Wess-Zumino-Witten Term In Noncommutative Gange Field Theory

In the thesis, after the introduc f the physics construction of \VZW term in the noncommutative two dimensions we study the l)1oPel ies of WZW term in the noncommutative four and high dimensions and construct the \VZW term in Geometry in lifferent niethocls in nonconimut at lye four and high dimensions. In noncommutati ye gauge field t heory( NC F T) ve still define the WZW term as the logarithm of the fermion determinant as in the commutative gauge field theory. In Chapter 2. :3. we introduce some l)ackgroundl knowledge including the knowledge of noncommutative gauge flekl theory( NC4FT) and WZW term. In Chapter 4. we introduce the WZW term in noncommutative two cliniensions. First, we calculate the cbiral anomaly in Fujikaw&s va: then we make a gauge transformation of fermion fields of U( I) at last we can get the WZW term by use of the melat ion between the changes of fermion cletermi nant a 11(1 Fuj ikawa .1 accoi)i and rela t ion I)etween Fuj ikawa .Jaccobi aicl the chiral anomaly. The result and method can be generalized to U( ii). In Chapter .3. we study the WZW term in noncommutative four dimensions. Due to the (lifflculties to integrate the anomaly, we construct the \VZ\V term by use of its geometry topological I)rol)erty i.e. WZW term is the closed 1ocbain of gauge trans- formation group. In the lower. 4 dimensions. with the notation of the 1st toj)oiogical i)roPel y of the gauge group derive from the one in the higher by 2. 6 dimensions. we study the (lecent relationship of the cohomologv from 6 (himensions 1w the A ?ci )iOfll])leN and get the the closed 1 ochain of the gauge group i.e. \VZXV term in noncomninut at ive -1 dlin)ensions But the method can not i)e geneiaiizecl to higher di- mensions. s?In chal)ter 6. we get the dosed n ocbain of the gauge group from tile cohoinologv gtoup of the connection space vhich cati l)e calculated easily. 1w use of tile relat ion l)etWeeil the co1ioiiio1oc v oro connect ups of gauge grout) and iou sl)ace. Of course we also get the WZW- ternm. Xoncomnmutation should be not iced all above. By 1)rodiuct ve a iso get the consistent anomalies in four ditnensiomis. \vliicli is coiisistent with tile result of Bonora etc.