Full Logarithmic Conformal Field theory — an Attempt at a Status Report

Fuchs, Jürgen; Schweigert, Christoph

doi:10.1002/prop.201910018

Cited by 6 publications

(6 citation statements)

References 37 publications

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“…They also do arise in conformal field theories, termed logarithmic (in place of rational). See, for example, Fuchs-Schweigert [FS19]. But now by Remark 7.2.3 the components of a weight zero vector-valued modular form with bounded denominators can certainly be transcendental over C(λ).…”

Section: If One Drops the Semisimplicity Stipulation On ρmentioning

confidence: 99%

The Unbounded Denominators Conjecture

Calegari¹,

Dimitrov²,

Tang³

2021

Preprint

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We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of SL 2 (Z). Our result includes also Mason's generalization of the original conjecture to the setting of vector-valued modular forms, thereby supplying a new path to the congruence property in rational conformal field theory. The proof involves a new arithmetic holonomicity bound of a potential-theoretic flavor, together with Nevanlinna's second main theorem, the congruence subgroup property of SL 2 (Z[1/p]), and a close description of the Fuchsian uniformization D(0, 1)/Γ N of the Riemann surface C µ N .forms for SL 2 (Z), in particular resolving -in a sharper form, in fact -Mason's unbounded denominators conjecture [Mas12, KM08] on generalized modular forms.1.1. A sketch of the main ideas. Our proof of Theorem 1.0.1 follows a broad Diophantine analysis path known in the literature (see [Bos04,Bos13] or [Bos20, Chapter 10]) as the arithmetic algebraization method.1.1.1. The Diophantine principle. The most basic antecedent of these ideas is the following easy lemma:x which defines a holomorphic function on D(0, R) for some R > 1 is a polynomial.Lemma 1.1.1 follows upon combining the following two obsevations, fixing some 1 > η > R −1 :(1) The coefficients a n are either 0 or else ≥ 1 in magnitude.(2) The Cauchy integral formula gives a uniform upper bound |a n | = o(η n ). We shall refer to the first inequality as a Liouville lower bound, following its use by Liouville in his proof of the lower bound |α − p/q| ≫ 1/q n for algebraic numbers α = p/q of degree n ≥ 1. We shall refer to the second inequality as a Cauchy upper bound, following the example above where it comes from an application of the Cauchy integral formula. The first non-trivial generalization of Lemma 1.1.1 was Émile Borel's theorem [Bor94]. Dwork famously used a p-adic generalization of Borel's theorem in his p-adic analytic proof of the rationality of the zeta function of an algebraic variety over a finite field (see Dwork's account in the book [DGS94, Chapter 2]). The simplest non-trivial statement of Borel's theorem is that an integral formal power series f (x) ∈ Z x must already be a rational function as soon as it has a meromorphic representation as a quotient of two convergent complex-coefficients power series on some disc D(0, R) of a radius R > 1. The subject of arithmetic algebraization blossomed at the hands of many authors, including most prominently Carlson,

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Section: If One Drops the Semisimplicity Stipulation On ρmentioning

confidence: 99%

The Unbounded Denominators Conjecture

Calegari¹,

Dimitrov²,

Tang³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In Section 3, we first state and prove our main result and then explain why this result generalizes our previous one from [LMSS1]. In fact, our present result was already mentioned in [LMSS1], and it was also described in [FS2]. Here, we are now supplying proofs for our claims.…”

Section: Introductionmentioning

confidence: 54%

Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups

Lentner¹,

Mierach²,

Schweigert³

et al. 2020

Preprint

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Given a finite modular tensor category, we associate with each compact surface with boundary a cochain complex in such a way that the mapping class group of the surface acts projectively on its cohomology groups. In degree zero, this action coincides with the known projective action of the mapping class group on the space of chiral conformal blocks. In the case that the surface is a torus and the category is the representation category of a factorizable ribbon Hopf algebra, we recover our previous result on the projective action of the modular group on the Hochschild cohomology groups of the Hopf algebra.

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“…and setting 97), though not in an entirely obvious way. Some care must be taken in interpreting the isomorphism (2.135).…”

Section: Hochschild Cohomology Centers and Drinfeld-reshetikhin Mapmentioning

confidence: 97%

“…By now, one means by a logarithmic conformal field theory a theory that has representations that are reducible but indecomposable, and one calls a module logarithmic if the Virasoro zero-mode does not act semisimply. An introduction to the topic is [96] and a status report on the understanding of conformal blocks and the modular functor in the logarithmic setting is [97]. The symmetry algebra of a conformal field theory is a vertex operator algebra and so one calls the VOA of a logarithmic theory a logarithmic VOA.…”

Section: Logarithmic Voa'smentioning

confidence: 99%

A QFT for non-semisimple TQFT

Creutzig¹,

Dimofte²,

Garner³

et al. 2021

Preprint

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We construct a family of 3d quantum field theories T A n,k that conjecturally provide a physical realization -and derived generalization -of non-semisimple mathematical TQFT's based on the modules for the quantum group U q (sl n ) at an even root of unity q = exp(iπ/k). The theories T A n,k are defined as topological twists of certain 3d N = 4 Chern-Simons-matter theories, which also admit string/M-theory realizations. They may be thought of as SU (n) k−n Chern-Simons theories, coupled to a twisted N = 4 matter sector (the source of non-semisimplicity). We show that T A n,k admits holomorphic boundary conditions supporting two different logarithmic vertex operator algebras, one of which is an sl n -type Feigin-Tipunin algebra; and we conjecture that these two vertex operator algebras are related by a novel logarithmic level-rank duality. (We perform detailed computations to support the conjecture.) We thus relate the category of line operators in T A n,k to the derived category of modules for a boundary Feigin-Tipunin algebra, and -using a logarithmic Kazhdan-Lusztiglike correspondence that has been established for n = 2 and expected for general n -to the derived category of U q (sl n ) modules. We analyze many other key features of T A n,k and match them from quantum-group and VOA perspectives, including deformations by flat P GL(n, C) connections, one-form symmetries, and indices of (derived) genus-g state spaces.

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Full Logarithmic Conformal Field theory — an Attempt at a Status Report

Cited by 6 publications

References 37 publications

The Unbounded Denominators Conjecture

The Unbounded Denominators Conjecture

Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups

A QFT for non-semisimple TQFT

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