View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server DIAS-STP-95-19 FSUJ TPI 4/95 July 1995 Generalized Thirring Mo dels y z I. Sachs and A. Wipf y Dublin Institute for Advanced Studies 10 Burlington Road, D 4, Ireland z Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universitat, Max-Wien-Platz 1, 07743 Jena, Germany Abstract The Thirring mo del and various generalizations of it are analyzed in detail. The four-Fermi interaction mo di es the equation of state. Chemical p otentials and twisted b oundary conditions b oth result in complex fermionic determinants which are analyzed. The non-minimal coupling to gravity do es deform the conformal algebra which in par- ticular contains the minimal mo dels. We compute the central charges, conformal weights and nite size e ects. For the gauged mo del we derive the partition functions and the ex- plicit expression for the chiral condensate at nite temp erature and curvature. The Bosonization in compact curved space-times is also investigated. < 1 Intro duction The resp onse of physical systems to a change of external conditions is of eminent imp ortance in physics. In particular the dep endence of exp ectation values on tem- p erature, the particle density, the space region, the imp osed b oundary conditions or external elds has b een widely studied [1]. Nevertheless, many prop erties of such systems are p o orly understo o d. The massless Thirring mo del [2], which is among the simplest interacting eld theories, has already led to considerable confusion ab out its thermo dynamic prop erties in the literature [3, 4, 5]. The reason is two-fold: Firstly, the computation of the fermionic determinant in the presence of a chemical p otential and/or non-trivial b oundary conditions is delicate, b ecause the eigenvalues of the Dirac op erator are generically complex. In section 3:1we prop ose a regularization scheme via analytic continuation. We argue that the so-obtained determinant, which di ers from previous results [4], leads to the correct equation of state. The second complication originates in the infrared-sector. An elegant infrared regu- larization, which is particularly well suited for the study of thermo dynamic prop erties, is to quantize the mo del on a torus. Harmonic contributions to the current arise then naturally and taking them into account turns out to b e crucial for a correct quanti- zation. In particular the so-obtained results di er from those gotten earlier [3] using b osonization. This is explained in section 3:2. On another front there has b een much e ort to quantize self-interacting eld theories in a background gravitational eld [6]. For example, one is interested whether a black hole still emits thermal radiation when self-interaction is included. Due to general arguments by Gibb ons and Perry [7] this question is intimately connected with the universality of the second law of thermo dynamics. The Thirring mo del (including the gauged version of it) is still solvable in curved space-time and we can study its prop erties in a background gravitational eld. This provides us in particular with an elegant approach to the study of its conformal structure: Correlation functions with current- and stress-tensor insertions, which are gotten by functional di erentiation with resp ect to the gauge- and gravitational elds, contain the necessary information to characterize the underlying symmetry algebras. To familiarize the reader with our approachwe rst rederive the conformal structure of the original Thirring mo del in section 3:3. We then showhow a non-minimal coupling to gravity leads in a natural way to a mo di cation of the conformal structure. In particular, very much as for a free scalar eld the central charge in the fermionic formulation of the Thirring mo del is not unique. Furthermore, we nd that the equivalence b etween nite size scaling and central charge of the Virasoro algebra holds only for a particular treatmentof 1 the zero-mo de sector in whichacharge at in nity is generated automatically. This charge combines in a non-trivial way with the Weyl-anomaly of the determinantof the uctuation op erators to reconcile the equivalence of the nite size scaling and the central charge. For certain values of the non-minimal coupling we obtain minimal mo dels from interacting fermions. This is the sub ject of sections 3.4 and 3.5. The gauged Thirring mo del, which contains the Schwinger mo del (QE D )asa 2 particular limit, is no longer conformally invariant but has a mass gap: The 'photon' 1 2 2 2 acquires a mass m = e =( + g ) via the Schwinger mechanism. It p ossesses a 2 2 non-trivial vacuum structure which promotes it to an attractivetoy mo del to mimic the complex vacuum structure in 4-d gauge theories. From our exp erience with the Schwinger mo del [8], which is supp osed to share certain asp ects with one- avour QC D [9], we exp ect that gauge elds with winding numb ers are resp onsible for the non- vanishing chiral condensate and in particular its temp erature dep endence. Con gura- tions with windings, so called instantons, exist only for nite volumes and minimize the Euclidean action. They lead to chirality violating vacuum exp ectation values. For example, a non-zero chiral condensate develops which only for high temp erature and large curvature vanishes exp onentially. Since for particular choices of the coupling constants the mo del reduces to well known and well studied exactly soluble mo dels there are many earlier works which are related to ours. Some of them concentrated more on the gauge sector and investigated the renormalization of the electric charge by the four-Fermi interaction [10 ] or the non-trivial vacuum structure in the Schwinger mo del [8, 11]. Others concentrated on the un-gauged conformal sector. Freedman and Pilch calculated the partition function of the un-gauged Thirring mo del on arbitrary Riemann surfaces [4]. We do not agree with their result and in particular show that there is no holomorphic factorization for general fermionic b oundary conditions. Also we deviate from Destri and deVega [5] whichinvestigated the un-gauged mo del on the cylinder with twisted b oundary conditions. We comment on these discrepancies in section 3:2. Section 2 contains intro ductory material and in particular the classical structure of the mo del. Other pap ers are dealing with di erent asp ects of certain limiting cases of the mo del considered here. In particular in [3], the thermo dynamics of the Thirring mo del has b een studied and the Hawking radiation has b een derived in [12 ]. The equivalence of the massive Thirring mo del and the Sine-Gordon mo del in curved space has b een shown in [13 ]. Partition functions for scalar elds with twisted b oundary conditions have b een computed in [14 ] and more recently in [15 ]. 2 2 Classical Theory The gauged Thirring mo del in curved space-time has the Lagrangian 2 g 1 L [A ; ; ]= i r j j F F , j = ; (1) Thir 4 4 where the gamma-matrices in curved space-time are related to the ones in Minkowski a space-time as = e ^ , r = @ + i! ieA is the co ordinate- and gauge covariant a derivative and F is the electromagnetic eld strength. The gravitational eld g a (or rather the 2-b ein e , since the theory contains fermions) is treated as classical background eld, whereas the 'photons' A and fermions will b e quantized. The classical theory is invariant under U (1) gauge- and axial transformations and corresp ondingly p ossesses conserved vector and axial-vector currents 5 j and j = = j : (2) 5 p Here = g denotes the totally antisymmetric tensor. In fact, the conservation laws together with the relation (2) b etween the vector- and axial currents imply that the currents are free elds 2 2 5 (3) r j = r j =0; which is the reason that accounts for the solubility of the mo del [16], even in the presence of gauge- and gravitational elds. Of course, for any gauge invariant regu- larization the axial current p ossesses an anomalous divergence in the quantized mo del. Thus the normal U (1) Ward identities in the un-gauged Thirring mo del [10 ] b ecome A anomalous when the fermions couple to a gauge eld. The solution of the equations of motion is most easily presented byintro ducing auxiliary scalar- and pseudo-scalar elds, in terms of which the action takes the form Z h i p 1 S = g F F + i (r ig @ + ig @ ) 1 2 4 (4) i +g (@ @ + @ @ ) : Note that for later use wehave allowed for di erent couplings of the fermionic currents to the scalar- and pseudo-scalar auxiliary elds and , resp ectively. The original Thirring mo del is recovered for g = g = g , since then 1 2 3 1 2 L = F + i r + gj B + g B B ; B = @ @ (5) 4 is classically and quantum-mechanically equivalent to (1), after elimination of the multiplier eld B . By decomp osing the gauge eld similarly as the B - eld as p 2 A = @ @ '; so that F = g r '; (6) 01 2 and cho osing isothermal co ordinates for which g = e , the generalized Dirac op erator reads 3 1 iF i G iF i G+ 5 5 2 2 D= = e @=e ; where (7) F = g + e and G = g + e' : 1 2 Hence, if (x) solves the free Dirac equation in at Minkowski space time, then 0 1 iF +i G 5 2 (x) e (8) 0 solves the Dirac equation of the interacting theory on curved space-time.