# Macdonald Indices for Four-dimensional $\mathcal{N}=3$ Theories

Prarit Agarwal<EMAIL_ADDRESS>Enrico Andriolo<EMAIL_ADDRESS>Gergely Kántor<EMAIL_ADDRESS>Constantinos Papageorgakis
<EMAIL_ADDRESS>CRST and School of Physics and Astronomy, Queen
Mary University of London,
Mile End Road, London E1 4NS, UK

###### Abstract

We brute-force evaluate the vacuum character for $\mathcal{N}=2$ vertex
operator algebras labelled by crystallographic complex reflection groups
$G(k,1,1)=\mathbb{Z}_{k}$, $k=3,4,6$, and $G(3,1,2)$. For $\mathbb{Z}_{3,4}$
and $G(3,1,2)$ these vacuum characters have been conjectured to respectively
reproduce the Macdonald limit of the superconformal index for rank one and
rank two S-fold $\mathcal{N}=3$ theories in four dimensions. For the
$\mathbb{Z}_{3}$ case, and in the limit where the Macdonald index reduces to
the Schur index, we find agreement with predictions from the literature.

††preprint: QMUL-PH-21-07

##  I Introduction

In recent years there has been intense activity pertaining to the study of
superconformal theories (SCFTs) that do not admit a Lagrangian description.
Theories with $\mathcal{N}\geq 2$ superconformal symmetry are ideal for such
explorations. Despite the lack of perturbative control, one can still extract
nontrivial data by exploiting the large amount of symmetry, e.g. by employing
the power of dualities Argyres and Seiberg (2007); Gaiotto (2012),
implementing the bootstrap programme Rattazzi et al. (2008); Beem et al.
(2016), or evaluating superconformal indices Kinney et al. (2007).

In this context, “pure” $\mathcal{N}=3$ SCFTs—$\mathcal{N}=3$ theories which
do not automatically enhance to $\mathcal{N}=4$—were envisioned in Ferrara et
al. (1999); Aharony and Evtikhiev (2016) and engineered in string theory
through the Sk-fold constructions of García-Etxebarria and Regalado (2016);
Aharony and Tachikawa (2016). These are isolated, holographic SCFTs
($a_{\mathrm{4D}}=c_{\mathrm{4D}}$) with an F-theory dual on an
$\mathrm{AdS}_{5}\times(\mathrm{S}^{5}\times\mathrm{T}^{2})/\mathbb{Z}_{k}$
background. The gravity description was used in Imamura and Yokoyama (2016);
Arai and Imamura (2019) to evaluate the superconformal index in the large-rank
limit. Candidates for additional rank-one and rank-two $\mathcal{N}=3$
examples were presented in Argyres et al. (2016, 2020), by constructing
corresponding Coulomb-branch geometries via gaugings of $\mathcal{N}=4$
theories by a discrete subgroup of the R-symmetry and electromagnetic duality
groups. The “Coulomb” limit of the superconformal index Gadde et al. (2013)
and the Higgs-branch Hilbert series for these models were evaluated in Bourton
et al. (2018); see also Evtikhiev (2020).

As $\mathcal{N}=3$ theories are automatically $\mathcal{N}=2$, a concrete
computational handle can be established through the description of a “Schur”
Bogomol’nyi–Prasad–Sommerfield (BPS) subsector of any $\mathcal{N}=2$ 4D
theory Gadde et al. (2013) by a (non-unitary) vertex operator algebra (VOA)
Beem et al. (2015). VOAs for $\mathcal{N}=3$ theories were initially
constructed in Nishinaka and Tachikawa (2016); Lemos et al. (2017) culminating
in the work of Bonetti et al. (2019). In that reference, it was conjectured
that certain VOAs labelled by non-Coxeter crystallographic complex reflection
groups encode the Schur subsector of the known $\mathcal{N}=3$ S-fold
theories. In particular, Bonetti et al. (2019) gave a prescription for an
elegant free-field realisation of such VOAs, along the lines of Adamović
(2016). By constructing the latter, one is able to recover the “Macdonald”
limit of the superconformal index Gadde et al. (2013) for $\mathcal{N}=3$
S-fold theories, from the VOA vacuum character. See also Song (2017) for an
alternate prescription on implementing a Macdonald grading of the chiral
algebra.

Albeit concrete, implementing the findings of Bonetti et al. (2019) in
practice quickly becomes computationally intensive. It is difficult to write
down the explicit free-field realisation of the relevant VOAs in all but the
simplest of cases, and also to evaluate the corresponding vacuum characters in
a fugacity expansion for increasing conformal weights. The goal of this short
note is to show how far one can get by implementing a brute-force approach
using mathematical software, for the VOAs labelled by the complex reflection
groups $G(k,1,1)=\mathbb{Z}_{k}$, $k=3,4,6$ ($\mathbb{Z}_{3,4}$ label rank-one
S-fold models) and $G(3,1,2)$ (labels a rank-two S-fold model). We employ the
$G(k,p,N)$ notation of Shephard and Todd (1954), where $p$ is a divisor of
$k$; general complex reflection groups are denoted as $\mathsf{G}$.

Towards that end, we reconstruct and explicitly exhibit the free-field
realisations of Bonetti et al. (2019) for the theories of interest. We then
provide algorithms for automating the process of finding null states and for
evaluating the VOA vacuum characters. Our code, appended to this letter, can
in principle be executed to obtain the corresponding Macdonald index at
arbitrary orders in a fugacity expansion. Note however that the vacuum
character computation time increases exponentially as a function of the
conformal weight. Our code is also customisable—and we have clearly signposted
how to do so—for the reader interested in extending it to the evaluation of
vacuum characters for VOAs labelled by other complex reflection groups, once
the complete free-field realisation of the VOA has been found.

Our results, all of which have been collected in the ancillary
“vacuum_characters_summary.nb” for quick reference, can be used to check the
conjecture of Bonetti et al. (2019) against independent calculations of the
Macdonald index of 4D $\mathcal{N}=3$ S-fold theories and vice versa. For
example, a proposal for the Schur limit of the superconformal index—a special
case of the Macdonald index—for the rank-one $\mathbb{Z}_{3}$ S-fold theory
was put forward in Zafrir (2021). In that limit, their and our findings are in
complete agreement.

##  II From Vacuum Characters to Indices

We begin with a minimal introduction to the results of Bonetti et al. (2019),
to which we refer the reader for a full account. 4D $\mathcal{N}=2$ SCFTs,
$\mathcal{T}$, contain a BPS subsector that is isomorphic to a VOA,
$\chi[\mathcal{T}]$ Beem et al. (2016). In this correspondence, the VOA
central charge $c$ is related to the type-B Weyl-anomaly coefficient in four
dimensions as $c=-12c_{\mathrm{4D}}$, while the VOA vacuum character
reproduces the Schur limit of the 4D superconformal index. The Schur subsector
of pure $\mathcal{N}=3$ 4D S-fold SCFTs was conjectured to be isomorphic to
“$\mathcal{N}=2$ VOAs”—i.e. VOAs containing the 2D $\mathcal{N}=2$
superconformal algebra (SCA) as a subalgebra—$\mathcal{W}_{\mathsf{G}}$
labelled by non-Coxeter crystallographic complex reflection groups. In fact,
Bonetti et al. (2019) proposed a free-field realisation of
$\mathcal{W}_{\mathsf{G}}$ in terms of a subalgebra of
$\textrm{rank}(\mathsf{G})$ copies of the $\beta\gamma bc$ ghost system. This
subalgebra was identified with the kernel of a screening operator,
$\mathbb{S}=\int\mathsf{J}$, acting on the $\beta\gamma bc$ systems. The
action of $\mathbb{S}$ is defined as $\mathbb{S}\cdot
X=\\{\mathsf{J}X\\}_{1}$, where $\\{\mathsf{J}X\\}_{1}$ denotes the
coefficient of the order-one pole in the holomorphic OPE of the screening
current $\mathsf{J}$ with some operator $X$.

A nice feature of the free-field description, which we will use extensively in
the evaluation of the vacuum characters, is that null states built out of
strong generators are identically zero. The free-field realisation also
facilitates the introduction of a corresponding R-filtration, inherited from
the R-symmetry of the 4D $\mathcal{N}=2$ SCFT.

The vacuum character of the R-filtered VOA is

$\displaystyle\chi_{{\mathcal{W}}_{\mathcal{G}}}(q,\xi,z):=\mathrm{Tr}(-1)^{F}q^{h}\xi^{R+r}z^{m}\;,$
(1)

where $F$ is the fermion number, $h$ is the conformal dimension, and $r,m$ are
associated with the $\mathsf{gl(1)}$ outer automorphism and $\mathsf{gl(1)}$
subalgebra of the 2D $\mathcal{N}=2$ SCA $\mathsf{osp(2|2)}$ respectively. $R$
is the weight under the R-filtration and the vacuum character is normalised so
as to start with a “1” in its $q$ expansion. This vacuum character can be
further refined by taking $\xi^{R+r}\to y^{R}v^{r}$. In the free-field
realisation the $\beta\gamma bc$ fields carry the quantum numbers presented in
Tab. 1.

| $h$ | $m$ | $r$ | $R$
---|---|---|---|---
$\beta_{\ell}$ | $\frac{1}{2}p_{\ell}$ | $\frac{1}{2}p_{\ell}$ | 0 | $\frac{1}{2}p_{\ell}$
$b_{\ell}$ | $\frac{1}{2}\left(p_{\ell}+1\right)$ | $\frac{1}{2}\left(p_{\ell}-1\right)$ | +$\frac{1}{2}$ | $1-\frac{1}{2}p_{\ell}$
$c_{\ell}$ | $-\frac{1}{2}\left(p_{\ell}-1\right)$ | $-\frac{1}{2}\left(p_{\ell}-1\right)$ | $-\frac{1}{2}$ | $\frac{1}{2}p_{\ell}$
$\gamma_{\ell}$ | $1-\frac{1}{2}p_{\ell}$ | $-\frac{1}{2}p_{\ell}$ | 0 | $1-\frac{1}{2}p_{\ell}$
$\partial$ | 1 | 0 | 0 | 0
Table 1:  Quantum numbers for the $\beta_{\ell}\gamma_{\ell}b_{\ell}c_{\ell}$
ghost systems used in VOA free-field realisations. The
$\ell=1,\ldots,\mathrm{rank}(\mathsf{G})$ labels the ghost-system species and
$p_{\ell}$ are the degrees of the fundamental invariants of $\mathsf{G}$.
These are given for $\mathbb{Z}_{k}$ by $p_{1}=k$ and for $G(3,1,2)$ by
$(p_{1},p_{2})=(3,6)$.

Eq. (1) was conjectured to correspond to the Macdonald limit of the 4D
superconformal index of a theory for which $\mathcal{W}_{\mathsf{G}}$ is the
associated VOA, $\mathcal{W}_{\mathsf{G}}=\chi[\mathcal{T}]$. This is defined
through Gadde et al. (2013)

$\displaystyle\mathcal{I}^{\mathcal{T}}_{\mathrm{Macdonald}}(q,t,z):=\mathrm{Tr}(-1)^{F}q^{E-2R-r}t^{R+r}z^{m}\;,$
(2)

where the trace is taken over the set of Schur operators of the 4D SCFT. Here
$E$ is the 4D conformal dimension, while $R$ and $r$ are charges for the
Cartan generators of the $SU(2)_{R}$ and $U(1)_{r}$ R-symmetry groups
respectively. Note that while from the point of view of this 4D
$\mathcal{N}=2$ Macdonald index $m$ is a quantum number for a global
$U(1)_{F}$, in the full $\mathcal{N}=3$ description it is part of the
$U(3)\supset SU(2)_{R}\times U(1)_{r}\times U(1)_{F}$ R-symmetry group. We
should emphasise that in order to connect with (1) one needs to redefine
$t\to\xi q$ so that

$\displaystyle\mathcal{I}^{\mathcal{T}}_{\mathrm{Macdonald}}(q,\xi,z)=\mathrm{Tr}(-1)^{F}q^{E-R}\xi^{R+r}z^{m}\;.$
(3)

##  III Implementation

We now describe the strategy behind our code, while detailed results for each
case are presented in subsequent subsections. We are interested in the
evaluation of the vacuum character for VOAs labelled by crystallographic, non-
Coxeter complex reflection groups $\mathbb{Z}_{3,4,6}$ and $G(3,1,2)$, and
interpreting them as Macdonald indices for 4D $\mathcal{N}=3$ S-fold theories.
To do so one needs to consider (1) and trace over all the states created by
acting only with normal-ordered products and derivatives of the strong
generators of the VOA on the $\mathsf{sl(2)}$-invariant vacuum, up to a given
conformal weight, while removing the contributions from null states. To speed
up the calculation, we have taken advantage of the symmetry of the spectrum
under conjugation, by only constructing positively-charged states under the
two $\mathsf{gl(1)}$ symmetries of the VOA.

The identification of null states is usually laborious and this is where the
free-field realisation becomes helpful. The construction algorithm of Bonetti
et al. (2019) starts with a simple free-field prescription for the generators
of the 2D $\mathcal{N}=2$ SCA and the chiral strong generators of the
$\mathcal{W}_{\mathsf{G}}$-algebra. One then introduces an ansatz for the
remaining bosonic strong generators constructed out of all possible super-
Virasoro primary, free-field combinations with the requisite quantum numbers
and undetermined coefficients, and fixes the latter by closing the VOA under
the OPE.

We have diverged slightly from this recipe in the following way. For the VOAs
$\chi[\mathcal{T}]$ that have appeared thus far in the literature, the OPE
coefficients can be completely fixed by writing down the most general
expressions for the expected set of generators and then imposing associativity
by solving the Jacobi identities Lemos et al. (2017). The W-algebra
presentations of the $\mathcal{W}_{\mathsf{G}}$ VOAs of interest to us were
already given in Bonetti et al. (2019) and inserting the free-field ansatze
into the existing OPEs straightforwardly fixes the undetermined coefficients
for all remaining strong generators (using the OPEdefs Thielemans (1994)
and/or SOPEN2defs Krivonos and Thielemans (1996) Mathematica packages); we
will expand upon this point on a case-by-case basis when discussing our
results.

Equipped with this information we proceed to our main algorithm. In summary,
all null states at a given conformal weight can be identified by looking for
all possible combinations of states with the same quantum numbers that are
identically zero upon using the free-field realisation. This last step
requires manipulating normal-ordered products of $\beta\gamma bc$ ghosts,
making heavy use of the OPEdefs Thielemans (1994) and ope.math Fujitsu (1994)
Mathematica packages. The null states that we find contain all those predicted
in Bonetti et al. (2019). It is then straightforward to write down the vacuum
character for given values of quantum numbers. Note that by unrefining in the
$R$ fugacity (e.g. if one were interested in the Schur index) the algorithm
becomes faster; we have a dedicated section in our code for this special case.

In addition to the above method, we have cross-checked our refined
$\mathbb{Z}_{3,4}$ results using a second algorithm that makes no connection
to the W-algebra presentation. This procedure constructs the VOA spectrum
using all states in $\mathrm{rank}(\mathsf{G})$ copies of the $\beta\gamma bc$
ghost system that lie in the kernel of a screening operator, $\mathbb{S}$. For
$\mathsf{G}=\mathbb{Z}_{k}$ this is given by Bonetti et al. (2019)

$\displaystyle\mathsf{J}=b\;e^{\left(k^{-1}-1\right)(\chi+\phi)}\;,$ (4)

where $\chi,\phi$ are chiral bosons

$\displaystyle\beta=e^{\chi+\phi},\quad\gamma=\partial\chi\;e^{-\chi-\phi}\;,$
(5)

and all expressions should be considered as normal-ordered. Such an approach
is conceptually more straightforward—construct all states using free fields
and then keep those in the kernel of $\mathbb{S}$—but is computationally more
expensive as can be seen from Fig 1. E.g. at $h=9/2$ one already needs to
check 941 and 881 terms for $\mathbb{Z}_{3}$ and $\mathbb{Z}_{4}$
respectively.

Figure 1:  Computation times for carrying out the calculation of the vacuum
character for VOAs at different conformal weights. Several different VOAs are
shown on the graph for comparison, along with the $\mathbb{Z}_{3}$ theory
computed using the kernel of the screening operator. We used a desktop PC with
an Intel Core i7-6700K CPU clocked at 4GHz, and 32GB RAM.

##  IV Results: $\mathsf{G}=\mathbb{Z}_{3}$

This is a rank-one VOA with central charge $c=-15$. Its W-algebra presentation
involves the following strong generators: $\mathcal{T}$, $\mathcal{J}$,
$\mathcal{G}$ and $\widetilde{\mathcal{G}}$ from the 2D $\mathcal{N}=2$ SCA,
as well as the chiral and anti-chiral generators $\mathcal{W}_{3}$, $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{3}$ and their
superpartners $\mathcal{G}_{\mathcal{W}_{3}}$ and
$\widetilde{\mathcal{G}}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{3}}$. Here
$\mathcal{G}_{\mathcal{W}_{3}}:=\\{\mathcal{G}\mathcal{W}_{3}\\}_{1}$ and so
on. For the explicit free-field realisation in terms of a single $\beta\gamma
bc$ ghost system one starts with a prescription for the $\mathcal{N}=2$ SCA
generators as well as for $\mathcal{W}_{3}$, $\mathcal{G}_{\mathcal{W}_{3}}$.
The ansatz for $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{3}$ contains
8 undetermined coefficients. We always count these before imposing the super-
Virasoro primary constraint. Through the OPEs from the W-algebra presentation
one can use it to also determine the free-field realisation of
$\widetilde{\mathcal{G}}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{3}}$. We have
calculated the fully-refined vacuum character (1) up to $O(q^{8})$ with the
accompanying “null states.nb” and have cross-checked this result using the
screening-operator approach in “screening.nb” up to $O(q^{4})$. The full
expression can be found in “vacuum_characters_summary.nb”.

This VOA is expected to encode the Schur sector of a rank one, 4D S-fold
$\mathcal{N}=3$ SCFT, with a Coulomb-branch operator of dimension $\Delta=3$
and trace-anomaly coefficient $c_{\mathrm{4D}}=\frac{5}{4}$. Through (3) the
Macdonald index of this S-fold theory—including the global $U(1)_{F}$
fugacity—can be identified with the refined vacuum character. Below we only
present the simpler, Schur limit of these expressions for brevity, where
$\xi\to 1$, $z\to 1$. Then:

$\displaystyle\mathcal{I}^{\mathbb{Z}_{3}}_{\mathrm{Schur}}$
$\displaystyle=1+q+q^{2}+2q^{3}-2q^{7/2}+3q^{4}-2q^{9/2}+4q^{5}-4q^{11/2}$
$\displaystyle+6q^{6}-6q^{13/2}+8q^{7}-8q^{15/2}+11q^{8}+O(q^{17/2}).$ (6)

It is interesting to observe that the Schur index for this theory matches the
expansion of the following closed-form expression up to $O(q^{10})$, although
we currently have neither a derivation for it nor a justification for why it
should hold to all orders:

$\displaystyle\frac{1}{3}\sum_{\epsilon\in\mathbb{Z}_{3}}\frac{\epsilon}{\sqrt{q}}$
$\displaystyle\text{P.E.}\left[\frac{1}{2}i_{\mathcal{N}=4}(q)\left(\epsilon+\frac{1}{\epsilon}\right)\right]\;.$
(7)

Here $\text{P.E.}[f(z)]:=\exp[\sum_{n=1}^{+\infty}\frac{1}{n}f(z^{n})]$ is the
plethystic exponential, while
$i_{\mathcal{N}=4}(q)=\frac{2q^{\frac{1}{2}}(1-q^{\frac{1}{2}})}{1-q}$
coincides with the single-letter Schur index of $\mathcal{N}=4$ super-
Yang–Mills.

In Zafrir (2021), an independent argument for determining the Schur index of
the $\mathsf{G}=\mathbb{Z}_{3}$ S-fold theory was presented. This entailed
starting from an $\mathcal{N}=1$ 4D UV Lagrangian theory, and flowing to an
interacting $\mathcal{N}=1$ SCFT in the IR, which can also be reached from the
$\mathbb{Z}_{3}$ S-fold theory via an $\mathcal{N}=1$ preserving marginal
deformation. Upon relabelling $q\to p^{2}$, our result (IV) agrees with that
of Zafrir (2021)—listed to $O(q^{7})$—providing a strong consistency check of
both calculations.

##  V Results: $\mathsf{G}=\mathbb{Z}_{4}$

This is a rank-one VOA with central charge $c=-21$. Its W-algebra presentation
involves the following strong generators: $\mathcal{T}$, $\mathcal{J}$,
$\mathcal{G}$ and $\widetilde{\mathcal{G}}$ from the 2D $\mathcal{N}=2$ SCA,
as well as the chiral and anti-chiral generators $\mathcal{W}_{4}$, $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{4}$ and their
superpartners $\mathcal{G}_{\mathcal{W}_{4}}$ and
$\widetilde{\mathcal{G}}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{4}}$. For the
explicit free-field realisation in terms of a single $\beta\gamma bc$ ghost
system one starts with a prescription for the $\mathcal{N}=2$ SCA generators
as well as for $\mathcal{W}_{4}$, $\mathcal{G}_{\mathcal{W}_{4}}$. The ansatz
for $\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern
1.5mu_{4}$ contains 19 undetermined coefficients. Through the OPEs in the
W-algebra presentation one can use it to also determine the free-field
realisation of $\widetilde{\mathcal{G}}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{4}}$. We have
calculated the fully-refined vacuum character (1) up to $O(q^{8})$ with the
accompanying “null states.nb” and exhibited it in
“vacuum_characters_summary.nb”. We have also cross-checked this result using
the screening-operator approach in “screening.nb” up to $O(q^{4})$.

This VOA is expected to encode the Schur sector of a rank one 4D S-fold
$\mathcal{N}=3$ SCFT, with a Coulomb-branch operator of dimension $\Delta=4$
and trace-anomaly coefficient $c_{\mathrm{4D}}=\frac{7}{4}$. Through (3) the
Macdonald index of this S-fold theory can be identified with the vacuum
character. The Schur limit of these expressions yields:

$\displaystyle\mathcal{I}^{\mathbb{Z}_{4}}_{\mathrm{Schur}}$
$\displaystyle=1+q-2q^{3/2}+5q^{2}-6q^{5/2}+10q^{3}-16q^{7/2}+27q^{4}$ (8)
$\displaystyle-38q^{9/2}+56q^{5}-86q^{11/2}+129q^{6}-178q^{13/2}$
$\displaystyle+251q^{7}-362q^{15/2}+511q^{8}+O(q^{17/2})\;.$ (9)

In this case the W-algebra construction is such that the bosonic states always
appear with integer while the fermionic ones with half-integer conformal
weights. Therefore there are no cancellations between bosonic and fermionic
states at each level and the chiral algebra partition function reproduces the
partition function of Schur operators in the corresponding 4D $\mathcal{N}=3$
theory.

##  VI Results: $\mathsf{G}=\mathbb{Z}_{6}$

This is a rank-one VOA with central charge $c=-33$. Its W-algebra presentation
involves the following strong generators: $\mathcal{T}$, $\mathcal{J}$,
$\mathcal{G}$ and $\widetilde{\mathcal{G}}$ from the $\mathcal{N}=2$ SCA, as
well as the chiral and anti-chiral generators $\mathcal{W}_{6}$, $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{6}$ and their
superpartners $\mathcal{G}_{\mathcal{W}_{6}}$ and
$\widetilde{\mathcal{G}}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{6}}$. For the
explicit free-field realisation in terms of a single $\beta\gamma bc$ ghost
system one starts with a prescription for the 2D $\mathcal{N}=2$ SCA
generators as well as for $\mathcal{W}_{6}$, $\mathcal{G}_{\mathcal{W}_{6}}$.
The ansatz for $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{6}$ contains
87 undetermined coefficients. Through the OPEs in the W-algebra presentation
one can use it to also determine the free-field realisation of
$\widetilde{\mathcal{G}}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{6}}$. We have
calculated the fully-refined vacuum character (1) up to $O(q^{8})$ with the
accompanying “nullstates.nb”. The full result can be found in
“vacuum_characters_summary.nb”. In the limit $\xi\to 1$, $z\to 1$ this reads:

$\displaystyle\chi_{\mathcal{W}_{\mathbb{Z}_{6}}}$
$\displaystyle=1+q-2q^{3/2}+3q^{2}-4q^{5/2}+8q^{3}-12q^{7/2}+19q^{4}$ (10)
$\displaystyle-26q^{9/2}+38q^{5}-58q^{11/2}+85q^{6}-116q^{13/2}$
$\displaystyle+165q^{7}-236q^{15/2}+326q^{8}+O(q^{17/2})\;.$ (11)

No known S-fold theory is associated with this VOA Aharony and Tachikawa
(2016).

##  VII Results: $\mathsf{G}=G(3,1,2)$

This is our only rank 2 example, with central charge $c=-48$. Its W-algebra
presentation involves the following strong generators: $\mathcal{T}$,
$\mathcal{J}$, $\mathcal{G}$ and $\widetilde{\mathcal{G}}$ from the 2D
$\mathcal{N}=2$ SCA, the chiral and anti-chiral generators $\mathcal{W}_{3}$,
$\mathcal{W}_{6}$, $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{3}$, $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{6}$, plus
their superpartners $\mathcal{G}_{\mathcal{W}_{3}}$,
$\mathcal{G}_{\mathcal{W}_{6}}$ and $\widetilde{\mathcal{G}}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{3}}$,
$\widetilde{\mathcal{G}}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{6}}$, as well
as $\mathcal{U}$—which is self conjugate—and its superpartners
$\mathcal{G}_{\mathcal{U}}$, $\widetilde{\mathcal{G}}_{\mathcal{U}}$ and
$\mathcal{G}_{\widetilde{\mathcal{G}}_{\mathcal{U}}}$. One also needs non-
chiral $\mathcal{O}$, $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{O}\mkern-1.5mu}\mkern 1.5mu$ and their
superpartners $\mathcal{G}_{\mathcal{O}}$,
$\widetilde{\mathcal{G}}_{\mathcal{O}}$, $\mathcal{G}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{O}\mkern-1.5mu}\mkern 1.5mu}$,
$\widetilde{\mathcal{G}}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{O}\mkern-1.5mu}\mkern 1.5mu}$,
$\mathcal{G}_{\widetilde{\mathcal{G}}_{\mathcal{O}}}$,
$\mathcal{G}_{\widetilde{\mathcal{G}}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{O}\mkern-1.5mu}\mkern 1.5mu}}$.

The free-field realisation requires two ghost systems,
$\beta_{\ell}\gamma_{\ell}b_{\ell}c_{\ell}$ with $\ell=1,2$. One starts with a
prescription for the 2D $\mathcal{N}=2$ SCA generators as well as for
$\mathcal{W}_{3}$, $\mathcal{W}_{6}$, $\mathcal{G}_{\mathcal{W}_{3}}$ and
$\mathcal{G}_{\mathcal{W}_{6}}$. The ansatz for $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{3}$ contains
84 undetermined coefficients. It turns out that through the OPEs in the
W-algebra presentation, one can fix the coefficients of $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{3}$ and by
doing so also determine the free-field realisation of all remaining
generators. We have calculated the fully-refined vacuum character (1) up to
$O(q^{9/2})$ with the accompanying “null states.nb”. The full result can be
found in “vacuum_characters_summary.nb”.

This VOA is expected to encode the Schur sector of a rank-two 4D S-fold
$\mathcal{N}=3$ SCFT, with Coulomb-branch operators of dimension $\Delta=3,6$
and trace anomaly coefficient $c_{\mathrm{4D}}=4$. Through (3) the Macdonald
index of this S-fold theory can be identified with the vacuum character. If
for simplicity one considers the limit $z\to 1$:

$\displaystyle\mathcal{I}^{G(3,1,2)}_{\mathrm{Macdonald}}$
$\displaystyle=1+q\xi+q^{3/2}\left(-\sqrt{\xi}+\xi^{3/2}\right)+q^{2}\left(\xi+\xi^{2}\right)$
(12)
$\displaystyle+q^{5/2}\left(-\sqrt{\xi}-\xi^{3/2}+2\xi^{5/2}\right)+q^{3}\left(\xi-\xi^{2}+2\xi^{3}\right)$
(13)
$\displaystyle+q^{7/2}\left(-\sqrt{\xi}-2\xi^{3/2}+2\xi^{5/2}+\xi^{7/2}\right)$
(14) $\displaystyle+q^{4}\left(2\xi-\xi^{3}+3\xi^{4}\right)$ (15)
$\displaystyle+q^{9/2}\left(-\sqrt{\xi}-3\xi^{3/2}+\xi^{5/2}+\xi^{7/2}+2\xi^{9/2}\right)$
(16) $\displaystyle+O(q^{5})\;.$ (17)

##  VIII Conclusions

In this letter we have calculated vacuum characters of rank-one and rank-two
VOAs labelled by non-Coxeter, crystallographic complex reflection groups. This
involved a brute-force implementation of the algorithms presented in Bonetti
et al. (2019) and leads to the Macdonald index of certain 4D $\mathcal{N}=3$
S-fold SCFTs. Our results were given as an expansion in the fugacity that
keeps track of the conformal weight, and were truncated to orders that require
short computation times when using a desktop computer; they can be pushed to
arbitrary higher orders by allocating appropriate resources. As they stand,
they can already be used as new data for $\mathcal{N}=3$ SCFTs. E.g. the
$\mathsf{G}=\mathbb{Z}_{3}$ result agrees in the Schur limit with Zafrir
(2021).

Our code is customisable. We have clearly signposted where changes would need
to be made to return vacuum characters of VOAs labelled by different complex
reflection groups, for which the free-field realisation is known. In
particular, it would be very interesting to extend this approach to the
$\mathcal{N}=3$ S-fold SCFT of rank two associated with $G(4,1,2)$ and the
rank-three example $G(3,3,3)$; a proposal for the Schur index of the latter
was also given in Zafrir (2021). Unfortunately, finding the free-field
realisation for both these VOAs—already needed before identifying the null
states—is a challenging task: the simplest anti-chiral strong generator
ansatze involve 425 and 2265 undetermined coefficients respectively. It would
perhaps be more promising to use the screening-operator approach, upon
determining $\mathbb{S}$. Although our screening-operator code is currently
more expensive to run, it could benefit from optimisations that parallelise
the computations, hence making it significantly faster on multi-core clusters.
It will also be interesting to check these results by directly studying the
BPS-states of $\mathcal{N}=3$ theories. One way to do so would be to study
three-string junctions in S-fold backgrounds as in Agarwal and Amariti (2016).
We hope to return to some of these questions in the near future.

##  IX Acknowledgements

We would like to thank B. Ergun, T. França, M. Martone, P. Longhi, C.
Meneghelli, A. Pini, E. Pomoni, F. Schaposnik Massolo and J. Song for useful
discussions and comments. We acknowledge financial support by the Royal
Society URF\R\180009, RGF\EA\181049, RGF\EA\180073 and the STFC ST/P000754/1.

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#  $\Gamma$-convergence for a class of action functionals induced by gradients
of convex functions

Luigi Ambrosio Scuola Normale Superiore, Pisa. E-mail<EMAIL_ADDRESS>Aymeric Baradat111Institut Camille Jordan, Lyon. E-mail<EMAIL_ADDRESS>lyon1.fr_ Yann Brenier222École Normale Supérieure, Paris. E-mail:
<EMAIL_ADDRESS>
###### Abstract

Given a real function $f$, the rate function for the large deviations of the
diffusion process of drift $\nabla f$ given by the Freidlin-Wentzell theorem
coincides with the time integral of the energy dissipation for the gradient
flow associated with $f$. This paper is concerned with the stability in the
hilbertian framework of this common action functional when $f$ varies. More
precisely, we show that if $(f_{h})_{h}$ is uniformly $\lambda$-convex for
some $\lambda\in\mathbb{R}$ and converges towards $f$ in the sense of Mosco
convergence, then the related functionals $\Gamma$-converge in the strong
topology of curves.

##  1 Introduction

Action functionals of the form

$I_{f}(\gamma):=\int_{0}^{1}\Big{\\{}|\dot{\gamma}(t)|^{2}+|\nabla
f|^{2}(\gamma(t))\Big{\\}}\mathrm{d}t,$

and the closely related ones (since they differ by a null lagrangian, the term
$2f(\gamma(1))-2f(\gamma(0))$)

$\int_{0}^{1}|\dot{\gamma}(t)-\nabla f(\gamma(t))|^{2}\mathrm{d}t,$ (1)

appear in many areas of Mathematics, for instance in the Freidlin-Wentzell
theory of large deviations for the SDE $\mathrm{d}X_{t}^{\epsilon}=\nabla
f(X^{\epsilon}_{t})\mathrm{d}t+\sqrt{\epsilon}\mathrm{d}B_{t}$ (see for
instance [9]) or in the variational theory of gradient flows pioneered by De
Giorgi, where they correspond to the integral form of the energy dissipation
(see [4]). In this paper, we investigate the stability of the action
functionals $I_{f}$ with respect to $\Gamma$-convergence of the functions $f$
(actually with respect to the stronger notion of Mosco convergence, see
below). More precisely, we are concerned with the case when the functions
under consideration are $\lambda$-convex and defined in a Hilbert space $H$.
In this case, the functional $I_{f}$ is well defined if we understand $\nabla
f(x)$ as the element with minimal norm in the subdifferential $\partial f(x)$:
this choice, very natural in the theory of gradient flows, grants the joint
lower semicontinuity property of $(x,f)\mapsto|\nabla f|(x)$ that turns out to
be very useful when proving stability of gradient flows, see [12], [5] and the
more recent papers [10], [11] where emphasis is put on the convergence of the
dissipation functionals. In more abstract terms, we are dealing with
autonomous Lagrangians $L(x,p)=|p|^{2}+|\nabla f|^{2}(x)$ that are unbounded
and very discontinuous with respect to $x$, and this is a source of difficulty
in the construction of recovery sequences, in the proof of the $\Gamma$-limsup
inequality.

Our interest in this problem comes from [3], where we dealt with the
derivation of the discrete Monge-Ampère equation from the stochastic model of
a Brownian point cloud, using large deviations and Freidlin-Wentzell theory,
along the lines of [6]. In that case $H=\mathbb{R}^{Nd}$ was finite
dimensional,

$f(x):=\max_{\sigma\in\mathfrak{S}_{N}}\langle x,A^{\sigma}\rangle,$

(with $A=(A_{1},\ldots,A_{N})\in\mathbb{R}^{Nd}$ given and
$A^{\sigma}=(A_{\sigma(1)},\ldots,A_{\sigma(N)})$ for all
$\sigma\in\mathfrak{S}_{N}$, the set of all permutations of $\llbracket
1,N\rrbracket$), and the approximating functions $f_{\epsilon}$ were given by

$f_{\epsilon}(t,x)=\epsilon
t\log\biggl{[}\frac{1}{N{\rm!}}\sum_{\sigma\in\Sigma_{N}}\exp\bigg{(}\frac{\langle
x,A^{\sigma}\rangle}{\varepsilon t}\biggr{)}\biggr{]}.$

In that case, our proof used some simplifications due to finite
dimensionality, and a uniform Lipschitz condition. In this paper, building
upon some ideas in [3], we provide the convergence result in a more general
and natural context. For the sake of simplicity, unlike [3], we consider only
the autonomous case. However it should be possible to adapt our proof to the
case when time-dependent $\lambda$-convex functions $f(t,\cdot)$ are
considered, under additional regularity assumptions with respect to $t$, as in
[3].

In the infinite-dimensional case, Mosco convergence (see Definition 4.1) is
stronger and more appropriate than $\Gamma$-convergence, since it ensures
convergence of the resolvent operators (under equi-coercitivity assumptions,
the two notions are equivalent). Also, since in the infinite-dimensional case,
the finiteness domains of the functions can be pretty different, the addition
of the endpoint condition is an additional source of difficulties, that we
handle with an interpolation lemma which is very much related to the structure
of monotone operators, see Lemma 3.1.

Defining the functionals $\Theta_{f,x_{0},x_{1}}:C([0,1];H)\to[0,\infty]$ by

$\Theta_{f,x_{0},x_{1}}(\gamma):=\left\\{\begin{aligned}
&I_{f}(\gamma)&&\text{if $\gamma\in AC([0,1];H)$, $\gamma(0)=x_{0}$,
$\gamma(1)=x_{1}$;}\\\ &+\infty&&\text{otherwise,}\end{aligned}\right.$ (2)

our main result reads as follows:

######  Theorem 1.1.

If $(f_{h})_{h}$ is uniformly $\lambda$-convex for some
$\lambda\in\mathbb{R}$, if $f_{h}\to f$ w.r.t. Mosco convergence, and if

$\lim_{h\to\infty}x_{h,i}=x_{i},\qquad\qquad\sup_{h}|\nabla
f_{h}|(x_{h,i})<\infty,\qquad i=0,\,1,$

then $\Theta_{f_{h},x_{h,0},x_{h,1}}$ $\Gamma$-converge to
$\Theta_{f,x_{0},x_{1}}$ in the $C([0,1];H)$ topology.

As a byproduct, under an additional equi-coercitivity assumption our theorem
grants convergence of minimal values to minimal values and of minimizers to
minimizers. Obviously the condition $x_{h,i}\to x_{i}$ is necessary, and we
believe that at least some (possibly more refined) bounds on the gradients at
the endpoints are necessary as well. If we ask also that $x_{h,i}$ are
recovery sequences, i.e. $f_{h}(x_{h,i})\to f(x_{i})$, then the result can
also be read in terms of the functionals (1).

As a final comment, it would be interesting to investigate this type of
convergence results also in a non-Hilbertian context, as it happened for the
theory of gradient flows. For instance, a natural context would be the space
of probability measures with finite quadratic moment. Functionals of this
form, where $f$ is a constant multiple of the logarithmic entropy, appear in
the so-called entropic regularization of the Wasserstein distance (see [8] and
the references therein).

Acknowledgements. We dedicate this paper to Edoardo Vesentini, a great
mathematician and a former President of the Accademia dei Lincei. As Director
of the Scuola Normale, he has been the pioneer of many projects that shaped
the Scuola Normale for many years to come. The first author acknowledges the
support of the PRIN 2017 project “Gradient flows, Optimal Transport and Metric
Measure Structures”. This work was prepared during the stay of the second
author at the Max Planck Institute for Mathematics in the Sciences in Leipzig,
that he would like to thank for its hospitality.

##  2 Preliminaries

Let $H$ be a Hilbert space. For a function $f:H\to(-\infty,\infty]$ we denote
by $D(f)$ the finiteness domain of $f$. We say that $f$ is $\lambda$-convex if
$x\mapsto f(x)-\tfrac{\lambda}{2}|x|^{2}$ is convex. It is easily seen that
$\lambda$-convex functions satisfy the perturbed convexity inequality

$f\bigl{(}(1-t)x+ty\bigr{)}\leq(1-t)f(x)+tf(y)-\frac{\lambda}{2}t(1-t)|x-y|^{2},\qquad
t\in[0,1].$

We denote by $\partial f(x)$ the Gateaux subdifferential of $f$ at $x\in
D(f)$, namely the set

$\partial f(x):=\left\\{p\in H:\ \liminf_{t\to
0^{+}}\frac{f(x+th)-f(x)}{t}\geq t\langle h,p\rangle\,\,\,\forall h\in
H\right\\}.$

It is a closed convex set, possibly empty. We denote by $D(\partial f)$ the
domain of the subdifferential.

In the case when $f$ is $\lambda$-convex, the monotonicity of difference
quotients gives the equivalent, non asymptotic definition:

$\partial f(x):=\left\\{p\in H:\ f(y)\geq f(x)+\langle
y-x,p\rangle+\frac{\lambda}{2}|y-x|^{2}\,\,\,\forall y\in H\right\\}.$ (3)

For any $x\in D(\partial f)$ we consider the vector $\nabla f(x)$ as the
element with minimal norm of $\partial f(x)$. We agree that $|\nabla
f(x)|=\infty$ if either $x\notin D(f)$ of $x\in D(f)$ and $\partial
f(x)=\emptyset$. For $\lambda$-convex functions, relying on (3), it can be
easily proved that $\partial f(x)$ is not empty if and only if

$\sup_{y\neq
x}\frac{\bigl{[}f(x)-f(y)+\frac{\lambda}{2}|x-y|^{2}\bigr{]}^{+}}{|x-y|}<\infty$
(4)

and that $|\nabla f|(x)$ is precisely equal to the supremum (see for instance
Theorem 2.4.9 in [4]).

For $\tau>0$ we denote by $f_{\tau}$ the regularized function

$f_{\tau}(x):=\min_{y\in H}f(y)+\frac{|y-x|^{2}}{2\tau}$ (5)

and we denote by $J_{\tau}=(\operatorname{Id}+\tau\partial f)^{-1}:H\to
D(\partial f)$ the so-called resolvent map associating to $x$ the minimizer
$y$ in (5). When $f$ is proper, $\lambda$-convex and lower semicontinuous,
existence and uniqueness of $J_{\tau}(x)$ follow by the strict convexity of
$y\mapsto f(y)+|y-x|^{2}/(2\tau)$, as soon as $\tau<-1/\lambda$ when
$\lambda<0$, and for all $\tau>0$ otherwise (we shall call admissible these
values of $\tau$). We also use the notation $J_{f,\tau}$ to emphasize the
dependence on $f$.

Now we recall a few basic and well-known facts (see for instance [7], [4]),
providing for the reader’s convenience sketchy proofs.

######  Theorem 2.1.

Assume that $f:H\to(-\infty,\infty]$ is proper, $\lambda$-convex and lower
semicontinuous. For all admissible $\tau>0$ one has:

  1. (i)

$f_{\tau}$ is differentiable everywhere, and for all $x\in H$,

$\nabla f_{\tau}(x)=\frac{x-J_{\tau}(x)}{\tau}\in\partial f(J_{\tau}(x)).$ (6)

  2. (ii)

$J_{\tau}$ is $(1+\lambda\tau)^{-1}$-Lipschitz, and $f_{\tau}\in C^{1,1}(H)$
with ${\rm Lip}(\nabla f_{\tau})\leq 3/\tau$ as soon as there holds
$(1+\tau\lambda)^{-1}\leq 2$.

  3. (iii)

For all $x\in D(\partial f)$,

$\nabla f_{\tau}(x+\tau\nabla f(x))=\nabla f(x).$ (7)

  4. (iv)

The following monotonicity properties hold for all $x\in H$:

$|\nabla f|(J_{\tau}(x))\leq|\nabla
f_{\tau}|(x)=\frac{|x-J_{\tau}(x)|}{\tau}\leq\frac{1}{1+\lambda\tau}|\nabla
f|(x).$ (8)

###### Proof.

The inclusion in (6) follows from performing variations around $J_{\tau}(x)$
in (5).

Before proving the equality in (6), let us prove the Lipschitz property for
$J_{\tau}$ given in (ii). Recall that the convexity of
$g=f-\frac{\lambda}{2}|\cdot|^{2}$ yields that $\partial f$ is
$\lambda$-monotone, namely

$\langle\xi-\eta,a-b\rangle\geq\lambda|a-b|^{2}\qquad\forall\xi\in\partial
f(a),\,\,\eta\in\partial f(b).$

Given $x$ and $y$, we apply this property to $a:=J_{\tau}(x)$,
$b:=J_{\tau}(y)$, $\xi:=(x-J_{\tau}(x))/\tau$ and
$\eta:=(y-J_{\tau}(y))/\tau$. (Thanks to the inclusion in (6), we have
$\xi\in\partial f(a)$ and $\eta\in\partial f(b)$.) By rearranging the terms,
we get

$\langle
x-y,J_{\tau}(x)-J_{\tau}(y)\rangle\geq(1+\lambda\tau)|J_{\tau}(x)-J_{\tau}(y)|^{2}.$

Hence, by the Cauchy-Schwarz inequality, $J_{\tau}$ is
$(1+\lambda\tau)^{-1}$-Lipschitz.

Let us go back to proving the equality in (6). For any $x$ and $z$, one has
(using $y=J_{\tau}(x)$ as an admissible competitor in the definition of
$f_{\tau}(x+z)$)

$f_{\tau}(x+z)-f_{\tau}(x)\leq\frac{|J_{\tau}(x)-(x+z)|^{2}}{2\tau}-\frac{|J_{\tau}(x)-x|^{2}}{2\tau}=\left\langle
z,\frac{x-J_{\tau}(x)}{\tau}\right\rangle+\frac{|z|^{2}}{2\tau}$

and, reversing the roles of $x$ and $x+z$,

$f_{\tau}(x)-f_{\tau}(x+z)\leq\left\langle-z,\frac{x+z-J_{\tau}(x+z)}{\tau}\right\rangle+\frac{|z|^{2}}{2\tau}.$

These two identities together with the continuity of $J_{\tau}$ imply that
$f_{\tau}$ is differentiable at $x$ and provides the equality in (6) and hence
the one in (8). The Lipschitz property for $\nabla f_{\tau}$ announced follows
directly from this identity and the Lipschitz property for $J_{\tau}$.

To get (7), it suffices to remark that for all $x\in D(\partial f)$, $0$
belongs to the subdifferential of the strictly convex function

$y\mapsto f(y)+\frac{|x+\tau\nabla f(x)-y|^{2}}{2\tau}$

at $y=x$. Hence, $x$ is the minimizer of this function, and
$J_{\tau}(x+\tau\nabla f(x))=x$. Then, we deduce (6) from (7).

The first inequality in (8) follows from the inclusion in (6). In order to
prove the second inequality, we perform a variation along the affine curve
joining $x$ to $J_{\tau}(x)$, namely, $\gamma(t):=(1-t)x+tJ_{\tau}(x)$. Since

$\displaystyle f(J_{\tau}(x))+\frac{1}{2\tau}|x-J_{\tau}(x)|^{2}$
$\displaystyle\leq f(\gamma(t))+\frac{1}{2\tau}|x-\gamma(t)|^{2}$
$\displaystyle\leq(1-t)f(x)+tf(J_{\tau}(x))+\frac{t}{2\tau}\big{(}t-\lambda\tau(1-t)\big{)}|x-J_{\tau}(x)|^{2}$

for all $t\in[0,1]$, taking the left derivative at $t=1$ gives

$\left(\frac{\lambda}{2}+\frac{1}{\tau}\right)|x-J_{\tau}(x)|^{2}\leq
f(x)-f(J_{\tau}(x)),$

so that the representation formula (4) for $|\nabla f|(x)$ gives

$\left(\frac{\lambda}{2}+\frac{1}{\tau}\right)|x-J_{\tau}(x)|^{2}\leq|\nabla
f|(x)|x-J_{\tau}(x)|-\frac{\lambda}{2}|x-J_{\tau}(x)|.$

By rearranging the terms, this leads to the second inequality in (8). ∎

Another remarkable property of $|\nabla f|$, for $f$ $\lambda$-convex and
lower semicontinuous, is the upper gradient property, namely,

$f(\gamma(0)),f(\gamma(\delta))<\infty\quad\text{and}\quad|f(\gamma(\delta))-f(\gamma(0))|\leq\int_{0}^{\delta}|\nabla
f|(\gamma(t))|\dot{\gamma}(t)|\mathrm{d}t$

for any $\delta>0$ and any absolutely continuous $\gamma:[0,\delta]\to H$
(with the convention $0\times\infty=0$), whenever $\gamma$ is not constant and
the integral in the right hand side is finite (see for instance Corollary
2.4.10 in [4] for the proof).

##  3 A class of action functionals

For $\delta>0$ and $f:H\to(-\infty,\infty]$ proper, $\lambda$-convex and lower
semicontinuous, we consider the autonomous functionals
$I_{f}^{\delta}:C([0,\delta];H)\to[0,\infty]$ defined by

$I^{\delta}_{f}(\gamma):=\int_{0}^{\delta}\Big{\\{}|\dot{\gamma}|^{2}+|\nabla
f|^{2}(\gamma)\Big{\\}}\mathrm{d}t,$

set to $+\infty$ on $C([0,\delta];H)\setminus AC([0,\delta];H)$. Notice also
that $I_{f}^{\delta}(\gamma)<\infty$ implies $\gamma\in D(\partial f)$ a.e. in
$(0,\delta)$.

Identity (4) ensures the lower semicontinuity of $|\nabla f|$; hence, under a
coercitivity assumption of the form $\\{f\leq t\\}$ compact in $H$ for all
$t\in\mathbb{R}$, the infimum

$\Gamma_{\delta}(x_{0},x_{\delta}):=\inf\left\\{I^{\delta}_{f}(\gamma):\
\gamma(0)=x_{0},\,\,\gamma(\delta)=x_{\delta}\right\\}\qquad
x_{0},\,x_{\delta}\in H$ (9)

is always attained whenever finite.

Also, by the Young inequality and the upper gradient property of $|\nabla f|$,
one has that $I_{f}^{\delta}(\gamma)<\infty$ implies
$\gamma(0),\,\gamma(\delta)\in D(f)$ and
$2|f(\gamma(\delta))-f(\gamma(0))|\leq I_{f}^{\delta}(\gamma)$. The same
argument shows that we may add to $I_{f}^{\delta}$ a null Lagrangian. Namely,
as done in [3], we can consider the functionals

$\int_{0}^{\delta}|\dot{\gamma}-\nabla f(\gamma)|^{2}\mathrm{d}t$

which differ from $I^{\delta}_{f}$ precisely by the term
$2f(\gamma(\delta))-2f(\gamma(0))$, whenever $\gamma$ is admissible in (9)
with $I_{f}^{\delta}(\gamma)<\infty$.

Because of the lack of continuity of $x\mapsto\nabla f(x)$, very little is
known in general about the regularity of minimizers in (9), even when $H$ is
finite-dimensional. However, one may use the fact that $I^{1}_{f}$ is
autonomous to perform variations of type
$\gamma\mapsto\gamma\circ(\operatorname{Id}+\epsilon\phi)$, $\phi\in
C^{\infty}_{c}(0,\delta)$, to obtain the Dubois-Reymond equation (see for
instance [2])

$\frac{\mathrm{d}}{\mathrm{d}t}\bigl{[}|\dot{\gamma}|^{2}-|\nabla
f|^{2}(\gamma))\bigr{]}=0\qquad\text{in the sense of distributions in
$(0,\delta)$}.$

It implies Lipschitz regularity of the minimizers when, for instance, $|\nabla
f|$ is bounded on bounded sets (an assumption satisfied in [3], but obviously
too strong for some applications in infinite dimension).

We will need the following lemma, estimating $\Gamma_{\delta}$ from above, to
adjust the values of the curves at the endpoints. The heuristic idea is to
interpolate on the graph of $f_{\tau}$ and then read back this interpolation
in the original variables. This is related to Minty’s trick (see [1] for an
extensive use of this idea): a rotation of $\pi/4$ maps the graph of the
subdifferential onto the graph of an entire $1$-Lipschitz function; here we
use only slightly tilted variables, of order $\tau$.

######  Lemma 3.1 (Interpolation).

Let $f:H\to(-\infty,\infty]$ be a proper, $\lambda$-convex and lower
semicontinuous function and let $\tau>0$ be such that
$(1+\tau\lambda)^{-1}\leq 2$. For all $\delta>0$ and all $x_{0}\in D(\partial
f)$, $x_{\delta}\in D(\partial f)$, with $\Gamma_{\delta}$ as in (9), one has

$\Gamma_{\delta}(x_{0},x_{\delta})\leq 2\delta\min_{i\in\\{0,\delta\\}}|\nabla
f|^{2}(x_{i})+\biggl{(}\frac{40}{\delta}+\frac{12\delta}{\tau^{2}}\biggr{)}|x_{\delta}-x_{0}|^{2}+\biggl{(}12\delta+\frac{40\tau^{2}}{\delta}\biggr{)}|\nabla
f(x_{\delta})-\nabla f(x_{0})|^{2}.$

###### Proof.

We use Theorem 2.1 to interpolate between $x_{\delta}$ and $x_{0}$ as follows:
set

$\tilde{\gamma}(t):=\left(1-\frac{t}{\delta}\right)(x_{0}+\tau\nabla
f(x_{0}))+\frac{t}{\delta}(x_{\delta}+\tau\nabla
f(x_{\delta})),\qquad\xi(t):=\nabla f_{\tau}(\tilde{\gamma}(t)),$

and

$\gamma(t):=J_{\tau}(\tilde{\gamma}(t))=\tilde{\gamma}(t)-\tau\xi(t),$

where the second equality follows from (6).

Since $\xi(0)=\nabla f_{\tau}(x_{0}+\tau\nabla f(x_{0}))=\nabla f(x_{0})$ and
a similar property holds at time $\delta$, the path $\gamma$ is admissible.
Let us now estimate the action of the path $\gamma$.

Kinetic term (we use our Lipschitz bound for $\nabla f_{\tau}$ to deduce that
$|\dot{\xi}(t)|\leq\frac{3}{\tau}|\dot{\tilde{\gamma}}(t)|$):

$\displaystyle\int_{0}^{\delta}|\dot{\gamma}|^{2}\mathrm{d}t$
$\displaystyle\leq
2\int_{0}^{\delta}|\dot{\tilde{\gamma}}|^{2}\mathrm{d}t+2\tau^{2}\int_{0}^{\delta}|\dot{\xi}|^{2}\mathrm{d}t$
$\displaystyle\leq
20\int_{0}^{\delta}|\dot{\tilde{\gamma}}|^{2}\mathrm{d}t=\frac{20}{\delta}|(x_{\delta}+\tau\nabla
f(x_{\delta}))-(x_{0}+\tau\nabla f(x_{0}))|^{2}$
$\displaystyle\leq\frac{40}{\delta}|x_{\delta}-x_{0}|^{2}+\frac{40\tau^{2}}{\delta}|\nabla
f(x_{\delta})-\nabla f(x_{0})|^{2}.$

Gradient term (we use the first inequality in (8), our Lipschitz bound for
$\nabla f_{\tau}$, and finally (7)):

$\displaystyle\int_{0}^{\delta}|\nabla f|^{2}(\gamma)\mathrm{d}t$
$\displaystyle\leq\int_{0}^{\delta}|\nabla
f_{\tau}|^{2}(\tilde{\gamma})\mathrm{d}t$
$\displaystyle\leq\int_{0}^{\delta}\left(|\nabla
f_{\tau}|(\tilde{\gamma}(0))+\frac{3}{\tau}|\tilde{\gamma}(t)-\tilde{\gamma}(0)|)\right)^{2}\mathrm{d}t$
$\displaystyle\leq\int_{0}^{\delta}\left\\{2|\nabla
f|^{2}(x_{0})+\frac{18}{\tau^{2}}\frac{t^{2}}{\delta^{2}}|(x_{\delta}+\tau\nabla
f(x_{\delta}))-(x_{0}+\tau\nabla f(x_{0}))|^{2}\right\\}\mathrm{d}t$
$\displaystyle\leq 2\delta|\nabla
f|^{2}(x_{0})+\frac{6\delta}{\tau^{2}}|(x_{\delta}+\tau\nabla
f(x_{\delta}))-(x_{0}+\tau\nabla f(x_{0}))|^{2}$ $\displaystyle\leq
2\delta|\nabla
f|^{2}(x_{0})+\frac{12\delta}{\tau^{2}}|x_{\delta}-x_{0}|^{2}+12\delta|\nabla
f(x_{\delta})-\nabla f(x_{0})|^{2}.$

We get the result by gathering these two estimates, and by remarking that in
the second line, we could have controlled $|\nabla
f|_{\tau}(\tilde{\gamma}(t))$ by its value at time $\delta$ instead of its
value at time $0$.

∎

Choosing $\delta=\tau$, bounding $|\nabla f|(x_{i})$, $i=0,1$ by the max of
these two values, and using $|\nabla f(x_{\delta})-\nabla f(x_{0})|^{2}\leq
4\max_{i\in\\{0,1\\}}|\nabla f|^{2}(x_{i})$, we will apply the interpolation
lemma in the form

$\Gamma_{\delta}(x_{0},x_{\delta})\leq\frac{52}{\tau}|x_{\delta}-x_{0}|^{2}+210\tau\max_{i\in\\{0,\delta\\}}|\nabla
f|^{2}(x_{i}).$ (10)

##  4 Proof of the main result

In this section, $f_{h}$, $f$ denote generic proper, $\lambda$-convex and
lower semicontinuous functions from $H$ to $(-\infty,\infty]$.

Mosco convergence is a particular case of $\Gamma$-convergence, where the
topologies used for the $\limsup$ and the $\liminf$ inequalities differ.

######  Definition 4.1 (Mosco convergence).

We say that $f_{h}$ Mosco converge to $f$ whenever:

  * (a)

for all $x\in H$ there exist $x_{h}\to x$ strongly with

$\limsup_{h}f_{h}(x_{h})\leq f(x);$

  * (b)

for all sequences $(x_{h})\subset H$ weakly converging to $x$, one has

$\liminf_{h}f_{h}(x_{h})\geq f(x).$

It is easy to check that for sequences of $\lambda$-convex functions, Mosco
convergence implies the pointwise convergence of $J_{f_{h},\tau}$ to
$J_{f,\tau}$, contrarily to usual $\Gamma$-convergence. Indeed, for $\tau>0$
admissible, (a) grants

$\limsup_{h\to\infty}\min_{y\in
H}f_{h}(y)+\frac{|y-x|^{2}}{2\tau}\leq\min_{y\in
H}f(y)+\frac{|y-x|^{2}}{2\tau},$

while (b) grants

$\liminf_{h\to\infty}\min_{y\in
H}f_{h}(y)+\frac{|y-x|^{2}}{2\tau}\geq\min_{y\in
H}f(y)+\frac{|y-x|^{2}}{2\tau},$

and the weak convergence of minimizers $y_{h}$ to the minimizer $y$.
Eventually, the convergence of the energies together with

$\liminf_{h\to\infty}f_{h}(y_{h})\geq
f(y)\qquad\text{and}\qquad\liminf_{h\to\infty}|y_{h}-x|^{2}\geq|y-x|^{2}$

grants that both liminf are limits, and that the convergence of $y_{h}$ is
strong.

Recall that given $x_{h,0},\,x_{h,1}\in H$, the functionals
$\Theta_{f_{h},x_{h,0},x_{h,1}}$ defined in (2), are obtained from
$I_{f_{h}}^{1}$ by adding endpoints constraints. $\Theta_{f,x_{0},x_{1}}$ is
defined analogously.

We say that $\Theta_{f_{h},x_{h,0},x_{h,1}}$ $\Gamma$-converge to
$\Theta_{f,x_{0},x_{1}}$ in the $C([0,1];H)$ topology if

  * (a)

for all $\gamma\in C([0,1];H)$ there exist $\gamma_{h}\in C([0,1];H)$
converging to $\gamma$ with

$\limsup_{h\to\infty}\Theta_{f_{h},x_{h,0},x_{h,1}}(\gamma_{h})\leq\Theta_{f,x_{0},x_{1}}(\gamma);$

  * (b)

for all sequences $(\gamma_{h})\subset C([0,1];H)$ converging to $\gamma$ one
has

$\liminf_{h\to\infty}\Theta_{f_{h},x_{h,0},x_{h,1}}(\gamma_{h})\geq\Theta_{f,x_{0},x_{1}}(\gamma).$

In connection with the proof of property (a) it is useful to introduce the
functional

$\Gamma-\limsup_{h\to\infty}\Theta_{f_{h},x_{h,0},x_{h,1}}(\gamma):=\inf\left\\{\limsup_{h\to\infty}\Theta_{f_{h},x_{h,0},x_{h,1}}(\gamma_{h}):\gamma_{h}\to\gamma\right\\}$

so that (a) is equivalent to
$\Gamma-\limsup_{h}\Theta_{f_{h},x_{h,0},x_{h,1}}\leq\Theta_{f,x_{0},x_{1}}$.
Recall also that the $\Gamma-\limsup$ is lower semicontinuous, a property that
can be achieved, for instance, by a diagonal argument.

###### Proof of Theorem 1.1.

It is clear that the endpoint condition passes to the limit with respect to
the $C([0,1];H)$ topology, since $x_{h,i}$ converge to $x_{i}$. Also, it is
well known that the action functional is lower semicontinuous in $C([0,1];H)$.
Hence, the $\Gamma$-liminf inequality, namely property (b), follows
immediately from Fatou’s lemma and the variational characterization (4) of
$|\nabla f|$. Indeed, for all $y\neq x$ and all sequences $x_{h}\to x$

$\displaystyle\liminf_{h\to\infty}|\nabla
f_{h}|^{2}(x_{h})\geq\liminf_{h\to\infty}\frac{[f_{h}(x_{h})-f_{h}(y_{h})+\frac{\lambda}{2}|x_{h}-y_{h}|^{2}\bigr{]}^{+}}{|x_{h}-y_{h}|}\geq\frac{[f(x)-f(y)+\frac{\lambda}{2}|x-y|^{2}\bigr{]}^{+}}{|x-y|}.$

where $y_{h}$ is chosen as in (a) of Definition 4.1. Passing to the supremum,
we get the inequality $\liminf_{h}|\nabla f_{h}|(x_{h})\geq|\nabla f|(x)$, and
this grants the lower semicontinuity of the gradient term in the functionals.
Notice that this part of the proof works also if we assume only that
$\Gamma$-$\liminf_{h}f_{h}\geq f$, for the strong topology of $H$, but the
stronger property (namely (b) in Definition 4.1) is necessary because we will
need in the next step convergence of the resolvents.

So, let us focus on the $\Gamma$-limsup one, property (a). Fix a path $\gamma$
with $\Theta_{f,x_{0},x_{1}}(\gamma)<\infty$, $\tau>0$ (with
$(1+\tau\lambda^{-1})\leq 2$ if $\lambda<0$) and consider the perturbed paths
$\gamma^{\tau}_{h}(t)=J_{f_{h},\tau}(\gamma(t))$,
$\gamma^{\tau}(t)=J_{f,\tau}(\gamma(t))$; using the
$(1+\tau\lambda)^{-1}$-Lipschitz property of the maps $J_{f,\tau}$, the first
inequality in (8), the convergence of $\gamma^{\tau}_{h}$ to $\gamma^{\tau}$
and eventually the second inequality in (8) one gets

$\displaystyle\limsup_{h\to\infty}\int_{0}^{1}\left\\{|\dot{\gamma}^{\tau}_{h}|^{2}+|\nabla
f_{h}|^{2}(\gamma^{\tau}_{h})\right\\}\mathrm{d}t$
$\displaystyle\leq\limsup_{h\to\infty}\int_{0}^{1}\left\\{(1+\tau\lambda)^{-2}|\dot{\gamma}|^{2}+\frac{|\gamma-\gamma^{\tau}_{h}|^{2}}{\tau^{2}}\right\\}\mathrm{d}t$
$\displaystyle\leq\int_{0}^{1}\left\\{(1+\tau\lambda)^{-2}|\dot{\gamma}|^{2}+\frac{|\gamma-\gamma^{\tau}|^{2}}{\tau^{2}}\right\\}\mathrm{d}t$
$\displaystyle\leq(1+\tau\lambda)^{-2}\int_{0}^{1}\left\\{|\dot{\gamma}|^{2}+|\nabla
f|^{2}(\gamma)\right\\}\mathrm{d}t.$

Also, the convergence of resolvents gives

$\lim_{h\to\infty}J_{f_{h},\tau}(x_{i})=J_{f,\tau}(x_{i}).$

Finally, using again the inequalities (8) and once more the convergence of
resolvents, we get

$\limsup_{h\to\infty}\\!|\nabla
f_{h}|(J_{f_{h},\tau}(x_{i}))\leq\frac{|J_{f,\tau}(x_{i})-x_{i}|}{\tau}\leq(1+\tau\lambda)^{-1}|\nabla
f|(x_{i})\leq 2|\nabla f|(x_{i}).$

Since the endpoints have been slightly modified by the composition with
$J_{f_{h},\tau}$, we argue as follows. Denoting by $S$ an upper bound for
$|\nabla f_{h}|(x_{h,i})$ and $2|\nabla f|(x_{i})$, we apply twice the
construction of Lemma 3.1, with $\delta=\tau$, to $f_{h}$ with endpoints
$x_{h,i}$, $J_{f_{h},\tau}(x_{i})$, to extend the curves $\gamma^{\tau}_{h}$,
still denoted $\gamma_{h}^{\tau}$, to the interval $[-\tau,1+\tau]$, in such a
way that (we use (10) in the first inequality, and the second inequality in
(8) in the second one)

$\displaystyle\limsup_{h\to\infty}\int_{-\delta}^{1+\delta}\left\\{|\dot{\gamma}_{h}^{\tau}|^{2}+|\nabla
f_{h}|^{2}(\gamma_{h}^{\tau})\right\\}\mathrm{d}t$
$\displaystyle\qquad\leq(1+\tau\lambda)^{-2}\int_{0}^{1}\left\\{|\dot{\gamma}|^{2}+|\nabla
f|^{2}(\gamma)\right\\}\mathrm{d}t+420\tau
S^{2}+\frac{52}{\tau}\Big{\\{}|x_{0}-J_{f,\tau}(x_{0})|^{2}+|x_{1}-J_{f,\tau}(x_{1})|^{2}\Big{\\}}$
$\displaystyle\qquad\leq(1+\tau\lambda)^{-2}\left(\int_{0}^{1}\left\\{|\dot{\gamma}|^{2}+|\nabla
f|^{2}(\gamma)\right\\}\mathrm{d}t+472\tau S^{2}\right)$

and the endpoint condition is satisfied at $t=-\tau$ and $t=1+\tau$. The limit
of the curves $\gamma^{\tau}_{h}$ in $[-\tau,1+\tau]$, still denoted
$\gamma^{\tau}$, is the one obtained applying the construction of Lemma 3.1
with $x_{i}$ and $J_{f,\tau}(x_{i})$ in the intervals $[-\tau,0]$ and
$[1,1+\tau]$, and which coincides with $J_{f,\tau}(\gamma(t))$ on $[0,1]$.

By a linear rescaling of the curves $\gamma_{h}^{\tau}$ and $\gamma^{\tau}$ to
$[0,1]$ we obtain curves $\tilde{\gamma}_{h}^{\tau}$ converging to
$\tilde{\gamma}^{\tau}$ in $C([0,1];H)$, with $\tilde{\gamma}^{\tau}$
convergent to $\gamma$ as $\tau\to 0$ and

$\displaystyle\Gamma-\limsup_{h\to\infty}\Theta_{f_{h},x_{h,0},x_{h,1}}(\tilde{\gamma}^{\tau})$
$\displaystyle\leq\limsup_{h\to\infty}\Theta_{f_{h},x_{h,0},x_{h,1}}(\tilde{\gamma}_{\tau}^{h})$
$\displaystyle\leq(1+O(\tau))\int_{0}^{1}\left\\{|\dot{\gamma}|^{2}+|\nabla
f|^{2}(\gamma)\right\\}\mathrm{d}t+O(\tau).$

Eventually, the lower semicontinuity of the $\Gamma$-upper limit and the
convergence of $\tilde{\gamma}^{\tau}$ to $\gamma$ provide:

$\Gamma-\limsup_{h\to\infty}\Theta_{f_{h},x_{h,0},x_{h,1}}(\gamma)\leq\int_{0}^{1}\left\\{|\dot{\gamma}|^{2}+|\nabla
f|^{2}(\gamma)\right\\}\mathrm{d}t.$

∎

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