\ _L Z; Q - A S UO °E> It 0 3 RIMS-953 Two-Dimensional Gravity and Nonlinear Gauge Theory Noriaki IKEDA December 1993 IIIIIIIIIIIIIIIIIIIIIIIIIIIEI pmmzammm 7§’?¤§I<i<% i.¥$u¥?I¥¤I&H1i9EPIi RESEARCH INSTITUTE FOR MATHEMATICAI. SCIENCES KYOTO UNIVERSITY, Ky0t0,Japan OCR Output OCR OutputTwo-Dimensional Gravity and Nonlinear Gauge Theory N0n1AK1 IKEDA Research Institute for Mathematical Sciences Kyoto University, Kyoto 606-01, Japan ABSTRACT We construct a gauge theory based on nonlinear Lie algebras, which is an extension of the usual gauge theory based on Lie algebras. It is a new approach to generalization of the gauge theory. The two—dimensional gravity is derived from nonlinear Poincaré algebra, which is the new Yang-Mills like formulation of the gravitational theory. As typical examples, we investigate RZ gravity with dynamical torsion and generic form of ’dilaton’ gravity. The supersymmetric extension of this theory is also discussed. OCR Output The gauge theory is fundamental object in quantum field theory. In this paper, we consider a generalization of the usual gauge theory based on Lie algebra and its application. There are many studies to extend it by modifying symmetry structure, such as super symmetry:" quantum grouplWil algebralHowever al in this paper, we consider quite new approach to extend the usual gauge theory. We construct a two-dimensional gauge theory based on nonlinear extension of Lie algebras as a generalization of the usual nonabelian gauge theory with internal gauge symmetry. We call it nonlinear gauge theory, which is a new generalization of nonabelian gauge theory. Nonlinear extension of Lie algebras which is called nonlinear Lie algebra has the following commutation relation: [TA»TBl = WAH(T)» where WAB(T) denotes a polynomial in Tg. The details of definition are presented in section 2.2. Quadratic nonlinear Lie algebras in the context of quantum field theory were first introduced by K. Schoutens, A. Sevrin and P. van Nieuwenhuizenlm They mainly analyzed the W-gravity as an application of nonlinear Lie algebra to quantum field theory. Our nonlinear gauge theory is different realization of nonlinear Lie algebra in quantum field theory, and presents a new category of constrainted systems. The two—dimensional nonlinear gauge theory is closely related to two-dimensional gravity. The investigation of dilfeomorphism-invariant field theories is expected to provide indispens able information for the quest of quantum gravity. At the classical level, the Einstein theory field theory of metric tensor with general covariance — is quite successful in describing gravitational phenomena. The general covariance in gravitation theory is external gauge symmetry, which is space time symmetry. It may be called gauge symmetry of the Utiyama typel°' On the other hand, the internal symmetry such as in QED or QCD is called the one of the Yang-Mills typelsl The connection to gauge symmetry of the Utiyama type with the Yang-Mills type has been investigated on various authors, Witten’s three-dimensional Einstein gravity (Chern Simons-Witten gravity) can be regarded as one example for the above descriptionl" In two dimension, Jackiw-Teitelboim’s modelm is a typical example that a. gravitational theory and a gauge theory have classically the same actionlil One of the reasons for investigating the OCR Output connection between two types of symmetries is that symmetry of the Yang—Mills type is easier to treat in quantization problems than that of the Utiyama type. In this paper, we introduce new examples which have such properties and are related to nonlinear gauge theory. The particular feature of two-dimensional nonlinear gauge theory is that it includes two dimensional gravitation theory. When the nonlinear algebra is Lorentz—covariant extension of the Poincaré algebra, the theory turns out to be the Yang—Mills-like formulation of R2 gravity with dynamical torsionlormum generic form of ‘dilaton’ gravitylTheml hidden symmetry of their gravitation theory is clarified. It yields new examples of the connection between the two types of gauge symmetries. Moreover a gauge theory of nonlinear Lie superalgebra is related to supergravity. When nonlinear superalgebra is nonlinear and Lorentz-covariant extension of super»Poincaré algebra, the theory turn out to be dilaton supergravitylm The remaining part of the paper is organized as follows: In chapter 2, our action for ‘nonlinear’ gauge theory is constructed in quite a general manner. We also consider nonlinear Lie algebras as a background structure for the ‘nonlinea1·’ gauge theory. The BRS formulation of 11onlinear gauge theory is formulated. The chapter 3 provides Lorentz-covariant quadratic extension of the Poincaré algebra and our nonlinear gauge tl1eory based on it. It turn out that the resulting theory is the Yang—Mills-like formulation of R2 gravity with dynamical torsion. In chapter 4, the relevance of this ‘nonlinear’ gauge theory to the ‘dilaton` gravity is clarified. In chapter 5, our action for ‘nonlinear’ gauge theory is extended to the supergauge theory. The connection to the dilaton supergravity of ’nonlinear’ supergauge theory is analyzed. Chapter 6 concludes the paper. 2. ‘Nonlinear’ Gauge Theory In this chapter, we explain a generic approach to the construction of a ‘nonlinear’ gauge theory — gauge theory based on nonlinear extension of Lie algebrasl m 2.1. iNONLINEARi GAUGE TRANSFORMATION We introduce a ‘coadjoint’ scalar field <DA in addition to a vector field hf for the purpose of constructing a ‘nonlinear’ gauge theory. Ilere A denotes an internal index. We consider OCR Output s(c)h,gZ* 0,,M + 11gC(<r>)h.{fc,° (2 1) 6(c)¢A = WgA(<I>)cU, where 6(c) is a gauge transformation with the gaugetransforrnation parameter cA, and Uf}C(¢) and WAg(®) are smooth functions of the field (DA to be determined below. In the CASE of usual nonabelian gauge theory, U£C : fgc and l/VBA = fgpjbg, where denote structure constants in some Lie algebra. Since the above gauge transformation should reduce to the usual nonabelian one if Ugc is constant and l/{/BA is a linear function of (DC, we assume Ugc : —UgB and lll/BA : —WAB. We investigate the commutator algebra of these gauge transformations with gauge transformation parameters cl and cz. We require a closed algebra on the scalar field (DA l6(¤1l»6(¢2ll‘I’A = 6(¢i)‘I’A, (22) First we consider the case that the gauge parameters cl and cg are independent of <I>A. In this case, the right—hand side of (2.2) becomes 8W ~ BW i6<¤i>.6<¤2>1¤>A(5,,§egi) Z »—¤rB» + rvmi mi In case of nonabelian gauge theory, since WBA : fg/,<I>C, we can rewrite (2.3) as i6<¤>,6<c2>14>A = <r£,i.r§D + rAB1§D>¤>E¤F¤€.’ ‘ fcfr/iE€C»gg‘I’i}g (2-4) : 6(Cf3)®/D by using the Jacobi identity of Lie algebra.: j,},,,jgD’ Z ri, (2.5) where [ABC] denotes antisyrnmetrization in the indices A, B, and C, and C? = fi}¤¢f¤€»’ In order to extend the above construction to (2.3), we replace the structure constant ff;/1 by OCR Output @5%,%*- and fg/4QC by WBA in (2.4). Then (2.4) becomes (2.3), and (2.5) is generalized to 8W[AB —W Z . 8,1,,) cw 0 2.6 ( ) Owing to this generalized Jacobi identity, (2.3) becomes BVV ")l/V i l6(¢1),6(¢2)l‘I’A(%fgi) = WB0 +D wcvCIV?D (2.7) 8W = l/VDA C?Cg:. This assumption results in the composite parameter 6W CSA = C{?Cg , (2.8) and the gauge algebra becomes the required form (2.2) on QA. In the 11ext section, we show that the expression (2.6) can be derived as the Jacobi identity for a nonlinear Lie algebra with structure functions WAB. Since the right-hand side of (2.8) involves QA, we have to consider the case that the gauge parameters C4 and c5 are dependent on QA in [6(c4),6(c5)]QA. In this case, we have BW BW 1·s<cr>.·$<c5>1<r>A(g;§,§) = row + w0DCEC? 0D 0 Dé [(WDAWBC gécf ~ égcgn (2-9) 0wBC B C 00C 0cC W »——[ 6q,D- w -- - DAC4 D C5D 54 BCGQB8,1,B C4 Cs r by using (2.6), and if we set 0w cg/1 Z *··0 A" 0 A y the gauge algebra satisfies lD(¢4)»D(¢s)lq’A = 6(¢E)‘DA· (210) On the other hand, as the commutator of two transformations on the vector field hg, we require [6(c1),6(c2)]hf : 6(e3)hf + ··~, (2.11) where the ellipsis indicates a term which is irrelevant here. lf we calculate the right-hand OCR Output side of (2.11) explicitly, it is obtained that A A B c (mii e l6(¢1)»6(¢2)lh,i = 6»(Unc¢1€z)· ;i,‘8u‘I’D11 + l UBGUEE +a00//IB5“` Ul2EU(?Bw * +%EQ) WED + DWCDh»·€i°i» 12 (2.12) where we assume that cy and cg are independent of <I>,q. From the first. term of left-hand side of (2.12), the composite parameter cg should be cf? == Ugccfcg.y (2,13) We require cg = c3, that is, we henceforth put BWi A ZBc() U‘I’ 2.14 ( l MM in order for the commutator algebra to represent a consistent composition law of Lagrangian symmetry. Then (2.12) becomes l6(¢1)»6(¢2)lh,i8W(3,;/?iB$ 62W= Gp ¢¢) ·' @i·‘I’D 8WDC UWM BWDE OWCB azvl/BC @2WEB B E C ——————- ——————————W —————W h + + ED + Cpl ”°‘ C2 0<r>,, 0<1>,, 0<1>,, ao,) 0<1>,,0<1>D 3¢I>A8<I>p 02w Z »v»s%»¤(£*?)*’·aw C’¤+aw aw s c?¤%’”° BZW — ·‘*C?Cg D“®B, (2.15) by using (2.6).