How I understand the Higgs mechanism is that, although it is called “spontaneously breaking of the gauge symmetry”, the gauge symmetry is actually not broken in the higgs mechanism and it is the global symmetry that is broken. The gauge invariance makes the Goldstone boson disappear, but it is the interaction between the gauge boson and the higgs boson and the spontaneously breaking of the global symmetry which gives a non-zero vacuum expectation value of the higgs field that make the gauge boson acquire a mass.
We know that Mermin-Wagner theorem (Coleman theorem) says that there is no spontaneously breaking of global continuous symmetry in dimension 1+1D. So does the Mermin-Wagner theorem apply to the Higgs mechanism? Why or why not?