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Register a new userThird and Higher Order NFPA Twisted Constructions of Conformal Field Theories from Lattices
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P.S. Montague
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We investigate orbifold constructions of conformal field theories from lattices by no-fixed-point automorphisms (NFPAs) $Z_p$ for $p$ prime, $p>2$, concentrating on the case $p=3$. Explicit expressions are given for most of the relevant vertex operators, and we consider the locality relations necessary for these to define a consistent conformal field theory. A relation to constructions of lattices from codes, analogous to that found in earlier work in the $p=2$ case which led to a generalisation of the triality structure of the Monster module, is also demonstrated.
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An account is given of the structure and representations of chiral bosonic meromorphic conformal field theories (CFTs), and, in particular, the conditions under which such a CFT may be extended by a representation to form a new theory. This general approach is illustrated by considering the untwisted and $Z_2$-twisted theories, $H(Lambda)$ and $tilde H(Lambda)$ respectively, which may be constructed from a suitable even Euclidean lattice $Lambda$. Similarly, one may construct lattices $Lambda_C$ and $tildeLambda_C$ by analogous constructions from a doubly-even binary code $C$. In the case when $C$ is self-dual, the corresponding lattices are also. Similarly, $H(Lambda)$ and $tilde H(Lambda)$ are self-dual if and only if $Lambda$ is. We show that $H(Lambda_C)$ has a natural ``triality structure, which induces an isomorphism $H(tildeLambda_C)equivtilde H(Lambda_C)$ and also a triality structure on $tilde H(tildeLambda_C)$. For $C$ the Golay code, $tildeLambda_C$ is the Leech lattice, and the triality on $tilde H(tildeLambda_C)$ is the symmetry which extends the natural action of (an extension of) Conways group on this theory to the Monster, so setting triality and Frenkel, Lepowsky and Meurmans construction of the natural Monster module in a more general context. The results also serve to shed some light on the classification of self-dual CFTs. We find that of the 48 theories $H(Lambda)$ and $tilde H(Lambda)$ with central charge 24 that there are 39 distinct ones, and further that all 9 coincidences are accounted for by the isomorphism detailed above, induced by the existence of a doubly-even self-dual binary code.
Holographic entanglement entropy and the first law of thermodynamics are believed to decode the gravity theory in the bulk. In particular, assuming the Ryu-Takayanagi (RT)cite{ryu-takayanagi} formula holds for ball-shaped regions on the boundary around CFT vacuum states impliescite{Nonlinear-Faulkner} a bulk gravity theory equivalent to Einstein gravity through second-order perturbations. In this paper, we show that the same assumptions can also give rise to second-order Lovelock gravity. Specifically, we generalize the procedure in cite{Nonlinear-Faulkner} to show that the arguments there also hold for Lovelock gravity by proving through second-order perturbation theory, the entropy calculated using the Wald formulacite{Wald_noether} in Lovelock also obeys an area law (at least up to second order). Since the equations for second-order perturbations of Lovelock gravity are different in general from the second-order perturbation of the Einstein-Hilbert action, our work shows that the holographic area law cannot determine a unique bulk theory even for second-order perturbations assuming only RT on ball-shaped regions. It is anticipated that RT on all subregions is expected to encode the full non-linear Einstein equations on asymptotically AdS spacetimes.
We study an example of higher-order field-theoretic model with an eighth-degree polynomial potential -- the $varphi^8$ model. We show that for some certain ratios of constants of the potential, the problem of finding kink-type solutions in $(1+1)$-dimensional space-time reduces to solving algebraic equations. For two different ratios of the constants, which determine positions of the vacua, we obtained explicit formulas for kinks in all topological sectors. The properties of the obtained kinks are also studied -- their masses are calculated, and the excitation spectra which could be responsible for the appearance of resonance phenomena in kink-antikink scattering are found.
We explore a conformal field theoretic interpretation of the holographic entanglement of purification, which is defined as the minimal area of entanglement wedge cross section. We argue that in AdS3/CFT2, the holographic entanglement of purification agrees with the entanglement entropy for a purified state, obtained from a special Weyl transformation, called path-integral optimizations. By definition, this special purified state has the minimal path-integral complexity. We confirm this claim in several examples.
It is widely expected that at sufficiently high temperatures order is always lost, e.g. magnets loose their ferromagnetic properties. We pose the question of whether this is always the case in the context of quantum field theory in $d$ space dimensions. More concretely, one can ask whether there exist critical points (CFTs) which break some global symmetry at arbitrary finite temperature. The most familiar CFTs do not exhibit symmetry breaking at finite temperature, and moreover, in the context of the AdS/CFT correspondence, critical points at finite temperature are described by an uncharged black brane which obeys a no-hair theorem. Yet, we show that there exist CFTs which have some of their internal symmetries broken at arbitrary finite temperature. Our main example is a vector model which we study both in the epsilon expansion and arbitrary rank as well as the large rank limit (and arbitrary dimension). The large rank limit of the vector model displays a conformal manifold, a moduli space of vacua, and a deformed moduli space of vacua at finite temperature. The appropriate Nambu-Goldstone bosons including the dilaton-like particle are identified. Using these tools we establish symmetry breaking at finite temperature for finite small $epsilon$. We also prove that a large class of other fixed points, which describe some of the most common quantum magnets, indeed behave as expected and do not break any global symmetry at finite temperature. We discuss some of the consequences of finite temperature symmetry breaking for the spectrum of local operators. Finally, we propose a class of fixed points which appear to be possible candidates for finite temperature symmetry breaking in $d=2$.
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