A New 2d/4d Duality via Integrability

A New 2d/4d Duality via Integrability

Heng-Yu Chen

Department of Physics National Taiwan University

February 28, 2012/ NTU String Seminar, Taipei

Based on1104.3021, withNick Dorey and Sungjay Lee (DAMTP) and Tim Hollowood (Swansea).

A New 2d/4d Duality

I Theory I: Four dimN = 2 SQCD with G = SU(L), plus L fundamental hypermultiplets of masses ~mF= (m1, . . . , mL) and L anti-fundamental hypermultiplets of masses ~mAF= ( ˜m1, . . . , ˜mL).

The complex gauge coupling is τ = ^4πi_g2 +^Θ_2π^4D

I Theory I is subjected to Ω-deformation with (₁, ₂) = (, 0), which preservesN = (2, 2) SUSY in x⁰− x¹plane. The coulomb branch of undeformed theory is lifted, only discrete points remain:

~a = ~mF− ~n , ~n = (n1, . . . , nL)∈ Z^L. (1)

I The low energy dynamics are governed by twisted superpotential W^{(I )}(~a, ), which is inherited from Nekrasov partition function Z(~a, 1, ₂) as:

W^{(I )}(~a, ) = lim

1→,2→0[₂Z(~a, 1, ₂)] + quantized fluxes (2)

I Theory II: Two dim N = (2, 2) SYM with G = U(N), plus L fundamental chiral multiplet of twisted masses ~MF= (M1, . . . , ML), L anti-fundamental chirals of twisted masses ~MAF= ( ˜M1, . . . , ˜ML);

and an adjoint chiral multiplet of twisted mass . The complex gauge coupling is ˆτ = ir +^Θ_2π^2D.

I This is the world volume theory of 4 dim “vortex/surface operator”.

Its low energy dynamics is also governed by effective twisted superpotential W^{(II )}({λk}) from an one-loop computation.

I W^{(II )}({λk}) is a “Yang-Yang” potential so that the F-term equation dλ_jW^{(II )}= 0 coincides with the Bethe Ansatz Equation (BAE) of SL(2,R) spin chain:

YL l =1

λ_j− Ml

λj− ˜Ml

=−q YN k=1

λ_j− λk− 

λj− λk+ , q = (−1)^N+1e^{2π ˆτ .} (3)

I The solution “Bethe Roots”{λj ≡ λ(ls)} are given by:

λ(ls) = Ml− (s − 1) + O(q) , s = 1, . . . , ˆnl, N = XL

l =1

ˆ nl. (4)

I The Conjectured Duality states that, the on-shell values of the twisted superpotentials for Theory I/II coincide:

W^{(I )}(a_l= m_l− nl)− W^{(I )}(a_l= m_l− ) = W^{(II )}({ˆnl}) , (5) if we make following identification of parameters:

ˆ

τ = τ +1

2(N + 1), ˆn_l = n_l− 1, Ml = m_l−3

2, ˜M_l= ˜m_l−1 2. (6)

I The VEVs of Chiral ring of Theory IOk = Trϕ^k are also mapped the conserved charges of SL(2,R) spin chain arising from Theory II.

I The exact perturbative matching, and first few instanton checks were performed earlier. Here we shall prove the duality exactly, by saddle point analysis of Z(~a, 1,2), such that SL(2,R) BAE appears andW^{(I )} andW^{(II )} match on-shell. The steps can be easily generalized for proving the duality in wide range of set-ups.

BAE from Nekrasov Instanton Partition Function

I We begin with the Gamma-function representation of Nekrasov Partition function[Nekrasov-Okounkov]:

Zinst=X

{~Y}

q^|~Y|Zvec( ~Y ) Y2L n=1

Zhyp( ~Y , µn) , q = e^{2πi τ} (7)

where Zvec( ~Y ) andZhyp( ~Y , µn) are:

Zvec( ~Y ) = Y

(li )6=(nj)

Γ ⁻¹₂ (xli − xnj− 1)

Γ ⁻¹₂ (x_li− xnj)
· Γ ⁻¹₂ (x_li⁽⁰⁾− xnj⁽⁰⁾)
Γ ⁻¹₂ (x_li⁽⁰⁾− xnj⁽⁰⁾− 1)
, Zhyp( ~Y , µn) =Y

li

Γ ⁻¹₂ (xli+ µn)
Γ ⁻¹₂ (x_li⁽⁰⁾+ µn)
.

xli = al+ (i− 1)1+ 2kli , x_li⁽⁰⁾ = al+ (i− 1)1. (8) with kli being the length of i -th row in the Young Tableau Yl.

I Now if we take the limit (₁, ₂)→ (, 0)[Nekrasov-Shatashvili], Stirling’s approximation for Γ(x ) yields:

Zinst=Z Y

li

dxli exp[⁻¹₂ Hinst(xli, x_li⁽⁰⁾)] , Hinst(xli) =Y(xli)−Y(xli⁽⁰⁾) , (9) where

Y xli

= log qX

(li )

xli+X

(li ),n

f (xli + ˜mn) + f (xli− mn+ )

+ 1

2 X

(li )6=(kj)

f (x_li− xkj− ) − f (xli− xkj + )

, (10)

with f (x ) = x log x− x and Y(xli⁽⁰⁾) =Y(xli → xli⁽⁰⁾).

I As 2→ 0, the instanton positions condense and become constant on the intervals:

I =[

li

[x_li⁽⁰⁾, xli] . (11)

I We can re-expressHinst in terms of instanton density ρ(x ):

Hinst[ρ] =−1 2 Z

I×Idx dy ρ(x )G(x−y)ρ(y)+

Z

Idx ρ(x ) log q R(x )
, (12) where the kernels are:

G(x ) = d dx log

x−  x + 



, R(x ) = A(x )D(x + ) P(x )P(x + ) ,

A(x ) = YL l =1

(x− ˜ml) , D(x ) = YL l =1

(x− ml) , P(x ) = YL l =1

(x− al) .

I In the ₂→ 0 limit, the functional integral is dominated by “saddle point equation”:

δHinst[ρ]

δx_j =− Z

Idy G(xj− y)ρ(y) + log q R(xj)

= 0 , (13)

I Integrating and exponetiating the saddle point equation, we obtained infinite set of equations for{xli}:

Q(xli+ )Q⁽⁰⁾(x_li − )

Q(xli− )Q⁽⁰⁾(xli + )=−q R(xli) , (14) Q(x ) =

YL k=1

Y∞ j =1

(x− xkj) , Q⁽⁰⁾(x ) = YL k=1

Y∞ j =1

(x− x_kj⁽⁰⁾) .

I To see SL(2,R) spin chain appearing, the infinite equations (14) can be truncated to finite set, if we impose the “quantization condition”:

al = ml−nl , nl∈ Z > 0 , −→ xli = x_li⁽⁰⁾= al+(i−1) , for i ≥ nl. (15)

One Slide Proof for (15)

I We can consider the following equality:

W(x + )−(1 + q)

2 W(x ) T (x )

P(x + ) =−qR(x)W(x − ) , (16) where

W(x ) = Q(x )

Q⁽⁰⁾(x ), T (x ) = 2 (1 + q)

Q(x + )

Q(x ) + qA(x )D(x )Q(x− ) Q(x )

 ,

T (x ) is a degree L polynomial in x .

I Now is the quantization condition al= ml− nl is imposed, the simple pole at x = al+ (nl− 1) in W(x − ) on RHS of (16) coincides with a zero of R(x ), this implies W(x ) cannot have simple pole at x = al+ (nl− 1) either. The argument can be repeated continuously for i ≥ nl, and only possible if x_li = x_li⁽⁰⁾, i ≥ nl, hence we obtain (15).

I Having truncated the infinite set of equations by quantization condition, we arrive at:

D(x_li + 2)

A(xli) =−qQ(xˆ _li− )

Q(xˆ li+ ) , Q(x ) =ˆ YL l =1

nYl−1 i =1

(x− xli) , (17) substituting in the identifications of parameters (6) and

xli = λli−^₂, we finally see that (17) precisely coincides with the SL(2,R) BAE (3).

I To complete the proof, we can now evaluateHinst[ρ] with the truncation/quantization condition imposed, and obtain:

Winst^(I)(ml− nl)− Winst^(I)(ml− ) = ˆY xli

− ˆY(xli⁽⁰⁾

, (18)

Y(xˆ li) = log q XN (li )=1

x_li + XN (li )=1

XL n=1



f (x_li− ˜m_n)− f (xli − mn+ 2)



+1 2

XN (li )6=(mj)=1



f (x_li − xmj− ) − f (xli − xmj+ )



. (19)

Again after matching the parameters, this precisely matches with the ˆ

q/instanton-dependent part ofW^{(II )}({ˆnl}) and completes our proof.

Simple Generalization: Linear Quiver Gauge Theories Here we provide a simple generalization to Ap-linear quiver gauge theories and their associated spin chains.

I Theory I: Four dim N = 2 with G = SU(L)^p, plus bi-fundamental hypermultiplets between adjacent nodes of mass µ_I I = 1, . . . , p− 1, the last (first) node has L (anti)-fundamental hypermultiplets of masses−mk+  (− ˜ml). Each SU(LI) has τI =^4πi_g2

I

+^Θ_2π^4DI .

I Theory II: Two dimN = (2, 2) SYM with G =Qp

I =1U(NI), with matter content of one adjoint of twisted mass  for each U(NI), bi-fundamentals of twisted mass /2 under U(NI)× U(NI +1). The U(N₁) node also has L fundamentals of ~M_F= (M₁, . . . , M_L) and L anti-fundamentals of ~M_AF= ( ˜M₁, . . . , ˜M_L). The complex gauge couplings are ˆτI = irI+^Θ_2π^2DI I = 1, . . . , p.

...

NS5

.. ... ..

... ..

x

x x

7

6 4,5 m₁ m₂

m₃

m_L m_L−1

˜ m3

˜ m₂

˜ m₁

˜ m_L

˜ m_L−1

... ..

.. .

NS5 NS5 NS5

a⁽¹⁾

L−1

a⁽¹⁾₃ a⁽¹⁾₁ a⁽¹⁾₂ a^(p)₁

a^(p)₂

a^(p)

L−1

a^(p)

L a⁽¹⁾

L

a^(p)₃

D4

Figure: The IIA-brane construction for Theory I in the linear quiver case.

D4

NS5

... ..

x

x x

7

6 4,5

... ..

D2

ˆ n⁽²⁾₃ ˆ n⁽²⁾₂ ˆ n⁽²⁾₁

ˆ n⁽²⁾

L ˆ n⁽²⁾

L−1

NS5

NS5

NS5

m₁ m₂ m3

m_L m_L−1

˜ m₃

˜ m₂

˜ m₁

˜ m_L

˜ m_L−1

Figure: The IIA-brane brane construction for Theory II in the linear quiver case.

I From the explicit brane set-up, we see that N_I =Pp J=I

PL l =1ˆn_l^(J), where ˆn_l^(J)is the number of D2s between l -th D4 and J-th NS5.

I The F-term equation of Theory II is identified with the BAE of SL(p + 1,R) spin chain (CIJ = Cartan matrix of SL(p + 1,R)):

−qI

Yp J=1

QJ(λ^{(I )}_j −¹₂CIJ) QJ(λ^{(I )}_j +¹₂CIJ)

=





d (λ⁽¹⁾_j )

a(λ⁽¹⁾_j ) I = 1 1 I > 1 ,

(20)

I The duality in this generalization states that:

W^(I)

ml−n^{(I )}_l −

p−1

X

J=I

µJ

−W^(I)

ml−−

p−1

X

J=I

µJ



=W^(II) {n^{(I )}_l }
,

(21) with the following identification of parameters:

x^{(I )}= λ^{(I )}−

p−1

X

J=I

µJ−1 2)−1

2 , qˆI = (−1)^NI+1qI

M_l= m_l−p + 2

2  , M˜_l= ˜m_l+

p−1

X

J=1

µ_J−1 2) +1

2 .

Future Directions

I Generalization of duality to other gauge groups SO(N) etc., or to other dimensions, compactifications from higher dimensions.

I Quantizing other more interesting integrable systems, such elliptic Calogero-Moser, Toda, Hitchin, Ruijsenaar-Schneider systems etc.?

I How do electromagnetic duality and mirror symmetry affect our duality/correspondence?

I Connections with matrix models and topological strings from instanton partition functions.

I Connections with wall-crossing phenomena in both 2 dim and 4 dim supersymmetric gauge theories?