^†^†institutetext: Physical Sciences, Kingsborough Community College, The City University of New York,
2001 Oriental Boulevard, Brooklyn, NY 11235-2398, USA

R-symmetries and curvature constraints in A-twisted heterotic Landau-Ginzburg models

Richard S. Garavuso [email protected]

Abstract

In this paper, we discuss various aspects of a class of A-twisted heterotic Landau-Ginzburg models on a Kähler variety $X$ . We provide a classification of the R-symmetries in these models which allow the A-twist to be implemented, focusing on the case in which the gauge bundle is either a deformation of the tangent bundle of $X$ or a deformation of a sub-bundle of the tangent bundle of $X$ . Some anomaly-free examples are provided. The curvature constraint imposed by supersymmetry in these models when the superpotential is not holomorphic is reviewed. Constraints of this nature have been used to establish properties of analogues of pullbacks of Mathai-Quillen forms which arise in the correlation functions of the corresponding A-twisted or B-twisted heterotic Landau-Ginzburg models. The analogue most relevant to this paper is a deformation of the pullback of a Mathai-Quillen form. We discuss how this deformation may arise in the class of models studied in this paper. We then comment on how analogues of pullbacks of Mathai-Quillen forms not discussed in previous work may be obtained. Standard Mathai-Quillen formalism is reviewed in an appendix. We also include an appendix which discusses the deformation of the pullback of a Mathai-Quillen form.

1 Introduction

A Landau-Ginzburg model is a nonlinear sigma model with a superpotential. For a heterotic Landau-Ginzburg model Witten:Phases ; DistlerKachru:0-2-Landau-Ginzburg ; AdamsBasuSethi:0-2-Duality ; MelnikovSethi:Half-twisted ; GuffinSharpe:A-twistedheterotic ; MelnikovSethiSharpe:Recent-Developments ; GaravusoSharpe:Analogues , the nonlinear sigma model possesses only $(0,2)$ supersymmetry and the superpotential is a Grassmann-odd function of the superfields which may or may not be holomorphic.

Heterotic Landau-Ginzburg models have field content consisting of $(0,2)$ bosonic chiral superfields

\Phi^{i}=(\phi^{i},\psi^{i}_{+})

and $(0,2)$ fermionic chiral superfields

\Lambda^{a}=\left(\lambda^{a}_{-},H^{a},E^{a}\right),

along with their conjugate antichiral superfields

\Phi^{\overline{\imath}}=\left(\phi^{\overline{\imath}},\psi^{\overline{\imath}}_{+}\right)

and

\Lambda^{\overline{a}}=\left(\lambda^{\overline{a}}_{-},\overline{H}^{\overline{a}},\overline{E}^{\overline{a}}\right).

The $\phi^{i}$ are local complex coordinates on a Kähler variety $X$ . The $E^{a}$ are local smooth sections of a Hermitian vector bundle $\mathcal{E}$ over $X$ , i.e. $E^{a}\in\Gamma(X,\mathcal{E})$ . The $H^{a}$ are nonpropagating auxiliary fields. The fermions couple to bundles as follows:

	$\displaystyle\psi^{i}_{+}\in\Gamma\left(K^{1/2}_{\Sigma}\otimes\Phi^{*}\!\left(T^{1,0}X\right)\right),$	$\displaystyle\lambda^{a}_{-}\in\Gamma\left(\overline{K}^{1/2}_{\Sigma}\otimes\left(\Phi^{*}\overline{\mathcal{E}}\right)^{\vee}\right),$
	$\displaystyle\psi^{\overline{\imath}}_{+}\in\Gamma\left(K^{1/2}_{\Sigma}\otimes\left(\Phi^{*}\!\left(T^{1,0}X\right)\right)^{\vee}\right),$	$\displaystyle\lambda^{\overline{a}}_{-}\in\Gamma\left(\overline{K}^{1/2}_{\Sigma}\otimes\Phi^{*}\overline{\mathcal{E}}\right),$

where $\Phi:\Sigma\rightarrow X$ and $K_{\Sigma}$ is the canonical bundle on the worldsheet $\Sigma$ .

In GuffinSharpe:A-twistedheterotic , heterotic Landau-Ginzburg models with superpotential of the form

W=\Lambda^{a}\,F_{a}\,,

(1.1)

where $F_{a}\in\Gamma\left(X,\mathcal{E}^{\vee}\right)$ were considered. It was claimed in GaravusoSharpe:Analogues that, when the superpotential (1.1) is not holomorphic, supersymmetry imposes a constraint which relates the nonholomorphic parameters of the superpotential to the Hermitian curvature. The details supporting that claim were worked out in Garavuso:Curvature ; Garavuso:Nonholomorphic for the case $E^{a}\equiv 0$ . This curvature constraint has been used in GaravusoSharpe:Analogues to establish properties of analogues of pullbacks of Mathai-Quillen forms. These analogues arise in the correlation functions of the corresponding A-twisted or B-twisted heterotic Landau-Ginzburg models.

In this paper, we will study certain aspects of A-twisted heterotic Landau-Ginzburg models with superpotential (1.1) and $E^{a}\equiv 0$ . Such models yield the A-twisted $(2,2)$ Landau-Ginzburg models of GuffinSharpe:A-twisted when $\mathcal{E}=TX$ and $\Lambda^{i}\,F_{i}=\Lambda^{i}\,\partial_{i}W^{(2,2)}$ , where $W^{(2,2)}$ is the (2,2) superpotential. Although R-symmetries for (2,2) Landau-Ginzburg models have been classified, this has not been done for heterotic Landau-Ginzburg models. Furthermore, for (2,2) Landau-Ginzburg models, a classification has been given only for the case of holomorphic superpotentials KachruWitten:Computing . We will provide a classification of the R-symmetries which allow the A-twist to be implemented, focusing on the case in which $\mathcal{E}$ is either a deformation of $TX$ or a deformation of a sub-bundle of $TX$ . The curvature constraint imposed by supersymmetry in these models when the superpotential is not holomorphic will be reviewed. The corresponding analogue of the pullback of a Mathai-Quillen form is a deformation of the pullback of a Mathai-Quillen form. We will discuss how this deformation may arise in the class of models studied in this paper. We will then comment on how analogues of pullbacks of Mathai-Quillen forms not discussed in previous work may be obtained.

This paper is organized as follows: The A-twist will be discussed in section 2. A classification of the corresponding R-symmetries, along with some anomaly-free examples, will be given in section 3. The curvature constraint imposed by supersymmetry when the superpotential is not holomorphic will be reviewed in section 4. In section 5, we will discuss how an analogue of a pullback of a Mathai-Quillen form may arise in the class of heterotic Landau-Ginzburg models discussed in this paper. In section 6, we will summarize our results and comment on how analogues of pullbacks of Mathai-Quillen forms not discussed in previous work may be obtained. Appendix A will review standard Mathai-Quillen formalism MathaiQuillen:Superconnections ; BerlinGetzlerVergne:Heat ; Kalkman:BRST ; Blau:The-Mathai-Quillen ; Wu:On-the-Mathai-Quillen ; CordesMooreRamgoolam:Lectures ; Wu:Mathai-Quillen . Finally, appendix B will discuss the analogue that is most relevant to this paper, i.e. a deformation of the pullback of a Mathai-Quillen form.

2 A-twist

Let $X$ be a Kähler variety with metric $g$ , antisymmetric tensor $B$ , local real coordinates $\phi^{\mu}$ , and local complex coordinates $\phi^{i}$ with complex conjugates $\phi^{\overline{\imath}}$ . Furthermore, let $\mathcal{E}$ be a vector bundle over $X$ with Hermitian fiber metric $h$ . We consider the action GuffinSharpe:A-twistedheterotic of an A-twisted heterotic Landau-Ginzburg model on $X$ with gauge bundle $\mathcal{E}$ :

	$\displaystyle S$	$\displaystyle=2t\int_{\Sigma}d^{2}z\left[\frac{1}{2}\left(g_{\mu\nu}+iB_{\mu\nu}\right)\partial_{z}\phi^{\mu}\partial_{\overline{z}}\phi^{\nu}+ig_{\overline{\imath}i}\psi_{+}^{\overline{\imath}}\overline{D}_{\overline{z}}\psi_{+}^{i}+ih_{a\overline{a}}\lambda_{-}^{a}D_{z}\lambda_{-}^{\overline{a}}\right.$
		$\displaystyle\phantom{=2t\int_{\Sigma}d^{2}z\left[\right.}+\biggl{.}F_{i\overline{\imath}a\overline{a}}\,\psi_{+}^{i}\psi_{+}^{\overline{\imath}}\lambda_{-}^{a}\lambda_{-}^{\overline{a}}+h^{a\overline{a}}F_{a}\overline{F}_{\overline{a}}+\psi_{+}^{i}\lambda_{-}^{a}D_{i}F_{a}+\psi_{+}^{\overline{\imath}}\lambda_{-}^{\overline{a}}\overline{D}_{\overline{\imath}}\overline{F}_{\overline{a}}\biggr{]}.$		(2.1)

Here, $t$ is a coupling constant, $\Sigma$ is a Riemann surface, $d^{2}z=-i\,dz\wedge d{\overline{z}}$ , $F_{a}\in\Gamma\left(X,\mathcal{E}^{\vee}\right)$ , and

$\displaystyle\overline{D}_{\overline{z}}\,\psi^{i}_{+}$	$\displaystyle=\overline{\partial}_{\overline{z}}\,\psi_{+}^{i}+\overline{\partial}_{\overline{z}}\,\phi^{j}\,\Gamma^{i}_{jk}\psi^{k}_{+}\,,$	$\displaystyle D_{z}\lambda^{\overline{a}}_{-}$	$\displaystyle=\partial_{z}\lambda_{-}^{\overline{a}}+\partial_{z}\phi^{\overline{\imath}}A^{\overline{a}}_{\overline{\imath}\overline{b}}\,\lambda^{\overline{b}}_{-}\,,$
$\displaystyle D_{i}F_{a}$	$\displaystyle=\partial_{i}F_{a}-A^{b}_{ia}F_{b}\,,$	$\displaystyle\overline{D}_{\overline{\imath}}\overline{F}_{\overline{a}}$	$\displaystyle=\overline{\partial}_{\overline{\imath}}\,\overline{F}_{\overline{a}}-A^{\overline{b}}_{\overline{\imath}\,\overline{a}}\,\overline{F}_{\overline{b}}\,,$
$\displaystyle A^{b}_{ia}$	$\displaystyle=h^{b\overline{b}}\,h_{\overline{b}a,i}\,,$	$\displaystyle A^{\overline{b}}_{\overline{\imath}\,\overline{a}}$	$\displaystyle=h^{\overline{b}b}\,h_{b\overline{a},\overline{\imath}}\,,$
$\displaystyle\Gamma^{i}_{jk}$	$\displaystyle=g^{i\overline{\imath}}\,g_{\overline{\imath}k,j}\,,$	$\displaystyle F_{i\overline{\imath}a\overline{a}}$	$\displaystyle=h_{a\overline{b}}\,A^{\overline{b}}_{\overline{\imath}\,\overline{a},i}\,.$

The A-twist is defined by choosing the fermions couple to bundles as follows:

	$\displaystyle\psi^{i}_{+}\in\Gamma\left(\Phi^{*}\!\left(T^{1,0}X\right)\right),$	$\displaystyle\lambda^{a}_{-}\in\Gamma\left(\overline{K}_{\Sigma}\otimes\left(\Phi^{*}\overline{\mathcal{E}}\right)^{\vee}\right),$
	$\displaystyle\psi^{\overline{\imath}}_{+}\in\Gamma\left(K_{\Sigma}\otimes\left(\Phi^{*}\!\left(T^{1,0}X\right)\right)^{\vee}\right),$	$\displaystyle\lambda^{\overline{a}}_{-}\in\Gamma\left(\Phi^{*}\overline{\mathcal{E}}\right),$

where $\Phi:\Sigma\rightarrow X$ and $K_{\Sigma}$ is the canonical bundle on $\Sigma$ . Anomaly cancellation requires GuffinSharpe:A-twistedheterotic ; KatzSharpe:Notes ; Sharpe:Notes

\Lambda^{\textrm{top}}\mathcal{E}^{\vee}\simeq K_{X}\,,\qquad\textrm{ch}_{2}\left(\mathcal{E}\right)=\textrm{ch}_{2}\left(TX\right).

(2.2)

The action (2) is invariant on-shell under the supersymmetry transformations

$\displaystyle\delta\phi^{i}$	$\displaystyle=i\alpha_{-}\psi^{i}_{+}\,,$	(2.3)
$\displaystyle\delta\phi^{\overline{\imath}}$	$\displaystyle=0\,,$
$\displaystyle\delta\psi^{i}_{+}$	$\displaystyle=0\,,$
$\displaystyle\delta\psi^{\overline{\imath}}_{+}$	$\displaystyle=-\alpha_{-}\partial_{z}\phi^{\overline{\imath}}\,,$
$\displaystyle\delta\lambda^{a}_{-}$	$\displaystyle=-i\alpha_{-}\psi^{j}_{+}\,A^{a}_{jb}\,\lambda^{b}_{-}+i\alpha_{-}h^{a\overline{a}}\,\overline{F}_{\overline{a}}\,,$
$\displaystyle\delta\lambda^{\overline{a}}_{-}$	$\displaystyle=0$

up to a total derivative. Since we have integrated out the auxiliary fields $H^{a}$ , one may use the $\lambda^{a}_{-}$ equation of motion

\lambda^{a}_{-}:\quad ih_{a\overline{a}}D_{z}\lambda^{\overline{a}}_{-}+F_{i\overline{\imath}a\overline{a}}\,\psi^{i}_{+}\psi^{\overline{\imath}}_{+}\lambda^{\overline{a}}_{-}-\psi^{i}_{+}D_{i}F_{a}=0\,,

(2.4)

to show Garavuso:Curvature that the action (2) can be written

S=it\int_{\Sigma}d^{2}z\,\left\{Q,V\right\}+t\int_{\Sigma}\Phi^{*}(K)+2t\int_{\Sigma}d^{2}z\left(\psi_{+}^{\overline{\imath}}\lambda_{-}^{\overline{a}}\overline{D}_{\overline{\imath}}\overline{F}_{\overline{a}}-\psi_{+}^{i}\lambda_{-}^{a}D_{a}F_{a}\right),

(2.5)

where

$\displaystyle\left\{Q,\phi^{i}\right\}$	$\displaystyle=-\psi^{i}_{+}\,,\qquad$	$\displaystyle\left\{Q,\phi^{\overline{\imath}}\right\}$	$\displaystyle=0\,,$
$\displaystyle\left\{Q,\psi^{i}_{+}\right\}$	$\displaystyle=0\,,\qquad$	$\displaystyle\left\{Q,\psi^{\overline{\imath}}_{+}\right\}$	$\displaystyle=-i\partial_{z}\phi^{\overline{\imath}}\,,$
$\displaystyle\left\{Q,\lambda^{a}_{-}\right\}$	$\displaystyle=\psi^{j}_{+}A^{a}_{jb}\lambda^{b}_{-}-h^{a\overline{a}}\,\overline{F}_{\overline{a}}\,,\qquad$	$\displaystyle\left\{Q,\lambda^{\overline{a}}_{-}\right\}$	$\displaystyle=0$

are the BRST transformations ( $\delta f=-i\alpha_{-}\{Q,f\}$ , where $f$ is any field),

V=2\left(g_{\overline{\imath}i}\psi^{\overline{\imath}}_{+}\overline{\partial}_{\overline{z}}\phi^{i}+i\lambda^{a}_{-}F_{a}\right),

and

\int_{\Sigma}\Phi^{*}(K)=\int_{\Sigma}d^{2}z\left(g_{i\overline{\imath}}+iB_{i\overline{\imath}}\right)\left(\partial_{z}\phi^{i}\,\overline{\partial}_{\overline{z}}\phi^{\overline{\imath}}-\overline{\partial}_{\overline{z}}\phi^{i}\partial_{z}\phi^{\overline{\imath}}\right)

is the integral over the worldsheet $\Sigma$ of the pullback to $\Sigma$ of the complexified Kähler form

K=-i\left(g_{i\overline{\imath}}+iB_{i\overline{\imath}}\right)d\phi^{i}\wedge d\phi^{\overline{\imath}}\,.

3 R-symmetries

Let us now discuss the R-symmetries which allow the A-twist described in section 2 to be obtained. A classification of these R-symmetries will be given in section 3.1. Some anomaly-free examples will be given in section 3.2.

3.1 Classification

For $F_{a}\equiv 0$ , the twisting is achieved by tensoring the fields with

K^{-Q_{R}/2}_{\Sigma}\otimes\overline{K}^{\,Q_{L}/2}_{\Sigma}\,,

where the fields have charges $Q_{L}$ and $Q_{R}$ , given in table 3.1,

Field	$Q_{L}$	$Q_{R}$
$\phi^{i}$	0	0
$\phi^{\overline{\imath}}$	0	0
$\psi^{i}_{+}$	0	1
$\psi^{\overline{\imath}}_{+}$	0	$-1$
$\lambda^{a}_{-}$	1	0
$\lambda^{\overline{a}}_{-}$	$-1$	0

Table 3.1: Charges when

F_{a}\equiv 0

under $U(1)_{L}$ and $U(1)_{R}$ R-symmetries, respectively. These R-symmetries defined by $Q_{L}$ and $Q_{R}$ are broken when $F_{a}\not\equiv 0$ .

Let us consider an $F_{a}$ of the form

F_{a}=\partial_{a}G+G_{a}\,.

(3.1)

Here, $G$ is quasihomogeneous and meromorphic, i.e.

G\left(\lambda^{n_{i}}\phi^{i},\lambda^{m_{\overline{\imath}}}\phi^{\overline{\imath}}\right)=\lambda^{d}\,G\left(\phi^{i},\phi^{\overline{\imath}}\right),

(3.2)

where $\lambda\in\mathbf{C}^{\times}$ , $n_{i}=-m_{\overline{\imath}}$ and $d$ are integers, and the deformation $G_{a}$ is chosen to be

G_{a}=\partial_{a}\left[\frac{1}{d}\sum_{i}n_{i}\left(\phi^{i}\right)^{\frac{d}{n_{i}}}\right]=\left(\phi^{a}\right)^{\frac{d}{n_{a}}-1}\,.

(3.3)

For an $F_{a}$ of this form, we can define new charges $Q^{\prime}_{L}=Q_{L}-Q$ and $Q^{\prime}_{R}=Q_{R}-Q$ , given in table 3.2,

Field	$Q_{L}$	$Q_{R}$	$Q$	$Q^{\prime}_{L}=Q_{L}-Q$	$Q^{\prime}_{R}=Q_{R}-Q$
$\phi^{i}$	0	0	$\alpha_{i}$	$-\alpha_{i}$	$-\alpha_{i}$
$\phi^{\overline{\imath}}$	0	0	$-\alpha_{i}$	$\alpha_{i}$	$\alpha_{i}$
$\psi^{i}_{+}$	0	1	$\alpha_{i}$	$-\alpha_{i}$	$1-\alpha_{i}$
$\psi^{\overline{\imath}}_{+}$	0	$-1$	$-\alpha_{i}$	$\alpha_{i}$	$\alpha_{i}-1$
$\lambda^{a}_{-}$	1	0	$\alpha_{a}$	$1-\alpha_{a}$	$-\alpha_{a}$
$\lambda^{\overline{a}}_{-}$	$-1$	0	$-\alpha_{a}$	$\alpha_{a}-1$	$\alpha_{a}$
$D_{i}$	0	0	$-\alpha_{i}$	$\alpha_{i}$	$\alpha_{i}$
$\overline{D}_{\overline{\imath}}$	0	0	$\alpha_{i}$	$-\alpha_{i}$	$-\alpha_{i}$
$\partial_{a}$	0	0	$-\alpha_{a}$	$\alpha_{a}$	$\alpha_{a}$
$\overline{\partial}_{\overline{a}}$	0	0	$\alpha_{a}$	$-\alpha_{a}$	$-\alpha_{a}$
$G$	0	0	$1$	$-1$	$-1$
$\overline{G}$	0	0	$-1$	$1$	$1$
$G_{a}$	0	0	$1-\alpha_{a}$	$\alpha_{a}-1$	$\alpha_{a}-1$
$\overline{G}_{\overline{a}}$	0	0	$\alpha_{a}-1$	$1-\alpha_{a}$	$1-\alpha_{a}$
$F_{a}=\partial_{a}G+G_{a}$	0	0	$1-\alpha_{a}$	$\alpha_{a}-1$	$\alpha_{a}-1$
$\overline{F}_{\overline{a}}=\overline{\partial}_{\overline{a}}\,\overline{G}+\overline{G}_{\overline{a}}$	0	0	$\alpha_{a}-1$	$1-\alpha_{a}$	$1-\alpha_{a}$

Table 3.2: Charges when

F_{a}=\partial_{a}G+G_{a}

expressed in terms of the parameters

\alpha_{i}=n_{i}/d=-m_{\overline{\imath}}/d\,,\qquad\alpha_{a}=n_{a}/d=-m_{\overline{a}}/d\,,

(3.4)

which yield a $U(1)_{L}\times U(1)_{R}$ -invariant action. On the (2,2) locus, we have

	$\displaystyle Q^{\prime}_{L}\bigl{(}\phi^{i}\bigr{)}$	$\displaystyle=Q^{\prime}_{L}\bigl{(}\psi^{i}_{+}\bigr{)},\quad Q^{\prime}_{L}\bigl{(}\lambda^{i}_{-}\bigr{)}=Q^{\prime}_{L}\bigl{(}\phi^{i}\bigr{)}+1\,,$
	$\displaystyle Q^{\prime}_{R}\bigl{(}\phi^{i}\bigr{)}$	$\displaystyle=Q^{\prime}_{R}\bigl{(}\lambda^{i}_{-}\bigr{)},\quad Q^{\prime}_{R}\bigl{(}\psi^{i}_{+}\bigr{)}=Q^{\prime}_{R}\bigl{(}\phi^{i}\bigr{)}+1\,.$

Off of this locus, although one has a pair of $U(1)$ symmetries, only $U(1)_{R}$ is an R-symmetry. The twisting is achieved by tensoring the fields with

K^{-Q^{\prime}_{R}/2}_{\Sigma}\otimes\overline{K}^{\,Q^{\prime}_{L}/2}_{\Sigma}\,.

Recall that for a Riemann surface $\Sigma$ of genus $g$ , the degree of the canonical bundle is $2g-2$ . It follows that, for the bundles $K^{-Q^{\prime}_{R}/2}_{\Sigma}$ and $\overline{K}^{\,Q^{\prime}_{L}/2}_{\Sigma}$ to be well-defined, $d$ must divide $g-1$ , i.e.

g=1+kd\,,\qquad k=0,1,2,\ldots\,.

(3.5)

This genus issue is well understood; more details can be found in GuffinSharpe:A-twisted ; Witten:Algebraic ; Witten:The-N-matrix . Table 3.2 gives a classification of the R-symmetries in the models we are discussing in terms of the charges $Q^{\prime}_{L}$ and $Q^{\prime}_{R}$ .

3.2 Examples

Let us consider some examples in which $\mathcal{E}$ is a deformation of $TX$ . For such examples, the anomaly cancellation conditions (2.2) are satisfied.

As a first example, consider the case in which $X$ is a complex affine space and $G$ is a Fermat polynomial:

Example 3.1.

Let $X=\mathbf{C}^{d}$ , and

G=\left(\phi^{1}\right)^{d}+\cdots+\left(\phi^{d}\right)^{d}\,.

Thus,

\left(n_{1},\ldots,n_{d}\right)=\left(1,\ldots,1\right)\,,

G_{a}=\left(\phi^{a}\right)^{d-1}\,,\qquad a=1,\ldots,d\,,

and

\alpha_{a}=\frac{n_{a}}{d}=\frac{1}{d}\,,\qquad a=1,\ldots,d\,.

The twist described in this example can be defined on worldsheets of genus $g$ given by (3.5).

As a second example, consider the case in which $X$ is a complex projective space and $G$ is a Fermat polynomial with zero locus defining a hypersurface in $X$ :

Example 3.2.

Let $X=\mathbf{CP}^{d-1}$ , and

G=\left(\phi^{1}\right)^{d}+\cdots+\left(\phi^{d}\right)^{d}\,.

Thus,

\left\{G=0\right\}\in\mathbf{CP}^{d-1}[d]\,,

\left(n_{1},\ldots,n_{d}\right)=\left(1,\ldots,1\right)\,,

G_{a}=\left(\phi^{a}\right)^{d-1}\,,\qquad\phi^{a}\,G_{a}=0\,,\qquad a=1,\ldots,d\,,

and

\alpha_{a}=\frac{n_{a}}{d}=\frac{1}{d}\,,\qquad a=1,\ldots,d\,.

The twist described in this example can be defined on worldsheets of genus $g$ given by (3.5).

As a final example, consider the case in which $X$ is a weighted complex projective space and the zero locus of $G$ is a hypersurface in that space:

Example 3.3.

Let $X=\mathbf{WCP}^{3}_{12,8,7,9}$ , and

G=\left(\phi^{1}\right)^{3}+\phi^{1}\left(\phi^{2}\right)^{3}+\phi^{2}\left(\phi^{2}\right)^{4}+\left(\phi^{4}\right)^{4}\,.

Thus,

\left\{G=0\right\}\in\mathbf{WCP}^{3}_{n_{1},n_{2},n_{3},n_{4}}[d]=\mathbf{WCP}^{3}_{12,8,7,9}[36]\,,

G_{a}=\left(\phi^{a}\right)^{\frac{d}{n_{a}}-1}\,,\qquad\phi^{a}\,G_{a}=0\,,\qquad a=1,\ldots,4\,,

and

	$\displaystyle\alpha_{1}$	$\displaystyle=\frac{n_{1}}{d}=\frac{12}{36}=\frac{1}{3}\,,$
	$\displaystyle\alpha_{2}$	$\displaystyle=\frac{n_{2}}{d}=\frac{8}{36}=\frac{2}{9}\,,$
	$\displaystyle\alpha_{3}$	$\displaystyle=\frac{n_{3}}{d}=\frac{7}{36}\,,$
	$\displaystyle\alpha_{4}$	$\displaystyle=\frac{n_{4}}{d}=\frac{9}{36}=\frac{1}{4}\,.$

The twist described in this example can be defined on worldsheets of genus $g$ given by (3.5):

	$\displaystyle g=1+kd$	$\displaystyle=1+k\left(36\right)\,,\qquad k=0,1,2,\ldots$
		$\displaystyle=1,37,73,109,\ldots\,.$

4 Curvature constraints

The action (2.5) is invariant on-shell under the supersymmetry transformations (2.3) up to a total derivative. It was claimed in GaravusoSharpe:Analogues that requiring this invariance when the superpotential (1.1) is not holomorphic imposes a constraint which relates the nonholomorphic parameters of the superpotential to the Hermitian curvature. This curvature constraint, along with an additional constraint imposed by supersymmetry, was derived in Garavuso:Curvature ; Garavuso:Nonholomorphic . Let us now briefly review the key steps of this derivation; see Garavuso:Curvature ; Garavuso:Nonholomorphic for more details.

Since $\delta f=-i\alpha_{-}\{Q,f\}$ , where $f$ is any field, the $Q$ -exact part of (2.5) is $\delta$ -exact and hence $\delta$ -closed. For the non-exact term of (2.5) involving $\Phi^{*}(K)$ , note that

\int_{\Sigma}\Phi^{*}(K)=\int_{\Phi(\Sigma)}K=\int_{\Phi(\Sigma)}\left[-i\left(g_{i\overline{\imath}}+iB_{i\overline{\imath}}\right)\right]d\phi^{i}\wedge d\phi^{\overline{\imath}}

and $K$ satisfies

\partial K=-i\,\partial_{k}\left(g_{i\overline{\imath}}+iB_{i\overline{\imath}}\right)d\phi^{k}\wedge d\phi^{i}\wedge d\phi^{\overline{\imath}}=0\,.

Thus,

\delta\left[\Phi^{*}(K)\right]=\left[\Phi^{*}(K)\right]_{k}\delta\phi^{k}=0\,.

(4.1)

It remains to consider the non-exact expression of (2.5) involving

\psi_{+}^{\overline{\imath}}\lambda_{-}^{\overline{a}}\overline{D}_{\overline{\imath}}\overline{F}_{\overline{a}}-\psi_{+}^{i}\lambda_{-}^{a}D_{i}F_{a}\,.

First, we compute

	$\displaystyle\delta\left(\psi^{\overline{\imath}}_{+}\lambda^{\overline{a}}_{-}\overline{D}_{\overline{\imath}}\overline{F}_{\overline{a}}\right)$	$\displaystyle=\left(-\alpha_{-}\partial_{z}\phi^{\overline{\imath}}\right)\lambda^{\overline{a}}_{-}\,\overline{D}_{\overline{\imath}}\,\overline{F}_{\overline{a}}-\psi_{+}^{\overline{\imath}}\lambda_{-}^{\overline{a}}A^{\overline{b}}_{\overline{\imath}\,\overline{a},k}\left(i\alpha_{-}\psi^{k}_{+}\right)\overline{F}_{\overline{b}}$
		$\displaystyle\phantom{=}+\psi^{\overline{\imath}}_{+}\lambda_{-}^{\overline{a}}\left(\partial_{i}\overline{D}_{\overline{\imath}}\overline{F}_{\overline{a}}+F_{i\overline{\imath}\,a\overline{a}}\,h^{a\overline{b}}\,\overline{F}_{\overline{b}}\right)\left(i\alpha_{-}\psi^{i}_{+}\right).$		(4.2)

Now, we compute

$\displaystyle\delta\left(-\psi^{i}_{+}\lambda^{a}_{-}D_{i}F_{a}\right)$	$\displaystyle=-\,\alpha_{-}\overline{F}_{\overline{a}}\,D_{z}\lambda^{\overline{a}}_{-}+\left(i\alpha_{-}h^{a\overline{b}}\,\overline{F}_{\overline{b}}\right)F_{i\overline{\imath}a\overline{a}}\,\psi^{i}_{+}\psi^{\overline{\imath}}_{+}\lambda^{\overline{a}}_{-}$
	$\displaystyle=\left(\alpha_{-}\partial_{z}\phi^{\overline{\imath}}\right)\lambda^{\overline{a}}_{-}\,\overline{D}_{\overline{\imath}}\,\overline{F}_{\overline{a}}-\alpha_{-}\partial_{z}\!\left(\overline{F}_{\overline{a}}\,\lambda^{\overline{a}}_{-}\right)+\alpha_{-}\overline{F}_{\overline{a},k}\,\partial_{z}\phi^{k}\lambda^{\overline{a}}_{-}$
	$\displaystyle\phantom{=}+\psi^{\overline{\imath}}_{+}\lambda^{\overline{a}}_{-}\,A^{\overline{b}}_{\overline{\imath}\,\overline{a},k}\,\left(i\alpha_{-}\psi^{k}_{+}\right)\overline{F}_{\overline{b}}\,,$	(4.3)

where we have used the $\lambda^{a}_{-}$ equation of motion (2.4) in the first step and $F_{i\overline{\imath}a\overline{a}}=h_{a\overline{b}}\,A^{\overline{b}}_{\overline{\imath}\,\overline{a},i}$ in the last step. It follows that (4) cancels (4) up to a total derivative, i.e.

\delta\left(-\psi^{i}_{+}\lambda^{a}_{-}D_{i}F_{a}\right)=-\,\delta\left(\psi^{\overline{\imath}}_{+}\lambda^{\overline{a}}_{-}\,\overline{D}_{\overline{\imath}}\overline{F}_{\overline{a}}\right)-\alpha_{-}\partial_{z}\!\left(\overline{F}_{\overline{a}}\,\lambda^{\overline{a}}_{-}\right),

(4.4)

when both the curvature constraint

\partial_{i}\overline{D}_{\overline{\imath}}\overline{F}_{\overline{a}}+F_{i\overline{\imath}\,a\overline{a}}\,h^{a\overline{b}}\,\overline{F}_{\overline{b}}=0

(4.5)

and the constraint

\overline{F}_{\overline{a},k}\,\partial_{z}\phi^{k}\lambda^{\overline{a}}_{-}=0

(4.6)

are satisfied.

Curvature constraints have been used in GaravusoSharpe:Analogues to establish properties of analogues of pullbacks of Mathai-Quillen forms. These analogues arise in the correlation functions of the corresponding A-twisted or B-twisted heterotic Landau-Ginzburg models. The analogue most relevant to this paper, i.e. a deformation of the pullback of a Mathai-Quillen form, is discussed in appendix B.

5 Physical realization of deformation of the pullback of a Mathai-Quillen form

Let us now describe how the deformation $\omega_{\delta s}(\mathcal{G},\nabla)$ , given by (B.1), of the pullback $s^{*}u(\mathcal{G},\nabla)$ , given by (A.5), of a Mathai-Quillen form $u(\mathcal{G},\nabla)$ , given by (A.1), may arise in the class of A-twisted heterotic Landau-Ginzburg models discussed in this paper.

Mathematically, the tangent bundle to $Y\equiv\{s=0\}$ , $s=(s_{p})$ , is defined by the kernel in the short exact sequence

0\>\longrightarrow\>TY\>\longrightarrow\>TM|_{Y}\>\stackrel{{\scriptstyle(D_{i}s_{p})}}{{\longrightarrow}}\>{\cal G}|_{Y}\>\longrightarrow\>0.

A deformation of the tangent bundle above is defined by

0\>\longrightarrow\>{\mathcal{E}^{\prime}}\>\longrightarrow\>TM|_{Y}\>\stackrel{{\scriptstyle(D_{i}s_{p}+(\delta s)_{ip})}}{{\longrightarrow}}\>{\cal G}|_{Y}\>\longrightarrow\>0,

where the $(\delta s)_{ip}$ define the deformation.

The action of the A-twisted heterotic Landau-Ginzburg model that RG flows to a nonlinear sigma model with tangent bundle deformation above is given by GuffinSharpe:A-twistedheterotic

	$\displaystyle S$	$\displaystyle=2t\int_{\Sigma}d^{2}z\left[\frac{1}{2}\left(g_{\mu\nu}+iB_{\mu\nu}\right)\partial_{z}\phi^{\mu}\overline{\partial}_{\overline{z}}\phi^{\nu}+ig_{\overline{a}a}\psi^{\overline{a}}_{+}\overline{D}_{\overline{z}}\psi^{a}_{+}+ig_{b\overline{b}}\lambda^{b}_{-}D_{z}\lambda^{\overline{b}}_{-}\right.$
		$\displaystyle\phantom{=2t\int_{\Sigma}d^{2}z\left[\right.}\biggl{.}+R_{a\overline{a}b\overline{b}}\psi^{a}_{+}\psi^{\overline{a}}_{+}\lambda^{b}_{-}\lambda^{\overline{b}}_{-}+g^{a\overline{a}}F_{a}\overline{F}_{\overline{a}}+\psi^{a}_{+}\lambda^{b}_{-}D_{a}F_{b}+\psi^{\overline{a}}_{+}\lambda^{\overline{b}}_{-}\overline{D}_{\overline{a}}\overline{F}_{\overline{b}}\biggr{]},$		(5.1)

with target space

X\>=\>{\rm Tot}\left({\cal G}^{*}\>\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\>M\right)

and gauge bundle ${\cal E}=TX$ , where

F_{a}=(F_{p},F_{i})=\left(s_{p},\phi^{p}(D_{i}s_{p}+(\delta s)_{ip})\right)\,,\qquad\overline{F}_{\overline{a}}=\left(\overline{F}_{\overline{p}},\overline{F}_{\overline{\imath}}\right)=\left(\overline{s}_{\overline{p}},\phi^{\overline{p}}\left(\overline{D}_{\overline{\imath}}\overline{s}_{\overline{p}}+(\delta\overline{s})_{\overline{\imath}\,\overline{p}}\right)\right)\,,

	$\displaystyle D_{a}F_{b}$	$\displaystyle=\partial_{a}F_{b}-\Gamma^{c}_{ab}F_{c}\,,\quad$	$\displaystyle\overline{D}_{\overline{a}}\overline{F}_{\overline{b}}$	$\displaystyle=\overline{\partial}_{\overline{a}}\overline{F}_{\overline{b}}-\Gamma^{\overline{c}}_{\overline{a}\overline{b}}\overline{F}_{\overline{c}}\,,$
	$\displaystyle\overline{D}_{\overline{z}}\psi^{a}_{+}$	$\displaystyle=\overline{\partial}_{\overline{z}}\psi^{a}_{+}+\overline{\partial}_{\overline{z}}\phi^{b}\Gamma^{a}_{bc}\psi^{c}_{+}\,,\quad$	$\displaystyle D_{z}\lambda^{\overline{b}}_{-}$	$\displaystyle=\partial_{z}\lambda^{\overline{b}}_{-}+\partial_{z}\phi^{\overline{a}}\Gamma^{\overline{b}}_{\overline{a}\,\overline{c}}\lambda^{\overline{c}}_{-}\,,$

and

$\displaystyle\psi^{i}_{+}$	$\displaystyle\equiv\chi^{i}\!$	$\displaystyle\in$	$\displaystyle\Gamma\left(\Phi^{*}\!\left(T^{1,0}M\right)\right),\quad$	$\displaystyle\lambda^{i}_{-}$	$\displaystyle\equiv\lambda^{i}_{\overline{z}}\!$	$\displaystyle\in$	$\displaystyle\Gamma\left(\overline{K}_{\Sigma}\otimes\left(\Phi^{*}\!\left(T^{0,1}M\right)\right)\!^{\vee}\right),$
$\displaystyle\psi^{\overline{\imath}}_{+}$	$\displaystyle\equiv\psi^{\overline{\imath}}_{z}\!$	$\displaystyle\in$	$\displaystyle\Gamma\left(K_{\Sigma}\otimes\left(\Phi^{*}\!\left(T^{1,0}M\right)\right)\!^{\vee}\right),\quad$	$\displaystyle\lambda^{\overline{\imath}}_{-}$	$\displaystyle\equiv\lambda^{\overline{\imath}}\!$	$\displaystyle\in$	$\displaystyle\Gamma\left(\Phi^{*}\!\left(T^{0,1}M\right)\right),$
$\displaystyle\psi^{p}_{+}$	$\displaystyle\equiv\psi^{p}_{z}\!$	$\displaystyle\in$	$\displaystyle\Gamma\left(K_{\Sigma}\otimes\Phi^{*}\,T^{1,0}_{\pi}\right),\quad$	$\displaystyle\lambda^{p}_{-}$	$\displaystyle\equiv\lambda^{p}\!$	$\displaystyle\in$	$\displaystyle\Gamma\left(\left(\Phi^{*}\,T^{0,1}_{\pi}\right)\!^{\vee}\right),$
$\displaystyle\psi^{\overline{p}}_{+}$	$\displaystyle\equiv\chi^{\overline{p}}\!$	$\displaystyle\in$	$\displaystyle\Gamma\left(\left(\Phi^{*}\,T^{1,0}_{\pi}\right)\!^{\vee}\right),\quad$	$\displaystyle\lambda^{\overline{p}}_{-}$	$\displaystyle\equiv\lambda^{\overline{p}}_{\overline{z}}\!$	$\displaystyle\in$	$\displaystyle\Gamma\left(\overline{K}_{\Sigma}\otimes\Phi^{*}\,T^{0,1}_{\pi}\right),$
$\displaystyle\phi^{p}$	$\displaystyle\equiv p_{z}\!$	$\displaystyle\in$	$\displaystyle\Gamma\left(K_{\Sigma}\otimes\Phi^{*}\,T^{1,0}_{\pi}\right),\quad$	$\displaystyle\phi^{\overline{p}}$	$\displaystyle\equiv\overline{p}_{\overline{z}}\!$	$\displaystyle\in$	$\displaystyle\Gamma\left(\overline{K}_{\Sigma}\otimes\Phi^{*}\,T^{0,1}_{\pi}\right).$

If we restrict to zero modes on a genus zero worldsheet, in the degree zero sector we find the following interactions among zero modes:

	$\displaystyle g^{\overline{p}p}$	$\displaystyle\overline{F}_{\overline{p}}F_{p}+\chi^{i}\lambda^{p}D_{i}F_{p}+\chi^{\overline{p}}\lambda^{\overline{\imath}}\,\overline{D}_{\overline{p}}\overline{F}_{\overline{\imath}}+R_{i\overline{p}p\overline{\imath}}\chi^{i}\chi^{\overline{p}}\lambda^{p}\lambda^{\overline{\imath}}$
		$\displaystyle=g^{\overline{p}p}\overline{s}_{\overline{p}}s_{p}+\chi^{i}\lambda^{p}D_{i}s_{p}+\chi^{\overline{p}}\lambda^{\overline{\imath}}\left(\overline{D}_{\overline{\imath}}\overline{s}_{\overline{p}}+(\delta\overline{s})_{\overline{\imath}\overline{p}}\right)+R_{i\overline{p}p\overline{\imath}}\chi^{i}\chi^{\overline{p}}\lambda^{p}\lambda^{\overline{\imath}}\,.$

If we now complex conjugate so as to relate the heterotic expression above to standard mathematics conventions, we find

$\displaystyle g^{p\overline{p}}$	$\displaystyle s_{p}\overline{s}_{\overline{p}}+\chi^{\overline{\imath}}\lambda^{\overline{p}}\,\overline{D}_{\overline{\imath}}\overline{s}_{\overline{p}}+\chi^{p}\lambda^{i}\left(D_{i}s_{p}+(\delta s)_{ip}\right)+R_{\overline{\imath}p\overline{p}i}\chi^{\overline{\imath}}\chi^{p}\lambda^{\overline{p}}\lambda^{i}$
	$\displaystyle=g^{p\overline{p}}s_{p}\overline{s}_{\overline{p}}+\rho^{p}Ds_{p}+\overline{D}\overline{s}_{\overline{p}}\,\rho^{\overline{p}}+\rho^{\overline{p}}\mathcal{R}_{\overline{p}p}\rho^{p}+\rho^{p}d\phi^{i}\,(\delta s)_{ip}$
	$\displaystyle=\left(s_{p}e^{p},\overline{s}_{\overline{p}}e^{\overline{p}}\right)_{\cal G}+\left\langle\rho^{p^{\prime}}f_{p^{\prime}},Ds_{p}e^{p}\right\rangle_{\cal G}+\left\langle\overline{D}\overline{s}_{\overline{p}}e^{\overline{p}},\rho^{\overline{p}^{\prime}}f_{\overline{p}^{\prime}}\right\rangle_{\cal G}$
	$\displaystyle\phantom{=}+\left(\rho^{\overline{p}^{\prime}}f_{\overline{p}^{\prime}},f^{\overline{p}}\left(\mathcal{R}_{\overline{p}p}f^{p},\rho^{p}f_{p}\right)_{{\cal G}^{\vee}}\right)_{{\cal G}^{\vee}}+\left\langle\rho^{p^{\prime}}f_{p^{\prime}},d\phi^{i}\,(\delta s)_{ip}e^{p}\right\rangle_{\cal G}$
	$\displaystyle={\cal A}+\left\langle\rho^{p^{\prime}}f_{p^{\prime}},d\phi^{i}\,(\delta s)_{ip}e^{p}\right\rangle_{\cal G}$
	$\displaystyle={\cal A}_{\delta s}\,,$	(5.2)

which is minus the exponent of (B.1). The deformation $\omega_{\delta s}(\mathcal{G},\nabla)$ will appear in the corresponding correlation functions. Explicitly, from the discussion in appendix B, we see that

\langle\widetilde{\mathcal{O}}_{1}\cdots\widetilde{\mathcal{O}}_{k}\rangle\propto\int_{X}\widetilde{\mathcal{O}}_{1}\wedge\cdots\wedge\widetilde{\mathcal{O}}_{k}\wedge\omega_{\delta s}(\mathcal{G},\nabla)=\int_{Y}\mathcal{O}_{1}\wedge\cdots\wedge\mathcal{O}_{k}\,,

(5.3)

where

\widetilde{\mathcal{O}}_{1}\wedge\cdots\wedge\widetilde{\mathcal{O}}_{k}\in H^{\dim{M}-\textrm{rk}\,\mathcal{G}}\left(M,\wedge^{\textrm{rk}\,TM-\textrm{rk}\,\mathcal{G}}\mathcal{G}^{\vee}\right)

and $\widetilde{\mathcal{O}}\in H^{\bullet}\left(M,\wedge^{\bullet}(TM)^{\vee}\right)$ is a lift of $\mathcal{O}\in H^{\bullet}\left(Y,\wedge^{\bullet}\mathcal{\mathcal{E}}^{\prime\vee}\right)$ .

6 Summary and outlook

We have studied certain aspects of A-twisted heterotic Landau Ginzburg models on a Kähler variety $X$ with gauge bundle $\mathcal{E}$ , superpotential (1.1)

W=\Lambda^{a}F_{a}\,,

and $E^{a}\equiv 0$ . Table 3.2 provides a classification of the R-symmetries which allow the A-twist to be implemented when $\mathcal{E}$ is either a deformation of $TX$ or a deformation of a sub-bundle of $TX$ . Some anomaly-free examples were provided in section 3.2. When the superpotential is not holomorphic, supersymmetry imposes the curvature constraint (4.5)

\partial_{i}\overline{D}_{\overline{\imath}}\overline{F}_{\overline{a}}+F_{i\overline{\imath}\,a\overline{a}}\,h^{a\overline{b}}\,\overline{F}_{\overline{b}}=0

and the constraint (4.6)

\overline{F}_{\overline{a},k}\,\partial_{z}\phi^{k}\lambda^{\overline{a}}_{-}=0\,.

The curvature constraint (4.5) was used in GaravusoSharpe:Analogues to establish properties of the deformation $\omega_{\delta s}(\mathcal{G},\nabla)$ , given by (B.1), of the pullback $s^{*}u(\mathcal{G},\nabla)$ , given by (A.5), of a Mathai-Quillen form $u(\mathcal{G},\nabla)$ , given by (A.1). In section 5, we described how $\omega_{\delta s}(\mathcal{G},\nabla)$ may arise in the class of heterotic Landau-Ginzburg models studied in this paper.

It would be interesting to consider A-twisted and B-twisted heterotic Landau-Ginzburg models with more general Grassmann-odd superpotentials. For example, one may consider the superpotential GuffinSharpe:A-twisted

W=\Lambda^{a}\Lambda^{b}\Lambda^{c}F_{abc}\,.

If this more general superpotential is not holomorphic, then supersymmetry should impose a curvature constraint analogous to (4.5) and a constraint analogous to (4.6). These new constraints could be derived using arguments similar to those used in Garavuso:Curvature ; Garavuso:Nonholomorphic . Furthermore, using arguments similar to those used in GaravusoSharpe:Analogues , the new curvature constraint could then be used to establish properties of new analogues of pullbacks of Mathai-Quillen forms which arise in the correlation functions of the corresponding A-twisted or B-twisted heterotic Landau-Ginzburg models. We leave a detailed study of this to future work.

Acknowledgements.

The author thanks Mauricio Romo for useful discussions.

Appendix A Review of Mathai-Quillen formalism

Consider an oriented vector bundle $\mathcal{G}\overset{\pi}{\longrightarrow}M$ of real rank $r=2m$ , with standard fiber $V$ , where $M$ is an oriented closed manifold of real dimension $n\geq r$ . Suppose that $\mathcal{G}$ has Euclidean metric $(\cdot,\cdot)_{\mathcal{G}}$ and compatible connection $\nabla$ . Under these circumstances, the Mathai-Quillen formalism MathaiQuillen:Superconnections ; BerlinGetzlerVergne:Heat ; Kalkman:BRST ; Blau:The-Mathai-Quillen ; Wu:On-the-Mathai-Quillen ; CordesMooreRamgoolam:Lectures ; Wu:Mathai-Quillen provides an explicit representative $u(\mathcal{G},\nabla)$ of the Thom class of $\mathcal{G}$ . Furthermore, the pullback $s^{*}u(\mathcal{G},\nabla)$ of $u(\mathcal{G},\nabla)$ by any section $s:M\rightarrow\mathcal{G}$ of $\mathcal{G}$ is a representative of the Euler class of $\mathcal{G}$ . Let us review the formalism in more detail.

A.1 Conventions

Our conventions for $M$ , $\mathcal{G}$ , and the dual $\mathcal{G}^{\vee}$ of $\mathcal{G}$ are as follows. The exterior derivatives on $M$ and $\mathcal{G}$ are respectively denoted by $d$ and $d^{\mathcal{G}}$ . We choose local coordinates $\phi^{I}$ on $M$ , where $I=1,\ldots,n$ . The connection on $\mathcal{G}$ is then given by $\nabla=d\phi^{I}\nabla_{I}$ . In terms of this connection, the curvature 2-form on $\mathcal{G}$ is given by $\mathcal{R}=\nabla^{2}$ . We choose a local oriented orthonormal frame $\{e_{A}\}$ for $\mathcal{G}$ and let $\{f^{A}\}$ be the dual coframe, where $A=1,\ldots,r$ . The section $s$ may thus be expressed as $s=s^{A}e_{A}$ . Similarly, we write $\rho=\rho_{A}f^{A}$ , where the $\rho_{A}$ are anticommuting orthonormal coordinates on $\mathcal{G}^{\vee}$ . The dual pairing on $\mathcal{G}$ is denoted by $\langle\cdot,\cdot\rangle_{\mathcal{G}}$ . Finally, the metric on $\mathcal{G}^{\vee}$ is denoted by $(\cdot,\cdot)_{\mathcal{G}^{\vee}}$ .

Now, consider the pullback bundle $\pi^{*}\mathcal{G}\rightarrow\mathcal{G}$ , i.e. the bundle over $\mathcal{G}$ whose fiber at $g\in\mathcal{G}$ is $(\pi^{*}\mathcal{G})_{g}=\mathcal{G}_{\pi(g)}$ . This bundle has Euclidean metric $\pi^{*}(\cdot,\cdot)_{\mathcal{G}}\equiv(\cdot,\cdot)_{\pi^{*}\mathcal{G}}$ , compatible connection $\pi^{*}\nabla\equiv\widetilde{\nabla}$ , curvature 2-form $\pi^{*}\mathcal{R}\equiv\widetilde{\mathcal{R}}$ , local oriented orthonormal frame $\{\pi^{*}e_{A}\}\equiv\{\tilde{e}_{A}\}$ , and tautological section $\tilde{x}=\tilde{x}^{A}\tilde{e}_{A}$ . (The tautological section of $\pi^{*}\mathcal{G}\rightarrow\mathcal{G}$ is the section which maps a point $g\in\mathcal{G}$ to $(g,g)\in\pi^{*}\mathcal{G}$ .) The dual bundle $(\pi^{*}\mathcal{G})^{\vee}\rightarrow\mathcal{G}$ has coframe $\{(\pi^{\vee})^{*}f^{A}\}\equiv\{\tilde{f}^{A}\}$ and metric $(\cdot,\cdot)_{(\pi^{*}\mathcal{G})^{\vee}}$ . We write $\tilde{\rho}=\tilde{\rho}_{A}\tilde{f}^{A}$ , where the $\tilde{\rho}_{A}\equiv(\pi^{\vee})^{*}\rho_{A}$ are anticommuting orthonormal coordinates on $(\pi^{*}\mathcal{G})^{\vee}$ . The dual pairing on $\pi^{*}E$ is denoted by $\langle\cdot,\cdot\rangle_{\pi^{*}\mathcal{G}}$ .

A.2 Mathai-Quillen Thom class representative

Consider the Mathai-Quillen form

u(\mathcal{G},\nabla)=a_{r}\int d\tilde{\rho}\,\exp\left(-\tilde{\mathcal{A}}\right),

(A.1)

where

a_{r}=\frac{(-1)^{\frac{r(r+1)}{2}}}{(2\pi)^{\frac{r}{2}}}

(A.2)

and

\tilde{\mathcal{A}}=\frac{1}{2}\Bigl{(}\tilde{x},\tilde{x}\Bigr{)}_{\pi^{*}\mathcal{G}}+\left\langle\widetilde{\nabla}\tilde{x},\tilde{\rho}\right\rangle_{\pi^{*}\mathcal{G}}+\frac{1}{2}\left(\tilde{\rho},\widetilde{\mathcal{R}}\tilde{\rho}\right)_{(\pi^{*}\mathcal{G})^{\vee}}.

(A.3)

We wish to show that this form satisfies the following definition.

Definition A.1.

A representative of the Thom class of $\mathcal{G}$ is a $d^{\mathcal{G}}$ -closed differential form $u(\mathcal{G})\in\Omega^{r}(\mathcal{G})$ such that $\int_{V}u(\mathcal{G})=1$ .

Proposition A.2.

The Mathai-Quillen form $u(\mathcal{G},\nabla)$ satisfies

(i)

$u(\mathcal{G},\nabla)\in\Omega^{r}(\mathcal{G})\,,$
(ii)

$d^{\mathcal{G}}u(\mathcal{G},\nabla)=0\,,$
(iii)

$\int_{V}u(\mathcal{G},\nabla)=1$

and hence is a representative of the Thom class of $\mathcal{G}$ .

Proof.

(i)

Since

\tilde{\mathcal{A}}\in\overset{2}{\underset{i=0}{\oplus}}\Omega^{i}\left(\mathcal{G},\Lambda^{i}\left(\pi^{*}\mathcal{G}\right)^{\vee}\right),

it follows that

\exp\left(-\tilde{\mathcal{A}}\right)\in\overset{r}{\underset{i=0}{\oplus}}\Omega^{i}\left(\mathcal{G},\Lambda^{i}\left(\pi^{*}\mathcal{G}\right)^{\vee}\right).

However, only the component of $e^{-\tilde{\mathcal{A}}}$ in $\Omega^{r}\left(\mathcal{G},\Lambda^{r}\left(\pi^{*}\mathcal{G}\right)^{\vee}\right)$ contributes to $u(\mathcal{G},\nabla)$ . Thus, $u(\mathcal{G},\nabla)\in\Omega^{r}(\mathcal{G})$ .

(ii)

Since $\widetilde{\nabla}$ is compatible with the metric $\left(\cdot,\cdot\right)_{\pi^{*}\mathcal{G}}$ , it follows that

d^{\mathcal{G}}\int d\tilde{\rho}\,\tilde{\alpha}=\int d\tilde{\rho}\,\widetilde{\nabla}\tilde{\alpha}\,,

where $\tilde{\alpha}\in\Omega\left(\mathcal{G},\Lambda\left(\pi^{*}\mathcal{G}\right)^{\vee}\right)$ . Furthermore,

$\displaystyle\left(\widetilde{\nabla}+\tilde{x}_{A}\frac{\partial}{\partial\tilde{\rho}_{A}}\right)\tilde{\mathcal{A}}$	$\displaystyle=\left(\widetilde{\nabla}\tilde{x},\tilde{x}\right)_{\pi^{}\mathcal{G}}+\left\langle\widetilde{\mathcal{R}}\tilde{x},\tilde{\rho}\right\rangle_{\pi^{}\mathcal{G}}-\frac{1}{2}\left(\tilde{\rho},\widetilde{\nabla}\widetilde{\mathcal{R}}\tilde{\rho}\right)_{\left(\pi^{*}\mathcal{G}\right)^{\vee}}$
	$\displaystyle\phantom{=}-\left(\widetilde{\nabla}\tilde{x},\tilde{x}\right)_{\pi^{}\mathcal{G}}-\left\langle\widetilde{\mathcal{R}}\tilde{x},\tilde{\rho}\right\rangle_{\pi^{}\mathcal{G}}$
	$\displaystyle=0\,,$	(A.4)

where we have used the Bianchi identity $\widetilde{\nabla}\,\widetilde{\mathcal{R}}=0$ . From these results, we obtain

	$\displaystyle d^{\mathcal{G}}u(\mathcal{G},\nabla)$	$\displaystyle=a_{r}\,d^{\mathcal{G}}\int d\tilde{\rho}\,\exp\left(-\tilde{\mathcal{A}}\right)$
		$\displaystyle=a_{r}\int d\tilde{\rho}\,\widetilde{\nabla}\exp\left(-\tilde{\mathcal{A}}\right)$
		$\displaystyle=a_{r}\int d\tilde{\rho}\,\left(\widetilde{\nabla}+\tilde{x}_{A}\frac{\partial}{\partial\tilde{\rho}_{A}}\right)\exp\left(-\tilde{\mathcal{A}}\right)$
		$\displaystyle=a_{r}\int d\tilde{\rho}\,\left[-\left(\widetilde{\nabla}+\tilde{x}_{A}\frac{\partial}{\partial\tilde{\rho}_{A}}\right)\tilde{\mathcal{A}}\right]\exp\left(-\tilde{\mathcal{A}}\right)$
		$\displaystyle=0\,.$

Here, the third equality holds because $\tilde{x}_{A}\left(\partial/\partial\tilde{\rho}_{A}\right)e^{-\tilde{\mathcal{A}}}$ contributes nothing to the Grassmann integral.

(iii)

	$\displaystyle\int_{V}u(\mathcal{G},\nabla)$	$\displaystyle=a_{r}\int_{V}\exp\left[-\frac{1}{2}\left(\tilde{x},\tilde{x}\right)_{\pi^{*}\mathcal{G}}\right]\int d\tilde{\rho}\,\frac{\left(-d\tilde{x}^{A}\tilde{\rho}_{A}\right)^{r}}{r!}$
		$\displaystyle=\frac{1}{(2\pi)^{\frac{r}{2}}}\int_{V}d\tilde{x}^{1}\wedge\cdots\wedge d\tilde{x}^{r}\,\exp\left[-\frac{1}{2}\left(\tilde{x},\tilde{x}\right)_{\pi^{*}\mathcal{G}}\right]$
		$\displaystyle=1\,.$

Thus, by definition A.1, $u(\mathcal{G},\nabla)$ is a representative of the Thom class of $\mathcal{G}$ . ∎

A.3 Mathai-Quillen Euler class representative

Now, consider the pullback of the Mathai-Quillen form $u(\mathcal{G},\nabla)$ by any section $s$ of $\mathcal{G}$ . We write this as

s^{*}u(\mathcal{G},\nabla)=a_{r}\int d\rho\,\exp\left(-\mathcal{A}\right),

(A.5)

where $a_{r}$ is given by (A.2) and

\mathcal{A}=\frac{1}{2}\left(s,s\right)_{\mathcal{G}}+\left\langle\nabla s,\rho\right\rangle_{\mathcal{G}}+\frac{1}{2}\left(\rho,\mathcal{R}\rho\right)_{\mathcal{G}^{\vee}}.

(A.6)

Proposition A.3.

The form $s^{*}u(\mathcal{G},\nabla)$ satisfies

(i)

$s^{*}u(\mathcal{G},\nabla)\in\Omega^{r}(M)\,,$
(ii)

$ds^{*}u(\mathcal{G},\nabla)=0\,.$

Proof.

(i)

The proof is similar to that of proposition A.2 (i) and uses the fact that

$\mathcal{A}\in\overset{2}{\underset{i=0}{\oplus}}\Omega^{i}\left(\mathcal{G},\Lambda^{i}\mathcal{G}^{\vee}\right).$
(ii)

The proof is similar to that of proposition A.2 (ii) and uses the results

$d\int d\rho\,\alpha=\int d\rho\,\nabla\alpha\,,$

where $\alpha\in\Omega\left(\mathcal{G},\Lambda\mathcal{G}^{\vee}\right)$ , and

$\left(\nabla+s_{A}\frac{\partial}{\partial\rho_{A}}\right)\mathcal{A}=0\,.$ (A.7)

∎

Proposition A.4.

The $d$ -cohomology class of $s^{*}u(\mathcal{G},\nabla)$ is independent of the section $s$ .

Proof.

Let $s_{\tau}=s+\tau s^{\prime}$ be an affine one-parameter family of sections of $\mathcal{G}$ and let

\mathcal{A}_{\tau}=\frac{1}{2}\left(s_{\tau},s_{\tau}\right)_{\mathcal{G}}+\left\langle\nabla s_{\tau},\rho\right\rangle_{\mathcal{G}}+\frac{1}{2}\left(\rho,\mathcal{R}\rho\right)_{\mathcal{G}^{\vee}}.

Then

	$\displaystyle\frac{d}{d\tau}s^{*}_{\tau}u(\mathcal{G},\nabla)$	$\displaystyle=a_{r}\,\frac{d}{d\tau}\int d\rho\,\exp\left(-\mathcal{A}_{\tau}\right)$
		$\displaystyle=-a_{r}\int d\rho\,\Bigl{[}\left(s^{\prime},s_{\tau}\right)_{\mathcal{G}}+\left\langle\nabla s^{\prime},\rho\right\rangle_{\mathcal{G}}\Bigr{]}\exp\left(-\mathcal{A}_{\tau}\right)$
		$\displaystyle=-a_{r}\,\int d\rho\,\left\{\left[\nabla+\left(s_{\tau}\right)_{A}\frac{\partial}{\partial\rho_{A}}\right]\left\langle s^{\prime},\rho\right\rangle_{\mathcal{G}}\right\}\exp\left(-\mathcal{A}_{\tau}\right)$
		$\displaystyle=-a_{r}\,\int d\rho\,\left[\nabla+\left(s_{\tau}\right)_{A}\frac{\partial}{\partial\rho_{A}}\right]\Bigl{[}\left\langle s^{\prime},\rho\right\rangle_{\mathcal{G}}\,\exp\left(-\mathcal{A}_{\tau}\right)\Bigr{]}$
		$\displaystyle=-a_{r}\,d\int d\rho\,\left\langle s^{\prime},\rho\right\rangle_{\mathcal{G}}\,\exp\left(-\mathcal{A}_{\tau}\right).$

It follows that

s^{*}_{\tau_{2}}u(\mathcal{G},\nabla)-s^{*}_{\tau_{1}}u(\mathcal{G},\nabla)=-a_{r}\,d\int^{\tau_{2}}_{\tau_{1}}d\tau\int d\rho\,\left\langle s^{\prime},\rho\right\rangle_{\mathcal{G}}\,\exp\left(-\mathcal{A}_{\tau}\right).

Thus, for arbitrary sections $s_{\tau_{1}}$ and $s_{\tau_{2}}$ of $\mathcal{G}$ , the $d$ -closed forms $s^{*}_{\tau_{1}}u(\mathcal{G},\nabla)$ and $s^{*}_{\tau_{2}}u(\mathcal{G},\nabla)$ differ by a $d$ -exact form and hence are cohomologous. ∎

Corollary A.5.

The form $s^{*}u(\mathcal{G},\nabla)$ is cohomologous to the Euler form

e(\mathcal{G},\nabla)=\frac{1}{(2\pi)^{\frac{r}{2}}}\int d\rho\,\exp\left[\frac{1}{2}\left(\rho,\mathcal{R}\rho\right)_{\mathcal{G}^{\vee}}\right]=\mathrm{Pfaff}\left(\frac{\mathcal{R}}{2\pi}\right)

and hence is a representative of the Euler class of $\mathcal{G}$ .

Proof.

This follows from proposition A.4 upon choosing $s$ to be the zero section. ∎

Remark A.6.

The top Chern class of a complex vector bundle is equal to the Euler class of the underlying real vector bundle.

Remark A.7.

If $s$ intersects the zero section of $\mathcal{G}$ transversely, then $s^{*}u(\mathcal{G},\nabla)$ is Poincaré dual to $s^{-1}(0)$ , i.e.

\int_{M}\omega\wedge s^{*}u(\mathcal{G},\nabla)=\int_{s^{-1}(0)}\omega\,,

(A.8)

where $\omega\in\Omega^{n-r}(M)$ is $d$ -closed.

Remark A.8.

When $n=r$ , integrating $s^{*}u(\mathcal{G},\nabla)$ over $M$ yields the Euler number of $\mathcal{G}$ .

Appendix B Deformation of the pullback of a Mathai-Quillen form

Various analogues of pullbacks of Mathai-Quillen forms were proposed in GaravusoSharpe:Analogues . Let us now discuss the analogue that is most relevant to this paper, i.e. a deformation of the pullback of a Mathai-Quillen form.

Consider deforming $s^{*}u(\mathcal{G},\nabla)$ to

\omega_{\delta s}(\mathcal{G},\nabla)=a_{r}\int d\rho\,\exp\left(-\mathcal{A}_{\delta s}\right),

(B.1)

where

\mathcal{A}_{\delta s}=\mathcal{A}+\left\langle\rho^{p^{\prime}}f_{p^{\prime}},d\phi^{i}\left(\delta s\right)_{ip}e^{p}\right\rangle_{\mathcal{G}}.

(B.2)

Here, $\mathcal{A}$ is given by (A.6) and

\left(\delta s\right)_{ip}\in\Gamma\left(\pi^{*}\mathcal{G}\otimes\pi^{*}TM\right).

(B.3)

The deformation $\omega_{\delta s}(\mathcal{G},\nabla)$ and $s^{*}u(\mathcal{G},\nabla)$ are special cases of the analogue $\omega_{\textrm{K1}}$ of $s^{*}u(\mathcal{G},\nabla)$ proposed in GaravusoSharpe:Analogues . Briefly,

\omega_{\textrm{K1}}\propto{\textstyle\int\left[{\textstyle\prod}_{\overline{x}}\,d\lambda^{\overline{x}}\,\right]\left[\prod_{r}d\chi^{r}\right]}\exp\left(-\mathcal{A_{\textrm{K1}}}\right)\in H^{\,\textrm{rk}\,\mathcal{G}}\left(M,\wedge^{\textrm{rk}\,\mathcal{F}_{2}}\mathcal{F}^{\vee}_{2}\otimes\det{\mathcal{G}^{\vee}}\otimes\det{\mathcal{F}_{2}}\right),

where $\mathcal{F}_{1}$ and $\mathcal{F}_{1}$ are holomorphic vector bundles on $M$ and

\mathcal{A_{\textrm{K1}}}=h^{x\overline{x}}s_{x}\overline{s}_{\overline{x}}+\chi^{\overline{\imath}}\lambda^{\overline{x}}\,\overline{D}_{\overline{\imath}}\,\overline{s}_{\overline{x}}\,+\chi^{r}\lambda^{\gamma}\widetilde{F}_{r\gamma}+F_{\overline{\imath}r\overline{x}\gamma}\chi^{\overline{\imath}}\chi^{r}\lambda^{\overline{x}}\lambda^{\gamma}\,.

Here, $x$ indexes local coordinates along the fibers of $\mathcal{G}$ , $\gamma$ indexes local coordinates along the fibers of $\mathcal{F}_{1}$ , $r$ indexes local coordinates along the fibers of $\mathcal{F}^{\vee}_{2}$ , and $i$ indexes local coordinates on $M$ . $s\in\Gamma{(\mathcal{G})}$ . The map $\widetilde{F}:\mathcal{F}_{1}\rightarrow\mathcal{F}_{2}$ is smooth and surjective. The curvature term $F_{\overline{\imath}r\overline{x}\gamma}\chi^{\overline{\imath}}\chi^{r}\lambda^{\overline{x}}\lambda^{\gamma}$ is subject to the constraint

\overline{\partial}_{\overline{\imath}}\widetilde{F}_{r\gamma}=h^{x\overline{x}}s_{x}F_{\overline{\imath}r\gamma\overline{x}}=-h^{x\overline{x}}s_{x}F_{\overline{\imath}r\overline{x}\gamma}\,

which is imposed physically by supersymmetry. Note that this constraint is consistent with the curvature 2-form being $\overline{\partial}$ -closed by virtue of the Bianchi identity. One may show that

\left(\overline{D}+h^{x\overline{x}}s_{x}\frac{\partial}{\partial\lambda^{\overline{x}}}\right)\mathcal{A_{\textrm{K1}}}=\chi^{i}\chi^{r}\lambda^{\gamma}\left(\overline{\partial}_{\overline{\imath}}\widetilde{F}_{r\gamma}+h^{x\overline{x}}s_{x}F_{\overline{\imath}r\overline{x}\gamma}\right)=0\,,

where $\overline{D}=\chi^{\overline{\imath}}\,\overline{\partial}_{\overline{\imath}}$ . It follows that $\overline{\partial}\,\omega_{\textrm{K1}}=0$ . Let $Y\equiv\left\{s=0\right\}\subset M$ and let $\mathcal{E}^{\prime}$ be the restriction to $Y$ of the kernel of the map $\widetilde{F}$ . Then

\int_{Y}\mathcal{O}_{1}\wedge\cdots\wedge\mathcal{O}_{k}=\int_{M}\widetilde{\mathcal{O}}_{1}\wedge\cdots\wedge\widetilde{\mathcal{O}}_{k}\wedge\omega_{\textrm{K1}}\,,

where $\widetilde{\mathcal{O}}_{1}\wedge\cdots\wedge\widetilde{\mathcal{O}}_{k}\in H^{\dim{M}-\textrm{rk}\,\mathcal{G}}\left(M,\wedge^{\textrm{rk}\,\mathcal{F}_{1}-\textrm{rk}\,\mathcal{F}_{2}}\mathcal{F}^{\vee}_{2}\right)$ and $\widetilde{\mathcal{O}}\in H^{\bullet}\left(M,\wedge^{\bullet}\mathcal{F}^{\vee}_{1}\right)$ is a lift of $\mathcal{O}\in H^{\bullet}\left(Y,\wedge^{\bullet}\mathcal{\mathcal{E}}^{\prime\vee}\right)$ . For this reason, $\omega_{\textrm{K1}}$ is called in GaravusoSharpe:Analogues the (first) kernel construction. See GaravusoSharpe:Analogues for further details. One recovers $\omega_{\delta s}(\mathcal{G},\nabla)$ and (when $\delta s=0$ ) $s^{*}u(\mathcal{G},\nabla)$ in the special case that $\mathcal{F}_{1}=TM$ and $\mathcal{F}_{2}=\mathcal{G}$ with the map $\widetilde{F}:\mathcal{F}_{1}\rightarrow\mathcal{F}_{2}$ defined by

F_{ip}=D_{i}s_{p}+\left(\delta s\right)_{ip}\,,

(B.4)

where $s_{p}$ is a holomorphic section of $\mathcal{G}$ . This corresponds to $\mathcal{E}^{\prime}$ being a deformation of $TY$ , with the deformation determined by $\delta s$ . If $\delta s=0$ , then $\mathcal{E}^{\prime}=TY$ . Note that

\overline{\partial}_{\overline{\imath}}\,F_{ip}=\overline{\partial}_{\overline{\imath}}\left(D_{i}s_{p}+\left(\delta s\right)_{ip}\right)=\left[\,\overline{D}_{\overline{\imath}}\,,D_{i}\right]s_{p}=R_{\overline{\imath}ip\overline{p}}\,g^{p\overline{p}}s_{p}\,.

Proposition B.1.

The form $\omega_{\delta s}(\mathcal{G},\nabla)$ satisfies

\overline{\partial}\,\omega_{\delta s}(\mathcal{G},\nabla)=0\,.

Proof.

For $\mathcal{A}$ given by (A.6), we have that $\overline{D}\mathcal{R}=0$ and hence

$\displaystyle\left(\overline{D}+s^{\overline{p}}\frac{\partial}{\partial\rho^{\overline{p}}}\right)\mathcal{A}$	$\displaystyle=-\left(s_{p}e^{p},\overline{D}s_{\overline{p}}\,e^{\overline{p}}\right)_{\mathcal{G}}+\left\langle\mathcal{R}s_{p^{\prime}}e^{p^{\prime}},\rho^{p}f_{p}\right\rangle_{\mathcal{G}}-\frac{1}{2}\left(\rho,\overline{D}\mathcal{R}\rho\right)_{\mathcal{G}^{\vee}}$
	$\displaystyle\phantom{=}+\left(s_{p}e^{p},\overline{D}s_{\overline{p}}\,e^{\overline{p}}\right)_{\mathcal{G}}-\left\langle\mathcal{R}s_{p^{\prime}}e^{p^{\prime}},\rho^{p}f_{p}\right\rangle_{\mathcal{G}}$
	$\displaystyle=0\,.$	(B.5)

It follows that

\left(\overline{D}+s^{\overline{p}}\frac{\partial}{\partial\rho^{\overline{p}}}\right)\mathcal{A}_{\delta s}=\left(\overline{D}+s^{\overline{p}}\frac{\partial}{\partial\rho^{\overline{p}}}\right)\left[\mathcal{A}+\left\langle\rho^{p^{\prime}}f_{p^{\prime}},d\phi^{i}\left(\delta s\right)_{ip}e^{p}\right\rangle_{\mathcal{G}}\right]=0\,.

(B.6)

Using this result, we obtain

	$\displaystyle\overline{\partial}\,\omega_{\delta s}(\mathcal{G},\nabla)$	$\displaystyle=a_{r}\,\overline{\partial}\int d\rho\,\exp\left(-\mathcal{A}_{\delta s}\right)$
		$\displaystyle=a_{r}\int d\rho\,\overline{D}\exp\left(-\mathcal{A}_{\delta s}\right)$
		$\displaystyle=a_{r}\int d\rho\,\left(\overline{D}+s^{\overline{p}}\frac{\partial}{\partial\rho^{\overline{p}}}\right)\exp\left(-\mathcal{A}_{\delta s}\right)$
		$\displaystyle=a_{r}\int d\rho\,\left[-\left(\overline{D}+s^{\overline{p}}\frac{\partial}{\partial\rho^{\overline{p}}}\right)\mathcal{A}_{\delta s}\right]\exp\left(-\mathcal{A}_{\delta s}\right)$
		$\displaystyle=0\,.$

∎

Proposition B.2.

The $\overline{\partial}$ -cohomology class of $\omega_{\delta s}(\mathcal{G},\nabla)$ is unchanged by antiholomorphic deformations of $s$ .

Proof.

Let $s_{\alpha}=s+\alpha\,s^{\prime}_{\overline{p}}\,e^{\overline{p}}$ be an affine one parameter family of sections of $\mathcal{G}$ and let

\mathcal{A}_{\delta s,\alpha}=\frac{1}{2}\left(s_{\alpha},s_{\alpha}\right)_{\mathcal{G}}+\left(\nabla s_{\alpha},\rho\right)_{\mathcal{G}}+\frac{1}{2}\left(\rho,\mathcal{R}\rho\right)_{\mathcal{G}^{\vee}}+\left\langle\rho^{p^{\prime}}f_{p^{\prime}},d\phi^{i}\left(\delta s\right)_{ip}e^{p}\right\rangle_{\mathcal{G}}.

Then

	$\displaystyle\frac{d}{d\alpha}\omega_{\delta s,\alpha}(\mathcal{G},\nabla)$	$\displaystyle=a_{r}\,\frac{d}{d\alpha}\int d\rho\,\exp\left(-\mathcal{A}_{\delta s,\alpha}\right)$
		$\displaystyle=-a_{r}\int d\rho\,\left[\left(s^{\prime}_{\overline{p}}\,e^{\overline{p}},s_{p}e^{p}\right)_{\mathcal{G}}+\left\langle\rho^{\overline{p}}f_{\overline{p}},\overline{D}s^{\prime}_{\overline{p}^{\prime}}\,e^{\overline{p}^{\prime}}\right\rangle_{\mathcal{G}}\right]\exp\left(-\mathcal{A}_{\delta s,\alpha}\right)$
		$\displaystyle=-a_{r}\int d\rho\,\left[\left(\overline{D}+s^{\overline{p}}\frac{\partial}{\partial\rho^{\overline{p}}}\right)\left\langle\rho^{\overline{p}}f_{\overline{p}},s^{\prime}_{\overline{p}^{\prime}}\,e^{\overline{p}^{\prime}}\right\rangle_{\mathcal{G}}\right]\exp\left(-\mathcal{A}_{\delta s,\alpha}\right)$
		$\displaystyle=-a_{r}\int d\rho\,\left(\overline{D}+s^{\overline{p}}\frac{\partial}{\partial\rho^{\overline{p}}}\right)\left[\left\langle\rho^{\overline{p}}f_{\overline{p}},s^{\prime}_{\overline{p}^{\prime}}\,e^{\overline{p}^{\prime}}\right\rangle_{\mathcal{G}}\exp\left(-\mathcal{A}_{\delta s,\alpha}\right)\right]$
		$\displaystyle=-a_{r}\,\overline{\partial}\int d\rho\,\left\langle\rho^{\overline{p}}f_{\overline{p}},s^{\prime}_{\overline{p}^{\prime}}\,e^{\overline{p}^{\prime}}\right\rangle_{\mathcal{G}}\,\exp\left(-\mathcal{A}_{\delta s,\alpha}\right).$

It follows that

\omega_{\delta s,\alpha_{2}}(\mathcal{G},\nabla)-\omega_{\delta s,\alpha_{1}}(\mathcal{G},\nabla)=-a_{r}\,\overline{\partial}\int^{\alpha_{2}}_{\alpha_{1}}d\alpha\int d\rho\,\left\langle\rho^{\overline{p}}f_{\overline{p}},s^{\prime}_{\overline{p}^{\prime}}\,e^{\overline{p}^{\prime}}\right\rangle_{\mathcal{G}}\exp\left(-\mathcal{A}_{\delta s,\alpha}\right),

which establishes that the $\overline{\partial}$ -cohomology class of $\omega_{\delta s}(\mathcal{G},\nabla)$ is unchanged by antiholomorphic deformations of $s$ . ∎

Remark B.3.

The $\overline{\partial}$ -cohomology class of $\omega_{\delta s}(\mathcal{G},\nabla)$ does seem to depend on the choice of the $\left(\delta s\right)_{ip}$ , at least naively. Let $\left(\delta s\right)_{ip,\gamma}=\left(\delta s\right)_{ip}+\gamma\left(\delta s\right)^{\prime}_{ip}$ and $\mathcal{A}_{\delta s,\gamma}=\mathcal{A}+\left\langle\rho^{p^{\prime}}f_{p^{\prime}},d\phi^{i}\left(\delta s\right)^{\gamma}_{ip}e^{p}\right\rangle_{\mathcal{G}}.$ Then

	$\displaystyle\frac{d}{d\gamma}\omega_{\delta s,\gamma}(\mathcal{G},\nabla)$	$\displaystyle=a_{r}\,\frac{d}{d\gamma}\int d\rho\,\exp\left(-\mathcal{A}_{\delta s,\gamma}\right)$
		$\displaystyle=-a_{r}\int d\rho\,\left\langle\rho^{p^{\prime}}f_{p^{\prime}},d\phi^{i}\left(\delta s\right)^{\prime}_{ip}e^{p}\right\rangle_{\mathcal{G}}\exp\left(-\mathcal{A}_{\delta s,\gamma}\right).$

It follows that

\omega_{\delta s,\gamma_{2}}(\mathcal{G},\nabla)-\omega_{\delta s,\gamma_{1}}(\mathcal{G},\nabla)=-a_{r}\int^{\gamma_{2}}_{\gamma_{1}}d\gamma\int d\rho\,\left\langle\rho^{p^{\prime}}f_{p^{\prime}},d\phi^{i}\left(\delta s\right)^{\prime}_{ip}e^{p}\right\rangle_{\mathcal{G}}\exp\left(-\mathcal{A}_{\delta s,\gamma}\right),

which is at least not obviously $\overline{\partial}$ -exact. The physical meaning of this result is commented on in GaravusoSharpe:Analogues .

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