Localization of Vortex Partition Functions in $\mathcal{N}=(2,2) $ Super Yang-Mills theory

0.16 MB·English

by  Yutaka Yoshida

KEK-TH 1433
Localization of Vortex Partition Functions in
= (2, 2) Super Yang-Mills theory
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Yutaka Yoshida

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J KEK Theory Center, Institute of Particle and Nuclear Studies,

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High Energy Accelerator Research Organization

] Tsukuba, Ibaraki 305-0801, Japan
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E-mail address: [e-mail]
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8 Abstract
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.
1 In this article, we study the localizaiton of the partition function
0
of BPS vortices in = (2,2) U(N) super Yang-Mills theory with
1 N
1 N-flavor on R2. The vortex partition function for = (2,2) su-
N
: per Yang-Mills theory is obtained from the one in = (4,4) super
v N
i Yang-Mills theory by mass deformation. We show that the partition
X
function can be written as Q-exact form and integration in the par-
r
a tition functions is localized to the fixed points which are related to
N-tuple one dimensional partitions of positive integers.
1 Introduction
The vortices in two dimensional field theory are instanton like objects and
have finite energy and action. We can consider the partition functions of
vortices. In supersymmetric theories with eight or four supercharges, our
insterest is the vortices described by Bogomoln’yi equation which preserve
half of the supersymmetries. The vortex moduli spaces in p-dimesional the-
ories with eight supercharges are constructed in D(p-2)-Dp system in [1], as
the Ka¨hler quotient spaces. Especially, the vortex partition functions for
= (4,4) super Yang-Mills theories with vortex number k are U(k) gauged
N
matrix model with four supercharges [1], [2]. But, it is difficult to perform
multi-variable integrations directly in vortex partition functions.
Another road to evaluate the vortex partitions is the reduction from in-
stanton partition functions in four dimensional = 2 super Yang-Mills
N
theory with surface operators. The instanton partition functions with sur-
face operators are expanded in double series with respect to vortex number
and instanton number. When surface operators are half-BPS in U(N) = 2
N
super Yang-Mills theory, the theory on the plane where the surface opera-
tors define becomes = (2,2) U(1) gauge theory with matter fields. The
N
insertion of surface operators in four dimensional gauge theory correspond
to introduction of open string in topological A-model amplitudes in toric
Calabi-Yau 3-folds [3] which can be calculated by refined topological ver-
tex [4]. The authors of [5] calculated abelian vortex partition functions in
= (2,2) SQED by taking decoupling limit of instaton parts.
N
In this article, we consider the evaluation of the vortex partition func-
tions for = (2,2) U(N) gauge theory in two dimensions with N = N
f
N
fundamental chiral multiplets and with generic twisted masses. This arti-
cle is organized as follow: In section two, we first explain the Bogomoln’yi
equation in 2d = (4,4) supersymmetric gauge theory and present vor-
N
tex partition functions. In section three, deforming the = (4,4) theory
N
by adding a superpotential term and taking large mass limit, we obtain the
vortex partition functions in = (2,2) theory. In section four, we evalu-
N
ate the vortex partitions by equivariant localization method. In section five,
1
we also calculate K-theoretic vortex partition function from the equivariant
character of the Ka¨hler quotient space. In section six, we discuss our results.
2 Vortex in = (4,4) U(N) super Yang-Mills
N
theories
In this section, we review = (4,4) U(N ) super Yang-Mills theory in two
c
N
dimensions [6] with the number of hypermultiplets N = N = N. We can
f c
construct = (4,4) multiplets by combining a pair of = (2,2) multiplets.
N N
The vector multiplet in = (4,4) consists of a pair of = (2,2) superfields
N N
(Σ,Φ).
Σ = σ +i√2θ+λ¯ i√2θ¯−λ +√2θ+θ¯−(D3 iF )+ ,
+ − 12
− − ···
1
Φ = σ˜ +√2θ+λ˜ +√2θ−λ˜ + θ+θ−(D1 iD2)+ , (2.1)
+ −
√2 − ···
where Σ is thetwisted chiral multiplet andΦ is the chiral multiplet in adjoint
˜
representation in U(N ). Here σ and σ˜ are scalars, λ and λ are fermions.
c
Di(i = 1,2,3) are auxiliary fields. The hypermultiplet in = (4,4) consists
N
of a pair of = (2,2) chiral superfields (Q ,Q˜ ),(i = 1, ,N ).
i i f
N ···
Q = q +√2θ+ψ +√2θ−ψ +θ+θ−F + ,
i i +i −i i
···
Q˜ = q˜ +√2θ+ψ˜ +√2θ−ψ˜ +θ+θ−F˜ + , (2.2)
i i +i −i i
···
where Q are chiral multiplets in fundamental representation for U(N ) and
c
N flavor. Q˜ are chiral multiplets in anti-fundamental representation for
f
U(N ) and N flavor.
c f
The Lagrangian for 2d = (4,4) super Yang-Mills theory with gauge
N
group U(N) and N flavor consists of
S = d2xd4θ(TrΣ¯Σ+TrΦ¯eVΦe−V +Q†e−2VQ +Q˜†e2VQ˜ ),
K i i i i
Z
S = d2xd2θQ˜ ΦQ . (2.3)
V i i
Z
We can also include the Fayet-Iliopoulos terms and the bosonic part in the
Lagrangian becomes
1 1 1 e2
L = Tr F Fµν + D σDµσ¯ + D σ˜Dµσ¯˜ + (q q† q˜†q˜ rI )2
boson − 2e2 µν 2e2 µ 2e2 µ 2 i i − i i − N
h (cid:16) (cid:17)
2
+D q†Dµq +D q˜†Dµq˜ +q˜† σ¯,σ q˜ + . (2.4)
µ i i µ i i i{ } i ···
i
When the Fayet-Iliopoulos parameter r is positive, the vacuum is unique up
to Weyl symmetry,
qa = √rδa, q˜a = 0, σ = 0, σ˜ = 0, (2.5)
i i i
wherewewritecolorindicesa(a = 1, ,N)andflavorindicesi(i = 1, ,N)
··· ···
explicitly. In this phase, gauge and flavor symmetry group breaks to little
symmetry group
U(N) SU(N) SU(N) . (2.6)
G F diag
× →
In this case, half-BPS equation exist which minimize the energy. This half-
BPS solutions of equation(Bogomoln’yi equation) are defined by
e2
F a = (qaqi† rδa),
12b 2 i b − b
(D q)a = 0, (2.7)
z i
where we define covariant derivative D = D iD .
z 1 2
−
The vortex instanton number is given by the first Chern number k;
1
k := TrF. (2.8)
2π
Z
The bosonic solution with vortex number-k has action
S = 2π(r +iθ)k. (2.9)
k
Next we consider the k-vortex partition function for = (4,4) super
N
Yang-Mills theories. In two dimensions, the vortex partition function in
= (4,4) Yang-Mills theories is constructed by k D0-branes and N D2-
N
branes system. This is called the vortex matrix model [2].
This model is obtained by dimensional reduction of = 1 supersymmet-
N
ric action in four dimensions to zero dimension or equivalently dimensional
reduction of = (2,2) supersymmetric theory in two dimensions to zero
N
dimension.
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This matrix model is given by
1 1
N=(4,4)
Z = Xexp( S S S ), (2.10)
k,N Vol(U(k)) D −g2 G − m − A
Z
where
1 1
S = Tr [ϕ,ϕ¯]2 [φ,φ¯]2 +[ϕ¯,φ¯][ϕ,φ]+[ϕ,φ¯][ϕ¯,φ]+D2 +g2ζD
G
−2 − 2
(cid:16)
+√2(λ¯ [ϕ¯,λ ]+λ¯ [ϕ,λ ] λ¯ [φ¯,λ ] λ¯ [φ,λ ]) ,
+ + − − + − − +
− −
N (cid:17)
S = I† ϕ,ϕ¯ I I† φ,φ¯ I +I†DI
m − i{ } i − i{ } i
Xi=1(cid:16)
+√2(µ† ϕ¯µ +µ† ϕµ µ† φ¯µ µ† φµ )
+i +i −i −i − +i −i − −i +i
+i√2[I†i(λ µ λ µ )+( µ† λ¯ µ† λ¯ )I ] ,
− +i − + −i − +i − − −i + i
(cid:17)
S = Tr [ϕ,B†] 2 + [φ,B†] 2 +D[B,B†]
A
| | | |
√2(cid:16)( [ϕ,ρ† ]ρ [ϕ¯,ρ† ]ρ ) [φ,ρ† ]ρ [φ¯,ρ† ]ρ
− − − − + + − + + − + +
+i√2 B([ρ† ,λ¯ ]+[ρ† ,λ¯ ])+B†([λ ,ρ ]+[λ ,ρ ]) . (2.11)
− + + − + − − +
¯ ¯ h i(cid:17)
(ϕ,ϕ¯,φ,φ,λ,λ,D)comefromthe2d = (2,2)vector multipletinthereduc-
N
tion. (I ,µ ),(i = 1, ,N) appear through dimensional reduction of the 2d
i ±i
···
chiral multiplets to zero dimension, which belong to fundamental represen-
tation of U(k). (B,ρ ) is the dimensional reduction of 2d chiral multiplet to
±
zero dimension, adjoint representation of U(k). X is the integration mea-
D
sure. The Fayet-Illioporos parameter in the vortex matrix model is related
to 2d gauge coupling
2π
ζ = . (2.12)
e2
In order to decouple gravity from the gauge theories, the gauge coupling
constant g in vortex matrix model goes to infinity. Then, the fields λ and D
become the Lagrangian multipliers and produce the constraint;
N
[B,B†]+ I I† = ζI . (2.13)
i i k
i=1
X
Solutions of this equation characterize the moduli space for k-vortex
N
= (B,I ) [B,B†]+ I I† = ζI /U(k). (2.14)
Mk,N { i | i i k}
i=1
X
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We can define the vortex partition function by using (2.10)
∞
ZN=(4,4) = 1+ ZN=(4,4)e2π(r+iθ)k. (2.15)
k,N
k=1
X
So far we have considered only N = N case, but vortex partition functions
c f
forgeneral N = N < N isalreadygiven in[1], [2]. Ingeneral N flavor case,
c f f
thevortexpartitionfunctionsaredescribedbyintroducingadditionalN N
f c
−
dimensional reduced 2d chiral multiplets to zero dimension. (I˜,µ˜ ),(j =
j j
1, ,N N¸). The moduli space for k-vortex is modified to
f
··· −
Nc Nf−Nc
= (B,I ,I˜) [B,B†]+ I I† I˜I˜† = ζI /U(k).
Mk,Nc,Nf { i j | i i − j j k}
i=1 j=1
X X
(2.16)
But, for simplicity, we restrict our attention to N = N type vortices.
c f
3 = (2,2) vortex partition function
N
In order to obtain = (2,2) vortex partition functions, we add a superpo-
N
ˆ
tential W(Φ) to the 2d theory in (2.3).
d2xd2θTrWˆ (Φ). (3.1)
Z
This breaks the = (4,4) supersymmetry to = (2,2) in two dimensions.
N N
The superpotential contains the quadratic term
Wˆ (Φ) = MΦ2, (3.2)
In the heavy mass limit, the bosonic part of Lagrangian in two dimensions is
obtained by setting Φ = 0 in (2.3), The vacuum in Higgs branch is still given
by
qa = √rδa, q˜a = 0, σ = 0. (3.3)
i i i
Apparently, the Q˜ fields are trivial, the vortex equation is equivalently to
vortex for the following action
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1 1 1
L = Tr F Fµν + D σDµσ¯ + [σ,σ¯]2
boson µν µ
− 2e2 2e2 2e2
he2 (cid:16)
+ (q q† rI )2 +iθF +D q†Dµq +q† σ,σ¯ q .
2 i i − N 12 µ i i i{ } i
(cid:17) i (3.4)
ThisisbosonicpartsoftheLagrangianfor = (2,2)U(N) superYang-Mills
N
with N fundamental chiral multiplets.
Let us consider how the superpotential deformation (3.2) affects the vor-
tex matrix model (2.11). The vortices preserve the half of supersymmetry,
so the vortex matrix model is deformed to preserve half of the = (2,2)
N
supersymmetry. As discussed in [7], in the presence of superpotential, vortex
matrix model is deformed by adding
∂Wˆ (Σ(0,2))
Tr dθ+Ξ +(h.c)
∂Σ(0,2) |θ¯+=0
Z
∂Wˆ (φ) ∂2Wˆ (φ)
¯
= G +ρ λ +(h.c). (3.5)
∂φ − ∂φ2 +
Here Σ(0,2) comes from the dimensional reduction of 2d = (0,2) chiral
N
multiplet,
Σ(0,2) = φ i√2θ+λ + . (3.6)
+
− ···
Ξ appears in the dimensional reduction of 2d = (0,2) fermi multiplet,
N
Ξ = ρ √2θ+G+ , (3.7)
−
− ···
where G is the auxiliary field in the fermi multiplet.
The left moving fermions µ ,µ† ,ρ and ρ† are absent in the mass de-
− − − −
formation. Moreover in the heavy mass limit, Ξ and Σ(0,2) multiplets are
decoupled from the vortex matrix model. Thus the vortex partition func-
tions for = (2,2) theory consist of three pieces:
N
1
S′ = Tr [ϕ,ϕ¯]2 +D2 +g2ζD+2λ¯ [ϕ,λ ] ,
G − −
2
N(cid:16) (cid:17)
S′ = I† ϕ,ϕ¯ I +I†DI
m − i{ } i i i
Xi=1(cid:16)
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√2µ† ϕ¯µ +i√2[I†λ µ µ† λ¯ I ] ,
− +i +i i − +i − +i − i
(cid:17)
S′ = Tr [ϕ,B†] 2 +D[B,B†]
A
| |
(cid:16)
√2[ϕ¯,ρ† ]ρ +i√2 B[ρ† ,λ¯ ]+B†[λ ,ρ ] . (3.8)
− + + + − − +
h i(cid:17)
The k-vortex partition function for = (2,2) super Yang-Mills theory is
N
1 1
Z = ϕ ϕ¯ B I χ η µ ρ exp( S′ S′ S′ ).
k,N Vol(U(k)) D D D D D D D +D + −g2 G − m − A
Z
(3.9)
4 Localization
In this section, we compute the vortex partition function in = (2,2) super
N
Yang-Mills theories by equivariant localization formula for supersymmetric
system [8], [9], [10]. The vortex partition function given by (3.8) is invariant
under the following supersymmetric transformation,
Q ϕ = 0,
ǫ
Q ϕ¯ = η, Q η = [ϕ,ϕ¯],
ǫ ǫ
Q D = [ϕ,χ], Q χ = D,
ǫ ǫ
Q I = µ , Q µ = ϕI ,
ǫ i +i ǫ +i i
Q I† = µ† , Q µ† = I†ϕ,
ǫ i − +i ǫ +i i
Q B = ρ , Q ρ = [ϕ,B] ǫB,
ǫ + ǫ +
−
Q B† = ρ† , Q ρ† = [ϕ,B†] ǫB†. (4.1)
ǫ − + ǫ + −
Here we defined η = i(λ + λ¯ )/√2, χ = i(λ λ¯ )/√2 and rescaled
− − − −
− − −
ϕ √2ϕ. The vortex partition function (3.9) can be written in Q -exact
ǫ
→ −
form. For the abelian vortex case, we can apply the localization formula
straightforwardly. But, Q does not generate appropriate fixed points for
ǫ
the nonabelian case. We recall that vev of adjoint scalars in the vector
multiplet play a crucial role to localization formula work well and generate
appropriate fixed points in Nekrasov partition functions in = 2 super
N
Yang-Mills theory. The twisted masses in chiral multiplets play a role of
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vev of adjoint scalars in the vortex partition functions at Higgs branch. We
introduce the generic twisted masses m ,(i = 1, ,N), then the Lagrangian
i
···
of 2d = (2,2) super Yang-Mills theory is modified as
N
1 1 1 e2
L = Tr F Fµν + D σDµσ¯ + [σ,σ¯]2 + (qq† rI )2
boson µν µ N
− 2e2 2e2 2e2 2 −
h (cid:16) (cid:17)
+D q†Dµq +q† σ m ,σ¯ m∗ q . (4.2)
µ i i i{ − i − i} i
i
The vacuum is labeled by
qa = √rδa, σa = maδi. (4.3)
i i b i b
The introduction of twisted mass terms also introduce twisted masses in
vortex partitions. The supersymmetry transformation is modified as
Q µ = ϕI Q µ = (ϕ m )I ,
ǫ +i i ǫ +i i i
→ −
Q µ† = I†ϕ Q µ† = I†( ϕ m ). (4.4)
ǫ +i i → ǫ +i i − − i
The vortex partition function (3.8) can be written in Q -exact form
ǫ
1
S′ = Q Tr [ϕ,ϕ¯]η +Dχ+g2ζχ ,
G ǫ
4
(cid:18) (cid:19)
Nc
S′ = Q µ†i(ϕ¯ m∗)Ii +Ii†(ϕ¯ m∗)µi +Ii†χIi ,
m ǫ + − i − i +
Xi=1 h i
S′ = Q Tr B†[ϕ¯,ρ ]+ρ† [ϕ¯,B]+χ[B,B†] . (4.5)
A ǫ − + +
h i
Q is nilpotent up to infinitesimal gauge transformation and with flavor ro-
ǫ
tation.
Let us consider the localization method. First, we introduce the vector
field Q∗ which acts on the fields and generates the supersymmetry transfor-
mation
∂ ∂ ∂ ∂
Q∗ = [ϕ,χ] +η +µ +ρ
+i +
∂D ∂ϕ¯ ∂I ∂B
i
∂ ∂ ∂ ∂
+D +[ϕ,ϕ¯] +(ϕ m )I +([ϕ,B] ǫB)
i i
∂χ ∂η − ∂µ − ∂ρ
+ +
∂ ∂
= Q∗i +Q∗i . (4.6)
F∂ B ∂
i i
B F
8
The critical points Q∗ = 0 is given by
(ϕ m )I = 0,
I i Ii
−
(ϕ ϕ ǫ)B = 0. (4.7)
I J IJ
− −
All the other fields are zero. The B and I are all zero except for B
i,i−1
and I ,I , ,I . Here k ’s are the partitions of N integers (i =
1,1 k1+1,2 ··· kN+1,N i
1, ,N) and satisfy a relation k = k .
··· i i
The localization formula [10] is expressed as contour integral
P
k
(ϕ ϕ )
I J
Z = dϕ − . (4.8)
k I
Sdet
I I=1 I6=J L
Y Y
The superdeterminant of is defined by
L
∂Q∗i ∂Q∗i
B B
Sdet = Sdet ∂Bj ∂Fj , (4.9)
L ∂Q∗Fi ∂Q∗Fi !
∂Bj ∂Fj
and
A B
Sdet = det(A BD−1C)det(D)−1. (4.10)
C D −
(cid:18) (cid:19)
Thus, the vortex partition function Z for = (2,2) super Yang-Mills
k
N
with twisted mass terms becomes
k k N
1 1 ϕ ϕ
I J
Z = dϕ − (4.11)
k k!(2πiǫ)k I ϕ m ϕ ϕ ǫ
I i I J
I YI=1 (cid:16)YI=1Yi=1 − (cid:17)IY6=J − −
For N = N = 1 and m = 0, the vortex partition function becomes
c f 1
k k
1 1 ϕ ϕ
I J
Z = dϕ − (4.12)
k k!(2πiǫ)k I ϕ ϕ ϕ ǫ
I I J
I YI=1 (cid:16)YI=1 (cid:17)IY6=J − −
This reproduces the vortex partition function of abelian k-vortex in refined
topological A-model amplitude of resolved conifold ( 1) ( 1) CP1
O − ⊕O − →
in [5]. We do not directly evaluate the contour integral of (4.11). Instead, in
the next section, we calculate the equivariant character which reproduce the
residues of (4.11) [11].
9