5. Emergent gravity

γ0 ð Þ<sup>x</sup> � �<sup>0</sup>

ð Þ<sup>x</sup> ð Þ <sup>a</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; <sup>0</sup> and the auxiliary gauge fields <sup>A</sup><sup>a</sup><sup>0</sup>

Φ<sup>t</sup> x !; t � �

) Φ !0 x !; t � �

) Φ<sup>0</sup> <sup>t</sup> x !; t � �

e a

dΦ<sup>i</sup> ð Þx <sup>2</sup><sup>π</sup> <sup>¼</sup> ni

On the other hand, within the representation of <sup>Γ</sup><sup>i</sup> <sup>¼</sup> <sup>γ</sup>~<sup>0</sup> we have

e 0 dΦ<sup>a</sup> ð Þx <sup>2</sup><sup>π</sup> <sup>¼</sup> <sup>n</sup>~<sup>a</sup>

ð Þ¼ x dtð Þ¼ x

ð Þ<sup>x</sup> and <sup>A</sup><sup>i</sup><sup>0</sup>

e i

ð Þ¼ <sup>x</sup> dxa

Φ ! x !; t � �

For the smoothly deformed vector-fields n<sup>i</sup>

Thus, the relationship between ei

For a non-uniform zero-lattice, we have

ea

vierbein fields e<sup>a</sup>

56 Superfluids and Superconductors

have

the 3D space.

the 4D space-time.

and

<sup>¼</sup> <sup>U</sup><sup>b</sup> ð Þ<sup>x</sup> <sup>γ</sup><sup>0</sup> <sup>U</sup><sup>b</sup> ð Þ<sup>x</sup>

� ��<sup>1</sup>

by using gauge description. There exists intrinsic relationship between the geometry fields

¼ Φ ! x !; t � �

On deformed zero-lattice, the "lattice distances" become dynamic vector fields. We define the

and to behave as ordinary vectors under local entanglement transformation along xa-direction,

ð Þ¼ x a π dΦ<sup>a</sup>

ð Þ¼ <sup>x</sup> tr <sup>γ</sup><sup>0</sup>

<sup>¼</sup> <sup>A</sup><sup>i</sup><sup>0</sup>

ð Þ� <sup>x</sup> ð Þ <sup>2</sup><sup>a</sup> Ai<sup>0</sup>

According to this relationship, the changing of entanglement of the vortex-membranes curves

ð Þ¼ <sup>x</sup> tr <sup>γ</sup>~<sup>0</sup>

<sup>¼</sup> <sup>A</sup><sup>~</sup> <sup>i</sup><sup>0</sup>

a π

According to this relationship, the changing of entanglement of the vortex-membranes curves

dγ~<sup>a</sup> ð Þ<sup>x</sup> � �

ð Þx , i ¼ j, k, 0,

¼ Φ<sup>t</sup> x !; t � �

<sup>¼</sup> <sup>X</sup> a γa na

> þ δ Φ ! x !; t � � ,

þ δΦ<sup>t</sup> x !; t � � :

ð Þx .

ð Þx that are supposed to transform homogeneously under the local symmetry,

dγ<sup>i</sup> ð Þ<sup>x</sup> � �

ð Þx , i ¼ 1; 2; 3:

ð Þx is obtained as

ð Þx (104)

ð Þx : (106)

ð Þx : (108)

<sup>d</sup>Φtð Þ¼ <sup>x</sup> ð Þ <sup>2</sup><sup>a</sup> <sup>A</sup><sup>~</sup> <sup>30</sup>ð Þ<sup>x</sup> : (110)

ð Þ<sup>x</sup> <sup>≪</sup> 1, within the representation of <sup>Γ</sup><sup>5</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup> we

(105)

(107)

(109)

Gravity is a natural phenomenon by which all objects attract one another including galaxies, stars, human-being and even elementary particles. Hundreds of years ago, Newton discovered the inverse-square law of universal gravitation, <sup>F</sup> <sup>¼</sup> GMm <sup>r</sup><sup>2</sup> where G is the Newton constant, r is the distance, and M and m are the masses for two objects. One hundred years ago, the establishment of general relativity by Einstein is a milestone to learn the underlying physics of gravity that provides a unified description of gravity as a geometric property of space-time. From Einstein's equations <sup>R</sup>μν � <sup>1</sup> <sup>2</sup> Rgμν ¼ 8πGTμν, the gravitational force is really an effect of curved space-time. Here Rμν is the 2nd rank Ricci tensor, R is the curvature scalar, gμν is the metric tensor, and Tμν is the energy-momentum tensor of matter.

In this section, we point out that there exists emergent gravity for knots on zero-lattice.

#### 5.1. Knots as topological defects

## 5.1.1. Knot as SO(4)/SO(3) topological defect in 3 + 1D space-time

A knot corresponds to an elementary object of a knot-crystal; a knot-crystal can be regarded as composite system of multi-knot. For example, for 1D knot, people divide the knot-crystal into N identical pieces, each of which is just a knot.

From point view of information, each knot corresponds to a zero between two vortexmembranes along the given direction. For a knot, there must exist a zero point, at which ξAð Þx is equal to ξBð Þx . The position of the zero is determined by a local solution of the zeroequation, Fθð Þ¼ x 0 or ξA,θð Þ¼ x ξB,θð Þx .

From point view of geometry, a knot (an anti-knot) removes (or adds) a projected zero of zerolattice that corresponds to removes (or adds) half of "lattice unit" on the zero-lattice according to

$$
\Delta \mathbf{x}\_i = \pm a\_{\text{eff}} \left( \overrightarrow{\mathbf{x}}, t \right) \simeq \pm a. \tag{111}
$$

As a result, a knot looks like a special type of edge dislocation on 3 + 1D zero-lattice. The zerolattice is deformed and becomes mismatch with an additional knot.

From point view of entanglement, a knot becomes topological defect of 3 + 1D winding spacetime: along x-direction, knot is anti-phase changing denoted by e<sup>i</sup>Γ1�ΔΦ<sup>x</sup> , ΔΦx ¼ π; along y-direction, knot is anti-phase changing denoted by eiΓ2�ΔΦ<sup>y</sup> , ΔΦ<sup>y</sup> ¼ π; along z-direction, knot is anti-phase changing denoted by e<sup>i</sup>Γ3�ΔΦ<sup>z</sup> , ΔΦ<sup>z</sup> ¼ π; along t-direction, knot is anti-phase changing denoted by e<sup>i</sup>Γ5�ΔΦ<sup>t</sup> , ΔΦ<sup>t</sup> ¼ π. Figure 4(a) and (b) shows an illustration a 1D knot.

In mathematics, to generate a knot at x0; y0; z0; t<sup>0</sup> , we do global topological operation on the knot-crystal, i.e.,

$$e^{i\Gamma^1 \cdot \Delta \otimes \mathbf{x}}(\mathbf{x})|0\rangle$$

tangential entanglement matrices in Figure 4(c) and (d). In Figure 4(d), local tangential entanglement matrices induced by an extra (unified) knot shows vortex-like topological configuration in projected 2D space (for example, x-y plane). As a result, local tangential entanglement matrices induced by an extra knot can be exactly mapped onto that of an orientable sphere

To characterize the topological property of 3 + 1D zero-lattice with an extra (unified) knot, we

ðð <sup>e</sup>jkeijkFjk

jk � dSi (117)

http://dx.doi.org/10.5772/intechopen.72809

59

� �A<sup>i</sup><sup>0</sup> <sup>∧</sup> Aj<sup>0</sup> (118)

Topological Interplay between Knots and Entangled Vortex-Membranes

jk label the local entanglement matrices on the

jk � dSi just indicates the local entanglement matrices on the

jk denote the spatial direction. The non-zero

x ¼ qm (119)

4π

<sup>F</sup>ij <sup>¼</sup> dAij <sup>þ</sup> <sup>A</sup>ik <sup>∧</sup> Akj

tangential sub-space Ai<sup>0</sup> ∧ A<sup>j</sup><sup>0</sup> to be the local frame of an orientable sphere with fixed chirality.

As a result, the entanglement pattern with an extra 3D knot is topologically deformed and the 3D knot becomes SO(3)/SO(2) magnetic monopole. From the point view of gauge description, a

Next, we study the spatial entanglement deformation and define <sup>Γ</sup><sup>i</sup> <sup>¼</sup> <sup>γ</sup>~0. Under this gauge description, we can only study the effect of a knot on 2D spatial zero-lattice and 1D tempo

Due to the spatial-tempo rotation symmetry, the knot also becomes SO(3)/SO(2) magnetic

monopole and traps a "magnetic charge" of the auxiliary gauge field A~ jk, i.e.,

Ψd<sup>3</sup>

jk � dSi is the "magnetic" charge of auxiliary gauge field Ajk. For single

ΔΦi ¼ π,ΔΦ<sup>j</sup> ¼ π,ΔΦ<sup>t</sup> ¼ π: (120)

<sup>ð</sup> ffiffiffiffiffiffi �<sup>g</sup> <sup>p</sup> <sup>Ψ</sup>†

apply gauge description and write down the following constraint

ψ. The upper indices of Fjk

knot traps a "magnetic charge" of the auxiliary gauge field, i.e.,

5.1.3. Knot as SO(3)/SO(2) magnetic monopole in 2 + 1D space-time

In the 2 + 1D space-time, a knot also leads to π-phase changing,

NF ¼

tangential sub-space and the lower indices of Fjk

ÐÐ ejkeijkFjk

4π

<sup>∭</sup> <sup>r</sup>FdV <sup>¼</sup> <sup>1</sup>

with fixed chirality.

where

and r<sup>F</sup> ¼ ffiffiffiffiffiffi

where qm <sup>¼</sup> <sup>1</sup>

zero-lattice.

4π

ÐÐ ejkeijkFjk

knot NF ¼ 1, the "magnetic" charge is qm ¼ 1.

�<sup>g</sup> <sup>p</sup> <sup>ψ</sup>†

Gaussian integrate <sup>1</sup>

with ΔΦx ¼ 0, x < x<sup>0</sup> and ΔΦx ¼ π, x ≥ x0;

$$e^{i\Gamma^2 \cdot \Delta \Phi\_y(\mathbf{x})}|0\rangle$$

with ΔΦ<sup>y</sup> ¼ 0, y < y<sup>0</sup> and ΔΦ<sup>y</sup> ¼ π, y ≥ y0;

$$e^{i\Gamma^3 \cdot \Delta \Phi\_z(\mathbf{x})}|\mathbf{0}\rangle\tag{114}$$

with ΔΦ<sup>z</sup> ¼ 0, z < z<sup>0</sup> and ΔΦ<sup>z</sup> ¼ π, z ≥ x0;

$$e^{i\Gamma^{\flat} \cdot \Delta \Phi\_l(\mathbf{x})}|0\rangle$$

with ΔΦ<sup>t</sup> ¼ 0, t < t<sup>0</sup> and ΔΦ<sup>t</sup> ¼ π, t ≥ t0. As a result, due to the rotation symmetry in 3 + 1D space-time, a knot becomes SO(4)/SO(3) topological defect. Along arbitrary direction, the local entanglement matrices around a knot at center are switched on the tangential sub-space-time.

#### 5.1.2. Knot as SO(3)/SO(2) magnetic monopole in 3D space

To characterize the topological property of a knot on the 3 + 1D zero-lattice, we use gauge description. We firstly study the tempo entanglement deformation and define <sup>Γ</sup><sup>5</sup> <sup>¼</sup> <sup>γ</sup>0. Under this gauge description, we can only study the effect of a knot on three spatial zero-lattice.

When there exists a knot, the periodic boundary condition of knot states along arbitrary direction is changed into anti-periodic boundary condition,

$$
\Delta\Phi\mathbf{x} = \pi, \Delta\Phi\_y = \pi, \Delta\Phi\_z = \pi. \tag{116}
$$

Consequently, along given direction (for example x-direction), the local entanglement matrices on the tangential sub-space are switched by eiΓ1�ΔΦ<sup>x</sup> ð Þ <sup>Δ</sup>Φ<sup>x</sup> <sup>¼</sup> <sup>π</sup> . Along <sup>x</sup>-direction, in the limit of <sup>x</sup> ! �∞, we have the local entanglement matrices on the tangential sub-space as <sup>Γ</sup><sup>2</sup> and <sup>Γ</sup><sup>3</sup> ; in the limit of x ! ∞, we have the local entanglement matrices on the tangential sub-space as <sup>e</sup><sup>i</sup>Γ1�ΔΦ<sup>x</sup> <sup>Γ</sup><sup>2</sup> <sup>e</sup>�iΓ1�ΔΦ<sup>x</sup> ¼ �Γ<sup>2</sup> and <sup>e</sup><sup>i</sup>Γ1�ΔΦ<sup>x</sup> <sup>Γ</sup><sup>3</sup> <sup>e</sup>�iΓ1�ΔΦ<sup>x</sup> ¼ �Γ<sup>3</sup> .

Because we have similar result along xi -direction for the system with an extra knot, the system has generalized spatial rotation symmetry. Due to the generalized spatial rotation symmetry, when moving around the knot, the local tangential entanglement matrices (we may use indices j, k to denote the sub space) must rotate synchronously. See the red arrows that denote local tangential entanglement matrices in Figure 4(c) and (d). In Figure 4(d), local tangential entanglement matrices induced by an extra (unified) knot shows vortex-like topological configuration in projected 2D space (for example, x-y plane). As a result, local tangential entanglement matrices induced by an extra knot can be exactly mapped onto that of an orientable sphere with fixed chirality.

To characterize the topological property of 3 + 1D zero-lattice with an extra (unified) knot, we apply gauge description and write down the following constraint

$$\iiint \rho\_{\rm F}dV = \frac{1}{4\pi} \iiint \epsilon\_{\rm jk} \epsilon\_{ijk} F\_{jk}^{\rm jk} \cdot dS\_{i} \tag{117}$$

where

is anti-phase changing denoted by e<sup>i</sup>Γ3�ΔΦ<sup>z</sup>

with ΔΦx ¼ 0, x < x<sup>0</sup> and ΔΦx ¼ π, x ≥ x0;

with ΔΦ<sup>y</sup> ¼ 0, y < y<sup>0</sup> and ΔΦ<sup>y</sup> ¼ π, y ≥ y0;

with ΔΦ<sup>z</sup> ¼ 0, z < z<sup>0</sup> and ΔΦ<sup>z</sup> ¼ π, z ≥ x0;

5.1.2. Knot as SO(3)/SO(2) magnetic monopole in 3D space

direction is changed into anti-periodic boundary condition,

<sup>e</sup><sup>i</sup>Γ1�ΔΦ<sup>x</sup> <sup>Γ</sup><sup>2</sup> <sup>e</sup>�iΓ1�ΔΦ<sup>x</sup> ¼ �Γ<sup>2</sup> and <sup>e</sup><sup>i</sup>Γ1�ΔΦ<sup>x</sup> <sup>Γ</sup><sup>3</sup> <sup>e</sup>�iΓ1�ΔΦ<sup>x</sup> ¼ �Γ<sup>3</sup>

Because we have similar result along xi

In mathematics, to generate a knot at x0; y0; z0; t<sup>0</sup>

e <sup>i</sup>Γ1�ΔΦ<sup>x</sup>

e

e

e

with ΔΦ<sup>t</sup> ¼ 0, t < t<sup>0</sup> and ΔΦ<sup>t</sup> ¼ π, t ≥ t0. As a result, due to the rotation symmetry in 3 + 1D space-time, a knot becomes SO(4)/SO(3) topological defect. Along arbitrary direction, the local entanglement matrices around a knot at center are switched on the tangential sub-space-time.

To characterize the topological property of a knot on the 3 + 1D zero-lattice, we use gauge description. We firstly study the tempo entanglement deformation and define <sup>Γ</sup><sup>5</sup> <sup>¼</sup> <sup>γ</sup>0. Under this gauge description, we can only study the effect of a knot on three spatial zero-lattice.

When there exists a knot, the periodic boundary condition of knot states along arbitrary

Consequently, along given direction (for example x-direction), the local entanglement matrices on the tangential sub-space are switched by eiΓ1�ΔΦ<sup>x</sup> ð Þ <sup>Δ</sup>Φ<sup>x</sup> <sup>¼</sup> <sup>π</sup> . Along <sup>x</sup>-direction, in the limit of <sup>x</sup> ! �∞, we have the local entanglement matrices on the tangential sub-space as <sup>Γ</sup><sup>2</sup> and <sup>Γ</sup><sup>3</sup>

in the limit of x ! ∞, we have the local entanglement matrices on the tangential sub-space as

has generalized spatial rotation symmetry. Due to the generalized spatial rotation symmetry, when moving around the knot, the local tangential entanglement matrices (we may use indices j, k to denote the sub space) must rotate synchronously. See the red arrows that denote local

changing denoted by e<sup>i</sup>Γ5�ΔΦ<sup>t</sup>

58 Superfluids and Superconductors

knot-crystal, i.e.,

, ΔΦ<sup>z</sup> ¼ π; along t-direction, knot is anti-phase

ð Þx j i0 (112)

<sup>i</sup>Γ2�ΔΦyð Þ<sup>x</sup> j i<sup>0</sup> (113)

<sup>i</sup>Γ3�ΔΦzð Þ<sup>x</sup> j i<sup>0</sup> (114)

<sup>i</sup>Γ5�ΔΦtð Þ<sup>x</sup> j i<sup>0</sup> (115)

ΔΦx ¼ π,ΔΦ<sup>y</sup> ¼ π,ΔΦ<sup>z</sup> ¼ π: (116)


.

;

, we do global topological operation on the

, ΔΦ<sup>t</sup> ¼ π. Figure 4(a) and (b) shows an illustration a 1D knot.

$$\begin{aligned} F^{\ddagger} &= d A^{\ddagger} + A^{\ddagger} \wedge A^{kj} \\ &\equiv -A^{\ddagger0} \wedge A^{j0} \end{aligned} \tag{118}$$

and r<sup>F</sup> ¼ ffiffiffiffiffiffi �<sup>g</sup> <sup>p</sup> <sup>ψ</sup>† ψ. The upper indices of Fjk jk label the local entanglement matrices on the tangential sub-space and the lower indices of Fjk jk denote the spatial direction. The non-zero Gaussian integrate <sup>1</sup> 4π ÐÐ ejkeijkFjk jk � dSi just indicates the local entanglement matrices on the tangential sub-space Ai<sup>0</sup> ∧ A<sup>j</sup><sup>0</sup> to be the local frame of an orientable sphere with fixed chirality.

As a result, the entanglement pattern with an extra 3D knot is topologically deformed and the 3D knot becomes SO(3)/SO(2) magnetic monopole. From the point view of gauge description, a knot traps a "magnetic charge" of the auxiliary gauge field, i.e.,

$$N\_{\rm F} = \int \sqrt{-\mathcal{g}} \Psi^{\dagger} \Psi d^{3} \mathbf{x} = q\_{m} \tag{119}$$

where qm <sup>¼</sup> <sup>1</sup> 4π ÐÐ ejkeijkFjk jk � dSi is the "magnetic" charge of auxiliary gauge field Ajk. For single knot NF ¼ 1, the "magnetic" charge is qm ¼ 1.

### 5.1.3. Knot as SO(3)/SO(2) magnetic monopole in 2 + 1D space-time

Next, we study the spatial entanglement deformation and define <sup>Γ</sup><sup>i</sup> <sup>¼</sup> <sup>γ</sup>~0. Under this gauge description, we can only study the effect of a knot on 2D spatial zero-lattice and 1D tempo zero-lattice.

In the 2 + 1D space-time, a knot also leads to π-phase changing,

$$
\Delta\Phi\mathbf{i} = \pi, \Delta\Phi\_{\mathbf{j}} = \pi, \Delta\Phi\_{\mathbf{t}} = \pi. \tag{120}
$$

Due to the spatial-tempo rotation symmetry, the knot also becomes SO(3)/SO(2) magnetic monopole and traps a "magnetic charge" of the auxiliary gauge field A~ jk, i.e.,

$$N\_{\rm F} = \int \sqrt{-g} \Psi^{\dagger} \Psi d^{3} \mathbf{x} = \tilde{\eta}\_{m} \tag{121}$$

matrix around a knot, along which the entanglement matrix does not change. Thus, we use the dual field ϖ<sup>0</sup><sup>i</sup> to enforce the topological constraint in Eq. (117). That is, to denote the upper index of Fjk that is the local tangential entanglement matrices, we set antisymmetric property of upper index of ϖ<sup>0</sup><sup>i</sup> and that of Fjk. Because ϖ<sup>0</sup><sup>i</sup> and ω<sup>0</sup><sup>i</sup> have the same SO(3,1)

In the path-integral formulation, to enforce such topological constraint, we may add a topo-

¼ � <sup>1</sup> 4π ð e0ijkR<sup>0</sup><sup>i</sup>

<sup>R</sup><sup>0</sup><sup>i</sup> <sup>¼</sup> <sup>d</sup>ω<sup>0</sup><sup>i</sup> <sup>þ</sup> <sup>ω</sup><sup>0</sup><sup>j</sup>

2 A<sup>j</sup><sup>0</sup> ∧ A<sup>k</sup><sup>0</sup>

4πð Þ 2a 2 ð e0ijkR<sup>0</sup><sup>i</sup>

e0ijk e<sup>0</sup>νλκR~ <sup>0</sup><sup>i</sup>

4πð Þ 2a 2 ð eijk0R~0<sup>i</sup>

The upper index of R~0<sup>i</sup> denotes entanglement transformation along given direction in winding space-time. We unify the index order of space-time into 1ð Þ ; 2; 3; 0 ST and reorganize the upper

logical description of Einstein-Hilbert action is proposed by S. W. MacDowell and F. Mansouri. The topological mutual BF term proposed in this paper is quite different from the MacDowell-

<sup>4</sup>πð Þ <sup>2</sup><sup>a</sup> <sup>2</sup> Ð

where <sup>R</sup>~0<sup>i</sup> <sup>¼</sup> <sup>d</sup>ω<sup>~</sup> <sup>0</sup><sup>i</sup> <sup>þ</sup> <sup>ω</sup><sup>~</sup> <sup>0</sup><sup>j</sup> <sup>∧</sup> <sup>ω</sup><sup>~</sup> ji. From <sup>F</sup>~k<sup>0</sup> � �A<sup>~</sup> kj <sup>∧</sup> <sup>A</sup><sup>~</sup> <sup>j</sup><sup>0</sup> and <sup>~</sup>e<sup>i</sup> <sup>∧</sup> <sup>~</sup>e<sup>j</sup> <sup>¼</sup> ð Þ <sup>2</sup><sup>a</sup> <sup>2</sup>

<sup>S</sup>MBF2 <sup>¼</sup> <sup>1</sup>

Next, we use Lagrangian approach to characterize the deformation of a knot (an SO(3)/SO(2)

0νF~jk λκd<sup>4</sup>

SMBF1 is linear in the conventional strength in R<sup>0</sup><sup>i</sup> and Fjk. This term is becomes

<sup>S</sup>MBF1 <sup>¼</sup> <sup>1</sup>

, i.e., <sup>ω</sup><sup>0</sup><sup>i</sup> ! <sup>ω</sup><sup>0</sup><sup>i</sup> � �<sup>0</sup>

4π ð

<sup>S</sup>MBF1 ¼ � <sup>1</sup>

=2 � �, due to SO(3,1) Lorentz invariance we can do Lorentz transformation and

e0ijk e<sup>0</sup>νλκR<sup>0</sup><sup>i</sup>

<sup>¼</sup> <sup>ω</sup><sup>0</sup><sup>i</sup> <sup>þ</sup> <sup>ϖ</sup><sup>0</sup><sup>i</sup>

0νFjk λκd<sup>4</sup> x

∧ e j ∧ e k

<sup>x</sup> ¼ � <sup>1</sup> 4π ð e0ijkR~0<sup>i</sup>

> ∧ ~e<sup>j</sup> ∧ ~e<sup>k</sup>

∧ Fjk

Topological Interplay between Knots and Entangled Vortex-Membranes

. As a result, the dual field ϖ<sup>0</sup><sup>i</sup> is

http://dx.doi.org/10.5772/intechopen.72809

∧ ωji: (127)

: (128)

A~ <sup>j</sup><sup>0</sup> ∧ A~ <sup>k</sup><sup>0</sup>

: (130)

eijk0Rij ∧ e<sup>k</sup> ∧ e0. In Ref. [16–19], a topo-

∧ F~jk (129)

, this term

. The induced topological mutual BF term

~m. Using the similar approach, we derive

(126)

61

generator γ<sup>0</sup><sup>i</sup>

replaced by ω<sup>0</sup><sup>i</sup>

where

becomes

Mansouri action.

absorb the dual field ϖ<sup>0</sup><sup>i</sup> into ω<sup>0</sup><sup>i</sup>

.

logical mutual BF term SMBF in the action that is

From Fjk � �A<sup>j</sup><sup>0</sup> <sup>∧</sup> Ak<sup>0</sup> and <sup>e</sup><sup>i</sup> <sup>∧</sup> <sup>e</sup><sup>j</sup> <sup>¼</sup> ð Þ <sup>2</sup><sup>a</sup>

topological defect) on 2 + 1D space-time, NF ¼ q

<sup>S</sup>MBF2 ¼ � <sup>1</sup>

index. The topological mutual BF term becomes <sup>1</sup>

another topological mutual BF term SMBF2 in the action that is

4π ð

where q <sup>~</sup><sup>m</sup> is the "magnetic" charge of auxiliary gauge field <sup>A</sup><sup>~</sup> ij. Remember that the correspondence between index of γ~<sup>i</sup> and index of space-time x<sup>i</sup> is γ~<sup>1</sup> ⇔ y, γ~<sup>2</sup> ⇔ z, γ~<sup>3</sup> ⇔ t.

To characterize the induced magnetic charge, we write down another constraint

$$\iiint \rho\_F dV = \frac{1}{4\pi} \iiint \epsilon\_{\vec{ij}} \epsilon\_{ijk} \tilde{F}^{\vec{ij}}\_{jk} \cdot dS\_i \tag{122}$$

where

$$\begin{split} \tilde{F}^{\dagger j} &= d\tilde{A}^{\dagger j} + \tilde{A}^{\dagger j} \wedge \tilde{A}^{\dagger j} \\ &\equiv -\tilde{A}^{\dagger 0} \wedge \tilde{A}^{\dagger 0} . \end{split} \tag{123}$$

The upper indices of <sup>F</sup>~ij <sup>¼</sup> dF~ij <sup>þ</sup> <sup>F</sup>~ik <sup>∧</sup> <sup>F</sup>~kj denote the local entanglement matrices on the tangential sub-space-time and the lower indices of F~ij jk denote the spatial direction. Therefore, according to above equation, the 2 + 1D zero-lattice is globally deformed by an extra knot.

In general, due to the hidden SO(4) invariant, for other gauge descriptions <sup>γ</sup>~<sup>0</sup> <sup>¼</sup> <sup>α</sup>Γ<sup>1</sup> <sup>þ</sup> <sup>β</sup>Γ<sup>2</sup> <sup>þ</sup> <sup>γ</sup>Γ<sup>3</sup> <sup>þ</sup>δΓ<sup>5</sup> , a knot also play the role of SO(3)/SO(2) magnetic monopole and traps a "magnetic charge" of the corresponding auxiliary gauge field.

#### 5.2. Einstein-Hilbert action as topological mutual BF term for knots

It is known that for a given gauge description, a knot is an SO(3)/SO(2) magnetic monopole and traps a "magnetic charge" of the corresponding auxiliary gauge field. For a complete basis of entanglement pattern, we must use four orthotropic SO(4) rotors Γ<sup>1</sup> � �<sup>0</sup> ð Þ<sup>x</sup> ; <sup>Γ</sup><sup>2</sup> � �<sup>0</sup> ð Þ<sup>x</sup> ; <sup>Γ</sup><sup>3</sup> � �<sup>0</sup> ð Þx ; �

<sup>Γ</sup><sup>5</sup> ð Þ0 ð ÞÞ x and four different gauge descriptions to characterize the deformation of a knot (an SO(4)/ SO(3) topological defect) on a 3 + 1D zero-lattice.

Firstly, we use Lagrangian approach to characterize the deformation of a knot (an SO(3)/SO(2) topological defect) on a 3D spatial zero-lattice, NF ¼ qm. The topological constraint in Eq. (117) can be re-written into

$$\frac{i}{4}\text{tr}\sqrt{-g}\overline{\Psi}\gamma^{i}\left(\gamma^{0i}/2\right)\Psi = \epsilon\_{\not k}\epsilon\_{\not k}\frac{1}{4\pi}\hat{D}\_{i}F^{\not k}\_{\not k} \tag{124}$$

or

$$\frac{i}{4}\text{tr}\sqrt{-\mathcal{g}}\overline{\Psi}\sigma\_0^{0i}\gamma^i(\gamma^{0i}/2)\Psi = i\epsilon\_{0ijk}\epsilon\_{0ijk}\sigma\_0^{0i}\frac{1}{4\pi\tau}\widehat{D}\_iF\_{jk}^{\vec{k}}\tag{125}$$

where Db <sup>i</sup> ¼ i <sup>b</sup>∂<sup>i</sup> <sup>þ</sup> <sup>i</sup>ω<sup>i</sup> is covariant derivative in 3 + 1D space-time. <sup>ϖ</sup><sup>0</sup><sup>i</sup> is a field that plays the role of Lagrangian multiplier. The upper index i of ϖ<sup>0</sup><sup>i</sup> denotes the local radial entanglement

matrix around a knot, along which the entanglement matrix does not change. Thus, we use the dual field ϖ<sup>0</sup><sup>i</sup> to enforce the topological constraint in Eq. (117). That is, to denote the upper index of Fjk that is the local tangential entanglement matrices, we set antisymmetric property of upper index of ϖ<sup>0</sup><sup>i</sup> and that of Fjk. Because ϖ<sup>0</sup><sup>i</sup> and ω<sup>0</sup><sup>i</sup> have the same SO(3,1) generator γ<sup>0</sup><sup>i</sup> =2 � �, due to SO(3,1) Lorentz invariance we can do Lorentz transformation and absorb the dual field ϖ<sup>0</sup><sup>i</sup> into ω<sup>0</sup><sup>i</sup> , i.e., <sup>ω</sup><sup>0</sup><sup>i</sup> ! <sup>ω</sup><sup>0</sup><sup>i</sup> � �<sup>0</sup> <sup>¼</sup> <sup>ω</sup><sup>0</sup><sup>i</sup> <sup>þ</sup> <sup>ϖ</sup><sup>0</sup><sup>i</sup> . As a result, the dual field ϖ<sup>0</sup><sup>i</sup> is replaced by ω<sup>0</sup><sup>i</sup> .

In the path-integral formulation, to enforce such topological constraint, we may add a topological mutual BF term SMBF in the action that is

$$\begin{split} S\_{\text{MBF1}} &= -\frac{1}{4\pi} \int \epsilon\_{0ijk} \epsilon\_{0\nu\lambda\kappa} R^{0i}\_{0\nu} F^{jk}\_{\lambda\kappa} d^4x \\ &= -\frac{1}{4\pi} \int \epsilon\_{0ijk} R^{0i} \wedge F^{jk} \end{split} \tag{126}$$

where

NF ¼

dence between index of γ~<sup>i</sup> and index of space-time x<sup>i</sup> is γ~<sup>1</sup> ⇔ y, γ~<sup>2</sup> ⇔ z, γ~<sup>3</sup> ⇔ t.

To characterize the induced magnetic charge, we write down another constraint

<sup>∭</sup> <sup>r</sup>FdV <sup>¼</sup> <sup>1</sup>

to above equation, the 2 + 1D zero-lattice is globally deformed by an extra knot.

5.2. Einstein-Hilbert action as topological mutual BF term for knots

of entanglement pattern, we must use four orthotropic SO(4) rotors Γ<sup>1</sup> � �<sup>0</sup>

<sup>0</sup> γ<sup>i</sup> γ<sup>0</sup><sup>i</sup>

4π ðð

<sup>F</sup>~ij <sup>¼</sup> dA<sup>~</sup> ij <sup>þ</sup> <sup>A</sup><sup>~</sup> ij <sup>∧</sup> <sup>A</sup><sup>~</sup> ij

The upper indices of <sup>F</sup>~ij <sup>¼</sup> dF~ij <sup>þ</sup> <sup>F</sup>~ik <sup>∧</sup> <sup>F</sup>~kj denote the local entanglement matrices on the tangen-

In general, due to the hidden SO(4) invariant, for other gauge descriptions <sup>γ</sup>~<sup>0</sup> <sup>¼</sup> <sup>α</sup>Γ<sup>1</sup> <sup>þ</sup> <sup>β</sup>Γ<sup>2</sup> <sup>þ</sup> <sup>γ</sup>Γ<sup>3</sup>

It is known that for a given gauge description, a knot is an SO(3)/SO(2) magnetic monopole and traps a "magnetic charge" of the corresponding auxiliary gauge field. For a complete basis

ð ÞÞ x and four different gauge descriptions to characterize the deformation of a knot (an SO(4)/

Firstly, we use Lagrangian approach to characterize the deformation of a knot (an SO(3)/SO(2) topological defect) on a 3D spatial zero-lattice, NF ¼ qm. The topological constraint in Eq. (117)

<sup>=</sup><sup>2</sup> � �<sup>Ψ</sup> <sup>¼</sup> <sup>e</sup>jkeijk

<sup>=</sup><sup>2</sup> � �<sup>Ψ</sup> <sup>¼</sup> <sup>i</sup>e0ijke0ijkϖ<sup>0</sup><sup>i</sup>

role of Lagrangian multiplier. The upper index i of ϖ<sup>0</sup><sup>i</sup> denotes the local radial entanglement

<sup>b</sup>∂<sup>i</sup> <sup>þ</sup> <sup>i</sup>ω<sup>i</sup> is covariant derivative in 3 + 1D space-time. <sup>ϖ</sup><sup>0</sup><sup>i</sup> is a field that plays the

1 4π <sup>D</sup><sup>b</sup> iFjk

> 0 1 4π <sup>D</sup><sup>b</sup> iFjk

, a knot also play the role of SO(3)/SO(2) magnetic monopole and traps a "magnetic charge"

� �A<sup>~</sup> <sup>i</sup><sup>0</sup> <sup>∧</sup> <sup>A</sup><sup>~</sup> <sup>j</sup><sup>0</sup>

:

<sup>~</sup><sup>m</sup> is the "magnetic" charge of auxiliary gauge field <sup>A</sup><sup>~</sup> ij

tial sub-space-time and the lower indices of F~ij

of the corresponding auxiliary gauge field.

SO(3) topological defect) on a 3 + 1D zero-lattice.

i 4 tr ffiffiffiffiffiffi �<sup>g</sup> <sup>p</sup> <sup>Ψ</sup>γ<sup>i</sup> <sup>γ</sup><sup>0</sup><sup>i</sup>

i 4 tr ffiffiffiffiffiffi �<sup>g</sup> <sup>p</sup> <sup>Ψ</sup>ϖ<sup>0</sup><sup>i</sup>

where q

60 Superfluids and Superconductors

where

<sup>þ</sup>δΓ<sup>5</sup>

<sup>Γ</sup><sup>5</sup> ð Þ0

or

where Db <sup>i</sup> ¼ i

can be re-written into

<sup>ð</sup> ffiffiffiffiffiffi �<sup>g</sup> <sup>p</sup> <sup>Ψ</sup>†

Ψd<sup>3</sup> x ¼ q

eijeijkF~ij

~<sup>m</sup> (121)

jk � dSi (122)

jk denote the spatial direction. Therefore, according

�

ð Þ<sup>x</sup> ; <sup>Γ</sup><sup>2</sup> � �<sup>0</sup>

jk (124)

jk (125)

ð Þ<sup>x</sup> ; <sup>Γ</sup><sup>3</sup> � �<sup>0</sup>

ð Þx ;

. Remember that the correspon-

(123)

$$\mathcal{R}^{0i} = d\omega^{0i} + \omega^{0j} \wedge \omega^{ji}. \tag{127}$$

From Fjk � �A<sup>j</sup><sup>0</sup> <sup>∧</sup> Ak<sup>0</sup> and <sup>e</sup><sup>i</sup> <sup>∧</sup> <sup>e</sup><sup>j</sup> <sup>¼</sup> ð Þ <sup>2</sup><sup>a</sup> 2 A<sup>j</sup><sup>0</sup> ∧ A<sup>k</sup><sup>0</sup> . The induced topological mutual BF term SMBF1 is linear in the conventional strength in R<sup>0</sup><sup>i</sup> and Fjk. This term is becomes

$$S\_{\rm MBF1} = \frac{1}{4\pi \left(2a\right)^2} \left[ \epsilon\_{0ijk} R^{0i} \wedge e^j \wedge e^k. \tag{128}$$

Next, we use Lagrangian approach to characterize the deformation of a knot (an SO(3)/SO(2) topological defect) on 2 + 1D space-time, NF ¼ q ~m. Using the similar approach, we derive another topological mutual BF term SMBF2 in the action that is

$$\mathcal{S}\_{\text{MBF2}} = -\frac{1}{4\pi} \left[ \epsilon\_{0\text{ijk}} \epsilon\_{0\text{v\lambda x}} \tilde{R}\_{0\nu}^{\ 0 \ \tilde{}} \tilde{F}\_{\lambda x}^{\ \dot{k}} d^{4}x = -\frac{1}{4\pi} \right] \epsilon\_{0\text{ijk}} \tilde{R}^{0i} \wedge \tilde{F}^{\text{jk}} \tag{129}$$

where <sup>R</sup>~0<sup>i</sup> <sup>¼</sup> <sup>d</sup>ω<sup>~</sup> <sup>0</sup><sup>i</sup> <sup>þ</sup> <sup>ω</sup><sup>~</sup> <sup>0</sup><sup>j</sup> <sup>∧</sup> <sup>ω</sup><sup>~</sup> ji. From <sup>F</sup>~k<sup>0</sup> � �A<sup>~</sup> kj <sup>∧</sup> <sup>A</sup><sup>~</sup> <sup>j</sup><sup>0</sup> and <sup>~</sup>e<sup>i</sup> <sup>∧</sup> <sup>~</sup>e<sup>j</sup> <sup>¼</sup> ð Þ <sup>2</sup><sup>a</sup> <sup>2</sup> A~ <sup>j</sup><sup>0</sup> ∧ A~ <sup>k</sup><sup>0</sup> , this term becomes

$$S\_{\rm MBF2} = \frac{1}{4\pi \left(2a\right)^2} \int \epsilon\_{\vec{\eta}\lambda\hat{0}} \tilde{\mathcal{R}}^{0\hat{\imath}} \wedge \tilde{\mathcal{e}}^{\hat{\jmath}} \wedge \tilde{\mathcal{e}}^{\hat{k}}.\tag{130}$$

The upper index of R~0<sup>i</sup> denotes entanglement transformation along given direction in winding space-time. We unify the index order of space-time into 1ð Þ ; 2; 3; 0 ST and reorganize the upper index. The topological mutual BF term becomes <sup>1</sup> <sup>4</sup>πð Þ <sup>2</sup><sup>a</sup> <sup>2</sup> Ð eijk0Rij ∧ e<sup>k</sup> ∧ e0. In Ref. [16–19], a topological description of Einstein-Hilbert action is proposed by S. W. MacDowell and F. Mansouri. The topological mutual BF term proposed in this paper is quite different from the MacDowell-Mansouri action.

According to the diffeomorphism invariance of winding space-time, there exists symmetry between the entanglement transformation along different directions. Therefore, with the help of a complete set of definition of reduced Gamma matrices γμ, there exist other topological mutual BF terms <sup>S</sup>MBF,i. For the total topological mutual BF term <sup>S</sup>MBF <sup>¼</sup> <sup>P</sup> <sup>i</sup> SMBF,i that characterizes the deformation of a knot (an SO(4)/SO(3) topological defect) on a 3 + 1D zero-lattice, the upper index of the topological mutual BF term Rij ∧ e<sup>k</sup> ∧ e<sup>l</sup> must be symmetric, i.e., i, j, k, l ¼ 1; 2; 3; 0.

By considering the SO(3,1) Lorentz invariance, the topological mutual BF term SMBF turns into the Einstein-Hilbert action SEH as

$$\begin{split} S\_{\text{MBF}} &= S\_{\text{EH}} = \frac{1}{16\pi (a)^2} \int \epsilon\_{ijkl} \mathbf{R}^{ij} \wedge \mathbf{e}^k \wedge \mathbf{e}^l \\ &= \frac{1}{16\pi \mathbf{G}} \int \sqrt{-\overline{g}} \mathbf{R} d^4 \mathbf{x} \end{split} \tag{131}$$

6. Discussion and conclusion

unification of matter and space-time, i.e.,

crystal based on entangled vortex-lines in <sup>4</sup>

Firstly, we consider two straight vortex-lines in <sup>4</sup>

Modern physics Knot physics

Table 1. The relationship between modern physics and knot physics.

on quantized vortex-lines in <sup>4</sup>

could be simulated.

curves space-time (3 + 1D zero-lattice) via a topological way.

In this paper, we pointed out that owing to the existence of local Lorentz invariance and diffeomorphism invariance there exists emergent gravity for knots on 3 + 1D zero-lattice. In knot physics, the emergent gravity theory is really a topological theory of entanglement deformation. For emergent gravity theory in knot physics, a topological interplay between 3 + 1D zero-lattice and the knots appears: on the one hand, the deformation of the 3 + 1D zerolattice leads to the changes of knot-motions that can be denoted by curved space-time; on the other hand, the knots trapping topological defects deform the 3 + 1D zero-lattice that indicates matter may curve space-time. The Einstein-Hilbert action SEH becomes a topological mutual BF term SMBF that exactly reproduces the low energy physics of the general relativity. In

Table 1, we emphasize the relationship between modern physics and knot physics.

In addition, this work would help researchers to understand the mystery in gravity. In modern physics, matter and space-time are two different fundamental objects and matter may move in (flat or curved) space-time. In knot physics, the most important physics idea for gravity is the

One can see that matter (knots) and space-time (zero-lattice) together with motion of matter are unified into a simple phenomenon—entangled vortex-membranes and matter (knots)

In the end of the paper, we address the possible physical realization of a 1D knot-crystal based

topological interplay between zero-lattice and knots, there is no Einstein gravity on a 1D knot-

system. Then, we rotate one vortex line around another by a rotating velocity ω0. Now, the

Space-time 3 + 1D zero-lattice of projected entangled vortex-membranes

Gravity Topological interplay between 3 + 1D zero-lattice and knots

Matter Knot: a topological defect of 3 + 1 D zero-lattice Motion Changing of the distribution of knot-pieces Mass Angular frequency for leapfrogging motion

Inertial reference system A knot under Lorentz boosting Coordinate translation Entanglement transformation

Curved space-time Deformed 3 + 1D zero-lattice

Matter knots ð Þ ⇔ Space–time zero ð Þ –lattice : (134)

Topological Interplay between Knots and Entangled Vortex-Membranes

http://dx.doi.org/10.5772/intechopen.72809

63

He superfluid. Because the emergent gravity in knot physics is

He superfluid. However, the curved space-time

He superfluid between opposite points on the

where <sup>G</sup> is the induced Newton constant which is <sup>G</sup> <sup>¼</sup> <sup>a</sup>2. The role of the Planck length is played by lp ¼ a, that is the "lattice" constant on the 3 + 1D zero-lattice.

Finally, from above discussion, we derived an effective theory of knots on deformed zerolattice in continuum limit as

$$\begin{split} \mathcal{S} &= \mathcal{S}\_{\text{zero-lattice}} + \mathcal{S}\_{\text{EH}} \\ &= \int \sqrt{-g(\mathbf{x})} \overline{\Psi} \Big( \boldsymbol{\varepsilon}\_{\text{a}}^{\mu} \gamma^{a} \hat{D}\_{\mu} - m\_{\text{knot}} \Big) \Psi \mathbf{d}^{4} \mathbf{x}. \end{split} \tag{132}$$
 
$$ + \frac{1}{16\pi \mathbf{G}} \int \sqrt{-\overline{\mathbf{g}}} \mathbf{R} \mathbf{d}^{4} \mathbf{x}$$

where Db <sup>μ</sup> ¼ i b∂<sup>μ</sup> þ iωμ. The variation of the action S via the metric δgμν gives the Einstein's equations

$$R\_{\mu\nu} - \frac{1}{2} R \mathcal{g}\_{\mu\nu} = 8\pi G T\_{\mu\nu}.\tag{133}$$

As a result, in continuum limit a knot-crystal becomes a space-time background like smooth manifold with emergent Lorentz invariance, of which the effective gravity theory turns into topological field theory.

For emergent gravity in knot physics, an important property is topological interplay between zero-lattice and knots. From the Einstein-Hilbert action, we found that the key property is duality between Riemann curvature Rij and strength of auxiliary gauge field Fkl: the deformation of entanglement pattern leads to the deformation of space-time.

In addition, there exist a natural energy cutoff ℏω<sup>0</sup> and a natural length cutoff a. In high energy limit (Δω � ω0) or in short range limit (Δx � a), without well-defined 3 + 1D zero-lattice, there does not exist emergent gravity at all.
