Non-holomorphic Deformations of Special Geometry and Their Applications

Abstract

The aim of these lecture notes is to give a pedagogical introduction to the subject of non-holomorphic deformations of special geometry. This subject was first introduced in the context of $N=2$ BPS black holes, but has a wider range of applicability. A theorem is presented according to which an arbitrary point-particle Lagrangian can be formulated in terms of a complex function $F$, whose features are analogous to those of the holomorphic function of special geometry. A crucial role is played by a symplectic vector that represents a complexification of the canonical variables, i.e. the coordinates and canonical momenta. We illustrate the characteristic features of the theorem in the context of field theory models with duality invariances. The function $F$ may depend on a number of external parameters that are not subject to duality transformations. We introduce duality covariant complex variables whose transformation rules under duality are independent of these parameters. We express the real Hesse potential of $N=2$ supergravity in terms of the new variables and expand it in powers of the external parameters. Then we relate this expansion to the one encountered in topological string theory. These lecture notes include exercises which are meant as a guidance to the reader.

Lectures given by G. L. Cardoso at the 2011 Frascati School on Black Objects in Supergravity, 9–13 May 2011

ITP-UU-12/20, Nikhef-2012-010.

Appendices

A   Symplectic Reparametrizations

In Sect. 1.2.1 we introduced the $2n$-vector $(x^i, F_i)$ and discussed its behavior under symplectic transformations. Here we consider derivatives of $F_i$ and show how they transform under symplectic transformations. We use the resulting expressions to give an alternative proof of integrability of the Eq. (1.10). In addition, we show that $\partial _{\eta } F$ transforms as a function under symplectic transformations.

We begin by recalling some of the elements of Sect.  1.2.1. The $2n$-vector $(x^i, F_i)$ is constructed using $F(x, \bar{x}) = F^{(0)} (x) + 2 \mathrm{i } \varOmega (x, \bar{x})$. Under symplectic transformations, it transforms as,

$$\begin{aligned} \tilde{x}^i=&\, U^i{}_j\,x^j + Z^{ij} [F^{(0)}_j(x) +2\mathrm{i } \varOmega _j(x,\bar{x})],\nonumber \\ \tilde{F}_i (\tilde{x},\bar{\tilde{x}}) =&\, V_i{}^j\,[F^{(0)}_j(x) +2\mathrm{i } \varOmega _j(x,\bar{x})] + W_{ij} \,x^j, \end{aligned}$$

(1.158)

where $U, V, Z$ and $W$ are the $n \times n$ submatrices (1.5) that define a symplectic transformation belonging to $\mathrm{{Sp}} (2n, \mathbb{R })$. Without loss of generality, we decompose $\tilde{F}_i$ as

$$\begin{aligned} \tilde{F}_i (\tilde{x},\bar{\tilde{x}}) = \tilde{F}^{(0)}_{i}(\tilde{x}) + 2\mathrm{i } \tilde{\varOmega }_i(\tilde{x},\bar{\tilde{x}}). \end{aligned}$$

(1.159)

This decomposition, which a priori is arbitrary, can be related to the decomposition of $F_i = F_i^{(0)} + 2 \mathrm{i } \varOmega _i$ in the following way. The symplectic transformation (1.158) is specified by the matrices $U, V, W$ and $Z$. Consider applying the same transformation (specified by these matrices) to the vector $(x^i, F_i^{(0)})$ alone. This yields the vector $(\hat{x}^i, {\tilde{F}}_i^{(0)} (\hat{x}))$, which is expressed in terms of ${\hat{x}}^i = \tilde{x}^i - 2\mathrm{i } Z^{ij} \varOmega _j(x,\bar{x})$ instead of $\tilde{x}^i$,

$$\begin{aligned} \hat{x}^i=&\, U^i{}_j\,x^j + Z^{ij} F^{(0)}_j(x),\nonumber \\ \tilde{F}^{(0)}_{i}(\hat{x}) =&\, V_i{}^j\,F^{(0)}_j(x) + W_{ij} \,x^j. \end{aligned}$$

(1.160)

Thus, by demanding that $\tilde{F}_i^{(0)}$ follows from the same symplectic transformation applied on $F^{(0)}_i$ alone, we relate the decomposition of $\tilde{F}_i$ to the decomposition of $F_i$. Then, the second equation of (1.158) can be written as

$$\begin{aligned} \tilde{\varOmega }_{i}(\tilde{x},\bar{\tilde{x}}) =&\, V_i{}^j\,\varOmega _j(x,\bar{x}) - \tfrac{1}{2}\mathrm{i } [\tilde{F}^{(0)}_{i}(\hat{x}) - \tilde{F}^{(0)}_{i}(\tilde{x})] \\ =&\, V_i{}^j\,\varOmega _j(x,\bar{x}) \nonumber \\&{} + \tfrac{1}{2}\mathrm{i } \sum _{m=1}^{\infty } \frac{(2\mathrm{i })^m}{m!} Z^{j_1k_1}\varOmega _{k_1}(x,\bar{x})\cdots Z^{j_mk_m}\varOmega _{k_m}(x,\bar{x}) \; \tilde{F}^{(0)}_{i j_1\cdots j_m} \nonumber (\hat{x}), \end{aligned}$$

(1.161)

where the $\tilde{F}^{(0)}_{i j_1\cdots j_m } (\hat{x})$ denote multiple derivatives of $\tilde{F}^{(0)}_i(\tilde{x})$ evaluated at $\hat{x}$. The right-hand side of (1.161) can be written entirely in terms of functions of $x$ and $\bar{x}$, upon expressing $\tilde{F}^{(0)}_{i j_1\cdots j_m } (\hat{x})$ in terms of derivatives of $F^{(0)}_i(x)$ using (1.160). We give the first few derivatives,

$$\begin{aligned} \tilde{F}^{(0)}_{ij} (\hat{x}) =&\,(V_i{}^l F^{(0)}_{lk} + W_{ik})\, [\mathcal{S }_0^{-1}]^k{}_j,\\ \tilde{F}^{(0)}_{ijk} (\hat{x}) =&\, [\mathcal{S }_0^{-1}]^l{}_i \,[\mathcal{S }_0^{-1}]^m{}_j\, [\mathcal{S }_0^{-1}]^n{}_k \,F^{(0)}_{lmn},\nonumber \\ \tilde{F}^{(0)}_{ijkl} (\hat{x}) =&\, [\mathcal{S }_0^{-1}]^m{}_{i} \,[\mathcal{S }_0^{-1}]^n{}_{j}\, [\mathcal{S }_0^{-1}]^p{}_k \,[\mathcal{S }_0^{-1}]^q{}_{l} \Big [F^{(0)}_{mnpq} - 3\, F^{(0)}_{r(mn} \mathcal{Z }_0^{rs} \, F^{(0)}_{pq)s} \Big ], \nonumber \end{aligned}$$

(1.162)

where we used the definitions

$$\begin{aligned} \mathcal{S }_0^{\,i}{}_j =&\, U^i{}_j +Z^{ik} F^{(0)}_{kj}, \nonumber \\ \mathcal{Z }_0^{ij} =&\, [\mathcal{S }_0^{-1}]^i{}_k \, Z^{kj}. \end{aligned}$$

(1.163)

Let us consider the first expression of (1.162). While $F^{(0)}_{ij}$ is manifestly symmetric in $i,j$, this appears not to be the case for $\tilde{F}^{(0)}_{ij}$. However, using the properties (1.5) of the matrices $U, V, W$ and $Z$, it follows that $\tilde{F}^{(0)}_{ij}$ is symmetric in $i,j$. Using this, we obtain

$$\begin{aligned} \tilde{F}^{(0)}_{ij} (\hat{x}) \, Z^{jk} = V_i{}^k - [\mathcal{S}_0^{-1,T}]_i{}^k. \end{aligned}$$

(1.164)

Exercise 18:

Verify (1.164) by computing $V^T \mathcal{{S}}_0$.

The symmetry of $\tilde{F}^{(0)}_{ij}$ implies that ${\tilde{F}}^{(0)}_i (\hat{x})$ can be integrated, i.e. ${\tilde{F}}^{(0)}_i (\hat{x}) = \partial {\tilde{F}}^{(0)} ( \hat{x}) / \partial \hat{x}^i$, with $\tilde{F}^{(0)} (\hat{x})$ given by the well-known expression [5],

$$\begin{aligned} \tilde{F}^{(0)} (\hat{x}) =&\, F^{(0)}(x) - \tfrac{1}{2} x^i F_i^{(0)} + \tfrac{1}{2} (U^T W)_{ij} \, x^i x^j + \tfrac{1}{2} (U^T V + W^T Z )_{i}{}^j x^i F_j^{(0)} \nonumber \\&\,+ \tfrac{1}{2} (Z^T V)^{ij} F_i^{(0)} F_j^{(0)}, \end{aligned}$$

(1.165)

up to a constant and up to terms linear in $\hat{x}^i$.

In addition to (1.163), we will also need the combinations $\mathcal{S }$ and $\hat{\mathcal{S }}$ given in (1.167) and (1.169) below, which are related to $\mathcal{S }_0$ by

$$\begin{aligned} \mathcal{S }^i{}_j =&\, \mathcal{S }_0^{\,i}{}_j +2\mathrm{i } Z^{ik} \varOmega _{kj},\nonumber \\ \hat{\mathcal{S }}^i{}_j =&\, \mathcal{S }_0^{\,i}{}_j +Z^{ik} \big [2\mathrm{i }\varOmega _{kj} -4\, \varOmega _{k\bar{l}} \bar{\mathcal{Z }}^{\bar{l}\bar{m}} \varOmega _{\bar{m} j} \big ], \nonumber \\ \mathcal{Z}^{ij} =&\, [ \mathcal{S}^{-1}]^i{}_k \, Z^{kj}. \end{aligned}$$

(1.166)

Observe that the matrices $\mathcal{Z }_0$, $\mathcal{Z }$ and $\hat{\mathcal{Z}} = \hat{\mathcal{S}}^{-1} \, Z$ are symmetric matrices by virtue of the fact that $Z U^T$ is a symmetric matrix [5].

Next we consider the transformation behavior of the derivatives $F_{ij} = \partial F_i / \partial x^j$ and $F_{i \bar{\jmath }} = \partial F_i / \partial {\bar{x}}^{\bar{\jmath }}$. First we observe that

$$\begin{aligned} \frac{\partial \tilde{x}^i}{\partial x^j} \equiv \mathcal{S}^i{}_j = U^i{}_j + Z^{ik} F_{kj} \,, \;\;\;\; \frac{\partial \tilde{x}^i}{\partial \bar{x}^{\bar{\jmath }}} \equiv Z^{ik} F_{k \bar{\jmath }}. \end{aligned}$$

(1.167)

Applying the chain rule to (1.158) yields the relation

$$\begin{aligned} F_{ij} \rightarrow {\tilde{F}}_{ij} = \left( V_i{}^l \hat{F}_{lk} + W_{ik} \right) [ \hat{\mathcal{S}}^{-1} ]^k{}_j, \end{aligned}$$

(1.168)

where $\tilde{F}_{ij} = \partial \tilde{F}_i / \partial \tilde{x}^j$ and

$$\begin{aligned} \hat{F}_{ij} =&\, F_{ij} - F_{i \bar{k}} \,\mathcal{Z}^{\bar{k} \bar{l}} \, \bar{F}_{\bar{l} j} = F^{(0)}_{ij} +2\mathrm{i }\varOmega _{ij} -4\, \varOmega _{i\bar{k}} \, \bar{\mathcal{Z }}^{\bar{k}\bar{l}} \, \varOmega _{\bar{l} j},\nonumber \\ \hat{\mathcal{S }}^i{}_j =&\, U^i{}_j + Z^{ik} \hat{F}_{kj}. \end{aligned}$$

(1.169)

Exercise 19:

Derive (1.168) by differentiating the second equation of (1.158) with respect to either $x$ or $\bar{x}$. Then combine the two resulting equations to arrive at (1.168).

Then, using the first equation of (1.162) as well as (1.164) in (1.168) yields,

$$\begin{aligned} \tilde{\varOmega }_{ij}(\tilde{x},\bar{\tilde{x}}) =&\, \tfrac{1}{2} \mathrm{i } \big [\tilde{F}^{(0)}_{ij}(\tilde{x}) -\tilde{F}^{(0)}_{ij}(\tilde{x}^k-2\mathrm{i }Z^{kl}\varOmega _l(x,\bar{x})) \big ] \\&\,+[\hat{\mathcal{S }}^{-1}]^k{}_i\,[\hat{\mathcal{S }}^{-1}]^l{}_j \Big [\varOmega _{kl} + 2\mathrm{i }\varOmega _{k\bar{m}}\bar{\mathcal{Z }}^{\bar{m}\bar{n}} \varOmega _{\bar{n} l} \nonumber \\&\,{} +2\mathrm{i }(\varOmega _{km} + 2\mathrm{i }\varOmega _{k\bar{p}}\bar{\mathcal{Z }}^{\bar{p}\bar{r}} \varOmega _{\bar{r} m}) \mathcal{Z }_0^{mn} (\varOmega _{nl} + 2\mathrm{i }\varOmega _{n\bar{q}} \bar{\mathcal{Z }}^{\bar{q}\bar{s}} \varOmega _{\bar{s} l})\Big ], \nonumber \end{aligned}$$

(1.170)

which is symmetric by virtue of the symmetry of ${\tilde{F}}^{(0)}_{ij}$, $\varOmega _{ij}$, ${ \mathcal Z}^{mn}$ and $\mathcal{Z}_0^{mn}$.

Subsequently we derive the following result from (1.161) [21],

$$\begin{aligned} \tilde{\varOmega }_{i\bar{\jmath }} = [\hat{\mathcal{S }}^{-1}]^k{}_i\,[{\bar{\mathcal{S }}}^{-1}]^{\bar{l}}{}_{\bar{\jmath }}\, \varOmega _{k\bar{l}} =[{\mathcal{S }}^{-1}]^k{}_i[\bar{\hat{\mathcal{S }}}^{-1}]^{\bar{l}}{}_{\bar{\jmath }}\, \varOmega _{k\bar{l}}. \end{aligned}$$

(1.171)

Exercise 20:

Deduce (1.171) by taking the first line of (1.161) and differentiating it with respect to $\bar{x}$. Use the relation (1.164) in the form

$$\begin{aligned} V_i{}^j = [\mathcal{S}_0^{-1,T}]_i{}^k + (V_i{}^l F_{lk}^{(0)} + W_{ik} ) \mathcal{Z}_0^{kj}, \end{aligned}$$

(1.172)

together with (1.166).

The relation (1.171) establishes that $\tilde{\varOmega }_{i\bar{\jmath }} = \overline{({\tilde{\varOmega }}_{j \bar{\imath }})}.$ Using this as well as (1.15), and recalling that $\tilde{\varOmega }_{i\bar{\jmath }} = \partial \tilde{\varOmega }_i/\partial {\bar{\tilde{x}}}^{ \bar{\jmath }}$, we obtain $\tilde{\varOmega }_{i\bar{\jmath }} = \overline{({\tilde{\varOmega }}_{j \bar{\imath }})} = \overline{(\partial \tilde{\varOmega }_j/\partial {\bar{\tilde{x}}}^{ \bar{\imath }})} = \partial (\overline{\tilde{{\varOmega }_{j}}})/\partial {\tilde{x}}^{i} = \partial \tilde{\varOmega }_{\bar{\jmath }}/\partial {\tilde{x}}^{i} \equiv \tilde{\varOmega }_{\bar{\jmath } i}$. This, together with the symmetry of $\tilde{\varOmega }_{ij}$, ensures the integrability of (1.158), as follows.

We consider the 1-form $\tilde{A} = \tilde{\varOmega }_i \, d {\tilde{x}}^i + \tilde{\varOmega }_{\bar{\imath }} \, d \bar{\tilde{x}}^{\bar{\imath }}$, which is real by virtue of $\tilde{\varOmega }_{\bar{\imath }} = (\overline{\tilde{{\varOmega }_i} })$. Its field strength reads $\tilde{\mathfrak{F }} = d \tilde{A} = \tilde{\varOmega }_{ij} \, d {\tilde{x}}^j \wedge \, d {\tilde{x}}^i + \left( \tilde{\varOmega }_{i \bar{\jmath }} - \tilde{\varOmega }_{\bar{\jmath } i} \right) \, d \bar{\tilde{x}}^{\bar{\jmath }} \wedge d \tilde{x}^i + \tilde{\varOmega }_{\bar{\imath } \bar{\jmath }} \, d \bar{\tilde{x}}^{\bar{\jmath }} \wedge d \bar{\tilde{x}}^{\bar{\imath }}$. Then, using $\tilde{\varOmega }_{ij} = \tilde{\varOmega }_{ji}$ as well as $\tilde{\varOmega }_{i \bar{\jmath }} = \tilde{\varOmega }_{\bar{\jmath } i}$, we conclude that $\tilde{\mathfrak{F }}=0$, which establishes that locally $\tilde{A} = d \tilde{\varOmega }$, with a real $\tilde{\varOmega }$.

Hence we conclude that the Eq. (1.158) are integrable and the decomposition (1.7) is preserved, i.e. the transformed $2n$-vector $(\tilde{x}^i, \tilde{F}_i)$ is constructed from a new function $\tilde{F} (\tilde{x}, \bar{\tilde{x}}) = \tilde{F}^{(0)} (\tilde{x}) + 2 \mathrm{i } \tilde{\varOmega }(\tilde{x}, \bar{\tilde{x}})$ with a real $\tilde{\varOmega }(\tilde{x}, \bar{\tilde{x}})$. This was established in Sect. 1.2.1 by relying on the Hamiltonian.

Next, let us assume that the function $F$ depends on a auxiliary real parameter $\eta$ that is inert under symplectic transformation, i.e. $F(x, \bar{x}; \eta )$, and let us consider partial derivatives with respect to it. A little calculation shows that $\partial _\eta F_i$ transforms in the following way,

$$\begin{aligned} \partial _\eta \tilde{F}_i = [\hat{\mathcal{S}}{}^{-1}]^j{}_i \left[ \partial _\eta F_j - F_{j\bar{k}} \,\bar{\mathcal{Z }}{}^{\bar{k}\bar{l}} \,\partial _\eta \bar{F}_{\bar{l}}\right] , \end{aligned}$$

(1.173)

where $\tilde{x}$ and $\bar{\tilde{x}}$ are kept fixed under the $\eta$-derivative in $\partial _\eta \tilde{F}_i(\tilde{x},\tilde{\bar{x}};\eta )$, while in $\partial _\eta F_i(x,\bar{x};\eta )$ the arguments $x$ and $\bar{x}$ are kept fixed.

Exercise 21:

Verify (1.173) by differentiating the second equation of (1.158) with respect to $\eta$, keeping $x$ and $\bar{x}$ fixed. Subsequently, use (1.159), (1.168) and (1.171) to arrive at (1.173).

Let us first consider (1.173) in the case of a holomorphic function $F$, so that $\varOmega =0$. In that case (1.173) implies that the derivative with respect to $x^i$ of $\partial _\eta \tilde{F}- \partial _\eta F$ must vanish. Therefore it follows that $\partial _\eta F$ transforms as a function under symplectic transformations (possibly up to an $x$-independent expression, which is irrelevant in view of the same argument that led to the equivalence (1.8)).

When $\varOmega \not =0$ one derives the following result using (1.173),

$$\begin{aligned} \frac{\partial \tilde{x}^j}{\partial x^i}\,\partial _\eta \tilde{F}_j - \frac{\partial \bar{\tilde{x}}^{\bar{\jmath }}}{\partial x^i}\,\partial _\eta (\overline{{{\tilde{F}}_{j}}} ) = \partial _\eta F_i. \end{aligned}$$

(1.174)

Exercise 22:

Deduce (1.174) by suitably combining (1.173) with its complex conjugate, and using the relation

$$\begin{aligned} \bar{\mathcal{Z}}^{\bar{\imath } \bar{\jmath }} \, {\bar{F}}_{\bar{\jmath } k} \, [\hat{\mathcal{S}}^{-1} \mathcal{S}] ^k{}_l = [ \bar{\hat{\mathcal{S }}}^{-1} {\bar{\mathcal{S}}} ]^{\bar{\imath }}{}_{\bar{\jmath }} \, \bar{\mathcal{Z}}^{\bar{\jmath } \bar{k}} \, \bar{F}_{\bar{k} l}. \end{aligned}$$

(1.175)

Next, we assume without loss of generality that the dependence of $\tilde{F}$ on $\eta$ is entirely contained in $\tilde{\varOmega }$. Then, using (1.15), it follows that

$$\begin{aligned} \partial _\eta (\overline{{{\tilde{F}}_{j}}} ) = - \partial _\eta \tilde{F}_{\bar{\jmath }}, \end{aligned}$$

(1.176)

and the relation (1.174) simplifies. Namely, the left hand side of (1.174) becomes equal to $\partial ( \partial _{\eta } {\tilde{F}} ) / \partial x^i$, where we used the existence of the new function $\tilde{F}$. Thus, we obtain from (1.174),

$$\begin{aligned} \frac{\partial }{\partial x^i} \left( \partial _{\eta } \tilde{F} - \partial _{\eta } F \right) = 0. \end{aligned}$$

(1.177)

This equation, together with its complex conjugate equation, implies that $\partial _\eta \tilde{F}-\partial _\eta F$ vanishes upon differentiation with respect to $x$ and $\bar{x}$, so that $\partial _\eta F$ transforms as a function under symplectic transformations (possibly up to an irrelevant term that is independent of $x$ and $\bar{x}$).

B   The Covariant Derivative $\mathcal{D}_{\eta }$

The modified derivative (1.97) acts as a covariant derivative for symplectic transformations. Here we verify this explicitly by showing that, given a quantity $G(x, \bar{x}; \eta )$ that transforms as a function under symplectic transformations, also $\mathcal{D}_{\eta } G$ transforms as a function.

To establish this, we need the transformation law of $\hat{N}^{ij}$ that enters in (1.97). Under symplectic transformations, $\hat{N}_{ij}$ given in (1.98) transforms as

$$\begin{aligned} \tilde{\hat{N}}_{ij} =\;&[\hat{\mathcal{S}}^{-1}]^k{}_i \, [{\bar{\hat{\mathcal{S}}}}^{-1} ]^{\bar{l}}{}_{\bar{\jmath }} \left[ \hat{N}_{kl} + \mathrm{i } \, F_{k \bar{m}} \, {\bar{\mathcal{Z}}}^{\bar{m} \bar{n}} \, {\bar{F}}_{\bar{n} p } \left( \delta ^p_l - \mathcal{Z}^{pq} \, F_{q \bar{l}} \right) \right. \nonumber \\&{}- \mathrm{i }\left. \,{\bar{F}}_{\bar{k} m} \, \mathcal{Z}^{mn} \, {F}_{n \bar{p}} \left( \delta ^{\bar{p}}_{\bar{l}} - {\bar{\mathcal{Z}}}^{\bar{p} \bar{q}} \, {\bar{F}}_{\bar{q} l } \right) \right] \nonumber \\&{}+ \mathrm{i } \, [\hat{\mathcal{S}}^{-1}]^k{}_i \, [{\bar{\hat{\mathcal{S}}}}^{-1}]^{\bar{l}}{}_{\bar{\jmath }} \, {\bar{F}}_{\bar{k} m} \, [\mathcal{S}^{-1} \, {\hat{\mathcal{S}}}]^m{}_l - \mathrm{i } [{\bar{\hat{\mathcal{S}}}}^{-1} ]^{\bar{k}}{}_{\bar{\imath }} \, {\bar{F}}_{\bar{k} l} \, [\mathcal{S}^{-1}]^l{}_j, \end{aligned}$$

(1.178)

where $\mathcal{S}, \hat{\mathcal{S}}$ and $\mathcal{Z}$ are defined in (1.166).

Exercise 23:

Verify (1.178) using (1.168) and (1.171).

Then, it follows that the inverse matrix $\hat{N}^{ij}$ transforms as

$$\begin{aligned} \tilde{\hat{N}}^{ij} = \left( \mathcal{S}^i{}_l - Z^{in}\, F_{n \bar{l}} \right) \, \hat{N}^{lk} \, \left( \mathcal{S}^j{}_k - Z^{jm}\, F_{m \bar{k}} \right) - \mathrm{i } \, \mathcal{S}^i{}_k \, \mathcal{Z}^{kl} \, \mathcal{S}^j{}_l. \end{aligned}$$

(1.179)

Since the matrix $\mathcal{Z} = \mathcal{S}^{-1} \, Z$ is symmetric [5], so is $\tilde{\hat{N}}^{ij}$. Observe that it can also be written as

$$\begin{aligned} \tilde{\hat{N}}^{ij} = \left( \mathcal{\bar{S}}^{\bar{\imath }}{}_{\bar{l}} - Z^{in}\, {\bar{F}}_{{\bar{n}} l} \right) \, \hat{N}^{lk} \, \left( \mathcal{S}^j{}_k - Z^{jm}\, F_{m \bar{k}} \right) - \mathrm{i } \, Z^{il} \, Z^{jm} \, F_{l \bar{m}}. \end{aligned}$$

(1.180)

Establishing the transformation behavior (1.179) turns out to be a tedious exercise, which we relegate to end of this appendix.

Now consider a quantity $G(x, {\bar{x}}; \eta )$ that transforms as a function under symplectic transformations, i.e. $G(x, {\bar{x}}; \eta ) = {\tilde{G}}({\tilde{x}}, {\bar{\tilde{x}}}; \eta )$. We then calculate the behavior of $\mathcal{D}_{\eta } G$ under symplectic transformations. First we establish

$$\begin{aligned} G_{\eta } = {\tilde{G}}_{\eta } + {\tilde{G}}_i \, Z^{ij} \, F_{\eta j} + {\tilde{G}}_{\bar{\imath }} \,Z^{ij} \, {\bar{F}}_{\eta {\bar{\jmath }}}, \end{aligned}$$

(1.181)

where, on the right hand side, the tilde quantities are differentiated with respect to the tilde variables, while those without a tilde are differentiated with respect to the original variables. Similarly,

$$\begin{aligned} G_i - G_{\bar{\imath }} = \left( {\tilde{G}}_j - {\tilde{G}}_{\bar{\jmath }}\right) \left( \mathcal{S}^j{}_i - Z^{jk} \, F_{k \bar{\imath } }\right) + \mathrm{i } \, {\tilde{G}}_{\bar{\jmath }} \, Z^{jk} \, \hat{N}_{ki}, \end{aligned}$$

(1.182)

as well as

$$\begin{aligned} F_{\eta j} = {\tilde{F}}_{\eta i} \, \mathcal{S}^i{}_j + {\tilde{F}}_{\eta \bar{\imath }} \, Z^{ik} \, {\bar{F}}_{{\bar{k}} j}, \end{aligned}$$

(1.183)

where we used that $F_{\eta }$ transforms as a symplectic function, as established in (1.177).

Exercise 24:

Verify (1.181) and (1.182) using $G(x, {\bar{x}}; \eta ) = {\tilde{G}}({\tilde{x}}, {\bar{\tilde{x}}}; \eta )$.

Then, inserting (1.181) and (1.182) into (1.97) yields,

$$\begin{aligned} \mathcal{D}_{\eta } G = {\tilde{G}}_{\eta } + \left( {\tilde{G}}_i - {\tilde{G}}_{\bar{\imath }}\right) Z^{ij} \, F_{\eta j} + \mathrm{i } \, \hat{N}^{ij} \left( F_{\eta j} + {\bar{F}}_{\eta \bar{\jmath }} \right) \left( {\tilde{G}}_k - {\tilde{G}}_{\bar{k}}\right) \left( \mathcal{S}^k{}_i - Z^{kl} \, F_{l \bar{\imath } }\right) . \end{aligned}$$

(1.184)

Next, using (1.183), we compute

$$\begin{aligned} \left( F_{\eta j} + {\bar{F}}_{\eta \bar{\jmath }} \right) =&\, \left( {\tilde{F}}_{\eta k} + {\bar{\tilde{F}}}_{\eta \bar{k}} \right) \left( \mathcal{S}^k{}_j - Z^{kl} \, F_{l \bar{\jmath } }\right) - \mathrm{i } \, {\bar{\tilde{F}}}_{\eta \bar{l}} \, Z^{lk} \, \hat{N}_{kj} \nonumber \\&\, + \left( {\tilde{F}}_{\eta l} + {\bar{\tilde{F}}}_{\eta l} \right) Z^{lk} \, F_{k \bar{j}} + \left( {\tilde{F}}_{\eta \bar{l}} + {\bar{\tilde{F}}}_{\eta \bar{l}} \right) Z^{lk} \, {\bar{F}}_{\bar{k} j}. \end{aligned}$$

(1.185)

Using that $\tilde{F}$ has the decomposition

$$\begin{aligned} {\tilde{F}} ({\tilde{x}}, {\bar{\tilde{x}}};\eta ) = {\tilde{F}}^{(0)} ({\tilde{x}}) + 2 \mathrm{i } \, {\tilde{\varOmega }}({\tilde{x}}, {\bar{\tilde{x}}};\eta ) \end{aligned}$$

(1.186)

with ${\tilde{\varOmega }}$ real, it follows that the second line of (1.185) vanishes. Inserting the first line of (1.185) into (1.184) and using $F_{i \bar{\jmath }} = - {\bar{F}}_{\bar{\jmath } i}$ as well as $\mathcal{S} \, Z^T = Z \, \mathcal{S}^T$, we obtain

$$\begin{aligned} \mathcal{D}_{\eta } G = {\tilde{G}}_{\eta } + \mathrm{i } \, {\tilde{N}}^{ij} \left( {\tilde{F}}_{\eta j} + \bar{\tilde{F}}_{\eta \bar{\jmath }} \right) \left( \tilde{G}_{i} - \tilde{G}_{\bar{\imath }} \right) = \widetilde{\left( \mathcal{D}_{\eta } G \right) }, \end{aligned}$$

(1.187)

which shows that $\mathcal{D}_{\eta } G$ transforms as a function under symplectic transformations.

Now we return to the transformation behavior of ${\hat{N}}^{ij}$ given in (1.179) and verify that it is the inverse of (1.178), i.e. $\tilde{\hat{N}}^{-1} \, \tilde{\hat{N}} = \mathbb I$. We use the decomposition $F(x, \bar{x}; \eta ) = F^{(0)}(x) + 2 \mathrm{i } \varOmega (x, \bar{x} ; \eta )$. We find it useful to introduce the following matrix notation,

$$\begin{aligned} {\bar{\mathcal{S}}}^{-1} \, \mathcal{S} =&\, \mathbb{I } + {\bar{\mathcal{Z}}} \left( F_{\cdot \cdot } - {\bar{F}}_{--} \right) , \nonumber \\ \mathcal{S}^{-1} \, {\hat{\mathcal{S}}} =&\, \mathbb{I } - X \;\;\;,\;\;\; X = \mathcal{Z} \, F_{\cdot \,-} \, {\bar{\mathcal{Z}}} {\bar{F}}_{- \cdot } = 4 \, \mathcal{Z} \, \varOmega _{\cdot \,-} \, {\bar{\mathcal{Z}}} \, {\varOmega }_{- \cdot } ,\nonumber \\ {\hat{\mathcal{S}}}^{-1} \mathcal{S} =&\, \left( \mathbb{I } - X \right) ^{-1} = \sum _{n=0}^{\infty } X^n, \nonumber \\ {\bar{\hat{\mathcal{S}}}} =&\, \mathcal{S} \left[ \mathbb{I } - X - \mathcal{Z} \left( {\hat{F}}_{\cdot \cdot } - {\bar{\hat{F}}}_{-- } \right) \right] = \left[ \mathbb{I } - \mathcal{Z} \left( F_{\cdot \cdot } - {\bar{F}}_{--} \right) - 4 \,\mathcal{Z} \, \varOmega _{- \cdot } \, \mathcal{Z} \, \varOmega _{\cdot -} \right] , \nonumber \\ \mathcal{Z} - {\bar{\mathcal{Z}}} =&\, - {\bar{\mathcal{Z}}} \left( F_{\cdot \cdot } - {\bar{F}}_{--} \right) \mathcal{Z} = - \mathcal{Z} \left( F_{\cdot \cdot } - {\bar{F}}_{--} \right) {\bar{\mathcal{Z}}}, \end{aligned}$$

(1.188)

where we assume that the power series expansion of $\mathcal{S}^{-1} \, {\hat{\mathcal{S}}}$ is convergent. Here $F_{\cdot \cdot } \;,\; F_{--} \;,\; F_{\cdot -}$ denote entries of the type $F_{ij}, F_{\bar{\imath } \bar{\jmath }}, F_{i \bar{\jmath }}$, respectively. Then, using (1.178), we compute

$$\begin{aligned} \mathcal{S}^T \, \tilde{\hat{N}} \, {\bar{\hat{\mathcal{S}}}} =&\, \sum _{n=0}^{\infty } \left( X^n\right) ^T \left( \hat{N} + 4 \mathrm{i } \, \varOmega _{\cdot -} \, {\bar{\mathcal{Z}}} \, \varOmega _{- \cdot } - 4 \mathrm{i } \varOmega _{- \cdot } \, \mathcal{Z} \, \varOmega _{\cdot -} + 2 \varOmega _{\cdot -} \, {\bar{X}} + 2 \varOmega _{-\cdot } \right) \\&\, - 2 \left( {\bar{\mathcal{S}}}^{-1} \, \mathcal{S} \right) ^T \sum _{n=0}^{\infty } \left( {\bar{X}}^n\right) ^T \, \varOmega _{- \cdot } \left[ \mathbb I - \mathcal{Z} \left( F_{\cdot \cdot } - {\bar{F}}_{--} \right) - 4 \,\mathcal{Z} \, \varOmega _{- \cdot } \, \mathcal{Z} \, \varOmega _{\cdot -} \right] . \nonumber \end{aligned}$$

(1.189)

Multiplying this with $\tilde{\hat{N}}^{-1} \, \mathcal{S}^{-1,T}$ from the left and requiring the resulting expression to equal ${\bar{\hat{\mathcal{S}}}}$ yields the relation

$$\begin{aligned}&\left[ \hat{N}^{-1} - 2 \mathrm{i } \, \hat{N}^{-1} \, \varOmega _{- \cdot } \, \mathcal{Z} - 2 \mathrm{i } \, \mathcal{Z} \, \varOmega _{\cdot -} \, \hat{N}^{-1} - 4 \mathcal{Z} \, \varOmega _{\cdot -} \, \hat{N}^{-1} \, \varOmega _{- \cdot } \, \mathcal{Z} - \mathrm{i } \mathcal{Z} \right] \nonumber \\&\qquad \quad \left[ \sum _{n=0}^{\infty } \left( X^n\right) ^T \left( \hat{N} + 4 \mathrm{i } \, \varOmega _{\cdot -} \, {\bar{\mathcal{Z}}} \, \varOmega _{- \cdot } - 4 \mathrm{i } \varOmega _{- \cdot } \, \mathcal{Z} \, \varOmega _{\cdot -} + 2 \varOmega _{\cdot -} \, {\bar{X}} + 2 \varOmega _{- \cdot } \right) \right. \nonumber \\&\qquad \left. {}-2 \left( {\bar{\mathcal{S}}}^{-1} \, \mathcal{S} \right) ^T \sum _{n=0}^{\infty } \left( {\bar{X}}^n\right) ^T \, \varOmega _{- \cdot } \left[ \mathbb{I } - \mathcal{Z} \left( F_{\cdot \cdot } - {\bar{F}}_{--} \right) - 4 \,\mathcal{Z} \, \varOmega _{- \cdot } \, \mathcal{Z} \, \varOmega _{\cdot -} \right] \right] \nonumber \\&{}= \left[ \mathbb I - \mathcal{Z} \left( F_{\cdot \cdot } - {\bar{F}}_{--} \right) - 4 \,\mathcal{Z} \, \varOmega _{- \cdot } \, \mathcal{Z} \, \varOmega _{\cdot -} \right] . \end{aligned}$$

(1.190)

Thus, checking $\tilde{\hat{N}}^{-1} \, \tilde{\hat{N}} = \mathbb{I }$ amounts to verifying the relation (1.190). To do so, we write (1.190) as a power series in $\mathcal{Z}$ by converting $\bar{\mathcal{Z}}$ into $\mathcal{Z}$ using the last relation in (1.188). Introducing the expressions

$$\begin{aligned} \sigma =&\, 4 \, \varOmega _{- \cdot } \, \mathcal{Z} \, \varOmega _{\cdot -} \, \mathcal{Z}, \nonumber \\ \varDelta =&\, \sum _{n=1}^{\infty } \left[ \left( F_{\cdot \cdot } - {\bar{F}}_{--} \right) \mathcal{Z} \right] ^n, \end{aligned}$$

(1.191)

we obtain

$$\begin{aligned} {\bar{X}} =&\, 4 \, \mathcal{Z} \left( \mathbb I + \varDelta \right) \, \varOmega _{- \cdot } \, \mathcal{Z} \, \varOmega _{\cdot -}, \nonumber \\ {\bar{X}}^T \, \varOmega _{- \cdot } =&\, \varOmega _{- \cdot } \, X , \nonumber \\ \sum _{n=0}^{\infty } \left( {\bar{X}}^n\right) ^T \, \varOmega _{- \cdot } =&\, \varOmega _{- \cdot } \, \sum _{n=0}^{\infty } X^n, \nonumber \\ X^n =&\, 4 \, \mathcal{Z} \, \varOmega _{\cdot -} \, \mathcal{Z} \left[ \left( \mathbb{I } + \varDelta \right) \sigma \right] ^{n-1} (\mathbb{I } + \varDelta ) \, \varOmega _{- \cdot } ,\;\;\;\;\;\; n \geqslant 1, \nonumber \\ \left( X^n\right) ^T =&\, 4 \, \varOmega _{\cdot -} ( \mathbb{I } + \varDelta ^T) \left[ \sigma ^T (\mathbb{I } + \varDelta ^T) \right] ^{n-1} \mathcal{Z} \, \varOmega _{- \cdot } \, \mathcal{Z} ,\;\;\;\;\;\; n \geqslant 1 ,\nonumber \\ \left( {\bar{\mathcal{S}}}^{-1} \, \mathcal{S} \right) ^T =&\, \mathbb{I } + \left( F_{\cdot \cdot } - {\bar{F}}_{--} \right) \mathcal{Z} \left( \mathbb{I } + \varDelta \right) . \end{aligned}$$

(1.192)

Then, (1.190) becomes

$$\begin{aligned}&\left[ \mathbb{I } - 2 \mathrm{i } \, \varOmega _{- \cdot } \, \mathcal{Z} - 2 \mathrm{i } \, \hat{N} \, \mathcal{Z} \, \varOmega _{\cdot -} \, \hat{N}^{-1} - 4 \, \hat{N} \, \mathcal{Z} \, \varOmega _{\cdot -} \, \hat{N}^{-1} \, \varOmega _{- \cdot } \, \mathcal{Z} - \mathrm{i } \, \hat{N} \, \mathcal{Z} \right] \nonumber \\&\left[ \sum _{n=0}^{\infty } \left( X^n\right) ^T \left[ \hat{N} + 4 \mathrm{i } \, \varOmega _{\cdot -} \, \mathcal{Z} \, \left( \mathbb I + \varDelta \right) \varOmega _{- \cdot } - 4 \mathrm{i } \varOmega _{- \cdot } \, \mathcal{Z} \, \varOmega _{\cdot -} \right. \right. \nonumber \\&\left. {}+ 8 \varOmega _{\cdot -} \, \mathcal{Z} \left( \mathbb I + \varDelta \right) \varOmega _{- \cdot } \, \mathcal{Z} \, \varOmega _{\cdot -} + 2 \varOmega _{- \cdot } \Big ] \right. \nonumber \\&\left. - 2 \left[ \mathbb I + \left( F_{\cdot \cdot } - {\bar{F}}_{--} \right) \mathcal{Z} \left( \mathbb I + \varDelta \right) \right] \varOmega _{-\cdot } \, \sum _{n=0}^{\infty } X^n \left[ \mathbb I - \mathcal{Z} \left( F_{\cdot \cdot } - {\bar{F}}_{--} \right) \right. \right. \nonumber \\&\left. {} - 4 \,\mathcal{Z} \, \varOmega _{- \cdot } \, \mathcal{Z} \, \varOmega _{\cdot -} \right] \Big ] = \hat{N} \, \left[ \mathbb I - \mathcal{Z} \left( F_{\cdot \cdot } - {\bar{F}}_{--} \right) - 4 \,\mathcal{Z} \, \varOmega _{- \cdot } \, \mathcal{Z} \, \varOmega _{\cdot -} \right] ,\qquad \quad \end{aligned}$$

(1.193)

where $X^n$ (with $n\geqslant 1$) is expressed in terms of $\mathcal{Z}$ according to (1.192). Now we proceed to check that (1.193) is indeed satisfied, order by order in $\mathcal{Z}$. Observe that the right hand side of (1.193) is quadratic in $\mathcal{Z}$, so first we check the cancellation of the terms up to order $\mathcal{Z}^2$. Then we proceed to check the terms at order $n$ with $n\geqslant 3$. Here we use the relations

$$\begin{aligned} F_{\cdot \cdot } - {\bar{F}}_{--} =&\, \mathrm{i } \hat{N} + 2 \mathrm{i } \varOmega _{\cdot -} + 2 \mathrm{i } \varOmega _{- \cdot }, \nonumber \\ \varDelta ^T \, \mathcal{Z} =&\, \mathcal{Z} \, \varDelta , \nonumber \\ \left[ \sigma ^T \left( \mathbb I + \varDelta ^T \right) \right] ^n \mathcal{Z} =&\, \mathcal{Z} \left[ \sigma \left( \mathbb I + \varDelta \right) \right] ^n , \end{aligned}$$

(1.194)

and we organize the terms at order $n$ into those that end on either $N$ (introduced in (1.100)), $\varOmega _{\cdot -}$ or $\varOmega _{- \cdot }$. It is then straightforward, but tedious, to check that at order $n$ in $\mathcal{Z}$ all these terms cancel out. This establishes the validity of the transformation law (1.179).

C   The Holomorphic Anomaly Equation in Big Moduli Space

The holomorphic anomaly Eq. (1.157) of perturbative topological string theory [35, 36] can be suscintly derived in the wave function approach [22] to the latter [23–26]. In this approach, the topological string partition function $Z$ is represented by a wavefunction,

$$\begin{aligned} Z(t; t_B, {\bar{t}}_B ) = \int d\phi \, \mathrm{{e}}^{-S(\phi ,t; t_B, {\bar{t}}_B)/\hbar } \, Z(\phi ), \end{aligned}$$

(1.195)

where $S(\phi ,t; t_B, {\bar{t}}_B)$ denotes the generating function (1.114) of canonical transformations⁶. We take the background dependent constant $c(t_B, {\bar{t}}_B)$ appearing in $S$ to be given by [23–26]

$$\begin{aligned} c(t_B, {\bar{t}}_B) = - \frac{\hbar }{2} \, \ln \det N_{IJ}(t_B, {\bar{t}}_B), \end{aligned}$$

(1.196)

with $N_{IJ}$ as in (1.132).

Differentiating (1.195) with respect to the background field ${\bar{t}}_B$ on the one hand, and with respect to the fluctuations $t$ on the other hand, yields the relation [24],

$$\begin{aligned} \frac{\partial Z(t; t_B, {\bar{t}}_B)}{\partial {\bar{t}}_B^L} = \frac{\hbar }{2} \, {\bar{F}}_{\bar{L}}{}^{IJ} \, \frac{\partial }{\partial t^I} \frac{\partial }{\partial t^J} Z(t; t_B, {\bar{t}}_B). \end{aligned}$$

(1.197)

Here ${\bar{F}}_{\bar{L}}{}^{IJ}$ is evaluated on the background, and is given by ${\bar{F}}_{\bar{L}}{}^{IJ} = {\bar{F}}_{\bar{L} \bar{M} \bar{O}} N^{MI} N^{OJ}$. Assigning scaling dimension $1$ to both $t_B$ and $t$ (and to their complex conjugates) and scaling dimension $2$ to $\hbar$, we see that (1.197) has scaling dimension $-1$. Setting

$$\begin{aligned} Z(t; t_B, {\bar{t}}_B) = \mathrm{{e}}^{W(t; t_B, {\bar{t}}_B)/\hbar }, \end{aligned}$$

(1.198)

we obtain from (1.197)

$$\begin{aligned} \frac{ \partial W(t; t_B, {\bar{t}}_B)}{\partial {\bar{t}}_B^L} = \tfrac{1}{2} \, {\bar{F}}_{\bar{L}}{}^{IJ} \, \left( \hbar \, \frac{\partial ^2 W }{\partial t^I \, \partial t^J} + \frac{\partial W}{\partial t^I} \, \frac{\partial W}{\partial t^J} \right) , \end{aligned}$$

(1.199)

which has scaling dimension $1$. The BCOV-solution [36] is obtained by making the ansatz [38]

$$\begin{aligned} W = \sum _{g=0,n=0}^{\infty } \frac{\hbar ^g}{n!} \, C^{(g)}_{I_1 \dots I_n} (t_B, {\bar{t}}_B) \, t^{I_1} \dots t^{I_n}, \end{aligned}$$

(1.200)

with

$$\begin{aligned} C^{(g)}_{I_1 \dots I_n} = 0 ,\;\;\;\;\;\; 2g - 2 + n \leqslant 0. \end{aligned}$$

(1.201)

The $C^{(g)}_{I_1 \dots I_n}$ are symmetric in $I_1, \dots , I_n$ and have scaling dimension $2 - 2g -n$. Inserting the ansatz (1.200) into (1.199), equating the terms of order $\hbar ^g$ for $g \geqslant 2$ and setting $t=0$ gives,

$$\begin{aligned} \partial _{\bar{L}} C^{(g)}(t_B, {\bar{t}}_B) = \tfrac{1}{2} \, {\bar{F}}_{\bar{L}}{}^{IJ} \, \left( C^{(g-1)}_{IJ} + \sum _{r=1}^{g-1} C^{(r)}_{I} \, C^{(g-r)}_{J} \right) ,\;\;\;\;\;\; g \geqslant 2. \end{aligned}$$

(1.202)

Exercise 25:

Verify (1.202).

Now we set [38]

$$\begin{aligned} C^{(g)}_{I_1 \dots I_n} = D_{I_1} \dots D_{I_n} F^{(g)} ,\;\;\;\;\;\; g\geqslant 1, \end{aligned}$$

(1.203)

where $D_L$ is given by

$$\begin{aligned} D_L V_M = \partial _L V_M + \mathrm{i } \, N^{PI} F_{ILM} V_P. \end{aligned}$$

(1.204)

$D_L$ acts as a covariant derivative for symplectic reparametrizations $V_M \rightarrow \left( \mathcal{S}_0^{-1} \right) ^P{}_M V_P$, since $N^{IJ}$ transforms as $N^{IJ} \rightarrow [\mathcal{S}_0 \, N^{-1} \, \mathcal{S}_0 ]^{IJ} - \mathrm{i } [\mathcal{S}_0 \, \mathcal{Z}_0 \, \mathcal{S}_0]^{IJ}$ (see (1.179)). The $F^{(g)}$ have scaling dimension $2 - 2g$ and transform as functions under symplectic transformations. Inserting (1.203) into (1.202) yields the holomorphic anomaly equation in big moduli space [38],

$$\begin{aligned} \partial _{\bar{L}} F^{(g)}(t_B, {\bar{t}}_B) = \tfrac{1}{2} \, {\bar{F}}_{\bar{L}}{}^{IJ} \, \left( D_I \partial _J F^{(g-1)} + \sum _{r=1}^{g-1} \partial _I F^{(r)}\, \partial _J F^{(g-r)} \right) ,\;\;\;\;\;\; g \geqslant 2. \end{aligned}$$

(1.205)

As an example, consider solving (1.205) for $g=2$. We need $F_I^{(1)} = \partial _I F^{(1)} (t_B,\bar{t}_B)$, which is non-holomorphic and given by⁷

$$\begin{aligned} \partial _I F^{(1)}(t_B, \bar{t}_B) = \partial _I f^{(1)}(t_B) + \tfrac{1}{2} \, \mathrm{i } \, F_{IJK} \, N^{JK}. \end{aligned}$$

(1.206)

Then, solving (1.205) for $F^{(2)}$ yields [25, 38]

$$\begin{aligned} F^{(2)}(t_B, \bar{t}_B) =&\, f^{(2)}(t_B) + \tfrac{1}{2} \, \mathrm{i } \, {N}^{IJ} \left( D_I {F}_J^{(1)} + {F}^{(1)}_I {F}^{(1)}_J \right) \nonumber \\&\, + \tfrac{1}{2} {N}^{IJ} {N}^{KL} \left( \tfrac{1}{4} F_{IJKL} + \tfrac{1}{3} \mathrm{i } \, N^{MN} F_{IKM} F_{JLN} + F_{IJK} {F}^{(1)}_L \right) . \end{aligned}$$

(1.207)

In this expression, all the terms are evaluated on the background $(t_B, \bar{t}_B)$.

Exercise 26:

Verify that (1.207) solves (1.205).

Observe that (1.206) transforms covariantly under symplectic transformations, provided that $f^{(1)}$ transforms as $f^{(1)} \longrightarrow f^{(1)} - \tfrac{1}{2} \ln \det \mathcal{S }_0$ in order to compensate for the transformation behavior $N_{IJ} \longrightarrow N_{KL} \, [ \bar{\mathcal{S }_0}^{-1}]^K{}_I \, [ \mathcal{S }_0^{-1} ]^L{}_J$ [5], so that

$$\begin{aligned} f_I^{(1)}&\longrightarrow \left( f_J^{(1)}- \tfrac{1}{2} \mathcal{Z}_0^{PQ} \, F_{PQJ} \right) \left( \mathcal{S}_0^{-1} \right) ^J{}_I, \nonumber \\ f_{IJ}^{(1)}&\longrightarrow \left( \mathcal{S}_0^{-1} \right) ^Q{}_J \, \partial _Q \left[ \left( f_L^{(1)}- \tfrac{1}{2} \mathcal{Z}_0^{PQ} \, F_{PQL} \right) \left( \mathcal{S}_0^{-1} \right) ^L{}_I \right] . \end{aligned}$$

(1.208)

Exercise 27:

Determine the transformation behavior of $f^{(2)}(t_B)$ under symplectic transformations (1.128) that ensures that $F^{(2)}(t_B, \bar{t}_B)$ transforms as a function. A useful transformation law is,

$$\begin{aligned} F_{IJKL} \longrightarrow&\, \left( \mathcal{S}_0^{-1} \right) ^M{}_I \, \partial _M \left[ F_{NOP} \left( \mathcal{S}_0^{-1} \right) ^N{}_J \, \left( \mathcal{S}_0^{-1} \right) ^O{}_K \,\left( \mathcal{S}_0^{-1} \right) ^P{}_L \right] \nonumber \\&\, = \left( \mathcal S _0^{-1}\right) ^M{}_I \left( \mathcal S _0^{-1}\right) ^N{}_J \left( \mathcal S _0^{-1}\right) ^O{}_K \left( \mathcal S _0^{-1}\right) ^P{}_L \Big [F_{MNOP} \nonumber \\&\, \qquad - F_{MPS} \mathcal{Z}_0^{SR} F_{RNO} - F_{OPS} \mathcal{Z}_0^{SR} F_{RMN} - F_{NPS} \mathcal{Z}_0^{SR} F_{ROM} \Big ]. \end{aligned}$$

(1.209)

D   The Functions $\mathcal H ^{(a)}_i$ for $a\geqslant 2$

Here we collect the explicit results for the various functions $\mathcal{H }^{(a)}_i$ (with $a \geqslant 2$) that appear in (1.143). These functions can be determined by iteration. We present the functions up to order $\mathcal{O }(\varOmega ^4)$. We use the notation $(N \varOmega )^I = N^{IJ} \varOmega _J \,,\, (N \bar{\varOmega })^I = N^{IJ} \varOmega _{\bar{J}}$. The symmetrization $F_{R(IJ} N^{RS} F_{KL)S}$ is defined with a symmetrization factor $1/(4!)$.

$$\begin{aligned} \mathcal{H }^{(2)}=\;&8\, N^{IJ}\varOmega _I \varOmega _{\bar{J}} -16\,\big [\varOmega _{IJ} (N\bar{\varOmega })^I (N\varOmega )^J +\,\varOmega _{I\bar{J}} (N\bar{\varOmega })^I (N\bar{\varOmega })^J + \mathrm{{h.c.}} \big ] \nonumber \\&{} - 8 \mathrm{i } \,\big [F_{IJK} (N\bar{\varOmega })^I (N\varOmega )^J (N\varOmega )^K - \mathrm{{h.c.}} \big ] \nonumber \\&{} + \tfrac{16}{3} \mathrm{i } \left[ \left( F_{IJKL} + 3 \mathrm{i } F_{IJR} N^{RS} F_{SKL} \right) \, (N\varOmega )^I (N\varOmega )^J (N\varOmega )^K (N \bar{\varOmega })^L\right. \nonumber \\&{} - \mathrm{{h.c.}}\left. \right] + 16\, \big [ \varOmega _{IJK} \, (N\varOmega )^I (N\varOmega )^J (N \bar{\varOmega })^K + \mathrm{{h.c.}} \big ] \nonumber \\&{} + 16\, \big [ \left( \varOmega _{IJ\bar{K}} + \mathrm{i } F_{IJP} N^{PQ} \varOmega _{Q \bar{K}} \right) \, \left( (N\varOmega )^I (N\varOmega )^J (N \varOmega )^K \right. \nonumber \\&{}+2 \left. (N\varOmega )^I (N \bar{\varOmega })^J (N \bar{\varOmega })^K \right) + \mathrm{{h.c.}} \big ] \nonumber \\&{} + 32\, \Big [\varOmega _{IQ} \, N^{QR} \, \varOmega _{RJ} \, (N \varOmega )^I (N \bar{\varOmega })^J + \mathrm{{h.c.}} \Big ] \nonumber \\&{} + 32\, \varOmega _{IQ} \, N^{QR} \, {\varOmega }_{\bar{R} \bar{J}} \, (N \varOmega )^I (N \bar{\varOmega })^J \nonumber \\&{} + 16 \mathrm{i } \Big [ F_{IJK} \, N^{KP} \, \varOmega _{PQ} \, \left( (N \varOmega )^I (N \varOmega )^J (N \bar{\varOmega })^Q \right. \nonumber \\&{}+ 2 \left. (N \varOmega )^Q (N \varOmega )^I (N \bar{\varOmega })^J \right) - \mathrm{{h.c.}} \Big ] \nonumber \\&{} + 16 \mathrm{i } \Big [ F_{IJK} \, N^{KP} \, \varOmega _{\bar{P} \bar{Q}} \, (N \varOmega )^I (N \varOmega )^J (N \bar{\varOmega })^Q - \mathrm{{h.c.}} \Big ] \nonumber \\&{} +8\, (N \varOmega ) ^I \, (N \varOmega )^J \, F_{IJQ} N^{QR} {\bar{F}}_{\bar{R} \bar{K} \bar{L}} (N \bar{\varOmega })^K (N \bar{\varOmega })^L \nonumber \\&{} + 32 \Big [(N \varOmega )^I \, \varOmega _{IJ} \, N^{JK} \varOmega _{K \bar{L}} (N \varOmega )^L + \mathrm{{h.c.}} \Big ] \nonumber \\&{} + 32 \Big [(N \bar{\varOmega })^I \, \varOmega _{IJ} \, N^{JK} \varOmega _{K \bar{L}} (N \bar{\varOmega })^L + \mathrm{{h.c.}} \Big ] \nonumber \\&{} + 32 \Big [(N \varOmega )^I \, \varOmega _{IJ}\, N^{JK} \varOmega _{\bar{K} L} (N \varOmega )^L + \mathrm{{h.c.}} \Big ] \nonumber \\&{} + 16 \mathrm{i } \Big [ (N\varOmega )^I (N\varOmega )^J F_{IJK} N^{KL} \varOmega _{\bar{L} P} (N\varOmega )^P - \mathrm{{h.c.}} \Big ] \nonumber \\&{} + 32 \Big [(N \varOmega )^I \varOmega _{I\bar{J}} \, N^{JK} \, \varOmega _{\bar{K} L} \, (N \bar{\varOmega })^L + \mathrm{{h.c.}} \Big ] \nonumber \\&{} + 32 \Big [(N \varOmega )^I \varOmega _{I\bar{J}} \, N^{JK} \,\varOmega _{K \bar{L}}\, (N \bar{\varOmega })^L \Big ],\end{aligned}$$

(1.210)

$$\begin{aligned} \mathcal{H }^{(3)}_1=&\,- \tfrac{8}{3}\mathrm{i } F_{IJK} (N\bar{\varOmega })^I(N\bar{\varOmega })^J(N\bar{\varOmega })^K \nonumber \\&{} +8 \mathrm{i }\, F_{IJK} (N\bar{\varOmega })^J (N\bar{\varOmega })^K \, N^{IP} \left[ 2 \varOmega _{\bar{P} \bar{Q}} (N \bar{\varOmega })^Q + 2 \varOmega _{\bar{P} Q} (N \varOmega )^Q \right. \nonumber \\&{} - \mathrm{i }\left. \bar{F}_{\bar{P} \bar{Q} \bar{R}} (N \bar{\varOmega })^Q (N \bar{\varOmega })^R \right] ,\end{aligned}$$

(1.211)

$$\begin{aligned} \mathcal{H }^{(3)}_2=\;&8\, \left( \varOmega _{IJ} + \mathrm{i } F_{IJK} (N \varOmega )^K \right) (N \bar{\varOmega })^I (N \bar{\varOmega })^J \nonumber \\&{}-\tfrac{4}{3} \mathrm{i } \left( F_{IJKL} + 3 \mathrm{i } F_{R(IJ} N^{RS} F_{KL)S} \right) \left( 6 (N \varOmega )^I (N \varOmega )^J (N \bar{\varOmega })^K (N \bar{\varOmega })^L \right. \nonumber \\&{}-4 \left. (N \bar{\varOmega })^I (N \bar{\varOmega })^J (N \bar{\varOmega })^K (N \varOmega )^L \right) \Big ] \nonumber \\&{}-\tfrac{16}{3} \, \varOmega _{IJK} \left( 3\,(N \bar{\varOmega })^I (N \bar{\varOmega })^J (N \varOmega )^K - (N \bar{\varOmega })^I (N \bar{\varOmega })^J (N \bar{\varOmega })^K\right) \nonumber \\&{} - 16\,\varOmega _{IJ\bar{K}} (N \bar{\varOmega })^I (N \bar{\varOmega })^J (N \bar{\varOmega })^K \nonumber \\&{} - 16 \mathrm{i } \,F_{IJK} N^{KP} \,\varOmega _{PQ} \big [- (N \bar{\varOmega })^I ( N \bar{\varOmega })^J (N \bar{\varOmega })^Q\nonumber \\&{} + (N \bar{\varOmega })^I ( N \bar{\varOmega })^J (N \varOmega )^Q + 2 (N \bar{\varOmega })^I ( N \varOmega )^J (N \bar{\varOmega })^Q \big ] \nonumber \\&{} - 16 \, (N \bar{\varOmega })^P \, \varOmega _{PQ} \, N^{QR} \varOmega _{RK} \, (N \bar{\varOmega })^K \nonumber \\&{} - 32 \, (N \varOmega )^I \, \left( \varOmega _{IJ} + \mathrm{i } F_{IJP} (N \varOmega )^P \right) \, N^{JK} \left( \varOmega _{\bar{K} \bar{L}} - \mathrm{i } {\bar{F}}_{\bar{K} \bar{L} \bar{M}} (N \bar{\varOmega })^M \right) (N \varOmega )^L\nonumber \\&{} + 16 \mathrm{i } (N \varOmega )^I (N \varOmega )^J F_{IJP}N^{PK} \left( {\varOmega }_{\bar{K} \bar{L}} - \mathrm{i } {\bar{F}}_{ \bar{K} \bar{L} \bar{Q}} (N \bar{\varOmega })^{\bar{Q}} \right) (N \varOmega )^L \nonumber \\&{} - 16\, (N \varOmega )^P \varOmega _{\bar{P} Q} N^{QR}\varOmega _{R \bar{K}} (N \varOmega )^K \nonumber \\&{} - 32 \, (N \bar{\varOmega })^I \left( \varOmega _{IJ} + \mathrm{i } F_{IJK} (N \varOmega )^K \right) N^{JL} \varOmega _{\bar{L} M} (N \varOmega )^M \nonumber \\&{} - 16 \mathrm{i } \,( N \bar{\varOmega })^I \, (N \bar{\varOmega })^J F_{IJK}N^{KP} \varOmega _{P \bar{Q}} (N \bar{\varOmega })^Q, \end{aligned}$$

(1.212)

$$\begin{aligned} \mathcal{H }^{(3)}_3=\;&16\,\varOmega _{I\bar{J}} (N\bar{\varOmega })^I (N\varOmega )^J \nonumber \\&{}- 16\,\Big [ 2 (N \bar{\varOmega })^K (N \varOmega )^L \left( \varOmega _{KM} N^{MN} \varOmega _{N \bar{L}} + \varOmega _{K \bar{L} Q} (N \varOmega )^Q\right) \nonumber \\&{} + (N \bar{\varOmega })^K \varOmega _{K \bar{L}} N^{LP} \left( \mathrm{i } F_{PMN} (N \varOmega )^M (N \varOmega )^N + 2 \varOmega _{PJ}(N \varOmega )^J \right. \nonumber \\&{}+2\left. \varOmega _{P \bar{J}} (N \bar{\varOmega })^J \right) + 2 \mathrm{i } (N \bar{\varOmega })^I (N \varOmega )^J F_{IJK} N^{KP} \varOmega _{P \bar{Q}} (N \varOmega )^Q + \mathrm{{h.c.}} \Big ], \end{aligned}$$

(1.213)

$$\begin{aligned} \mathcal{H }^{(4)}_1=\;&32 \, (N \bar{\varOmega })^I \left( \varOmega _{IJ} + \mathrm{i } {F}_{IJK} (N \varOmega )^K \right) N^{JP} \left( \varOmega _{\bar{P} \bar{Q}} - \mathrm{i } {\bar{F}}_{\bar{P} \bar{Q} \bar{R}} (N \bar{\varOmega })^R\right) (N \varOmega )^Q, \end{aligned}$$

(1.214)

$$\begin{aligned} \mathcal{H }^{(4)}_2=\;&32 \, (N \varOmega )^P \, \varOmega _{\bar{P} Q} \, N^{QR} \, \varOmega _{\bar{R} K}\, (N \bar{\varOmega })^K \end{aligned}$$

(1.215)

$$\begin{aligned} \mathcal{H }^{(4)}_3=\;&8 \, F_{IJR} N^{RS} {\bar{F}}_{\bar{S} \bar{K} \bar{L}} \, (N \bar{\varOmega })^I (N \bar{\varOmega })^J (N\varOmega )^K (N \varOmega )^L, \end{aligned}$$

(1.216)

$$\begin{aligned} \mathcal{H }^{(4)}_4=\;&- \tfrac{4}{3} \mathrm{i } \left( F_{IJKL} + 3 \mathrm{i } \, F_{IJR} N^{RS} F_{SKL} \right) (N \bar{\varOmega })^I (N \bar{\varOmega })^J (N \bar{\varOmega })^K (N\bar{\varOmega })^L, \end{aligned}$$

(1.217)

$$\begin{aligned} \mathcal{H }^{(4)}_5=\;&- 16 \mathrm{i } \, F_{IJK} N^{KL} \varOmega _{\bar{L} Q} \, (N \bar{\varOmega })^Q (N \bar{\varOmega })^I (N \bar{\varOmega })^J,\end{aligned}$$

(1.218)

$$\begin{aligned} \mathcal{H }^{(4)}_6=\;&- 16 \mathrm{i }\, F_{IJK} N^{KP}\left( \varOmega _{\bar{P} \bar{Q}} - \mathrm{i } {\bar{F}}_{\bar{P} \bar{Q} \bar{R}} (N \bar{\varOmega })^R \right) (N \bar{\varOmega })^I (N \bar{\varOmega })^J (N \varOmega )^Q,\end{aligned}$$

(1.219)

$$\begin{aligned} \mathcal{H }^{(4)}_7=\;&16 \,\left( \varOmega _{IJ\bar{K}} + \mathrm{i } F_{IJP} \, N^{PQ} \, \varOmega _{Q \bar{K}}\right) \, (N \bar{\varOmega })^I (N \bar{\varOmega })^J (N \varOmega )^K, \end{aligned}$$

(1.220)

$$\begin{aligned} \mathcal{H }^{(4)}_8 =&\, 32 \, (N \bar{\varOmega })^I \left( \varOmega _{IJ} + \mathrm{i } {F}_{IJK} (N \varOmega )^K \right) N^{JP} \varOmega _{\bar{P} Q} \, (N \bar{\varOmega })^Q, \end{aligned}$$

(1.221)

$$\begin{aligned} \mathcal{H }^{(4)}_9 =&\, - 16 \mathrm{i } \, (N \bar{\varOmega })^I (N \bar{\varOmega })^J F_{IJK} N^{KL} \varOmega _{\bar{L} P} (N \bar{\varOmega })^P. \end{aligned}$$

(1.222)

E   Transformation Laws by Iteration

The Hesse potential in Sect. 1.4 depends on $\varOmega$, whose behavior under symplectic transformations can be determined by iteration. Here we summarize the result for the transformation behavior of derivatives of $\varOmega$ (expressed in terms of the covariant variables of Sect. 1.3), up to a certain order. We use the conventions of Sect. 1.4 and suppress the superscript of $F^{(0)}$.

$$\begin{aligned} {\tilde{\varOmega }}_I=\;&[\mathcal{S }_0^{-1}]^J{}_I \Big [ \varOmega _J +\mathrm{i } F_{JKL} \,\left( \mathcal{Z }_0 \varOmega \right) ^K\,(\mathcal{Z }_0 \varOmega )^L -2\mathrm{i } \varOmega _{JK} (\mathcal{Z }_0 \varOmega )^K\nonumber \\&\,+2\mathrm{i } \varOmega _{J\bar{K}} (\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{K}} + \tfrac{2}{3} F_{JKLP} (\mathcal{Z }_0 \varOmega )^K (\mathcal{Z }_0 \varOmega )^L (\mathcal{Z }_0 \varOmega )^P\nonumber \\&\, + 2 F_{KLP} (\mathcal{Z }_0 \varOmega )^K{}_J (\mathcal{Z }_0 \varOmega )^L (\mathcal{Z }_0 \varOmega )^P \nonumber \\&\, + 4 F_{JKL} (\mathcal{Z }_0 \varOmega )^K (\mathcal{Z }_0 \varOmega )^L{}_{P} (\mathcal{Z }_0 {\varOmega })^{P}\nonumber \\&\,- 4 F_{JKL} (\mathcal{Z }_0 \varOmega )^K (\mathcal{Z }_0 \varOmega )^L{}_{\bar{P}} (\mathcal{\bar{Z} }_0 {\bar{\varOmega }})^{\bar{P}} \nonumber \\&\, - 2 F_{JKL} \mathcal{Z }_0^{LP} F_{PQS} (\mathcal{Z }_0 \varOmega )^K (\mathcal{Z }_0 \varOmega )^Q (\mathcal{Z }_0 \varOmega )^S\nonumber \\&\,+ 2 {\bar{F}}_{\bar{K} \bar{L} \bar{P}} (\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{K}}{}_J (\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{L}} (\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{P}} \nonumber \\&\, - 2 \varOmega _{JKL} (\mathcal{Z }_0 \varOmega )^K (\mathcal{Z }_0 \varOmega )^L- 4 \varOmega _{KL} (\mathcal{Z }_0 \varOmega )^K{}_J (\mathcal{Z }_0 \varOmega )^L \nonumber \\&\, - 2 \varOmega _{J \bar{K} \bar{L}} (\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{K}} (\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{L}} - 4 \varOmega _{\bar{K} \bar{L}} (\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{K}}{}_J (\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{L}}\nonumber \\&\, + 4 \varOmega _{JK \bar{L}} (\mathcal{Z }_0 \varOmega )^K (\bar{\mathcal{Z }_0 } \bar{\varOmega })^{\bar{L}} + 4 \varOmega _{K \bar{L}} (\mathcal{Z }_0 \varOmega )^K{}_J (\bar{\mathcal{Z }_0 } \bar{\varOmega })^{\bar{L}} \nonumber \\&\, + 4 \varOmega _{K \bar{L}} (\mathcal{Z }_0 \varOmega )^K (\bar{\mathcal{Z }_0 } \bar{\varOmega })^{\bar{L}}{}_J \Big ] +\mathcal{O }(\varOmega ^4)\,,\nonumber \\ \tilde{\varOmega }_{IJ}=\;&[\mathcal{S }_0^{-1}]^K{}_I [\mathcal{S }_0^{-1}]^L{}_J \Big [ \varOmega _{KL} -F_{KLM} \,\mathcal{Z }_0^{MN} \varOmega _N \nonumber \\&\, - \mathrm{i } F_{KLP} \mathcal{Z}_0^{PM} F_{MQR} (\mathcal{Z}_0 \varOmega )^{Q} (\mathcal{Z}_0 \varOmega )^R + 2 \mathrm{i } F_{KLP} (\mathcal{Z}_0 \varOmega )^{P}{}_{Q} (\mathcal{Z}_0 \varOmega )^Q \nonumber \\&\, - 2 \mathrm{i } F_{KLP} (\mathcal{Z}_0 \varOmega )^{P}{}_{\bar{Q}} (\bar{\mathcal{Z}}_0 {\bar{\varOmega }})^{\bar{Q}} + \mathrm{i } F_{KLMN} (\mathcal{Z }_0 \varOmega )^M (\mathcal{Z }_0 \varOmega )^N \nonumber \\&\, + \mathrm{i } F_{KMN} (\mathcal{Z }_0 \varOmega )^M{}_L (\mathcal{Z }_0 \varOmega )^N + \mathrm{i } F_{KMN} (\mathcal{Z }_0 \varOmega )^N{}_L (\mathcal{Z }_0 \varOmega )^M \nonumber \\&\, - 2 \mathrm{i } F_{KMN} \mathcal{Z }_0^{MP} F_{PQL} (\mathcal{Z }_0 \varOmega )^Q (\mathcal{Z }_0 \varOmega )^N \nonumber \\&- 2 \mathrm{i } \varOmega _{KLP} (\mathcal{Z }_0 \varOmega )^P - 2 \mathrm{i } \varOmega _{KP} (\mathcal{Z }_0 \varOmega )^P{}_L + 2 \mathrm{i } \varOmega _{KP} \mathcal{Z }_0^{PQ} F_{QLS} (\mathcal{Z }_0 \varOmega )^S \nonumber \\&\,+ 2 \mathrm{i } \varOmega _{KL \bar{P}} (\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{P}} + 2 \mathrm{i } \varOmega _{K \bar{P}} (\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{P}}{}_L \Big ] +\mathcal{O }(\varOmega ^3) \,,\nonumber \\ \tilde{\varOmega }_{I\bar{J}}=\;&[\mathcal{S }_0^{-1}]^K{}_I [\bar{\mathcal{S }}_0^{-1}]^{\bar{L}}{}_{\bar{J}} \,\Big [ \varOmega _{K\bar{L}} + 2 \mathrm{i } F_{KMN} (\mathcal{Z }_0 \varOmega )^M{}_{\bar{L}} (\mathcal{Z }_0 \varOmega )^N \nonumber \\&\, - 2 \mathrm{i } {\bar{F}}_{\bar{L} \bar{P} \bar{N}} (\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{N}}{}_{K} (\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{P}} \nonumber \\&\, - 2 \mathrm{i } \varOmega _{KM{\bar{L}}} (\mathcal{Z }_0 \varOmega )^M - 2 \mathrm{i } \varOmega _{KM} (\mathcal{Z }_0 \varOmega )^M{}_{\bar{L}} + 2 \mathrm{i } \varOmega _{K \bar{L} \bar{M}} (\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{M}} \nonumber \\&\, + 2 \mathrm{i } \varOmega _{K \bar{M} } (\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{M}}{}_{\bar{L}} \Big ] +\mathcal{O }(\varOmega ^3) \,, \nonumber \\ {\tilde{\varOmega }}_{IJL}=\;&[\mathcal{S }_0^{-1}]^M{}_I [\mathcal{S }_0^{-1}]^N{}_J [\mathcal{S }_0^{-1}]^K{}_L \left[ \varOmega _{MNK} - F_{MNKP} (\mathcal{Z}_0 \varOmega )^P \right. \nonumber \\&{}- F_{MNP}\left. (\mathcal{Z}_0 \varOmega )^P{}_K - F_{KMP} (\mathcal{Z}_0 \varOmega )^P{}_N - F_{NKP} (\mathcal{Z}_0 \varOmega )^P{}_M \right. \nonumber \\&{}+\left. F_{MNP} \mathcal{Z}_0^{PQ} F_{KQR} (\mathcal{Z}_0 \varOmega )^R + F_{KMP} \mathcal{Z}_0^{PQ} F_{QNR} (\mathcal{Z}_0 \varOmega )^R \right. \nonumber \\&{}+\left. F_{NKP} \mathcal{Z}_0^{PQ} F_{QMR} (\mathcal{Z}_0 \varOmega )^R \right] +\mathcal{O }(\varOmega ^2) \,, \nonumber \\ {\tilde{\varOmega }}_{IJ \bar{K}}=\;&[\mathcal{S }_0^{-1}]^M{}_I [\mathcal{S }_0^{-1}]^N{}_J [\bar{\mathcal{S }}_0^{-1}]^{\bar{L}}{}_{\bar{K}} \left[ \varOmega _{M N \bar{L}} - F_{MNQ} (\mathcal{Z}_0 \varOmega )^Q{}_{\bar{L}} \right] \nonumber \\&\, +\mathcal{O }(\varOmega ^2), \end{aligned}$$

(1.223)

where $(\mathcal{Z }_0 \varOmega )^M = \mathcal{Z }_0^{MN} \varOmega _N$, $(\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{M}} = \bar{\mathcal{Z }}_0^{\bar{M} \bar{N}} {\varOmega }_{\bar{N}} \,,\, (\mathcal{Z }_0 \varOmega )^M{}_{\bar{L}} = \mathcal{Z }_0^{MN} \varOmega _{N \bar{L}} \,,\, (\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{P}}{}_L = \bar{\mathcal{Z }}_0^{\bar{P} \bar{N}} {\varOmega }_{\bar{N} L}$, $(\mathcal{Z }_0 \varOmega )^L{}_{\bar{P}} = \mathcal{Z }_0^{LK} \varOmega _{K {\bar{P}}}$, $(\bar{\mathcal{Z }}_0 \bar{\varOmega })^{\bar{P}}{}_{\bar{L}} = \bar{\mathcal{Z }}_0^{\bar{P} \bar{N}} {\varOmega }_{\bar{N} \bar{L}}$.