K-theoretic AGT conjecture

Shintaro Yanagida (RIMS/OCAMI)

Brain Circulation Joint Meeting of Mathematics and Physics

March 31, 2014

Summary of the research in this project

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• Period of stay

2012/12/18 – 2013/3/7 Higher School of Economics (Moscow)2013/05/07 – 07/29 Kings College London2013/08/22 – 10/07 University of Toronto2013/11/16 – 12/16 Higher School of Economics (Moscow)2014/01/13 – 03/24 Higher School of Economics (Moscow)

• Contents

(1) Algebraic Geometry (papers [1], [2])moduli spaces of stable sheaves on abelian/K3 surfaces

(2) Representation theory (papers [3], [4])Hall algebras, the (deformed) Virasoro/W algebra

(3) Geometric Representation Theory (paper [5])AGT conjecture(s)

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• Papers

[1] S. Yanagida and K. Yoshioka, Bridgeland’s stabilities on abeliansurfaces, Mathematische Zeitschrift 276 (2014), Issue 1-2, pp 571-610.

[2] H. Minamide, S. Yanagida and K. Yoshioka, Some moduli spaces ofBridgeland’s stability conditions, to appear in Int. Math. Res. Notices.

[3] S. Yanagida, Bialgebra structure on Bridgeland’s Hall algebra oftwo-periodic complexes, arXiv:1304.6970.

[4] S. Yanagida, Classical and Quantum Conformal Field Theories,arXiv:1402.2943.

[5] S. Yanagida, Whittaker vector of deformed Virasoro algebra andMacdonald symmetric functions, arXiv:1402.2946.

Abstract

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• Main objects of this talk:1. K-theoretic Nekrasov partition functionthe generating function of “volumes” of instanton moduli spacessatisfying Zamolodchikov-type recursive formula

2. Why does such a formula hold?hidden symmetry: the deformed Virasoro algebraa quantum deformation of the Virasoro Lie algebra, which describes thealgebraic system underlying in the 2-dim conformal field theory

• This topic stems from the so-called (K-theoretic) AGT conjecture.

Contents

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§1. K-theoretic Nekrasov partition function

§2. Zamolodchikov-type recursive formula

§3. Virasoro algebra

§4. AGT conjecture

§5. K-theoretic analogue

§6. Explicit formula of Whittaker vector

§1 Nekrasov partition function

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§1.1 Combinatorial expression [Nekrasov, 2002]

r ∈ Z≥1 : rankx, ~, ǫ1, ǫ2,

−→a = (a1, ... , ar) : indeterminant

Z(x; ~; ǫ1, ǫ2;−→a ):=

∑

−→Y

x|−→Y |

∏1≤α,β≤r N

−→Yα,β

∈ Q(e~ǫi, e~aα)[[x]]

−→Y = (Y1, ... , Yr) : r-tuple of partitions |−→Y | := |Y1| + · · · + |Yr|

N−→Yα,β :=

∏

�∈Yα

(1 − exp[~{ℓYβ(�)ǫ1 − (aYα(�) + 1)ǫ2 − aβ + aα}])

∏

�∈Yβ

(1 − exp[~{−(ℓYβ(�) + 1)ǫ1 + aYα(�)ǫ2 − aβ + aα}])

§1.2 Partition, Young diagram

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• Partition λ = (λ1,λ2, ... ,λk) : empty sequence ∅ = () = (0) or

non-increasing sequence of natural number (λ1 ≥ · · ·λk ≥ 1)

|λ| := λ1 + · · · + λk , ℓ(λ) := k : length

λ ⊢ ndef⇐⇒ λ is a partition s.t. |λ| = n.

• Young diagram of the partition λ:

λ1 boxes

λ2 boxes

λℓ−1

λℓ

✲

❄

j

i

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• (relative) arm and leg

λ : partition, � = (i, j) : box located at (i.j)

aλ(�) := λi − j : arm, ℓλ(�) := λ∨j − i : leg,

where λ∨ is the transpose of λ, λi :=

{λi (i ≤ ℓ(λ))

0 (i > ℓ(λ))

E.g. λ = (3, 2, 1, 1) λ∨ = (4, 2, 1)

(3, 2, 1, 1) (4, 2, 1)

� = (1, 1) aλ(�) = λ1 − 1 = 3 − 1 = 2 aλ(�) = λ1 − 1 = 2, ℓλ(�)= λ∨

1 − 1 = 4 − 1 = 3 ℓλ(�) = λ∨1 − 1 = 3

� = (4, 3) aλ(�) = λ4 − 3 = −2, ℓλ(�) = λ∨3 − 4 = −3

§1.3 Geometric definition

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• M(r,n) : moduli space of framed torsion free sheaves on CP2

with rank = r, c2 = nsmooth quasi-projective variety, dim = 2rn

• ADHM (Quiver) description :

M(r, n) =

(B1, B2, j, k)

∣∣∣∣∣

B1, B2 ∈ End(Cn),j ∈ Hom(Cr,Cn),k ∈ Hom(Cn,Cr)

s.t. (1)&(2)

/GLn(C)

(1)[B1, B2] + jk = 0

(2)there is no proper subspace S ⊂ Cn s.t. Bi(S) ⊂ S and Im(j) ⊂ S

• Action of T := T2 × Tr (T := C×):

(B1, B2, j, k) 7→ (t1B1, t2B2, js−1, t1t2sk)

ti = e−~ǫi ∈ T, s = (sα = e−~aα)rα=1 ∈ Tr

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• Partition function of ”5-dim pure SU(r)-gauge theory”

ZNf=0,instrank=r (x; ~; ǫ1, ǫ2;

−→a ) = Z(x; ~; ǫ1, ǫ2;−→a )

:=

∞∑

n=0

xn2nr∑

i=0

(−1)ichT[Hi(M(r, n),O)]

=

∞∑

n=0

xnchT[H0(M(r, n),O)]

• Localization Theorem for T-equivariant Grothendieck group gives

Z(x; ~; ǫ1, ǫ2;−→a ) =

∑

−→Y

x|−→Y |

∧−1([T−→

Y]),

where−→Y = (Y1, ... , Yr) parameterize the T-fixed points.

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• RemarkThe limit ~ → 0 gives the (cohomological) Nekrasov partition function.

Zcoh(x; ǫ1, ǫ2;−→a ) :=

∞∑

n=0

xn lim~→0

2nr∑

i=0

(−1)ichT[Hi(M(r, n),O)]

=∑

−→Y

x|−→Y |

∏1≤α,β≤r n

−→Yα,β(ǫ1, ǫ2,

−→a ),

n−→Yα,β(ǫ1, ǫ2,

−→a ) :=∏

�∈Yα

[−ℓYβ(�)ǫ1 + (aYα

(�) + 1)ǫ2 + aβ − aα]

×∏

�∈Yβ

[(ℓYα(�) + 1)ǫ1 − aYβ(�)ǫ2 + aβ − aα].

The localization theorem means

∏1≤α,β≤r n

−→Yα,β(ǫ1, ǫ2,

−→a ) = EuT(T−→Y).

§1.4 Examples of Nekrasov partition function

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• Abbreviation:

Zcoh−→Y

(ǫ1, ǫ2;−→a ) := [

∏

1≤α,β≤r

n−→Yα,β(ǫ1, ǫ2;

−→a )]−1,

Zcohn (ǫ1, ǫ2,

−→a ) :=∑

|−→Y |=n

Zcoh−→Y

(ǫ1, ǫ2,−→a ).

E.g. r = 1 : ZcohY =

1

nY11(ǫ1, ǫ2)

nY11 =∏

�∈Y[−ℓY(�)ǫ1 + (aY(�) + 1)ǫ2][(ℓY(�) + 1)ǫ1 − aY(�)ǫ2]

Zcoh0 = Zcoh

(∅) = 1, Zcoh1 = Zcoh

(1) =1

ǫ1ǫ2

Zcoh2 = Zcoh

(2) + Zcoh(1,1) =

1

2(ǫ1 − ǫ2)ǫ1ǫ22

+1

2(ǫ2 − ǫ1)ǫ21ǫ2

=1

2ǫ21ǫ22

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Zcoh3 = Zcoh

(3) + Zcoh(2,1) + Zcoh

(1,1,1) =1

6ǫ1ǫ22(ǫ1 − ǫ2)(ǫ1 − 2ǫ2)

+

1

(2ǫ1 − ǫ2)(2ǫ2 − ǫ1)ǫ1ǫ21

+1

6ǫ21ǫ2(ǫ1 − ǫ2)(2ǫ1 − ǫ2)=

1

6ǫ31ǫ32

• If r = 1, then

Zcoh(x; ǫ1, ǫ2) =∑

n=0

xnZcohn (ǫ1, ǫ2) = exp(

x

ǫ1ǫ2)

E.g. 2. r = 2

Zcoh1 = Zcoh

(1),∅ + Zcoh∅,(1)

=1

ǫ1ǫ2(a1 − a2)(ǫ1 + ǫ2 + a2 − a1)+

1

(ǫ1 + ǫ2 + a1 − a2)(a2 − a1)ǫ2ǫ1

Zcoh2 = Zcoh

(2),∅ + Zcoh(1,1),∅ + Zcoh

(1),(1) + Zcoh∅,(1,1) + Zcoh

∅,(2)

§2 Zamolodchikov-type recursion formula

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Hereafter we only consider the case r = 2.

§2.1 Cohomological case

Theorem[Fateev-Litvinov 2010]

Set a := a1 − a2. Then Zn(ǫ1, ǫ2, a) satisfies the next recursion formula:

Zcohn (ǫ1, ǫ2, a) = δn,0 +

∑

r,s∈Z≥1

1≤rs≤n

Rcohr,s (ǫ1, ǫ2)Z

cohn−rs(ǫ1, ǫ2, (rǫ1 − sǫ2)/2)

4a2 − (rǫ1 + sǫ2)2,

Rcohr,s (ǫ1, ǫ2) := 2

∏

1−r≤j≤r1−s≤k≤s(r,s) 6=(0,0)

(jǫ1 + kǫ2)−1.

Remark

The proof uses an integral expression of Zn.

§2.2 K-theoretic case

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Set Q := e~(a2−a1), q := e−~ǫ1 and t := e~ǫ2 .Then Z(x; ~; ǫ1, ǫ2;

−→a ) =∑

n≥0 xnZn(q, t; Q) with Zn(q, t; Q) ∈ Q(q, t, Q).

Theorem [Y. 2010]

K-theoretic partition function Z(x; ~; q, t; Q) =∑

n xnZn(q, t; Q) satisfies

Zn(q, t; Q) = δn,0 +∑

r,s∈Z1≤rs≤n

Rr,s(q, t)Zn−rs(q, t; qrts)

Q − qrt−s

with

Rr,s(q, t) := sgn(−r)qrt−s∏

1−|r|≤j≤|r|1−|s|≤k≤|s|(j,k) 6=(0,0)

(1 − qjt−k)−1

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• Key of the proof: the integral formula of ZKn

q1 := q, q2 := t−1, q3 := q−1t

Zn(q, t, Q) =1

n!

(1 − q−1

3

(1 − q1)(1 − q2)

)n

×∮

· · ·∮ n∏

i=1

dxi

2π√−1

n∏

k=1

P(xk; Q1/2, q3)

∏

i<j

ω(xj/xi; q1, q2, q3)

where

P(x; a, q) :=x

(x − a)(x − a−1)(x − qa)(x − qa−1)

ω(y; q1, q2, q3) :=(y − 1)2(y − q3)(y − q−1

3 )

(y − q1)(y − q−11 )(y − q2)(y − q−1

2 )

§3 Virasoro algebra

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§3.1 Definition• Virc : Lie algebra with central extension

c ∈ C : central charge,

generators : Ln (n ∈ Z)

relations : [Ln, Lm] = (n − m)Ln+m +1

12cn(n2 − 1)δn+m,0

• Triangular decomposition : Virc = Virc,+ ⊕ Virc,0 ⊕ Virc,−

Virc,± := ⊕±n∈Z>0CLn, Virc,0 := CL0 ⊕ C

• PBW basis of U(Virc): {L−λLn0Lµ | n ∈ Z≥0, λ,µ : partition}

with

Lλ := Lλℓ· · · Lλ1, L−λ := L−λ1 · · · L−λℓ

for a parition λ = (λ1, ... ,λℓ)

§3.2 Verma module Mc,h

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• h ∈ C : highest weight

Cc,h := C|c, h〉 : one-dimensional (Virc,+ ⊕ Virc,0) representation

Ln |c, h〉 = 0 (n > 0), L0 |c, h〉 = h |c, h〉• Mc,h := IndVircVirc,+⊕Virc,0

Cc,h

• weight decomposition :

Mc,h = ⊕n∈Z≥0M

(n)c,h, M

(n)c,h := {v ∈ Mc,h | L0v = (h − n)v}

• basis of Mh,n : {L−λ |h〉 ;λ ⊢ n}

Dual Verma module M∗c,h := IndVircVirc,−⊕Virc,0

C∗h

• C∗h := C〈c, h| : 1-dim (Virc,− ⊕ Virc,0) rep.

• 〈c, h| Ln = 0 (n < 0), 〈c, h| L0 = h 〈c, h|• M∗

c,h = ⊕n∈Z≥0M

∗,(n)c,h , M

∗,(n)c,h := {v ∈ M∗

c,h | vL0 = (h + n)v}

• basis of M∗,(n)c,h : {〈c, h| Lλ;λ ⊢ n}

§3.3 Shapovalov form

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• · : M∗c,h × Mc,h → C : bilinear form

〈c, h| · |c, h〉 := 1, 〈c, h| Lλ · L−µ |c, h〉 :=∑

ν1,ν2,n

δν1,∅δν2,∅cν1,ν2,nhn

(LλL−µPBW=

∑

ν1,ν2:partitionn∈Z≥0

cν1,ν2,nL−ν1Ln0Lν2)

• Then uLn · v = u · Lnv (u ∈ M∗c,h, v ∈ Mc,h).

Below we simply write 〈c, h| LλL−µ |c, h〉 := 〈c, h| Lλ · L−µ |c, h〉, uv := u · v.

• 〈c, h| LλL−µ |c, h〉 = 0 unless |λ| = |µ|• 〈c, h| LλL−µ |c, h〉 = 〈c, h| L−µLλ |c, h〉.

§3.4. Kac determinant

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• Define a matrix Kn := (〈c, h| LλL−µ |c, h〉)λ,µ⊢n. Then

det Kn =∏

r,s∈Z≥1

1≤rs≤n

(h − hr,s)p(n−rs),

where p(m) := #{λ | λ ⊢ m},

hr,s :=1

48[(13 − c)(r2 + s2) − 24rs − 2(1 − c)

+√(1 − c)(25 − c)(r2 − s2)]

Proof needs free field realization (Feigin-Fuchs, 1980’s.)

• Single pole phenomena [K. Brown, J. Algebra (2003)]

Each element of K−1n has at most simple poles with respect to h.

Remark Similar phenomena holds for finite dimensional Lie algebra.(Ostapenko, J. Algebra 147 (1992))

§4. AGT conjecture

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• The original AGT conjecture [Alday-Gaiotto-Tachikawa, 2009] claims

the conformal block (four-point correlation function of CFT) coincides

with the Nekrasov partition function (with Nf = 4 matters).

• Here we only mention a degenerate version [Gaiotto 2009]

§4.1 Whittaker vector

• Λ ∈ C fix.

Whittaker vector wcoh is an element of Mc,h :=∏

n≥0 M(n)c,h satisfying :

L1wcoh = Λ2wcoh, Lnw

coh = 0 (n ≥ 2), wcoh = |h〉 + · · ·

• Dual Whittaker vector w∗,coh ∈ M∗c,h is similarly defined:

w∗,cohL−1 = Λ2w∗,coh, w∗,cohL−n = 0 (n ≥ 2), w∗,coh = 〈h| + · · ·

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• Under the expansion

wcoh = |c, h〉 + Λ2wcoh1 + Λ4wcoh

2 + Λ6wcoh3 + · · · , wcoh

n ∈ Mh,n,

one can calculate

wcoh1 =

1

2hL−1 |h〉 ,

wcoh2 =

(8h + 8c)L2−1 − 12hL−2

4h(16h2 + 2ch − 10h + c)|h〉 ,

wcoh3 =

1

24h(3h2 + ch − 7h + 2)(16h2 + 2ch − 10h + c)

[12h(7h − c − 3)L−3

− 12(9h2 + 3ch − 7h + c)L−2L−1 + (24h2 − 26h + 11ch + 8c + c2)L3−1

]

· |h〉 ,wcoh

4 = · · ·

• wcohn = L1w

cohn+1, wcoh

d vn = 0 (d ≥ 2).

§4.2 AGT relation for pure SU(2)-gauge theory

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Theorem [Y. 2011]

〈w∗,coh, wcoh〉 = Zcoh,r=2(x; ǫ1, ǫ2,−→a ).

Here the parameters are related as :

Virasoro Nekrasov

c 13 + 6(ǫ1/ǫ2 + ǫ2/ǫ1)h (ǫ1/ǫ2 + ǫ2/ǫ1 + 2)/4 − (a2 − a1)

2/ǫ1ǫ2Λ x1/4/(ǫ1ǫ2)

Restatement [Marshakov-Mironov-Morozov, 2009]

One has⟨w∗,coh | wcoh

⟩=∑∞

n=0 Λ4n(K−1

n )(1n,1n), so that

(K−1n )(1n,1n) = (ǫ1ǫ2)

4nZn(ǫ1, ǫ2;−→a )

§4.3. A proof of the pure SU(2) AGT relation

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• To show the coincidence of Zn and (K−1n )(1n,1n), it is enough to prove that

both equations satisfy the same recursive formula.

Theorem [Y. 2011]

fn(h, c) = δn,0 +∑

r,s∈Z≥1

1≤rs≤n

Rcohr,s (c)fn−rs(c, hr,−s)

h − hr,s

for fn(c, c) := (Kn)−1(1n,1n) on Mh,c.

§4.4 What is AGT conjecture ?

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[Alday-Gaiotto-Tachikawa, 2009]

Formulation in physics:

4 dimensional N = 2 SU(2) super Yang-Mills theory

=?2 dimensional Liouville conformal field theory

(Partial) mathematical formulation:

T-equivariant cohomology of instanton moduli space(algebraic geometry)

=?Verma module of Virasoro algebra

(representation theory)

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moduli space gauge group algebra proved?

M(2, n) G = SL(2,C) Vir YM(r, n) G = SL(r,C) W(g) Y [1,2]UnG G: general W(Lg) △ [3]

UnG,P G: general, P = B L

gaff Y [4]

G, P: general W(Lg, e) N

Table 1: AGT conjectures

[1] Shiffmann-Vassserot (2012)[2] Maulik-Okounkov (2012)[3] Braverman-Finkelberg-Nakajima (2014?) for G = ADE[4] Braverman-Etingof (2005)

§5 K-theoretic analogue

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In the K-theoretic analogue, the symmetry should be described by thedeformed Virasoro algebra Virq,t.

§5.1 Virq,t [Shiraishi-Kubo-Awata-Odake (1996)]

• q, t ∈ C, p := qt−1

generator : Tn(n ∈ Z),

relation : [Tn, Tm] = −∞∑

ℓ=1

fℓ(Tn−ℓTm+ℓ − Tm−ℓTn+ℓ)

− (1 − q)(1 − t−1)

1 − p(pn − p−n)δn+m,0,

where∞∑

k=0

fkzk = exp[

∞∑

n=1

(1 − qn)(1 − t−n)

1 + pnzn

n].

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• In the limit ~ → 0 with t = qβ and q = e~,T(z) :=

∑n∈Z Tnz

−n has the ~-expansion

T(z) = 2 + β~2(z2L(z) +(1 − β)2

4β) + O(~4) (L(z) =

∑

n∈Z

Lnz−n−2)

gives the relations of Virc among Ln with c = 1 − 6(1 − β)2/β.

§5.2 Whittaker vector

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• h ∈ C: highest weight

Mh : Verma module of Virq,t

: generated by |h〉 s.t. Tn |h〉 = 0 (n > 0), T0 |h〉 = h |h〉.Mh is graded as Mh = ⊕n∈Z≥0

Mh,n by deg Tn := −n, deg 〈h| = 0.

• The dual Verma modules M∗h generated by 〈h| is similarly defined.

• The Whittaker vector w ∈ Mh is an element satisfying

T1w = Λ2w, Tnw = 0 (n ≥ 2), w = |h〉 + · · ·

• The dual vector w∗ is similarly defined.

§5.3. K-theoretic AGT relation

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Theorem [Y]

〈w∗ | w〉 = Zr=2(x; ~, ǫ1, ǫ2,−→a ),

where the parameters are related as

Virq,t K theoretic Nekrasov

q e−~ǫ1

t e~ǫ2

h e~(a1−a2)/2 + e~(a2−a1)/2

Λ x1/4

Remark (1) Conjectured by Awata and Yamada (2009).

(2) In the higher rank case (r ≥ 3), the deformed W algebra Wq,t(slr)comes into the game.

(3) It is also related to the r-tensored Fock representation of theDing-Iohara-Miki algebra, conjectured byAwata-Feigin-Hoshino-Kanai-Shiraishi-Y. (2011).

§6 Explicit formula of Whittaker vector

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Using the Fock representation of Virq,t, we have an explicit formula of w.

• Fock representation of Virq,t [Shiraishi-Kubo-Awata-Odake (1996)]

T(z) = Λ1(z) + Λ2(z),

Λ1(z) = p1/2 exp[−∞∑

n=1

1 − tn

1 + pnb−n

nt−np−n/2zn]

× exp[−∞∑

n=1

(1 − tn)bn

npn/2z−n]tb0

Λ2(z) = p−1/2 exp[∞∑

n=1

1 − tn

1 + pnb−n

nt−npn/2zn]

× exp[

∞∑

n=1

(1 − tn)bn

np−n/2z−n]t−b0

bn are the Heisenberg generators s.t. [bn, bm] = n1 − q|n|

1 − t|n|δn+m,0b0.

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• Fock representation

H: the Heisenberg algebra generated by bn and 1.

F0: the representation of H generated by |0〉 with bn |0〉 = 0 (n ≥ 0).

Then Virq,t acts on F0 as T0 |0〉 = h |0〉, Tn |0〉 = 0 (n ≥ 0).

For generic q, t, h, we have an isomorphism

Mh∼−−→ F0, |0〉 7−→ 1,

intertwining the actions of Virq,t and H.

• F0 is identified with the space of symmetric functions byb−λ |0〉 7−→ pλ,

where pλ = pλ1pλ2 · · · is the power-sum symmetric function.

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Recall the Whittaker vector w =∑

n≥0 Λ2nwn of Virq,t.

Under the above isomorphisms, wn is an element of degree n symmetricfunction with coefficient in Q(q, t, h).

Theorem [Y]

wn =∑

λ⊢n

Pλ(q, t)∏

(i,j)∈λ

Q1/2

1 − Qqjt−i

qλi−j

1 − qλi−j+1tλ∨j −i

Pλ(q, t): Macdonald symmetric function.h = Q1/2 + Q−1/2.

Remark

(1) Conjectured by Awata and Yamada (2009).

(2) In the cohomological limit, we have an explicit formula of the Whittakervector wcoh of the Virasoro Lie algebra (originally proved by [Y. 2010]).