We separate the system of the financial market into two parts, the first is the state part *X*_{t} of the hidden demand state and the second is the space part *Y*_{t} 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*

$${s}^{2}=0\leftrightarrow {H}^{-14}\mathrm{(}\mathcal{A}\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}=0.$$(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

$$\begin{array}{l}{H}^{-14}\mathrm{(}\mathcal{A}\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}=\{{\displaystyle \int {\alpha}_{k}\mathrm{(}{\Phi}_{i}^{+}\mathrm{(}{y}_{t}\mathrm{)}\mathrm{)}+{\beta}_{k}\mathrm{(}{\Phi}_{j}\mathrm{(}{x}_{t}\mathrm{)}\mathrm{)}|{\Phi}_{i}^{+}\mathrm{(}{y}_{t}\mathrm{}\mathrm{)}\mathrm{,}\text{\hspace{0.17em}}}\\ \text{\hspace{1em}}{\Phi}_{j}\mathrm{(}{x}_{t}\mathrm{)}\in \mathcal{O}\subset \mathcal{A}\}\mathrm{,}\end{array}$$(27)

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

$${\int}_{{H}^{-14}\mathrm{(}\mathcal{A}\mathrm{,}s\mathrm{)}}S\mathrm{(}{\Phi}_{1}\mathrm{,}\text{\hspace{0.17em}}{\Phi}_{2}\mathrm{,}\dots ,{\Phi}_{14}\mathrm{)}\text{d}t}\mathrm{,}\text{\hspace{0.17em}}\left\{{\displaystyle \int S\text{d}t}\mathrm{,}\text{\hspace{0.17em}}{\displaystyle \int S\text{d}t}\right\}=0.$$(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

$${H}^{-k}\mathrm{(}\mathcal{A}\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}=0$$(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

$${y}_{t}={\alpha}_{t}+{\beta}_{t}{x}_{t}+{\u03f5}_{t}$$(30)

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

$${\Phi}_{i}\mathrm{(}\mathrm{(}{y}_{t}-{\alpha}_{t}\mathrm{)}-{\beta}_{t}{x}_{t}\simeq {\u03f5}_{t}\mathrm{}\mathrm{)}\mathrm{.}$$(31)

Let *ϵ*_{t} be a real present shock from economics and ${\u03f5}_{t}^{\ast}$ be the expected shock in the future. In the equilibrium, we assume the steady state of macroeconomics with no shock, i.e. ${\u03f5}_{t}^{2}=0.$ Let 0 be a space of the equilibrium and let it be equivalent to the moduli state space model of the supply space *Y*_{t} and the demand space *X*_{t} of the market with ${\u03f5}_{t}^{2}=<{\u03f5}_{t}\mathrm{,}\text{\hspace{0.17em}}{\u03f5}_{t}^{\ast}>\simeq {Y}_{t}\mathrm{/}{X}_{t}.$ Consider a short exact sequence of the macroeconomics in the general equilibrium with the market risk *β*_{t} of the sudden shock in the demand side and transfer it into the supply side of the economics and let systematics risk *α*_{t} 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.

$$0\to \mathbb{Z}\mathrm{/}2\mathrm{\to}{X}_{t}\stackrel{{\beta}_{t}}{\to}{Y}_{t}\stackrel{{\alpha}_{t}}{\to}{Y}_{t}\mathrm{/}{X}_{t}\to \mathrm{0,}$$(32)

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

$$0\to \mathbb{Z}\mathrm{/}2\mathrm{\to}Spin\mathrm{(}n\mathrm{)}:={X}_{t}\stackrel{{\beta}_{t}}{\to}SO\mathrm{(}n\mathrm{)}:={Y}_{t}\stackrel{{\alpha}_{t}}{\to}{Y}_{t}\mathrm{/}{X}_{t}\to 0.$$(33)

From this comparison between two exact sequences, we obtain the moduli state space model *X*_{t} 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, ≪[*s*_{1}], [*s*_{2}], [*s*_{3}], [*s*_{4}]≫:=*Y*_{t}. In the macroeconomics model of the market, this space can be defined by the supply side of the market with the Lie group *Y*_{t}:=*SO*(*n*)=*Spin*(*n*)/ℤ_{2}. The moduli group ℤ_{2} 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 *β*_{t} and *α*_{t}.

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 ℝ^{−n} and a complex number with the negative dimension 𝒞^{−n} and we define *S*^{−1}*S*^{1*}, 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 *S*^{0} be a superpoint in superspace underlying time series data of the moduli state space model (*x*_{t}, *y*_{t}). We define *S*^{0} by the interaction between the unit cycle *S*^{1} 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.

$${S}^{0}={S}^{-1}\vee {S}^{1}={S}^{-2}\vee {S}^{2}\cdots {S}^{-k}\vee {S}^{k}$$(34)

**Definition 12:** Let *S*^{−k} be a superspace underlying time series data of the moduli state space model (*x*_{t}, *y*_{t}). 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 *H*^{−k} (𝒜, *s*) by using a homotopy class as a functor from TOP to GROUP.

$${H}^{-k}\mathrm{(}\mathcal{A}\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}=\mathrm{[}\mathrm{(}\mathcal{A}\mathrm{,}\text{\hspace{0.17em}}s\mathrm{}\mathrm{)}\mathrm{,}\text{\hspace{0.17em}}{S}^{-k}\mathrm{}\mathrm{]}\mathrm{.}$$(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,

$$\begin{array}{ccccccccccc}0& \to & \mathrm{(}\mathcal{A}\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}& \to & {S}^{14}& \to & {S}^{11}& \to & {S}^{11}/{S}^{14}\sim {S}^{11}\vee {S}^{-14}\sim {S}^{-3}& \underset{j}{\to}& 0~{\u03f5}_{t}^{\ast}\\ \downarrow & & \downarrow {\Phi}_{i}& & \downarrow & & \downarrow & & \downarrow {S}^{3}\vee {S}^{-3}\sim {S}^{0}& & \downarrow <{\u03f5}_{t},{\u03f5}_{t}^{\ast}>\\ 0& \to & \mathbb{Z}/2& \to & {X}_{t}& \stackrel{{\beta}_{t}}{\to}& {Y}_{t}& \stackrel{{\alpha}_{t}}{\to}& {Y}_{t}/{X}_{t}\sim {S}^{3}& \underset{i}{\to}& 0~{\u03f5}_{t}\end{array}$$(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. *ϵ*_{t}~0,

$$j\mathrm{:}{S}^{-3}\to {\u03f5}_{t}^{\ast}\mathrm{,}\text{\hspace{0.17em}}j\mathrm{:}{S}^{3}\to {\u03f5}_{t}\mathrm{,}$$(37)

$$<i\mathrm{,}\text{\hspace{0.17em}}j>\mathrm{:}{S}^{3}\vee {S}^{-3}\sim {S}^{0}\to {\u03f5}_{t}^{2}\mathrm{.}$$(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,

$$0\to \mathbb{Z}\mathrm{/}2\mathrm{\to}{X}_{t}\to {Y}_{t}\to {Y}_{t}\mathrm{/}{X}_{t}\to 0$$(39)

This sequence induces an infinite sequence

$$\begin{array}{l}0\to \mathbb{Z}\mathrm{/}2\mathrm{\to}{X}_{t}\to {Y}_{t}\to {Y}_{t}\mathrm{/}{X}_{t}\to {H}^{1}\mathrm{(}\mathbb{Z}/2;\mathbb{Z}/2\mathrm{)}\\ \text{\hspace{1em}}\to {H}^{1}\mathrm{(}{X}_{t}\mathrm{;}\mathbb{Z}/2\mathrm{)}\to {H}^{1}\mathrm{(}{Y}_{t}\mathrm{;}\mathbb{Z}/2\mathrm{)}\to {H}^{1}\mathrm{(}{Y}_{t}\mathrm{/}{X}_{t}\mathrm{;}\mathbb{Z}/2\mathrm{)}\\ \text{\hspace{1em}}\to {H}^{2}\mathrm{(}\mathbb{Z}/2;\mathbb{Z}/2\mathrm{)}\to {H}^{2}\mathrm{(}{X}_{t}\mathrm{;}\mathbb{Z}/2\mathrm{)}\to {H}^{2}\mathrm{(}{Y}_{t}\mathrm{;}\mathbb{Z}/2\mathrm{)}\\ \text{\hspace{1em}}\to {H}^{2}\mathrm{(}{Y}_{t}\mathrm{/}{X}_{t}\mathrm{;}\mathbb{Z}/2\mathrm{)}\to {H}^{3}\mathrm{(}\mathbb{Z}/2;\mathbb{Z}/2\mathrm{)}\to {H}^{3}\mathrm{(}{X}_{t}\mathrm{;}\mathbb{Z}/2\mathrm{)}\\ \text{\hspace{1em}}\to {H}^{3}\mathrm{(}{Y}_{t}\mathrm{;}\mathbb{Z}/2\mathrm{)}\to {H}^{3}\mathrm{(}{Y}_{t}\mathrm{/}{X}_{t}\mathrm{;}\mathbb{Z}/2\mathrm{)}\to \cdots \to {H}^{11}\mathrm{(}\mathbb{Z}/2;\mathbb{Z}/2\mathrm{)}\\ \text{\hspace{1em}}\to {H}^{11}\mathrm{(}{X}_{t}\mathrm{;}\mathbb{Z}/2\mathrm{)}\to {H}^{11}\mathrm{(}{Y}_{t}\mathrm{;}\mathbb{Z}/2\mathrm{)}\to {H}^{11}\mathrm{(}{Y}_{t}\mathrm{/}{X}_{t}\mathrm{;}\mathbb{Z}/2\mathrm{)}\to \cdots \end{array}$$(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

$$\mathrm{(}\mathcal{A}\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}\to {H}^{-1}\mathrm{(}\mathcal{A}\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}\to {H}^{-2}\mathrm{(}\mathcal{A}\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}\to \cdots \to {H}^{-14}\mathrm{(}\mathcal{A}\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}\to \cdots \mathrm{.}$$(41)

We also have an exact sequence

$$0\to \mathrm{(}A\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}\to {S}^{14}\to {S}^{11}\to {S}^{11}\mathrm{/}{S}^{14}\sim {S}^{11}\vee {S}^{-14}\sim {S}^{-3}\to 0$$(42)

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

$$\begin{array}{l}0\to {H}^{-1}\mathrm{(}\mathrm{(}A\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}\mathrm{)}\to {H}^{-1}\mathrm{(}{S}^{14}\mathrm{)}\to {H}^{-1}\mathrm{(}{S}^{11}\mathrm{)}\to {H}^{-1}\mathrm{(}{S}^{-3}\mathrm{)}\\ \text{\hspace{1em}}\to {H}^{-2}\mathrm{(}\mathrm{(}A\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}\mathrm{)}\to {H}^{-2}\mathrm{(}{S}^{14}\mathrm{)}\to {H}^{-2}\mathrm{(}{S}^{11}\mathrm{)}\to \cdots \end{array}$$(43)

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

The BV cohomology group is

$$\cdots \to {H}^{-7}\mathrm{(}\mathcal{A}\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}\to {H}^{-8}\mathrm{(}\mathcal{A}\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}\to \to {H}^{-9}\mathrm{(}\mathcal{A}\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}\cdots \to {H}^{-14}\mathrm{(}\mathcal{A}\mathrm{,}\text{\hspace{0.17em}}s\mathrm{)}$$(44)

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

Let *x*_{t}∈*X*_{t} be a moduli state space of a D-brane and anti-D-brane. Let *y*_{t}∈*Y*_{t} be an observation space of the financial market.

$$\begin{array}{l}{H}^{-14}\mathrm{(}{x}_{t}\sim {y}_{t}\mathrm{)}=0={\alpha}_{t}{y}_{t}-{\beta}_{t}{x}_{t}\\ \text{\hspace{1em}}\mathrm{:}={\displaystyle \sum _{i=1}^{2}}{n}_{i}{g}_{i}\mathrm{:}\text{\hspace{0.17em}}=pp\mathrm{(}{\u03f5}_{t}^{2}\mathrm{)}:\text{\hspace{0.17em}}=\text{\hspace{0.17em}}<{\alpha}_{t}\mathrm{,}\text{\hspace{0.17em}}{\beta}_{t}>\text{\hspace{0.17em}}<{x}_{t}\mathrm{,}\text{\hspace{0.17em}}{y}_{t}{>}^{\ast}\mathrm{,}\end{array}$$(45)

where *α*_{t}y_{t} is a group operation of the reversed direction of the translation, i.e. *α*_{t}y_{t}:=*g*(*y*_{t})=*y*_{t}^{−}*α*_{t} and *β*_{t}x_{t} 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 *S*^{3} as torus *S*^{1}×*S*^{1} 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 *X*_{t}, the general equilibrium of the market exists as knots of the market risk *β*_{t} and the systematics risk *α*_{t}. We induce the general equilibrium point of the market as the superpoint in the superspace of the financial market with *S*^{0}~*S*^{−3}*∨S*^{3}. 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.

$${\Phi}_{i}\mathrm{:}{Y}_{t}\times I\to {S}^{11}\mathrm{,}\text{\hspace{0.17em}}{\Phi}_{i}^{+}\mathrm{:}{X}_{t}\times I\to {S}^{-14}$$(46)

where *S*^{−1} is a unit sphere with a negative dimension (nonoriented supermanifold of its parity of *S*^{1}) *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 *α*_{t} and a business cycle *β*_{t}. Thus, we obtain the result

$${Y}_{t}\mathrm{/}{X}_{t}=\frac{{\alpha}_{t}\mathrm{[}{y}_{t}\mathrm{]}}{{\beta}_{t}\mathrm{(}[\mathrm{}{x}_{t}]\mathrm{)}}\simeq \mathrm{[}{\u03f5}_{t}{}^{\ast}\mathrm{]}\in {S}^{-3}\mathrm{.}$$(47)

We explicitly define 14 ghost and 11 antighost fields in the financial market with the demand side of the state space with *x*_{t}∈*H*^{−14}(*X*_{t}):=[*X*_{t}, *S*^{−14}] and with the supply side *y*_{t}∈*H*^{11}(*Y*_{t})=[*Y*_{t}, *S*^{11}]. In the market equilibrium, we have

$${S}^{3}{\mathrm{|}}_{{\u03f5}_{t}}\vee {S}^{11}\vee {S}^{-14}{\mathrm{|}}_{{\u03f5}_{t}^{\ast}}\sim {S}^{3}\vee {S}^{-3}\simeq {S}^{0}\u220d{\u03f5}_{t}^{2}=0.$$(48)

If the market is in nonequilibrium, then there exists a cocycle (*α*_{t}, *β*_{t}) and *ϵ*_{t}≠0 such that

$${y}_{t}={\alpha}_{t}+{\beta}_{t}{x}_{t}+{\u03f5}_{t}\mathrm{.}$$(49)

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