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Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model

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arxiv 2111.01106 v3 pith:YYZFCO2H submitted 2021-11-01 hep-th math-phmath.MP

classification hep-thmath-phmath.MP
keywords symmetrytheorycasesconformalfour-pointfunctionssolutionsblocks
verification ladder T0 review T1 audit T2 compute T3 formal
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abstract

We define the two-dimensional $O(n)$ conformal field theory as a theory that includes the critical dilute and dense $O(n)$ models as special cases, and depends analytically on the central charge. For generic values of $n\in\mathbb{C}$, we write a conjecture for the decomposition of the spectrum into irreducible representations of $O(n)$. We then explain how to numerically bootstrap arbitrary four-point functions of primary fields in the presence of the global $O(n)$ symmetry. We determine the needed conformal blocks, including logarithmic blocks, including in singular cases. We argue that $O(n)$ representation theory provides upper bounds on the number of solutions of crossing symmetry for any given four-point function. We study some of the simplest correlation functions in detail, and determine a few fusion rules. We count the solutions of crossing symmetry for the $30$ simplest four-point functions. The number of solutions varies from $2$ to $6$, and saturates the bound from $O(n)$ representation theory in $21$ out of $30$ cases.

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Cited by 5 Pith papers

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