Representations of shifted quantum affine algebras

Abstract

We develop the representation theory of shifted quantum affine algebras U µ q (ĝ) and of their truncations which appeared in the study of quantized K-theoretic Coulomb branches of 3d N = 4 SUSY quiver gauge theories. Our direct approach is based on relations that we establish with the category O of representations of the quantum affine Borel subalgebra Uq( b) of the quantum affine algebra Uq(ĝ) and on associated quantum integrable models we have previously studied. We introduce the category O µ of representations of U µ q (ĝ) and we classify its simple objects. For g = sl2 we prove the existence of evaluation morphisms to q-oscillator algebras. We establish the existence of a fusion product for shifted quantum affine algebras and we get a ring structure on the sum of the Grothendieck groups K0(O µ ). We introduce induction and restriction functors to the category O of Uq( b). As a by product we classify simple finite-dimensional representations of U µ q (ĝ) and we obtain a cluster algebra structure on the Grothendieck ring of finitedimensional representations. We establish a necessary condition for a simple representation to descend to a truncation, which is also sufficient for g = sl2. We introduce a related partial ordering on simple modules and we prove a truncation has only a finite number of simple representations. We state a conjecture on the parametrization of simple modules of a non simply-laced truncation in terms of the Langlands dual Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations proved by using the Baxter polynomiality of quantum integrable models.

Engineering 3D $\mathcal{N}=2$ theories using the quantum affine $\mathfrak{sl}(2)$ algebra

“…with the notation q c (1) = q c ⊗ 1, q c (2) = 1 ⊗ q c , and ∆(q c ) = q c ⊗ q c . Shifted algebra The asymptotic algebra U a is closely related to another deformation of the algebra U, namely the shifted algebra U µ with parameters µ = (µ + , µ − ) ∈ Z × Z [29]. Shifted quantum affine algebras have been introduced by Finkelberg and Tsymbaliuk in their study of (Ktheoretic) Coulomb branches of 3D N = 4 supersymmetric gauge theories [46].…”

“…Moreover, when µ ± ≤ 0, the shifted algebra is a special case of the asymptotic algebra U a . For any µ, the Drinfeld coproduct 3.5 defines a homomorphism of algebras U µ+µ ′ → U µ ⊗ U µ ′ (after completion) [29].…”

“…15 It explains the relevance of the quantum affine sl(2) algebra with quantum group parameter q 2 = e Rε in the simpler case of the omega-background R 2 ε × S 1 R . It should be noted that the limit q 1 → ∞ with q 2 fixed is equivalent to q 1 → 0 with q 2 fixed thanks to the algebras morphism σ V defined in [15] and σ : X ± (z) → X ∓ (z), q c → q −c considered in [29], both inverting the parameters q, q 1 , q 2 . We start our discussion with the Higgsing method used in [27,28], and for which the string theory description has been briefly recalled in section 2.2.…”

“…It is also slightly more general as it is possible to introduce any number of fundamental/antifundamental chiral matter multiplets with arbitrary masses. On the other hand, we find it necessary to employ a shifted [29] (or asymptotic [30]) version of the quantum affine sl(2) algebra. These shifts naturally arise in the presence of fundamental/antifundamental matter multiplets, and such fields are required in order to obtain a non-trivial vortex partition function.…”

Folded quantum integrable models and deformed W-algebras

“…• The specialization Π q = Π t=1,α=0 : we obtain elements of Im(χ L q ), i.e. q-characters of modules over the Langlands dual quantum affine algebra U q ( L g), as defined in [H5,Section 12] (in the context of the parametrization of simple representations of shifted quantum affine algebras). So, in the total, we have 5 interesting specializations of the refined ring of interpolating (q, t)-characters:…”