Applications Of Statistical Physics Methods To Gromov-Witten Type Theories

Gromov-Witten type theory is the general name of field theory coupled with two-dimensional topological gravity.The most basic example of Gromov-Witten type theories is two-dimensional topological gravity itself,which corresponds to the intersection theory of Deligne-Mumford moduli space Mg,n in mathematics,also known as Gromov-Witten theory of a point.The famous Witten conjecture and recent developments about this type of theory put the Gromov-Witten type theory into a significant role in the connection between quantum field theory and various branches of mathematics.In general Gromov-Witten type theories,we study their free energy functions or partition functions,expressed as formal power series in infinitely many formal variables.The infinity makes it hard to compute or to get closed formulas.Therefore,it is an important topic to explore approaches to study the functions with infinite variables.Three approaches have been proposed in the literature to dealing with infinite variables.The first is an ansatz introduced by Itzykson and Zuber in the study of the Kont-sevich integral.They introduced a new set of variables and transformed the problem of infinite variables into the problem of finite variables by assuming that the higher genus free energy of two-dimensional topological gravity is polynomial under these new variables.Zhou deduced analogues of Itzykson-Zuber ansatz in one-dimensional topological gravity and Hermitian 1-matrix model,and interpreted these analogues by Wilson's renormalization theory in statistical physics.The second is the mean field theory introduced by Dijkgraaf and Witten in the topological field theory coupling to two-dimensional topological gravity.By establishing constitutive relations,the computation problem of the two-point functions of genus zero in big phase space is transformed into the computation problem in small phase space.Thirdly,Zhou introduced the emergent geometry inspired by the emergent phenomenon in statistical physics to explore geometric structures that are hard to see in the space with finite variables but emerge naturally in the space with infinite variables.We will further apply the above methods to the study of Gromov-Witten type theories.By combining the renormalization theory with the Virasoro constraint,we proved the Itzykson-Zuber ansatz.The recursive formulas for calculating the higher genus free energies of one-dimensional topological gravity,Hermitian 1-matrix model and two-dimensional topological gravity are given.In particular,this result solved the computation problem of free energies of two-dimensional topological gravity.By extending the mean field theory introduced by Dijkgraaf and Witten in topological field theory,we established the constitutive relations in the open intersection theory,Grothendieck's dessins d'enfants,the Hermitian matrix model with even couplings and the generalized Brezin-Gross-Witten model.As a result,closed formulas of one-point functions of genus zero are given.When applying the renormalization theory and the mean field theory to the special deformation theory in emergent geometry,on one hand,ghost variables appear naturally,on the other hand,the special deformations of the spectral curves of above theories are expressed in a unified form and this derives the dualities between above theories in big phase space.