Modular Invariant Partition Functions Research Articles

On modular invariant partition functions for tensor products of conformal field theories

We give two results concerning the construction of modular invariant partition functions for conformal field theories constructed by tensoring together other conformal field theories. First we show how the possible modular invariants for the tensor product theory are constrained if the allowed modular invariants of the individual conformal field theory factors have been classified. We illustrate the use of these constraints for theories of the type SU(2) K A ⊗SU(2) K B , finding all consistent theories for K A , K B odd. Second we show how known diagonal modular invariants can be used to construct some inherently asymmetric ones where the holomorphic and antiholomorphic theories do not share the same chiral algebra. Some explicit examples are given.

Operator content of the fusion cyclic solid-on-solid model

The fusion model of the L-state cyclic solid-on-solid (CSOS) model is considered at criticality. Each ZL-charge sector of the fusion CSOS model is related to a sum of total spin sectors of the fusion vertex model with a seam on which the vertex weights are modified by a phase factor. The latter in turn becomes the higher-spin XXZ quantum chain with a twisted boundary condition in the extremely anisotropic limit. Using the known operator content of the higher-spin XXZ chain, the authors deduce that of the fusion CSOS model. The modular invariant partition function of the fusion CSOS at fusion level k is a sum of products of the Z(k) parafermionic and free boson sectors with the effective coupling g'=L2(k-lambda/pi) for L odd and g"=L2(1/k-lambda/pi)/4 for L even, where lambda is the crossing parameter. Where L is odd and a multiple of k, or when L/2 is a multiple of k for L even, the modular invariant partition function becomes a simple product of the Z(k) parafermionic and the Gaussian partition function.

RG flows of non-unitary minimal CFTs

In this paper we study the renormalization group flow of the ( p, q) minimal (non-unitary) CFT perturbed by the Ф 1, 3 operator with a positive coupling. In the perturbative region q a ̊ (q-p) , we find a new IR fixed point which corresponds to the (2 p- q, p) minimal CFT. The perturbing field near the new IR fixed point is identified with the irrelevant Ф 3, 1 operator. We extend this result to show that the non-diagonal ((A, D)-type) modular invariant partition function of the ( p, q) minimal CFT flows into the (A, D)-type partition function of the (2 p- q, p) minimal CFT and the diagonal partition function into the diagonal.

Construction of modular branching functions from Bethe's equations in the 3-state Potts chain

We use the single particle excitation energies and the completeness rules of the 3-state anti-ferromagnetic Potts chain, which have been obtained from Bethe's equation, to compute the modular invariant partition function. This provides a fermionic construction for the branching functions of the $D_4$ representation of $Z_4$ parafermions which complements the previous bosonic constructions. It is found that there are oscillations in some of the correlations and a new connection with the field theory of the Lee-Yang edge is presented.

D-E classfiication of the local extensions ofsu 2 current algebras

A method is developed for constructing single valued rational 4-point functions of primary fields for su2 conformal current algebra satisfying the Knizhnik-Zamolodchikov equation. For integer conformal dimensions Δ these rational solutions are proven to be in one-to-one correspondence with non-diagonal modular invariant partition functions of the D-even and E-even series of the ADE classification.

MODULAR INVARIANTS OF KAČ-MOODY ALGEBRAS FROM SELF-DUAL LATTICES

A general method for the construction of modular invariant partition functions of Kač-Moody algebras using a covariant lattice approach is presented. The A-D-E series for [Formula: see text] conformal invariant theories and partition functions of [Formula: see text] are constructed explicitly to demonstrate the technique.

NEW EXCEPTIONAL $(A_1^{(1)})^{\oplus r} $ INVARIANTS AND THE ASSOCIATED GEPNER MODELS

New modular invariant partition functions for tensor products of [Formula: see text] affine Lie algebras are presented. These exceptional modular invariants can be understood in terms of automorphisms of the fusion rules of the affine algebra or of its extensions. There are three isolated cases, as well as an infinite series of new invariants. As an application, the new modular invariants are employed to produce new Gepner type compactifications of the heterotic string.

Complete classification of simple current modular invariants for RCFT's with a center (Z p ) k

Simple currents have been used previously to construct various examples of modular invariant partition functions for given rational conformal field theories. In this paper we present for a large class of such theories (namely those with a center that decomposes into factors Zp,p prime) thecomplete set of modular invariants that can be obtained with simple currents. In addition to the fusion rule automorphisms classified previously forany center, this includes all possible left-right combinations of all possible extensions of the chiral algebra that can be obtained with simple currents, for all possible current-current monodromies. Formulas for the number of invariants of each kind are derived. Although the number of invariants in each of these subsets depends on the current-current monodromies, the total number of invariants depends rather surprisingly only onp and the number ofZp factors.

Spectral flow between conformal field theories in 1 + 1 dimensions

Consider the φ 1,3-perturbed minimal unitary conformal field theories M p (with diagonal modular invariant partition function) of central charge c p = 1 − ( 6 p(p + 1) ), p = 3, 4,… It is believed that for one sign of the perturbing parameter they are massive scattering theories of kinks, whereas for the other they are massless but not scale-invariant quantum field theories interpolating between M p in the UV and M p−1 in the IR limit. We propose integral equations for the exact finite-volume energies of the first and kth excited states in φ 1,3- perturbed M 2k + 2 on the cylinder (of circumference R). These integral equations are similar to the thermodynamic Bethe Ansatz equations recently proposed by Al. Zamolodchikov for the ground-state energy in these theories. The behaviour of the conjectured expressions for the finite-volume energies at small and large R, which we investigate analytically and numerically, is in excellent agreement with predictions of conformal perturbation theory around the UV and IR fixed point, respectively, providing strong support for our proposal. In particular, we study in some detail the flow of the spin field ( dimension d = 3 40 ) of the tricritical Ising model M 4 to the spin field (d = 1 8 ) of the critical Ising model M 3 . For φ 1,3- perturbed M 4 we also compare the small- R behaviour of the conjectured first energy-gap with results from the “truncated conformal-space approach”, and discuss the limitations of the latter method when conformal perturbation theory has UV divergencies.

Primary operators of coset conformal field theories

The nature of primary operators of coset conformal field theories is discussed. Some primary states of typical coset conformal field theory are explicitly calculated that have correct conformal dimensions and dynamics. The nature of modular invariant partition functions from the diagonal sum of absolute squares of characters of coset primary states is studied.

Coulomb-gas representation of minimal models on Riemann surfaces.

In this paper we use the Coulomb-gas approach to construct the minimal-model partition function and correlation functions on Riemann surfaces. The background-charge term in the action of the nonchiral scalar field is carefully studied and the various modular-invariant partition functions are shown to be reproducible from a treatment of this term. These results can be understood as a generalization of those of the microscopic lattice-model approach.

Fractional superstrings with space-time critical dimensions four and six.

We propose possible new string theories based on local world-sheet symmetries corresponding to extensions of the Virasoro algebra by fractional spin currents. They have critical central charges $c=6(K+8)/(K+2)$ and Minkowski space-time dimensions $D=2+16/K$ for $K\geq2$ an integer. We present evidence for their existence by constructing modular invariant partition functions and the massless particle spectra. The dimension $4$ and $6$ strings have space-time supersymmetry.

SU(2)k MODULAR-INVARIANT PARTITION FUNCTIONS FROM ORBIFOLDS

We present a method which generates the modular-invariant partition functions of the ADE series of SU(2)k. Dividing the diagonal theory by discrete subgroups of the conformal group, we construct all the modular-invariant partition functions, thus proving that orbifold construction generates all the partition functions of SU(2)k.

TOWERS OF ALGEBRAS IN RATIONAL CONFORMAL FIELD THEORIES

Jones fundamental construction is applied to rational conformal field theories. The Jones algebra which emerges in this application is realized in terms of duality operations. The generators of the algebra are an open version of Verlinde’s operators. The polynomial equations appear in this context as sufficient conditions for the existence of Jones algebra. The ADE classification of modular invariant partition functions is put in correspondence with Jones classification of subfactors.

New exceptional modular invariant partition functions for simple Kac-Moody algebras

Using numerical and analytical methods, we obtain many new exceptional modular invariant partition functions due to automorphisms of extensions of Kac-Moody algebras. We then conjecture the existence of such an invariant for all Kac-Moody algebras with fixed points under either a simple current of order 2 and spin 4, or currents of order 3 and spins (3, 3), or currents of order 5 and spins (2, 3, 3, 2). In particular, the first condition is satisfied for two infinite series: B ̂ n and D ̂ n at level k = 8, and three finite series involving C ̂ n , D ̂ n and SU(N) algebras. Exceptional invariants of other types are also presented: they occur for G 2, k = 4; F 4, k = 3; SU(9), k = 3; etc.

Modular invariants for affine $$\widehat{SU}(3)$$ theories at prime heights

A proof is given for the existence of two and only two modular invariant partition functions in affine\(\widehat{SU}(3)_k \) theories at heightsn=k+3 which are prime numbers. Arithmetic properties of the ring of algebraic integers ℤ(ω) which is related toSU(3) weights are extensively used.

The equivalence of NSR and GS four-dimensional twisted strings

Using a general construction of the modular invariant partition function for four-dimensional strings with twisted, orbifold-like, boundary conditions, we discuss the equivalence of their Neveu-Schwarz-Ramond and Green-Schwarz formulations. Four-dimensional GS strings are found to be equivalent to the class of NSR theories on orbifolds based on the canonical, free supercurrent, with underlying N = 2 linear world-sheet supersymmetry. This equivalence provides interesting insights on the consistency conditions for all order modular invariance to be applied on four-dimensional Green-Schwarz strings.

The branching rules of conformal embeddings

After recalling the main properties of a conformal embedding of Lie algebrasg⊃p, which is defined by the equality of the Sugawara central charges on both sides, we launch a systematic study of their branching rules. The bulk of the paper is devoted to the proof of a general formula in the casesu(mn) 1⊃su(m) n⊕su(n) m. At the end we give some applications to the construction of modular invariant partition functions.

Modular transformations of c⩾1 Virasoro characters

We study the action of the modular group on the unitary Virasoro characters when c⩾1. In the case of a discrete distribution of primary fields, the characters cannot transform linearly unless pathological convergence properties are allowed. In the continuous case, a unique linear transformation exists, but the vacuum character falls outside the transformed spectrum except at c=1; in this case we explicitly construct a continuous modular-invariant partition function, identified with that of a free uncompactified boson. We discuss possible physical implications of our findings.

SIMPLE CURRENTS, MODULAR INVARIANTS AND FIXED POINTS

We review the use of simple currents in constructing modular invariant partition functions and the problem of resolving their fixed points. We present some new results, in particular regarding fixed point resolution. Additional empirical evidence is provided in support of our conjecture that fixed points are always related to some conformal field theory. We complete the identification of the fixed point conformal field theories for all simply laced and most non-simply laced Kac-Moody algebras, for which the fixed point CFT’s turn out to be Kac-Moody algebras themselves. For the remaining non-simply laced ones we obtain spectra that appear to correspond to new non-unitary conformal field theories. The fusion rules of the simplest unidentified example are computed.