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

We study a system of n Abelian vector fields coupled to 12n(n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2} n(n+1)$$\end{document} complex scalars parametrising the Hermitian symmetric space Sp(2n,R)/U(n).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsf{Sp}}(2n, {\mathbb {R}})/ {\textsf{U}}(n).$$\end{document} This model is Weyl invariant and possesses the maximal non-compact duality group Sp(2n,R).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsf{Sp}}(2n, {\mathbb {R}}).$$\end{document} 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 Sp(2n,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsf{Sp}}(2n, {\mathbb {R}})$$\end{document} invariance. The resulting conformal higher-derivative σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-model on Sp(2n,R)/U(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsf{Sp}}(2n, {\mathbb {R}})/ {\textsf{U}}(n)$$\end{document} 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 σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-model Lagrangian generates a Weyl anomaly satisfying the Wess–Zumino consistency condition.


Introduction
A unique feature of the Weyl multiplet of N = 4 conformal supergravity [1] is the presence of a dimensionless complex scalar field φ that parametrises the Hermitian symmetric space SL(2, R)/SO (2). 1 The most general family of invariant actions for N = 4 conformal supergravity was derived only a few years ago by Butter, Ciceri, de Wit and Sahoo, [2,3].Such an action is uniquely determined by a holomorphic function H(φ) which accompanies the terms quadratic in the Weyl tensor in the Lagrangian.
For the special choice H = const, in which case the N = 4 conformal supergravity action proves to be invariant under rigid SL(2, 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, Pletnev and Tseytlin [5] as an "induced action", obtained by integrating out an Abelian N = 4 vector multiplet coupled to external N = 4 conformal supergravity. 2The purely φ-dependent part of the Lagrangian is a higher-derivative σ-model of the form [6]: g mn R)∇ m φ∇ n φ + 1 12(Im φ) 4 α∇ m φ∇ m φ∇ n φ∇ n φ + β∇ m φ∇ m φ∇ n φ∇ n φ , where and α and β are numerical parameters.In the case of N = 4 conformal supergravity, these coefficients are [5]: α = 1 2 β = 1.The Lagrangian (1.1) is invariant under SL(2, R) transformations acting on the upper half-plane Im φ > 0 with metric (1.4) The functional d 4 x √ −g L proves to be invariant under Weyl transformations g mn (x) → e 2σ(x) g mn (x) , since the scalar field φ is inert under such transformations.The higher-derivative σ-model (1.1) possesses the N = 1 supersymmetric extension [7] which relates the parameters α and β.Both parameters are completely fixed if N = 2 supersymmetry is required [7][8][9].
The conformal higher-derivative σ-model (1.1) admits a nontrivial generalisation that is obtained by replacing the Hermitian symmetric space SL(2, R)/SO (2) with an arbitrary n-dimensional Kähler manifold M n , with n the complex dimension.We assume that M n is parametrised by n local complex coordinates φ I and their conjugates φĪ .Let K(φ, φ) be the corresponding Kähler potential such that the Kähler metric g I J (φ, φ) is given by Associated with M n is a higher-derivative sigma model of the form where R mn is the spacetime Ricci tensor, with Γ I JK being the Christoffel symbols for the Kähler metric g I J .Finally, F IJ K L and G IJ K L are tensor fields on the target space, which are constructed from the Kähler metric g I J , Riemann tensor R I JK L and, in general, its covariant derivatives.We recall that the Christoffel symbols Γ I JK and the curvature tensor R I JK L are given by the expressions3 A typical expression for F IJ K L is The possible structure of G IJ K L is analogous.It should be pointed out that actions of the form (1.6) naturally emerge at the component level in N = 2 superconformal higherderivative σ-models [8] (see also [9]), and in N = 1 ones [7].
By construction, the action (1.6) is invariant under arbitrary holomorphic isometries of M n .A nontrivial observation is that (1.6) is also invariant under arbitrary Weyl transformations of spacetime provided the scalars φ I are inert under these transformations.
Choosing M n = C n and g I J (φ, φ) = δ I J in (1.6) and integrating by parts, one obtains the Fradkin-Tseytlin (FT) operator [13] ∆ which is conformal when acting on dimensionless scalar fields. 4Given a Weyl inert scalar field ϕ, the Weyl transformation (1.5) acts on ∆ 0 ϕ as An action of the form d 4 x √ −g L(φ, φ), with L given by (1.1), naturally emerges as an induced action in Maxwell's electrodynamics coupled to a dilaton ϕ and an axion a with Lagrangian (1.12) Here F mn = 1 2 ε mnrs F rs is the Hodge dual of the electromagnetic field strength , with ε 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 αβ = F βα and its conjugate F α β .
More precisely, if one considers the effective action, Γ[φ, φ], obtained by integrating out the quantum gauge field in the model (1.12), then the logarithmically divergent part of Γ[φ, φ] is given by d 4 x √ −g L(φ, φ), as demonstrated by Osborn [6].An important question arises: why is the induced action Weyl and SL(2, R) invariant?
We recall that the group of electromagnetic duality rotations of free Maxwell's equations is U(1).More than forty years ago, it was shown by Gaillard and Zumino [16,17] that the non-compact group Sp(2n, R) is the maximal duality group of n Abelian vector field strengths F mn = (F mn,i ), with i = 1, . . ., n, in the presence of a collection of complex scalars φ ij = φ ji parametrising the homogeneous space Sp(2n, R)/U(n), with i, j = 1, . . ., n.In the absence of such scalars, the largest duality group proves to be U(n), the maximal compact subgroup of Sp(2n, R).These results admit a natural extension to the case when the pure vector field part L(F ) of the Lagrangian L(F ; φ, φ) is a nonlinear U(1) duality invariant theory [18][19][20][21][22] (see [23][24][25] for reviews), for instance Born-Infeld theory.However, in the case that L(F ) is quadratic, the F -dependent part of L(F ; φ, φ) is also invariant under the Weyl transformations in curved space.Then, computing the path integral over the gauge fields leads to an effective action, Γ[φ, φ], such that its logarithmically divergent part is invariant under Weyl and rigid Sp(2n, R) transformations, see, e.g., [26,27] 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 = (A m,i ), i = 1, . . ., n, coupled to a complex field φ = (φ ij ) and its conjugate φ = ( φīj ) parametrising the homogeneous space The corresponding Lagrangian is where we have also introduced the real matrices Ξ and Υ defined by with Ξ being positive definite.The model described by (1.14) has two fundamental properties: (i) its duality group is Sp(2n, R) (see, e.g.[23] for the technical details); and (ii) it is Weyl invariant.The induced action must respect these properties.
This paper is organised as follows.In section 2 we compute the logarithmically divergent part of the effective action obtained by integrating out the vector fields in the model

Computing the induced action
In this section we compute the logarithmically divergent part of the effective action, Γ[φ, φ], defined by Here S[A; φ, φ] is the classical action corresponding to (1.14), χ(A) denotes a gauge fixing condition, ∆ gh the corresponding Faddeev-Popov operator [28], and η an arbitrary background field.Since the effective action is independent of η, this field can be integrated out with some weight that we choose to be In general, the logarithmically divergent part of the effective action has the form where (a 2 ) total denotes the appropriate sum of diagonal DeWitt coefficients.We identify the induced action with d 4 x √ −g (a 2 ) total , modulo an overall numerical coefficient.

Quantisation
We choose the simplest gauge-fixing condition which leads to the ghost operator with 1 the n × n unit matrix.Integrating the right-hand side of (2.1) with the weight functional (2.3) leads to the gauge-fixing term As a result, the gauge-fixed action becomes where here we have introduced hatted indices corresponding to a pair of spacetime and internal indices A m := (A mi ), ∆ mn := (∆ 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 ∆ mn is defined as: From here onward matrix indices will be suppressed, unless there may be ambiguity or confusion.The one-loop effective action is specified by Since Ξ is symmetric and positive definite, due to (1.13), its inverse Ξ −1 , square root Ξ 1/2 and inverse square root Ξ −1/2 are well-defined.We perform a local field redefinition in the path integral: so that the operator which appeared in (2.8) becomes5 Inserting the explicit form of ∆ mn from (2.9a) and (2.9b), the ∆ mn operator is now minimal: After our field redefinition the one-loop effective action is given by Γ (1) [φ, φ] = i 2 Tr ln ∆ − iTr ln ∆ gh . (2.14)

Heat kernel calculations
Since the operator ∆ mn defined by (2.13a) is minimal, we can proceed with the standard heat kernel technique in curved space, by bringing it to the form: The generalised covariant derivative ∇m introduced above is defined to act on a column (2.16) The generalised covariant derivatives have no torsion, meaning ∇p , ∇q with R mnpq some generalised curvature anti-symmetric in p, q.Explicitly it has the form Using the standard Schwinger-DeWitt formalism [29][30][31][32][33] in curved spacetime for an operator of the form (2.15a), in the coincidence limit the DeWitt coefficient traced over matrix indices, (a 2 ) ∆(x, x), is given by where 'Tr' denotes the matrix trace.Similarly for the ghost operator (2.6), the corre- where the contributions T 1 , . . ., T 20 are listed in appendix C. We have also introduced: The total DeWitt coefficient corresponding to the logarithmic divergence of the effective action (2.14) is given by where (a 2 ) ∆ gh (x, x) was given in (2.20).Recalling the expression for Ξ and Υ in terms of the original fields φ and its conjugate φ (1.15), and defining the total DeWitt coefficient is given by where F is the square of the Weyl tensor, G is the Euler density, and we have removed the total derivative pieces Tr ∇ m Y m and Tr ✷Z since they do not contribute to the induced action d 4 x √ −g (a 2 ) total .The ✷R in (2.26) is also a total derivative and can be omitted.
We make the standard choice for Kähler potential K(φ ij , φīj ) on Sp(2n, R)/U(n) which is well defined since Ξ is a positive definite matrix.The group Sp(2n, R) acts on Sp(2n, R)/U(n) by fractional linear transformations (A.18).Given such a transformation, the Kähler potential changes as in accordance with (A.20).Therefore, the Kähler metric is invariant under arbitrary Sp(2n, R) transformations.
The Kähler metric is given by 6 where (i . (2.32) In accordance with (1.8), the Christoffel symbols are given by and the Riemann curvature tensor is Noting that the inverse Kähler metric of (2.31) is one can calculate the Christoffel symbols, Riemann curvature tensor and Ricci tensor for the metric considered in (2.31) and we find: (2.36a) 6 The partial derivatives with respect to symmetric matrices φ = (φ ij ) and φ = ( φīj ) are defined by dK(φ, φ) = dφ ij ∂K(φ, φ) ∂φ īj , and therefore ∂φ kl ∂φ ij = δ k (i δ l j) .Symmetrisation of n indices includes a 1/n! factor.Vertical bars are notation to exclude indices contained between them from a separate symmetrisation, for example, (i The latter relation means that Sp(2n, R)/U(n) is an Einstein space.
As pointed out at the beginning of this subsection, the complex variables φ and their conjugates φ can be viewed either as symmetric matrices φ = (φ ij ) and φ = ( φīj ) or as vector columns φ = (φ I ) and φ = ( φĪ ), with I, Ī = 1, . . ., 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: where Every isometry transformation (A.18) acts on ∇φ and D 2 φ as follows: It is now seen that the induced action defined by (2.37) is invariant under the isometry transformations on Sp(2n, R)/U(n).

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 It is well known that the purely gravitational part of this variation satisfies the Wess-Zumino consistency condition [34] [ see, e.g., [35][36][37] for a review. 7The φ-dependent part of the Weyl anomaly will be discussed below.
As a generalisation of (1.6), we can introduce a conformal higher-derivative σ-model associated with a Riemannian manifold (W d , g) parametrised by local coordinates ϕ µ .
The action is where and F µνσρ (ϕ) is a tensor field of rank (0, 4) on W d .The Weyl invariance of the above action follows from In the case that the target space is Kähler, eq.(1.6), the relation (3.4) takes the form Choosing W d to be R, specifying g µν (ϕ) and F µνσρ (ϕ) to be constant, respectively, and restricting the background spacetime to be flat, the action (3.3) turns into with f a coupling constant, which is the model studied recently by Tseytlin [38].
Assuming the model (3.3) originates from an induced action of some theory, the ϕdependent part of the Weyl anomaly should have the form 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 where L (n) (F ; φ, φ) is given by (1.14), the complex scalar fields φ I and their conjugates φĪ parametrise Sp(2n, R)/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 Sp(2n, R) terms that are quartic in ∂φ and ∂ φ, 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 [39,40].All couplings constants in (3.8) are dimensionless, and the freedom to choose them is dictated by Sp(2n, 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 U(1) duality invariant deformation [41,42]; it was called the ModMax theory in [41].Using the methods developed in [19][20][21], it can be coupled to the dilatonaxion field (1.12) to result in a conformal and SL(2, R) duality invariant model described by the Lagrangian [43] L where and γ is a non-negative coupling constant [41].For γ = 0 the model (3.9) reduces to (1.12).A challenging problem is to compute an induced action generated by (3.9).

B Alternative field redefinition
Here we describe an alternative calculation of Tr ln ∆ compared with that given in section 2.2.Consider a path integral over two vector fields with the operator ∆ mn given in (2.9a).We perform an alternative local field definition in the path integral which was not possible for the original quadratic action (2.8).Now the operator which appeared in (B.1) has the form which, once expanded explicitly from (2.9a) and (2.9b), is minimal: 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 Ξ, Ξ −1 and Υ, rather than Ξ 1/2 and Ξ −1/2 .Following the same procedure as section 2.2, the final result for (a 2 ) total (including total derivative contributions) is

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.

( 1 .
14).Generalisations of our analysis and open problems are briefly discussed in section 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 Sp(2n, R)/U(n).Appendix B provides an alternative calculation of the induced action compared with that given in subsection 2.2.Appendix C provides a complete list of the structures introduced in (2.22).