The Bethe/Gauge Correspondence

2.1. Algebraic Bethe Ansatz 21on our Hilbert space H. Here, the c j are some (unimportant) coefficients, and λ ∈ C is knownas the spectral parameter. The operators τ(λ) are called transfer matrices and by constructiongenerate the conserved charges:I j ∝ djdλ j log τ(λ) ∣∣∣λ=λ0. (2.1)For naive quantum integrability, we want the I j to be conserved and in involution. This iscertainly the case if we can express the Hamiltonian H in terms of τ and if the equation[τ(λ), τ(µ)] = 0 (2.2)holds for all λ, µ ∈ C.To be able to model different physical systems, we need to build in some more room inthe formalism. In addition to the physical Hilbert space H = ⊗ H l we introduce an auxiliaryspace V a and require that the transfer matrix is equal to a traceτ(λ) = tr a T (λ)over V a of some new operator T (λ) that acts on the product V a ⊗ H l . T (λ) is called themonodromy matrix. For example, if we take V a = C 2 , we can write T (λ) as a block matrix inauxiliary space,( )A(λ) B(λ)T (λ) =, (2.3)C(λ) D(λ)whose matrix elements A(λ), · · ·, D(λ) are operators on H. In this caseτ(λ) = tr a T (λ) = A(λ) + D(λ) (2.4)Fundamental commutation relations. Let’s rewrite equation (2.2) in terms of the monodromymatrix T (λ). Consider two distinct auxiliary spaces V a and V b . WriteT a (λ) := T (λ) ⊗ 1 b and T b (λ) := 1 a ⊗ T (λ) .These are operators on V a ⊗ V b ⊗ H, and the subscript denotes the auxiliary space on which theoperator acts nontrivially.The trace tr ab over a tensor product V a ⊗ V b is equal to the product of the traces over theindividual spaces V a and V b , sothat is,tr ab [T a (λ), T b (µ)] = [tr a T a (λ), tr b T b (µ)] = [τ(λ), τ(µ)] = 0 ,tr ab T a (λ) T b (µ) = tr ab T b (µ) T a (λ) . (2.5)This means that there exists a similarity transformation R ab (λ, µ) on V a ⊗ V b such thatR ab (λ, µ) T a (λ) T b (µ) R −1ab (λ, µ) = T b(µ) T a (λ) .Indeed, due to the cyclic property of the trace, this clearly satisfies (2.5). R ab (λ, µ) is ratherunimaginatively called the R-matrix.If we multiply the last equation by R ab (λ, µ) from the right we arrive at the fundamentalcommutation relation for the monodromy matrix T (λ):R ab (λ, µ) T a (λ) T b (µ) = T b (µ) T a (λ) R ab (λ, µ) . (2.6)Since (2.6) is equivalent to (2.2), it ensures that the charges (2.1) are in involution. We canrephrase (2.6) in more mathematical terms by saying that the R-matrix is an intertwining