emptyset degree leftmoon rightmoon fullmoon newmoon diameter bigtimes langle
rangle triangleright triangleleft not otherlangle otherrangle
othertriangleright othertriangleleft othernot otheremptyset 11institutetext:
Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

# Shades of Iteration: from Elgot to Kleene

Sergey Goncharov

###### Abstract

Notions of iteration range from the arguably most general _Elgot iteration_ to
a very specific _Kleene iteration_. The fundamental nature of Elgot iteration
has been extensively explored by Bloom and Esik in the form of _iteration
theories_ , while _Kleene iteration_ became extremely popular as an integral
part of (untyped) formalisms, such as automata theory, regular expressions and
Kleene algebra. Here, we establish a formal connection between Elgot iteration
and Kleene iteration in the form of Elgot monads and Kleene monads,
respectively. We also introduce a novel class of _while-monads_ , which like
Kleene monads admit a relatively simple description in algebraic terms. Like
Elgot monads, while-monads cover a large variety of models that meaningfully
support while-loops, but may fail the Kleene algebra laws, or even fail to
support a Kleen iteration operator altogether.

##  1 Introduction

Iteration is fundamental in many areas of computer science, such as semantics,
verification, theorem proving, automata theory, formal languages,
computability theory, compiler optimisation, etc. An early effort to
identifying a generic notion of iteration is due to Elgot [Elgot75], who
proposed to consider an _algebraic theory_ induced by a notion of abstract
machine (motivated by _Turing machines_ , and their variants) and regard
iteration as an operator over this algebraic theory.

Roughly speaking, an algebraic theory carries composable spaces of morphisms
$L(n,m)$, indexed by natural numbers $n$ and $m$ and including all functions
from $n$ to $m$111Here we identify numbers $n\in\mathbb{N}$ with finite
ordinals $\\{0,\ldots,n-1\\}$., called _base morphisms_. For example,
following Elgot, one can consider as $L(n,m)$ the space of all functions
$n\times S\to m\times S$ representing transitions from a machine state ranging
over $n$ to a machine state ranging over $m$, and updating the background
store over $S$ (e.g. with $S$ being the Turing machine’s tape) in the
meanwhile. In modern speech, $L(n,m)$ is essentially the space of Kleisli
morphisms $n\to Tm$ of the _state monad_ $T=(\operatorname{-\\!-}\times
S)^{S}$. Then a machine over $m$ halting states and $n$ non-halting states is
represented by a morphism in $L(n,{m+n})$, and the iteration operator is meant
to compute a morphism in $L(n,m)$, representing a run of the machine, obtained
by feedbacking all non-halting states. This perspective has been extensively
elaborated by Bloom and Esik [BloomEsik93] who identified the ultimate
equational theory of Elgot iteration together with plenty other examples of
algebraic theories induced by existing semantic models, for which the theory
turned out to be sound and complete.

By replacing natural numbers with arbitrary objects of a category with finite
coproducts and by moving from purely equational to a closely related and
practically appealing quasi-equational theory of iteration, one arrives at
_(complete) Elgot monads_ [AdamekMiliusEtAl10, GoncharovSchroderEtAl18], which
are monads $T$, equipped with an iteration operator

$\displaystyle\textbf{}~{}~{}\frac{f\colon X\to T(Y+X)}{f^{\dagger}\colon X\to
TY}$ ($\dagger$)

In view of the connection between computational effects and monads, pioneered
by Moggi [Moggi91], Elgot monads provide arguably the most general model of
iteration w.r.t. functions carrying computational effects, such as mutable
store, non-determinism, probability, exceptional and abnormal termination,
input-output actions of process algebra. The standard way of semantics via
_domain theory_ yields a general (least) fixpoint operator, which sidelines
Elgot iteration and overshadows its fundamental role. This role becomes
material again when it comes to the cases when the standard scenario cannot be
applied or is difficult to apply, e.g. in constructive setting [Goncharov21],
for deterministic hybrid system semantics [GoncharovJakobEtAl18], and infinite
trace semantics [LevyGoncharov19].

In contrast to Elgot iteration, _Kleene iteration_ , manifested by _Kleene
algebra_ , is rooted in logic and automata theory [Kleene56], and crucially
relies on non-determinism. The laws of Kleene algebra are from the outset
determined by a rather conservative observation model, describing discrete
events, coming one after another in linear order and in finite quantities.
Nevertheless, Kleene algebra and thus Kleene iteration proved to be extremely
successful (especially after the celebrated complete algebraic axiomatization
of Kleene algebra by Kozen [Kozen94]) and have been accommodated in various
formalizations and verification frameworks from those for concurrency
[HoareMollerEtAl09] to those for modelling hybrid systems [Platzer08]. A
significant competitive advantage of Kleene iteration is that it needs no
(even very rudimental) type grammar for governing well-definedness of
syntactic constructs, although this cannot be avoided when extending Kleene
algebra with standard programming features [Kozen02, Aboul-HosnKozen06,
KozenMamouras13]. Semantically, just as Elgot iteration, Kleene iteration can
be reconciled with computational effects, leading to _Kleene monads_
[Goncharov10], which postulate Kleene iteration with the type profile:

$\displaystyle\textbf{}~{}~{}\frac{f\colon X\to TX}{f^{\ast}\colon X\to TX}$
($\ast$)

Given $f$, $f^{\ast}$ self-composes it non-deterministically indefinitely many
times. In contrast to Elgot monads, the stock of computational effects
modelled by Kleene monads is rather limited, which is due to the fact that
many computational effects are subject to laws, which contradict the Kleene
algebra laws. For a simple example, consider the computational effect of
exception raising, constrained by the law, stating that postcomposing an
exception raising program by another program is ineffective. Together with the
Kleene algebra laws, we obtain a havoc:

$\displaystyle\operatorname{\mathsf{raise}}e_{1}=\operatorname{\mathsf{raise}}e_{1};\
\bot=\bot=\operatorname{\mathsf{raise}}e_{2};\
\bot=\operatorname{\mathsf{raise}}e_{2},$

where $\bot$ is the unit of non-deterministic choice. This and similar issues
led to a number of proposals to weaken Kleene algebra laws [FokkinkZantema94,
Moller07, McIverRabehajaEtAl11, GomesMadeiraEtAl17] (potentially leading to
other classes of monads, somewhere between Elgot and Kleene), although not
attempting to identify the weakest set of such laws from the foundational
perspective. At the same time, it seems undebatable that Kleene iteration and
the Kleene algebra laws yield the most restricted notion of iteration.

We thus obtain a spectrum of potential notions of iteration between Elgot
monads and Kleene monads. The goal of the present work is, on the one hand to
explore this spectrum, and on the other hand to contribute into closing the
conceptual gap between Kleene iteration and Elgot iteration. To that end, we
introduce _while-monads_ , which capture iteration in the conventional form of
while-loops. Somewhat surprisingly, despite extensive work on axiomatizing
iteration in terms of ($\dagger$ ‣ 1), a corresponding generic axiomatization
in terms of “while” did not seem to be available. We highlight the following
main technical contributions of the present work:

  * $\bullet$

We provide a novel axiomatization of Kleene algebra laws, which is effective
both for Kleene algebras and Kleene monads (2);

  * $\bullet$

We show that the existing axiomatization of Elgot monads is minimal
(LABEL:pro:mini);

  * $\bullet$

We establish a connection between Elgot monads and while-monads
(LABEL:thm:main);

  * $\bullet$

We render Kleene monads as Elgot monads with additional properties
(LABEL:thm:elg-klee).

##  2 Preliminaries

We rely on rudimentary notions and facts of category theory, as used in
semantics, most notably _monads_ [Awodey10]. For a (locally small) category
$\mathbf{C}$ we denote by $|\mathbf{C}|$ the class of its objects and by
$\mathbf{C}(X,Y)$ the set of morphisms from $X\in|\mathbf{C}|$ to
$Y\in|\mathbf{C}|$. We often omit indices at components of natural
transformations to avoid clutter. $\mathbf{Set}$ will denote the category of
classical sets and functions, i.e. sets and functions formalized in a
classical logic with the law of excluded middle (we will make no use of the
axiom of choice). By $\langle f,g\rangle\colon X\to Y\times Z$ we will denote
the _pairing_ of two morphisms $f\colon X\to Y$ and $g\colon X\to Z$ (in a
category with binary products), and dually, by $[f,g]\colon X+Y\to Z$ we will
denote the _copairing_ of $f\colon X\to Z$ and $g\colon Y\to Z$ (in a category
with binary coproducts). By $\operatorname{!}\colon X\to 1$ we will denote
terminal morphisms (if $1$ is an terminal object).

An ($F$-)algebra for an endofunctor $F\colon\mathbf{C}\to\mathbf{C}$ is a pair
$(A,a\colon FA\to A)$. Algebras form a category under the following notion of
morphism: $f\colon A\to B$ if a morphism from $(A,a)$ to $(B,b)$ if
$bf=(Ff)\,a$. The _initial algebra_ is an initial object of this category
(which may or may not exit). We denote this object $(\mu
F,\operatorname{\mathsf{in}})$. ($F$-)coalgebras are defined dually as pairs
of the form $(A,\operatorname{}\linebreak[1]a\colon{A\to FA})$. The final
coalgebra will be denoted $(\nu
F,\operatorname{}\linebreak[1]\operatorname{\mathsf{out}})$. By Lambek’s Lemma
[Lambek68], both $\operatorname{\mathsf{in}}$ and
$\operatorname{\mathsf{out}}$ are isomorphisms, and we commonly make use of
their inverses $\operatorname{\mathsf{in}}^{\text{\kern
0.5pt-}\mathsf{\kern-0.5pt1}}$ and $\operatorname{\mathsf{out}}^{\text{\kern
0.5pt-}\mathsf{\kern-0.5pt1}}$.

##  3 Monads for Computation

We work with monads represented by _Kleisli triples_
$(T,\eta,(\operatorname{-\\!-})^{\sharp})$ where $T$ is a map
$|\mathbf{C}|\to|\mathbf{C}|$, $\eta$ is the family $(\eta_{X}\colon X\to
TX)_{X\in|\mathbf{C}|}$ and $(\operatorname{-\\!-})^{\sharp}$ sends $f\colon
X\to TY$ to $f^{\sharp}\colon TX\to TY$ in such a way that the standard _monad
laws_

$\displaystyle\eta^{\sharp}=\mathsf{id},$ $\displaystyle f^{\sharp}\eta=f,$
$\displaystyle(f^{\sharp}g)^{\sharp}=f^{\sharp}g^{\sharp}$

hold true. It is then provable that $T$ extends to a functor with $Tf=(\eta
f)^{\sharp}$ and $\eta$ to a _unit_ natural transformation. Additionally, we
can define the _multiplication_ natural transformation $\mu\colon TT\to T$
with $\mu_{X}=\mathsf{id}^{\sharp}$ (thus extending $T$ to a monoid in the
category of endufunctors). We preferably use bold letters, e.g.
${\bm{\mathsf{T}}}$, for monads, to contrast with the underlying functor $T$.
The axioms of monads entail that the morphisms of the form $X\to TY$ determine
a category, called _Kleisli category_ , and denoted
$\mathbf{C}_{{\bm{\mathsf{T}}}}$, under _Kleisli composition_ $f\cdot
g=f^{\sharp}\,g$ with $\eta$ as the identity morphism. Intuitively, Kleisli
category is the category of (generalized) effectful programs w.r.t.
$\mathbf{C}$ as the category of “pure”, or effectless, programs. More
precisely, we will call _pure_ those morphisms in
$\mathbf{C}_{{\bm{\mathsf{T}}}}$ that are of the form $\eta\,f$. We thus use
diagrammatic composition $g;f$ alongside and equivalently to functional
composition $f\cdot g$, as the former fits with the sequential composition
operators of traditional programming languages.

A monad ${\bm{\mathsf{T}}}$ is _strong_ if it comes with a natural
transformation $\tau_{X,Y}\colon X\times TY\to T(X\times Y)$ called _strength_
and satisfying a number of coherence conditions [Moggi91]. Any monad on
$\mathbf{Set}$ is canonically strong [Kock72].

######  Example 1 (Monads)

Recall some computationally relevant monads on $\mathbf{Set}$ (all monads on
$\mathbf{Set}$ are strong [Moggi91]).

  1. 1.

_Maybe-monad_ : $TX=X+1$, $\eta(x)=\operatorname{\mathsf{inl}}x$,
$f^{\sharp}(\operatorname{\mathsf{inl}}x)=f(x)$,
${f^{\sharp}(\operatorname{\mathsf{inr}}\star)=\operatorname{\mathsf{inr}}\star}$.

  2. 2.

_Powerset monad_ : $TX={\mathcal{P}}X$, $\eta(x)=\\{x\\}$,
$f^{\sharp}(S\subseteq X)=\\{y\in f(x)\mid x\in S\\}$.

  3. 3.

$TX=\\{S\mid S\in{\mathcal{P}}^{\scriptscriptstyle\mathbf{+}}(X+1),\text{if
$S$ is infinite then~{}}\operatorname{\mathsf{inr}}\star\in S\\}$ where
${\mathcal{P}}^{\scriptscriptstyle\mathbf{+}}$ is the non-empty powerset
functor, ${\eta(x)=\\{\operatorname{\mathsf{inl}}x\\}}$,
$f^{\sharp}(S\subseteq X)=\\{y\in f(x)\mid{\operatorname{\mathsf{inl}}x\in
S}\\}\cup(\\{\operatorname{\mathsf{inr}}\star\\}\cap S)$.

  4. 4.

_Exception monad_ : $TX=X+E$ where $E$ is a fixed (unstructured) non-empty set
of exceptions, $\eta(x)=\operatorname{\mathsf{inl}}x$,
$f^{\sharp}(\operatorname{\mathsf{inl}}x)=f(x)$,
${f^{\sharp}(\operatorname{\mathsf{inr}}e)=\operatorname{\mathsf{inr}}e}$.

  5. 5.

_Non-deterministic writer monad_ : $TX={\mathcal{P}}(M\times X)$ where
$(M,\operatorname{}\linebreak[1]{\operatorname{\epsilon}},\operatorname{}\linebreak[1]\bullet)$
is any monoid, $\eta(x)=\\{(e,x)\\}$, $f^{\sharp}(S\subseteq M\times
X)=\\{(n\bullet m,y)\mid(m,x)\in S,\operatorname{}\linebreak[1](n,y)\in
f(x)\\}$.

  6. 6.

_Discrete sub-distribution monad_ : $TX={\\{d\colon[0,1]\to X\mid\sum_{x\in
X}d(x)\leq 1\\}}$ (the _supports_ of $d$, $\\{x\in X\mid d(x)>0\\}$ are
necessarily countable – otherwise the sum $\sum_{x\in X}d(x)$ would diverge),
$\eta(x)$ is the _Dirac distribution_ $\delta_{x}$, centred in $x$, i.e.
$\delta_{x}(y)=1$ if $x=y$, $\delta_{x}(y)=0$ otherwise, $({f\colon
X\to\mathcal{D}Y)^{\sharp}}(d)(y)={\sum_{x\in X}f(x)(y)\cdot d(x)}$.

  7. 7.

_Partial state monad:_ $TX=(X\times S+1)^{S}$, where $S$ is a fixed set of
global states, $\eta(x)(s)=\operatorname{\mathsf{inl}}(x,s)$,
$f^{\sharp}(g\colon S\to X\times S+1)(s)=\operatorname{\mathsf{inr}}\star$ if
$g(s)=\operatorname{\mathsf{inr}}\star$ and $f^{\sharp}(g\colon S\to Y\times
S+1)(s)=f(x)(s^{\prime})$ if $g(s)=\operatorname{\mathsf{inl}}(x,s^{\prime})$.

  8. 8.

_Partial interactive input:_ $TX=\nu\gamma.\,((X+\gamma^{I})+1)$, where $I$ is
a set of input values, $\eta(x)=\operatorname{\mathsf{out}}^{\text{\kern
0.5pt-}\mathsf{\kern-0.5pt1}}(\operatorname{\mathsf{inl}}\operatorname{\mathsf{inl}}x)$,
$(f\colon X\to TY)^{\sharp}$ is the unique such morphism $f^{\sharp}\colon
TX\to TY$ that (eliding the isomorphisms
$T\mathbin{\cong}(\operatorname{-\\!-}+T^{I})+1$)

$\displaystyle
f^{\sharp}(\operatorname{\mathsf{inl}}\operatorname{\mathsf{inl}}x)=\;$
$\displaystyle f(x),$ $\displaystyle
f^{\sharp}(\operatorname{\mathsf{inl}}\operatorname{\mathsf{inr}}h)=\;$
$\displaystyle\operatorname{\mathsf{inl}}\operatorname{\mathsf{inr}}(f^{\sharp}\,h),$
$\displaystyle f^{\sharp}(\operatorname{\mathsf{inr}}\star)=\;$
$\displaystyle\operatorname{\mathsf{inr}}\star.$

Intuitively, $p\in TX$ is a computation that either finishes and gives a
result in $X$, or takes an input from $I$ and continues recursively, or
(unproductively) diverges.

  9. 9.

_Partial interactive output:_ $TX=\nu\gamma.\,((X+\gamma\times O)+1)$, where
$O$ is a set of output values,
$\eta(x)=\operatorname{\mathsf{out}}^{\text{\kern
0.5pt-}\mathsf{\kern-0.5pt1}}(\operatorname{\mathsf{inl}}\operatorname{\mathsf{inl}}x)$,
$(f\colon X\to TY)^{\sharp}$ is the unique such morphism $f^{\sharp}\colon
TX\to TY$ that (eliding the isomorphisms
$T\mathbin{\cong}(\operatorname{-\\!-}+T\times O)+1$)

$\displaystyle
f^{\sharp}(\operatorname{\mathsf{inl}}\operatorname{\mathsf{inl}}x)=\;$
$\displaystyle f(x),$ $\displaystyle
f^{\sharp}(\operatorname{\mathsf{inl}}\operatorname{\mathsf{inr}}(p,o))=\;$
$\displaystyle\operatorname{\mathsf{inl}}\operatorname{\mathsf{inr}}(f^{\sharp}(p),o),$
$\displaystyle f^{\sharp}(\operatorname{\mathsf{inr}}\star)=\;$
$\displaystyle\operatorname{\mathsf{inr}}\star.$

The behaviour of $p\in TX$ is as in the previous case, except that it outputs
to $O$ instead of expecting an input from $I$ in the relevant branch.

Kleisli categories are often equivalent to categories with more familiar
independent descriptions. For example, the Kleisli category of the maybe-monad
is equivalent to the category of partial functions and the Kleisli category of
the powerset monad is equivalent to the category of relations. Under the
monads-as-effects metaphor, partial functions can thus be regarded as possibly
non-terminating functions and relations as non-deterministic functions.

The above examples can often be combined. E.g. non-deterministic stateful
computations are obtained as $TA=S\to{\mathcal{P}}(A\times S)$. The Java monad
of [JacobsPoll03],

$TX=S\to(X\times S+E\times S)+1$

with $S$ the set of states and $E$ the set of exceptions.

##  4 Kleene Monads

Idempotent semiring laws: idempotence: $\displaystyle f\lor f=f$
commutativity: $\displaystyle f\lor g=g\lor f$ neutrality of $\bot$:
$\displaystyle f\lor\bot=f$ associativity of $\lor$: $\displaystyle
f\lor(g\lor h)=(f\lor g)\lor h$ associativity of ; : $\displaystyle
f;(g;h)=(f;g);h$ right strictness: $\displaystyle f;\bot=\bot$ right
neutrality of $\eta$: $\displaystyle f;\eta=f$ right distributivity:
$\displaystyle(f\lor g);h=f;h\lor g;h\hskip 70.0001pt$ left strictness:
$\displaystyle\bot;f=\bot$ left neutrality of $\eta$: $\displaystyle\eta;f=f$
left distributivity: $\displaystyle f;(g\lor h)=f;g\lor f;h$ Iteration laws:
right unfolding: $\displaystyle f^{\ast}=\eta\lor f;f^{\ast}$ right induction:
$\displaystyle f;g\leq f\implies f;g^{\ast}\leq f$ left unfolding:
$\displaystyle f^{\ast}=\eta\lor f^{\ast};f$ left induction: $\displaystyle
f;g\leq g\implies f^{\ast};g\leq g$ Figure 1: Axioms Kleene algebras/monads.

A Kleene algebra can be concisely defined as an idempotent semiring
$(S,\operatorname{}\linebreak[1]\bot,\operatorname{}\linebreak[1]\eta,\operatorname{}\linebreak[1]\lor,\operatorname{}\linebreak[1]\,;\,)$
equipped with an operator $(\operatorname{-\\!-})^{\ast}\colon S\to S$, such
that

  * $\bullet$

$g;f^{\ast}$ is the least (pre-)fixpoint of $g\lor(\operatorname{-\\!-});f$,

  * $\bullet$

$f^{\ast};h$ is the least fixpoint of $h\lor f;(\operatorname{-\\!-})$,

where the order is induced by $\lor$: $f\leq g$ if $f\lor g=g$. We assume here
and henceforth that sequential composition ; binds stronger than $\lor$.

More concretely, a Kleene algebra
$(S,\operatorname{}\linebreak[1]\bot,\operatorname{}\linebreak[1]\eta,\operatorname{}\linebreak[1]\lor,\operatorname{}\linebreak[1]\,;\,,(\operatorname{-\\!-})^{\ast})$
is an algebraic structure, satisfying the laws in Figure 1. A categorical
version of Kleene algebra emerges as a class of monads, called  _Kleene
monads_ [GoncharovSchroderEtAl09], which can be used for interpreting
effectful languages with iteration and non-determinism.

######  Definition 1 (Kleene-Kozen Category/Kleene Monad)

We say that a category $\mathbf{C}$ is a _Kleene-Kozen category_ if
$\mathbf{C}$ is enriched over bounded (i.e. possessing a least element) join-
semilattices and strict join-preserving morphisms, and there is _Kleene
iteration_ operator

$(\operatorname{-\\!-})^{\ast}\colon\mathbf{C}(X,X)\to\mathbf{C}(X,X),$

such that, given $f\colon Y\to Y$, $g\colon Y\to Z$ and ${h\colon X\to Y}$,
$g\,f^{\ast}$ is the least (pre-)fixpoint of $g\lor(\operatorname{-\\!-})\,f$
and $f^{\ast}\,h$ is the least (pre-)fixpoint of $h\lor
f\,(\operatorname{-\\!-})$.

A monad ${\bm{\mathsf{T}}}$ is a _Kleene monad_ if
$\mathbf{C}_{{\bm{\mathsf{T}}}}$ is a Kleene-Kozen category.

Recall that a monoid is nothing but a single-object category, whose morphisms
are identified with monoid elements, and whose identity morphisms and morphism
composition are identified with monoidal unit and composition. This suggests a
connection between Kleene-Kozen categories and Kleene algebras.

###### Proposition 1

A Kleene algebra is precisely a Kleene-Kozen category with one object.

We record the following characterization of Kleene-Kozen categories (hence,
also of Kleene algebras by 1).

###### Proposition 2

A category $\mathbf{C}$ is a Kleene-Kozen category iff

  * $\bullet$

$\mathbf{C}$ is enriched over bounded join-semilattices and strict join-
preserving morphisms;

  * $\bullet$

there is an operator
$(\operatorname{-\\!-})^{\ast}\colon\mathbf{C}(X,X)\to\mathbf{C}(X,X)$, such
that

    1. 1.

$f^{\ast}=\mathsf{id}\lor f^{\ast}\,f$;

    2. 2.

$\mathsf{id}^{\ast}=\mathsf{id}$;

    3. 3.

$f^{\ast}=(f\lor\mathsf{id})^{\ast}$;

    4. 4.

$h\,f=f\,g$ implies $h^{\ast}\,f=f\,g^{\ast}$.

###### Proof

Let us show _necessity_.

  1. 1.

The law $f^{\ast}=\mathsf{id}\lor f^{\ast}\,f$ holds by assumption.

  2. 2.

Since $\mathsf{id}=\mathsf{id}\lor\mathsf{id}\,\mathsf{id}$, $\mathsf{id}$ is
a fixpoint of $f\mapsto\mathsf{id}\lor f\,\mathsf{id}$, and thus
$\mathsf{id}^{\ast}\leq\mathsf{id}$. Also
$\mathsf{id}^{\ast}=\mathsf{id}\lor\mathsf{id}^{\ast}\,\mathsf{id}\geq\mathsf{id}$.
Hence $\mathsf{id}=\mathsf{id}^{\ast}$ by mutual inequality.

  3. 3.

To show that $(f\lor\mathsf{id})^{\ast}\leq f^{\ast}$, note that
$f^{\ast}=\mathsf{id}\lor f^{\ast}\,f=\mathsf{id}\lor f^{\ast}\,f\lor
f^{\ast}=\mathsf{id}\lor f^{\ast}\,(f\lor\mathsf{id})$, and use the fact that
$(f\lor\mathsf{id})^{\ast}$ is the least fixpoint. The opposite inequality is
shown analogously, by exploiting the fact that $f^{\ast}$ is a least fixpoint.

  4. 4.

Suppose that $h\,f=f\,g$, and show that $h^{\ast}\,f=f\,g^{\ast}$. Note that

$f\lor(h^{\ast}\,f)\,g=f\lor h^{\ast}\,f\,g=f\lor h^{\ast}\,h\,f=h^{\ast}\,f,$

i.e. $h^{\ast}\,f$ satisfies the fixpoint equation for $f\,g^{\ast}$, and
therefore $h^{\ast}\,f\geq f\,g^{\ast}$. By a symmetric argument,
$h^{\ast}\,f\leq f\,g^{\ast}$, hence $h^{\ast}\,f=f\,g^{\ast}$.

We proceed with _sufficiency_. Suppose that $(\operatorname{-\\!-})^{\ast}$ is
as described in the second clause of the present proposition. Observe that by
combining assumptions 1 and 4 we immediately obtain the dual version of 1,
which is $f^{\ast}=\mathsf{id}\lor f\,f^{\ast}$. Now, fix $f\colon Y\to TY$
and $g\colon Y\to TZ$, and show that $f^{\ast}\,g$ is the least fixpoint of
$g\lor f\,(\operatorname{-\\!-})$ – we omit proving the dual property, since
it follows by a dual argument. From $f^{\ast}=\mathsf{id}\lor f^{\ast}\,f$ we
obtain $f^{\ast}\,g=g\lor f\,(f^{\ast}\,g)$, i.e. $f^{\ast}\,g$ is a fixpoint.
We are left to show that it is the least one. Suppose that $h=g\lor f\,h$ for
some $h$, which entails

$(\mathsf{id}\lor f)\,h=h\lor f\,h=g\lor f\,h\lor f\,h=h=h\,\mathsf{id}.$

Since $g\leq h$, using assumptions 2, 3 and 4, we obtain

$f^{\ast}\,g\leq(\mathsf{id}\lor f)^{\ast}\,g\leq(\mathsf{id}\lor
f)^{\ast}\,h=h\,\mathsf{id}^{\ast}=h,$

as desired.

The axioms of Kleene monads do not in fact need a monad, and can be
interpreted in any category. We focus on Kleisli categories for two reasons:
(i) in practice, Kleene-Kozen categories are often realized as Kleisli
categories, and monads provide a compositional mechanism for constructing more
Kleene-Kozen categories by generalities; (ii) we will relate Kleene monads and
Elgot monads, and the latter are defined by axioms, which do involve both
general Kleisli morphisms and the morphisms of the base category.

###### Example 2

Let us revisit 1. Many monads therein fail to be Kleene simply because they
fail to support binary non-determinism. 1.6 is an interesting case, since we
can define the operation of probabilistic choice
$+_{p}\colon\mathbf{C}(X,TY)\times\mathbf{C}(X,TY)\to\mathbf{C}(X,TY)$ indexed
by $p\in[0,1]$, meaning that $x+_{p}y$ is resolved to $x$ with probability $p$
and to $y$ with probability $1-p$. For every ${x\in X}$, $f(x)+_{p}g(x)$ is a
_convex sum_ of the distributions $f(x)$ and $g(x)$. This operation satisfies
the axioms of _barycentric algebras_ (or, _abstract convex sets_ [Stone49]),
which are somewhat similar to those of a monoid, but with the multiplication
operator indexed over $[0,1]$. To get rid of this indexing, one can remove the
requirement that probabilities sum up to at most $1$ and thus obtain spaces of
_valuations_ [VaraccaWinskel06] instead of probability distributions.
Valuations can be conveniently added pointwise, and thus defined addition
satisfies monoidal laws, but fails idempotence, hence still does not yield a
Kleene monad. Given two valuations $v$ and $w$, we also can define $v\lor w$
as the pointwise maximum. This satisfies the axioms of semilattices, but fails
both distributivity laws.

1.3 is the $\mathbf{Set}$-reduct or _Plotkin powerdomain_ [Plotkin76] over a
flat domain. It supports proper non-deterministic choice, but the only
candidate for $\bot$ is not a unit for it.

Kleene monads of 1 are only 2 and 5. The non-deterministic state monad over
$TX=({\mathcal{P}}(X\times S))^{S}$ obtained by adapting 1.7 in the obvious
way is also Kleene.

Except for the powerset monad, our examples of Kleene monads are in fact
obtained by generic patterns.

###### Proposition 3

Let ${\bm{\mathsf{T}}}$ be a Kleene monad. Then so are

  1. 1.

the state monad transformer $(T(\operatorname{-\\!-}\times S))^{S}$ for every
$S$;

  2. 2.

the writer monad transformer $T(M\times\operatorname{-\\!-})$ for every monoid
$(M,{\operatorname{\epsilon}},\bullet)$ if ${\bm{\mathsf{T}}}$ is strong and
strength $\tau_{X,Y}\colon X\times TX\to T(X\times Y)$ respects the Kleene
monad structure, as follows:

$\displaystyle\begin{gathered}\tau\,(\mathsf{id}\times\bot)=\bot,\qquad\tau\,(\mathsf{id}\times
f^{\ast})=(\tau\,(\mathsf{id}\times f))^{\ast},\\\\[4.30554pt]
\tau\,(\mathsf{id}\times(f\lor g))=\tau\,(\mathsf{id}\times
f)\lor\tau\,(\mathsf{id}\times g).\end{gathered}$ (3)

###### Proof

  1. 1.

By definition, the Kleisli category of the state transformer is equivalent to
the full subcategory of $\mathbf{C}_{{\bm{\mathsf{T}}}}$ over the objects of
the form $X\times S$ (using the isomorphism $\mathbf{C}(X,(T(Y\times
S))^{S})\mathbin{\cong}\mathbf{C}(X\times S,T(Y\times S))$). The enrichment,
the iteration operator and the axioms are clearly restricted along the induced
inclusion functor.

  2. 2.

The semilattice structure for every $\mathbf{C}(X,T(M\times Y))$ is inherited
from $\mathbf{C}_{{\bm{\mathsf{T}}}}$, but to show enrichment, the strictness
and the distributivity laws must be verified manually. For every $f\colon X\to
T(M\times Y)$, let $f^{\circ}\colon M\times X\to T(M\times Y)$ be as follows

$\displaystyle M\times X\,\xrightarrow{\tau\,(\mathsf{id}\times
f)}\,T(M\times(M\times Y))\,\mathbin{\cong}\,T((M\times M)\times
Y)\,\xrightarrow{T(\bullet\times\mathsf{id})}\,T(M\times Y).$

The assumptions (3) entail the following identities:


$\sum_{j=1}^{r}\epsilon_{j}k_{j}=k,\quad\bigg{|}\sum_{j=1}^{r}\epsilon_{j}|k_{j}|^{2}-\beta\bigg{|}\leq\delta^{-1}L^{-2\gamma},$
(A.23)

where $k\in\mathbb{Z}_{L}^{d}$ and $\beta\in\mathbb{R}$ are fixed parameters.
Let the number of solutions to this system be $\mathfrak{E}_{r}$, then we have

$\mathfrak{E}_{2}\lesssim_{1}\delta^{-1}\min(L^{d},L^{d+1-2\gamma}),\quad\,\,\mathfrak{E}_{2}\lesssim_{1}\delta^{-1}L^{d-\gamma-\eta}\,\,\mathrm{if}\,\,|r|\geq
L^{-\gamma+\eta};\quad\,\,\mathfrak{E}_{3}\lesssim_{1}\delta^{-1}L^{2(d-\gamma)}.$
(A.24)

Moreover, suppose we have four vector variables $k_{j}\,(1\leq j\leq 4)$ as
above, that (A.23) with $r=3$ holds for $(k_{1},k_{2},k_{3})$, and that the
same equation holds also for the three vectors $(k_{1},k_{2},k_{4})$ with the
same $\epsilon_{j}$ and possibly different $(k,\beta)$, then the number of
solutions is bounded by

$\mathfrak{E}_{4}\lesssim_{1}\delta^{-1}L^{2(d-\gamma)}\mathfrak{X}^{-1},$
(A.25)

where $\mathfrak{X}$ is defined as in Proposition 10.5 from $P$, and
$P\sim_{0}|k_{3}-k_{4}|$.

###### Proof.

This follows from (parts of) Lemma A.3 of [18]. ∎

##  Appendix B Failure of a physical space $L^{1}$ bound

Consider the reduced Fourier coefficients $a_{k}(t)$ in (3.2). In the Gibbs
measure case of [48], these coefficients are not independent, but their
cumulants satisfy the (discrete analog of) physical space $L^{1}$ bound,
namely

$\int_{(\mathbb{T}_{L}^{d})^{3}}\big{|}\mathbb{K}\big{(}\check{a}(0,t),\overline{\check{a}(x_{1},t)},\check{a}(x_{2},t),\overline{\check{a}(x_{3},t)}\big{)}\big{|}\,\mathrm{d}x_{1}\mathrm{d}x_{2}\mathrm{d}x_{3}\leq
C_{0}L^{-\gamma}$ (B.1)

for $t=0$ (and any $t>0$ due to invariance), where $\check{a}(x,t)$ is the
inverse Fourier transform of $a$ as defined in (1.1). The bound (B.1) is
crucial in the derivation in [48], see the discussions in Section 1.4; however
it is not true for $t>0$, in the off-equilibrium setting in this paper. To see
this, just choose $t=1$, expand $a_{k}(1)$ into Taylor series consisting of
multilinear expressions of $a_{k}(0)$, and take lowest order terms. By using
(3.3)–(3.4) and Lemma A.1, we get that the cumulant
$\mathbb{K}\big{(}a_{k_{1}}(1),\overline{a_{k_{2}}(1)},a_{k_{3}}(1),\overline{a_{k_{4}}(1)}\big{)}$
contains a leading contribution

$\mathbb{K}_{0}(k_{1},k_{2},k_{3},k_{4}):=\frac{-\delta
i}{2L^{d-\gamma}}\mathbf{1}_{k_{1}-k_{2}+k_{3}-k_{4}=0}\int_{0}^{1}e^{-\pi
i\cdot\delta L^{2\gamma}\Omega s}\,\mathrm{d}s\\\
\times\varphi_{\mathrm{in}}(k_{1})\varphi_{\mathrm{in}}(k_{2})\varphi_{\mathrm{in}}(k_{3})\varphi_{\mathrm{in}}(k_{4})\bigg{(}\frac{1}{\varphi_{\mathrm{in}}(k_{1})}-\frac{1}{\varphi_{\mathrm{in}}(k_{2})}+\frac{1}{\varphi_{\mathrm{in}}(k_{3})}-\frac{1}{\varphi_{\mathrm{in}}(k_{4})}\bigg{)},$
(B.2)

where

$\Omega:=|k_{1}|^{2}-|k_{2}|^{2}+|k_{3}|^{2}-|k_{4}|^{2}$

and we have omitted the $\epsilon$ factor in (3.5) (as (B.2) vanishes if
$\epsilon\neq 1$). Note the other contributions contain either higher powers
of $\delta$ or more negative powers of $L$, so we may ignore these other terms
and replace $\mathbb{K}$ by $\mathbb{K}_{0}$.

Now, choose $k_{j}^{*}$ such that
$k_{1}^{*}-k_{2}^{*}+k_{2}^{*}-k_{4}^{*}=|k_{1}^{*}|^{2}-|k_{2}^{*}|^{2}+|k_{3}^{*}|^{2}-|k_{4}^{*}|^{2}=0$,
but

$\frac{1}{\varphi_{\mathrm{in}}(k_{1}^{*})}-\frac{1}{\varphi_{\mathrm{in}}(k_{2}^{*})}+\frac{1}{\varphi_{\mathrm{in}}(k_{3}^{*})}-\frac{1}{\varphi_{\mathrm{in}}(k_{4}^{*})}\neq
0;$

this is possible due to the off-equilibrium setting. Using the definition of
$\check{a}$, we can calculate

$\int_{(\mathbb{T}_{L}^{d})^{3}}\mathbb{K}\big{(}\check{a}(0,t),\overline{\check{a}(x_{1},t)},\check{a}(x_{2},t),\overline{\check{a}(x_{3},t)}\big{)}\cdot
e^{-2\pi i(k_{2}^{*}\cdot x_{1}-k_{3}^{*}\cdot x_{2}+k_{4}^{*}\cdot
x_{3})}\,\mathrm{d}x_{1}\mathrm{d}x_{2}\mathrm{d}x_{3}\\\
=L^{d}\sum_{k_{1}}\mathbb{K}_{0}(k_{1},k_{2}^{*},k_{3}^{*},k_{4}^{*})=L^{d}\mathbb{K}_{0}(k_{1}^{*},k_{2}^{*},k_{3}^{*},k_{4}^{*})\sim
L^{\gamma},$ (B.3)

which means that (B.1) cannot be true.

##  Appendix C Table of notations

In this appendix we collect some important notations used in this paper. Table
1 contains notations about trees, couples and gardens. Table 2 contains
notations about molecules. Table 3 contains notations about layerings and
others, which are specific to this paper.

custom-line = letter = I , tikz = line width=0.08em, width=0.08em , Il c lI
Concept Symbol Where defined

Tree $\mathcal{T}$ Definition 3.3

Root, node, leaf $\mathfrak{r},\mathfrak{n},\mathfrak{l}$ Definition 3.3

Couple, garden $\mathcal{Q},\mathcal{G}$Definition 3.4

Paired tree — Definition 3.4

Lone leaf $\mathfrak{l}_{\mathrm{lo}}$ Definition 3.4

Leaf set $\mathcal{L}$ Definitions 3.3, 3.4

Branching node set$\mathcal{N}$ Definitions 3.3, 3.4

Order $n(\mathcal{T})$ etc. Definitions 3.3, 3.4

Decoration (trees etc.) $\mathscr{D},\mathscr{I}$ Definition 3.5

Regular couple, regular tree — Definition 4.1

Dominant couple — Proposition 4.4

Prime couple — Proposition 4.5

Skeleton $\mathcal{G}_{\mathrm{sk}}$ Proposition 4.5

Table 1. Notations about trees, couples and gardens.

Il c lI Concept Symbol Where defined

Molecule, atom, bond $\mathbb{M},v,\ell$ Definition 7.1

Circular rank $\chi$ Definition 7.1

Molecule from garden $\mathbb{M}(\mathcal{G})$ Definition 7.2

Decoration (molecules) — Definition 7.4

Block, realization $\mathbb{B},\mathcal{G}[\mathbb{B}]$ Definition 7.5, Remark
7.9

(CL) and (CN) blocks, splicing — Proposition 7.7

Vines (I)–(VIII), ladder $\mathbb{V},\mathbb{L}$ Definition 7.11

HV, VC, HVC, DV, vine-like object $\mathbb{U}$ Definition 7.11, Lemma 10.1

Vines (I-a), (II-b) etc. — Proposition 7.12

Twist — Definition 7.14

Gap, SG, LG, ZG $r$ Definition 10.3

(CL) vines set, result after splicing
$\mathscr{V},\mathscr{V}_{0},\mathcal{G}_{\mathrm{sb}}$ Definition 10.2,
Section 10.2

Degenerate and tame atoms — Proposition 10.6 (3)

Cut, $\alpha$\- and $\beta$-atoms — Definition 11.1

Table 2. Notations about molecules.

Il c lI Concept Symbol Where defined

Cumulant $\mathbb{K}$ Definition 3.1

Layering, pre-layering $\mathfrak{L}[\mathcal{G}]$ Definition 3.6

Canonical layered garden $\mathscr{G}_{p}$ Definition 3.7

Canonical layered couple $\mathscr{C}_{p}$ Definition 3.7

Expression for layered objects $\mathcal{K}_{\mathcal{G}}$ etc. Definition 3.8

Coherent regular objects — Definition 4.7

Incoherency index (regular objects) $g$ Definition 4.7

Layered full (LF) twist — Definition 7.18

Coherent vines/ladders — Definition 8.1

Incoherency index (vines/ladders) $g$ Definition 8.1

Expression for (CL) vine chains $\mathcal{K}^{\mathbb{U}}$,
$\widetilde{\mathcal{K}}^{\mathbb{U}}$ etc. Proposition 9.5

Table 3. Notations about layerings and others.

## References

  * [1] I. Ampatzoglou, C. Collot and P. Germain. Derivation of the kinetic wave equation for quadratic dispersive problems in the inhomogeneous setting. arXiv:2107.11819.
  * [2] B. J. Benney and P. Saffman. Nonlinear interaction of random waves in a dispersive medium. _Proc. R. Soc. A_ 289 (1966), 301–320.
  * [3] B. J. Benney and A. C. Newell. Random wave closures. _Studies. Appl. Math._ 48 (1969), 29.
  * [4] T. Bodineau, I. Gallagher, L. Saint-Raymond and S. Simonella. Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations. arXiv:2008.10403.
  * [5] T. Bodineau, I. Gallagher, L. Saint-Raymond and S. Simonella. Long-time correlations for a hard-sphere gas at equilibrium. arXiv:2012.03813.
  * [6] T. Bodineau, I. Gallagher, L. Saint-Raymond and S. Simonella. Long-time derivation at equilibrium of the fluctuating Boltzmann equation. arXiv:2201.04514.
  * [7] L. Boltzmann, Weitere Studien uber das Warme gleichgenicht unfer Gasmolakular. Sitzungsberichte der Akademie der Wissenschaften 66 (1872), 275–370.
  * [8] T. Buckmaster, P. Germain, Z. Hani and J. Shatah. Onset of the wave turbulence description of the long-time behavior of the nonlinear Schrödinger equation. _Invent. Math._ 225 (2021), 787–855.
  * [9] S. Cai and X Lu. The spatially Homogeneous Boltzmann Equation for Bose-Einstein Particles: Rate of Strong Convergence to Equilbrium. _J. Stat. Phys._ 175 (2019), 289–350.
  * [10] C. Collot, H. Dietert and P. Germain. Stability and cascades for thr Kolmogorov-Zakharov spectrum of wave turbulence. arXiv:2208.00947.
  * [11] C. Collot and P. Germain. On the derivation of the homogeneous kinetic wave equation. arXiv:1912.10368.
  * [12] C. Collot and P. Germain. Derivation of the homogeneous kinetic wave equation: longer time scales. arXiv:2007.03508.
  * [13] R. Davidson. _Methods in Nonlinear Plasma Theory_ , First Edition, Academic Press, 1972.
  * [14] Y. Deng and Z. Hani. On the derivation of the wave kinetic equation for NLS. _Forum of Math. Pi._ 9 (2021), e6.
  * [15] Y. Deng and Z. Hani. Full derivation of the wave kinetic equation. _Invent. Math._ 233 (2023), no. 2, 543–724.
  * [16] Y. Deng and Z. Hani. Propagation of chaos and higher order statistics in wave kinetic theory. _J. Eur. Math. Soc. (JEMS)_ , to appear.
  * [17] Y. Deng and Z. Hani. Rigorous justification of wave kinetic theory. arXiv:2207.08358.
  * [18] Y. Deng and Z. Hani. Derivation of the wave kinetic equation: full range of scaling laws. arXiv:2301.07063.
  * [19] Y. Deng, Z. Hani and X. Ma. Long time justification of the Boltzmann equation. In preparation.
  * [20] Y. Deng, A. Nahmod and H. Yue. Optimal local well-posedness for the periodic derivative nonlinear Schrödinger equation. _Comm. Math. Phys._ 384 (2021), no. 2, 1061–1107.
  * [21] A. Dymov and S. Kuksin. Formal expansions in stochastic model for wave turbulence 1: kinetic limit. _Comm. Math. Phys._ 382 (2021), 951–1014.
  * [22] A. Dymov and S. Kuksin. Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition. _J. Stat. Phys._ 190 (2023), article number: 3.
  * [23] L. Erdös, M. Salmhofer and H-T. Yau. Quantum diffusion of the random Schrödinger evolution in the scaling limit. _Acta Math._ 200 (2008), no. 2, 211–277.
  * [24] L. Erdös, B. Schlein, and H-T. Yau. Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. math. 167, 515–614 (2007).
  * [25] L. Erdös and H-T. Yau. Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. _Comm. Pure Appl. Math._ 53 (2000), 667–735.
  * [26] M. Escobedo and J. Velázquez. Finite time blow-up and condensation for the bosonic Nordheim equation. _Invent. Math._ 200 (2015), 761–847.
  * [27] M. Escobedo and J. Velázquez. On the theory of Weak Turbulence for the Nonlinear Schrödinger Equation. _Mem. Amer. Math. Soc._ 238 (2015), number 1124.
  * [28] E. Faou. Linearized Wave Turbulence Convergence Results for Three-Wave Systems. _Comm. Math. Phys._ 378, 807–849 (2020).
  * [29] A. A. Galeev and R. Z. Sagdeev. _Review of plasma physics_. In: Leontovich, M.A. (ed.) vol. 7 (1979), p. 307. Consultants Bureau, New York.
  * [30] I. Gallagher, L. Saint-Raymond, and B. Texier. From Newton to Boltzmann: the case of hard-spheres and short-range potentials, ZLAM (2014).
  * [31] P. Germain, A. Ionescu and M-B. Tran. Optimal local well-posedness theory for the kinetic wave equation. _J. Funct. Anal._ 279 (2020), no. 4, 108570.
  * [32] I. Gallagher and I. Tristani. On the convergence of smooth solutions from Boltzmann to Navier-Stokes. _Ann. Henri Lebesgue_ 3 (2020), 561–614.
  * [33] J. Guioth, F. Bouchet, and G. Eyink. Path Large Deviations for the Kinetic Theory of Weak Turbulence. _Journal of Statistical Physics_ , Volume 189, Issue 2, 2022.
  * [34] F. Golse and L. Saint-Raymond. The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. _Invent. Math._ 155 (2004), 81–161.
  * [35] F. Golse. On the dynamics of large particle systems in the mean field limit. Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity, 1–144. _Lect. Notes Appl. Math. Mech._ 3 Springer, 2016.
  * [36] Felipe Hernández. Quantum Diffusion via an Approximate Semigroup Property. arXiv 2206.12998.
  * [37] M. Hauray, P.-E. Jabin. Particle approximation of Vlasov equations with singular forces: propagation of chaos. _Ann. Sci. Èc. Norm. Supér._ (4) 48 (2015), no.4, 891–940.
  * [38] D. Hilbert. Mathematical Problems. _Bulletin of the American Mathematical Society_. 8 (1901), 437–479.
  * [39] Z. Hani, J. Shatah, and H. Zhu. Inhomogeneous turbulence for Wick NLS. _Preprint_.
  * [40] A. Hannani, M. Rosenzweig, G. Staffilani and M.-B. Tran. On the wave turbulence theory for a stochastic KdV type equation - generalization for the inhomogeneous kinetic limit. arXiv:2210.17445.
  * [41] K. Hasselmann. On the nonlinear energy transfer in a gravity wave spectrum part 1. _J. Fluid Mech._ 12 (1962), 481–500.
  * [42] K. Hasselmann. On the nonlinear energy transfer in a gravity wave spectrum. Part 2. _J. Fluid Mech._ 15 (1963), 273–281.
  * [43] P.-E. Jabin. A review of the mean field limits for Vlasov equations. _Kinet. Relat. Models_ 7(2014), no.4, 661–711.
  * [44] P. A. Janssen, Progress in ocean wave forecasting. _Journal of Computational Physics_ , 227 (2008), 3572–3594.
  * [45] F. King, BBGKY hierarchy for positive potentials, Ph.D. dissertation, Dept. Mathematics, Univ. California, Berkeley, 1975.
  * [46] S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys. 279 (2008), no. 1, 169–185.
  * [47] O. E. Lanford. Time Evolution of Large Classical Systems, ed. J. Moser, _Lecture Notes in Physics_ Vol. 38 (1975), Springer, Heidelberg, 1–111.
  * [48] J. Lukkarinen and H. Spohn. Weakly nonlinear Schrödinger equation with random initial data. _Invent. Math._ 183 (2011), 79–188.
  * [49] X. Ma. Almost sharp wave kinetic theory of multidimensional KdV type equations with $d\geq 3$. arXiv:2204.06148.
  * [50] S. Nazarenko. Wave turbulence. _Lecture Notes in Physics_ Vol. 825 (2011), Springer, Heidelberg.
  * [51] L. Nordheim. On the kinetic method in the new statistics and application in the electron theory of conductivity. _Proc. Roy. Soc. London Ser. A_ 119 (1928), 689–698.
  * [52] S. Olla, S. R. S. Varadhan, H.-T. Yau. Hydrodynamical limit for a Hamiltonian system with weak noise. _Comm. Math. Phys._ 155(3): 523–560 (1993).
  * [53] R. E. Peierls. Zur kinetischen Theorie der Wärmeleitung in Kristallen. Annalen Physik 3 (1929), 1055–1101.
  * [54] M. Pulvirenti and S. Simonella. The Boltzmann-Grad limit of a hard sphere system: analysis of the correlation error. _Inventiones Math._ Volume 207, pages 1135–1237 (2017).
  * [55] L. Saint-Raymond. Hydrodynamic limits of the Boltzmann equation. _Lecture Notes in Mathematics_. Springer-Verlag Berlin Heidelberg 2009.
  * [56] B. Schlein. Derivation of effective evolution equations from many-body quantum mechanics. _Riv. Math. Univ. Parma (N.S.)_ 8(2017), no.1, 83–108.
  * [57] _Guide to Wave Analysis and Forecasting_ , Secretariat of the World Meteorological Organization, Geneva, Switzerland 1998.
  * [58] A. Soffer and M.-B. Tran. On the Energy Cascade of 3-Wave Kinetic Equations: Beyond Kolmogorov-Zakharov Solutions. _Comm. Math. Phys._ 376 (2020), 2229–2276.
  * [59] H. Spohn. Derivation of the transport equation for electrons moving through random impurities. _J. Statist. Phys._ 17 (1977), no.6, 385–412.
  * [60] H. Spohn. Kinetic equations from Hamiltonian dynamics. _Rev. Mod. Phys._ 52 (1980), no. 3, 569–615.
  * [61] H. Spohn. On the Boltzmann equation for weakly nonlinear wave equations, in _Boltzmann’s Legacy, ESI Lectures in Mathematics and Physics_ (2008), ISBN print 978-3-03719-057-9, 145–159.
  * [62] G. Staffilani and M.-B. Tran. On the wave turbulence theory for the stochastic and random multidimensional KdV type equations. arXiv:2106.09819.
  * [63] C. Sulem and P.-L. Sulem. The nonlinear Schrödinger equation: Self-Focusing and Wave Collapse. Springer Applied Sciences Series. Volume 139. ISBN-13: 978-0387986111.
  * [64] E. Uehling and G. Uhlenbeck. Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases. _Phys. Rev._ 43 (1933), 552–561.
  * [65] A. A. Vedenov. Theory of weakly turbulent plasma. In: Leontovich, M.A. (ed.) _Reviews of Plasma Physics_ , vol. 3 (1967), 229. Consultants Bureau, New York.
  * [66] V. E. Zakharov. Weak turbulence in media with decay spectrum. _Zh. Priklad. Tech. Fiz._ 4 (1965), 5–39.
  * [67] V.E. Zakharov, V.S. L’vov and and G. Falkovich. _Kolmogorov Spectra of Turbulence: I Wave Turbulence_. Springer, Berlin, 1992.
  * [68] G. M. Zaslavskii and R. Z. Sagdeev. _Sov. Phys. JETP_ 25 (1967), 718.


i}\,\mathcal{D}^{i}\,\sigma^{j}\Bigr{)}\,,\\\
s_{\text{g-R}}^{i}&\;=\;\frac{1}{\pi^{2}}\,\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,\Bigl{(}\epsilon^{\mu\nu\rho}\mathcal{A}^{i}_{\mu}\,\partial_{\nu}\,\bigl{(}\mathcal{A}^{(R)}_{\rho}-\tfrac{1}{2}v_{\rho}\bigr{)}+{\rm
i}\mathcal{D}^{i}H+{\rm
i}\,\frac{\sigma^{i}}{4}\bigl{(}R^{(3)}+2v_{\mu}v^{\mu}+2H^{2}\bigr{)}\Bigr{)}\,.\end{split}$
(148)

Here $i,j$ are Cartan labels for the gauge group, $\mathcal{A}$ is the three-
dimensional gauge field, $\mathcal{D}$ is the $D$-term auxiliary scalar of the
three-dimensional $\mathcal{N}=2$ vector multiplet, $\sigma$ is the (Coulomb
branch) scalar of the three-dimensional $\mathcal{N}=2$ vector multiplet,
$\mathcal{A}^{(R)}$ is the three-dimensional background gauge field for the
$R$ current, $v^{\mu}=-{\rm
i}\,\varepsilon^{\mu\nu\rho}\,\partial_{\nu}\,c_{\rho}$ is the Hodge dual of
the graviphoton $c_{\mu}$ (the background $U(1)_{\text{KK}}$ gauge field), $H$
is the scalar in the supergravity multiplet, and $R^{(3)}$ is the Ricci scalar
of $\mathcal{M}_{3}$.

Then we have the background R-R CS terms and the gravitational CS term for the
spin connection $\omega$,

$\begin{split}R_{\mathcal{M}_{3}}&\;=\;\frac{1}{\pi^{2}}\,\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,\Bigl{(}\epsilon^{\mu\nu\rho}\,\bigl{(}\mathcal{A}^{(R)}_{\mu}-\tfrac{1}{2}v_{\mu}\bigr{)}\partial_{\nu}\bigl{(}\mathcal{A}^{(R)}_{\rho}-\tfrac{1}{2}v_{\rho}\bigr{)}+{\rm
i}\,\frac{H}{2}\left(R^{(3)}+2v_{\mu}v^{\mu}+2H^{2}\right)\Bigr{)}\,,\\\
G_{\mathcal{M}_{3}}&\;=\;\frac{1}{\pi^{2}}\,\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,\Bigl{(}\epsilon^{\mu\nu\rho}\operatorname{Tr}\bigl{(}\omega_{\mu}\,\partial_{\nu}\,\omega_{\rho}-\tfrac{2}{3}\omega_{\mu}\,\omega_{\nu}\,\omega_{\rho}\bigr{)}+4\,\epsilon^{\mu\nu\rho}\bigl{(}\mathcal{A}^{{(R)}}_{\mu}-\tfrac{3}{2}v_{\mu}\bigr{)}\partial_{\nu}\bigl{(}\mathcal{A}^{(R)}_{\rho}-\tfrac{3}{2}v_{\rho}\bigr{)}\Bigr{)}\,.\end{split}$
(149)

Finally we have the CS actions involving the graviphoton. These are gauge-KK,
R-KK, and KK-KK, whose bosonic parts read DiPietro:2016ond

$\begin{split}s_{\text{g-KK}}^{i}&\;=\;\frac{1}{\pi^{2}}\,\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,\Bigl{(}\epsilon^{\mu\nu\rho}\mathcal{A}^{i}_{\mu}\,\partial_{\nu}\,c_{\rho}-{\rm
i}\,\mathcal{D}^{i}+{\rm i}\,\sigma^{i}\,H\Bigr{)}\,,\\\
s_{\text{R-KK}}&\;=\;\frac{1}{\pi^{2}}\,\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,\Bigl{(}{\rm
i}\,v^{\mu}\,\bigl{(}\mathcal{A}^{(R)}_{\mu}-\frac{1}{2}v_{\mu}\bigr{)})-\frac{{\rm
i}}{2}\,v^{\mu}\,v_{\mu}+\frac{{\rm i}}{2}\,H^{2}-\frac{{\rm
i}}{4}\,R^{(3)}\Bigr{)}\,,\\\ s_{\text{KK-
KK}}&\;=\;\frac{1}{\pi^{2}}\,\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,\Bigl{(}{\rm
i}\,v^{\mu}\,c_{\mu}-2\,{\rm
i}\,H\Bigr{)}\;=\;\frac{1}{\pi^{2}}\,\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,\Bigl{(}\epsilon^{\mu\nu\rho}\,c_{\mu}\,\partial_{\nu}\,c_{\rho}-2\,{\rm
i}\,H\Bigr{)}\,.\end{split}$ (150)

The equations (148), (149), (150) together make up the complete list of CS
terms. As we explain below, when we have a KK reduction these actions can be
combined together into a succinct expression in a natural manner.

The coefficients of the above actions are obtained by calculating the
coefficients of the CS pieces, which are obtained by integrating out all
massive fermions that couple to the corresponding gauge fields. Integrating
out a fermion f with real-mass $m_{\text{f}}$ and charges
$e^{\text{f}}_{x},e^{\text{f}}_{y}$ under the gauge fields
$\mathcal{A}_{x},\mathcal{A}_{y}$ generates the term
$\frac{1}{\pi^{2}}\,\int_{\mathcal{M}_{3}}\mathcal{A}_{x}\wedge\mathrm{d}\mathcal{A}_{y}$
with coefficient given by the one-loop exact formula (we follow the
conventions of DiPietro:2014bca )

$-\frac{{\rm
i}\pi}{8}\,\sum_{\text{f}}\mathrm{sgn}(m_{\text{f}})\,e^{\text{f}}_{x}\,e^{\text{f}}_{y}\,.$
(151)

The contribution of the fermion to the coefficient of the gravitational CS
term is given by (see Appendix A of Closset:2018ghr )

$-\frac{{\rm i}\,\pi}{192}\,\sum_{\text{f}}\mathrm{sgn}(m_{\text{f}})\,.$
(152)

The full effective action of the theory is the sum of the actions (148), (149)
(150) with coefficients obtained by summing (151), (152) over all the massive
fermions in the theory. (The actions with $x$ and $y$ different appear twice
in the final action—as $x$-$y$ and $y$-$x$—and therefore need to be multiplied
by a factor of two.)

The situation of interest in Section 3 is the Kaluza-Klein reduction of a
four-dimensional theory on a circle of radius $R$. The bosonic fields in the
three-dimensional vector multiplet are written in term of the 4d fields in
(90), and the three-dimensional fermions are the reduction of the
corresponding 4d fermions. Consider a fermion of R-charge $r_{\text{f}}$
transforming in a representation of weight $\rho_{\text{f}}$ under the gauge
group. The tree-level real-mass of a KK mode of this fermion is given by (in
the convention of DiPietro:2014bca )

$m_{\text{f}}\;=\;-(\rho_{\text{f}}\cdot
A_{Y}-p_{Y}-r_{\text{f}}\,A^{\text{nm}}_{Y}-\frac{1}{2}V^{\text{nm}}_{Y})\,,$
(153)

where $A^{\text{nm}}$, $V^{\text{nm}}$ are the 4d background R-gauge fields
given in (72). Note that $p_{Y}$ also enters (151) as the charge of the
fermion under $U(1)_{\text{KK}}$.252525 In the Euclidean context that we
discuss here, the background fields $A^{\text{nm}}$, $V^{\text{nm}}$, and the
effective radius $R$ are complex. The definition (153) is thought of as an
analytic continuation, and is read off from the coupling of the fermion to the
other fields and parameters in the component of the covariant derivative along
the KK direction $Y$. The quantity $m_{\text{f}}$ appears in the main text in
formulas for the one-loop correction From the expression e.g. (97), we see
that it appears only as the function $\mbox{sgn}(m_{\text{f}})$. Then the the
sgn function needs to be appropriately defined. As discussed below Equation
(100), we can do that by defining it for real $R$, and then continuing the
formulas to complex $R$. Since we take $R\to 0$ at the end of the
calculations, it is enough to keep only the singular pieces in the formula
(153).

Using these relations we proceed to write the three-dimensional effective
action directly in terms of the dynamical 4d fields. The contribution of the
actions coming from (148) and (150) to the full effective action can be
written as the sum of the following two actions,

$\begin{split}\widetilde{S}^{\,\text{f}}_{\text{g-g}}&\;=\;-{\rm
i}\pi\,\frac{\mathrm{sgn}(m_{\text{f}})}{8}\Bigl{(}\rho_{\text{f}}^{i}\,\rho_{\text{f}}^{j}\,s^{ij}_{\text{g-g}}+2\,p_{Y}\,\rho_{\text{f}}^{i}\,s^{i}_{\text{g-KK}}+p_{Y}^{2}\,s_{\text{KK-
KK}}\Bigr{)}\\\ &\;=\;-{\rm
i}\pi\,\frac{\mathrm{sgn}(m_{\text{f}})}{8}\biggl{(}\bigl{(}\rho_{\text{f}}\cdot
A_{Y}-p_{Y}\bigr{)}^{2}\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,\bigl{(}{\rm
i}\,v^{\mu}c_{\mu}-2{\rm i}\,H\bigr{)}\\\ &\kern
60.00009pt~{}~{}~{}~{}+2\bigl{(}\rho_{\text{f}}\cdot
A_{Y}-p_{Y}\bigr{)}\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,\bigl{(}-{\rm
i}\,v^{\mu}\,(\rho_{\text{f}}\cdot A_{\mu})+{\rm i}\,(\rho_{\text{f}}\cdot
D)\bigr{)}\\\ &\kern
60.00009pt~{}~{}~{}~{}+\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,\bigl{(}\epsilon^{\mu\nu\rho}(\rho_{\text{f}}\cdot
A_{\mu})\,\partial_{\nu}\,(\rho_{\text{f}}\cdot
A_{\rho})\biggr{)}\,,\end{split}$ (154)
$\begin{split}2\,\widetilde{S}^{\,\text{f}}_{\text{g-R}}&\;=\;-2\,{\rm
i}\pi\,\frac{\mathrm{sgn}(m_{\text{f}})}{8}\,r_{\text{f}}\,\Bigl{(}\rho_{\text{f}}^{i}\,s^{i}_{\text{g-R}}+p_{Y}\,s_{\text{R-KK}}\Bigr{)}\\\
&\;=\;-2\,{\rm i}\pi\,\frac{\mathrm{sgn}(m_{\text{f}})}{8}\,r_{\text{f}}\\\
&\kern 20.00003pt~{}~{}\biggl{(}\bigl{(}\rho_{\text{f}}\cdot
A_{Y}-p_{Y}\bigr{)}\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,\Bigl{(}-{\rm
i}\,v^{\mu}\big{(}\mathcal{A}^{(R)}_{\mu}-\frac{1}{2}v_{\mu}\big{)}+{\rm
i}\,\frac{1}{2}v^{\mu}\,v_{\mu}-{\rm i}\,{\frac{1}{2}}H^{2}+{\rm
i}\,{\frac{1}{4}}R^{(3)}\Bigr{)}\\\ &\kern
60.00009pt~{}~{}+\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,\Bigl{(}\epsilon^{\mu\nu\rho}(\rho_{\text{f}}\cdot
A_{\mu})\,\partial_{\nu}\,\bigl{(}\mathcal{A}^{(R)}_{\rho}-{\frac{1}{2}}v_{\rho}\bigr{)}+{\rm
i}\,(\rho_{\text{f}}\cdot D)H\Bigr{)}\biggr{)}\,.\end{split}$ (155)

Finally we specialize to the BPS configurations considered in the main text,
given in (92). The above two terms take the following value

$\begin{split}\widetilde{S}^{\text{f}}_{\text{g-g}}&\;=\;-{\rm
i}\pi\,\frac{\mathrm{sgn}(m_{\text{f}})}{8R^{2}}\,\bigl{(}\rho_{\text{f}}\cdot\underline{u}-k_{Y}\bigr{)}^{2}\,A_{\mathcal{M}_{3}}\,,\\\
2\,\widetilde{S}^{\text{f}}_{\text{g-R}}&\;=\;-{\rm
i}\pi\,\frac{\mathrm{sgn}(m_{\text{f}})}{8R}\,2\,r_{\text{f}}\,\bigl{(}\rho_{\text{f}}\cdot\underline{u}-k_{Y}\bigr{)}\;L_{\mathcal{M}_{3}}\,,\end{split}$
(156)

where $A_{\mathcal{M}_{3}}$ and $L_{\mathcal{M}_{3}}$ are functions of the
three-dimensional background262626To compare with DiPietro:2016ond  note that
$A_{\mathcal{M}_{3}}=-A_{\mathcal{M}_{3}}^{\text{there}}$ and
$L_{\mathcal{M}_{3}}={\rm i}\,L_{\mathcal{M}_{3}}^{\text{there}}.$,

$\begin{split}A_{\mathcal{M}_{3}}&\;=\;\frac{1}{\pi^{2}}\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,\bigl{(}{\rm
i}\,v^{\mu}\,c_{\mu}-2\,{\rm i}\,H\bigr{)}\,,\\\
L_{\mathcal{M}_{3}}&\;=\;\frac{1}{\pi^{2}}\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,\Bigl{(}-{\rm
i}\,v^{\mu}\mathcal{A}^{(R)}_{\mu}+{\rm i}\,v^{\mu}\,v_{\mu}-{\rm
i}\,{\frac{1}{2}}H^{2}+{\rm i}\,{\frac{1}{4}}R^{(3)}\Bigr{)}\,.\end{split}$
(157)

We now turn to the remaining terms in the full action, namely those coming
from the terms in (149),

$\begin{split}S^{\text{f}}_{\text{R-R}}&\;=\;-{\rm
i}\pi\,\frac{\mathrm{sgn}(m_{\text{f}})}{8}\,\bigl{(}r_{\text{f}}^{2}-\frac{1}{6}\bigr{)}\,R_{\mathcal{M}_{3}}\,,\\\
S^{\text{f}}_{\text{grav}}&\;=\;-{\rm
i}\pi\,\frac{\mathrm{sgn}(m_{\text{f}})}{192}\,G_{\mathcal{M}_{3}}\,.\end{split}$
(158)

Note that both $R_{\mathcal{M}_{3}}$ and $G_{\mathcal{M}_{3}}$ contain
$\mathcal{A}^{(R)}\wedge\mathrm{d}\mathcal{A}^{(R)}$ terms, and it is the sum
of the corresponding coefficients that is fixed by (151). Since the
coefficient in $S^{\text{f}}_{\text{grav}}$ is fixed by (152), the shift
$-1/6$ in the coefficient of $S^{\text{f}}_{\text{R-R}}$ serves to cancel the
$\mathcal{A}^{(R)}\wedge\mathrm{d}\mathcal{A}^{(R)}$ term coming from
$S^{\text{f}}_{\text{grav}}$. The final result for the action of the BPS
configurations up to $\text{O}(R^{0})$ obtained by integrating out a fermion f
is given by the sum of the actions in (156), (158).

##  Appendix C Values of supersymmetrized Chern-Simons actions

In this appendix we record the values of various terms in the supersymmetrized
actions of Appendix B evaluated on the configurations discussed in Section 3.
We first recall from §3 the values of the various fields entering the actions.
The three-dimensional metric is

$\mathrm{d}s_{3}^{2}\;=\;\mathrm{d}\theta^{2}+\sin^{2}\theta\,\mathrm{d}\phi_{1}^{2}+\cos^{2}\theta\,\mathrm{d}\phi_{2}^{2}-c^{2}\,,$
(159)

the graviphoton and its Hodge dual are

$c\;=\;-{\rm
i}\frac{\Omega}{\sqrt{1-\Omega^{2}}}\,\bigl{(}\sin^{2}\theta\,\mathrm{d}\phi_{1}+\cos^{2}\theta\,\mathrm{d}\phi_{2}\bigr{)}\,,\qquad
v\;=\;\frac{2\,{\rm i}}{\sqrt{1-\Omega^{2}}}\,c\,.$ (160)

The auxiliary background supergravity multiplet fields are

$H\;=\;-\frac{{\rm
i}}{\sqrt{1-\Omega^{2}}}\,,\qquad\mathcal{A}^{R}_{\mu}\;=\;\Bigl{(}\frac{\tau}{R}-\frac{n_{0}}{2R}+\frac{{\rm
i}}{\sqrt{1-\Omega^{2}}}\Bigr{)}\,c_{\mu}\,.$ (161)

The four-dimensional gauge fields are

$A^{i}_{Y}\;=\;\frac{u_{i}}{R}\,,\qquad D^{i}\;=\;0\,,$ (162)

The Chern-Simons action for $c$

$S^{\text{CS}}(c)\;=\;\int_{\mathcal{M}_{3}}\,c\wedge\mathrm{d}c\;=\;{\rm
i}\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,v^{\mu}\,c_{\mu}\,,$ (163)

evaluates to

$\frac{1}{4\pi^{2}}\,S^{\text{CS}}(c)\;=\;\;=\;\frac{\Omega^{2}}{1-\Omega^{2}}\;=\;-1+\text{O}(\gamma)\,.$
(164)

The other building blocks for the actions of the background fields in the
three-dimensional theory are given below, including their limiting behavior as
$\gamma\to 0$ with $\tau$ fixed (i.e. as $\Omega\to\infty$),

$\frac{1}{4\pi^{2}}\,S^{(H)}\;=\;\frac{{\rm
i}}{4\pi^{2}}\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,H\;=\;\frac{1}{2(1-\Omega^{2})}\;=\;\text{O}(\gamma)\,,$
(165)
$\frac{1}{4\pi^{2}}\,S^{(v)}\;=\;\frac{1}{4\pi^{2}}\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,v^{\mu}\,v_{\mu}\;=\;\frac{\Omega^{2}}{(1-\Omega^{2})^{\frac{3}{2}}}\;=\;\text{O}(\gamma)\,,$
(166)
$\frac{1}{4\pi^{2}}\,S^{(H^{2})}\;=\;\frac{1}{4\pi^{2}}\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,H^{2}\;=\;-\frac{1}{2(1-\Omega^{2})^{\frac{3}{2}}}\;=\;\text{O}(\gamma)\,,$
(167)
$\frac{1}{4\pi^{2}}\,S^{(R)}\;=\;\frac{1}{4\pi^{2}}\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{g}\,R^{(3)}\;=\;-\frac{-6+8\Omega^{2}}{2(1-\Omega^{2})^{\frac{3}{2}}}\;=\;\text{O}(\gamma)\,.$
(168)

##  Appendix D Dimensional reduction for the case $\Omega_{1}\neq\Omega_{2}$

We begin by writing the background configuration in (72) as a KK
compactification to three dimensions, i.e. a circle fibration on a 3-manifold
$M_{3}$. We have

$\mathrm{d}s_{4}^{2}\;=\;\mathrm{d}s_{3}^{2}+{\rm
e}^{2\phi}(\mathrm{d}t_{E}+\widetilde{c}\,)^{2}\,,$ (169)

where the metric on $M_{3}$ is

$\begin{split}\mathrm{d}s_{3}^{2}\;=\;\widetilde{g}_{\mu\nu}\,\mathrm{d}x^{\mu}\,\mathrm{d}x^{\nu}\;=\;\mathrm{d}\theta^{2}+\sin^{2}\theta\,\mathrm{d}\phi_{1}^{2}+\cos^{2}\theta\,\mathrm{d}\phi_{2}^{2}-{\rm
e}^{2\phi}\,\widetilde{c}^{\,2}\,,\\\ \end{split}$ (170)

and the graviphoton and KK scalar are

$\begin{split}{\rm
e}^{2\phi}&\;=\;1-\Omega_{1}^{2}\,\sin^{2}\theta-\Omega_{2}^{2}\,\cos^{2}\theta\,,\\\
\widetilde{c}\;=\;\widetilde{c}_{\mu}\,\mathrm{d}x^{\mu}&\;=\;-{\rm i}\,{\rm
e}^{-2\phi}\bigl{(}\Omega_{1}\,\sin^{2}\theta\,\mathrm{d}\phi_{1}+\Omega_{2}\,\cos^{2}\theta\,\mathrm{d}\phi_{2}\bigr{)}\,.\end{split}$
(171)

For the case $\Omega_{1}=\Omega_{2}=\Omega$, we have that
$e^{2\phi}=1-\Omega^{2}$, so that the graviphoton $c$ defined in (80) is
related to $\widetilde{c}$ as $c={\rm e}^{\phi}\,\widetilde{c}$. The magnitude
of the volume form in three dimensions is

$\sqrt{\widetilde{g}}\;=\;\frac{1}{2}\,{\rm e}^{-\phi}\,\sin 2\theta\,.$ (172)

The associated Chern-Simons action

$S^{\text{CS}}(\,\widetilde{c}\,)\;=\;\int_{\mathcal{M}_{3}}\,\widetilde{c}\wedge\mathrm{d}\widetilde{c}\;=\;{\rm
i}\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\,\sqrt{\widetilde{g}}\;\widetilde{v}^{\mu}\,\widetilde{c}_{\mu}$
(173)

(where $\widetilde{v}=-{\rm i}*\mathrm{d}\widetilde{c}$ is the Hodge dual)
evaluates to

$S^{\text{CS}}(\,\widetilde{c}\,)\;=\;4\pi^{2}\,\Omega_{1}\,\Omega_{2}\int_{0}^{\pi/2}\frac{\sin
2\theta}{(1-\Omega_{1}^{2}\sin^{2}\theta-\Omega_{2}^{2}\cos^{2}\theta)^{2}}\,\mathrm{d}\theta\;=\;4\pi^{2}\frac{\Omega_{1}\Omega_{2}}{(1-\Omega_{1}^{2})(1-\Omega_{2}^{2})}\,.$
(174)

For the identification between the four-dimensional and the three-dimensional
fields, we follow the treatment of Assel:2014paa  applied to the metric (169).
The result is

$\frac{1}{2}\,{\rm
e}^{\phi}\,\widetilde{v}_{\mu}\;=\;V^{\text{nm}}_{\mu}-V^{\text{nm}}_{t_{E}}\,\widetilde{c}_{\mu}\,,\qquad\widetilde{H}\;=\;{\rm
e}^{-\phi}\,V_{t_{E}}\,,\qquad\mathcal{A}^{R}_{\mu}\;=\;A^{\text{nm}}_{\mu}-A^{\text{nm}}_{t_{E}}\;\widetilde{c}_{\mu}+\frac{1}{2}\,{\rm
e}^{\phi}\,\widetilde{v}_{\mu}\,.$ (175)

The values of these fields are

${\rm
e}^{\phi}\,\widetilde{v}_{\mu}\;=\;-2\,V^{\text{nm}}_{Y}\,c_{\mu}\;=\;2\,{\rm
i}\,\widetilde{c}_{\mu}\,,\qquad\widetilde{H}\;=\;-{\rm i}\,e^{-\phi}\,,$
(176) $\mathcal{A}^{R}_{\mu}\;=\;{\rm
i}\bigl{(}\frac{1}{2}(\Omega_{1}+\Omega_{2})-1\bigr{)}\,\widetilde{c}_{\mu}+\frac{1}{2}\,{\rm
e}^{\phi}\,\widetilde{v}_{\mu}\;=\;\frac{{\rm
i}}{2}(\Omega_{1}+\Omega_{2})\,\widetilde{c}_{\mu}\,.$ (177)

We can now calculate the various actions as in Appendix C, and we find that,
in the $\gamma\to 0$, $\Omega\to\infty$ limit we have the effective
replacement

$\frac{1}{R^{2}}\to\frac{1}{\tau\,\sigma}\,,\qquad\frac{1}{R}\to\frac{\tau+\sigma}{2\tau\,\sigma}$
(178)

in the effective potential (104).

## References

  * (1) B. Sundborg, _The Hagedorn transition, deconfinement and N=4 SYM theory_ , _Nucl. Phys. B_ 573 (2000) 349 [hep-th/9908001].
  * (2) O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, _The Hagedorn - deconfinement phase transition in weakly coupled large N gauge theories_ , _Adv. Theor. Math. Phys._ 8 (2004) 603 [hep-th/0310285].
  * (3) J. Kinney, J. M. Maldacena, S. Minwalla and S. Raju, _An Index for 4 dimensional super conformal theories_ , _Commun. Math. Phys._ 275 (2007) 209 [hep-th/0510251].
  * (4) S. M. Hosseini, K. Hristov and A. Zaffaroni, _An extremization principle for the entropy of rotating BPS black holes in AdS 5_, _JHEP_ 07 (2017) 106 [1705.05383].
  * (5) A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, _Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS 5 black holes_, _JHEP_ 10 (2019) 062 [1810.11442].
  * (6) S. Choi, J. Kim, S. Kim and J. Nahmgoong, _Large AdS black holes from QFT_ , 1810.12067.
  * (7) F. Benini and P. Milan, _Black Holes in 4D $\mathcal{N}$=4 Super-Yang-Mills Field Theory_, _Phys. Rev. X_ 10 (2020) 021037 [1812.09613].
  * (8) S. Choi, J. Kim, S. Kim and J. Nahmgoong, _Comments on deconfinement in AdS/CFT_ , 1811.08646.
  * (9) M. Honda, _Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula_ , _Phys. Rev._ D100 (2019) 026008 [1901.08091].
  * (10) A. Arabi Ardehali, _Cardy-like asymptotics of the 4d $\mathcal{N}=4$ index and AdS5 blackholes_, _JHEP_ 06 (2019) 134 [1902.06619].
  * (11) J. Kim, S. Kim and J. Song, _A 4d $\mathcal{N}$ = 1 Cardy Formula_, _JHEP_ 01 (2021) 025 [1904.03455].
  * (12) A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, _The asymptotic growth of states of the 4d $\mathcal{N}=1$ superconformal index_, _JHEP_ 08 (2019) 120 [1904.05865].
  * (13) A. Cabo-Bizet and S. Murthy, _Supersymmetric phases of 4d $\mathcal{N}$ = 4 SYM at large $N$_, _JHEP_ 09 (2020) 184 [1909.09597].
  * (14) F. Benini, E. Colombo, S. Soltani, A. Zaffaroni and Z. Zhang, _Superconformal indices at large $N$ and the entropy of AdS5 $\times$ SE5 black holes_, _Class. Quant. Grav._ 37 (2020) 215021 [2005.12308].
  * (15) A. Amariti, I. Garozzo and G. Lo Monaco, _Entropy function from toric geometry_ , 1904.10009.
  * (16) A. González Lezcano and L. A. Pando Zayas, _Microstate counting via Bethe Ansätze in the 4d $\mathcal{N}$ = 1 superconformal index_, _JHEP_ 03 (2020) 088 [1907.12841].
  * (17) A. Lanir, A. Nedelin and O. Sela, _Black hole entropy function for toric theories via Bethe Ansatz_ , _JHEP_ 04 (2020) 091 [1908.01737].
  * (18) M. David, J. Nian and L. A. Pando Zayas, _Gravitational Cardy Limit and AdS Black Hole Entropy_ , _JHEP_ 11 (2020) 041 [2005.10251].
  * (19) A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, _The large- $N$ limit of the 4d $\mathcal{N}$ = 1 superconformal index_, _JHEP_ 11 (2020) 150 [2005.10654].
  * (20) S. Murthy, _The growth of the $\frac{1}{16}$-BPS index in 4d $\mathcal{N}=4$ SYM_, 2005.10843.
  * (21) P. Agarwal, S. Choi, J. Kim, S. Kim and J. Nahmgoong, _AdS black holes and finite N indices_ , 2005.11240.
  * (22) A. Cabo-Bizet, _From multi-gravitons to Black holes: The role of complex saddles_ , 2012.04815.
  * (23) C. Copetti, A. Grassi, Z. Komargodski and L. Tizzano, _Delayed Deconfinement and the Hawking-Page Transition_ , 2008.04950.
  * (24) K. Goldstein, V. Jejjala, Y. Lei, S. van Leuven and W. Li, _Residues, modularity, and the Cardy limit of the 4d $\mathcal{N}=4$ superconformal index_, 2011.06605.
  * (25) L. Di Pietro and Z. Komargodski, _Cardy formulae for SUSY theories in $d=$ 4 and $d=$ 6_, _JHEP_ 12 (2014) 031 [1407.6061].
  * (26) L. Di Pietro and M. Honda, _Cardy Formula for 4d SUSY Theories and Localization_ , _JHEP_ 04 (2017) 055 [1611.00380].
  * (27) J. L. Cardy, _Operator Content of Two-Dimensional Conformally Invariant Theories_ , _Nucl. Phys._ B270 (1986) 186.
  * (28) S. Garoufalidis, S. Murthy and D. Zagier, “Work in progress.”.
  * (29) E. Shaghoulian, _Modular Invariance of Conformal Field Theory on $S^{1}\times S^{3}$ and Circle Fibrations_, _Phys. Rev. Lett._ 119 (2017) 131601 [1612.05257].
  * (30) A. Gadde, _Modularity of supersymmetric partition functions_ , 2004.13490.
  * (31) S. S. Razamat, _On a modular property of N=2 superconformal theories in four dimensions_ , _JHEP_ 10 (2012) 191 [1208.5056].
  * (32) C. Romelsberger, _Counting chiral primaries in $N=1$, $d=4$ superconformal field theories_, _Nucl. Phys._ B747 (2006) 329 [hep-th/0510060].
  * (33) S. Nawata, _Localization of $N$=4 Superconformal Field Theory on $S^{1}\times S^{3}$ and Index_, _JHEP_ 11 (2011) 144 [1104.4470].
  * (34) B. Assel, D. Cassani and D. Martelli, _Localization on Hopf surfaces_ , _JHEP_ 08 (2014) 123 [1405.5144].
  * (35) C. Closset, H. Kim and B. Willett, _$\mathcal{N}$ = 1 supersymmetric indices and the four-dimensional A-model_, _JHEP_ 08 (2017) 090 [1707.05774].
  * (36) F. Benini and P. Milan, _A Bethe Ansatz type formula for the superconformal index_ , _Commun. Math. Phys._ 376 (2020) 1413 [1811.04107].
  * (37) A. Arabi Ardehali, J. Hong and J. T. Liu, _Asymptotic growth of the 4d $\mathcal{N}$ = 4 index and partially deconfined phases_, _JHEP_ 07 (2020) 073 [1912.04169].
  * (38) F. Benini and G. Rizi, _Superconformal index of low-rank gauge theories via the Bethe Ansatz_ , 2102.03638.
  * (39) A. G. Lezcano, J. Hong, J. T. Liu and L. A. P. Zayas, _The Bethe-Ansatz approach to the $\mathcal{N}=4$ superconformal index at finite rank_, 2101.12233.
  * (40) O. Aharony, F. Benini, O. Mamroud and P. Milan, “A gravity interpretation for the bethe ansatz expansion of the $N=4$ SYM index.” as quoted in talk by O. Aharony at http://scgp.stonybrook.edu/archives/32982, Nov., 2020.
  * (41) A. González Lezcano, J. Hong, J. T. Liu and L. A. Pando Zayas, _Sub-leading Structures in Superconformal Indices: Subdominant Saddles and Logarithmic Contributions_ , _JHEP_ 01 (2021) 001 [2007.12604].
  * (42) A. Amariti, M. Fazzi and A. Segati, _The superconformal index of $\mathcal{N}=4$ $USp(2N_{c})$ and $SO(N_{c})$ SYM as a matrix integral_, 2012.15208.
  * (43) A. Amariti, M. Fazzi and A. Segati, _Expanding on the Cardy-like limit of the superconformal index of 4d $\mathcal{N}=1$ ABCD SCFTs_, 2103.15853.
  * (44) S. N. M. Ruijsenaars, _First order analytic difference equations and integrable quantum systems_ , _Journal of Mathematical Physics_ 38 (1997) 1069.
  * (45) G. Felder and A. Varchenko, _The elliptic gamma function and $\text{SL}(3,\mathbb{Z})\ltimes\mathbb{Z}^{3}$_, _Advances in Mathematics_ 156 (2000) 44.
  * (46) E. M. Rains, _Limits of elliptic hypergeometric integrals_ , _Ramanujan J._ 18 (2007) 257 [math/0607093].
  * (47) A. P. Kels and M. Yamazaki, _Lens elliptic gamma function solution of the Yang–Baxter equation at roots of unity_ , _J. Stat. Mech._ 1802 (2018) 023108 [1709.07148].
  * (48) A. Arabi Ardehali, _High-temperature asymptotics of supersymmetric partition functions_ , _JHEP_ 07 (2016) 025 [1512.03376].
  * (49) S. Garoufalidis and M. Marino, _On Chern-Simons matrix models_ , math/0601390.
  * (50) D. Gang, _Chern-Simons Theory on $L(p,q)$ Lens Spaces and Localization_, _J. Korean Phys. Soc._ 74 (2019) 1119 [0912.4664].
  * (51) N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla and T. Sharma, _Constraints on Fluid Dynamics from Equilibrium Partition Functions_ , _JHEP_ 09 (2012) 046 [1203.3544].
  * (52) K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz and A. Yarom, _Towards hydrodynamics without an entropy current_ , _Phys. Rev. Lett._ 109 (2012) 101601 [1203.3556].
  * (53) S. Kim, _The Complete superconformal index for N=6 Chern-Simons theory_ , _Nucl. Phys. B_ 821 (2009) 241 [0903.4172].
  * (54) K. Intriligator and N. Seiberg, _Aspects of 3d N=2 Chern-Simons-Matter Theories_ , _JHEP_ 07 (2013) 079 [1305.1633].
  * (55) C. Closset, H. Kim and B. Willett, _Seifert fibering operators in 3d $\mathcal{N}=2$ theories_, _JHEP_ 11 (2018) 004 [1807.02328].
  * (56) A. Kapustin, B. Willett and I. Yaakov, _Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter_ , _JHEP_ 03 (2010) 089 [0909.4559].
  * (57) F. A. Dolan and H. Osborn, _Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories_ , _Nucl. Phys. B_ 818 (2009) 137 [0801.4947].
  * (58) V. P. Spiridonov and G. S. Vartanov, _Elliptic Hypergeometry of Supersymmetric Dualities_ , _Commun. Math. Phys._ 304 (2011) 797 [0910.5944].
  * (59) V. P. Spiridonov and G. S. Vartanov, _Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots, and vortices_ , _Commun. Math. Phys._ 325 (2014) 421 [1107.5788].
  * (60) G. Felder and A. Varchenko, _The elliptic gamma function and SL(3,Z) x Z^3_ , _ArXiv Mathematics e-prints_ (1999) [math/9907061].
  * (61) B. Assel, D. Cassani, L. Di Pietro, Z. Komargodski, J. Lorenzen and D. Martelli, _The Casimir Energy in Curved Space and its Supersymmetric Counterpart_ , _JHEP_ 07 (2015) 043 [1503.05537].
  * (62) A. Arabi Ardehali, L. Cassia and Y. Lü, _From Exact Results to Gauge Dynamics on $\mathbb{R}^{3}\times S^{1}$_, _JHEP_ 08 (2020) 053 [1912.02732].
  * (63) E. Witten, _Quantum Field Theory and the Jones Polynomial_ , _Commun. Math. Phys._ 121 (1989) 351.
  * (64) A. Arabi Ardehali, J. T. Liu and P. Szepietowski, _High-Temperature Expansion of Supersymmetric Partition Functions_ , _JHEP_ 07 (2015) 113 [1502.07737].
  * (65) G. Festuccia and N. Seiberg, _Rigid Supersymmetric Theories in Curved Superspace_ , _JHEP_ 06 (2011) 114 [1105.0689].
  * (66) M. F. Sohnius and P. C. West, _An Alternative Minimal Off-Shell Version of N=1 Supergravity_ , _Phys. Lett._ 105B (1981) 353.
  * (67) M. Sohnius and P. C. West, _The Tensor Calculus and Matter Coupling of the Alternative Minimal Auxiliary Field Formulation of $N=1$ Supergravity_, _Nucl. Phys._ B198 (1982) 493.
  * (68) S. M. Kuzenko and G. Tartaglino-Mazzucchelli, _Three-dimensional N=2 (AdS) supergravity and associated supercurrents_ , _JHEP_ 12 (2011) 052 [1109.0496].
  * (69) S. M. Kuzenko, U. Lindstrom and G. Tartaglino-Mazzucchelli, _Off-shell supergravity-matter couplings in three dimensions_ , _JHEP_ 03 (2011) 120 [1101.4013].
  * (70) S. M. Kuzenko, U. Lindstrom and G. Tartaglino-Mazzucchelli, _Three-dimensional (p,q) AdS superspaces and matter couplings_ , _JHEP_ 08 (2012) 024 [1205.4622].
  * (71) S. M. Kuzenko, U. Lindstrom, M. Rocek, I. Sachs and G. Tartaglino-Mazzucchelli, _Three-dimensional $\mathcal{N}=$ 2 supergravity theories: From superspace to components_, _Phys. Rev. D_ 89 (2014) 085028 [1312.4267].
  * (72) C. Closset, T. T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, _Comments on Chern-Simons Contact Terms in Three Dimensions_ , _JHEP_ 09 (2012) 091 [1206.5218].
  * (73) C. Closset, T. T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, _Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories_ , _JHEP_ 10 (2012) 053 [1205.4142].
  * (74) C. Closset, T. T. Dumitrescu, G. Festuccia and Z. Komargodski, _Supersymmetric Field Theories on Three-Manifolds_ , _JHEP_ 05 (2013) 017 [1212.3388].
  * (75) B. Willett, _Localization on three-dimensional manifolds_ , _J. Phys. A_ 50 (2017) 443006 [1608.02958].
  * (76) O. Aharony, S. S. Razamat, N. Seiberg and B. Willett, _3d dualities from 4d dualities_ , _JHEP_ 07 (2013) 149 [1305.3924].
  * (77) C. Closset, T. T. Dumitrescu, G. Festuccia and Z. Komargodski, _The Geometry of Supersymmetric Partition Functions_ , _JHEP_ 01 (2014) 124 [1309.5876].
  * (78) F. A. H. Dolan, V. P. Spiridonov and G. S. Vartanov, _From 4d superconformal indices to 3d partition functions_ , _Phys. Lett. B_ 704 (2011) 234 [1104.1787].
  * (79) V. P. Spiridonov and G. S. Vartanov, _Elliptic hypergeometric integrals and ’t Hooft anomaly matching conditions_ , _JHEP_ 06 (2012) 016 [1203.5677].
  * (80) V. Niarchos, _Seiberg dualities and the 3d/4d connection_ , _JHEP_ 07 (2012) 075 [1205.2086].
  * (81) A. Gadde and W. Yan, _Reducing the 4d Index to the $S^{3}$ Partition Function_, _JHEP_ 12 (2012) 003 [1104.2592].
  * (82) Y. Imamura, _Relation between the 4d superconformal index and the $S^{3}$ partition function_, _JHEP_ 09 (2011) 133 [1104.4482].
  * (83) F. Benini, T. Nishioka and M. Yamazaki, _4d Index to 3d Index and 2d TQFT_ , _Phys. Rev. D_ 86 (2012) 065015 [1109.0283].
  * (84) L. F. Alday, M. Fluder and J. Sparks, _The Large N limit of M2-branes on Lens spaces_ , _JHEP_ 10 (2012) 057 [1204.1280].
  * (85) D. Cassani and Z. Komargodski, _EFT and the SUSY Index on the 2nd Sheet_ , 2104.01464.
  * (86) V. Jejjala, Y. Lei, S. Van Leuven and W. Li, _$SL(3,\mathbb{Z})$ Modularity and New Cardy Limits of the $\mathcal{N}=4$ Superconformal Index_, 2104.07030.