Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Quiver

New quantum toroidal algebras
from supersymmetric gauge theories
Jean-Emile Bourgine
Korea Institute for Advanced Study

New trends in Integrable Systems
Osaka City University
2019-09-14

JEB, Saebyeok Jeong [arXiv:1906.01625]

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverWhat are quantum toroidal algebras?
In a nutshell
Toroidal algebras are central extensions of 2-loop algebras (= double affine algebras).
Quantum toroidal algebras are affine versions of quantum affine algebras.

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverWhat are quantum toroidal algebras?
In a nutshell
Toroidal algebras are central extensions of 2-loop algebras (= double affine algebras).
Quantum toroidal algebras are affine versions of quantum affine algebras.

Main applications:
Construction of integrable systems using the Hopf algebra structure[Feigin, Jimbo, Miwa, Mukhin 2015]

Non-perturbative symmetries of supersymmetric (SUSY) gauge theories
From string theory realizations: (p,q)-branes web or topological strings[Awata, Feigin, Shiraishi 2011]. Main motivation for introducing our deformation of quantum toroidal gl(p).
Correspondence with W-algebras (AGT correspondence)[Awata, Feigin, Hoshino, Kanai, Shiraishi, Yanagida 2011]Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Outline1

Introduction

2

Quantum toroidal algebras

3

Deformation of quantum toroidal gl(p)4Representations and application to gauge theories

5

Perspectives

6

General approach to quiver representations

Representations

Perspectives

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Toroidal algebras
Consider a simple Lie algebra g with Chevalley basis xω+ , xω− , hω (ω = 1 · · · rank).
!
Notations: xω+ ≡ eω ≡ eαω , xω− ≡ fω ≡ e−αω .
Example: sl(2) generators x ± , h: [h, x ± ] = ±2x ± , [x + , x − ] = h.

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Toroidal algebras
Consider a simple Lie algebra g with Chevalley basis xω+ , xω− , hω (ω = 1 · · · rank).
!
Notations: xω+ ≡ eω ≡ eαω , xω− ≡ fω ≡ e−αω .
Example: sl(2) generators x ± , h: [h, x ± ] = ±2x ± , [x + , x − ] = h.
Affinization is achieved by one of two equivalent methods:1

Add an extra root to obtain a generalized Cartan matrix Cωω0d xω± , hω (ω = 0, 1),
Example sl(2):±[hω , xω0 ] = ±Cωω0 xω±0 , [xω+ , xω−0 ] = δωω0 hω0 .

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Toroidal algebras
Consider a simple Lie algebra g with Chevalley basis xω+ , xω− , hω (ω = 1 · · · rank).
!
Notations: xω+ ≡ eω ≡ eαω , xω− ≡ fω ≡ e−αω .
Example: sl(2) generators x ± , h: [h, x ± ] = ±2x ± , [x + , x − ] = h.
Affinization is achieved by one of two equivalent methods:
12

Add an extra root to obtain a generalized Cartan matrix Cωω0d xω± , hω (ω = 0, 1),
Example sl(2):±[hω , xω0 ] = ±Cωω0 xω±0 , [xω+ , xω−0 ] = δωω0 hω0 .
Central extension of the loop algebra: C[t, t −1 ] ⊗ g ⊕ Cc.
d x ± = t k ⊗ x ± , hk = t k ⊗ h,
Example sl(2):k±±[hk , xl ] = ±2xk+l, [xk+ , xl− ] = hk+l + ckδk+l , [hk , hl ] = ckδk+l

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Toroidal algebras
Consider a simple Lie algebra g with Chevalley basis xω+ , xω− , hω (ω = 1 · · · rank).
!
Notations: xω+ ≡ eω ≡ eαω , xω− ≡ fω ≡ e−αω .
Example: sl(2) generators x ± , h: [h, x ± ] = ±2x ± , [x + , x − ] = h.
Affinization is achieved by one of two equivalent methods:
12

Add an extra root to obtain a generalized Cartan matrix Cωω0d xω± , hω (ω = 0, 1),
Example sl(2):±[hω , xω0 ] = ±Cωω0 xω±0 , [xω+ , xω−0 ] = δωω0 hω0 .
Central extension of the loop algebra: C[t, t −1 ] ⊗ g ⊕ Cc.
d x ± = t k ⊗ x ± , hk = t k ⊗ h,
Example sl(2):k±±[hk , xl ] = ±2xk+l, [xk+ , xl− ] = hk+l + ckδk+l , [hk , hl ] = ckδk+lP
P
Introduce the currents h(z) = k∈Z z −k hk , x ± (z) = k∈Z z −k xk± .
P[h(z), x ± (w )] = ±2δ(z/w )x ± (z), with δ(z) = k∈Z z k .

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Toroidal algebras
Consider a simple Lie algebra g with Chevalley basis xω+ , xω− , hω (ω = 1 · · · rank).
!
Notations: xω+ ≡ eω ≡ eαω , xω− ≡ fω ≡ e−αω .
Example: sl(2) generators x ± , h: [h, x ± ] = ±2x ± , [x + , x − ] = h.
Affinization is achieved by one of two equivalent methods:
12

Add an extra root to obtain a generalized Cartan matrix Cωω0d xω± , hω (ω = 0, 1),
Example sl(2):±[hω , xω0 ] = ±Cωω0 xω±0 , [xω+ , xω−0 ] = δωω0 hω0 .
Central extension of the loop algebra: C[t, t −1 ] ⊗ g ⊕ Cc.
d x ± = t k ⊗ x ± , hk = t k ⊗ h,
Example sl(2):k±±[hk , xl ] = ±2xk+l, [xk+ , xl− ] = hk+l + ckδk+l , [hk , hl ] = ckδk+lP
P
Introduce the currents h(z) = k∈Z z −k hk , x ± (z) = k∈Z z −k xk± .
P[h(z), x ± (w )] = ±2δ(z/w )x ± (z), with δ(z) = k∈Z z k .
Toroidal algebras were formulated by combining these two methods.
[Moody, Rao, Yokonuma 1990]

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Quantization. Quantization is used to define a non-trivial coalgebraic structure.
Reminder: coalgebras have a coproduct ∆ : A → A ⊗ A, a counit  : A → C
Hopf algebra: antipode S : A → A such that ∇(S ⊗ 1)∆ = ∇(1 ⊗ S)∆ = .
(R-matrix, Yang-Baxter equation, quantum integrable systems,...)

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverQuantization. Quantization is used to define a non-trivial coalgebraic structure.
Reminder: coalgebras have a coproduct ∆ : A → A ⊗ A, a counit  : A → C
Hopf algebra: antipode S : A → A such that ∇(S ⊗ 1)∆ = ∇(1 ⊗ S)∆ = .
(R-matrix, Yang-Baxter equation, quantum integrable systems,...). Replace the Cartan sector with operators ψω± in the universal envelopping algebra.
A parameter q ∈ C× is also introduced.
Example Uq (sl(2)): generators x ± , ψ ± = q ±h ,
ψ + x ± = q ±2 x ± ψ + ,ψ − x ± = q ∓2 x ± ψ − ,∆(x + ) = x + ⊗ 1 + ψ − ⊗ x + ,S(x + ) = −(ψ − )−1 x + ,(x ± ) = 0,

(ψ ± ) = 1.

[x + , x − ] =ψ+ − ψ−q − q −1∆(x − ) = 1 ⊗ x − + x − ⊗ ψ + ,S(x − ) = −x − (ψ + )−1 ,∆(ψ ± ) = ψ ± ⊗ ψ ±S(ψ ± ) = (ψ ± )−1 ,Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverQuantization. Quantization is used to define a non-trivial coalgebraic structure.
Reminder: coalgebras have a coproduct ∆ : A → A ⊗ A, a counit  : A → C
Hopf algebra: antipode S : A → A such that ∇(S ⊗ 1)∆ = ∇(1 ⊗ S)∆ = .
(R-matrix, Yang-Baxter equation, quantum integrable systems,...). Replace the Cartan sector with operators ψω± in the universal envelopping algebra.
A parameter q ∈ C× is also introduced.
Example Uq (sl(2)): generators x ± , ψ ± = q ±h ,
ψ + x ± = q ±2 x ± ψ + ,ψ − x ± = q ∓2 x ± ψ − ,∆(x + ) = x + ⊗ 1 + ψ − ⊗ x + ,S(x + ) = −(ψ − )−1 x + ,(x ± ) = 0,

[x + , x − ] =ψ+ − ψ−q − q −1∆(x − ) = 1 ⊗ x − + x − ⊗ ψ + ,S(x − ) = −x − (ψ + )−1 ,∆(ψ ± ) = ψ ± ⊗ ψ ±S(ψ ± ) = (ψ ± )−1 ,(ψ ± ) = 1.

Obtain quantum toroidal algebras [Ginzburg, Kapranov, Vasserot 1995]

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Quantum toroidal gl(p) : definition
When g = gl(p), an extra parameter κ ∈ C× can be introduced. We use:
q1 = q −1 κ,

q2 = q −1 κ−1 ,q3 = q 2⇒

q1 q2 q3 = 1.

The algebra is formulated in terms of a central element c and 4p Drinfeld currents
X −k ±X ∓k ±xω± (z) =z xω,k , ψω± (z) =z ψω,±k .
k∈Zk≥0

It has a second central element c̄ obtained as∓ 1 c̄

q3 2 =

p−1

Y
ω=0

±
ψω,0
.

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverQuantum toroidal gl(p) : definition
The algebraic relations read±c/4

ψω+ (z)xω±0 (w ) = gωω0 (q3z/w )±1 xω±0 (w )ψω± (z),∓c/4ψω− (z)xω±0 (w ) = gωω0 (q3[ψω± (z), ψω±0 (w )] = 0,z/w )±1 xω±0 (w )ψω− (z),

+
−
−
+
ψω,0
ψω,0
= ψω,0ψω,0=1

c/2
ψω−0 (w )ψω+ (z),
−c/2
0
z/w )
gωω (q3xω± (z)xω±0 (w ) = gωω0 (z/w )±1 xω±0 (w )xω± (z),ψω+ (z)ψω−0 (w ) =gωω0 (q3 z/w )[xω+ (z), xω−0 (w )] =

δω,ω0
1/2
−1/2
q3 − q3

h

−c/2

δ(q3

−c/4

z/w )ψω+ (q3

i
c/2
c/4
z) − δ(q3 z/w )ψω− (q3 z) ,together with the Serre relations

i
X h ±
1/2
−1/2 ±
±
±
±
±
±
±
±
xω (zσ(1) )xω (zσ(2) )xω±1 (w ) − (q3 + q3
)xω (zσ(1) )xω±1 (w )xω (zσ(2) ) + xω±1 (w )xω (zσ(1) )xω (zσ(2) ) = 0.
σ∈S2Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverQuantum toroidal gl(p) : definition
The structure functions gωω0 (z) are defined as (δω,ω0 is the Kronecker delta mod p)gωω0 (z) =q3−1

1 − q3 z
1 − q3−1 zδω,ω0 δω,ω0 −1 δω,ω0 +1
1/2 1 − q1 z
1/2 1 − q2 z
q3
q3
1 − q2−1 z
1 − q1−1 z

Example: For p = 6,
1 − q3 z
q −1
 3 1 − q −1 z3
 1/2 1 − q2 zq
 31 − q1−1 z



0




0





0



 1/2 1 − q1 z
q3
1 − q2−1 z


1/2

q3

1 − q1 z

1 − q2−1 z−1 1 − q3 z

q3

1 − q3−1 z
1/2 1 − q2 z

q3

1 − q1−1 z
0

0
1/2

q3

1 − q1 z

1 − q2−1 z−1 1 − q3 z

q3

1/2
q3

1 − q3−1 z
1 − q2 z
1 − q1−1 z

0

0

0

0

1/2
q3

q3−1
1/2

q3

0

0

0

0

1 − q1 z
1 − q2−1 z
1 − q3 z
1 − q3−1 z
1 − q2 z
1 − q1−1 z
0

0
1/2
q3

q3−1
1/2

q3

1 − q1 z
1 − q2−1 z
1 − q3 z
1 − q3−1 z
1 − q2 z
1 − q1−1 z

1/2

q3

1 − q2 z 

1 − q1−1 z 



0





0




0




1/2 1 − q1 z 
q3

−1 
1 − q2 z 
1 − q3 z q3−1
1 − q3−1 zIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverQuantum toroidal gl(p): coalgebraic structure
The algebra has the structure of a Hopf algebra with the Drinfeld coproduct
c

/4

c

/2

∆(xω+ (z)) = xω+ (z) ⊗ 1 + ψω− (q3(1) z) ⊗ xω+ (q3(1) z),
c

/2

c

/4

∆(xω− (z)) = xω− (q3(2) z) ⊗ ψω+ (q3(2) z) + 1 ⊗ xω− (z),
±c

/4

∓c

/4

∆(ψω± (z)) = ψω± (q3 (2) z) ⊗ ψω± (q3 (1) z),
the counit (xω± (z)) = 0, (ψω± (z)) = 1, and the antipode−c/4

S(xω+ (z)) = −ψω− (q3−c/2z)−1 xω+ (q3

z),

−c/2

S(xω− (z)) = −xω− (q3−c/4z)ψω+ (q3z)−1 ,

S(ψω± (z)) = ψω± (z)−1 .

Remarks:
F We denoted c(1) = c ⊗ 1, c(2) = 1 ⊗ c, and ∆(c) = c(1) + c(2) , (c) = 0, S(c) = −c.
F We recover the Ding-Iohara-Miki algebra when p = 1.
[Ding, Iohara 1997 - Miki 2007]

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Outline1

Introduction

2

Quantum toroidal algebras

3

Deformation of quantum toroidal gl(p)4Representations and application to gauge theories

5

Perspectives

6

General approach to quiver representations

Representations

Perspectives

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Motivation• In applications to SUSY gauge theories, two types of representations are used.
In both representations, the following functions play a central role:Sωω0 (z) =(1 − q1 z)δω,ω0 −1 (1 − q2 z)δω,ω0 +1,(1 − z)δω,ω0 (1 − q1 q2 z)δω,ω0−1C

gωω0 (z) = q3 2 ωω0Sωω0 (z)Sωω0 (q3 z)

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverMotivation• In applications to SUSY gauge theories, two types of representations are used.
In both representations, the following functions play a central role:Sωω0 (z) =(1 − q1 z)δω,ω0 −1 (1 − q2 z)δω,ω0 +1,(1 − z)δω,ω0 (1 − q1 q2 z)δω,ω0−1C

gωω0 (z) = q3 2 ωω0Sωω0 (z)Sωω0 (q3 z)• Some SUSY gauge theories observables can be constructed purely algebraically.
For quantum toroidal gl(p), theories are defined on the spacetime S 1 × (C × C)Zp .
The Zp -action (θ, z1 , z2 ) ∈ S 1 × C × C → (θ, e 2iπ/p z1 , e −2iπ/p z2 ) defines an orbifold.
[Awata, Kanno, Mironov, Morozov, Suetake, Zenkevich 2017]

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverMotivation• In applications to SUSY gauge theories, two types of representations are used.
In both representations, the following functions play a central role:Sωω0 (z) =(1 − q1 z)δω,ω0 −1 (1 − q2 z)δω,ω0 +1,(1 − z)δω,ω0 (1 − q1 q2 z)δω,ω0−1C

gωω0 (z) = q3 2 ωω0Sωω0 (z)Sωω0 (q3 z)• Some SUSY gauge theories observables can be constructed purely algebraically.
For quantum toroidal gl(p), theories are defined on the spacetime S 1 × (C × C)Zp .
The Zp -action (θ, z1 , z2 ) ∈ S 1 × C × C → (θ, e 2iπ/p z1 , e −2iπ/p z2 ) defines an orbifold.
[Awata, Kanno, Mironov, Morozov, Suetake, Zenkevich 2017]• The Zp -action can be generalized with two integers (ν1 , ν2 ) ∈ Zp × Zp ,(θ, z1 , z2 ) ∈ S 1 × C × C → (θ, e 2iπν1 /p z1 , e 2iπν2 /p z2 ).
This deformation of the orbifold leads to (ν1 , ν2 )-dependent functions
δSωω0 (z) =δ(1 − q1 z) ω,ω0 −ν1 (1 − q2 z) ω,ω0 −ν2(1 − z)δω,ω0(1 − q1 q2 z)δω,ω0 −ν −ν12.

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Definition of the (ν1 , ν2 )-deformed algebra. How is the deformation of quantum toroidal gl(p) defined?

Perspectives

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Definition of the (ν1 , ν2 )-deformed algebra. How is the deformation of quantum toroidal gl(p) defined?
The structure functions for the deformed algebra are defined asgωω0 (z) =Sωω0 (z),Sω0 ω (z −1 )(ω, ω 0 ∈ Zp ).

The algebraic relations between currents needs to be deformed into
ψω+ (z)xω±0 (w ) = gωω0 (z/w )±1 xω±0 (w )ψω+ (z),ψω− (z)xω−0 (w ) = gωω0 (z/w )−1 xω−0 (w )ψω− (z),ψω− (z)xω+0 (w ) = gω−ν3 c ω0 (q3−c z/w )xω+0 (w )ψω− (z),ψω+ (z)ψω−0 (w ) =gωω0 −ν3 c (q3c z/w ) −ψω0 (w )ψω+ (z),gωω0 (z/w )[ψω± (z), ψω±0 (w )] = 0,xω± (z)xω±0 (w ) = gωω0 (z/w )±1 xω±0 (w )xω± (z),−[xω+ (z), xω−0 (w )] = Ω δω,ω0 δ(z/w )ψω+ (z) − δω,ω0 −ν3 c δ(q3c z/w )ψω+ν
(q3c z) .
3c
where we have introduced ν3 = −ν1 − ν2 .

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Important remarks±,
The Cartan currents now contain some zero modes aω,0±

ψω± (z) = z ∓aω,0

X
k≥0

±
z ∓k ψω,±k.

Perspectives

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Important remarks±,
The Cartan currents now contain some zero modes aω,0±

ψω± (z) = z ∓aω,0X±
z ∓k ψω,±k.

k≥0

The algebra does not quite reduce to quantum toroidal gl(p) as ν1 = −ν2 = 1.

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverImportant remarks±,
The Cartan currents now contain some zero modes aω,0±

ψω± (z) = z ∓aω,0X±
z ∓k ψω,±k.

k≥0

The algebra does not quite reduce to quantum toroidal gl(p) as ν1 = −ν2 = 1.
The coincidence between shifts in the arguments q3±c z and indices ω ± ν3 c follows
from the definition of the Zp -action in the gauge theory.

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverImportant remarks±,
The Cartan currents now contain some zero modes aω,0±

ψω± (z) = z ∓aω,0X±
z ∓k ψω,±k.

k≥0

The algebra does not quite reduce to quantum toroidal gl(p) as ν1 = −ν2 = 1.
The coincidence between shifts in the arguments q3±c z and indices ω ± ν3 c follows
from the definition of the Zp -action in the gauge theory.
Well defined only if ρ(c) ∈ Z.

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverImportant remarks±,
The Cartan currents now contain some zero modes aω,0±

ψω± (z) = z ∓aω,0X±
z ∓k ψω,±k.

k≥0

The algebra does not quite reduce to quantum toroidal gl(p) as ν1 = −ν2 = 1.
The coincidence between shifts in the arguments q3±c z and indices ω ± ν3 c follows
from the definition of the Zp -action in the gauge theory.
Well defined only if ρ(c) ∈ Z.
The comparison with quantum toroidal gl(p) leads to the following conjecture:
this algebra is equivalent to the quantum toroidal algebra built upon a
Kac-Moody algebra with the non-symmetrizable Cartan matrixCωω0 = δωω0 + δωω0 +ν1 +ν2 − δωω0 +ν1 − δωω0 +ν2 .

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Typically,


1

0

−1


Cωω0 =  0

−1


0
1

1
1
0
−1
0
−1
0

0
1
1
0
−1
0
−1

−1
0
1
1
0
−1
0

0
−1
0
1
1
0
−1

−1
0
−1
0
1
1
0


0

−1

0


−1

0


1
1

Perspectives

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverCoalgebraic structure
The (ν1 , ν2 )-deformed algebra has the structure of a Hopf algebra with the coproductc−∆(xω+ (z)) = xω+ (z) ⊗ 1 + ψω+ν(q3(1) z) ⊗ xω+ (z),3 c(1)−c−c+−∆(xω− (z)) = xω− (z) ⊗ ψω−ν(q3 (1) z) + 1 ⊗ xω−ν(q3 (1) z),
3 c(1)
3 c(1)−c+∆(ψω+ (z)) = ψω+ (z) ⊗ ψω−ν(q3 (1) z),
3 c(1)−c−c−−∆(ψω− (z)) = ψω−ν(q3 (2) z) ⊗ ψω−ν(q3 (1) z),
3 c(2)
3 c(1)

the counit (xω± (z)) = 0, (ψω± (z)) = 1, and the antipode−(q3c z)−1 xω+ (z),S(xω+ (z)) = −ψω+ν
3c+S(ψω+ (z)) = ψω+ν
(q3c z)−1 ,3c−+S(xω− (z)) = −xω+ν
(q3c z)ψω+ν
(q3c z)−1 ,3c3c−S(ψω− (z)) = ψω+2ν
(q32c z)−1 ,3c

The coproduct has been twisted to make manifest the coincidence between shifts inarguments and indices.

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Outline1

Introduction

2

Quantum toroidal algebras

3

Deformation of quantum toroidal gl(p)4Representations and application to gauge theories

5

Perspectives

6

General approach to quiver representations

Representations

Perspectives

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Vertical representations• Deform the highest weight modules of quantum toroidal gl(p)(analogous to finite dimensional representations of quantum affine algebras)[Feigin, Jimbo, Miwa, Mukhin 2012]

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Vertical representations• Deform the highest weight modules of quantum toroidal gl(p)(analogous to finite dimensional representations of quantum affine algebras)[Feigin, Jimbo, Miwa, Mukhin 2012]• Levels ρV (c) = 0, ρV (c̄) = n (⇒ ψ + (z) and ψ − (w ) commute!)

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Vertical representations• Deform the highest weight modules of quantum toroidal gl(p)(analogous to finite dimensional representations of quantum affine algebras)[Feigin, Jimbo, Miwa, Mukhin 2012]• Levels ρV (c) = 0, ρV (c̄) = n (⇒ ψ + (z) and ψ − (w ) commute!)• Highest state |∅ii determined by n weights v = (vα )nα=1 and n colors cα ∈ Zp .

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Vertical representations• Deform the highest weight modules of quantum toroidal gl(p)(analogous to finite dimensional representations of quantum affine algebras)[Feigin, Jimbo, Miwa, Mukhin 2012]• Levels ρV (c) = 0, ρV (c̄) = n (⇒ ψ + (z) and ψ − (w ) commute!)• Highest state |∅ii determined by n weights v = (vα )nα=1 and n colors cα ∈ Zp .
• The action of the Cartan reads"#−1/2pω−ν3 (q3z)
± −1ψω (z ) |∅ii =
|∅ii,
1/2
pω (q3 z)∓pω (z) =

n
Y

−1/2

(1 − zq3vα )δcα ,ω ,α=1where pω (z) is the Drinfeld polynomial and [f (z)]± denotes an expansion of f (z) in
powers of z ∓1 .

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Quiver

Vertical representations• Vertical modules have a basis of states |λii labelled by n-tuples Young diagrams λ.
To each box = (α, i, j) ∈ λ of coordinates (i, j) ∈ λ(α) , we associate:a position χ = vα q1i−1 q2j−1 ∈ C× ,a color c( ) = cα + (i − 1)ν1 + (j − 1)ν2 ∈ Zp .

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Quiver

Vertical representations• Vertical modules have a basis of states |λii labelled by n-tuples Young diagrams λ.
To each box = (α, i, j) ∈ λ of coordinates (i, j) ∈ λ(α) , we associate:a position χ = vα q1i−1 q2j−1 ∈ C× ,a color c( ) = cα + (i − 1)ν1 + (j − 1)ν2 ∈ Zp .
• The action of the Drinfeld currents on this basis readXρV (xω+ (z)) |λii =δ(z/χ )Yω[λ+ ] (χ ) |λ + ii,∈Aω (λ)∗[λ− ]X

ρV (xω− (z)) |λii =δ(z/χ )Yω+ν1 +ν2 (q3−1 χ ) |λ − ii,∈Rω (λ)"ρV (ψω± (z)) |λii =∗[λ]Yω+ν1 +ν2 (q3−1 z)

#
|λii.

[λ]

Yω (z)±• Aω (λ)/Rω (λ) = set of boxes of color c( ) = ω that can be added/removed to λ.
[λ]∗[λ]• Matrix elements are written in terms of Y-observables Yω (z), Yω (z).
• The highest state |∅ii corresponds to empty Young diagrams.

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Horizontal representations• Deform the vertex representations of quantum toroidal gl(p). [Saito 1996]

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Horizontal representations• Deform the vertex representations of quantum toroidal gl(p). [Saito 1996]• Levels ρH (c) = 1, ρH (c̄) = n
Representations depend on p weights uω ∈ C× and p integers nω ∈ Z.

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverHorizontal representations• Deform the vertex representations of quantum toroidal gl(p). [Saito 1996]• Levels ρH (c) = 1, ρH (c̄) = n
Representations depend on p weights uω ∈ C× and p integers nω ∈ Z.
• Formulated in terms of p coupled Heisenberg algebras with modes αω,k ,hik/2[αω,k , αω0 ,l ] = kδk+l q3
δωω0 + q3−k δω ω0 −ν3 − q1k δω ω0 +ν1 − q2k δω ω0 +ν2 ,(k > 0).

and the zero modes Pω (z), Qω (z) (written with 2p finite Heisenberg algebras),Pω (z)Qω0 (w ) = Fωω0 w Cωω0 z −Cωω0 Qω0 (w )Pω (z).
−δω,ω0 −ν

with Fωω0 = (−1)δωω0 (−q3 )3(−q1 )−δωω0 +ν1(−q2 )−δωω0 +ν2.

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverHorizontal representations• Deform the vertex representations of quantum toroidal gl(p). [Saito 1996]• Levels ρH (c) = 1, ρH (c̄) = n
Representations depend on p weights uω ∈ C× and p integers nω ∈ Z.
• Formulated in terms of p coupled Heisenberg algebras with modes αω,k ,hik/2[αω,k , αω0 ,l ] = kδk+l q3
δωω0 + q3−k δω ω0 −ν3 − q1k δω ω0 +ν1 − q2k δω ω0 +ν2 ,(k > 0).

and the zero modes Pω (z), Qω (z) (written with 2p finite Heisenberg algebras),Pω (z)Qω0 (w ) = Fωω0 w Cωω0 z −Cωω0 Qω0 (w )Pω (z).
−δω,ω0 −ν

with Fωω0 = (−1)δωω0 (−q3 )3(−q1 )−δωω0 +ν1(−q2 )−δωω0 +ν2.

• Define the vacuum |∅i such that αω,k>0 |∅i = 0, Pω (z) |∅i = |∅i.
Standard PBW basis obtained by acting with αω,k<0 and Qω (z).

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverHorizontal representations
Drinfeld currents are represented in terms of vertex operators,ρH (xω+ (z)) = uω z −nω Qω (z) exp 

X zk
k>0

k





αω,−k  exp −

k>0


−1 nω
ρH (xω− (z)) = uωz Qω (z)−1 Pω−ν3 (q3−1 z) exp −

X zk
k>0


+
ρH (ψω(z)) = F −1/2 Pω−ν3 (q3−1 z) exp −X z −kX z −k
k>0

k

k

−k/2

(q3

k


−k/2
q3
αω,k  ,αω,−k  exp 

k>0
k/2

αω,k − q3kαω−ν3 ,k ) ,uω−ν3 nω−ν3 nω −nω−ν Qω−ν3 (q3−1 z)3
q3
Pω−ν3 (q3−1 z)zuωQω (z)
X zk−k× exp (q αω−ν3 ,−k − αω,−k ) .
k 3
k>0

−
ρH (ψω(z)) = F 1/2

X z −k
k/2q3

αω−ν3 ,k Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverIntertwining operators
Intertwining operators introduced as a generalization of Virasoro vertex operators.
( quantum Knizhnik-Zamolodchikov equations) [Frenkel, Reshetikhin 1992]

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverIntertwining operators
Intertwining operators introduced as a generalization of Virasoro vertex operators.
( quantum Knizhnik-Zamolodchikov equations) [Frenkel, Reshetikhin 1992]
d symmetry.
They were used to solve the XXZ model in relation with Uq (sl(2))[Davies, Foda, Jimbo, Miwa, Nakayashiki 1992]

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverIntertwining operators
Intertwining operators introduced as a generalization of Virasoro vertex operators.
( quantum Knizhnik-Zamolodchikov equations) [Frenkel, Reshetikhin 1992]
d symmetry.
They were used to solve the XXZ model in relation with Uq (sl(2))[Davies, Foda, Jimbo, Miwa, Nakayashiki 1992]

Intertwiners Φ : V ⊗ H → H and Φ∗ : H → V ⊗ H are identified with the
topological vertex used to compute the gauge theory partition functions.
[Awata, Feigin, Shiraishi, 2011][Awata, Kanno, Mironov, Morozov, Suetake, Zenkevich 2017]

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverIntertwining operators
Intertwining operators introduced as a generalization of Virasoro vertex operators.
( quantum Knizhnik-Zamolodchikov equations) [Frenkel, Reshetikhin 1992]
d symmetry.
They were used to solve the XXZ model in relation with Uq (sl(2))[Davies, Foda, Jimbo, Miwa, Nakayashiki 1992]

Intertwiners Φ : V ⊗ H → H and Φ∗ : H → V ⊗ H are identified with the
topological vertex used to compute the gauge theory partition functions.
[Awata, Feigin, Shiraishi, 2011][Awata, Kanno, Mironov, Morozov, Suetake, Zenkevich 2017]

These operators are obtained by solving the following equation:ρH (e)Φ = Φ (ρV ⊗ ρH ∆(e)) , orρV ⊗ ρH ∆0 (e) Φ∗ = Φ∗ ρH (e),for every element e = xω± (z), ψω± (z), c of the algebra.
(∆0 is the opposite coproduct obtained by permutation.)
H

Φ

V

V

H
H

Φ∗

H

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Application to gauge theories
The solutions have been found, they decompose on the vertical basisXX ∗Φ=Φλ hhλ| , Φ∗ =
Φλ |λiiλλwhere Φλ and Φ∗λ are vertex operators acting on horizontal modules.

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Application to gauge theories
The solutions have been found, they decompose on the vertical basisXX ∗Φ=Φλ hhλ| , Φ∗ =
Φλ |λiiλλwhere Φλ and Φ∗λ are vertex operators acting on horizontal modules.
We build the operator T by gluing the intertwiners Φ and Φ∗ .
The gluing rules follow from the string theory realization
H

Φ

H
V

H

Φ∗

H

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Application to gauge theories
The solutions have been found, they decompose on the vertical basisXX ∗Φ=Φλ hhλ| , Φ∗ =
Φλ |λiiλλwhere Φλ and Φ∗λ are vertex operators acting on horizontal modules.
We build the operator T by gluing the intertwiners Φ and Φ∗ .
The gluing rules follow from the string theory realization
Φ

H

H
V

H

Φ∗

H

Along the vertical direction, gluing is done by a scalar product:X
T [U(N)] =Φλ ⊗ Φ∗λ : H ⊗ H → H ⊗ H
λQuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Application to gauge theories
The solutions have been found, they decompose on the vertical basisXX ∗Φ=Φλ hhλ| , Φ∗ =
Φλ |λiiλλwhere Φλ and Φ∗λ are vertex operators acting on horizontal modules.
We build the operator T by gluing the intertwiners Φ and Φ∗ .
The gluing rules follow from the string theory realization
Φ

H

H
V

H

Φ∗

H

Along the vertical direction, gluing is done by a scalar product:X
T [U(N)] =Φλ ⊗ Φ∗λ : H ⊗ H → H ⊗ H
λThe vacuum expectation value reproduces the gauge theory partition function:
Z = (h∅| ⊗ h∅|) T (|∅i ⊗ |∅i) .

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Outline1

Introduction

2

Quantum toroidal algebras

3

Deformation of quantum toroidal gl(p)4Representations and application to gauge theories

5

Perspectives

6

General approach to quiver representations

Representations

Perspectives

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Summary of the results
From the mathematics perspective:
Define a new algebra that deforms the quantum toroidal gl(p) algebra.
Show that it has the structure of a Hopf algebra.
Provide a highest weight representation on Young diagrams.
Provide the vertex representation (or level one representation).
Constuct the intertwining operators between these two.

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverSummary of the results
From the mathematics perspective:
Define a new algebra that deforms the quantum toroidal gl(p) algebra.
Show that it has the structure of a Hopf algebra.
Provide a highest weight representation on Young diagrams.
Provide the vertex representation (or level one representation).
Constuct the intertwining operators between these two.

From the physics perspective:
Define a colored topological vertex pertaining to the spacetime orbifold.
Extend the algebraic construction of SUSY gauge theories’ partitions functions.
Include the case ν1 = 1, ν2 = 0 corresponding to the insertion of a surface defect.
Construct the qq-characters of the gauge theories (other type of observable).
Observe two inequivalent fundamental qq-characters.

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Quiver

Open questions
From the mathematics perspective:
Show that Serre relations are obeyed in both horizontal and verticalrepresentations.
Prove the conjectured equivalence with a quantum toroidal algebra defined upon
the Cartan matrix Cωω0 . (quantum affine algebra?)
Look for Miki’s automorphism mapping (c, c̄) → (−c̄, c) (S-duality).
Find more representations (MacMahon modules?) and classify them.

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Quiver

Open questions
From the mathematics perspective:
Show that Serre relations are obeyed in both horizontal and verticalrepresentations.
Prove the conjectured equivalence with a quantum toroidal algebra defined upon
the Cartan matrix Cωω0 . (quantum affine algebra?)
Look for Miki’s automorphism mapping (c, c̄) → (−c̄, c) (S-duality).
Find more representations (MacMahon modules?) and classify them.

From the physics perspective:
Find a duality with q-deformed W-algebras

AGT-correspondence!

Construct the associated quantum integrable systems.

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Outline1

Introduction

2

Quantum toroidal algebras

3

Deformation of quantum toroidal gl(p)4Representations and application to gauge theories

5

Perspectives

6

General approach to quiver representations

Representations

Perspectives

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Main idea
Combine different ingredients to derive a general approach to SUSY gauge theories:
I. ADHM construction
Description of the non-perturbative sector of SUSY gauge theories.
(instanton moduli space)[Atiyah-Drinfeld-Hitchin-Manin, Kronheimer, Nakajima,...]

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Main idea
Combine different ingredients to derive a general approach to SUSY gauge theories:
I. ADHM construction
Description of the non-perturbative sector of SUSY gauge theories.
(instanton moduli space)[Atiyah-Drinfeld-Hitchin-Manin, Kronheimer, Nakajima,...]

II. Cohomological Hall algebra
Quiver representation = set of vector spaces and linear maps.
Associate an algebra to the quivers.
[Vasserot, Schiffmann 2012] [Rapcak, Soibelman, Yang, Zhao 2018]

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Main idea
Combine different ingredients to derive a general approach to SUSY gauge theories:
I. ADHM construction
Description of the non-perturbative sector of SUSY gauge theories.
(instanton moduli space)[Atiyah-Drinfeld-Hitchin-Manin, Kronheimer, Nakajima,...]

II. Cohomological Hall algebra
Quiver representation = set of vector spaces and linear maps.
Associate an algebra to the quivers.
[Vasserot, Schiffmann 2012] [Rapcak, Soibelman, Yang, Zhao 2018]

III. Algebraic approach to topological strings
Construct the intertwining operator from a Hopf algebra.
(generalized topological vertex).
Compute amplitudes via a diagrammatic technique.
Identify amplitudes with gauge theory observables.

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Main idea
Combine different ingredients to derive a general approach to SUSY gauge theories:
I. ADHM construction
Description of the non-perturbative sector of SUSY gauge theories.
(instanton moduli space)[Atiyah-Drinfeld-Hitchin-Manin, Kronheimer, Nakajima,...]

II. Cohomological Hall algebra
Quiver representation = set of vector spaces and linear maps.
Associate an algebra to the quivers.
[Vasserot, Schiffmann 2012] [Rapcak, Soibelman, Yang, Zhao 2018]

III. Algebraic approach to topological strings
Construct the intertwining operator from a Hopf algebra.
(generalized topological vertex).
Compute amplitudes via a diagrammatic technique.
Identify amplitudes with gauge theory observables.
. The simplest context is given by N = 2 Super Yang-Mills on R4 ' C2 .

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverN = 2 Super Yang-Mills on C2

N = 2 Super Yang-Mills on C2[ADHM]

ADHM quiver[Vasserot, Schiffmann 2012]

Affine Yangian of gl(1)(SHc algebra)

q-deformation

Quantum toroidal gl(1)(Ding-Iohara-Miki algebra)[Awata, Feigin, Shiraishi 2011]

Topological vertex

Diagrammatic technique
Topological strings amplitudes= gauge theory observables(partition function, qq-character)Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverN = 2 Super Yang-Mills on C2 /Zp
Case ν1 = −ν2 = 1

N = 2 Super Yang-Mills on C2 /Zp[Nakajima]

Cyclic quiverAffine Yangian of gl(p)q-deformation

Quantum toroidal gl(p)[AKMMSZ 2017]

Colored topological vertex[Chaimanowong, Foda 2018]

Diagrammatic technique
Topological strings amplitudes= gauge theory observables(partition function, qq-character)Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Hints for a very general construction...

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

General mathematical construction
Quiver representation(brane configuration)

Group action
Fixed points
Tangent space

Cohomological Hall
algebra (COHA)

q-deformation

q-COHA (K-theory)

Coproduct
Representations

Intertwining operator

Diagrammatic techniqueCorrelators/Amplitudes

Perspectives

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives. How is the algebra built out of the quiver representation?

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

ADHM quiver: definition
N

J

B1

I

K

B2

• Definition: N and K are vector spaces of dimension n and k.
Take B1 , B2 ∈ End(K ), I ∈ Hom(N, K ), J ∈ Hom(K , N) and consider:
Mk (n) = {B1 , B2 , I , JµC = 0, C[B1 , B2 ]IN = K }/GL(K ),with the moment map µC = [B1 , B2 ] + IJ.

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

ADHM quiver: definition
N

J

B1

I

K

B2

• Definition: N and K are vector spaces of dimension n and k.
Take B1 , B2 ∈ End(K ), I ∈ Hom(N, K ), J ∈ Hom(K , N) and consider:
Mk (n) = {B1 , B2 , I , JµC = 0, C[B1 , B2 ]IN = K }/GL(K ),with the moment map µC = [B1 , B2 ] + IJ.
• Consider the action of U(1)n+2 ⊂ GL(N) × SO(4)(h, t1 , t2 ) ∈ U(1)n × U(1) × U(1) : (B1 , B2 , I , J) → (t1 B1 , t2 B2 , Ih, t1 t2 h−1 J).
The fixed points are labelled by n-tuple partitions λ = (λ(1) , · · · , λ(n) ).

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

ADHM quiver: Nekrasov factor• Construct the tangent space (δB1 , δB2 , δI , δJ)
modulo δµC = 0 and gl(K ) transformation.

Perspectives

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

ADHM quiver: Nekrasov factor• Construct the tangent space (δB1 , δB2 , δI , δJ)
modulo δµC = 0 and gl(K ) transformation.
• At a fixed point λ, the tangent space supports an action of U(1)n+2 .
We determine the character Xλ of this action.

Perspectives

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Quiver

ADHM quiver: Nekrasov factor• Construct the tangent space (δB1 , δB2 , δI , δJ)
modulo δµC = 0 and gl(K ) transformation.
• At a fixed point λ, the tangent space supports an action of U(1)n+2 .
We determine the character Xλ of this action.
• The Nekrasov factor is obtained by taking the plethystic exponential,N(v , λ|v 0 , λ0 ) = I[X ] =YS(χ /χ ) ×

∈λ
∈λ0

with
!

S(z) =n Y Y
α=1 ∈λ0

1−

vα
χ

0


×

n 
YY
∈λ α=1

1 − q1 q2χvα0(1 − q1 z)(1 − q2 z).
(1 − z)(1 − q1 q2 z)

For two different fixed points λ 6= λ0 , need another mathematical construction.

,Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Quiver

ADHM quiver: Y-observables• From the Nekrasov factor, we determine the Y-observable
0N(v , λ|v 0 , λ0 + )N(v , λ + |v 0 , λ0 ), Y ∗[λ ] (q1 q2 χ ) =00N(v , λ|v , λ )N(v , λ|v 0 , λ0 )

n 
n 
Y
Y
vα  Yz
Y [λ] (z) =1−×−S(χ /z), Y ∗[λ] (z) = Y [λ] (z) ×
z
vα
α=1
α=1

Y [λ] (χ ) =⇒∈λIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Quiver

ADHM quiver: Y-observables• From the Nekrasov factor, we determine the Y-observable
0N(v , λ|v 0 , λ0 + )N(v , λ + |v 0 , λ0 ), Y ∗[λ ] (q1 q2 χ ) =00N(v , λ|v , λ )N(v , λ|v 0 , λ0 )

n 
n 
Y
Y
vα  Yz
Y [λ] (z) =1−×−S(χ /z), Y ∗[λ] (z) = Y [λ] (z) ×
z
vα
α=1
α=1

Y [λ] (χ ) =⇒∈λ. Y-obervables, and the function S(z), determine the algebra!

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Quiver

ADHM quiver: Y-observables• From the Nekrasov factor, we determine the Y-observable
0N(v , λ|v 0 , λ0 + )N(v , λ + |v 0 , λ0 ), Y ∗[λ ] (q1 q2 χ ) =00N(v , λ|v , λ )N(v , λ|v 0 , λ0 )

n 
n 
Y
Y
vα  Yz
Y [λ] (z) =1−×−S(χ /z), Y ∗[λ] (z) = Y [λ] (z) ×
z
vα
α=1
α=1

Y [λ] (χ ) =⇒∈λ. Y-obervables, and the function S(z), determine the algebra!
What happens in the case of a Zp -orbifold?

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Cyclic quiver• We consider first the standard Zp -action ν1 = −ν2 = 1.

Representations

Perspectives

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverCyclic quiver• We consider first the standard Zp -action ν1 = −ν2 = 1.
• We can take the long path and consider the cyclic quiver
N0

KO

Np−1

Kp−1

K1

N1

K2

N2

with fixed point given by colored partitions, c( ) = cα + i − j ∈ Zp for= (i, j) ∈ λ(α) .

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverCyclic quiver• We consider first the standard Zp -action ν1 = −ν2 = 1.
• We can take the long path and consider the cyclic quiver
N0

KO

Np−1

Kp−1

K1

N1

K2

N2

with fixed point given by colored partitions, c( ) = cα + i − j ∈ Zp for
Or we can take a shortcut and consider the Zp -invariant character!!!

= (i, j) ∈ λ(α) .

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives(ν1 , ν2 )-deformed quiver• For the (ν1 , ν2 )-deformed Zp -action, the coloring isc( ) = cα + (i − 1)ν1 + (j − 1)ν2 ∈ Zp ,for

= (i, j) ∈ λ(α)

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives(ν1 , ν2 )-deformed quiver• For the (ν1 , ν2 )-deformed Zp -action, the coloring isc( ) = cα + (i − 1)ν1 + (j − 1)ν2 ∈ Zp ,= (i, j) ∈ λ(α)

for

• Projecting the character on its Zp -invariant part, we deduceN(v , λ|v 0 , λ0 ) =Y

Sc( )c( ) (χ /χ ) ×YY∈λ0 α∈Cc( ) (n)



vα
1−
,
χ

δ

Sωω0 (z) =Y∈λ α∈Cc( )+ν1 +ν2 (n0 )

∈λ
∈λ0

×

Y

δ

(1 − q1 z) ω,ω0 −ν1 (1 − q2 z) ω,ω0 −ν2(1 − z)δω,ω0(1 − q1 q2 z)δω,ω0 −ν −ν
1

2

.


1−

χ
q3 vα0

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Quiver(ν1 , ν2 )-deformed quiver• For the (ν1 , ν2 )-deformed Zp -action, the coloring isc( ) = cα + (i − 1)ν1 + (j − 1)ν2 ∈ Zp ,= (i, j) ∈ λ(α)

for

• Projecting the character on its Zp -invariant part, we deduceN(v , λ|v 0 , λ0 ) =Y

Sc( )c( ) (χ /χ ) ×YY∈λ0 α∈Cc( ) (n)

δω,ω0

χ
q3 vα0δ(1 − q1 z) ω,ω0 −ν1 (1 − q2 z) ω,ω0 −ν2(1 − z)

1−



vα
1−
,
χ

δ

Sωω0 (z) =Y∈λ α∈Cc( )+ν1 +ν2 (n0 )

∈λ
∈λ0

×

Y

(1 − q1 q2 z)δω,ω0 −ν −ν
1

.

2

[λ]

∗[λ]

• With the same method as before, we derive the Y-observables Yω (z) and Yω(z).

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Quiver(ν1 , ν2 )-deformed quiver• For the (ν1 , ν2 )-deformed Zp -action, the coloring isc( ) = cα + (i − 1)ν1 + (j − 1)ν2 ∈ Zp ,= (i, j) ∈ λ(α)

for

• Projecting the character on its Zp -invariant part, we deduceN(v , λ|v 0 , λ0 ) =Y

Sc( )c( ) (χ /χ ) ×YY∈λ0 α∈Cc( ) (n)

δω,ω0

χ
q3 vα0δ(1 − q1 z) ω,ω0 −ν1 (1 − q2 z) ω,ω0 −ν2(1 − z)

1−



vα
1−
,
χ

δ

Sωω0 (z) =Y∈λ α∈Cc( )+ν1 +ν2 (n0 )

∈λ
∈λ0

×

Y

(1 − q1 q2 z)δω,ω0 −ν −ν
1

.

2

[λ]

∗[λ]

• With the same method as before, we derive the Y-observables Yω (z) and Yω
Now, we have all the information needed to define our algebra!!!

(z).

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Construction of the algebra. The algebra is determined following these few steps:
I. Define an action on states parameterized by fixed points
The matrix elements are given by the Y-observables.
This is the vertical representation = Cohomological Hall algebra action.

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Construction of the algebra. The algebra is determined following these few steps:
I. Define an action on states parameterized by fixed points
The matrix elements are given by the Y-observables.
This is the vertical representation = Cohomological Hall algebra action.
II. Define a set of vertex operators
Impose normal-ordering relations of the typeηω (z)ηω0 (w ) = Sωω0 (z/w ) : ηω (z)ηω0 (w ) : .
They will form the horizontal representation.

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Construction of the algebra. The algebra is determined following these few steps:
I. Define an action on states parameterized by fixed points
The matrix elements are given by the Y-observables.
This is the vertical representation = Cohomological Hall algebra action.
II. Define a set of vertex operators
Impose normal-ordering relations of the typeηω (z)ηω0 (w ) = Sωω0 (z/w ) : ηω (z)ηω0 (w ) : .
They will form the horizontal representation.
III. Combine these two actions to form the algebra
Vertical/Horizontal are identified with representations of level c = 0/c = 1.

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Construction of the algebra. The algebra is determined following these few steps:
I. Define an action on states parameterized by fixed points
The matrix elements are given by the Y-observables.
This is the vertical representation = Cohomological Hall algebra action.
II. Define a set of vertex operators
Impose normal-ordering relations of the typeηω (z)ηω0 (w ) = Sωω0 (z/w ) : ηω (z)ηω0 (w ) : .
They will form the horizontal representation.
III. Combine these two actions to form the algebra
Vertical/Horizontal are identified with representations of level c = 0/c = 1.
IV. Add a co-algebraic structureShow that it defines a Hopf algebra.

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Open questions
We can associate an algebra to a quiver representation, but can we do more?

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Open questions
We can associate an algebra to a quiver representation, but can we do more?
When does the full AFS construction of the topological vertex holds?
(Non-symplectic quivers? How to combine x + , ψ − and x − , ψ + Borel
subalgebras?)

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Open questions
We can associate an algebra to a quiver representation, but can we do more?
When does the full AFS construction of the topological vertex holds?
(Non-symplectic quivers? How to combine x + , ψ − and x − , ψ + Borel
subalgebras?)
What is the interpretation of the amplitudes?
(topological invariant? branes configuration?)

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Open questions
We can associate an algebra to a quiver representation, but can we do more?
When does the full AFS construction of the topological vertex holds?
(Non-symplectic quivers? How to combine x + , ψ − and x − , ψ + Borel
subalgebras?)
What is the interpretation of the amplitudes?
(topological invariant? branes configuration?)
Can we further associate an integrable system? W-algebras? symmetricpolynomials?

QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

Open questions
We can associate an algebra to a quiver representation, but can we do more?
When does the full AFS construction of the topological vertex holds?
(Non-symplectic quivers? How to combine x + , ψ − and x − , ψ + Borel
subalgebras?)
What is the interpretation of the amplitudes?
(topological invariant? branes configuration?)
Can we further associate an integrable system? W-algebras? symmetricpolynomials?
Consider more general examples...
Quiver with fixed points labelled by half-partitions? [in progress...]QuiverIntroduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverConclusion

Supersymmetric gauge theories offer a new horizon forthe study of quantum groups, guiding our exploration ofquantum toroidal algebras.

Introduction

Quantum toroidal algebras

Deformation of quantum toroidal gl(p)Representations

Perspectives

QuiverConclusion

Supersymmetric gauge theories offer a new horizon forthe study of quantum groups, guiding our exploration ofquantum toroidal algebras.

Thank you !!!