KIAS - Q19005

YITP-SB-19-18

New quantum toroidal algebras

from 5D $= N 1$ instantons on orbifolds

Jean-Emile Bourgine$†$, Saebyeok Jeong$∗$

[.4cm] $†$Korea Institute for Advanced Study (KIAS)

Quantum Universe Center (QUC)

85 Hoegiro, Dongdaemun-gu, Seoul, South Korea

bourgine@kias.re.kr

[.4cm] $∗$C.N. Yang Institute for Theoretical Physics

Stony Brook University

Stony Brook, NY 11794-3840, USA

saebyeok.jeong@gmail.com

[.4cm]

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 $= N 1$ supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold $/ × S 1 C 2 Z p$ where the action of $Z p$ is determined by two integer parameters $( ν 1 , ν 2 )$. The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of $⁢ g l ( p )$. We show that it has the structure of a Hopf algebra, and present two representations, called vertical and horizontal, obtained by deforming respectively the Fock representation and Saito’s vertex representations of the quantum toroidal algebra of $⁢ g l ( p )$. We construct the vertex operator intertwining between these two types of representations. This object is identified with a $( ν 1 , ν 2 )$-deformation of the refined topological vertex, allowing us to reconstruct the Nekrasov partition function and the $⁢ q q$-characters of the quiver gauge theories.

Contents
1 section 1 1 §1 <tag close=" ">1</tag>Introduction

Non-perturbative dynamics of supersymmetric gauge theories is a prolific research subject in theoretical physics. Since the innovative work [], direct microscopic studies on four-dimensional gauge theories with $= N 2$ supersymmetry became largely accessible, from exact computations of their partition functions on the (non-commutative) $C 2$. The divergence in the partition function coming from the non-compactness of $C 2$ is properly regularized by introducing the $Ω$-background [], effectively localizing the four-dimensional theory to the origin. In turn, the path integral reduces to an equivariant integration on the finite dimensional framed moduli space of non-commutative instantons, for which equivariant localization can be applied for its exact evaluation. The Nekrasov partition function has been a powerful tool to investigate the correspondences of four-dimensional $= N 2$ supersymmetric quiver gauge theories with other objects in mathematical physics, i.e., quantum integrable systems [], two-dimensional CFTs [], flat connections on Riemann surfaces [], and isomonodromic deformations of Fuchsian systems [].

Very rich algebraic structures lie at the heart of these correspondences []. For instance, the AGT correspondence [] between Nekrasov partition functions and conformal blocks of Liouville/Toda 2D CFTs can be understood algebraically as the action of W-algebras on the cohomology of instantons moduli space []. In this context, the W-algebra currents are coupled to an infinite Heisenberg algebra, and the total action is formulated in terms of a quantum algebra, namely the Spherical Hecke central algebra [] (isomorphic to the affine Yangian of $⁢ g l ( 1 )$ []). The coupling to an Heisenberg algebra is essential for the definition of a coalgebraic structure, thus emphasizing the underlying quantum integrability since the coproduct provides the R-matrix satisfying the celebrated quantum Yang-Baxter equation.

A closely related but different connection with quantum algebras arises from the type IIB strings theory realization of the five-dimensional uplifts of 4D $= N 2$ gauge theories, that is the 5D $= N 1$ quiver gauge theories compactified on $S 1$. In this construction, $= N 1$ gauge theories emerge as the low-energy description of the dynamics of 5-branes webs []. Here, each brane carries the charges $( p , q )$, generalizing D5-branes (charge $( 1 , 0 )$) and NS5-branes (charge $( 0 , 1 )$). Their world-volume include the five-dimensional gauge theory spacetime, together with an extra line segment in the 56-plane. Individual branes’ line segments are joined by trivalent vertices, and form the $( p , q )$-branes web. Alternatively, the $( p , q )$-brane web can be seen as the toric diagram of a Calabi-Yau threefold on which topological strings can be compactified []. The trivalent vertices are then identified with the (refined) topological vertex, thereby leading to a very efficient method of computing 5D Nekrasov partition functions as topological strings amplitudes [].

Awata, Feigin and Shiraishi observed in [] that a specific representation of the quantum toroidal $⁢ g l ( 1 )$ algebra (or Ding-Iohara-Miki algebra []) can be associated to each edge of the $( p , q )$-branes web. The charges $( p , q )$ are identified with the values of the two central charges while the brane position define the weight of the representation. As such, the D5-branes correspond to the so-called vertical representation while an horizontal representation is associated to NS5-branes (possibly dressed by extra D5-branes). 1 footnote 1 1 footnote 1 In fact, the vertical representation is simply the q-deformation of the affine Yangian action mentioned previously, it is expected to describe a quantum toroidal action on the K-equivariant cohomology of the quiver variety describing the instanton moduli space. The equivalent of the horizontal representation can also be defined for 4D $= N 2$ theories, thus extending the whole algebraic construction of the Nekrasov partition functions []. However, for this purpose, it is necessary to consider the central extension of the Drinfeld double of the affine Yangian following from the construction given in []. The refined topological vertex is then identified with an intertwiner between vertical and horizontal representations, that is in fact the toroidal version of the vertex operator introduced in [] for the quantum group $⁢ U q ( ^ ⁢ s l ( 2 ) )$. In this way, the Nekrasov partition function is written as a purely algebraic object using the quantum toroidal algebra, just like conformal blocks with W-algebras []. This algebraic construction turns out to be useful in probing various properties of the partition function, e.g. in addressing the (q-deformed) AGT correspondence [], or in studying strings’ S-duality [].

In [], an important class of half-BPS observables, called $⁢ q q$-characters, were defined, whose characteristic property is the regularity of their gauge theory expectation values. This regularity property encodes efficiently an infinite set of constraints on the partition function called non-perturbative Dyson-Schwinger equations []. The algebraic nature of these constraints was observed in []. Actually, the constraints take an even more elegant form in the algebraic construction described above as they express the invariance of an operator $T$ under the adjoint action of the quantum toroidal algebra []. This operator is obtained by gluing intertwiners along the edges of the $( p , q )$-branes web, and its vacuum expectation value reproduces the 5D Nekrasov instanton partition.

A natural question is how to generalize the algebraic construction to gauge theories on more complicated manifolds. Among other manifolds, the $Z p$-orbifolded $C 2$ 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 $Z p$-action []. The generalization of the algebraic construction is not entirely straightforward since it is necessary to introduce the information of the coloring corresponding to the $Z p$-action of the orbifolding. In this scope, deformations of the quantum toroidal $⁢ g l ( 1 )$ algebra must be considered. A special case of the $Z p$-orbifolded $C 2$ is the (un-resolved) $A p$-type ALE spaces. The ALE instantons were introduced by Kronheimer in [], and the ALE instanton moduli spaces were constructed as quiver varieties in []. The algebraic construction of the corresponding 5D Nekrasov partition functions has been realized recently using an underlying quantum toroidal $⁢ g l ( p )$ algebra. There, the index carried by the Drinfeld currents renders the $Z p$-coloring due to the orbifolding. Incidentally, the vertical representation of this quantum toroidal algebra should coincide with the q-deformation of the affine Yangian of $⁢ g l ( p )$ acting on the cohomology of the moduli space of ALE instantons, extending by further affinization the algebraic actions discovered in [].

In this work, we extend the algebraic construction of 5D Nekrasov partition functions to a more general $Z p$-orbifolding depending on two integer parameters $( ν 1 , ν 2 )$. 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 $⁢ q q$-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 $⁢ g l ( p )$), (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 $= N 1$ gauge theories on the spacetime $/ C 2 Z p$. Finally, the section four is dedicated to the algebraic construction of gauge theories observables, giving the expression of the $( ν 1 , ν 2 )$-colored refined topological vertex and a few examples of application.

2 section 2 2 §2 <tag close=" ">2</tag>Instantons on orbifolds 2.1 subsection 2.1 2.1 §2.1 <tag close=" ">2.1</tag>Action of the abelian group <Math mode="inline" tex="\mathbb{Z}_{p}" text="Z _ p" xml:id="S2.SS1.m1"> <XMath> <XMApp> <XMTok role="SUBSCRIPTOP" scriptpos="post4"/> <XMTok font="blackboard" role="UNKNOWN">Z</XMTok> <XMTok font="italic" fontsize="70%" role="UNKNOWN">p</XMTok> </XMApp> </XMath> </Math> on the ADHM data

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 [] involves only two vector spaces $M$ and $K$ of dimension $m$ and $k$ respectively, where $k$ is the instanton number. Introducing the four matrices $: B 1 , B 2 → K K$ $: I → M K$ and $: J → K M$, the instanton moduli space is identified with the quiver variety (see, for instance, [])

(2.1) Equation 2.1 2.1 $M k = { B 1 , B 2 , I , J ╱ [ B 1 , B 2 ] + I J = 0 , C [ B 1 ; B 2 ] I ( M ) = K } ╱ GL ( K ) .$

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

(2.2) Equation 2.2 2.2 $→ ( B 1 , B 2 , I , J ) ( ⁢ t 1 B 1 , ⁢ t 2 B 2 , ⁢ I t , ⁢ t - 1 J t 1 t 2 ) , ∈ ( t , t 1 , t 2 ) × ( C × ) m ( C × ) 2 .$

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 $= λ ( λ ( 1 ) , ⁢ ⋯ λ ( m ) )$ of the integer $k$, here identified with the $m$-tuple Young diagrams with $= | λ | k$ boxes. At the fixed point, the vector space $K$ is decomposed into

(2.3) Equation 2.3 2.3 $= K ⊕ = α 1 m ⊕ ∈ ( i , j ) λ ( α ) ⁢ B 1 - i 1 B 2 - j 1 I ( M α ) ,$

where $M α$ denotes the one-dimensional vector spaces generated by the basis vectors of $M$. Thus, each box of the $m$-tuple partition $λ$ with coordinate $∈ ( i , j ) λ ( α )$ corresponds to a one-dimensional vector space $⁢ B 1 - i 1 B 2 - j 1 I ( M α )$. We further associate to the box the complex variable called instanton position or, sometimes, the box content of . The parameters $a 1 , ⋯ , a m$ are the Coulomb branch vevs of the gauge theory. We also define the exponentiated quantities $= v α e ⁢ R a α$, $= ( q 1 , q 2 ) ( e ⁢ R ε 1 , e ⁢ R ε 2 )$ and .

In this paper, gauge theories are considered on the 5D orbifolded omega-background $/ × S R 1 ( × C ε 1 C ε 2 ) Z p$ where $= Z p ⁢ / Z p Z$ is a subgroup of the torus $⊂ ⁢ U ( 1 ) 2 ⁢ S O ( 4 )$. The action of the group $Z p$ on the spacetime is parameterized by two integers $ν 1 , ν 2$,

(2.4) Equation 2.4 2.4 $( θ , z 1 , z 2 ) ∈ × S R 1 C ε 1 C ε 2 → ( θ , ⁢ e / ⁢ 2 i π ν 1 p z 1 , ⁢ e / ⁢ 2 i π ν 2 p z 2 ) , ∈ with ( ν 1 , ν 2 ) × Z p Z p .$

Furthermore, it is possible to combine it with a global gauge transformation in the subgroup $⊂ ⁢ U ( 1 ) m ⁢ U ( m )$. As a result, we obtain an action of $Z p$ on the ADHM data by specialization of the $( + m 2 )$-torus action (), taking

(2.5) Equation 2.5 2.5 $= t ⁢ diag ( e / ⁢ 2 i π c α p ) = α 1 , ⋯ , m , = t 1 e / ⁢ 2 i π ν 1 p , = t 2 e / ⁢ 2 i π ν 2 p .$

This action of the abelian group $Z p$ is parameterized by the $+ m 2$ integers $( c α , ν 1 , ν 2 )$ considered modulo $p$. The transformation of the vector spaces in the decomposition () of $K$ leads to associate to each box , in addition to the complex variables and , the integer such that

(2.6) Equation 2.6 2.6

We call color any integer parameter defined modulo $p$. For short, we also say that $c α$ and $ν 1 , ν 2$ are respectively color of the Coulomb branch vevs, and of the parameters $q 1 , q 2$. The map $: c → λ Z p$ defines a coloring of the $m$-tuple partitions $λ$, and $K$ has a natural decomposition into sectors of a given color ,

(2.7) Equation 2.7 2.7 $= K ⊕ ∈ ω Z p ⁢ K ω ( λ ) .$
Notations

We denote $⁢ C ω ( m )$ the subset of $[ [ 1 , m ] ]$ such that the Coulomb branch vevs $a α$ (or $v α$) with $∈ α ⁢ C ω ( m )$ have color $= c α ω$ (or, equivalently, that the box $∈ ( 1 , 1 ) λ ( α )$ with $∈ α ⁢ C ω ( m )$ is of color $⁢ c ( α , 1 , 1 ) = c α = ω$). Similarly, $⁢ K ω ( λ )$ denotes the set of boxes of the $m$-tuple colored partition $λ$ that carry the color . Besides, in the generic case $≠ + ν 1 ν 2 0$, the shift of color indices $ω$ by the quantity $+ ν 1 ν 2$ appears in many formulas. To simplify these expressions, we introduce the notation $= ¯ ω + ω ν 1 ν 2$ for the shifted indices, along with the map . Finally, we also introduce the extra variables $q 3$ and $ν 3$ such that $= ⁢ q 1 q 2 q 3 1$ and $= + ν 1 ν 2 ν 3 0$. Due to the fact that the $Z p$-action coincides with a subgroup of the torus action, in all formulas the shift of color indices $+ ω ν i$ coincide with a factor $q i$ multiplying the parameters associated to instanton positions in the moduli space.

McKay subgroups in <Math mode="inline" tex="SO(4)" text="S * O * 4" xml:id="S2.SS1.SSS0.Px2.m1"> <XMath> <XMApp> <XMTok meaning="times" role="MULOP">⁢</XMTok> <XMTok font="italic" role="UNKNOWN">S</XMTok> <XMTok font="italic" role="UNKNOWN">O</XMTok> <XMDual> <XMRef idref="S2.SS1.SSS0.Px2.m1.1"/> <XMWrap> <XMTok role="OPEN" stretchy="false">(</XMTok> <XMTok meaning="4" role="NUMBER" xml:id="S2.SS1.SSS0.Px2.m1.1">4</XMTok> <XMTok role="CLOSE" stretchy="false">)</XMTok> </XMWrap> </XMDual> </XMApp> </XMath> </Math>

Although we are considering here a different problem, it is interesting to make a short parallel with the action of $⊂ ⁢ × ⁢ S U ( 2 ) L S U ( 2 ) R ⁢ S O ( 4 )$ on the omega-background (see, for instance, []). This action takes a simpler form if we employ the quaternionic coordinates

(2.8) Equation 2.8 2.8 $= Z ( z 1 - ¯ z 2 z 2 ¯ z 1 ) , ∈ ( z 1 , z 2 ) × C ε 1 C ε 2 .$

Then the $× 2 2$ matrices $∈ ( G L , G R ) ⁢ × ⁢ S U ( 2 ) L S U ( 2 ) R$ act on the quaternions as $→ Z ⁢ G L Z G R$. The McKay subgroups of $⁢ S U ( 2 )$ possess an ADE-classification. For instance, the $A - p 1$ series corresponds to the action of $Z p$, it is generated (multiplicatively) by the diagonal matrices

(2.9) Equation 2.9 2.9 $= G ( e / ⁢ 2 i π p 0 0 e - / ⁢ 2 i π p ) .$

Considering only the action of the $A - p 1$ subgroup on the left, the background coordinates transform as $→ ( z 1 , z 2 ) ( ⁢ e / ⁢ 2 i π p z 1 , ⁢ e - / ⁢ 2 i π p z 2 )$. This transformation can be recovered from the action () of $Z p$ by choosing $ν 1 = - ν 2 = 1$. The orbifold of the spacetime under this action of $Z p$ reproduces the ALE space constructed in []. Instantons of $= N 1$ gauge theories defined on ALE spacetimes have been extensively studied []. In [], their contributions to the gauge theory partition functions have been reproduced using algebraic techniques based on the quantum toroidal algebra of $⁢ g l ( p )$. The generalization to DE-type McKay subgroups with only left action is expected to involve quantum toroidal algebras based on either $⁢ s o ( p )$ or $⁢ s p ( p )$ Lie algebras [].

It is also possible to consider simultaneously the action of two McKay subgroups $A - p 1 1$ and $A - p 2 1$, with one acting on the left, the other on the right. As a result, coordinates now transform as

(2.10) Equation 2.10 2.10 $→ ( z 1 , z 2 ) ( ⁢ e / ⁢ 2 i π ( + p 1 p 2 ) ( ⁢ p 1 p 2 ) z 1 , ⁢ e / ⁢ 2 i π ( - p 1 p 2 ) ( ⁢ p 1 p 2 ) z 2 ) .$

We recognize here another particular case of the $Z p$-action defined in (), albeit more general than before. It is simply obtained by the specialization $= ν 1 + p 1 p 2$, $= ν 2 - p 1 p 2$ and $= p ⁢ p 1 p 2$. Thus, the action () leads to a particularly rich context. Moreover, taking $= ν 1 0$, the first coordinate $z 1$ 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 []. We build here a general algebraic framework to address this kind of problems. It may be possible to further generalize our approach to the action of DE-type McKay subgroups with both left and right actions, but this is beyond the scope of this paper.

2.2 subsection 2.2 2.2 §2.2 <tag close=" ">2.2</tag>Instantons partition function

The computation of the Nekrasov instanton partition function on such $Z p$-orbifolds has been performed in []. 2 footnote 2 2 footnote 2 Strictly speaking, in [], the authors consider the instantons on the minimal resolution of the orbifold. Instead, here, following [], we simply consider the $Z p$-invariant part of the instanton moduli space $M k$. Both approaches should provide the same result []. For simplicity, we do not introduce fundamental matter multiplets, those being obtained in the limit $→ q 0$ of the gauge coupling parameters. Furthermore, we only discuss the case of linear quiver gauge theories $A r$, with $⁢ U ( m ( i ) )$ gauge groups at each node $= i ⁢ 1 ⋯ r$. Thus, the node $i$ carries the following parameters:

item  1st item

a set of colored exponentiated gauge couplings $q ω , i$,

item  2nd item

a $p$-vector of colored Chern-Simons levels $= κ ( i ) ( κ ω ( i ) ) ∈ ω Z p$,

item  3rd item

an $m ( i )$-vector of Coulomb branch vevs $= a ( i ) ( a α ( i ) ) = α 1 m ( i )$ defining the exponentiated parameters $= v ( i ) ( v α ( i ) ) = α 1 m ( i )$ with $= v α ( i ) e ⁢ R a α ( i )$,

item  4th item

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

In addition, each link $→ i j$ between two nodes $i$ and $j$ represent a chiral multiplet of matter fields in the bifundamental representation of the gauge group $⁢ × ⁢ U ( m ( i ) ) U ( m ( j ) )$, with mass $∈ μ ⁢ i j C$. For linear quivers, all bifundamental masses can be set to $q 3 - 1$ 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 $λ ( i )$ 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:

(2.11) Equation 2.11 2.11

Vector and bifundamental contributions are written in terms of the Nekrasov factor $⁢ N ( v , | λ ⁢ μ v ′ , λ ′ )$. 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 $⁢ N ( v , | λ ⁢ μ v ′ , λ ′ )$ given in [], we need to introduce the equivariant character $M v$ and $K λ$ of the vector spaces $M$ and $K$ associated to each node,

(2.12) Equation 2.12 2.12

A linear involutive operation $∗$ acts on such characters by flipping the sign of $R$: $= ( e ⁢ R a α ) ∗ e - ⁢ R a α$, $= ( q 1 ∗ , q 2 ∗ ) ( q 1 - 1 , q 2 - 1 )$ and thus (see [] for more details on these notations). Introducing $= S λ - M ⁢ P 12 K λ$ with $= P 12 ⁢ ( - 1 q 1 ) ( - 1 q 2 )$, the Nekrasov factor writes

(2.13) Equation 2.13 2.13 $⁢ N ( v , | λ v ′ , λ ′ ) = ⁢ I [ - ⁢ M v M v ′ ∗ ⁢ S λ S λ ′ ∗ P 12 ∗ ] Z p = ⁢ I [ - + ⁢ M v K λ ′ * ⁢ q 3 - 1 M v ′ * K λ ⁢ P 12 K λ K λ ′ * ] Z p ,$

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

(2.14) Equation 2.14 2.14 $= ⁢ I [ - ∑ ∈ i I + e ⁢ R w i ∑ ∈ i I - e ⁢ R w i ] - ∏ ∈ i I + 1 e ⁢ R w i - ∏ ∈ i I - 1 e ⁢ R w i ,$

and $[ ⋯ ] Z p$ denotes the operation of keeping only the $Z p$-invariant parts. In particular, the RHS of () involves a coloring function $: c → ⁢ Z [ a α , ε 1 , ε 2 ] Z p$ defined on weights $w i$ as the linear map taking the values $= ⁢ c ( a α ) c α$, $= ⁢ c ( ε 1 ) ν 1$ and $= ⁢ c ( ε 2 ) ν 2$ so that (justifying our slight abuse of notations). The $[ ⋯ ] Z p$ projects on $Z p$-invariant factors.

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

(2.15) Equation 2.15 2.15

The function $⁢ S ⁢ ω ω ′ ( z )$ is sometimes called the scattering function, it carries two color indices $ω , ω ′$:

(2.16) Equation 2.16 2.16 $= ⁢ S ⁢ ω ω ′ ( z ) ⁢ ( - 1 ⁢ q 1 z ) δ ω , - ω ′ ν 1 ( - 1 ⁢ q 2 z ) δ ω , - ω ′ ν 2 ⁢ ( - 1 z ) δ ω , ω ′ ( - 1 ⁢ q 1 q 2 z ) δ ω , - ω ′ ν 1 ν 2 .$

In this expression, the non-zero matrix elements have been expressed in a compact way using the delta function $δ ω , ω ′$ defined modulo $p$ (i.e. $= δ ω , ω ′ 1$ iff $= ω ω ′$ modulo $p$, zero otherwise). In fact, $⁢ S ⁢ ω ω ′ ( z )$, and more generally all the matrices of size $× p p$ with indices $ω , ω ′$ appearing in this paper, are circulant matrices: their matrix elements only depend on the difference $- ω ω ′$ of row and column indices. In particular, $= ⁢ S + ω ⁢ ν ω ′ ( z ) ⁢ S - ⁢ ω ω ′ ν ( z )$ for all $∈ ν Z p$. Finally, the function $⁢ S ⁢ ω ω ′ ( z )$ satisfies a sort of crossing symmetry,

(2.17) Equation 2.17 2.17 $= ⁢ S ⁢ ω ¯ ω ′ ( / q 3 z ) ⁢ f ⁢ ω ω ′ ( z ) S ⁢ ω ′ ω ( z ) ,$

with the function $= ⁢ f ⁢ ω ω ′ ( z ) ⁢ F ⁢ ω ω ′ z β ⁢ ω ω ′$ defined by 3 footnote 3 3 footnote 3 The function $⁢ f ⁢ ω ω ′ ( z )$ also controls the asymptotics of the scattering function since $⁢ S ⁢ ω ω ′ ( z ) ∼ 0 1$ and $⁢ S ⁢ ω ω ′ ( z ) ∼ ∞ f ⁢ ω ′ ω ( z ) - 1$. It obeys an important reflection symmetry $= ⁢ f ⁢ ¯ ω ω ′ ( / q 3 z ) ⁢ f ⁢ ω ′ ω ( z ) - 1$ coming from $= ⁢ F ⁢ ω ω ′ F ⁢ ¯ ω ′ ω q 3 - β ⁢ ω ω ′$ and $= β ⁢ ¯ ω ′ ω β ⁢ ω ω ′$.

(2.18) Equation 2.18 2.18 $= β ⁢ ω ω ′ - + δ ⁢ ω ω ′ δ + ⁢ ω ω ′ ν 1 ν 2 δ + ⁢ ω ω ′ ν 1 δ + ⁢ ω ω ′ ν 2 , = F ⁢ ω ω ′ ⁢ ( - 1 ) δ ⁢ ω ω ′ ( - q 3 ) - δ ω , - ω ′ ν 3 ( - q 1 ) - δ + ⁢ ω ω ′ ν 1 ( - q 2 ) - δ + ⁢ ω ω ′ ν 2 .$
2.3 subsection 2.3 2.3 §2.3 <tag close=" ">2.3</tag><Math mode="inline" tex="{\mathcal{Y}}" text="Y" xml:id="S2.SS3.m1"> <XMath> <XMTok font="caligraphic" role="UNKNOWN">Y</XMTok> </XMath> </Math>-observables

A new class of BPS-observables for supersymmetric gauge theories was introduced in [], they are called $⁢ q q$-characters. As the name suggests, they correspond to a natural deformation of the $q$-characters of Frenkel-Reshetikhin [] from the gauge theory point of view []. They were defined in [] as particular combinations of chiral ring observables in such a way that their expectation values exhibit an important regularity property []. This regularity property encodes an infinite set of constraints called non-perturbative Dyson-Schwinger equations. From a different viewpoint, $⁢ q q$-characters in 5D gauge theories can also be defined in terms of Wilson loops [] (see also [] for a string theory perspective). 4 footnote 4 4 footnote 4 In [], the $⁢ q q$-characters of 4D $= N 2$ gauge theories with the insertion of surface defects were considered. In this case, the non-perturbative Dyson-Schwinger equations produce either Knizhnik - Zamolodchikov equations or BPZ equations that are satisfied by the surface defect partition functions []. These surface defect partition functions were investigated in the context of Bethe/gauge correspondence in [], and in their relation to the oper submanifold of the moduli space of flat connections on Riemann surfaces in [].

The $⁢ q q$-characters are half-BPS observables written as combinations of $Y$-observables. In the case of a $Z p$-orbifold, it is natural to introduce two inequivalent $Y$-observables $= ⁢ Y ω [ λ ] ( z ) ⁢ I [ ⁢ e - ⁢ R ζ S λ ] Z p$ and $= ⁢ Y ω [ λ ] ⁣ ∗ ( z ) ⁢ I [ ⁢ e ⁢ R ζ S λ ∗ ] Z p$ where $= z e ⁢ R ζ$ and $= ⁢ c ( ζ ) ω$. These two observables encode the recursion relations satisfied by Nekrasov factors,

(2.19) Equation 2.19 2.19

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

(2.20) Equation 2.20 2.20

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

(2.21) Equation 2.21 2.21 $= ⁢ Y ω [ λ ] ⁣ ∗ ( z ) ⁢ f ω [ λ ] ( z ) Y ω [ λ ] ( z ) ,$ $= ⁢ Y ω [ λ ] ⁣ ∗ ( z ) ⁢ f ω [ λ ] ( z ) Y ω [ λ ] ( z ) ,$

with

(2.22) Equation 2.22 2.22

It will allow us to express all the equations below in terms of $⁢ Y ω [ λ ] ( z )$ only. 5 footnote 5 5 footnote 5 The presence of the function $⁢ f ω [ λ ] ( z )$ can be interpreted as follows. Note that $= ⁢ I ( X ∗ ) ⁢ ( - 1 ) ⁢ rk X ∗ det ⁢ X ∗ I ( X )$, for $= X - ∑ ∈ i I + e ⁢ R w i ∑ ∈ i I - e ⁢ R w i$, $= ⁢ rk X - | I + | | I - |$, and $= det X ∏ ∈ i I + / e ⁢ R w i ∏ ∈ i I - e ⁢ R w i$. Applying this reflection relation to $= X [ ⁢ e - ⁢ R ζ S λ ] Z p$, we recover the relation () with $⁢ f ω [ λ ] ( z )$ given in () identified with $⁢ ( - 1 ) ⁢ rk X ∗ det X ∗$. Furthermore, the $Y$-observables possess an alternative expression following from the shell formula derived in appendix ,

(2.23) Equation 2.23 2.23

Here, the sets $⁢ A ω ( λ )$ and $⁢ R ω ( λ )$ denote respectively the set of boxes of color $ω$ that can be added to or removed from the $m$-tuple Young diagram $λ$. 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.

3 section 3 3 §3 <tag close=" ">3</tag>New quantum toroidal algebras

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 $q 1 , q 2$ and the rank $∈ p Z > 0$, this algebra will depend on the integers $( ν 1 , ν 2 )$ modulo $Z p$. Taking $ν 1 = - ν 2 = 1$, the $Z p$-action () reduces to the standard action defining ALE spaces. Thus, in this limit the $( ν 1 , ν 2 )$-deformed algebra should reduce to the quantum toroidal algebra of $⁢ g l ( p )$. 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 $⁢ g l ( p )$ is given in appendix , it includes its two main representations called, in the gauge theory context, vertical and horizontal representations.

The key ingredient to define the deformation of the quantum toroidal algebra of $⁢ g l ( p )$ is the scattering function $⁢ S ⁢ ω ω ′ ( z )$ defined in (). Indeed, this function plays an essential role in the two elementary representations involved in the algebraic engineering of partition functions and $⁢ q q$-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 $× p p$ matrix $β ⁢ ω ω ′$ that could be identified with the underlying Cartan matrix of the deformed quantum toroidal algebra (see the subsection ). Note that the matrix $β ⁢ ω ω ′$ naturally reduces to the generalized Cartan matrix of the Kac-Moody algebra $^ ⁢ g l p$ when $ν 1 = - ν 2 = 1$. In general, it is non-symmetrizable, yet, like in the case of $^ ⁢ g l p$, it is a circulant matrix. Its eigenvectors $= v j ( 1 , Ω j , Ω j 2 , ⋯ , Ω j - p 1 )$ are written in terms of the $p$th root of unity $= Ω j e / ⁢ 2 i π j p$, and the corresponding eigenvalues read

(3.1) Equation 3.1 3.1 $= λ j - ⁢ 4 e / ⁢ i π ν 3 j p sin ( / ⁢ π ν 1 j p ) sin ( / ⁢ π ν 2 j p ) .$

In particular, the eigenvector $= v 0 ( 1 , 1 , ⋯ , 1 )$ has the eigenvalue zero which relates $β ⁢ ω ω ′$ to the Cartan matrix of affine Lie algebras, and thus justifies the designation toroidal of the deformed algebra.

3.1 subsection 3.1 3.1 §3.1 <tag close=" ">3.1</tag>Definition of the algebra

Like in the case of $⁢ g l ( p )$, the $( ν 1 , ν 2 )$-deformed quantum toroidal algebra is defined in terms of a central element $c$ and $⁢ 4 p$ Drinfeld currents, denoted $⁢ x ω ± ( z )$ and $⁢ ψ ω ± ( z )$, with $∈ ω Z p$. The currents $⁢ ψ ω ± ( z )$ (together with $c$) generate the Cartan subalgebra, while the currents $⁢ x ω ± ( z )$ deform the notion of Chevalley generators $e ω , f ω$. The algebraic relations obeyed by the currents resemble those defining the quantum toroidal algebra of $⁢ g l ( p )$ in (), the main difference being the presence of shifts in the indices $ω$ by the product $⁢ ν 3 c$: 6 footnote 6 6 footnote 6 Comparing with the standard definition of quantum toroidal algebras, the Drinfeld currents have been redefined as follows: $→ ⁢ x ± ( z ) ⁢ x ± ( ⁢ q 3 ± / c 4 z )$, $→ ⁢ ψ ω + ( z ) ⁢ ψ ω + ( z )$ and $→ ⁢ ψ ω - ( z ) ⁢ ψ ω - ( ⁢ q 3 - / c 2 z )$. This redefinition makes the coincidence between shifts of indices $± ω ⁢ ν 3 c$ and spectral parameters $⁢ z q 3 ± c$ manifest. In fact, this asymmetric form of the algebraic relations appears naturally in the construction of a central extension of the Yangian double [].

(3.2) Equation 3.2 3.2 $= ⁢ x ω ± ( z ) x ω ′ ± ( w ) ⁢ g ⁢ ω ω ′ ( / z w ) ± 1 x ω ′ ± ( w ) x ω ± ( z ) , = ⁢ ψ + ω ( z ) x ± ω ′ ( w ) ⁢ g ⁢ ω ω ′ ( / z w ) ± 1 x ± ω ′ ( w ) ψ + ω ( z ) , = ⁢ ψ - ω ( z ) x + ω ′ ( w ) ⁢ g - ω ⁢ ν 3 c ω ′ ( / ⁢ q 3 - c z w ) x + ω ′ ( w ) ψ - ω ( z ) , = ⁢ ψ - ω ( z ) x - ω ′ ( w ) ⁢ g ⁢ ω ω ′ ( / z w ) - 1 x - ω ′ ( w ) ψ - ω ( z ) , = ⁢ ψ ω + ( z ) ψ ω ′ - ( w ) ⁢ ⁢ g - ⁢ ω ω ′ ⁢ ν 3 c ( / ⁢ q 3 c z w ) ⁢ g ⁢ ω ω ′ ( / z w ) ψ ω ′ - ( w ) ψ ω + ( z ) , = [ ⁢ ψ ω ± ( z ) , ⁢ ψ ω ′ ± ( w ) ] 0 , = [ ⁢ x ω + ( z ) , ⁢ x ω ′ - ( w ) ] ⁢ Ω [ - ⁢ δ ω , ω ′ δ ( / z w ) ψ ω + ( z ) ⁢ δ ω , - ω ′ ⁢ ν 3 c δ ( / ⁢ q 3 c z w ) ψ - + ω ⁢ ν 3 c ( ⁢ q 3 c z ) ] .$ $= ⁢ x ω ± ( z ) x ω ′ ± ( w ) ⁢ g ⁢ ω ω ′ ( / z w ) ± 1 x ω ′ ± ( w ) x ω ± ( z ) , = ⁢ ψ + ω ( z ) x ± ω ′ ( w ) ⁢ g ⁢ ω ω ′ ( / z w ) ± 1 x ± ω ′ ( w ) ψ + ω ( z ) , = ⁢ ψ - ω ( z ) x + ω ′ ( w ) ⁢ g - ω ⁢ ν 3 c ω ′ ( / ⁢ q 3 - c z w ) x + ω ′ ( w ) ψ - ω ( z ) , = ⁢ ψ - ω ( z ) x - ω ′ ( w ) ⁢ g ⁢ ω ω ′ ( / z w ) - 1 x - ω ′ ( w ) ψ - ω ( z ) , = ⁢ ψ ω + ( z ) ψ ω ′ - ( w ) ⁢ ⁢ g - ⁢ ω ω ′ ⁢ ν 3 c ( / ⁢ q 3 c$