for 1 ≤ i ≤ N − l N = s . Now ¯ l i s can be negative and thecorresponding YDs will have anti-boxes [15]. In terms of these extended YDs the quadraticCasimir C ( R ) is given by C ( R ) = N (cid:88) i =1 ¯ l i (¯ l i − i + N + 1) . (6) A representation R of u ( N ) k is an integrable representation if ≤ ¯ l N ≤ · · · ≤ ¯ l ≤ k [10]. III. CHERN-SIMONS THEORY AS UNITARY MATRIX MODEL
For an integrable representation R the hook numbers h i = ¯ l i + N − i satisfy 0 < h N < · · ·
1. Before we discussthe large k, N phases in this theory we take a pause to study the eigenvalue density of thelevel-rank dual theory and its connection to ρ ( θ ). IV. LEVEL-RANK DUALITY N ↔ k duality in U ( N ) CS theory implies that the dominant YDs in two theories, dual toeach other are related by a transposition followed by a shift. In order to prove this statementwe write down the partition function of U ( N ) k CS theory in terms of number of boxes in5ifferent columns in a YD. A YD corresponding to an integrable representation of u ( N ) k can be characterised by ¯ v µ - the number of boxes in µ th column of a YD R where 1 ≤ µ ≤ k and ¯ v ≤ N . { ¯ v µ } is the set of box numbers in different rows of ˜ R , where ˜ R is transposeof R . The quadratic Casimir C ( R ) can be written in terms of ¯ v µ . Also the S modulartransform matrix (3) is invariant under transposition. We introduce new variables φ µ = 2 πk + N (cid:18) w µ − k + N − (cid:19) , where w µ = ¯ v µ + k − µ. (12)Since 0 ≤ ¯ v µ ≤ N , φ µ s are distributed in a range of 2 π . The partition function (2) canbe written in terms of φ µ s and it turns out that the effective action is symmetric under φ µ → π − φ µ . Hence for any classical solution φ µ s are distributed symmetrically about φ = π from 0 to 2 π . In the continuum limit we define a distribution functions for φ µ s˜ ρ ( φ ) = 1 k k (cid:88) µ =1 δ ( φ − φ µ ) (13)and the partition function is given by Z pN,k = (cid:90) [ dφ ] e − ( N + k ) ˜ S eff [˜ ρ ] (14)where, ˜ S eff [ ˜ ρ ] = p ˜ λπ (cid:90) ˜ ρ ( φ ) (cid:18) π − ( φ − π ) (cid:19) dφ + pπλ ˜ λ − ˜ λ (cid:90) − (cid:90) ˜ ρ ( φ ) ˜ ρ ( φ (cid:48) ) log (cid:18) φ − φ (cid:48) (cid:19) dφdφ (cid:48) (15)and ˜ λ = 1 − λ . The saddle point equation for ˜ ρ ( φ ) is given by − (cid:90) π ˜ ρ ( φ (cid:48) ) cot (cid:18) φ − φ (cid:48) (cid:19) dφ (cid:48) = p π ˜ λ ( π − φ ) . (16)Comparing (10) and (16) we find˜ ρ ( φ ) = 12 π ˜ λ − λ ˜ λ ρ ( φ + π ) . (17)This relation establishes the fact that under N ↔ k duality the dominant YDs in U ( N ) k and U ( k ) N CS theories are related by a transposition with a shift N + k . Using (17) we alsosee that S [ ρ ] = ˜ S [ ˜ ρ ]. The relation (17) is similar to what found in [11] in the context ofmatter CS theory on S × S . 6 . LARGE N PHASES
The unitary matrix model (9) was studied in [9, 17]. It was observed that the system has agapped phase in the large k, N limit and the eigenvalue distribution is given by, ρ ( θ ) = p π λ tanh − (cid:34)(cid:115) − e − πλ cos θ (cid:35) . (18)Since ρ ( θ ) ≥
0, this implies eigenvalues are distributed over the range − − e − πλ < θ < − e − πλ . (19)See fig.1 for the eigenvalue distribution. / ( π λ ) - π π θ ρ ( θ ) FIG. 1: ρ ( θ ) for one-gap phase.We calculate the PF (8) on this solution (18) and check that after suitable analytic contin-uation p → ip the PF exactly matches with T S for all values of 0 ≤ λ ≤
1. Hence thisphase is equivalent to the dominant phase obtained in [5–8]. However, due to the constraint(11) on ρ ( θ ) the eigenvalue density saturates the upper bound at λ = 1 /π log cosh( π/p ) ≡ λ ∗ [9]. Therefore the gapped phase is not valid anymore for λ > λ ∗ .7 . Cap-gap phase For λ > λ ∗ the eigenvalue density develops a cap about θ = 0. To find that phase we takethe following ansatz for ρ ( θ ) ρ ( θ ) = (cid:40) πλ for − θ < θ < θ ˆ ρ ( θ ) for − θ < θ < − θ and θ < θ < θ . (20)Using the map z = e iθ , the saddle point equation for ˆ ρ ( θ ) is given by − (cid:90) ˆ ρ ( z (cid:48) ) z + z (cid:48) z − z (cid:48) dz (cid:48) = p log( z )2 πiλ − πλ (cid:90) z (cid:48) z + z (cid:48) z − z (cid:48) dz (cid:48) . (21)Following [11] we define a resolvent function Φ( z )Φ( z ) = (cid:90) ˆ ρ ( z (cid:48) ) iz (cid:48) z + z (cid:48) z − z (cid:48) dz (cid:48) = h ( z ) H ( z ) , where (22) h ( z ) = (cid:112) ( z − z cos θ + 1)( z − z cos θ + 1) . From the normalization of eigenvalue density it follows that,Φ( z → ∞ ) ∼ − πλ (cid:90) dωiω . (23)The resolvent Φ( z ) has branch cut in complex z plane. The eigenvalue density ˆ ρ ( z ) isobtained from the discontinuity of Φ( z )Φ + ( z ) − Φ − ( z ) = 4 π ˆ ρ ( z ) . (24)Following [18] the function H ( z ) can be evaluated as H ( z ) = i (cid:73) dw πi p log( w )2 πiλ − πλ (cid:82) s w + sw − s dsh ( w )( w − z ) . (25)Plugging this expression in (22) we expand the r.h.s. for large z and comparing the expressingwith (23) we find the following two constraint p πλ (cid:73) dz πi log( z ) h ( z ) + iπλ (cid:90) dωh ( ω ) = 0 , p πλ (cid:73) dz πi z log( z ) h ( z ) + iπλ (cid:90) ωdωh ( ω ) = 0 . (26)8sing the formula given in appendix, we numerically solve these two equations to find theendpoints θ and θ . From the discontinuity Φ( z ) we compute the eigenvalue densityˆ ρ ( θ ) = − | sin φ | π λ (cid:113) (sin φ − sin θ )(sin θ − sin φ ) (cid:112) (1 + cos θ )(1 − cos θ ) (cid:32) p (cid:0) cos θ Π( ψ, n , m ) − cos φ F ( ψ, m ) (cid:1) (1 + cos φ )(cos φ − cos θ ) − (cid:0) Π( n , m ) − sin φ K ( m ) (cid:1) sin φ (cid:33) (27)where m , m , n , n , ψ are given in (A3). The eigenvalue density in cap-gap phase is plottedin fig. 2. The free energies for these two phases as a function of λ are plotted in fig. 3. From π λ - π π θρ ( θ ) FIG. 2: ρ ( θ ) for cap-gap phase.this figure we see that the cap-gap phase has free energy less than that of gapped phase andhence dominant over the gapped phase for λ > λ ∗ . However the on-shell PF on the cap-gapphase differs from T S . VI. DISCUSSION
In order to obtain a real saddle point equation (10) we use an analytic continuation in p .However if we use an analytic continuation in λ [9], we would also get a real saddle pointequation for YD distribution u ( h ) in h plane with a coth kernel similar to what consideredin [7]. Solution of this equation renders a YD distribution which never crosses the maximumvalue 1 for p = 1. The YD distribution is different from the one-gap eigenvalue distribution9 λ F ( λ ) FIG. 3: Free energy as a function of λ . The solid blue line is the free energy for the gappedphase for λ < λ ∗ . The dashed blue line is the extension of the same beyond λ ∗ . The redline depicts the free energy for the cap-gap phase.obtained in this paper. But with a proper analytic continuation of p and λ one can relatethe two [9]. The YD distribution obtained in [5, 7, 9] violates the integrability bound forsome value of λ between 0 and 1. In strictly k, N → ∞ limit the sum over R in (2) isunrestricted. Hence one should not expect any phase transition in the system for p = 1.But when we consider the level and rank to be large but finite and take the double scalinglimit properly we observe the phase transition. In the double scaling limit θ i s are defined insuch a way (7) that they have range between 0 and 2 π and the dominant representations arealways integrable. But this change of variables imposes a cap on the eigenvalue distributionwhich triggers a phase transition in the theory.The ’t Hooft expansion of the PF of SU ( N ) CS theory on S is proposed to be dual totopological closed string theory on the S blow up of the conifold geometry [19] for arbitrary λ and all orders of 1 /N . In canonical framing the CS PF is equal to S and an exactfunction of λ which matches with string theory side. In Seifert framing, we observe that thePF of CS theory in the gapped phase is equal to that in the string theory side. But the PF incap-gap phase differs from S for λ > λ ∗ . Dependence of phase on the choice of framing isbit puzzling here. The question is why a new phase pops up in the theory when we take the10ouble scaling limit. The saddle equation (10) also admits multi-cut solutions, which wererelated to some non-perturbative D-instantons [20], are different than the cap-gap phase. Itwould be interesting to understand the meaning of this new phase in the string theory sideas well.We explicitly check the level-rank duality in CS theory on S . The theory admits threetypes of phases. For λ < λ ∗ one has gapped phase and capped phase. These two phases arelevel-rank dual to each other. For λ > λ ∗ the theory admits a cap-gap phase which is levelrank dual to itself. There is a third order phase transition at λ ∗ . The phase structure issimilar to that of CS-matter theory on S × S [9, 11] except that here we do not have anygap less phase.The partition function of q -deformed U ( N ) Yang-Mills on a generic Riemann surface with zero θ term is equal to the PF of CS theory on M ( g,p ) up to a phase factor for q = e πiN + k and k, p ∈ Z [10]. Thus our analysis shows that the q -deformed Yang-Mills undergoes a phasetransition even for p = 1 unlike [7]. Acknowledgments:
We thank Arghya Chattopadhyay and Neetu for working on this problemat the initial stage. We are grateful to Rajesh Gopakumar and Dileep Jatkar for readingour manuscript and giving their valuable comments. The work of SD is supported by the
MATRICS grant (no.
MTR/2019/000390 , the Department of Science and Technology,Government of India). We are indebted to people of India for their unconditional supporttoward the researches in basic science.
Appendix A: Useful formulae
We use the following useful results in our calculations.( i ) . (cid:73) dz πi log( z ) h ( z ) = 2 F ( ψ, m ) (cid:112) (1 + cos θ )(1 − cos θ ) , ( ii ) . (cid:73) dz πi z log( z ) h ( z ) = 2 cos θ F ( ψ, m ) (cid:112) (1 + cos θ )(1 − cos θ ) + 2 βv (cid:48) ( β ) v ( β )+ 12 log (cid:18) (1 − cos θ )(1 + cos θ )4 K ( m ) (cid:19) − log (cid:20) v (2 β ) v (cid:48) (0) (cid:21) (A1)11here, ψ = sin − (cid:114) − cos θ , β = F ( ψ, m )2 K ( m ) , q = e − π K (cid:48) ( m K ( m , K (cid:48) ( m ) = K ( (cid:113) − m ) . ( iii ) . (cid:90) e iθ e − iθ dωh ( ω ) = 2 iK ( m ) (cid:112) (1 − cos θ )(1 + cos θ ) , ( iv ) . (cid:90) e iθ e − iθ ωdωh ( ω ) = 2 i (2Π ( n, m ) − K ( m )) (cid:112) (1 − cos θ )(1 + cos θ ) (A2)where, m = (cid:115) θ − cos θ )(1 + cos θ )(1 − cos θ ) , m = (cid:115) (1 − cos θ )(1 + cos θ )(1 − cos θ )(1 + cos θ ) ; n = cos θ −
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