Abstract
We study monopole operators at the conformal critical point of the $$ \mathbb{C}{\mathrm{\mathbb{P}}}^{N_b-1} $$ theory in 2+1 spacetime dimensions. Using the state-operator correspondence and a saddle point approximation, we compute the scaling dimensions of the operators that insert one or two units of magnetic flux to next-to-leading order in 1/N b . We compare our results to numerical studies of quantum antiferromagnets on two-dimensional lattices with SU(N b ) global symmetry, using the mapping of the monopole operators to valence bond solid order parameters of the lattice antiferromagnet. For the monopole operators that insert three or more units of magnetic flux, we find that the rotationally-symmetric saddle point is unstable; in order to obtain the scaling dimensions of these operators, even at leading order in 1/N b , one should consider non-spherically-symmetric saddles.
Highlights
In 2 + 1 dimensions, pure U(1) gauge theory confines [1]
We study monopole operators at the conformal critical point of the CPNb−1 theory in 2+1 spacetime dimensions
We compare our results to numerical studies of quantum antiferromagnets on two-dimensional lattices with SU(Nb) global symmetry, using the mapping of the monopole operators to valence bond solid order parameters of the lattice antiferromagnet
Summary
In 2 + 1 dimensions, pure U(1) gauge theory confines [1]. One can prevent confinement by introducing a sufficiently large number N of massless matter fields, in which case the infrared dynamics is believed to be governed by a non-trivial interacting conformal field theory (CFT). In the case at hand, the S2 ground state energy in the presence of 4πq magnetic flux is computed at leading order in Nb, where the Lagrange multiplier field λ and the gauge field don’t fluctuate and assume a saddle point configuration that minimizes this energy. To obtain the correct leading order results at large Nb, one would have to first find the dominant (non-spherically symmetric) saddle point configuration for the gauge field and the Lagrange multiplier field λ. We leave this task to future work. Our main task is to determine the 1/Nb expansion of this ground state energy, which we write as
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