Seminar at YITP - 東京大学matsuo/file/YITP2019.pdfSeminar at YITP Quantum toroidal algebra Relation with integrability and application to AGT conjecture Yutaka Matsuo The University

Seminar at YITP

Quantum toroidal algebraRelation with integrability and application to AGT conjecture

Yutaka Matsuo

The University of Tokyo

April 5, 2019

based on works arXiv:1110.5255 (Zhang), 1306.1523 (Kanno, Zhang), 1405.3141 (Rim, Zhang), 1504.04150 (Zhu), 1512.02492

(Bourgine, Zhang), 1606.08020 (Bourine, Fukuda, Zhang, Zhu), 1810.08512 (Harada)

Yutaka Matsuo (The University of Tokyo) Quantum toroidal algebra April 5, 2019 1 / 46

Introduction

Introduction

Virasoro algebra: 2D conformal symmetry, gauge symmetry of stringtheory, critical phenomena in 2D statistical models:

[Ln, Lm] = (n −m)Ln+m +c

12(n3 − n)δn+m,0

It was very important subject during 80s and still remains activelystudied.

W1+∞ and quantum toroidal algebra: Universal symmetry thatcontains Virasoro and many other chiral algebras and theirq-deformed versions. They are essential to understand subjectsrelated to higher dimensional physics, such asAlday-Gaiotto-Tachikawa conjecture for N = 2 D = 4 SYM,(quantum) Seiberg-Witten curve, topological strings and so on.


Introduction

Classical limit

Virasoro: 2D conformal algebra δz = −zn+1. Infinitesimaltransformation is expressed as the first order derivative δn = zn+1∂z .

W1+∞: defined by higher order derivatives

δn,m = znDm, D = z∂z , (n ∈ Z,m ≥ 0)

It is related to area preserving diffeomorphism in two dimensions.

Toroidal algebra:

δn,m = UnVm, n,m ∈ Z, U = qD ,V = z ,D = z∂z

U,V satisfy quantum torus algebra UV = qVU. There is SL(2,Z)automorphism U ′ = UaV b,V ′ = UcV d with ad − bc = 1.


Introduction

Quantum deformation

Quantum deformation of W1+∞ and toroidal algebra was developed,not in the context of string theory, but in the algebraic studies ofintegrable models such as Calogero-Sutherland and its q-deformedversion (Ruijsenaar-Schneider) with various motivations bymathematicians.

Because of such historical reason, the quantum symmetries has manydifferent names.

W1+∞: SHc (from shuffle algebra), Affine Yangian (Yangiansymmetry associated with affine algebra gl(1))

Quantum toroidal algebra: also referred to as ”Ding-Iohara-Miki(DIM)”

Further deformation of quantum toroidal algebra is called ”ellipticDIM” and so on.

Since W1+∞ and quantum toroidal algebra has many names, I willsometimes use U to refer these universal symmetries in general.


Introduction

Deformation parameters and reductions

Virasoro algebra has a deformation parameter c

W1+∞ has one parameter β. Sometimes it is convenient to express itby ε1, ε2, ε3 with

∑3i=1 εi = 0 with β = −ε1/ε2. By using scale degree

of freedom, it may contain additional reduction parameter n. When itis positive integer, the algebra reduces to Wn-algebra+U(1)symmetry with central charge,

c = (n − 1)(1− Q2n(n + 1)) + 1, Q =√β − 1/

√β

DIM has additional parameter t. One may sometimes use q1 = q,q2 = t−1, q3 = (q1q2)

−1. W1+∞ is obtained from DIM by taking alimit q = e~β, t = e~ in ~ → 0 limit. It may have n reduction also. Inthat case, we obtain quantum Wn algebra with U(1) factor.

Recently, more general reduction (p, q) reduction was found (cornerVOA).


Introduction

Plan of the talk

Proof of Alday-Gaiotto-Tachikawa conjecture by ULimitation of conventional CFT approachRelation with integrable modelsProof of AGT conjecture by U

Derivation of second quantized SW curve (qq-character)

New reduction of U : corner VOA and Web of W-algebra


AGT conjecture

1 Introduction

2 AGT conjecture

3 CFT approach to AGT conjecture

4 Integrable model and SHc and DIM

5 Recent developments of DIMqq-charactercorner VOA and WoW

6 Conclusion


AGT conjecture

AGT conjecture

Original form of AGT conjecture (2009)

Instanton partition function of D = 4, N = 2, Nf = 4 superYang-Mills with SU(2) gauge group = four point function of Virasoroconformal block with U(1) factor.∑

~λ

q|~λ|Zvec(~λ, ~a)Zfd(~λ, ~a, ~m) = 〈Vα1(0)Vα2(q)Vαe (1)Vα4(∞)〉

Instanton partition function of D = 4, N = 2∗ super Yang-MillsSU(2) gauge group= one point torus conformal block function


AGT conjecture

~λ = (λ1, λ2) is a pair of Young diagrams which label the localizationfixed points in the moduli space of SU(2) gauge theory.|~λ| = |λ1|+ |λ2| gives the number of boxes of two Young diagramswhich is identical to the number of instantons.

The contribution of instanton partition function is computed(regularized) by omega background with the deformation parametersε1, ε2.

~a ∈ C⊗2: VEV of scalar component of vector multiplet, ~m ∈ C⊗4:mass of four hyper multiplets.

CFT side is described c = 1 + 6(√b +

√1/b)2 Virasoro algebra with

b = ε1/ε2.

q ∈ C is the instanton expansion parameter q = e2πiτ , τ = θ + i2πg .

Vα(z) is the primary field of CFT with conformal dimension related toα ∈ C. αi is linearly related to ~a, ~m.


AGT conjecture

Higher rank and quiver gauge theories

Higher rank generalization (Wyllard, Morozon-Mironov):

AGT for SU(n) gauge group = Wn algebra + U(1) factor conformalblock

Wn algebra: chiral algebra generated by spin 2, 3, · · · , n(Zamolodchikov, Fateev, Lukyanov)

CFT side is computed by Toda field theory of rank n. Correlationfunction was given by Fateev and Litvinov.

Linear quiver gauge theories:

gauge group SU(n1)⊗ · · · ⊗ SU(nl): l + 3 point functions

Quiver diagram (Figure in the next slide)

When n1 = · · · = n` = n, the conformal block function is given byl + 3 point function 〈Vα1 · · ·Vαl+3

〉 of Wn-algebra (Toda field theory)


AGT conjecture

Quiver diagram and Nekrasov partition function

We consider a linear quiver gauge theories of the following type.

N1

~a1 ~a2 ~aQ

N2 NQm12 m23

N1 N2 NQ

fund. ~m1 fund. ~m2 fund. ~mQ

Partition function is written as ”matrix multiplication” with index ~Y :

Zinst. =∑

~Y1,···~YQ

Q∏i=1

q|~Yi |i Zvect.(~ai , ~Yi )Zfund.(~ai , ~Yi ; ~mi )

·∏

i→j∈EQ

Zbfd.(~ai , ~Yi ; ~aj , ~Yj |mij)Q∏i=1

(ZCS( ~Yi ))κi


AGT conjecture

Nekrasov factor and its recursive properties

The explicit form of partition function: bifundamental rep.

Zbfd.(~a, ~Y ; ~b, ~W |m12) =N1∏`=1

N2∏`′=1

NY`,W`′ (a` − b`′ −m12)

NY ,W (t) =∏

(i,j)∈Y

F (t,W ′j − i ,Yi − j + 1)

∏(i,j)∈W

F (t,−Y ′j + i − 1,−Wi + j) ,

F (t, n,m) = t + ε1m − ε2n (4d), 1− tq−n1 qm2 (5d)

The contribution from the other representations are derived from it.Nekrasov factor satisfies simple recursion formulae

NY+b,W (t)

NY ,W (t)=

∏c∈A(W ) F (t, y(c)− y(b), x(b)− x(c)∏

y∈R(W ) F (t, y(c)− y(b) + 1, x(b)− x(c) + 1).

and so on. Namely, the addition/subtraction of a box is represented byproduct of factors on the surface. It is the most important lemma whichgives our main theorem on the characterization of Gaiotto state andintertwiner which will appear later.


AGT conjecture

Correspondences between diagram and building blocks

Such partition function is written by combining ”building block” written inthe states/operators in SHc

〈G , ~a| |G , ~a〉

Gaiotto state

U( ~m)

Flavor vertex

V12(m)

Intertwiner


AGT conjecture

Comparison of both sides

Conformal block function

〈V (z1)V (z2) · · ·V (zl+3)〉 = U · Z1 · · ·Zl · U

We use the decomposition of unity in the intermediate channel1 =

∑~λ|~a, ~λ〉〈~a, ~λ| where |~a, ~λ〉 is an orthonormal basis of the

Wn + U(1) module.

Z~λ1,~λ2= 〈~a1, ~λ1|Vµ(1)|~a2, ~λ2〉, U~λ

= Z~λ,∅, U~λ= Z∅,~λ

Nekrasov partition function has very similar structure where

Z~λ1,~λ2= Zbf (~a1, ~λ1; ~a2, ~λ2|µ)Zvec(~a1, ~λ1)

1/2Zvec(~a2, ~λ2)1/2


AGT conjecture

What should be proved

Proof of AGT conjecture is boiled down to

Find a proper orthonormal basis of Wn + U(1) module |~λ〉Show the identity

〈~λ1|Vµ(1)|~λ2〉 = Zbf (~a1, ~λ1; ~a2~λ2|µ)Zvec(~a1, ~λ1)1/2Zvec(~a2, ~λ2)

1/2


CFT approach to AGT conjecture

1 Introduction

2 AGT conjecture




6 Conclusion




W-algebra: a nonlinear algebra that contains Virasoro

TT ∼ c + T

TW = W

WW = c + T + T 2

Algebra is explicitly written for W (3). For higher W (n), it is toocomplicated to express all the OPE. Instead of writing OPE amongcurrents, we use free field realization (Fateev-Lukyanov) by quantumMiura transformation,

R = (Q∂ − ∂(~h1~φ)) · · · (Q∂ − ∂(~hn~φ)) =n∑

`=0

Qn−`U(`)(z)(∂z)n−l



The structure of the Hilbert space is described by free field Fock spacemodulo the screening currents

Q(±)j =

∫dz

2πiV

(±)j (z), V

(±)j =: eα±(~ej ~φ)

where ~ej is jth simple root and α+ = β, α− = 1/β.A standard method to evaluate the conformal block function is to use theDotsenko-Fateev integral,

〈V (z1) · · ·V (zl)〉 = 〈V (z1) · · ·V (zl)n−1∏a=1

(Q+a )Na〉Fock

It reduces to Selberg integral of Jack symmetric polynomial (eigenfunctionof Calogero-Sutherland model). For n = 2, the integration can bemanaged (derived by Kadell) to give AGT conjecture (Morozov-Mironov).For higher n



Shapovalov matrix

The other way to check AGT conjecture is to construct the decompositionof unity,

1 =∑~λ1,~λ2

|~λ1〉(S−1)~λ1

~λ2〈~λ2|, S~λ1~λ2

= 〈~λ1|~λ2〉.

The state |~λ〉 is a basis which is constructed as

J−n1 · · · J−n`1L−m1 · · · L−m`2

· · · |~a〉

The inner product 〈~λ1|~λ2〉 is called “Shapovalov matrix”. It is blockdiagonal and we need inner product between the basis with the fixedlevels. There is only a finite number of state for each level. So this iswell-defined problem to compute order by order by computer. It was usedby many authors to check AGT at lower levels. However, it does not give asystematic analytic proof which holds to all orders.



Connection between CFT and integrable model

The integrable model which is relevant to study CFT isCalogero-Sutherland model whose Hamiltonian is,

H =N∑i=1

(Di )2 + β

∑i<j

xi + xj(xi − xj

(Di − Dj), Di = xi∂xi

This Hamiltonian is known to be integrable, namely we have infinitenumber of commuting operators H2 = H,H3,H4, · · · . Their eigenfunctionsare called Jack polynomial which is labeled by Young diagram.In the large N limit, one may rewrite the coordinates xi by the bosonicoscillators defined by, (for n > 0)

a−n ↔ pn =N∑i=1

(xi )n, an =

n

β

∂

∂pn

Through this correspondence, one may map Hilbert space of CS system tothe Fock space of free boson theory.



AFLT proof

AFLT managed to give an algorithm to construct orthogonal basis |~λ〉 forSU(2) gauge group.

When one of the component is empty, |(λ, ∅)〉 = Jλ(a)|0〉 ⊗ |0〉, whereJλ(a) is Jack polynomial (eigenfunction of Calogero-Sutherland)

For generic ~λ, they proposed a recursive method which indirectlyconstruct the basis. They show that it satisfies,

〈~λ1|Vµ(1)|~λ2〉 = Zbf (~λ1, ~λ2|µ)Zvec(~λ1)1/2Zvec(~λ2)

1/2

This is a proof of AGT conjecture. There remained some questions

Why Calogero-Sutherland is relevant?

Construction of general basis remains non-direct.


Integrable model and SHc and DIM

1 Introduction

2 AGT conjecture




6 Conclusion



Calogero-Sutherland and CFT

Rewriting Calogero Hamiltonian

H =N∑i=1

(yi )2, yi = Di + β(

∑j>i

xjσij +∑j<i

xiσij)∂j

The operator yi is called as Dunkl operators and commute with eachother [yi , yj ] = 0. The algebra generated by xi , yi , and permutationσ ∈ SN is called degenerate double affine Hecke algebra. (DDAHA)

Higher charges of Calogero system are written as higher power sum ofyi . The eigenstate (Jack polynomial) is also written in terms of xi , yias well.One may use different set of operator Yi = qDi · · · (differenceoperator) and consider the algebra generated by (Xi )

±1, (Yi )±1,Ri .

This is a q-deformation of DDAHA and referred to as double affineHecke algebra (DAHA). It is a symmetry behind q-deformed Calogero(Ruijsnaar-Schneider model)



SHc (W1+∞, affine Yangian)

The algebra generated by SN symmetric combinations of DDAHA is calledSHc . We consider a combination (roughly), D1,n ∼

∑i xiy

ni ,

D−1,n ∼∑

i (xi )−1yni , D0,n ∼

∑i (yi )

n. One may combine them in theform of Drinfeld current,

D±1(ζ) =∞∑n=0

D±1,n ζ−n−1, D0(ζ) =

∞∑n=0

D0,nζ−n−1.

These generators satisfy a simple action on the Jack polynomials,

D±1(ζ)Jλ =∑

x∈A/R(λ)

Λx(λ)

ζ − φxJλ+x , D0(ζ)Jλ =

∑x∈λ

1

ζ − φxJλ,

where φx = ε1(i − 1) + ε2(j − 1).



A(Y ) and R(Y ): Figure

Y ∈ A(Y )

∈ R(Y )



SHc continued

SHc contains two parameters ε1,2 (ε3 = −ε+) of omega background.Indeed only the ratio β = −ε1/ε2 is relevant. In holomorphic realization,the algebra is written compactly,

[D0(u),D0(v)] = 0 [D0(u),D±1(v)] = ±D±1(v)− D±1(u)

u − v,

h(u − v)D1(u)D1(v) ∼ D1(v)D1(u)h(v − u), h(u) =3∏

i=1

(u + εi ),

[D−1(u),D1(v)] =E (v)− E (u)

u − vε−1+ , E (u) := Y(u + ε+)Y(u)−1

Y(u) := ec(u)eΦ(u−ε1)eΦ(u−ε2)e−Φ(u)e−Φ(u−ε+)

where ε+ = ε1 + ε2, c(u) = c0 log(u)−∑∞

n=1cnnun are ”central charges” of

SHc .



Relation between SHc and W-algebra

Figure

Horizontal line: U(1) current, Virasoro, W

Vertical line: Drinfeld currents

D0,2 ∼ H = β2∑n,m

(a−na−man+m + a−n−manam) + β(1− β)∞∑n=1

na−nan

=√2β

∑n>0

a−nL(β)n + diagonal

e0 ∼ a−1, f0 ∼ a1

D0,2, a±1 may be used to generate all the AY currents.



coproduct

Important property of SHc : allows to take co-product (= analog of tensorproduct representation). While the original representation is described byone Fock space F , one may construct more general representation F⊗n.Since SHc is generated by D0,2 and D±1,0, what we need is the expressionform them,

∆(D±1,0) = D±1,0 ⊗ 1 + 1⊗ D±1,0 = δ(D±1,0)

∆(D0,2) = δ(D0,2) + (β − 1)∞∑n=1

nβn−1a−n ⊗ an

The last term represents the mixing due to the deformation. Coproduct ofAY has similar representation

D±1(ζ)|~λ〉 =∑

x∈A/R(~λ)

Λx(~λ)

ζ − φx|~λ± x〉, D0(ζ)|~λ〉 =

∑x∈~λ

1

ζ − φx|~λ〉

where φx = aα + ε1(i − 1) + ε2(j − 1), α ∈ (1, · · · , n).Yutaka Matsuo (The University of Tokyo) Quantum toroidal algebra April 5, 2019 28 / 46


Equivalence of Wn + U(1) module and SHc

By taking coproduct n − 1 times, D0,2 ∼ H and D±n,0 are written interms of n-bosons.

Wn algebra + U(1) currents are written in terms of n-bosons,

(Q∂z − ∂zφ1) · · · (Q∂z − ∂zφn) =n∑

i=0

Qn−iW (i)(z)∂n−iz .

Schiffmann and Vasserot showed that these two Hilbert space areequivalent. This identification remains true even when the Hilbertspace is degenerate (such as minimal model)(Fukuda-Nakamura-M-Zhu).

Compared with W -algebra basis, SHc description has a definiteadvantage that it is expressed in terms of orthogonal basis labeled byn-tuple Young diagrams. Indeed, |~λ〉 can be identified with AFLTbasis.



Construction of Gaiotto state

AGT conjecture for pure super-SU(n) Yang-Mills is given by,

〈G |Λ2L0 |G 〉 =∑~λ

Zvect(~a, ~λ)Λ2|~λ|

where |G 〉 is called as Gaiotto state satisfying conherent state conditionfor W algebra,

W(d)1 |G 〉 = λ

(d)1 |G 〉, 1 ≤ d ≤ N,

Shiffmann and Vasserot claimed such |G 〉 is written as,

|G 〉 =∑~λ

|~a, ~λ〉,

where we used the normalization 〈~a, ~λ|~a, ~µ〉 = Zvect(~a, ~λ)δ~λ,~µ. It is easy tosee that this state satisfies reproduces the instanton partition function.



Proof of other gauge theories

Inclusion of fundamental matter [M-Rim-Zhang]

|G , ~m〉 =∑~λ

Zfd( ~m, ~λ)|~λ〉

satisfies (generalized) coherent state condition.

Description of quiver gauge theory [Kanno-M-Zhang, Negut]The vertex operator in CFT is expressed as,

Vµ =∑~λ,~ν

|~a, ~λ〉〈~b, ~ν|Zbf(~a, ~λ; ~b, ~ν|µ)

The right hand side behaves as a “primary field” under conformaltransformation. (cf. Carlsson and Okounkov)


Recent developments of DIM qq-character

1 Introduction

2 AGT conjecture




6 Conclusion



Seiberg-Witten curve and qq-character

Seiberg-Witten curve for SU(N) gauge theory:

y +1

y= u(z), u(z) :=

N∏i=1

(z − ai ), y ∈ C∗, z ∈ C or C∗ .

Can we derive it from second quantized system?

Yes! It reduces to a simple property of Gaiotto state![Bourgine-Matsuo-Zhang 2015]

D−1(z)|G , ~a〉 ∝1

Y(z)|G , ~a〉, D1(z)|G , ~a〉 ∝ P−

z Y(z + ε+)|G , ~a〉

where Y(z) is an operator which is diagonal w.r.t. |~λ〉,Y(z)|~λ〉 = Y(z , ~λ)|~λ〉.



Derivation of qq-character

For gauge theory with the coupling to fundamental matter, by evaluating

〈G |D−1(z)qD |G 〉,

in two ways, one obtains a generating function of the correlation functionof the form:

P−z

⟨Y(z + ε+) +

q

Y(z)

⟩= 0,

where

〈O〉 := 〈G , ~a|OqD |G , ~a〉〈G , ~a|qD |G , ~a〉

It implies ⟨Y(z + ε+) + qY(z)−1

⟩= χ(z)

χ(z) is N-th order polynomial which is a deformation of u(z). Nekrasovcalled it qq-character.



First and second quantization of SW curve

Classical: Seiberg-Witten curve for SU(N) gauge theory:

y +1

y= u(z), u(z) :=

N∏i=1

(z − ai ), y ∈ C∗, z ∈ C or C∗ .

Quantum: replace y = e~∂z , Schrodinger eq. gives quantum curve

(e~∂z + e−~∂z + u(z))ψ(z) = ψ(z + ~) + ψ(z − ~) + u(z)ψ(z) = 0

which looks like Baxter TQ relation (NS limit ε1 = ~, ε2 = 0).

Double quantum: ⟨Y(z + ε+) + qY(z)−1

⟩= χ(z)

SHc and quantum toroidal algebra knows how to describe quantizedgeometry from the representation theory. It also applies to toricCalabi-Yau which is described by topological vertex.


Recent developments of DIM corner VOA and WoW

1 Introduction

2 AGT conjecture




6 Conclusion



Plane partition realization of Affine Yangian

Plane partition realization: |Λ〉 the orthogonal basis labelled by a planepartition Λ, (recursion formula for Nekrasov factor)

ψ(u) |Λ〉 = ψΛ(u) |Λ〉 ,

e(u) |Λ〉 =∑∈Λ+

1

u − q − h

√− 1

σ3resu→q+h ψΛ(u) |Λ + 〉 ,

f (u) |Λ〉 =∑∈Λ−

1

u − q − h

√− 1

σ3resu→q+h ψΛ− (u) |Λ− 〉 ,

ψΛ(u) = ψ0(u − q)∏∈Λ

ϕ(u − q − h )

with h = h1x + h2y + h3z when the box is located at (x , y , z). Planepartition appears in the context of Calabi-Yau geometry through meltingcrystal picture.



Illustration of plane partition

When it is sliced in one direction, it is decomposed into a set of partitionsλ1 � · · · � λn. Plane partition may represent random surface or ”meltingcrystal”.



Corner VOA: more general truncation of AY

Corner VOA YLMN appears as a truncation of the affine Yangianthrough constraints on its parameters. (Prochazka and Rapcak)

L

λ1+

M

λ2+

N

λ3= 1

With this constraint, the plane partition contains a null state (pit) at(L+ 1,M + 1,N + 1). Note: there is a shift symmetry(L,M,N) → (L+ k ,M + k ,N + k) and one may set one of L,M,Nto be zero.

When (L,M,N) = (0, 0,N), the height of the plane partition islimited by N. Such diagram can be sliced horizontally to describe NYoung diagrams λ1 � λ2 � · · · � λN , which can be identified withthe Hilbert space of WN algebra (plus U(1) boson). This agrees withthe horizontal picture.



Plane partition with infinite legs

Plane partition may have nonvanishing asymptotic Young diagrams inthree directions, say λ, µ, ρ. The partition function for original planepartition is MacMahon function,

∞∏n=1

(1− qn)−n = M(q)

With the infinite legs attached, the partition function is proportional to,

Cµνρ(q)M(q)

where Cµνρ(q) is the (unrefined) topological vertex. It describes anontrivial representation of AY. The conformal weight and U(1) chargebecomes also nonvanishing. They are computed from the coefficients ofψ(u). This function remains finite since the contributions of infinite boxescancel with each other and only finite factor remains.



Connecting two Y -algebras: Web of W (WoW)

One may connect two corner VOAs by sharing one asymptotic Youngdiagram with another one.

L

M

N

R

We have two AYs acting on two plane partitions depicted as two circles. Inaddition to them, we need to introduce extra generators which changesthe shape of the asymptotic Young diagram. Thus, WoW consists oftensor product of two W1+∞ with extra generators.



WoW: generalization

By changing L,M,N,R, we obtain different types of VOA.

L

M

N

R

(L,M,N,R) = (0, 1, 1, 0) bcβγ system.

(L,M,N,R) = (0, 1, 2, 0) N = 2 SCA

(L,M,N,R) = (0, 1, 3, 0) Polyakov-Bershadsky VOA

One may connect the third vertex (WoWoW) and so on, and it describesinfinitely many unknown algebras. The treatment of negative Youngdiagrams may be more complicated for the general models.



Double truncation

In the following, we consider a special case where we put the secondconstraint on the parameters.

L1λ1

+M1

λ2+

N1

λ3= 1 ,

L2λ1

+M2

λ2+

N2

λ3= 1 .

In this case, we seem to have two pits at (Li ,Mi ,Ni ) (i = 1, 2). This is,however, too naive. With such parametrization, we have periodicity in hfunction,

hx+L1,y+M1,z+N1 = hx+L2,y+M2,z+N2

Since h governs the representation of the plane partition, above relationimplies that the plane partition becomes degenerate, namely the boxes inthe different locations should be identified. We will denote YL1M1N1:L2M2N2

to describe such algebra. With two constraints, there is no free parameter.



We illustrate the degeneration phenomena in the simplest case Y120:001.

1

2

3

1

2

3

⇔

The algebra may be identified as Y001 described by one Young diagram orY120 by plane partition with a pit at (2, 3, 1). One may construct thelatter diagram by cutting Young diagram into hooks. One needs to imposethat both diagrams are consistent as plane partition which imposes tightconstraints on both.



Double truncation and minimal models [Harada-M]

Theorem (Condition for Minimal models in corner VOA and WoW)

Double truncation of corner VOA and WoW describes their minimalmodels. The primary fields are labeled by asymptotic Young diagrams inthe free legs.

While we use ”theorem”, it remains a conjecture.

We checked it for Wn algebra and N = 2 SCA. It works perfectly.

In the conventional method, minimal models of VOA are obtained byexamining the null states and modular duality. Here we obtainedthem from a simple geometrical requirement.


Conclusion

Summary

The algebra U (quantum toroidal algebra, DIM, SHc , Affine Yangian..) provide a universal picture of conformal field theory and integrablemodels.

It manages to describe higher dimensional physics such as superYang-Mills in 4D and 5D and topological strings.

It also gives the second quantized picture of SW curve andCalabi-Yau geometry.

There appears a new truncation of U which gives a new family ofVOA.

Double truncation of U gives a geometrical picture of minimal modelsof VOA.


Documents

Seminar at YITP - 東京大学matsuo/file/YITP2019.pdfSeminar at YITP Quantum toroidal algebra Relation with integrability and application to AGT conjecture Yutaka Matsuo The University