An Alternative Topological Field Theory of Generalized Complex Geometry
Noriaki Ikeda, Tatsuya Tokunaga
TL;DR
This paper introduces a novel topological field theory based on generalized complex geometry, extending existing models by utilizing AKSZ formulation to unify symplectic and complex structures.
Contribution
It presents a new topological field theory that generalizes Zucchini's model, incorporating both symplectic and complex structures within the AKSZ framework.
Findings
New topological field theory for generalized complex geometry
Unifies A and B models as special cases
Provides a framework for future geometric and physical applications
Abstract
We propose a new topological field theory on generalized complex geometry in two dimension using AKSZ formulation. Zucchini's model is model in the case that the generalized complex structuredepends on only a symplectic structure. Our new model is model in the case that the generalized complex structure depends on only a complex structure.
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YITP-04-15
**An Alternative Topological Field Theory
of Generalized Complex Geometry **
Noriaki IKEDA1***E-mail address: [email protected] and Tatsuya TOKUNAGA2 †††E-mail address: [email protected]
1*Department of Mathematical Sciences, Ritsumeikan University
Kusatsu, Shiga 525-8577, Japan
2Yukawa Institute for Theoretical Physics, Kyoto University
Kyoto 606-8502, Japan
We propose a new topological field theory on generalized complex geometry in two dimension using AKSZ formulation. Zucchini’s model is model in the case that the generalized complex structure depends on only a symplectic structure. Our new model is model in the case that the generalized complex structure depends on only a complex structure.
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 -form , 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 theory 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 summarized. In appendix B, the AKSZ formulation of the Batalin-Vilkovisky formalism in general dimensions is reviewed.
2 Model, Model and Zucchini Model
In this section, we review the AKSZ formulation of topological sigma models such as model, model and the Zucchini model.
2.1 Model and Model
model and model are defined on the graded bundle
[TABLE]
Here , and in the general graded bundles (100). Local coordinates are written by superfields on this bundle: (\mbox{\boldmath\phi}^{i},\mbox{\boldmathB}_{1i},\mbox{\boldmathA}_{1}{}^{i},\mbox{\boldmathB}_{0,i}). \mbox{\boldmath\phi}^{i} is a map \mbox{\boldmath\phi}^{i}:\Pi T\Sigma\rightarrow M, and \mbox{\boldmathB}_{1i} is a basis of sections of \Pi T^{*}\Sigma\otimes\mbox{\boldmath\phi}^{*}(T^{*}[1]M). \mbox{\boldmathA}_{1}{}^{i} is a basis of sections of \Pi T^{*}\Sigma\otimes\mbox{\boldmath\phi}^{*}(T[1]M), and \mbox{\boldmathB}_{0i} is a basis of sections of \Pi T^{*}\Sigma\otimes\mbox{\boldmath\phi}^{*}(T^{*}[0]M). The antibracket on this bundle (1) is
[TABLE]
from (102).
The model action with a symplectic form in [34] is
[TABLE]
where is a superderivative . where the integration means the integration on the supermanifold, . This action is consistent if and only if the -form satisfies the symplectic condition , namely
[TABLE]
This model is rewritten by the AKSZ formulation on the graded bundle . We introduce \mbox{\boldmathA}_{1}^{i}, \mbox{\boldmathB}_{0i} and \mbox{\boldmathB}_{1i} as auxiliary fields, and rewrite the action using the first order formalism. The action in AKSZ formulation is
[TABLE]
We can check that if and only if the -form satisfies the symplectic condition (4).
Also, model action with a Poisson bivector is
[TABLE]
which is called the Poisson sigma model [35][36]. The consistency condition is satisfied if and only if is a Poisson bivector field i.e.
[TABLE]
model with a complex structure is
[TABLE]
which is a covariant form of model action in [6], but is different from the action in [37]. We can check that the consistency condition is satisfied if and only if satisfies the integrability condition for the complex structure
[TABLE]
2.2 Zucchini Model
In [1], Zucchini has proposed a topological sigma model with a generalized complex structure on a two dimensional worldsheet . Although he called this model ”the Hitchin sigma model”, here we call it the Zucchini model.
First we consider case. The action of the Zucchini’s model is
[TABLE]
The master equation is satisfied if , and 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 . If in the action (10), the action reduces to the summation of two realizations of model such that (3) + (6). However, if , the action (10) does not reduce to the model action (8). So we can not easily see whether the Zucchini model can be related to model.
Also, we can consider -transformation property of this model [1]. The -transformation is defined by (77), (83) and
[TABLE]
The -transformation produces the field term such as
[TABLE]
This suggests that the Zucchini action with should have a Wess-Zumino term
[TABLE]
where is a three dimensional worldvolume such that is a two dimensional boundary of .
3 3D Topological Field Theory with Generalized Complex 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 case
Let be a three dimensional worldvolume with a coordinate for , and be a two dimensional boundary of . First we consider case.
By using the Stokes theorem, we can see the action (10) as
[TABLE]
where is a three dimensional derivative . \mbox{\boldmath\phi}^{i} and \mbox{\boldmathB}_{1i} can be extended to those on such that \mbox{\boldmath\phi}^{i}:\Pi TX\rightarrow M and \mbox{\boldmathB}_{1i} is a basis of sections of \Pi T^{*}X\otimes\mbox{\boldmath\phi}^{*}(T^{*}[1]M). We introduce a superfield \mbox{\boldmathA}_{1}^{i} with total degree one, which is a basis of a section of \Pi T^{*}X\otimes\mbox{\boldmath\phi}^{*}(T[1]M) such that \mbox{\boldmathA}_{1}^{i}=d\mbox{\boldmath\phi}^{i}, and a superfield \mbox{\boldmathB}_{2i} with total degree two, which is a basis of a section of \Pi T^{*}X\otimes\mbox{\boldmath\phi}^{*}(T^{*}[2]M) such that \mbox{\boldmathB}_{2i}=-d\mbox{\boldmathB}_{1i}. Moreover, we introduce two Lagrange multiplier fields \mbox{\boldmathY}_{2i} and \mbox{\boldmathZ}_{1}^{i} in order to realize two equations such as \mbox{\boldmathA}_{1}^{i}=d\mbox{\boldmath\phi}^{i} and \mbox{\boldmathB}_{2i}=-d\mbox{\boldmathB}_{1i} by the equations of motion. The superfield \mbox{\boldmathY}_{2i} with total degree two is a section of \Pi T^{*}X\otimes\mbox{\boldmath\phi}^{*}(T^{*}[2]M), and the superfield \mbox{\boldmathZ}_{1}^{i} with total degree one is a section of \Pi T^{*}X\otimes\mbox{\boldmath\phi}^{*}(T[1]M). The 3D action (14) is equivalent to
[TABLE]
We define \mbox{\boldmathY}^{\prime}_{2i}=\mbox{\boldmathY}_{2i}-\frac{1}{2}\mbox{\boldmathB}_{2i} and \mbox{\boldmathZ}_{1}^{\prime i}=\mbox{\boldmathZ}_{1}^{i}-\frac{1}{2}\mbox{\boldmathA}_{1}^{i}. The action (15) is rewritten as
[TABLE]
where is independent of a generalized complex structure. can be written as
[TABLE]
which is analogical with the model action (8).
The antibracket (P-structure) on , which is induced from the antibracket (2) on , for \mbox{\boldmath\phi}^{i}, \mbox{\boldmathB}_{2,i}, \mbox{\boldmathA}_{1}{}^{i} and \mbox{\boldmathB}_{1,i} is given by the antibracket (102) in . In order to define the antibrackets for \mbox{\boldmathY}^{\prime}_{2i} and \mbox{\boldmathZ}_{1}^{\prime i}, we introduce two antibracket conjugate fields \mbox{\boldmathX}^{i}, which are maps from to , and \mbox{\boldmathV}_{1i}, which are sections of \Pi T^{*}X\otimes\mbox{\boldmath\phi}^{*}(T^{*}[1]M). The model is defined on the graded bundle of the direct product of and . The second bundle is represented by auxiliary fields. The antibracket is
[TABLE]
We can check that satisfies the master equation if , and are components of the generalized complex structure (72). We can take the proper boundary conditions ;
[TABLE]
such that the total derivative terms on the master equation vanish. Here means that we take the components which are tangent to the boundary .
Also, because , satisfies the master equation
[TABLE]
where , , and are defined in Appendix A. Therefore, we can see as a three dimensional AKSZ action with generalized complex structure. We discuss why the condition is not but in subsection 3.3.
We call three dimensional generalized complex sigma model.
We consider 3D -transformation property from the 2D -transformations (11) and the conditions \mbox{\boldmathA}_{1}^{i}=d\mbox{\boldmath\phi} and {\mbox{\boldmathB}}_{2i}=-d\mbox{\boldmathB}_{1i}. 3D -transformations are
[TABLE]
We can see that 3D action (16) is invariant under the -transformation such that
[TABLE]
3.2 case I :Action induced from the Zucchini model
In the similar way, we can consider the case of a twisted generalized complex structure with . From the Zucchini model with (13), a three dimensional action is derived as
[TABLE]
This action (23) satisfies the master equation , if , , and are components of a twisted generalized complex structure (87). However, this action is not -invariant under the -transformation (21), (77) and (83). The action (23) transforms under the -transformation as
[TABLE]
which has been expected from -transformation property (12) in the two dimensional model.
Since is closed, from the Poincaré Lemma, we can locally write with a -form on such as
[TABLE]
The \frac{\partial b_{jk}}{\partial\phi^{i}}\mbox{\boldmathA}_{1}^{i}\mbox{\boldmathA}_{1}^{j}\mbox{\boldmathA}_{1}^{k} term in (23) can be absorbed to by a local -transformation in the action (23), and we obtain just the action (16). In other words, the terms in (23) are consistent up to -exact terms as a global theory, and this model is meaningful only as a cohomology class in . It is a gerbe gauge transformation dependence [1].
If we set in (13), we obtain the AKSZ formulation of the WZ-Poisson sigma model [38]:
[TABLE]
From (23), the 3D topological sigma model equivalent to (26) is
[TABLE]
3.3 case II : -invariant action
We can construct a -invariant action with in three dimensional manifold . We introduce other terms.
[TABLE]
satisfies the master equation under the antibracket (18) if and only if , , and are components of a twisted generalized complex structure. Namely, the master equation gives
[TABLE]
where , , and are defined in Appendix A. The integrability condition is not but because the action is -transformation invariant, has -transformation ambiguity by (83), and is defined as a cohomology class in in a twisted generalized complex structure.
Since does not depend on a twisted generalized complex structure, is satisfied under the condition (29). We can introduce the coupling constants by redefining {\mbox{\boldmathY}}^{\prime}_{2i} and {\mbox{\boldmathZ}_{1}}^{\prime i} to g_{1}{\mbox{\boldmathY}}^{\prime}_{2i} and g_{2}\mbox{\boldmathZ}_{1}^{\prime i}. If we take the limits that and , then and a twisted generalized complex structure does not change. We call this model a three dimensional twisted generalized complex sigma model.
We can change the -transformation so that the action is invariant, though the action (28) is not invariant under the original -transformation (21). The -transformations for \mbox{\boldmathB}_{2i} and {\mbox{\boldmathY}}^{\prime}_{2i} are changed to
[TABLE]
and -transformations for the other fields are the same as (21). Then we can check 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
First we consider the case. We consider a dimensional reduction, which can keep a generalized complex structure, from a three dimensional worldvolume to a two dimensional manifold . is compactified to . Then is compactified to . It should be noticed that is generally a different manifold from .
Here, we take X=\Sigma\times\mbox{\boldmathR}^{+}, where has a local coordinate and \mbox{\boldmathR}^{+}=[0,\infty) has a local coordinate . The second component is compactified such that {\Sigma^{\prime}}=L\times\mbox{\boldmathR}^{+}, whose local coordinate is , where is a manifold in one dimension. We formulate the dimensional reduction from a general three dimensional manifold to a general two dimensional manifold . 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 dimension 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, , reduces to the graded bundle for the two dimensional model, . Under the dimensional reduction , the fields are reduced as follows.
[TABLE]
where {\mbox{\boldmath\tilde{\phi}{-1}}}^{i} has the total degree , {\mbox{\boldmath\tilde{\phi}{}}}^{i}, {\mbox{\boldmath\tilde{\alpha{0}}{}}}^{i} and {\mbox{\boldmath\tilde{\beta_{0}}{}}}_{i} have the total degree [math], {\mbox{\boldmath\tilde{A_{1}}{}}}^{a}, {\mbox{\boldmath\tilde{B_{1}}{}}}_{i} and {\mbox{\boldmath\tilde{\beta_{1}}{}}}_{i} have the total degree , and {\mbox{\boldmath\tilde{B_{2}}{}}}_{i} has the total degree . All these superfields do not depend on .
The antibracket induced from three dimensions is
[TABLE]
We take a three dimensional AKSZ action (16) with a generalized complex structure. The existence of the negative total degree superfield {\mbox{\boldmath\tilde{\phi}{-1}}}^{i} 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 {\mbox{\boldmath\tilde{\phi}{-1}}}^{i} 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
[TABLE]
where , etc. After setting all the antifield with negative ghost numbers to zero, the following non-BV action is
[TABLE]
Since by the dimensional reduction, the fields reduce to
[TABLE]
the action (34) reduces to
[TABLE]
up to total derivative terms. Therefore the action of a 2D topological field theory is
[TABLE]
Next we formulate the action by the AKSZ formulation. We define where and are AKSZ actions for and , respectively. is easily derived after substituting (31) to (16);
[TABLE]
up to total derivative terms. The condition comes from (38) and . We introduce an negative total degree, which is defined as one for , and zero for the other fields. We can expand for the negative total degree such as , where
[TABLE]
are the negative total degree terms. Therefore
[TABLE]
Here we write the first two actions and with the negative total degree zero and one by substituting (31) to (16),
[TABLE]
for are recursively derived from the master equation . It should be noticed that since a target space has finite dimensions, is nonzero for only a finite number of . This action is a special case of a nonlinear gauge theory with -forms (a generalization of the Poisson sigma model) analyzed in the paper [39][40][41].
4.2
Here we consider case. A 2D topological field theory of twisted generalized complex geometry is derived in a similar way in subsection 4.1 from -terms in section 3.2:
[TABLE]
and for are recursively derived from .
Also, from -invariant -terms in section 3.3,
[TABLE]
and for are recursively derived from .
5 Two Special Reductions to Complex Geometry and Symplectic Geometry
In this section, we consider two special reductions related to complex geometry and of symplectic geometry.
5.1 Complex geometry
First we consider our model in complex geometry, which is the case that in the action (40). We redefine superfields as
[TABLE]
where is a constant. After this redefinition, the action (40) is
[TABLE]
and has at least the higher order of than because {\mbox{\boldmath\tilde{\phi}{-1}}}^{i}=\lambda{\mbox{\boldmath\tilde{\phi}{-1}^{\prime}}}^{i}. We can take the limit with preserving the complex structure. for reduces to zero, and the 2D action is
[TABLE]
This action is nothing but the model action (8) up to a total derivative and the all over factor , which depends on only . The master equation impose the condition that is a complex structure.
We make a comment about the difference between the action (50) with a finite and the B model action (54) with . 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 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 {\mbox{\boldmath\tilde{\phi}{}}}^{i}, which is independent of {\mbox{\boldmath\tilde{\phi}{-1}}}^{i}, because there is no metric with fermionic indices on the fermionic space of {\mbox{\boldmath\tilde{\phi}{-1}}}^{i}. So these deformed parts can be seen as a topological theory with only B field-like couplings on the fermionic space of {\mbox{\boldmath\tilde{\phi}_{-1}}}^{i}. 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 in the action (40). We redefine superfields as
[TABLE]
where is a constant. After this redefinition, the 2D action (40) reduces to
[TABLE]
and is at least the higher order of than because {\mbox{\boldmath\tilde{\phi}{-1}}}^{i}=\mu{\mbox{\boldmath\tilde{\phi}{-1}^{\prime}}}^{i}.
After taking the limit with preserving the symplectic structure, for reduces to zero, and the 2D action is
[TABLE]
The BV condition is satisfied if and only if 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 , this action is a different realization of the Poisson structure from the 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 supersymmetric sigma model with a non-trivial 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 be a manifold of even dimension with a local coordinate . We consider the vector bundle . We denote a section as where and .
is equipped with a natural indefinite metric of signature defined by
[TABLE]
for , where is an interior product with a vector field . In the Cartesian coordinate , The metric is written as follows:
[TABLE]
We define a Courant bracket on as follows:
[TABLE]
with , where denotes Lie derivation with respect a vector field and is the exterior differential of . This bracket is antisymmetric but do not satisfy the Jacobi identity. We may consider a so called Dorfman bracket as follows:
[TABLE]
which satisfies the Jacobi identity but is not antisymmetric. Antisymmetrization of a Dorfman bracket coincides with a Courant bracket.
A generalized almost complex structure is a section of , which is an isometry of the metric , and satisfies
[TABLE]
A -transformation is an isometry defined by
[TABLE]
where is a –form. A Courant bracket is covariant under the -transformation
[TABLE]
if the –form is closed. The -transform of is defined by
[TABLE]
has the eigenbundles because , In order to divide to each eigenbundle, we need complexification of , . The projectors on the eigenbundles are defined by
[TABLE]
The generalized almost complex structure is integrable if
[TABLE]
for any , where the bracket is the Courant bracket. Then is called a generalized complex structure. Integrability is equivalent to the single statement
[TABLE]
for all , where is the generalized Nijenhuis tensor defined by
[TABLE]
The -transform of a generalized complex structure is a generalized complex structure if the –form is closed.
We decompose a generalized almost complex structure in coordinate form as follows
[TABLE]
where , , .
Then the conditions and derive
[TABLE]
where
[TABLE]
The integrability condition (69) is equivalent to the following condition
[TABLE]
where
[TABLE]
Here is a differentiation with respect to . The –transform is
[TABLE]
where .
The usual complex structures is embedded in generalized complex structures as the special form
[TABLE]
Indeed, one can check this form satisfies conditions, (73) and (75) if and only if is a complex structure. Similarly, the usual symplectic structures is obtained as the special form of generalized complex structures
[TABLE]
This satisfies (73) and (75) if and only if 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 can be modified by a closed –form. Let be a closed –form. We define the twisted Courant brackets by
[TABLE]
where . Under the -transform with a closed –form,
[TABLE]
holds with the brackets replaced by . For a non closed , one has
[TABLE]
So, the -transformation shifts by the exact –form :
[TABLE]
One can define an twisted generalized Nijenhuis tensor as follows
[TABLE]
by using the brackets instead of . A generalized almost complex structure is integrable if
[TABLE]
for all . Then we call an twisted generalized complex structure.
The integrability conditions is as follows:
[TABLE]
where
[TABLE]
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 be a smooth manifold in dimensions. If we consider We define a supermanifold . Mathematically, , whose bosonic part is , is defined as a cotangent bundle with reversed parity of the fiber. That is, a base manifold has a Grassman even coordinate and the fiber of has a Grassman odd coordinate. We introduce a grading called total degrees, which is denoted for a function . The coordinates of the base manifold have grade zero and the coordinates of the fiber have grade one. Similarly, we can define for a tangent bundle . is also called a supermanifold.
We must consider more general assignments for the degree of the fibers of or . For an integer , we define , which is called a graded cotangent bundle. is a cotangent bundle, whose fiber has the degree . 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 . If is odd, the fiber is Grassman odd, and if is even, the fiber is Grassman even. We define a graded tangent bundle in the same way.
We consider a vector bundle . A graded vector bundle is defined in the similar way. is a vector bundle whose fiber has a shifted degree by . Note that only the degree of fiber is shifted, and the degree of base space is not shifted.
We consider a Poisson manifold 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) from . Then a Poisson structure shifts to a graded Poisson structure by grading of . 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 satisfies the following identities:
[TABLE]
where and are functions on , and and are total degrees of the functions, respectively. The graded Poisson structure is also called P-structure. If , the antibracket is equivalent to the Schouten bracket. For higher , the antibracket is equivalent to the Loday bracket [43] with the degree .
Typical examples of Poisson manifold are a cotangent bundle and a vector bundle . First we consider a cotangent bundle . Since has a natural symplectic structure, we can define a Poisson bracket induced from the symplectic structure. If we take a local coordinate on and a local coordinate of the fiber, we can define a Poisson bracket as follows:
[TABLE]
where and are functions on , and and are the right and left differentiations with respect to , respectively. Here we shift the degree of fiber by , i.e. the space . Then a Poisson structure shifts to a graded Poisson structure. The corresponding graded Poisson bracket is called antibracket, . Let \mbox{\boldmath\phi}^{i} be a local coordinate of and \mbox{\boldmathB}_{n-1,i} a basis of the fiber of . The antibracket on a cotangent bundle is expressed as:
[TABLE]
The total degree of the antibracket is . This antibracket satisfies the property (88) for .
Next, we consider a vector bundle . There is a natural Poisson structure on the fiber of induced from a paring of and . If we take a local coordinate on the fiber of and on the fiber of , we can define
[TABLE]
where and are functions on . We shift the degrees of fibers of and like , where and are positive integers. The Poisson structure changes to a graded Poisson structure . Let \mbox{\boldmathA}_{p}{}^{a} be a basis of the fiber of and \mbox{\boldmathB}_{q,a} a basis of the fiber of . The antibracket is represented as
[TABLE]
The total degree of the antibracket is . This antibracket satisfies the property (88) for .
We define a Q-structure. A Q-structure is a function on a graded manifold which satisfies the classical master equation . is called a Batalin-Vilkovisky action. We require that satisfy the compatibility condition
[TABLE]
where and are arbitrary functions. 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 be a base manifold in dimensions, with or without boundary, and be a target manifold in dimensions. We denote a smooth map from to .
We consider a supermanifold , whose bosonic part is . is defined as a tangent bundle with reversed parity of the fiber. We take a local coordinate of , , where is a coordinate on the base space and is a super coordinate on the fiber and . We extend a smooth function to a function on the supermanifold \mbox{\boldmath\phi}:\Pi TX\rightarrow M. is called a superfield and an element of . We introduce a new non-negative integer grading on . A coordinate on a base manifold has zero and a coordinate on the fiber has one. This grading is called the form degree. We denote the form degree of the function . The total degree defined in the previous section is a grading with respect to , on the other hand The form degree is a grading with respect to . We define a ghost number such that . W assign the ghost numbers of and zero. Thus has the total degree zero and has total degree one.
We consider a P-structure on . We take to construct a Batalin-Vilkovisky structure in a topological sigma model on a general dimensional worldvolume. We consider for an -dimensional base manifold . Let a superfield \mbox{\boldmath\phi}^{i} be local a coordinate of , where are indices of the local coordinate on . Let a superfield \mbox{\boldmathB}_{n-1,i} be a basis of sections of \Pi T^{*}X\otimes\mbox{\boldmath\phi}^{*}(T^{*}[n-1]M). Expansions to component fields of the superfields are the following:
[TABLE]
where is the ghost number of the component field.
From (90) in the previous subsection, we define an antibracket on a cotangent bundle as
[TABLE]
where and are functions of \mbox{\boldmath\phi}^{i} and \mbox{\boldmathB}_{n-1,i}. The total degree of the antibracket is . If and are functionals of \mbox{\boldmath\phi}^{i} and \mbox{\boldmathB}_{n-1,i}, we understand an antibracket is defined as
[TABLE]
where the integration means the integration on the supermanifold, . 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 . In a topological sigma model in dimension worldvolume, we assign the total degree of and such that . The total graded bundle is , where . Let \mbox{\boldmathA}_{p}{}^{a_{p}} be a basis of sections of \Pi T^{*}X\otimes\mbox{\boldmath\phi}^{*}(E[p]) and \mbox{\boldmathB}_{n-p-1,a_{p}} a basis of the fiber of \Pi T^{*}X\otimes\mbox{\boldmath\phi}^{*}(E^{*}[n-p-1]). Expansions to component fields of the superfields are
[TABLE]
From (92), we define the antibracket as
[TABLE]
We need to consider various grading assignments for , because each assignment induces different Batalin-Vilkovisky structures. In order to consider all independent assignments, we define the following bundle. Let be series of vector bundles, where . We consider a direct sum of each bundle :
[TABLE]
and we can define a P-structure on the graded vector bundle
[TABLE]
which is isomorphic to the graded bundle
[TABLE]
[TABLE]
where \mbox{\boldmathA}_{0}{}^{a_{0}}=\mbox{\boldmath\phi}^{i}, that is component is the antibracket (95) on the graded cotangent bundle . Note that all terms of the antibracket have the total degree , and we can confirm that the antibracket (102) satisfies the identity (88).
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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