1
The Hilbert scheme as a symplectic quotient
We would like to argue that HilbN (Cm ) is isomorphic to the space MN,m of N ×
N complex mutually commuting matrices Z i ’s together with complex N -dimensional
vector Q satisfying
M
X
(1.1)
[Z A , Z A† ] + QQ† = m1N ,
A=1
modded out by the U (N ) action, following Nekrasov.
HilbN (Cm ) is the set of length N ideals of C[x1 , · · · , xm ]. It is also the moduli space
of N noncommutative solitons on Cm (Gopakumar-Spradlin-Headrick). The latter is
described by a projection operator P on a Hilbert space H, with P H N -dimensional.
H is constructed as the Fock space with creation and annihilation operators aA , a†A ,
A = 1, · · · , m, with states of the form f (a†A )|0i for polynomial f . P is required to
satisfy
(1.2)
(1 − P )aA P = 0, for all A.
Given Z A ’s and Q, we can construct the N × N Hermitian matrix
X
X
G = h0| exp(
Z A aA ) QQ† exp(
Z B† a†B )|0i
Using (1.1), it is easy to see that
" m
#
X
X
X
G = h0| exp(
Z B aB )
(Z A† − aA )(ZA − a†A ) exp(
Z C† a†C )|0i
(1.3)
(1.4)
A=1
Let V be the N -dimensional vector space on which Z A act. (Z A − a†A ) does not
annihilate any state in H ⊗ V (with a finite number of excitations), and hence G is
positive definite. One can then construct the operator
X
X
P = Q† exp(
Z A† a†A )|0iG−1 h0| exp(
Z B aB )Q
(1.5)
on H. This is a projection operator with property (1.2). Thus we have shown that every
solution to (1.1) corresponds to a point in HilbN (Cm ), namely MN,m ,→ HilbN (Cm ).
Going the other way around is much more difficult. Suppose we have the projection
operator P : H → Ve ⊂ H satisfying (1.2), Ve being N -dimensional. We can pick a set
of polynomial functions fi , i = 1, · · · , N , such that |fi i ≡ P fi (a†A )|0i span Ve . If P were
constructed from Z A and Q as (1.5), we can consider a set of vectors vi = fi (Z A )Q that
span V (not necessarily orthonormal). The matrices G, Z A , and vector Q are related
by
vi† G−1 vj = hfi |fj i,
vi† G−1 h(Z A )vj = hfi |h(a†A )|fj i,
vi† G−1 Q = hfi |0i.
1
(1.6)
These relations, however, can only determine the matrices up to the GL(N, C) action
Z A → ΩZ A Ω−1 ,
Q → ΩQ,
vi → Ωvi ,
(1.7)
†
G → ΩGΩ .
We will, however, give a physical argument that MN,m should be isomorphic to
HilbN (Cm ), at least for m = 3, 4. This is done by considering D0-D6 or D0-D8 system
in B-field background. With sufficiently large B field, the former has a supersymmetric
bound state that preserves 1/8 supersymmtries, whereas the latter has a bound state
that preserves 1/16 supersymmetries. In the limit α0 → 0, α02 B = θ held fixed, the D6
or D8-brane world volume theory becomes the noncommutative gauge theory, and the
D0-branes are described by noncommutative instantons. The length scale of the NCYM
is set by the noncommutative parameter θ. We denote by gN C the gauge coupling at
1
the length scale θ 2 . It is related to the closed string coupling gs by
2
gN
C = gs (
θ
).
α0
(1.8)
From the D0-brane point of view, the B-field determines the Fayet-Illiopoulos term
in the 0 + 1 dimensional gauge theory on the D0-branes. In the limit of small α0 and
α0 B → ∞, the FI parameter ζ is given by
ζ
θ
1
√ =
= 0 2
gs
α gN C
α0
(1.9)
It is known that in the strict noncommutative limit, with gN C large, the moduli
space of N noncommutative instantons is given by HilbN (Cm ).
2
The D1-D7 gauge theory
We will consider the system of N D1-branes lying on a single D7-brane in type IIB
string theory in flat space. The system is supersymmetric when there is a certain
critical amount of B-field along the D7-brane world volume and transverse to the
D1-branes (Witten). When the B-field is below the critical value, there is no supersymmetric bound state of D1-D7. When the B-field is above the critical value, there
is a supersymmetric bound state.
The low energy 1+1 dimensional world volume theory is the (8, 8) U (N ) superYang-Mills theory on the D1-branes coupled to a fundamental N = 2 chiral field Q,
coming from the 1-7 strings. We will denote the (scalar components) N = 2 chiral
2
fields in the N = 8 multiplet by Z i , i = 1, 2, 3. There is a Fayet-Illiopoulos term,
corresponding to the value of the B-field. The scalar potential is (Hori-Tong)
1
1
1
† 2
i†
†
i
i†
†
i
V = Tr − 2 [σ, σ ] + [Z , σ ][σ, Z ] + [Z , σ][σ , Z ] + Q† {σ, σ † }Q
2
gY M
2
!2
(2.1)
X
gY2 M
i
i† 2
†
2
i
j
i†
j†
+
Tr
[Z , Z ] + QQ − ζef f 1N
− gY M Tr[Z , Z ][Z , Z ]
2
i
The FI parameter ζ is renormalized to
ζef f = ζ + Tr ln
σ
µ
(2.2)
where the coefficient of the logarithmic correction is given by the number of flavors
Nf , which is 1 in our case. In the deep IR, ζef f becomes negative, there is no solution
to the D-flatness condition. This is expected since the U (1)A symmetry is anomalous,
and there cannot be any nontrivial IR (2, 2) SCFT.
If ζef f = 0, the vanishing of scalar potential will set Q = 0, and all the Z i ’s and
Z i† ’s, as well as σ, σ † , commute with one another. The moduli space of classical vacua
is simply SymN (R8 ) in this case.
The most interesting case is ζef f > 0. The D-flatness condition will force Q to be
nonzero and Q† Q = N ζef f . The vacua are given by the solutions of
[Z i , Z j ] = 0, [σ, σ † ] = [σ, Z i ] = [σ † , Z i ] = 0,
X
[Z i , Z i† ] + QQ† = ζef f 1N ,
σQ = σ † Q = 0,
(2.3)
i
up to the U (N ) gauge equivalence. It follows that σ can be diagonalized. If σ had
generic distinct eigenvalues (one of them being zero corresponding to the eigenvector
Q), then Z i could be simultaneously diagonalized as well, and we would have moduli
space SymN −1 (R8 ). In general, if σ had only one zero eigenvalue (acting on Q), then
the Z i ’s must be block diagonal, consisting of an (N − 1) × (N − 1) block and a 1 × 1
block, the latter corresponding to the vector Q. This could not satisfy the D-flatness
condition. The more interesting situation is when σ = 0.1 Now the Z i ’s are general
commuting complex matrices satisfying the D-flatness condition only. They may not
be diagonalizable, and may carry more than order N degrees of freedom. In fact,
the resulting U (N ) quotient is the Hilbert scheme of N points on C3 , HilbN (C3 ). It is
4/3
known that HilbN (C3 ) has a branch whose dimension grows at least as fast as 316 N 4/3 .
The size of this branch is parameterized by ζef f .
1
σ = 0 might appear to be in conflict with ζef f > 0. One can get around this by considering the
theory at finite temperature, which is what we will be interested in.
3
For the gauge theory to be well approximated by the non-linear sigma model, we
need to consider temperature T gY M . If ζef f (σ ∼ gY M ) is large, we expect a phase in
which the D1-D7 system is described by the (2, 2) nonlinear sigma model on (a branch
of) HilbN (C3 ), and has order N 4/3 degrees of freedom.
3
A (0, 2) linear model for HilbN (C4)
The Hilbert scheme of N points on C4 can be realized as the moduli space of noncommutative instantons in a eight-dimensional noncommutative U (1) gauge theory. It is
however not obvious how to embed this system into string theory, or how to relate it
to a 1+1 dimensional gauge theory. There is, however, a 1+1 dimensional (0, 2) gauge
theory, which can be thought of as a deformation of (8, 8) U (N ) super-Yang-Mills, that
lead to the nonlinear sigma model on HilbN (C4 ) at low energies. We shall now describe
this theory.
The (8, 8) U (N ) super-Yang-Mills consists of a (2, 2) gauge multiplet and three (2, 2)
chiral multiplets Φi , i = 1, 2, 3. The (2, 2) gauge multiplet can be futher decomposed
into a (0, 2) gauge multiplet V and a (0, 2) chiral multiplet Σ. The (2, 2) chiral multiplet
Φi can be decomposed into a (0, 2) chiral multiplet Y i and a (0, 2) fermi multiplet Λi .
The corresponding superfields are
V = v− − 2iθ+ λ̄− − 2iθ̄+ λ− + 2θ+ θ̄+ D,
√
Σ = σ − i 2θ+ λ̄+ − iθ+ θ̄+ D+ σ,
√
i
− iθ+ θ̄+ D+ Z i ,
Y i = Z i + 2θ+ ψ+
√
√
Λi = λi− − 2θ+ Gi − iθ+ θ̄+ D+ λi− − 2θ̄+ E i .
The E i ’s are themselves chiral superfields, determined by Σ and Y i via
√
E i = i 2[Σ, Y i ]
(3.1)
(3.2)
A subtlety is that, the off-shell action of the nonabelian (2, 2) theory is not exactly the
same as the action constructed with the standard kinetic terms for the corresponding
(0, 2) multiplets. For example, the (0, 2) kinetic term for Σ contains the coupling to D
field, TrD[σ, σ † ], which is not present in the (2, 2) theory. The scalar potential (with
FI term) for the (0, 2) theory is

!2 


2
X
gY M
i
j
j†
i†
i
i†
i†
i
i†
†
V =
Tr [Z , Z ][Z , Z ] + [σ, Z ][Z , σ ] +
[Z , Z ] + [σ, σ ] − ζ1N


2
i
(3.3)
4
Despite that the D-term is different from the (2, 2) theory, when ζ = 0 the scalar
potential is in fact the same as that of the (2, 2) theory. This theory does not have
classical supersymmetric vacua for nonzero ζ. To fix this we will introduce a (0, 2) chiral
multiplet Q in the fundamental representation. Writing Z A = (Z i , σ), A = 1, 2, 3, 4,
the scalar potential is then

!2 


2
X
gY M
A
B
A†
B†
A
A†
†
V =
Tr [Z , Z ][Z , Z ] +
[Z , Z ] + QQ − ζ1N
(3.4)


2
A
Setting V to zero, and modding out by the U (N ) gauge action, we see that the moduli
space of classical vacua is HilbN (C4 ). When ζ 1, for temperature T in the range
gY M e−ζ T gY M ,
(3.5)
the gauge theory is effectively described by the nonlinear sigma model on HilbN (C4 ).
In this phase the theory has order N 3/2 degrees of freedom. In the deep IR the theory
reduces to the orbifold CFT on SymN (R8 ), which has order N degrees of freedom. As
we will see from the gravity dual, the (8, 8) super-Yang-Mills theory is expected to have
∼ N 3/2 degrees of freedom at temperature T ∼ gY M (and also reduces to the orbifold
CFT in the deep IR).
A
Hilbert Scheme of Points
In general, the Hilbert scheme HilbN (Cm ) is the space of length N ideals of the ring
C[x1 , · · · , xm ]. When m > 2, HilbN (Cm ) is not smooth for sufficiently large N . It is
in fact not irreducible: there are components of different dimensions, and in particular
there are branches whose dimension grow faster than linear in N at large N . A perhaps
more useful description of HilbN (Cm ) is the space of M mutually commuting complex
matrices Z A ,
(A.1)
[Z A , Z B ] = 0,
modded out by the GL(N, C) action Z A → ΩZ A Ω−1 . More precisely, one can parameterize open charts of the Hilbert scheme by commuting Z A ’s that obey the constraint
fa (Z 1 , · · · , Z m )e1 = ea ,
a = 1, · · · , N.
(A.2)
Here fa are N independent polynomials in the Z A ’s, with f1 = 1. ei are a fixed basis
of Cm . For illustration, there are two important choices of the fa ’s. The first choice is
fa (Z) = (Z 1 )a−1 , a = 1, · · · , N . This leads to the chart that covers generic points on
5
SymN (Cm ), with dimension mN . The second choice is fa being the N lowest degree
monomials in the Z A ’s, up to a certain degree d, determined by
d+m−1
d+m
<N ≤
m
m
Namely, f1 = 1, f2 = Z 1 , f3 = Z 2 , · · · , fN = Z i1 Z i2 · · · Z id . This chart covers a branch
of the Hilbert scheme where all N points coincide on Cm , and the dimension of this
branch grows at least as fast as (math/0506161)
m2
16
m!
2
− m2
for sufficiently large N .
6
2
N 2− m
(A.3)