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Cosmological correlator for the 2-site chain

Differential equations for correlation functions in cosmology are studied in [ABHJLP]. Therein, a basis of master integrals is constructed from the underlying hyperplane arrangement via canonical forms in the setup of positive geometries, which results in a matrix differential equation for the cosmological correlator that is in $\varepsilon$-factorized form. For the $2$-site chain (mathematically, the path graph on $2$ vertices), the underlying $D$-ideal was investigated in [FPSW].

Here we revisit the $D$-ideal $I = \langle \nabla_1+\nabla_3,\nabla_2+\nabla_3,H\rangle$ (see Equation (11) in [FPSW]), and carry out the gauge transformation to write the connection matrices in $\varepsilon$-factorized form. This form is especially useful, as it allows for the construction of formal power series solutions in the variable $\varepsilon$ of such systems via the ``path-ordered exponential formalism.’’

i1 : w = {1,1,1};
i2 : D = makeWeylAlgebra(frac(QQ[ϵ])[x,y,z], w);
i3 : delta1 = (x^2-z^2)*dx^2+2*(1-ϵ)*x*dx-ϵ*(1-ϵ);
i4 : delta2 = (y^2-z^2)*dy^2+2*(1-ϵ)*y*dy-ϵ*(1-ϵ);
i5 : delta3 = (x+z)*(y+z)*dx*dy-ϵ*(x+z)*dx-ϵ*(y+z)*dy+ϵ^2;
i6 : h = x*dx+y*dy+z*dz-2*ϵ;
i7 : I = ideal(delta1+delta3, delta2+delta3, h)

             2  2    2  2                                        2          
o7 = ideal (x dx  - z dx  + x*y*dx*dy + x*z*dx*dy + y*z*dx*dy + z dx*dy + (-
     ------------------------------------------------------------------------
                                                2                           
     3ϵ + 2)x*dx - ϵ*z*dx - ϵ*y*dy - ϵ*z*dy + 2ϵ  - ϵ, x*y*dx*dy + x*z*dx*dy
     ------------------------------------------------------------------------
                    2         2  2    2  2                                   
     + y*z*dx*dy + z dx*dy + y dy  - z dy  - ϵ*x*dx - ϵ*z*dx + (- 3ϵ + 2)y*dy
     ------------------------------------------------------------------------
                  2
     - ϵ*z*dy + 2ϵ  - ϵ, x*dx + y*dy + z*dz - 2ϵ)

o7 : Ideal of D

First, we check that the system has finite holonomic rank using holonomicRank.

i8 : assert(holonomicRank I == 4)

Then, we compute the system in connection form and verify that it meets the integrability conditions.

i9 : elapsedTime A = connectionMatrices I;
 -- 2.86622s elapsed
i10 : elapsedTime assert isIntegrable A
 -- 4.75529s elapsed
i11 : netList A

      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
o11 = || 2ϵ/x                                                                                                                                                                                                                                                                                                                                                                                                                                    (-y)/x                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (-z)/x                                                                                                                                                                                                                                                                                                                                                                                                                                                0                                                                                                                             |                                                                                                                                                                                                                                                                                                                                     |
      || (4x2y2ϵ^2+4xy2zϵ^2-2x2z2ϵ^2-2y2z2ϵ^2-4xz3ϵ^2+x3zϵ-3xy2zϵ+2xz3ϵ)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                                                                                                                                                                                                                                                                               (2x3y2ϵ-2x2y3ϵ+2x3yzϵ-2xy3zϵ-x3z2ϵ+x2yz2ϵ-xy2z2ϵ+y3z2ϵ-2x3yz+2xy3z)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                                                                                                                                                                                                                                                                                                          (-2x2y2zϵ-x3z2ϵ-3xy2z2ϵ+x2z3ϵ+y2z3ϵ+4xz4ϵ+2xy2z2-2xz4)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                                                                                                                                                                                                                                                                                                      (-xyz+xz2+yz2-z3)/(2x2y+2xy2-x2z-2xyz-y2z)                                                                                    |                                                                                                                                                                                                                                                                                                                                     |
      || (-2xyz2ϵ^2-2y2z2ϵ^2-4yz3ϵ^2+2x2y2ϵ+x2yzϵ+xy2zϵ+2y2z2ϵ+2yz3ϵ)/(2x3y2z+x3yz2+x2y2z2-x3z3-xy2z3-x2z4-xyz4)                                                                                                                                                                                                                                                                                                                                 (x2yz2ϵ+2xy2z2ϵ+y3z2ϵ+2xyz3ϵ+2y2z3ϵ-2x2y3-x2y2z-xy3z-x2yz2-y3z2-xyz3-y2z3)/(2x3y2z+x3yz2+x2y2z2-x3z3-xy2z3-x2z4-xyz4)                                                                                                                                                                                                                                                                                                                                                  (2x2y2ϵ+x2yzϵ+xy2zϵ-2x2z2ϵ+xyz2ϵ+y2z2ϵ-2xz3ϵ+2yz3ϵ-2x2y2-x2yz-xy2z+x2z2-y2z2+xz3-yz3)/(2x3y2+x3yz+x2y2z-x3z2-xy2z2-x2z3-xyz3)                                                                                                                                                                                                                                                                                                                         (-yz+z2)/(2xy-xz-yz)                                                                                                          |                                                                                                                                                                                                                                                                                                                                     |
      || (-2x3yzϵ^3+2xy3zϵ^3-2x2yz2ϵ^3+8xy2z2ϵ^3+2y3z2ϵ^3+8xyz3ϵ^3+8y2z3ϵ^3+8yz4ϵ^3-6x3y2ϵ^2-6x2y3ϵ^2+x4zϵ^2-4x3yzϵ^2-21x2y2zϵ^2-4xy3zϵ^2+x3z2ϵ^2-8x2yz2ϵ^2-15xy2z2ϵ^2-6y3z2ϵ^2+4x2z3ϵ^2-8xyz3ϵ^2-16y2z3ϵ^2+4xz4ϵ^2-12yz4ϵ^2+2x4yϵ+10x2y3ϵ+x4zϵ+2x3yzϵ+21x2y2zϵ+2xy3zϵ+x3z2ϵ+8x2yz2ϵ+5xy2z2ϵ+4y3z2ϵ-2x2z3ϵ+8y2z3ϵ-2xz4ϵ+4yz4ϵ)/(2x5y3+3x5y2z+x4y3z+x4y2z2-3x3y3z2-x5z3-x4yz3-5x3y2z3-x2y3z3-x4z4-x3yz4-x2y2z4+xy3z4+x3z5+x2yz5+2xy2z5+x2z6+xyz6) (x4yzϵ^2+x3y2zϵ^2-x2y3zϵ^2-xy4zϵ^2+x3yz2ϵ^2-3x2y2z2ϵ^2-5xy3z2ϵ^2-y4z2ϵ^2-4x2yz3ϵ^2-8xy2z3ϵ^2-4y3z3ϵ^2-4xyz4ϵ^2-4y2z4ϵ^2+2x4y2ϵ+6x3y3ϵ+4x2y4ϵ-x4yzϵ+9x3y2zϵ+13x2y3zϵ+3xy4zϵ+x3yz2ϵ+9x2y2z2ϵ+11xy3z2ϵ+3y4z2ϵ+4x2yz3ϵ+12xy2z3ϵ+8y3z3ϵ+6xyz4ϵ+6y2z4ϵ-4x4y2-8x2y4-2x4yz-4x3y2z-16x2y3z-2xy4z-2x3yz2-8x2y2z2-4xy3z2-2y4z2-2x2yz3-2xy2z3-4y3z3-2xyz4-2y2z4)/(2x5y3+3x5y2z+x4y3z+x4y2z2-3x3y3z2-x5z3-x4yz3-5x3y2z3-x2y3z3-x4z4-x3yz4-x2y2z4+xy3z4+x3z5+x2yz5+2xy2z5+x2z6+xyz6) (2x3y2zϵ^2-2x2y3zϵ^2-x4z2ϵ^2+3x3yz2ϵ^2-3x2y2z2ϵ^2-3xy3z2ϵ^2-x3z3ϵ^2+3x2yz3ϵ^2-9xy2z3ϵ^2-y3z3ϵ^2+4x2z4ϵ^2-4xyz4ϵ^2-4y2z4ϵ^2+4xz5ϵ^2-4yz5ϵ^2-2x4yzϵ+2x3y2zϵ+10x2y3zϵ-x4z2ϵ-x3yz2ϵ+23x2y2z2ϵ+5xy3z2ϵ-x3z3ϵ+3x2yz3ϵ+13xy2z3ϵ+3y3z3ϵ-10x2z4ϵ+8y2z4ϵ-10xz5ϵ+6yz5ϵ-8x2y3z-16x2y2z2-2xy3z2-4x2yz3-4xy2z3-2y3z3+4x2z4+2xyz4-4y2z4+4xz5-2yz5)/(2x5y3+3x5y2z+x4y3z+x4y2z2-3x3y3z2-x5z3-x4yz3-5x3y2z3-x2y3z3-x4z4-x3yz4-x2y2z4+xy3z4+x3z5+x2yz5+2xy2z5+x2z6+xyz6) (2x2y2ϵ-2xy2zϵ-2xyz2ϵ+2y2z2ϵ+2yz3ϵ-2z4ϵ-2x2y2+x2yz+x2z2-2y2z2-yz3+3z4)/(2x3y2+x3yz-x2y2z-x3z2-x2yz2-2xy2z2-xyz3+y2z3+xz4+yz4) |                                                                                                                                                                                                                                                                                                                                     |
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      || 0                                                                                                                                                                                                                                                                                                                                                                              1                                                                                                                                                                                                                                                                                                                                                                                                                                     0                                                                                                                                                                                                                                                                                                                                                             0                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
      || (-4xyϵ^2-4z2ϵ^2+2x2ϵ+2xyϵ+2yzϵ+2z2ϵ)/(2x2y2+2xy3+x2yz-y3z-x2z2-2xyz2-y2z2)                                                                                                                                                                                                                                                                                                     (2x2yϵ+6xy2ϵ+2xyzϵ-2y2zϵ-4x2y-4xy2)/(2x2y2+2xy3+x2yz-y3z-x2z2-2xyz2-y2z2)                                                                                                                                                                                                                                                                                                                                                             (-2x2zϵ+2yz2ϵ+4z3ϵ-2yz2-2z3)/(2x2y2+2xy3+x2yz-y3z-x2z2-2xyz2-y2z2)                                                                                                                                                                                                                                                                                            (x2z-z3)/(2x2y+2xy2-x2z-2xyz-y2z)                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
      || (2xz2ϵ^2+2yz2ϵ^2+4z3ϵ^2-2x2yϵ-x2zϵ-xyzϵ-2yz2ϵ-2z3ϵ)/(2x2y2z+x2yz2+xy2z2-x2z3-y2z3-xz4-yz4)                                                                                                                                                                                                                                                                                     (-x2z2ϵ-2xyz2ϵ-y2z2ϵ-2xz3ϵ-2yz3ϵ+2x2y2+x2yz+xy2z+x2z2+y2z2+xz3+yz3)/(2x2y2z+x2yz2+xy2z2-x2z3-y2z3-xz4-yz4)                                                                                                                                                                                                                                                                                                                            (2x2yϵ+x2zϵ+xyzϵ-xz2ϵ-3yz2ϵ-4z3ϵ+2yz2+2z3)/(2x2y2+x2yz+xy2z-x2z2-y2z2-xz3-yz3)                                                                                                                                                                                                                                                                                (-xz+z2)/(2xy-xz-yz)                                                                                                          |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
      || (-2x2yzϵ^3+2y3zϵ^3-2x2z2ϵ^3-8xyz2ϵ^3+2y2z2ϵ^3-8xz3ϵ^3-8yz3ϵ^3-8z4ϵ^3+2x3yϵ^2+6x2y2ϵ^2+4xy3ϵ^2+2x3zϵ^2+5x2yzϵ^2+14xy2zϵ^2-y3zϵ^2+7x2z2ϵ^2+10xyz2ϵ^2+3y2z2ϵ^2+16xz3ϵ^2+12yz3ϵ^2+16z4ϵ^2-6x3yϵ-6xy3ϵ-2x3zϵ-11x2yzϵ-6xy2zϵ-3y3zϵ-3x2z2ϵ-8xyz2ϵ-3y2z2ϵ-4xz3ϵ-6yz3ϵ-6z4ϵ)/(2x3y4+x3y3z+3x2y4z-3x3y2z2+x2y3z2-x3yz3-5x2y2z3-xy3z3-y4z3+x3z4-x2yz4-xy2z4-y3z4+2x2z5+xyz5+y2z5+xz6+yz6) (x3yzϵ^2+x2y2zϵ^2-xy3zϵ^2-y4zϵ^2+x3z2ϵ^2+5x2yz2ϵ^2+3xy2z2ϵ^2-y3z2ϵ^2+4x2z3ϵ^2+8xyz3ϵ^2+4y2z3ϵ^2+4xz4ϵ^2+4yz4ϵ^2-4x3y2ϵ-6x2y3ϵ-2xy4ϵ-3x3yzϵ-13x2y2zϵ-9xy3zϵ+y4zϵ-3x3z2ϵ-11x2yz2ϵ-9xy2z2ϵ-y3z2ϵ-8x2z3ϵ-12xyz3ϵ-4y2z3ϵ-6xz4ϵ-6yz4ϵ+8x3y2+4xy4+2x3yz+16x2y2z+4xy3z+2y4z+2x3z2+4x2yz2+8xy2z2+2y3z2+4x2z3+2xyz3+2y2z3+2xz4+2yz4)/(2x3y4+x3y3z+3x2y4z-3x3y2z2+x2y3z2-x3yz3-5x2y2z3-xy3z3-y4z3+x3z4-x2yz4-xy2z4-y3z4+2x2z5+xyz5+y2z5+xz6+yz6) (-2x3yzϵ^2+2x2y2zϵ^2-2x3z2ϵ^2-2x2yz2ϵ^2+2xy2z2ϵ^2-2y3z2ϵ^2-4x2z3ϵ^2+6xyz3ϵ^2-2y2z3ϵ^2+4xz4ϵ^2+8yz4ϵ^2+8z5ϵ^2+6x3yzϵ-4x2y2zϵ-4xy3zϵ+2x3z2ϵ+10x2yz2ϵ-10xy2z2ϵ+2x2z3ϵ-6xyz3ϵ-2y2z3ϵ-12xz4ϵ-14yz4ϵ-16z5ϵ+4xy3z+4xy2z2+2y3z2+4xyz3+2y2z3+4xz4+6yz4+6z5)/(2x3y4+x3y3z+3x2y4z-3x3y2z2+x2y3z2-x3yz3-5x2y2z3-xy3z3-y4z3+x3z4-x2yz4-xy2z4-y3z4+2x2z5+xyz5+y2z5+xz6+yz6) (2x2y2ϵ-2x2yzϵ+2x2z2ϵ-2xyz2ϵ+2xz3ϵ-2z4ϵ-2x2y2+xy2z-2x2z2+y2z2-xz3+3z4)/(2x2y3-x2y2z+xy3z-2x2yz2-xy2z2-y3z2+x2z3-xyz3+xz4+yz4) |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      || 0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              0                                                                                                                                                                                                    ||
      || (2xz2ϵ^2+2yz2ϵ^2+4z3ϵ^2-2x2yϵ-x2zϵ-xyzϵ-2yz2ϵ-2z3ϵ)/(2x2y2z+x2yz2+xy2z2-x2z3-y2z3-xz4-yz4)                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (-x2z2ϵ-2xyz2ϵ-y2z2ϵ-2xz3ϵ-2yz3ϵ+2x2y2+x2yz+xy2z+x2z2+y2z2+xz3+yz3)/(2x2y2z+x2yz2+xy2z2-x2z3-y2z3-xz4-yz4)                                                                                                                                                                                                                                                                                                                                                                            (2x2yϵ+x2zϵ+xyzϵ-xz2ϵ-3yz2ϵ-4z3ϵ+2yz2+2z3)/(2x2y2+x2yz+xy2z-x2z2-y2z2-xz3-yz3)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (-xz+z2)/(2xy-xz-yz)                                                                                                                                                                                 ||
      || 0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              1                                                                                                                                                                                                    ||
      || (4x3y2zϵ^3-4xy4zϵ^3+8x2y2z2ϵ^3-8xy3z2ϵ^3+4x2yz3ϵ^3-4y3z3ϵ^3+8xyz4ϵ^3-8y2z4ϵ^3-2x4y2ϵ^2+2x2y4ϵ^2-3x4yzϵ^2-5x3y2zϵ^2+7x2y3zϵ^2+9xy4zϵ^2+x4z2ϵ^2-10x3yz2ϵ^2-18x2y2z2ϵ^2+22xy3z2ϵ^2+5y4z2ϵ^2+x3z3ϵ^2-21x2yz3ϵ^2-9xy2z3ϵ^2+13y3z3ϵ^2+4x2z4ϵ^2-12xyz4ϵ^2+8y2z4ϵ^2+4xz5ϵ^2+4yz5ϵ^2+4x4y2ϵ-4x2y4ϵ+3x4yzϵ+3x3y2zϵ-5x2y3zϵ-5xy4zϵ+x4z2ϵ+2x3yz2ϵ+10x2y2z2ϵ-6xy3z2ϵ-7y4z2ϵ+x3z3ϵ+11x2yz3ϵ+3xy2z3ϵ-7y3z3ϵ-2x2z4ϵ+4xyz4ϵ-2y2z4ϵ-2xz5ϵ-2yz5ϵ)/(2x4y4z+x4y3z2+x3y4z2-3x4y2z3-3x2y4z3-x4yz4-2x3y2z4-2x2y3z4-xy4z4+x4z5+4x2y2z5+y4z5+x3z6+2x2yz6+2xy2z6+y3z6-x2z7-y2z7-xz8-yz8) (-2x4y2zϵ^2-2x3y3zϵ^2+2x2y4zϵ^2+2xy5zϵ^2-4x3y2z2ϵ^2+4xy4z2ϵ^2-2x3yz3ϵ^2-2x2y2z3ϵ^2+2xy3z3ϵ^2+2y4z3ϵ^2-4x2yz4ϵ^2+4y3z4ϵ^2+2x4y3ϵ-2x2y5ϵ+6x4y2zϵ+6x3y3zϵ-6x2y4zϵ-6xy5zϵ+2x4yz2ϵ+16x3y2z2ϵ-16xy4z2ϵ-2y5z2ϵ+6x3yz3ϵ+6x2y2z3ϵ-6xy3z3ϵ-6y4z3ϵ+2x2yz4ϵ-2y3z4ϵ-4x4y3+4x2y5-4x4y2z-4x3y3z+4x2y4z+4xy5z-4x4yz2-4x3y2z2+4xy4z2+4y5z2-4x3yz3-4x2y2z3+4xy3z3+4y4z3)/(2x4y4z+x4y3z2+x3y4z2-3x4y2z3-3x2y4z3-x4yz4-2x3y2z4-2x2y3z4-xy4z4+x4z5+4x2y2z5+y4z5+x3z6+2x2yz6+2xy2z6+y3z6-x2z7-y2z7-xz8-yz8) (2x4y2ϵ^2-4x3y3ϵ^2+2x2y4ϵ^2+3x4yzϵ^2-x3y2zϵ^2+x2y3zϵ^2+5xy4zϵ^2-x4z2ϵ^2+6x3yz2ϵ^2-14x2y2z2ϵ^2+10xy3z2ϵ^2-y4z2ϵ^2-x3z3ϵ^2-9x2yz3ϵ^2-7xy2z3ϵ^2+y3z3ϵ^2+4x2z4ϵ^2-12xyz4ϵ^2+8y2z4ϵ^2+4xz5ϵ^2+4yz5ϵ^2-4x4y2ϵ+2x3y3ϵ-6x2y4ϵ-3x4yzϵ-x3y2zϵ-7x2y3zϵ-9xy4zϵ-x4z2ϵ+36x2y2z2ϵ-16xy3z2ϵ-3y4z2ϵ-x3z3ϵ+27x2yz3ϵ+21xy2z3ϵ-7y3z3ϵ-10x2z4ϵ+14xyz4ϵ-12y2z4ϵ-10xz5ϵ-10yz5ϵ+4x2y4+4x2y3z+4xy4z-16x2y2z2+4xy3z2+4y4z2-12x2yz3-8xy2z3+4y3z3+4x2z4-4xyz4+4y2z4+4xz5+4yz5)/(2x4y4+x4y3z+x3y4z-3x4y2z2-3x2y4z2-x4yz3-2x3y2z3-2x2y3z3-xy4z3+x4z4+4x2y2z4+y4z4+x3z5+2x2yz5+2xy2z5+y3z5-x2z6-y2z6-xz7-yz7) (2x3y2ϵ+2x2y3ϵ-6x3yzϵ-6xy3zϵ+2x3z2ϵ+2y3z2ϵ+12xyz3ϵ-4xz4ϵ-4yz4ϵ-x3y2-x2y3+6x3yz+6xy3z-x3z2-3x2yz2-3xy2z2-y3z2-12xyz3+5xz4+5yz4)/(2x3y3-x3y2z-x2y3z-2x3yz2-2xy3z2+x3z3+x2yz3+xy2z3+y3z3+2xyz4-xz5-yz5) ||
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+

Next, we use gauge matrix for changing base to a base given by suitable set of standard monomials, and compute the gauge transform with respect to this gauge matrix.

i12 : F = baseFractionField D;
i13 : B = {1_D,dx,dy,dx*dy};
i14 : elapsedTime g = gaugeMatrix(I, B);
 -- .625402s elapsed

              4      4
o14 : Matrix F  <-- F
i15 : elapsedTime A1 = gaugeTransform(g, A);
 -- 1.33443s elapsed
i16 : elapsedTime assert isIntegrable A1
 -- .92329s elapsed
i17 : netList A1

      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
o17 = || 0                            1                      0                                         0                                                      |                                                                         |
      || (-2ϵ^2+ϵ)/(x2-z2)            (3xϵ+zϵ-2x)/(x2-z2)    (yϵ+zϵ)/(x2-z2)                           (-y-z)/(x-z)                                           |                                                                         |
      || 0                            0                      0                                         1                                                      |                                                                         |
      || (-2ϵ^3+ϵ^2)/(x3+x2y-xz2-yz2) (2ϵ^2-ϵ)/(x2+xy-xz-yz) (-xϵ^2+yϵ^2+2zϵ^2+xϵ-zϵ)/(x3+x2y-xz2-yz2) (2x2ϵ-2xzϵ-2yzϵ-2z2ϵ-2x2-xy+xz+yz+z2)/(x3+x2y-xz2-yz2) |                                                                         |
      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      || 0                            0                                        1                      0                                                      |                                                                          |
      || 0                            0                                        0                      1                                                      |                                                                          |
      || (-2ϵ^2+ϵ)/(y2-z2)            (xϵ+zϵ)/(y2-z2)                          (3yϵ+zϵ-2y)/(y2-z2)    (-x-z)/(y-z)                                           |                                                                          |
      || (-2ϵ^3+ϵ^2)/(xy2+y3-xz2-yz2) (xϵ^2-yϵ^2+2zϵ^2+yϵ-zϵ)/(xy2+y3-xz2-yz2) (2ϵ^2-ϵ)/(xy+y2-xz-yz) (2y2ϵ-2xzϵ-2yzϵ-2z2ϵ-xy-2y2+xz+yz+z2)/(xy2+y3-xz2-yz2) |                                                                          |
      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      || 2ϵ/z                                             (-x)/z                                           (-y)/z                                           0                                                                          ||
      || (2xϵ^2-xϵ)/(x2z-z3)                              (-x2ϵ-xzϵ-2z2ϵ+x2+z2)/(x2z-z3)                   (-xyϵ-xzϵ)/(x2z-z3)                              (x+y)/(x-z)                                                                ||
      || (2yϵ^2-yϵ)/(y2z-z3)                              (-xyϵ-yzϵ)/(y2z-z3)                              (-y2ϵ-yzϵ-2z2ϵ+y2+z2)/(y2z-z3)                   (x+y)/(y-z)                                                                ||
      || (2xyϵ^3-2z2ϵ^3-xyϵ^2+z2ϵ^2)/(x2y2z-x2z3-y2z3+z5) (-xyϵ^2-yzϵ^2+2z2ϵ^2+yzϵ-z2ϵ)/(xy2z-y2z2-xz3+z4) (-xyϵ^2-xzϵ^2+2z2ϵ^2+xzϵ-z2ϵ)/(x2yz-x2z2-yz3+z4) (2x2yzϵ+2xy2zϵ-2xz3ϵ-2yz3ϵ+x2y2-x2yz-xy2z+xz3+yz3-z4)/(x2y2z-x2z3-y2z3+z5) ||
      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+

Now we are ready to perform the gauge transform from this basis to the $\epsilon$-factorized form.

i18 : changeEps = transpose((1/(2*z*ϵ^2)) * matrix {
              {2*z*ϵ^2, -ϵ^2*(x-z), -ϵ^2*(y-z), -ϵ^2*(x+y)},
              {0, ϵ*(x^2-z^2), 0, ϵ*(x+y)*(x+z)},
              {0, 0, ϵ*(y^2-z^2), ϵ*(x+y)*(y+z)},
              {0, 0, 0, -(x+y)*(x+z)*(y+z)}});

              4      4
o18 : Matrix F  <-- F
i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
 -- .381495s elapsed
i20 : elapsedTime assert isIntegrable A2
 -- .707192s elapsed
i21 : netList A2

      +---------------------------------------------------------------------------------------------+
o21 = || ϵ/(x+z) 2zϵ/(x2-z2) 0       0                      |                                       |
      || 0       ϵ/(x-z)     0       ϵ/(x+y)                |                                       |
      || 0       0           ϵ/(x+z) (-yϵ+zϵ)/(x2+xy+xz+yz) |                                       |
      || 0       0           0       2ϵ/(x+y)               |                                       |
      +---------------------------------------------------------------------------------------------+
      || ϵ/(y+z) 0       2zϵ/(y2-z2) 0                      |                                       |
      || 0       ϵ/(y+z) 0           (-xϵ+zϵ)/(xy+y2+xz+yz) |                                       |
      || 0       0       ϵ/(y-z)     ϵ/(x+y)                |                                       |
      || 0       0       0           2ϵ/(x+y)               |                                       |
      +---------------------------------------------------------------------------------------------+
      || (xϵ+yϵ+2zϵ)/(xy+xz+yz+z2) (-2xϵ)/(x2-z2)            (-2yϵ)/(y2-z2)             0          ||
      || 0                         (xϵ-yϵ-2zϵ)/(xy+xz-yz-z2) 0                          (-ϵ)/(y+z) ||
      || 0                         0                         (-xϵ+yϵ-2zϵ)/(xy-xz+yz-z2) (-ϵ)/(x+z) ||
      || 0                         0                         0                          0          ||
      +---------------------------------------------------------------------------------------------+

Finally, we verify that only the last system is in the $\epsilon$-factorized form using isEpsilonFactorized.

i22 : assert isEpsilonFactorized(A2, ϵ)
i23 : assert not isEpsilonFactorized(A1, ϵ)
i24 : assert not isEpsilonFactorized(A, ϵ)

References

[FPSW] C. Fevola, G. L. Pimentel, A.-L. Sattelberger, and T. Westerdijk. Algebraic Approaches to Cosmological Integrals. Preprint arXiv:2410.14757. To appear in Le Matematiche.

[ABHJLP] N. Arkani-Hamed, D. Baumann, A. Hillman, A. Joyce, H. Lee, and G. L. Pimentel. Differential Equations for Cosmological Correlators. Preprint arXiv:2312.05303.

See also


The source of this document is in ConnectionMatrices/examples.m2:71:0.