 By K. Behrend, C. Gomez, V. Tarasov, G. Tian, P.de Bartolomeis, B. Dubrovin, C. Reina

The publication gathers the lectures given on the C.I.M.E. summer time tuition "Quantum Cohomology" held in Cetraro (Italy) from June thirtieth to July eighth, 1997. The lectures and the next updating conceal a wide spectrum of the topic at the box, from the algebro-geometric standpoint, to the symplectic strategy, together with contemporary advancements of string-branes theories and q-hypergeometric capabilities.

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Additional resources for Quantum cohomology: lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 30-July 8, 1997

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E graph. i-th, JPo,. 2. ,P,}, -+ position; of T on F', point -y the of the of three y consists b) S, is d. = we use the total stable every = a) is not -, b, for a) d(v) b) EvE V d(v) 3. bounds) it unstable edges containing equal to the set of stable = 2. genus is 3. The stabilization Let 1. 4) (pr where the vertical V(P' d; ^ ), 'r, incident with = y(i) i, fe(j) the mark of the is closed a Stacks defines . is the d) -r, fo(j) indexed by a in a has -r, d) element for with xi, we words, maps as in .

1. 1, Tpl(-O - - on H'(71, L = f *Tp,(-O dAO. We also oo)(oo). and Tp,(Pl) To calculate - Ao)ioo weights has oo)). - The torus need the of weights these, that note (Ai -Aj)j:Aj. (-00) changes The the by (-0) Twisting But has respectively. Tpi(oo), Tpi(O) weights by Tpi(O) weight AldA has weight and Tpi(oo) oo) are AOdA'. 1 at (oo). A, + uj)it-l Ao (Ai (,))jOo at (0) and (Aj holds same f*. 16) - for ( (1-d). 1. 18) e(g*N) II E V. 3). 18). 12) is Ci2 - replace -ci, for an unstable flag i E F, by the weight of T on where f i, jJ is the edge containing i.

D) (k). Again, morphism M(r') covering followed by 6tale to = the desired obtain is also for in the E V,, v we Pl: = (0, 1), = and done in families C, on oo A) defines This or -+ this d). 2. roots 1. p /-td(v) where Ad(v) i on V(-rs). 'l --+ (4-6) d). 6). 3. 3 Propositions the d) acts Aut(T, d) p the on acts on mor- -H(,rS) immersion V(-r') Putting in Aut (T, d). 7) V(P') V(-r-')IG which is a closed T7 d) / Vg,n (Pr d), (r, d) Aut , immersion. 4. Proposition classes Of Vg,n(]Pr, the image Of 4 (7-,d,-y) d)(k).