The Kernel of the Modular Representation and the Galois Action in RCFT

Bantay, P.

doi:10.1007/s00220-002-0760-x

“…We follow [13], 4 with most of the representation-theory part collected in Appendix B. Calculations with the Appell functions are given in Secs.…”

Section: Modular Transformations Of Sℓ(2|1) Charactersmentioning

confidence: 99%

“…Strictly speaking, this applies to rational conformal field theories, where modular properties of the characters and the structure of the Verlinde formula are known, at least in principle (for a discussion of the modular transformation properties of characters and other quantities and for further references, see [3,4,5,6,7]). …”

Section: Introductionmentioning

confidence: 99%

Higher-Level Appell Functions, Modular Transformations, and Characters

Semikhatov

¹

,

Taorimina

²

,

Tipunin

³

2005

Commun. Math. Phys.

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Bore Fe ginu po sluqa ego p tides tileti ABSTRACT. We study modular transformation properties of a class of indefinite theta series involved in characters of infinite-dimensional Lie superalgebras. The level-ℓ Appell functions K ℓ satisfy open quasiperiodicity relations with additive theta-function terms emerging in translating by the "period." Generalizing the well-known interpretation of theta functions as sections of line bundles, the K ℓ function enters the construction of a section of a rank-(ℓ + 1) bundle V ℓ,τ . We evaluate modular transformations of the K ℓ functions and construct the action of an SL(2, Z) subgroup that leaves the section of V ℓ,τ constructed from K ℓ invariant.Modular transformation properties of K ℓ are applied to the affine Lie superalgebra sℓ(2|1) at rational level k > −1 and to the N = 2 super-Virasoro algebra, to derive modular transformations of "admissible" characters, which are not periodic under the spectral flow and cannot therefore be rationally expressed through theta functions. This gives an example where constructing a modular group action involves extensions among representations in a nonrational conformal model.

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“…In fact, for a quantum field theory A in Minkowski space of d ≥ 2 + 1 dimensions and a certain category DHR(A) of representations [16] -admittedly too small to be physically realistic -the following have been shown [19]: (1) DHR(A) is symmetric monoidal, semisimple and rigid, (2) there exists a compact group G such that DHR(A) ≃ Rep G, (3) there exists a QFT F on which G acts by global symmetries and such that (4) F G ∼ = A, (5) the vacuum representation of F , restricted to A, contains all irreducible representations in DHR(A), (6) all intermediate theories A ⊂ B ⊂ F are of the form B = F H for some closed H ✁ G, and (7) DHR(F ) is trivial. All this should be understood as a Galois theory for quantum fields.…”

Section: Introductionmentioning

confidence: 99%

Conformal Orbifold Theories and Braided Crossed G-Categories

Mueger

¹

2005

Commun. Math. Phys.

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The aim of the paper is twofold. First, we show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category G − LocA of twisted representations. This category is a braided crossed G-category in the sense of Turaev [60]. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of G-manifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT ❀ modular category ❀ 3-manifold invariant.Secondly, we study the relation between G−LocA and the braided (in the usual sense) representation category Rep A G of the orbifold theory A G . We prove the equivalence Rep A G ≃ (G−LocA) G , which is a rigorous implementation of the insight that one needs to take the twisted representations of A into account in order to determine Rep A G . In the opposite direction we have G− LocA ≃ Rep A G ⋊ S, where S ⊂ Rep A G is the full subcategory of representations of A G contained in the vacuum representation of A, and ⋊ refers to the Galois extensions of braided tensor categories of [44,48].Under the assumptions that A is completely rational and G is finite we prove that A has g-twisted representations for every g ∈ G and that the sum over the squared dimensions of the simple g-twisted representations for fixed g equals dim Rep A. In the holomorphic case (where Rep A ≃ Vect C ) this allows to classify the possible categories G−LocA and to clarify the rôle of the twisted quantum doubles D ω (G) in this context, as will be done in a sequel. We conclude with some remarks on non-holomorphic orbifolds and surprising counterexamples concerning permutation orbifolds. * Supported by NWO.

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“…4.3 are in contradiction to what can be derived from certain statements in [2], which in turn follow from [1]. This claim was wrong, being based on an erroneous deduction from the statements in [1,2]. I regret this mistake.…”

mentioning

confidence: 90%

Conformal Orbifold Theories and Braided Crossed G-Categories

Müger

¹

2005

Commun. Math. Phys.

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The aim of the paper is twofold. First, we show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category G − LocA of twisted representations. This category is a braided crossed G-category in the sense of Turaev [60]. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of G-manifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT ❀ modular category ❀ 3-manifold invariant.Secondly, we study the relation between G−LocA and the braided (in the usual sense) representation category Rep A G of the orbifold theory A G . We prove the equivalence Rep A G ≃ (G−LocA) G , which is a rigorous implementation of the insight that one needs to take the twisted representations of A into account in order to determine Rep A G . In the opposite direction we have G− LocA ≃ Rep A G ⋊ S, where S ⊂ Rep A G is the full subcategory of representations of A G contained in the vacuum representation of A, and ⋊ refers to the Galois extensions of braided tensor categories of [44,48].Under the assumptions that A is completely rational and G is finite we prove that A has g-twisted representations for every g ∈ G and that the sum over the squared dimensions of the simple g-twisted representations for fixed g equals dim Rep A. In the holomorphic case (where Rep A ≃ Vect C ) this allows to classify the possible categories G−LocA and to clarify the rôle of the twisted quantum doubles D ω (G) in this context, as will be done in a sequel. We conclude with some remarks on non-holomorphic orbifolds and surprising counterexamples concerning permutation orbifolds.

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The Kernel of the Modular Representation and the Galois Action in RCFT

Cited by 80 publications

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Higher-Level Appell Functions, Modular Transformations, and Characters

Higher-Level Appell Functions, Modular Transformations, and Characters

Conformal Orbifold Theories and Braided Crossed G-Categories

Conformal Orbifold Theories and Braided Crossed G-Categories

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