  A familiar anomaly affects SU(2) gauge theory in four dimensions: a theory
with an odd number of fermion multiplets in the spin 1/2 representation of the
gauge group, and more generally in representations of spin 2r+1/2, is
inconsistent. We describe here a more subtle anomaly that can affect SU(2)
gauge theory in four dimensions under the condition that fermions transform
with half-integer spin under SU(2) and bosons with integer spin. Such a theory,
formulated in a way that requires no choice of spin structure, and with an odd
number of fermion multiplets in representations of spin 4r+3/2, is
inconsistent. The theory is consistent if one picks a spin or spin_c structure.
Under Higgsing to U(1), the new SU(2) anomaly reduces to a known anomaly of
"all-fermion electrodynamics." Like that theory, an SU(2) theory with an odd
number of fermion multiplets in representations of spin 4r+3/2 can provide a
boundary state for a five-dimensional gapped theory whose partition function on
a closed five-manifold Y is $(-1)^{\int_Y w_2w_3}$. All statements have analogs
with SU(2) replaced by Sp(2N). There is also an analog in five dimensions.
