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I'm reading these notes on infinite-dimensional Lie algebras. On page 5, author defines Heisenberg algebra and shows that certain infinite sums of elements of Heisenberg algebra (I'm being a little imprecise here, but it's all in the notes) obey Virasoro algebra relations.

I'm ignorant of physics involved here, so this construction looks very strange to me. Can someone explain to me why would anyone consider these sums in the first place? I've skimmed through the Kac-Raina book (because it's referenced in notes), but I've found no answer.

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This construction comes out of string theory. You can find it described in your favorite string theory textbook. Green, Schwarz and Witten and Polchinski are standard references.

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