c
2004 International Press
Adv. Theor. Math. Phys. 7 (2004) 831–864
SEIBERG-WITTEN
PREPOTENTIAL FROM
INSTANTON COUNTING
Nikita A. Nekrasov
1
Institut des Hautes Etudes Scientifiques, Le Bois-Marie
Bures-sur-Yvette, F-91440 France
nikita@ihes.fr
Abstract
Direct evaluation of the Seiberg-Witten prepotential is accom-
plished following the localization programme suggested in [1]. Our
results agree with all low-instanton calculations available in the litera-
ture. We present a two-parameter generalization of the Seiberg-Witten
prepotential, which is rather natural from the M-theory/five dimen-
sional perspective, and conjecture its relation to the tau-functions of
KP/Toda hierarchy.
To Arkady Vainshtein on his 60th anniversary
e-print archive: http://lanl.arXiv.org/abs/hep-th/0206161
ITEP-TH-22/02, IHES/P/04/22
1
on leave of absence from ITEP, 117259, Moscow, Russia
832
M.Jourdain: Par ma foi!
il y a plus de quarante ans
que je dis de la prose
sans que j’en suisse rien ;
et je vous suis le plus oblig´e du monde
de m’avoir appris cela....
Le Bourgeois gentilhomme.
J. -B. Moli`ere
1 Introduction
The dynamics of gauge theories is a long and fascinating subject. The dy-
namics of supersymmetric gauge theories is a subject with shorter history.
However, more facts are known about susy theories, and with better preci-
sion [2] yet with rich enough applications both in physics and mathematics.
In particular, the solution of Seiberg and Witten [3] of N = 2 gauge theory
using the constraints of special geometry of the moduli space of vacua led
to numerous achievements in understanding of the strong coupling dynam-
ics of gauge theory and as well as string theory backgrounds of which the
gauge theories in question arise as low energy limits. The low energy effec-
tive Wilsonian action for the massless vector multiplets (a
l
) is governed by
the prepotential F(a; Λ), which receives one-loop perturbative and instanton
non-perturbative corrections (here Λ is the dynamically generated scale):
F(a; Λ) = F
pert
(a; Λ) + F
inst
(a; Λ) (1.1)
In spite of the fact that these instanton corrections were calculated in
many indirect ways, their gauge theory calculation is lacking beyond two
instantons[4][5]. The problem is that the instanton measure seems to get
very complicated with the growth of the instanton charge, and the integrals
are hard to evaluate.
The present paper attempts to solve this problem via the localization
technique, proposed long time ago in [1][6][7]. Although we tried to make
the paper readable to both mathematicians and physicists we don’t expect
it to be quite understandable without some background material, which we
suggest to look up in [3][8][9].
Notations. Let G be a semi-simple Lie group, T is maximal torus, g =
Lie(G) its Lie algebra, t = Lie(T ) its Cartan subalgebra, W = N (T )/T
denote its Weyl group, U = (t ⊗
C)/W denotes the complexified space of
833
conjugacy classes in g. We consider the moduli space M
k
(G) of framed
G-instantons: the anti-self-dual gauge fields A, F
+
A
= 0, in the principal
G-bundle P over the 4-sphere S
4
= IR
4
∪ ∞ with
k = −
1
8hπ
2
Z
IR
4
Tr F
A
∧ F
A
(1.2)
considered up to the gauge transformations g : A 7→ g
−1
Ag + g
−1
dg, s.t.
g(∞) = 1. We also consider several compactifications of the space M
k
(G):
the Uhlenbeck compactification
˜
M
k
(G) and the Gieseker compactification
f
M
k
for G = U(N) or SU(N). In the formula (1.2) we use the trace in the
adjoint representation, and h stands for the dual Coxeter number of G.
Field theory description. We calculate vacuum expectation value of
certain gauge theory observables. These observables are annihilated by
a combination of the supercharges, and their expectation value is not
sensitive to various parameters, the energy scale in particular. Hence,
one can do the calculation in the ultraviolet, where the theory is weakly
coupled and the instantons dominate. Or, one can do the calculation in
the infrared, and relate the answer to the prepotential of the effective
low-energy theory. By equating these two calculations we obtain the
desired formula.
Mathematical description. We study G × T
2
equivariant coho-
mology of the moduli space
f
M
k
, where G acts by rotating the gauge
orientation of the instantons at infinity, and T
2
is the maximal torus
of SO(4)– the group of rotations of IR
4
which also acts naturally on
the moduli space
2
. Let p :
f
M
k
→ pt be the map collapsing the moduli
space to a point. We consider the following generating function:
Z
inst
(a,
1
,
2
; q) =
∞
X
k=0
q
k
I
f
M
k
1 (1.3)
where
H
1 denotes the localization of the pushforward p
∗
1 of 1 ∈
H
∗
G×T
2
(
f
M
k
) in H
∗
G×T
2
(pt) = C[U,
1
,
2
]. We denote the coordinates
on t by a and the coordinates on the Lie algebra of T
2
by
1
,
2
. In
explicit calculations
3
we represent 1 by a cohomologically equal form
which allows to replace
H
1 by an ordinary integral:
2
Throughout the paper we mostly consider the SU (N ) instantons (or U (N ) noncom-
mutative instantons). We use the notation
f
M
k,N
when we want to emphasize that the
gauge group is U (N ).
3
For G = SU(N) we actually use a = (a
1
, . . . , a
N
) s.t.
P
l
a
l
= 0
834
I
g
M
k
1 =
Z
g
M
k
exp [ω + µ
G
(a) + µ
T
2 (
1
,
2
)] (1.4)
where ω is a symplectic form on
g
M
k
, invariant under the G×T
2
action,
and µ
G
, µ
T
2
are the corresponding moment maps.
Our first claim is
Z
inst
(a,
1
,
2
; q) = exp
F
inst
(a,
1
,
2
; q)
1
2
(1.5)
where the function F
inst
is analytic in
1
,
2
near
1
=
2
= 0.
We also have the following explicit expression for Z in the case
4
1
= −
2
= ~
for
5
G = SU(N):
Z
inst
(a, ~, −~; q) =
X
~
k
q
|k|
Y
(l,i)6=(n,j)
a
ln
+ ~ (k
l,i
− k
n,j
+ j − i)
a
ln
+ ~ (j − i)
(1.6)
Here a
ln
= a
l
− a
n
, the sum is over all colored partitions:
~
k = (k
1
, . . . , k
N
),
k
l
= {k
l,1
≥ k
l,2
≥ . . . k
l,n
l
≥ k
l,n
l
+1
= k
l,n
l
+2
= . . . = 0},
|
~
k| =
X
l,i
k
l,i
,
and the product is over 1 ≤ l, n ≤ N, and i, j ≥ 1.
Already (1.6) can be used to make rather powerful checks of the Seiberg-
Witten solution. But the checks are more impressive when one considers
the theory with fundamental matter. To get there one studies the bundle V
over
g
M
k
of the solutions of the Dirac equation in the instanton background.
Let us consider the theory with N
f
flavors. It can be shown that the gauge
theory instanton measure calculates in this case (cf. [11]):
Z(a, m,
1
,
2
; q) =
X
k
q
k
I
g
M
k
Eu
G×T
2
×U(N
f
)
(V ⊗ M) (1.7)
4
in the general case we also have a formula, but it looks less transparent
5
a simple generalization to SO and Sp cases will be presented in [10]
835
where M = C
N
f
is the flavor space, where acts the flavor group U(N
f
),
m = (m
1
, . . . , m
N
f
) are the masses = the coordinates on the Cartan subal-
gebra of the flavor group Lie algebra, and finally Eu
G×T
2
×U(N
f
)
denotes the
equivariant Euler class.
The formula (1.6) generalizes in this case to:
Z
inst
(a, m,
1
,
2
; q) =
X
~
k
q~
N
f
|k|
Y
(l,i)
N
f
Y
f=1
Γ(
a
l
+m
f
~
+ 1 + k
l,i
− i)
Γ(
a
l
+m
f
~
+ 1 − i)
×
×
Y
(l,i)6=(n,j)
a
ln
+ ~ (k
l,i
− k
n,j
+ j − i)
a
ln
+ ~ (j − i)
(1.8)
Again, we claim that
F
inst
(a, m,
1
,
2
; q) =
1
2
logZ
inst
(a, m,
1
,
2
; q) (1.9)
is analytic in
1,2
.
The formulae (1.6)explctm were checked against the Seiberg-Witten solution
[12]. Namely, we claim that
F
inst
(a, m,
1
,
2
)|
1
=
2
=0
=the instanton part of the prepotential of the
low-energy effective theory of the N = 2 gauge theory with the gauge group
G and N
f
fundamental matter hypermultiplets.
Mathematical formulation. The latter statement means that F
inst
is related to periods of a family of curves. More precisely, consider the
following family of curves
6
Σ
u
(here we formulate things for G = SU(N)
but the generalization to general G is well-known [12] ):
w +
Λ
2N−N
f
Q(λ)
w
= P(λ) =
N
Y
l=1
(λ − α
l
) (1.10)
where Q(λ) =
Q
N
f
f=1
(λ+m
f
). The base of the family (1.10) is the space
U = C
N−1
3 u of the polynomials P(we set
P
l
α
l
= 0). Consider the
6
Λ, m
f
are fixed for the family