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# Zeitschrift für Naturforschung A

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# Anomaly on Superspace of Time Series Data

Salvatore Capozziello
• Corresponding author
• Dipartimento di Fisica, Università di Napoli “Federico II”, Via Cinthia, I-80126 Napoli, Italy
• Istituto Nazionale di Fisica Nucleare, Sez. di Napoli, Via Cinthia, Napoli, Italy
• Gran Sasso Science Institute, Via F. Crispi 7, I-67100 L’Aquila, Italy
• Email
• Other articles by this author:
/ Richard Pincak
• Institute of Experimental Physics, Slovak Academy of Sciences, Watsonova 47, 043 53 Kosice, Slovak Republic
• Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia
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• Other articles by this author:
/ Kabin Kanjamapornkul
• Department of Survey Engineering, Faculty of Engineering, Chulalongkorn University, 254 Phyathai Road, Bangkok, Thailand
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Published Online: 2017-11-16 | DOI: https://doi.org/10.1515/zna-2017-0274

## Abstract

We apply the G-theory and anomaly of ghost and antighost fields in the theory of supersymmetry to study a superspace over time series data for the detection of hidden general supply and demand equilibrium in the financial market. We provide proof of the existence of a general equilibrium point over 14 extradimensions of the new G-theory compared with the M-theory of the 11 dimensions model of Edward Witten. We found that the process of coupling between nonequilibrium and equilibrium spinor fields of expectation ghost fields in the superspace of time series data induces an infinitely long exact sequence of cohomology from a short exact sequence of moduli state space model. If we assume that the financial market is separated into two topological spaces of supply and demand as the D-brane and anti-D-brane model, then we can use a cohomology group to compute the stability of the market as a stable point of the general equilibrium of the interaction between D-branes of the market. We obtain the result that the general equilibrium will exist if and only if the 14th Batalin–Vilkovisky cohomology group with the negative dimensions underlying 14 major hidden factors influencing the market is zero.

## 1 Introduction

The algebraic defect of mathematical modeling of the financial market over a real number field is a source of extra dimensions in Kolmogorov space of underlying time series data. We cannot observe 14 hidden extra dimensions in time series data because all extra dimensions in 14-dimensional G-theory [1, 2] of underlying time series data are canceled and they disappear in the analogy to the Feynmann model canceled halfway by the positive metric the longitudinal gauge particle, which is a real field. The source of the cancelation of the hidden dimensions is induced by the invariant property of the nonoriented supermanifold, which is the so-called parity anomaly over supermanifold of a ghost field [3]. An arbitrage is an evolution feedback between the hidden behaviour of the trader from the ghost field of supply coupled with the antighost field induced from the demand side, which vanishes when the market is in general equilibrium. With this application of the cohomology theory in economics as the quantum fluctuation in the business cycle and the new G-theory, we produce a new market risk-cocycle coupled modeling over superspace in time series data. In this new model, we can explicitly define an arbitrary opportunity as the Chern–Simon market anomaly over the induced ghost and antighost fields in the financial market.

Recently, most scientists [4] and economists [5] turned their interest to the use of a new mathematical supergeometry [6] and superalgebra [7] of Poisson bracket [8], a nonoriented supermanifold [9], superpoint [10], and superstatistics [11, 12] for modeling the anomalous equilibrium state [4] in hidden dimensions of the financial market as the parallel world [13] of the financial market in the general equilibrium theory. Typical economical modeling is based on the real number field in which each time series data of the record is embedded in the Euclidean plane without the supersymmetric property of the hidden ghost field as a supermanifold.

The supermanifold [9] is a highly promising mathematical object in mathematical physics, which applies the mathematics of the Lie superalgebras to the behaviour of bosons and fermions. The driving force in the formation of the supermanifold is a spinor field in mathematics and physics, and the work on the characteristic class of the second cohomology group by Chern and Simon, who also made a major contribution with the use an anomaly current of the Yang–Mills field as the field with the hidden cancelation in the mirror symmetry modeling of the grand unified theory [14]. In the central concept of supermathematics [15], the objects of the study include the supersymmetry, the supermanifolds, the BV cohomology [16], and the superlagrangian [17], namely, in the context of the superstring M-theory and G-theory [1]. Besides using the dynamic stochastic general equilibrium (DSGE) model [18, 19] for the modeling of macroeconomics without taking into account the market microstructure, we can apply the supersymmetry theory from the superalgebra of the sheave sequence of the resolution theory from the BV cohomology [16] and the unified theory of the anomaly group SO(32) and the grand unified E8×E8 [20] to unify the modeling of the microeconomics with the macroeconomics theory as the interaction of the D-brane and the anti-D-brane of investiment saving-liquidity money (IS-LM) and DSGE in the supermanifold.

There exists a new approach [21] in econophysics and economic modeling [22] for the investigation of the general equilibrium modeling as a master equation for the orderbook in the stock market with the string [23, 24] and the D-brane theory approach [25]. The orderbook model based on the stochastic process [26] and defining a buy and sell operator as the time series model observes data directly, not from a hidden state. Most researchers in the financial market equilibrium [25] come from the signal processing, statistics, computer science, and quantum physics areas [27, 28], some of them come from the econometric research area studying the general equilibrium model [5] based on the DSGE model [18, 19], which is based on the stochastic control theory. We use an alternative approach in the cohomology [16] and the grand unified theory approaches [20] to find a master equation for both microeconomics model lagrangian of the utility function throughout the macroeconomic model in the hidden superspace of the time series data. We called this equation a new version of the modified Yang–Mill–Chern–Simon master equation for the financial market over the ghost field in financial market modeling.

In the space of time series data [29], there is a source of hidden information wherein we record data into the time series in a superspace, the so-called financial hidden ghost field. In nature, all the physical quantities that have pairs are so-called parity quantities. In quantum physics, the charge time parity (CPT) theory is an example of active research beyond the superstatistics in which the theory is induced from the supermanifold. The generalisation of the supermanifold is the hypermanifold. Hyperstatistics is one of the examples of an active research for the M-theory and the unified theory with the hypersurface, namely, Calabi–Yau space. In this work, we use this approach for the study of a new modeling of the quantum financial market in the supersymmetry theory of 14 hidden negative dimensions, the G-model. In this work, we develop a new definition of the ghost field in the nonorientated state in the supermanifold modeling of the financial market. We introduce a new cohomology theory with a negative dimension with a unifying equation for the financial market in the superspace modeling.

The article is organised as follows: in Section 2, we specify the basic definition of the superspace in time series data and of the Lie superalgebra with the Poisson bracket in the master equation. We define 14 ghost fields in the financial market in this section. In Section 3, we define the least action for the Yang–Mills field in the Chern–Simon theory in financial market from the behaviour of the traders in the financial market. We define a new BV cohomology in the financial market with a new Yang–Mills–Chern–Simon master equation for the financial market. In this section, we provide a source for the proof of existence of the general equilibrium model of the market in the grand unified theory anomaly model in 14 hidden dimensions by using coupling between 14 ghost fields in the BV cohomology theory. In Section 4, we discuss the result of the proof and provide a conclusion for future work.

## 2 Superspace of Time Series Data

In the time series induced from the financial market, most researchers found that the nonlinear and nonstationary states in normal distribution cannot be used while the market crashes. The source of the abnormal state in the financial time series might be a source of the hidden dimension in the Kolmogorov space of time series in which ghost and antighost fields of supply and demand in the financial market exists. Then, we use the supersymmetry theory over the supermanifold of time series data. We can define a parity of the supply and demand by using the Poisson bracket. If we suppose that the supply S is an inverse of the demand D in the superspace of the financial market S, D∈𝒜, we can define a Poisson bracket {S, D} in the financial market by using the adjoint and coadjoint representations. We first introduce the classic master equation into the population-based models in the standard probabilistic model. Then, we extend the master equation to the superspace of time series data over the supermanifold with the induced ghost and antighost fields in time series data.

## 2.1 Chern–Simon (2+1) Dimensions in 14-Dimensional Modeling of Financial Markets

In macroeconomics, for the control of the inflation target, DSGE uses πt of the superspace of macroeconomics as the financial market superspace in our model 𝒜 the well-known instrument of the monetary policy interest rate, i=it. In contrast to the microeconomics theory, inflation and interest rates in the utility and the production functions do not exist. Only the price and the quantity are defined there. The classic macroeconomic model is based on the stochastic model of the IS-LM model and the Phillips curve, the so-called DSGE. The IS-LM model is based on the mathematical structure of a flat Euclidean plane without any curvature of the spacetime and with mirror symmetry. We can visualise the IS-LM model (see Fig. 1) in the superspace of time series data by using the interaction between the D-brane and the anti-D-brane model in which the ghost field exists. One D-brane of the market is a supply side and the other anti-D-brane is the demand side of the market. With this model, the equilibrium point in the IS-LM model will be connected by components that imply that the supply side and the demand side are on each side of the Mobius strip of a nonoriented supermanifold (see Fig. 2).

Figure 1:

IS-LM and the DSGE model in the D-brane and the anti-D-brane diagram model. This is not the model with the same classic macroeconomics because we work on the level of the topological space underlying all the parameters in economics with an extra property of the spontaneous supersymmetry breaking. The equation of the IS-LM in the classic macroeconomic model is expressed by Y=(YT)+I(r*)+G+NX(E). The meaning of the variable is shown in the table with its associated ghost and antighost fields. The space of the demand side is a hidden space and denoted by Xt and Yt is the supply side of economics in the IS-LM, AS-AD, and DSGE model. All the demand and supply sides of economics have a hidden dimension of the underlying topological space defined by the supermanifold (𝒜, s) in which the element of the supermanifold is a ghost or antighost field of all the IS-LM parameters defined from classic economics. This superspace of time series data of economics is defined by using the tangent space of the supermanifold of the underlying classic DSGE model.

Figure 2:

Four-dimensional model with Hopf fibration S3 of the IS-LM. We notice from the model of the superspace of time series data that the IS curve is a connected component with the LM curve and cannot be completely separated from the supply and demand sides of the market. In the supply exists a demand and in the demand also exists a supply. When the market is in equilibrium, the supply and the demand can be separated by using the market cocycles αt and βt in the torus S1×S1 as underlying space of the interaction between the D-brane and the anti-D-brane. S−11 is defined as a hidden space with 11 factors influencing the macroeconomics on the supply side and S−14 is a space with 14 dimensions influencing the market factor on the demand side. The details for these factors are shown in Table 1 with its superspace of time series data in the ghost and antighost fields.

In the history of the discovery of the ghost field, not many theoretical physicists have noticed the existence of a ghost field from the formulas as a massless nonoriented ghost propagator in the Feynmann work on loop space in the theoretical physics of the supersymmetry theory, except for Faddeev and Popov [30] using the Yang–Mills theory. The hidden behaviour of the traders on the expectation of the future market parameters is a source of the supersymmetry theory in the financial market in which the ghost fields over the superspace of the market state in the financial market are defined over 14 hidden dimensions. There exists ghost and antighost fields in the financial market in which the hidden behaviour of the traders is induced from the existence of the well-known market parameters (Tab. 1). In the supersymmety theory, we have two mirror symmetries with an embedded nonoriented supermanifold between the two interaction planes of the D-brane and the anti-D-brane for the financial market (see Fig. 3).

Table 1:

Definition of the first 14 ghost fields in the financial market from the market microstructure of the underlying superspace in time series data.

Figure 3:

Market hypersurface of the supply side of the D-brane and the demand side as the anti-D-brane. It is a superspace of time series data with the supersymmetry property of the mirror symmetry. When the market is in general equilibrium, we have a twist between the supply and the demand by using the duality property of the market as the duality map in the algebraic topology. This is the source of the nonoriented supermanifold in the financial market with the induce spinor field in time series data as the ghost and the antighost fields in the underlying extra dimension of the superspace in time series data.

The other source of the ghost field over the superspace of time series data is an algebraic defect of the real number field and the real vector space in economic modeling. The superspace of the vector space over the real number field can be generalised by using the supermanifold model in which the ghost field exists. Macroeconomics uses the Hamiltonian for the modeling of the macroeconomics model as the dynamics of the lagrangian of the market in the supply and the demand side.

In theoretical physics, U(1) gauge theory in spacetime coupled with particle physics containing the massless charge one is the most highly promising theoretical structure for the grand unified theory or the M-theory for financial market modeling. In the Kolmogorov space in time series data, we can introduce a single massless Dirac fermion model for a new quantity (behaviour of trader) as an extra dimension in the Chern–Simons term with (2+1) dimensions [31] or what we called the Hopf term with the least action over the Yang–Mills field. In theoretical physics and string theory, we know that the Chern–Simons terms violate both the parity (P) and the time inversion (T) symmetries, which are useful for the modeling of the gravitational field and superstring theory. This problem gives rise to a new theory – the so-called parity anomaly on a nonoriented manifold. The application of this new theory now plays an important role in the topological superconductors, M-theory [10], and other applications to machine learning and the financial market modeling.

## 2.2 Ghost Field in the Financial Market

A ghost field in the superspace of time series data can be visualised as the smallest unseparated component of time series data, which we cannot separate and embed in the Kolmogorov space of time series data as a Calabi–Yau space (see Fig. 4).

Figure 4:

Calabi–Yau space in time series data. It contains the ghost and antighost fields in the superspace underlying financial time series data induced from the financial market. We did not notice a Calabi–Yau space in time series data because the scale of the Euclidean plane is too large to detect it. It curls into the extra dimensions of 11 dimensions and 14 dimensions of underlying superspace of time series data. It is induced from the interaction of the ghost and antighost fields of the behaviour of the trader with the expectation fields of all the parameters in economics that influences the market risk. If we work on the BV cohomology level of the superspace of time series data, we can define the Calabi–Yau space in time series data embedded into the D-brane and anti-D-brane of the financial market microstructure.

Let xt, yt be two financial time series from the supply side and the demand side. We can induce a field of a pair time series in the equilibrium state by a lag operator gG of the Lie group translation

$Adg[xt, yt]=[Adgxt, Adgyt],$(1)

$ad=Ad∗e:g→Endg,adxt=ddt|t=0Adetxt,adxtyt=[xt, yt].$(2)

Let 𝒜j be a superspace of time series data. The dimension of the superspace is denoted by the superscript dim(Aj)=j=gh(A) and can be plus and minus depending on the predefined ghost field in financial time series data.

Definition 1: Every trader in the financial market has its ghost field to make a decision for buying and selling stock as a decision field for sending the command for the selling or buying message. We define two major types of this hidden decision field into two categories – the so-called ghost field Φi for buying operator (minimum state in the superspace of time series data) and the antighost field ${\Phi }_{i}^{+}$ for selling operator (maximum state in the superspace of time series data).

$Φ:(A, s)→ℤ/2={1, −1},$(3)

where 𝒜 is a supermanifold of the financial market with the master equation s as a hidden ground field. Next, [s4] stands for the maximum state in the physiology of time series data as a ground field and [s4] stands for the minimum state as a ground field. The master equation is defined by the Poisson bracket of the superlagrangian of ghost and antighost fields

$s:={∫S dt,−}=0$(4)

where S is the least action of the superlagrangian of the ghost field in the superspace of time series data.

Let L be a lagrangian of the maximum utility function in the classic microeconomics model. We induce a superlagrangian ℒ over the ghost field Φ(p) of price p. We assume that the ghost field is induced by the expectation price from the behaviour of a trader in the financial market. If we are in general equilibrium of financial economics, we use the Batalin–Vilkovisky (BV) formalism for doubling the number of fields in the financial market with parity (P). Let a price p spinor field be denoted as Φ(p). Let Φ(p)+ be an antifield. The parity of the antifield Φ+ is according to the relationship $p\left({\Phi }_{i}\right)=1-p\left({\Phi }_{i}^{+}\right)$ [16], where pi)∈{0, 1} mod 2. Let ghi) be a ghost number of Φi. If ghi(pt))=0, we come to the normal field of the price pt~N(0, 1) in a stationary state in which it satisfies the random walk model. If ghi)>0, we induce a ghost field of price. We can naturally assume that the market has two sides, the supply and demand sides. In the superlagrangian model of the financial market, the ghost field will be on the supply side and the antighost field can be denoted as a hidden field on the demand side of the financial market in such a way that ghi)+ghi)+=−1. We use the BV cohomology for the financial market to formulate a Poisson structure between the ghost fields of the supply and the demand sides of the market. We define a Poisson bracket of degree 1 by [8]

${Φi,S,Φj,D+}=−{Φj,D+,Φi,S}=δij.$(5)

Let Φi,STgG be a Lie superalgebra of the financial market and Φj,DTgG* be a dual tangent supermanifold of financial market. We define the adjoint representation map for financial market by

$adΦi,S*Φj,D={Φi,S, Φj,D+}.$(6)

We have

$adΦi,S[Φ(p), Φj,D]=[adΦi,SΦ(p), adΦi,SΦj,D]$(7)

We introduce 14 dimensional modelings of the financial market from both microeconomic and macroeconomic points of view with the anomaly gauge group in the SO(32) G-theory. The source of the 14 dimensions is defined as a hidden state for 12 dimensions and two states of the observed dimension in the spacetime of the pricetime of time series model. The following is a definition for all the ghost fields in finance:

• We use Φi(π), Φi(r), ghi(π)+ghi(r))=−1 for parity of ghost field for inflation and interest rate.

• The source of more than six hidden dimensions induced from the ghost field of the Yang–Mills field in the financial market between the behaviour of the traders in the market from the supply and the demand sides and from the hidden supply and demand sides. We denote Φi(Fμν)+Φi(*Fμν)=−1, μ=1, 2, 3, ν=1, 2, 3.

• In the market, we have the supply (S) and demand (D) sides, as the two-dimensional model of the market is influenced by market factors. We define their ghost field by Φi(S), Φi(D), ghi(S))+ghi(D))=−1.

• The last two dimensions of the 14-dimensional model of the financial market appear as time series data of buying and selling spinor operator in spinor field. We denote the ghost field by (θν, xν)=(Φi(B), Φi(S)), ghi(B))+ghi(S))=−1.

All predefined ghost fields carry their antighost field in the CPT theory with parity $p\left({\Phi }_{i}\right)=1-p\left({\Phi }_{i}^{+}\right).$ We have a moduli state space model of the ghost field and antighost field mod 2 of its parity. We define a canonical coordinate for the BV antibracket by

${Φi, Φj+}=−{Φj+, Φi}=δij$(8)

Let S be the least active of all 14 ghost and antighost fields in the financial market. For the first six ghost fields for the behaviour of the trader in the market, we can write down an explicit form of S, a ghost field in the financial market by using a path integral to the least action borrowed from the formula for the gravitational Chern–Simons term in (2+1) dimensional spacetime. It is one part of the grand unified theory in theoretical physics (see Fig. 5 for details of the 14 ghost fields in G-theory). However, in finance, unified theory plays a very important role in the connection ${\Gamma }_{ij}^{k}$ of the Dirac spinor field from the theory of supersymmetry.

Figure 5:

Icosahedral group E8×E8 of the model from the grand unified theory for the financial market in the hidden 11-dimensional model of the supermanifold of the financial market as a D-brane with the ghost field of the cocycle βtH−14(Xt) quantum entanglement state with the anti-D-brane with S11 with the induced ghost field of the cocycle αtH11(Yt):=[Yt, S11]. If we consider the anomaly cancelation of the dimension from 14–11=3, we will obtain a state space model in spinor fields, S3 as an error from the forecasting model in the econometric. In this way, we obtain a spinor field in time series data as a unified theory for the financial market.

## 3.1 Yang–Mills Equation in the Financial Market

We study an arbitrary opportunity in the financial market by the induced hidden ghost field as the market cycle and cocycle αk, βkH−14(𝒜, s). The 14 hidden ghost fields belong to the superalgebras of the supermanifold for the financial market 𝒜, which we cannot separate. A ground field s of H−14(𝒜, s) is a master equation of the transition superprobability of coupling two market cocycles as a spinor field in financial time series data (see Fig. 6). The way in which the cocycles couple each other in the hidden dimension cannot be observed in space of time series data. The coupling induces a market quantum nonconserved chiral current ${J}_{A}^{\mu }$ and varies from an arbitrary opportunity in the financial market in which it is defined from the behaviour of the trader in the Eric transactional model to the behaviour of the trader in the new cohomology theory in the financial market [32].

Figure 6:

Spin-orbit coupling between the ghost and antighost field cocycles and localised in the reverse direction of the timescale. The lower box is the anti-D-brane of the financial market and the upper box is the D-brane of the financial market. We did not notice the arbitron like the arbitrary opportunity particle because the collision and the dynamics of the superprobability transition is very fast and exists at a very small scale, the so-called Calabi–Yau space of time series data.

We define a Yang–Mills field for the financial market by the interaction of the Chern–Simon–Eric field 𝒜 induced by the interaction of two sides of the market of the behaviour of the traders Ai, i=1, 2, 3.

We are going to prove the existence of the tensor field gij in αt and the tensor field gij in βt as an interaction between two D-branes of an induced third tensor field in orbifold ${g}_{i}^{k}$ as an ingredient of the connection in the financial market.

Consider the interaction between the ghost and antighost fields in the financial market.

Definition 2: The tensor field of the Berazian coordinate transformation in the financial market is defined by the change of the state of the ghost and antighost fields in finance (see Fig. 7):

Figure 7:

Interaction between the berazian coordinate in the supermanifold of time series data Xt and Yt. We simplify its berazian by borrowing a notation of the Jacobian field as the new definition of the tensor ghost field in financial time series data. An arbitrary opportunity is directly defined by changing the tensor ghost field using the interaction of the behaviour of the trader in the market. We can use the Poincare disk model to visualise the market equilibrium by denoting one circle of the supplied field. The other induced cycle is the demand field. If these two cycles are in contact with each other properly, the market will be in equilibrium. If the circle is not in the proper position, all fields will be induced and push each other as the market cocycle model.

$gij=∂Φi∂Φj, gij=∂Φi+∂Φj+, gji=∂Φi∂Φj+.$(9)

Definition 3: The connection over the ghost field in the financial market is defined by

$Γijm=12gml(∂jgil+∂iglj−∂lgji).$(10)

Definition 4: The Ricci tensor in time series data is a contraction of the curvature tensor defined by ${R}_{ik}={R}_{ikl}^{j}$ with respect to the natural frame of an arbitrary opportunity (connection),

$Rik=∂kΓjij−∂jΓkij+ΓkmjΓjim−ΓjmjΓkim.$(11)

Definition 5: Let Fμν be a Yang–Mills field for the financial market and let Aμ be a Chern–Simon field for the financial market. Let ${H}_{DR}^{2}\left(\mathcal{A}\right)$ be a De Rham cohomology for the financial market 𝒜. We have a supersymmetry of antiself dual between the brane and the antibrane in the financial market defined by

$∗F∇[si]=∮HDR2(A)FμνkAν=F∇[si*]$(12)

Definition 6: We have the anomaly or the arbitrary opportunity after the quantisation of the market by the behaviour of the trader Fμν with the anti-D-sister ${F}^{\nabla \left[{s}_{i}\right]}$ with the Chern–Simon current induced in 14 hidden extra dimensions of the financial market:

$JAμ=∂μFμν+[Aμ, Fμν].$(13)

$14<∗FμνFμν> =∂Kμ,$(14)

where

$Kμ=ϵμαβγ<12Aα∂βAγ+13AαAβAγ>,$(15)

where a connection ${\Gamma }_{\alpha \nu }^{\mu }={\left({A}_{\alpha }\right)}_{\nu }^{\mu },$ in which A is a Chern–Simon term in the financial market. An arbitrage or arbitron is a Chern–Simons anomaly, Kμ or an anomaly current induced from the behaviour of the trader as the twist ghost field between two sides of the market. This field of the financial market induces a Ricci curvature in the financial market as an arbitrary opportunity for each connection in the coupling ghost field of the behaviour of the trader. This curvature blends the Euclidean plane of the space of the observation to twist with the mirror plane behind the Euclidean plane and wraps to each other as the D-brane and the anti-D-brane interaction of the superspace of financial time series data (see Fig. 1).

$Rναβμ(Γ)=eαμRbαβaeνb.$(16)

Definition 7: Let 𝒞 be a configuration space of the subsystem of the financial market and denote the probability of finding a system in the configuration of the supply side 𝒞 at time t. Let 𝒞′ be a configuration space of the subsystem and denote the probability of finding a system in the configuration of the demand side of the financial market 𝒟 at time t by 𝒫𝒟(t). A master equation is the dynamics of the financial market system of the transition probability between the supply and the demand state defined by

$dPD(t)dt=∑SiλSDPS(t)−∑SiλDSPD(t)$(17)

where λ𝒮𝒟 is a transition probability between the supply and demand market state in the financial market.

Definition 8: The classic master equation in the Batalin–Vilkovisky cohomology for the financial market is defined by the solution of the classic master

${∫PS(t)dt, ∫PD(t)dt}=:{Φi(xt), Φj+(yt)}=0,$(18)

where 𝒫𝒮(t)∈𝒜0,0 is a differential expression in the ghost and antighost fields with gh(S)=0, a ghost number and a parity number p(S)=0. Here, xt is the demand side of the market and yt is the supply side of the market.

## 3.2 Anomaly on a Nonoriented Supermanifold of the Financial Market

A coordinate of the space of time series is invariant under the group transformation – we call it the translational and rotational invariant, but not for the reflection of the timescaled (R) and the time reversal (T) symmetry.

Let xtX with the dimension of X in (2+1) dimension, we quantise the gauge Chern–Simon field in time series data xt by the least action as a ghost field S.

Let 𝒜 be a superspace, where the financial market of the measurement is induced by its ghost field. Let f∈𝒜∈ℱ, we have a probabilistic density in the superstatistics p(xt)=∫fdt in Φi. Let xi be a financial time series of the observation in moduli state space model of the supermanifold. Let G be the ghost field of buying and selling operators. Let S be the least action of the financial market as the ghost field. We have the relationship

${{G, S}, xt}=∂xt.$(19)

The BV cohomology for the financial market is defined by the differential with the exact sequence with s2=0. We define a canonical coordinate of the financial market in the ghost field of 14 dimensions by using the Poisson bracket as the codifferential complex in the BV cohomology of the master equation in the financial market,

$s={∫Sdt,−}.$(20)

We induce a BV cohomology as a long exact sequence of the financial market as the following:

$⋯→H−11(A, s)→⋯⋯→H−1(A, s)→∂H−1(A, s) →H−1(ℱ, s)→H0(A/C, s)→H0(A, s)→H0(ℱ, s) →H1(A, s)→∂H1(ℱ, s)→⋯→H3(A, s)→⋯$(21)

Definition 9: We define a differential of the ghost and antighost fields with parity by

$sΦ=(−1)p(Φi)+1δSδΦi+,$(22)

$sΦ+=(−1)p(Φi)δSδΦi.$(23)

## 3.3 Superstatistics of Superpoint and Ordinary Least Square Regression

We start from the normal situation of the normal distribution of the shock in the economics ϵ~N(μ, σ2) in the ordinary statistics in the sigma field,

$yt=αt+βtxt+ϵt,$(24)

where we have a new integration over the cocycle αt and βt in the BV cohomology.

This is a new kind of a supermathematical theory in the probability theory with a new integral sign over the tangent of the supermanifold under the Berezin coordinate transformation. The structure of the supergeometry of the superpoint [10] induces the superstatistics of the hidden ghost field in the financial market. We let yt be an observable variable in the state space model and xt be ahidden variable of the state. We denote Φi(yt)∈𝒜 as a ghost field and ${\Phi }_{i}^{+}\left({x}_{t}\right)$ as an antighost field in finance with the relation of its ghost number,

$gh(Φi(yt))+gh(Φi+(xt))=−1.$(25)

We have a parity $p\left({\Phi }_{i}\left({y}_{t}\right)\right)=1-p\left({\Phi }_{i}^{+}\left({y}_{t}\right)\right)$ and $p\left({\Phi }_{i}\left({x}_{t}\right)\right)=1-p\left({\Phi }_{i}^{+}\left({x}_{t}\right)\right)$ for both state and space variables in the ghost and antighost fields.

Pair trading is the root of a superstatistical arbitrage theory which has a deep connection with the new model of DSGE in the superstatistic theory. The source of the financial market equilibrium comes from a pair of similar stock (xt, yt), which is based on a pattern of mean-reversion or OU-equation or Fokker–Planck equation in statistical physics.

## 3.4 Moduli State Space Model in 14-Dimension Superstatistical Theory

We separate the system of the financial market into two parts, the first is the state part Xt of the hidden demand state and the second is the space part Yt of the observation of the supply space of the state space model. In this work, we use an algebraic equation from the algebraic topology and the differential geometry as a main tool for the definition of a new mathematical object for an arbitrary opportunity in the DSGE system of macroeconomics.

Theorem 1: When the market is in equilibrium, we have

$s2=0↔H−14(A, s)=0.$(26)

Proof:

We are in the 14 hidden dimensional modeling in which each element of the cohomology group cancels each other when the market in equilibrium. In the study by Getzler [16], we have an element of the BV cohomology group as a superprobability of the transition nonoriented superstate of the ghost fields, f, g in financial market as

$H−14(A, s)={∫αk(Φi+(yt))+βk(Φj(xt))|Φi+(yt), Φj(xt)∈O⊂A},$(27)

where αk, βk are cocycles of the ghost and antighost fields ${\Phi }_{j}\left({x}_{t}\right),\text{\hspace{0.17em}}{\Phi }_{i}^{+}\left({y}_{t}\right)\in \mathcal{A}.$ We have a master equation in the financial suppermanifold 𝒜 as

$∫H−14(A,s)S(Φ1, Φ2,…,Φ14)dt, {∫Sdt, ∫Sdt}=0.$(28)

Let 𝒜 be a supermanifold of the financial market with the master equation (𝒜, s). The associate invariant algebraic group property of the market with higher dimensions influencing the factor of the market can be expressed by using the BV cohomology in the algebraic equation

$H−k(A, s)=0$(29)

We divide the market into two separate sheets of the D-brane and the anti-D-brane of the embedded indifference curve of supply and the utility curve of demand. The interaction of two D-branes is induced from the trade between the supply and the demand as the general equilibrium point. We define the D-brane sheet of the market in the real dimension and the antiself-duality (AdS) of the D-brane to the anti-D-brane is induced from the duality map from the supply to the demand.

Let IS-LM be written by

$yt=αt+βtxt+ϵt$(30)

Take a ghost functor ${\Phi }_{i},\text{\hspace{0.17em}}{\Phi }_{i}^{+}:X\to \left(\mathcal{A},\text{\hspace{0.17em}}s\right)\simeq \left[X,\text{\hspace{0.17em}}{S}^{±k}\right],$

$Φi((yt−αt)−βtxt≃ϵt).$(31)

Let ϵt be a real present shock from economics and ${ϵ}_{t}^{\ast }$ be the expected shock in the future. In the equilibrium, we assume the steady state of macroeconomics with no shock, i.e. ${ϵ}_{t}^{2}=0.$ Let 0 be a space of the equilibrium and let it be equivalent to the moduli state space model of the supply space Yt and the demand space Xt of the market with ${ϵ}_{t}^{2}=<{ϵ}_{t},\text{\hspace{0.17em}}{ϵ}_{t}^{\ast }>\simeq {Y}_{t}/{X}_{t}.$ Consider a short exact sequence of the macroeconomics in the general equilibrium with the market risk βt of the sudden shock in the demand side and transfer it into the supply side of the economics and let systematics risk αt be a shock on both sides with price sticky market. The shock in economics induces a business cycle that we can notice from the empirical data of the US unemployment from 1945 to present (see Fig. 8).

Figure 8:

Left panel, the unemployment rate for the US macroeconomy from 1945 to 2017, monthly data. We notice a business cycle from a time series data of the unemployment rate. Right panel, we use the moduli state space model of (IMF-ITD)-chain-(1, 4) transformation [32] to detect a business cocycle.

$0→ℤ/2→Xt→βtYt→αtYt/Xt→0,$(32)

where we use the well-known exact sequence of the spinor

$0→ℤ/2→Spin(n):=Xt→βtSO(n):=Yt→αtYt/Xt→0.$(33)

From this comparison between two exact sequences, we obtain the moduli state space model Xt as a behaviour of the trader on the demand side defined by the spin group and the Pauli matrix. The observation space of the supply side of the market is a physiological space of time series data that is spanned by basis of the equivalent class of spin, ≪[s1], [s2], [s3], [s4]≫:=Yt. In the macroeconomics model of the market, this space can be defined by the supply side of the market with the Lie group Yt:=SO(n)=Spin(n)/ℤ2. The moduli group ℤ2 defines a state above and below the underlying financial time series data. When the market is in equilibrium, the short exact sequence will induce an infinite exact sequence of the market cocycles βt and αt.

Now, we define an expected ghost field in the superspace of time series data and the superpoint in time series data of the underlying financial time series data.

Definition 10: Let S−1 be a space of unit circle with one hidden dimension. The minus sign means that this space is a future space and not a real space. It is induced from the expectation field underlying one factor as one dimension of the market factor. We explicitly define an associate real number with negative rank ℝn and a complex number with the negative dimension 𝒞n and we define S−1S1*, where ∗ is the mirror symmetry operation in the supersymmetry. The algebraic operation is the same with the positive dimension but the quantity of data underlying that space is not real but induced from the forecasting system or from the expectation of the traders in the financial market. When we come to present time, all negative dimensions will interact with positive real dimensions and fuse to the superpoint in the superspace in time series data.

Definition 11: Let S0 be a superpoint in superspace underlying time series data of the moduli state space model (xt, yt). We define S0 by the interaction between the unit cycle S1 of the space of the present value of time series data and S−1 of the future expectation value of the superspace of time series data.

$S0=S−1∨S1=S−2∨S2⋯S−k∨Sk$(34)

Definition 12: Let Sk be a superspace underlying time series data of the moduli state space model (xt, yt). Let 𝒜 be a superspace of time series data with its ground field of the master equation s={∫Sdt, −} We define a BV cohomology group for the superspace in time series data a with negative k dimension Hk (𝒜, s) by using a homotopy class as a functor from TOP to GROUP.

$H−k(A, s)=[(A, s), S−k].$(35)

The meaning of the cohomology group of the negative dimension is the superdistribution of the underlying superspace in time series data as the probability superdistribution of the future events.

We define the superspace of the financial market as the hidden layer of the extra dimensions of the Kolmogorov space in time series data. By this definition, given a short exact sequence of the moduli state space model above, we induce an associate short exact sequence in the superspace layer of the sheave cohomology,

$0→(A, s)→S14→S11→S11/S14∼S11∨S−14∼S−3→j0~ϵt∗↓↓Φi↓↓↓S3∨S−3∼S0↓<ϵt,ϵt∗>0→ℤ/2→Xt→βtYt→αtYt/Xt∼S3→i0~ϵt$(36)

where i, j are the projection maps from the expected observation superspace S±3 of spinor field in time series data to shock and expected shock in time series data; when the equilibrium holds, i.e. ϵt~0,

$j:S−3→ϵt∗, j:S3→ϵt,$(37)

$:S3∨S−3∼S0→ϵt2.$(38)

To compute the market on the level of the invariance of the dimension of the cohomology of the sphere, we use the fact from the sheave cohomology that a short exact sequence induces an infinitely long exact sequence. The meaning of this infinite sequence comes from the homotopy path of the embedded point to the sphere and from the embedded sphere to torus and so on. We consider a short exact sequence of the moduli state space,

$0→ℤ/2→Xt→Yt→Yt/Xt→0$(39)

This sequence induces an infinite sequence

$0→ℤ/2→Xt→Yt→Yt/Xt→H1(ℤ/2;ℤ/2) →H1(Xt;ℤ/2)→H1(Yt;ℤ/2)→H1(Yt/Xt;ℤ/2) →H2(ℤ/2;ℤ/2)→H2(Xt;ℤ/2)→H2(Yt;ℤ/2) →H2(Yt/Xt;ℤ/2)→H3(ℤ/2;ℤ/2)→H3(Xt;ℤ/2) →H3(Yt;ℤ/2)→H3(Yt/Xt;ℤ/2)→⋯→H11(ℤ/2;ℤ/2) →H11(Xt;ℤ/2)→H11(Yt;ℤ/2)→H11(Yt/Xt;ℤ/2)→⋯$(40)

In the hidden layer of the superspace in time series data, we induce an infinitely long exact sequence in the BV cohomology of a negative dimension

$(A, s)→H−1(A, s)→H−2(A, s)→⋯→H−14(A, s)→⋯.$(41)

We also have an exact sequence

$0→(A, s)→S14→S11→S11/S14∼S11∨S−14∼S−3→0$(42)

in which we induce an infinite exact sequence of the cohomology of a negative dimension of a sphere,

$0→H−1((A, s))→H−1(S14)→H−1(S11)→H−1(S−3) →H−2((A, s))→H−2(S14)→H−2(S11)→⋯$(43)

When the market is in general equilibrium, we have Hk (𝒜, s)=0 for all k>0. For details of the proof, see the study by Getzler [16].

The BV cohomology group is

$⋯→H−7(A, s)→H−8(A, s)→→H−9(A, s)⋯→H−14(A, s)$(44)

for the pullback functor of the effect of the Calabi–Yau manifold.

Let xtXt be a moduli state space of a D-brane and anti-D-brane. Let ytYt be an observation space of the financial market.

$H−14(xt∼yt)=0=αtyt−βtxt :=∑i=12nigi: =pp(ϵt2): = <αt, βt> ∗,$(45)

where αtyt is a group operation of the reversed direction of the translation, i.e. αtyt:=g(yt)=ytαt and βtxt is a group action of the spinor rotational group. The equilibrium is the anomaly cancelation and the net area will be contractible to the point. The curvature in the area of the market in the equilibrium state cancels each other and the net sum is zero. The meaning of the zeros in the BV cohomology group is the analogy with the zero cohomology group of a plane without an obstruction component. We explicitly define a ghost field of the financial market in the loop space of time series data (see Fig. 9) by using a fundamental group over a homotopy path

Figure 9:

Left panel, two layers of the market. The upper layer is a four-dimensional model of the financial market S3 as torus S1×S1 with the Hopf fibration. The lower layer is a hidden layer of the market as a superspace in time series data. This layer is associated with an expectation layer of the market in which the induction from the interaction between 14 ghost fields of the behaviour of the traders in the financial market on the expected 14 market factors influence the market future equilibrium point in the physiology of time series data. In this proof, we need to define a spectral sequence of the space – so-called negative dimension unit sphere S−1 as a space of the expectation ghost field in the financial market. In the economic modeling of the supermanifold, the negative dimension S−1 is not yet defined because no one has yet started to use the supersymmetry theory and the BV cohomology in the economy. On the right panel, where we use the BV cohomology group of the negative order 3 to compute the space of Xt, the general equilibrium of the market exists as knots of the market risk βt and the systematics risk αt. We induce the general equilibrium point of the market as the superpoint in the superspace of the financial market with S0~S−3∨S3. In this model, we can generally associate the hidden sphere with any dimension up to the market factors influencing the underlying financial market by using the sheave cohomology as the main tool for computing the market equilibrium.

$Φi:Yt×I→S11, Φi+:Xt×I→S−14$(46)

where S−1 is a unit sphere with a negative dimension (nonoriented supermanifold of its parity of S1) mod 2.

We explicitly define an element of a ghost field by using a group action over a cotangent bundle of the supermanifold as a cocycle αt and a business cycle βt. Thus, we obtain the result

$Yt/Xt=αt[yt]βt([xt])≃[ϵt∗]∈S−3.$(47)

We explicitly define 14 ghost and 11 antighost fields in the financial market with the demand side of the state space with xtH−14(Xt):=[Xt, S−14] and with the supply side ytH11(Yt)=[Yt, S11]. In the market equilibrium, we have

$S3|ϵt∨S11∨S−14|ϵt∗∼S3∨S−3≃S0∍ϵt2=0.$(48)

If the market is in nonequilibrium, then there exists a cocycle (αt, βt) and ϵt≠0 such that

$yt=αt+βtxt+ϵt.$(49)

## 4 Discussion and Conclusion

In this work, we provided an explanation for the asymmetric model of the financial market and its duality. We explained why the financial market equilibrium will give zeros for the BV cohomology group of 14 negative dimensions. We use a new G-theory to explain the financial market in 14 dimensions. Our model is not the E8×E8 and SO(32) model of the financial market over 11 dimensions, M-theory. We assume that under the record of financial time series data, there exists a superspace as a supermanifold of the ghost and antighost fields with the parity mode 2 induced from the supply and demand ghost fields of the market. These fields wrap each other similar to the D-brane interaction with the anti-D-brane model in the unified theory. We study the ghost and antighost fields in the superspace of the market in the mirror symmetry with the anomaly as an arbitrary opportunity. We define a new BV cohomology for the financial market with its market supersymmetry and cohomology as its duality between the supply and demand as a superspace of time series data. We use supermanifold modeling in the space of time series data when the space is in the nonoriented hidden ghost field. We define a new equation for the financial market in which one can unify the microeconomic and macroeconomic theories by using the Chern–Simon and Yang–Mills theories with the BV cohomology. We use the G-theory approach in the unified theory in the 14 dimensional model of the birth of the universe to unify the economic parameters in the superspace of time series data with the transition superprobability for the quantised ghost field in the anomaly Chern–Simon current induced from the coupling between the market cocycle of the behaviour of the traders in the financial market. We write a Yang–Mills–Chern–Simon master equation in general from the superlagrangian with the least action principle. We use the BV cohomogy to prove the existence of the DSGE point in the financial market when the cohomology is in the exact sequence and the 14 hidden dimensions are vanishing with arbitrary opportunity existing as a curvature of an embedded superspace of time series data in the form of the Calabi–Yau space in time series data in the reverse direction of the timescale. In future work, it is possible to follow the model and use computer simulation to visualise the geometry of the market in equilibrium and to perform some empirical analyses over 14 major hidden factors in the supply side and 11 factors in the demand side according to our high dimensional economic modeling. To find a market general equilibrium as arbitrary opportunity, we use the connection and the covariant derivative to error of the forecasting market of all the hidden factors and take it to zeros according to our providing theorem of the 14th BV cohomology group. We conclude that the disappearance of the cohomology group will satisfy the stability of the topological property of the market and the empirical analysis will show us the precise system of complicated equations for the prediction of all parameters from the economic factors in which one can influence an equilibrium point of the market. Our research shows that the market equilibrium will produce a spinor field of the systematic risk and the market risk as the coupling market cocycle and vanishing as an induced spinor field from the general equilibrium point of the financial market in the form of an arbitrary opportunity over the market mirror symmetric property. We need exactly 14 dimensions for the real economic modeling of the global financial market and for the real computer simulation results on the real data to visualise the general equilibrium model as our future work to verify our theoretical modeling of the cohomology group of the superspace of the market.

## Acknowledgment

S. Capozziello acknowledges the financial support of the INFN (iniziative specifiche TEONGRAV and QGSKY). K. Kanjamapornkul is supported by a scholarship from the 100th Anniversary Chulalongkorn University Fund for Doctoral Scholarship. The work was partly supported by VEGA grant no. 2/0009/16. R. Pincak would like to thank the TH division at CERN for hospitality. We are particularly grateful to Erik Bartoš for the important comments and observations.

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Accepted: 2017-10-11

Published Online: 2017-11-16

Published in Print: 2017-11-27

Citation Information: Zeitschrift für Naturforschung A, Volume 72, Issue 12, Pages 1077–1091, ISSN (Online) 1865-7109, ISSN (Print) 0932-0784,

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