gaugeTransform -- computes the gauge transform of a system of connection matrices

• Usage:

  gaugeTransform(M, A, D)

  gaugeTransform(M, A)

• Inputs:
  • M, a matrix, change of basis matrix
  • A, a list, a system of connection matrices
  • D, a ring, the Weyl algebra (if omitted, it will be inferred from first element of A)
• Outputs:
  • L, a list, the system of connection matrices $\widetilde{A}_i = M A_i M^{-1} + (\partial_i \bullet M)M^{-1}$ after the gauge transformation with respect to $M$.

i1 : D = makeWeylAlgebra(QQ[x,y]);

i2 : I = ideal(x*dx^2-y*dy^2+2*dx-2*dy, x*dx+y*dy+1);

o2 : Ideal of D

i3 : A = connectionMatrices I;

i4 : M = matrix {{x,0}, {0,y}};

             2      2
o4 : Matrix D  <-- D

i5 : gaugeTransform(M, A, D)

o5 = {| 0 -1 |, | 0 x/y  |}
      | 0 0  |  | 0 -1/y |

o5 : List