Polynomial codimension growth and the Specht problem

We construct a continuous family of algebras over a field of characteristic zero with slow codimension growth bounded by a polynomial of degree 4. This is achieved by building, for any real number α∈(0,1)α∈(0,1) a commutative nonassociative algebra AαAα whose codimension sequence cn(Aα)cn(Aα), n=1,2,…n=1,2,… , is polynomially bounded and lim⁡logn⁡cn(Aα)=3+αlim⁡logn⁡cn(Aα)=3+α.As an application we are able to construct a new example of a variety with an infinite basis of identities.