In the broad sense of the word, a topological string is a 2-dimensional TQFT. In its refined form this goes by the name TCFT. The “C” standing for conformal field theory points to what historically was the main inspiration and still is the default meaning of topological strings: the A-model and B-model 2d TQFTs, which are each obtained by a “topological twisting” of 2d SCFTs.
Accordingly, much of “physical” string theory has its analogs in topological string theory. Notably the toplogical analogs of the D-branes of the physical string – the A-branes and B-branes – have been studied in great (mathematical) detail, giving rise to homological mirror symmetry and, eventually, the notion of TCFT itself.
Also the perspective of string theory as the dimensional reduction of a conjectured UV-completion of 11-dimensional supergravity – “M-theory” – has its analog for topological strings, going, accordingly, by the term topological M-theory.
Review includes
Andrew Neitzke, Cumrun Vafa, Topological strings and their physical applications, hep-th/0410178
I. Antoniadis, S. Hohenegger, Topological Amplitudes and Physical Couplings in String Theory, Nucl.Phys.Proc.Suppl.171:176-195,2007 (arXiv:hep-th/0701290)
Marcel Vonk, A mini-course on topological strings (arXiv:hep-th/0504147)
Andrew Neitzke, Nonperturbative topological strings, 2005 (pdf)
The relation to topological M-theory/the topological membrane is discussed for instance in
See also
wikipedia topological string theory
Lotte Hollands, Topological Strings and Quantum Curves (arXiv:0911.3413)
Mina Aganagić, Cumrun Vafa, Large N duality, mirror symmetry, and a Q-deformed A-polynomial for knots, arxiv/1204.4709
Disucssion of black holes in string theory via the topological string’ Gopakumar-Vafa invariants:
Rajesh Gopakumar, Cumrun Vafa, M-Theory and Topological Strings–I (arXiv:hep-th/9809187)
Rajesh Gopakumar, Cumrun Vafa, M-Theory and Topological Strings–II (arXiv:hep-th/9812127)
The following includes discussion of superstring string scattering amplitudes in terms of topological string scattering amplitudes (for review see NeitzkeVafa04, section 6 and Antoniadis-Hohenegger 07:
M. Bershadsky, S. Cecotti, Hirosi Ooguri, Cumrun Vafa, Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes, Commun.Math.Phys.165:311-428,1994 (arXiv:hep-th/9309140)
I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor, Topological Amplitudes in String Theory, Nucl.Phys. B413 (1994) 162-184 (arXiv:hep-th/9307158)
K.S. Narain, N. Piazzalunga, A. Tanzini, Real topological string amplitudes, JHEP (2017) 2017:80 (arXiv:1612.07544)