SciPost Phys. 8, 018 (2020) ·
published 4 February 2020
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We discuss symmetry fractionalization of the Lorentz group in (2+1)d non-spin quantum field theory (QFT), and its implications for dualities. We prove that two inequivalent non-spin QFTs are dual as spin QFTs if and only if they are related by a Lorentz symmetry fractionalization with respect to an anomalous $\mathbb{Z}_2$ one-form symmetry. Moreover, if the framing anomalies of two non-spin QFTs differ by a multiple of 8, then they are dual as spin QFTs if and only if they are also dual as non-spin QFTs. Applications to summing over the spin structures, time-reversal symmetry, and level/rank dualities are explored. The Lorentz symmetry fractionalization naturally arises in Chern-Simons matter dualities that obey certain spin/charge relations, and is instrumental for the dualities to hold when viewed as non-spin theories.
SciPost Phys. 7, 065 (2019) ·
published 26 November 2019
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The existence of higher-spin quantum conserved currents in two dimensions
guarantees quantum integrability. We revisit the question of whether
classically-conserved local higher-spin currents in two-dimensional sigma
models survive quantization. We define an integrability index $\mathcal{I}(J)$
for each spin $J$, with the property that $\mathcal{I}(J)$ is a lower bound on
the number of quantum conserved currents of spin $J$. In particular, a positive
value for the index establishes the existence of quantum conserved currents.
For a general coset model, with or without extra discrete symmetries, we derive
an explicit formula for a generating function that encodes the indices for all
spins. We apply our techniques to the $\mathbb{CP}^{N-1}$ model, the $O(N)$
model, and the flag sigma model $\frac{U(N)}{U(1)^{N}}$. For the $O(N)$ model,
we establish the existence of a spin-6 quantum conserved current, in addition
to the well-known spin-4 current. The indices for the $\mathbb{CP}^{N-1}$ model
for $N>2$ are all non-positive, consistent with the fact that these models are
not integrable. The indices for the flag sigma model $\frac{U(N)}{U(1)^{N}}$
for $N>2$ are all negative. Thus, it is unlikely that the flag sigma models are
integrable.
SciPost Phys. 6, 017 (2019) ·
published 4 February 2019
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We study (1+1)-dimensional non-linear sigma models whose target space is the
flag manifold $U(N)\over U(N_1)\times U(N_2)\cdots U(N_m)$, with a specific
focus on the special case $U(N)/U(1)^{N}$. These generalize the well-known
$\mathbb{CP}^{N-1}$ model. The general flag model exhibits several new elements
that are not present in the special case of the $\mathbb{CP}^{N-1}$ model. It
depends on more parameters, its global symmetry can be larger, and its 't Hooft
anomalies can be more subtle. Our discussion based on symmetry and anomaly
suggests that for certain choices of the integers $N_I$ and for specific values
of the parameters the model is gapless in the IR and is described by an
$SU(N)_1$ WZW model. Some of the techniques we present can also be applied to
other cases.
