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By "Gross-Neveu like" I mean non-supersymmetric QFTs whose partition function/beta-function (or any n-point function) is somehow exactly solvable in the large $N_c$ or $N_f$ or 't Hooft limit.

(..supersymmetric examples would also be helpful to know of if in case there are no other theories like the above..)

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There are many quantum field theory models which are exactly solvable in the Large $N$ limit, such that the $\mathbb{C}P^N$ model, the Thirring model, the $O(N)$ vector model etc. Please see the following review by Moshe Moshe and Jean Zinn-Justin covering many of these models. The main idea is that Feynman diagrams (for example the vacuum diagrams in the case of the partition function) are proportional to certain powers of $N$ depending on the number of vertices, lines and loops, and the leading order diagrams can be summed. There are other methods which lead to the same results such as variational computations. When the fields in the model belong to the fundamental vector representations of $O(N)$ or $U(N)$, the computation of the large $N$ limit (summation of the leading diagrams) is quite easy, however, when the fields belong to the adjoint representation (for example, the gluons in QCD), the analysis becomes more complicated. The case of large $N$ QCD was solved by t'Hooft, please see the following review by Aneesh Manohar.

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Thanks for the reference! –  user6818 Dec 5 '12 at 22:20
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