Introduction to Floer Homology and its relation with TQFT

Floer theory is one of the most active fields in mathematical physics. In this expository paper, we will discuss where this theory comes from and what it is as well as its relation with TQFT. §1 Foundation of Symplectic Geometry and Morse Homology Historically, Eliashberg, Conley-Zehnder, Gromov respectively proved the Arnold conjecture for Riemann surfaces, 2n-torus, the existence of at least one fixed point under π2(M) = 0. Then Floer [F1,F2,F3,F4] made a breakthrough toward the Arnold conjecture. He first established the Arnold conjecture for Lagrangian intersections and symplectic fixed points (still under π2(M) = 0). Then he used the variational method of Conley and Zehnder as well as the elliptic techniques of Gromov and Morse-Smale-Witten complex to develop his infinite dimensional approach to the Morse theory. Now this method is widely recognized as Floer theory. We first give the background of the Arnold conjecture. Let (M,ω) be a compact symplectic manifold. The symplectic form ω determines an isomorphism between T ∗M and TM . Thus we can get a Hamiltonian vector field XH : M → TM from the exact form dH : M → T ∗M , where H : M → R is called a Hamiltonian function. We can write the above relation explicitly as ι(XH)ω = dH. Then we extend the Hamiltonian function to a smooth time dependent 1-periodic family of Hamiltonian functions Ht+1 = Ht :M → R, for t ∈ R. Consider the Hamiltonian differential equation (1) ẋ(t) = XHt(x(t)) The solution of (1) generates a family of symplectomorphisms ψt :M →M , s.t. d dt ψt = Xt ◦ ψt, ψ0 = id. We find the fixed points of the map ψ1 are in 1-1 correspondence with the 1-periodic solutions of (1) and denote such kind of solutions by P(H) = {x : R/Z →M |ẋ(t) = XHt(x(t))} Definition 1.1 A periodic solution x is called non − degenerate if the following identity holds. (2) det(I − dψ1(x(0))) 6= 0. Now we can state the Arnold conjecture. Conjecture 1.2 (Arnold) Let (M,ω) be a compact symplectic manifold and Ht = Ht+1 : M → R be a smooth time dependent 1-periodic Hamiltonian function. Suppose that the 1-periodic solutions of (1) are all non-degenerate. Then