Automated Trading Systems: Developed and Emerging Capital Markets

O. Hudak

Department of Corporate Finance, Faculty of Finance, Cesta na amfiteater 1, SK
- 974 01 Banska Bystrica, Slovak Republic

## Abstract

Automated trading systems on developed and emerging capital markets are
studied in this paper. The standard for developed market is automated trading
system with 40-days simple moving average. We tested it for the index SIX
Industrial for 1000 and 730 trading days. The Buy and Hold trading system was
7.80 times more profitable than this etalon trading system for active trading.
Simple taking of profitable standard trading system from a developed capital
market does not lead to optimal results on the emerging capital markets. We
then studied optimized standard trading system based on the simple moving
average. The parameter of optimization was the number of weeks. We took the
range from 1 to 100 weeks. An optimal system was that with 5 weeks. This
trading system has some of its characteristics comparable with the etalon
trading system on the NYSE Composite Index. The emerging market is more risky
than the developed market. The profit on the emerging market is also higher.
The range of optimized system parameter is quite robust. Observed was increase
of number of trades in the range from the 21 weeks to the 25 weeks. This
indicates creation of a new optimal middle range trading system. Results of
testing for liquid shares are quantitatively similar.

##  1 Introduction

Can active management of a portfolio of securities add value? That was the
question [1] which was discussed for developed capital markets. Those capital
markets have their properties, which are studied and understood to quite a
large extension [2]. There exist algorithmic prescriptions, called automated
trading systems, for trading on these markets. It means that there exist
algorithmic prescriptions generating buy and sell signals. These algorithmic
prescriptions are unambiguously defined rules based on fundamental and
technical analysis. They are taking into account investment characteristics of
an investor. While development in prices in future is different from that in
history, reaction of buyers and sellers on period of similar development on
the market is similar. To define these algorithmic prescriptions for the
nearest period it is necessary to test and to optimize these algorithmic
prescriptions on recent historical data. From similarity of reaction of buyers
and sellers on similar situations we obtain for using these algorithmic
prescriptions with tested and optimized parameters consistency. They are
robust. In longer periods it is necessary to combine these algorithmic
prescriptions with reoptimalisation of them, using for example a method of
shifting window. Main methods of using automated trading systems are as
analysis of development on the capital market and as an investment strategy,
where buyers and sellers are acting after every buy or sell signal.

This paper has the following sections: Automated Trading Systems, Etalon -
Developed Capital Market, Emerging Capital Markets, Automated Trading Systems
and Emerging Capital Markets, and Summary. We studied here automated trading
systems on developed and emerging capital markets. Main results are the
following ones. The standard for developed markets automated trading system
with 40-days simple moving average was used for the index SIX Industrial from
the emerging slovak capital market. It was tested for 1000 and 730 trading
days. It is not an optimal trading system for the emerging market due to the
fact that the Buy and Hold trading system was 7.80 times more profitable than
this etalon trading system for active trading. Simple taking of profitable
standard trading system from a developed capital market thus does not lead to
optimal results on the emerging capital markets. We then studied optimized
standard trading system based on the simple moving average. The parameter of
optimization was the number of weeks in the moving average. We took the range
from 1 to 100 weeks. An optimal system was that with 5 weeks. This trading
system then has some of its characteristics comparable with the etalon trading
system 40-days simple moving average trading on the NYSE Composite Index. The
emerging market is more risky than the developed market. The profit on the
emerging market is also higher. The range of optimized system parameter is
quite robust. Observed was increase of number of trades in the range of the 21
weeks to the 25 weeks. This indicates creation of new optimal middle range
trading system. Results of testing for liquid shares are quantitatively
similar to the results of testing for the SIX 30 Industrial index.

##  2 Automated Trading Systems

Advantages of automated trading systems are, [2], in automated decision
process, in discipline, in higher consistency of results with elimination of
emotions, automated elimination of loss to lower levels and automated using of
every profitable trend, and in finding every important trend. Disadvantages
are that no system is working universally good for every conditions on the
market, as the market develops its characteristics are changing and thus we
should not relay on historical data only. It may happen that buy and sell
signals may not be realized due to some other reasons (low liquidity, broker
inability to realize the trade, realization of the trade at a price completely
different from that signaled by the trading system, and other random events
which may lead to the situation in which buy and sell signals cannot be
realized). There are complementary methods to lower disadvantages of the
trading system, they are based on the following of and analyzing of several
automated trading systems and indicators simultaneously (almost all of these
trading systems should give the same trading signal and none of them should
give an opposite signal), new algorithms are looking for and tested
(artificial intelligence, etc.). To use automated trading systems it is
necessary to satisfy conditions like liquidity of the market with securities,
large set of historical data as concerning price (at least 5 years for middle
period range of trading or to have at least 30 generated trades during the
period which historical data cover) [3]. Normality of distribution of returns
[4] on emerging and developed markets should be verified. A user of the
automated trading systems should have defined its characteristics like
investment horizon, whether he/she is an aggressive or conservative investor.
It is necessary to know rate of returns for investment of free (not invested
on the capital market) finances to banks, and to be able to reinvest to the
same securities. Effective commission fees should be known. Stop-loss
characteristics for an investment should be known. An investor should has
enough reserve funds to survive the worst situations on the market. The
automated trading system should be defined unambiguously and should be simple.
Defining the algorithm of the trading system one should preserve basic
principles of financial modeling, [5]. There is a principle of independence, a
principle of minimization of constrains and a principle of top-down approach.

Automated trading systems should be compared with etalons of automated trading
systems. It is necessary to compare risk and performance of these systems.
System based on passive strategy Buy and Hold (BH) is one etalon. Another
etalon on developed capital markets is automated trading system based on
40-days moving average indicator. There are such criteria of risk for a given
trading system like the longest and the deepest fall of the price in an open
position (this is connected with reserve funds), number of trades (commission
fees), number of win trades (small number of win trades with large profit does
not mean that such a system would be consistent in the future).

##  3 Etalon - Developed Capital Market

As an etalon for a developed capital market we will use standard passive
reference automated system Buy and Hold. Standard active etalon automatic
trading system for developed capital market is 40-days simple moving average
system, [2]. We used 19-years (January 1977 - December 1996) period for index
NYSE Composite. For 40-days simple moving average automated trading system we
obtained profit 94.44 for this period. The etalon Buy and Hold system gives us
for the same period only 85.01 points. Number of trades for 40-days simple
moving average automated trading system was 57. It means 3.20 trades per year.
35 percents were profitable. The largest loss in points was 10.60 percents
from the level 145.32 of the index in the year 1986. The largest loss in
percents was 22.53 percents from the level 60.24. The index of risk for this
automated trading system defined as the ratio of total profit to the largest
loss has the value 6.13.

##  4 Emerging Capital Markets

An efficient capital market is that market in which every security price
equals to its investment value (NPV - Net Present Value) at all times.
Presence of the (log-) normality distribution of returns on effective markets
most of the time is that point which forms a base for analyzing returns on
diversified portfolios when the holding period is relatively short. The only
correct measures of returns and risk are mean return per a given period and
the standard deviation. It is an empirical evidence that the (log-) normal
distribution of returns is not present all the time with the same parameters
on efficient markets (trends are changing), and it is not true that such a
distribution of returns is present on efficient markets as such for given
every time periods. The aim of many econophysical models is to explain both
phenomena. However simple tests of presence of normality of distribution of
returns on efficient capital markets should be done first. It is not known
whether an emerging capital market is or it is not an efficient market in the
sense in which every security price equals its investment value at all times.
In the case of the emerging capital market testing of the presence of the
(log-) normal distribution is even more important than in the case of the
efficient capital market. It is necessary to find the answer to the questions:
what are the correct measures of mean returns and risk on emerging markets?
Recently we tested normality of distribution of returns on the slovak capital
market, which is an emerging capital market, within the period of years 1993
(the starting year of capital market trading) - 1996. We studied presence [4]
of normally distributed returns on the slovak capital market from the short
time point of view as concerning main capital market indexes, industrial
indexes, and for shares. We used a simple sign-test which was appropriately
modified. As an etalon we considered the capital market of the U.S.A. We have
found that main indexes of the slovak capital market (SAX, SEVIS 100, RMS F,
RMS I, SIX ALL, SIX 50, SIX 30 Industrial) and indexes of industries on the
same market, as well as those shares with the highest market capitalization,
have comparable values of the index measuring presence of normality of
distribution with those values of this index for the etalon market. On the
emerging market we have found an industrial structure as concerning presence
of the normality of returns. With increasing time and liquidity of shares on
the emerging market the presence of the normality of returns increases. From
our results it follows that the risk may be described by the standard
deviation of returns for main indexes of the slovak capital market and for
some of industrial indexes of the same market, as well as for those shares
which have the highest market capitalization.

##  5 Automated Trading Systems and Emerging Capital Markets

As an emerging capital market we used the slovak capital market. We tested
this market. Our aim was to find its characteristics for analysis, and for
trading (where however number of trading days is smaller for the emerging
market with respect to the period of 19 years of the etalon above). We used
for testing of automated trading systems the methodology from [3]. As an
market index we used the index SIX 30 Industrial. Further we tested a set of
liquid shares from shares traded on Bratislava Stock Exchange (where in the
period which we studied 777 different share titles were traded) and a set of
liquid shares from the RMS Slovakia over the counter market (where in the
period studied 639 different share titles were traded). We used two time
periods for the index and shares from the Bratislava Stock Exchange (1000 and
730 trading days) and one time period for shares traded on the RMS Slovakia
(730 trading days, there were not 1000 trading days on this market). We used
data from the beginning of trading on the Bratislava Stock Exchange in 1993.
End of the time period is on 31st August 1996. The time period 730 trading
days does not contain data from the beginning of trading until the spring of
1994\. We considered Long positions traded on the closing price last value in
a given week for shares and on the last value of the index for a given week in
trades only. Thus trading was on the Closing price for shares and on the value
of the index. Number of weeks which we used to wait to confirm the signal buy
or the signal sell was 0 weeks. Thus we traded aggressively. We considered
zero commission fees. We assumed that there existed possibility to reinvest
all of the investment again into the same share title or index.

Main results which we obtained were the following. For the standard automated
trading system with 40-days simple moving average for the index SIX Industrial
(1000 trading days) we have found rate of return (per annum) 3.68 percents.
There were 4 trades, one of them was profitable. The largest loss was 71.87
points. The ratio of total profit to the largest loss is 1.43 . The Buy and
Hold trading system was 7.80 times more profitable. Thus we can see that
simple procedure of taking profitable standard trading system from a developed
capital market and to use it on an emerging capital market does not lead to
optimal results on the emerging capital market. Or even it does not lead to
acceptable results on this market, see number of trades and number of win
trades which is very low.

We then optimized this standard trading system based on the simple moving
average. The parameter of optimization was the number of weeks. We took the
range from 1 to 100 weeks for the SIX 30 Industrial (1000 trading days trading
period). An optimal system was that with 5 weeks. It is 25 trading days. The
rate of return (per annum) was 89.59 percents. Number of trades was 13. There
were 4 win trades, e.i. 30.08 percents of trades. This number is comparable
with the etalon, the NYSE Composite Index. Number of trades per year is 3.71 .
This number is also comparable with the NYSE Composite etalon. The largest
loss was 120.30 points. The ratio of the total profit to the largest loss is
20.63, which is better than on the etalon market. The emerging market is
however more risky than the developed market. The Buy and Hold trading system
was 0.32 times less profitable. As the optimal range for active trading using
trading system based on the simple moving average we have found range from 3
to 7 weeks. This range is quite robust. We have also observed increase of
number of trades in the range of the 21 weeks to the 25 weeks using trading
system based on the simple moving average. This increase indicates creation of
a new optimal middle range trading system. Testing the same system on the same
index for 730 trading days we obtained quantitatively similar characteristics
as for 1000 trading days period. Results for liquid shares are also similar
quantitatively.

##  6 Discussion

In this paper we discussed basic automated trading systems for developed and
emerging capital markets. Can active management of a portfolio of securities
add value? That question [1] was discussed for developed capital markets.
Active management of portfolio uses usually automated trading systems. They
are algorithmic prescriptions generating buy and sell signals which may add
value. These rules are taking into account investment characteristics of an
investor. We used an empirical fact known from developed capital markets that
while development in prices in future is different from that in history,
reaction of buyers and sellers on period of similar development on the market
is similar. Thus it has sense to define these algorithmic prescriptions for
the nearest period on the market by testing and optimizing these algorithmic
prescriptions on recent historical data. From similarity of reaction of buyers
and sellers on similar situations we obtain prescriptions with parameters
which are consistent and robust. In longer periods these algorithmic
prescriptions are reoptimized, using the method of shifting window. Automated
trading systems are used mainly to analyze development on the capital market
and as an investment strategy, where buyers and sellers are acting after every
buy or sell signal. In section Automated Trading Systems we discussed
advantages and disadvantages of these systems. Automated trading systems may
be used when conditions like liquidity of the market with securities and large
set of historical data as concerning price are fulfilled. Normality of
distribution of returns on developed capital markets is usually present.
However it should be verified. It is known that this distribution is only
roughly describing distribution of returns, see for example [6]. We discuss
properties of developed capital markets as well as of emerging capital
markets. The empirical analysis of developed capital markets lead to
description of distribution of return on these markets by Truncated L vy
Flight distributions. It is not known whether this distribution is present on
emerging capital markets and it is difficult to verify presence of this
distribution on emerging capital markets due to their specific properties. We
used properties of normal distribution of returns which is most of the time
present on the developed and in some time periods on emerging capital markets,
as we verified this fact. A user of the automated trading systems has its
characteristics like investment horizon and whether he/she is an aggressive or
conservative investor. In test we are using in general rate of returns for
investment of free (not invested on the capital market) finances into banks.
It is assumed that there is present possibility to reinvest return into the
same securities. Effective commission fees should be known. We used in our
tests zero commission fees. We did not used stop-loss characteristics for
trading systems because the standard 40-days moving average method on
developed capital markets does not use them. It is assumed that an investor
has enough reserve funds to survive the worst situations on the market.
Automated trading system tested in our paper was compared with the etalon of
automated trading systems, with Buy and Hold system. Another etalon on
developed capital markets is automated trading system based on 40-days moving
average indicator mentioned above. We tested this etalon for a developed
capital market and for an emerging capital market. We have found that on the
emerging capital market this etalon is not a good etalon. The parameter of 40
days should be changed. Criteria of risk for a given trading system like the
longest and the deepest fall of the price in an open position, number of
trades and number of win trades may be used on emerging capital markets. We
used 19-years (January 1977 - December 1996) period for the NYSE Composite
Index. For 40-days simple moving average automated trading system we obtained
profit 94.44. The etalon Buy and Hold system gives us for the same period only
85.01 points. Thus active trading on developed capital market gives a value.
Number of trades for 40-days simple moving average automated trading system
was 57 during all the period. Thus 3.20 trades per year were found. 35
percents of the trades were profitable. The largest loss was 10.60 percents
from the level 145.32 of the index in the year 1986. The largest loss in
percents was 22.53 percents from the level 60.24. For this automated trading
system we have found the ratio of total profit to the largest loss with the
value 6.13.

On the efficient capital market every security price equals to its investment
value (NPV) almost all the time. This is not true on emerging capital markets.
Tests of presence of the (log-) normal distribution of returns on effective
markets and on emerging markets show that these markets are different. On
developed capital markets most of the time this presence is a base for
analyzing returns on diversified portfolios when the holding period is
relatively short (CAPM and similar models). Then the only correct measures of
returns and risk are mean return per a given period and the standard deviation
of returns. It is an empirical evidence that the (log-) normal distribution of
returns on developed capital markets is not present (with 95 percent
probability for a given time period, or with a similar probability) all the
time. Trends are changing on developed capital markets and thus the same
parameters of the normal distribution of returns on efficient markets cannot
be used for longer time periods and moreover it is not true that such as a
distribution of returns ((log-) normal)is present on efficient markets as such
for given time periods. Simple tests of presence of normality of distribution
of returns on efficient capital markets should be thus done first also. Many
of indicators in technical analysis and many models in fundamental analysis
are using the fact of presence of this distribution on developed capital
markets (in practical sense). On emerging capital markets security price does
not equal to its investment value (NPV) at time periods which are much more
frequent than those for the developed capital market. In the case of the
emerging capital market testing of the presence of the (log-) normal
distribution of returns is necessary for the same reasons as for developed
capital market. Moreover it is necessary to find the answer to such questions
like what are the correct measures of mean returns and risk on emerging
markets? Tests of presence of normal distribution of returns on the emerging
slovak capital market within the period of years 1993 (starting year of
capital market trading) to 1996 were performed in this paper. In this period
the market was certainly an emerging market, thus we do not study latter years
on this market in which it is approaching developed market properties more.
Results of our tests of presence of normally distributed returns, from the
short time point of view, as concerning main capital market indexes,
industrial indexes, and for shares for slovak capital market using a simple
sign-test of hypothesis of presence of this distribution with 95 percent
probability which was appropriately modified have shown that main indexes of
the slovak capital market (SAX, SEVIS 100, RMS F, RMS I, SIX ALL SIX 50, SIX
30 Industrial), and indexes of industries of the same market, as well as those
shares with the highest market capitalization, are reaching comparable values
of the index measuring presence of normality with those values of this index
for the etalon market (NYSE in USA). With increasing time and liquidity of
shares on the emerging market the presence of the normality of returns
increases. From our results it follows that the risk may be described by the
standard deviation of returns for main indexes of the slovak capital market
and for some of indexes for industries of the same market, as well as for
those shares which have the highest market capitalization. We did not tested
presence of the TLF distribution for the slovak capital market. Thus the
question whether an active management of a portfolio of securities can add
value may be extended to emerging capital markets. This was done in this
paper. As an emerging capital market we used the slovak capital market, as it
was mentioned. For testing we used the methodology used for testing developed
capital markets, [3]. We used the index SIX 30 Industrial as an market index.
We tested also a set of liquid shares from shares traded on Bratislava Stock
Exchange and a set of liquid shares from the RMS Slovakia over the counter
market. Two time periods for the index and shares from the Bratislava Stock
Exchange (1000 and 730 trading days) and one time period for shares traded on
the RMS Slovakia market (730 trading days) were studied. Data from the
beginning of trading on the Bratislava Stock Exchange in 1993 were used. The
time period 730 trading days does not contain data from the beginning of
trading until the spring of 1994\. Thus we eliminated trading days at the
beginning of trading on the market when prices on the market were developing
from those resulted from the coupon privatization to those resulting from
trading on the emerging capital market. We considered Long positions in trades
on the last Close price value in a given week for shares and on the last value
of the index for a given week in trades only. Short position trading is
forbidden on slovak capital market. Trading using software was done on the
Closing price for shares and on the value of the index. We traded as
aggressive investors, number of weeks which we used to wait to confirm the
signal buy or the signal sell was 0 weeks. This corresponds to the etalon for
active trading systems on developed capital markets which is 40-days simple
moving average trading system. Zero commission fees were considered. It was
assumed that there existed possibility to reinvest all of the investment
returns from a given period (from the period with open Long position) again
into the same share title or index. In its simplest form we did not gave money
into bank when Long position was closed. This simplification also correspond
to the mentioned etalon.

The standard etalon automated trading system with 40-days simple moving
average on the index SIX Industrial (1000 trading days) lead to the rate of
return (per annum) 3.68 percents. There were 4 trades, only one of them was
profitable. The largest loss was 71.87 points. The ratio of total profit to
the largest loss is 1.43 . The Buy and Hold trading system was 7.80 times more
profitable. The passive trading is more profitable than the active trading
using the standard etalon automated trading system from developed capital
markets. We see that it is not possible simply to take standard trading system
etalon from a developed capital markets and to use it for emerging capital
markets. It does not lead to optimal results on the emerging capital market.
It leads even to such unacceptable results on this market as low number of
trades and low number of win trades.

We then optimized this standard trading system. The parameter of optimization
was the number of weeks in the simple moving average. We took the range from 1
week to 100 weeks for the parameter for the SIX 30 Industrial index with 1000
trading days trading period. An optimal system was that with 5 weeks which
corresponds to 25 trading days. The rate of return (per annum) was 89.59
percents for this optimized trading system for this period. This is larger
than the rate of returns using Buy and Hold trading system. Number of trades
was 13. There were 4 win trades, e.i. 30.08 percents of trades. This number is
comparable with the etalon for active trading on the developed market on the
NYSE Composite Index. Number of trades per year is 3.71 . Again this number is
also comparable with the NYSE Composite etalon. The largest loss was 120.30
points. The ratio of the total profit to the largest loss is 20.63, which is
larger than that for trading using the etalon for developed market and trading
with this etalon system on the SIX 30 Industrial. The ratio is larger than
that on the developed market. The emerging market is more risky than the
developed market which corresponds to higher rate of returns on the emerging
market than on the developed market. The Buy and Hold trading system was 0.32
times less profitable than the trading using the optimized trading system. The
optimal range of the parameter for active trading using trading system based
on the optimized simple moving average is that range for which the results are
near the results for the optimized system. We have found that the optimal
range of the parameter is from 3 to 7 weeks with maximum for 5 weeks. This
range is quite robust around the optimal range of 5 weeks. As we so above the
40-days simple moving average trading systems is unacceptable in this case.

Observed increase of the number of trades in the range of 21 weeks to 25 weeks
using trading system based on simple moving average indicates creation of a
new optimal for middle range trading system. The same system tested on the
same index for 730 trading days gave quantitatively similar characteristics as
for 1000 trading days period. Thus characteristics of the emerging market were
similar whether we took into account the period of beginning of trading or
not. Results of testing for liquid shares are also quantitatively similar to
the results of testing optimized trading system on the index SIX 30
Industrial. This index contains in its portfolio liquid shares, so the
observed similarity is not surprising. Automated trading systems known from
developed capital markets do not have the same properties on emerging capital
markets. It is necessary for emerging capital markets to test these systems
again.

## Acknowledgements

Part of the results presented here was preliminary presented in the authors
talk given in slovak on the seminar Infofinex 1997 in Banska Bystrica. The
author would like to express his thanks to the organizers of seminar for
invitation. The paper was supported by the grant VEGA No. 1/0495/03.

## References

  * [1] Ambachtsheer K.P. and Farrell J.L., ”Can Active Management Add Value?”, Financial Analysts Journal (nov.-dec. 1979)]
  * [2] Pring M.J., Technical Analysis Explained, 3rd ed., McGraw-Hill, 1991
  * [3] Colby R.W. and Meyers T.A., The Encyclopedia of Technical Market Indicators, Business One IRWIN, Homewood, 1991
  * [4] Hudak O., ”Tests of normal distribution of returns presence on capital markets”, proc. ”Statistical methods in banking, finances and insurance”, VIII. Slovak Statistical Conference, Donovaly, 1996
  * [5] Miller R.M., Computer-Aided Financial Analysis, Addison-Wesley Pub.Comp., 1991
  * [6] Mantegna R.N. and Stanley H.E., An Introduction to Econophysics, Correlations and Complexity in Finance, Cambridge University Press, Cambridge, 2000


# Multiple products of meromorphic functions

A. Zuevsky  Institute of Mathematics
Czech Academy of Sciences
Zitna 25, Prague
Czech Republic<EMAIL_ADDRESS>
###### Abstract.

Let $\mathfrak{g}$ be an infinite-dimensional Lie algebra, and $G$ be the
algebraic completion of a $\mathfrak{g}$-module. Using the geometric model of
Schottky uniformization of Riemann sphere to obtain a higher genus Riemann
surface, we construct a family of parametric extensions of coboundary
operators for the double complexes of meromorphic functions depending on
elements of $G$, and having prescribed analytic properties.

AMS Classification: 53C12, 57R20, 17B69, 46L87, 46M20, 47B34

###### Key words and phrases:

Cochain complexes, meromorphic functions with defined analytic properties,
Schottky uniformization

Conflict of Interest and Data availability Statements:

The authors state that:

1.) The paper does not contain any potential conflicts of interests.

2.) The paper does not use any datasets. No dataset were generated during
and/or analysed during the current study.

3.) The paper includes all data generated or analysed during this study.

4.) Data sharing is not applicable to this article as no datasets were
generated or analysed during the current study.

5.) The data of the paper can be shared openly.

6.) No AI was used to write this paper.

##  1\. Introduction

The ordinary notion of cohomology of infinite-dimensional Lie algebras was
described in [13]. The purpose of this paper is to enlarge the set of
coboundary operators for double complex spaces of meromorphic functions in
several complex parameters having specific analytic properties. The
construction developed here enriches the structure of cohomology of manifolds.
We propose new formula for the coboundary operators depending on $\kappa$
parameters and brings about to another sophisticated structure of cohomology
spaces. The family of extensions presented leads to a development of the
characteristic classes theory for smooth complex manifolds and their
foliations [2, 3].

The spaces for double complexes we use are formed by the set of meromorphic
functions depending on elements of the algebraic completion of representation
spaces for infinite-dimensional Lie algebras. For meromorphic functions of
double complex spaces we axiomatically require certain analytic properties
with respect to a number complex parameters defined on a configurations space.
In the cohomology theory for vertex algebras [5, 10, 23, 20, 16] the similar
role is played by rational functions given by matrix elements that correspond
to the simplest auxiliary geometry model given by Riemann sphere. In that
correspondence, parameters of matrix elements are associated with local
coordinates on the geometric model.

In order to introduce geometrically motivated parametric extensions of chain
complexes of meromorphic functions it is natural to pass to more advanced
geometric model provided by Schottky uniformization procedure [4, 9, 24, 31]
to form a higher genus Riemann surface. The main result of the paper contains
in Proposition 1 where properties of the above mentioned extensions are
proven. A family of coboundary operators is constructed, and its convergence
is proven, according to geometric property of this model. In the geometric
model, similar to the case of the self-sewing procedure [30] to obtain a torus
from Riemann sphere, multiple handles are attached to construct a genus
$\kappa$ Riemann surface. The extended coboundary operators act on the same
space of meromorphic functions, and are then given by sums of products of the
action of certain operators on meromorphic functions. In this procedure, the
sum of meromorphic functions depending on a number of $\mathfrak{g}$-module
extension elements transform to trace functions representing corresponding
characters [20].

It has now been shown in [14] that sewn conformal blocks are absolutely and
locally uniformly convergent for simple self-dual $C_{2}$-cofinite vertex
operator algebra of CFT type. The problem of convergence of meromorphic
functions with specified properties is important in many problems of
functional analysis and operator theory. It appears in considerations of
recent results related to conformal field theory [14, 12, 27], non-commutative
geometry, foliation theory [2, 3, 15, 7, 1], cohomology of smooth manifolds
[8, 11], applications of infinite-dimensional Lie algebras [28, 25, 6],
K-theory of operator algebras [22, 7], noncommutative differential geometry
[22], and noncommutative dynamical systems [15, 1, 19]. In particular, it is
possible to compute higher cohomology classes for foliations of smooth
manifolds using multiple products introduced in this paper.

##  2\. Meromorphic functions with predetermined analytic properties

In this Section the space of predetermined meromorphic functions is
introduced. Defined as meromorphic functions with specific analytic properties
on a configuration space, that functions depend on certain series in several
complex variables.

Let $\mathfrak{g}$ be an infinite-dimensional Lie algebra, (in particular, a
Kac-Moody affine Lie algebra [20]). Let us assume that $\mathfrak{g}$ has a
representation $W$ (e.g., vertex operators representation) in terms of Laurent
series expanded in powers of a complex parameter $z\in\mathbb{C}$ with
coefficients $w(l)\in W$, where $W=\bigoplus_{l\in\mathbb{C}}W_{(l)}$ is
$\mathbb{C}$-graded with respect to a grading operator $K$. Let us assume that
$W_{(l)}=0$, when $l$ is sufficiently negative, and $\dim W_{(l)}<\infty$.
Following [16], we consider the algebraic completion
$G=\prod_{l\in\mathbb{C}}W_{(l)}$ of a $\mathfrak{g}$-representation space
$W$, which is a direct product allowing infinite sums. For $G$ we assume
existence of the highest state $\mathbf{1}_{G}\in G$ and a non-degenerate
bilinear pairing $(.,.)$ normalized by the property
$(\mathbf{1}_{G},\mathbf{1}_{G})=1$. Let
$G^{\prime}=\coprod_{l\in\mathbb{C}}W_{(l)}^{*}$ denote the dual with respect
to a non-degenerate pairing $(.,.)$. An element dual to $g\in G$ will be
denoted by $\overline{g}$. Assume that $G$ is equipped with a map
$\omega_{g}:G\to G[[(z,z^{-1}]]$, i.e., for each $g\in G$ and
$z\in\mathbb{C}$, we associate a complex series
$g\mapsto\omega_{g}(z)=\sum_{l\in\mathbb{Z}}g(l)\;z^{-l-1}$, $g(l)\in G$. For
$g$, $\widetilde{g}\in G$, $\omega_{g}(z)\widetilde{g}$ contains only finitely
many negative power terms, that is, $\omega_{g}(z)\widetilde{g}\in G$. For
$g\in G$, $g$, $\widetilde{g}\in G$, $z\in\mathbb{C}$, $g_{l}\widetilde{g}=0$,
$l\gg 0$, $\omega_{\bf 1}(z)={\rm Id}$. Denote by $T$ the translation operator
[20] satisfying the derivation property
$\omega_{Tg}(z)=\frac{d}{dz}\omega_{g}(z)$. To prove convergence, we would
also require that the codimension of the subspace
$\widetilde{W}_{(2)}=\left\\{g_{(-2)}\widetilde{g}\right\\}$ in $W$ for all
$g$, $\widetilde{g}\in W$, is finite. In particular, for Kac-Moody algebras
[5, 21], the structure and properties above have been absorbed into a standard
construction of a $C_{2}$-cofinite vertex algebra of CFT type with a Kac-Moody
algebra endowed with a graded Fock space generated by the universal enveloping
algebra $U({\mathfrak{g}})$ acting on a highest weight vector. Denote by
$F_{n}\mathbb{C}$ the configuration space of $n\geq 1$ ordered coordinates in
$\mathbb{C}^{n}$,
$F_{n}\mathbb{C}=\\{(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}\;|\;z_{i}\neq
z_{j},i\neq j\\}$. For each $(z_{1},\dots,z_{n})\in F_{n}\mathbb{C}$, and
$g_{1},\dots,g_{n}\in G$, and a linear function
$f(g_{1},z_{1};\ldots;g_{n},z_{n})\in G[[(z,z^{-1}]]$, we consider functions
$\mathcal{F}:G[[(z,z^{-1}]]\to\mathbb{C}$,
$\mathcal{F}(g_{1},z_{1};\ldots;g_{n},z_{n})=\mathcal{F}(f)$.

Similar to the case of matrix elements for grading-restricted vertex algebras,
we require the following properties for $\mathcal{F}$ described below. Poles
of $\mathcal{F}$ are only allowed at at $z_{i}=z_{j}$, $i\neq j$. It is
assumed that $\mathcal{F}(g_{1},z_{1};\ldots;g_{n},z_{n})$, is absolutely
convergent in the region $|z_{1}|>\ldots>|z_{n}|>0$, to a function in
$(z_{1},\dots,z_{n})$, The locality and associativity properties are assumed
for $\mathcal{F}$. I.e., for $g_{1}$, $g_{2}$, $g\in G$, the meromorphic
functions $\mathcal{F}\left(g_{1},z_{1};g_{2},z_{2}\right)$,
$\mathcal{F}\left(g_{2},z_{2};g_{1},z_{1}\right)$,
$\mathcal{F}\left(\omega_{g_{1}}(z_{1}-z_{2})g_{2},z_{2}\right)$, are
absolutely convergent in the regions $|z_{1}|>|z_{2}|>0$, $|z_{2}|>|z_{1}|>0$,
$|z_{2}|>|z_{1}-z_{2}|>0$, respectively, to a common meromorphic function in
$z_{1}$ and $z_{2}$. The poles are only allowed at $z_{1}=0=z_{2}$,
$z_{1}=z_{2}$.

Let us introduce the notation of the action of an operator $V$ on
$\mathcal{F}$ as $V$. $\mathcal{F}(g_{1}$,
$z_{1};\ldots;g_{n},z_{n})=\mathcal{F}(V.g_{1},V.z_{1};\ldots;V.g_{n},V.z_{n})$.
It is assumed that an operator $V$ may act both on $G$-elements and $z_{i}$
$1\leq i\leq n$. Denote by $(V)_{i}.$ the operator acting on the $i$-th entry
of $\mathcal{F}$. For $V=\left\\{e^{zT},z^{K},\sigma\right\\}$, for $\sigma\in
S_{n}$, and as follows. For $V=\sigma\in S_{n}$, an element of the symmetric
group, let us define
$\sigma.\mathcal{F}(g_{1},z_{1};\ldots;g_{n},z_{n})=\mathcal{F}\left(g_{\sigma(1)},z_{\sigma(1)};\ldots;g_{\sigma(n)},z_{\sigma(n)}\right)$.
For $z\in\mathbb{C}$, let

$\displaystyle
e^{zT}\mathcal{F}(g_{1},z_{1};\ldots;g_{n},z_{n})=\mathcal{F}(g_{1},z_{1}+z;\ldots;g_{n},z_{n}+z),$
(2.1)

where it is assumed that both sides of the expression above are absolutely
convergent on the open disk $|z|<\min_{i\neq j}\\{|z_{i}-z_{j}|\\}$, and equal
as power series expansions in $z$. A meromorphic $G$-function has $K$-property
if for $z\in\mathbb{C}^{\times}$ satisfies $(zz_{1},\ldots,zz_{n})\in
F_{n}\mathbb{C}$, and

$\displaystyle
z^{K}\mathcal{F}(x_{1},\ldots,x_{n})=\mathcal{F}\left(z^{K}g_{1},zz_{1};\ldots;z^{K}g_{n},zz_{n}\right).$
(2.2)

We then define the action of partial derivatives on an element
$\mathcal{F}(g_{1},z_{1};\ldots;g_{n},z_{n})$

$\displaystyle\partial_{z_{i}}\mathcal{F}(g_{1},z_{1};\ldots;g_{n},z_{n})$
$\displaystyle=$
$\displaystyle\mathcal{F}((T)_{i}\;(g_{1},z_{1};\ldots;g_{n},z_{n})),$
$\displaystyle\sum\limits_{i\geq
1}\partial_{z_{i}}\mathcal{F}(g_{1},z_{1};\ldots;g_{n},z_{n})$
$\displaystyle=$ $\displaystyle
T.\mathcal{F}(g_{1},z_{1};\ldots;g_{n},z_{n}).$ (2.3)

For $s\geq 1$ and $1\leq q\leq s-1$, let $J_{s;q}$ be the set of elements of
the group $S_{s}$ which preserve the order of the first $p$ numbers and the
order of the last $s-q$ numbers, i.e., $J_{s,q}=\\{\sigma\in
S_{s}\;|\;\sigma(1)<\cdots<\sigma(q),\;\sigma(q+1)<\cdots<\sigma(s)\\}$.
Denote by $J_{s;q}^{-1}=\\{\sigma\;|\;\sigma\in J_{m;q}\\}$. In what follows,
we apply the condition

$\sum_{\sigma\in
J_{n;q}^{-1}}(-1)^{|\sigma|}\mathcal{F}(g_{\sigma(1)},z_{\sigma(1)};\ldots;g_{\sigma(n)},z_{\sigma(n)})=0.$
(2.4)

on certain meromorphic functions.

Denote by $P_{k}:G\to W_{(k)}$, $k\in\mathbb{C}$, the projection of $G$ on
$W_{(k)}$. Following [16] (see also [17]), we formulate the following
definition. For $i$, $j=1,\dots,(l+k)n$, $k\geq 0$, $i\neq j$, $1\leq
l^{\prime},l^{\prime\prime}\leq n$, let $(l_{1},\ldots,l_{n})$ be a partition
of $(l+k)n=\sum\limits_{i\geq 1}l_{i}$, and $k_{i}=\sum_{j=1}^{i-1}l_{j}$. For
$\zeta_{i}\in\mathbb{C}$, define
$h_{i}=f(g_{k_{1}+l_{1}},z_{k_{1}+l_{1}}-\zeta_{i};\ldots
g_{k_{l}+l_{l}},z_{k_{l}+l_{l}}-\zeta_{i};\mathbf{1}_{G})$, for $i=1,\dots,l$.
In addition to that, it is assumed that the function
$\sum\limits_{(r_{1},\ldots,r_{l})\in\mathbb{Z}^{l}}\mathcal{F}(P_{r_{1}}h_{1},\zeta_{1};\ldots;P_{r_{l}}h_{l},\zeta_{l})$,
is absolutely convergent to an analytically extension in
$(z_{1},\ldots,z_{n})_{l+k}$ in the domains
$|z_{k_{i}+p}-\zeta_{i}|+|z_{k_{j}+q}-\zeta_{j}|<|\zeta_{i}-\zeta_{j}|$, for
$i$, $j=1,\dots,k$, $i\neq j$, and for $p=1,\dots$, $l_{i}$, $q=1$, $\dots$,
$l_{j}$. The convergence and analytic extension do not depend on complex
parameters $(\zeta)_{l}$. On the diagonal of $(z_{1},\ldots,z_{n})_{l+k}$ the
order of poles is bounded from above by prescribed positive numbers
$\beta(g_{l^{\prime},i},g_{l^{\prime\prime},j})$. For $g_{1},\ldots,g_{l+k}\in
G$, $z_{i}\neq z_{j}$, $i\neq j$ $|z_{i}|>|z_{s}|>0$, for $i=1,\dots,k$,
$s=k+1,\dots,l+k$ the sum $\sum_{q\in\mathbb{C}}$
$\mathcal{F}(\omega_{g_{1}}(z_{1})\ldots\omega_{g_{k}}(z_{k})P_{q}(\omega_{g_{1+k}}(z_{1+k})\ldots\omega_{g_{l+k}}(z_{l+k})))$,
is absolutely convergent and analytically extendable to a meromorphic function
in variables $(z_{1},\ldots,z_{n})_{l+k}$. The order of pole that is allowed
at $z_{i}=z_{j}$ is bounded from above by the numbers
$\beta(g_{l^{\prime},i},g_{l^{\prime\prime},j})$.

A particular class of $G$-meromorphic functions is given by matrix elements
$\mathcal{F}(g_{1},z_{1}$ ; $\ldots g_{n},z_{n})$ $=$
$\left(\overline{g},f(g_{1},z_{1};\ldots;g_{n},z_{n})\right)$ for an arbitrary
$\overline{g}$ in the dual space. It is well-known that every vertex algebra
module matrix element is a rational function on Riemann sphere [10, 23]. We
call an $n$ complex parameters $(z_{1},\ldots,z_{n})$ $G$-meromorphic function
$\mathcal{F}(g_{1},z_{1};\ldots;g_{n},z_{n})$ with analytical behavior on a
$F_{n}\mathbb{C}$-domain defined above a predetermined meromorphic function.

Finally, for $\zeta\in\mathbb{C}$ we introduce the action of $G$-elements $f$
of the form $f(g_{1},z_{1}$ ; $\ldots;g_{m},z_{m};w,\zeta)$ on a predetermined
meromorphic function $\mathcal{F}$ by the action of $\omega_{g}$-series with
complex parameters associated to coordinates of insertion of a handle to the
sphere:

$\displaystyle
f(g_{1},z_{1};\ldots;g_{m},z_{m};w,\zeta)*\mathcal{F}(g_{m+1},z_{m+1};\cdots;g_{m+n},z_{m+n})$
(2.5)
$\displaystyle\quad=\mathcal{F}(\omega_{g_{1}}(z_{1})\ldots\omega_{g_{m}}(z_{m})\;f(g_{m+1},z_{m+1};\ldots;g_{m+n},z_{m+n};w,\zeta)).$

For two vectors ${\bf Y}=(Y_{1},Y_{2},Y_{3})$ and $\widetilde{\bf
Y}=(\widetilde{Y}_{1},\widetilde{Y}_{2},\widetilde{Y}_{3})$ let

${\bf Y}*\widetilde{\bf
Y}=\left(Y_{1}*\widetilde{Y}_{1},Y_{2}*\widetilde{Y}_{2},Y_{3}*\widetilde{Y}_{3}\right),$
(2.6)

be the multiplication with respect to $*$. The action (2.5) allows to define
the coboundary operators for double complexes constructed an infinite Lie
algebra module $G$.

For fixed $m$, and $n\in\mathbb{Z}_{+}$, let
$C_{m}^{n}=C_{m}^{n}(\mathfrak{g},G)$ be the vector space of all predetermined
meromorphic functions $\mathcal{F}(g_{1},z_{1};\ldots;g_{n},z_{n})$ compatible
with $m$ $G$-valued series $\omega_{g^{\prime}_{j}}(z^{\prime}_{j})$, $1\leq
j\leq m$, for $(g^{\prime}_{1},\ldots,g^{\prime}_{m})\in G$,
$(z^{\prime}_{1},\ldots,z^{\prime}_{m})\in\mathbb{C}$, and satisfying
conditions (2.1), (2.2), and (2.4). Let $C_{m}^{0}=G$. Then we have
$C_{m}^{n}\subset C_{m-1}^{n}$. Let us introduce further notations
$f^{(1)}(w_{p})$ $=$
$f(\overline{w}_{p},\eta_{1,p};g_{1},z_{1};w_{p},\eta_{2,p})$,
$f^{(2)}(w_{p})$ $=$ $f(\overline{w}_{p},\eta_{1,p};w_{p},\eta_{2,p})$,
$\sigma_{n+1,1,\ldots,n}$ $f^{(1)}(w_{p})=(-1)^{n+1}$
$f\left(\overline{w}_{p},\eta_{1,p};g_{n+1},z_{n+1}\right.$;
$\left.w_{p},\eta_{2,p}\right)$.

##  3\. The extension of the meromorphic function cochain complex

In this Section we use the auxiliary geometric model of the Schottky
uniformization of Riemann sphere to introduce an extension of the classical
coboundary operator to a family of coboundary operators for the cohomology of
predetermined meromorphic functions defined on $F_{n}\mathbb{C}$.

Let us recall the set up for self-sewing the complex sphere [30]. Consider the
construction of a torus $\Sigma^{(1)}$ formed by self-sewing a handle to
Riemann sphere $\Sigma^{(0)}$. This is given by Yamada formalism [30], or so-
called $\rho$-formalism. Let $z_{1}$, $z_{2}$ be local coordinates in the
neighborhood of two separated points $p_{1}$ and $p_{2}$ on the sphere.
Consider two disks $\left|z_{a}\right|\leq r_{a}$, for $r_{a}>0$ and $a=1$,
$2$. Note that $r_{1}$, $r_{2}$ must be sufficiently small to ensure that the
disks do not intersect. Introduce a complex parameter $\rho$ where $|\rho|\leq
r_{1}r_{2}$, and excise the disks
$\\{z_{a}:\,\left|z_{a}\right|<|\rho|r_{\bar{a}}^{-1}\\}\subset\Sigma^{(0)}$,
to form a twice-punctured sphere
$\widehat{\Sigma}^{(0)}=\Sigma^{(0)}\backslash\bigcup_{a=1,2}\\{z_{a}:\,\left|z_{a}\right|<|\rho|r_{\bar{a}}^{-1}\\}$.
We use the convention $\bar{1}=2$, $\bar{2}=1$. We define annular regions
$\mathcal{A}_{a}\subset\widehat{\Sigma}^{(0)}$ with
$\mathcal{A}_{a}=\left\\{z_{a}:\;|\rho|r_{\bar{a}}^{-1}\leq\left|z_{a}\right|\leq
r_{a}\right\\}$ and identify them as a single region
$\mathcal{A}=\mathcal{A}_{1}\simeq\mathcal{A}_{2}$, via the sewing relation

$z_{1}z_{2}=\rho,$ (3.1)

to form a torus
$\Sigma^{(1)}=\widehat{\Sigma}^{(0)}\backslash\\{\mathcal{A}_{1}\cup\mathcal{A}_{2}\\}\cup\mathcal{A}$.
The sewing relation (3.1) is a parameterization of a cylinder connecting the
punctured Riemann surface to itself.

A genus $\kappa$ Riemann surface is formed by sewing of $\kappa$ handles to
the complex sphere. To give an example of this, recall the Schottky
uniformization procedure. We identify $\kappa$ pairs of non-intersecting
annuli centered at points $A_{\pm p}\in\widehat{\mathbb{C}}$, for $1\leq
p\leq\kappa$, and sewing parameters $\rho_{p}$ satisfying the sewing condition

$(z-A_{-p})(\widetilde{z}-A_{p})=\rho_{p}.$ (3.2)

As we will see below, the extension of the cochain complexes by a parametric
family of coboundary operators acting on the space of predetermined
meromorphic functions on $F_{n}\mathbb{C}$ is motivated by geometric nature of
Schottky uniformization procedure of forming a genus $\kappa$ Riemann surface.
Consider the following function
$\widetilde{\delta}^{n}_{m}\mathcal{F}=\mathcal{F}(\delta^{n}_{m}f)$,
$\mathcal{F}(f)\in C^{n}_{m}$, corresponding to the action of the coboundary
operator $\delta^{n}_{m}$ on $f\in G$:

$\displaystyle\widetilde{\delta}^{n}_{m}\mathcal{F}(f)$ $\displaystyle=$
$\displaystyle\sum\limits_{{w_{p}\in W_{(k)};\;k\geq
0}}\;\;\prod_{p=1}^{\kappa}\rho^{k}_{p}\;\left(f^{(1)}(w_{p}),(-1)^{n+1}\sigma_{n+1,1,\ldots,n}\;f^{(1)}(w_{p}),f^{(2)}(w_{p})\right)$
(3.3)
$\displaystyle\;\;*\left(f,f,\sum_{i=1}^{n}(-1)^{i}f(\ldots;\omega_{g_{i}}(z_{i}-\zeta_{i})\;\omega_{g_{i+1}}(z_{i+1}-\zeta_{i})\mathbf{1}_{G};\ldots)\right),$

where the product $*$ of two vectors was defined in (2.6). In the proof of
Proposition 1 we will see that (3.3) gives a predetermined meromorphic
function on $F_{n+1}\mathbb{C}$. In (3.3) $w_{p}\in W_{(n)}$,
$p=1,\ldots,\kappa$, and $\eta_{1,p}$, $\eta_{2,p}\in\mathbb{C}$. The
parameters $\eta_{a,p}$, $a=1$, $2$ correspond to coordinates of the auxiliary
geometric model of $p$-th pair of points on the sphere where $p$-th cylinder
is attached to form a genus $\kappa$ Riemann surface in Schottky procedure.

Here we have to make the following remarks. In this paper we consider the
simplest way to obtain a family of coboundary operators of the form (3.3) with
equal powers of $\rho_{p}^{k}$ for $1\leq p\leq\kappa$ in the sum. Note that
in (3.3) is not performed over a basis of $W_{(k)}^{\otimes\kappa}$ but over
just one copy of the basis of $W_{(k)}$. That allows us to remain in the frame
of the geometric model we have chosen. In (3.3) we make use the recipy of a
coboundary of operator construction based on the geometric model related to so
called canonical Schottky uniformization (3.2) (see, e.g., [29]). The
dependence of $\widetilde{\delta}^{n}_{m}\mathcal{F}$ on $A_{\pm p}$
parameters is somewhat implicit. The convergence for the coboundary operator
(3.3) discussed in [16] (see also [18]) for the case $\kappa=1$, generalizes
for $\kappa>1$. In the proof of convergence of (3.3) to a well-defined
meromorphic function with predetermined properties we use the Schottky
uniformization model to insure well-defined behavior of the resulting
function. A more general situation discussing convergence and well-behavior
problem for products of of the classical coboundary operators, the main
approach is the construction of differential equations that products and
approximations by using Jacobi identity. For the cohomology theory of vertex
algebras, such techniques do not work because cochains do not satisfy Jacobi
identity. We will apply the general constructions of [14, 18] to study
properties of products of coboundary operators somewhere else. More advanced
construction related to standard Schottky parameter construction (see, e.g.,
[28]) containing summation over $\kappa$ separate basis in the components of
$G^{\otimes\kappa}$ with grading parameters $k_{i}$, $1\leq i\leq\kappa$, and
corresponding powers $k_{i}$ in $\rho_{i}^{k_{i}}$, as well as a proof of
convergence, will be given elsewhere.

Let $g_{1},\dots,g_{n+1}\in G$, and $(z_{1},\dots,z_{n+1})\in
F_{n+1}\mathbb{C}$, and let us denote
$\varphi_{1,n}=f(g_{1},z_{1};\ldots;g_{n},z_{n})$,
$\varphi_{i}=f(g_{1},z_{1};\ldots;g_{i-1},z_{i-1};\omega_{g_{i}}(z_{i}-\zeta_{i})\;\omega_{g_{i+1}}(z_{i+1}-\zeta_{i})\mathbf{1}_{G})$;
$g_{i+2},z_{i+1};\ldots;g_{n+1},z_{n+1})$,
$\varphi_{2,n+1}=f(g_{2},z_{2};\ldots;g_{n+1},z_{n+1})$.

We consider (3.3) with the action defined in (2.5) with $\varphi_{i}$
independent of $\zeta_{i}$. When we take $\zeta_{i}=z_{i+1}$ for
$i=1,\dots,n$, we obtain
$\widetilde{\delta}^{n}_{m}\mathcal{F}=\sum\limits_{w_{p}\in W_{(k)},\;k\geq
0}\;\prod_{p=1}^{\kappa}\rho_{p}^{k}\;\mathcal{F}(\phi)$, with

$\displaystyle\phi=\omega_{h_{p}}(\eta_{1,p})(\omega_{g_{1}}(z_{1}-\eta_{2,p})e^{-\eta_{2,p}T}\varphi_{2,n+1}e^{\eta_{2,p}T}+\sum_{i=1}^{n}(-1)^{i}e^{-\eta_{2,p}T}\varphi_{i}e^{\eta_{2,p}T}$
$\displaystyle\qquad\qquad+(-1)^{n+1}\omega_{g_{n+1}}(z_{n+1}-\eta_{2,p})e^{-\eta_{2,p}T}\varphi_{1,n}e^{\eta_{2,p}T})\omega_{\mathbf{1}_{G}}(-\eta_{2,p})w_{p}).$

Since (3.3) contains a sum over just one copy copies of $W_{(k)}$-basis, by
performing summation for all $w_{p}\in G_{(k)}$ to obtain trace function over
$G$. Using property (2.1) of maps $\mathcal{F}$ we finally find that

$\displaystyle\widetilde{\delta}^{n}_{m}\mathcal{F}=\sum\limits_{k\geq
0}\prod_{p=1}^{\kappa}\rho_{p}^{k}\;\mathrm{Tr}_{G}\;\delta^{n}_{m}f_{p},$
(3.4)
$\displaystyle\delta^{n}_{m}f_{p}=f(\omega_{g_{1}}(z_{1}-\eta_{2,p});\varphi_{2,n+1})$
(3.5)
$\displaystyle\qquad+\sum_{i=1}^{n}(-1)^{i}f(\varphi_{i})+(-1)^{n+1}f(\omega_{g_{n+1}}(z_{n+1}-\eta_{2,p});\varphi_{1,n}).$

Thus, we see that the correspondence $f\mapsto\mathcal{F}:\mathcal{F}(f)$
transforms to the correspondence (3.4). We now notate the action of operators
$V$ on $\widetilde{\delta}^{n}_{m}\mathcal{F}$ in the following form. For
$V=\left\\{e^{zT},z^{K}\right\\}$ we define

$V.\left(\widetilde{\delta}^{n}_{m}\mathcal{F}\right)=\sum\limits_{k\geq
0}\;\prod_{p=1}^{\kappa}\rho_{p}^{k}\;\mathrm{Tr}_{G}\left(V_{\delta_{V,\sigma}\sigma_{p}}\right).\delta^{n}_{m}f_{p}.$
(3.6)

where $(V_{p(k)}).$ acts on one consequent multiplicand in
$\delta^{n}_{m}f_{p}$ only. Then the action of partial derivatives on
$\delta^{n}_{m}\mathcal{F}$ remains the same as in (2). We then formulate the
main result of this paper

######  Proposition 1.

For $(\eta_{1,p}$, $\eta_{2,p})\in\mathbb{C}$, $p=1,\ldots,\kappa$,
$\zeta_{i}\in\mathbb{C}$, $i=1,\ldots,n$, $\zeta_{1,p}\zeta_{2,p}=\rho_{p}$,
the parametric family (3.3) of coboundary operators
$\widetilde{\delta}^{n}_{m}=\delta^{n}_{m}(\rho_{1},\ldots,\rho_{\kappa};\eta_{2,1}\ldots,\eta_{2,\kappa})$
for $\mathcal{F}\in C_{m}^{n}(\mathfrak{g},G)$ extends the chain double
complex $(C^{n}_{m},\delta^{n}_{m})$ to
$(C^{n}_{m},\widetilde{\delta}^{n}_{m})$ with
$\widetilde{\delta}^{n}_{m}:C_{m}^{n}\to C_{m-1}^{n+1}$, and
$\widetilde{\delta}^{n+1}_{m-1}\circ\widetilde{\delta}^{n}_{m}=0$.

With the coboundary operator (3.3) we define the ${}^{n}_{m}$-th cohomology
$H^{n}_{m}(\mathfrak{g},G)$ of the complex
$\left(C_{m}^{n},\widetilde{\delta}^{n}_{m}\right)$ in the standard way. In
[14] it was established that the correlation functions for a $C_{2}$-cofinite
vertex operator algebra of CFT-type are absolutely and locally uniformly
convergent on the sewing domain since it is a multiple sewing of correlation
functions associated with genus zero conformal blocks. Though one can make use
of the results of [14] to prove Proposition (1) we give here its alternative
proof.

###### Proof.

We first prove that $\widetilde{\delta}^{n}_{m}\mathcal{F}$ belongs to
$C^{n+1}_{m-1}$. Let us first show that (3.3) is convergent to a predetermined
meromorphic function on $F_{n+1}\mathbb{C}$ compatible with $(m-1)$
$\omega$-series. For that purpose we use the construction of Schottky
uniformization of Riemann sphere to form a genus $\kappa$ Riemann surface. It
is an auxiliary geometric model space defining the coboundary operator (3.3).
Let us choose $\kappa$ pairs of points on initial Riemann sphere and denote by
$x_{p}$, $x^{\prime}_{p}$ the centers of non-intersecting annuli
$\mathcal{A}_{1,p}$ and $\mathcal{A}_{2,p}$. Complex parameters $\eta_{1,p}$,
$\eta_{2,p}$ in $1\leq p\leq\kappa$-multipliers in (3.3) correspond in the
auxiliary geometric model to local coordinates for insertion of $\kappa$
handles. The form of the coboundary operator (3.3) and the procedure of
described in this proof above shows the algebraic side of the procedure of
self-serving Riemann sphere to form a torus. According to [20], the converging
functions represent characters for $\mathfrak{g}$ are traces over $W_{(k)}$,
$k\geq 0$ basis. Then, the sum of $\kappa$-powers of traces in (3.5) converges
to a meromorphic function. The meromorphic function depending on
$(g_{1},\ldots,g_{n+1})$ and $(z_{1},\ldots,z_{n+1})$ corresponds to the
$\kappa$ pairs of insertions of handles to form multiply self-sewn Riemann
sphere obtained in the Schottky uniformization procedure. We show now that the
result of action of the coboundary operator (3.3) on elements of the space
$C^{n}_{m}$ corresponds to an absolutely converging in $\rho_{p}$, $1\leq
p\leq\kappa$, predetermined meromorphic function on $F_{n+1}\mathbb{C}$ with
only possible poles at $z_{i}=z_{j}$, and $1\leq i$, $j\leq n+1$. In order to
prove this we use the geometric interpretation of the coboundary operator
(3.3) in terms of the Schottky uniformization of Riemann sphere as a auxiliary
geometric model.

In the geometric model, complex parameters $(z_{1},\ldots,z_{n+1})$ correspond
to the local coordinates of $(n+1)$ points on original Riemann sphere with
excised annuli ${\mathcal{A}}_{a,p}$. Recall the sewing parameter condition
$\zeta_{1,p}\zeta_{2,p}=\rho_{p}$ of the sewing procedure. The complex
parameters $\zeta_{a,p}$, $a=1$, $2$ correspond to coordinates inside
identified annuluses ${\mathcal{A}}_{a,p}$, and $|\zeta_{a,p}|\leq\rho_{p}$.
Recall that it is assumed the the subspaces $W_{(k)}$ are finite-dimensional,
and the codimension of the subspace $\widetilde{W}_{(2)}$ is finite. Thus, in
(3.4), for any $k\geq 0$ we have a finite number of terms in the corresponding
power $k$ of $\rho_{p}$. Since for $g_{1},\dots,g_{n}\in G$,
$\mathcal{F}(g_{1},z_{1};\ldots;g_{n},z_{n})$, is absolutely convergent in the
region $|z_{1}|>\ldots>|z_{n}|>0$, to a function in $(z_{1},\dots,z_{n})$, the
$k$-th coefficients of the expansions in $\zeta_{a,p}$,
${\mathcal{R}}_{k}(z_{1},\ldots,z_{n+1},\zeta_{a,p})={\rm
coeff}\;\mathcal{F}(g_{1},z_{1};\ldots;g_{n+1},z_{n+1};\zeta_{a,p})$, are
absolutely convergent in powers of $\rho_{p}$ with some radia of convergence
$R_{a,p}\leq\rho_{p}$, and $|\zeta_{a,p}|\leq R_{a,p}$. Due to the sewing
condition on $\zeta_{a,p}$ and $\rho_{p}$, the dependence of the above
coefficients on $\rho_{p}$ is expressed via $\zeta_{a,p}$, $a=1$, $2$. Let us
rewrite the action of the coboundary operator (3.3) as

$\delta^{m}_{n}\mathcal{F}=\sum_{l\in\mathbb{Z},\;g\in
W_{(l)},\;k\in\mathbb{Z}}\rho_{p}^{l-k-1}\;{\mathcal{R}}_{k}(z_{1},\ldots,z_{n+1};\zeta_{a,p}),$
(3.7)

as a complex series in $\rho_{p}$ for $|\zeta_{a,p}|\leq R_{a,p}$, where and
$|\rho_{p}|\leq r$ for $r_{p}<r_{1,p}r_{2,p}$. We apply Cauchy’s inequality to
coefficients in $\zeta_{a,p}$ above and find

$\left|{\mathcal{R}}_{k}(z_{1},\ldots,z_{n+1};\zeta_{a,p})\right|\leq
M_{a,p}\;R_{a,p}^{-k},$ (3.8)

with $M_{a,p}=\sup_{\left|\zeta_{a,p}\right|\leq
R_{a,p},\left|\rho_{p}\right|\leq
r_{p}}\left|{\mathcal{R}}(z_{1},\ldots,z_{n+1})\right|$. Using (3.8) we obtain
for (3.7)
$\left|\mathcal{F}_{l}\right|\leq\left|{\mathcal{R}}_{k}(z_{1},\ldots,z_{n+1};\zeta_{a,p})\right|\leq
M_{a,p}R_{a,p}^{-k}$. Thus, for $M=\min\left\\{M_{a,p}\right\\}$ and
$R=\max\left\\{R_{a,p}\right\\}$,
$\left|\mathcal{R}_{l}(z_{1},\ldots,z_{n+1})\right|\leq MR^{-l+k+1}$. We infer
that (3.3) is absolute convergent to a meromorphic function $\mathcal{F}$ as a
complex series in $\rho_{p}$ for $|\zeta_{a,p}|\leq r_{a,p}$, and
$|\rho_{p}|\leq r_{p}$ for $r_{p}<r_{1,p}r_{2,p}$.

Now, suppose that points $(q_{1},\ldots,q_{n+1})$ with local coordinates
corresponding to the complex parameters $(z_{1},\ldots,z_{n+1})$ of
$\delta^{n}_{m}\mathcal{F}$ are situated on initial Riemann sphere outside
sewing annuli. An element $f$ is a $G$-valued meromorphic function defined on
the configuration space $F_{n}\mathbb{C}$. Each summand inside the trace in
(3.4), is the classical coboundary operator $\delta^{n}_{m}\mathcal{F}$ [13],
defines for every $w_{p}\in W_{(k)}$ a $G$-valued meromorphic function on the
configuration space $F_{n+1}\mathbb{C}$. Since (3.4) is by construction a
power series in $\rho_{p}$, $1\leq p\leq\kappa$ of
$\widetilde{\delta}^{n}_{m}\mathcal{F}$ elements, it constitutes a map
$F_{n+1}\mathbb{C}\to G$,
$(z_{1},\dots,z_{n+1})\mapsto\widetilde{\mathcal{F}}(z_{1},\dots,z_{n+1})$,
where $\widetilde{\mathcal{F}}$ is a meromorphic function with no further
poles appearing other then poses at $z_{i}=z_{j}$, $1\leq i,j\leq n+1$.

Recall the sewing relation (3.2) for the Schottky uniformization parameters
that identifies local coordinates for each $1\leq p\leq\kappa$ pair of annuli
to form $\kappa$ handles. By construction, it is assumed that annular regions
$\mathcal{A}_{N,p}$, $N=1,2$, do not intersect. Now let us consider the case
when a pair of points $q_{i}$, $q_{j}$, $1\leq i,j\leq n+1$, belongs to open
domains $\mathcal{U}_{i}$, $\mathcal{U}_{j}$ in two different annuli described
above. Suppose local coordinates of $q_{i}$, $q_{j}$ of the geometric model
correspond to parameters $z_{i}$, $z_{j}$ among parameters of
$\delta^{n}_{m}\mathcal{F}$. The sewing condition (3.2) defines [30] an
analytical extension of the $G$-meromorphic function given by (3.4) from
domains covering initial Riemann sphere to domains covering the resulting
genus $\kappa$ Riemann surface obtained in the Schottky uniformization. Since
points $q_{i}$ and $q_{j}$ belong to two different annuli, thus analytical
extension of (3.4) to the genus $\kappa$ Riemann surface is automatically
defined on the configuration space $F_{n+1}\mathbb{C}$ due to the condition of
non-intersection of annuli. In the opposite case, suppose $q_{i}$ and $q_{j}$
belong to the same annular region, and $z_{i}$, $z_{j}$ satisfy the condition
of configuration space $F_{n+1}\mathbb{C}$. Denote by $z^{\prime}_{i}$,
$z^{\prime}_{j}$ the parameters related with the parameters $z_{i}$, $z_{j}$
through the sewing conditions. Let us substitute $z_{i}$, $z_{j}$ into the
sewing condition (3.2) and express for $z^{\prime}_{i}$, $z^{\prime}_{j}$. It
results in $z_{i}=z_{j}$ that contradicts the configuration space
$F_{n+1}\mathbb{C}$ condition. Thus, we infer that in the Schottky
uniformization model the configuration space $F_{n+1}\mathbb{C}$ condition is
preserved for (3.4) . Suppose a point $q_{i}$ belongs to $\mathcal{A}_{1,p}$
of the $p$-th pair of annuli, and corresponding parameter $z_{i}$ coincides
with $\eta_{1,p}$. Since $\mathcal{F}(\phi)$ is a meromorphic function of
specific form, then its only extra poles can occur at $z_{i}=\eta_{2,p}$. This
is impossible due to the sewing condition identifying the parameters
$\eta_{1,p}=\eta_{2,p}$. The same happens when $q_{i}$ belongs to the second
annular region $\mathcal{A}_{2,p}$. Taking into account that the
approximations (3.7) to the action of the coboundary operator (3.3) are
considered on attaching handles in the geometric Schottky uniformization model
and are well-defined meromorphic functions, therefore, the limiting function
of (3.4) is a $G$-meromorphic function with only possible poles at
$z_{i}=z_{j}$, $1\leq i<j\leq n+1$.

It is easy to check that (3.3) preserves properties (2.1), (2.2), and (2.4)
for $\widetilde{\delta}^{n}_{m}\mathcal{F}$. Indeed, since
$\delta^{n}_{m}f_{p}$, $1\leq p\leq\kappa$ is defined on $\mathcal{F}\in
C^{n+1}_{m-1}$, we have $\delta^{n}_{m}f_{p}(V.)=V.\delta^{n}_{m}f_{p}$.
Taking into account (3.6) we obtain the invariance, i.e.,

$\displaystyle\sum\limits_{k\geq
0}\;\prod_{p=1}^{\kappa}\;\rho_{p}^{k}\;\mathrm{Tr}_{G}(f_{p}(V.))=\sum\limits_{k\geq
0}\;\prod_{p=1}^{\kappa}\;\rho_{p}^{k}\;\mathrm{Tr}_{G}V.f_{p}=V.\sum\limits_{k\geq
0}\;\prod_{p=1}^{\kappa}\;\rho_{p}^{k}\;\mathrm{Tr}_{G}f_{p}=V.\widetilde{\delta}^{n}_{m}\mathcal{F}.$

In particular, consider

$\sum_{\sigma\in
J_{n;s}^{-1}}(-1)^{|\sigma|}\sigma.(\delta^{n}_{m}\mathcal{F})=\sum\limits_{k\geq
0}\;\prod_{p=1}^{\kappa}\rho_{p}^{k}\;\mathrm{Tr}_{G}\sum_{\sigma_{p}(k)\in
J_{n;s}^{-1}}(-1)^{|\sigma(p(k))|}(\sigma_{p(k)}).\delta f_{p}.$

Since $\delta f_{p}$ represents the standard coboundary operator which maps
$C^{n}_{m}\to C^{n+1}_{m-1}$, then, $\mathcal{F}(\delta f)\in C^{n+1}_{m-1}$.
According to (3.6) for $V=\sigma$, in each separate multiplier the condition
(2.4) is satisfied, therefore the last expression vanishes. Using the
Proposition 3.10 of [16], we see that (3.5) in (3.3) is compatible with
$(m-1)$ $\omega_{g^{\prime}_{j}}(z^{\prime}_{j})$-series for $1\leq j\leq
m-1$, for $(z^{\prime}_{1},\ldots,z^{\prime}_{m-1})\in\mathbb{C}$. The proof
of the chain property for the standard operator (3.5) is obvious. It is
provided by cancellation of combinations of $f$-elements as in [16] in the
summands of (3.4) in (3.3). Thus, we obtain the statement of the proposition.
∎

## Acknowledgment

The author is supported by the Institute of Mathematics, the Academy of
Sciences of the Czech Republic (RVO 67985840).

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