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Localizing gauge theories

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N. Nekrasov at Stony Brook University
N. Nekrasov
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Abstract

We study {N} = 2 supersymmetric gauge theories on toric four dimensional manifolds. We recall the notion of Omega-background and calculate the partition function of gauge theory in this background. As a result, we get generalizations of the formulae of the author for the theories on R4 and Nakajima-Yoshioka for the theories on R4 blown up at one point. In the case of compact toric space our results give an alternative derivation of Donaldson invariants, and generalize the results of Göttche-Zagier.
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September 23, 2014 16:1 WSPC/Trim Size: 9.75in x 6.5in for Proceedings Lisbonne
LOCALIZING GAUGE THEORIES I.
NIKITA A. NEKRASOV
∗
Institut des Hautes Etudes Scientifiques
35 route de Chartres,
91440 Bures-sur-Yvette, FRANCE
E-mail: nikita@ihes.fr
We study N = 2 supersymmetric gauge theories on toric four dimensional manifolds. We
recall the notion of Ω-background and calculate the partition function of gauge theory
in this background. As a result, we get generalizations of the formulae of the author for
the theories on R
4
and Nakajima-Yoshioka for the theories on R
4
blown up at one point.
In the case of compact toric space our results give an alternative derivation of Donaldson
invariants, and generalize the results of G¨ottche-Zagier.
1. Introduction
In the past years some progress has been achieved in the quantum field theory
calculations of the low-energy effective actions including non-perturbative
effects. N = 2 supersymmetric Yang-Mills theory has simple perturbation
theory and non-trivial instanton dynamics. By utilizing supersymmetry one
is able to calculate the instanton corrections to the low-energy effective ac-
tion [23]. This is done by placing N = 2 gauge theory in a non-trivial
geometric background, involving R-symmetry currents. In this paper we
shall generalize the results of [23]. Namely, we study N = 2 gauge theory
on a toric four-manifold X. The motivations for this work are manifold.
First, we hope to learn about the gauge theory applications of the holomor-
phic anomaly, which is well-known in the context of topological strings [1,3].
Secondly, the similar formulae are expected to hold in the higher dimensional
generalizations of the supersymmetric gauge theories. There, the applica-
tions to the black hole counting [30] are expected. Also, the topological
vertex and Donaldson-Thomas theory also behave in the similar way. The
crucial ingredient missing in all these problems is the detailed understanding
of the stability conditions. Since in the case of Donaldson theory lots of in-
formation is already available, e.g. in the work [8], we may hope to advance
∗
On leave of absence from Institute for Theoretical and Experimental Physics
1
September 23, 2014 16:1 WSPC/Trim Size: 9.75in x 6.5in for Proceedings Lisbonne
2 Nikita Nekrasov
along this route.
2. N = 2 gauge theory
We study pure N = 2 gauge theory. Generalizations involving matter are
straightforward. It is well-known that the theory admits the so-called twist-
ing [29], which permits supersymmetric compactifications on arbitrary four-
manifold X. The field content of the twisted theory (which turns out to be
the integral representation of Donaldson theory) is the following: the gauge
field A, the complex adjoint scalar φ, and the adjoint fermions: one-form ψ,
self-dual two-form χ and a scalar η.
On generic X the twisted theory admits one nilpotent supercharge Q, also
called topological charge. If the holonomy of X is reduced then there are
extra charges, coming from the original supercharges of the physical theory
of the same chirality as Q. If the manifold X admits isometries, then there
are extra conserved supercharges G of opposite chirality. Mathematically,
the theory on X studies equivariant cohomology of the space of gauge fields.
In addition to the gauge group, the equivariance with respect to isometries
of X is quite useful in analyzing the theory.
The bosonic part of the N = 2 theory action is given by:
S
bos
=
Z
Tr
kF k
2
+ kD
A
φk
2
+ k[φ,
¯
φ]k
2
(1)
3. Ω-background, local theory: R
4
We start with the theory on R
4
. Suppose Ω,
¯
Ω are two commuting infinites-
imal rotations of Euclidean space R
4
, [Ω,
¯
Ω] = 0, Ω = kΩ
µν
k. These are
generated by the vector fields V,
¯
V , V
µ
= Ω
µ
ν
x
ν
, and the indices are lifted
by the Euclidean metric δ
µν
. We skip the details and the motivation of
the construction [23]. The bosonic part of the action of the theory in the
Ω-background is given by:
S
bos
=
Z
Tr
kF k
2
+ kD
A
φ − ι
V
F k
2
+ k[φ,
¯
φ] − ι
¯
V
D
A
φ + ι
V
D
A
¯
φk
2
(2)
4. Instanton partition function on R
4
In [23] we calculated the partition function of the gauge theory in the Ω-
background. It depends on the skew eigen-values
1
,
2
of the matrix Ω and
the vev a of the Higgs field φ.
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Localizing gauge theory 3
4.1. Higher times
The action Eq. (2) is the bosonic part of the integral of ch
2
=
1
2
Trφ
2
over
the superspace. Formally one can consider the integral of any function
f(ch
1
, ch
2
, ch
3
, . . .) where
ch
p
=
1
p
Tr
φ
2πi
p
(3)
(only N of ch
p
are independent). We expand:
f =
X
p
τ
p
ch
p
+
X
p
1
p
2
τ
p
1
p
2
ch
p
1
ch
p
2
+ . . .
The parameters τ = (τ
p
|τ
p
1
p
2
| . . .) are called times. All but τ
2
should be
viewed as formal parameters.
4.2. Partition function
The calculation in [10, 22, 23] gives:
Z(a, τ,
1
,
2
) =
X
~
k
µ
~
k
(a,
1
,
2
)expf
~τ
( ˆm
p
(a,
k,
1
,
2
)) (4)
ˆm
p
(a,
k,
1
,
2
) =
X
l
a
p
l
+
+
X
l,i
(a
l
+
1
(i −1))
p
− (a
l
+
1
i)
p
− (a
l
+
1
(i −1) +
2
k
li
)
p
+ (a
l
+
1
i +
2
k
li
)
p
where
k = (k
1
, . . . , k
N
) is the N-tuple of partitions,
k
l
= (k
l1
≥ k
l2
≥ k
l3
≥ . . . ≥ k
lk
0
l1
> 0)
Finally µ
~
k
(a,
1
,
2
) is the measure on partitions, derived in [23]:
µ
~
k
(a,
1
,
2
) = exp
Z
dxdyf
00
~
k
(x)f
00
~
k
(y)γ
1
,
2
(x − y) (5)
and the function f
~
k
(the profile of the colored partition
k) , and the kernel γ
are defined in [22].
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4 Nikita Nekrasov
4.3. Expansion of the partition function on R
4
In order to proceed with more general four-manifolds we need the partition
function on R
4
to be evaluated for the most general parameters, both toric
1
,
2
and the couplings τ
~n
. Expansion in
1
,
2
has the following form:
Z(a,
1
,
2
; τ
~n
) = exp
F
0
1
2
+
1
+
2
1
2
H
1
2
+ F
1
+
(
1
+
2
)
2
1
2
G
1
+ O(
1
,
2
) (6)
where F
0
, F
1
, H
1
2
, G
1
are functions of a, τ
~n
.
5. Ω-background, global theory
The standard Donaldson theory in four dimensions can be viewed as the
partially twisted six dimensional super-Yang-Mills theory, compacitifed on
T
2
× X, where X is our four-fold, and the torus T
2
has vanishing size.
For X with isometries we modify this construction. Consider flat X-
bundle over T
2
, which can be viewed as an orbifold of R
2
× X by the action
of Z
2
, which acts by shifts in R
2
direction, and by two commuting isometries
g
1
, g
2
in the X direction:
(z, ¯z, x) 7→ (z + n + mτ, ¯z + n + m¯τ, g
n
1
g
m
2
· x) (7)
By compactifying the gauge theory on this background, with the partial
twist along X, in the limit of vanishing volume of T
2
we end up with the
theory in the Ω-background. The bosonic part of its action coincides with
Eq. (2).
5.1. Toric preliminaries
In this section we collect all the necessary information about toric varieties.
We use slightly unconventional language, which is more familiar to physicists.
Let M, d be non-negative integers, and T
M
, T
d
, T
M+d
the standard tori
U(1)
M,d,M+d
of the corresponding dimensions. In what follows a runs from
a to M + d, α runs from 1 to M, µ runs from 1 to d, and l runs from 1 to
N. The torus T
M+d
acts in the standard fashion on the space C
M+d
:
Z = (Z
a
) 7→ (Z
a
exp iθ
a
)
Any integral matrix Q : Z
M
→ Z
M+d
defines a homomorphism of T
M
into
T
M+d
, and the corresponding action of T
M
on C
M+d
:
Z = (Z
a
) 7→
Z
a
exp i
X
α
Q
a
α
ϕ
α
!
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Localizing gauge theory 5
This action preserves the norms: p
a
=
1
2
|Z
a
|
2
and the symplectic form
ω =
X
i
dp
a
∧ dϑ
a
Z
a
=
p
2p
a
e
iϑ
a
Now, fix a collection r of M real numbers r
1
, . . . , r
M
– Fayet-Illiopoulos
terms in the physical language – and consider the Hamiltonian reduction of
C
M+d
with respect to T
M
at the level of the moment map
m = (m
1
, . . . , m
M
), m
α
=
X
a
Q
a
α
p
a
given by r. Explicitly, we consider the space
X =
m
−1
(r)/T
M
This is our toric variety. By construction, it comes endowed with the map:
p : X → R
M+d
+
which factors through the projection
X → X/T
d
= ∆
X
where T
d
= T
M+d
/T
M
. We neglect possible torsion part (it is absent for
generic Q). The image of ∆
X
under p is the convex d-dimensional polyhedron
(perhaps non-compact), which fits some d-dimensional affine subspace bL
d
in R
M+d
+
. Another canonical structure on X is a set of complex line bundles
L
α
and L
a
. More precisely, any character χ of the torus T
M
defines a line
bundle, which is associated with the help of χ to the principal T
M
bundle:
m
−1
(r) → X (8)
All of the above is said under the assumption of genericity of r, so that the
Eq. (8) indeed defines a principal bundle. The line bundles L
a
are associated
to the characters χ
M
a
= expiQ
a
α
ϕ
α
, while L
α
are associated to χ
M
α
= expiϕ
α
.
Hence, topologically:
L
a
≈ ⊗
α
L
⊗Q
a
α
α
(9)
The torus T
M+d
acts on X. Of course, only T
d
acts faithfully. The full torus
T
M+d
becomes visible when we start looking at the line bundles over X.
Then L
a
become distinct line bundles as T
M+d
equivariant bundles. They
correspond to the characters χ
M+d
a
= e
iθ
a
. In what follows we shall denote
χ
M+d
simply by χ.
The fixed points v of T
d
action on X are the vertices of ∆
X
. Generically,
these vertices have d edges emanating from them. These edges correspond
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6 Nikita Nekrasov
to fixed complex projective lines in X (if X is non-compact the edges may
go all the way to infinity, in this case they correspond to the fixed complex
lines in X).
The tangent space T
v
X to X at any fixed point is a representation of T
d
,
and, consequently, a representation of T
M+d
. The representation splits as a
sum of d one-dimensional irreducibles:
T
v
X =
d
M
µ=1
χ
v,µ
where we took the liberty of identifying the characters with the one dimen-
sional representations. The characters χ
v,µ
define the weights of the T
M+d
action on T
v
X:
χ
v,µ
= expa
X
a
w
v,µ;a
θ
a
(10)
which clearly obey:
X
a
w
v,µ;a
Q
a
α
= 0, ∀α
Each fixed point v defines a subset I
v
of the set of indices {1, . . . , M + d} of
cardinality d, such that for any
∀a ∈ I
v
Z
a
(v) = 0
It is natural to label the elements of I
v
by the index µ = 1, . . . , d. In other
words, I
v
coincides with the set of weight subspaces of T
v
X.
The remaining M coordinates Z
i
are completely fixed by the moment map
equations, up to the action of T
M
. Let Z
M
v
⊂ (Z
M+d
)
t
denote the sub-lattice
{(n
1
, . . . , n
M+d
)|n
a
= 0, a ∈ I
v
}, and R
M
v
= Z
M
v
⊗ R. Then
∂(v) ∈ R
M
v
. Let
Q
v
denote the restriction of Q
t
onto Z
M
v
. It must be invertible over Z in
order for v to be isolated fixed point. Then:
w
v,µ;a
=
δ
µ;a
− Q
µ
α
(Q
−1
v
)
α
a
(11)
5.1.1. Second cohomology
The line bundles L
a
generate the K-group of X. Their first Chern classes
generate H
2
(X, Z). In the T
M+d
-equivariant cohomology they are repre-
sented by the equivariantly closed forms: φ
a
+ c
1
(L
a
). In the localized
cohomology, which is spanned by the fixed points v ∈ X
T
, each point con-
tributing a copy of C((ε
1
, . . . , ε
M+d
)), the Chern classes localized to φ
(v)
a
,
which are linear functions of ε
a
.
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Localizing gauge theory 7
In the applications of gauge theory to four-manifold invariants one studies
correlators of the 2-observables, say for single trace operators: O
(2)
J
. We
evaluate integrated 2-observables using localization formula:
Z
Σ
O
(2)
f
=
X
v
H
(v)
Σ
f(ch
p
(E
v
)) (12)
where: Σ is the 2-homology class, and H
Σ
+P.D.[Σ] is the equivariant version
of its Poincare dual, H
Σ
is a function on X, linear in p
i
, and H
(v)
Σ
its value
at the fixed point v. Finally, E
v
is the localized universal sheaf, i.e. ι
∗
v
E,
where ι
v
: M × {v} → M × X is the inclusion, where M is the moduli space
of sheaves (compactified instanton moduli space).
5.2. Equivariant bundles
In gauge theory with gauge group G we usually fix the principal G-bundle P,
integrate over the space of all connections on P and then sum over topological
classes of P . When we formulate the theory on some curved manifold X we
have to fix the metric on X. If the manifold X is a toric variety the metric is
uniquely specified by the matrix of charges Q and the set of Fayet-Iliopoulos
terms r. If we want to turn on the Ω-background an extra data is needed,
namely the lift of the action of the torus T
M+d
on P . For the gauge group
G = SU(N) this is the same as the choice of N-dimensional representation
E of T
M+d
. The latter is specified by the matrix
k
l,a
, l = 1, . . . , N , a = 1, . . . M + d (13)
of weights. The rank N vector bundle splits as a sum of N line bundles L
l
,
of Chern classes:
c
1
(L
l
) =
X
a
k
l,a
Q
a
α
c
1
(L
α
)
5.3. Examples
(1) Let us consider X = CP
d
, which corresponds to M = 1. In this case
∆
X
is just a simplex,
X
a
p
a
= r
(if r > 0, of course). The fixed points v are in one-to-one correspon-
dence with the vertices of the simplex, i.e. we can set v = 1, . . . M + 1,
p
v
= (0, . . . , r, . . . , 0)
v
September 23, 2014 16:1 WSPC/Trim Size: 9.75in x 6.5in for Proceedings Lisbonne
8 Nikita Nekrasov
(2) The next example is non-compact. X is the total space of the vector
bundle O(q
1
) ⊕ O(q
d−1
) over P
1
. It corresponds to M = 1, the vector
of charges:
Q = (q
1
, q
2
, . . . , q
d−1
, 1, 1)
t
There are two fixed points v - they belong to the zero section of the
vector bundle, and are the North and South poles of the sphere.
5.4. Stability and holomorphic anomaly
The partition function of the gauge theory in Ω-background is analytic in
the vev of the Higgs field. However the expansion in the parameters
1
,
2
in the limit → 0 leads to holomorphic functions in a, F
0
(a; Λ), F
1
(a; Λ)
etc which are actually not single-valued. They can be made single valued
at the expense of adding some non-holomorphic terms (the example of the
second Eisenstein series E
2
is helpful here). The precise expression of these
non-holomorphic terms is not known. We are proposing an approach below
which should fix them, at least partly.
Another, more profound, reason to expect them is to recall the Gromov-
Witten interpretation of the N = 2 partition functions. At least in the
case
1
= −
2
the expansion coefficients F
g
(a; Λ) are (the limits of) the
generating functions of the genus g Gromov-Witten invariants of certain
(local) Calabi-Yau manifolds, see Ref. [28]. These are well-known to be the
large ¯a limits of non-holomorphic automorphic forms on the moduli spaces
of Kahler structures of these Calabi-Yaus, see Refs. [1, 3].
6. MASTER FORMULA
We are now almost ready to state our main formula.
We first fix the vacuum expectation value of the Higgs field Φ,
hΦi = a ≡ diag(a
1
, . . . , a
N
) (14)
In the case of compact X we should take the integral over a, to take into
account the tunneling between different vacua.
The second cohomology group of X is isomorphic to Z
d
, generically. The
U(N) gauge bundle is reduced to the U(1)
N
bundle, in the presence of
the Higgs vev. We should sum over all equivalence classes of the U(1)
N
bundles. They are classified by the vectors
k
α
= (k
α,l
), α = 1, . . . , d. In
the SU(N )/Z
N
gauge theory one fixes the traces: {
k
α
} = w
α
=
P
N
l=1
k
α,l
.
Actually, in the absence of the charged matter, everything depends only on
September 23, 2014 16:1 WSPC/Trim Size: 9.75in x 6.5in for Proceedings Lisbonne
Localizing gauge theory 9
w
a
modN. Equivariantly, as we discussed above, topologically equivalent line
bundles may differ. So in fact we might sum over equivalence classes of
equivariant line bundles. These are labelled by vectors
k
e
∈ Z
N
, for each
edge e of the toric diagram ∆
X
. Equivalent information is contained in the
vectors
k
a
= (k
a,l
) from Eq. (13). However, the choice of equivariant bundle
does not correspond to the fluctuating parameter. So the real sum goes over
k
α
only. Moreover, the actual set of vectors
k
α
is even smaller, because of
the so-called stability conditions, to be mentioned below.
The partition function of the N = 2 theory in Ω-background on the toric
manifold X, in the presence of 2-observables,
hexp
X
~n
Z
t
~p
O
(2)
ch
p
1
ch
p
2
...
i, t
~n
∈ H
2
c
(X, Z)
is given by:
Z
~w
(a, ε; τ
~n
, t
~n
) =
X
~
k
a
∈Z
N
,{
~
k
α
}=w
α
Y
v
Z(a + ε ·
k
v
, w
v1
, w
v2
; τ
~n
+ t
~n
H
v
) (15)
where ε ·
k
v
is given by:
(
X
a
w
ia
ε
a
p
a
)
6.1. Non-equivariant limit
In the case of compact X the partition function (15) has a finite non-
equivariant limit, i.e. the limit where ε → 0. Indeed, the singular terms
in the logarithm of partition function vanish thanks to the identity:
X
v
1
w
v1
w
v2
=
Z
X
1 = 0,
X
v
φ
(v)
a
(ε)
w
v1
w
v2
=
Z
X
c
1
(L
a
) = 0 (16)
X
v
w
v1
+ w
v2
w
v1
w
v2
=
Z
X
c
1
(X) = 0
and the finite part reduces to:
Z
~w
(a; τ
~n
, t
~n
) =
X
~
k
a
∈Z
N
,{
~
k
a
}=w
a
exp
Z
X
F
0
(a +
X
a
k
a
c
1
(L
a
))+ (17)
+
Z
X
c
1
(X)H
1
2
(a +
X
a
k
a
c
1
(L
a
)) + χ(X)F
1
(a) + σ(X)G
1
(a)
September 23, 2014 16:1 WSPC/Trim Size: 9.75in x 6.5in for Proceedings Lisbonne
10 Nikita Nekrasov
If we now integrate over a then the resulting function of τ ,
t (for τ =
(logΛ, 0, 0, . . .)) is, in fact, the generating function of Donaldson invariants
of X, for some specific choice of the metric on X. This choice is correlated
with the choice of contour for the a integral. Put another way, the sum over
k
a
’s goes over some subset of all possible first Chern classes. This subset is
singled out by some stability condition, which involves explicitly the Kahler
class of X (i.e. the choice of r).
6.2. Non-holomorphic speculations
Looking at the non-equivariant limit of the partition function we recognize
the holomorphic approach formulae of Ref. [13], see also Ref. [8] . There,
the low-energy effective theory of N = 2 super Yang-Mills theory was ex-
ploited in order to write down a contribution of the Coulomb branch to the
Donaldson invariants of manifolds with b
+
2
= 1 (all toric manifolds have this
property). More systematic approach, taking into account full multiplet of
N = 2 susy leads to the non-holomorphic integrand, see Ref. [13,19], whose
integral can be converted to the contour integral of a holomorphic form, sim-
ilar to what we have above. We hope to infer from this the non-holomorphic
modular completion of our master formula.
6.3. Relation to the results of Nakajima-Yoshioka
This section is devoted to the simplest non-trivial example of the application
of the formula (15): X =
ˆ
C
2
, the blowup of a point on C
2
. It is a toric
manifold, M = 1, d = 2. In this case the formula (15) reduces to:
Z
X,w
(a,
1
,
2
,
3
; τ
~n
, t
~n
) = (18)
X
~
k∈Z
N
,{
~
k}=w
Z(a +
k
1
,
1
+
3
,
2
−
1
; τ
~n
+ t
~n
(
1
+
3
))×
Z(a +
k
2
,
1
−
2
,
2
+
3
; τ
~n
+ t
~n
(
2
+
3
))
which, for
3
= 0 coincides with the result of Ref. [21].
Acknowledgments
I would like to thank the organizers of the Congress for the beautiful con-
ference, and for inviting me to give a lecture.
I also wish to thank A. Losev and A. Okounkov for useful discussions.
Research was partially supported by 01-01-00549 grant from RFFI.
September 23, 2014 16:1 WSPC/Trim Size: 9.75in x 6.5in for Proceedings Lisbonne
Lo calizing gauge theory 11
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hep-th/0405146
... First of all, M-strings give an indirect way of capturing features of (2, 0) theories, that can be exploited to probe certain conjectural properties of these theories. In this paper in particular we will see that indeed the ALE partition function of M-strings agrees with the 5d uplift of the equivariant ALE partition function of a 4d N = 2 * pure super Yang-Mills theory of type U(r) [19][20][21][22], which gives a very non-trivial consistency check for several claims in the literature about these systems [23]. Secondly, the Ω-deformed flat background, the T 2 × S 4 partition function, and the S 1 × S 5 superconformal index only capture information on the local operators of these theories, while JHEP04(2024)018 the ALE partition functions we consider in this paper are capable of probing finer aspects of their global structure, in particular their relative (chiral) nature [24,25]. 1 In fact, as we will see in details below, our results depend on a slightly more refined collection of 6d quantum numbers which give the six-dimensional counterpart of the choice of flat connection at infinity in gauge theory. ...
... We present two consistency checks for our proposal. One one hand, we formulate a conjectural 6d N = (1, 0) version of the Nekrasov master formula [19], which relates the partition function T 2 × C 2 /Z n (where X denotes the blow-up of X at the origin) to a product of n copies of the partition function on T 2 × C 2 . This elliptic version of the Nekrasov master formula can be viewed as a generalization of the elliptic version of the Nakajima-Yoshioka blow-up equations [41,42] for T 2 × C 2 [43][44][45][46] (see also [47,48] for additional related works). ...
... (4. 19) In fact, it is possible to give a combinatorial formula for the sum over residues in equation (4.10), building upon [76] as well as [77]. Denote by Y κ (a) the set of Young diagrams Y (a) of size nκ (a) . ...
The ALE partition functions of M-strings
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Full-text available
  • Apr 2024
  • J HIGH ENERGY PHYS
We compute the equivariant partition function of the six-dimensional M-string SCFTs on a background with the topology of a product of a two-dimensional torus and an ALE singularity. We determine the result by exploiting BPS strings probing the singularity, whose worldvolume theories we determine via a chain of string dualities. A distinguished feature we observe is that for this class of background the BPS strings’ worldsheet theories become relative field theories that are sensitive to finer discrete data generalizing to 6d the familiar choices of flat connections at infinity for instantons on ALE spaces. We test our proposal against a conjectural 6d \( \mathcal{N} \) = (1, 0) generalization of the Nekrasov master formula, as well as against known results on ALE partition functions in four dimensions.
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... However, this is not the full story, as four-manifolds B with non-trivial H 2 (B; Z) also admit gauge configurations where the field strength has flux on non-contractible cycles. It was conjectured in [13,16] that the flux contributions enter as a shift of the Coulomb branch parameter in each copy of the Nekrasov partition function. The full partition function is then obtained as a sum over these flux sectors. ...
... The full partition function is then obtained as a sum over these flux sectors. In the literature [7,[9][10][11][12][13][14][16][17][18][19][20], the sum is over equivariant fluxes which are then constrained by stability equations. ...
... In particular, we allow the theories to localise to different distributions of (anti-)instantons at the torus fixed points (also known as Pestunization). We explicitly compute the one-loop partition function of these theories around each flux sector, whose factorised form 1 agrees with the form conjectured in [13,16]. From this form we can read out the shifts of the Coulomb branch parameter at each torus fixed point by the fluxes and write down the Coulomb branch partition function 2 , including instanton and classical parts. ...
From 5d Flat Connections to 4d Fluxes (the Art of Slicing the Cone)
Preprint
Full-text available
  • May 2023
We compute the Coulomb branch partition function of the 4d $\mathcal{N}=2$ vector multiplet on closed simply-connected quasi-toric manifolds $B$. This includes a large class of theories, localising to either instantons or anti-instantons at the torus fixed points (including Donaldson-Witten and Pestun-like theories as examples). The main difficulty is obtaining flux contributions from the localisation procedure. We achieve this by taking a detour via the 5d $\mathcal{N}=1$ vector multiplet on closed simply-connected toric Sasaki-manifolds $M$ which are principal $S^1$-bundles over $B$. The perturbative partition function can be expressed as a product over slices of the toric cone. By taking finite quotients $M/\mathbb{Z}_h$ along the $S^1$, the locus picks up non-trivial flat connections which, in the limit $h\to\infty$, provide the sought-after fluxes on $B$. We compute the one-loop partition functions around each topological sector on $M/\mathbb{Z}_h$ and $B$ explicitly, and then factorise them into contributions from the torus fixed points. This enables us to also write down the conjectured instanton part of the partition function on $B$.
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... Based on a number of previous results and conjectures in the literature (see e.g. [23,35,42]), we expect an explicit formula of the form: ...
... The CB partition function (1.7) is a holomorphic function of a. The schematic formula (1.8) is inspired by the 'holomorphic approach' to the DW twist [43], and by a conjecture of Nekrasov for toric four-manifolds [42]. By contrast, the Moore-Witten u-plane integral approach deals with a non-holomorphic integrand, which renders the vector-multiplet integration better defined. ...
... Partition functions of 4d N = 2 field theories on toric four-manifolds can be computed in terms of the partition functions on toric patches C 2 [42], and similarly for the 5d uplift. On each patch, one considers the so-called Nekrasov partition function on C 2 × S 1 with the Ω-background, which is obtained by the identification (z 1 , z 2 , x 5 ) ∼ (e 2πiτ 1 z 1 , e 2πiτ 2 z 2 , x 5 + β) , (6.1) where (z 1 , z 2 , x 5 ) are the C 2 × S 1 coordinates, and we also introduced the dimensionless Ω-deformation parameters: ...
Partition functions and fibering operators on the Coulomb branch of 5d SCFTs
Article
Full-text available
  • Jan 2023
  • J HIGH ENERGY PHYS
A bstract We study 5d $$ \mathcal{N} $$ N = 1 supersymmetric field theories on closed five-manifolds $$ \mathcal{M} $$ M 5 which are principal circle bundles over simply-connected Kähler four-manifolds, $$ \mathcal{M} $$ M 4 , equipped with the Donaldson-Witten twist. We propose a new approach to compute the supersymmetric partition function on $$ \mathcal{M} $$ M 5 through the insertion of a fibering operator, which introduces a non-trivial fibration over $$ \mathcal{M} $$ M 4 , in the 4d topologically twisted field theory. We determine the so-called Coulomb branch partition function on any such $$ \mathcal{M} $$ M 5 , which is conjectured to be the holomorphic ‘integrand’ of the full partition function. We precisely match the low-energy effective field theory approach to explicit one-loop computations, and we discuss the effect of non-perturbative 5d BPS particles in this context. When $$ \mathcal{M} $$ M 4 is toric, we also reconstruct our Coulomb branch partition function by appropriately gluing Nekrasov partition functions. As a by-product of our analysis, we provide strong new evidence for the validity of the Lockhart-Vafa formula for the five-sphere partition function.
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... 16 Note that there always is a locally free S 1 ⊂ T 3 ; precisely the one that agrees with the remaining, non-degenerate S 1 at the vertices of the polygon, cf. figure 1. 17 Note that γ ∈ Aut(π) is an isometry since the covering π is Riemannian. 18 Here, gP denotes the associated bundle gP = P × ad g of the gauge bundle P . ...
... Note that (6.7) has no flux-dependence 31 which we anticipate, since already S cl on X is independent of the topological class of flat connections. The instanton piece is obtained as the standard product of Nekrasov partition functions on C 2 ϵ i 1 ,ϵ i 2 over the fixed points, applying the appropriate shifts to a [14,17]. For x such that the cohomologically twisted background on B localises to instantons at r of the m fixed points and anti-instantons at the remaining ones, we obtain ...
From 5d flat connections to 4d fluxes (the art of slicing the cone)
Article
Full-text available
  • Oct 2023
  • J HIGH ENERGY PHYS
A bstract We compute the Coulomb branch partition function of the 4d $$ \mathcal{N} $$ N = 2 vector multiplet on closed simply-connected quasi-toric manifolds B . This includes a large class of theories, localising to either instantons or anti-instantons at the torus fixed points (including Donaldson-Witten and Pestun-like theories as examples). The main difficulty is to obtain flux contributions from the localisation procedure. We achieve this by taking a detour via the 5d $$ \mathcal{N} $$ N = 1 vector multiplet on closed simply-connected toric Sasaki-manifolds M which are principal S ¹ -bundles over B . The perturbative partition function can be expressed as a product over slices of the toric cone. By taking finite quotients M /ℤ h along the S ¹ , the locus picks up non-trivial flat connections which, in the limit h → ∞, provide the sought-after fluxes on B . We compute the one-loop partition functions around each topological sector on M /ℤ h and B explicitly, and then factorise them into contributions from the torus fixed points. This enables us to also write down the conjectured instanton part of the partition function on B .
View
... These investigations ultimately allowed the direct evaluation of the JHEP02(2023)110 on-shell action of all supersymmetry-preserving asymptotically locally AdS backgrounds in terms of their nuts and bolts, as shown in [10] in minimal (two derivative) gauged 4d N = 2 supergravity. 1 In an a priori far removed (both in time and in scope) development, observables in conformal and supersymmetric (non-gravitational) field theories have been shown to factorize in terms of basic building blocks in many cases, see [11][12][13][14][15] and references thereof, as often shown via supersymmetric localization [16,17]. Based on the AdS/CFT correspondence, the 3d SCFT factorization in terms of holomorphic blocks, [14], inspired the construction of gravitational blocks in (two derivative) 4d N = 2 supergravity coupled to matter, [18], applied to supersymmetric black hole backgrounds. ...
... 12 Note that the conjecture was originally presented in Lorentzian conventions, where the prepotential standarly comes with an additional factor of i that we have stripped off here. 13 Here we reinsert the length scale L, which was set to 1 in [1]. ...
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Full-text available
  • Feb 2023
  • J HIGH ENERGY PHYS
A bstract We initiate a systematic study of supersymmetric backgrounds in 4d 𝒩 = 2 Euclidean supergravity in the presence of infinite towers of higher derivative corrections. Adopting a Gibbons-Hawking view towards the evaluation of the action in terms of nuts and bolts, we consider the two maximally symmetric vacua ℝ ⁴ and ℍ ⁴ (Euclidean AdS 4 ) and their unique supersymmetric deformations with (anti-) self-dual Maxwell tensors corresponding to a single nut at the center. These are the Omega background of Nekrasov-Okounkov, Ωℝ ⁴ , and its generalization with a cosmological constant of Martelli-Passias-Sparks, denoted Ωℍ ⁴ (also known as the gravity dual of the U(1) × U(1) squashed sphere). We write down the BPS configurations in the superconformal formalism in the presence of vector multiplets and derive the corresponding off- and on-shell actions. Our results provide a rigorous proof for important parts of the conjecture in [1] and its holographic corollary in [2], which we discuss in detail along with extensions such as the addition of hypermultiplets and the presence of conical defects.
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... Moreover, using various ideas from string theory [17], one can generalize these matrix models so as to describe more non-trivial spatial backgrounds, such as ALE spaces, conifolds or K3 manifolds [38]. One can observe, that the intuition of quantum field theory on curved spacetime, in particular the cluster decomposition, which is a useful method of computations in topological field theories, persists in the matrix model approach [27]. ...
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  • Jun 2023
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... , d e r ). The Nekrasov partition function of the pure N = 2 U(r) gauge theory on M 2 is then given by the "master formula" [115,116] ...
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... In more technical detail, our results are a direct application of the conjecture of [1] about gravitational building blocks at full quantum level. It proposes a gluing mechanism in analogy to the 4d N = 2 field theory factorization in terms of Nekrasov partition functions [3,4]. The gravitational block serves to determine the bulk on-shell action for JHEP10(2022)190 various asymptotically AdS 4 backgrounds with fixed points of the canonical isometry, as first noticed in [5] and [6]. ...
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  • Oct 2022
  • J HIGH ENERGY PHYS
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... If M 4 is a toric variety then it can be glued from C 2 pieces. The corresponding 4d master formula for non-compact toric varieties [32][33][34] takes the form admits different refinements, for example we can fix the holonomy at infinity, in case a boundary of the toric space has non-trivial topology (allowing different flat connections at infinity). We aren't interested in such refinements and leave them aside. ...
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  • COMMUN MATH PHYS
Motivated by M-theory, we study rank n K-theoretic Donaldson–Thomas theory on a toric threefold X . In the presence of compact four-cycles, we discuss how to include the contribution of D4-branes wrapping them. Combining this with a simple assumption on the (in)dependence on Coulomb moduli in the 7d theory, we show that the partition function factorizes and, when X is Calabi–Yau and it admits an ADE ruling, it reproduces the 5d master formula for the geometrically engineered theory on $$A_{n-1}$$ A n - 1 ALE space, thus extending the usual geometric engineering dictionary to $$n>1$$ n > 1 . We finally speculate about implications for instanton counting on Taub-NUT.
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Blowups in BPS/CFT Correspondence, and Painlevé VI
Article
  • Apr 2023
We study four-dimensional supersymmetric gauge theory in the presence of surface and point-like defects (blowups) and propose an identity relating partition functions at different values of \(\Omega \)-deformation parameters \(({\varepsilon }_{1}, {\varepsilon }_{2})\). As a consequence, we obtain the formula conjectured in 2012 by O. Gamayun, N. Iorgov, and O. Lysovyy, relating the tau-function \({\tau }_{PVI}\) to \(c=1\) conformal blocks of Liouville theory and propose its generalization for the case of Garnier–Schlesinger system. To this end, we clarify the notion of the quasiclassical tau-function \({\tau }_{PVI}\) of Painlevé VI and its generalizations. We also make some remarks about the sphere partition functions, the boundary operator product expansion in the \({{\mathcal {N}}}=(4,4)\) sigma models related to four-dimensional \({{\mathcal {N}}}=2\) theories on toric manifolds, discuss crossed instantons on conifolds, elucidate some aspects of the BPZ/KZ correspondence, and applications to quantization.
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Lectures on Hilbert Schemes of Points on Surfaces
Article
Full-text available
  • Jan 2014
This paper is based on author's lectures at Kyoto University in 2010 Summer, and in the 6th MSJ-SI `Development of Moduli Theory' at RIMS in June 2013. The purpose of lectures was to review several results on Hilbert schemes of points which were obtained after author's lecture note was written. Among many results, we choose those which are about equivariant homology groups $H^T_*(X^{[n]})$ of Hilbert schemes of points on the affine plane $X = \mathbb C^2$ with respect to the torus action.
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Gromov–Witten theory and Donaldson–Thomas theory, I
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Full-text available
  • Sep 2006
  • COMPOS MATH
We conjecture an equivalence between the GromovSimons theory of Donaldson and Thomas. For CalabiWitten theory and $q$ is the Euler characteristic parameter of DonaldsonYau toric surfaces.
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Quantum Foam and Topological Strings
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Full-text available
  • Apr 2008
We find an interpretation of the recent connection found between topological strings on Calabi-Yau threefolds and crystal melting: Summing over statistical mechanical configuration of melting crystal is equivalent to a quantum gravitational path integral involving fluctuations of Kähler geometry and topology. We show how the limit shape of the melting crystal emerges as the average geometry and topology of the quantum foam at the string scale. The geometry is classical at large length scales, modified to a smooth limit shape dictated by mirror geometry at string scale and is a quantum foam at area scales ~ gsα'.
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D -Particle Bound States and Generalized Instantons
Article
Full-text available
  • Jan 2000
We compute the principal contribution to the index in the supersymmetric quantum mechanical systems which are obtained by reduction to 0 + 1 dimensions of , D= 4,6,10 super-Yang–Mills theories with gauge group SU(N). The results are: for D=4,6, for D=10. We also discuss the D=3 case.
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Hook formula and related identities
Article
  • May 1992
A new proof of the hook formula for the dimension of representations of the symmetric group is given with the help of identities which are of independent interest. A probabilistic interpretation of the proof and new formulas relating the parameters of the Young diagrams are given.
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Bott R.: The moment map and equivariant cohomology. Topology. 23, 1-28
Article
  • Dec 1984
  • TOPOLOGY
THE PURPOSE of this note is to present a de Rham version of the localization theorems of equivariant cohomology, and to point out their relation to a recent result of Duistermaat and Heckman and also to a quite independent result of Witten. To a large extent all the material that we use has been around for some time, although equivariant cohomology is not perhaps familiar to analysts. Our contribution is therefore mainly an expository one linking together various points of view. The paper of Duistermaat and Heckman ( 1 l) which was our initial stimulus concerns the moment mapf:M+R' for the action of a torus T' on a compact symplectic manifold M. Their theorem asserts that the push-forward byfof the symplectic (or Liouville) measure on A4 is a piece-wise polynomial measure on R'. An equivalent version is that the Fourier transform of this measure (which is the integral over A4 of e-i(c,fi) is exactly given by stationary phase approximation. For example when I = 1 (so that T'is the circle S) and the fixed points of the action are isolated points P, we have the exact formula , where (u is the symplectic 2-form on M and the e(P) are certain integers attached to the infinitesimal action of S at P. This principle, that stationary-phase is exact when the "Hamiltonian"fcomes from a circle action, is such an attractive result that it seemed to us to deserve further study.
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Symmetric Functions and Hall Polynomials
Book
  • Jan 1995
A classical monograph on symmetric functions.
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