Shifted Quantum Groups and Matter Multiplets in Supersymmetric Gauge Theories

Abstract

The notion of shifted quantum groups has recently played an important role in algebraic geometry. This subtle modification of the original definition brings more flexibility in the representation theory of quantum groups. The first part of this paper presents new mathematical results for the shifted quantum toroidal $\mathfrak {gl}(1)$ and quantum affine $\mathfrak {sl}(2)$ algebras (resp. denoted ${\ddot{U}}_{q_1,q_2}^{\varvec{\mu }}(\mathfrak {gl}(1))$ and ${\dot{U}}_q^{\varvec{\mu }}(\mathfrak {sl}(2))$). It defines several new representations, including finite dimensional highest $\ell$-weight representations for the toroidal algebra, and a vertex representation of ${\dot{U}}_q^{\varvec{\mu }}(\mathfrak {sl}(2))$ acting on Hall–Littlewood polynomials. It also explores the relations between representations of ${\dot{U}}_q^{\varvec{\mu }}(\mathfrak {sl}(2))$ and ${\ddot{U}}_{q_1,q_2}^{\varvec{\mu }}(\mathfrak {gl}(1))$ in the limit $q_1\rightarrow \infty$ ($q_2$ fixed), and present the construction of several new intertwiners. These results are used in the second part to construct BPS observables for 5d ${{\mathcal {N}}}=1$ and 3d ${{\mathcal {N}}}=2$ gauge theories. In particular, it is shown that 5d hypermultiplets and 3d chiral multiplets can be introduced in the algebraic engineering framework using shifted representations, and the Higgsing procedure is revisited from this perspective.

Appendices

Example of a Finite Dimensional Shifted Representation

In this appendix, we provide some explicit formulas for the action of the Drinfeld currents defining the algebra ${{\mathcal {E}}}^{(0,3)}$ on the five dimensional module ${{\mathcal {M}}}_\mu$ indexed by the Young diagram $\mu =21$. The representation is obtained as $\iota _{P_\mu }\rho _v^{(0,1)}$ with

$$\begin{aligned} P_\mu (z)=(1-vq_1^2/z)(1-vq_2^2/z)(1-vq_1q_2/z) \end{aligned}$$

(A.1)

In the basis $e_0=\vert \emptyset \rangle \!\rangle$, $e_1=\vert 1\rangle \!\rangle$, $e_2=\vert 2\rangle \!\rangle$, $e_3=\vert 1^2\rangle \!\rangle$ and $e_4=\vert 21\rangle \!\rangle$, the currents $x^\pm (z)$ have the form of $5\times 5$ matrices

$$\begin{aligned} x^+(z)= \begin{pmatrix} 0 &{} \alpha _1 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} \alpha _2 &{} \alpha _3 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} \alpha _4\\ 0 &{} 0 &{} 0 &{} 0 &{} \alpha _5\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 \end{pmatrix},\quad x^-(z)=\begin{pmatrix} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ \beta _1 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} \beta _2 &{} 0 &{} 0 &{} 0\\ 0 &{} \beta _3 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} \beta _4 &{} \beta _5 &{} 0 \end{pmatrix}. \end{aligned}$$

(A.2)

The expression of the coefficients shortens if we introduce the notation $p_{i,j}=1-q_1^iq_2^j$,

$$\begin{aligned} \begin{aligned} \alpha _1&=\delta (v/z)q_3^{-1/2}p_{-1,-1}p_{2,0}p_{0,2}p_{1,1},\quad \beta _1=\delta (v/z)p_{1,1}p_{-1,0}p_{0,-1}\\ \alpha _2&=\delta (vq_2/z)q_3^{-1/2}p_{-1,-1}p_{2,-1}p_{0,1}p_{1,0}^2p_{1,-1}^{-1},\quad \beta _2=\delta (vq_2/z)p_{1,1}p_{-1,0}p_{0,-2}\\ \alpha _3&=\delta (vq_1/z)q_3^{-1/2}p_{-1,-1}p_{-1,2}p_{0,1}^2p_{1,0}p_{-1,1}^{-1},\quad \beta _3=\delta (vq_1/z)p_{1,1}p_{-2,0}p_{0,-1}\\ \alpha _4&=\delta (vq_1/z)q_3^{-1/2}p_{-1,-1}p_{0,1}p_{1,0}p_{0,2},\quad \beta _4=\delta (vq_1/z)p_{1,1}p_{-1,0}p_{0,-1}p_{-2,1}p_{-1,1}^{-1}\\ \alpha _5&=\delta (vq_2/z)q_3^{-1/2}p_{-1,-1}p_{0,1}p_{1,0}p_{2,0}\quad \beta _5=\delta (vq_2/z)p_{1,1}p_{-1,0}p_{0,-1}p_{1,-2}p_{1,11}^{-1}. \end{aligned} \end{aligned}$$

(A.3)

The action of the Cartan currents on the state $e_i$ is diagonal, with the eigenvalue $q_3^{-1/2}[{\tilde{\psi }}_i(z)]_\pm$ obtained by expanding the following rational functions,

$$\begin{aligned} \begin{aligned} {\tilde{\psi }}_0(z)&=\dfrac{(z-vq_1^2)(z-vq_2^2)(z-vq_1q_2)(z-vq_1^{-1}q_2^{-1})}{z^3(z-v)},\\ {\tilde{\psi }}_1(z)&=\dfrac{(z-vq_1^2)(z-vq_2^2)(z-vq_1q_2)^2(z-vq_1^{-1})(z-vq_2^{-1})}{z^3(z-v)(z-vq_1)(z-vq_2)},\\ {\tilde{\psi }}_2(z)&=\dfrac{(z-vq_1^2)(z-vq_1q_2)(z-vq_1q_2^2)(z-vq_2^{-1})(z-vq_1^{-1}q_2)}{z^3(z-vq_1)(z-vq_2)},\\ {\tilde{\psi }}_3(z)&=\dfrac{(z-vq_2^2)(z-vq_1q_2)(z-vq_1^2q_2)(z-vq_1^{-1})(z-vq_1q_2^{-1})}{z^3(z-vq_1)(z-vq_2)},\\ {\tilde{\psi }}_4(z)&=\dfrac{(z-v)(z-vq_1^2q_2)(z-vq_1q_2^2)(z-vq_1q_2^{-1})(z-vq_1^{-1}q_2)}{z^3(z-vq_1)(z-vq_2)}. \end{aligned} \end{aligned}$$

(A.4)

Fig. 17

Quiver variety describing the moduli space of instantons

Equivariant Characters

In this appendix, we analyze the 5d hypermultiplets and the Higgsing procedure from the point of view of the equivariant characters of instantons and vortex moduli spaces. The moduli space of U(n) framed instantons of charge k on the $\Omega$-deformed non-commutative ${{\mathbb {R}}}^4$ follows from the ADHM construction [78,79,80].³¹ It is a quiver variety with a single node $K={{\mathbb {C}}}^k$ framed by $N={{\mathbb {C}}}^n$,

$$\begin{aligned} {{\mathcal {M}}}^I_{k,n}=\{B_1,B_2\in {{\,\mathrm{\text {End}}\,}}(K), I\in {{\,\mathrm{\text {Hom}}\,}}(N,K), J\in {{\,\mathrm{\text {Hom}}\,}}(K,N)\diagup \mu _{{\mathbb {R}}}=\zeta \mathbb {1}_k, \mu _{{\mathbb {C}}}=0\}\diagup U(k),\nonumber \\ \end{aligned}$$

(B.1)

with the moment maps

$$\begin{aligned} 2\mu _{{\mathbb {R}}}=II^\dagger -J^\dagger J+[B_1,B_1^\dagger ]+[B_2,B_2^\dagger ],\quad \mu _{{\mathbb {C}}}=[B_1,B_2]+IJ. \end{aligned}$$

(B.2)

The corresponding quiver is represented on Fig. 17. When $\zeta >0$, the real moment map can be replaced by the stability condition ${{\mathbb {C}}}[B_1,B_2]IN=K$ provided that we quotient by the action of GL(k) instead of U(k). The torus ${\mathfrak {t}}=U(1)_{\varepsilon _1}\times U(1)_{\varepsilon _2}\times U(1)^n$ acts on this moduli space, and the fixed point are labeled by n-tuples of partitions ${\varvec{\lambda }}$ with $|{\varvec{\lambda }}|=k$ boxes. At the fixed point ${\varvec{\lambda }}$, we have the following decomposition of N and K into one-dimensional subspaces

$$\begin{aligned} N=\bigoplus _{l=1}^n N_l,\qquad K=\bigoplus _{(l,i,j)\in {\varvec{\lambda }}}B_1^{i-1}B_2^{j-1}IN_l. \end{aligned}$$

(B.3)

As a result, the characters for the action of the torus ${\mathfrak {t}}$ on the spaces N and K are given by

(B.4)

The characters $v_l$ and are resp. identified with the exponentiated Coulomb branches and the box contents $\chi _{(l,i,j)}=v_lq_1^{i-1}q_2^{j-1}$.

The Nekrasov instanton partition function is a sum over the fixed points ${\varvec{\lambda }}$, the summands are obtained from the character $\chi _{{\varvec{\lambda }},{\varvec{\lambda }}}$ of the torus action on the tangent space $T_{\varvec{\lambda }}{{\mathcal {M}}}^I_{k,N}$ at the fixed point, with³²

$$\begin{aligned} \chi _{{\varvec{\lambda }},{\varvec{\lambda }}'}=-(1-q_1)(1-q_2){{\mathcal {K}}}_{\varvec{\lambda }}{{\mathcal {K}}}_{{\varvec{\lambda }}'}^\vee +{{\mathcal {N}}}{{\mathcal {K}}}_{{\varvec{\lambda }}'}^\vee +q_1q_2{{\mathcal {K}}}_{\varvec{\lambda }}{{\mathcal {N}}}^\vee , \end{aligned}$$

(B.5)

with the linear operation $v_l^\vee =v_l^{-1}$, $q_\alpha ^\vee =q_\alpha ^{-1}$. The vector contribution is obtained by applying the index functor to this expression,

$$\begin{aligned} {{\mathbb {I}}}\left[ \sum _{i\in I_+}e^{w_i}-\sum _{i\in I_-}e^{w_i}\right] =\dfrac{\prod _{i\in I_+} 1-e^{w_i}}{\prod _{i\in I_-} 1-e^{w_i}}\quad \Rightarrow \quad {{\mathbb {I}}}[\chi _{{\varvec{\lambda }},{\varvec{\lambda }}}]={\mathcal {Z}}^v({\varvec{\lambda }},{{\varvec{v}}})^{-1}, \end{aligned}$$

(B.6)

and the Nekrasov factor corresponds to the case $n=1$: ${{\mathcal {N}}}_{\lambda ,\lambda '}(v/v')={{\mathbb {I}}}[\chi _{\lambda ,\lambda '}]$.

1.1 Massive hypermultiplets

In the presence of $N^f$ fundamental and $N^{{\bar{f}}}$ antifundamental hypermultiplets, the equivariant character is replaced by,

$$\begin{aligned} \chi _{{\varvec{\lambda }},{\varvec{\lambda }}'}=-(1-q_1)(1-q_2){{\mathcal {K}}}_{\varvec{\lambda }}{{\mathcal {K}}}_{{\varvec{\lambda }}'}^\vee +({{\mathcal {N}}}-{\bar{{{\mathcal {M}}}}}){{\mathcal {K}}}_{{\varvec{\lambda }}'}^\vee +q_1q_2{{\mathcal {K}}}_{\varvec{\lambda }}({{\mathcal {N}}}^\vee -{{\mathcal {M}}}^\vee ) \end{aligned}$$

(B.7)

with ${{\mathcal {M}}}=\sum _{a=1}^{N^f}\mu _a$ and $\overline{{\mathcal {M}}}=\sum _{a=1}^{N^{{\bar{f}}}}{\bar{\mu }}_a$ [82] (neglecting the one-loop terms). The summands of the instanton partition function are again obtained as

$$\begin{aligned} {{\mathbb {I}}}[\chi _{{\varvec{\lambda }},{\varvec{\lambda }}}]^{-1}={\mathcal {Z}}^v({\varvec{\lambda }},{{\varvec{v}}}){\mathcal {Z}}^{f}({\varvec{\lambda }},{{\varvec{v}}},{\varvec{\mu }}){\mathcal {Z}}^{\bar{f}}({\varvec{\lambda }},{{\varvec{v}}},\bar{{\varvec{\mu }}}). \end{aligned}$$

(B.8)

The extra terms $-{\bar{{{\mathcal {M}}}}}{{\mathcal {K}}}_{{\varvec{\lambda }}'}^\vee -q_1q_2{{\mathcal {K}}}_{\varvec{\lambda }}{{\mathcal {M}}}^\vee$ in the character also induce an extra factor in the ${{\mathcal {Y}}}$-observables. These observables were introduced by Nekrasov in [56], they can be obtained as a variation of the equivariant character (see [44]),³³

(B.9)

From the bilinear expression of the character B.7, we deduce that the r.h.s. of these formulas is indeed independent of the second Young diagram. The ${{\mathcal {Y}}}$-observables are the unique rational functions satisfying these constraints for the infinite set of points $\chi _{(l,i,j)}=v_lq_1^{i-1}q_2^{j-1}$. The extra factors in the ${{\mathcal {Y}}}$-observables coming from the deformation of the character in the presence of hypermultiplets are simply the inverse of the polynomials defined in 5.4 and involved in the shift of representations,e

(B.10)

Using the shell formula, it is seen that the ${{\mathcal {Y}}}$-observable ${{\mathcal {Y}}}_\lambda (z)$ coincides with the function 3.19 defining the vertical Fock representation. Thus, restricting ourselves to U(1) instantons, the algebra ${{\mathcal {E}}}$ acts on the fixed point $\vert \lambda \rangle \!\rangle$ of the torus action by the vertical Fock representation given in 3.20. This action is the (K-theoretic) (double) Cohomological Hall algebra of the instanton moduli space. Matrix elements of the currents are expressed in terms of the ${{\mathcal {Y}}}$-observables, and so they are expected to be modified in the presence of hypermultiplets as follows,

(B.11)

Comparing with 3.20, we deduce that antifundamental hypermultiplets lead to the shift $\iota _P\rho _v^{(0,1)}$ of the vertical representation with the polynomial $P=P_{\bar{{\varvec{\mu }}}}^{{\bar{f}}}$. Naively, fundamental hypermultiplets would introduce a similar shift $\iota _{P^{-1}}^*\rho _v^{(0,1)}$ with $P=P_{\varvec{\mu }}^f$. However, in this case $P(z)^{-1}$ is no longer a finite Laurent series, it introduce extra poles in the action of the Cartan current that cannot be accounted for by the commutator $[x^+(z),x^-(w)]$. Thus, it is not clear how to define properly the vertical action there. This is a limitation of our simplified approach, and this issue could be resolved using the more involved geometric techniques employed e.g. in [81, 83].

Fig. 18

Quiver variety describing the vortex moduli space

1.2 Higgsing and the vortex moduli space

The relation between the instanton moduli space and the vortex moduli space has been investigated in [84] from the point of view of the ${{\mathcal {N}}}=(2,2)$ quantum mechanics on the worldvolume of D1-branes. By weakly gauging a U(1) subgroup of the total symmetry group, it is possible to give a large mass to some of the fields and decouple them. It corresponds to imposing $B_1=J=0$ in the ADHM construction, and we find the vortex moduli space given by the quiver variety of Fig. 18,

$$\begin{aligned} {{\mathcal {M}}}_{k,n}^V=\{B_2\in {{\,\mathrm{\text {End}}\,}}(K), I\in {{\,\mathrm{\text {Hom}}\,}}(N,K)\diagup \mu _{{\mathbb {R}}}=\zeta \mathbb {1}_k\}/U(k), \end{aligned}$$

(B.12)

with the real moment map $2\mu _{{\mathbb {R}}}=II^\dagger +[B_2,B_2^\dagger ]$. The torus $U(1)_{\varepsilon _2}\times U(1)^n$ acts on ${{\mathcal {M}}}_{k,n}^V$ and the fixed point are labeled by n-tuple of positive integers ${{\varvec{k}}}=(k_1,\ldots ,k_n)$ such that $|{{\varvec{k}}}|=\sum _lk_l=k$. At each fixed point, we have the following decomposition of K and N and the corresponding characters ${{\mathcal {K}}}_{{{\varvec{k}}}}$ and ${{\mathcal {N}}}$,

$$\begin{aligned} N=\bigoplus _{l=1}^n N_l,\quad K=\bigoplus _{l=1}^n\bigoplus _{j=1}^{k_l}B_2^{j-1}IN_l\quad \Rightarrow \quad {{\mathcal {N}}}=\sum _{l=1}^nv_l,\quad {{\mathcal {K}}}_{{\varvec{k}}}=\sum _{l=1}^nv_l\sum _{j=1}^{k_l}q_2^{j-1}.\nonumber \\ \end{aligned}$$

(B.13)

The equivariant character of the tangent space of the vortex moduli space has been computed in [66], it corresponds to $\chi ^\text {3d}_{{{\varvec{k}}},{{\varvec{k}}}}$ with

$$\begin{aligned} \chi ^\text {3d}_{{{\varvec{k}}},{{\varvec{k}}}'}=-(1-q^{2}){{\mathcal {K}}}_{{{\varvec{k}}}}{{\mathcal {K}}}_{{{\varvec{k}}}'}^\vee +{{\mathcal {N}}}{{\mathcal {K}}}_{{{\varvec{k}}}'}^\vee ,\quad \text {with}\quad q^2=q_2. \end{aligned}$$

(B.14)

Taking the index functor, this character produces the 3d version of the Nekrasov factor defined in 6.3, $N_{k,k'}(v/v')={{\mathbb {I}}}[\chi _{k,k'}]$.

The vortex moduli space studied here corresponds to a non-commutative Hilbert scheme [85], but unfortunately we were unable to find a derivation of the K-theoretic COHA for this variety. A possible approach could be to start from the K-theoretic COHA derivation of the finite dimensional module of quantum affine algebras performed in [86,87,88]. Specializing to the Kirillov–Reshetikhin representation of dimension N for the quantum affine $\mathfrak {sl}(2)$ algebra, it might be possible to consider the limit $N\rightarrow \infty$ of this geometric construction. Here, again, we will be more modest and simply consider the variation of the equivariant character. In this way, we define 3d analog of Nekrasov’s ${{\mathcal {Y}}}$-observables,

$$\begin{aligned} Y_{{\varvec{k}}}(v_lq^{2k_l})={{\mathbb {I}}}[\chi ^\text {3d}_{{{\varvec{k}}},{{\varvec{k}}}'+\delta _l}-\chi ^\text {3d}_{{{\varvec{k}}},{{\varvec{k}}}'}],\quad Y_{{{\varvec{k}}}'}^*(v_lq^{2k_l+2})={{\mathbb {I}}}[\chi ^\text {3d}_{{{\varvec{k}}}+\delta _l,{{\varvec{k}}}'}-\chi ^\text {3d}_{{{\varvec{k}}},{{\varvec{k}}}'}], \end{aligned}$$

(B.15)

where $\delta _l=(0,\ldots ,1,0,\ldots )\in {{\mathbb {Z}}}^n$ is a vector with a 1 in the lth row and zero elsewhere. It gives indeed the functions defined in [11],

$$\begin{aligned} \begin{aligned} Y_{{{\varvec{k}}}}(z)=\prod _{l=1}^n\left( 1-\dfrac{v_lq^{2k_l}}{z}\right) ,\quad Y_{{{\varvec{k}}}}^*(q^2z)=\prod _{l=1}^nq^{2k_l}\dfrac{z-v_lq^{-2}}{z-v_lq^{2k_l-2}}. \end{aligned} \end{aligned}$$

(B.16)

In the case of U(1) vortices, the prefundamental representation acts on a module spanned by states $\vert k\rangle \!\rangle$ labeled by the fixed point of the equivariant action on the vortex moduli space, its action can be expressed using ${{\mathcal {Y}}}$-observables as

$$\begin{aligned} \begin{aligned} {\tilde{X}}^+(z)\vert k\rangle \!\rangle&=\delta (v q^{2k}/z)\mathop {\text {Res}}_{z=v q^{2k}}\dfrac{1}{z Y_k(z)}\vert k+1\rangle \!\rangle ,\\ {\tilde{X}}^-(z)\vert k\rangle \!\rangle&=\delta (v q^{2k-2}/z)\mathop {\text {Res}}_{z=v q^{2k-2}}z^{-1}Y_k^*(q^2z)\vert k-1\rangle \!\rangle ,\\ {\tilde{\Psi }}^\pm (z)\vert k\rangle \!\rangle&=\left[ \dfrac{Y_k^*(q^2z)}{Y_k(z)}\right] _\pm \vert k\rangle \!\rangle . \end{aligned} \end{aligned}$$

(B.17)

It reproduces the prefundamental action 2.18 up to a change in the normalization of the states and a rescaling of the currents by harmless factors.

Higgsing Recall that the 3d gauge theories are obtained from a two steps procedure that consists in (1) adjusting the mass of n hypermultiplets to a critical value and (2) taking the limit $q_1\rightarrow \infty$ to decouple the adjoint chiral multiplet. Once the mass takes its critical value, the set of fixed points is restricted to n-tuples of single column Young diagrams identified with the n positive integers ${{\varvec{k}}}=(k_1,\ldots ,k_n)$, and the characters ${{\mathcal {K}}}_{\varvec{\lambda }}$ should be replaced by ${{\mathcal {K}}}_{{\varvec{k}}}$. Furthermore, adjusting the mass of n antifundamental hypermultiplets to $\bar{\mu }_l=q_1v_l$ corresponds to setting ${\bar{{{\mathcal {M}}}}}=q_1{{\mathcal {N}}}$ and ${{\mathcal {M}}}=0$ in the expression B.7 of the equivariant character, and we find

$$\begin{aligned} \chi _{{{\varvec{k}}},{{\varvec{k}}}'}^\text {5d}=(1-q_1)\chi _{{{\varvec{k}}},{{\varvec{k}}}'}^\text {3d}+q_1q_2{{\mathcal {N}}}^\vee {{\mathcal {K}}}_{{{\varvec{k}}}}. \end{aligned}$$

(B.18)

In the limit $q_1\rightarrow \infty$, only the finite terms in $\chi _{{{\varvec{k}}},{{\varvec{k}}}'}^\text {5d}$ will contribute once the index functor is applied. Examining the term of power $q_1^0$, we find indeed $\left[ \chi _{{{\varvec{k}}},{{\varvec{k}}}'}^\text {5d}\right] _0=\chi _{{{\varvec{k}}},{{\varvec{k}}}'}^\text {3d}$. We would arrive at the same conclusion if we had adjusted instead the mass of fundamental hypermultiplet, i.e. setting ${{\mathcal {M}}}=q_1^2q_2{{\mathcal {N}}}$ and ${\bar{{{\mathcal {M}}}}}=0$ in B.7. This relation between the equivariant characters justifies our approach of the limit of highest $\ell$-weight representation, and we can check that the ${{\mathcal {Y}}}$-observables have indeed the proper limit (the factor $q^{-2k}$ can be absorbed in a renormalization of the states)

$$\begin{aligned} \begin{aligned} {{\mathcal {Y}}}_k(z)&\mathop {\longrightarrow }\limits _{LII} q^{-2k}Y_k(z),\quad {{\mathcal {Y}}}_k(q_3^{-1}z)\mathop {\longrightarrow }\limits _{LII} q^{-2k}Y_k^*(q^2z)\\&\quad \Rightarrow \quad \psi _k(z)\mathop {\longrightarrow }\limits _{LII} q^{2k}\dfrac{z(z-vq^{-2})}{(z-vq^{2k})(z-vq^{2k-2})}. \end{aligned} \end{aligned}$$

(B.19)