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We present explicit computations and conjectures for 2 → 2 scattering matrices in large N U(N ) Chern-Simons theories coupled to fundamental bosonic or fermionic matter to all orders in the 't Hooft coupling expansion. The bosonic and fermionic S-matrices map to each other under the recently conjectured Bose-Fermi duality after a level-rank transposition. The S-matrices presented in this paper may be regarded as relativistic generalization of Aharonov-Bohm scattering. They have unusual structural features: they include a non-analytic piece localized on forward scattering, and obey modified crossing symmetry rules. We conjecture that these unusual features are properties of S-matrices in all Chern-Simons matter theories. The S-matrix in one of the exchange channels in our paper has an anyonic character; the parameter map of the conjectured Bose-Fermi duality may be derived by equating the anyonic phase in the bosonic and fermionic theories.

It has been recently demonstrated that the thermal partition function of any large N Chern-Simons gauge theories on S 2 × S 1 , coupled to fundamental matter, reduces to a capped unitary matrix model. The matrix models corresponding to several specific matter Chern-Simons theories at temperature T were determined in [1]. The large N saddle point equations for these theories were determined in the same paper, and were solved in the low temperature phase. In this paper we find exact solutions for these saddle point equations in three other phases of these theories and thereby explicitly determine the free energy of the corresponding theories at all values of T 2 /N . As anticipated on general grounds in [1], our results are in perfect agreement with conjectured level rank type bosonization dualities between pairs of such theories.

We comment on the relationships between several supersymmetric lattice models; the "orbifold lattice theory" by Cohen-Kaplan-Katz-Unsal (CKKU), lattice regularization of the topological field theory by Sugino and the "geometrical approach" by Catterall. We point out that these three models have close relationships; the N = (2, 2) model by Catterall [1] and the two-dimensional N = (2, 2) lattice theory being similar to Sugino's construction [2] can be derived by appropriate truncation of fields in the two-dimensional N = (4, 4) orbifold lattice theory by CKKU [3]. Catterall's N = (2, 2) description possesses extra degrees of freedom compared to the target N = (2, 2) theory. If we remove those extra degrees of freedom in a way keeping supersymmetry on the lattice, Catterall's description reduces to a model of the Sugino type.There are several types of the model: The series of models proposed by Cohen-Kaplan-Katz-Unsal-Endres [3][4][5][6][7] are "orbifold lattices" which are constructed from reduced supersymmetric matrix models by the orbifold projection [26] and the deconstruction [27]. In their way, the orbifold projection generates the lattice theory with preserved subset of supersymmetry of the target theory. The deconstruction dynamically generates space-time by the vacuum expectation value 1 √ 2a of bosonic link fields, where a denotes the lattice spacing. The other approach, proposed by Sugino [2,[9][10][11][12], are lattice regularizations of the "topological field theory action" which is equivalent to the extended supersymmetric gauge theory. In his approach, the BRST-like supercharges are preserved on the lattice because such charges do not generate the infinitesimal translation.Catterall proposed models [1,13,14] which are based on the Kahler-Dirac formalism and the lattice analogue of differential forms [28]. In his models, the 1-form and 2-form fields have to be complex because they are in the bi-fundamental representation of the lattice gauge group and the hermiticity cannot be maintained under gauge transformations. Since the counterparts of these 1-and 2-form fields in the target theory are hermitian, Catterall's models have extra degrees of freedom which we have to discard in the path-integral. If one performs such truncation in a naive way, supersymmetry on the lattice would be broken.Seemingly, these three types of model are quite different. There exist, however, close relationships between them. We will clarify such relationships in this paper. This investigation of the relationships would be very useful to develop the lattice formulations of supersymmetric theories. First, in section 2, we show that Catterall's N = (2, 2) action [1] can be embedded in CKKU's N = (4, 4) action [3] under suitable field truncation. Then, in section 3, we explain the relationship between Catterall's "complexified" N = (2, 2) lattice theory and Sugino's theory of ref. [2]. For Catterall's model to contain the correct numbers of degrees of freedom compared to the target N = (2, 2) theory, we have to truncate extra deg...

We continue our investigation on the Nambu-Poisson description of M5-brane in a large constant C-field background (NP M5-brane theory) constructed in Refs. [1,2]. In this paper, the low energy limit where the NP M5-brane theory is applicable is clarified. The background independence of the NP M5-brane theory is made manifest using the variables in the BLG model of multiple M2-branes. An all order solution to the Seiberg-Witten map is also constructed.1