DERIVED KZ EQUATIONS

Vadim Schechtman111Institut de Mathématiques de Toulouse, Université Paul
Sabatier, 118 route de Narbonne, 31062 Toulouse, France and Alexander
Varchenko222Department of Mathematics, University of North Carolina at Chapel
Hill, Chapel Hill, NC 27599-3250, USA; Faculty of Mathematics and Mechanics,
Lomonosov Moscow State University, Leninskiye Gory 1, 119991 Moscow GSP-1,
Russia; Moscow Center of Fundamental and Applied Mathematics Leninskiye Gory
1, 119991 Moscow GSP-1, Russia

December 27, 2020

To the 30-th anniversary of [SV]

Abstract

In this note we strengthen the results of [SV] by presenting their derived
version. Namely, we define a "derived Knizhnik - Zamolodchikov connection" and
identify it with a "derived Gauss - Manin connection".

§1. Introduction

1.0. Brief review of the paper. The main result of [SV] provided a realization
of Knizhnik - Zamolodchikov equations arising in physics as equations on
horizontal sections for a Gauss - Manin connection.

More explicitly, without going into details to be given below, the KZ
connection acts on a space of functions depending on $\mathbf{z}\in B$ where
$B$ is a domain in $\mathbb{C}^{n}$ with values in a homology group
$H_{0}(\mathfrak{n},M)$ where $\mathfrak{n}$ is a certain Lie algebra, and $M$
a (maybe infinite dimensional) $\mathfrak{n}$-module. In other words the KZ
connection acts on the trivial vector bundle over $B$ with a fiber
$H_{0}(\mathfrak{n},M)$, this vector bundle to be denoted
$\mathcal{H}_{0}(\mathfrak{n},M)$.

All homology spaces $H_{i}(\mathfrak{n},M)$ are $\Lambda$-graded

$H_{i}(\mathfrak{n},M)=\oplus_{\lambda\in\Lambda}H_{i}(\mathfrak{n},M)_{\lambda}$

where $\Lambda$ is certain lattice. For a given $\lambda$ only a finite number
of spaces $H_{i}(\mathfrak{n},M)_{\lambda}$, $0\leq i\leq N$, are different
from $0$. Let us pick $\lambda$.

On the other hand one has introduced in op. cit. a fibration (a smooth
surjective map)

$p_{\lambda}:X_{\lambda}\longrightarrow B$

and a $\mathcal{D}$-module $\mathcal{L}_{\lambda}$ over $X_{\lambda}$, and a
finite group $\Sigma_{\lambda}$ (a product of symmetric groups) which acts on
$X_{\lambda}$ and $\mathcal{L}_{\lambda}$.

One has constructed an isomorphism of the bundle
$\mathcal{H}_{0}(\mathfrak{n},M)_{\lambda}$ equipped with the KZ connection
with the bundle $(R^{N}p_{\lambda*}\mathcal{L}_{\lambda})^{\Sigma_{\lambda}}$
equipped with the GM connection.

In fact in [SV] for all $0\leq i\leq N$ there were established isomorphisms

$\beta_{i,\lambda}:\
\mathcal{H}_{i}(\mathfrak{n},M)_{\lambda}\overset{\sim}{\longrightarrow}(R^{N-i}p_{\lambda*}\mathcal{L}_{\lambda})^{\Sigma_{\lambda}}$
$None$

of vector bundles over $B$. However the question of identification of a
connection on $\mathcal{H}_{i}(\mathfrak{n},M)_{\lambda}$ corresponding to the
GM connection on
$(R^{N-i}p_{\lambda*}\mathcal{L}_{\lambda})^{\Sigma_{\lambda}}$ was left open
for $i>0$, although a natural candidate has been given.

In the present note we establish this remaining point. To do this we start
from the remark that by its very definition in op. cit. isomorphisms (1.0.1)
are induced by a map of complexes

$\eta_{\lambda}=(\eta_{\lambda,i}):\
\mathcal{C}_{\bullet}(\mathfrak{n},M)_{\lambda}\longrightarrow\Omega_{X_{\lambda}/B}^{N-\bullet}(\mathcal{L}_{\lambda})^{\Sigma_{\lambda}}$
$None$

where $\mathcal{C}_{\bullet}(\mathfrak{n},M)_{\lambda}$ is the
$\lambda$-homogeneous part of the Chevalley chain complex, and
$\Omega_{X_{\lambda}/B}^{\bullet}(\mathcal{L}_{\lambda})$ is certain complex
of differential form on $X_{\lambda}$, the relative de Rham complex of
$\mathcal{L}_{\lambda}$.

A naive expectation would be that:

$(a)$ for the KZ part:

each term $\mathcal{C}_{i}(\mathfrak{n},M)_{\lambda}$ comes equipped with an
integrable connection, these connections are compatible with differentials and
thus induce a connection on the cohomology
$\mathcal{H}_{i}(\mathfrak{n},M)_{\lambda}$;

$(b)$ for the GM part:

similarly, each term $\Omega_{X_{\lambda}/B}^{j}(\mathcal{L}_{\lambda})$ comes
equipped with an integrable $\Sigma_{\lambda}$-equivariant connection, these
connections are compatible with differentials and thus induce a connection on
the cohomology $R^{j}p_{\lambda*}\mathcal{L}_{\lambda}$;

$(c)$ the map $\eta_{\lambda}$ is compatible with the connections in (a), (b),
and therefore the isomorphisms $\beta_{i,\lambda}$ (1.0.1) identify two
connections.

In reality, $(a)$ is literally true (and easy); this is present in [SV].

Point $(b)$ is more delicate: there is no natural connection on the complex
$\Omega_{X_{\lambda}/B}^{\bullet}(\mathcal{L}_{\lambda})$. Happily, to define
a connection on the cohomology a weaker structure is sufficient:

$(b^{\prime})$ there exists a filtered complex such that the term $E_{1}$ of
the corresponding spectral sequence (recalled in Appendix) coincides with the
de Rham complex of the GM connection on
$R^{\bullet}p_{\lambda*}\mathcal{L}_{\lambda}$.

This filtered complex is described below: it is a generalization of the Katz -
Oda construction for the GM connection, [KO].

Accordingly, $(c)$ should be replaced by

$(c^{\prime})$ the map $\eta_{\lambda}$ may be extended to a map of filtered
complexes which, after passing to $E_{1}$-terms, induces a map from the de
Rham complex of the KZ connection to the de Rham complex of the GM connection.

Now we will describe some details of what was said above.

1.1. Knizhnik - Zamolodchikov connection. Let $\mathfrak{g}$ be a complex Lie
algebra equipped with an element

$\Omega\in\mathfrak{g}\otimes\mathfrak{g}$

having the following property:

1.1.1. Let $M_{1},M_{2}$ be arbitrary $\mathfrak{g}$-modules. The actions of
$\Omega$ and $\mathfrak{g}$ on $M_{1}\otimes M_{2}$ commute.

1.1.2. Example. Let $\mathfrak{g}$ be finite dimensional, equipped with a non-
degenerate invariant symmetric bilinear form $(,)$. Denote

$\Omega=\sum_{i}x_{i}\otimes x^{i}\in\mathfrak{g}\otimes\mathfrak{g}$

where $\\{x_{i}\\}\subset\mathfrak{g}$ is any $\mathbb{C}$-base, and
$\\{x^{i}\\}$ is the dual base, i.e. $(x_{i},x^{i})=\delta_{ij}$. This element
("the Casimir") does not depend on a choice of a base and satisfies 1.1.1.

Let $M_{1},\ldots,M_{n}$ be $\mathfrak{g}$-modules, $n\geq 1$. Denote
$M=M_{1}\otimes\ldots\otimes M_{n}$.

For a smooth affine complex333in what follows the base field $\mathbb{C}$ of
complex numbers may be replaced by any field of characteristics $0$ algebraic
variety $U$, $\Omega^{\bullet}(U)$ will denote the space of global sections
for its algebraic de Rham complex $\Omega^{\bullet}_{U}$. Thus
$\Omega^{0}_{U}=\mathcal{O}_{U}$ is the sheaf of functions, etc.

If $M$ is a vector space, we denote

$\Omega^{\bullet}(U;M):=\Omega^{\bullet}(U)\otimes M.$

Let $n\geq 1$ be an integer. Let $M_{1},\ldots,M_{n}$ be
$\mathfrak{g}$-modules; set $M=M_{1}\otimes\ldots\otimes M_{n}$. For each
$i\neq j$ we have an operator

$\Omega_{ij}:M\longrightarrow M$

acting as $\Omega$ on $M_{i}\otimes M_{j}$ and as identity on the other
factors.

Denote

$U_{n}=\\{\mathbf{z}=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}|\ z_{i}\neq z_{j}\
\text{for all}\ i\neq j\\}$ $None$

Thus $U_{1}=\mathbb{C}$.

The KZ connection is an operator

$\nabla_{KZ}:\Omega^{0}(U_{n};M)=\mathcal{O}(U_{n})\otimes
M\longrightarrow\Omega^{1}(U_{n};M)$

given by

$\nabla_{KZ}=d_{DR}+\Omega_{KZ}:=d_{DR}-\frac{1}{\kappa}\sum_{i<j}\frac{\Omega_{ij}(dz_{i}-dz_{j})}{z_{i}-z_{j}}$
$None$

where $d_{DR}$ is the de Rham differential. Here $\kappa\in\mathbb{C}^{*}$ is
a complex parameter.

Thus $\nabla_{KZ}=d_{DR}$ if $n=1$.

This connection is integrable: if we define, starting from $\nabla_{KZ}$,
operators

$\nabla_{KZ}:\ \Omega^{i}(U_{n};M)\longrightarrow\Omega^{i+1}(U_{n};M)$

for all $i$ in the usual way then $\nabla_{KZ}^{2}=0$ (this amounts to the
classical YB equation for the differential form $\Omega_{KZ}$).

In other words, $\nabla_{KZ}$ is an integrable connection (i.e. it defines a
structure of a $\mathcal{D}_{U_{n}}$-module) on the trivial bundle
$\mathcal{M}$ over $U_{n}$ with fiber $M$.

1.2. The Chevalley complex and the derived KZ. Let
$\mathfrak{n}\subset\mathfrak{g}$ be a Lie subalgebra.

We will be interested in Chevalley chain complexes

$C_{\bullet}(\mathfrak{n},M):\
\ldots\longrightarrow\Lambda^{2}\mathfrak{n}\otimes
M\longrightarrow\mathfrak{n}\otimes M\longrightarrow M\longrightarrow 0$

where $d(g\otimes x)=gx$,

$d(g_{1}\wedge g_{2}\otimes x)=g_{1}\otimes g_{2}x-g_{2}\otimes
g_{1}x-[g_{1},g_{2}]\otimes x,$

etc.

Let $\mathcal{C}_{\bullet}(\mathfrak{n},M)$ denote the trivial vector bundle
over $U_{n}$ with a fiber $C_{\bullet}(\mathfrak{n},M)$, so it is a complex of
vector bundles.

We define the derived KZ connection as an integrable connection on
$\mathcal{C}_{\bullet}(\mathfrak{n},M)$ given by the same formula as above,

$\nabla_{KZ}=d_{DR}+\Omega_{KZ}:=d_{DR}-\frac{1}{\kappa}\sum_{i<j}\frac{\Omega_{ij}(dz_{i}-dz_{j})}{z_{i}-z_{j}}$
$None$

where now the operators

$\Omega_{ij}:\ C_{l}(\mathfrak{n},M)=\Lambda^{l}\mathfrak{n}\otimes
M\longrightarrow C_{l}(\mathfrak{n},M)$

are acting through the factor $M$.

Whence we get the corresponding de Rham complex

$\Omega_{KZ}^{\bullet}(U_{n},C_{\bullet}(\mathfrak{n},M))=DR(\mathcal{C}_{\bullet}(\mathfrak{n},M),\nabla_{KZ})(U_{n}).$
$None$

It is a double complex: the commutation of the Chevalley differential with
$\nabla_{KZ}$ follows from 1.1.1.

We call it the KZ-Chevalley complex.

In fact this complex appears avant la lettre already in [SV] 7.2.3.

1.3. Derived Gauss - Manin connection. Let $N\geq 0$ be an integer. Consider
the affine space $\mathbb{C}^{n+N}$ with coordinates
$z_{1},\ldots,z_{n},t_{1},\ldots,t_{N}$, and inside it an open subspace

$U_{n,N}=\\{z_{i}\neq z_{j},\ z_{i}\neq t_{a},\ t_{a}\neq t_{b}\\}.$

We have an obvious projection

$p:U_{n,N}\longrightarrow U_{n}.$

The de Rham algebra $\Omega^{\bullet}(U_{n,N})$ is the total complex of a
bicomplex

$\Omega^{\bullet}(U_{n,N})=\operatorname{Tot}\Omega^{\bullet\bullet}(U_{n,N})$

where $\Omega^{pq}(U_{n,N})$ is the space of forms containing $p$
differentials $dt_{i}$ and $q$ differentials $dz_{m}$, the full de Rham
differential being the sum

$d_{DR}=d_{z}+d_{t}.$

The relative de Rham complex is by definition

$\Omega^{\bullet}(U_{n,N}/U_{n})=(\Omega^{0\bullet}(U_{n,N}),d_{t});$

one has a projection

$p:\ \Omega^{\bullet}(U_{n,N})\longrightarrow\Omega^{\bullet}(U_{n,N}/U_{n})$

Let $\mathcal{L}$ be a $\mathcal{D}_{U_{n,N}}$-module, i.e. a quasicoherent
$\mathcal{O}_{U_{n,N}}$-module equipped with an integrable connection

$\nabla:\ \mathcal{L}\longrightarrow\Omega^{1}_{U_{n,N}}\otimes\mathcal{L};$

its de Rham complex is

$DR(\mathcal{L}):\
0\longrightarrow\mathcal{L}\overset{\nabla}{\longrightarrow}\Omega^{1}_{U_{n,N}}\otimes\mathcal{L}\overset{\nabla}{\longrightarrow}\Omega^{2}_{U_{n,N}}\otimes\mathcal{L}\longrightarrow\ldots$

By definition, the de Rham complex of the derived Gauss - Manin connection on
the direct image $Rp_{*}\mathcal{L}$, to be denoted $DR(Rp_{*}\mathcal{L})$,
is the same complex $DR(\mathcal{L})$ equipped with a decreasing filtration

$F_{z}^{0}DR(\mathcal{L})=DR(\mathcal{L})\supset
F_{z}^{1}DR(\mathcal{L})\supset\ldots$ $None$

where $F_{z}^{i}DR(\mathcal{L})$ is the subcomplex containing $\geq i$
differentials $dz_{a}$.

Note that the utmost left column of $F^{0}/F^{1}$ is the relative de Rham
complex representing $Rp_{*}\mathcal{L}$ whose cohomology are the sheaves
$R^{i}p_{*}\mathcal{L}$. These sheaves carry the usual GM connections
$\nabla^{i}$.

The complexes $E^{i}(DR(Rp_{*}\mathcal{L}),F^{\bullet}_{z})$ defined in the
Appendix, A2.1 (the components of the $E_{1}$ term of the spectral sequence
for our filtered complex) are nothing else but their de Rham complexes of
$R^{i}p_{*}\mathcal{L}$:

$E^{i}(DR(Rp_{*}\mathcal{L}),F^{\bullet}_{z})\overset{\sim}{=}DR(R^{i}p_{*}\mathcal{L},\nabla^{i}).$

This isomorphism justifies the above definition.

1.3.1. Remark. For the case of a trivial connection on $\mathcal{O}_{U_{n,N}}$
the above construction is nothing else but the Katz - Oda definition of the
usual GM connection, [KO].

1.4. Coulomb $\mathcal{D}$-modules. Let $V$ be a finite dimensional complex
vector space equipped with a symmetric bilinear form $(,)$. Let

$\mu=(\mu_{1},\ldots,\mu_{n})\in V^{n},\
\alpha=(\alpha_{1},\ldots,\alpha_{N})\in V^{N},$

and $\kappa\in\mathbb{C}^{*}$.

We associate to these data a $\mathcal{D}$-module $\mathcal{L}(\mu,\alpha)$,
to be called a Coulomb 444”Loi fondamentale de l’Élictricité. La force
répulsive des deux petits globes électrisés de la même nature d’électricité,
est en raison inverse du carré de la distance du centre de deux globes.”
Charles-Augustin de Coulomb, Premier Mémoire sur l’Électricité et le
Magnétisme, 1785. $\mathcal{D}$-module, over $U_{n,N}$: by definition it is
the structure sheaf $\mathcal{O}_{U_{n,N}}$ equipped with a connection

$\nabla(\mu,\alpha)=d_{DR}+\frac{1}{\kappa}\omega(\mu,\alpha)$

where

$\omega(\mu,\alpha)=\sum_{i<j}(\mu_{i},\mu_{j})d\ln(z_{i}-z_{j})-$
$-\sum_{i,a}(\mu_{i},\alpha_{a})d\ln(z_{i}-t_{a})+\sum_{a<b}(\alpha_{a},\alpha_{b})d\ln(t_{a}-t_{b}).$

1.5. On the other hand we can associate with the data $(V,\mu,\alpha)$ above a
Lie algebra $\mathfrak{g}=\mathfrak{g}(\alpha)$ ("a Kac-Moody algebra without
Serre relations") and a collection of "contragradient Verma"
$\mathfrak{g}$-modules $M(\mu_{1})^{c},\ldots,M(\mu_{n})^{c}$.

For example if $V$ is one-dimensional and $\alpha_{1}=\ldots=\alpha_{N}$ then
$\mathfrak{g}=\mathfrak{sl}_{2}$.

Let

$M=M(\mu_{1})^{c}\otimes\ldots\otimes M(\mu_{n})^{c},$

and consider the total complex of the de Rham complex (1.2.2)
$\operatorname{Tot}\Omega^{\bullet}(U_{n},C_{\bullet}(\mathfrak{n},M))$. It is
$\Lambda$-graded where $\Lambda=\sum_{i}\mathbb{Z}\alpha_{i}\subset V$, and it
carries a decreasing filtration

$F^{\bullet}_{z}\operatorname{Tot}\Omega^{\bullet}(U_{n},C_{\bullet}(\mathfrak{n},M))$

where

$F^{\bullet}_{z}\operatorname{Tot}\Omega^{\bullet}(U_{n},C_{\bullet}(\mathfrak{n},M))\subset\operatorname{Tot}\Omega^{\bullet}(U_{n},C_{\bullet}(\mathfrak{n},M))$

is the subcomplex of differential forms containing $\geq i$ differentials
$dz_{a}$.

Let $\lambda=\sum_{i}\alpha_{i}\in\Lambda$. Our main result defines a map from
the $\lambda$-homogeneous component of this filtered complex to the filtered
complex $(DR(Rp_{*}\mathcal{L}(\mu,\alpha),F^{\bullet}_{z})$.

For details see Theorem 3.8 and Corollary 3.9.

Plan of the paper

In the next $\S 2$ we discuss in detail the case
$\mathfrak{g}=\mathfrak{sl}_{2}$. The general case is discussed in $\S 3$. In
the Appendix we recall some standard homological algebra of filtered
complexes.

1.6. Acknowledgements. We are grateful to B.Toen and D.Gaitsgory for useful
conversations. A. Varchenko was supported in part by NSF grant DMS-1954266.

§2. The case $\mathfrak{g}=\mathfrak{sl}_{2}$

2.0. Setup. We consider the Lie algebra $\mathfrak{g}=\mathfrak{sl}_{2}$ with
standard generators $e,f,h$; let
$\mathfrak{n}:=\mathbb{C}f\subset\mathfrak{g}$ (resp.
$\mathfrak{n}_{+}:=\mathbb{C}e$) be the lower (resp. upper) triangular
subalgebra. We will identify $\mathfrak{n}_{+}$ with $\mathfrak{n}^{*}$, with
$e$ being dual to $f$.

The Casimir element is

$\Omega=\frac{1}{2}h\otimes h+e\otimes f+f\otimes e.$

2.0.1. Invariance lemma. Let $M_{1},M_{2}$ be arbitrary
$\mathfrak{g}$-modules. The actions of $\Omega$ and $\mathfrak{g}$ on
$M_{1}\otimes M_{2}$ commute.

Proof: exercise for the reader.

2.1. Chevalley complex.

If $M$ is a $\mathfrak{g}$-module,
$C_{\bullet}(\mathfrak{n}^{*},M^{*})=C_{\bullet}(\mathfrak{n}_{+},M^{*})$ will
denote the Chevalley chain complex

$0\longrightarrow\mathfrak{n}^{*}\otimes
M^{*}\overset{d^{*}}{\longrightarrow}M^{*}\longrightarrow 0$ $None$

living in degrees $-1,0$. Here the action of $\mathfrak{n}^{*}$ on the dual
space $M^{*}$ is given by

$(f^{*}\alpha)(x)=\alpha(ex),\ x\in M,\ \alpha\in M^{*}$ $None$

where $f^{*}\in\mathfrak{n}^{*}$ is defined by $f^{*}(f)=1$.

Next, $C^{\bullet}(\mathfrak{n},M^{c})$ will denote the dual complex

$0\longrightarrow M\overset{d}{\longrightarrow}\mathfrak{n}\otimes
M\longrightarrow 0,\ d(x)=f\otimes ex$ $None$

living in degrees $0,1$.

For $m\in\mathbb{C}$, $M(m)$ will denote the Verma module with a vacuum vector
$v$ such that $hv=mv,ev=0$. It is $\mathbb{N}$-graded:

$M(m)=\oplus_{k\geq 0}M(m)_{k}$

where $M(m)_{k}=\mathbb{C}f^{k}v$.

Fix a natural $n\geq 1$ and an $n$-tuple
$\mathbf{m}=(m_{1},\ldots,m_{n})\in\mathbb{C}^{n}$, and consider the tensor
product

$M(\mathbf{m})=M(m_{1})\otimes\ldots\otimes M(m_{n})$

The above grading on each $M(m_{i})$ gives rise to an $\mathbb{N}$-grading on
$M(\mathbf{m})$:

$M(\mathbf{m})=\oplus_{k\geq 0}M(\mathbf{m})_{k}$

where

$M(\mathbf{m})_{k}=\oplus_{k_{1}+\ldots+k_{n}=k}M(m_{1})_{k_{1}}\otimes\ldots\otimes
M(m_{n})_{k_{n}}.$

For a multi-index $a=(k_{1},\ldots,k_{n})$ we denote

$|a|=\sum_{i=1}^{n}\ a_{i},$ $None$

and

$f^{a}v:=f^{a_{1}}v_{1}\otimes\ldots\otimes f^{a_{n}}v_{n}\in
M(\mathbf{m})_{|a|}.$ $None$

The Chevalley complex acquires a grading as well:

$C^{\bullet}(\mathfrak{n},M(\mathbf{m})^{c})=\oplus_{k\geq
0}C^{\bullet}(\mathfrak{n},M(\mathbf{m})^{c})_{k},$

with

$C^{\bullet}(\mathfrak{n},M(\mathbf{m})^{c})_{k}:0\longrightarrow
M(\mathbf{m})_{k}\longrightarrow\mathfrak{n}\otimes
M(\mathbf{m})_{k-1}\longrightarrow 0.$ $None$

2.2. Logarithmic forms. Recall that $\kappa\in\mathbb{C}^{*}$ is fixed.

We fix an integer $N\geq 0$ and consider the space $U_{n,N}$ (see 1.4 above).

We are going to define certain logarithmic forms on this space. For a function
$u$ we denote

$d\ln u:=\frac{du}{u}$

The symmetric group $\Sigma_{N}$ acts on forms from
$\Omega^{\bullet}(U_{n,N})$ by permuting variables $t_{1},\dots,t_{N}$.

For a differential form $w$ we define by $\operatorname{Alt}\,w$ the skew-
symmetrization of $w$ with respect to the $\Sigma_{k}$-action,

$\displaystyle\operatorname{Alt}\,w(t_{1},\dots,t_{N})=\sum_{\sigma\in\Sigma_{N}}(-1)^{\sigma}w(t_{\sigma(1)},\dots,t_{\sigma(N)}).$

All forms appearing in our constructions are skew-symmetric. They are given by
the following formulas. For $a=(a_{1},\dots,a_{n})\in\mathbb{N}^{n}$,
$|a|:=\sum a_{i}=N$, we define

$\displaystyle w_{a}=\frac{1}{a_{1}!\dots a_{n}!}\operatorname{Alt}\,u_{a},$

where

$\displaystyle u_{a}$ $\displaystyle=$ $\displaystyle
d\ln(t_{1}-z_{1})\wedge\dots\wedge d\ln(t_{a_{1}}-z_{1})+$
$\displaystyle+\,d\ln(t_{a_{1}+1}-z_{2})\wedge\dots\wedge
d\ln(t_{a_{1}+a_{2}}-z_{2})+$
$\displaystyle\dots+\,d\ln(t_{a_{1}+\dots+a_{n-1}+1}-z_{n})\wedge\dots\wedge
d\ln(t_{k}-z_{n}).$

Similarly, for $b=(b_{1},\dots,b_{n})$, $|b|=N-1$, we define

$\displaystyle w_{b}=\frac{1}{b_{1}!\dots b_{n}!}\operatorname{Alt}\,u_{b},$

where

$\displaystyle u_{b}$ $\displaystyle=$ $\displaystyle-\kappa\biggl{(}\,\
d\ln(t_{2}-z_{1})\wedge\dots\wedge d\ln(t_{b_{1}+1}-z_{1})+$
$\displaystyle+\,d\ln(t_{b_{1}+2}-z_{2})\wedge\dots\wedge
d\ln(t_{b_{1}+b_{2}+1}-z_{2})+$
$\displaystyle\dots+\,d\ln(t_{b_{1}+\dots+b_{n-1}+2}-z_{n})\wedge\dots\wedge
d\ln(t_{N}-z_{n})\biggr{)}.$

In this formula we start from the variable $t_{2}$ and have the factor
$-\kappa$ in front of the exterior product.

For example if $N=2$, $a=(2,0)$, $b=(1,0)$, then

$\displaystyle w_{a}$ $\displaystyle=$ $\displaystyle d\ln(t_{1}-z_{1})\wedge
d\ln(t_{2}-z_{1})$ $\displaystyle w_{b}$ $\displaystyle=$
$\displaystyle-\kappa(d\ln(t_{2}-z_{1})+d\ln(t_{1}-z_{2})).$

2.3. Coulomb $\mathcal{D}$-module. Define a "Coulomb interaction" closed
$1$-form

$\omega_{\mathbf{m}}:=\sum_{1\leq s<u\leq
n}\frac{m_{s}m_{u}}{2}\,d\ln(z_{s}-z_{u})+\sum_{1\leq i<j\leq
N}2\,d\ln(t_{i}-t_{j})-$
$-\sum_{i=1}^{N}\sum_{s=1}^{n}m_{s}\,d\ln(t_{i}-z_{s})\in\Omega^{1}(U_{n,N})$
$None$

Define a differential $\nabla_{\mathbf{m}}$ on the graded space
$\Omega^{\bullet}(U_{n,N})$

$\nabla_{\mathbf{m}}:=d_{DR}+\frac{1}{\kappa}\omega_{\mathbf{m}}:\
\Omega^{i}(U_{n,N})\longrightarrow\Omega^{i+1}(U_{n,N})$

Note that $\nabla_{\mathbf{m}}^{2}=0$ since $d_{DR}\omega_{\mathbf{m}}=0$.

We will denote by $\Omega_{\mathbf{m}}^{\bullet}(U_{n,N})$ the space
$\Omega^{\bullet}(U_{n,N})$ equipped with the differential
$\nabla_{\mathbf{m}}$.

This is nothing else but the complex of global sections for the de Rham
complex $DR(\mathcal{L}(\mathbf{m},N))$ of the Coulomb $\mathcal{D}$-module
$\mathcal{L}(\mathbf{m})=\mathcal{L}(\mathbf{m},N)$ over $U_{n,N}$ which is by
definition the structure sheaf $\mathcal{O}_{U_{n,N}}$ equipped with a
connection $\nabla_{\mathbf{m}}:=d_{DR}+\frac{1}{\kappa}\omega_{\mathbf{m}}$.

2.4. Coulomb - KZ - Chevalley complex and a canonical $N$-cocycle in it.
Recall a Chevalley complex $C^{\bullet}(\mathfrak{n},M(\mathbf{m})^{c})_{N}$.

Consider a double complex which as a bigraded vector space is a tensor product

$C^{\bullet\bullet}_{\mathbf{m},N}:=\\{C^{pq}_{\mathbf{m},N}\\}$

where

$C^{pq}_{\mathbf{m},N}:=\Omega^{p}(U_{n,N})\otimes
C^{q}(\mathfrak{n},M(\mathbf{m})^{c})_{N}$

Note that along $q$-axis it has only two nontrivial components: $0\leq q\leq
1$.

By definition it is equipped with two differentials:

— the horizontal one is a KZ - Coulomb differential

$\nabla_{{\operatorname{KZ}},\operatorname{Coul}}=d_{DR}+\frac{1}{\kappa}\omega_{\mathbf{m}}-\frac{1}{\kappa}\omega_{{\operatorname{KZ}}}$

where

$\omega_{{\operatorname{KZ}}}:=\sum_{1\leq i<j\leq
n}\Omega_{ij}d\ln(z_{i}-z_{j})$ $None$

It acts on the index $p$:

$\nabla_{{\operatorname{KZ}},\operatorname{Coul}}:\Omega^{p}(U_{n,N})\otimes
C^{q}(\mathfrak{n},M(\mathbf{m})^{c})_{N}\longrightarrow\Omega^{p+1}(U_{n,N})\otimes
C^{q}(\mathfrak{n},M(\mathbf{m})^{c})_{N}$

— the vertical one is the Chevalley differential $d_{{\operatorname{Ch}}}$
acting on the second factor.

We will be interested in the associated total complex

$C^{\bullet}_{\mathbf{m},N}:=\operatorname{Tot}C^{\bullet\bullet}_{\mathbf{m},N}.$

Recall the notations (2.1.4).

Define elements

$\displaystyle\mathcal{I}_{0}$ $\displaystyle:=$
$\displaystyle\sum_{|a|=N}w_{a}\otimes f^{(a)}v\in C^{N0}_{\mathbf{m},N}$
$\displaystyle\mathcal{I}_{1}$ $\displaystyle:=$
$\displaystyle\sum_{|b|=N-1}w_{b}\otimes(f\otimes f^{(b)}v)\in
C^{N-1,1}_{\mathbf{m},N}$ $\displaystyle\mathcal{I}$ $\displaystyle:=$
$\displaystyle\mathcal{I}_{0}+\mathcal{I}_{1}\in C^{N}_{\mathbf{m},N}$

2.5. Theorem. $\mathcal{I}$ is a cocycle in $C^{\bullet}_{\mathbf{m},N}$ of
total degree $N$. In components:

$\nabla_{{\operatorname{KZ}},\operatorname{Coul}}\mathcal{I}_{0}=0$ $None$
$d_{{\operatorname{Ch}}}\mathcal{I}_{0}+\nabla_{{\operatorname{KZ}},\operatorname{Coul}}\mathcal{I}_{1}=0.$
$None$

Proof. We deduce Theorem 2.5 from the two main results in [SV]. The first of
them is [SV, Theorem 6.16.2] on the relation between the Lie algebra
differential and the de Rham differential. The second is [SV, Theorem
$7.2.5^{\prime\prime}$] on the relation between the KZ equations and the
Gauss-Manin connection.

Since all forms $w_{a}$ are closed the equation (2.5.1) may be rewritten as

$\frac{1}{\kappa}\biggl{(}\omega_{\mathbf{m}}-\omega_{KZ}\biggr{)}\mathcal{I}_{0}=0.$

This equation is the statement of [SV, Theorem 7.2.5’’] applied to the
$\mathfrak{sl}_{2}$ case.

Equation (2.5.2) may be rewritten as

$\frac{1}{\kappa}\biggl{(}\omega_{\mathbf{m}}-\omega_{KZ}\biggr{)}\mathcal{I}_{1}+d_{Ch}\mathcal{I}_{0}=0$

and it can be split into two equations.

One of these equations follows from [SV, Theorem 7.2.5’’] applied to the
situation with $N-1$ of $t$-variables instead of the $N$ variables
$t_{1},\dots,t_{N}$, and the other equation follows from [SV, Theorem 6.16.2].

More precisely, consider the splitting

$\frac{1}{\kappa}\omega_{\mathbf{m}}\mathcal{I}_{1}=\mathcal{P}_{1}+\mathcal{P}_{2},$

where $\mathcal{P}_{1},\mathcal{P}_{2}$ are defined as follows. We have

$\displaystyle\mathcal{I}_{1}=\sum_{|b|=N-1}\sum_{\sigma\in
S_{N}}(-1)^{\sigma}u_{b}(t_{\sigma(2)},\dots,t_{\sigma(N)})\otimes\Big{(}f\otimes
f^{(b)}v\Big{)}$

and $\omega_{\mathbf{m}}$ is the sum of 1-forms,
$\omega_{\mathbf{m}}=\sum_{\alpha}\omega_{\alpha}$, see (2.3.1). We say that a
summand

$\displaystyle(-1)^{\sigma}\omega_{\alpha}\wedge
u_{b}(t_{\sigma(2)},\dots,t_{\sigma(k)})\otimes\Big{(}f\otimes
f^{(b)}v\Big{)}$

belongs to $\mathcal{P}_{1}$ if $\omega_{\alpha}$ does not have the variable
$t_{\sigma(1)}$, otherwise it belongs to $\mathcal{P}_{2}$.

2.5.1. Lemma. We have

$\mathcal{P}_{1}-\frac{1}{\kappa}\omega_{{\operatorname{KZ}}}\mathcal{I}_{1}=0,$
$None$ $\mathcal{P}_{2}+d_{{\operatorname{Ch}}}\mathcal{I}_{0}=0.$ $None$

Proof of the Lemma. Equation (2.5.1.2) follows from [SV], Theorem 6.16.2.
Equation (2.5.1.1) follows from [SV], Theorem $7.2.5^{\prime\prime}$.
$\square$

This implies (2.5.2) and achieves the proof of 2.5. $\square$

2.6. Interpretation of the cocycle $\mathcal{I}$ as a map $\eta:DR($KZ)
$\longrightarrow DR$(GM). Note that the Coulomb de Rham complex
$\Omega^{\bullet}_{\mathbf{m}}(U_{n,N})$ is a dg-module over the de Rham
algebra $\Omega^{\bullet}(U_{n,N})$ which in turn is a
$\Omega^{\bullet}(U_{n})$-algebra due to the projection
$p:U_{n,N}\longrightarrow U_{n}$.

Consider the trivial vector bundle $\mathcal{M}(\mathbf{m})$ over $U_{n}$ with
a fiber $M(\mathbf{m})$; it carries the integrable KZ connection

$\nabla_{{\operatorname{KZ}}}=d_{z}-\frac{1}{\kappa}\omega_{{\operatorname{KZ}}}$
$None$

which makes of it a $\mathcal{D}_{U_{n}}$-module. The space of global sections
of its de Rham complex will be

$DR(\mathcal{M}(\mathbf{m}))(U_{n})=\Omega^{\bullet}(U_{n})\otimes_{\mathbb{C}}M(\mathbf{m}).$

As usual this object is $\mathbb{N}$-graded.

Next we can pass to Chevalley chains and consider a complex of vector bundles

$\mathcal{C}_{\bullet}(\mathfrak{n}^{*},M(\mathbf{m})^{*}_{N})=C_{\bullet}(\mathfrak{n}^{*},\mathcal{M}(\mathbf{m})^{*}_{N}),$

whose dual will be

$\mathcal{C}^{\bullet}(\mathfrak{n},M(\mathbf{m})^{c}_{N})=C^{\bullet}(\mathfrak{n},\mathcal{M}(\mathbf{m})^{c}_{N}).$

Both complexes carry KZ connections induced by (2.7.1); therefore we may
consider their de Rham complexes which are $\Omega^{\bullet}_{U_{n}}$-modules.

Our main hero, the KZ - Coulomb - Chevalley complex may be rewritten in a form

$C^{\bullet\bullet}_{\mathbf{m},N}=DR(\mathcal{C}^{\bullet}(\mathfrak{n},M(\mathbf{m})^{c}_{N})(U_{n})\otimes_{\Omega^{\bullet}(U_{n})}\Omega^{\bullet}_{\mathbf{m}}(U_{n,N})$

By linear algebra, to give a $0$-cocycle

$Z\in\operatorname{Tot}(A^{\bullet}\otimes B^{\bullet})^{0}$

in the total complex of a tensor product of two complexes $A^{\bullet}\otimes
B^{\bullet}$ is equivalent to giving a map of complexes

$\eta(Z):\ A^{\bullet*}\longrightarrow B^{\bullet}.$

Therefore our cocycle $\mathcal{I}$ gives rise to a map between two complexes

$\eta=\eta(\mathcal{I}):DR(\mathcal{C}_{\bullet}(\mathfrak{n}^{*},M(\mathbf{m})^{*}_{N})(U_{n})\longrightarrow\Omega^{\bullet}_{\mathbf{m}}(U_{n,N})[N]$
$None$

Both complexes are filtered:

namely, we define

$F^{i}DR(\mathcal{C}_{\bullet}(\mathfrak{n}^{*},M(\mathbf{m})^{*}_{N})(U_{n})\subset
DR(\mathcal{C}_{\bullet}(\mathfrak{n}^{*},M(\mathbf{m})^{*}_{N})(U_{n})$

to be the subcomplex of forms of degree $\geq i$, and

$F_{z}^{i}\Omega^{\bullet}_{\mathbf{m}}(U_{n,N})\subset\Omega^{\bullet}_{\mathbf{m}}(U_{n,N})$

to be the subcomplex of forms containing $\geq i$ differentials $dz_{a}$.

2.6.1. Key fact. The map $\eta$ is compatible with the filtrations. $\square$

As a corollary , the induced map of $E_{1}$-terms of the corresponding
spectral sequences gives rise to maps between the de Rham complexes

$\eta^{i}:\
DR(H_{i}(\mathfrak{n}^{*},\mathcal{M}(\mathbf{m})^{*}_{N}),\nabla_{\operatorname{KZ}})\longrightarrow
DR(Rp_{*}^{N-i}\mathcal{L}(\mathbf{m},N),\nabla_{\operatorname{GM}}),$

$0\leq i\leq 1$, cf 1.3.

By construction these maps land in the subsheaves of anti-invariants

$\eta^{i}:\
DR(H_{i}(\mathfrak{n}^{*},\mathcal{M}(\mathbf{m})^{*}_{N}),\nabla_{\operatorname{KZ}})\longrightarrow
DR(Rp_{*}^{N-i}\mathcal{L}(\mathbf{m},N)^{\Sigma_{N},-},\nabla_{\operatorname{GM}}),$

Let us sum up our results.

2.7. Theorem. The map  (2.6.2) is a morphism of filtered complexes. The
induced map of $E_{1}$ terms for the corresponding spectral sequences is a
pair of morphisms

$\eta^{i}:\
DR(H_{i}(\mathfrak{n}^{*},\mathcal{M}(\mathbf{m})^{*}_{N}),\nabla_{\operatorname{KZ}})\longrightarrow
DR(Rp_{*}^{N-i}\mathcal{L}(\mathbf{m},N)^{\Sigma_{N},-},\nabla_{\operatorname{GM}}),$

$0\leq i\leq 1$.

Here $\eta^{i}$ is a map from the de Rham complex of
$H_{i}(\mathfrak{n}^{*},\mathcal{M}(\mathbf{m})^{*}_{N})$ equipped with the KZ
connection to the de Rham complex of
$Rp_{*}^{N-i}\mathcal{L}(\mathbf{m},N)^{\Sigma_{N},-}$ equipped with the
Gauss-Manin connection, or, which is the same, a morphism of lisse
$\mathcal{D}$-modules over $U_{n}$:

$\eta^{i}:\
(H_{i}(\mathfrak{n}^{*},\mathcal{M}(\mathbf{m})^{*}_{N}),\nabla_{\operatorname{KZ}})\longrightarrow(Rp_{*}^{N-i}\mathcal{L}(\mathbf{m},N)^{\Sigma_{N},-},\nabla_{\operatorname{GM}}).$

These maps are isomorphisms for generic $\kappa$.

2.8. Corollary: integral solutions for higher KZ. Let us return to the
notations of 2.1.

Consider the complex $C^{\bullet}(\mathfrak{n},M(\mathfrak{m})^{c})_{N}$, see
(2.1.3)

$0\longrightarrow M(\mathfrak{m})_{N}\longrightarrow\mathfrak{n}\otimes
M(\mathfrak{m})_{N-1}\longrightarrow 0$

which we denote here for brevity

$C^{\bullet}:\ 0\longrightarrow
C^{0}\overset{d}{\longrightarrow}C^{1}\longrightarrow 0,$

and the dual complex

$C^{*\bullet}:\ 0\longrightarrow
C^{1*}\overset{d^{*}}{\longrightarrow}C^{0*}\longrightarrow 0$

In this subsection we consider the analytic version of our varieties and
$\mathcal{D}$-modules.

For any $\mathbf{z}=(z_{1},\ldots,z_{n})\in U_{n}$ we denote by
$F(\mathbf{z})$ the fiber

$F(\mathbf{z}):=p^{-1}(\mathbf{z})=\\{(t_{1},\ldots,t_{N})\in\mathbb{C}^{N}|\
t_{i}\neq t_{j};t_{i}\neq z_{a}\\}\subset\mathbb{C}^{N}.$

We will deal with the analytic Coulomb $\mathcal{D}$-module
$\mathcal{L}^{{\operatorname{an}}}(\mathfrak{m})$ over $U_{n,N}$. Consider its
de Rham complex

$\Omega^{{\operatorname{an}}\bullet}_{\mathfrak{m}}:=DR(\mathcal{L}^{{\operatorname{an}}}(\mathfrak{m})).$

For each $\mathbf{z}\in U_{n}$ let
$\Omega_{\mathfrak{m}}^{\bullet}(\mathbf{z})$ denote the restriction of
$\Omega^{{\operatorname{an}}\bullet}_{\mathfrak{m}}$ to the fiber
$F(\mathbf{z})$; inside it we have the skew-symmetric part

$\Omega^{\bullet}_{\mathfrak{m}}(\mathbf{z})^{\Sigma_{N},-}\subset\Omega_{\mathfrak{m}}^{\bullet}(\mathbf{z})$

Next, inside $\Omega^{\bullet}_{\mathfrak{m}}(\mathbf{z})^{\Sigma_{N},-}$
consider the finite-dimensional Aomoto subcomplex of differential forms with
logarithmic singularities along all hyperplanes $t_{i}=t_{j}$ and
$t_{i}=z_{a}$; let us denote this subcomplex

$A^{\bullet}(\mathbf{z}):\ 0\longrightarrow
A^{N-1}(\mathbf{z})\overset{d_{A}(\mathbf{z})}{\longrightarrow}A^{N}(\mathbf{z})\longrightarrow
0,$

the differential $d_{A}(\mathbf{z})$ being the multiplication by the one-form

$\frac{1}{\kappa}\omega_{\mathfrak{m}}(\mathbf{z})=\frac{1}{\kappa}\biggl{(}\sum_{1\leq
i<j\leq
N}2\,d\ln(t_{i}-t_{j})-\sum_{i=1}^{N}\sum_{s=1}^{n}m_{s}\,d\ln(t_{i}-z_{s})\biggr{)}\in\Omega^{1}(F(\mathbf{z}))$

cf. (2.3.1). This subcomplex will have only two nontrivial components living
in degrees $N-1$ and $N$.

We denote by

$W^{i}(\mathbf{z}):=H^{i}(A^{\bullet}(\mathbf{z})),\ i=N-1,N,$

its cohomology.

Global maps $\eta^{i}$

The space

$C^{0}=M(\mathfrak{m})_{N}$

admits a base $\\{f^{a}v,\ |a|=N\\}$; let us denote by $\\{f^{a\vee}\\}$ the
dual base of $C^{0*}$.

Similarly, the space

$C^{1}=\mathfrak{n}\otimes M(\mathfrak{m})_{N-1}$

admits a base $\\{f\otimes f^{b}v,\ |b|=N-1\\}$; let us denote
$\\{f^{\vee}\otimes f^{b\vee}\\}$ the dual base of $C^{1*}$.

Define two maps

$\eta^{i}:\ C^{i*}\longrightarrow\Omega^{N-i}(U_{n,N}),\ i=0,1,$

by

$\eta^{0}(f^{a\vee})=w_{a},$

and

$\eta^{1}(f^{\vee}\otimes f^{b\vee})=w_{b}.$

Denote

$\mathcal{A}^{i}:=\eta^{i}(C^{i*})\subset\Omega^{N-i}(U_{n,N}).$

Let $\mathbf{z}\in U_{n}$. The restriction to the fiber $F(\mathbf{z})$
induces maps

$\mathcal{A}^{N-i}\longrightarrow A^{N-i}(\mathbf{z});$

composing them with the maps $\eta_{i}$ we get maps

$\eta^{i}(\mathbf{z}):\ C^{i*}\longrightarrow A^{N-i}(\mathbf{z})$

According to [SV] these maps are isomorphisms; moreover, they induce an
isomorphism of complexes

$\eta^{\bullet}(\mathbf{z}):C^{*\bullet}\overset{\sim}{\longrightarrow}A^{\bullet}(\mathbf{z})[N]$

where on the left we have the Chevalley differential whereas on the right we
have the twisted de Rham differential in the de Rham complex of the fiber.

Chains of the Betti realization

For each $\mathbf{z}$ let $\mathcal{L}_{\mathfrak{m}}(\mathbf{z})$ denote the
restriction of $\mathcal{L}(\mathfrak{m})$ to $F(\mathbf{z})$; let

$\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee}=\mathcal{H}om(\mathcal{L}_{\mathfrak{m}}(\mathbf{z}),\mathcal{O}^{\operatorname{an}}_{F(\mathbf{z})})$

be the dual $D$-module. Let

$\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}}\subset\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee}$

be the subsheaf of horizontal sections; it isa locally constant sheaf over
$F(z)$.

Let
$C_{\bullet}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})$
denote the complex

$0\longrightarrow
C_{2N}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})\longrightarrow\ldots\longrightarrow
C_{0}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})\longrightarrow
0$

of finite singular chains with coefficients in
$\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}}$. We will be dealing
with subspaces of $i$-cycles

$Z_{i}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})\subset
C_{i}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})$

and with homology spaces
$H_{i}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})$.

2.8.1. GM connection: Betty realization. When $\mathbf{z}$ varies, these
complexes form a complex of (infinite dimensional) vector bundles over
$U_{n}$, denoted by
$\mathcal{C}_{\bullet}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})$.
Each term
$\mathcal{C}_{i}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})$
carries a flat connection.

Indeed, given $\mathbf{z}_{0}$ and a finite singular chain

$\gamma(\mathbf{z}_{0})\in
C_{i}(F(\mathbf{z}_{0}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z}_{0})^{\vee\text{hor}}),$

when can move $\mathbf{z}$ in a small neighbourhood $V\ni\mathbf{z}_{0}$ such
that nothing changes topologically; this provides a parallel transport of
$\gamma(\mathbf{z}_{0})$ over $V$, i.e. a flat family of chains

$\\{\gamma(\mathbf{z})\in
C_{i}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})\\}_{\mathbf{z}\in
V}.$ $None$

These connections are obviously compatible with boundary, i.e. we get a flat
connection on the complex
$\mathcal{C}_{\bullet}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})$.
This is the Betty incarnation of the derived GM connection.

It induces flat connections on the bundles of cycles
$\mathcal{Z}_{i}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})$
and on the homology
$\mathcal{H}_{i}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})$.

$\square$

We can integrate $i$-forms against $i$-chains, i.e. we have pairings

$\int:\
C_{i}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})\otimes\Omega^{i}_{\mathfrak{m}}(\mathbf{z})\longrightarrow\mathbb{C}.$

Let

$\\{\gamma_{i}(\mathbf{z})\in
Z_{i}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})\\}_{\mathbf{z}\in
V}$ $None$

be a flat family of cycles over a small open $V\subset U_{n}$ whose classes in
$H_{i}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})$
form a flat section of the GM connection.

Let $i=N$. For each $x\in C^{0*}$ and $\mathbf{z}\in V$ we get a number

$\int_{\gamma_{N}(\mathbf{z})}\eta^{0}(\mathbf{z})(x)\in\mathbb{C};$

it is linear with respect to $x$, so we’ve got an element

$\int_{\gamma_{N}(\mathbf{z})}\eta^{0}(\mathbf{z})(\bullet)\in(C^{0*})^{*}=C^{0}=M(\mathfrak{m})_{N}.$
$None$

Note that if $x=d^{*}y$ then $\eta^{0}(\mathbf{z})(x)$ is a coboundary in
$\Omega_{\mathfrak{m}}^{\operatorname{an}}(\mathbf{z})$, so the integral is
zero since $\gamma_{N}(\mathbf{z})$ is a cycle. This means that (2.8.2)
belongs to the subspace of "singular vectors"

$(\operatorname{Coker}(d^{*}))^{*}=\operatorname{Ker}d=\operatorname{Ker}(e:M(\mathfrak{m})_{N}\longrightarrow
M(\mathfrak{m})_{N-1}).$

Similarly if $i=N-1$ then for each $x\in C^{1*}$ and $\mathbf{z}\in V$ we get
a number

$\int_{\gamma_{N-1}(\mathbf{z})}\eta^{1}(\mathbf{z})(x)\in\mathbb{C};$

which is linear with respect to $x$, so we’ve got an element

$\int_{\gamma_{N-1}(\mathbf{z})}\eta^{1}(\mathbf{z})(\bullet)\in(C^{1*})^{*}=C^{1}=M(\mathfrak{m})_{N-1}.$

Its image in

$(\operatorname{Ker}(d^{*}))^{*}=C^{1}/dC^{0}=\operatorname{Coker}(e:M(\mathfrak{m})_{N}\longrightarrow
M(\mathfrak{m})_{N-1})$

depends only on the homology class

$\overline{\gamma_{N-1}(\mathbf{z})}\in
Z_{i}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}}).$

2.8.1. Theorem. (a) For any local flat family of $N$-cycles

$\\{\gamma_{N}(\mathbf{z})\in
Z_{N}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})\\}_{z\in
V},\ V\subset U_{n},$

the linear map $\int_{\gamma(\mathbf{z})}$ defines a solution of the KZ
equations with values in $Ker(d)$, i.e. in the weight component of

$\operatorname{Ker}e:\ M(\mathfrak{m})\longrightarrow M(\mathfrak{m})$

of weight $\sum_{s=1}^{n}m_{i}-2N$. For generic $\kappa$ any solution of the
KZ equations in this space is given by a suitable family
$\gamma_{N}(\mathbf{z})$.

(b) For any local flat family of $(N-1)$-cycles

$\\{\gamma_{N-1}(\mathbf{z})\in
Z_{N-1}(F(\mathbf{z}),\mathcal{L}_{\mathfrak{m}}(\mathbf{z})^{\vee\text{hor}})\\}_{z\in
V},\ V\subset U_{n},$

the linear map $\int_{\gamma(\mathbf{z})}$ defines a solution of the KZ
equations with values in $\operatorname{Coker}(d)$, i.e. in the weight
component of $M(\mathfrak{m})/eM(\mathfrak{m})$ of weight
$\sum_{s=1}^{n}m_{i}-2(N-1)$. For generic $\kappa$ any solution of the KZ
equations in this space is given by a suitable family
$\gamma_{N-1}(\mathbf{z})$.

Proof. Part (a) is proved in [SV], whereas part (b) is new and follows from
Theorem 2.7 $\square$

See [CV], where the dimensions of the spaces $\operatorname{Ker}d$ and
$\operatorname{Coker}d$ are calculated for nonnegative integers
$m_{1},\ldots,m_{n}$.

2.9. Exotic (dual) KZ equations.

Let $N=1,n=2$. Let us look up more attentively at the KZ - Coulomb part of our
cocycle.

So we have a $2$-dimensional subspace

$M_{1}\subset M(m_{1})\otimes M(m_{2})$

with a base $\\{fv_{1}\otimes v_{2},\ v_{1}\otimes fv_{2}\\}$ whose elements
we will write as columns.

The Casimir $\Omega$ acts on this subspace by the matrix

$\Omega=\left(\begin{matrix}(m_{1}-2)m_{2}/2&m_{2}\\\
m_{1}&m_{1}(m_{2}-2)/2\end{matrix}\right)$

Consider a double Coulomb - KZ complex $\Omega^{\bullet\bullet}(M)$: as a
graded space

$\Omega^{\bullet\bullet}(M_{1}):=\Omega^{\bullet\bullet}(U_{2,1})\otimes
M_{1}$

where $\Omega^{ij}$ are differential forms in $z,t$, of degree $i$ (resp. $j$)
with respect to $z$ (resp. to $t$).

The first (horizontal) differential is a KZ connection

$d^{\prime}=\nabla_{KZ}=d_{z}-\frac{1}{\kappa}\frac{\Omega(dz_{1}-dz_{2})}{z_{1}-z_{2}}=d_{z}+A_{1}dz_{1}+A_{2}dz_{2}$

where $d_{z}$ means de Rham with respect to $z$, whereas the second (vertical)
differential

$d^{\prime\prime}=d_{t}$

(de Rham with respect to $t$)

The identity $\nabla_{KZ}^{2}=0$ means that the KZ connection is integrable.

In coordinates:

$\partial_{z_{2}}A_{1}-\partial_{z_{1}}A_{2}-[A_{1},A_{2}]=0,$

in our case $[A_{1},A_{2}]=0$.

Now we will descibe the relevant part of the cocycle $\mathcal{I}$ from
Theorem 2.5.

Consider a form

$\omega^{01}=I=\left(\begin{matrix}(t-z_{1})^{-1}\Phi dt\\\ (t-z_{2})^{-1}\Phi
dt\end{matrix}\right)=\left(\begin{matrix}I_{1}\\\
I_{2}\end{matrix}\right)\in\Omega^{01}(M_{1})$

2.9.1. Claim. We have

$d^{\prime\prime}\omega^{01}=0,$

(obvious), and

$d^{\prime}\omega^{01}=d^{\prime\prime}\omega^{10}$ $None$

where

$\omega^{10}=J_{1}dz_{1}+J_{2}dz_{2}\in\Omega^{10}(M_{1}),$

with

$J_{1}=\left(\begin{matrix}-(t-z_{1})^{-1}\Phi\\\ 0\end{matrix}\right),\
J_{2}=\left(\begin{matrix}0\\\ -(t-z_{2})^{-1}\Phi\end{matrix}\right).$

2.9.2. Claim. We have

$d^{\prime}\omega^{10}=0,$

in coordinates

$-\partial_{z_{2}}J_{1}+\partial_{z_{1}}J_{2}-\frac{1}{\kappa}\frac{\Omega
J_{2}}{z_{1}-z_{2}}+\frac{1}{\kappa}\frac{\Omega J_{1}}{z_{1}-z_{2}}=0.$
$None$

The last differential equation is called the dual KZ equation: it is a system
of two linear differential equations on two functions (nonzero coordinates of
vectors $J_{1},J_{2}$).

The equation does not depend on $t$, whereas our vectors $J_{1},J_{2}$ do. For
all $t$ the couple

$(J_{1}(t,z),J_{2}(t,z))$

is a solution of (2.8.2).

§3. Kac-Moody case

3.1. Kac-Moody algebras without Serre relations. We start with the data from
[SV], 6.1. Let $\mathfrak{h}$ be a finite-dimensional vector space equipped
with a non-degenerate symmetric bilinear form $(,)$.

We fix a finite set of non-zero covectors
$\\{\alpha_{1},\ldots,\alpha_{r}\\}\subset\mathfrak{h}^{*}$ whose elements are
called simple roots; let $B=(b_{ij})$ where $b_{ij}=(\alpha_{i},\alpha_{j})$
(this is "the symmetrized Cartan matrix").

We denote by

$h_{i}=b(\alpha_{i})$

where $b:\mathfrak{h}^{*}\overset{\sim}{\longrightarrow}\mathfrak{h}$ is the
isomorphism induced by $(,)$.

We define $\mathfrak{g}=\mathfrak{g}(B)$ as a Lie algebra with generators
$e_{i},f_{i},1\leq i\leq r$, and $\mathfrak{h}$ and relations

$[e_{i},f_{j}]=\delta_{ij}h_{i},$ $[h,e_{i}]=\alpha_{i}(h)e_{i},\
[h,f_{i}]=-\alpha_{i}(h)f_{i},$ $[h,h^{\prime}]=0,\
h,h^{\prime}\in\mathfrak{h}$

We denote by $\mathfrak{n}=\mathfrak{n}_{-}\subset\mathfrak{g}$ (resp. by
$\mathfrak{n}_{+}$) the Lie subalgebra generated by all elements $f_{i}$
(resp. $e_{i}$); it is a free Lie algebra with these generators.

We have the triangular decomposition

$\mathfrak{g}=\mathfrak{n}_{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{+}$

Root lattice

Let

$\Lambda=\sum_{i}\mathbb{Z}\alpha_{i}\subset\mathfrak{h}^{*}$

denote the abelian subgroup generated by $\alpha_{i}$.

We will use the notations for "positive" and "negative" submonoids:

$\Lambda_{\geq 0}:=\sum_{i=1}^{r}\mathbb{Z}_{\geq 0}\alpha_{i}\subset\Lambda,\
\Lambda_{>0}:=\Lambda_{\geq 0}\setminus\\{\text{\bf 0}\\};$ $\Lambda_{\leq
0}:=-\Lambda_{\leq 0},\Lambda_{<0}:=\Lambda_{\leq 0}\setminus\\{\text{\bf
0}\\}$

Principal gradation

Our algebra $\mathfrak{g}$ is $\Lambda$-graded:

$\mathfrak{g}=\oplus_{\lambda\in\Lambda}\ \mathfrak{g}_{\lambda}$

where

$\mathfrak{g}_{\lambda}=\\{x\in\mathfrak{g}|\ [h,x]=\lambda(h)x\ \text{for
all}\ h\in\mathfrak{h}\\}$

with

$\mathfrak{h}=\mathfrak{g}_{\text{\bf 0}},\ $
$\mathfrak{n}:=\mathfrak{n}_{-}=\oplus_{\lambda\in\Lambda_{<0}}\
\mathfrak{g}_{\lambda}=\oplus_{\lambda\in\Lambda_{<0}}\
\mathfrak{n}_{\lambda}$ $\mathfrak{n}_{+}=\oplus_{\lambda\in\Lambda_{>0}}\
\mathfrak{g}_{\lambda}=\oplus_{\lambda\in\Lambda_{>0}}\
\mathfrak{n}_{\lambda}$

Verma modules

For $\mu\in\mathfrak{h}^{*}$ $M(\mu)$ will denote a $\mathfrak{g}$-module with
one generator $v=v_{\mu}$ and relations

$hv_{\mu}=\mu(h)e_{\mu},\ e_{i}v_{\mu}=0.$

It is $(\mu+\Lambda_{\leq 0})$-graded:

$M(\mu)=\oplus_{\lambda\in\mu+\Lambda_{\leq 0}}M(\mu)_{\lambda}$

where

$M(\mu)_{\lambda}=\\{x\in M(\mu)|\ hx=\lambda(h)x\\}$

A map

$U\mathfrak{n}\longrightarrow M(\mu),\ x\mapsto xv_{\mu}$

is an isomorphism of vector spaces.

Notation: duals for $\Lambda$-graded spaces

In the sequel we will be dealing with various $\Lambda$-graded spaces
$V=\oplus_{\lambda\in\Lambda}V_{\lambda}$ with finite dimentional components
$V_{\lambda}$. In that case $V_{\lambda}^{*}$ will denote the restricted dual:

$V^{*}=\oplus_{\lambda\in\Lambda}V^{*}_{\lambda}.$

Double

The Borel Lie subalgebra

$\mathfrak{b}:=\mathfrak{n}\oplus\mathfrak{h}\subset\mathfrak{g}$

carries a structure of a Lie bialgebra (see [D]) described in [SV], 6.14.1.
This means in particular that we have a cobracket map

$\mathfrak{b}\longrightarrow\mathfrak{b}\wedge\mathfrak{b}$

which gives, after the passage to duals, a Lie algebra structure on the space
$\mathfrak{b}^{*}$. The projection $\mathfrak{b}\longrightarrow\mathfrak{n}$
induces an embedding $\mathfrak{n}^{*}\hookrightarrow\mathfrak{b}^{*}$, and
the subspace $\mathfrak{n}^{*}$ is a Lie subalgebra of $\mathfrak{b}^{*}$.

This allows one to define its Drinfeld double
$\tilde{\mathfrak{g}}=D(\mathfrak{b})$; it is a Lie algebra which as a vector
space is

$\tilde{\mathfrak{g}}=\mathfrak{b}\oplus\mathfrak{b}^{*}=\mathfrak{n}\oplus\mathfrak{h}\oplus\mathfrak{h}^{*}\oplus\mathfrak{n}^{*}.$

If $M$ is a Verma module, one introduces a structure of a
$\mathfrak{b}^{*}$-module on $M^{*}$ which, together with an obvious structure
of a a $\mathfrak{b}$-module gives rise to a $D(\mathfrak{b})$-module
structure on $M^{*}$, see [SV], 6.16.

3.2. Chevalley complexes. We fix $n\geq 1$ and an $n$-tuple of weights

$\mu=(\mu_{1},\ldots,\mu_{n})\in\mathfrak{h}^{*n}$

Consider

$M(\mu)=M(\mu_{1})\otimes\ldots\otimes M(\mu_{n})$

The $\Lambda$-gradations on each $M(\mu_{i})$ gives rise to a
$\Lambda$-gradation on their tensor product $M(\mu)$.

Each $M(\mu_{i})^{*}$ is a $\tilde{\mathfrak{g}}$-module, whence the tensor
product

$M(\mu)^{*}=M(\mu_{1})^{*}\otimes\ldots\otimes M(\mu_{n})^{*}$

is a $\tilde{\mathfrak{g}}$-module as well. In particular due to the
inclusions of Lie algebras

$\mathfrak{n}^{*}\subset\mathfrak{b}^{*}\subset
D(\mathfrak{b}^{*})=\tilde{\mathfrak{g}}$

$M(\mu)^{*}$ is a $\mathfrak{n}^{*}$-module.

We will be interested in Chevalley homology complexes:

$C_{\bullet}(\mathfrak{n}^{*},M(\mu)^{*}):\
\ldots\longrightarrow\Lambda^{2}\mathfrak{n}^{*}\otimes
M(\mu)^{*}\longrightarrow\mathfrak{n}^{*}\otimes M(\mu)^{*}\longrightarrow
M(\mu)^{*}\longrightarrow 0$ $None$

They are analogues of (2.1.1).

They carry a $\Lambda$-grading induced by gradings on $\mathfrak{n}$ and
$M(\mu)$:

$C_{\bullet}(\mathfrak{n}^{*},M(\mu)^{*})=\oplus_{\lambda\in\Lambda}C_{\bullet}(\mathfrak{n}^{*},M(\mu)^{*})_{\lambda}$

where we denote by $C_{\bullet}(\mathfrak{n}^{*},M(\mu)^{*})_{\lambda}$ the
subcomplex of weight $|\mu|+\lambda$, $|\mu|:=\sum_{a=1}^{n}\mu_{a}$.

3.3. The Casimir element and KZ equation.

We have an invariant Casimir element

$\Omega\in\tilde{\mathfrak{g}}\hat{\otimes}\tilde{\mathfrak{g}}^{*}$ $None$

defined in [SV], 7.2. Namely,

$\Omega:=\sum_{\lambda\in\Lambda_{<0}}\Omega_{\lambda}+\Omega_{0}+\sum_{\lambda\in\Lambda_{>0}}\Omega_{\lambda}\in$
$\in\mathfrak{n}\otimes\mathfrak{n}^{*}\oplus\mathfrak{h}\otimes\mathfrak{h}^{*}\oplus\mathfrak{h}^{*}\otimes\mathfrak{h}\oplus\mathfrak{n}^{*}\otimes\mathfrak{n}$

where

$\Omega_{0}=\frac{1}{2}(\Omega_{\mathfrak{h}}+\Omega_{\mathfrak{h}^{*}})$

and
$\Omega_{\lambda}\in\mathfrak{n}_{\lambda}\otimes\mathfrak{n}_{\lambda}^{*}$
for $\lambda<0$ (resp.
$\in\mathfrak{n}^{*}_{\lambda}\otimes\mathfrak{n}_{\lambda}$ for $\lambda>0$)
are canonical elements.

Recall the space $U_{n}$ from 1.1.

Let
$\mathcal{C}_{\bullet}(\mathfrak{n}^{*},M(\mu)^{*})=C_{\bullet}(\mathfrak{n}^{*},\mathcal{M}(\mu)^{*})$
be the trivial vector bundle over $U_{n}$ with a fiber
$C_{\bullet}(\mathfrak{n}^{*},M(\mu)^{*})$; it is a $\Lambda$-graded complex
of vector bundles:

$C_{\bullet}(\mathfrak{n}^{*},\mathcal{M}(\mu)^{*})=\oplus_{\lambda\in\Lambda}C_{\bullet}(\mathfrak{n}^{*},\mathcal{M}(\mu)^{*})_{\lambda}$

For

$\lambda=-\sum_{i=1}^{r}k_{i}\alpha_{i}\in\Lambda_{\leq 0}$

with $\sum_{i=1}^{r}k_{i}=N$ the complex
$\mathcal{C}_{\bullet}(\mathfrak{n}^{*},M(\mu)^{*})_{\lambda}$ leaves in
degrees $[-N,0]$.

The invariant Casimir element allows one to define the KZ connection on each
$\mathcal{C}_{\bullet}(\mathfrak{n}^{*},M(\mu)^{*})_{\lambda}$, see 1.2.

3.4. A Coulomb $\mathcal{D}$-module and its de Rham complex. Pick

$\lambda=-\sum_{i=1}^{r}k_{i}\alpha_{i}\in\Lambda_{\leq 0};$

let $N=\sum k_{i}$.

Consider the space $\mathbb{C}^{n,N}=\mathbb{C}^{n+N}$ with coordinates
$z_{1},\ldots,z_{n},t_{1},\ldots,t_{N}$ and a subspace

$U_{n,N}=\\{(\mathbf{z},\mathbf{t})\in\mathbb{C}^{n,N}|\ z_{i}\neq
z_{j},t_{i}\neq t_{j},z_{i}\neq t_{j}\\}$

We have a projection

$p=p_{n,N}:\ U_{n,N}\longrightarrow U_{n}$

We shall use a notation $[k]=\\{1,\ldots,k\\}$.

Pick a map of sets

$\pi:[N]\longrightarrow[r]$

such that

$|\pi^{-1}(i)|=k_{i},\ i\in[r].$

We will denote by

$\Sigma_{\pi}\overset{\sim}{=}\Sigma_{k_{1}}\times\ldots\times\Sigma_{k_{r}}$

a subgroup of the symmetric group respecting all the fibers $\pi^{-1}(i)$.

A Coulomb $\mathcal{D}$-module $\mathcal{L}(\mu,\lambda)$

By definition $\mathcal{L}(\mu,\lambda)$ is a $\mathcal{D}_{U_{n,N}}$-module
which is $\mathcal{O}_{U_{n,N}}$ equipped with an integrable connection

$\nabla_{\mu,\lambda}=d_{DR}+\frac{1}{\kappa}\omega_{\mu,\lambda}$

where $\omega_{\mu,\lambda}$ is a closed differential $1$-form

$\omega_{\mu,\lambda}=\sum_{1\leq i<j\leq
n}(\mu_{i},\mu_{j})\frac{dz_{i}-dz_{j}}{z_{i}-z_{j}}-\sum_{i\in[n],k\in[N]}(\mu_{i},\alpha_{\pi(k)})\frac{dz_{i}-dt_{k}}{z_{i}-t_{k}}+$
$+\sum_{1\leq k<l\leq
N}(\alpha_{\pi(k)},\alpha_{\pi(l)})\frac{dt_{k}-dt_{l}}{t_{k}-t_{l}}$ $None$

It gives rise to the de Rham complex

$\Omega^{\bullet}_{\mu,\lambda}(U_{n,N}):=DR(\mathcal{L}(\mu,\lambda))(U_{n,N})=(\Omega^{\bullet}(U_{n,N}),\nabla_{\mu,\lambda})$

We will be interested in the subcomplex of $\Sigma_{\pi}$-skew-invariants

$\Omega^{\bullet}_{\mu,\lambda}(U_{n,N})^{\Sigma_{\pi},-}\subset\Omega^{\bullet}_{\mu,\lambda}(U_{n,N})$

3.5. Relative de Rham complexes and derived Gauss-Manin.

(a) The de Rham complex $\Omega^{\bullet}(U_{n,N})$ is the total complex of a
bicomplex

$\Omega^{\bullet}(U_{n,N})=\operatorname{Tot}\Omega^{\bullet\bullet}(U_{n,N})$

where $\Omega^{pq}(U_{n,N})$ is the space of forms containing $p$
differentials $dt_{i}$ and $q$ differentials $dz_{m}$, the full de Rham
differential being the sum

$d_{DR}=d_{z}+d_{t}.$

The relative de Rham complex is by definition

$\Omega^{\bullet}(U_{n,N}/U_{n})=(\Omega^{0\bullet}(U_{n,N}),d_{t});$

one has a projection

$p:\ \Omega^{\bullet}(U_{n,N})\longrightarrow\Omega^{\bullet}(U_{n,N}/U_{n})$

(b) Coulomb twisting

Similarly the form

$\omega_{\mu,\lambda}=\omega_{\mu,\lambda,z}+\omega_{\mu,\lambda,t}$

with

$\omega_{\mu,\lambda,t}=\sum_{i\in[n],k\in[N]}(\mu_{i},\alpha_{\pi(k)})\frac{dt_{k}}{z_{i}-t_{k}}+\sum_{1\leq
k<l\leq N}(\alpha_{\pi(k)},\alpha_{\pi(l)})\frac{dt_{k}-dt_{l}}{t_{k}-t_{l}}$

which is $d_{t}$-closed.

We define

$\Omega^{\bullet}_{\mu,\lambda}(U_{n,N}/U_{n}):=(\Omega^{\bullet}(U_{n,N}/U_{n}),d_{t}+\frac{1}{\kappa}\omega_{\mu,\lambda,t})$

We have an epimorphism of complexes

$p:\
\Omega^{\bullet}_{\mu,\lambda}(U_{n,N})\longrightarrow\Omega^{\bullet}_{\mu,\lambda}(U_{n,N}/U_{n})$
$None$

(c) Derived Gauss - Manin connection.

The complex $\Omega^{\bullet}_{\mu}(U_{n,N})$ is the total complex of a double
complex

$\Omega^{\bullet\bullet}_{\mu,\lambda}(U_{n,N}):=(\Omega^{\bullet\bullet}(U_{n,N}),\nabla_{\mu,\lambda,z}+\nabla_{\mu,\lambda,t})$

where

$\nabla_{\mu,\lambda,z}=d_{z}+\frac{1}{\kappa}\omega_{\mu,\lambda,z},\
\nabla_{\mu,\lambda,t}=d_{z}+\frac{1}{\kappa}\omega_{\mu,\lambda,t}.$

We shall write the differential $\nabla_{\mu,z}$ (resp. $\nabla_{\mu,t}$)
horizontally (resp. vertically).

The map $p$ (3.5.1) is nothing but the projection to the utmost left vertical
component.

We can identify $\Omega^{\bullet\bullet}_{\mu,\lambda}(U_{n,N})$ with the de
Rham complex of the connection $\nabla_{\mu,\lambda,z}$ on the complex
$\Omega^{\bullet}_{\mu,\lambda}(U_{n,N}/U_{n})$:

$\nabla_{\mu,\lambda,z}:\
\Omega^{\bullet}_{\mu,\lambda}(U_{n,N}/U_{n})=\Omega^{0\bullet}_{\mu,\lambda}(U_{n,N})\longrightarrow\Omega^{\bullet}_{\mu,\lambda}(U_{n,N}/U_{n})\otimes\Omega^{1}(U_{n})=\Omega^{1\bullet}_{\mu,\lambda}(U_{n,N})$

This is the derived GM connection on the complex
$Rp_{*}\mathcal{L}(\mu,\lambda)(U_{n})$.

3.7. A map $\eta$ and its lifting $\tilde{\eta}$.

In [SV] a map of complexes

$\eta:C_{\bullet}(\mathfrak{n}^{*},\mathcal{M}(\mu)^{*})_{\lambda}\longrightarrow\Omega^{\bullet}_{\mu,\lambda}(U_{n,N}/U_{n})^{\Sigma_{\pi},-}[N]$
$None$

has been defined, see op. cit. (7.2.4).

Here we consider both complexes appearing in (3.7.1) as cohomological
complexes concentrated in degrees $[-N,0]$.

For each $\mathbf{z}\in U_{n}$ consider a fiber

$U_{\mathbf{z}}=U_{n,N;\mathbf{z}}:=p^{-1}(\mathbf{z})$

We can compose $\eta$ with the restriction map

$r_{\mathbf{z}}:\Omega^{\bullet}_{\mu,\lambda}(U_{n,N}/U_{n})\longrightarrow\Omega^{\bullet}_{\mu,\lambda}(U_{n,N;\mathbf{z}})$

to get

$\eta_{\mathbf{z}}=r_{\mathbf{z}}\circ\eta:C_{\bullet}(\mathfrak{n}_{-},\mathcal{M}(\mu)^{*})_{\lambda}\longrightarrow\Omega^{\bullet}_{\mu,\lambda}(U_{n,N;\mathbf{z}})^{\Sigma_{\pi},-}[N]$

A remarkable feature of the mappings $\eta_{\mathbf{z}}$ is the following:

for generic values of $\kappa$ the maps $\eta_{\mathbf{z}}$ are quasi-
isomorphisms for all $\mathbf{z}\in U_{n}$.

Here "generic" means $\kappa\in\mathbb{C}\setminus$ (an explicitly given
discrete countable subset).

Main result

We start with is a definition of a map of graded $\mathcal{O}(U_{n})$-modules

$\tilde{\eta}:C_{\bullet}(\mathfrak{n}^{*},\mathcal{M}(\mu)^{*})_{\lambda}\longrightarrow\Omega^{\bullet}_{\mu,\lambda}(U_{n,N})^{\Sigma_{\pi},-}[N]$
$None$

which lifts $\eta$, i.e. such that

$\eta=p\circ\tilde{\eta}.$

Here is a picture:

$\begin{matrix}&&\Omega^{\bullet}_{\mu,\lambda}(U_{n,N})^{\Sigma_{\pi},-}[N]\\\
&\tilde{\eta}\nearrow&\downarrow p\\\
C_{\bullet}(\mathfrak{n}^{*},\mathcal{M}(\mu)^{*})_{\lambda}&\overset{\eta}{\longrightarrow}&\Omega^{\bullet}_{\mu,\lambda}(U_{n,N}/U_{n})^{\Sigma_{\pi},-}[N]\end{matrix}$

The definition of $\tilde{\eta}$ is a modification of that of $\eta$. Namely,
for a monomial

$x\in C_{i}(\mathfrak{n}^{*},\mathcal{M}(\mu)^{*})_{\lambda}$

the corresponding differential form

$\eta(x)\in\Omega^{N-i}_{\mu,\lambda}(U_{n,N}/U_{n})$

contains fractions of the form $dt_{i}/(t_{i}-z_{p})$. To obtain
$\tilde{\eta}(x)$ we replace all these fractions by $d\ln(t_{i}-z_{p})$.
That’s it.

Confer the definition of forms $u_{a},u_{b}$ in 2.2 for
$\mathfrak{g}=\mathfrak{sl}_{2}$.

The map $\tilde{\eta}$ induces a map of $\Omega^{\bullet}(U_{n,N})$-modules

$\tilde{\eta}:\Omega^{\bullet}(U_{n,N})\otimes_{\Omega^{\bullet}(U_{n})}C_{\bullet}(\mathfrak{n}^{*},\mathcal{M}(\mu)^{*})_{\lambda}\longrightarrow\Omega^{\bullet}_{\mu,\lambda}(U_{n,N})^{\Sigma_{\pi},-}[N]$
$None$

The space on the left is the underlying space of the De Rham complex for the
derived KZ; it carries the KZ differential $\nabla_{KZ}$.

3.8. Theorem. (a) The map $\tilde{\eta}$ (3.7.3) commutes with the
differentials on both sides.

(b) Both sides of (3.7.3) carry natural decreasing filtrations in $z$
direction, and the map $\tilde{\eta}$ respects these filtrations.

In other words, we’ve got a map of filtered complexes

$\begin{matrix}DR(\text{derived
KZ})&\overset{\tilde{\eta}}{\longrightarrow}&DR(\text{derived GM})\\\
\parallel&&\parallel\\\
\operatorname{Tot}\Omega_{KZ}^{\bullet}(U_{n},C_{\bullet}(\mathfrak{n}^{*},\mathcal{M}(\mu)^{*})_{\lambda})&\overset{\tilde{\eta}}{\longrightarrow}&\operatorname{Tot}(\Omega^{\bullet\bullet}_{\mu,\lambda})(U_{n,N})^{\Sigma_{\pi},-}[N]\end{matrix}$
$None$

The proof is similar to that of Section 2.

3.9. Corollary. The map $\tilde{\eta}$ induces maps of
$\mathcal{D}_{U_{n}}$-modules (which are isomorphisms for generic $\kappa$)

$\eta_{i}:\
(\mathcal{H}_{i}(\mathfrak{n}^{*},\mathcal{M}(\mu)^{*})_{\lambda},\nabla_{KZ})\longrightarrow(R^{N-i}p_{*}\mathcal{L}(\mu,\lambda)^{\Sigma_{\pi},-},\nabla_{GM})$

for all $0\leq i\leq N$.

For $i=0$ such a mapping has been constructed in [SV].

As in Section 2, we get from this an integral representation for the
solutions.

Appendix

We recall here some standard constructions from homological algebra.

A.1. Bicomplexes. A a bicomplex  in an abelian category $\mathcal{C}$ is a
collection of objects $A^{\bullet\bullet}=\\{A^{pq},\ p,q\in\mathbb{Z}\\}$,
and arrows

$d_{h}^{pq}:\ A^{pq}\longrightarrow A^{p+1,q},\ d_{h}^{p,q+1}:\
A^{pq}\longrightarrow A^{p,q+1}$

such that

$d_{h}^{2}=d_{v}^{2}=0,\ d_{h}d_{v}=d_{v}d_{h}$

One associates to it a simple complex, to be denoted

$A^{\bullet}=\operatorname{Tot}A^{\bullet\bullet}$

with components

$A^{i}=\oplus_{p+q=i}A^{pq}$

and a differential $d:A^{i}\longrightarrow A^{i+1}$ with components

$d^{pq}=d_{h}^{pq}+(-1)^{p}d_{v}^{pq}:\ A^{pq}\longrightarrow A^{p+1,q}\oplus
A^{p,q+1}.$

A.2. Filtered complexes. Let $A^{\bullet}$ be a simple complex. Consider a
decreasing filtration by subcomplexes on it:

$F^{0}A^{\bullet}=A^{\bullet}\supset F^{1}A^{\bullet}\supset\ldots.$

We associate to it a collection of complexes

$E(A^{\bullet},F)^{i}:\ 0\longrightarrow
H^{i}(F^{0}A^{\bullet}/F^{1}A^{\bullet})\longrightarrow
H^{i+1}(F^{1}A^{\bullet}/F^{2}A^{\bullet})\longrightarrow\ldots,$ $None$

$i\geq 0$, where a differential

$H^{i+p}(F^{p}A^{\bullet}/F^{p+1}A^{\bullet})\longrightarrow
H^{i+p+1}(F^{p+1}A^{\bullet}/F^{p+2}A^{\bullet})$

is the boundary map for the short exact sequence

$0\longrightarrow F^{p+1}A^{\bullet}/F^{p+2}A^{\bullet}\longrightarrow
F^{p}A^{\bullet}/F^{p+2}A^{\bullet}\longrightarrow
F^{p}A^{\bullet}/F^{p+1}A^{\bullet}\longrightarrow 0$

(This is nothing else but the $E_{1}$ term of the spectral sequences for
$(A^{\bullet},F^{\bullet})$.)

A.3. Example. Suppose that $A^{\bullet}=\operatorname{Tot}A^{\bullet\bullet}$
with $A^{pq}=0$ for $p<0$, and a filtration is defined by

$F^{i}A^{j}=\oplus_{p\geq i,p+q=j}A^{pq}.$

Then a $p$-th graded piece

$F^{p}A^{\bullet}/F^{p+1}A^{\bullet}=\\{A^{pq},\ q\in\mathbb{Z}\\},$

and the differential induced by $d$ on it coincides with the vertical
differential $d_{v}$.

It follows that a complex $E(A^{\bullet},F^{\bullet})^{i}$ is identified with

$0\longrightarrow H^{i}_{v}(A^{\bullet 0})\longrightarrow H^{i}_{v}(A^{\bullet
1})\longrightarrow\ldots,$

with a differential induced by $d_{h}$.

References

[CV] D.Cohen, A.Varchenko, Resonant local systems on complements of
discriminantal arrangements and $\mathfrak{sl}_{2}$ representations, Geom.
Dedicata 101 (2003), 217–233.

[D] V.Drinfeld, Quantum groups, Proceedings of the International Congress of
Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 798–820, AMS, Providence,
RI, 1987.

[G] A.Grothendieck, Crystals and de Rham cohomology of schemes, Notes by I.
Coates and O. Jussila, Adv. Stud. Pure Math. 3, Dix exposés sur la cohomologie
des schémas, 306–358, North-Holland, Amsterdam, 1968.

[K] V.G.Kac, Infinite dimensional Lie algebras, Cambridge University Press,
Cambridge, 1990.

[KO] N.Katz, T.Oda, On the differentiation of de Rham cohomology classes with
respect to parameters. J. Math. Kyoto Univ. 8 (1968), 199–213.

[SV] V.Schechtman, A.Varchenko, Arrangements of hyperplanes and Lie algebra
homology, Invent. Math. 106 (1991), 139–194.

[S] J.-P. Serre, Lie algebras and Lie groups, 1964 lectures given at Harvard
University. Second edition. Lecture Notes in Mathematics, 1500, Springer-
Verlag, Berlin, 1992.


corresponding big (dual-)exponential maps after Perrin-Riou-Nakamura. This
happens at the infinite level of the corresponding generalized admissible
abelian tower in our setting, which generalizes the corresponding setting
considered in [Nakamura1].

######  Setting 5.1.

We consider an abelian $p$-adic Lie group $\Gamma^{\prime}$ such that the
Fréchet-Stein algebra $\Pi^{\infty}(\Gamma^{\prime})$ in the sense of [ST1]
could be written as a family of affinoid algebras
$\Pi^{\infty}(\Gamma^{\prime})_{k},k=0,1,...$ in the sense of rigid analytic
geometry:

$\Pi^{\infty}(\Gamma^{\prime})\overset{\sim}{\longrightarrow}\varprojlim_{k}\Pi^{\infty}(\Gamma^{\prime})_{k}.$

If we use the notation $X$ to denote the Fréchet-Stein space attached to the
algebra $\Pi^{\infty}(\Gamma^{\prime})$ which could be written as:

$X\overset{\sim}{\longrightarrow}\varinjlim_{k}X_{k}:$

with:

$\mathcal{O}_{X}\overset{\sim}{\longrightarrow}\varprojlim_{k}\mathcal{O}_{X_{k}}:$

Now we start to consider the corresponding relative Robba rings where our
Iwasawa theory happens:

######  Definition 5.2.

Many definitions of the Robba rings are actually compatible with [Nakamura1],
the only difference is that we will deform over some general algebras. We use
the notations $\Pi^{r}_{\mathrm{an,con},K}$ and $\Pi_{\mathrm{an,con},K}$ to
denote the corresponding Robba rings as in [Nakamura1, Section 2.1]. Now we
consider the corresponding Robba rings
$\Pi^{r}_{\mathrm{an,con},K,\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})}$
and
$\Pi_{\mathrm{an,con},K,\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})}$
in the relative sense:

(5.1)
$\displaystyle\Pi^{r}_{\mathrm{an,con},K,\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})}:=\Pi^{r}_{\mathrm{an,con},K}\widehat{\otimes}\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime}),$
(5.2)
$\displaystyle\Pi_{\mathrm{an,con},K,\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})}:=\Pi_{\mathrm{an,con},K}\widehat{\otimes}\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime}).$

One also has the corresponding definitions for Robba rings with respect to
some interval $[s,r]$. And be very careful that in our current development the
corresponding Robba ring $\Pi^{?}_{\mathrm{an,con},K}$ is the ring
$\mathrm{B}^{?}_{\mathrm{rig},K}$ where $?=,r,[s,r]$, instead of the ring
$\Pi^{?}_{\mathrm{an,con}}(\pi_{K})$ considered in [KPX, Definition 2.2.2].

######  Setting 5.3.

Our motivic objects are still usual $(\varphi,\Gamma)$-modules over the Robba
ring $\Pi_{\mathrm{an,con},K}$. Namely we consider those modules finite
locally free over this big non-noetherian ring carrying commuting semilinear
action from the $\varphi$ and $\Gamma$.

######  Remark 5.4.

The ideas we presented here actually are initiated in [KP]. One replaces the
corresponding $p$-adic Lie group $\Gamma$ by a pro-étale site, and considers
the deformation of the sheaves (both the corresponding motivic objects and the
corresponding Berger’s $p$-adic differential equations attached in whatever
reasonable context like Lubin-Tate setting or some else) by the corresponding
Iwasawa deformations, and then considers the corresponding big exponential
maps by using the connections and their duals.

To study Iwasawa theory we look at those geometric $(\varphi,\Gamma)$-modules,
namely those geometric ones in the sense of $p$-adic Hodge theory after
Berger’s celebrated thesis. We do not go beyond [Nakamura1] on this, but we
would like to recall the basic construction on the corresponding $p$-adic
differential equation attached to a de Rham $(\varphi,\Gamma)$-module $M$.

######  Setting 5.5.

Recall the basic definition of the corresponding $p$-adic differential
equation attache to a de Rham $(\varphi,\Gamma)$-module $M$ after Berger. As
in [Nakamura1, Section 3.2] (we will use some different notations from
[Nakamura1, Section 3.2]), we consider the following constructions. Let $M$ be
a $(\varphi,\Gamma)$-module $M$ over $\Pi_{\mathrm{an,con},K}$. In our
situation we work with at least de Rham ones in the sense of Berger’s
celebrated paper [Ber1]. Then we have the corresponding relation for any
integer $k\geq l^{\prime}$ for some $l^{\prime}(K,M)\geq 0$ fixed which is
assumed to be sufficiently large depending on $K,M$:

(5.4) $\displaystyle
F_{\mathrm{dif},k}(M)\overset{\sim}{\longrightarrow}F_{\mathrm{dR},K}(M)\otimes
K_{k}((t))$

with the corresponding lattice $F_{\mathrm{dR},K}(M)\otimes K_{k}[[t]]$ inside
with reasonable rank. Then we have the corresponding continuous differential:

$\widehat{\Omega}_{\Pi_{\mathrm{an,con},K}/K_{0}^{\mathrm{ur}}}=\widehat{\Omega}_{\Pi_{\mathrm{an,con},K}}dT.$

The corresponding differential equation in the sense of Berger is the
following construction which is denoted by $\mathrm{N}_{\mathrm{dif}}(M)$, we
will denote this by $\Theta_{\mathrm{Ber,dif}}(M)$ which is actually defined
in the following way. Recall as in [Nakamura1, Theorem 3.5] by [Ber1, 5.10]
and [Ber2, 3.2.3], first we have $\Theta_{\mathrm{Ber,dif},k}(M)$ which is
defined to be the subset of all the elements in $M^{(k)}[1/t]$ such that the
corresponding image of them under each map $\iota_{k}^{\prime}$ for all
$k^{\prime}\geq k$ lives in $F_{\mathrm{dR},K}(M)\otimes K_{k}^{\prime}[[t]]$.
Recall from [Nakamura1, Section 3.2] we have then the corresponding set
$\Theta_{\mathrm{Ber,dif}}(M)=\varinjlim_{k}\Theta_{\mathrm{Ber,dif},k}(M)$
which could be endowed with furthermore the structure of
$(\varphi,\Gamma)$-module over the Robba ring $\Pi_{\mathrm{an,con},K}$. And
furthermore we have the corresponding properties that
$d_{0}(\Theta_{\mathrm{Ber,dif}}(M))\subset t\Theta_{\mathrm{Ber,dif}}(M)$
where the differential operator $d_{0}$ is defined to be
$\frac{\mathrm{log}(\gamma)}{\mathrm{log}(\chi(\gamma))}$ living in in our
notation $\Pi^{\infty}(\Gamma_{K})$.

Therefore recall from [Nakamura1, Section 3.2] then we can define the
connection $\partial:=\frac{1}{t}d_{0}$ mapping the corresponding differential
$(\varphi,\Gamma)$-module considered above to itself. The differential
structures and the corresponding Frobenius structures and the corresponding
Lie group action structures are compatible in the following sense:

(5.5) $\displaystyle\partial\varphi-p\varphi\partial=0,$ (5.6)
$\displaystyle\partial\gamma-\chi(\gamma)\gamma\partial=0,$

and:

(5.7) $\displaystyle\varphi(dT)-pdT=0,$ (5.8)
$\displaystyle\gamma(dT)-\chi(\gamma)dT=0.$

######  Lemma 5.6.

Now we set $d_{i}:=d_{0}-i$ for $i\geq 0$ integer, we have for any $m$-th
($m\geq 0$) Hodge filtration of $M$ (which is assumed to be de Rham) which
gives the full module:

$d_{m-1}d_{m-2}d_{m-3}...d_{1}d_{0}(\Theta_{\mathrm{Ber,dif}}(M))\subset M.$

###### Proof.

This is just [Nakamura1, Lemma 3.6]. ∎

We now define the corresponding equivariant big exponential map after Perrin-
Riou-Nakamura.

######  Assumption 5.7.

Now we assume that there is a quotient $G_{K}\rightarrow\Gamma^{\prime}$ in
our situation among all the other requirements as above.

######  Definition 5.8.

Take any sufficiently large $m\geq 0$ as in the previous lemma. First we
consider the differential equation $\Theta_{\mathrm{Ber,dif}}(M)$ attached to
$M$ which is assumed to be de Rham. We consider the
$\Gamma^{\prime}$-deformation of $M$ over the algebra
$\Pi^{\infty}(\Gamma^{\prime})$. This happens by considering the corresponding
quotient $G_{K}\rightarrow\Gamma^{\prime}$ which defines a corresponding
Galois representation with value in $\Pi^{\infty}(\Gamma^{\prime})$, which by
corresponding embedding fully faithful gives rise to a corresponding
$(\varphi,\Gamma)$-module over the relative Robba ring
$\Pi_{\mathrm{an,con},K}\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})$.
Denote this by $\mathrm{Dfm}_{\Gamma^{\prime}}$ which could be written as the
projective limit of the families
$\\{\mathrm{Dfm}_{\Gamma^{\prime},p}\\}_{p\geq 0}$ over
$\Pi_{\mathrm{an,con},K}\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})_{p}$
for all $p\geq 0$. Then we consider the product
$\mathrm{Dfm}_{\Gamma^{\prime},p}\widehat{\otimes}M$ then take the
corresponding projective limit over all $p\geq 0$. Then we consider the
following definition which is an equivariant version of [Nakamura1, Definition
3.7]:

$\mathrm{Exp}^{\Pi^{\infty}(\Gamma^{\prime})}_{M,m}:(\varprojlim\mathrm{Dfm}_{\Gamma^{\prime},p}\widehat{\otimes}\Theta_{\mathrm{Ber,dif}}(M))^{\psi=1}\rightarrow(\varprojlim\mathrm{Dfm}_{\Gamma^{\prime},p}\widehat{\otimes}M)^{\psi=1}$

which is just defined by extension of $d_{m-1}d_{m-2}d_{m-3}...d_{1}d_{0}$.
This is now
$\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})$-linear
instead of just $\Pi^{\infty}(\Gamma_{K})$. And the corresponding
$(\varphi,\Gamma)$-modules here could be basically geometrized by the
corresponding Hodge-Iwasawa construction above. We consider the corresponding
family of vector bundle $\varprojlim\mathcal{F}(M)_{p}$ and
$\varprojlim\mathcal{F}(\Theta_{\mathrm{Ber,dif}}(M))_{p}$ over the
corresponding adic version of Fargues-Fontaine curve deformed by
$\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})$. We
have now that the geometrized big exponential map by taking the corresponding
sheaf cohomology:

$\mathrm{Exp}^{\Pi^{\infty}(\Gamma^{\prime})}_{\mathcal{F},m}:R^{1}\Gamma_{\mathrm{sheaf}}(\varprojlim\mathcal{F}(\Theta_{\mathrm{Ber,dif}}(M))_{p})\rightarrow
R^{1}\Gamma_{\mathrm{sheaf}}(\varprojlim\mathcal{F}(M)_{p}),$

and the corresponding derived version:

$\mathrm{Exp}^{\Pi^{\infty}(\Gamma^{\prime}),\bullet}_{\mathcal{F},m}:R^{\bullet}\Gamma_{\mathrm{sheaf}}(\varprojlim\mathcal{F}(\Theta_{\mathrm{Ber,dif}}(M))_{p})\rightarrow
R^{\bullet}\Gamma_{\mathrm{sheaf}}(\varprojlim\mathcal{F}(M)_{p}).$

###  5.2. Characteristic Ideal Sheaves

Now we follow the corresponding idea in [Nakamura1] to consider the
corresponding characteristic ideal and the corresponding determinant of a
given derived maps with respect to the corresponding equivariant Iwasawa
cohomology. For this we consider the corresponding map defined above, and we
look at the corresponding equivariant Iwasawa cohomology, i.e. the
corresponding more general commutative deformation in our current context.

First in our situation we make the following assumption:

######  Assumption 5.9.

We now assume that the corresponding deformation happens over Krull noetherian
rings. This could basically allow us to define the corresponding
characteristic ideal over the sheaf
$\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}X_{\Pi^{\infty}(\Gamma^{\prime})}$
of any torsion module $M$, which we will denote it by
$\mathrm{char}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}X_{\Pi^{\infty}(\Gamma^{\prime})}}(M)$
which is again a sheaf of module over
$\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}X_{\Pi^{\infty}(\Gamma^{\prime})}$.

In the current generalized context, following Nakamura [Nakamura1, Section
3.4] we define the corresponding determinant of a general map $f:M\rightarrow
N$ of coadmissible modules over the sheaf
$\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}X_{\Pi^{\infty}(\Gamma^{\prime})}$.
First as above we have the corresponding characteristic ideals
$\mathrm{char}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}X_{\Pi^{\infty}(\Gamma^{\prime})}}(M_{\mathrm{torsion}})$
and
$\mathrm{char}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}X_{\Pi^{\infty}(\Gamma^{\prime})}}(N_{\mathrm{torsion}})$.
Then quotienting out the corresponding torsion part we have:

(5.9) $\displaystyle\overline{f}:M/M_{\mathrm{torsion}}\rightarrow
N/N_{\mathrm{torsion}}.$

We will regard this as a corresponding map on the sheaves over the the space
$\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}X_{\Pi^{\infty}(\Gamma^{\prime})}$,
then for any $Y=\mathrm{Sp}(\Pi^{\infty}(\Gamma^{\prime})_{p})$ quasicompact
of $X_{\Pi^{\infty}(\Gamma^{\prime})}$ we consider the corresponding ring
$\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})_{p}$,
then we have the notion of determinant of $\overline{f}$ which we will denote
it by
$\mathrm{det}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})_{p}}(\overline{f})$.
We then organize this construction to get the determinant sheaf
$\mathrm{det}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}X_{\Pi^{\infty}(\Gamma^{\prime})}}(\overline{f})$.
Then now we define the determinant of $f$ as:

(5.10)
$\displaystyle\mathrm{det}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}X_{\Pi^{\infty}(\Gamma^{\prime})}}$
$\displaystyle(f)$ (5.11)
$\displaystyle:=\mathrm{det}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}X_{\Pi^{\infty}(\Gamma^{\prime})}}(\overline{f})\mathrm{char}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}X_{\Pi^{\infty}(\Gamma^{\prime})}}(M_{\mathrm{torsion}})\mathrm{char}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}X_{\Pi^{\infty}(\Gamma^{\prime})}}(N_{\mathrm{torsion}})^{-1}.$

On the corresponding derived level we consider after [Nakamura1, Section 3.4]
the following construction for the big exponential map:

$\mathrm{Exp}^{\Pi^{\infty}(\Gamma^{\prime}),\bullet}_{\mathcal{F},m}:R^{\bullet}\Gamma_{\mathrm{sheaf}}(\varprojlim\mathcal{F}(\Theta_{\mathrm{Ber,dif}}(M))_{p})\rightarrow
R^{\bullet}\Gamma_{\mathrm{sheaf}}(\varprojlim\mathcal{F}(M)_{p}).$

For each $p$ we consider the localized map:

$\mathrm{Exp}^{\Pi^{\infty}(\Gamma^{\prime})_{p},\bullet}_{\mathcal{F},m}:R^{\bullet}\Gamma_{\mathrm{sheaf}}(\mathcal{F}(\Theta_{\mathrm{Ber,dif}}(M))_{p})\rightarrow
R^{\bullet}\Gamma_{\mathrm{sheaf}}(\mathcal{F}(M)_{p}).$

Then what we can do is to define the corresponding determinant of the this by:

(5.12)
$\displaystyle\mathrm{det}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})_{p}}$
$\displaystyle(\mathrm{Exp}^{\Pi^{\infty}(\Gamma^{\prime})_{p},\bullet}_{\mathcal{F},m}):=$
(5.13)
$\displaystyle\mathrm{det}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})_{p}}(R^{1}\Gamma_{\mathrm{sheaf}}(\mathcal{F}(\Theta_{\mathrm{Ber,dif}}(M))_{p})\rightarrow
R^{1}\Gamma_{\mathrm{sheaf}}(\mathcal{F}(M)_{p}))\cdot$ (5.14)
$\displaystyle\mathrm{det}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})_{p}}(R^{2}\Gamma_{\mathrm{sheaf}}(\mathcal{F}(\Theta_{\mathrm{Ber,dif}}(M))_{p})\rightarrow
R^{2}\Gamma_{\mathrm{sheaf}}(\mathcal{F}(M)_{p}))^{-1}.$

Then we can organize this to be the corresponding determinant sheaf:

(5.15)
$\displaystyle\mathrm{det}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})}$
$\displaystyle(\mathrm{Exp}^{X_{\Pi^{\infty}(\Gamma^{\prime})},\bullet}_{\mathcal{F},m})=$
(5.16)
$\displaystyle\mathrm{det}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})}(R^{1}\Gamma_{\mathrm{sheaf}}(\mathcal{F}(\\{\Theta_{\mathrm{Ber,dif}}(M))_{p}\\}_{p})\rightarrow
R^{1}\Gamma_{\mathrm{sheaf}}(\\{\mathcal{F}(M)_{p}\\}_{p}))\cdot$ (5.17)
$\displaystyle\mathrm{det}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})}(R^{2}\Gamma_{\mathrm{sheaf}}(\mathcal{F}(\\{\Theta_{\mathrm{Ber,dif}}(M))_{p}\\}_{p})\rightarrow
R^{2}\Gamma_{\mathrm{sheaf}}(\\{\mathcal{F}(M)_{p}\\}_{p}))^{-1}$ (5.18)
$\displaystyle=\mathrm{det}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})}(R^{1}\Gamma_{\mathrm{sheaf}}(\mathcal{F}(\\{\Theta_{\mathrm{Ber,dif}}(M))_{p}\\}_{p})\rightarrow
R^{1}\Gamma_{\mathrm{sheaf}}(\\{\mathcal{F}(M)_{p}\\}_{p}))\cdot$ (5.19)
$\displaystyle\mathrm{char}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})}(R^{2}\Gamma_{\mathrm{sheaf}}(\mathcal{F}(\\{\Theta_{\mathrm{Ber,dif}}(M))_{p}\\}_{p}))^{-1}\cdot$
(5.20)
$\displaystyle\mathrm{char}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})}(R^{2}\Gamma_{\mathrm{sheaf}}(\\{\mathcal{F}(M)_{p}\\}_{p})).$

The following conjecture reads that in our current situation we have the
corresponding equivariant Iwasawa main conjecture. But we will only consider
the situation where $K=\mathbb{Q}_{p}$ and the corresponding full admissible
$p$-adic Lie extension after [LZ1] namely we consider
$K_{\infty}=\mathbb{Q}^{\mathrm{ur}}_{p}(\mu_{p^{\infty}})$. We first recall
Nakamura’s result in [Nakamura1]:

######  Theorem 5.10.

(Nakamura’s Iwasawa Main Conjecture) In our current notations, let
$\Gamma^{\prime}$ be trivial. Consider a de Rham $(\varphi,\Gamma)$ module $M$
over the usual Robba ring. We assume that this module has rank $g$ with $g$
Hodge-Tate weights $a_{1},...a_{g}$. Then we have the following formula
linking the determinant of the Perrin-Riou-Nakamura big exponential map and
the characteristic ideals in the equivariant way:

(5.21)
$\displaystyle\prod_{i}\prod^{g-a_{i}-1}_{b_{i}}\nabla_{a_{i}+b_{i}}=\mathrm{det}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})}$
$\displaystyle(R^{1}\Gamma_{\mathrm{sheaf}}(\mathcal{F}(\\{\Theta_{\mathrm{Ber,dif}}(M))_{p}\\}_{p})\rightarrow
R^{1}\Gamma_{\mathrm{sheaf}}(\\{\mathcal{F}(M)_{p}\\}_{p}))\cdot$ (5.22)
$\displaystyle\mathrm{char}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})}(R^{2}\Gamma_{\mathrm{sheaf}}(\mathcal{F}(\\{\Theta_{\mathrm{Ber,dif}}(M))_{p}\\}_{p}))^{-1}\cdot$
(5.23)
$\displaystyle\mathrm{char}_{\Pi^{\infty}(\Gamma_{K})\widehat{\otimes}\Pi^{\infty}(\Gamma^{\prime})}(R^{2}\Gamma_{\mathrm{sheaf}}(\\{\mathcal{F}(M)_{p}\\}_{p})).$

Then in this situation:

######  Conjecture 5.11.

(Equivariant Iwasawa Main Conjecture, after Nakamura) Consider a de Rham
$(\varphi,\Gamma)$ module $M$ over the usual Robba ring. We assume that this
module has rank $g$ with $g$ Hodge-Tate weights $a_{1},...a_{g}$. Then we
conjecture we have the corresponding analog of the formula linking the
determinant of the Perrin-Riou-Nakamura big exponential map and the
characteristic ideals in the equivariant way.

######  Remark 5.12.

This is also inspired by the work of [LVZ] and [LZ1], while we work in the
generality after [Nakamura1]. Although we have not mentioned explicitly but we
definitely conjecture that one has the parallel statement for Lubin-Tate
differential equations after [BerFou] in the equivariant setting.

### Acknowledgements

The paper grew along our study on the Hodge-Iwasawa theory initiated closely
after [KP], [KL1] and [KL2], which benefits from different perspectives from
the ideas and the inspiration from the essential innovations included in these
previous works. We would like to thank Professor Kedlaya for the corresponding
discussion along our preparation in different periods on deep foundations and
various possible extensions of [KP], [KL1] and [KL2].

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