Weyl invariance, non-compact duality and conformal higher-derivative sigma models

Abstract

We study a system of n Abelian vector fields coupled to $\frac{1}{2} n(n+1)$ complex scalars parametrising the Hermitian symmetric space ${\textsf{Sp}}(2n, {\mathbb {R}})/ {\textsf{U}}(n).$ This model is Weyl invariant and possesses the maximal non-compact duality group ${\textsf{Sp}}(2n, {\mathbb {R}}).$ Although both symmetries are anomalous in the quantum theory, they should be respected by the logarithmic divergent term (the “induced action”) of the effective action obtained by integrating out the vector fields. We compute this induced action and demonstrate its Weyl and ${\textsf{Sp}}(2n, {\mathbb {R}})$ invariance. The resulting conformal higher-derivative $\sigma$-model on ${\textsf{Sp}}(2n, {\mathbb {R}})/ {\textsf{U}}(n)$ is generalised to the cases where the fields take their values in (i) an arbitrary Kähler space; and (ii) an arbitrary Riemannian manifold. In both cases, the $\sigma$-model Lagrangian generates a Weyl anomaly satisfying the Wess–Zumino consistency condition.

1 Introduction

A unique feature of the Weyl multiplet of ${\mathcal {N}}=4$ conformal supergravity [1] is the presence of a dimensionless complex scalar field $\phi$ that parametrises the Hermitian symmetric space $\textsf{SL}(2,{\mathbb {R}}) /\textsf{SO}(2).$¹ The most general family of invariant actions for ${\mathcal {N}}=4$ conformal supergravity was derived only a few years ago by Butter et al. [2, 3]. Such an action is uniquely determined by a holomorphic function ${\mathcal {H}}(\phi )$ which accompanies the terms quadratic in the Weyl tensor in the Lagrangian.

For the special choice ${\mathcal {H}}= {\textrm{const}},$ in which case the ${\mathcal {N}}=4$ conformal supergravity action proves to be invariant under rigid $\textsf{SL}(2,{\mathbb {R}})$ transformations, the corresponding action was constructed in 2015 by Ciceri and Sahoo [4] to second order in fermions. The bosonic sector of the latter action had been computed in 2012 by Buchbinder et al. [5] as an “induced action”, obtained by integrating out an Abelian ${\mathcal {N}}=4$ vector multiplet coupled to external ${\mathcal {N}}= 4$ conformal supergravity.² The purely $\phi$-dependent part of the Lagrangian is a higher-derivative $\sigma$-model of the form [6]:

$$\begin{aligned}{} & {} {\mathcal {L}}(\phi , \bar{\phi }) = \frac{1}{({{\textrm{Im}}}\, \phi )^2} \Bigg [ {\mathcal {D}}^2 \phi {\mathcal {D}}^2 \bar{\phi }\nonumber \\{} & {} \qquad \qquad \qquad - 2 (R^{mn}- \frac{1}{3} g^{mn} R) \nabla _m \phi \nabla _n \bar{\phi }\Bigg ] \nonumber \\{} & {} \qquad \qquad \qquad + \frac{1}{12 ( {\textrm{Im}}\, \phi )^4} \Bigg [ \alpha \nabla ^m \phi \nabla _m \phi \nabla ^n \bar{\phi }\nabla _n \bar{\phi }\nonumber \\{} & {} \qquad \qquad \qquad +\beta \nabla ^m \phi \nabla _m \bar{\phi }\nabla ^n \phi \nabla _n \bar{\phi }\Bigg ], \end{aligned}$$

(1.1)

where $R_{mn}$ is the spacetime Ricci tensor,

$$\begin{aligned} {\mathcal {D}}^2 \phi := \nabla ^m \nabla _m \phi + \frac{\text {i}}{{\textrm{Im}}\, \phi } \nabla ^m \phi \nabla _m \phi , \end{aligned}$$

(1.2)

and $\alpha$ and $\beta$ are numerical parameters. In the case of ${\mathcal {N}}=4$ conformal supergravity, these coefficients are [5]: $\alpha = \frac{1}{2}\beta =1.$ The Lagrangian (1.1) is invariant under $\textsf{SL}(2,{\mathbb {R}})$ transformations

$$\begin{aligned} \phi \rightarrow \phi ' = \frac{a\phi + b}{c\phi +d},\quad \left( \begin{array}{cc} a &{} \quad b\\ c &{} \quad d \end{array} \right) \in \textsf{SL}(2,{\mathbb {R}}) \end{aligned}$$

(1.3)

acting on the upper half-plane ${\textrm{Im}}\, \phi >0$ with metric

$$\begin{aligned} \text {d}{\mathfrak s}^2 = \frac{ \text {d}\phi \,\text {d}\bar{\phi }}{ ( {\textrm{Im}}\, \phi )^2 } \quad \Longrightarrow \quad \Gamma ^\phi _{\phi \phi } = \frac{\text {i}}{{\textrm{Im}}\, \phi }, \quad \Gamma ^{\bar{\phi }}_{\bar{\phi }\bar{\phi }} = - \frac{\text {i}}{{\textrm{Im}}\, \phi }.\nonumber \\ \end{aligned}$$

(1.4)

The functional $\int \text {d}^4 x \, \sqrt{-g}\, {\mathcal {L}}$ proves to be invariant under Weyl transformations

$$\begin{aligned} g_{mn} (x) \rightarrow \text {e}^{2\sigma (x) }g_{mn}(x), \end{aligned}$$

(1.5)

since the scalar field $\phi$ is inert under such transformations. The higher-derivative $\sigma$-model (1.1) possesses the ${\mathcal {N}}=1$ supersymmetric extension [7] which relates the parameters $\alpha$ and $\beta .$ Both parameters are completely fixed if ${\mathcal {N}}=2$ supersymmetry is required [7,8,9].

The conformal higher-derivative $\sigma$-model (1.1) admits a nontrivial generalisation that is obtained by replacing the Hermitian symmetric space $\textsf{SL}(2,{\mathbb {R}}) / \textsf{SO}(2)$ with an arbitrary n-dimensional Kähler manifold ${\mathfrak {M}}^{n},$ with n the complex dimension. We assume that ${\mathfrak {M}}^{n}$ is parametrised by n local complex coordinates $\phi ^I$ and their conjugates $\bar{\phi }^{\bar{I}}.$ Let ${\mathfrak {K}}(\phi , \bar{\phi })$ be the corresponding Kähler potential such that the Kähler metric ${\mathfrak {g}}_{I\bar{J}}(\phi ,\bar{\phi })$ is given by ${\mathfrak {g}}_{I{\bar{J}}} = \partial _I \partial _{{\bar{J}}} {\mathfrak {K}}.$ Associated with ${\mathfrak {M}}^n$ is a higher-derivative sigma model of the form

$$\begin{aligned} S&=\int \!\!\text {d}^4x\,\sqrt{-g} \,\bigg \{ {\mathfrak {g}}_{I\bar{J}}(\phi ,\bar{\phi })\Bigg [{\mathcal {D}}^2\phi ^{I}{\mathcal {D}}^2\bar{\phi }^{\bar{J}}\nonumber \\&\quad -2\Bigg (R^{mn}-\frac{1}{3}Rg^{mn}\Bigg )\nabla _m\phi ^{I}\nabla _n\bar{\phi }^{\bar{J}}\Bigg ] \nonumber \\&\quad + {\mathfrak {F}}_{IJ \bar{K}\bar{L}} (\phi ,\bar{\phi }) \nabla ^m\phi ^{I} \nabla _m\phi ^{J} \nabla ^n\bar{\phi }^{\bar{K}} \nabla _n\bar{\phi }^{\bar{L}} \nonumber \\&\quad +\Bigg [ {\mathfrak {G}}_{IJ \bar{K}\bar{L}} (\phi ,\bar{\phi }) \nabla ^m\phi ^{I} \nabla ^n\phi ^{J} \nabla _m\bar{\phi }^{\bar{K}} \nabla _n\bar{\phi }^{\bar{L}} +{\mathrm{c.c.}} \Bigg ] \bigg \} , \end{aligned}$$

(1.6)

where

$$\begin{aligned} {\mathcal {D}}^2\phi ^I:=\nabla ^m \nabla _m \phi ^I+\Gamma ^I_{{~}JK}(\phi , \bar{\phi }) \nabla ^m\phi ^J\nabla _m\phi ^K, \end{aligned}$$

(1.7)

with $\Gamma ^I_{{~}JK}$ being the Christoffel symbols for the Kähler metric ${\mathfrak {g}}_{I\bar{J}}.$ Finally, ${\mathfrak {F}}_{IJ \bar{K}\bar{L}}$ and ${\mathfrak {G}}_{IJ \bar{K}\bar{L}}$ are tensor fields on the target space, which are constructed from the Kähler metric ${\mathfrak {g}}_{I\bar{J}},$ Riemann tensor ${\mathfrak {R}}_{I \bar{J}K \bar{L}}$ and, in general, its covariant derivatives. We recall that the Christoffel symbols $\Gamma ^I_{{~}JK}$ and the curvature tensor ${\mathfrak {R}}_{I {\bar{J}} K {\bar{L}}}$ are given by the expressions³

$$\begin{aligned} \Gamma ^I_{{~}JK}= & {} {\mathfrak {g}}^{I {\bar{L}}} \partial _J \partial _K \partial _{{\bar{L}}} {\mathfrak {K}},\nonumber \\ {\mathfrak {R}}_{I {\bar{J}} K {\bar{L}}}= & {} \partial _I \partial _K \partial _{{\bar{J}}} \partial _{{\bar{L}}} {\mathfrak {K}} -{\mathfrak {g}}^{M {\bar{N}}} \partial _I \partial _K \partial _{{\bar{N}}} {\mathfrak {K}} \partial _{{\bar{J}}} \partial _{{\bar{L}}} \partial _M {\mathfrak {K}}. \end{aligned}$$

(1.8)

A typical expression for ${\mathfrak {F}}_{IJ \bar{K}\bar{L}}$ is

$$\begin{aligned} {\mathfrak {F}}_{IJ \bar{K}\bar{L}} = \alpha _1 {\mathfrak {R}}_{(I \bar{K}J)\bar{L}} +\alpha _2 {\mathfrak {g}}_{(I \bar{K}} {\mathfrak {g}}_{J) \bar{L}} +\cdots \end{aligned}$$

(1.9)

The possible structure of ${\mathfrak {G}}_{IJ \bar{K}\bar{L}}$ is analogous. It should be pointed out that actions of the form (1.6) naturally emerge at the component level in ${\mathcal {N}}=2$ superconformal higher-derivative $\sigma$-models [8] (see also [9]), and in ${\mathcal {N}}=1$ ones [7].

By construction, the action (1.6) is invariant under arbitrary holomorphic isometries of ${\mathfrak {M}}^{n}.$ A nontrivial observation is that (1.6) is also invariant under arbitrary Weyl transformations of spacetime provided the scalars $\phi ^I$ are inert under these transformations. Choosing ${\mathfrak {M}}^{n}={\mathbb {C}}^n$ and ${\mathfrak {g}}_{I\bar{J}}(\phi ,\bar{\phi }) = \delta _{I\bar{J}}$ in (1.6) and integrating by parts, one obtains the Fradkin–Tseytlin (FT) operator [13, 14]

$$\begin{aligned} \Delta _0 = (\nabla ^m \nabla _m)^2 + 2 \nabla ^m \big ( {R}_{mn} \,\nabla ^n - \frac{1}{3} {R} \,\nabla _m \big ), \end{aligned}$$

(1.10)

which is conformal when acting on dimensionless scalar fields.⁴ Given a Weyl inert scalar field $\varphi ,$ the Weyl transformation (1.5) acts on $\Delta _0 \varphi$ as

$$\begin{aligned} \Delta _0 \varphi ~ \rightarrow ~\Delta _0 \varphi = \text {e}^{-4\sigma } \Delta _0 \varphi . \end{aligned}$$

(1.11)

An action of the form $\int \text {d}^4 x \, \sqrt{-g}\, {\mathcal {L}}(\phi , \bar{\phi }),$ with ${\mathcal {L}}$ given by (1.1), naturally emerges as an induced action in Maxwell’s electrodynamics coupled to a dilaton $\varphi$ and an axion $\mathfrak a$ with Lagrangian

$$\begin{aligned} L(F; \phi , \bar{\phi })= & {} - \frac{1}{4} \text {e}^{-\varphi } F^{mn}F_{mn} -\frac{1}{4} {\mathfrak {a}} {\tilde{F}}^{mn} F_{mn} \nonumber \\= & {} \frac{\text {i}}{2} \phi F^{\alpha \beta } F_{\alpha \beta } + {\mathrm{c.c.}}, \quad \phi = {\mathfrak {a}} +\text {i}e^{-\varphi }. \end{aligned}$$

(1.12)

Here ${\tilde{F}}^{mn}= \frac{1}{2}\varepsilon ^{mnrs} F_{rs}$ is the Hodge dual of the electromagnetic field strength $F_{mn}=2\nabla _{[m} A_{n]} = 2\partial _{[m} A_{n]},$ with $\varepsilon ^{mnrs}$ the Levi-Civita tensor. The second form of the Lagrangian (1.12) is written using two-component spinor notation, where the field strength $F_{mn} = - F_{nm}$ is replaced with a symmetric rank-2 spinor $F_{\alpha \beta } = F_{\beta \alpha }$ and its conjugate ${\bar{F}}_{\dot{\alpha }\dot{\beta }}.$ More precisely, if one considers the effective action, $\Gamma [\phi , \bar{\phi }],$ obtained by integrating out the quantum gauge field in the model (1.12), then the logarithmically divergent part of $\Gamma [\phi , \bar{\phi }]$ is given by $\int \text {d}^4 x \, \sqrt{-g}\, {\mathcal {L}}(\phi , \bar{\phi }),$ as demonstrated by Osborn [6]. An important question arises: why is the induced action Weyl and $\textsf{SL}(2,{\mathbb {R}})$ invariant?

We recall that the group of electromagnetic duality rotations of free Maxwell’s equations is ${\textsf{U}}(1).$ More than forty years ago, it was shown by Gaillard and Zumino [17, 18] that the non-compact group ${\textsf{Sp}}(2n, {\mathbb {R}})$ is the maximal duality group of n Abelian vector field strengths $F_{mn}= (F_{mn, i}),$ with $i =1,\ldots , n,$ in the presence of a collection of complex scalars $\phi ^{ij}=\phi ^{ji}$ parametrising the homogeneous space ${\textsf{Sp}}(2n, {\mathbb {R}})/{\textsf{U}}(n),$ with $i, j = 1,\ldots , n.$ In the absence of such scalars, the largest duality group proves to be ${\textsf{U}}(n),$ the maximal compact subgroup of ${\textsf{Sp}}(2n, {\mathbb {R}}).$ These results admit a natural extension to the case when the pure vector field part L(F) of the Lagrangian $L(F;\phi , \bar{\phi })$ is a nonlinear ${\textsf{U}}(1)$ duality invariant theory [19,20,21,22,23] (see [24,25,26] for reviews), for instance Born–Infeld theory. However, in the case that L(F) is quadratic, the F-dependent part of $L(F; \phi , \bar{\phi })$ is also invariant under the Weyl transformations in curved space. Then, computing the path integral over the gauge fields leads to an effective action, $\Gamma [\phi , \bar{\phi }],$ such that its logarithmically divergent part is invariant under Weyl and rigid ${\textsf{Sp}}(2n, {\mathbb {R}})$ transformations, see, e.g., [27, 28] for formal arguments. Both symmetries are anomalous at the quantum level, but the logarithmically divergent part of the one-loop effective action is invariant under these transformations.

In this paper we demonstrate that an action of the type (1.6) emerges as an induced action in a model for n Abelian gauge fields $A_{m}=\left( A_{m,i}\right) ,$ $i=1,\ldots ,n,$ coupled to a complex field $\phi =(\phi ^{ij})$ and its conjugate $\bar{\phi }=(\bar{\phi }^{\,\bar{i}\bar{j}})$ parametrising the homogeneous space ${\textsf{Sp}}(2n,{\mathbb {R}})/{\textsf{U}}(n),$

$$\begin{aligned} \phi =\phi ^{\text {T}}\in \textsf{Mat}(n, {\mathbb {C}}) ,\quad \text {i}(\bar{\phi }-\phi )>0. \end{aligned}$$

(1.13)

The corresponding Lagrangian is

$$\begin{aligned} L^{(n)}(F;\phi ,\bar{\phi })= & {} -\frac{1}{4} \Bigg \{\left( F^{mn}\right) ^\text {T}\Xi F_{mn}+\left( F^{mn}\right) ^\text {T}\Upsilon \tilde{F}_{mn}\Bigg \} \nonumber \\= & {} \frac{\text {i}}{2}\left( F^{\alpha \beta }\right) ^\text {T}\phi F_{\alpha \beta } + {\mathrm{c.c.}}, \end{aligned}$$

(1.14)

where we have also introduced the real matrices $\Xi$ and $\Upsilon$ defined by

$$\begin{aligned} \phi = \Upsilon + \text {i}\,\Xi , \end{aligned}$$

(1.15)

with $\Xi$ being positive definite. The model described by (1.14) has two fundamental properties: (i) its duality group is ${\textsf{Sp}}(2n, {\mathbb {R}})$ (see, e.g. [24] for the technical details); and (ii) it is Weyl invariant. The induced action must respect these properties.

This paper is organised as follows. In Sect. 2 we compute the logarithmically divergent part of the effective action obtained by integrating out the vector fields in the model (1.14). Generalisations of our analysis and open problems are briefly discussed in Sect. 3. The main body of the paper is accompanied by three technical appendices. In Appendix A we collect necessary facts about the Hermitian symmetric space ${\textsf{Sp}}(2n, {\mathbb {R}} )/ {\textsf{U}}(n).$ Appendix B provides an alternative calculation of the induced action compared with that given in Sect. 2.2. Appendix C provides a complete list of the structures introduced in (2.22).

2 Computing the induced action

In this section we compute the logarithmically divergent part of the effective action, $\Gamma [\phi , \bar{\phi }],$ defined by

$$\begin{aligned} \textrm{e}^{\text {i}\Gamma [\phi , \bar{\phi }]}&=\int [{\mathfrak D} A] \, \delta \Bigg ( \eta - \chi (A) \Bigg ) \, \textrm{Det} \,(\Delta _{\textrm{gh}})\, \text {e}^{\text {i}S[A; \phi , \bar{\phi }] }. \end{aligned}$$

(2.1)

Here $S[A;\phi ,\bar{\phi }]$ is the classical action corresponding to (1.14),

$$\begin{aligned} S[A;\phi ,\bar{\phi }]=\int \!\!\text {d}^4x\,\sqrt{-g} \,L^{(n)}(F;\phi ,\bar{\phi }), \end{aligned}$$

(2.2)

$\chi (A)$ denotes a gauge fixing condition, $\Delta _{\textrm{gh}}$ the corresponding Faddeev–Popov operator [29], and $\eta$ an arbitrary background field. Since the effective action is independent of $\eta ,$ this field can be integrated out with some weight that we choose to be

$$\begin{aligned} \textrm{exp} \Bigg (-\frac{\text {i}}{2} \int \!\!\text {d}^4x\,\sqrt{-g}\,\eta ^\text {T}\Xi \eta \Bigg ). \end{aligned}$$

(2.3)

In general, the logarithmically divergent part of the effective action has the form

$$\begin{aligned} \Gamma _{\infty } = \frac{\ln \Lambda }{(4\pi )^2} \int \!\!\text {d}^4x\,\sqrt{-g}\, (a_2)_{\textrm{total}}, \end{aligned}$$

(2.4)

where $(a_2)_{\textrm{total}}$ denotes the appropriate sum of diagonal DeWitt coefficients and $\Lambda$ is a UV cutoff. We identify the induced action with $\int \!\!\text {d}^4x\,\sqrt{-g}\, (a_2)_{\textrm{total}},$ modulo an overall numerical coefficient.

2.1 Quantisation

We choose the simplest gauge-fixing condition

$$\begin{aligned} \chi (A)=\nabla ^mA_m, \end{aligned}$$

(2.5)

which leads to the ghost operator

$$\begin{aligned} \Delta _{\textrm{gh}}:=\Box \mathbb {1}, \end{aligned}$$

(2.6)

with $\mathbb {1}$ the $n\times n$ unit matrix. Integrating the right-hand side of (2.1) with the weight functional (2.3) leads to the gauge-fixing term

$$\begin{aligned} S_{\text {G.F.}}[A;\phi ,\bar{\phi }]{=}{-}\frac{1}{2}\int \!\!\text {d}^4x\,\sqrt{{-}g}\,(\nabla ^mA_m)^\text {T}\Xi (\nabla ^nA_n). \end{aligned}$$

(2.7)

As a result, the gauge-fixed action becomes

$$\begin{aligned} S_{\text {quadratic}}[A;\phi ,\bar{\phi }]=\frac{1}{2}\int \!\!\text {d}^4x\,\sqrt{-g}\,A_{\hat{m}} \Delta ^{\hat{m}\hat{n}}A_{\hat{n}}, \end{aligned}$$

(2.8)

where here we have introduced hatted indices corresponding to a pair of spacetime and internal indices $A_{\hat{m}}:=(A_{mi}),$ $\Delta ^{\hat{m}\hat{n}}:=(\Delta ^{mi,nj}).$ Contractions over hatted indices encode summations over both indices, however the position of the hatted indices (up or down) indicates only the position of the spacetime indices, internal indices are always understood as matrix multiplication. The non-minimal operator $\Delta ^{\hat{m}\hat{n}}$ is defined as:

$$\begin{aligned}&\Delta ^{mi,nj}:=\Xi ^{ij}g^{mn}\Box +V^{mi,p,nj}\nabla _p-R^{mn}\Xi ^{ij}, \qquad \Box :=\nabla ^m\nabla _m, \end{aligned}$$

(2.9a)

$$\begin{aligned}&V^{mi,p,nj}:=(\nabla ^p\Xi ^{ij})g^{mn}-(\nabla ^n\Xi ^{ij})g^{mp}\nonumber \\&\quad +(\nabla ^m\Xi ^{ij})g^{pn}-(\nabla _q\Upsilon ^{ij})\varepsilon ^{mpnq}. \end{aligned}$$

(2.9b)

From here onward matrix indices will be suppressed, unless there may be ambiguity or confusion. The one-loop effective action is specified by

$$\begin{aligned} \Gamma ^{(1)}[\phi ,\bar{\phi }]=\frac{\text {i}}{2} \text {Tr}\ln {\Delta } -\text {i}\text {Tr}\ln \Delta _\textrm{gh}. \end{aligned}$$

(2.10)

Since $\Xi$ is symmetric and positive definite, due to (1.13), its inverse $\Xi ^{-1},$ square root $\Xi ^{1/2}$ and inverse square root $\Xi ^{-1/2}$ are well-defined. We perform a local field redefinition in the path integral:

$$\begin{aligned} A_{m} \rightarrow \Xi ^{-1/2}A_{m}, \end{aligned}$$

(2.11)

so that the operator which appeared in (2.8) becomes⁵

$$\begin{aligned} \tilde{\Delta }^{\hat{m}\hat{n}}=\Xi ^{-1/2}\Delta ^{mn}\Xi ^{-1/2}. \end{aligned}$$

(2.12)

Inserting the explicit form of $\Delta ^{\hat{m}\hat{n}}$ from (2.9a) and (2.9b), the $\tilde{\Delta }^{\hat{m}}_{{~~}\hat{n}}$ operator is now minimal:

$$\begin{aligned} \tilde{\Delta }^{\hat{m}}_{{~~}\hat{n}}&=\mathbb {1}\,\delta ^m_{{~}n}\Box +Q^{\hat{m}}_{{~~}p\hat{n}}\nabla ^p+T^{\hat{m}}_{{~~}\hat{n}}, \end{aligned}$$

(2.13a)

$$\begin{aligned} Q^{\hat{m}}_{{~~}p\hat{n}}\!\!&:=-2\left( \nabla _p\Xi ^{1/2}\right) \Xi ^{-1/2}\delta ^m_{{~}n}+\Xi ^{-1/2}\,V^m_{{~}pn}\,\Xi ^{-1/2}, \end{aligned}$$

(2.13b)

$$\begin{aligned} T^{\hat{m}}_{{~~}\hat{n}}&:\!\!{=}{-}\!\!\left( \Box \Xi ^{1/2}\right) \!\!\delta ^m_{{~}n} {+}2\!\!\left( \nabla ^p\Xi ^{1/2}\right) \!\!\Xi ^{{-}1/2}\!\!\left( \nabla _p\Xi ^{1/2}\right) \!\!\Xi ^{{-}1/2}\delta ^m_{{~}n}\nonumber \\&\quad {-}\Xi ^{{-}1/2}\,V^m_{{~}pn}\Xi ^{{-}1/2}\left( \nabla ^p\Xi ^{1/2}\right) \Xi ^{{-}1/2}{-}R^m_{{~}n}\mathbb {1}. \end{aligned}$$

(2.13c)

After our field redefinition the one-loop effective action is given by

$$\begin{aligned} \Gamma ^{(1)}[\phi ,\bar{\phi }]=\frac{\text {i}}{2}\text {Tr}\ln \tilde{\Delta }-\text {i}\text {Tr}\ln \Delta _\textrm{gh}. \end{aligned}$$

(2.14)

2.2 Heat kernel calculations

Since the operator $\tilde{\Delta }^{\hat{m}}_{{~~}\hat{n}}$ defined by (2.13a) is minimal, we can proceed with the standard heat kernel technique in curved space, by bringing it to the form:

$$\begin{aligned} \tilde{\Delta }^{\hat{m}}_{{~~}\hat{n}}&=\big (\hat{\nabla }^p\hat{\nabla }_p\big )^{\hat{m}}_{{~~}\hat{n}}+\hat{P}^{\hat{m}}_{{~~}\hat{n}}, \end{aligned}$$

(2.15a)

$$\begin{aligned} \hat{P}^{\hat{m}}_{{~~}\hat{n}}&:=-\frac{1}{2}\nabla ^pQ^{\hat{m}}_{{~~}p\hat{n}}-\frac{1}{4}\,Q^{\hat{m}}_{{~~}q{\hat{p}}}Q^{\hat{p}q}_{{~~~}\hat{n}}+T^{\hat{m}}_{{~~}\hat{n}}. \end{aligned}$$

(2.15b)

The generalised covariant derivative $\hat{\nabla }_m$ introduced above is defined to act on a column matrix $A^{\hat{m}} =(A^m_{{~}i})$ as

$$\begin{aligned} \big (\hat{\nabla }_pA\big )^{\hat{m}}:=\nabla _pA^{\hat{m}}+\frac{1}{2}\,Q^{\hat{m}}_{{~}p\hat{n}}A^{\hat{n}}. \end{aligned}$$

(2.16)

The generalised covariant derivatives have no torsion, meaning

$$\begin{aligned} \big [\hat{\nabla }_p,\hat{\nabla }_q\big ] A^{\hat{m}}=\hat{R}^{\hat{m}}_{{~~}\hat{n}pq}A^{\hat{n}}, \end{aligned}$$

(2.17)

with $\hat{R}^{\hat{m}}_{{~~}\hat{n}pq}$ some generalised curvature anti-symmetric in p, q. Explicitly it has the form

$$\begin{aligned} \hat{R}^{\hat{m}}_{{~}\hat{n}pq}= & {} R^m_{{~}npq}\mathbb {1}+\frac{1}{2}\nabla _pQ^{\hat{m}}_{{~~}q\hat{n}}-\frac{1}{2}\nabla _qQ^{\hat{m}}_{{~~}p\hat{n}}\nonumber \\{} & {} +\frac{1}{4}\,Q^{\hat{m}}_{{~~}p\hat{r}}Q^{\hat{r}}_{{~~}q\hat{n}}-\frac{1}{4}\,Q^{\hat{m}}_{{~~}q\hat{r}}Q^{\hat{r}}_{{~~}p\hat{n}}. \end{aligned}$$

(2.18)

Using the standard Schwinger–DeWitt formalism [30,31,32,33,34] in curved spacetime for an operator of the form (2.15a), in the coincidence limit the DeWitt coefficient traced over matrix indices, $(a_2)^{\tilde{\Delta }}(x,x),$ is given by

$$\begin{aligned} (a_2)^{\tilde{\Delta }}(x,x)&=\bigg (\frac{1}{45}R^{mnpq}R_{mnpq}-\frac{1}{45}R^{mn}R_{mn}\nonumber \\&\quad +\frac{1}{18}R^2+\frac{2}{15}\Box R\bigg )\text {Tr}\mathbb {1}\nonumber \\&\quad +\frac{1}{12}\hat{R}^{\hat{m}\hat{n}pq}\hat{R}_{\hat{n}\hat{m}pq}+\frac{1}{6}\big (\hat{\nabla }^p\hat{\nabla }_p\hat{P}\big )^{\hat{m}}_{{~~}\hat{m}}\nonumber \\&\quad +\frac{1}{2}\big (\hat{P}^2\big )^{\hat{m}}_{{~~}\hat{m}}+\frac{1}{6}R\hat{P}^{\hat{m}}_{{~~}\hat{m}}, \end{aligned}$$

(2.19)

where ‘$\text {Tr}$’ denotes the matrix trace. Similarly for the ghost operator (2.6), the corresponding traced DeWitt coefficient $(a_2)^{\Delta _{gh}}(x,x)$ (noting that the generalised curvature vanishes) is

$$\begin{aligned} (a_2)^{\Delta _{gh}}(x,x){} & {} =\bigg (\frac{1}{180}R^{mnpq}R_{mnpq}-\frac{1}{180}R^{mn}R_{mn}\nonumber \\{} & {} \quad +\frac{1}{72}R^2+\frac{1}{30}\Box R\bigg )\text {Tr}\mathbb {1}, \end{aligned}$$

(2.20)

which contains purely gravitational components. Armed with the set of Eqs. (2.15a–2.18), we expand out $(a_2)^{\tilde{\Delta }}(x,x)$ (2.19) explicitly in terms of $Q^{\hat{m}}_{{~~}p\hat{n}}$ (2.13b) and $T^{\hat{m}}_{{~~}\hat{n}}$ (2.13c)

$$\begin{aligned} (a_2)^{\tilde{\Delta }}(x,x){} & {} =\bigg (-\frac{11}{180}R^{mnpq}R_{mnpq}-\frac{1}{45}R^{mn}R_{mn}\nonumber \\{} & {} \quad +\frac{1}{18}R^2+\frac{2}{15}\Box R\bigg )\text {Tr}\mathbb {1}\nonumber \\{} & {} \quad +\frac{1}{6}R^{p{~}qr}_{{~}m}\left( \nabla _qQ^{mi}_{{~~}r,pi}\right) \nonumber \\{} & {} \quad +\frac{1}{12}R^{p{~}qr}_{{~}m}Q^{mi}_{{~~}q\hat{s}}Q^{\hat{s}}_{{~~}r,pi}-\frac{1}{12}R\left( \nabla ^pQ^{\hat{m}}_{{~~}p\hat{m}}\right) \nonumber \\{} & {} \quad -\frac{1}{24}R\,Q^{\hat{m}}_{{~~}q\hat{p}}Q^{\hat{p}q}_{{~~~}\hat{m}}+\frac{1}{6}R\,T^{\hat{m}}_{{~~}\hat{m}}\nonumber \\{} & {} \quad -\frac{1}{12}\Box \nabla ^pQ^{\hat{m}}_{{~~}p\hat{m}}\nonumber \\{} & {} \quad -\frac{1}{12}\left( \Box Q^{\hat{m}}_{{~~}q\hat{p}}\right) Q^{\hat{p}q}_{{~~~}\hat{m}}\nonumber \\{} & {} \quad -\frac{1}{24}\left( \nabla _qQ^{\hat{m}}_{{~~}r\hat{p}}\right) \big (\nabla ^qQ^{\hat{p}r}_{{~~~}\hat{m}}\big )\nonumber \\{} & {} \quad -\frac{1}{24}\left( \nabla ^qQ^{\hat{m}}_{{~~}r\hat{p}}\right) \big (\nabla ^rQ^{\hat{p}}_{{~~}q\hat{m}}\big )\nonumber \\{} & {} \quad +\frac{1}{8}\left( \nabla ^rQ^{\hat{m}}_{{~~}r\hat{p}}\right) \big (\nabla ^sQ^{\hat{p}}_{{~~}s\hat{m}}\big )\nonumber \\{} & {} \quad +\frac{1}{24}\left( \nabla ^qQ^{\hat{m}}_{{~~}r\hat{p}}\right) Q^{\hat{p}}_{{~~}q\hat{s}}Q^{\hat{s}r}_{{~~~}\hat{m}}\nonumber \\{} & {} \quad -\frac{1}{24}\left( \nabla ^qQ^{\hat{m}}_{{~~}r\hat{p}}\right) Q^{\hat{p}r}_{{~~~}\hat{s}}Q^{\hat{s}}_{{~~}q\hat{m}}\nonumber \\{} & {} \quad +\frac{1}{8}\left( \nabla ^rQ^{\hat{m}}_{{~~}r\hat{p}}\right) Q^{\hat{p}}_{{~~}t\hat{s}}Q^{\hat{s}t}_{{~~~}\hat{m}}\nonumber \\{} & {} \quad +\frac{1}{96}\,Q^{\hat{m}}_{{~~}q\hat{s}}Q^{\hat{s}}_{{~~}r\hat{p}}Q^{\hat{p}q}_{{~~~}{{\hat{t}}}}Q^{{{\hat{t}}}r}_{{{~~~~}r}\hat{m}}\nonumber \\{} & {} \quad +\frac{1}{48}\,Q^{\hat{m}}_{{~~}q\hat{p}}Q^{\hat{p}q}_{{~~~}\hat{r}}Q^{\hat{r}}_{{~~}s{{\hat{t}}}}Q^{{{\hat{t}}}s}_{{{~~~}s}\hat{m}}\nonumber \\{} & {} \quad -\frac{1}{2}\left( \nabla ^pQ^{\hat{m}}_{{~~}p\hat{q}}\right) T^{\hat{q}}_{{~~}\hat{m}}\nonumber \\{} & {} \quad -\frac{1}{4}T^{\hat{m}}_{{~~}\hat{r}}Q^{\hat{r}}_{{~~}p\hat{q}}Q^{\hat{q}p}_{{~~~}\hat{m}}+\frac{1}{6}\Box T^{\hat{m}}_{{~~}\hat{m}}+\frac{1}{2}(T^2)^{\hat{m}}_{{~~}\hat{m}}. \nonumber \\ \end{aligned}$$

(2.21)

Using the definitions of $Q^{\hat{m}}_{{~~}p\hat{n}}$ (2.13b) and $T^{\hat{m}}_{{~~}\hat{n}}$ (2.13c), we perform the laborious task of expanding $(a_2)^{\tilde{\Delta }}(x,x)$ in terms of the matrices $\Xi$ and $\Upsilon .$ It reduces to the following form:

$$\begin{aligned} (a_2)^{\tilde{\Delta }}(x,x){} & {} =\text {Tr}\bigg [\frac{7}{24}T_1{+}\frac{1}{48}T_2{-}\frac{5}{12}T_3{+}\frac{1}{12}T_4\nonumber \\{} & {} \quad {+}\frac{7}{12}T_5{+}\frac{5}{24}T_6{-}\frac{1}{24}T_7\nonumber \\{} & {} \quad {+}\frac{5}{24}T_8{+}\frac{7}{24}T_{9}{+}\frac{1}{48}T_{10}{+}\frac{1}{4}T_{11}{+}\frac{1}{4}T_{12}\nonumber \\{} & {} \quad {-}\frac{1}{2}T_{13}{+}\frac{1}{2}T_{14}{-}\frac{1}{2}T_{15}{-}\frac{1}{2}T_{16}{-}\frac{1}{2}T_{17}{-}\frac{1}{2}T_{18}\nonumber \\{} & {} \quad {+}\frac{1}{6}T_{19}{+}\frac{1}{6}T_{20}{+}{\mathcal {X}}\mathbb {1}+\nabla _m{\mathcal {Y}}^m{+}\Box {\mathcal {Z}}\bigg ],\nonumber \\ \end{aligned}$$

(2.22)

where the contributions $T_1,\ldots , T_{20}$ are listed in Appendix C. We have also introduced:

$$\begin{aligned} {\mathcal {X}}:{} & {} {=}{-}\frac{11}{180}R^{mnpq}R_{mnpq}{+}\frac{43}{90}R^{mn}R_{mn}\nonumber \\{} & {} \quad -\frac{1}{9}R^2{-}\frac{1}{30}\Box R, \nonumber \\ \end{aligned}$$

(2.23a)

$$\begin{aligned} {\mathcal {Y}}^m:{} & {} =-\frac{1}{4}\Xi ^{-1}(\nabla ^m\Xi )\Xi ^{-1}(\nabla ^n\Xi )\Xi ^{-1}(\nabla _n\Xi )\nonumber \\{} & {} \quad +\frac{1}{4}\Xi ^{-1}(\nabla ^m\Xi )\Xi ^{-1}(\nabla ^n\Upsilon )\Xi ^{-1}(\nabla _n\Upsilon )\nonumber \\{} & {} \quad +\frac{1}{12}\Xi ^{-1}(\nabla ^n\Xi )\Xi ^{-1}(\nabla ^m\Upsilon )\Xi ^{-1}(\nabla _n\Upsilon )\nonumber \\{} & {} \quad +\frac{1}{12}\Xi ^{-1}(\nabla ^n\Xi )\Xi ^{-1}(\nabla _n\Upsilon )\Xi ^{-1}(\nabla ^m\Upsilon )\nonumber \\{} & {} \quad {+}\frac{1}{6}\Xi ^{{-}1}(\nabla ^m\Xi )\Xi ^{{-}1}(\Box \Xi )\nonumber \\{} & {} \quad {-}\frac{1}{6}\Xi ^{{-}1}(\nabla ^m\Upsilon )\Xi ^{{-}1}(\Box \Upsilon )\nonumber \\{} & {} \qquad -\frac{1}{3}\Bigg (R^{mn}-\frac{1}{2}Rg^{mn}\Bigg )\Xi ^{-1}(\nabla _n\Xi ), \end{aligned}$$

(2.23b)

$$\begin{aligned} {\mathcal {Z}}:{} & {} =-\frac{1}{6}\Xi ^{-1}(\nabla ^m\Upsilon )\Xi ^{-1}(\nabla _m\Upsilon )-\frac{1}{3}\Xi ^{-1}(\Box \Xi )\nonumber \\{} & {} \quad +\frac{4}{3}\Xi ^{-1/2}(\Box \Xi ^{1/2})\nonumber \\{} & {} \quad +\frac{4}{3}\Xi ^{-1}(\nabla _n\Xi )\Xi ^{-1/2}(\nabla ^n\Xi ^{1/2}). \end{aligned}$$

(2.23c)

The total DeWitt coefficient corresponding to the logarithmic divergence of the effective action (2.14) is given by

$$\begin{aligned} (a_2)_{\textrm{total}}=(a_2)^{\tilde{\Delta }}(x,x)-2(a_2)^{\Delta _{gh}}(x,x), \end{aligned}$$

(2.24)

where $(a_2)^{\Delta _{gh}}(x,x)$ was given in (2.20). Recalling the expression for $\Xi$ and $\Upsilon$ in terms of the original fields $\phi$ and its conjugate $\bar{\phi }$ (1.15), and defining

$$\begin{aligned}&{\mathcal {D}}^2\phi :=\Box \phi +\text {i}(\nabla ^m\phi )\Xi ^{-1}(\nabla _m\phi ),\nonumber \\&{\mathcal {D}}^2\bar{\phi }:=\Box \bar{\phi }-\text {i}(\nabla ^m\bar{\phi })\Xi ^{-1}(\nabla _m\bar{\phi }), \end{aligned}$$

(2.25)

the total DeWitt coefficient is given by

$$\begin{aligned} (a_2)_{\textrm{total}}&=n\bigg (\frac{1}{10}F-\frac{31}{180}G-\frac{1}{10}\Box R\bigg )+\frac{1}{4}\text {Tr}\Bigg [\Xi ^{-1}({\mathcal {D}}^2\phi ) \Xi ^{-1}({\mathcal {D}}^2\bar{\phi })\nonumber \\&\quad -2\Bigg (R^{mn}-\frac{1}{3}Rg^{mn}\Bigg )\Xi ^{-1}(\nabla _m\phi )\Xi ^{-1}(\nabla _n\bar{\phi })\Bigg ]\nonumber \\&\quad +\frac{1}{24}\text {Tr}\Bigg [\Xi ^{-1}(\nabla ^m\phi )\Xi ^{-1}(\nabla _m\bar{\phi })\Xi ^{-1}(\nabla ^n\phi )\Xi ^{-1}(\nabla _n\bar{\phi })\Bigg ]\nonumber \\&\quad +\frac{1}{48}\text {Tr}\Bigg [\Xi ^{-1}(\nabla ^m\phi )\Xi ^{-1}(\nabla ^n\bar{\phi })\Xi ^{-1}(\nabla _m\phi )\Xi ^{-1}(\nabla _n\bar{\phi })\Bigg ], \end{aligned}$$

(2.26)

where F is the square of the Weyl tensor, G is the Euler density,

$$\begin{aligned}{} & {} F=R^{mnpq}R_{mnpq}-2R^{mn}R_{mn}+\frac{1}{3}R^2, \nonumber \\{} & {} G=R^{mnpq}R_{mnpq}-4R^{mn}R_{mn}+R^2, \end{aligned}$$

(2.27)

and we have removed the total derivative pieces $\text {Tr}\big [\nabla _m{\mathcal {Y}}^m\big ]$ and $\text {Tr}\big [\Box {\mathcal {Z}}\big ]$ since they do not contribute to the induced action $\int \!\!\text {d}^4x\,\sqrt{-g}\,(a_2)_{\textrm{total}}.$ The $\Box R$ in (2.26) is also a total derivative and can be omitted.

Setting $n=1$ in (2.26) yields the expected result derived in [5, 6]

$$\begin{aligned} (a_2)_{\textrm{total}}&=\frac{1}{10}F-\frac{31}{180}G-\frac{1}{10}\Box R \nonumber \\&\quad {+}\frac{1}{4({{\textrm{Im}}}\phi )^2}\Bigg [{\mathcal {D}}^2\phi {\mathcal {D}}^2\bar{\phi }{-}2\Bigg (R^{mn}{-}\frac{1}{3}Rg^{mn}\Bigg )\nonumber \\&\quad \times \nabla _m\phi \nabla _n\bar{\phi }\Bigg ]+\frac{1}{48({{\textrm{Im}}}\phi )^4}\Bigg [ \nabla ^m\phi \nabla _m\phi \nabla ^n\bar{\phi }\nabla _n\bar{\phi }\nonumber \\&\quad +2\nabla ^m\phi \nabla _m\bar{\phi }\nabla ^n\phi \nabla _n\bar{\phi }\Bigg ]. \end{aligned}$$

(2.28)

2.3 Geometric expression for the induced action

To recast (2.26) in terms of geometric objects defined on the Hermitian symmetric space ${\textsf{Sp}}(2n,{\mathbb {R}})/{\textsf{U}}(n),$ here we analyse the dependence of (2.26) on the symmetric matrix $\phi =\left( \phi ^{ij}\right) =\left( \phi ^{ji}\right) \equiv (\phi ^I)$ and its conjugate $\bar{\phi }=\left( \bar{\phi }^{\,\bar{i}\bar{j}}\right) =\left( \bar{\phi }^{\,\bar{j}\bar{i}}\right) \equiv (\bar{\phi }^{{\bar{I}}}).$

We make the standard choice for Kähler potential ${\mathfrak {K}}(\phi ^{ij},\bar{\phi }^{\,\bar{i}\bar{j}})$ on ${\textsf{Sp}}(2n,{\mathbb {R}})/{\textsf{U}}(n)$

$$\begin{aligned} {\mathfrak {K}}(\phi ,\bar{\phi }):=-4\text {Tr}\ln \Xi , \end{aligned}$$

(2.29)

which is well defined since $\Xi$ is a positive definite matrix. The group ${\textsf{Sp}}(2n,{\mathbb {R}})$ acts on ${\textsf{Sp}}(2n,{\mathbb {R}})/{\textsf{U}}(n)$ by fractional linear transformations (A.18). Given such a transformation, the Kähler potential changes as

$$\begin{aligned} {\mathfrak {K}}(\phi ,\bar{\phi })\rightarrow {\mathfrak {K}}(\phi ,\bar{\phi })+\Lambda (\phi )+\bar{\Lambda }(\bar{\phi }), \end{aligned}$$

(2.30)

in accordance with (A.20). Therefore, the Kähler metric is invariant under arbitrary ${\textsf{Sp}}(2n,{\mathbb {R}})$ transformations.

The Kähler metric is given by⁶

$$\begin{aligned} {\mathfrak {g}}_{ij,\bar{k}\bar{l}}={\mathfrak {g}}_{(ij),(\bar{k}\bar{l})}=\frac{\partial ^2 {\mathfrak {K}}}{\partial \phi ^{ij}\partial \bar{\phi }^{\,\bar{k}\bar{l}}}=(\Xi ^{-1})_{i (\bar{k}}(\Xi ^{-1})_{\bar{l})j}, \end{aligned}$$

(2.31)

where $(i_1\cdots i_n)$ denotes symmetrisation in indices $i_1,\ldots ,i_n.$ Note that pairs of indices are symmetrised over due to $\phi$ being symmetric (1.13). Here and in what follows, we use the notation

$$\begin{aligned}{} & {} {\mathfrak {K}}_{i_1i_2,\ldots ,i_{2p-1}i_{2p},\bar{i}_1\bar{i}_2,\ldots ,\bar{i}_{2q-1}\bar{i}_{2q}}\nonumber \\{} & {} \quad =\frac{\partial ^{p+q} {\mathfrak {K}}}{\partial \phi ^{i_1i_2}\cdots \partial \phi ^{i_{2p-1}i_{2p}}\partial \bar{\phi }^{\,\bar{i}_1\bar{i}_2}\cdots \partial \bar{\phi }^{\,\bar{i}_{2q-1}\bar{i}_{2q}}}. \end{aligned}$$

(2.32)

In accordance with (1.8), the Christoffel symbols are given by

$$\begin{aligned}{} & {} \Gamma ^{i_1i_2}_{\quad i_3i_4,i_5i_6}={\mathfrak {g}}^{i_1i_2,\bar{i}_7\bar{i}_8}{\mathfrak {K}}_{i_3i_4,i_5i_6,\bar{i}_7\bar{i}_8},\nonumber \\{} & {} \qquad \Gamma ^{\bar{i}_1\bar{i}_2}_{\quad \bar{i}_3\bar{i}_4,\bar{i}_5\bar{i}_6}=(\Gamma ^{i_1i_2}_{\quad i_3i_4,i_5i_6})^*, \end{aligned}$$

(2.33)

and the Riemann curvature tensor is

$$\begin{aligned}{} & {} {\mathfrak {R}}_{i_1i_2,\bar{i}_3\bar{i}_4,i_5i_6,\bar{i}_7\bar{i}_8}={\mathfrak {K}}_{i_1i_2,i_5i_6,\bar{i}_3\bar{i}_4,\bar{i}_7\bar{i}_8}\nonumber \\{} & {} \quad -{\mathfrak {g}}^{i_9i_{10},\bar{i}_{11}\bar{i}_{12}}{\mathfrak {K}}_{i_1i_2,i_5i_6,\bar{i}_{11}\bar{i}_{12}}{\mathfrak {K}}_{i_9i_{10},\bar{i}_3\bar{i}_4,\bar{i}_7\bar{i}_8}. \end{aligned}$$

(2.34)

Noting that the inverse Kähler metric of (2.31) is

$$\begin{aligned} {\mathfrak {g}}^{ij,\bar{k}\bar{l}}=\Xi ^{i(\bar{k}}\,\Xi ^{\bar{l}) j}, \end{aligned}$$

(2.35)

one can calculate the Christoffel symbols, Riemann curvature tensor and Ricci tensor for the metric considered in (2.31) and we find:

$$\begin{aligned}&\Gamma ^{i_1i_2}_{\quad i_3i_4,i_5i_6}=\text {i}\delta ^{(i_1}_{\quad (i_3}(\Xi ^{-1})_{i_4)(i_5}\delta ^{i_2)}_{\quad i_6)}, \end{aligned}$$

(2.36a)

$$\begin{aligned}&\Gamma ^{\bar{i}_1\bar{i}_2}_{\quad \bar{i}_3\bar{i}_4,\bar{i}_5\bar{i}_6}=-\text {i}\delta ^{(\bar{i}_1}_{\quad (\bar{i}_3}(\Xi ^{-1})_{\bar{i}_4)(\bar{i}_5}\delta ^{\bar{i}_2)}_{\quad \bar{i}_6)}, \end{aligned}$$

(2.36b)

$$\begin{aligned}&{\mathfrak {R}}_{i_1i_2,\bar{i}_3\bar{i}_4,i_5i_6,\bar{i}_7\bar{i}_8}\nonumber \\&\quad =\frac{1}{2}(\Xi ^{-1})_{(\bar{i}_8|(i_1}(\Xi ^{-1})_{i_2)(\bar{i}_3}(\Xi ^{-1})_{\bar{i}_4)(i_5}(\Xi ^{-1})_{i_6)|\bar{i}_7)}, \end{aligned}$$

(2.36c)

$$\begin{aligned}&{\mathfrak {R}}_{i_1i_2,\bar{i}_3\bar{i}_4}=-\frac{n+1}{4}(\Xi ^{-1})_{i_1(\bar{i}_3}(\Xi ^{-1})_{\bar{i}_4)i_2}\nonumber \\&\quad =-\frac{n+1}{4}{\mathfrak {g}}_{i_1i_2,\bar{i}_3\bar{i}_4}. \end{aligned}$$

(2.36d)

The latter relation means that ${\textsf{Sp}}(2n,{\mathbb {R}})/{\textsf{U}}(n)$ is an Einstein space.

As pointed out at the beginning of this subsection, the complex variables $\phi$ and their conjugates $\bar{\phi }$ can be viewed either as symmetric matrices $\phi =(\phi ^{ij})$ and $\bar{\phi }=( \bar{\phi }^{\bar{i}\bar{j}})$ or as vector columns $\phi = (\phi ^I)$ and $\bar{\phi }=( \bar{\phi }^{\bar{I}}),$ with $I,\bar{I}=1,\ldots ,\frac{1}{2}{n(n+1)}.$ Resorting to the latter notation, the geometric structures (2.31) and (2.36a–2.36c) can be used to recast (2.26) in the form:

$$\begin{aligned} (a_2)_{\textrm{total}}&=n\bigg (\frac{1}{10}F-\frac{31}{180}G-\frac{1}{10}\Box R\bigg ) \nonumber \\&\quad +\frac{1}{4}{\mathfrak {g}}_{I\bar{J}}(\phi ,\bar{\phi })\Bigg [{\mathcal {D}}^2\phi ^{I}{\mathcal {D}}^2\bar{\phi }^{\bar{J}}\nonumber \\&\quad -2\Bigg (R^{mn} -\frac{1}{3}Rg^{mn}\Bigg )\nabla _m\phi ^{I}\nabla _n\bar{\phi }^{\bar{J}}\Bigg ]\nonumber \\&\quad +\frac{1}{24}{\mathfrak {R}}_{I\bar{J}K \bar{L}}(\phi ,\bar{\phi })\Bigg [2\nabla ^m\phi ^{I}\nabla _m\bar{\phi }^{\bar{J}}\nabla ^n\phi ^{K}\nabla _n\bar{\phi }^{\bar{L}}\nonumber \\&\quad +\nabla ^m\phi ^{I}\nabla ^n\bar{\phi }^{\bar{J}}\nabla _m\phi ^{K}\nabla _n\bar{\phi }^{\bar{L}}\Bigg ], \end{aligned}$$

(2.37a)

where

$$\begin{aligned}&{\mathcal {D}}^2\phi ^I=\Box \phi ^I+\Gamma ^I_{{~}JK}\nabla ^m\phi ^J\nabla _m\phi ^K,\nonumber \\&{\mathcal {D}}^2\bar{\phi }^{\bar{I}}=\Box \bar{\phi }^{\bar{I}}+\Gamma ^{\bar{I}}_{{~~}\bar{J}\bar{K}}\nabla ^m\bar{\phi }^{\bar{J}}\nabla _m\bar{\phi }^{\bar{K}}. \end{aligned}$$

(2.37b)

Every isometry transformation (A.18) acts on $\nabla \phi$ and ${\mathcal {D}}^2 \phi$ as follows:

$$\begin{aligned} \nabla ^m\phi '&=\big ( (C\phi +D)^{-1}\big )^{\text {T}}(\nabla ^m\phi )(C\phi +D)^{-1}, \end{aligned}$$

(2.38a)

$$\begin{aligned} {\mathcal {D}}^2\phi '&=\big ( (C\phi +D)^{-1}\big )^{\text {T}}({\mathcal {D}}^2\phi )(C\phi +D)^{-1}. \end{aligned}$$

(2.38b)

It is now seen that the induced action defined by (2.37) is invariant under the isometry transformations on ${\textsf{Sp}}(2n,{\mathbb {R}})/{\textsf{U}}(n).$

3 Generalisations and open problems

Relation (2.37), which constitutes the induced action, is our main result. The same structure also determines the Weyl anomaly of the effective action

$$\begin{aligned} \delta _\sigma \Gamma \propto \frac{1}{(4\pi )^2} \int \!\!\text {d}^4x\,\sqrt{-g}\, \sigma \,(a_2)_{\textrm{total}}. \end{aligned}$$

(3.1)

It is well known that the purely gravitational part of this variation satisfies the Wess–Zumino consistency condition [35]

$$\begin{aligned}{}[\delta _{\sigma _2}, \delta _{\sigma _1} ]\Gamma =0, \end{aligned}$$

(3.2)

see, e.g., [36,37,38] for a review.⁷ The $\phi$-dependent part of the Weyl anomaly will be discussed below.

As a generalisation of (1.6), we can introduce a conformal higher-derivative $\sigma$-model associated with a Riemannian manifold $({\mathcal {W}}^d, {\mathfrak {g}})$ parametrised by local coordinates $\varphi ^\mu .$ The action is

$$\begin{aligned} S&=\int \!\!\text {d}^4x\,\sqrt{-g} \,\bigg \{ {\mathfrak {g}}_{\mu \nu }(\varphi )\Bigg [{\mathcal {D}}^2 \varphi ^{\mu }{\mathcal {D}}^2 \varphi ^{\nu }\nonumber \\&\quad -2\Bigg (R^{mn}-\frac{1}{3}Rg^{mn}\Bigg )\nabla _m\varphi ^\mu \nabla _n\varphi ^{\nu }\Bigg ] \nonumber \\&\quad + {\mathfrak {F}}_{\mu \nu \sigma \rho } (\varphi ) \nabla ^m\varphi ^{\mu } \nabla _m\varphi ^{\nu } \nabla ^n\varphi ^{\sigma } \nabla _n\varphi ^{\rho } \bigg \} , \end{aligned}$$

(3.3a)

where

$$\begin{aligned} {\mathcal {D}}^2 \varphi ^\mu := \Box \varphi ^\mu + \Gamma ^\mu _{{~}\nu \sigma } \nabla ^m \varphi ^\nu \nabla _m \varphi ^\sigma ,\quad \Box = \nabla ^m \nabla _m, \end{aligned}$$

(3.3b)

and ${\mathfrak {F}}_{\mu \nu \sigma \rho } (\varphi )$ is a tensor field of rank (0, 4) on ${\mathcal {W}}^d.$ The Weyl invariance of the above action follows from

$$\begin{aligned}{} & {} \delta _\sigma \bigg \{\sqrt{-g} \, {\mathfrak {g}}_{\mu \nu }(\varphi )\Bigg [{\mathcal {D}}^2 \varphi ^{\mu }{\mathcal {D}}^2 \varphi ^{\nu }\nonumber \\{} & {} \qquad -2\Bigg (R^{mn} -\frac{1}{3}Rg^{mn}\Bigg )\nabla _m\varphi ^\mu \nabla _n\varphi ^{\nu }\Bigg ] \bigg \}\nonumber \\{} & {} \quad =4 \sqrt{-g} \nabla _m \bigg \{ {\mathfrak {g}}_{\mu \nu }(\varphi ) \Bigg [ \nabla _n \sigma \nabla ^m\varphi ^\mu \nabla ^n \varphi ^\nu \nonumber \\{} & {} \qquad -\frac{1}{2}\nabla ^m \sigma \nabla ^n \varphi ^\mu \nabla _n \varphi ^\nu \Bigg ] \bigg \}. \end{aligned}$$

(3.4)

In the case that the target space is Kähler, Eq. (1.6), the relation (3.4) takes the form

$$\begin{aligned}&\delta _{\sigma }\bigg \{\sqrt{-g}\,{\mathfrak {g}}_{I\bar{J}}(\phi ,\bar{\phi })\Bigg [{\mathcal {D}}^2\phi ^{I}{\mathcal {D}}^2\bar{\phi }^{\bar{J}}\nonumber \\&\qquad -2\Bigg (R^{mn}-\frac{1}{3}Rg^{mn}\Bigg )\nabla _m\phi ^{I}\nabla _n\bar{\phi }^{\bar{J}}\Bigg ]\bigg \}\nonumber \\&\qquad =2\sqrt{-g}\,\nabla _m\bigg \{ {\mathfrak {g}}_{I\bar{J}}(\phi ,\bar{\phi })\Bigg [\nabla _n\sigma \Bigg (\nabla ^m\phi ^{I}\nabla ^n\bar{\phi }^{\bar{J}}+\nabla ^n\phi ^{I}\nabla ^m\bar{\phi }^{\bar{J}}\Bigg )\nonumber \\&\qquad -\nabla ^m\sigma \nabla ^n\phi ^{I}\nabla _n\bar{\phi }^{\bar{J}}\Bigg ]\bigg \}. \end{aligned}$$

(3.5)

Choosing ${\mathcal {W}}^d$ to be $\mathbb {R},$ specifying ${\mathfrak {g}}_{\mu \nu }(\varphi )$ and ${\mathfrak {F}}_{\mu \nu \sigma \rho } (\varphi )$ to be constant, respectively, and restricting the background spacetime to be flat, the action (3.1) turns into

$$\begin{aligned} S = \int \text {d}^4x \, L, \quad L = (\partial ^2 \varphi )^2 +f (\partial ^m \varphi \partial _m \varphi )^2, \end{aligned}$$

(3.6)

with f a coupling constant, which is the model studied recently by Tseytlin [39].

Assuming the model (3.3) originates from an induced action of some theory, the $\varphi$-dependent part of the Weyl anomaly should have the form

$$\begin{aligned} \delta _\sigma \Gamma&\propto \int \!\!\text {d}^4x\,\sqrt{-g} \, \sigma \bigg \{ {\mathfrak {g}}_{\mu \nu }(\varphi )\Bigg [{\mathcal {D}}^2 \varphi ^{\mu }{\mathcal {D}}^2 \varphi ^{\nu }\nonumber \\&\quad -2\Bigg (R^{mn} -\frac{1}{3}Rg^{mn}\Bigg )\nabla _m\varphi ^\mu \nabla _n\varphi ^{\nu }\Bigg ] \nonumber \\&\quad + {\mathfrak {F}}_{\mu \nu \sigma \rho } (\varphi ) \nabla ^m\varphi ^{\mu } \nabla _m\varphi ^{\nu } \nabla ^n\varphi ^{\sigma } \nabla _n\varphi ^{\rho } \bigg \} . \end{aligned}$$

(3.7)

The anomaly satisfies the Wess–Zumino consistency condition (3.2) as a consequence of the relation (3.4).

It would be interesting to study renormalisation properties of a higher-derivative theory in Minkowski space with Lagrangian of the form

$$\begin{aligned} {\mathcal {L}}(F, \phi , \bar{\phi })&=L^{(n)}(F;\phi ,\bar{\phi }) +f_1 {\mathfrak {g}}_{I\bar{J}}(\phi ,\bar{\phi }){\mathcal {D}}^2\phi ^{I}{\mathcal {D}}^2\bar{\phi }^{\bar{J}} \nonumber \\&\quad +{\mathfrak {R}}_{I\bar{J}K \bar{L}}(\phi ,\bar{\phi })\Bigg [f_2 \partial ^m\phi ^{I}\partial _m\bar{\phi }^{\bar{J}}\partial ^n\phi ^{K}\partial _n\bar{\phi }^{\bar{L}}\nonumber \\&\quad +f_3 \partial ^m\phi ^{I}\partial ^n\bar{\phi }^{\bar{J}}\partial _m\phi ^{K}\partial _n\bar{\phi }^{\bar{L}}\Bigg ] +\dots , \end{aligned}$$

(3.8)

where $L^{(n)}(F;\phi ,\bar{\phi })$ is given by (1.14), the complex scalar fields $\phi ^I$ and their conjugates $\bar{\phi }^{{\bar{I}}}$ parametrise ${\textsf{Sp}}(2n, {\mathbb {R}} )/ {\textsf{U}}(n),$ and $f_1, f_2$ and $f_3$ are dimensionless coupling constants. All the structures in the F-independent part of (3.8) appear in the induced action (2.37). The ellipsis in (3.8) denotes other ${\textsf{Sp}}(2n, {\mathbb {R}})$ terms that are quartic in $\partial \phi$ and $\partial \bar{\phi },$ such as the Kähler metric squared structure in (1.9). Such terms are possible for $n>1.$ The renormalisation of the most general fourth-order sigma models with dimensionless couplings in four dimensions was studied in [40, 41]. All couplings constants in (3.8) are dimensionless, and the freedom to choose them is dictated by ${\textsf{Sp}}(2n, {\mathbb {R}}).$ This implies that the theory with classical Lagrangian (3.8) is renormalisable at the quantum level.

It was discovered two years ago that Maxwell’s theory possesses a one-parameter conformal and ${\textsf{U}}(1)$ duality invariant deformation [42, 43]; it was called the ModMax theory in [42]. Using the methods developed in [20,21,22], it can be coupled to the dilaton-axion field (1.12) to result in a conformal and $\textsf{SL}(2,{\mathbb {R}})$ duality invariant model described by the Lagrangian [44]

$$\begin{aligned}{} & {} L_\gamma (F; \phi , \bar{\phi }) = - \frac{1}{2}\text {e}^{-\varphi } \big ( {\omega } + {\bar{\omega }}\big ) (\cosh \gamma -1) \nonumber \\{} & {} \quad + \text {e}^{-\varphi } {\sqrt{\omega \bar{\omega }} }\,{\sinh \gamma } +\frac{\text {i}}{2} \big (\phi \,\omega - \bar{\phi }\,\bar{\omega }\big ), \end{aligned}$$

(3.9a)

where

$$\begin{aligned}{} & {} \omega =\alpha +\text {i}\beta = F^{\alpha \beta } F_{\alpha \beta }, \nonumber \\{} & {} \alpha = \frac{1}{4} \, F^{ab}F_{ab}, \quad \beta = \frac{1}{4} \, F^{ab} \tilde{F}_{ab}, \end{aligned}$$

(3.9b)

and $\gamma$ is a non-negative coupling constant [42]. For $\gamma = 0$ the model (3.9) reduces to (1.12). A challenging problem is to compute an induced action generated by (3.9).

Appendices

Appendix A: Hermitian symmetric space ${\textsf{Sp}}(2n, {\mathbb {R}} )/ {\textsf{U}}(n)$

In this appendix we collect necessary facts about the Hermitian symmetric space ${\textsf{Sp}}(2n, {\mathbb {R}} )/ {\textsf{U}}(n).$ Here the symplectic group is defined by

$$\begin{aligned} {\textsf{Sp}}(2n, {\mathbb {R}} ) {=} \left\{ g \in \textsf{GL}(2n, {\mathbb {R}}), \quad g^\textrm{T} J g {=} J, \quad J {=}\left( \begin{array}{cc} 0 &{} \quad {\mathbb {1}}_n\\ {-}{\mathbb {1}}_n &{} \quad 0 \end{array} \right) \right\} .\nonumber \\ \end{aligned}$$

(A.1)

Its maximal compact subgroup, $H = {\textsf{Sp}}(2n, {\mathbb {R}} ) \cap \textsf{SO}(2n),$ proves to be isomorphic to ${\textsf{U}}(n).$ One way to see this is to make use of the isomorphism

$$\begin{aligned} {\textsf{Sp}}(2n, {\mathbb {R}} ) \cong {\textsf{Sp}}(2n, {\mathbb {C}} ) \cap \textsf{SU}(n,n) \equiv \underline{G}, \end{aligned}$$

(A.2)

which is obtained by considering the bijective map

$$\begin{aligned} \varphi : g ~\rightarrow ~\underline{g} = {\mathbb {A}}^{-1} g {\mathbb {A}}, \quad {\mathbb {A}} = \frac{1}{\sqrt{2}} \left( \begin{array}{rc} {\mathbb {1}}_n ~&{} ~{\mathbb {1}}_n\\ -\text {i}{\mathbb {1}}_n ~&{} ~ \text {i}{\mathbb {1}}_n \end{array} \right) , \end{aligned}$$

(A.3)

for any $g \in {\textsf{Sp}}(2n, {\mathbb {R}} ).$ Each complex matrix $\underline{g} = \varphi (g)$ is characterised by the properties

$$\begin{aligned} \underline{g}^{\textrm{T}} J \underline{g} = J, \quad \underline{g}^\dagger I_{n,n} \underline{g} = I_{n,n}, \quad I_{n,n}= \left( \begin{array}{cc} {\mathbb {1}}_n ~&{} 0 \\ 0 ~&{} - {\mathbb {1}}_n \end{array} \right) , \end{aligned}$$

(A.4)

and thus $\underline{g} \in {\textsf{Sp}}(2n, {\mathbb {C}} ) \cap \textsf{SU}(n,n).$ The inverse map $\varphi ^{-1}$ takes every group element $\underline{g} \in {\textsf{Sp}}(2n, {\mathbb {C}} ) \cap \textsf{SU}(n,n)$ to some $g \in \textsf{GL}(2n, {\mathbb {R}}).$ For every element h from the maximal compact subgroup, $h \in H = {\textsf{Sp}}(2n, {\mathbb {R}} ) \cap \textsf{SO}(2n),$ its image $\underline{h} = \varphi (h)$ is unitary, $\underline{h}^\dagger \underline{h} ={\mathbb {1}}_{2n}.$ Simple calculations show that every $h \in H = {\textsf{Sp}}(2n, {\mathbb {R}} ) \cap \textsf{SO}(2n)$ has the form

$$\begin{aligned} h = \left( \begin{array}{rc} A &{} \quad B \\ -B &{} \quad A \end{array} \right) ,\quad A A^\textrm{T} +B B^\textrm{T } = {\mathbb {1}}_n, \quad A B^\textrm{T} = B A^{\textrm{T}},\nonumber \\ \end{aligned}$$

(A.5)

and for its image $\underline{h} =\varphi (h)$ we obtain

$$\begin{aligned} \varphi (h) = \left( \begin{array}{cc} A -\text {i}B~&{} 0 \\ 0 ~&{} A +\text {i}B \end{array} \right) , \quad (A +\text {i}B )(A +\text {i}B)^\dagger = {\mathbb {1}}_n. \nonumber \\ \end{aligned}$$

(A.6)

The group ${\textsf{Sp}}(2n, {\mathbb {R}} )$ naturally acts on ${\mathbb {C}}^{2n}.$ This action is extended to that on a complex Grassmannian $\textrm{Gr}_{m, 2n} ({\mathbb {C}}),$ with $0<m<2n,$ the space of m-planes through the origin in ${\mathbb {C}}^{2n}.$ Of special interest is the Grassmannian $\textrm{Gr}_{n, 2n} ({\mathbb {C}}).$ Given an n-plane ${\mathcal {P}}\in \textrm{Gr}_{n, 2n} ({\mathbb {C}}),$ it can be identified with a $2n \times n$ matrix of rank n

$$\begin{aligned} {\mathcal {P}}= \left( \begin{array}{c} M \\ N \end{array} \right) , \end{aligned}$$

(A.7a)

defined modulo equivalence transformations

$$\begin{aligned} \left( \begin{array}{c} M \\ N \end{array} \right) \sim \left( \begin{array}{c} M R\\ NR \end{array} \right) ,\quad R \in \textsf{GL}(n, {\mathbb {C}}). \end{aligned}$$

(A.7b)

We denote by ${\mathfrak {X}} \subset \textrm{Gr}_{n, 2n} ({\mathbb {C}})$ the collection of all n-planes satisfying the two conditions:

$$\begin{aligned} {\mathcal {P}}^\textrm{T} J {\mathcal {P}}= & {} 0; \end{aligned}$$

(A.8a)

$$\begin{aligned} {\mathcal {P}}^\dagger (\text {i}J ) {\mathcal {P}}> & {} 0. \end{aligned}$$

(A.8b)

Condition (A.8b) means that the Hermitian matrix ${\mathcal {P}}^\dagger (\text {i}J ) {\mathcal {P}}$ is positive definite. Condition (A.8a) means that ${\mathcal {P}}$ is a Lagrangian subspace of ${\mathbb {C}}^{2n}$ with respect to the symplectic structure J.

By construction, the group ${\textsf{Sp}}(2n, {\mathbb {R}} )$ naturally acts on ${\mathfrak {X}}.$ It turns out that this action is transitive, and $\mathfrak {X}$ can be identified with ${\textsf{Sp}}(2n, {\mathbb {R}} )/ {\textsf{U}}(n).$ The simplest way to see this is to make use of the realisation ${\underline{G}}$ of ${\textsf{Sp}}(2n, {\mathbb {R}} ),$ Eq. (A.2). The picture changing transformation (A.3) is accompanied with

$$\begin{aligned} {\mathcal {P}}~\rightarrow ~ \underline{{\mathcal {P}}}= {\mathbb A}^{-1} {\mathcal {P}}=\left( \begin{array}{c} \underline{M} \\ \underline{N} \end{array} \right) . \end{aligned}$$

(A.9)

For the transformed n-plane $\underline{{\mathcal {P}}}$ the conditions (A.8) take the form

$$\begin{aligned} \underline{{\mathcal {P}}}^\textrm{T} J \underline{{\mathcal {P}}}= & {} 0; \end{aligned}$$

(A.10a)

$$\begin{aligned} \underline{{\mathcal {P}}}^\dagger I_{n,n} \underline{{\mathcal {P}}}> & {} 0. \end{aligned}$$

(A.10b)

The latter condition tells us that the matrix $\underline{M}$ in (A.9) is nonsingular, and therefore

$$\begin{aligned} \left( \begin{array}{c} \underline{M} \\ \underline{N} \end{array} \right) ~\sim ~ \left( \begin{array}{c} {\mathbb {1}}_n \\ \underline{N} \underline{M}^{-1} \end{array} \right) ~\equiv ~ \left( \begin{array}{c} {\mathbb {1}}_n \\ \psi \end{array} \right) . \end{aligned}$$

(A.11)

In terms of the $n\times n$ matrix $\psi ,$ the conditions (A.10) are equivalent to

$$\begin{aligned} \psi ^\textrm{T} = \psi , \quad {\mathbb {1}}_n > \psi ^\dagger \psi . \end{aligned}$$

(A.12)

The complex $n\times n$ matrix $\psi$ constrained by (A.12) and its conjugate $\bar{\psi }$ provide a global coordinate system for ${\mathfrak {X}}.$ Associated with $\psi$ and $\bar{\psi }$ is the group element

$$\begin{aligned} {\mathfrak {S}} (\psi , \bar{\psi }){} & {} = \left( \begin{array}{cc} s~&{} \bar{\psi }{\bar{s}} \\ \psi s ~&{} {\bar{s}} \end{array} \right) \in {\textsf{Sp}}(2n, {\mathbb {C}} ) \cap \textsf{SU}(n,n), \nonumber \\ s{} & {} = \big ( {\mathbb {1}}_n - \bar{\psi }\psi \big )^{-\frac{1}{2}}. \end{aligned}$$

(A.13)

Its important property is

$$\begin{aligned} {\mathfrak {S}} (\psi , \bar{\psi }) {\mathcal {P}}_0 = \left( \begin{array}{c} s \\ \psi s \end{array} \right) \sim \left( \begin{array}{c} {\mathbb {1}}_n \\ \psi \end{array} \right) , \quad {\mathcal {P}}_0:= \left( \begin{array}{c} {\mathbb {1}}_n \\ 0 \end{array} \right) . \end{aligned}$$

(A.14)

Thus ${\mathfrak {S}} (\psi , \bar{\psi })$ maps the “origin” ${\mathcal {P}}_0$ to the point (A.11) of ${\mathfrak {X}},$ and therefore the group ${\textsf{Sp}}(2n, {\mathbb {C}} ) \cap \textsf{SU}(n,n)$ acts transitively on ${\mathfrak {X}}.$ The isotropy subgroup of ${\mathcal {P}}_0$ can be seen to consist of the group elements (A.6), which span ${\textsf{U}}(n).$ We conclude that

$$\begin{aligned} {\mathfrak {X}} = \frac{ {\textsf{Sp}}(2n, {\mathbb {C}} ) \cap \textsf{SU}(n,n) }{{\textsf{U}}(n)} =\frac{ {\textsf{Sp}}(2n, {\mathbb {R}} ) }{{\textsf{U}}(n)}. \end{aligned}$$

(A.15)

Making use of (A.9)–(A.11), we can reconstruct a generic element of $\mathfrak X$ in the original real realisation (A.1).

$$\begin{aligned} {\mathcal {P}}= {\mathbb {A}} \left( \begin{array}{c} {\mathbb {1}}_n \\ \psi \end{array} \right) \sim \left( \begin{array}{c} {\mathbb {1}}_n +\psi \\ -\text {i}({\mathbb {1}}_n -\psi ) \end{array} \right) . \end{aligned}$$

(A.16)

Since the matrices ${\mathbb {1}}_n \pm \psi$ are non-singular, ${\mathcal {P}}$ is equivalent to

$$\begin{aligned} {\mathcal {P}}\sim \left( \begin{array}{c} \phi \\ {\mathbb {1}}_n \end{array} \right) , \quad \phi = \text {i}\frac{ {\mathbb {1}}_n +\psi }{ {\mathbb {1}}_n -\psi }. \end{aligned}$$

(A.17a)

The properties of $\phi$ follow from (A.8):

$$\begin{aligned} \phi ^{\textrm{T}} = \phi , \quad \text {i}\big (\bar{\phi }- \phi \big ) >0. \end{aligned}$$

(A.17b)

In this paper we make use of the $\phi$-parametrisation (A.17) of the coset space (A.15).

The group ${\textsf{Sp}}(2n,{\mathbb {R}})$ acts on ${\textsf{Sp}}(2n,{\mathbb {R}})/{\textsf{U}}(n)$ by fractional linear transformations

$$\begin{aligned} \phi ~\rightarrow ~ \phi '= (A\phi +B)(C\phi +D)^{-1}, \begin{pmatrix} A &{} \quad B\\ C &{} \quad D \end{pmatrix}\in {\textsf{Sp}}(2n,{\mathbb {R}}), \end{aligned}$$

(A.18)

and therefore

$$\begin{aligned} \text {d}\phi '=\big ( (C\phi +D)^{-1}\big )^{\text {T}}\text {d}\phi \,(C\phi +D)^{-1}. \end{aligned}$$

(A.19)

Using the definition of symplectic matrices, Eq. (A.1), one can show that the positive-definite matrix $\Xi ,$ which is defined by (1.15), transforms as follows:

$$\begin{aligned}&(\Xi ')^{-1}=(C\phi +D)\Xi ^{-1}(C\bar{\phi }+D)^\text {T}\nonumber \\&\quad =(C\bar{\phi }+D)\Xi ^{-1}(C\phi +D)^\text {T}. \end{aligned}$$

(A.20)

Let ${\mathcal {P}}_1$ and ${\mathcal {P}}_2$ be two points in ${\textsf{Sp}}(2n,{\mathbb {R}})/{\textsf{U}}(n).$ We associate with them the following two-point function:

$$\begin{aligned}{} & {} {\mathfrak s}^2 ({\mathcal {P}}_1, {\mathcal {P}}_2) = - 4 {\textrm{Tr}} \Bigg [ \big ({\mathcal {P}}_1^\dagger J \bar{\mathcal {P}}_2\big ) \big ({\mathcal {P}}_2^\textrm{T} J \bar{\mathcal {P}}_2\big )^{-1} \nonumber \\{} & {} \quad \times \big ({\mathcal {P}}_2^\textrm{T} J \bar{\mathcal {P}}_1\big ) \big ({\mathcal {P}}_1^\dagger J \bar{\mathcal {P}}_1\big )^{-1} \Bigg ]. \end{aligned}$$

(A.21)

By construction, it is invariant under arbitrary ${\textsf{Sp}}(2n,{\mathbb {R}})$ transformations

$$\begin{aligned} {\mathcal {P}}_{1,2} ~\rightarrow ~ g {\mathcal {P}}_{1,2}, \quad g \in {\textsf{Sp}}(2n,{\mathbb {R}}). \end{aligned}$$

(A.22)

It is also invariant under arbitrary equivalence transformations (A.7),

$$\begin{aligned} {\mathcal {P}}_{1,2} ~\rightarrow ~ {\mathcal {P}}_{1,2} R_{1,2}, \quad R_{1,2} \in \textsf{GL}(n, {\mathbb {C}}). \end{aligned}$$

(A.23)

Therefore, ${\mathfrak {s}}^2 ({\mathcal {P}}_1, {\mathcal {P}}_2)$ is a two-point function of ${\textsf{Sp}}(2n,$ ${\mathbb {R}})/{\textsf{U}}(n)$ which is invariant under the isometry group ${\textsf{Sp}}(2n,{\mathbb {R}}).$ Due to the invariance of ${\mathfrak {s}}^2 ({\mathcal {P}}_1, {\mathcal {P}}_2)$ under arbitrary right shifts (A.23), both ${\mathcal {P}}_1$ and ${\mathcal {P}}_2$ can be chosen to have the form (A.17). In the case that ${\mathcal {P}}_1$ and ${\mathcal {P}}_2$ are infinitesimally separated, ${\mathfrak {s}}^2 ({\mathcal {P}}_1, {\mathcal {P}}_2)$ becomes

$$\begin{aligned} \text {d}{\mathfrak {s}}^2 = {\textrm{Tr}} \Bigg [ \text {d}\bar{\phi }\,\Xi ^{-1} \text {d}\phi \,\Xi ^{-1} \Bigg ], \end{aligned}$$

(A.24)

which is the Kähler metric on ${\textsf{Sp}}(2n,{\mathbb {R}})/{\textsf{U}}(n),$ Eq. (2.31).

Appendix B: Alternative field redefinition

Here we describe an alternative calculation of $\text {Tr}\ln {\Delta }$ compared with that given in Sect. 2.2. Consider a path integral over two vector fields

$$\begin{aligned} {\textrm{Det}} \,\Delta ^{-1} =N\int [{\mathfrak {D}} B] [{\mathfrak {D}} A]\, \text {e}^{\text {i}\int \!\!\text {d}^4x\,\sqrt{-g}\,B_{\hat{m}}\Delta ^{\hat{m}\hat{n}}A_{\hat{n}}} \end{aligned}$$

(B.1)

with the operator $\Delta ^{\hat{m}\hat{n}}$ given in (2.9a). We perform an alternative local field definition in the path integral

$$\begin{aligned} A^{\hat{m}} \rightarrow A^{\hat{m}},\quad B_{\hat{m}} \rightarrow B_m \Xi ^{-1}, \end{aligned}$$

(B.2)

which was not possible for the original quadratic action (2.8). Now the operator which appeared in (B.1) has the form

$$\begin{aligned} \tilde{\Delta }^{\hat{m}}_{{~}\hat{n}}:=\Xi ^{-1}\Delta ^m_{{~}n}, \end{aligned}$$

(B.3)

which, once expanded explicitly from (2.9a) and (2.9b), is minimal:

$$\begin{aligned} \tilde{\Delta }^{\hat{m}}_{{~}\hat{n}}&=\mathbb {1}\,\delta ^m_{{~}n}\Box +Q^m_{{~}pn}\nabla ^p+T^m_{{~}n}, \end{aligned}$$

(B.4a)

$$\begin{aligned} Q^{\hat{m}}_{{~}p\hat{n}}&:=\Xi ^{-1}\left( \nabla _p\Xi \right) \delta ^m_{{~}n}-\Xi ^{-1}\left( \nabla _n\Xi \right) \delta ^m_{{~}p}\nonumber \\&\quad +\Xi ^{-1}\left( \nabla ^m\Xi \right) g_{pn}-\Xi ^{-1}\left( \nabla ^q\Upsilon \right) \epsilon ^m_{{~}pnq}, \end{aligned}$$

(B.4b)

$$\begin{aligned} T^{\hat{m}}_{{~}\hat{n}}&:=-R^m_{{~}n}\mathbb {1}. \end{aligned}$$

(B.4c)

Although both approaches of obtaining a minimal operator will lead to equivalent logarithmic divergences up to total derivative, the alternative field definition proves to be much easier to manage computationally, since it solely involves derivatives of $\Xi ,$ $\Xi ^{-1}$ and $\Upsilon ,$ rather than $\Xi ^{1/2}$ and $\Xi ^{-1/2}.$ Following the same procedure as Sect. 2.2, the final result for $(a_2)_\textrm{total}$ (including total derivative contributions) is

$$\begin{aligned} (a_2)_{\textrm{total}}&=n\bigg (\frac{1}{10}F-\frac{31}{180}G-\frac{1}{10}\Box R\bigg )\nonumber \\&\quad +\frac{1}{4}\text {Tr}\Bigg [\Xi ^{-1}({\mathcal {D}}^2\phi ) \Xi ^{-1}({\mathcal {D}}^2\bar{\phi })\nonumber \\&\quad -2\Bigg (R^{mn}-\frac{1}{3}Rg^{mn}\Bigg )\Xi ^{-1}(\nabla _m\phi )\Xi ^{-1}(\nabla _n\bar{\phi })\Bigg ]\nonumber \\&\quad +\frac{1}{24}\text {Tr}\Bigg [\Xi ^{-1}(\nabla ^m\phi )\Xi ^{-1}(\nabla _m\bar{\phi })\Xi ^{-1}(\nabla ^n\phi )\Xi ^{-1}(\nabla _n\bar{\phi })\Bigg ]\nonumber \\&\quad +\frac{1}{48}\text {Tr}\Bigg [\Xi ^{-1}(\nabla ^m\phi )\Xi ^{-1}(\nabla ^n\bar{\phi })\Xi ^{-1}(\nabla _m\phi )\Xi ^{-1}(\nabla _n\bar{\phi })\Bigg ]\nonumber \\&\quad +\text {Tr}\big [\nabla _m {\mathcal {Y}}^m\big ]+\text {Tr}\big [\Box \tilde{{\mathcal {Z}}}\big ]. \end{aligned}$$

(B.5)

Compared to the original field redefinition, the total derivative contributions are the same for ${\mathcal {Y}}^m$ (2.23b) and differ from ${\mathcal {Z}}$ (2.23c)

$$\begin{aligned}&\tilde{{\mathcal {Z}}}:=-\frac{1}{6}\Xi ^{-1}(\nabla ^m\Upsilon )\Xi ^{-1}(\nabla _m\Upsilon )\nonumber \\&\qquad -\frac{1}{3}\Xi ^{-1}(\Box \Xi )+\frac{1}{3}\Xi ^{-1}(\nabla ^m\Xi )\Xi ^{-1}(\nabla _m\Xi ). \end{aligned}$$

(B.6a)

Indeed the two results for $(a_2)_{\textrm{total}}$ (2.26) and (B.5) differ only by a total derivative, which does not contribute to the induced action $\int \!\!\text {d}^4x\,\sqrt{-g}\, (a_2)_{\textrm{total}}.$

Appendix C: Curved space basis structures

Included below is a complete list of the basis structures introduced in (2.22). Note that under the trace over matrix indices some of these structures are equivalent to one another (via their transpose), however, since these structures are generated directly during the computation we have left them distinct for ease of computational reproducibility.

$$\begin{aligned} T_1&:=\Xi ^{-1}(\nabla ^m\Xi )\Xi ^{-1}(\nabla _m\Xi )\Xi ^{-1}(\nabla ^n\Xi )\Xi ^{-1}(\nabla _n\Xi ) ,\\ T_2&:=\Xi ^{-1}(\nabla ^m\Xi )\Xi ^{-1}(\nabla ^n\Xi )\Xi ^{-1}(\nabla _m\Xi )\Xi ^{-1}(\nabla _n\Xi ),\\ T_3&:=\Xi ^{-1}(\nabla ^m\Xi )\Xi ^{-1}(\nabla _m\Xi )\Xi ^{-1}(\nabla ^n\Upsilon )\Xi ^{-1}(\nabla _n\Upsilon ) ,\\ T_4&:=\Xi ^{-1}(\nabla ^m\Xi )\Xi ^{-1}(\nabla ^n\Xi )\Xi ^{-1}(\nabla _m\Upsilon )\Xi ^{-1}(\nabla _n\Upsilon ),\\ T_5&:=\Xi ^{-1}(\nabla ^m\Xi )\Xi ^{-1}(\nabla ^n\Xi )\Xi ^{-1}(\nabla _n\Upsilon )\Xi ^{-1}(\nabla _m\Upsilon ),\\ T_6&:=\Xi ^{-1}(\nabla ^m\Xi )\Xi ^{-1}(\nabla _m\Upsilon )\Xi ^{-1}(\nabla ^n\Xi )\Xi ^{-1}(\nabla _n\Upsilon ),\\ T_7&:=\Xi ^{-1}(\nabla ^m\Xi )\Xi ^{-1}(\nabla ^n\Upsilon )\Xi ^{-1}(\nabla _m\Xi )\Xi ^{-1}(\nabla _n\Upsilon ),\\ T_8&:=\Xi ^{-1}(\nabla ^m\Xi )\Xi ^{-1}(\nabla ^n\Upsilon )\Xi ^{-1}(\nabla _n\Xi )\Xi ^{-1}(\nabla _m\Upsilon ),\\ T_9&:=\Xi ^{-1}(\nabla ^m\Upsilon )\Xi ^{-1}(\nabla _m\Upsilon )\Xi ^{-1}(\nabla ^n\Upsilon )\Xi ^{-1}(\nabla _n\Upsilon ),\\ T_{10}&:=\Xi ^{-1}(\nabla ^m\Upsilon )\Xi ^{-1}(\nabla ^n\Upsilon )\Xi ^{-1}(\nabla _m\Upsilon )\Xi ^{-1}(\nabla _n\Upsilon ),\\ T_{11}&:=\Xi ^{-1}(\Box \Xi )\Xi ^{-1}(\Box \Xi ),\\ T_{12}&:=\Xi ^{-1}(\Box \Upsilon )\Xi ^{-1}(\Box \Upsilon ),\\ T_{13}&:=\Xi ^{-1}(\Box \Xi )\Xi ^{-1}(\nabla ^m\Xi )\Xi ^{-1}(\nabla _m\Xi ),\\ T_{14}&:=\Xi ^{-1}(\Box \Xi )\Xi ^{-1}(\nabla ^m\Upsilon )\Xi ^{-1}(\nabla _m\Upsilon ),\\ T_{15}&:=\Xi ^{-1}(\Box \Upsilon )\Xi ^{-1}(\nabla ^m\Xi )\Xi ^{-1}(\nabla _m\Upsilon ),\\ T_{16}&:=\Xi ^{-1}(\Box \Upsilon )\Xi ^{-1}(\nabla ^m\Upsilon )\Xi ^{-1}(\nabla _m\Xi ),\\ T_{17}&:=R_{mn}\,\Xi ^{-1}(\nabla ^m\Xi )\Xi ^{-1}(\nabla ^n\Xi ),\\ T_{18}&:=R_{mn}\,\Xi ^{-1}(\nabla ^m\Upsilon )\Xi ^{-1}(\nabla ^n\Upsilon ),\\ T_{19}&:=R\,\Xi ^{-1}(\nabla ^m\Xi )\Xi ^{-1}(\nabla _m\Xi ),\\ T_{20}&:=R\,\Xi ^{-1}(\nabla ^m\Upsilon )\Xi ^{-1}(\nabla _m\Upsilon ). \end{aligned}$$