# Energy bounds for vertex operator algebra extensions

March 24, 2023

Let V be a simple unitary vertex operator algebra and U be a (polynomially)
energy-bounded unitary subalgebra containing the conformal vector of V. We give
two sufficient conditions implying that V is energy-bounded. The first
condition is that U is a compact orbifold for some compact group G of unitary
automorphisms of V. The second condition is that V is exponentially
energy-bounded and it is a finite direct sum of simple U-modules. As
consequence of the second condition, we prove that if U is a regular
energy-bounded unitary subalgebra of a simple unitary vertex operator V, then
$V$ is energy-bounded. In particular, every simple unitary extension (with the
same conformal vector) of a simple unitary affine vertex operator algebra
associated with a semisimple Lie algebra is energy-bounded.

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