Abstract
We present an approach to evaluate the full operatorial Q-system of all mathfrak{u}left(p,qBig|r+sright) -invariant spin chains with representations of Jordan-Schwinger type. In particular, this includes the super spin chain of planar mathcal{N}=4 super Yang-Mills theory at one loop in the presence of a diagonal twist. Our method is based on the oscillator construction of Q-operators. The Q-operators are built as traces over Lax operators which are degenerate solutions of the Yang-Baxter equation. For non-compact representations these Lax operators may contain multiple infinite sums that conceal the form of the resulting functions. We determine these infinite sums and calculate the matrix elements of the lowest level Q-operators. Transforming the Lax operators corresponding to the Q-operators into a representation involving only finite sums allows us to take the supertrace and to obtain the explicit form of the Q-operators in terms of finite matrices for a given magnon sector. Imposing the functional relations, we then bootstrap the other Q-operators from those of the lowest level. We exemplify this approach for non-compact spin −s spin chains and apply it to mathcal{N}=4 at the one-loop level using the BMN vacuum as an example.
Highlights
Non-compact spin chains are quantum integrable models that appear in certain limits of four-dimensional quantum field theories [1,2,3,4]
We outlined the derivation of the Lax operators on which this construction is based, and defined the Q-operators with their functional relations
For non-compact spin chains with infinitedimensional state spaces, these Lax operators are given in terms of infinite sums which hides the analytic properties of the resulting Q-system and complicates their evaluation
Summary
Non-compact spin chains are quantum integrable models that appear in certain limits of four-dimensional quantum field theories [1,2,3,4]. In contrast to their compact counterparts the physical or quantum space of non-compact spin chains is infinite-dimensional. Due to the Yang-Baxter equation the transfer matrices built in this way commute for different representations in the auxiliary space. They satisfy certain functional equations which arise from fusion in the auxiliary space, see e.g. They satisfy certain functional equations which arise from fusion in the auxiliary space, see e.g. [8] for an overview
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