An Alternative Topological Field Theory of Generalized Complex Geometry
arXiv:0704.1015v3  [hep-th]  26 Aug 2007
YITP-04-15
An Alternative Topological Field Theory
of Generalized Complex Geometry
Noriaki IKEDA1∗ and Tatsuya TOKUNAGA2 †
1Department of Mathematical Sciences, Ritsumeikan University
Kusatsu, Shiga 525-8577, Japan
2Yukawa Institute for Theoretical Physics, Kyoto University
Kyoto 606-8502, Japan
October 27, 2018
Abstract
We propose a new topological field theory on generalized complex geometry in two
dimension using AKSZ formulation. Zucchini’s model is A model in the case that the
generalized complex structure depends on only a symplectic structure. Our new model
is B model in the case that the generalized complex structure depends on only a complex
structure.
∗E-mail address: ikeda@yukawa.kyoto-u.ac.jp
†E-mail address: tokunaga@yukawa.kyoto-u.ac.jp
http://arxiv.org/abs/0704.1015v3
1 Introduction
In [1][2], Zucchini has constructed a two dimensional topological sigma model on generalized
complex geometry [3] [4] [5] by the AKSZ formulation [6] (also see [7]), which is a general
geometrical framework to construct a topological sigma model by the Batalin-Vilkovisky
formalism [8]. Also, there are many recent papers [9]-[32] on this topic. Zucchini’s model is a
generalization of the Poisson sigma model and is similar to A model in [6]. However B model
looks different from the Zucchini model because B model has more fields than the Zucchini
model has.
In this paper, we propose an alternative realization of generalized complex geometry by
a topological field theory by the AKSZ formulation. Our model is similar to B model, not
A model in the sense of AKSZ, as a worldsheet action of a topological sigma model with
superifields on a supermaifold. Our model is the first candidate which naturally includes B
model and may be related to a topological string theory on generalized Calabi-Yau geometry
[23] [24].
First we construct a three dimensional topological field theory of generalized complex
geometry with a nontrivial 3-form H , which has Zucchini’s model as a boundary action. This
topological field theory is a reconstruction by the AKSZ formulation of the model proposed
in the paper [33]. Next after a dimensional reduction, we derive a topological field theory
of generalized complex geometry in two dimensions from three dimensions. We can see that
this model has a generalized complex structure as a consistency condition of a topological
BV action. If the generalized complex structure is a complex structure, our model has one
parameter marginal deformation of the model without changing a complex structure, and
reduces to B model in a limit of the deformation. If the generalized complex structure is a
symplectic structure, our model becomes a new 2D topological sigma model with a symplectic
structure.
The paper is organized as follows. In section 2, the AKSZ actions of A model, B model
and the Zucchini model are reviewed. In section 3, three dimensional topological field the-
ory of generalized complex geometry is rederived in the AKSZ formulation. In section 4, we
derive a two dimensional topological field theory of generalized complex geometry and check
its properties. In section 5, our model is reduced in two special ways. Section 6 includes
conclusion and discussion. In appendix A, a generalized complex structure is briefly summa-
rized. In appendix B, the AKSZ formulation of the Batalin-Vilkovisky formalism in general
n dimensions is reviewed.
2 A Model, B Model and Zucchini Model
In this section, we review the AKSZ formulation of topological sigma models such as A model,
B model and the Zucchini model.
2.1 A Model and B Model
A model and B model are defined on the graded bundle
T ∗[1]M ⊕ (T [1]M ⊕ T ∗[0]M) . (1)
Here E = TM , n = 2 and p ≥ 1 in the general graded bundles (100). Local coordinates
are written by superfields on this bundle: (φi,B1i,A1
i,B0,i). φ
i is a map φi : ΠTΣ →
M , and B1i is a basis of sections of ΠT
∗Σ ⊗ φ∗(T ∗[1]M). A1i is a basis of sections of
ΠT ∗Σ ⊗ φ∗(T [1]M), and B0i is a basis of sections of ΠT ∗Σ ⊗ φ∗(T ∗[0]M). The antibracket
on this bundle (1) is
(F,G) ≡ F
∂B1,i
∂B1,i
∂B0,i
∂B0,i
G (2)
from (102).
The A model action with a symplectic form Qij in [34] is
SAQ =
Qij(φ)dφ
idφj , (3)
where d is a superderivative d = θµ∂µ. where the integration
ΠTΣ means the integration on the
supermanifold,
ΠTΣ d
2θd2σ. This action is consistent if and only if the 2-formQ = 1
Qijdφ
satisfies the symplectic condition dMQ = 0, namely
∂kQij + ∂iQjk + ∂jQki = 0. (4)
This model is rewritten by the AKSZ formulation on the graded bundle T ∗[1]M⊕(T [1]M ⊕ T ∗[0]M).
We introduce Ai1, B0i and B1i as auxiliary fields, and rewrite the action using the first order
formalism. The action in AKSZ formulation is
SAQ =
B1idφ
i −B0idAi1 −B1iAi1 +
Qij(φ)A
. (5)
We can check that (SAQ, SAQ) = 0 if and only if the 2-formQ satisfies the symplectic condition
Also, A model action with a Poisson bivector P ij is
SAP =
B1idφ
i −B0idAi1 +
P ij(φ)B1iB1j , (6)
which is called the Poisson sigma model [35][36]. The consistency condition (SAP , SAP ) = 0
is satisfied if and only if P ij is a Poisson bivector field i.e.
P il∂lP
jk + P jl∂lP
ki + P kl∂lP
ij = 0. (7)
B model with a complex structure J ij is
B1idφ
i −B0idAi1 + J ij(φ)B1iA
∂J ik
(φ)B0iA
1, (8)
which is a covariant form of B model action in [6], but is different from the action in [37]. We
can check that the consistency condition (SB, SB) = 0 is satisfied if and only if J
j satisfies
the integrability condition for the complex structure
J li∂lJ
j − J lj∂lJki − Jkl∂iJ lj + Jkl∂jJ li = 0. (9)
2.2 Zucchini Model
In [1], Zucchini has proposed a topological sigma model with a generalized complex structure
on a two dimensional worldsheet Σ. Although he called this model ”the Hitchin sigma model”,
here we call it the Zucchini model.
First we consider H = 0 case. The action of the Zucchini’s model is
B1idφ
P ij(φ)B1iB1j +
Qij(φ)dφ
idφj + J ij(φ)B1idφ
j . (10)
The master equation (SZ , SZ) = 0 is satisfied if P , Q and J satisfy the conditions for a
generalized complex structure (73), (74) and (75). We can see that the Batalin-Vilkovisky
structure of this model defines a generalized complex structure on a target manifold M . If
J ij = 0 in the action (10), the action reduces to the summation of two realizations of A model
such that (3) + (6). However, if P ij = Qij = 0, the action (10) does not reduce to the B
model action (8). So we can not easily see whether the Zucchini model can be related to B
model.
Also, we can consider b-transformation property of this model [1]. The b-transformation
is defined by (77), (83) and
= φi,
B̂1i = B1i + bijdφ
j . (11)
The b-transformation produces the b field term such as
ŜZ = SZ −
bijdφ
idφj. (12)
This suggests that the Zucchini action with H 6= 0 should have a Wess-Zumino term
SZH =
B1idφ
P ijB1iB1j +
Qijdφ
idφj + J ijB1idφ
Hijkdφ
idφjdφk,(13)
where X is a three dimensional worldvolume such that Σ = ∂X is a two dimensional boundary
of X .
3 3D Topological Field Theory with Generalized Com-
plex Structures from 2D Zucchini Model
In this section, we review a three dimensional topological field theory with a generalized
complex structure from the Zucchini model in two dimensions. Here this topological field
theory is redefined by the AKSZ formulation, which was not explicitly written in [33].
3.1 H = 0 case
Let X be a three dimensional worldvolume with a coordinate (σM) for M = 1, 2, 3, and
Σ = ∂X be a two dimensional boundary of X . First we consider H = 0 case.
By using the Stokes theorem, we can see the action (10) as
B1idφ
P ijB1iB1j +
Qijdφ
idφj + J ijB1idφ
dB1idφ
∂P ij
dφkB1iB1j + P
ijdB1iB1j +
dφkdφidφj
∂J ij
dφkB1idφ
j + J ijdB1idφ
j , (14)
where d is a three dimensional derivative d = θM∂M . φ
i and B1i can be extended to those
on X such that φi : ΠTX → M and B1i is a basis of sections of ΠT ∗X ⊗ φ∗(T ∗[1]M). We
introduce a superfield Ai1 with total degree one, which is a basis of a section of ΠT
∗(T [1]M) such that Ai1 = dφ
i, and a superfield B2i with total degree two, which is a basis
of a section of ΠT ∗X ⊗ φ∗(T ∗[2]M) such that B2i = −dB1i. Moreover, we introduce two
Lagrange multiplier fields Y 2i and Z
1 in order to realize two equations such as A
1 = dφ
and B2i = −dB1i by the equations of motion. The superfield Y 2i with total degree two is a
section of ΠT ∗X ⊗ φ∗(T ∗[2]M), and the superfield Zi1 with total degree one is a section of
ΠT ∗X ⊗ φ∗(T [1]M). The 3D action (14) is equivalent to
−B2iAi1 +
∂P ij
1B1iB1j − P ijB2iB1j +
∂J ij
1B1iA
1 − J ijB2iA
1 + (A
1 − dφ
i)Y 2i + (B2i + dB1i)Z
1. (15)
We define Y ′2i = Y 2i − 12B2i and Z
1 = Z
1 − 12A
1. The action (15) is rewritten as
SZ = Sa + Sb + total derivative ;
−Y ′2idφi + dB1iZ ′i1 + Y ′2iAi1 +B2iZ ′i1 ,
B2idφ
B1idA
1 − J ijB2iA
1 − P ijB2iB1j +
1B1k +
∂P jk
1B1jB1k. (16)
where Sa is independent of a generalized complex structure. Sb can be written as
〈0 +B2, d(φ+ 0)〉+
〈A1 +B1, d(A1 +B1)〉
−〈0 +B2,J (A1 +B1)〉 −
〈A1 +B1,Ai1
(A1 +B1)〉+ total derivative,(17)
which is analogical with the B model action (8).
The antibracket (P-structure) on X , which is induced from the antibracket (2) on Σ, for
i, B2,i, A1
i and B1,i is given by the antibracket (102) in n = 3. In order to define the
antibrackets for Y ′2i and Z
1 , we introduce two antibracket conjugate fields X
i, which are
maps from ΠTX to M , and V 1i, which are sections of ΠT
∗X ⊗ φ∗(T ∗[1]M). The model
is defined on the graded bundle of the direct product of T ∗[2]M ⊕ (T [1]M ⊕ T ∗[1]M) and
(T [0]M⊕T ∗[2]M)⊕ (T [1]M ⊕ T ∗[1]M). The second bundle is represented by auxiliary fields.
The antibracket is
(F,G) ≡ F
∂B2,i
∂B2,i
∂B1,i
∂B1,i
∂Y ′2,i
∂Y ′2,i
∂Z1′i
∂V 1,i
∂V 1,i
∂Z1′i
G. (18)
We can check that SZ satisfies the master equation (SZ , SZ) = 0 if J , P and Q are components
of the generalized complex structure (72). We can take the proper boundary conditions
Σ = ∂X ;
//|∂X = 0,B2i//|∂X = 0,Y ′2i//|∂X = 0,Z1
//|∂X = 0, (19)
such that the total derivative terms on the master equation (SZ , SZ) vanish. Here // means
that we take the components which are tangent to the boundary ∂X .
Also, because (Sa, Sa) = (Sa, Sb) = 0, Sb satisfies the master equation (Sb, Sb) = 0
Aijk = Bijk = Cijk = 0,
∂iDjkl + (ijkl cyclic) = 0, (20)
where Aijk, Bijk, Cijk and Djkl are defined in Appendix A. Therefore, we can see Sb as a three
dimensional AKSZ action with generalized complex structure. We discuss why the condition
is not Djkl = 0 but ∂iDjkl + (ijkl cyclic) = 0 in subsection 3.3.
We call Sb three dimensional generalized complex sigma model.
We consider 3D b-transformation property from the 2D b-transformations (11) and the
conditions Ai1 = dφ and B2i = −dB1i. 3D b-transformations are
= φi,
= Ai1,
B̂1i = B1i + bijA
B̂2i = B2i − d(bijAj1),
2i = Y
bijdA
−J lk
1 − P lk
1B1k + d(J
kbliA
1) + d(P
lkbliB1k),
= Z ′i1 . (21)
We can see that 3D action (16) is invariant under the b-transformation such that
ŜZ = SZ . (22)
3.2 H 6= 0 case I :Action induced from the Zucchini model
In the similar way, we can consider the case of a twisted generalized complex structure with
H 6= 0. From the Zucchini model with H 6= 0 (13), a three dimensional action is derived as
SZH = Sa + SHb + total derivative ;
−Y ′2idφ
i + dB1iZ
1 + Y
i +B2iZ
SHb =
B2idφ
B1idA
1 − J ijB2iA
1 − P ijB2iB1j +
Hijk +
1B1k +
∂P jk
1B1jB1k. (23)
This action (23) satisfies the master equation (SZH, SZH) = 0, if J , P , Q and H are compo-
nents of a twisted generalized complex structure (87). However, this action is not b-invariant
under the b-transformation (21), (77) and (83). The action (23) transforms under the b-
transformation as
ŜZH = SZH −
1 = SZH −
(dMb)[ijk]A
1, (24)
which has been expected from b-transformation property (12) in the two dimensional model.
Since H is closed, from the Poincaré Lemma, we can locally write H with a 2-form q on
M such as
Hijk =
. (25)
1 term in (23) can be absorbed to Q by a local b-transformation qij = bij in
the action (23), and we obtain just the H = 0 action (16). In other words, the H terms in
(23) are consistent up to H-exact terms as a global theory, and this model is meaningful only
as a cohomology class in H3(M). It is a gerbe gauge transformation dependence [1].
If we set Qij = J
j = 0 in (13), we obtain the AKSZ formulation of the WZ-Poisson sigma
model [38]:
SWZP =
B1idφ
P ijB1iB1j +
Hjkldφ
idφjdφk. (26)
From (23), the 3D topological sigma model equivalent to (26) is
SWZP = Sa + SWZPb ;
−Y ′2idφ
i + dB1iZ
1 + Y
1 +B2iZ
SWZPb =
B2idφ
dB1iA
1 − P ijB2iB1j +
HijkA
∂P jk
1B1jB1k. (27)
3.3 H 6= 0 case II : b-invariant action
We can construct a b-invariant action with H 6= 0 in three dimensional manifold X . We
introduce other H terms.
SI = Sa + SIb ;
−Y ′2idφ
i + dB1iZ
1 + Y
1 +B2iZ
SIb =
B2idφ
B1idA
1 − J ijB2iA
1 − P ijB2iB1j +
J liHjkl +
−P klHijl −
1B1k +
∂P jk
1B1jB1k. (28)
SI satisfies the master equation (SI , SI) = 0 under the antibracket (18) if and only if J , P ,
Q and H are components of a twisted generalized complex structure. Namely, the master
equation (SI , SI) = 0 gives
AHijk = BHijk = CHijk = 0,
∂iDHjkl + (ijkl cyclic) = 0, (29)
where AHijk, BHijk, CHijk and DHjkl are defined in Appendix A. The integrability condition
is not DHijk = 0 but ∂iDHjkl + (ijkl cyclic) = 0 because the action SI is b-transformation
invariant, H ijk has b-transformation ambiguity by (83), and H is defined as a cohomology
class in H3(M) in a twisted generalized complex structure.
Since SIa does not depend on a twisted generalized complex structure, (SIb, SIb) = 0 is
satisfied under the condition (29). We can introduce the coupling constants by redefining Y ′2i
and Z1
′i to g1Y
2i and g2Z
1 . If we take the limits that g1 → 0 and g2 → 0, then SI → SIb
and a twisted generalized complex structure does not change. We call this model SIb a three
dimensional twisted generalized complex sigma model.
We can change the b-transformation so that the action SI is invariant, though the action
(28) is not invariant under the original b-transformation (21). The b-transformations for B2i
and Y ′2i are changed to
B̂2i = B2i −
2i = Y
1 − bijdZ
1, (30)
and b-transformations for the other fields are the same as (21). Then we can check ŜI = SI
after short calculation.
4 2D Topological Field Theory of Generalized Complex
Geometry
In this section, we propose a new two dimensional topological field theory of generalized
complex geometry using the 3D topological field theory. First, only a part of the 3D BV
formalism action is dimensionally reducted to in two dimension, and next this is modified
in the 2D BV formalism such that the master equations determine just generalized complex
structures. One important reason to have to take this unusual way is that generally, master
equations of BV formalisms are not kept by a dimensional reduction.
4.1 H = 0
First we consider the H = 0 case. We consider a dimensional reduction, which can keep a
generalized complex structure, from a three dimensional worldvolume X to a two dimensional
manifold Σ′. X is compactified to Σ′ × S1. Then ΠTX is compactified to ΠTΣ′ × ΠTS1. It
should be noticed that Σ′ is generally a different manifold from Σ.
Here, we take X = Σ×R+, where Σ has a local coordinate (σ1, σ2) and R+ = [0,∞) has
a local coordinate (σ3). The second component (σ2) is compactified such that Σ′ = L×R+,
whose local coordinate is (σ1, σ3), where L is a manifold in one dimension. We formulate
the dimensional reduction from a general three dimensional manifold X to a general two
dimensional manifold Σ′. Here we ignore Kaluza-Klein modes and consider only massless
sectors, because we will see that the consistent BV action can be constructed in two di-
mension even if these KK modes are omitted. It is not our purpose that we derive the
two dimensional model which is completely equivalent to the 3D topological field theory.
The target graded bundle for the three dimensional model, T ∗[2]M ⊕ (T [1]M ⊕ T ∗[1]M), re-
duces to the graded bundle for the two dimensional model, (T ∗[1]M ⊕ (T [−1]M ⊕ T ∗[2]M))⊕
((T [0]M ⊕ T ∗[1]M)⊕ (T [1]M ⊕ T ∗[0]M)). Under the dimensional reduction (σ1, σ2, σ3) →
(σ1, σ3), the fields are reduced as follows.
i(σ1, σ2, σ3) = φ̃
(σ1, σ3) + θ2φ̃−1
(σ1, σ3),
1, σ2, σ3) = Ã1
(σ1, σ3) + θ2α̃0
i(σ1, σ3),
B1i(σ
1, σ2, σ3) = B̃1i(σ
1, σ3) + θ2β̃0i(σ
1, σ3),
B2i(σ
1, σ2, σ3) = B̃2i(σ
1, σ3) + θ2β̃1i(σ
1, σ3), (31)
where φ̃−1
has the total degree −1, φ̃
, α̃0
i and β̃0i have the total degree 0, Ã1
, B̃1i and
β̃1i have the total degree 1, and B̃2i has the total degree 2. All these superfields do not
depend on θ2.
The antibracket induced from three dimensions is
(F,G) ≡ F
∂β̃1i
∂β̃1i
∂φ̃−1
∂B̃2i
∂B̃2i
∂φ̃−1
∂β̃0i
∂β̃0i
∂B̃1i
∂B̃1i
G. (32)
We take a three dimensional AKSZ action Sb (16) with a generalized complex structure. The
existence of the negative total degree superfield φ̃−1
complexifies the dimensional reduction
in the AKSZ formulation. Generally in [39], it is known that even if we substitute (31) to
(16), we do not obtain the correct AKSZ action in two dimensions, and we need more φ̃−1
terms.
In order to derive the correct AKSZ action, first we should consider the dimensional
reduction via the non-BV formalism. The superfields are expanded by the ghost numbers to
i = φ(0)i + φ(−1)i + φ(−2)i + φ(−3)i,
B1i = B
1,i +B
1,i +B
1,i +B
1,i ,
i = A
1 + A
1 + A
(−1)i
1 + A
(−2)i
B2,i = B
2,i +B
2,i +B
2,i +B
2,i , (33)
where φ(−1)i ≡ θMφ(−1)iM , etc. After setting all the antifield with negative ghost numbers to
zero, the following non-BV action is
2i dφ
(0)i +
1i dA
1 − J ijB
1 − P ijB
∂φ(0)i
(φ(0)i)A
∂φ(0)i
∂φ(0)j
(φ(0)i)A
∂P jk
∂φ(0)i
(φ(0)i)A
1k . (34)
Since by the dimensional reduction, the fields reduce to
φ(0)i(σ1, σ2, σ3) = φ̃
(σ1, σ2),
1, σ2, σ3) = Ã1
(σ1, σ3) + θ2α̃0
(0)i(σ1, σ3),
1i (σ
1, σ2, σ3) = B̃1
1, σ3) + θ2β̃0
1, σ3),
2i (σ
1, σ2, σ3) = B̃2
1, σ3) + θ2β̃1
1, σ3), (35)
the action (34) reduces to
i dφ̃
+ B̃1
i dα̃0
(0)i + Ã1
− J ijÃ1
i + P
ijB̃1
i β̃1
 α̃0
(0)k +
− ∂J
 β̃0
 Ã1
 α̃0
(0)j − ∂P
 Ã1
∂P jk
 B̃1
j B̃1
J ijα̃0
(0)j + P ijβ̃0
i , (36)
up to total derivative terms. Therefore the action S
R of a 2D topological field theory is
R = S
0 + S
i dφ̃
+ B̃1
i dα̃0
(0)i + Ã1
−J ijÃ1
i + P
ijB̃1
i β̃1
 α̃0
(0)k +
− ∂J
 β̃0
 Ã1
 α̃0
(0)j − ∂P
 Ã1
∂P jk
 B̃1
j B̃1
J ijα̃0
(0)j + P ijβ̃0
i . (37)
Next we formulate the action SR by the AKSZ formulation. We define SR = S0 + S1
where S0 and S1 are AKSZ actions for S
0 and S
1 , respectively. S0 is easily derived after
substituting (31) to (16);
β̃1idφ̃
− B̃2idφ̃−1
+ B̃1idα̃0
i + Ã1
dβ̃0i
up to total derivative terms. The condition (S1, S1) = 0 comes from (38) and (SR, SR) = 0.
We introduce an negative total degree, which is defined as one for φ̃−1, and zero for the other
fields. We can expand S1 for the negative total degree such as S1 =
p=0 S
1 , where
i1 · · · φ̃−1
ipL[p]i1···ip(φ̃, Ã1
, α̃0, B̃1i, β̃0i, B̃2i, β̃1i) (39)
are the negative total degree p terms. Therefore
SR = S0 +
1 . (40)
Here we write the first two actions S
1 and S
1 with the negative total degree zero and one
by substituting (31) to (16),
−J ijÃ1
β̃1i + P
B̃1iβ̃1j
 α̃0
 β̃0k
 Ã1
 α̃0
j − ∂P
 Ã1
B̃1k +
∂P jk
 B̃1jB̃1k
J ijα̃0
j + P ijβ̃0j
B̃2i, (41)
∂J ij
B̃2iÃ1
∂P ij
B̃2iB̃1j −
∂2Qjk
 Ã1
B̃1k −
∂2P jk
B̃1jB̃1k
. (42)
1 for p > 1 are recursively derived from the master equation (S1, S1) =
p=0{(S1, S1)}[p] =
0. It should be noticed that since a target space M has finite dimensions, S
1 is nonzero
for only a finite number of p. This action is a special case of a nonlinear gauge theory with
2-forms (a generalization of the Poisson sigma model) analyzed in the paper [39][40][41].
4.2 H 6= 0
Here we consider H 6= 0 case. A 2D topological field theory of twisted generalized complex
geometry is derived in a similar way in subsection 4.1 from H-terms in section 3.2:
SR = S0 +
1 ; (43)
−J ijÃ1
β̃1i + P
B̃1iβ̃1j
3Hijk +
 α̃0
 β̃0k
 Ã1
 α̃0
j − ∂P
 Ã1
B̃1k +
∂P jk
 B̃1jB̃1k
J ijα̃0
j + P ijβ̃0j
B̃2i, (44)
∂J ij
B̃2iÃ1
∂P ij
B̃2iB̃1j −
Hijk +
 Ã1
 Ã1
B̃1k −
∂2P jk
B̃1jB̃1k
, (45)
and S
1 for p > 1 are recursively derived from (SR, SR).
Also, from b-invariant H-terms in section 3.3,
SR = S0 +
1 ; (46)
−J ijÃ1
β̃1i + P
B̃1iβ̃1j
3J liHjkl +
 α̃0
−P klHjkl −
 β̃0k
 Ã1
P klHjkl +
 α̃0
j − ∂P
 Ã1
B̃1k +
∂P jk
 B̃1jB̃1k
J ijα̃0
j + P ijβ̃0j
B̃2i, (47)
∂J ij
B̃2iÃ1
∂P ij
B̃2iB̃1j −
JmiHjkm +
 Ã1
−P kmHjkm −
 Ã1
B̃1k −
∂2P jk
B̃1jB̃1k
, (48)
and S
1 for p > 1 are recursively derived from (SR, SR).
5 Two Special Reductions to Complex Geometry and
Symplectic Geometry
In this section, we consider two special reductions related to complex geometry and of sym-
plectic geometry.
5.1 Complex geometry
First we consider our model in complex geometry, which is the case that P = Q = H = 0 in
the action (40). We redefine superfields as
, φ̃−1
= λφ̃′
, α̃0
i = λα̃′
B̃1i = λB̃
, β̃0i = −β̃′0i,
B̃2i = λB̃
, β̃1i =
, (49)
where λ is a constant. After this redefinition, the action (40) is
SR = S0 +
1 , (50)
i − Ã′
+ B̃′
, (51)
J ijβ̃′1iÃ
− J ijα̃′0
 , (52)
λφ̃−1
∂J ij
∂2Jkj
, (53)
and S
1 has at least the higher order of λ than λ
p because φ̃−1
= λφ̃′
. We can take the
limit λ −→ 0 with preserving the complex structure. S [p]1 for p > 0 reduces to zero, and the
2D action is
SRJ =
β̃1idφ̃
− Ã1
dβ̃0i + J
jβ̃1iÃ1
∂J ik
β̃0iÃ1
. (54)
This action is nothing but the B model action (8) up to a total derivative and the all over
factor 1
, which depends on only J ij. The master equation (SbJ , SbJ) = 0 impose the condition
that J ij is a complex structure.
We make a comment about the difference between the action (50) with a finite λ and the
B model action (54) with λ → 0. Following the well-known method in [6], we can see that
the topological string theory has to be deformed by the other terms in (50) than in the B
model. In the calculation of [6], we may locally take the complex structure as a constant, and
the kinetic terms (51) and two terms in (52) without the derivatives of J ij are only different
parts from those in the B model. Here it should be noted that although these deformed parts
may seem to decouple to the B model part, the interactions between them can come from the
non-constant metric. These deformed parts couple to only the metric on the bosonic space
of φ̃
, which is independent of φ̃−1
, because there is no metric with fermionic indices on
the fermionic space of φ̃−1
. So these deformed parts can be seen as a topological theory
with only B field-like couplings on the fermionic space of φ̃−1
. Physically, we may assume
that there is no topological information along fermionic directions, although this situation
with no metric is special. Therefore in this assumption, we can see that our action (50) is
equivalent to topological string theory, called topological B model. As a future work, it would
be interesting to check this equivalence more carefully.
5.2 Symplectic geometry
Next we consider our model in symplectic geometry, which is the case that J = H = 0 in the
action (40). We redefine superfields as
, φ̃−1
= µφ̃′
= µÃ′
, α̃0
i = α̃′
B̃1i =
, β̃0i = −µβ̃′0i,
B̃2i = µB̃
, β̃1i =
, (55)
where µ is a constant. After this redefinition, the 2D action (40) reduces to
SR = S0 +
+ B̃′
− Ã′
P ijB̃′
∂P jk
 (57)
 α̃′
j − 1
∂P jk
+ P ijβ̃′
µφ̃−1
∂P ij
B̃2iB̃1j −
∂2Qjk
∂2P jk
,(58)
and S
1 is at least the higher order of λ than λ
p because φ̃−1
= µφ̃′
After taking the limit µ −→ 0 with preserving the symplectic structure, S [p]1 for p > 0
reduces to zero, and the 2D action is
SRP =
β̃1idφ̃
+ B̃1idα̃0
i + P ijB̃1iβ̃1j +
∂P jk
B̃1jB̃1k. (59)
The BV condition (SbP , SbP ) = 0 is satisfied if and only if P
ij is a Poisson structure (the
inverse of a symplectic structure) (7). It should be noticed that although this action (59)
depends on only a symplectic structure P ij, this action is a different realization of the Poisson
structure from the A model (6), because we can also check that this model is not equivalent
to topological string theory following the similar way as in [6].
6 Conclusions and Discussion
We have constructed a topological field theory with a generalized complex structure in three
dimensions and two dimensions using the AKSZ formulation. Our model reduces to B model
in a limit if the generalized complex structure is only a complex structure, although Zucchini
model reduces to A model in the limit that the generalized complex structure is only a
symplectic structure.
It would be interesting to check that the Zucchini model and our model are equivalent to
a topological string theory with a generalized complex structure [23][24], which is constructed
from the twisted N = (2, 2) supersymmetric sigma model with a non-trivial B field.
Appendix A. Generalized Complex Structure
In this appendix A, we summarize a generalized complex structure, based on description of
section 3 in [11] and section 2 in [1].
Let M be a manifold of even dimension d with a local coordinate {φi}. We consider
the vector bundle TM ⊕ T ∗M . We denote a section as X + ξ ∈ C∞(TM ⊕ T ∗M) where
X ∈ C∞(TM) and ξ ∈ C∞(T ∗M).
TM ⊕ T ∗M is equipped with a natural indefinite metric of signature (d, d) defined by
〈X + ξ, Y + η〉 = 1
(iXη + iY ξ), (60)
for X + ξ, Y + η ∈ C∞(TM ⊕ T ∗M), where iV is an interior product with a vector field V .
In the Cartesian coordinate (∂/∂φi, dφi), The metric is written as follows:
, (61)
We define a Courant bracket on TM ⊕ T ∗M as follows:
[X + ξ, Y + η] = [X, Y ] + LXη −LY ξ −
dM(iXη − iY ξ), (62)
with X + ξ, Y + η ∈ C∞(TM ⊕T ∗M), where LV denotes Lie derivation with respect a vector
field V and dM is the exterior differential of M . This bracket is antisymmetric but do not
satisfy the Jacobi identity. We may consider a so called Dorfman bracket as follows:
(X + ξ) ◦ (Y + η) = [X, Y ] + LXη − iY dξ, (63)
which satisfies the Jacobi identity but is not antisymmetric. Antisymmetrization of a Dorfman
bracket coincides with a Courant bracket.
A generalized almost complex structure J is a section of C∞(End(TM ⊕ T ∗M)), which is
an isometry of the metric 〈 , 〉, J ∗IJ = I, and satisfies
J 2 = −1. (64)
A b-transformation is an isometry defined by
exp(b)(X + ξ) = X + ξ + iXb, (65)
where b ∈ C∞(∧2T ∗M) is a 2–form. A Courant bracket is covariant under the b-transformation
[exp(b)(X + ξ), exp(b)(Y + η)] = exp(b)[X + ξ, Y + η], (66)
if the 2–form b is closed. The b-transform of J is defined by
Ĵ = exp(−b)J exp(b). (67)
J has the ±
−1 eigenbundles because J 2 = −1, In order to divide TM ⊕ T ∗M to each
eigenbundle, we need complexification of TM ⊕ T ∗M , (TM ⊕ T ∗M)⊗ C. The projectors on
the eigenbundles are defined by
−1J ). (68)
The generalized almost complex structure J is integrable if
Π∓[Π±(X + ξ),Π±(Y + η)] = 0, (69)
for any X + ξ, Y + η ∈ C∞(TM ⊕ T ∗M), where the bracket is the Courant bracket. Then J
is called a generalized complex structure. Integrability is equivalent to the single statement
N(X + ξ, Y + η) = 0, (70)
for all X + ξ, Y + η ∈ C∞(TM ⊕ T ∗M), where N is the generalized Nijenhuis tensor defined
N(X + ξ, Y + η) = [X + ξ, Y + η]− [J (X + ξ),J (Y + η)] + J [J (X + ξ), Y + η]
+J [X + ξ,J (Y + η)]. (71)
The b-transform Ĵ of a generalized complex structure J is a generalized complex structure
if the 2–form b is closed.
We decompose a generalized almost complex structure J in coordinate form as follows
, (72)
where J,K ∈ C∞(TM ⊗ T ∗M), P ∈ C∞(∧2TM), Q ∈ C∞(∧2T ∗M).
Then the conditions J ∗IJ = I, and J 2 = −1 derive
i = −J ij
J ikJ
j + P
ikQkj + δ
j = 0,
J ikP
kj + J jkP
ki = 0,
j +QjkJ
i = 0, (73)
where
P ij + P ji = 0,
Qij +Qji = 0. (74)
The integrability condition (69) is equivalent to the following condition
Aijk = Bijk = Cijk = Dijk = 0, (75)
where
Aijk = P il∂lP jk + P jl∂lP ki + P kl∂lP ij,
Bijk = J li∂lP jk + P jl(∂iJkl − ∂lJki) + P kl∂lJ j i − J j l∂iP lk,
Cijk = J li∂lJkj − J lj∂lJki − Jkl∂iJ lj + Jkl∂jJ li
+P kl(∂lQij + ∂iQjl + ∂jQli),
Dijk = J li(∂lQjk + ∂kQlj) + J lj(∂lQki + ∂iQlk)
+J lk(∂lQij + ∂jQli)−Qjl∂iJ lk −Qkl∂jJ li −Qil∂kJ lj . (76)
Here ∂i is a differentiation with respect to φ
i. The b–transform is
Ĵ ij = J
j − P ikbkj,
P̂ ij = P ij,
Q̂ij = Qij + bikJ
j − bjkJki + P klbkiblj . (77)
where bij + bji = 0.
The usual complex structures J is embedded in generalized complex structures as the
special form
0 −tJ
. (78)
Indeed, one can check this form satisfies conditions, (73) and (75) if and only if J is a
complex structure. Similarly, the usual symplectic structures Q is obtained as the special
form of generalized complex structures
0 −Q−1
. (79)
This satisfies (73) and (75) if and only if Q is a symplectic structure, i. e. it is closed. Other
exotic examples exist. There exists manifolds which cannot support any complex or symplectic
structure, but admit generalized complex structures.
The Courant bracket on TM ⊕ T ∗M can be modified by a closed 3–form. Let H ∈
C∞(∧3T ∗M) be a closed 3–form. We define the H twisted Courant brackets by
[X + ξ, Y + η]H = [X + ξ, Y + η] + iX iYH, (80)
where X + ξ, Y + η ∈ C∞(TM ⊕ T ∗M). Under the b-transform with b a closed 2–form,
[exp(b)(X + ξ), exp(b)(Y + η)] = exp(b)[X + ξ, Y + η], (81)
holds with the brackets [ , ] replaced by [ , ]H . For a non closed b, one has
[exp(b)(X + ξ), exp(b)(Y + η)]H−dM b = exp(b)[X + ξ, Y + η]H . (82)
So, the b-transformation shifts H by the exact 3–form dMb:
Ĥ = H − dMb. (83)
One can define an H twisted generalized Nijenhuis tensor NH as follows
N(X + ξ, Y + η) = [X + ξ, Y + η]H − [J (X + ξ),J (Y + η)]H + J [J (X + ξ), Y + η]H
+J [X + ξ,J (Y + η)]H , (84)
by using the brackets [ , ]H instead of [ , ]. A generalized almost complex structure J is H
integrable if
NH(X + ξ, Y + η) = 0, (85)
for all X + ξ, Y + η ∈ C∞(TM ⊕ T ∗M). Then we call J an twisted generalized complex
structure.
The H integrability conditions is as follows:
AHijk = BHijk = CHijk = DHijk = 0, (86)
where
AHijk = Aijk,
BHijk = Bijk + P jlP kmHilm
CHijk = Cijk − J liP kmHjlm + J ljP kmHilm,
DHijk = Dijk −Hijk + J liJmjHklm + J ljJmkHilm + J lkJmiHjlm. (87)
Appendix B. AKSZ Formulation of Batalin-Vilkovisky
Formalism
In the appendix B, we review the AKSZ formulation in any dimension [42]. In order to
construct and analyze topological field theories systematically, it is useful to use Batalin-
Vilkovisky formalism. The geometric structure of the AKSZ formulation is called Batalin-
Vilkovisky Structures.
B-1. Batalin-Vilkovisky Structures on Graded Vector Bundles
Let M be a smooth manifold in d dimensions. If we consider We define a supermanifold
ΠT ∗M . Mathematically, ΠT ∗M , whose bosonic part is M , is defined as a cotangent bundle
with reversed parity of the fiber. That is, a base manifold M has a Grassman even coordinate
and the fiber of ΠT ∗M has a Grassman odd coordinate. We introduce a grading called total
degrees, which is denoted |F | for a function F . The coordinates of the base manifold have
grade zero and the coordinates of the fiber have grade one. Similarly, we can define ΠTM for
a tangent bundle TM . ΠTM is also called a supermanifold.
We must consider more general assignments for the degree of the fibers of T ∗M or TM .
For an integer p, we define T ∗[p]M , which is called a graded cotangent bundle. T ∗[p]M is a
cotangent bundle, whose fiber has the degree p. This degree is also called the total degree.
A coordinate of the bass manifold have the total degree zero and a coordinate of the fiber
have the total degree p. If p is odd, the fiber is Grassman odd, and if p is even, the fiber is
Grassman even. We define a graded tangent bundle T [p]M in the same way.
We consider a vector bundle E. A graded vector bundle E[p] is defined in the similar way.
E[p] is a vector bundle whose fiber has a shifted degree by p. Note that only the degree of
fiber is shifted, and the degree of base space is not shifted.
We consider a Poisson manifold N with a Poisson bracket {∗, ∗}. If we shift the total
degree, we can construct a graded manifold (a graded cotangent bundle or a graded vector
bundle) Ñ from N . Then a Poisson structure {∗, ∗} shifts to a graded Poisson structure
by grading of Ñ . The graded Poisson bracket is called an antibracket and denoted by (∗, ∗).
(∗, ∗) is graded symmetric and satisfies the graded Leibniz rule and the graded Jacobi identity
with respect to grading of the manifold. The antibracket (∗, ∗) with the total degree −n + 1
satisfies the following identities:
(F,G) = −(−1)(|F |+1−n)(|G|+1−n)(G,F ),
(F,GH) = (F,G)H + (−1)(|F |+1−n)|G|G(F,H),
(FG,H) = F (G,H) + (−1)|G|(|H|+1−n)(F,H)G,
(−1)(|F |+1−n)(|H|+1−n)(F, (G,H)) + cyclic permutations = 0, (88)
where F,G and H are functions on Ñ , and |F |, |G| and |H| are total degrees of the functions,
respectively. The graded Poisson structure is also called P-structure. If n = 1, the antibracket
is equivalent to the Schouten bracket. For higher n, the antibracket is equivalent to the Loday
bracket [43] with the degree −n + 1.
Typical examples of Poisson manifold N are a cotangent bundle T ∗M and a vector bundle
E ⊕ E∗. First we consider a cotangent bundle T ∗M . Since T ∗M has a natural symplectic
structure, we can define a Poisson bracket induced from the symplectic structure. If we take
a local coordinate φi on M and a local coordinate Bi of the fiber, we can define a Poisson
bracket as follows:
{F,G} ≡ F
G, (89)
where F and G are functions on T ∗M , and
∂ /∂ϕ and
∂ /∂ϕ are the right and left differen-
tiations with respect to ϕ, respectively. Here we shift the degree of fiber by p, i.e. the space
T ∗[p]M . Then a Poisson structure shifts to a graded Poisson structure. The corresponding
graded Poisson bracket is called antibracket, (∗, ∗). Let φi be a local coordinate of M and
Bn−1,i a basis of the fiber of T
∗[p]M . The antibracket (∗, ∗) on a cotangent bundle T ∗[p]M
is expressed as:
(F,G) ≡ F
∂Bp,i
∂Bp,i
G. (90)
The total degree of the antibracket (∗, ∗) is −p. This antibracket satisfies the property (88)
for −p = −n + 1.
Next, we consider a vector bundle E ⊕ E∗. There is a natural Poisson structure on the
fiber of E ⊕ E∗ induced from a paring of E and E∗. If we take a local coordinate Aa on the
fiber of E and Ba on the fiber of E
∗, we can define
{F,G} ≡ F
G, (91)
where F and G are functions on E ⊕ E∗. We shift the degrees of fibers of E and E∗ like
E[p]⊕E∗[q], where p and q are positive integers. The Poisson structure changes to a graded
Poisson structure (∗, ∗). Let Apa be a basis of the fiber of E[p] and Bq,a a basis of the fiber
of E∗[q]. The antibracket is represented as
(F,G) ≡ F
∂Bq,a
G− (−1)pqF
∂Bq,a
G. (92)
The total degree of the antibracket (∗, ∗) is −p − q. This antibracket satisfies the property
(88) for −p− q = −n + 1.
We define a Q-structure. A Q-structure is a function S on a graded manifold Ñ which
satisfies the classical master equation (S, S) = 0. S is called a Batalin-Vilkovisky action. We
require that S satisfy the compatibility condition
S(F,G) = (SF ,G) + (−1)|F |+1(F, SG), (93)
where F andG are arbitrary functions. (S, F ) = δF generates an infinitesimal transformation,
which is a BRST transformation, which coincides with the gauge transformation of the theory.
The AKSZ formulation of the Batalin-Vilkovisky formalism is defined as a P-structure and
a Q-structure on a graded manifold.
B-2. Batalin-Vilkovisky Structures of Topological Sigma Models
In this subsection, we explain Batalin-Vilkovisky structures of topological sigma models. Let
X be a base manifold in n dimensions, with or without boundary, and M be a target manifold
in d dimensions. We denote φ a smooth map from X to M .
We consider a supermanifold ΠTX , whose bosonic part is X . ΠTX is defined as a tangent
bundle with reversed parity of the fiber. We take a local coordinate of ΠTX , (σµ, θµ), where σµ
is a coordinate on the base space and θµ is a super coordinate on the fiber and µ = 1, 2, · · · , n.
We extend a smooth function φ to a function on the supermanifold φ : ΠTX → M . φ is
called a superfield and an element of ΠT ∗X ⊗M . We introduce a new non-negative integer
grading on ΠT ∗X . A coordinate σµ on a base manifold has zero and a coordinate θµ on the
fiber has one. This grading is called the form degree. We denote degF the form degree of the
function F . The total degree defined in the previous section is a grading with respect to M ,
on the other hand The form degree is a grading with respect to X . We define a ghost number
ghF such that ghF = |F | − degF . W assign the ghost numbers of σµ and θµ zero. Thus σµ
has the total degree zero and θµ has total degree one.
We consider a P-structure on T ∗[p]M . We take p = n−1 to construct a Batalin-Vilkovisky
structure in a topological sigma model on a general n dimensional worldvolume. We consider
T ∗[n− 1]M for an n-dimensional base manifold X . Let a superfield φi be local a coordinate
of ΠT ∗X ⊗M , where i, j, k, · · · are indices of the local coordinate on M . Let a superfield
Bn−1,i be a basis of sections of ΠT
∗X ⊗φ∗(T ∗[n− 1]M). Expansions to component fields of
the superfields are the following:
i = φ(0)i + θµ1φ(−1)iµ1 +
θµ1θµ2φ(−2)iµ1µ2 + · · ·+
θµ1 · · · θµnφ(−n)iµ1···µn , (94)
Bn−1,i = B
(n−1)
n−1,i + · · ·+
(n− 1)!
θµ1 · · · θµn−1B(0)µ1···µn−1n−1,i +
θµ1 · · · θµnB(−1)µ1···µnn−1,i,
where (p) is the ghost number of the component field.
From (90) in the previous subsection, we define an antibracket (∗, ∗) on a cotangent bundle
T ∗[n− 1]M as
(F,G) ≡ F
∂Bn−1,i
∂Bn−1,i
G, (95)
where F and G are functions of φi and Bn−1,i. The total degree of the antibracket is −n+1.
If F and G are functionals of φi and Bn−1,i, we understand an antibracket is defined as
(F,G) ≡
∂Bn−1,i
∂Bn−1,i
G, (96)
where the integration
ΠTX means the integration on the supermanifold,
ΠTX d
nθdnσ. Through
this article, we always understand an antibracket on two functionals in a similar manner and
abbreviate this notation.
Next we consider a P-structure on E ⊕ E∗. In a topological sigma model in n dimension
worldvolume, we assign the total degree of p and q such that p+ q = n− 1. The total graded
bundle is E[p] ⊕ E∗[n − p − 1], where −n + 1 ≤ p ≤ n − 1, p 6= 0. Let Apap be a basis of
sections of ΠT ∗X ⊗φ∗(E[p]) and Bn−p−1,ap a basis of the fiber of ΠT ∗X ⊗φ∗(E∗[n− p− 1]).
Expansions to component fields of the superfields are
ap = A(p)app + θ
µ1A(p−1)apµ1p + · · ·++
(p− 1)!
θµ1 · · · θµ(p−1)A(0)apµ1···µ(p−1)p
+ · · ·+ 1
θµ1 · · · θµn , A(−n+p)apµ1···µnp (97)
Bn−p−1,ap = B
(n−p−1)
n−p−1,ap
+ θµ1B
(n−p−2)
µ1n−p−1,ap
+ · · ·+ 1
(n− p− 1)!
θµ1 · · · θµ(n−p−1)B(0)µ1···µ(n−p−1)n−p−1,ap
+ · · ·+ 1
θµ1 · · · θµnB(−p−1)µ1···µnn−p−1,ap,
From (92), we define the antibracket as
(F,G) ≡ F
∂Apap
∂Bn−p−1,ap
G− (−1)npF
∂Bn−p−1,ap
∂Apap
G. (98)
We need to consider various grading assignments for E ⊕ E∗, because each assignment
induces different Batalin-Vilkovisky structures. In order to consider all independent assign-
ments, we define the following bundle. Let Ep be series of vector bundles, where −n + 1 ≤
p ≤ n− 1. We consider a direct sum of each bundle Ep[p] :
p=−n+1,p 6=0
Ep[p], (99)
and we can define a P-structure on the graded vector bundle
T ∗[n− 1]M ⊕
p=−n+1,p 6=0
Ep[p]⊕ E∗p [n− p− 1]
 , (100)
which is isomorphic to the graded bundle
T ∗[n− 1]
p=−n+1,p 6=0
Ep[p]
 . (101)
as a sum of (95) and (98):
(F,G) ≡
p=−n+1
∂Apap
∂Bn−p−1 ap
G− (−1)npF
∂Bn−p−1 ap
∂Apap
G. (102)
where A0
a0 = φi, that is p = 0 component is the antibracket (95) on the graded cotangent
bundle T ∗[n− 1]M . Note that all terms of the antibracket have the total degree −n+1, and
we can confirm that the antibracket (102) satisfies the identity (88).
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