1. Gravity-Khovanov Homology-QFT
Presenter : Vijay Sharma

2. Background:
Manifolds with additional structure can be used to construct topological invariants
Additional structure can be connection on principal G-bundle for some Lie Group G
Procedure is
To take this additional structure and get a number (complex number)
Try to integrate the complex number over all possible choices of such an additional structure
Example
Chern-Simons number associated with the connection A as below
Invariant will be
Ref 1 2

3. Knot Polynomials
Invariants evaluated previously can be associated with Knot Polynomials
For G = SU(2), R = two-dimensional representations of SU(2) and W = S3
Expectation values of W(K,R) evaluated previously is equal to the Jones Polynomial
Ref 2 3

4. Knot Polynomials and Khovanov Homology
From Physics point of view four dimensional Khovanov Homology when compactified on S1 reduces to three dimensional theory that yields the Jones Polynomial
Further more, the Jones polynomial can be computed by counting the solutions of certain elliptic partial differential equations in 4 dimensions and Khovanov Homology
can be constructed by counting the solutions of related equations in 4+1 dimensions
Khovanov Homology associates a vector space as invariant to a knot rather than number. Using this vector space one can derive
Graded Euler characteristic of Khovanov Invariant H(L) = Jones Polynomial J(L)
Ref 3
Ref 4 4

5. Computing Khovanov Homology
- Given a link L in S3,
- Choose a planar diagram D for your link
- Construct the “cube of resolutions” CD
Apply a functor A to the cube to obtain the chain complex CKh(D)
Take the co-homology of this complex to get HKh(L)
The complete calculation for Trefoil Knot using above scheme has been done at
Ref 5 5

6. Gravity - Khovanov Homology
- Euler Characteristic is associate to Gauss Curvature
- Gauss curvature is related to Riemann curvature tensor
- Riemann Curvature tensor is constructed of Levi-Civita connection
Loosely speaking
Khovanov Invariant ~ Euler Characteristic ~ ∫Gauss Curvature ~ ∫Rμν
We can motivate to have Khovanov Invariant computed out of Levi-Civita Connection
Khovanov Invariant ~ ∫ 6

7. Gravity - Khovanov Homology - QFT
Gauge-Gravity
Khovanov Invariant ~ ∫
Jones Invariant ~ ∫ A
To Show
Khovanov Invariant can be computed of Riemann Tensor
To study Gauge – Gravity correspondence from here 7

8. Note: For ease I referenced all the documents at each slide also.
References
Note: For ease I referenced all the documents at each slide also.
[1] Dror-Brar Natan, “Perturbative aspects of Chern-Simons Topological Quantum Field
theory,” PhD Thesis, June 1991
[2] E. Witten, “Knot Invariants from Four-Dimensional Gauge Theory,”
arXiv: v1
[3] E. Witten, “Fivebranes And Knots,” arXiv:
[4] Dror Bar-Natan, “On Khovanov’s categorification of the Jones polynomial,” Algebraic
& Geometric Topology, Volume 2 (2002) 337–370
LSI Proprietary 8