Partition Functions And Brane Webs For Higher Dimensional Superconformal Field Theories And Little String Theories

In this thesis,the author mainly studied certain 5d_N=1,6d_N=（1,0）supercon-formal field theories and little string theories with（p,q）5-brane web constructions,the main focus is on computation of the partition functions using refined topological vertex formalism.The author first studied brane webs with ON-planes which can be used to construct D-type quiver gauge theories,those brane webs are not the usual toric brane webs,so the usual refined topological vertex formalism cannot be applied to compute their partition functions.The author found a reflection formula for refined topological vertex by intro-ducing a mirror vertex factor,and the computational rule of refined topological vertex with ON-planes can be derived from this reflection formula.In addition,based on the Dynkin quiver structure of the D-type quiver gauge theories the refined topological vertex with ON-planes should be able to cancel the contributions that mix disconnected gauge nodes in the partition function.Thus,the author summarized a set of rules for the refined topological vertex with ON-planes,and successfully computed the partition functions of5d SU（2）+8F,SU（3）_κ=7and SU（3）_κ=9theories using this set of rules.When computing these examples,the author also developped a series of techniques for computation of the partition functions of quiver gauge theories,which separates the full partition function into sub parts,the sub parts are computed first and then the computation results of sub parts are substituted back to the full partition function to do the further computation.This way of computation greatly simplified the computation process,and the author obtained exact results of one-instanton and two-instanton partition functions.Then,the author discussed the brane web construction of trivalent/quadrivalent glu-ing which is also non-toric.Trivalent/quadrivalent gluing is a new way of constructing brane webs in recent years,previous researchers have successfully constructed brane webs of 6d_N=（1,0）（D_N,D_N）,（E_N,E_N）conformal matter theories on S¹,based on these brane webs the author further compactified these theories on another S¹and obtained brane webs of the corresponding 6d_N=（1,0）D,E-type little string theories.After that,the au-thor considered the brane web constructions of the general rank case of D,E-type little string theories,and introduced the brane web of a general node.The author first com-puted the partition function of the general node using refined topological vertex,and then constructed the expression of partition function for general rank D_N,E_N-type little string theories using the partition function of the general node.This expression of partition func-tion for little string theories after taking the corresponding conformal field theory limit co-incides with the known expression of partition function of 6d_N=（1,0）superconformal field theories on S¹.Finally,the author continued to study the symmetries of the D,E-type little string theories,taking D₄,D₅,E₆-type little string theories as examples,and found that the ex-changing of the two external NS-charged 5-branes in each node of the brane web is ac-tually the Weyl reflection of each node,so the brane webs of D₄,D₅,E₆-type little string theories have affine D₄,D₅,E₆Weyl group symmetries.The author reduced the affine D₄,D₅,E₆Weyl group symmetries to D₄,D₅,E₆Weyl group symmetries,and found the corresponding Weyl invariant Coulomb branch parameters.After expanding the partition functions of D₄,D₅,E₆-type little string theories with respect to the invariant Coulomb branch parameters,the author saw the D₄,D₅,E₆characters in the expansion coefficients.