Short answer: 1+1=1/2+1/2+1=1/2+1/2+1=1/2+1/2+1/2+1/2=2.
Long answer: the Weyl invariance is (almost) manifest. The proof is essentially a simple dimension counting. As the Weyl symmetry is about rescaling of the metric hαβ, we have to make explicit the metric dependence in the various terms of the action. As the theory contains spinors, we have to use a 2d "tetrad" formalism with a "zweibein" eaα satisfying hαβ=eaαebβηab, eαaebβ=ηba where α, β... are curved 2d space-time coordinates indices and a, b... are 2d Minkowski coordinates indices. This formalism is implicit in the action written in the question but it is better to replace the curved metric dependant gamma matrices ρα by Minkowski, metric independant, gamma matrices ρa. So we rewrite the action as:
S=k∫dσ√−h(hαβ∂αXμ∂βXμ+2iˉψμρaeαa∂αψμ
−iˉχaρbρaψμeβb∂βXμ−14ˉχaρbρaψμˉχbψμ)
This action is invariant under the Weyl symmetry
eaα↦λeaα, X↦X, ψ↦λ−1/2ψ, χ↦λ−1/2χ
where λ is an arbitrary function on the 2d spacetime.
Indeed:
As eaα↦λeaα, we have hαβ↦λ2hαβ and so det(h)↦λ4det(h) because we are in 2d and so √det(h)↦λ2√det(h). So we have to show that each term between the brackets in the expression giving S is multiplied by λ−2 under the Weyl symmetry.
First term: term in hαβ∂X∂X. As hαβ↦λ2hαβ, we have hαβ↦λ−2hαβ hence the result because X↦X.
Third term: term in χψeβb. We have χ↦λ−1/2χ, ψ↦λ−1/2ψ, eβb↦λ−1eβb hence the result because −1/2−1/2−1=−2.
Fourth term: term in χψχψ. We have χ↦λ−1/2χ, ψ↦λ−1/2ψ, hence the result because −1/2−1/2−1/2−1/2=−2.
Second term: term in ¯ψρaeαa∂ψ. It is the unique slightly subtle point because this term contains a derivative of a field charged under the Weyl symmetry. In fact, it is a general but not completely trivial fact that in any spacetime dimension, a fermion minimally coupled to gravity is (classically) Weyl invariant: the Dirac operator is conformally invariant. Let's see what happens explicitely in our 2d case. Under , ψ↦λ−1/2ψ, eαa↦λ−1eαa, the term ¯ψρaeαa∂ψ gives
λ−2¯ψρaeαa∂ψ+λ−3/2¯ψρaeαaψ∂(λ−1/2).
We have to show that the undesired extra term vanishes:
¯ψρaψ=0,
which is a special property of Majorana fermions in 2d (and which was already implicitely used in the expression of the action S to put an apparently non covariant derivative instead of the full Dirac operator). There are many ways to prove this relation. One possibility: ¯ψρaψ=ψTρ0ρaψ because ψ is real; if a=0, we obtain something proportional to ψTψ=ψ2++ψ2−=0 because the components ψ± of ψ are anticommuting variables; if a=1, ρ0ρ1 is the 2d "γ5 matrix" and so we obtain something proportional to ψ2+−ψ2−=0 because the components ψ± of ψ are anticommuting variables.