In topological models, the horizon problem is solved simply because the locality, relevant in our universe in phase II, is not natural in phase I [1] . The light modes of phase I are non-local as viewed from phase II. A known example is the winding modes of the string gas cosmology [6] . Fluctuations visible in phase II are not part of the degrees of freedom of phase I.

How does phase I look from the perspective of phase II [1] ? In phase I there should not be any position dependent observables. Let us assume the state in

phase I is given by | I 〉 . We would expect n-point correlations of physical observables in this state

〈 I | O i 1 ( x 1 ) ⋯ O i n ( x n ) | I 〉 = A i 1 , ⋯ , i n (3.1)

to be position independent when all ∂ j A i 1 , ⋯ , i n = 0 . This is a key feature of a topological quantum field theory. While we view phase I as a topological phase from the perspective of frame II it is curious that the reverse is also true: phase II can be viewed from the perspective of frame I as a topological theory [1] . This is illustrated in Figure 2.

In topological field theories observables must be a measure of global features. Consequently, there are no propagating signals. This property is achieved in the Becchi-Rouet-Stora-Tyutin (BRST) [4] [5] formalism by the presence of a Grassmann odd charge operator Q.

This operator Q is nilpotent, hermitian, and it commutes with the Hamiltonian, [ H , Q ] = 0 . The action of the charge operator on fields Φ is given by

δ   Φ = i ε [ Q , Φ ] (3.2)

where ε is a Grassmann parameter, a supernumber that anticommutes with all other Grassmann variables. Q is also the Noether charge for the BRST symmetry. The action combines together bosonic and fermionic fields in a way similar to the pairing in supersymmetric theories. Physical states in the Hilbert space are Q-cohomology classes: these states are Q-closed (i.e. | ψ 〉 satisfying Q | ψ 〉 = 0 ) modulo Q-exact (i.e. | ψ 〉 such that | ψ 〉 = Q | χ 〉 for some | χ 〉 ). This latter requirement implies that the fermionic partners of bosonic fields are in fact ghosts so that all degrees of freedom cancel in the BRST sense.

If we assume that the vacuum is Q-invariant, then Q-exact operators have a vanishing expectation value 〈 [ Q , O ] 〉 = 0 . In topological field theories, the energy-momentum tensor (given by the variation of the action with respect to the metric) is Q-exact, i.e. T α β = { Q , λ α β } for some λ . This implies that the partition function is invariant under metric variations

δ Z = ∫ D   Φ   e − S ( − δ S ) = − ∫ D   Φ   e − S { Q , ∫ g δ g α β λ α β } = − 〈 { Q , ∫ g δ g α β λ α β } 〉 = 0

provided the integration measure is BRST invariant.

Another way to illuminate background independence in a topological theory in general is based on calculating Wilson loops in 3D Chern-Simons (CS) theory [7] ¹. Wilson loops give a natural class of gauge invariant observables that do not require a choice of metric. Let C be an oriented closed curve in M. Intrinsically C is simply a circle, but the topological classification of embeddings of a circle in M may be complicated, as we can imagine in Figure 3. Let R be an irreducible representation of G. One then defines the Wilson loop W R ( C ) to be the following functional of the connection A i . One computes the holonomy of A i around C, getting an element of G that is well-defined up to conjugacy, and then one takes the trace of this element in the representation R. Thus, the definition is

W R ( C ) = T r R P exp ∫ C     A i d x i (3.3)

The crucial property of this definition is that there is no need to introduce a metric, so general covariance is maintained.

Consider the partition function Z, defined as

Z = ∫ D A exp ( i L ) ∏ i W R i ( C i ) (3.4)

where D A represents Feynman integral over all gauge orbits, the C i are non-intersecting knots and R i representation assigned to C i . The partition function Z is thus automatically independent of any background metric. However, there is still a question of whether the theory contains local excitations.

We consider the topological theory by Fang and Gu [8] [9] [10] . The topological quantum field theory (TQFT) approach can not be easily generalized into 3 + 1D because consistency with Einstein’s gravity in 3 + 1D contains propagating a

mode, the graviton. Therefore it is obviously not a case for TQFT in the usual sense. Secondly, there is no Chern-Simons like action in 3 + 1D. Fang and Gu have shown that Einstein gravity might emerge by adding a topological mass term of the 2-form gauge field. Physically, such a phenomenological theory might describe a loop condensing phase, i.e. flux lines in the context of gauge theory.

Due to the recent developments in the classification of topological phases of quantum matter in higher dimensions [11] [12] [13] [14] [15] , new types of TQFT have been discovered in 3 + 1D to describe the three-loop-braiding statistics. It is argued that such types of TQFT are closely related to Einstein gravity and that gravitational field will disappear at extremely high energy scale. 3 + 1D quantum gravity would be controlled by a TQFT renormalization group fixed point. At intermediate energy scales, Einstein gravity and classical spacetime would emerge via loop (flux lines) condensation of the underlying TQFT. The uncondensed loop-like excitation is a natural candidate of dark matter. Such kind of dark matter will not contribute scalar curvature but will be a direct source of torsion. Normal matter, like Dirac fermions, will not contribute to torsion.

Let us begin with the topological gravity theory in 3 + 1D [16] . Consider the following topological invariant action

S t o p = k 1 4 π ∫ ε a b c d R a b ∧ e c ∧ e d + k 2 2 π ∫ B a b ∧ R a b + k 3 2 π ∫ B ˜ a ∧ T a (3.5)

where e is the tetrad field, R is the curvature tensor, T is the torsion tensor and B , B ˜ are 2-form gauge fields. Like in the CS theory, the values of k i are quantized. Without loss of generality, the following values can be chosen k 1 = k 2 = 2 and k 3 = 1 for convenience. The above action is invariant under the following (twisted) 1-form and 2-form gauge transformations, respectively:

e a → e a + D f a

B a b → B a b − k 3 2 k 2 ( B ˜ a f b − B ˜ b f a )

B ˜ a → B ˜ a − k 1 k 3 ε a b c d f b R c d , (3.6)

and

B a b → B a b + D ξ a b , (3.7)

B ˜ a → B ˜ a + D ξ ˜ a

B a b → B a b − k 3 2 k 2 ( ξ ˜ a ∧ e b − ξ ˜ b ∧ e a ) . (3.8)

Such an action can be regarded as the non-Abelian generalization of AAdA + BF type TQFT [17] [18] [19] of the Poincare gauge group. Physically, it has been shown that such kind of TQFT describes the three-loop-braiding statistics [20] [21] . As a TQFT, the action Equation (3.5) is a super-renormalizable theory. The coefficient quantization and canonical quantization of such a theory are discussed in [9] .

SUSY generalization of 3 + 1D topological gravity is discussed in [8] . One needs to introduce the gauge connection of super Poincare group and write the action as ∫ s T r [ A ∧ A ∧ ( d A + A ∧ A ) ] + ∫ s T r ( B ∧ F ) . For the N = 1 case, one can express A, B and F as follows

A μ ≡ 1 2 ω μ a b M a b + e μ a P a + ψ ¯ μ α Q α

B μ ν ≡ 1 2 B μ ν a b M a b + B ˜ μ ν a P a + B μ ν α Q α

F μ ν ≡ 1 2 R μ ν a b M a b + T μ ν a P a + R ¯ μ ν α Q α (3.9)

Here R ¯ μ ν α is the super curvature tensor defined as R ¯ μ ν α = D μ ψ ¯ ν α − D ν ψ ¯ μ α where D μ is the covariant derivative for spinor fields. Fermionic loops (flux lines) cannot be condensed. Therefore supersymmetry breaking happens at very high energy scale when bosonic loops condense and classical space-time emerges. More details are presented in [10] .

Although the total action S is super-renormalizable, it does not imply UV-complete quantum gravity theory due to explicit breaking of 2-form gauge

symmetries by the S θ = − θ 2 π ∫ B a b ∧ B a b term. The algebraic tensor 2-category

theory [22] [23] may provide an equivalent UV-complete description for a topological quantum gravity theory in 3 + 1D.

In [10] the authors give a more profound treatment. It includes a deformation parameter λ which represents the bare cosmological constant term. It plays a crucial role in this scenario. λ = 0 corresponds to a trivial universe with vanishing Riemann curvature, while λ ≠ 0 corresponds to a non-trivial universe where Einstein gravity arises at low energy. In this scenario SUSY does not survive at energy scale below Planck energy.

We disclose arguments for preons. The distinctive feature of our preons (called here chernons) is the treatment of SUSY as unbroken global symmetry with the particles in supermultiplets. The chiral and vector supermultiplets for three colors are given in Table 1 [2] .

Chernon interactions are 2 + 1 dimensional inside a 3 + 1D world. Chern-Simons-Maxwell (CSM) interaction models have been studied in condensed matter physics, e.g. [24] [25] [26] . In this note we extrapolate the CSM model a long way to particle physics phenomenology at high energy in the early universe.

We construct the visible matter of two fermionic chernons: one charged m⁻, one neutral m⁰, and the photon. The Wess-Zumino [27] type action [2] is supersymmetric as well as C symmetric. The chernons have zero (or very small) mass. The chernon baryon (B) and lepton (L) numbers are zero. Given these quantum numbers, quarks consist of three chernons, as indicated in Table A1².

In [26] a 2 + 1 dimensional Chern-Simons (CS) action [7] [28] was used to derive chernon-chernon interaction, which turns out to trigger the fist phase transition between O and II. In 2 + 1 dimensions, a fermionic field has its spin polarization fixed up by the sign of mass [29] [30] [31] [32] . The model includes two positive-energy spinors (two spinor families) and a complex scalar φ . The fermions obey Dirac equation, each one with one polarization state according to the sign of the mass parameter.

The chernon-chernon scattering amplitude in the non-relativistic approximation is obtained by calculating the t-channel exchange diagrams of the Higgs scalar and the massive gauge field. The propagators of the two exchanged particles and the vertex factors are calculated from the action [26] .

The gauge invariant effective potential for the scattering considered is obtained in [33] [34]

V MCS ( r ) = e 2 2 π [ 1 − θ m c h ] K 0 ( θ r ) + 1 m c h r 2 { l − e 2 2 π θ [ 1 − θ r K 1 ( θ r ) ] } 2 (3.10)

where K 0 ( x ) and K 1 ( x ) are the modified Bessel functions and l is the angular momentum ( l = 0 in this note). In (3.10) the first term [ ] corresponds to the electromagnetic potential, the second one { }² contains the centrifugal barrier ( l / m r 2 ), the Aharonov-Bohm term and the two photon exchange term.

One sees from (3.10) the first term may be positive or negative while the second term is always positive. The function K 0 ( x ) diverges as x → 0 and approaches zero for x → ∞ and K 1 ( x ) has qualitatively similar behavior. For our scenario we need negative potential between equal charge chernons. Being embarrassed of having no data points for several parameters in (3.10) we can give one relation between these parameter values for a binding potential. We must require the condition³

θ ≫ m c h (3.11)

The potential (3.10) also depends on v², the vacuum expectation value, and on y, the parameter that measures the coupling between fermions and Higgs scalar. Being a free parameter, v² indicates the energy scale of the spontaneous breakdown of the U(1) local symmetry.

A summary of the three phases and their properties is given in Table 2.