aa r X i v : . [ h e p - t h ] J un Lens space matter determinantsin the vector model
J.S.Dowker Theory Group,School of Physics and Astronomy,The University of Manchester,Manchester, England
A simple quadrature is given for the free energy (or logdet) of thematter singlet sector in the N -vector model on a lens–space. [email protected] , [email protected] . Introduction In a technical calculation, Radi˘cevi´c, [1], has evaluated the free energy of thesinglet sector vector model on three-dimensional (homogeneous) lens spaces. Fol-lowing Shenker and Yin, [2], Sundborg, [3], and Aharony et al , [4], the singletcondition involves coupling the U( n )–vector matter field to a Chern–Simons gaugetheory which was taken at infinite level in order to decouple the gauge and matterfields. The total partition function is then a sum over products of the matter andgauge partition functions, which can be computed separately. In this note, I wishto present a different evaluation of the matter sector to that in [1], specifically bygiving a more compact treatment of the relevant degeneracies. In doing this I willdraw together some earlier work.
2. The setup
The field theory under discussion falls into a quite general and well–knownclass that we have earlier termed ‘automorphic field theory’. Some references canbe found in [5]. In a computation that has many points of contact with [1] (and alsowith Banerjee et al , [6]) SU( n ) symmetry breaking by ‘Wilson loops’ on fixed pointfree factors of the three-sphere was investigated, [7,8,9]. In these works, we wereinterested in the free–energy of the theory on an Einstein–like universe and whetherany broken symmetry would be restored at finite temperature. Here, rather, it isthe effective action on just the factored spheres that one requires.There is no point in describing the general situation and I move to the specificcase under scrutiny here by stating that the quantum theories on S / Z q of a fieldbelonging to the fundamental representation of the group U( n ) are classified by thehomorphisms, ρ ∈ Hom (cid:0) Z q , U( n ) (cid:1) , which is an n × n representation of Z q (in U( n )).As such it can be decomposed into irreps with the different decompositions classi-fying the homomorphisms, [7]. As discussed in [7], an element of Hom (cid:0) Z q , U(n) (cid:1) is determined by the phases, φ i ( i = 1 , , . . . n ), of the diagonalised fundamentalmatrix, ρ ( γ ), where γ is the generator of Z q . Because γ q = E , these phases equal φ i = 2 πq r i , ≤ r i ≤ q − . showing that the homomorphisms, or ‘twistings’, are specified by the ordered n –vector r = { r i } where r ≥ r ≥ . . . ≥ r n . In other terminologies this is a This is the general classification holding whatever representation the field may actually belongto. n ) to Q q − r =0 U ( g r ) where g r is the number of repetitions of r in r .This is shown, for a more general situation, in [7] (the extension from SU( n ) toU( n ) being trivial) and is also given in [1].
3. The determinants
The direct sum structure of the homomorphism ρ means that any spectralquantity, S , (I have in mind the logdet) associated with ρ translates into an al-gebraic sum of the spectral quantities for every diagonal element which are justthe spectral quantities for the simpler homomorphisms Hom (cid:0) Z q , U(1) (cid:1) . I write thisdecomposition as, S q ( r ) = n X i =1 S q ( r i ) = q − X r =0 g r S q ( r ) ≡ S ( q, g ) . (1)The q –vector g ≡ { g r } can also be taken to label the twistings, P r g r = n .The particular case of the U(1) logdet has been discussed in [10], section 4,from a mainly numerical point of view and, since the details can be found in thisreference, I need give only the final, computable answer,log det ( q, r ) = Z ∞ dx Re (cid:18) cosh τ / τ ddτ cosh( qτ δ )2 sinh τ sinh qτ / (cid:19) , (2)where τ = x + iy , y lying between 0 and the first singularity of the integrand abovethe real axis. The parameter δ ≡ r/q − /
2. The logdet is effectively plotted inFig.3 of [10] as a function of r/q , for q = 5 , , Quadrature quickly gives numer-ical agreement with Radi˘cevi´c, [1] Table II, obtained by rather more complicatedmanipulations.Just to prove the arithmetic the numbers for q = 9 are [-3.034402,-1.620781,-0.024723,1.261412, 1.965100,1.965100,1.261412,-0.024723,-1.620781]It is a simple matter for any given twisting, i.e. r (or g ), to compute thecombination (1). Continuing r/q into the reals, as one can formally, in (2), the curves show a curious crossingalmost on zero logdet. See the next section. Obtained in 3 secs on a slowish machine. . Infinite q The graphs of logdet, for different q , against the ‘flux’, r/q , show an approx-imate crossing at roughly r/q ≈ . q increases, the crossingbecomes more exact and independent of α . This behaviour can be elucidated byconsidering the infinite q limit. For this I choose the same limit as in [7], that is q → ∞ such that r/q remains finite, tending to f , and that the (reinstated) radiusof S , a , also becomes infinite with a/q tending to, say, 1. The lens space thendegenerates into the product R × S with the circle being of unit radius and havinga threaded flux of f . I proceed informally by stating that the ζ –function on such amanifold has the structure, [11], Z ( s ) = | R | π s − Z (cid:12)(cid:12)(cid:12)(cid:12) f (cid:12)(cid:12)(cid:12)(cid:12) (2 s − , (3)in terms of the one–dimensional Epstein ζ –function related to the Hurwitz ζ –function by Z (cid:12)(cid:12)(cid:12)(cid:12) f (cid:12)(cid:12)(cid:12)(cid:12) (2 s −
2) = ζ H (2 s − , f ) + ζ H (2 s − , − f ) , but I do not need this as Z | | has the inversion formula, which, in general terms,relates the eigenfunction and image forms, [11], Z (cid:12)(cid:12)(cid:12)(cid:12) f (cid:12)(cid:12)(cid:12)(cid:12) (2 s ) = π s − / Γ(1 / − s )Γ( s ) Z (cid:12)(cid:12)(cid:12)(cid:12) − f (cid:12)(cid:12)(cid:12)(cid:12) (1 − s ) . Then (3) reads Z ( s ) = | R | π π s − / Γ(3 / − s )Γ( s ) Z (cid:12)(cid:12)(cid:12)(cid:12) − f (cid:12)(cid:12)(cid:12)(cid:12) (3 − s ) , (4)and so Z ′ (0) = − | R | π Z (cid:12)(cid:12)(cid:12)(cid:12) − f (cid:12)(cid:12)(cid:12)(cid:12) (3)= − | R | π Cl (2 πf ) , (5)in terms of the cosine Clausen function. This typically occurs in investigations ofsymmetry breaking on manifolds R m × S , the references for which are too numerousto list and so I give only Davies and McLachlan, [12], as a later one. Here canbe found plots of a few low Clausen functions, but these are easily generated bymachine. 3ooking at (5) as a function of the flux, f , one sees that the logdet is eitherpositively or negatively infinite depending on which side of the zeros of the Clausenfunction f sits. Figure 1 repeats the graphs in [10] with, for comparison, one of theClausen function, Cl (2 πf ). This has been (arbitrarily) scaled for display purposes.Infinitely scaled, according to (5), it represents the logdet at q = ∞ . The curvehas deformed into two vertical lines through the roots of Cl , which are at the fluxvalues f = f ≈ . f = 1 − f .Suitably scaled, the Clausen expression provides a reasonable approximationfor the logdet for q ≥ d /Z q , where, this time, the Clausen function, Cl d (2 πf ), appears. As d tends toinfinity, the root, f , tends to 1 / d , when f is a root of a Bernoulli poly-nomial.
5. Comments
The reason for the relatively simple expression, (2), is that the degeneraciesof the lens space twisted modes (which are quite involved, Unwin, [13], [1]) enteronly via their generating functions, which are explicit and compact. Denoting thedegeneracies by D l ( q, r ) I repeat the expression here ( l is the mode label), ∞ X l =1 D l ( q, r ) e − lτ = − ddτ cosh( qτ δ )2 sinh τ sinh qτ / . (6)4he fact that a closed expression for the logdet can be found, [1], means thatthe integral (2) can be done.A special case is when the repetitions, g r , are independent of r , say g = g c ≡ ( g, g, . . . , g ), for then a roots of unity argument shows that S ( q, g c ) = g q − X r =0 S q ( r ) = g S (0) (7)where S is the whole sphere quantity, for which there exists a simple Riemann ζ –function closed form, [14].This can be checked, at a later stage, by performing the r sum on (6)Such an integer vector g c is possible only if q divides n because q g = n and, al-though I do not intend to discuss the gauge sector partition function, it is interestingto note that, [1], the gauge preferred vacuum in this case is g c . References.
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