Residue Formulas for Prepotentials, InstantonExpansions and Conformal Blocks

P. Gavrylenko1, A. Marshakov: hep-th/1312.6382
111

Department of Mathematics, NRU HSE, Moscow, Russia
Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine
Department of Physics, Kyiv National University, Ukraine

P. Gavrylenko, A. Marshakov ()Residue Formulas for Prepotentials, Instanton Expansions
4 ìàðòàand2014
Conformalã.
Blocks1 / 13

Generalized Seiberg-Witten system-parametric family of the genus-g curves Σ: F (x, z; u1, ..., ug ) = 0with marked cycles {Ai , Bi } ∈ H1(Σ, Z)

gSeiberg-Witten 1-form dS = xdz
Such connestion ∇ that ∇ dS are holomorphic
Generalized Seiberg-Witten equations (Krichever(92),
Seiberg-Witten(94),...)∂∂ui

1ai =2πi

ZdS,AiP. Gavrylenko, A. Marshakov ()aiD =ZdS =∂F∂aii = 1, . . . , gBi

Residue Formulas for Prepotentials, Instanton Expansions
4 ìàðòàand2014
Conformal
ã.
Blocks2 / 13

Generalized Seiberg-Witten system∂F
1 X(dS)2=Res∂qi2 −1dzp(qi )Where qi are the poles of 2-dierential in the curve denition in theGaiotto description of the quiver gauge theories.
x 2 = φ2 (z)dS = xdz

Whereφ2 (z)dz 2 ≈miare mass parameters

mi2 dz 2ci (u1 , ..., ug )dz 2++ reg .
(z − zi )2z − zi{zi } = {z1 , ..., zg +3 } → {0, 1, ∞, q1 , ..., qg }P. Gavrylenko, A. Marshakov ()Residue Formulas for Prepotentials, Instanton Expansions
4 ìàðòàand2014
Conformal
ã.
Blocks3 / 13

3-rd dervative residue formula and WDVV equation
Theorem:identity(Krichever, P.G.-A.M.) Extended SW prepotential satises anX∂3F
d$I d$J d$K=Res∂XI ∂XJ ∂XKdxdz
dx=0

where {XI } = {ai } ∪ {qk } and {d$I } = {dωi } ∪ {dΩk }∂dSHere dωi = ∂dS∂a are holomorphic, and dΩi = ∂q are meromorphic.
ii(Consequence of the previous one): Prepotential of the quivergauge theory satises WDVV equation (Marshakov-Mironov-Morozov, 96)
Theorem:

FI FJ−1 FK = FK FJ−1 FI∂ F
Where matrices (FI )JK = ∂X ∂X∂X3IP. Gavrylenko, A. Marshakov ()JK

Residue Formulas for Prepotentials, Instanton Expansions
4 ìàðòàand2014
Conformal
ã.
Blocks4 / 13

Proof of the WDVV relation (point-counting argument)

Suppose that we have the formula
X

FIJK =res

f (z)=0rI (z)rJ (z)rK (z)R(z)dzf (z)Where the number of points f (λi ) = 0 equals to the number of(Marshakov '00)
Then introduce (commutative) algebra HS with the multiplication ∗Sindices(rI ∗ rJ )(λi ) = S(λi )rI (λi )rJ (λi )SThe homomorphismlS (r ) =NXi=1P. Gavrylenko, A. Marshakov ()R(λi )r (λi )0f (λi )S(λi )2Residue Formulas for Prepotentials, Instanton Expansions
4 ìàðòàand2014
Conformal
ã.
Blocks5 / 13

Proof of the WDVV relation (point-counting argument)

And the non-degenerate bilinear form
ηIJ = lS (rI ∗ rJ )Then

FIJK = (FI )JK = lS (rI ∗ rJ ∗ rK )Associativity of HS :FI η −1 FK = FK η −1 FI

We choose S(λi ) = r (λ1 ) , then rJ ∗ = id and η = FJ
JSo

P. Gavrylenko, A. Marshakov ()iFI FJ−1 FK = FK FJ−1 FI

Residue Formulas for Prepotentials, Instanton Expansions
4 ìàðòàand2014
Conformalã.
Blocks6 / 13

WDVV equation in the quiver gauge theories

General SU(2) quiver theory
x2 =The set

Pg −1 (z)Qz(z − 1) gi=1 (z − qi ){XI } = {a1 , ..., ag , q1 , ..., qg }The number of critical poinrs is 2g .
Constrained Zamolodchikov casex2 =The set

Qg̃2−1 (z)Q −1z(z − 1) 2g̃
i=1 (z − qi ){XI } = {a1 , ..., ag̃ , q1 , ..., q2g̃ −1 }The number of critical points is 3g̃ − 1
WDVV is satised in these both cases!
P. Gavrylenko, A. Marshakov ()Residue Formulas for Prepotentials, Instanton Expansions
4 ìàðòàand2014
Conformal
ã.
Blocks7 / 13

Relations for the period matrix

In the constrained Zamolodchikov caseg

1 X
F(a, q) =ai Tij (q)aj
2i,j=1

All relations  relations for the period matrix (a are parameters):
WDVV
Rauch formula
3-rd derivative residue
Their consequences

P. Gavrylenko, A. Marshakov ()Residue Formulas for Prepotentials, Instanton Expansions4 ìàðòàand2014
Conformalã.
Blocks8 / 13

Further directions

Field-theoretic calculations for the half-trifundamental
SU(N) quiver theories
New exact conformal blocks (for WN algebras) and simple check of
the AGT relation in Zamolodchikov-like limit.
By the Gamayun-Iorgov-Lisovy hypothesis (theorem for N = 2) each
exact conformal block can give the corresponding isomonodromicτ -function. So we can try to generalize this hypothesis to N > 2:
SL(N) case at the isomonodromic side and WN case at the CFT side.

Thank you for your attention!
P. Gavrylenko, A. Marshakov ()Residue Formulas for Prepotentials, Instanton Expansions
4 ìàðòàand2014
Conformal
ã.
Blocks9 / 13Equation: Nc = 2, Nf = 4 theoryx2 =q(q − 1) ∂F∂qz(z − 1)(z − q)

{F, q} =1 − q + q2
2q 2 (q − 1)2F(a, q) = −πa2

P. Gavrylenko, A. Marshakov ()K (1 − q)K (q)Residue Formulas for Prepotentials, Instanton Expansions
4 ìàðòàand2014
Conformal
ã.
10Blocks/ 13

Equation: Nc = 2, Nf = 4, 1 mass deformationx2 =2zq(q − 1) ∂F∂q − (z − q)mz 2 (z − 1)(z − q)
F 000 ++m2 F 0 [3q(2 − 3q)F 00 + 2(1 − 3q)F 0 ] − [3q 4 (q − 1)2 F 02 F 002 + q 2 (q 2 − q + 1)F 04 ]−2q 2 (q − 1)F 0 [q 2 (q − 1)F 02 + m2 q(q − 2)F 0 − m4 ]−m4 [(q − 1)2 q 2 F 003 + 6q(q − 1)2 F 0 F 002 + 3(q 2 + q − 1)F 00 F 02 + (3 + 2q)F 03 ]=02(q − 1)F 0 [q 2 (q − 1)F 02 + m2 q(q − 2)F 0 − m4 ]P. Gavrylenko, A. Marshakov ()Residue Formulas for Prepotentials, Instanton Expansions
4 ìàðòàand2014
Conformal
ã.
11Blocks/ 13

Equation: Nc = 2, Nf = 4, 1 mass deformation

Computations ising the non-linear equation:
22422

4

13a −14a m +m 2
F = Fpert (a; m) + (a2 − m2 ) log q + a −mq + ...
2 q+64a2

Nekrasov-Lukyanov-Litvinov-Zamolodchikov computation of thequasiclassical conformal block:
a2 +m2 −m2∞
1F1 =0 = a2 log q +q+2 2 2 242222 )22 )2a +m1 −m∞
a +2a (m1 +m∞ )−3(m12 −m∞a4 −(m12 −m∞++−q2 + . . .
416a264(a2 + 3 )4Answers dieres by the integer numbers like 34 and coincide in the a → ∞limit.

P. Gavrylenko, A. Marshakov ()Residue Formulas for Prepotentials, Instanton Expansions
4 ìàðòàand2014
Conformal
ã.
12Blocks/ 13

Equation: Nc = 2, Nf = 0r
dS =22Λ∂F∂Λ!

2− 16Λ

22(u − x)dxx 2 − Λ4

 23∂3F∂ F∂F+ Λ 2 −=0∂Λ3∂Λ∂ΛThe known equation (M. Matone), equivalent to the hypergeometric one.
P. Gavrylenko, A. Marshakov ()Residue Formulas for Prepotentials, Instanton Expansions
4 ìàðòàand2014
Conformalã.
13Blocks/ 13