KIAS - Q19005

YITP-SB-19-18

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$\u2020$

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Quantum toroidal algebras are obtained from quantum affine algebras by a further affinization, and, like the latter, can be used to construct integrable systems. These algebras also describe the symmetries of instanton partition functions for 5D $=N1$ supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold $/\times S1C2Zp$ where the action of $Zp$ is determined by two integer parameters $(\nu 1,\nu 2)$. The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of $gl(p)$. We show that it has the structure of a Hopf algebra, and present two representations, called

Non-perturbative dynamics of supersymmetric gauge theories is a prolific research subject in theoretical physics. Since the innovative work [

Very rich algebraic structures lie at the heart of these correspondences [

A closely related but different connection with quantum algebras arises from the type IIB strings theory realization of the five-dimensional uplifts of 4D $=N2$ gauge theories, that is the 5D $=N1$ quiver gauge theories compactified on $S1$. In this construction, $=N1$ gauge theories emerge as the low-energy description of the dynamics of 5-branes webs [

Awata, Feigin and Shiraishi observed in [

In [

A natural question is how to generalize the algebraic construction to gauge theories on more complicated manifolds. Among other manifolds, the $Zp$-orbifolded $C2$ are of a particular interest, since the partition functions on these spaces can be computed by simply projecting out the contributions which are not invariant under the $Zp$-action [

In this work, we extend the algebraic construction of 5D Nekrasov partition functions to a more general $Zp$-orbifolding depending on two integer parameters $(\nu 1,\nu 2)$. We propose an extended quantum toroidal algebra relevant to the construction, and prove its Hopf algebra structure. We define both horizontal and vertical representations, and derive the vertex operator which intertwines between these representations. Finally, using these ingredients, we give an algebraic construction of Nekrasov partition functions and $qq$-characters. The orbifolds considered in this work incorporate the case of codimension-two defect insertion, whose applications to BPS/CFT correspondence, Bethe/gauge correspondence, and Nekrasov-Rosly-Shatashvili correspondence have been largely investigated [

This paper is written in such a way that mathematicians interested only in the formulation of the extended algebra can focus on the reading of section three, together with the appendices (quantum toroidal $gl(p)$), (representations) and (automorphisms and gradings) for more details. Instead, the section two provides a brief description of the physical context in which the algebra emerges, i.e. instantons of 5D $=N1$ gauge theories on the spacetime $/C2Zp$. Finally, the section four is dedicated to the algebraic construction of gauge theories observables, giving the expression of the $(\nu 1,\nu 2)$-colored refined topological vertex and a few examples of application.

In order to derive the group action on the instanton moduli space, we focus first on the case of a pure $U(m)$ gauge theory. In this case, the ADHM construction of the moduli space [

The complexified global symmetry group $\times GL(M)SL(2,C)2$ acts on the ADHM matrices, preserving the quiver variety $Mk$. It contains an $(+m2)$-dimensional torus that acts follows,

The fixed points of this action parameterize the configurations of instantons with total charge $k$. They are in one-to-one correspondence with the $m$-tuple partitions $=\lambda (\lambda (1),\cdots \lambda (m))$ of the integer $k$, here identified with the $m$-tuple Young diagrams with $=|\lambda |k$ boxes. At the fixed point, the vector space $K$ is decomposed into

where $M\alpha $ denotes the one-dimensional vector spaces generated by the basis vectors of $M$. Thus, each box $=\n \n \n \n \n \n \n \n \n \n (\alpha ,i,j)$ of the $m$-tuple partition $\lambda $ with coordinate $\in (i,j)\lambda (\alpha )$ corresponds to a one-dimensional vector space $B1-i1B2-j1I(M\alpha )$. We further associate to the box the complex variable $=\varphi \n \n \n \n \n \n \n \n \n \n +a\alpha (-i1)\epsilon 1(-j1)\epsilon 2$ called

In this paper, gauge theories are considered on the 5D orbifolded omega-background $/\times SR1(\times C\epsilon 1C\epsilon 2)Zp$ where $=Zp/ZpZ$ is a subgroup of the torus $\subset U(1)2SO(4)$. The action of the group $Zp$ on the spacetime is parameterized by two integers $\nu 1,\nu 2$,

Furthermore, it is possible to combine it with a global gauge transformation in the subgroup $\subset U(1)mU(m)$. As a result, we obtain an action of $Zp$ on the ADHM data by specialization of the $(+m2)$-torus action (), taking

This action of the abelian group $Zp$ is parameterized by the $+m2$ integers $(c\alpha ,\nu 1,\nu 2)$ considered modulo $p$. The transformation of the vector spaces in the decomposition () of $K$ leads to associate to each box $\n \n \n \n \n \n \n \n \n \n =(\alpha ,i,j)\in \lambda $, in addition to the complex variables $\varphi \n \n \n \n \n \n \n \n \n \n $ and $\chi \n \n \n \n \n \n \n \n \n \n $, the integer $c(\n \n \n \n \n \n \n \n \n \n )$ such that

We call

We denote $C\omega (m)$ the subset of $[[1,m]]$ such that the Coulomb branch vevs $a\alpha $ (or $v\alpha $) with $\in \alpha C\omega (m)$ have color $=c\alpha \omega $ (or, equivalently, that the box $\in (1,1)\lambda (\alpha )$ with $\in \alpha C\omega (m)$ is of color $c(\alpha ,1,1)=c\alpha =\omega $). Similarly, $K\omega (\lambda )$ denotes the set of boxes $\in \n \n \n \n \n \n \n \n \n \n \lambda $ of the $m$-tuple colored partition $\lambda $ that carry the color $=c(\n \n \n \n \n \n \n \n \n \n )\omega $. Besides, in the generic case $\ne +\nu 1\nu 20$, the shift of color indices $\omega $ by the quantity $+\nu 1\nu 2$ appears in many formulas. To simplify these expressions, we introduce the notation $=\xaf\omega +\omega \nu 1\nu 2$ for the shifted indices, along with the map $=\xafc(\n \n \n \n \n \n \n \n \n \n )+c(\n \n \n \n \n \n \n \n \n \n )\nu 1\nu 2$. Finally, we also introduce the extra variables $q3$ and $\nu 3$ such that $=q1q2q31$ and $=+\nu 1\nu 2\nu 30$. Due to the fact that the $Zp$-action coincides with a subgroup of the torus action, in all formulas the shift of color indices $+\omega \nu i$ coincide with a factor $qi$ multiplying the parameters associated to instanton positions in the moduli space.

Although we are considering here a different problem, it is interesting to make a short parallel with the action of $\subset \times SU(2)LSU(2)RSO(4)$ on the omega-background (see, for instance, [

Then the $\times 22$ matrices $\in (GL,GR)\times SU(2)LSU(2)R$ act on the quaternions as $\to ZGLZGR$. The McKay subgroups of $SU(2)$ possess an ADE-classification. For instance, the $A-p1$ series corresponds to the action of $Zp$, it is generated (multiplicatively) by the diagonal matrices

Considering only the action of the $A-p1$ subgroup on the left, the background coordinates transform as $\to (z1,z2)(e/2i\pi pz1,e-/2i\pi pz2)$. This transformation can be recovered from the action () of $Zp$ by choosing $\nu 1=-\nu 2=1$. The orbifold of the spacetime under this action of $Zp$ reproduces the ALE space constructed in [

It is also possible to consider simultaneously the action of two McKay subgroups $A-p11$ and $A-p21$, with one acting on the left, the other on the right. As a result, coordinates now transform as

We recognize here another particular case of the $Zp$-action defined in (), albeit more general than before. It is simply obtained by the specialization $=\nu 1+p1p2$, $=\nu 2-p1p2$ and $=pp1p2$. Thus, the action () leads to a particularly rich context. Moreover, taking $=\nu 10$, the first coordinate $z1$ is invariant and the orbifolded spacetime can be reinterpreted as the insertion of a codimension-two defect in a 5D omega background with no orbifold [

The computation of the Nekrasov instanton partition function on such $Zp$-orbifolds has been performed in [

a set of colored exponentiated gauge couplings $q\omega ,i$,

a $p$-vector of colored Chern-Simons levels $=\kappa (i)(\kappa \omega (i))\in \omega Zp$,

an $m(i)$-vector of Coulomb branch vevs $=a(i)(a\alpha (i))=\alpha 1m(i)$ defining the exponentiated parameters $=v(i)(v\alpha (i))=\alpha 1m(i)$ with $=v\alpha (i)eRa\alpha (i)$,

an associated vector of colors $=c(i)(c\alpha (i))=\alpha 1m(i)$.

In addition, each link $\to ij$ between two nodes $i$ and $j$ represent a chiral multiplet of matter fields in the bifundamental representation of the gauge group $\times U(m(i))U(m(j))$, with mass $\in \mu ijC$. For linear quivers, all bifundamental masses can be set to $q3-1$ by a rescaling of the Coulomb branch vevs.

The instantons contribution to the gauge theories partition function is expressed as a sum over the content of $r$ $m(i)$-tuple Young diagrams $\lambda (i)$ describing the configuration of instantons at the $i$th node. Each term can be further decomposed into the contributions of vector (gauge) multiplets, bifundamental chiral (matter) multiplets, and Chern-Simons factors:

Vector and bifundamental contributions are written in terms of the Nekrasov factor $N(v,|\lambda \mu v\prime ,\lambda \prime )$. For a better readability, we drop the node indices in the following, and simply distinguish the two nodes involved in the definition of the Nekrasov factor with a prime. In order to write down the expression of $N(v,|\lambda \mu v\prime ,\lambda \prime )$ given in [

A linear involutive operation $\ast $ acts on such characters by flipping the sign of $R$: $=(eRa\alpha )\ast e-Ra\alpha $, $=(q1\ast ,q2\ast )(q1-1,q2-1)$ and thus $=(eR\varphi \n \n \n \n \n \n \n \n \n \n )\ast e-R\varphi \n \n \n \n \n \n \n \n \n \n $ (see [

where the $I$-symbol is the equivariant index functor,

and $[\cdots ]Zp$ denotes the operation of keeping only the $Zp$-invariant parts. In particular, the RHS of () involves a coloring function $:c\to Z[a\alpha ,\epsilon 1,\epsilon 2]Zp$ defined on weights $wi$ as the linear map taking the values $=c(a\alpha )c\alpha $, $=c(\epsilon 1)\nu 1$ and $=c(\epsilon 2)\nu 2$ so that $=c(\varphi \n \n \n \n \n \n \n \n \n \n )c(\n \n \n \n \n \n \n \n \n \n )$ (justifying our slight abuse of notations). The $[\cdots ]Zp$ projects on $Zp$-invariant factors.

Replacing the equivariant characters by their expressions (), the Nekrasov factor can be written in a more explicit form,

The function $S\omega \omega \prime (z)$ is sometimes called the

In this expression, the non-zero matrix elements have been expressed in a compact way using the delta function $\delta \omega ,\omega \prime $ defined modulo $p$ (i.e. $=\delta \omega ,\omega \prime 1$ iff $=\omega \omega \prime $ modulo $p$, zero otherwise). In fact, $S\omega \omega \prime (z)$, and more generally all the matrices of size $\times pp$ with indices $\omega ,\omega \prime $ appearing in this paper, are circulant matrices: their matrix elements only depend on the difference $-\omega \omega \prime $ of row and column indices. In particular, $=S+\omega \nu \omega \prime (z)S-\omega \omega \prime \nu (z)$ for all $\in \nu Zp$. Finally, the function $S\omega \omega \prime (z)$ satisfies a sort of

with the function $=f\omega \omega \prime (z)F\omega \omega \prime z\beta \omega \omega \prime $ defined by

A new class of BPS-observables for supersymmetric gauge theories was introduced in [

The $qq$-characters are half-BPS observables written as combinations of $Y$

Replacing the equivariant characters with the expressions (), we find the explicit formulas

Due to the crossing symmetry (), these two $Y$-observables satisfy the relation

with

It will allow us to express all the equations below in terms of $Y\omega [\lambda ](z)$ only.

Here, the sets $A\omega (\lambda )$ and $R\omega (\lambda )$ denote respectively the set of boxes of color $\omega $ that can be added to or removed from the $m$-tuple Young diagram $\lambda $. This expression arises from the cancellations of contributions by neighboring boxes, it plays an essential role in the definition of the vertical representation of the algebra.

In order to reconstruct the instanton partition functions on the general orbifold (), the definition of a new quantum toroidal algebra is necessary. In addition to the complex parameters $q1,q2$ and the rank $\in pZ>0$, this algebra will depend on the integers $(\nu 1,\nu 2)$ modulo $Zp$. Taking $\nu 1=-\nu 2=1$, the $Zp$-action () reduces to the standard action defining ALE spaces. Thus, in this limit the $(\nu 1,\nu 2)$-deformed algebra should reduce to the quantum toroidal algebra of $gl(p)$. In fact, this is true only up to a twist in the definition of the Drinfeld currents (see the subsection () of the appendix). A brief reminder on the quantum toroidal algebra of $gl(p)$ is given in appendix , it includes its two main representations called, in the gauge theory context,

The key ingredient to define the deformation of the quantum toroidal algebra of $gl(p)$ is the scattering function $S\omega \omega \prime (z)$ defined in (). Indeed, this function plays an essential role in the two elementary representations involved in the algebraic engineering of partition functions and $qq$-characters. In the vertical representation, it enters through the definition () of the $Y$-observables that describe the recursion relations among Nekrasov factors. Instead, in the horizontal representation, it expresses the normal-ordering relations between vertex operators. Thus, from the physics perspective, the scattering function is the natural object to consider for the deformation of the algebra. Moreover, through the crossing symmetry relation (), this function defines the $\times pp$ matrix $\beta \omega \omega \prime $ that could be identified with the underlying Cartan matrix of the deformed quantum toroidal algebra (see the subsection ). Note that the matrix $\beta \omega \omega \prime $ naturally reduces to the generalized Cartan matrix of the Kac-Moody algebra $^glp$ when $\nu 1=-\nu 2=1$. In general, it is non-symmetrizable, yet, like in the case of $^glp$, it is a circulant matrix. Its eigenvectors $=vj(1,\Omega j,\Omega j2,\cdots ,\Omega j-p1)$ are written in terms of the $p$th root of unity $=\Omega je/2i\pi jp$, and the corresponding eigenvalues read

In particular, the eigenvector $=v0(1,1,\cdots ,1)$ has the eigenvalue zero which relates $\beta \omega \omega \prime $ to the Cartan matrix of affine Lie algebras, and thus justifies the designation

Like in the case of $gl(p)$, the $(\nu 1,\nu 2)$-deformed quantum toroidal algebra is defined in terms of a central element $c$ and $4p$ Drinfeld currents, denoted $x\omega \pm (z)$ and $\psi \omega \pm (z)$, with $\in \omega Zp$. The currents $\psi \omega \pm (z)$ (together with $c$) generate the Cartan subalgebra, while the currents $x\omega \pm (z)$ deform the notion of Chevalley generators $e\omega ,f\omega $. The algebraic relations obeyed by the currents resemble those defining the quantum toroidal algebra of $gl(p)$ in (), the main difference being the presence of shifts in the indices $\omega $ by the product $\nu 3c$: