\ _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 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 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 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/,C, we can rewrite (2.3) as i6<¤>,6 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 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 ,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 In the case that C4 and c5 is dependent on A, the commutator of two transformations on h,/l is calculated by using (2.6) as follows: [.s(c.,),.s(C5)]h;} : 6(c6)h,q-62W CQCQEE)-;56%D,,B, ’‘?' / where cC?aw Z +al) * aw which is the same as (2.10) and gives the consistent gauge algebra. OCR Output The resulting form of the gauge transformation is given by @WBc(‘I’) 6 hA:8 .A+————-hg C(C) * "”” , 13A 1* (2.16) 6(¢)‘I’A = WBA(‘I’)¢ in terms of the functions WAB which satisfy the condition (2.6). This gauge transformation reduces to the usual nonabelian one when WBA() = fg/{DC. Then the gauge algebra is given by 62W) * ’‘ (6(C,),r(C2))11g} : 6(c3)h,j- c$c$%;-é&12,.<1>B, (2,7) l6(¢1)»6(¢2)l‘I’A = 6(¢s)‘I’A» where we have defined D,,A = 8,,A + WAB( 6(c)(D,,A) = (D,,<1>C)cOW &B (2.19) This reveals that the object D,) may be recognized as a covariant differentiation, D,,A transforming as a coadjoint vector. If the algebra is nondegenerate, it turns out that the gauge algebra is open as seen from (2.17) except for the case : 0, that is, the case of nonabelian gauge transformation. The commutator of two covariant diflerentiations provides lD1¢a Dvlq)A `; H/AB(q’)Rgi1» (220) where the curvature Rf, is defined by GW * * ‘i*‘ RQ, : 0,,1;- 0,,1),;+ £1i,{1i§, (2.21) and the covariant differentiation on D,. (DA has been so determined as the resultant expression D,,(D,,A) transforms like a coadjoint field, that is, D,,D,,A = 8,,D,,<1>A6l/V + —h,,D,,C. i€B (NPC Unfortunately, the curvature does not transform homogeneously because the gauge algebra (2.17) is open: 2s(c)1z;},9W : Eénfyc+((1>,,<1>D)5$B§I%1i§c-ic OZW C (,1 H 12)}, (2.22) which seems troublesome for tl1e construction of invariant actions. OCR Output This section is devoted to a brief explanation of an algebraic background for the ‘nonlin ear` gauge transformation defined in the previous section. The relevant object is the Jacobi identity for nonlinear Lie algebraslwhm We consider a vector space with a basis {TA} which is the polynomial algebra with a commutative product. On this algebra, we define a bracket product lTA»TBl = WANT), (223) as the second product structure, where lVAB(T) denotes a polynomial in TC which satisfies the antisymmetry property WAB = —-WBA: WAB(T) = w§,,§,°1C"” + I4/5,3Yb + w,(QTCTD + - ··, (2.24) where l/l'),g,0lc Wgg, etc. stand for structure constants in this algebra. We assume the bracket product is bilinear and antisymrnetric with respect to exchange A with B, satisfies the Jacobi identity and has the derivation nature [TA. TBTcl = [Tai 7`BlTc + TBlYl··1» Ykrl l7}iTB»7`cl I T/ilYb»Tcl + lTA»Tcl7`B We emphasize that the multiplication of TC and TD is dehned in the sense of the polyno mial algebra (i.e. not as the matrix, for instance), thus TCTD : TDTC. Generally speaking, it is not necessary to restrict consideration to the case of polynomial algebra, which is com mutative by definition. However, it is enough to treat the commutative case for the present purposes because we make use of t}1e structure functions WAB only in the form WAg() where represents a set of scalar fields C which do commute: ¢I>Cp = DC. We note that the zeroth—order term U/S2 may be recognized as a central element in the algebra (2.23). This definition includes the ordinary Lie algebras as a special case when WAB(T) is linear in TA, which makes the algebra (2.23) deserve the name of nonlinear Lie algebra. The typical examples of nonlinear algebra are W algebraandml quantum groupl uli The representation theories of quadratic extension of nonlinear SU(2) algebra and some interesting nonlinear algebras are investigated by several anthorsl ml * Strictly speaking, it is needed to extend WAg(T) to analytic functions of TA in the definition of nonlinear Lie algebra. OCR Output The Jacobi identity with respect to the bracket product (2.23) implies 3[/HAH GTI;. -WCW : 0, which is a generalization of the Jacobi identity for Lie algebras. The expression (2.17) can be regarded as a gauge algebra based on the nonlinear Lie algebra (2.23) in the sense that the functions l»VAB() satisfy the same constraint identity (2.6) as the Jacobi identity (2.25) for WAB(T) in the algebra (2.23). However, generally the field A is not a representation of its algebra different from the coadjoint field of Lie algebra. The relation of QA with the algebra should be more investigated. 2.3. AN INVARIANT Acrion We do not have a straightforward way of constructing invariant actions under the gauge transformation (2.16) owing to the inhornogeneous form of (2.22) as noticed in section 2.1. Hence we will instead seek for an action which provides covariant equations of motion under those transformations, and examine whether it is invariant under them. We determine of equations of motion as follows: The gauge algebra (2.17) should be closed, at least, on shell so as to originate from some Lagrangian symmetry. Hence the choice UHQ7,4 = O (2.26) seems an immediate guess for appropriate equations of motion, which are indeed covariant (2.19) under the transformation (2.16). Since we have the vector, hg, and the scalar, (DA, fields as our basic ones, as a. first trial for Lagrangian we consider LO : X""hQD,,A, (2.27) where X"" denotes an invertible constant tensor to be determined later. However, D,,A itself contains the fields hg`, and hence the variation of Lg with respect to hg does not lead to the covariant equations of motion (2.26), OCR Output I ·"*'' L = no - ;X·wAB1.,;1.,;, (2.28) which yields the desired equations of motion (2,26) provided X"" = —X"". Note that this Lagrangian can be rewritten as 12 -X»··#[q»AR{yBW g §,(Ei)+ WBC - q>A€’.hfhf] (2.29) up to a total derivative. If we require Poincaré invariance of the Lagrangian, the antisymmetric tensor Xt"' should be an invariant one X'"’ = J"', (2.30) where c”" denotes the Levi—Civita tensor in two dimensions. At this point, we are inevitably led to considering the two-dimensional situation. In summary, we have obtained a candidate action . S = /,11, L; 1L; ZBW -g*V 5[S(L%) QARV 4. WBC -.g?] QA-hh, (2.31) which comes out to be diffeomorphism invariant. In fact, the gauge transformation of L`, is sh); = 0,,BW wv(($’;{9) WBC - <1»,,-hc+ fc] aw [-a,,wBCh,c”° 8Wuc B C 6Wcz 18Wcv c n E <1> i W ——————W h h , The relation (2.6) enables us to show we = 0,, wv wm BW- i gcquhc, BA (2.32) which confirms the invariance of our action (2.31) under the ‘n0nlinear’ gauge transformation (2.16). The equations of motion which follow from (2.31) are given by the following: e""D,,A : 0, c’"’Rf,, = 0, (2.33) OCR Output which are indeed covariant due to the transformation law (2.19), (2.22). 10 Since the action (2.31) has the local gauge symmetry by the gauge transformation (2.16), there are some constraints. In order to quantize the theory in such system, it is necessary to fix gauge. When we make gauge—fixing and quantize gauge theory, use of BRS formalism is standard. In this section, we formulate BRS formalism of the ‘nonlinear’ gauge theory. We discuss how to construct the BRS transformation 6,, corresponding to the classical symmetry (2.16). Notice that the generator algebra for the gauge transformation (2.16) is openfasm can be seen from the gauge algebra (2.17). We first try to regard the gauge-transformation parameter cA as fermionic FP ghost, 6hf : 0,,.;+** U;,C(<1>)h§c, *C (2 34) WA = WBA(‘I’)¢°. where 6 is a candidate for the BRS transformation. In this level, cA is not a. gauge parameter like cA in the section 2.2, but a fundamental field. With the aid of the BRS nilpotency oondition 6‘A = O, we can uniquely determine the transformation law for the ghost whether cA depends on QA or not, which is different from the classical gauge transformation in section 2.1; 1 8W BC B C 6 A = ---—°°° 2.35 ( l 2 8A which satisfies 2A6c= 0. However, the naive definition (2.16) does not satisfy the nilpotency 6‘ = 0 on the field hg owing to the algebra non-closure: 62/LA = —cBcC———-£CLDBZW ,,p " 8¢I>A3D (2 36) I BC 62WBC _ CCE _ 3A8 6BST = O, and 0n—shell nilpotent on all fields under quantum equations of motion because Heisenberg operators always satisfy quantum equations of motion. OCR Output The desired BRS transformation 6,, and gauge—fixed Lagrangian LT can be obtained as follows: In view of the equation (2.36), we introduce a gauge—fixing fermionic function D to provide 6,, = 6 + 6,,, (2.37) where we define 02w so "‘B°`’£ 6 =——-—- "h** '°° 0<1>,,0<1>D""’61»;>’ 2.38 ( ) and 6,, = 0 on the other fields. The gauge-fixed Lagrangian is given by LT : L - i(6,, —§_6,,)S`Z. (2.39) Then 6,,h,/is21 calculated to be . OWBCr 6Bh,,= 6 ¢€€puHp 2A BC(L " llén _ Eébln (2 40) ‘ 0* . B C Fawm on ..... _...w _. a<1>,,0<1>D('°° C 3E EF ah) by using 69. Gu/DF F -— 66 -6 lU = 6,,l) 6 Q ————— 6hD( +2 bl(Jr bl +4sn,,G 0<1>G ° Since the Jacobi identity (2.6) provides the following identity W cBcCcF WEp = 0, it turns out that 6,2h,/1 vanishes under quantum equations of motion, that is, 6,, is on-shell nilpotent, and (2.37) yields a symmetry of the Lagrangian (2.39), as it should be: 6,,LT = 8,,8W [(QQ)ef"' WBC flc] — QA--hc 8(PA (2.41) OCR Output BZW 60 3 C BC 8 Q _ *‘l' ’*°° 0<1>,,a<1>D 6h}? 12 2.5. QUANTIZATION ON THE CYLINDER By means of the BRS formalism given in the previous section, we quantize the system on the cylindrical spacetime S] >< Rl in the temporal gauge. The gauge-fixing function for the temporal gauge is Q = EAhgA, (2.42) where we introduce FP antighosts EA and NL fields bA which satisfy the BRS transformation law 685,4 = {bl;. (2.43) The gauge-fixed Lagrairgian (2.39) now reads . . LT = L + l7Ah.()A + 1€A(CAHW ·l- lh()BCC), where the dot denotes a derivative with respect to the time 10. This Lagrangian reduces to the form of a free theory L; = ,4ii," + bQ,Ji0"‘+ 12,,8* (2.44) through field redefinition bi.; E 6.4 + @@,4 — .WABM+ OW @636?¢.B /C where we have discarded total derivatives w.r.t. xl in the Lagrangian (2.44). We can quantize explicitly (2.44) on S1 >< Rl and derive mode expansion as follows: h¤"(¤¤) = \/Z;1 .. ng /1nAe-gui L W ®A8—{,-xxl ®A(x) = " eA(x) = L W Z C:e‘“fy;x1 ’ J? 1t:_OO 6A(z) : `/57;L W 2€4.;e·*·»1R00 From the canonical commutation relations ofthe fundamental fields, we get the commutation 13 OCR Output relations of mode variables: [<¤.....h.?,1’ = —w£6.+..,¤, (2 45) mn. C5} = 636.+...0. As usual, it is assumed that 41,4,,, hf, EA,. and c': {or n 2 1 annihilate the Fock vacuum. Then nilpotency of the BRS charge in quantum level is necessary for the theory to be unitary and to be well—defined. The symmetry (2.37) yields a conserved charge Q = / dawg, (2.46) where its Noether current J: is defined by #:6 h -+6BL Mi<1>—-+6 -+a6L »—-+6 · 6Lb B BOL V BBLT ABc °cAB44L- A T 6(6,,h,,’*) 6(6,,¤I>A) 8(3,,cA) 8(8,,EA) 8(3,,bA) 4u/W __q)__;__h__® _;______·°OWBC Bi BC .A 0<1>,,)"°62WBC "*°° a<1>,,a<1>D6h,{>69 CBC The zeroth component of its Noether current. JB is given by J2 = (DAGIC+A J; r J : :{WBC’ h1B}CC ew-· % CB€ACC. (247) by using (2.42). Note that the operator ordering offers a matter for consideration in the definition of J written by mode variables. The ordering should have been so chosen as the BRS charge (2.46) to be hermitian like (2.47). This naive definition of the charge suffers from divergence difficulty, which can be re· moved by normal—ordering prescription of mode variables QB = :Q: Q, =i > n:A,.,cQ*": TI:-·O¤ . EZ 2]-2 Z Wim ‘ * ={¤>A.¤.... -—-<1>A..../»{...}¤-;_,..-,..C ’ g W(k)A¤···A»·.¢, __q, B- C _ 2 · BC · AHH ` Ax—i¤¤¤-icmCAlc-2;*4ni-m—l‘ * l=—-00 The canonical commutation relation (2.45) establishes the nilpotency Q; = 0 of the BRS operator straightforwardly. This enables us to define the quantum theory in terms of the BRS cohomologyl ml 14 OCR Output 3. R2 gravity with dynamical torsion In this chapter, we consider R2 gravity with dynamical torsion in two spacetime dimen sionsasm] the typical example of the nonlinear gauge theoryf m 3.1. Puma R2 Grmvirv We first make a brief exposition of R2 gravity with dynamical torsion in two spacetime dimensionsim When the spin connection w,,“° and the zweibein c,,“ are independent variables, the action, invariant under local Lorentz and general coordinate transformations 604.:,, = Bur — v8;,w,,’\A — (8,,v)w,\, (31) 6Ge,,“ = —·re“e,,b°’\ - v8,\e,,“ - (0,,1/`)eA“, genericallyx takes the following form: 2 ab uu 1 a pv 1S =% d I €(R)u, R ab — BET')., T n — 1), (3.2) where 1· and v" are local—Lorentz and general coordinate transformation parameters ,and a, B, 7 are constant parameters, and w,,c°Eb w,,,°b c E det(e,,°), RpvdEb 6”uJV_lIb avlvll, db T,,,,“ E 8,,e,,“ + w,,“°e,,;, - (n •—» v). Our convention is that the Levi-Civita antisyrnmetric tensor c satisfies em = cm = 1, and the fiat metric is given by ry = diag(+1, -1). The Lagrangian in the action (3.2) can be rewritten as G = ·(F01)+L2 ·T01°T01¤··€'Y» i 4eoz 4c,B (33) OCR Output where we have defined Fm, E 3,,44:,, — 3,,w,,. Various aspects of this theory were extensively studied in Ref.[19]. * We do not consider higher-derivative theoriesim 15 3.2. YANG-MILLS-LIKE Fomvrurxrion In order to interpret the theory defined by (3.2) as a nonlinear gauge theory, we consider a first-order Lagrangian Cc: = <#F01 — CCYWJ + ¢/>¤To1°—¢B¢>.»¢°— 67 (3-4) by introducing auxiliary fields up and ¢,,. This is equivalent to (3.3) in the region e gé 0. Namely, it coincides with the Lagrangian (3.3) when the auxiliary fields 66.0 = ·v”8iv>, (3.5) 6¤¢a : ’T(ab¢—b U A8,\¢a· Its equations of motion are as follows: 8plP `i' ¢u€nb€;4b : 01 6,»¢.. + w,mi¢—<..i€,.(¤s<¤+°°2 B¢¤¢° + *1) = 0. 260Lp —— Fg] = U, 2€,6¢n — Tmc = O. We note that the Lagrangian (3.4) in a weak—coupling limit or = B == 7 = O is none other than that of topological ISO(1, 1) gauge theorylu which is known to be equivalent to the gravity theory I = I dzmt/—g0R. (3.7) This observation suggests that the full Lagrangian (3.4) itself might have some symmetry of the Yang-Mills type which is related to the lSO(1, 1) gauge transformation. In order to find such symmetry, we use a trivial gauge symmetry proportional to the equations of motion (3.6), which Lagrangian system always has. We modify the local-Lorentz and general coordinate transformation (3.1) and (3.5) by the trivial gauge symmetry 6w,, = 60w,, + c,,,\v"[20ze¢p — Fm], (feud = 606/,° + 6,,,\v"[2,6€¢°— Toll`], (3 8) OCR Output W = 6aw + v[¤¢° + 7)]· 16 If we set t = ·r - ’\°’\vw,\, c= ·-vc,\“, we can explicitly write downl thc symmetry 6: 6w,, = 8,,t + 2aq,Cc°c,,°¢p, 6e,," = ——tc°°c,,;, + 8uc° + w,,c°cbbh + 2Bebccc,,°¢°, (3.9) 6w = ¢“°c¤ 4%, 6% = —f¢..b¢>+b <,.»¤(¤ 6Lg = 3,,[e””e,,“eabc(mp+b2 B¢c¢°— (3.10) The gauge transformations (3.1), (3.5), and (3.9) are reducible. In the following sections, we confirm (3.9) is a. special case of our nonlinear gauge trans formation introduced in the previous chapter. 3.3. QuArmAr1cALLv Nomrwmn Po1NcAnr5 A1,GnBnA The general-covariance property of our action (2.31) suggests that we might utilize it to construct two—dimensional gravitation theory. It seems a natural choice to adopt the Poincaré algebra in two dimensions as a base algebra on which the construction will be per· formed. Since the diffeomorphism invariance is guaranteed by the deHnition of (2.31), our requirement is that the theory should be local—Lorentz covariant. Hence we consider quadrat ically nonlinear extension of the Poincaré algebra which preserves the Lorentz structure of the genuine Poincaré algebra: J,J :0, J,1> ze °P l l [ ul ur, (3.11) calP¤.P1»l= —¤at(¤·U + BU PCP.: + ‘1I)» where ex, H, 7 are constants, rf" is the two-dimensional flat metric, and I is a central element added to the Poincaré algebra} Note added: Quite recently, this symmetry has been also obtained in Ref.[21] by means of Hamiltonian analysis of the action (3.2). Instead ofthe Poincaré group ISO(1, 1), we can deal with the de Sitter group SO(2, 1) by means of a constant shift in the field gp and an appropriate redelinition ofthe parameter 7. The right-hand side of (P,,,P;,] can contain a term proportional to cd], which results in nonlinear (anti-)de Sitter algebra. We omit that term just for simplicity. 17 OCR Output We set {TA} = {P,,J}, that is, Tg = P0, T; = P1, and T2 = J. Then the structure constants defined for the above algebra are given by Wg;} = ¢”n(6cl'602 — $0261}), Wg; = 0, gig? : _a(6Ck6D|€H), Wéignb = __B17ab(6Ck6Dl€H), Wé22)2u = Wé2ga2 = 0, W32 = -7<6C*6D’¢H>. ww = 0 {Or i 2 2., where tl1e notation of (2.24) is used. We also set gauge fields Ii: = (c,’:,w,,) and scalar fields QA = (¢,,,¢p) to obtain (3.4): 1 1 pv uv · n 2 a L1 Z -56 *PFuv " E6 ¢u]uv _ cmp " C/;¢a¢ " C7 as the Lagrangian (2.31). The gauge transformation law (2.16) now reads (3.9): {Sw), = But + 2orq,cc°e,,° 18 OCR Output 4. Generic Form of ‘Di1aton’ Gravity In this chapter, we consider generic form of ’dilaton’ gravity as the nonlinear gauge theorvlm 4.1. Tm: Acrion The general expression for the action of two-dimensional metric gw, coupled to a scalar field Q (without higher-derivative terms) is given by - - - I_ - — - /iS = d°¢ ~/—y(·y""¢'w)R — V(v>)) (4-1) We may replace L((®) by a linear function lm S = 6%it »/-:1 /§ (y"”6,»<¢>6»<¢> + iw? — W( through a field redefinition v¤ = U(¢)» an = ¢"§)··»» W( where rc is an arbitrary constant. In particular, we can set rc : 0 to obtain S = idzzz: LD; LD = (/—g(§ The action (4.4) is of course invariant under the following general coordinate transformation: 6c:9;w = “'UA6A9pv "‘ (8;¤v)!L\u/\A " (an/Ul9;¤»\» (4 5) %w=—W&w In the following section, we investigate a gauge-theoretical origin ol` the ‘dilat0n’ field gp in the action (4.4), which might be regarded as a “pure” gravity in two dimensions. + This field transformation is well-defined only locally in the field space of ep where (B11/81,6) 96 0. In particular, it is inapplicable to the case of Ll(¢) = constant, where the field ¢ is regarded as a scalar matter rather than the ‘dilaton' field. - lg OCR Output It seems a natural choice to adopt the Poincaré algebra in two dimensions as a base algebra on which tlic construction will be pcr|`orrnc We also set the vector field hf : (e,,“,w,,) and the scalar field A = (d>,,, cp) to obtain 1 1 LS : —§e""¢,pF),,, — -§e’"’q$,,T,,,,“— eW(g0) (4.8) as the Lagrangian (2.31), where we have defined Fw, E 3,,w,, — 3,,44:,,, T,,,," E Hue,/I + w,,c°°e,,b —- (p 4-+ 1/), (4.9) e E det(e,,“). The gauge transformation law (2.16) now reads BW 5wp : But + q,ccbe,,°—5@, 6e”“ : —te"°e“;, + 8,,6* + wuelwcb, (4.10) 6w = ¢“"c..¢». where we have put cA : (c“, t). { The following form of nonlinear Poincaré algebra is not the most general one that preserves the Lorentz structure. It is chosen so as to realize the torsion-free condition as an equation of motion in the resultant ‘nonlinear’ gauge theory. 20 OCR Output We can actually confirm 6LS = —8,,[e”"e,,“e,,;,eWBW " (5)- $], (4.11) which corresponds to (2.32) in the previous section. The equations of motion which follow from (4.8) are given by Bw + ¢a€nb€;sb : Ow b ‘?u‘f’¤ + °*’u‘¤b;$;’énbcxiliv(v = 0» Eéw-(iv + 65; = 0. Eflwnrua = 0» which of course corresponds to (2.33). Note that the gravitation theory (4.4) is obtained if one puts e,,“e,,°n,;, = gu., and integrates out the fields gb, and wl, in the Lagrangian (4.8) under the condition e gf 0. The reformulation (4.8) reveals that the theory (4.4) possesses hidden gauge symmetry (4.10) of the Yang-Mills type. This gauge symmetry includes difleomorphism, as is the case for the Yang—Mills—like formulation of R2 gravity with dynamical torsion in the previous chapter. 5. ‘Nonlinear’ Supergauge Theory In this chapter, we extend the previous consideration to the a ‘nonlinear’ supergauge theory i. e. gauge theory based on nonlinear extension of Lie superalgebras, and apply this gauge theory to two-dimensional dilaton supergravity. 5.1. ‘NONLINEAR’ GAUGE TRANSFORMATION We introduce a supermultiplet of gauge fields (hf,f,,”), where Inf is a bosonic field and fu" a fermionic one. A denotes an internal index of the even generators and oz the odd generators. As is the case for ‘nonlinear’ gauge theory exposed in Ref.[6, 11, 12], we need a ‘coadjoint” bosonic scalar field 45,; and a fermionic one gba. According to the same discussion as in Ref. [12], we determine the gauge transformation of gauge Helds hf and fn", auxiliary fields q$A and $0,. 21 OCR Output By the same discussion as the bosonic case in the chapter 2, the gauge transformation of hu", fu", ¢A and 1/:,,, is determined. The resulting form of the gauge transformation 6(c,·r) is given by 8W OU 8 8V ,8WBg C BUB 8UCp Bl/p 6, 7’<6·z»¤U18!} B___h ____l·y ;9_____hC__]_(mi" aa. wi. ”’ +°“" f"v¢,." +’“ + f 6(c,T)q$A =B’B cWgA + tTUAB, 6(C,T)'lJ)u ZBp CUBG + T`/pa, (5.1) where cA,·r" are gauge parameters and WAB(d>,1/1), UAp(¢, and V,,,,g(¢, are smooth functions of the fields ¢A and dwa to be determined below. Since the transformation includes the one in the case of the usual Lie superalgebra, we assume that WAB(¢, 1/2) = —Wg,4(¢,1/J) and V,,,p(¢, 1b) = V,g,,,(q$,1l¤). Similar to (2.6), WAH, UAB and Vmg should satisfy the condition 8WAB 6WBc . W +U ————+A t, A,B,C :0, cp A5 ¤¤ where Antisym(A,B,C) denotes antisymmetrization in the indices A, B and C, and Sym(oz,B,·y) symmetrization in the indices oi, B and ·y. The expression (5.2) can be derived from the Jacobi identity for a nonlinear Lie superalgebra with structure functions WAB, U Ap and Vup. The gauge algebra is given by 22 OCR Output OCR Outputwhere the curvatures Rf, and RZ, are defined by 8W .3U .8V A A BC ”* ”*" R;}. = 6..h.— 0.h.+ —£h,.h..+ »§(h,.:.— h.e..) + ¤;,i%c..c.*._ .R5:. = 3,.:.** — ae;6W - »—f»».h.+6U BV —$ 5.2. AN INVARIANT Acrrow We do not have a straightforward way of constructing invariant actions under the gauge transformation (5.1), as noticed from the inhomogeneous form of the transforms of the curvatures. Hence by similar discussion to chapter 2, we will instead seek an action which provides covariant equations of motion under those transformations, and examine whether it is invariant under them. The gauge algebra (5.3) should be closed, at least, on shell so as to originate from some Lagrangian symmetry. Hence the choice Dufé.4 Z Ov Dulpu : 0: (5-9) seems an immediate guess for appropriate equations of motion, which are indeed covariant (5.6) under the transformation (5.1). lf WAB, UAB and Vpu are nondegenerate, the equations (5.9) imply Rf, = O, Hg, = U, under the formula (5.7). The Lagrangian is constructed from our basic fields which are the vector ones hf and {M"' the scalar ones d>A and iba. Moreover sinoe it is natural that the desired Lagrangian produces (5.9) and (5.10) as the equations of motion, we are led to the Lagrangian BW BW ”C . c =1 —-XW,(: ins. + Mm:.(,j§%) + WBC - ¢..—- —¢..r»,.r». _ _ BUB7 +2U———7— 8UB h B1+ BVBV——;————— BVBB 1, A<1B7 ¢A3¢A ibn Gila!pn £v 2 B18¢A ¢A I/foralba gu gv (5.11) OCR Output which indeed yields the equations of motion (5.9) and (5.10), where X"" = —X" 24 lf we require Poincaré invariance of the Lagrangian, the antisymmetric tensor XW should be an invariant one X'"' = c’"', (5.12) where e"" denotes the Levi—Civita tensor in two dimensions. At this point, we are led to considering the two-dimensional situation. In summary, we have obtained a candidate action S = I dlx L, . . c = —;¢··” | mn;}. +($$§ mn;. + WBC8W —¢,.-@@)0 8W ¢..h.,.r». ’”° _ " Z ~g.#·~[¢,,(a,,1.,/-* 12,;.,,)**’C + wBCh,/1.,] ¢""[i*l>¤(<'?,.€»“— @»E,.°) + iUB-,(h,.£.»"BB — h.,£,.") + iVpq£,/35*]. which comes out to be diffeomorphism invariant. The relation (5.2) enables us to show 8W _0U C . 6(c,0)L = -8,,[{ BCc’"’cWBgh,,+ ZUB-ygyly <§££)——¢A -—h.,+ 8¢A z—Z§.," 6¢A _ 1%W 3 which confirms the invariance of our action (5.13) under the ‘nonlinear’ gauge transformation (5.1). The equations of motion which follow from (5.13) are given as follows: WD = O, MRA = U, 6 ”¢’° 6 “" (5.15) OCR Output e""D.,¢,, = 0, e"”Rjf,, = 0, which are indeed covariant due to the transformation law (5.6), (2.22). 25 As a typical example of nonlinear supergauge theory defined in previous section, we consider generic form of two-dimensional N = 1 dilaton supergravity. N = 1 dilaton super— gravity has been defined in Ref. [25]. The superfield formalism are very useful when we construct a supersymmetric field theoryiAlsom in this paper, we construct N = 1 dilaton supergravity by that method. Thus we introduce superspace (ac", O"), where 9 is Grassmann number and y and cv run over 0 and 1. In order to construct N = 1 supersymmetric extension of two-dimensional dilaton grav ity, we introduce a scalar superfield and a superzweibein Ep" as <1> E up + ilix + QGOF, E,," E e,,“ + i(;·y“(,, + %60e,,°A, where 0 and 6 are real Grassmann numbers and X and Q, are fermions. Since 0 and 6 are real, 9 term is not needed. Generic form of dilaton supergravity considered in this paper is the following a.cti0n:[mml SG = 2i I d.:z2 d0 E[S - Z()], (5.16) where Z() is a. smooth function of and F H W9 = 6 i-1+91 -Cin i`f‘ 1 00 -A +5Zg c C;¤'Y5Cv » S = A + il?53 + QGOG, E = _2cIW'Y5Dp G = ‘R “ '%C;¤'7I‘E + ic"'/C1¤75CvA ‘§A2» (517) OCR Output Eu" = eu" + i5·y"(,, + -}00e,,"A, u 0: ' u 1* ' cr1 ' cv u Et = EQ; —§(9·rs) wi. — Z(0m»)3i A — EZ09 Q A+ (1»E) r Ep" = i(0‘r“)p, Ep`! = dp`!(. 1 — 26 Here we denote cpu = c—]€;w R = 2e""3,,w,,, 5.18 ( ) - D}l The locally well-defined field redefinition enables us to have (5.16) from the action in Ref. [25]. The action (5.16) is invariant under the general coordinate, local-Lorentz and the following local supersymmetry transformation: 6s —li Z»x)C. — LFC,). " 2 " 2 2 ’ 2 (5.19) 6Se,," = i€·y“§,,, 1 1 éscp : 2(8y: + El-U;A'Y5)€ + 5,)/liA€s 6SA = {EE, where c is a gauge parameter. In the following section, we investigate a gauge-theoretical origin of the ‘dilaton’ super field in the action (5.16). 5.4. NoNuNr:A11 SUPER-Po1NcAm3 ALGEBRA In order to construct the action (5.16) as the nonlinear supergauge theory, we expand each field in terms of 0 and 9, and carry out the Grassmann integration of the action (5.16). Then (5.16) becomes as follows: . . _ . 1/2[E{_ l SG = da: — e"" 27 where we have written Z(¢I>) a.s Z(‘I’) = V( V, Ll and W being smooth functions of ep. Next we regard tl1e zweibein e,,“ and the spinconnection wu as the independent fields as the bosonic caseibyn] which the following term is added to the action: SA Z _':;€Iw l d2x¢a]Yfw Yiwu Z Guerra + w1•‘a€vbh“Zb + '}€b(€u'75'YCvl “ (U H ")· That is, if the auxiliary field Moreover since A and F are auxiliary fields, they are eliminated from the action (5.20) by using the equations of motion of A and F: A = U( The final expression for the action is as follows: Sp = I dla: Cp LP = ‘%6W |‘P(6;¤°°‘v _ avwul " iY'Y5(0uCv " Gvgn) + ¢S¤(8u€v“ _ Guerin) iu(*P)v( i>2w,.C» + §>2*v"¤‘.i»U··¢,/’C.» ééii` - 1(§vsV( This action (5.22) is classically equivalent to (5.16). Comparing (5.22) with (5.13), we find · 28 OCR Output the base nonlinear superalgebra has the following commutation relation: lJ1Pal = Cabnhcpca [Papa] = —§mU(J)V(J) - § lJ,Q¤l = é(Q‘rs)¤. (523) [1%.Qpl = 1; and the other commutation relations vanish. Here we set {TA} = {Pa, J}, that is, T0 = P0, Y] = P1, and T2 = J, and we also set the vector fields hg = (el/‘,w“), .f,,"' = (7l(,,)"' and the scalar field $,4 = (¢,,,¢p), 1/rc, = Xiu; ucd is the two-dimensional Minkowski metric. Note that the choice W(J) = V(J) : U(J) = 0 corresponds to the original super Poincaré algebra. Then the structure functions defined in (2.23) for the above algebra are given by W2a : 'Wn2 : Eaniébv W22 = Ov Wai = —§<¤z»U(<#)V( Uza = tin, (5.24) OCR Output X · " 1 U — u up — 4(xv W )p<»¤ ( The gauge transformation law (5.1) now reads 29 &uu . = But + €1,cC°€,,°u(‘P)v(‘P)6 1 % [Z “ Z?YXW(‘P) + §e:»¢(5<‘vC»»)‘5;' 8U ' + OL!§(>?1T) 6§u° = —§¢(·1sC,.)“ + ¤»c¢°¤»°(vsx)°W( 1 1 b 1 r 1 a b cv cr a u oa 6x = —§i(·vsx) + Z(·rs·v x) me U( where we have put cA = (c", t). The equations of motion which follow from (5.22) are given by 8u*P + ¢¤€“°€;:b ` Z 02 _ _ 0 6,.% +w,.¢..»¢b b + 1mc,. (—;U(¢)V(a¤)i§xx(v)> + W¤—¤..z»(x·v Z C»)U = 0. 1 1 1 · 1 “°“° 6,.54 — 5215% + Z>2·vs·r¤..t¢,.U —§C,»(5V — v¤..t¢) = 0, 1 - 5CuV l6#wV —_ al/OJ}!8 + 1cbCc[|€V5;{X(uv) ` ' 811 + _` %Cb6c}l(Y7CV)%' BV bCbc _ &%(Cl|'Y5 (5.26) ée“"T,,,“, = 0. They of course correspond to (5.15). Note that the gravitation theory (5.20) is obtained if one adds (5.21) to the Lagrangian (5.22) and integrates out the fields Qin and wp in the Lagrangian (5.22) under the condition e gé 0. The reformulation (5.22) reveals that the theory (5.16) possesses hidden gauge symmetry (5.25) of the Yang-Mills type. This gauge symmetry includes diffeomorphism, as is the case for the Yang—Mi1ls-like formulation of generic form of dilaton gravityi m 30 OCR Output We have made a systematic construction of the nonlinear gauge theory (2.31) with the gauge transformation (2.16) based on a general nonlinear Lie algebra. A new category of the gauge theory has been constructed as an extension of the gauge theory based on the usual Lie algebra. We have carried out the BRS quantization of nonlinear gauge theory. Since the gauge is open, we have had to modify usual quantization method. The BRS charge has been explicitly constructed without anomaly in the cylindrical spacetime S I x R1 in the temporal gauge. The algebraic structure of the above nonlinear gauge theory seems to be clear by con struction. However the field A is not a representation of nonlinear Lie algebra; it is different from the coadjoint field of usual Lie algebra. The relation of A with the algebra deserves further investigation. The clarification of its geometric structure is desirable in view of the rich structure present in the usual nonabelian gauge theory. However, the gauge algebra (2.17) is open, which implies non-existence of Lie-group-like object corresponding to the nonlinear Lie algebra. That might cause difficulty in the geometrical investigation of the nonlinear gauge theory. In the chapters 3 and 4, we have considered typical examples which have nonlinear gauge symmetry defined in previous chapter. We have considered Lorentz-covariant quadratic extension (4.6) of the Poincaré algebra in two dimensions in chapter 3. The nonlinear gauge theory based on it has turned out to be the Yang-Mills-like formulation of R2 gravity with dynamical torsion. Namely, the “def0rmation” we observed in Ref.[10] has been shown to be Lorentz·covaria.nt quadratically nonlinear extension described above. We have also considered general Lorentz-covariant nonlinear extension (4.6) of the Poincaré algebra in two dimensions in chapter 4. The nonlinear gauge theory based on its algebra has turned out to be the generic form of ‘dilaton’ gravity, which clarifies a gauge theoretical origin of the non-geometric scalar field in two-dimensional gravitation theory. We note that this theory is a generalization of Yang-Mills-like formulation of the Jackiw Teitelboirn’s modelm and the dilaton gravitylm because the latter corresponds to a particular choice W( 31 OCR Output We have also found that the generic form of ‘dilaton’ supergravity turns out to be the nonlinear supergauge theory based on the Lorentz—covariant nonlinear extension (5.23) of the super—Poincaré algebra in two dimensions. The reformulation (5.22) reveals that the theory (5.16) possesses hidden gauge symmetry (5.25) of the Yang-Mills type. We also can interpret two formulations of dilaton supergravity that the super structure of the base space in dilaton supergravity is transmuted into the super structure of target space in a nonlinear super-Poincaré algebra. This theory is a generalization of Yang-Mills-like formulation of the dilaton supergravitylm We expect that construction of the nonlinear gauge theory and its application to two dimensional gravity in this paper has offered a new method to analyze quantum Field theory. The analysis of this theory at the quantum level is still insuflicient. It might be interesting to compare the theory (5.16) of the Utiyama type and the one (5.22) of the Yang-Mills type in the aspects of their quantization. ACKNOWLEDGEMENTS The author would like to thank N. Nakanishi for valuable comments and careful reading of the manuscript. He thanks K. ·I. lzawa and for valuable discussion as his collaborator. He also thanks I. Ojima and M. Abe for valuable comments and discussions. 32 OCR Output REFERENCES for example, J. Wcss and J. Bagger, Supcrsymmetry and Supergravity (Princeton University Press, 1983). D. Bernard, Suppl. Progr. Theor. Phys. 102 (1990) 49; I. Ya,. Aref’eva and I. V. Volovich, Mod. Phys, Lett. 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