# When the Fourier transform is one loop exact?

Maxim Kontsevich, Alexander Odesskii

###### Abstract

We investigate the question: for which functions
$f(x_{1},...,x_{n}),~{}g(x_{1},...,x_{n})$ the asymptotic expansion of the
integral $\int
g(x_{1},...,x_{n})e^{{f(x_{1},...,x_{n})+x_{1}y_{1}+...+x_{n}y_{n}\over\hbar}}dx_{1}...dx_{n}$
consists only of the first term. We reveal a hidden projective invariance of
the problem which establishes its relation with geometry of projective
hypersurfaces of the form $\\{(1:x_{1}:...:x_{n}:f)\\}$. We also construct
various examples, in particular we prove that Kummer surface in
$\mathbb{P}^{3}$ gives a solution to our problem.

IHES, 35 route de Chartres, Bures-sur-Yvette, F-91440, France

and

Brock University, 1812 Sir Isaac Brock Way, St. Catharines, ON, L2S 3A1,
Canada

e-mails:

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Formal wave functions
  3. 3 Reformulation of the problem in terms of conical germs
    1. 3.1 Admissible pairs and projectively dual hypersurfaces
    2. 3.2 Projective invariance of the problem
    3. 3.3 Generalization to the projectively dual lower-dimensional cones
  4. 4 Reformulation of the problem in terms of constraints on wave functions
    1. 4.1 Abstract formalism
    2. 4.2 Explicit formulas in general case
    3. 4.3 Algebraic case
    4. 4.4 Explicit formulas in the algebraic case
    5. 4.5 A question about monodromic regular holonomic $D$-modules
  5. 5 Examples of admissible hypersurfaces and corresponding pairs
    1. 5.1 Quadratic hypersurfaces
    2. 5.2 Hypersurfaces admitting quadratic parametrization
    3. 5.3 Ruled surfaces in $\mathbb{P}^{3}$
    4. 5.4 Steiner Roman surface
    5. 5.5 Generalized Steiner Roman hypersurfaces
    6. 5.6 Kummer surfaces in $\mathbb{P}^{3}$
    7. 5.7 Extensions of admissible pairs and families of surfaces of degree four
    8. 5.8 Toric hypersurfaces
    9. 5.9 Segre cubic in $\mathbb{P}^{4}$
  6. 6 Classification of admissible pairs of functions in one variable
  7. 7 Toward a classification of admissible pairs of functions in two variables
  8. 8 A potential application to generalized Dirichlet series
  9. 9 Conjectures and open questions
  10. Appendix. Explicit formulas for equations
  11. Acknowledgements

##  1 Introduction

This paper is inspired by beautiful results of [1] where a number of
surprising identities for the Fourier transforms were discovered, for example

$\int_{\mathbb{R}^{2}}sign(x_{2})|x_{2}|^{-{2\over 3}}e^{i{x_{1}^{3}\over
x_{2}}}e^{i(x_{1}y_{1}+x_{2}y_{2})}dx_{1}dx_{2}={2\pi
i\over\sqrt{3}}\,sign(y_{2})|y_{2}|^{-{2\over 3}}e^{{i\over 27}{y_{1}^{3}\over
y_{2}}}.$

The problem studied in [1] is the following. Let $F$ be a local field, $\psi$
a nontrivial unitary additive character of $F$, and $\chi_{1},...,\chi_{k}$
multiplicative characters of $F$. A complex valued distribution of the form

$\psi(Q(x_{1},...,x_{n}))\prod_{j=1}^{k}\chi_{j}(P_{j}(x_{1},...,x_{n}))$

is called elementary if $Q$ is a rational function and $P_{j}$ are
polynomials. The problem is when the Fourier transform of such a distribution
is also elementary. It was observed in [1] that one can set formally
$\psi(x)=e^{{ix\over\hbar}}$ and use the formal stationary phase method. This
observation leads us to the following formal version of the problem where we
no longer consider actual integrals and deal with the asymptotic expansion of
oscillating integrals. Moreover, our functions $f,g$ (analogous of $Q,P_{j}$
from [1]) are no longer assumed to be rational and can be locally analytic or
just formal germs.

Let $f(x_{1},...,x_{n})$ be a function in $n$ variables such that its Hessian
is not identically zero:

$\det\Bigg{(}{\partial^{2}f\over\partial x_{i}\partial
x_{j}}\Bigg{)}_{i,j}\neq 0.$ (1.1)

Let $\vec{x}=\vec{t}$ be a critical point of the function111Here and in the
sequel we use vector notations like $\vec{x}=(x_{1},...,x_{n})$,
$\vec{t}=(t_{1},...,t_{n})$ etc.

$f(x_{1},...,x_{n})+x_{1}y_{1}+...+x_{n}y_{n}.$ (1.2)

It follows from (1.1) that the mapping $\vec{y}\mapsto\vec{t}$ has a non-
degenerate Jacobian at generic point. Indeed, equating to zero first
derivatives of (1.2) with respect to variables $x_{1},...,x_{n}$ we get

$y_{i}=-{\partial f(t_{1},...,t_{n})\over\partial t_{i}},~{}~{}~{}i=1,...,n$
(1.3)

and the Jacobian of this map is proportional to the Hessian (1.1) at
$\vec{x}=\vec{t}$. Let $g(x_{1},...,x_{n})$ be another function in
$x_{1},...,x_{n}$. By the formal Fourier transform of the function
$g(\vec{x})e^{{f(\vec{x})\over\hbar}}$ we mean the perturbative expansion of
the formal integral

$\int
g(x_{1},...,x_{n})e^{{f(x_{1},...,x_{n})+x_{1}y_{1}+...+x_{n}y_{n}\over\hbar}}dx_{1}...dx_{n}$

given by the stationary phase method at the critical point $\vec{x}=\vec{t}$.
Recall that this expansion has the form

$\int
g~{}e^{{f+x_{1}y_{1}+...+x_{n}y_{n}\over\hbar}}dx_{1}...dx_{n}=(2\pi\hbar)^{{n\over
2}}e^{{\hat{f}\over\hbar}}\det\Bigg{(}-{\partial^{2}f\over\partial
x_{i}\partial x_{j}}\Bigg{)}_{i,j}^{-{1\over
2}}\Bigg{\rvert}_{\vec{x}=\vec{t}}~{}\sum_{k=0}^{\infty}A_{k}\hbar^{k}$ (1.4)

where coefficients $A_{k}$ of formal power series in $\hbar$ can be written as
differential polynomials in $f(t_{1},...,t_{n}),~{}g(t_{1},...,t_{n})$ with
coefficients in $\mathbb{Q}$, divided by
$\det\Big{(}{\partial^{2}f\over\partial x_{i}\partial
x_{j}}\Big{)}_{i,j}^{3k}\Big{\rvert}_{\vec{x}=\vec{t}}\,$, see Appendix.

We want to study the following question: for which functions $f,g$ we have

$A_{k}=0\text{~{}~{}~{}~{}for all~{}~{}~{}~{}}k\geq 1.$ (1.5)

This condition can be written more explicitly in the form222Here and in the
sequel such identities for integrals with unspecified domain of integration
mean that the r.h.s. is the perturbative expansion of the l.h.s. given by
stationary phase method. Moreover, we often write the r.h.s. up to
multiplication by a fourth root of unity $\pm 1,\pm i$.

$\int
g(\vec{x})e^{{f(\vec{x})+\vec{x}\cdot\vec{y}\over\hbar}}d^{n}\vec{x}=(2\pi\hbar)^{{n\over
2}}\hat{g}(\vec{y})e^{{\hat{f}(\vec{y})\over\hbar}}.$ (1.6)

Here

$\hat{f}(\vec{y})=(f(\vec{x})+\vec{x}\cdot\vec{y})\Big{\rvert}_{\vec{x}=\vec{t}}$
(1.7)

is the Legendre transform of function $f$, i.e. the critical value of
$f(\vec{x})+\vec{x}\cdot\vec{y}$, and

$\hat{g}(\vec{y})=g(\vec{x})\cdot\det\Bigg{(}-{\partial^{2}f\over\partial
x_{i}\partial x_{j}}\Bigg{)}_{i,j}^{-{1\over
2}}\Bigg{\rvert}_{\vec{x}=\vec{t}}.$ (1.8)

By a loose analogy with the terminology from physics literature we call the
condition (1.6) the 1-loop exactness of the formal Fourier transform.333The
name is not totally precise as the Feynman graphs appearing in the expansion
(1.4) are not necessarily connected, see Appendix.

Definition 1.1. The pair of functions $f,g$ is called admissible if the
Hessian of $f$ is not identically zero (see (1.1)), the function $g$ is not
identically zero, and the identity (1.6) holds.

Notice that if we fix a function $f$, then the set of functions $g$ such that
the pair $f,g$ is admissible or $g=0$, is a vector space, which we denote by
$V_{f}$.

Definition 1.2. The rank of a function $f$ is the dimension of vector space of
functions

$V_{f}=\\{g~{}|~{}f,g~{}\text{is an admissible pair or}~{}g=0\\}.$

The rank of an admissible pair $f,g$ is the rank of $f$.

Definition 1.3. A function $f$ is admissible if its rank is larger than zero.
In other words, $f$ is admissible if there exists a (non-zero) function $g$
such that pair $f,g$ is admissible.

Sometimes, if $f$ is fixed and clear from the context, we will call $g$
admissible if $f,g$ is an admissible pair.

Remark 1.1. First, in the case when $f$ is a concave function defined globally
on $\mathbb{R}^{n}$, and satisfying the condition

$\lim_{|\vec{x}|\to\infty}{f(\vec{x})\over|\vec{x}|}=-\infty~{}~{}~{}(\text{e.g.}~{}~{}~{}f(\vec{x})=-{1\over
2}\sum_{i=1}^{n}x_{i}^{2}~{}),$

and $g$ is also defined globally,

$|g(\vec{x})|\leq C_{1}e^{C_{2}|\vec{x}|},$

the integral
$\int_{\mathbb{R}^{n}}g(\vec{x})e^{{f(\vec{x})+\vec{x}\cdot\vec{y}\over\hbar}}d\vec{x}$
is convergent for $\hbar>0$ and admits the asymptotic expansion as in (1.4).

In the general case where $f,g$ are germs of analytic functions, or even
formal power series with coefficients in a field ${\bf k}$ of characteristic
zero, by the formal integral we mean the expression (1.4) where all terms
$A_{i}$ are differential polynomials divided by an integer power of the
Hessian and therefore make sense, and the front factors are considered as
formal symbols.

Also, in the usual Fourier transform one uses $\sqrt{-1}$ in the exponent. We
omit it for our convenience, in order to simplify formulas.

In the sequel we will also replace $\det\Big{(}-{\partial^{2}f\over\partial
x_{i}\partial x_{j}}\Big{)}_{i,j}$ by $\det\Big{(}{\partial^{2}f\over\partial
x_{i}\partial x_{j}}\Big{)}_{i,j}$, which changes the result by a fourth root
of unity.

Remark 1.2. Here we explain an explicit procedure producing coefficients
$A_{i}$ above, see Appendix for details. The calculation of the integral in
the l.h.s. of (1.4) near non-degenerate critical point can be reduced (after
some shift of variables $x_{1},...,x_{n}$) to the following case.

Let $F=F_{2}+F_{3}+...$ be a formal power series in $n$ variables
$x_{1},...,x_{n}$ with coefficients in a field ${\bf k}\supset\mathbb{Q}$
where $F_{i}$ are homogeneous polynomials of degree $i$, and $F_{2}$ is a non-
degenerate quadratic form. We denote by $F^{\prime\prime}_{2}$ the
corresponding symmetric matrix. Let $g=1+...\in{\bf k}[[x_{1},...,x_{n}]]$ be
another power series (series $g$ starts with $1$ just for convenience). The
formal integral

${(\det F^{\prime\prime}_{2})^{{1\over 2}}\over(2\pi\hbar)^{{n\over 2}}}\int
g(\vec{x})e^{{F(\vec{x})\over\hbar}}dx_{1}...dx_{n}=1+...\in{\bf k}[[\hbar]]$

can be defined in the following way:

First we rescale variables $x_{i}=\sqrt{\hbar}~{}\tilde{x}_{i}$. After that we
have

${F(\vec{x})\over\hbar}=F_{2}(\tilde{x}_{1},...,\tilde{x}_{n})+\hbar^{{1\over
2}}F_{3}(\tilde{x}_{1},...,\tilde{x}_{n})+\hbar^{{2\over
2}}F_{4}(\tilde{x}_{1},...,\tilde{x}_{n})+...$

and

$g(\vec{x})e^{{F(\vec{x})\over\hbar}}=g(\hbar^{{1\over
2}}\vec{\tilde{x}})e^{F_{2}(\vec{\tilde{x}})}e^{\hbar^{{1\over
2}}F_{3}(\vec{\tilde{x}})+\hbar^{{2\over 2}}F_{4}(\vec{\tilde{x}})+...}$

The r.h.s. of this formula can be expanded as power series in $\hbar^{{1\over
2}}$ and $\tilde{x}_{1},...\tilde{x}_{n}$. In this way we reduced to computing
the integrals of the form $\int he^{F_{2}}d\tilde{x}_{1}...d\tilde{x}_{n}$
where $h$ is a monomial in $\tilde{x}_{1},...\tilde{x}_{n}$. This formal
integral is defined to be zero if degree of $h$ is odd, and as ${1\over
m!}\Delta^{m}_{F_{2}}(h)\int e^{F_{2}}d\tilde{x}_{1}...d\tilde{x}_{n}$ if
degree of $h$ is $2m$. Here $\Delta_{F_{2}}=-{1\over
2}\sum_{i,j}b_{ij}\partial_{\tilde{x}_{i}}\partial_{\tilde{x}_{j}}$ where
$(b_{ij})=(F^{\prime\prime}_{2})^{-1}$, and we formally declare $\int
e^{F_{2}}d\tilde{x}_{1}...d\tilde{x}_{n}:={(2\pi\hbar)^{{n\over 2}}\over(\det
F^{\prime\prime}_{2})^{{1\over 2}}}.$

Notice that the final expression $\sum_{i=0}^{\infty}A_{i}\hbar^{i}$ is a
power series in $\hbar$ because integrals with monomials of odd degree are
zero.

Remark 1.3. If $f,g$ is an admissible pair, then the pair $\hat{f},\hat{g}$ of
functions from the r.h.s. of (1.6) is also admissible, because the Fourier
transform is essentially an involution. Moreover, we have isomorphism of
vector spaces $V_{f}\cong V_{\hat{f}}$ given by $g\mapsto\hat{g}$. In
particular, $\dim V_{f}=\dim V_{\hat{f}}$ so $f$ and $\hat{f}$ have the same
rank.

Remark 1.4. Let $\Sigma\subset\mathbb{P}^{n+1}$ be projective hypersurface
locally defined by

$x_{n+1}=x_{0}f\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over x_{0}}\Big{)}.$ (1.9)

Then its projective dual hypersurface
$\widehat{\Sigma}\subset\mathbb{P}^{n+1}$ is locally defined by

$y_{0}=y_{n+1}\hat{f}\Big{(}{y_{1}\over y_{n+1}},...,{y_{n}\over
y_{n+1}}\Big{)}$ (1.10)

where $\hat{f}$ is the Legendre transform of $f$ given by (1.7) and
coordinates $(y_{0},...,y_{n+1})$ are dual to $(x_{0},...,x_{n+1})$. This
projective duality plays an important role in this paper.

Let us describe the content of the paper.

In Section 2 we introduce the formalism of formal wave functions which allows
to write expressions involving exponentials, integrals and delta functions in
a purely algebraic context over an arbitrary field $\bf k$ of characteristic
zero.

In Section 3 we reformulate our problem in terms of projectively dual
hypersurfaces defined by (1.9) and (1.10). We show that admissibility
condition does not depend on the choice of projective coordinates. In this way
we deduce a projective invariance of the original problem: if $f$ is
admissible and we make an arbitrary linear change of variables

$x_{i}=\sum_{j=0}^{n+1}a_{i,j}\tilde{x}_{j},~{}~{}~{}i=0,...,n+1$

in (1.9), solve with respect to $\tilde{x}_{n+1}$ writing the result in the
form

$\tilde{x}_{n+1}=\tilde{x}_{0}\tilde{f}\Big{(}{\tilde{x}_{1}\over\tilde{x}_{0}},...,{\tilde{x}_{n}\over\tilde{x}_{0}}\Big{)},$
(1.11)

then $\tilde{f}$ is also admissible for arbitrary non-degenerate matrix
$(a_{i,j})\in GL(n+2)$.

In Section 4 we explain how to reduce our problem in the case of algebraic
hypersurfaces to a finite system of differential equations for $f,g$. Notice
that in general our system of differential equations (1.5) is infinite. In the
special case when $g$ is a solution of a regular holonomic system of
differential equations, our question about admissible pairs essentially
reduces to the following interesting problem concerning algebraic holonomic
$D$-modules:

find regular holonomic $D$-modules $M$ on the affine space $\mathbb{A}^{n+2}$
which are monodromic (i.e. the action of the Euler operator
$\sum_{i=0}^{n+1}x_{i}\partial_{x_{i}}$ is locally finite) and such that the
singular support of $M$ does not contain conormal bundles to
$\mathbb{A}^{n+2}$ and $\\{0\\}\subset\mathbb{A}^{n+2}$.

In Section 5 we study numerous examples. In particular, we prove that Kummer
quartic surface in $\mathbb{P}^{3}$ and Segre cubic in $\mathbb{P}^{4}$ are
both admissible. We also construct a huge family of admissible functions for
arbitrary $n\geq 2$. These functions are defined parametrically as

$f={\phi_{n+1}\over\phi_{0}},~{}~{}~{}x_{i}={\phi_{i}\over\phi_{0}},~{}~{}~{}i=1,...,n,$
$\phi_{i}={1\over 2}\sum_{j,k=0}^{n}a_{i,j,k}u_{j}u_{k},~{}~{}~{}i=0,...,n+1$

where $u_{0},...,u_{n}$ are coordinates on the corresponding hypersurface,
$a_{i,j,k}\in{\bf k}$ are constants and $a_{i,k,j}=a_{i,j,k}.$ We prove that
any non-degenerate function of this family (i.e. with the generically non-
vanishing Hessian) is admissible and $\dim V_{f}\geq n+1$ for $n>2$. We also
construct examples of such functions for arbitrary $n\geq 2$ with the rank
equal to $(n+1)!$.

In Section 6 we present the classification of admissible pairs of functions in
one variable.

In Section 7 we show some classification results of admissible pairs of
functions in two variables.

In Section 8 we outline a potential application of our studies to the
construction of generalized Dirichlet series based on the Poisson summation
formula and the Mellin transform.

In Section 9 we formulate several conjectures and open questions. Notice that
some other conjectures and open questions are discussed in the main part of
the paper.

In Appendix we recall how to write explicitly the infinite system of
differential equations (1.5) for $f,g$ based on the stationary phase method
and the Feynman diagrams technique.

##  2 Formal wave functions

Here we introduce a rigorous language of wave functions which allows us to use
expressions like exponential functions depending on small parameter $\hbar$
and written as $\exp(f(x_{1},\dots,x_{n})/\hbar)$, or delta functions
$\delta(f(x_{1},\dots,x_{n}))$ etc, in a purely algebraic situation, when
neither coordinates nor functions take real or complex values. The whole
calculus makes sense over any field $\bf k$ of characteristic zero.

Let $M=\mathbb{A}^{2n}$ be an affine space of dimension $2n$ over $\bf k$,
endowed with a translationally-invariant symplectic structure. In other words,
we have global coordinates444In the sequel we will often use notation
$y_{1}=x_{n+1},...,y_{n}=x_{2n}$. $(x_{1},\dots,x_{2n})$ on $M$ defined up to
affine-symplectic transformations

$x_{i}\mapsto\sum_{j=1}^{2n}a_{ij}x_{j}+b_{j}$

where $(a_{ij})_{1\leq i,j,\leq 2n}$ is an invertible matrix preserving the
standard symplectic 2-form

$\omega=\sum_{i=1}^{n}dx_{i}\wedge dx_{i+n}\,.$

Denote by $(\gamma_{ij})_{1\leq i,j\leq 2n}$ the tensor for the inverse bi-
vector field:

$\gamma_{ij}:=\begin{cases}1&\mbox{ if }j=i+n,\quad 1\leq i\leq n,\\\
-1&\mbox{ if }i=j+n,\quad 1\leq j\leq n,\\\ 0&\mbox{ otherwise}\end{cases}$

Let $\hbar$ be a formal variable. We define the canonical Moyal star-product
on the vector space

${\bf k}[x_{1},\dots,x_{2n},\hbar]$

by the formula

$f\star g:=\sum_{k=0}^{\infty}{\hbar^{k}\over k!}\left[\left(\sum_{1\leq
i,j\leq 2n}{\gamma_{ij}\over 2}{\partial\over\partial
x_{i}^{(1)}}{\partial\over\partial
x_{j}^{(2)}}\right)^{k}(f((x_{i}^{(1)})_{1\leq i\leq 2n}\cdot
g((x_{i}^{(2)})_{1\leq i\leq
2n})\right]\Bigg{|}_{x_{i}^{(1)}=x_{i}^{(2)}=x_{i},\,\,\,\,i=1,\dots,2n}$

where in the square brackets we consider functions on $M\times M$ endowed
coordinates

$(x_{1}^{(1)},\dots,x_{2n}^{(1)}),~{}(x_{1}^{(2)},\dots,x_{2n}^{(2)})$

(i.e. two copies of the original coordinates on $M$). One can rewrite the
above formula as

$f\star g=\left[\exp\left({1\over\hbar}\sum_{1\leq i,j\leq 2n}{1\over
2}\gamma_{ij}{\partial\over\partial
x_{i}^{(1)}}\boxtimes{\partial\over\partial x_{j}^{(2}}\right)(f\boxtimes
g)\right]\Bigg{|}_{\text{diagonal}}\,.$

Algebra $({\bf k}[x_{1},\dots,x_{2n},\hbar],\star)$ over $\bf k$ is an
associative unital algebra generated by $(2n+1)$ elements
$(\hat{x}_{i})_{i=1,\dots,2n},\hbar$ where $\hat{x}_{i}$ correspond to
$x_{i}\in{\bf k}[x_{1},\dots,x_{2n},\hbar]$, satisfying the relations

$[\hat{x}_{i},\hat{x}_{j}]=\gamma_{ij}\hbar,\quad[\hat{x}_{i},\hbar]=0$

and can be identified with the algebra555We will see later that it is more
natural to consider differential operators acting on half-densities instead of
functions. of polynomial $\hbar$-differential operators in $n$ variables
$x_{1},\dots,x_{n}$ by

$\hat{x}_{i}\mapsto
x_{i},\quad\hat{x}_{n+i}\mapsto\hbar{\partial\over\partial{x_{i}}},\qquad\forall
i=1,\dots,n\,.$

The Moyal product is covariant with respect to the action of the group of
affine-symplectic automorphisms of $M$. It extends to the ${\bf
k}[[\hbar]]$-linear product on $A[[\hbar]]$ where $A$ is the algebra of
functions on a Zariski open subset of $M$, or analytic functions in a Stein
open domain in $M$ if ${\bf k}=\mathbb{C}$, or $C^{\infty}$-functions in an
open domain if ${\bf k}=\mathbb{R}$. Also, for general field $\bf k$ of
characteristic zero, one can take $A$ to be the algebra of formal power series
at a given point $m\in M$.

Let $L$ be a Lagrangian submanifold of $M$ in a broad sense, i.e. an algebraic
subvariety, or analytic/smooth in the case ${\bf k}=\mathbb{C}$ or ${\bf
k}=\mathbb{R}$, or a germ of such subvariety, or even a formal germ at some
point of $M$.

We assume that a spin-structure on $L$ is given, which means that we are given
a line bundle on $L$ whose tensor square is identified with the canonical
bundle $K_{L}$, i.e. the bundle $\wedge^{n}T^{*}_{L}$ of top-degree forms on
$L$. The chosen square root bundle we denote by $K_{L}^{\otimes 1/2}$. In the
case of a germ, or a formal germ at point $m\in M$, it is sufficient to choose
a square root of the fiber $(K_{L})_{Im}$ at the base point $m$.

Our main goal is the following

###### Construction 1

. With a given pair $(L,K_{l}^{\otimes 1/2})$ we associate a module (or
more, precisely, a sheaf of modules) ${\mathcal{W}F}_{L,K_{L}^{\otimes 1/2}}$
over the quantum algebra $A[[\hbar]]$ where $A$ is the algebra of functions in
the formal completion of $M$ at $L$.

Elements of the vector space ${\mathcal{W}F}_{L,K_{L}^{\otimes 1/2}}$ are
called formal wave functions supported on $L$.

There are several approaches to this constructions. The one presented below is
an explicit one in affine symplectic coordinates, but it is based on a
nontrivial consistency check666There is another approach (not described in
this paper) based on Gelfand-Kazhdan type formal geometry..

First, we give the definition of the module ${\mathcal{W}F}_{L,K_{L}^{\otimes
1/2}}$ in local coordinates. Let $(x_{1},\dots,x_{2n})$ be global affine
symplectic coordinates on $M$ such that the symplectic form is the standard
one, and such that locally near a point of $L$ the projection $\pi$ to the
coordinate space $\mathbb{A}^{n}$ by first $n$ coordinates
$(x_{1},\dots,x_{n})$ has a non-zero Jacobian. Hence, locally $L$ is a graph
of a closed 1-form on $\mathbb{A}^{n}$:

$x_{n+i}=\alpha_{i}(x_{1},\dots,x_{n})\quad\forall i=1,\dots,n,\qquad
d\alpha=0,\quad\alpha:=\sum_{i=1}^{n}\alpha_{i}dx_{i}\,.$

Let us also choose a generator of $K_{L}^{\otimes 1/2}$ whose tensor square is
$\pi^{*}(dx_{1}\wedge\dots\wedge dx_{n})$. We will denote in short this
generator by $(dx_{1}\wedge\dots\wedge dx_{n})^{1/2}$, it is well-defined up
to a sign.

After we make the choices from above (i.e. affine symplectic coordinate system
$(x_{1},\dots,x_{2n})$ and the generator $(dx_{1}\wedge\dots\wedge
dx_{n})^{1/2}$), we declare ${\mathcal{W}F}_{L,K_{L}^{\otimes 1/2}}$ to be the
${\bf k}[[\hbar]]$-module

$\mathcal{O}(L)[[\hbar]]\simeq\mathcal{O}[\pi(L)][[\hbar]]$

where $\mathcal{O}(L)$ is the algebra of functions on $L$, identified with
functions on open domain (or a formal germ) $\pi(L)\subset\mathbb{A}^{n}$,
with the action of generators $\hat{x}_{i},\quad i=1,\dots,2n$ of the quantum
algebra given by

$\hat{x}_{i}\mapsto
x_{i},\quad\hat{x}_{n+i}\mapsto\hbar{\partial\over\partial{x_{i}}}+\alpha_{i},\qquad\forall
i=1,\dots,n\,.$

It is a nontrivial fact that this action extends by continuity from polynomial
elements to the functions on the affine space $\mathbb{A}^{2n}\simeq M$
defined in the formal completion of $L\subset M$.

Notationally, it is convenient to choose locally a primitive of closed 1-form
$\alpha$, i.e. a function $f(x_{1},\dots,x)$ defined on the domain
$\pi(L)\subset\mathbb{A}^{n}$, such that

$\alpha=df\iff\alpha_{i}={\partial f\over\partial x_{i}}\quad\forall
i=1,\dots,n\,.$

Then the element of ${\mathcal{W}F}_{L,K_{L}^{\otimes 1/2}}$ corresponding to
a series

$g=g(x_{1},\dots,x_{n};\hbar)=\sum_{k\geq
0}g_{k}\hbar^{k}\in\mathcal{O}[\pi(L)][[\hbar]]$

we denote by the formal product

$\psi=e^{{f(x_{1},\dots,x_{n})\over\hbar}}g(x_{1},\dots,x_{n};\hbar)\cdot(dx_{1}\wedge\dots\wedge
dx_{n})^{1/2}\,.$

The choice of the primitive is irrelevant: if we shift it by a constant $f\to
f+const$ then we formally multiply the expression above by
$e^{{const\over\hbar}}$ without affecting series $g$.

Now we want to study the dependence of the description of
${\mathcal{W}F}_{L,K_{L}^{\otimes 1/2}}$ under the change of choices made.
First, for a given affine symplectic coordinate system $(x_{1},\dots,x_{2n})$,
if we change the square root $(dx_{1}\wedge\dots\wedge dx_{n})^{1/2}$ by sign,
then the isomorphism

${\mathcal{W}F}_{L,K_{L}^{\otimes
1/2}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}O[\pi(L)][[\hbar]]$

will be also changed by sign.

Next, let us change the affine symplectic coordinate system
$(x_{1},\dots,x_{2n})$ preserving first $n$ coordinates $x_{1},\dots,x_{n}$:

$x_{i}\to x_{i},\quad x_{n+i}\to
x_{n+i}+\sum_{j=1}^{n}b_{ij}x_{j}+c_{i},\qquad i=1,\dots,n,\quad
b_{ij}=b_{ji}\in\mathbf{k},\quad c_{i}\in\mathbf{k}\,.$ (2.12)

Then we multiply the corresponding formal wave function $\psi$ by

$\psi\to e^{{1\over\hbar}\left({1\over
2}\sum_{i,j}b_{ij}x_{i}x_{j}+\sum_{i}c_{i}x_{i}\right)}\psi$

If we apply an affine symplectic transformation associated with an invertible
$(n\times n)$-matrix $a=(a_{ij})_{1\leq i,j\leq n}$:

$x_{i}\to x_{i}^{\prime}=\sum_{j=1}^{n}a_{ij}x_{j},\quad x_{n+i}\to
x_{n+i}^{\prime}=\sum_{j=1}^{n}(a^{-1})_{ji}x_{n+j}\qquad\forall i=1,\dots,n$
(2.13)

then we change the wave function by

$e^{{1\over\hbar}f}g\to e^{{1\over\hbar}f^{\prime}}g^{\prime},\qquad
f^{\prime}(\vec{x}):=f(\vec{x}\,^{\prime}(\vec{x})),\quad
g^{\prime}(\vec{x};\hbar):=g(\vec{x}\,^{\prime}(\vec{x});\hbar)\cdot\det(a)^{-1/2}\,.$

We can now describe the dependence of the description of
${\mathcal{W}F}_{L,K_{L}^{\otimes 1/2}}$ under the (almost) general affine
symplectic transformation. Namely, let us assume that $(x_{1},\dots,x_{2n})$
and $(x_{1}^{\prime},\dots,x_{2n}^{\prime})$ are two affine symplectic
coordinate systems on $M$ such that both projections from $L$ to the affine
space $\mathbb{A}^{n}$ given by $(x_{1},\dots,x_{n})$ and
$(x_{1}^{\prime},\dots,x_{n}^{\prime})$ are open embeddings. Let us make an
assumption (which is an open condition) that
$(x_{1},\dots,x_{n},x_{1}^{\prime},\dots,x_{n}^{\prime})$ form a system of
coordinates on $M$. By applying the above modifications (2.12),(2.13), we may
assume that

$x_{i}^{\prime}=x_{n+i},\quad x_{n+i}^{\prime}=-x_{i},\qquad\forall
i=1,\dots,n$

Then we declare that the corresponding formal wave functions undergo the
formal Fourier transform:

$g(\vec{x};\hbar)e^{{f(\vec{x})\over\hbar}}\to\tilde{g}(\vec{y};\hbar)e^{{\tilde{f}(\vec{y})\over\hbar}}:={1\over(2\pi\hbar)^{n/2}}\int
g(\vec{x};\hbar)e^{{f(\vec{x})+\vec{x}\cdot\vec{y}\over\hbar}}d^{n}\vec{x}$

where the integral in the r.h.s. is understood as the asymptotic expansion,
calculated via the stationary phase method. In particular, the exponent
$\tilde{f}$ is the Legendre transform of $f$:

$\hat{f}(\vec{y})=\text{Critical value of }f(\vec{x})+\vec{x}\cdot\vec{y}\,.$

The consistency check mentioned before, says that for three affine symplectic
coordinate systems on $M$: $(x_{1},\dots,x_{2n})$,
$(x_{1}^{\prime},\dots,x_{2n}^{\prime})$ and
$(x_{1}^{\prime\prime},\dots,x_{2n}^{\prime\prime})$, the passage from the
first to the second, and then from the second to the third, coincides with the
passage from the first to the third. Here we will give the sketch of the proof
which is not purely algebraic and is based partially on analysis.

After making the assumption that the triple of coordinate systems under
consideration is sufficiently generic, the consistency question can be reduced
to the following equality

$\int\Bigg{(}e^{-{\vec{y}\cdot\vec{y}\over 2\hbar}}\Big{(}\int
g(\vec{x})e^{f(\vec{x})+\vec{x}\cdot\vec{y}\over\hbar}d^{n}\vec{x}\Big{)}\Bigg{)}e^{\vec{y}\cdot\vec{z}\over\hbar}d^{n}\vec{y}=(2\pi\hbar)^{n/2}e^{{\vec{z}\cdot\vec{z}\over
2\hbar}}\int e^{{1\over\hbar}(f(\vec{x})+{\vec{x}\cdot\vec{x}\over
2}+\vec{x}\cdot\vec{z})}d^{n}\vec{x}$ (2.14)

where $f(\vec{x})$ is a formal power series starting with quadratic terms

$f(\vec{x})=-{1\over 2}\sum_{ij}b_{ij}x_{i}x_{j}+\dots$ (2.15)

where symmetric matrix $(b_{ij})_{1\leq i,j\leq n}$ has no eigenvalues equal
to $0$ or $1$.

In the case $\mathbf{k}=\mathbb{R}$, function $f$ being a global strictly
concave smooth real-valued function on $\mathbb{R}^{n}$ such that
$f(\vec{x})\leq-{c\over 2}\vec{x}\cdot\vec{x}$ for some $c>1$ and any smooth
function $g$ of at most exponential growth at infinity, all the integrals in
(2.14) are absolutely convergent. The equality follows because the l.h.s can
be rewritten as

$\int\int
g(\vec{x})e^{{1\over\hbar}(f(\vec{x})+\vec{x}\cdot\vec{y}+\vec{y}\cdot\vec{z}-{\vec{y}\cdot\vec{y}\over
2})}d^{n}\vec{x}\,d^{n}\vec{y}$

and then identified with r.h.s. using the rewriting

$\vec{x}\cdot\vec{y}+\vec{y}\cdot\vec{z}-{\vec{y}\cdot\vec{y}\over
2}=-{\vec{w}\cdot\vec{w}\over 2}+{\vec{x}\cdot\vec{x}\over
2}+{\vec{z}\cdot\vec{z}\over
2}+\vec{x}\cdot\vec{z},\quad\vec{w}:=\vec{y}-\vec{x}-\vec{z}\,.$

Each term in the $\hbar$-expansion of the equality (2.14) is a polynomial
identity with rational coefficients involving finitely many Taylor
coefficients of $f,g$. The fact that it holds for real $C^{\infty}$ examples
as above giving Zariski dense subsets of possible Taylor coefficients of
finite jets of $f,g$ at $0$ implies that (2.14) holds for arbitrary formal
series $f,g$ with coefficients in any field $\mathbf{k}\supset\mathbb{Q}$.
This concludes the proof of the consistency check.

Remark 2.1. One can further generalize equality (2.14). Namely, each term in
$\hbar$-expansion is an equality of certain finite sums of numbers obtained by
contraction of upper and lower indices for certain symmetric tensors in
$n$-dimensional space. The tensors under consideration are Taylor coefficients
of series $f,g$ and the inverses to symmetric matrices $B$ and $B-{\bf 1}_{n}$
where $B=(b_{ij})_{1\leq i,j\leq n}$ is the (negative) Hessian of $f$ at $0$
(see (2.15)) and ${\bf 1}_{n}$ is the identity matrix (could be replaced by
any non-degenerate quadratic form). The fact that the equality holds in any
positive dimension $n\geq 0$ implies by the Weyl’s fundamental theorem in
invariant theory that it holds by purely formal reasons, as the cancellation
of linear combination of oriented graphs controlling the contraction of
indices. Therefore, the equality (2.14) makes sense and holds in arbitrary
$\mathbb{Q}$-linear rigid symmetric monoidal category, like, e.g. finite-
dimensional super vector spaces. Hence the construction

$(L,K_{L}^{\otimes 1/2})\leadsto{\mathcal{W}}F_{L,K_{L}^{\otimes 1/2}}$

can be extended to the case of super manifolds.

In what follows we will not specify the choice of $K_{L}^{\otimes 1/2}$ and
hence omit it from the notation.

Remark 2.2. Let $f,g$ be an admissible pair. The formal product

$\psi=g(x_{1},...,x_{n})e^{{1\over\hbar}f(x_{1},...,x_{n})}(dx_{1}\wedge...\wedge
dx_{n})^{{1\over 2}}$

is an element of ${\mathcal{W}F}_{L}$ where $L\subset\mathbb{A}^{2n}$ is the
graph of $df$, a germ of Lagrangian submanifold in
$\mathbb{A}^{2n}=T^{*}\mathbb{A}^{n}$. Simultaneously,
$L\subset\mathbb{A}^{2n}=T^{*}(\mathbb{A}^{n})^{*}$ is the graph of $d\hat{f}$
where $\hat{f}(y_{1},...,y_{n})$ is the Legendre transform of $f$, see (1.7).

The admissible pair $\hat{f},\hat{g}$ obtained by the Fourier transform from
$f,g$ gives the same element $\hat{\psi}=\psi\in{\mathcal{W}F}_{L}$. Our
initial question about finding admissible pairs can be reformulated as the
question about finding elements $\psi\in{\mathcal{W}F}_{L}$ such that in two
descriptions of ${\mathcal{W}F}_{L}$ corresponding to the projections either
to coordinates $x_{1},...,x_{n}$ or to $x_{n+1},...,x_{2n}$, the functions
$g(x_{1},...,x_{n},\hbar)$ (and $\hat{g}(x_{n+1},...,x_{2n},\hbar)$)
associated with $\psi=\hat{\psi}$ do not depend on $\hbar$.

Finally, we explain how to interpret in terms of formal wave functions
expressions involving delta functions. Let us assume that we have a (germ) of
$k$-dimensional submanifold in a $n$-dimensional affine space. After making an
affine change of coordinates, we may assume that the submanifold under the
consideration is given the graph of a map form an open domain in
$\mathbb{A}^{k}$ to $\mathbb{A}^{n-k}$, i.e. given by

$\displaystyle x_{k+1}$ $\displaystyle=\phi_{1}(x_{1},\dots,x_{k})$
$\displaystyle x_{k+2}$ $\displaystyle=\phi_{2}(x_{1},\dots,x_{k})$
$\displaystyle\dots$ $\displaystyle x_{n}$
$\displaystyle=\phi_{n-k}(x_{1},\dots,x_{k})\,.$

Assume that we are also given two functions $f(x_{1},\dots,x_{k})$ and
$g(x_{1},\dots,x_{k})$. We would like to make sense of the following
expression:

$g(x_{1},\dots,x_{k})e^{{1\over\hbar}f(x_{1},\dots,x_{k})}\prod_{i=1}^{n-k}\delta(x_{k+i}-\phi_{i}(x_{1},\dots,x_{k}))\,.$

which is an element of ${\mathcal{W}F}_{L}$ where $L\subset
M\simeq\mathbb{A}^{2n}=T^{*}\mathbb{A}^{n}$ is the conormal bundle to the
$k$-dimensional submanifold in $\mathbb{A}^{n}$ defined above. This can be
achieved by making the Fourier transform in variables $(x_{k+1},\dots,x_{n})$:

$\int
g(x_{1},\dots,x_{k})e^{{1\over\hbar}(f(x_{1},\dots,x_{k})+\sum_{i=1}^{n-k}x_{k+i}y_{i})}\prod_{i=1}^{n-k}\delta(x_{k+i}-\phi_{i}(x_{1},\dots,x_{k}))\prod_{i=1}^{n-k}dx_{k+i}=\\\
=g(x_{1},\dots,x_{k})e^{{1\over\hbar}(f(x_{1},\dots,x_{k})+\sum_{i=1}^{n-k}\phi_{i}(x_{1},\dots,x_{k})y_{i})}\,.$

So, we see that new exponent which is function in variables
$x_{1},\dots,x_{k};y_{1},\dots,y_{n-k}$

$f(x_{1},\dots,x_{k})+\sum_{i=1}^{n-k}\phi_{i}(x_{1},\dots,x_{k})y_{i}$

which happen to be a linear function in $(n-k)$ variables
$y_{1},\dots,y_{n-k}$.

In our formalism we have

$\delta(x)={1\over 2\pi\hbar}\int e^{{xy\over\hbar}}dy.$ (2.16)

Recall that all our identities hold up to fourth root of unity (see
Introduction). In the case of the actual Dirac distribution $\delta(x)$ on
$\mathbb{R}$, the exact formula is

$\delta(x)={1\over 2\pi\hbar}\int_{-\infty}^{\infty}e^{{xy\over
i\hbar}}dy,~{}~{}~{}\hbar>0.$

Remark 2.3. Algebra ${\bf k}[x_{1},...,x_{2n},\hbar]$ with Moyal star product
(as well as its completions associated with open domain in $\mathbb{A}^{2n}$
or formal completions) has a canonical derivation over $\bf k$ given by

$\tau(x_{i})={1\over 2}x_{i},~{}~{}~{}\tau(\hbar)=\hbar.$

If $L\subset\mathbb{A}^{2n}$ is conical, then $\tau$ admits a natural
extension $\tau_{L}$ to ${\mathcal{W}F}_{L}$. Locally, if $x_{1},...,x_{n}$
are coordinates, then

$L=\text{graph}~{}dF_{L}(x_{1},...,x_{n})$

where $F_{L}$ is homogeneous of degree 2. Let

$\psi=G_{L}(x_{1},...,x_{n},\hbar)e^{{F_{L}(x_{1},...,x_{n})\over\hbar}}(dx_{1}...dx_{n})^{{1\over
2}}\in{\mathcal{W}F}_{L}.$

We define

$\tau_{L}(\psi)=(\hbar\partial_{\hbar}G_{L}(x_{1},...,x_{n},\hbar)+{1\over
2}\sum_{i=1}^{n}x_{i}\partial_{x_{i}}G_{L}(x_{1},...,x_{n},\hbar))e^{{F_{L}(x_{1},...,x_{n})\over\hbar}}(dx_{1}...dx_{n})^{{1\over
2}}.$

##  3 Reformulation of the problem in terms of conical germs

###  3.1 Admissible pairs and projectively dual hypersurfaces

Definition 3.1.1. A germ of smooth hypersurface $\Sigma$ in projective space
$\mathbb{P}(V)$ is called non-degenerate if the Gauss map
$\Sigma\to\mathbb{P}(V^{*})$ given by $x\mapsto T_{x}\Sigma$ is an immersion.

Projective duality identifies germs of smooth non-degenerate hypersurfaces in
$\mathbb{P}(V)$ and $\mathbb{P}(V^{*})$.

Theorem 3.1.1. There is one to one correspondence between admissible pairs of
germs of functions $f,g$ in $n$ variables and germs of distributions in $n+2$
variables which are smooth densities on a conical germ of a hypersurface in
$\mathbb{A}^{n+2}$, independent of $\hbar$, satisfying certain genericity
constraints explained below, homogeneous of degree $-{n+2\over 2}$ and such
that their Fourier transform is (up to the formal factor
$(2\pi\hbar)^{{n+2\over 2}}$) a distribution with the same properties on the
dual space. This correspondence is given by

$(f,g)~{}~{}~{}\leftrightsquigarrow~{}~{}~{}G(x_{0},...,x_{n+1})=\delta\Big{(}{x_{n+1}\over
x_{0}}-f\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}\Big{)}~{}g\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}~{}x_{0}^{-{n+2\over 2}}$ (3.17)

and we have

$\int
G(x_{0},...,x_{n+1})e^{{1\over\hbar}(x_{0}y_{0}+...+x_{n+1}y_{n+1})}dx_{0}...dx_{n+1}=(2\pi\hbar)^{{n+2\over
2}}\hat{G}(y_{0},...,y_{n+1})$ (3.18)

where $\hat{G}$ is given by (3.19) for some germs of functions
$\hat{f},\hat{g}$.

The genericity constraints777For any non-degenerate hypersurface in
$\mathbb{P}(\mathbb{A}^{n+2})$ the corresponding germ of cones satisfies above
constraints at generic point. for a smooth conical germ $C$ at point
$p\in\mathbb{A}^{n+2}\backslash 0$ are the following:

1. The projectivization of $C$ is non-degenerate,

2. Coordinate $x_{0}$ of point $p$ is non-zero,

3. The tangent space $T_{x_{0}}C$ does not contain vector $(0,...,0,1)$.

Proof Let $f,g$ be admissible. Notice that the formula (3.17) for $G$
describes a germ of a general smooth distribution in $n+2$ variables supported
on a conical germ of a hypersurface in $\mathbb{A}^{n+2}$, and homogeneous of
degree $-{n+2\over 2}$. Let us check that Fourier transform of $G$ can be
represented by a similar formula multiplied by $(2\pi\hbar)^{{n+2\over 2}}$.
Indeed, we have

$\int
G(x_{0},...,x_{n+1})e^{{1\over\hbar}(x_{0}y_{0}+...+x_{n+1}y_{n+1})}dx_{0}...dx_{n+1}\stackrel{{\scriptstyle(1)}}{{=}}$
$\int\delta\Big{(}x_{n+1}-x_{0}f\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}\Big{)}~{}g\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}~{}x_{0}^{-{n\over
2}}e^{{1\over\hbar}(x_{0}y_{0}+...+x_{n+1}y_{n+1})}dx_{0}...dx_{n+1}\stackrel{{\scriptstyle(2)}}{{=}}$
$\int g\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}~{}x_{0}^{-{n\over
2}}e^{{1\over\hbar}(x_{0}y_{n+1}f\big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\big{)}+x_{0}y_{0}+...+x_{n}y_{n})}dx_{0}...dx_{n}\stackrel{{\scriptstyle(3)}}{{=}}$
$\int g\big{(}x_{1},...,x_{n}\big{)}~{}x_{0}^{{n\over
2}}e^{{x_{0}y_{0}\over\hbar}+{x_{0}y_{n+1}\over\hbar}\big{(}f(x_{1},...,x_{n})+x_{1}{y_{1}\over
y_{n+1}}+...+x_{n}{y_{n}\over
y_{n+1}}\big{)}}dx_{0}...dx_{n}\stackrel{{\scriptstyle(4)}}{{=}}$ $\int
x_{0}^{{n\over 2}}e^{{x_{0}y_{0}\over\hbar}}~{}\Bigg{(}{2\pi\hbar\over
y_{n+1}x_{0}}\Bigg{)}^{{n\over 2}}\hat{g}\Big{(}{y_{1}\over
y_{n+1}},...,{y_{n}\over
y_{n+1}}\Big{)}e^{{y_{n+1}x_{0}\over\hbar}\hat{f}\Big{(}{y_{1}\over
y_{n+1}},...,{y_{n}\over
y_{n+1}}\Big{)}}dx_{0}\stackrel{{\scriptstyle(5)}}{{=}}$
$(2\pi\hbar)^{{n+2\over 2}}\delta\Big{(}y_{0}+y_{n+1}\hat{f}\Big{(}{y_{1}\over
y_{n+1}},...,{y_{n}\over y_{n+1}}\Big{)}\Big{)}\hat{g}\Big{(}{y_{1}\over
y_{n+1}},...,{y_{n}\over y_{n+1}}\Big{)}y_{n+1}^{-{n\over
2}}\stackrel{{\scriptstyle(6)}}{{=}}$ $(2\pi\hbar)^{{n+2\over
2}}\delta\Big{(}{y_{0}\over y_{n+1}}+\hat{f}\Big{(}{y_{1}\over
y_{n+1}},...,{y_{n}\over y_{n+1}}\Big{)}\Big{)}\hat{g}\Big{(}{y_{1}\over
y_{n+1}},...,{y_{n}\over y_{n+1}}\Big{)}y_{n+1}^{-{n+2\over 2}}=$
$(2\pi\hbar)^{{n+2\over 2}}\hat{G}(y_{0},...,y_{n+1})$

where

$\hat{G}(y_{0},...,y_{n+1})=\delta\Big{(}{y_{0}\over
y_{n+1}}+\hat{f}\Big{(}{y_{1}\over y_{n+1}},...,{y_{n}\over
y_{n+1}}\Big{)}\Big{)}\hat{g}\Big{(}{y_{1}\over y_{n+1}},...,{y_{n}\over
y_{n+1}}\Big{)}y_{n+1}^{-{n+2\over 2}}$ (3.19)

and this formula is indeed similar to (3.17).

Here is an explanation of all steps in this calculation:

(1) We use the homogeneous property of delta function.

(2) We integrate by $x_{n+1}$ removing delta function.

(3) We make a change of variables $x_{i}\mapsto x_{0}x_{i},~{}i=1,...,n$.

(4) We integrate by $x_{1},...,x_{n}$ using our assumption that $f,g$ is
admissible, see (1.6).

(5) We integrate by $x_{0}$ using an integral representation of delta
function, see (2.16).

(6) We use the homogeneous property of delta function again.

Conversely, let us assume that the Fourier transform of $G$ is supported on a
conical germ and also is homogeneous of degree $-{n+2\over 2}$, i.e.

$\int\delta\Big{(}x_{n+1}-x_{0}f\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}\Big{)}~{}g\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}~{}x_{0}^{-{n\over
2}}e^{{1\over\hbar}(x_{0}y_{0}+...+x_{n+1}y_{n+1})}dx_{0}...dx_{n+1}=$
$(2\pi\hbar)^{{n+2\over 2}}\delta\Big{(}y_{0}+y_{n+1}\hat{f}\Big{(}{y_{1}\over
y_{n+1}},...,{y_{n}\over y_{n+1}}\Big{)}\Big{)}\hat{g}\Big{(}{y_{1}\over
y_{n+1}},...,{y_{n}\over y_{n+1}}\Big{)}y_{n+1}^{-{n\over 2}}$

where $f,g,\hat{f},\hat{g}$ are some germs of functions in $n$ variables
independent of $\hbar$. Integrating the l.h.s. by $x_{n+1}$ and making change
of variables $x_{i}\mapsto x_{0}x_{i}$, $~{}~{}y_{i}\mapsto
y_{n+1}y_{i},~{}i=1,...,n$ we obtain:

$\int g\big{(}x_{1},...,x_{n}\big{)}~{}x_{0}^{{n\over
2}}e^{{x_{0}y_{0}\over\hbar}+{x_{0}y_{n+1}\over\hbar}(f(x_{1},...,x_{n})+x_{1}y_{1}+...+x_{n}y_{n})}dx_{0}...dx_{n}=$
$(2\pi\hbar)^{{n+2\over
2}}\delta\Big{(}y_{0}+y_{n+1}\hat{f}\Big{(}y_{1},...,y_{n}\Big{)}\Big{)}\hat{g}\Big{(}y_{1},...,y_{n}\Big{)}y_{n+1}^{-{n\over
2}}.$

Finally, we multiply this equation by ${1\over 2\pi\hbar}e^{-{y_{0}\over\hbar
y_{n+1}}}$, integrate by $y_{0}$, and after subsequent integration of the
l.h.s. by $x_{0}$ we obtain (1.6). $\square$

Remark 3.1.1. One can check that if the densities $G$ and $\hat{G}$ are
related by the Fourier transform (3.18), then their supports are projectively
dual. See also Remark 1.4.

###  3.2 Projective invariance of the problem

Notice that the conditions on distribution $G$ from the Theorem 3.1.1
(ignoring the genericity constraints 2, 3 from Section 3.1) are invariant
under $GL(n+2)$ acting on $\mathbb{A}^{n+2}$.

Corollary 3.2.1. $GL(n+2)$ acts on germs of admissible pairs at generic point.

Definition 3.2.1. A cone $C$ (as well as its projectivization) is called
admissible if it is locally defined by (1.9) where function $f$ is admissible.
The rank of $C$ is the rank of any such $f$. We denote the rank of $C$ by
$rk(C)$. A cone $C$ is admissible iff $rk(C)>0$. We will also use notation
$rk(\Sigma)$ for $rk(C)$ where $\Sigma\subset\mathbb{P}^{n+1}$ is the
projectivization of $C$.

The admissibility of cones defined above does not depend on the choice of
projective coordinates.

In order to write explicit formulas for $GL(n+2)$ action (see also (1.11)), it
is convenient to describe $f,g$ parametrically as

$f=\phi_{0}(u_{1},...,u_{n}),~{}~{}~{}x_{i}=\phi_{i}(u_{1},...,u_{n}),~{}i=1,...,n,$
(3.20) $g=\psi(u_{1},...,u_{n}).$

Then the pair $\tilde{f}(x_{1},...,x_{n}),\tilde{g}(x_{1},...,x_{n})$ defined
parametrically by

$\tilde{f}=\tilde{\phi}_{0}(u_{1},...,u_{n}),~{}~{}~{}x_{i}=\tilde{\phi}_{i}(u_{1},...,u_{n}),~{}i=1,...,n,$
(3.21) $\tilde{g}=\tilde{\psi}(u_{1},...,u_{n})$

is also admissible, where

$\tilde{\phi}_{i}={\sum_{j=0}^{n}a_{i,j}\phi_{j}+a_{i,n+1}\over\sum_{j=0}^{n}a_{n+1,j}\phi_{j}+a_{n+1,n+1}},~{}i=0,...,n,$
(3.22) $\tilde{\psi}=\psi\cdot{\det\Big{(}{\partial\phi_{i}\over\partial
u_{j}}\Big{)}_{1\leq i,j\leq
n}\over\det\Big{(}{\partial\tilde{\phi}_{i}\over\partial u_{j}}\Big{)}_{1\leq
i,j\leq
n}}\cdot\Big{(}\sum_{j=0}^{n}a_{n+1,j}\phi_{j}+a_{n+1,n+1}\Big{)}^{-{n+2\over
2}}.$

Here $(a_{i,j})_{0\leq i,j\leq n+1}$ is an arbitrary non-degenerate constant
matrix.

Remark 3.2.1. The projective invariance of admissible pairs is proven by a
direct but not very transparent calculation. It can be explained in another
way. First, notice that the group $Aff(n+1)$ of affine transformations of
$\mathbb{A}^{n+1}$ acts on admissible pairs as

$\tilde{\phi}_{i}=\sum_{j=0}^{n}a_{i,j}\phi_{j}+b_{i},~{}i=0,...,n,$ (3.23)
$\tilde{\psi}=\psi\cdot{\det\Big{(}{\partial\phi_{i}\over\partial
u_{j}}\Big{)}_{1\leq i,j\leq
n}\over\det\Big{(}{\partial\tilde{\phi}_{i}\over\partial u_{j}}\Big{)}_{1\leq
i,j\leq n}}.$

Here $(a_{i,j})_{0\leq i,j\leq n}$ is an arbitrary non-degenerate constant
matrix and $b_{i}$ are arbitrary constants. Notice that the last equation can
be also written as888In invariant terms, we have a hypersurface in affine
space $\mathbb{A}^{n+1}$, endowed with a volume element.

$\tilde{\psi}~{}d\tilde{\phi}_{1}\wedge...d\tilde{\phi}_{n}=\psi~{}d\phi_{1}\wedge...\wedge
d\phi_{n}.$

Indeed, after the change of variables

$\hbar\mapsto{\hbar\over y_{0}},~{}y_{i}\mapsto{y_{i}\over y_{0}}$

the equation (1.6) can be written as

$\int
e^{{\phi_{0}y_{0}+\phi_{1}y_{1}+...+\phi_{n}y_{n}\over\hbar}}~{}\psi~{}d\phi_{1}\wedge...d\phi_{n}=(2\pi\hbar)^{{n\over
2}}\cdot y_{0}^{-{n\over 2}}~{}\hat{g}\Bigg{(}{y_{1}\over
y_{0}},...,{y_{n}\over y_{0}}\Bigg{)}\cdot
e^{{y_{0}\over\hbar}\hat{f}\big{(}{y_{1}\over y_{0}},...,{y_{n}\over
y_{0}}\big{)}}.$ (3.24)

The l.h.s. of the equation (3.24) is manifestly invariant (up to
multiplication by a constant independent of $u_{1},...,u_{n}$) with respect to
the affine action (3.23) and the dual action of $GL(n+1)$ on variables
$y_{0},...,y_{n}$, so that the form
$\phi_{0}y_{0}+\phi_{1}y_{1}+...+\phi_{n}y_{n}$ is invariant. This gives the
action (3.23) of the group $Aff(n+1)$ on the set of admissible pairs.

By definition, the Fourier transform acts as an involution (up to the
reflection $x_{i}\to-x_{i},i=1,\dots,n$) on the set of admissible pairs.
Conjugating by the Fourier transform the action of the group $Aff(n+1)$
described above, we obtain the second action of the same group on the set of
admissible pairs. One can check that these two actions generate the action of
$GL(n+2)$.

###  3.3 Generalization to the projectively dual lower-dimensional cones

It seems to be natural to generalize previous considerations in the following
way:

1\. Densities $G(x_{0},...,x_{n+1}),~{}\hat{G}(y_{0},...,y_{n+1})$ are
homogeneous, related by the Fourier transform but supported on projectively
dual cones $C,\hat{C}$ of lower dimensions, not necessarily hypersurfaces.

2\. Densities $G,\hat{G}$ are both finite linear combinations of derivatives
of delta functions, not necessarily just proportional to delta functions.

We want to write explicitly conditions for $G,\hat{G}$. To simplify formulas,
we assume that both $G,\hat{G}$ are proportional to delta functions, in the
general case computations are similar.

Suppose that our cone $C\subset\mathbb{A}^{n+2}$ is defined parametrically by

$x_{i}=\phi_{i}(u_{0},...,u_{m_{1}}),~{}~{}~{}i=0,...,n+1$

where $u_{0},...,u_{m_{1}}$ are coordinates on $C$, functions $\phi_{i}$ are
homogeneous of degree $d_{1}$, and $\dim C=m_{1}+1$.

Similarly, suppose that $\hat{C}\subset(\mathbb{A}^{n+2})^{*}$ is defined
parametrically by

$y_{i}=\psi_{i}(v_{0},...,v_{m_{2}}),~{}~{}~{}i=0,...,n+1$

where $v_{0},...,v_{m_{1}}$ are coordinates on $\hat{C}$, functions $\psi_{i}$
are homogeneous of degree $d_{2}$, and $\dim\hat{C}=m_{2}+1$.

We can write our densities $G,\hat{G}$ as

$G(x_{0},...,x_{n+1})=\int\prod_{i=0}^{n+1}\delta(x_{i}-\phi_{i}(u_{0},...,u_{m_{1}}))g(u_{0},...,u_{m_{1}})du_{0}...du_{m_{1}},$
$\hat{G}(y_{0},...,y_{n+1})=\int\prod_{i=0}^{n+1}\delta(y_{i}-\psi_{i}(v_{0},...,v_{m_{2}}))\hat{g}(v_{0},...,v_{m_{2}})dv_{0}...dv_{m_{2}}$

where $g,\hat{g}$ are also homogeneous with certain homogeneous degrees. The
condition (3.18) after the integration with respect to variables
$x_{0},...,x_{n+1}$ in the l.h.s. reads

$\int
g(u_{0},...,u_{m_{1}})e^{{1\over\hbar}\sum_{i=0}^{n+1}\phi_{i}(u_{0},...,u_{m_{1}})y_{i}}du_{0}...du_{m_{1}}=$
(3.25) $(2\pi\hbar)^{{n+2\over
2}}\int\prod_{i=0}^{n+1}\delta(y_{i}-\psi_{i}(v_{0},...,v_{m_{2}}))\hat{g}(v_{0},...,v_{m_{2}})dv_{0}...dv_{m_{2}}.$

The homogeneity properties of the Fourier transform imply

$\deg g+m_{1}+1={n+2\over 2}~{}d_{1},~{}~{}~{}\deg\hat{g}+m_{2}+1={n+2\over
2}~{}d_{2}.$

One can deal with condition (3.25) in the following way.

a) Choose a partition $\\{0,...,n+1\\}=I_{1}\sqcup I_{2}$ such that
$|I_{1}|=m_{2}+1$, such that $(y_{i})_{i\in I_{1}}$ form a local system of
coordinates on $\hat{C}$.

b) Perform integration in the r.h.s. with respect to $v_{0},...,v_{m_{2}}$
removing delta functions $\delta(y_{i}-\psi_{i}(v_{0},...,v_{m_{2}})),~{}i\in
I_{1}.$ This reduces the number of delta functions in the r.h.s. by $m_{2}+1$.

c) Multiply the equation (3.25) by
${1\over(2\pi\hbar)^{n-m_{2}+1}}e^{\sum_{i\in I_{2}}{y_{i}z_{i}\over\hbar}}$
and integrate with respect to $y_{i},~{}i\in I_{2}$. After doing that the
delta functions of the form $\prod_{i\in
I_{2}}\delta(z_{i}+\phi_{i}(u_{0},...,u_{m_{1}}))$ appear in the l.h.s., and
we can remove them by integrating with respect to variables $u_{j},~{}j\in J$
where $J\subset\\{0,...,m_{1}\\}$ and $|J|=n-m_{2}+1$. In this way we remove
all delta functions. Notice that we should have $n\leq m_{1}+m_{2}$, this is
always true for projectively dual cones.

d) After the removal of all delta functions from (3.25) one can take an
expansion of the l.h.s. at a critical point.

Example 3.3.1. Let $C\subset\mathbb{A}^{5}$ be defined parametrically by

$C=\\{(x_{0},x_{1},x_{2},x_{3}.x_{4})~{}|~{}x_{2}=h_{1}(u)x_{0}+h_{2}(u)x_{1},x_{3}=h_{3}(u)x_{0}+h_{4}(u)x_{1},x_{4}=h_{5}(u)x_{0}+h_{6}(u)x_{1}\\}$

where $u$ is a coordinate on $C$ and functions $h_{1},...,h_{6}$ satisfy
conditions

$h_{4}^{\prime}(u)h_{1}^{\prime}(u)=h_{2}^{\prime}(u)h_{3}^{\prime}(u),~{}~{}~{}h_{6}^{\prime}(u)h_{1}^{\prime}(u)=h_{2}^{\prime}(u)h_{5}^{\prime}(u)$

and certain genericity constraints. One can check that the dual cone $\hat{C}$
can also be defined parametrically as

$\hat{C}=\\{(y_{0},y_{1},y_{2},y_{3},y_{4})~{}|~{}y_{0}=(p_{1}(v)h_{1}(v)-h_{3}(v))y_{3}+(p_{2}(v)h_{1}(v)-h_{5}(v))y_{4},$
$y_{1}=(p_{1}(v)h_{2}(v)-h_{4}(v))y_{3}+(p_{2}(v)h_{2}(v)-h_{6}(v))y_{4},y_{2}=-p_{1}(v)y_{3}-p_{2}(v)y_{4}\\}$

where $v$ is a coordinate on $\hat{C}$ and

$p_{1}(v)={h_{3}^{\prime}(v)\over h_{1}^{\prime}(v)}={h_{4}^{\prime}(v)\over
h_{2}^{\prime}(v)},~{}~{}~{}p_{2}(v)={h_{5}^{\prime}(v)\over
h_{1}^{\prime}(v)}={h_{6}^{\prime}(v)\over h_{2}^{\prime}(v)}.$

We have $\dim C=\dim\hat{C}=3$ and therefore both $C$ and its dual $\hat{C}$
have codimension 2 in $\mathbb{A}^{5}$.

Admissibility condition (3.25) in this case can be written as

$\int
g(x_{0},x_{1},u)e^{{1\over\hbar}(x_{0}y_{0}+x_{1}y_{1}+(h_{1}(u)x_{0}+h_{2}(u)x_{1})y_{2}+(h_{3}(u)x_{0}+h_{4}(u)x_{1})y_{3}+(h_{5}(u)x_{0}+h_{6}(u)x_{1})y_{4}}dx_{0}dx_{1}du$
$=(2\pi\hbar)^{{5\over
2}}\int\hat{g}(y_{3},y_{4},v)\delta(y_{2}+p_{1}(v)y_{3}+p_{2}(v)y_{4})$
$\delta((p_{1}(v)h_{1}(v)-h_{3}(v))y_{3}+(p_{2}(v)h_{1}(v)-h_{5}(v))y_{4}-y_{0})$
$\delta((p_{1}(v)h_{2}(v)-h_{4}(v))y_{3}+(p_{2}(v)h_{2}(v)-h_{6}(v))y_{4}-y_{1})dv.$

Multiplying both sides by
${1\over(2\pi\hbar)^{2}}e^{-{1\over\hbar}(y_{0}z_{0}+y_{1}z_{1})}$,
integrating by $y_{0},y_{1}$, and in the l.h.s. integrating also by
$x_{0},x_{1}$ we obtain

$\int
g(z_{0},z_{1},u)e^{{1\over\hbar}((h_{1}(u)z_{0}+h_{2}(u)z_{1})y_{2}+(h_{3}(u)z_{0}+h_{4}(u)z_{1})y_{3}+(h_{5}(u)z_{0}+h_{6}(u)z_{1})y_{4})}du=$
$(2\pi\hbar)^{{1\over
2}}\int\hat{g}(y_{3},y_{4},v)\delta(y_{2}+p_{1}(v)y_{3}+p_{2}(v)y_{4})$
$e^{-{1\over\hbar}(((p_{1}(v)h_{1}(v)-h_{3}(v))y_{3}+(p_{2}(v)h_{1}(v)-h_{5}(v))y_{4})z_{0}+((p_{1}(v)h_{2}(v)-h_{4}(v))y_{3}+(p_{2}(v)h_{2}(v)-h_{6}(v))y_{4})z_{1})}dv.$

We can integrate by $v$ in the r.h.s. removing delta function, and integrate
by $u$ in the l.h.s. by taking expansion at a critical point. In this way we
obtain admissibility conditions for $C$ explicitly. It would be interesting to
classify such admissible cones and generalize these for higher dimensions.

##  4 Reformulation of the problem in terms of constraints on wave functions

###  4.1 Abstract formalism

In the previous Section we reformulated our original problem of finding
admissible pairs $f,g$ in $n$ variables in terms of finding distributions
$G,\hat{G}$ in $n+2$ variables, given by (3.17), (3.19), which are related by
the Fourier transform (3.18). Let us apply the formalism of wave functions
from Section 2 to these distributions.

Conical germs $C$ and $\hat{C}$ on which distributions $G$, $\hat{G}$ are
supported, are given by equations

${x_{n+1}\over x_{0}}-f\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}=0,~{}~{}~{}~{}~{}~{}~{}{y_{0}\over
y_{n+1}}+\hat{f}\Big{(}{y_{1}\over y_{n+1}},...,{y_{n}\over
y_{n+1}}\Big{)}=0.$ (4.26)

Denote by $L$ the conormal bundle to cone $C\subset\mathbb{A}^{n+2}$. It is a
Lagrangian subvariety of $\mathbb{A}^{2(n+2)}=T^{*}\mathbb{A}^{n+2}$ invariant
under the action of $GL(1)\times GL(1)$ on $\mathbb{A}^{2(n+2)}$ where the
first $GL(1)$ rescales coordinates $x_{0},...,x_{n+1}$ and the second $GL(1)$
rescales coordinates $y_{0},...,y_{n+1}$. In particular, $L$ is
conical999Generically, the invariance under the second copy of $GL(1)$ means
that $L$ is a conormal bundle to a subvariety in $\mathbb{A}^{n+2}$ of
arbitrary dimension. The invariance under the first copy of $GL(1)$ means that
this subvariety is conical..

Nonvanishing of the Hessian of germ $f$ (and hence of $\hat{f}$) can be
reformulated as the following genericity condition on the smooth $GL(1)\times
GL(1)$ invariant germ $L$ of Lagrangian submanifold at point
$p\in\mathbb{A}^{2(n+2)}$, similar to conditions 1, 2, 3 in Section 3.1:

1) coordinates $x_{0}$ and $y_{n+1}$ of $p$ are non-zero.

2) projections from $T_{p}L$ to $\mathbb{A}^{n+2}_{x_{0},...,x_{n},y_{n+1}}$
and to $\mathbb{A}^{n+2}_{y_{0},...,y_{n},x_{n+1}}$ are one-to-one.

The space $\mathbb{A}^{2(n+2)}$ with the symplectic structure
$\sum_{i=0}^{n+1}dx_{i}\wedge dy_{i}$ is simultaneously the cotangent space to
$\mathbb{A}^{n+2}_{x_{0},...,x_{n+1}}$ and to
$\mathbb{A}^{n+2}_{y_{0},...,y_{n+1}}$.

Projective duality between the projectivizations of cones $C$ and $\hat{C}$
can be reformulated as the property of $L$ to be simultaneously the conormal
bundle to $C$ and $\hat{C}$.

Let

$F_{1}(x_{0},...,x_{n+1})=0,~{}~{}~{}~{}~{}~{}F_{2}(y_{0},...,y_{n+1})=0$
(4.27)

be equations for the cones $C$ and $\hat{C}$ respectively. Here
$F_{1},~{}F_{2}$ are germs at non-zero points of homogeneous functions of some
homogeneous degrees which are proportional with an invertible factors to
l.h.s. of equations (4.26).

Distributions $G,\hat{G}$ can be understood as elements
$\psi,\hat{\psi}\in{\mathcal{W}F}_{L}$. Similarly to Remark 2.2 the fact that
$\hat{G}$ is the Fourier transform of $G$ can be written as equality
$\psi=\hat{\psi}$.

Let us interpret $F_{1},~{}F_{2}$ as elements of the quantum algebra acting on
${\mathcal{W}F}_{L}$. The property of $G$ to be a smooth density on $C$ can be
rewritten as101010Recall that $C$ is defined by equation $F_{1}=0$ and
therefore, $G$ is proportional to $\delta(F_{1})$. For example, we have
$P(x_{0},...,x_{n+1})\delta(P(x_{0},...,x_{n+1}))=0$ for an arbitrary
polynomial $P$.

1. $F_{1}\cdot\psi=0.$

Similarly the dual condition on $\hat{G}$ gives

${\bf 1^{\prime}.}$ $F_{2}\cdot\psi=0.$

The condition on $G,\hat{G}$ of being homogeneous of degree $-{n+2\over 2}$
can be rewritten as

2. $\Big{(}\sum_{i=0}^{n+1}y_{i}\star x_{i}\Big{)}\cdot\psi=0.$

Indeed, the homogeneity of $\psi$ (considered as a distribution in variables
$x_{0},...,x_{n+1}$) of degree $-{n+2\over 2}$ means

$\sum_{i=0}^{n+1}x_{i}\partial_{x_{i}}\psi+{n+2\over 2}\psi=0,$

therefore $\Big{(}\sum_{i=0}^{n+1}y_{i}\star x_{i}\Big{)}\cdot\psi={1\over
2}\Big{(}\sum_{i=0}^{n+1}\hbar\partial_{x_{i}}x_{i}+\hbar
x_{i}\partial_{x_{i}}\Big{)}\cdot\psi={\hbar\over
2}\sum_{i=0}^{n+1}\Big{(}2x_{i}\partial_{x_{i}}+1\Big{)}\psi=0.$

Finally, the property that $G,\hat{G}$ are given by densities independent of
$\hbar$ can be rewritten as

3. $\tau_{L}\psi=-{n\over 4}\psi$ where $\tau_{L}$ is defined in Remark 2.3 in Section 2. The eigenvalue $-{n\over 4}$ can be seen from the formula (4.29) in Section 4.2.

The discussion above can be summarized as

Proposition 4.1.1. Fix functions $f,\hat{f}$ and the corresponding cones
$C,\hat{C}$ given by equations (4.27). There is one-to-one correspondence
between the space of $g$ such that pair $f,g$ is admissible and non-zero
elements $\psi\in{\mathcal{W}F}_{L}$ satisfying properties ${\bf
1,~{}1^{\prime}~{},2,~{}3}$ where $L$ is a $GL(1)\times GL(1)$-invariant
Lagrangian germ satisfying the genericity condition above. $\square$

###  4.2 Explicit formulas in general case

Let us write equations on $\psi$ explicitly. Conditions 1, 2, 3 give

$\psi=g\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}~{}x_{0}^{-{n+2\over 2}}\delta\Big{(}{x_{n+1}\over
x_{0}}-f\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over x_{0}}\Big{)}\Big{)}$ (4.28)

for some function $g$. Indeed, condition 1 means that $\psi$ is proportional
to the delta function of cone $C$, condition 2 gives homogeneity condition for
the coefficient of proportionality, and condition 3 means that function $g$
does not depend on $\hbar$. However, condition ${\bf 1^{\prime}}$ becomes
singular in these coordinates because the action of
$F_{2}(\hbar\partial_{x_{0}},...,\hbar\partial_{x_{n+1}})$ in general is not
defined on $\psi$ given by (4.28). In order to overcome this problem, let us
make the Fourier transform with respect to the coordinate $x_{n+1}$.
Geometrically this means that we choose the projection of our cone $C$ to
coordinates $x_{0},...,x_{n},y_{n+1}$. After making the Fourier transform in
$x_{n+1}$ the element $\psi$ given by (4.28) takes the form

$\psi=x_{0}^{-{n\over 2}}e^{{x_{0}y_{n+1}\over\hbar}f\big{(}{x_{1}\over
x_{0}},...,{x_{n}\over x_{0}}\big{)}}g\Big{(}{x_{1}\over
x_{0}},...,{x_{n}\over x_{0}}\Big{)}$ (4.29)

and condition ${\bf 1^{\prime}}$ means that

$F_{2}(\hbar\partial_{x_{0}},...\hbar\partial_{x_{n}},y_{n+1})\cdot\psi=0$

and can be rewritten as an equation on $g$

$F_{2}\Big{(}\hbar\partial_{x_{0}}-{n\hbar\over
2x_{0}}+Q_{0},\hbar\partial_{x_{1}}+Q_{1},...,\hbar\partial_{x_{n}}+Q_{n},y_{n+1}\Big{)}\cdot
g\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over x_{0}}\Big{)}=0$ (4.30)

where operator $F_{2}$ in (4.30) is equal to

$\Bigg{(}x_{0}^{-{n\over 2}}e^{{x_{0}y_{n+1}\over\hbar}f\big{(}{x_{1}\over
x_{0}},...,{x_{n}\over x_{0}}\big{)}}\Bigg{)}^{-1}\cdot
F_{2}(\hbar\partial_{x_{0}},...\hbar\partial_{x_{n}},y_{n+1})\cdot
x_{0}^{-{n\over 2}}e^{{x_{0}y_{n+1}\over\hbar}f\big{(}{x_{1}\over
x_{0}},...,{x_{n}\over x_{0}}\big{)}}$

and, therefore, $Q_{0},...,Q_{n}$ are defined by

$\Bigg{(}x_{0}^{-{n\over 2}}e^{{x_{0}y_{n+1}\over\hbar}f\big{(}{x_{1}\over
x_{0}},...,{x_{n}\over
x_{0}}\big{)}}\Bigg{)}^{-1}\cdot\hbar\partial_{x_{0}}\cdot x_{0}^{-{n\over
2}}e^{{x_{0}y_{n+1}\over\hbar}f\big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\big{)}}=\hbar\partial_{x_{0}}-{n\hbar\over 2x_{0}}+Q_{0},$
$\Bigg{(}x_{0}^{-{n\over 2}}e^{{x_{0}y_{n+1}\over\hbar}f\big{(}{x_{1}\over
x_{0}},...,{x_{n}\over
x_{0}}\big{)}}\Bigg{)}^{-1}\cdot\hbar\partial_{x_{i}}\cdot x_{0}^{-{n\over
2}}e^{{x_{0}y_{n+1}\over\hbar}f\big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\big{)}}=\hbar\partial_{x_{i}}+Q_{i},~{}~{}~{}i=1,...,n.$

Explicitly, we have

$Q_{0}=y_{n+1}f\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}-{y_{n+1}\over x_{0}}\sum_{i=1}^{n}x_{i}f_{i}\Big{(}{x_{1}\over
x_{0}},...,{x_{n}\over x_{0}}\Big{)},$ $Q_{i}=y_{n+1}f_{i}\Big{(}{x_{1}\over
x_{0}},...,{x_{n}\over x_{0}}\Big{)},~{}~{}~{}i=1,...,n$

where by $f_{i}$ we denote partial derivative of $f$ with respect to its
$i-$th argument.

Notice that $Q_{0},...,Q_{n}$ are independent of $\hbar$.

The l.h.s. of (4.30) makes sense as a power series in $\hbar$. For example, if
we choose $F_{2}$ in the simplest form

$F_{2}(y_{0},...,y_{n+1})={y_{0}\over y_{n+1}}+\hat{f}\Big{(}{y_{1}\over
y_{n+1}},...,{y_{n}\over y_{n+1}}\Big{)}$

as in (4.26), and take Taylor expansion111111Computation of this Taylor series
is not a straightforward problem because while arguments of $F_{2}$ pairwise
commute, parts of these arguments at different powers of $\hbar$ does not, for
example $\hbar\partial_{x_{i}}$ does not commute with $Q_{i}$. of the l.h.s.
of (4.30) at $\hbar=0$ we obtain a power series in $\hbar$ where each term is
a differential operator (with coefficients written in terms of $f$) applied to
the function $g$. This can be written as a system of partial differential
equations for unknown functions $f,g$ linear in $g$. Notice that this system
is infinite in general.

###  4.3 Algebraic case

Consider the algebraic case where $f$ is an algebraic function and, therefore,
$\hat{f},~{}C,~{}\hat{C}$ are algebraic. Let us choose
$F_{1}(x_{0},...,x_{n+1}),~{}F_{2}(y_{0},...,y_{n+1})$ to be homogeneous
polynomials of certain degrees, such that

$F_{1}(1,x_{1},...,x_{n},f(x_{1},...,x_{n}))=0,~{}~{}~{}~{}~{}~{}~{}F_{2}(1,y_{1},...,y_{n},\hat{f}(y_{1},...,y_{n}))=0.$

In this special case one can eliminate parameter $\hbar$, and the problem of
finding an element $\psi$ satisfying to properties ${\bf
1,~{}1^{\prime},~{}2,~{}3}$ turns to the following purely algebraic question.

Find homogeneous polynomials
$F_{1}(x_{0},...,x_{n+1}),~{}F_{2}(y_{0},...,y_{n+1})$ in two dual groups of
variables (or more abstractly, in dual vector spaces) such that there exists a
non-zero cyclic module $M$ over the ring of polynomial differential operators

${\bf k}[x_{0},...,x_{n+1}][\partial_{x_{0}},...,\partial_{x_{n+1}}]$

generated by an element $\psi\in M$ such that

$\begin{split}&F_{1}(x_{0},...,x_{n+1})\cdot(\psi)=0,\\\
&F_{2}(\partial_{x_{0}},...,\partial_{x_{n+1}})\cdot(\psi)=0,\\\
&\Big{(}\sum_{i=0}^{n+1}x_{i}\partial_{x_{i}}+{n+2\over
2}\Big{)}\psi=0.\end{split}$ (4.31)

Notice that the first equation means that

$\psi=G_{1}\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}x_{0}^{m}\delta(F_{1}(x_{0},...,x_{n+1}))$

where the factor in front of the delta function is a homogeneous function of
degree $m$. The third equation means that

$m-\deg F_{1}=-{n+2\over 2}\,.$

The second equation gives a finite system of partial differential equations on
$F_{1},G_{1}$ linear in $G_{1}$.

One has a universal module $M_{\psi}$ generated by the cyclic vector $\psi$
satisfying the above equations.

Roughly speaking, the study of admissible pairs with algebraic function $f$
(or equivalently, $\hat{f}$) can be rephrased as the study of projectively
dual algebraic hypersurfaces in $\mathbb{P}^{n+1}$, $(\mathbb{P}^{n+1})^{*}$
such that the corresponding finitely generated module $M_{\psi}$ is not zero.

We will see in the sequel both holonomic and nonholonomic121212Holonomicity at
generic point means that the space of $G_{1}$ is finite-dimensional. examples
of modules $M_{\psi}$.

Remark 4.3.1. Let $\Sigma\subset\mathbb{P}^{n+1}$ and
$\widehat{\Sigma}\subset(\mathbb{P}^{n+1})^{*}$ be projectively dual
hypersurfaces. Let $\Sigma$ (resp, $\widehat{\Sigma}$) be defined by a
homogeneous irreducible polynomial $F_{1}(x_{0},...,x_{n+1})$ (resp.
$F_{2}(y_{0},...,y_{n+1})$). Given polynomial $F_{1}$ one can determine
polynomial $F_{2}$ (up to multiplication by a non-zero constant) from the
condition: $F_{2}$ is a homogeneous polynomial of smallest degree such that

$F_{2}(\partial_{x_{0}}F_{1},...,\partial_{x_{n+1}}F_{1})=0\mod F_{1},$

and similarly one can find polynomial $F_{1}$ if polynomial $F_{2}$ is given.
We can define polynomials $H_{1}\in{\bf k}[x_{0},...,x_{n+1}]$ and
$H_{2}\in{\bf k}[y_{0},...,y_{n+1}]$ by

$F_{1}(\partial_{y_{0}}F_{2},...,\partial_{y_{n+1}}F_{2})=(f_{2}-1)F_{2}(y_{0},...,y_{n+1})H_{2}(y_{0},...,y_{n+1}),$
(4.32)
$F_{2}(\partial_{x_{0}}F_{1},...,\partial_{x_{n+1}}F_{1})=(f_{1}-1)F_{1}(x_{0},...,x_{n+1})H_{1}(x_{0},...,x_{n+1})$

where $f_{1}=\deg F_{1},~{}f_{2}=\deg F_{2}$. Notice that

$\deg H_{1}=\deg H_{2}=f_{1}f_{2}-f_{1}-f_{2}.$

The mappings

$x_{i}\mapsto\partial_{y_{i}}F_{2},~{}~{}~{}y_{i}\mapsto\partial_{x_{i}}F_{1},~{}~{}~{}i=1,...,n$

define birational isomorphisms between projective hypersurfaces
$\Sigma,~{}\widehat{\Sigma}$. They are mutually inverse as birational mappings
of projective varieties. This can be written algebraically as

$\partial_{y_{i}}F_{2}~{}\Big{\rvert}_{y_{0}=\partial_{x_{0}}F_{1},...,y_{n+1}=\partial_{x_{n+1}}F_{1}}=x_{i}H_{1}~{}\mod~{}F_{1},~{}~{}~{}i=1,...,n$

and

$\partial_{x_{i}}F_{1}~{}\Big{\rvert}_{x_{0}=\partial_{y_{0}}F_{2},...,x_{n+1}=\partial_{y_{n+1}}F_{2}}=y_{i}H_{2}~{}\mod~{}F_{2},~{}~{}~{}i=1,...,n.$

These formulas can be verified by computing derivatives of equations (4.32),
reducing modulo $F_{1}$ or $F_{2}$ and using Euler’s homogeneous function
theorem.

Remark 4.3.2. Projective duality interchanges polynomials
$F_{1}(x_{0},...,x_{n+1}),~{}F_{2}(y_{0},...,y_{n+1})$. Dual element
$\hat{\psi}$ has a form (up to a constant)

$\hat{\psi}=G_{1}\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}x_{0}^{m}~{}\Big{\rvert}_{x_{0}=\partial_{y_{0}}F_{2},...,x_{n+1}=\partial_{y_{n+1}}F_{2}}\det\Big{(}\partial_{y_{i}}\partial_{y_{j}}F_{2}\Big{)}^{{1\over
2}}H_{2}^{-1}\delta(F_{2}(y_{0},...,y_{n+1}))$

where $H_{2}\in{\bf k}[y_{1},...,y_{n}]$ (and similar polynomial $H_{1}$) are
defined above in Remark 4.3.1.

Remark 4.3.3. The question of finding pairs of polynomials

$F_{1}\in{\bf k}[x_{0},...,x_{n+1}],~{}~{}~{}F_{2}\in{\bf
k}[\partial_{x_{0}},...,\partial_{x_{n+1}}]$

generating a non-zero cyclic module $M$ over the ring of differential
operators, can be generalized to an associative algebra with two maximal
commutative subalgebras. In the case of differential operators homogeneity of
$F_{1},~{}F_{2}$ follows automatically from the nonvanishing of $M$ (assuming
that both $F_{1},~{}F_{2}$ are irredusible). In other natural cases, such as
quantum torus or ring of difference operators we are not aware of any
nontrivial example.

Remark 4.3.4. The condition that both $F_{1},~{}F_{2}$ are irreducible
polynomials defining projectively dual non-degenerate hypersurfaces can be
generalized.

First, one can consider the case where $F_{1}$ is a power of irreducible
polynomial. In terms of distributions $G,\hat{G}$ it means that they are
finite linear combinations of derivatives of delta functions supported on
corresponding coves. In the original formulation it means that $g,\hat{g}$ are
polynomials in $\hbar$.

Second, for a pair of projectively dual projective manifolds one (or even
both) of them could have codimension larger than one, see Section 3.3.

###  4.4 Explicit formulas in the algebraic case

Let us choose $F_{1}$ in the non-polynomial form

$F_{1}=x_{n+1}-x_{0}f\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over x_{0}}\Big{)}$

where $f$ is an algebraic function, and

$F_{2}\in{\bf k}[\partial_{x_{0}},...,\partial_{x_{n+1}}]$

is an irreducible polynomials as above. In this case properties 1, 2, 3 give
(see (4.28))

$\psi=g\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}~{}x_{0}^{-{n\over
2}}\delta\Big{(}x_{n+1}-x_{0}f\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}\Big{)}$ (4.33)

and the property ${\bf 1^{\prime}}$ reads

$F_{2}(\partial_{x_{0}},...,\partial_{x_{n+1}})\cdot(\psi)=0$ (4.34)

where $\psi$ is given by (4.33). Cyclic $D$-module over the ring ${\bf
k}[\partial_{x_{0}},...,\partial_{x_{n+1}}]$ generated by $\psi$ is contained
in the bigraded vector space whose $(a,b)-$graded component is spent by
expressions of the form

$h\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}~{}x_{0}^{a}\delta^{(b)}\Big{(}x_{n+1}-x_{0}f\Big{(}{x_{1}\over
x_{0}},...,{x_{n}\over x_{0}}\Big{)}\Big{)}$

where $h$ is a function in $n$ variables, $a\in-{n\over 2}-\mathbb{Z}_{\geq
0}$, $b\in\mathbb{Z}_{\geq 0}$ We can identify the above bigraded vector space
as the module over ${\bf k}[\partial_{x_{0}},...,\partial_{x_{n+1}}]$ with
another one spent by

$h\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}~{}{x_{0}^{a}\over\Big{(}x_{n+1}-x_{0}f\Big{(}{x_{1}\over
x_{0}},...,{x_{n}\over x_{0}}\Big{)}\Big{)}^{b+1}}$

where $h,a,b$ are as above. The isomorphism is given by
$s\delta^{(b)}(t)\mapsto s{(-1)^{b}b!\over t^{b+1}}$.

The l.h.s. of (4.34) lies in the direct sum of bigraded components with
$a=-{n\over 2}-k,~{}b=\deg F_{2}-k$ where $k=0,...,\deg F_{2}$.

Introduce new variables $v_{0},...,v_{n+1}$ such that

$v_{i}={x_{i}\over
x_{0}},~{}i=1,...,n,~{}~{}~{}~{}v_{0}=x_{0},~{}~{}~{}~{}v_{n+1}=x_{n+1}-x_{0}f\Big{(}{x_{1}\over
x_{0}},...,{x_{n}\over x_{0}}\Big{)}.$

In these variables element $\psi$ given by (4.33) reads as

$\psi=g\big{(}v_{1},...,v_{n}\big{)}~{}v_{0}^{-{n\over 2}}v_{n+1}^{-1}.$
(4.35)

Partial derivatives $\partial_{x_{0}},...,\partial_{x_{n+1}}$ in new variables
become more complicated differential operators

$\partial_{x_{i}}\mapsto D_{i},~{}i=0,...,n+1$

where

$D_{0}=-{1\over
v_{0}}\sum_{j=1}^{n}v_{j}\partial_{v_{j}}+\partial_{v_{0}}+\Big{(}\sum_{j=1}^{n}v_{j}f_{v_{j}}-f\Big{)}\partial_{v_{n+1}}$
$D_{i}={1\over
v_{0}}\partial_{v_{i}}-f_{v_{i}}\partial_{v_{n+1}},~{}i=1,...,n,~{}~{}~{}~{}D_{n+1}=\partial_{v_{n+1}}$

where $f=f(v_{1},...,v_{n}),~{}f_{v_{i}}={\partial f\over\partial v_{i}}$.

The final conclusion is that in coordinates $v_{0},...,v_{n+1}$ the system of
equations for $f,g$ can be written as

$F_{2}(D_{0},...,D_{n+1})\cdot\Big{(}g(v_{1},...,v_{n})v_{0}^{-{n\over
2}}v_{n+1}^{-1}\Big{)}=0.$ (4.36)

To obtain a system of partial differential equations for $f,g$ one should
equate coefficients at all powers of $v_{0},v_{n+1}$ in (4.36) to zero. In
this way we get a system of $\deg F_{2}$ differential equations on $f,g$.

Notice that if our cone $C$ is given parametrically by (3.20), we can rewrite
our system (4.36) in parametric form by doing the corresponding change of
variables.

###  4.5 A question about monodromic regular holonomic $D$-modules

One can try to look for admissible pairs of a special type, when both
homogeneous distributions $G,~{}\hat{G}$ generate regular holonomic
$D$-modules.

Recall the classical result by J. L. Brylinski [2] which says that regular
holonomic $D$-modules on a vector space $V$ such that their Fourier transforms
are also regular holonomic are exactly those for which the action of Euler
vector field $E=\sum_{i}x_{i}\partial_{x_{i}}$ is locally finite. Such
$D$-modules are called monodromic regular holonomic.

The singular support $SS(M)$ of such a $D$-module $M$ is $GL(1)\times
GL(1)$-invariant (possibly reducible) Lagrangian cone $L\subset T^{*}V=V\oplus
V^{*}$, and it coincides with the singular support of its Fourier transform
$SS({\cal F}(M))$ under the identification $T^{*}V=V\oplus V^{*}=T^{*}V^{*}$.

Thus, we arrive to the following problem: study monodromic regular holonomic
$D$-modules $M$ such that $SS(M)$ does not contain $V\times\\{0\\}$ and
$\\{0\\}\times V^{*}$. Indeed, in this case any non-zero element $\psi\in M$
homogeneous with respect to $E$ is killed by some non-trivial homogeneous
polynomial $F_{1}(x_{i}),~{}F_{2}(\partial_{x_{i}})$.

Riemann-Hilbert correspondence identifies monodromic regular holonomic
$D$-modules with so-called monodromic perverse sheaves on $V$. Thus, one can
reformulate the above problem in purely topological terms, concerning finite-
dimensional representations of the fundamental group of $L\setminus L^{sing}$.

Finally, for an irreducible algebraic cone $C\subset V$, such that $C\neq 0,V$
there is a natural $GL(1)\times GL(1)$ invariant irreducible Lagrangian cone
$L$ which is the conormal bundle to $C$. Notice that $L=L_{C}\neq
V\times\\{0\\},~{}\\{0\\}\times V^{*}$. So $L_{C}$ is a natural candidate for
the singular support.

Question 4.5.1. For which $C$ there exists a monodromic regular holonomic
$D$-module whose singular support is $L_{C}$, may be with multiplicities?

##  5 Examples of admissible hypersurfaces and corresponding pairs

In this Section we assume that $\bf k=\bar{k}$.

###  5.1 Quadratic hypersurfaces

Lemma 5.1.1. Let $f(x_{1},...,x_{n})=-{1\over 2}(x_{1}^{2}+...+x_{n}^{2})$.
Then pair $f,g$ is admissible iff the function $g(x_{1},...,x_{n})$ is
harmonic, i.e.

${\partial^{2}g\over\partial x_{1}^{2}}+...+{\partial^{2}g\over\partial
x_{n}^{2}}=0.$

We have in this case

$\int g(x_{1},...,x_{n})e^{{1\over\hbar}(-{1\over 2}x_{1}^{2}-...-{1\over
2}x_{n}^{2}+x_{1}y_{1}+...+x_{n}y_{n})}dx_{1}...dx_{n}=(2\pi\hbar)^{{n\over
2}}g(y_{1},...,y_{n})e^{{1\over 2\hbar}(y_{1}^{2}+...+y_{n}^{2})}.$

Proof. For arbitrary function $g$ we have

$\int g(x_{1},...,x_{n})e^{{1\over\hbar}(-{1\over 2}x_{1}^{2}-...-{1\over
2}x_{n}^{2}+x_{1}y_{1}+...+x_{n}y_{n})}dx_{1}...dx_{n}=(2\pi\hbar)^{{n\over
2}}e^{{1\over
2\hbar}(y_{1}^{2}+...+y_{n}^{2})}\sum_{i=0}^{\infty}{\hbar^{i}\over
i!}\Delta^{i}g(y_{1},...,y_{n})$

where $\Delta={\partial^{2}\over\partial
y_{1}^{2}}+...+{\partial^{2}\over\partial y_{n}^{2}}$. The r.h.s. consists
only of the first term iff $\Delta g=0$. $\square$

Theorem 5.1.1. Let $Q=\sum_{i,j=0}^{n+1}a_{i,j}x_{i}x_{j}$ be a non-degenerate
quadric. Here $a_{i,j}=a_{j,i}$ and $\det(a_{i,j})\neq 0$. Then the
hypersurface in $\mathbb{P}^{n+1}$ defined by $Q=0$ is admissible. Its rank is
two if $n=1$ and infinity otherwise.

Proof. Any such quadric is $GL(n+2)$ equivalent to
$Q_{0}=2x_{0}x_{n+1}+x_{1}^{2}+...+x_{n}^{2}$. The projective hypersurface
$Q_{0}=0$ is a projectivization of affine hypersurface defined by the equation
$x_{n+1}=-{1\over 2}(x_{1}^{2}+...+x_{n}^{2})$ which is admissible by Lemma 1.
Therefore, the hypersurface $Q=0$ is also admissible by projective invariance.
$\square$

Remark 5.1.1. Solving the equation

$Q(1,x_{1},...,x_{n},f)=0$

for an arbitrary non-degenerate quadric $Q$ we obtain a family of functions
$f$ which depends on a lot of parameters $a_{i,j}$. One can see that by
applying group of transformations

$x_{i}\mapsto\sum_{j=1}^{n}q_{i,j}x_{j}+b_{j},~{}~{}~{}f\mapsto\lambda
f+\sum_{j=1}^{n}\mu_{j}x_{j}+\nu$

one can reduce $f$ to one of the following forms:

a) $f(x_{1},...,x_{n})=(x_{1}^{2}+...+x_{n}^{2}+1)^{{1\over 2}}$ ,

b) $f(x_{1},...,x_{n})=(x_{1}^{2}+...+x_{n-1}^{2}+x_{n})^{{1\over 2}}$ for
$n>1$, and $f(x_{1})=x_{1}^{{1\over 2}}$ for $n=1$,

c) $f(x_{1},...,x_{n})={x_{1}^{2}+...+x_{n-1}^{2}+1\over x_{n}}$ for $n>1$,
and $f(x_{1})={1\over x_{1}}$ for $n=1$,

d) $f(x_{1},...,x_{n})=-x_{1}^{2}-...-x_{n}^{2}$.

In each of the cases above pair $f,g$ is admissible iff $g$ satisfies a
certain second order partial differential equation which can be obtained from
harmonic equation in Lemma 5.1.1 by the action of $GL(n+2)$, see Section 3.2.

It would be interesting to lift, if possible, the corresponding formal
integrals of the form (1.6) to actual convergent integral identities. For
example, it is known [3] that if

$g(\vec{x})={1\over\sqrt{1+\vec{x}^{2}}\cdot(1+\sqrt{1+\vec{x}^{2}})^{{n-2\over
2}}}$

then

$\int_{\mathbb{R}^{n}}g(\vec{x})e^{-{1\over\hbar}\sqrt{1+\vec{x}^{2}}+{i\over\hbar}\vec{x}\vec{y}}d\vec{x}=(2\pi\hbar)^{{n\over
2}}g(\vec{y})e^{-{1\over\hbar}\sqrt{1+\vec{y}^{2}}}$

###  5.2 Hypersurfaces admitting quadratic parametrization

Let $C\subset\mathbb{A}^{n+2}$ be a cone given parametrically by

$x_{i}={1\over 2}\sum_{j,k=0}^{n}a_{i,j,k}u_{j}u_{k},~{}~{}~{}i=0,...,n+1$

where $a_{i,j,k}\in{\bf k}$ are constants and $a_{i,k,j}=a_{i,j,k}.$ The dual
cone $\hat{C}$ is given by

$\det\Bigg{(}\sum_{i=0}^{n+1}a_{i,j,k}y_{i}\Bigg{)}_{0\leq j,k\leq n}=0.$

Indeed, let $Q=x_{0}y_{0}+...+x_{n+1}y_{n+1}={1\over
2}\sum_{i=0}^{n+1}\sum_{j,k=0}^{n}a_{i,j,k}y_{i}u_{j}u_{k}.$ The point
$(y_{0},...,y_{n+1})\in\mathbb{A}^{n+2}$ belongs to the dual cone $\hat{C}$
iff the linear system $\partial_{u_{i}}Q=0,~{}i=0,...,n$ for $u_{0},...,u_{n}$
has a non-zero solution.

It follows from Theorem 3.1.1 that cone $C$ is admissible iff

$\int g_{1}\Big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\Big{)}~{}x_{0}^{-{n\over
2}}e^{{1\over\hbar}(x_{0}y_{n+1}f\big{(}{x_{1}\over x_{0}},...,{x_{n}\over
x_{0}}\big{)}+x_{0}y_{0}+...+x_{n}y_{n})}dx_{0}...dx_{n}=$
$(2\pi\hbar)^{{n+2\over 2}}\delta\Big{(}{y_{0}\over
y_{n+1}}+\hat{f}\Big{(}{y_{1}\over y_{n+1}},...,{y_{n}\over
y_{n+1}}\Big{)}\Big{)}g_{2}\Big{(}{y_{1}\over y_{n+1}},...,{y_{n}\over
y_{n+1}}\Big{)}y_{n+1}^{-{n+2\over 2}}$

for some functions $g_{1},g_{2}$, where $x_{0}y_{n+1}f\big{(}{x_{1}\over
x_{0}},...,{x_{n}\over x_{0}}\big{)}=x_{n+1}$ and $\hat{f}$ corresponds to the
dual cone $\hat{C}$. Substituting our quadratic parametrization in the l.h.s.
and taking into account the equation for the dual cone we get

$\int g(u_{0},...,u_{n})e^{{1\over
2\hbar}\sum_{i=0}^{n+1}\sum_{j,k=0}^{n}a_{i,j,k}y_{i}u_{j}u_{k}}du_{0}...du_{n}=$
$(2\pi\hbar)^{{n+2\over
2}}\delta\Bigg{(}\det\Bigg{(}\sum_{i=0}^{n+1}a_{i,j,k}y_{i}\Bigg{)}_{0\leq
j,k\leq n}\Bigg{)}\hat{g}\Big{(}{y_{1}\over y_{n+1}},...,{y_{n}\over
y_{n+1}}\Big{)}y_{n+1}^{{n\over 2}}$

where $g$ is a homogeneous function of degree 1.

Theorem 5.2.1. Any non-degenerate cone $C$ admitting a quadratic
parametrization is admissible with the rank larger or equal than $n+1$.
Moreover, any linear homogeneous function $g$ is admissible.

Proof. Without any loss of generality we can set $g(u_{0},...,u_{n})=u_{0}$
because of the $GL(n+1)$-action on variables $u_{0},...,u_{n}$. In order to
compute the integral

$\int u_{0}e^{{1\over
2\hbar}\sum_{i=0}^{n+1}\sum_{j,k=0}^{n}a_{i,j,k}y_{i}u_{j}u_{k}}du_{0}...du_{n}$
(5.37)

in our formalism we need the following

Lemma 5.2.1. The following identities hold

$\int e^{{1\over\hbar}({1\over
2}\sum_{i,j=1}^{n}a_{i,j}u_{i}u_{j}+\sum_{i=1}^{n}b_{i}u_{i})}du_{1}...du_{n}={(2\pi\hbar)^{{n\over
2}}\over(\det A)^{{1\over 2}}}e^{-{1\over 2\hbar}\vec{b}A^{-1}\vec{b}^{t}}$
$\vec{b}A^{-1}\vec{b}^{t}-c=-{\det\begin{pmatrix}A&\vec{b}^{t}\\\
\vec{b}&c\end{pmatrix}\over\det A}$

where $A=(a_{i,j})$ is a non-degenerate symmetric $n\times n$ matrix,
$\vec{b}=(b_{1},...,b_{n})$ is a vector and $c$ is a constant.

Proof. The first identity is a standard Gaussian integral up to multiplication
by $\pm i$. The second identity is obtained by the last row and the last
column expansion of the determinant in numerator in the r.h.s. $\square$

Lemma 5.2.2. The following identity holds

${1\over 2\pi\hbar}\int ue^{{au^{2}\over\hbar}}du=\delta(a).$

Proof. Multiply both sides of this identity by $e^{{av\over\hbar}}$ and
integrate by $a$. In the r.h.s. we get 1, and in the l.h.s.

${1\over 2\pi\hbar}\int
ue^{{au^{2}+av\over\hbar}}duda=\int\delta(u^{2}+v)udu=1.$

$\square$

To compute the integral (5.37) we first integrate with respect to
$u_{1},...,u_{n}$ using identities from Lemma 5.2.1 and get

${(2\pi\hbar)^{{n\over
2}}\over\Big{(}\det\Big{(}\sum_{i=0}^{n+1}a_{i,j,k}y_{i}\Big{)}_{1\leq j,k\leq
n}\Big{)}^{{1\over 2}}}\int u_{0}~{}e^{{u_{0}^{2}\over
2\hbar}{\det\Big{(}\sum_{i=0}^{n+1}a_{i,j,k}y_{i}\Big{)}_{0\leq j,k\leq
n}\over\det\Big{(}\sum_{i=0}^{n+1}a_{i,j,k}y_{i}\Big{)}_{1\leq j,k\leq
n}}}~{}du_{0}.$

Finally, we integrate by $u_{0}$ using Lemma 5.2.2 and get

$\int u_{0}e^{{1\over
2\hbar}\sum_{i=0}^{n+1}\sum_{j,k=0}^{n}a_{i,j,k}y_{i}u_{j}u_{k}}du_{0}...du_{n}=$
$(2\pi\hbar)^{{n+2\over
2}}\Big{(}\det\Big{(}\sum_{i=0}^{n+1}a_{i,j,k}y_{i}\Big{)}_{1\leq j,k\leq
n}\Big{)}^{{1\over
2}}2\delta\Bigg{(}\det\Bigg{(}\sum_{i=0}^{n+1}a_{i,j,k}y_{i}\Bigg{)}_{0\leq
j,k\leq n}\Bigg{)}.$

$\square$

One can reformulate the previous theorem using formalism from Section 3.3.
Namely, consider projectively dual cones

$C_{univ}=\\{X\in Mat((n+1)\times(n+1))~{}|~{}X^{t}=X,~{}rkX\leq
1\\}\subset\mathbb{A}^{{(n+1)(n+2)\over 2}},$ $\hat{C}_{univ}=\\{Y\in
Mat((n+1)\times(n+1))~{}|~{}Y^{t}=Y,~{}\det(Y)=0\\}\subset(\mathbb{A}^{{(n+1)(n+2)\over
2}})^{*}.$

Notice that $\dim C_{univ}=n+1$ while $\hat{C}_{univ}$ is a hypersurface. Cone
$C_{univ}$ admits quadratic parametrization $X=(x_{i,j})_{0\leq i,j\leq n}$
where $x_{i,j}=u_{i}u_{j}$ (passing to the projectivization we obtain the
Veronese map $\mathbb{P}^{n}\to\mathbb{P}^{{(n+1)(n+2)\over 2}-1}$ from the
classical algebraic geometry).

Theorem 5.2.2. The Fourier transform of the density $u_{0}du_{0}...du_{n}$ on
$C_{univ}$ is supported on $\hat{C}_{univ}$. More precisely

$\int e^{{1\over
2\hbar}\sum_{j,k=0}^{n}y_{j,k}u_{j}u_{k}}u_{0}du_{0}...du_{n}=(2\pi\hbar)^{{n+2\over
2}}(\det(y_{j,k})_{1\leq j,k\leq n})^{{1\over 2}}2\delta(\det(y_{j,k})_{0\leq
j,k\leq n}).$

The proof is omitted as it is essentially the same as for Theorem 5.2.1.

One can deduce geometrically Theorem 5.2.1 from Theorem 5.2.2 in the following
way. The choice of $(a_{i,j,k})$ gives a linear projection
$\mathbb{A}^{{(n+1)(n+2)\over 2}}\twoheadrightarrow\mathbb{A}^{n+2}$. The
image of $C_{univ}$ becomes a conical hypersurface in $\mathbb{A}^{n+2}$. The
Fourier transform of the density $u_{0}du_{0}...du_{n}$ on the image is the
restriction to the subspace
$(\mathbb{A}^{n+2})^{*}\hookrightarrow(\mathbb{A}^{{(n+1)(n+2)\over 2}})^{*}$
of the corresponding density on $\hat{C}_{univ}$ and is supported on
$(\mathbb{A}^{n+2})^{*}\cap\hat{C}_{univ}=\hat{C}$.

Remark 5.2.1. It would be interesting to study homogeneous densities supported
on conical orbit of reductive groups in finite-dimensional representations
such that their Fourier transforms are supported on hypersurfaces (or cones of
lower dimensions). By a similar consideration as above, this could lead to a
new class of admissible conical hypersurfaces beyond those admitting quadratic
parametrization.

Remark 5.2.2. One can write system of differential equations for admissible
functions $g(u_{0},...,u_{n})$ using results of Section 4.3, see for example
(4.36). This system depends on tensor $a_{i,j,k}$. It could be holonomic or
nonholonomic depending on $a_{i,j,k}$. If it is holonomic, then $rk(C)$ is
finite, otherwise it is infinite.

Conjecture 5.2.1. If $rk(C)$ is finite, then any admissible $g$ has a form

$g(u_{0},...,u_{n})={Q\over P_{1}...P_{k}},~{}~{}~{}k=0,1,...$

where $Q$ is a homogeneous polynomial in $u_{0},...u_{n}$ of degree $k+1$, and
$P_{1},...,P_{k}$ are homogeneous linear polynomials in $u_{0},...,u_{n}$.

Remark 5.2.3. Theorem 5.2.1 gives the lower bound $n+1$ for the rank $rk(C)$
of cone $C$ admitting quadratic parametrization. Experiments show that if
$n\geq 3$ and tensor $a_{i,j,k}$ is generic, then this bound is attained and
$rk(C)=n+1$. If $n=2$ and tensor $a_{i,j,k}$ is generic, then $rk(C)=6$, see
Section 5.4. On the other hand, $rk(C)$ could be infinite for $n\geq 2$. For
example, the standard smooth quadric hypersurface admits a quadratic
parametrization, and its rank is infinite for $n\geq 2$. It would be
interesting to find $rk(C)$ in terms of algebraic properties of tensor
$a_{i,j,k}$. In particular, it would be interesting to find conditions for
tensor $a_{i,j,k}$ equivalent to the property $rk(C)>n+1$.

Conjecture 5.2.2. If $rk(C)$ is finite, then $rk(C)\leq(n+1)!$.

It would be interesting to prove this Conjecture and classify all tensors
$a_{i,j,k}$ such that $rk(C)=(n+1)!$. See Section 5.4 where we construct an
example of such cone for arbitrary $n$.

###  5.3 Ruled surfaces in $\mathbb{P}^{3}$

Theorem 5.3.1. Let $\Sigma\subset\mathbb{P}^{3}$ be a surface defined
parametrically by

$x_{0}=1,~{}~{}~{}x_{1}=p_{1}(u_{2})+q_{1}(u_{2})u_{1},~{}~{}~{}x_{2}=p_{2}(u_{2})+q_{2}(u_{2})u_{1},~{}~{}~{}x_{3}=p_{3}(u_{2})+q_{3}(u_{2})u_{1}$

where $p_{1},p_{2},p_{3},q_{1},q_{2},q_{3}$ are arbitrary generic functions in
one variable. Then $\Sigma$ is admissible and its rank is infinity.

The corresponding admissible pairs $f(x_{1},x_{2}),g(x_{1},x_{2})$ have the
following parametrization

$f=p_{3}(u_{2})+q_{3}(u_{2})u_{1},~{}~{}~{}x_{1}=p_{1}(u_{2})+q_{1}(u_{2})u_{1},~{}~{}~{}x_{2}=p_{2}(u_{2})+q_{2}(u_{2})u_{1},$
(5.38) $g={h(u_{2})\over
q_{1}(u_{2})\big{(}p^{\prime}_{2}(u_{2})+q^{\prime}_{2}(u_{2})u_{1}\big{)}-q_{2}(u_{2})\big{(}p^{\prime}_{1}(u_{2})+q^{\prime}_{1}(u_{2})u_{1}\big{)}}$

where $h(u_{2})$ is an arbitrary function.

Proof. Substituting the parametric form (5.38) in the l.h.s. of (1.6) we get

$\int
h(u_{2})e^{{1\over\hbar}(p_{3}(u_{2})+q_{3}(u_{2})u_{1}+(p_{1}(u_{2})+q_{1}(u_{2})u_{1})y_{1}+(p_{2}(u_{2})+q_{2}(u_{2})u_{1})y_{2})}du_{1}du_{2}$

After integrating by $u_{1}$ we obtain

$2\pi\hbar\int\delta(q_{3}(u_{2})+q_{1}(u_{2})y_{1}+q_{2}(u_{2})y_{2})h(u_{2})e^{{1\over\hbar}(p_{3}(u_{2})+p_{1}(u_{2})y_{1}+p_{2}(u_{2})y_{2})}du_{2}$

and integrating by $u_{2}$ we obtain an expression of a form of the r.h.s. of
(1.6). $\square$

Remark 5.3.1. Notice that the parametric representation (5.38) admits the
change of variables $u_{1}\to a(u_{2})u_{1}+b(u_{2}),~{}u_{2}\to c(u_{2})$, so
the family of admissible functions (5.38) is parameterized by four functions
in one variable. We can set, for example,
$p_{1}(u_{2})=0,q_{1}(u_{2})=1,q_{2}(u_{2})=u_{2}$.

Remark 5.3.2. For a generic ruled surface the whole space $V_{f}$ of
admissible functions $g$ is given by (5.38). However, in special cases the
space $V_{f}$ of admissible functions $g$ can be bigger. For example it is
bigger if $f$ corresponds to a quadric. If $f(x_{1},x_{2})=x_{1}x_{2}$, then
$V_{f}$ consists of functions of the form
$g(x_{1},x_{2})=h_{1}(x_{1})+h_{2}(x_{2})$ where $h_{1},h_{2}$ are arbitrary
functions in one variable. It would be interesting to study in details for
which ruled surfaces the space $V_{f}$ is bigger than those given by (5.38).

###  5.4 Steiner Roman surface

Hypersurfaces admitting quadratic parametrization in the case $n=2$ are
equivalent to each other for the generic tensor $a_{i,j,k}$. The corresponding
projective variety is called Steiner Roman surface [4].

Theorem 5.4.1. Let $\Sigma\subset\mathbb{P}^{3}$ be a surface defined
parametrically by

$x_{i}=q_{i}(u_{1},u_{2}),~{}i=0,...,3$

where $q_{i}$ are generic nonhomogeneous quadratic polynomials in
$u_{1},u_{2}$. Then $\Sigma$ is admissible and its rank is 6.

Proof. It is known that any such surface is equivalent to projectivization of
the affine surface defined by $x_{3}=f(x_{1},x_{2})$ where

$f(x_{1},x_{2})=x_{1}x_{2}+{x_{1}\over x_{2}}+{x_{2}\over x_{1}}.$

In this case pair $f,g$ is admissible iff

$g(x_{1},x_{2})=c_{1}+{c_{2}\over x_{1}}+{c_{3}\over
x_{2}}+c_{4}~{}\Big{(}{1\over x_{1}^{2}}-{1\over
x_{2}^{2}}\Big{)}+c_{5}~{}\Big{(}x_{1}-{x_{1}\over
x_{2}^{2}}\Big{)}+c_{6}~{}\Big{(}x_{2}-{x_{2}\over x_{1}^{2}}\Big{)}$

where $c_{1},...,c_{6}$ are constants. One can check this by solving the
system of differential equations (4.36) with given function $f$. $\square$

###  5.5 Generalized Steiner Roman hypersurfaces

Let cone $C\subset\mathbb{A}^{n+2}$ for $n\geq 2$ be given by

$x_{0}^{{1\over 2}}+x_{1}^{{1\over 2}}+...+x_{n+1}^{{1\over 2}}=0.$

Remark 5.5.1. If $n=2$, then $C$ is equivalent to Steiner Roman surface
(Section 5.4).

Theorem 5.5.1. The cone $C$ (and therefore its dual $\hat{C}~{}$) is
admissible with the rank equal to $(n+1)!$.

Proof. The equation for the dual cone $\hat{C}$ can be written as

${1\over y_{0}}+{1\over y_{1}}+...+{1\over y_{n+1}}=0$

or, equivalently, as

$\sum_{i=0}^{n+1}y_{0}...\widehat{y}_{i}...y_{n+1}=0.$

Therefore, we can choose the formal wave function in the form

$\tilde{g}(x_{0},...,x_{n})\delta\Big{(}x_{0}^{{1\over 2}}+x_{1}^{{1\over
2}}+...+x_{n+1}^{{1\over 2}}\Big{)}$

and it should be annihilated by differential operator

$\sum_{i=0}^{n+1}\partial_{x_{0}}...\widehat{\partial}_{x_{i}}...\partial_{x_{n+1}}.$

Let us make the change of variables $x_{i}=u_{i}^{2},~{}~{}i=0,...,n+1$. In
the new variables our wave functions reads

$\psi=g(u_{0},...,u_{n})\delta(u_{0}+u_{1}+...+u_{n+1})$

and the differential operator (up to a coefficient and after multiplication by
$u_{0}u_{1}...u_{n+1}$) takes the form

$D=\sum_{i=0}^{n+1}u_{i}\partial_{u_{0}}...\widehat{\partial}_{u_{i}}...\partial_{u_{n+1}}.$

The equation $D(\psi)=0$ is equivalent to the following system of differential
equations for the function $g(u_{0},...,u_{n})$

$\sum_{i=0}^{n}u_{i}\partial_{u_{i}}g+(n+1)g=0$ $\sum_{0\leq i_{1}<i_{2}\leq
n}(u_{i_{1}}+u_{i_{2}})\partial_{u_{i_{1}}}\partial_{u_{i_{2}}}g+n\sum_{i=0}^{n}\partial_{u_{i}}g=0$
$........................................................................................................................................$
$\sum_{0\leq i_{1}<...<i_{k}\leq
n}(u_{i_{1}}+...+u_{i_{k}})\partial_{u_{i_{1}}}...\partial_{u_{i_{k}}}g+(n+2-k)\sum_{0\leq
j_{1}<...<j_{k-1}\leq n}\partial_{u_{j_{1}}}...\partial_{u_{j_{k-1}}}g=0$
(5.39)
$........................................................................................................................................$
$(u_{0}+...+u_{n})\partial_{u_{0}}...\partial_{u_{n}}g+\sum_{j=0}^{n}\partial_{u_{0}}...\widehat{\partial}_{u_{j}}...\partial_{u_{n}}g=0.$

To finish the proof we need the following

Lemma 5.5.1. The system (5.39) is holonomic and has no more than
$(n+1)!$-dimensional space of solutions at generic point $(u_{0},...,u_{n})$.

Proof. The system (5.39) defines a cyclic $D$-module $M$ generated by $g$. The
filtration $D^{\leq i}$ by the degree of differential operators, after
applying to the generator $g$ induces a good filtration on $M$. The associated
graded space $grM$ is a graded module over polynomial ring ${\bf
k}[u_{0},...,u_{n},v_{0},...,v_{n}]$ where $\deg u_{i}=0,~{}\deg v_{i}=1$ and
$v_{i}$ are images of $\partial_{u_{i}}$ in the polynomial ring. The module
$grM$ is also cyclic and defined by equations

$\Big{(}u_{0}v_{0}+...+u_{n}v_{n}\Big{)}h=0$ $\sum_{0\leq i_{1}<i_{2}\leq
n}(u_{i_{1}}+u_{i_{2}})v_{i_{1}}v_{i_{2}}h=0$
$.........................................................................$
$(u_{0}+...+u_{n})v_{0}...v_{n}h=0.$

One can check that for generic $u_{i}$ (e.g. $u_{0}=...=u_{n}=1$) the above
system gives a 0-dimensional scheme which is a complete intersection of length
$(n+1)!$, the product of degrees in $v_{i}$ of our equations. $\square$

Lemma 5.5.2. Let $\\{0,1,...,n+1\\}=P_{1}\sqcup...\sqcup
P_{m+1},~{}~{}m=1,...,n+1$ be an arbitrary partition. Set
$u_{n+1}=-u_{0}-...-u_{n}$. Then the following functions

$g={1\over u_{0}...u_{n+1}}\cdot{u_{i_{1}}...u_{i_{m+1}}\over\sum_{\alpha\in
P_{1}}u_{\alpha}\cdot\sum_{\alpha\in
P_{2}}u_{\alpha}\cdot...\cdot\sum_{\alpha\in
P_{m}}u_{\alpha}},~{}~{}~{}i_{1}\in P_{1},...,i_{m+1}\in P_{m+1}$

satisfy the system (5.39). The vector space spent by these functions is
$(n+1)!$-dimensional.

Proof. This can be proved by induction as follows. Without loss of generality
we can set $P_{1}=\\{0,...,k\\}$ and $i_{1}=k$ because of $S_{n+1}$ action on
the system (5.39). Write

$g(u_{0},...,u_{n})={g_{1}(u_{k+1},...,u_{n})\over
u_{0}...u_{k-1}(u_{0}+...+u_{k})}$

and check that $g$ satisfies (5.39) iff $g_{1}$ satisfies similar system with
$n-k$ variables $u_{k+1},...,u_{n}$. $\square$

Example 5.5.1. Let $n=4$,
$P_{1}=\\{0,1\\},~{}P_{2}=\\{2,3\\},~{}P_{3}=\\{4,5\\}$ and choose $0\in
P_{1},~{}2\in P_{2},~{}4\in P_{3}$. We have

$g={1\over
u_{0}u_{1}u_{2}u_{3}u_{4}u_{5}}\cdot{u_{0}u_{2}u_{4}\over(u_{0}+u_{1})(u_{2}+u_{3})}.$

Remark 5.5.2. We have $\deg C=2^{n}$ and $\deg\hat{C}=n+1$. In particular,
$\hat{C}$ is not equivalent to $C$ and gives another example of admissible
cone with the rank equal to $(n+1)!$.

###  5.6 Kummer surfaces in $\mathbb{P}^{3}$

Recall that the Kummer surface is a quartic surface in $\mathbb{P}^{3}$ with
16 double points. Abstractly it is the quotient of a principally polarized two
dimensional abelian variety by the antipodal involution.

Theorem 5.6.1. Any Kummer surface is admissible of rank 6.

Proof. We need to check (4.31) for the Kummer surfaces. Choose equation for
Kummer surface in the form [5]

$F_{1}=(x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+a(x_{0}x_{1}+x_{2}x_{3})+b(x_{0}x_{2}+x_{1}x_{3})+c(x_{0}x_{3}+x_{1}x_{2}))^{2}$
$-4(a^{2}+b^{2}+c^{2}-abc-4)x_{0}x_{1}x_{2}x_{3}.$

Computing equation for dual polynomial we get

$F_{2}=(4a-2bc)(\partial_{x_{0}}\partial_{x_{1}}\partial_{x_{2}}^{2}+\partial_{x_{0}}^{2}\partial_{x_{2}}\partial_{x_{3}}+\partial_{x_{1}}^{2}\partial_{x_{2}}\partial_{x_{3}}+\partial_{x_{0}}\partial_{x_{1}}\partial_{x_{3}}^{2})+$
$(4b-2ac)(\partial_{x_{0}}\partial_{x_{1}}^{2}\partial_{x_{2}}+\partial_{x_{0}}^{2}\partial_{x_{1}}\partial_{x_{3}}+\partial_{x_{1}}\partial_{x_{2}}^{2}\partial_{x_{3}}+\partial_{x_{0}}\partial_{x_{2}}\partial_{x_{3}}^{2})+$
$(4c-2ab)(\partial_{x_{0}}^{2}\partial_{x_{1}}\partial_{x_{2}}+\partial_{x_{0}}\partial_{x_{1}}^{2}\partial_{x_{3}}+\partial_{x_{0}}\partial_{x_{2}}^{2}\partial_{x_{3}}+\partial_{x_{1}}\partial_{x_{2}}\partial_{x_{3}}^{2})+$
$(a^{2}-4)(\partial_{x_{0}}^{2}\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2}\partial_{x_{3}}^{2})+(b^{2}-4)(\partial_{x_{0}}^{2}\partial_{x_{2}}^{2}+\partial_{x_{1}}^{2}\partial_{x_{3}}^{2})+(c^{2}-4)(\partial_{x_{0}}^{2}\partial_{x_{3}}^{2}+\partial_{x_{1}}^{2}\partial_{x_{2}}^{2})+$
$(4abc-2a^{2}-2b^{2}-2c^{2}-8)\partial_{x_{0}}\partial_{x_{1}}\partial_{x_{2}}\partial_{x_{3}}.$

Let us choose $g$ in the form

$g=x_{1}^{{1\over 2}}x_{2}^{{1\over
2}}\Big{(}(b\sqrt{c^{2}-4}+2a-bc)(x_{0}^{2}+cx_{0}x_{3}+x_{3}^{2})+2\sqrt{c^{2}-4}(x_{0}x_{2}+x_{1}x_{3})+$
$+(c\sqrt{c^{2}-4}-c^{2}+4)(x_{0}x_{1}+x_{2}x_{3})\Big{)}^{{1\over 2}}.$

One can check by direct calculation that

$F_{2}\Big{(}g\cdot\delta(F_{1})\Big{)}=0$

where $F_{2}$ is a differential operator and $g,F_{1}$ are expression written
above. Notice that $F_{1},F_{2}$ are invariant with respect to the action of
the group of permutation $S_{4}$ acting on $x_{0},...,x_{3}$ and
simultaneously on $a,b,c$. Applying this action to $g$ we obtain other
admissible functions. We can change sign of the square root $\sqrt{c^{2}-4}$
as well. One can check that in this way we obtain a six-dimensional space of
admissible functions $g$. On the other hand, we know (see Section 7) that if
an admissible surface is not ruled, then $\dim V_{f}\leq 6$. $\square$

The space $V_{f}$ for a Kummer surface can be constructed as follows. Let
$\mathcal{A}_{2}$ be a two-dimensional principally polarized abelian variety.
Let $\sigma:\mathcal{A}\to\mathcal{A}$ be its involution with a fixed point.
Construct the corresponding Kummer surface by
$\mathcal{K}=\mathcal{A}_{2}/\sigma.$ Let $\mathcal{L}$ be a line bundle of
degree 2 on $\mathcal{A}_{2}$ (hence with 4 sections) and such that
$\mathcal{L}^{\sigma}=\mathcal{L}$. It is known that $\sigma$ acts as identity
operator on the space $\Gamma(\mathcal{L})$ of sections of $\mathcal{L}$. We
can identify $x_{0},...,x_{3}$ with a basis in $\Gamma(\mathcal{L})$. It is
also known that $\dim\Gamma(\mathcal{L}^{2})=16$ and
$\Gamma(\mathcal{L}^{2})=\Gamma^{+}\oplus\Gamma^{-}$ where $\sigma$ acts as
identity on $\Gamma^{+}$ and as negative identity on $\Gamma^{-}$. We have
$\dim\Gamma^{+}=10$ and $x_{i}x_{j},~{}0\leq i\leq j\leq 3$ is a basis in
$\Gamma^{+}$. The space $V_{f}$ can be identified with $\Gamma^{-}$.

These spaces of sections can be constructed explicitly over $\mathbb{C}$ using
theta functions in two variables.

Define the functions $\theta_{a,b}(u_{1},u_{2}),~{}a,b\in\mathbb{Z}/2$ by

$\theta_{00}(u_{1},u_{2})=\sum_{i,j\in\mathbb{Z}}e^{2\pi\sqrt{-1}~{}\big{(}2iu_{1}+2ju_{2}+i(i-1)\tau_{11}+2ij\tau_{12}+j(j-1)\tau_{22}\big{)}},$
$\theta_{10}(u_{1},u_{2})=\sum_{i,j\in\mathbb{Z}}e^{2\pi\sqrt{-1}~{}\big{(}(2i+1)u_{1}+2ju_{2}+i^{2}\tau_{11}+(2i+1)j\tau_{12}+j(j-1)\tau_{22}\big{)}},$
$\theta_{01}(u_{1},u_{2})=\sum_{i,j\in\mathbb{Z}}e^{2\pi\sqrt{-1}~{}\big{(}2iu_{1}+(2j+1)u_{2}+i(i-1)\tau_{11}+i(2j+1)\tau_{12}+j^{2}\tau_{22}\big{)}},$
$\theta_{11}(u_{1},u_{2})=\sum_{i,j\in\mathbb{Z}}e^{2\pi\sqrt{-1}~{}\big{(}(2i+1)u_{1}+(2j+1)u_{2}+i^{2}\tau_{11}+(2ij+i+j)\tau_{12}+j^{2}\tau_{22}\big{)}}$

where $\tau_{00},\tau_{12},\tau_{22}$ are constants such that the matrix
$\begin{pmatrix}\tau_{11}&\tau_{12}\\\ \tau_{11}&\tau_{22}\end{pmatrix}$
belongs to Siegel upper half-space of genus two, i.e. its imaginary part is
positive. The functions $\theta_{a,b}(u_{1},u_{2})$ are holomorphic and
satisfy the following periodicity and quasiperiodicity properties

$\theta_{a,b}(u_{1}+1,u_{2})=\theta_{a,b}(u_{1},u_{2}),~{}~{}~{}\theta_{a,b}(u_{1},u_{2}+1)=\theta_{a,b}(u_{1},u_{2}),$
$\theta_{a,b}(u_{1}+\tau_{11},u_{2}+\tau_{12})=e^{-2\pi\sqrt{-1}~{}2u_{1}}\theta_{a,b}(u_{1},u_{2}),~{}~{}~{}\theta_{a,b}(u_{1}+\tau_{12},u_{2}+\tau_{22})=e^{-2\pi\sqrt{-1}~{}2u_{2}}\theta_{a,b}(u_{1},u_{2}).$

Moreover, any holomorphic function with these properties is a linear
combination of $\theta_{a,b}(u_{1},u_{2}),$ $a,b\in\mathbb{Z}/2$.

We can define Kummer surface and corresponding functions
$f(x_{1},x_{2}),g(x_{1},x_{2})$ parametrically by

$f={\theta_{00}(u_{1},u_{2})\over\theta_{11}(u_{1},u_{2})},~{}x_{1}={\theta_{10}(u_{1},u_{2})\over\theta_{11}(u_{1},u_{2})},~{}x_{2}={\theta_{01}(u_{1},u_{2})\over\theta_{11}(u_{1},u_{2})},$
$g={1\over\Delta}~{}\Bigg{(}c_{1}\Bigg{(}{\theta_{00}(u_{1},u_{2})\over\theta_{11}(u_{1},u_{2})}\Bigg{)}_{u_{1}}+c_{2}\Bigg{(}{\theta_{10}(u_{1},u_{2})\over\theta_{11}(u_{1},u_{2})}\Bigg{)}_{u_{1}}+c_{3}\Bigg{(}{\theta_{01}(u_{1},u_{2})\over\theta_{11}(u_{1},u_{2})}\Bigg{)}_{u_{1}}+$
$c_{4}\Bigg{(}{\theta_{00}(u_{1},u_{2})\over\theta_{11}(u_{1},u_{2})}\Bigg{)}_{u_{2}}+c_{5}\Bigg{(}{\theta_{10}(u_{1},u_{2})\over\theta_{11}(u_{1},u_{2})}\Bigg{)}_{u_{2}}+c_{6}\Bigg{(}{\theta_{01}(u_{1},u_{2})\over\theta_{11}(u_{1},u_{2})}\Bigg{)}_{u_{2}}\Bigg{)}$

where $c_{1},...,c_{6}$ are constants, indexes $u_{1},u_{2}$ stand for partial
derivatives and

$\Delta=\Bigg{(}{\theta_{10}(u_{1},u_{2})\over\theta_{11}(u_{1},u_{2})}\Bigg{)}_{u_{1}}~{}\Bigg{(}{\theta_{01}(u_{1},u_{2})\over\theta_{11}(u_{1},u_{2})}\Bigg{)}_{u_{2}}-\Bigg{(}{\theta_{01}(u_{1},u_{2})\over\theta_{11}(u_{1},u_{2})}\Bigg{)}_{u_{1}}~{}\Bigg{(}{\theta_{10}(u_{1},u_{2})\over\theta_{11}(u_{1},u_{2})}\Bigg{)}_{u_{2}}.$

Then pair $f,g$ is admissible.

Remark 5.6.1. Kummer surfaces have many degenerations which are not Kummer
surfaces anymore but also give examples of admissible surfaces. For example,
Steiner Roman surface can be obtained in this way.

Remark 5.6.2. It would be interesting to find a more transparent proof of
Theorem 5.6.1, for example, using parametrization of Kummer surface by theta
functions or using relation of Kummer surface with the quadratic line complex.
Notice that our space of sections $\Gamma^{-}$ described above has a basis
$y_{1},...,y_{6}$ such that

$y_{1}^{2}+...+y_{6}^{2}=0,$
$\lambda_{1}y_{1}^{2}+...+\lambda_{6}y_{6}^{2}=0,$
$\lambda_{1}^{2}y_{1}^{2}+...+\lambda_{6}^{2}y_{6}^{2}=0$

where $\lambda_{1},...,\lambda_{6}\in\mathbb{C}$ are constants. Therefore, the
space $\Gamma^{-}$ gives an embedding of the quadratic line complex to
$\mathbb{P}^{5}$.

###  5.7 Extensions of admissible pairs and families of surfaces of degree
four

Theorem 5.7.1. Let a pair of functions
$\tilde{f}(x_{1},...,x_{k},y_{k+1},...,y_{n}),~{}\tilde{g}(x_{1},...,x_{k},y_{k+1},...,y_{n})$
be admissible as functions in variables $x_{1},...,x_{k}$ where
$y_{k+1},...,y_{n}$ are considered as parameters, and let this pair be also
admissible as functions in variables $y_{k+1},...,y_{n}$ where
$x_{1},...,x_{k}$ are considered as parameters. Let us construct a new pair of
functions $f(x_{1},...,x_{n}),~{}g(x_{1},...,x_{n})$ by doing the Fourier
transform with respect to variables $y_{k+1},...,y_{n}$ and using our
admissibility assumption

$\int\tilde{g}(x_{1},...,x_{k},y_{k+1},...,y_{n})e^{{\tilde{f}(x_{1},...,x_{k},y_{k+1},...,y_{n})-x_{k+1}y_{k+1}-...-x_{n}y_{n}\over\hbar}}dy_{k+1}...dy_{n}=$
$(2\pi\hbar)^{{n-k\over
2}}g(x_{1},...,x_{n})e^{{f(x_{1},...,x_{n})\over\hbar}}.$

Then the pair of functions in $n$ variables $f,g$ is admissible.

Proof. Notice that the pair $f,g$ is also admissible with respect to variables
$x_{k+1},...,x_{n}$ as the Fourier transform of an admissible pair. We compute
the integral in the l.h.s. of (1.6) in two steps so that in each step the
integral is one loop exact. First, we integrate with respect to
$x_{k+1},...,x_{n}$ using the admissibility with respect to these variables.
Second, we integrate with respect to $x_{1},...,x_{k}$ using admissibility of
$\tilde{f},\tilde{g}$ with respect to these variables. $\square$

Example 5.7.1. Let us define functions
$\tilde{f}(x_{1},y_{2}),~{}\tilde{g}(x_{1},y_{2})$ by

$\tilde{f}(x_{1},y_{2})=\sqrt{x_{1}^{2}-1}\sqrt{y_{2}^{2}-1}+(a_{1}x_{1}+a_{2})\sqrt{y_{2}^{2}-1}+(a_{3}y_{2}+a_{4})\sqrt{x_{1}^{2}-1}+a_{5}x_{1}y_{2},$
$\tilde{g}(x_{1},y_{2})={c_{1}\over\sqrt{x_{1}+1}\sqrt{y_{2}+1}}+{c_{2}\over\sqrt{x_{1}-1}\sqrt{y_{2}+1}}+{c_{3}\over\sqrt{x_{1}+1}\sqrt{y_{2}-1}}+{c_{4}\over\sqrt{x_{1}-1}\sqrt{y_{2}-1}}$

where $a_{1},...,a_{5},c_{1},...,c_{4}$ are constants. It follows from results
of Section 6 (see Theorem 6.1) that the pair $\tilde{f},\tilde{g}$ is
admissible with respect to variable $x_{1}$ and with respect to variable
$y_{2}$ (but it is not admissible as a pair of functions in two variables).
After the Fourier transform with respect to $y_{2}$ we get the following
admissible pair of functions in two variables

$f(x_{1},x_{2})=a_{4}\sqrt{x_{1}^{2}-1}+$
$\sqrt{x_{2}-(a_{1}+a_{5})x_{1}-a_{2}-(a_{3}+1)\sqrt{x_{1}^{2}-1}}\sqrt{x_{2}+(a_{1}-a_{5})x_{1}+a_{2}-(a_{3}-1)\sqrt{x_{1}^{2}-1}},$
$g(x_{1},x_{2})={{c_{1}\over\sqrt{x_{1}+1}}+{c_{2}\over\sqrt{x_{1}-1}}\over\sqrt{x_{2}-(a_{1}+a_{5})x_{1}-a_{2}-(a_{3}+1)\sqrt{x_{1}^{2}-1}}}+$
${{c_{3}\over\sqrt{x_{1}+1}}+{c_{4}\over\sqrt{x_{1}-1}}\over\sqrt{x_{2}+(a_{1}-a_{5})x_{1}+a_{2}-(a_{3}-1)\sqrt{x_{1}^{2}-1}}}.$

A similar example of an admissible pair coming from Theorem 5.7.1 is given
below, where explicit formulas for $\tilde{f},\tilde{g}$ are omitted.

Theorem 5.7.2. Define functions $f(x_{1},x_{2}),g(x_{1},x_{2})$ parametrically
as

$f=u_{1}u_{2}^{4}+2r_{1}r_{2}u_{1}u_{2}^{3}+(r_{1}^{2}r_{2}^{2}+2)u_{1}u_{2}^{2}+2u_{1}u_{2}r_{1}r_{2}+u_{1}+{1\over
u_{1}},~{}x_{1}=u_{2}^{2}r_{2}-u_{2}r_{1}+r_{2}+{1\over
u_{1}},~{}x_{2}=u_{2}^{2}$ $g=\sqrt{u_{1}}~{}C_{1}+\sqrt{u_{1}}~{}{C_{2}\over
u_{2}}+u_{1}^{3/2}(r_{1}r_{2}u_{2}+u_{2}^{2}+1)C_{3}+u_{1}^{3/2}(r_{1}r_{2}u_{2}+u_{2}^{2}+1){C_{4}\over
u_{2}}$

where $r_{1},r_{2},C_{1},C_{2},C_{3},C_{4}$ are constants. Then the pair
$f(x_{1},x_{2}),g(x_{1},x_{2})$ is admissible and its rank is 4.

Proof. Since $f$ is an algebraic function, we just need to check (4.36) which
can be done by a direct calculation. Alternatively, notice that for any fixed
$x_{2}$ the pair $f,g$ is admissible as pair of functions in one variable
$x_{1}$. Moreover, the integral $\int
g(x_{1},x_{2})e^{{f(x_{1},x_{2})+x_{1}y_{1}+x_{2}y_{2}\over\hbar}}dx_{1}dx_{2}$
can be computed in two steps: first with respect to $x_{1}$, and than with
respect to $x_{2}$. In each step the integral with respect to one variable is
one loop exact. $\square$

Remark 5.7.1. One can check that Example 5.7.1 and Theorem 5.7.2 both give a
two parametric family of admissible cones in $\mathbb{P}^{3}$ of degree 4. In
both cases these cones have non-isolated singularities. We have not checked if
these cones are projectively equivalent.

Remark 5.7.2. One can generalize the construction from Theorem 5.7.1
considering for example functions in three or more groups of variables. It
might be interesting to study admissible pairs obtained in this way. On the
other hand, these admissible pairs look degenerate in a certain sense.

###  5.8 Toric hypersurfaces

Admissible pairs of the form
$f=x_{1}^{a_{1}}...x_{n}^{a_{n}},~{}g=x_{1}^{b_{1}}...x_{n}^{b_{n}}$ were
completely described in [1]. Here we just reformulate their results in terms
of our formalism.

Theorem 5.8.1. Let $C\subset\mathbb{P}^{n+1}$ be a toric projective
hypersurface defined by

$x_{0}^{p_{0}}...x_{m-1}^{p_{m-1}}=x_{m}^{p_{m}}...x_{n+1}^{p_{n+1}}$ (5.40)

where $p_{0},...,p_{n+1}>0$ are integers and
$p_{0}+...+p_{m-1}=p_{m+}+...+p_{n+1}$. Then $C$ is admissible if there exist
numbers $r_{0},...,r_{n+1}\in\mathbb{Q}$ such the following identity holds as
an identity for functions in one variable $t$

$\Gamma(p_{0}t+r_{0})...\Gamma(p_{m-1}t+r_{m-1})=ae^{bt}\Gamma(p_{m}t+r_{m})...\Gamma(p_{n+1}t+r_{n+1})$
(5.41)

where $a,b$ are constants.

We expect that the rank $C$ is equal to the number of vectors
$(r_{0},...,r_{n+1})\in\mathbb{Q}^{n+2}$ up to translation of the form
$r_{i}\to r_{i}+p_{i}u,~{}i=0,...,n+1$ for some $u$, such that the identity
(5.41) holds.

Remark 5.8.1. The transcendental condition (5.41) is equivalent to a
combinatorial one: the union of sets of poles (with multiplicities) in the
l.h.s. and in the r.h.s. of equation (5.41) coincide:

$\bigcup_{i=0}^{m-1}{1\over p_{i}}(\mathbb{Z}_{\geq
0}+r_{i})=\bigcup_{i=m}^{n+1}{1\over p_{i}}(\mathbb{Z}_{\geq 0}+r_{i}).$

Notice that here we multiply $t$ by $-1$.

Proof. Let $f(x_{1},...,x_{n})=x_{1}^{a_{1}}...x_{n}^{a_{n}}$ where
$a_{1},...,a_{n}\in\mathbb{Q}\setminus 0$. Let us set

$g(x_{1},...,x_{n})=x_{1}^{b_{1}}...x_{n}^{b_{n}}.$

Let

$I(\vec{y})=\int
x_{1}^{b_{1}}...x_{n}^{b_{n}}e^{{1\over\hbar}(x_{1}^{a_{1}}...x_{n}^{a_{n}}+x_{1}y_{1}+...+x_{n}y_{n})}.$
(5.42)

Lemma 5.8.1. The following equality of the formal wave functions holds

$e^{{x\over\hbar}}=\int{({x\over\hbar})^{{t\over\hbar}}\over\Gamma({t\over\hbar}+1)}{dt\over\hbar}.$
(5.43)

Proof. First, let us show that the r.h.s. of this equation makes sense as a
formal wave function. Let

$S(t)=\exp\Big{(}-{\zeta(-1)\over 1}t-{\zeta(-3)\over 3}t^{3}-{\zeta(-5)\over
5}t^{5}-...\Big{)}\in\mathbb{Q}[[t]]$

where $\zeta$ stands for the Riemann zeta function. Notice that

$S(t)S(-t)=1.$ (5.44)

Recall Stirling’s formula

$\Gamma(t+1)\sim(2\pi t)^{{1\over 2}}t^{t}e^{-t}S(t^{-1})$

where $\sim$ means the asymptotic expansion for $t\to\infty$. Since the r.h.s.
of (5.43) is also understood as an asymptotic expansion, we can replace the
gamma function with the r.h.s. of Stirling’s formula and rewrite the equation
(5.43) as

$(2\pi\hbar)^{{1\over 2}}e^{{x\over\hbar}}=\int t^{-{1\over
2}}S\Big{(}-{\hbar\over t}\Big{)}e^{{t\over\hbar}(1+\ln x-\ln t)}dt.$

The r.h.s. of this integral can be computed by the expansion at the critical
point $t=x$.

Now let us prove the identity (5.43). Let $\phi_{l}(x)$ (resp. $\phi_{r}(x)$)
be the l.h.s. (resp. the r.h.s.) of (5.43). One can check that these functions
satisfy the same differential equations
$\partial_{x}\phi_{l}(x)={x\over\hbar}\phi_{l}(x),~{}\partial_{x}\phi_{r}(x)={x\over\hbar}\phi_{r}(x)$.
Therefore, we have $\phi_{l}(x)=c\phi_{r}(x)$ where $c$ is independent of $x$.
To compute $c$ we just compute the first term of (5.43) by the stationary
phase method. $\square$

Using Lemma 5.8.1 we replace $e^{{1\over\hbar}x_{1}^{a_{1}}...x_{n}^{a_{n}}}$
in $I(\vec{y})$ with the corresponding r.h.s. of the identity (5.43) and
obtain

$I(\vec{y})=\int\prod_{i=1}^{n}x_{i}^{b_{i}+{a_{i}t\over\hbar}}e^{{x_{i}y_{i}\over\hbar}}~{}{1\over\Gamma({t\over\hbar}+1)\hbar^{{t\over\hbar}}}~{}{dt\over\hbar}~{}dx_{1}...dx_{n}.$
(5.45)

Lemma 5.8.2. Let $a>0$. The following equalities between formal wave functions
hold

$\int
x^{b+{a\over\hbar}}e^{{xy\over\hbar}}dx=\Big{(}-{y\over\hbar}\Big{)}^{-b-{a\over\hbar}-1}\Gamma(b+{a\over\hbar}+1),$
$\int
x^{b-{a\over\hbar}}e^{{xy\over\hbar}}dx=\Big{(}{y\over\hbar}\Big{)}^{-b+{a\over\hbar}-1}{2\pi
i\over\Gamma(-b+{a\over\hbar})}.$

Proof. For the first equation notice that if $x>0,~{}y<0,~{}\hbar>0$, then we
have the usual Euler integral representation of the gamma function in the form

$\int_{0}^{\infty}x^{b+{a\over\hbar}}e^{{xy\over\hbar}}dx=\Big{(}-{y\over\hbar}\Big{)}^{-b-{a\over\hbar}-1}\Gamma(b+{a\over\hbar}+1)$

which implies our equality for the formal wave functions. The second equation
makes sense only on the level of wave functions and follows from the first one
and the equation (5.44). $\square$

Using these formulas we rewrite $I(\vec{y})$ in (5.45) as

$I(\vec{y})=\int{1\over\Gamma({t\over\hbar}+1)\hbar^{{t\over\hbar}}}\prod_{i=1}^{n}\Big{(}\pm{y_{i}\over\hbar}\Big{)}^{-b_{i}-{a_{i}t\over\hbar}-1}\cdot{\prod_{i=1}^{k}\Gamma(b_{i}+{a_{i}t\over\hbar}+1)\over\prod_{i=k+1}^{n}\Gamma(-b_{i}-{a_{i}t\over\hbar})}\cdot{dt\over\hbar}$

where we assume $a_{1},...,a_{k}>0,~{}a_{k+1},...,a_{n}<0$. It follows from
this expression that the integral $I(\vec{y})$ is one loop exact if there
exist $a,b\in\mathbb{Q}$, $a\neq 0$ such that

${1\over\Gamma({t\over\hbar}+1)}\cdot{\prod_{i=1}^{k}\Gamma(b_{i}+{a_{i}t\over\hbar}+1)\over\prod_{i=k+1}^{n}\Gamma(-b_{i}-{a_{i}t\over\hbar})}=c_{1}c_{2}^{t}\Gamma(b+{at\over\hbar}+1)\text{~{}~{}~{}or~{}~{}~{}}{c_{1}c_{2}^{t}\over\Gamma(-b-{at\over\hbar})}.$

Notice that we must have $a=a_{1}+...+a_{n}-1$. Reformulating this condition
in terms of toric hypersurface (5.40) we obtain the condition (5.41).
$\square$

The statement and the proof of this theorem can be generalized to the field
$\mathbb{R}$ instead of $\mathbb{Q}$.

Remark 5.8.2. If an identity of the form (5.41) holds, then it holds by virtue
of Gauss multiplication formula. This can be shown by considering poles of the
l.h.s. and the r.h.s. of (5.41).

Example 5.8.1. Let $C\subset\mathbb{P}^{3}$ be a toric projective surface
defined by

$x_{0}x_{1}x_{2}=x_{3}^{3}.$

Any vector $(r_{0},r_{1},r_{2},r_{3})\in\mathbb{R}^{4}$ such that

$\Gamma(t+r_{0})\Gamma(t+r_{1})\Gamma(t+r_{2})=ae^{bt}\Gamma(3t+r_{3})$

is equal to $(0,{1\over 3},{2\over 3},0)$ up to translation by $(u,u,u,3u)$
and permutation of $r_{0},r_{1},r_{2}$. Therefore, $C$ is admissible with rank
$3!=6$.

Example 5.8.2. More generally, let $C\subset\mathbb{P}^{n+1}$ be a toric
projective surface defined by

$x_{0}...x_{n}=x_{n+1}^{n+1}.$

Then its rank is equal to $(n+1)!$.

Example 5.8.3. Let $C\subset\mathbb{P}^{3}$ be a toric projective surface
defined by

$x_{0}x_{1}x_{2}^{2}=x_{3}^{4}.$

Any vector $(r_{0},r_{1},r_{2},r_{3})\in\mathbb{R}^{4}$ such that

$\Gamma(t+r_{0})\Gamma(t+r_{1})\Gamma(2t+r_{2})=ae^{bt}\Gamma(4t+r_{3})$

is equal to either $(0,{1\over 2},{1\over 2},0)$ or $({1\over 4},{3\over
4},0,0)$ up to translation by $(u,u,2u,4u)$ and permutation of $r_{0},r_{1}$.
Therefore, $C$ is admissible with rank $2\cdot 2=4$.

Example 5.8.4. Let $C\subset\mathbb{P}^{3}$ be a toric projective surface
defined by

$x_{0}x_{1}^{p}=x_{2}x_{3}^{p}$

for some $p>0$. Any vector $(r_{0},r_{1},r_{2},r_{3})\in\mathbb{R}^{4}$ such
that

$\Gamma(t+r_{0})\Gamma(pt+r_{1})=ae^{bt}\Gamma(t+r_{2})\Gamma(pt+r_{3})$

is equal to $(0,v,0,v)$ for arbitrary $v$, up to translation by $(u,pu,u,pu)$.
Therefore, $C$ is admissible with infinite rank.

Remark 5.8.3. One can show that if $n=2$, then any admissible toric surface of
the form (5.40) is given by examples 5.8.1, 5.8.3, 5.8.4 (up to permutations
of $x_{0},...,x_{3}$ and multiplication of $p_{0},...,p_{3}$ by a common
factor.

###  5.9 Segre cubic in $\mathbb{P}^{4}$

Theorem 5.9.1. Define a cone in $\mathbb{A}^{5}$ by

$x_{0}x_{1}x_{2}+x_{1}x_{2}x_{3}+x_{2}x_{3}x_{4}+x_{3}x_{4}x_{0}+x_{4}x_{0}x_{1}=0.$

This cone is admissible with rank 24.

Proof. Let

$F_{1}=x_{0}x_{1}x_{2}+x_{1}x_{2}x_{3}+x_{2}x_{3}x_{4}+x_{3}x_{4}x_{0}+x_{4}x_{0}x_{1}.$

Computing dual polynomial we obtain

$F_{2}=\partial_{x_{0}}^{2}\partial_{x_{1}}^{2}+\partial_{x_{1}}^{2}\partial_{x_{2}}^{2}+\partial_{x_{2}}^{2}\partial_{x_{3}}^{2}+\partial_{x_{3}}^{2}\partial_{x_{4}}^{2}+\partial_{x_{4}}^{2}\partial_{x_{0}}^{2}+$
$+2\partial_{x_{0}}\partial_{x_{1}}\partial_{x_{2}}\partial_{x_{3}}+2\partial_{x_{1}}\partial_{x_{2}}\partial_{x_{3}}\partial_{x_{4}}+2\partial_{x_{2}}\partial_{x_{3}}\partial_{x_{4}}\partial_{x_{0}}+2\partial_{x_{3}}\partial_{x_{4}}\partial_{x_{0}}\partial_{x_{1}}+2\partial_{x_{4}}\partial_{x_{0}}\partial_{x_{1}}\partial_{x_{2}}$
$-2\partial_{x_{0}}\partial_{x_{1}}^{2}\partial_{x_{2}}-2\partial_{x_{1}}\partial_{x_{2}}^{2}\partial_{x_{3}}-2\partial_{x_{2}}\partial_{x_{3}}^{2}\partial_{x_{4}}-2\partial_{x_{3}}\partial_{x_{4}}^{2}\partial_{x_{0}}-2\partial_{x_{4}}\partial_{x_{0}}^{2}\partial_{x_{1}}.$

Let $g$ be a cyclic permutation of one of the following expressions

$\sqrt{{x_{0}x_{1}x_{2}\over
x_{3}x_{4}}},~{}\sqrt{{x_{0}(x_{1}+x_{3})(x_{1}+x_{4})\over
x_{1}x_{2}}},~{}\sqrt{{x_{3}(x_{2}+x_{0})(x_{2}+x_{4})\over x_{1}x_{2}}},$
$\sqrt{{x_{0}x_{4}(x_{0}+x_{2})\over
x_{1}(x_{0}+x_{3})}},~{}\sqrt{{x_{0}x_{1}(x_{0}+x_{3})\over
x_{4}(x_{0}+x_{2})}},~{}\sqrt{{x_{1}x_{2}x_{3}\over(x_{2}+x_{4})(x_{2}+x_{0})}}.$

Thus we have 30 possible functions $g$. One can check that the vector space
spent by these functions is 24-dimensional and

$F_{2}\Big{(}g\delta(F_{1})\Big{)}=0$

for any such $g$. One can also check that the system for $g$ is holonomic and
has a 24-dimensional space of solutions at a generic point. $\square$

Remark 5.9.1. Let $H=\det((\partial_{x_{i}}\partial_{x_{j}}F_{1})_{0\leq
i,j\leq 4})$. One can check that divisor on Segre surface $F_{1}=0$ defined by
$H=0$ is equivalent to the divisor defined by

$x_{0}x_{1}x_{2}x_{3}x_{4}(x_{0}+x_{2})(x_{1}+x_{3})(x_{2}+x_{4})(x_{3}+x_{0})(x_{4}+x_{1})=0.$

##  6 Classification of admissible pairs of functions in one variable

In the case of functions $f(x),g(x)$ in one variable, the classification of
admissible pairs $f,g$ can be done directly, by equating the higher loop
contributions to zero and solving the corresponding system of differential
equations.

Lemma 6.1. If $f(x)$ is admissible, then it has one of the following forms:

1) $f(x)=(a_{0}+a_{1}x+a_{2}x^{2})^{{1\over 2}}+b_{0}+b_{1}x$ where
$a_{0},a_{1},a_{2},b_{0},b_{1}$ are arbitrary constants such that the
polynomial $a_{0}+a_{1}x+a_{2}x^{2}$ has distinct roots. In particular, if
$a_{2}=0$, then $a_{1}\neq 0$ (root at infinity has multiplicity at most one).

2) $f(x)={1\over a_{0}+a_{1}x}+b_{0}+b_{1}x$ where $a_{0},a_{1},b_{0},b_{1}$
are arbitrary constants and $a_{1}\neq 0$.

3) $f(x)=b_{0}+b_{1}x+b_{2}x^{2}$ where $b_{0},b_{1},b_{2}$ are arbitrary
constants and $b_{2}\neq 0$.

Proof. Equating to zero the terms $A_{2},A_{3},A_{4}$ of the expansion (1.4)
one obtains a system of polynomial differential equations for $f(x),g(x)$.
Assuming that $f^{\prime\prime}(x),g(x)\neq 0$ one can check by direct
computations that this system is equivalent to the following:

$9f^{\prime\prime}(x)^{2}f^{(5)}(x)-45f^{\prime\prime}(x)f^{\prime\prime\prime}(x)f^{(4)}(x)+40f^{\prime\prime\prime}(x)^{3}=0,$
(6.46) $12f^{\prime\prime}(x)^{2}\cdot
g^{\prime\prime}(x)-12f^{\prime\prime}(x)f^{\prime\prime\prime}(x)\cdot
g^{\prime}(x)+(5f^{\prime\prime\prime}(x)^{2}-3f^{\prime\prime}(x)f^{(4)}(x))\cdot
g(x)=0.$

The first equation can be written as

${d^{3}\over dx^{3}}\Big{(}f^{\prime\prime}(x)^{-{2\over 3}}\Big{)}=0$

which gives

$f^{\prime\prime}(x)=(a_{0}+a_{1}x+a_{2}x^{2})^{-{3\over 2}}$

where $a_{0},a_{1},a_{2}$ are arbitrary constants such that the vector
$(a_{0},a_{1},a_{2})$ is non-zero. After the integration we obtain the
statement of Lemma. $\square$

Theorem 6.1. The following pairs $f(x),g(x)$ are admissible:

1)
$f(x)=\sqrt{x^{2}-1},~{}~{}~{}g(x)={c_{1}\over\sqrt{x+1}}+{c_{2}\over\sqrt{x-1}}$.

2) $f(x)={1\over x},~{}~{}~{}g(x)=c_{1}x^{-{1\over 2}}+c_{2}x^{-{3\over 2}}$.

3) $f(x)=x^{{1\over 2}},~{}~{}~{}g(x)=c_{1}+c_{2}x^{-{1\over 2}}$.

4) $f(x)=x^{2},~{}~{}~{}g(x)=c_{1}+c_{2}x$.

Moreover, any admissible pair is equivalent to one of these under
transformations

$x\mapsto\lambda_{0}+\lambda_{1}x,~{}~{}~{}f\mapsto\mu_{0}f+\mu_{1}+\mu_{2}x$

where $\lambda_{1},\mu_{0}\neq 0$. In particular, the rank of any admissible
pair is equal to 2.

Proof. In each of the cases 1), 2), 3), 4) the corresponding cone $C$ is
defined by a non-degenerate quadric. Therefore, all these cases are equivalent
with respect to the projective action. But the pair 4) is admissible because
it corresponds to a Gaussian integral. On the other hand, it follows from
Lemma 1 that any admissible pair is equivalent to one listed above. The
corresponding functions $g(x)$ can be found by solving the second equation of
the system (6.46) for a known function $f(x)$. $\square$

##  7 Toward a classification of admissible pairs of functions in two
variables

The following results are based on huge direct computations and therefore we
give only a rough scheme of the proof.

Theorem 7.1. Let $f(x_{1},x_{2})$ be admissible. Then its rank is infinite iff
the corresponding surface is ruled. If the corresponding surface is not ruled,
then $\dim V_{f}\leq 6$. Moreover, $\dim V_{f}$ cannot be equal to 5.

Theorem 7.2. Let $f(x_{1},x_{2})$ be admissible and $\dim V_{f}=6$. Then the
corresponding surface is either Kummer (including degenerations of Kummer
surfaces such as Steiner Roman surface) or a toric surface given by

$x_{0}x_{1}x_{2}=x_{3}^{3}.$

Proof. Consider first three equations (9.47) for $f,g$ (see Appendix) as a
system of linear equations for $g$ and denote its space of solutions by
$\widetilde{V}_{f}$. We have $V_{f}\subset\widetilde{V}_{f}$. Bringing this
system of three equations to an involutive form as a system for $g$ one can
see that it has an infinite-dimensional space of solutions iff $f$ corresponds
to a ruled surface. Moreover, if the surface is not ruled, then
$\dim\widetilde{V}_{f}\leq 6$ and $\dim\widetilde{V}_{f}=6$ only in the cases
listed above. On the other hand, we know that for ruled surfaces $\dim V_{f}$
is infinite and for Kummer surfaces as well as for a toric variety defined
above the dimension of $V_{f}$ is equal to 6. One can check that $\dim
V_{f}\neq 5$ in a similar way. $\square$

Remark 7.1. It would be interesting to obtain the full classification of
admissible pairs of functions in two variables. This means

1\. Find detailed classification of admissible pairs in the case of ruled
surfaces.

2\. Find all admissible pairs such that $\dim V_{f}=1,2,3,4$.

Remark 7.2. Equations (9.47) become much simpler if one writes them in terms
of local projective invariants. This can be done in the case $n=2$ using so-
called asymptotic coordinates on a hypersurface in $\mathbb{P}^{3}$ [6]. It
would be a good idea to use this approach for the classification of admissible
pairs in two variables.

##  8 A potential application to generalized Dirichlet series

In the special case when $f,g$ are both homogeneous and the equation (1.6) can
be lifted to an actual identity between distributions, one can try to imitate
the proof of the functional equation for the Riemann $\zeta$-function based on
the Poisson summation formula and the Mellin transform.

Let

$\int_{\mathbb{R}^{n}}g(\vec{x})e^{-{f(\vec{x})\over\hbar}+{i\over\hbar}\vec{x}\vec{y}}d\vec{x}=(2\pi\hbar)^{{n\over
2}}\hat{g}(\vec{y})e^{-{\hat{f}(\vec{y})\over\hbar}}.$

If the Poisson summation formula is applicable (possibly after some
regularization), we have

$\sum_{\vec{x}\in\mathbb{Z}^{n}}g(\vec{x})e^{-{f(\vec{x})\over\hbar}}=(2\pi\hbar)^{{n\over
2}}\sum_{\vec{y}\in\mathbb{Z}^{n}}\hat{g}(2\pi\hbar\vec{y})e^{-{\hat{f}(2\pi\hbar\vec{y})\over\hbar}}.$

Multiplying this equation by $\int_{0}^{\infty}\hbar^{s-1}d\hbar$, integrating
by $\hbar$ and making the change of variable $t={1\over\hbar}$ we get in the
l.h.s.

$\sum_{\vec{x}\in\mathbb{Z}^{n}}\int_{0}^{\infty}\hbar^{s-1}g(\vec{x})e^{-{f(\vec{x})\over\hbar}}d\hbar=\sum_{\vec{x}\in\mathbb{Z}^{n}}\int_{0}^{\infty}t^{-1-s}g(\vec{x})e^{-f(\vec{x})t}dt=\Gamma(-s)\sum_{\vec{x}\in\mathbb{Z}^{n}}g(\vec{x})f(\vec{x})^{s},$

and in the l.h.s. similarly, assuming that $\hat{f},\hat{g}$ are homogeneous,
$\deg\hat{f}=a,~{}\deg\hat{g}=b$, and with the change of variable
$t=\hbar^{a-1}$ we get

$(2\pi)^{{n\over
2}}\sum_{\vec{y}\in\mathbb{Z}^{n}}\int_{0}^{\infty}\hbar^{{n\over
2}}\hat{g}(2\pi\hbar\vec{y})e^{-{\hat{f}(2\pi\hbar\vec{y})\over\hbar}}d\hbar=(2\pi)^{{n\over
2}+b}\sum_{\vec{y}\in\mathbb{Z}^{n}}\int_{0}^{\infty}\hbar^{{n\over
2}+b+s}\hat{g}(\vec{y})e^{-\hat{f}(2\pi\vec{y})\hbar^{a-1}}{d\hbar\over\hbar}=$
${(2\pi)^{{n\over 2}+b}\over
a-1}\sum_{\vec{y}\in\mathbb{Z}^{n}}\hat{g}(\vec{y})\int_{0}^{\infty}t^{{{n\over
2}+b+s\over a-1}}e^{-\hat{f}(2\pi\vec{y})t}{dt\over t}={(2\pi)^{{n\over
2}+b}\over a-1}\Gamma\Bigg{(}{{n\over 2}+b+s\over
a-1}\Bigg{)}\sum_{\vec{y}\in\mathbb{Z}^{n}}\hat{g}(\vec{y})\hat{f}(2\pi\vec{y})^{-{{n\over
2}+b+s\over a-1}}.$

Equating the results of these calculations for the l.h.s. and the r.h.s. we
get

$\Gamma(-s)\sum_{\vec{x}\in\mathbb{Z}^{n}}g(\vec{x})f(\vec{x})^{s}={(2\pi)^{{n\over
2}+b}\over a-1}\Gamma\Bigg{(}{{n\over 2}+b+s\over
a-1}\Bigg{)}\sum_{\vec{y}\in\mathbb{Z}^{n}}\hat{g}(\vec{y})\hat{f}(2\pi\vec{y})^{-{{n\over
2}+b+s\over a-1}}.$

The above arguments do not make rigorous sense for non-smooth functions, and
one has to find a way to regularize sums and integrals above and get
potentially an example of a functional equation possibly with some correction
terms related with singularities.

For example, it would be interesting to extract a functional equation from the
actual identity in Schwartz space $S^{\prime}(\mathbb{R}^{2})$ of
distributions of moderate growth:

Let $\hbar>0$ and

$\phi(x_{1},x_{2})={1\over(x_{1}-i0)^{{3\over
2}}}e^{{i\over\hbar}(x_{1}+\sqrt{+ix_{1}}\sqrt{-ix_{2}})^{2}}.$

This function is the boundary value of a holomorphic function in $x_{1}$ with
$\text{Im}~{}x_{1}<0$ and in $x_{2}$ with $\text{Im}~{}x_{2}>0$. Then

$\iint_{\mathbb{R}^{2}}\phi(x_{1},x_{2})e^{{i\over\hbar}(x_{1}y_{1}+x_{2}y_{2})}dx_{1}dx_{2}=2\pi
i\hbar~{}\overline{\phi(y_{2},y_{1}+y_{2})}.$

##  9 Conjectures and open questions

Let $\Sigma\subset\mathbb{P}^{n+1}$ be a projective hypersurface (may be a
non-algebraic germ), non-degenerate at the generic point, and
$C\subset\mathbb{A}^{n+2}$ the corresponding cone. Assume that $n\geq 2$.

1. It would be interesting to determine when $rk(C)$ is infinite in terms of projective differential geometry of $\Sigma$. We know the answer in the simplest nontrivial case $n=2$: $rk(C)$ is infinite iff $\Sigma$ is a ruled surface, see Section 5.3. Notice that if $\Sigma\subset\mathbb{P}^{3}$ is a ruled surface, then its projective dual $\widehat{\Sigma}\subset\mathbb{P}^{3}$ is also ruled but the property of being ruled is not self dual if $n>2$. For arbitrary $n\geq 2$ we can only suggest the following

Conjecture 9.1. If $rk(C)$ is infinite, then both
$\Sigma\subset\mathbb{P}^{n+1}$ and $\widehat{\Sigma}\subset\mathbb{P}^{n+1}$
are ruled.

Notice that the Segre cubic has rank 24 and ruled, but its dual is not ruled,
see Section 5.9.

2. Let $rk(C)$ be finite. It would be interesting to understand which values $rk(C)$ can take. For example, if $n=2$, then we know examples with $rk(C)=4,~{}6$, we also know that $rk(C)\neq 5$ and $rk(C)\leq 6$.

Conjecture 9.2. If $rk(C)$ is finite, then $rk(C)\leq(n+1)!$.

It would be interesting to classify all $C$ with the finite largest possible
$rk(C)$ for given $n$.

Conjecture 9.3. If $rk(C)$ is finite, then $\Sigma$ is algebraic.

3. Let $\Sigma$ be algebraic and $rk(C)$ finite.

Conjecture 9.4. If $rk(C)>0$ and finite, then $\Sigma$ is singular.

In all interesting known examples $\Sigma$ has only isolated singularities
(double points). The only exception is described in Section 5.7, these
families of surfaces have one-dimensional singularity locus.

Let ${\mathfrak{S}}_{n,d}$ be the set of algebraic hypersurfaces
$\Sigma\subset\mathbb{P}^{n+1}$ of degree $d>2$ with the largest possible
number of double points. For some values of $n,d$ all hypersurfaces from
${\mathfrak{S}}_{n,d}$ are admissible (with rank $(n+1)!$). For example
${\mathfrak{S}}_{2,4}$ are Kummer surfaces (16 double points), and
${\mathfrak{S}}_{3,3}$ is the Segre cubic (10 double points). It would be
interesting to determine for which values of $n,d$ elements of
${\mathfrak{S}}_{n,d}$ are admissible.

4. Let $\Sigma\subset\mathbb{P}^{n+1}$ be algebraic and defined by irreducible polynomial $F_{1}(x_{0},...,x_{n+1})$, and its projective dual $\widehat{\Sigma}$ defined by an irreducible polynomial $F_{2}(y_{0},...,y_{n+1})$. Recall that the natural birational isomorphism $\sigma:~{}\Sigma\to\widehat{\Sigma}$ is given by $y_{i}=\partial_{x_{i}}F_{1},~{}i=0,...,n+1$ (see also Section 4,3, Remark 4.3.1). Let $H=\det((\partial_{x_{i}}\partial_{x_{j}}F_{1})_{0\leq i,j\leq n+1})$. Notice that $H$ is the determinant of the Jacobian of $\sigma$, and therefore the divisor $\cal D$ on $\Sigma$ defined by $H=0$ is the singularity locus of $\sigma$. In certain examples of admissible hypersurfaces $\Sigma$ the corresponding $D$-module is holonomic, has regular singularities on $\cal D$, and $\pi_{1}(\Sigma\setminus{\cal D})$ acts on the space of admissible $g$. See for example Section 5.9. It would be interesting to study this class of admissible hypersurfaces. See also Section 4.5.

5. The family of surfaces of degree four in Section 5.7, Theorem 5.7.2 was obtained as all possible deformations of the toric surface

$x_{0}x_{1}x_{2}^{2}=x_{3}^{4}$

in class of surfaces of rank 4. It would be interesting to study deformations
of other admissible toric hypersurfaces preserving its rank. We have checked
that the toric surface

$x_{0}x_{1}x_{3}=x_{4}^{3}$

with rank 6 (see Section 5.8) does not have such deformations.

6. It would be interesting to study admissible pairs of the form

$f(x_{1},...,x_{N})=\sum_{i=0}^{N-m}\phi(x_{i+1},...,x_{i+m}),~{}~{}~{}g(x_{1},...,x_{N})=\sum_{i=0}^{N-m}\psi(x_{i+1},...,x_{i+m}),~{}~{}~{}N\gg
m$

and similar with variables $x_{i_{1},...,i_{k}}$, $k>1$, $1\leq
i_{1},...,i_{k}\leq N$. Such admissible pairs could be regarded as ”integrable
lattice models” of certain type.

7. It would be interesting to prove or disprove the following

Conjecture 9.5. Let $f(x_{1},...,x_{n})$ be a function in $n$ variables such
that its Hessian is not identically zero and such that

$\int
g(\vec{x},\hbar)e^{{f(\vec{x})+\vec{x}\cdot\vec{y}\over\hbar}}d\vec{x}=(2\pi\hbar)^{{n\over
2}}\hat{g}(\vec{y},\hbar)e^{{\hat{f}(\vec{y})\over\hbar}}$

where $g(\vec{x},\hbar),~{}\hat{g}(\vec{y},\hbar)$ are both non-zero
polynomial in $\hbar$. Then function $f$ is admissible.

This conjecture is based on some calculation in the case $n=1$ and is not
supported by any calculations for $n\geq 2$. Notice that if $g$ does not
depend on $\hbar$ and $\hat{g}$ is a polynomial in $\hbar$, then pair $f,g$ is
not necessarily admissible even in the case $n=1$.

## Appendix. Explicit formulas for equations

Recall that $f,g$ are functions in variables
$\vec{x}=(x_{1},x_{2},\dots,x_{n})$ and we assume that the Hessian matrix
$\partial^{2}f:=(\partial_{i}\partial_{j}f)_{1\leq i,j\leq n}$ is non-
degenerate. Denote by

$(p^{ij})_{1\leq i,j\leq n}:=(\partial^{2}f)^{-1}$

the inverse matrix-valued function.

Main notation: for $k\geq 1$ (all summation variables and indices are assumed
to be integers),

$A_{k}:=\sum_{v\in[0,2k]}\sum_{\begin{subarray}{c}d_{0}\in[0,\infty);\\\
d_{1},\dots,d_{v}\in[3,\infty)\\\ \mbox{\small such that }\\\
\sum_{i}d_{i}=2(k+v),\\\ d_{1}\geq d_{2}\geq\dots\geq
d_{v}\end{subarray}}\sum_{\begin{subarray}{c}a_{ij}\in[0,\infty)\\\
\mbox{\small where }{0\leq i\leq j\leq v},\\\ \mbox{\small satisfying }\forall
i:\\\ d_{i}=\sum_{j<i}a_{ji}+\\\
+2a_{ii}+\sum_{j>i}a_{ij}\end{subarray}}{(-1)^{v}\over
Sym_{(d_{i}),(a_{ij})}}\sum_{\begin{subarray}{c}b_{ijl},c_{ijl}\in[1,n]\\\
\mbox{\small where }\\\ 1\leq i\leq j\leq v\\\ \mbox{\small and }1\leq l\leq
a_{ij}\end{subarray}}\left(\prod_{\begin{subarray}{c}i,j\in[1,v]\\\
l\in[1,a_{ij}]\\\ \mbox{\small such that}\\\ i\leq
j\end{subarray}}p^{b_{ijl},c_{ijl}}\right)\cdot\\\
\cdot\left[\prod_{l_{1}\in[1,a_{00}]}(\partial_{b_{00l_{1}}}\partial_{c_{00l_{1}}})\prod_{\begin{subarray}{c}i\in[1,v]\\\
l_{2}\in[1,a_{0i}]\end{subarray}}\partial_{b_{0il_{2}}}g\right]\cdot\prod_{i\in[1,v]}\left[\prod_{\begin{subarray}{c}j\in[0,i)\\\
l_{1}\in[1,a_{ji}]\end{subarray}}\partial_{c_{jil_{1}}}\prod_{l_{2}\in[1,a_{ii}]}(\partial_{b_{iil_{2}}}\partial_{c_{iil_{2}}})\prod_{\begin{subarray}{c}j\in(i,v]\\\
l_{3}\in[1,a_{ij}]\end{subarray}}\partial_{b_{ijl_{3}}}f\right]$

Here the symmetry factor is defined by

$Sym_{(d_{i}),(a_{ij})}=\prod_{i}m_{i}!\cdot\prod_{\begin{subarray}{c}i,j\in[0,v]\\\
\mbox{\small such that}\\\ i\leq
j\end{subarray}}{a_{ij}!}\cdot\prod_{i\in[0,v]}{2^{a_{ii}}}$

where $m_{1},m_{2},\dots\geq 1$ are multiplicities of the repeating terms in
sequence $(d_{1},d_{2},\dots,d_{v})$, i.e.

$d_{1}=\cdots=d_{m_{1}}>d_{m_{1}+1}=\cdots=d_{m_{1}+m_{2}}>d_{m_{1}+m_{2}+1}=\cdots$

Meaning: let us expand $f$ at some point $\vec{x}^{(0)}$ as

$f=f_{0}+f_{1}+f_{2}+f_{\geq 3}$

where $f_{0}=f(\vec{x}^{(0)})$ is a constant, $f_{1},f_{2}$ are homogeneous
polynomials in $\vec{x}-\vec{x}^{(0)}$ of degree 1 and 2 respectively, and
$f_{\geq 3}$ is a series in $\vec{x}-\vec{x}^{(0)}$ containing terms of
degrees $\geq 3$ only.

Then the formal Fourier transform, at point

$\vec{y}^{(0)}=\partial
f_{|\vec{x}^{(0)}}:=(\partial_{1}f,\dots,\partial_{n}f)_{|\vec{x}^{(0)}}$

is equal, after normalization, to

$\int g\,e^{-{f-f_{0}-f_{1}\over\hbar}}d^{n}\vec{x}=\int g\,e^{-{f_{2}+f_{\geq
3}\over\hbar}}d^{n}\vec{x}=\\\ =\sum_{v\geq 0}{(-1)^{v}\over v!}\hbar^{-v}\int
gf_{\geq 3}^{v}\,e^{-{f_{2}\over\hbar}}d^{n}\vec{x}=\\\
=(2\pi\hbar)^{n/2}\det(\partial^{2}f_{|\vec{x}^{(0)}})^{-1/2}\left(g_{|\vec{x}^{(0)}}+\sum_{k\geq
1}\hbar^{k}{A_{k}}_{|\vec{x}^{(0)}}\right)$

In terms of (not connected) Feynman graphs, $v\geq 0$ denotes the number of
vertices at which we put Taylor coefficients of $f_{\geq 3}$ (and at exactly
one exceptional vertex we put Taylor coefficients of $g$). We label vertices
by $\\{0,1,\dots,v\\}=[0,v]\cap\mathbb{Z}$ where $0$ corresponds to $g$, and
the rest to $f_{\geq 3}$. Moreover, we assume that the ordering of vertices is
chosen in such a way that $d_{1}\geq d_{2}\geq\cdots\geq d_{v}\geq 3$ where
for all $i\in[0,v]$ number $d_{i}$ is the degree (valency) of vertex labeled
by $i$. Denote by $a_{ij}\geq 0$ the number of edges connecting vertices $i$
and $j$. We enumerate edges connecting $i$ and $j$ by $\\{1,\dots,a_{ij}\\}$.
Then we put two space indices $b_{ijl},c_{ijl}\in[1,n]$ on two ends of the