Identifying isomorphisms Sample Clauses

Identifying isomorphisms. ‌ The essential element in tachyon condensation (section 3.7) is the use of a member of the fermionic cohomology of a matrix factorisation. The justification comes from trian- gulated categories in category theory and is outlined in [18] but the actual construction involves a direct sum of brane anti brane and a tachyonic string. The cohomology is key. Cone construction was also used in the method of double cones [20, 27] to show isomor- phism between the known indecomposable matrix factorisations of any model and the corresponding cone constructions [27, 18]. This provided a sufficient but not a necessary criterion in that it could only confirm isomorphism not disprove it. In addition it would be complicated to implement. By considering the bosonic cohomology we were able to find a novel straightforward way to construct an algorithm which immediately tells us whether or not any two matrix factorisations are isomorphic or not. Q1Q2 Consider the bosonic cohomology under the action of the d0 operator as in equation 3.46. d Q1Q2 : M B → M F | d0 (B) = Q1B − BQ2 . (3.78) { } The bosonic cohomology contains all closed (in the Kernel) but not exact forms (in the Image) and are computed in SINGULAR as a set {bi}, of not necessarily linearly inde- pendent cohomology representatives Bi which span the cohomology. For any Bi we have Q1Bi − BiQ2 = 0 . (3.79) We assume the Qi in this case are indecomposable and that Q2 arises from a similarity transformation of Q1: Q2 = UQ1U —1, where U = S 0 0 T , S, T ∈ C[[x]] invertible . (3.80) Q1Q1 The identity I is certainly in the cohomology of d0 0 . Therefore, for an element = d Q1 Σ in the cohomology of d0 suppose we have some linear combination of the Bi say B = Q1Q2 ciBi which is invertible, that is to say has a determinant which is expressible as a power series ∈ C[[x]], and has a constant term. Then we have Q1B = BQ2 ,
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