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Block Spin Transformations of 2D Sigma Model, Toward Solving a Millennium Problem K.R.Ito Abstract: We analyze 2D O(N) sigma model by block spin transformation: [ ] ∑ 1 g 0 exp − ⟨ϕ, G−1 (ϕ2 (x) − N β)2 0 ϕ⟩ − 2 2N x 2 2 where ϕ(x) = (ϕ1 (x), · · · , ϕN (x)), G−1 0 = −∆ + m0 and −∆ is the lattice laplacian on Z . Put ϕ0 = ϕ and recursively deﬁne block spins ϕn+1 (x) = 1 ∑ ϕn (Lx + ζ) L2 |ζ|<L/2 We represnt ϕn in terms of block-spin ϕn+1 of next order and ﬂuctuations zn . zn is a massive Gaussian but gets strong eﬀects of back ground ﬁeld ϕn+1 . We deﬁne a new notion of domain walls in the sigma model which has O(N) symmetry. The domain wall regions Dw have high energies, and on (Dw )c we can safely implement the block spin transformations. Thus we obtain the renormalization group ﬂow of the 2D sigma model, which enables us to prove our long-standing claim. 1