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Register a new userConformal field theory from lattice fermions
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Added by
Alexander Stottmeister
Publication date
2021
fields
Physics
and research's language is
English
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We provide a rigorous lattice approximation of conformal field theories given in terms of lattice fermions in 1+1-dimensions, focussing on free fermion models and Wess-Zumino-Witten models. To this end, we utilize a recently introduced operator-algebraic framework for Wilson-Kadanoff renormalization. In this setting, we prove the convergence of the approximation of the Virasoro generators by the Koo-Saleur formula. From this, we deduce the convergence of lattice approximations of conformal correlation functions to their continuum limit. In addition, we show how these results lead to explicit error estimates pertaining to the quantum simulation of conformal field theories.
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In generic conformal field theories with $W_3$ symmetry, we identify a primary field $sigma$ with rational Kac indices, which produces the full $mathbb{Z}_3$ charged and neutral sectors by the fusion processes $sigma times sigma$ and $sigma times sigma^*$, respectively. In this sense, this field generalises the $mathbb{Z}_3$ fundamental spin field of the three-state Potts model. Among the degenerate fields produced by these fusions, we single out a `parafermion field $psi$ and an `energy field $varepsilon$. In analogy with the Virasoro case, the exact curves for conformal dimensions $(h_sigma,h_psi)$ and $(h_sigma,h_varepsilon)$ are expected to give close estimates for the unitarity bounds in the conformal bootstrap analysis.
Linking numbers appear in local quantum field theory in the presence of tensor fields, which are closed two-forms on Minkowski space. Given any pair of such fields, it is shown that the commutator of the corresponding intrinsic (gauge invariant) vector potentials, integrated about spacelike separated, spatial loops, are elements of the center of the algebra of all local fields. Moreover, these commutators are proportional to the linking numbers of the underlying loops. If the commutators are different from zero, the underlying two-forms are not exact (there do not exist local vector potentials for them). The theory then necessarily contains massless particles. A prominent example of this kind, due to J.E. Roberts, is given by the free electromagnetic field and its Hodge dual. Further examples with more complex mass spectrum are presented in this article.
In this paper we discuss two constructions of an effective field theory starting from a local interaction functional. One relies on the well-established graphical combinatorics of the BPHZ algorithm to renormalize divergent Feynman amplitudes. The other, more recent and due to Costello, relies on an inductive construction of local counterterms that uses no graphical combinatorics whatsoever. We show that these two constructions produce the same effective field theory.
We review recent progress in operator algebraic approach to conformal quantum field theory. Our emphasis is on use of representation theory in classification theory. This is based on a series of joint works with R. Longo.
The idea of summing over all intermediate states that is central for implementing locality in quantum systems can be realized by coend constructions. In the concrete case of systems of conformal blocks for a certain class of conformal vertex algebras, one deals with coends in functor categories. Working with these coends involves quite a few subtleties which, even though they have in principle already been understood twenty years ago, have not been sufficiently appreciated by the conformal field theory community.
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