1. Hyun Seok Yang Center for Quantum Spacetime Sogang University
Topology Change of Spacetime and Resolution of Spacetime Singularity in Emergent Gravity
Hyun Seok Yang
Center for Quantum Spacetime
Sogang University

2. References Collaboration with S. Lee and R. Roychowdhury
1. Notes on emergent gravity, JHEP 09 (2012) 030, arXiv:
2. Test of emergent gravity, arXiv:
3. Topology change of spacetime and resolution of spacetime singularity
in emergent gravity, arXiv:1212.xxxx
4. Seiberg-Witten map, Darboux theorem and equivalence principle,
arXiv:1212.xxxx
5. Emergent gravity from bottom-up approach, in progress

3. Topology change in general relativity ?
Consider two 4-manifolds (𝑀 1 , 𝑔 1 ) and (𝑀 2 , 𝑔 2 ) with different topology.
Is it possible to realize a smooth transition from (𝑀 1 , 𝑔 1 ) to 𝑀 2 , 𝑔 2
in general relativity ?
Theorem for the topology change in general relativity:
R. Geroch (1967) & F. J. Tipler (1977)
The answer is No if Einstein equations hold on entire spacetime.
If topology change occurs within a finite region, spacetime singularities
necessarily arise for this change of topology.
Spacetime singularities at the center of black hole and very beginning
of our universe.

4. Darboux theorem=Seiberg-Witten map =Equivalence principle
Darboux theorem: A novel form of the equivalence principle
for electromagnetic force
The electromagnetic force can always be eliminated by a local coordinate transformation as far as U(1) gauge is defined on a spacetime with symplectic structure.
Introduce local coordinates 𝑥 𝑎 (𝑎=1, ⋯,4) on a local chart 𝑈⊂𝑀 where
the symplectic structure is represented by
ℱ= 𝐵 𝑎𝑏 + 𝐹 𝑎𝑏 𝑥 𝑑 𝑥 𝑎 ∧𝑑 𝑥 𝑏 (1)
One can always find a local coordinate transformation 𝜙:𝑥↦𝑦=𝑦(𝑥) to eliminate the electromagnetic force ℱ on 𝑈⊂𝑀 such that
ℱ ​ 𝑈 = 1 2 𝐵 𝜇𝜈 𝑑 𝑦 𝜇 ∧𝑑 𝑦 𝜈 (2)

5. Assume the coordinate transformation as
𝑥 𝜇 𝑦 = 𝑦 𝜇 + 𝜃 𝜇𝜈 𝐴 𝜈 𝑦 (3)
which plays the role of covariant dynamical coordinates in NC gauge theory.
Then we get an intriguing relation called the Seiberg-Witten map
𝐹 𝜇𝜈 𝑦 = 𝐹𝜃 𝐹 𝜇𝜈 𝑥 , (4)
𝑑 4 𝑦= 𝑑 4 𝑥 det⁡(1+𝐹𝜃) 𝑥 , (5)
where the field strength of “symplectic gauge fields” 𝐴 𝜇 𝑦 is given by
𝐹 𝜇𝜈 𝑦 = 𝜕 𝜇 𝐴 𝜈 𝑦 − 𝜕 𝜈 𝐴 𝜇 𝑦 +{̂̂̂ 𝐴 𝜇 , 𝐴 𝜈 } 𝜃 𝑦 (6)
NC gauge theory is defined by quantizing the covariant coordinates 𝑥 𝜇 𝑦 ∈ 𝐶 ∞ 𝑀 ↦ 𝑥 𝜇 𝑦 ∈ 𝒜 𝜃 with the Poisson structure 𝐵 −1 ≡𝜃= 1 2 𝜃 𝜇𝜈 𝜕 𝜇 ∧ 𝜕 𝜈 in which the coordinate generators of 𝒜 𝜃 are noncommuting with the Heisenberg algebra relation
𝑦 𝜇 , 𝑦 𝜈 ⋆ =𝑖 𝜃 𝜇𝜈 (7)

6. Noncommutative U(1) gauge theory
The action of NC U(1) gauge theory is given by
𝑆 = 1 4 ∫ 𝑑 4 𝑦 𝐹 𝜇𝜈 𝐹 𝜇𝜈 , (8)
with the NC field strength 𝐹 𝜇𝜈 ∈ 𝒜 𝜃 defined by
𝐹 𝜇𝜈 = 𝜕 𝜇 𝐴 𝜈 − 𝜕 𝜈 𝐴 𝜇 −𝑖 𝐴 𝜇 , 𝐴 𝜈 ⋆ (9)
Note that the field strength (9) of NC U(1) gauge fields is nonlinear and
so one can find a nontrivial solution of the self-duality equation defined by
𝐹 𝜇𝜈 𝑦 = ± 1 2 𝜀 𝜇𝜈 𝜌𝜎 𝐹 𝜌𝜎 𝑦 (10)
A solution of the NC self-duality equation (10) is called a NC U(1) instanton.
The self-duality equation (10) for general NC (1) instantons can beautifully be solved
by the ADHM construction.

7. ADHM construction
Introduce a linear Dirac operator 𝒟 † depending on 𝒛= 𝑧 1 = 𝑦 2 +𝑖 𝑦 1 , z 2 = y 4 + iy 3 𝒟 † (𝒛)= 𝜏 𝑧 𝜎 𝑧 † = 𝐵 2 − 𝑧 2 𝐵 1 − 𝑧 1 𝐼 − 𝐵 1 † + 𝑧 1 𝐵 2 † − 𝑧 2 𝐽 † (11) with 𝐵 1 , 𝐵 2 :𝑉→𝑉, 𝐼:ℂ→𝑉, 𝐽:𝑉→ℂ where 𝑉 is a complex vector space with dimension 𝑘. The ADHM construction requires the factorization condition 𝒟 † 𝒟= Δ 𝑘 ⊗ 1 2 (12) where Δ 𝑘 is a 𝑘×𝑘 matrix and 1 2 is a unit matrix quaternion space. The factorization condition (12) implies the key equations 𝜏 𝑧 𝜏 𝑧 † − 𝜎 𝑧 † 𝜎 𝑧 =0, 𝜏 𝑧 𝜎 𝑧 =0 (13) which can be written as the form 𝜇 ℝ ≡ 𝐵 1 , 𝐵 1 † + 𝐵 2 , 𝐵 2 † +𝐼 𝐼 † − 𝐽 † 𝐽=2 𝜁 ℝ = 𝜂 𝜇𝜈 3 𝜃 𝜇𝜈 , (14) 𝜇 ℂ ≡ 𝐵 1 , 𝐵 2 +𝐼𝐽= 𝜁 ℂ = 1 2 𝜂 𝜇𝜈 2 +𝑖 𝜂 𝜇𝜈 1 𝜃 𝜇𝜈 . (15)

8. NC U(1) instantons
In the ADHM construction, NC U(1) gauge fields with instanton number 𝑘 are given
in the form
𝐴 𝜇 𝑦 =𝑖 𝜓 † 𝑦 𝜕 𝜇 𝜓(𝑦) (16)
where 𝜓(𝑦) is a free module over 𝒜 𝜃 satisfying the equations
𝜓 † 𝜓=1, 𝒟 † 𝜓= (17)
The NC field strength (9) determined by the ADHM gauge fields (16) is necessarily
self-dual or anti-self-dual if 𝜓 and 𝒟 obey the completeness relation
𝜓 𝜓 † +𝒟 1 𝒟 † 𝒟 𝒟 † = 1 2𝑘 (18)
Therefore the NC generalization of ADHM construction provides the complete set of
NC U(1) instantons with arbitrary instanton number 𝑘.

9. Emergent gravity
Consider a commutative limit 𝜃 →0 where NC gauge fields reduce to symplectic
gauge fields. Using the SW map (4) and (5), the action (8) of NC U(1) gauge theory
in this limit can be written as
𝑆= 1 4 ∫ 𝑑 4 𝑥 𝐺 𝐺 𝜇𝜌 𝐺 𝜎𝜈 𝐹 𝜇𝜈 𝐹 𝜌𝜎 (19)
where we introduced an effective metric determined by U(1) gauge fields
𝐺 𝜇𝜈 = 𝛿 𝜇𝜈 + 𝐹𝜃 𝜇𝜈 , 𝐺 𝜇𝜈 ≡ 𝐺 −1 𝜇𝜈 = 𝐹𝜃 𝜇𝜈 (20)
The effective metric 𝐺 𝜇𝜈 is related to the gravitational metric 𝑔 𝜇𝜈 by
𝐺 𝜇𝜈 = 𝛿 𝜇𝜈 + 𝑔 𝜇𝜈 𝑥 (21)
One can show that the metric
𝑔 𝜇𝜈 𝑥 = 𝛿 𝜇𝜈 +2 𝐹𝜃 𝜇𝜈 (22)
is always K 𝑎 hler if the Bianchi identity is satisfied, i.e. 𝑑𝐹=0. Furthermore it was proved that it is a gravitational instanton, i.e. a hyper-K 𝑎 hler manifold if and only if 𝐹 𝜇𝜈 is coming from the SW map of NC U(1) instantons.

10. Four manifolds from symplectic gauge fields
In the emergent gravity approach, the spacetime geometry is determined by U(1) gauge fields on NC spacetime. Accordingly the topology of spacetime is determined by the topology of NC U(1) gauge fields which is nontrivial and rich.
It turns out that the topology change of spacetime is ample in emergent gravity
and the subsequent resolution of spacetime singularity is possible in NC spacetime.
To illuminate the issues, let us consider an explicit solution in general relativity
whose metric is assumed to be of the form
𝑑 𝑠 2 = 𝐴 2 𝑟 𝑑 𝑟 2 + 𝜎 𝐵 2 (𝑟)( 𝜎 𝜎 2 2 ) (23)
where 𝑟 2 = 𝑥 1 2 +⋯+ 𝑥 4 2 and we introduced a left-invariant coframe { 𝜎 𝑖 :𝑖=1,2,3}
for 𝕊 3 defined by
𝜎 𝑖 =− 1 𝑟 2 𝜂 𝜇𝜈 𝑖 𝑥 𝜇 𝑑 𝑥 𝜈 (24)
The above metric can be written as the form
𝑔 𝜇𝜈 𝑥 = 𝐴 2 + 𝐵 2 𝛿 𝜇𝜈 − 1 𝑟 2 𝐴 2 − 𝐵 𝜂 3 𝜂 𝑖 𝜇𝜈 𝑇 𝑖 (25)
where 𝑇 𝑖 (𝑖=1,2,3) are Hopf coordinates defined by the Hopf map 𝜋: 𝕊 3 → 𝕊 2 .

11. Topology change of spacetime
The U(1) field strength determined by the metric 𝑔 𝜇𝜈 (𝑥) is given by
𝐹 𝜇𝜈 (𝑥)= 𝑓 1 𝑟 𝜂 𝜇𝜈 3 + 𝑓 2 𝑟 𝜂 𝜇𝜈 𝑖 𝑇 𝑖 (26)
where
𝑓 1 𝑟 =1− 𝐴 2 + 𝐵 2 , 𝑓 2 𝑟 =− 1 𝑟 2 ( 𝐴 2 − 𝐵 2 ) (27)
If the functions A and B are given by
𝐴 2 𝑟 = 𝑟 𝑟 4 + 𝑡 4 = 𝐵 2 𝑟 , (28)
the above U(1) gauge fields describes symplectic U(1) instantons, obeying the self-duality equation (10). In this case the metric (22) is the Eguchi-Hanson space which
describes a non-compact, self-dual, ALE space on the cotangent bundle of 2-sphere 𝑇 ∗ 𝕊^2 with SU(2) holonomy group.
Now we can see how the smooth topology change of spacetime arises due to U(1) instantons. By turning off the dynamical gauge fields in (26), i.e. 𝐴=𝐵=1 in (27),
the metric (22) becomes flat, i.e. 𝑔 𝜇𝜈 = 𝛿 𝜇𝜈 and recovers the space ℝ 4 .

12. But, if the symplectic gauge fields in (26) are developed, the space evolves to a curved manifold with nontrivial topology whose metric is given by (23) or (25).
For example, for the case (28), we can compare the topological invariants before and after the transition:
ℝ 4 : 𝜒 𝑀 =1, 𝜏 𝑀 =0,
𝐸𝐻: 𝜒 𝑀 =2, 𝜏 𝑀 =1.
In the course of transition concomitant with the topology change of spacetime,
no spacetime singularity arises. This transition is simply described by introducing
generic (large) dynamical gauge fields and this process is completely well-defined in gauge theory.
I want to emphasize that the smooth topology change of spacetime is possible only
in NC spacetime. I will give an impressive example.

13. NC spacetime is crucial for everything
Let us try to solve the ADHM equations (14) and (15) on commutative ℂ 2 .
But, in order to keep the deformation intact, the ADHM data should be modified as
𝜏 𝑧 𝜏 𝑧 † − 𝜎 𝑧 † 𝜎 𝑧 =2 𝜁 ℝ , 𝜏 𝑧 𝜎 𝑧 = 𝜁 ℂ (29).
The corresponding U(1) gauge fields (16) defined by the deformed ADHM data (29)
On commuative ℂ 2 were already obtained by Braden and Nekrasov (BN) and we call them BN instantons. The field strength (26) is given by
𝐴 2 𝑟 = 𝑟 2 ( 𝑟 2 +2 𝑡 2 ) 𝑟 2 + 𝑡 , 𝐵 2 𝑟 = 𝑟 4 + 𝑟 2 𝑡 2 + 𝑡 4 𝑟 2 ( 𝑟 2 + 𝑡 2 ) (30)
However the four-manifold determined by the BN instanton exhibits a spacetime singularity. For example the Kretschmann scalar 𝐾= 𝑅 𝜇𝜈𝜌𝜎 𝑅 𝜇𝜈𝜌𝜎 is given by
𝐾 64 𝑡 8 = 2 𝑟 2 +3 𝑡 𝑟 4 𝑟 2 +2 𝑡 regular terms (31)
which blows up at 𝑟=0 indicating the presence of a spacetime singularity.

14. It can be shown that the BN instanton also brings about the same kind of topology change as the NS instanton. The topology change due to the BN instantons can
positively be supported by calculating the topological invariants which are given by χ(M) = 2, τ (M) = 1. The Euler number χ(M) = 2 stems from the bolt 𝕊 2 in
the metric (23) with the coefficients (30). It should be compared to χ( ℝ 4 ) = 1 and
𝜏 ℝ 4 =0 for ℝ 4 which is the case of completely turning off the dynamical gauge fields, i.e. A = B = 1 in Eq. (27).
But we observed in Eq. (31) that the spacetime geometry after the topology change becomes singular. It is important to recall that the topology change in this case
occurs in commutative spacetime and so the appearance of spacetime singularity
is rather consistent with the theorem for the topology change of spacetime
in general relativity. However the topology change due to the NS instantons
does not suffer any spacetime singularity because the spacetime geometry
after the topology change becomes the EH space that is manifestly free from spacetime singularity.