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By Edwin H. Spanier

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6) Spec ∆g = { i,j=1 gij pi pj ; pi ∈ Zd }. On the other hand, the energy of a winding string is (as one might expect intuitively) a function of its length. 7) 2 gij wi wj . L = i,j=1 Now, the only string theory input we need to bring in is that the total Hamiltonian contains both terms, H = ∆ g + L2 + · · · where the extra terms · · · express the energy of excited (or “oscillator”) modes of the string. Then, the inversion g → g−1 , combined with the interchange p ↔ w, leaves the spectrum of H invariant.

A maps into the center of Oaa . a a ∼ = θC (ιa (ψ) · φ) = θa (ψ · ιa (φ)) Figure 10. ιa is the adjoint of ιa . a b a b πb a : Oaa → Obb Figure 11. The double-twist diagram defines the map πb a : Oaa → Obb . {f ∈ Oab : pf = f } and Obc = {f ∈ Oba : f p = f }. 4 4A linear category in which idempotents split in this way is often called Karoubian. 4. 38 2. D-BRANES AND K-THEORY IN 2D TOPOLOGICAL FIELD THEORY ∼ = a b a b ∼ = Figure 12. The (generalized) Cardy condition expressing factorization of the double-twist diagram in the closed string channel.

3c. 9) ιa (φ)ψ = ψιb (φ) for all φ ∈ C and ψ ∈ Oab . 3d. 10) for all ψ ∈ Oaa . θC (ιa (ψ)φ) = θa (ψιa (φ)) 32 2. D-BRANES AND K-THEORY IN 2D TOPOLOGICAL FIELD THEORY 3e. ”2 Define πb a : Oaa → Obb as follows. Since Oab and Oba are in duality (using θa or θb ), if we let ψµ be a basis for Oba then there is a dual basis ψ µ for Oab . 11) ψµ ψψ µ , µ and we have the “Cardy condition”: πb a = ιb ◦ ιa . 12) C→C C⊗C →C C →C⊗C C→C Figure 2. Four diagrams defining the Frobenius structure in a closed 2d TFT.

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