From Vertex Operator Algebras to Conformal Nets and Back

Abstract

We consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. We present a general procedure which associates to every strongly local vertex operator algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a conformal net <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A Subscript upper V"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mi>V</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {A}_V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acting on the Hilbert space completion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and prove that the isomorphism class of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A Subscript upper V"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mi>V</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {A}_V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> does not depend on the choice of the scalar product on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that the class of strongly local vertex operator algebras is closed under taking tensor products and unitary subalgebras and that, for every strongly local vertex operator algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W right-arrow from bar script upper A Subscript upper W"> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo stretchy="false"> ↦ </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mi>W</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">W\mapsto \mathcal {A}_W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> gives a one-to-one correspondence between the unitary subalgebras <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the covariant subnets of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A Subscript upper V"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mi>V</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {A}_V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Many known examples of vertex operator algebras such as the unitary Virasoro vertex operator algebras, the unitary affine Lie algebras vertex operator algebras, the known <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c equals 1"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">c=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> unitary vertex operator algebras, the moonshine vertex operator algebra, together with their coset and orbifold subalgebras, turn out to be strongly local. We give various applications of our results. In particular we show that the even shorter Moonshine vertex operator algebra is strongly local and that the automorphism group of the corresponding conformal net is the Baby Monster group. We prove that a construction of Fredenhagen and Jörß gives back the strongly local vertex operator algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from the conformal net <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A Subscript upper V"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mi>V</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {A}_V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and give conditions on a conformal net <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" math