Generalized Thirring Models

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DIAS-STP-95-19

FSUJ TPI 4/95

July 1995

Generalized Thirring Mo dels

y z

I. Sachs and A. Wipf

Dublin Institute for Advanced Studies

10 Burlington Road, D 4, Ireland

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

particular limit, is no longer conformally invariant but has a mass gap: The 'photon'

2 2 2

acquires a mass m = e =( + g ) via the Schwinger mechanism. It p ossesses a

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

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

space-time as = e ^ , r = @ + i! ieA is the co ordinate- and gauge covariant

derivative and F is the electromagnetic eld strength. The gravitational eld g

(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

j and j = = j : (2)

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

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

h i

S = g F F + i (r ig @ + ig @ )

1 2

(4)

+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

L = F + i r + gj B + g B B ; B = @ @

(5)

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

A = @ @ '; so that F = g r ';

(6)

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

iF +i G

(x) e (8)

solves the Dirac equation of the interacting theory on curved space-time. The vector

currents are related as

j = = ^ e p j :

0 0

The same relation holds for the axial vector current.

Di eomorphism invariance then leads covariantly conserved energy-momentum

tensor

2 S

T :

(9)

Applying the variational identities in App endix A one obtains after a lengthy but

straightforward computation

i 1

( ) ( )

g F F F F + [ D (D ) ] T =

4 2

+2r r g r r + ( $ ) (10)

j (g r g r ) + ( $ ) +

1 2

2 4

( )

r ; +g g j r 2g j

2 2

( )

where wehaveintro duced the symmetrization A B = (A B + A B ). The rst

two lines are just the energy momentum of the electromagnetic eld, charged fermions

and free neutral (pseudo-) scalars. The remaining terms re ect the interaction b e-

tween the fermionic and auxiliary elds. On shell T is conserved as required by

general covariance. Using the eld equations for and its trace reads

T = F F :

(11)

In particular for A = 0 it vanishes, and the theory b ecomes Weyl-invariant.

Symplectic structure: In the presence of b oth fermions and b osons it is convenient

to exploit the graded Poisson structure [17]

! !

B(y) A(x) B(y) A(x)

: fA(x);B(y)g dz

0 0

x =y

O(z) (z) (z) O(z)

O O

The sum is over all fundamental elds O (x) in the theory . The sign is minus if one

or b oth of the elds A and B are b osonic (even) and it is plus if b oth are fermionic

(o dd) elds. The momentum densities (x) conjugate to the O - elds are given by

functional left-derivatives. A simple calculation yields the following momenta

y 5

= i ; = g j +2@ and = g j +2@ :

2 0 1 0 0

In the following sections we are lead to consider the Euclidean version of the

mo del. Then one must replace the Lorentzian ;g and ! by their Euclidean

counterparts. For example, with our conventions the relation (2) b ecomes

j = i j

and the generalized Dirac op erator in Euclidean space-time b ecomes

2 f f

D= = e e @=e; where f = iF + G +

(12)

(see (7) for the de nition of F and G), instead of (7). Also, to recover the Euclidean

2 2 2

Thirring mo del as particular limit of (4) wemust set g = g = g .

2 1 5

3 Thermo dynamic- and conformal prop erties

In this section we analyze the quantum theory corresp onding to the classical action

(4) without gauge elds, in at and curved space-time. The gauged mo del is then

considered in the next section. Here we calculate the partition function, ground state

energy, equation of state and determine the conformal structure of the un-gauged

mo del.

To allow for a non-vanishing U(1)-charge we couple this conserved charge to a

chemical p otential .For the nite temp erature mo del the imaginary time must vary

from zero to the inverse temp erature and the b osonic and fermionic elds must

ob ey p erio dic and anti-p erio dic b oundary conditions, resp ectively. We enclose the

system in a spatial b ox with length L to avoid infrared divergences.

We shall determine the dep endence of the partition function and correlators on

the metric. This provides us with an alternative approach to the conformal structure

and its relation to nite size-e ects. Also, it enables us to study the e ect of non-

minimal coupling to gravity in section 3:3. Hence we allow for an arbitrary metric or

2-b ein e with Euclidean signature. We can cho ose (quasi) isothermal co ordinates

and a Lorentz frame such that

0 1

e = e e^ e

a a

0 1

(13)

j j

2 2 2

g = e g^ e ; g = e ;

where = + i is the Teichmueller parameter and the gravitational Liouville

1 0

eld. Space-time is then a square of length L and has volume V = d x g .We

allow for the general twisted b oundary conditions for the fermions

0 1 2i( + ) 0 1

0 0 5

(x + L; x ) = e (x ;x )

(14)

0 1 2i( + ) 0 1

1 1 5

(x ;x + L) = e (x ;x ):

The parameters and represent vectorial and chiral twists, resp ectively.We could

i i

allow for twisted b oundary conditions for the (pseudo) scalars as well [14, 15], e.g.

0 1 1 0

(x + nL; x + mL)=(x ;x )+2(m+n). However, to recover the Thirring mo del

for equal couplings wemust assume that these elds are p erio dic. For =0, =i =L

in which case V = L, and for = = 0 the partition function has the usual

0 0

thermo dynamical interpretation. Its logarithm is prop ortional to the free energy at 6

temp erature T =1= .

3.1 Fermionic Generating Functional

Twisted b oundary conditions as in (14) require some care in the fermionic path in-

tegral. The subtleties are not related to the unavoidable ultra-violet divergences but

to the transition from Minkowski- to Euclidean space-time. To see that more clearly

let S denote the space of fermionic elds in Minkowski space-time with chirality 1.

Since b oth the commutation relations and the action do not connect S and S we

can consistently imp ose di erent b oundary conditions on S and S . On the other

hand, in the Euclidean path-integral for the generating functional

R R

p p

y y

y g iD= + g ( + )

Z [; ]= D D e ;

(15)

the Dirac op erator

0 D

D= =

D 0

exchanges the twochiral comp onents of , i.e. D= : S !S .Thus, in contrast to

the situation in Minkowski space the twochiral sectors are related in the action. Of

course, the eigenvalue problem for iD= is then not well de ned. This is the origin of

the ambiguity in the de nition of the determinant. It is related to the ambiguities

one encounters when one quantizes chiral fermions [18 ]. Here we reformulate this

problem in sucha way that the determinant with chiral twists ( 6= 0) can b e obtained

by analytic continuation. The resulting determinants do not factorize into (anti-)

holomorphic pieces. In app endix B we give further arguments in favour of our result

by calculating the determinants in a di erentway.

Let us now study the generating functional for fermions in an external gravita-

tional and auxiliary eld. For that we observe that on the torus wemust add a

harmonic piece to the auxiliary elds to which the fermionic current couples in (4).

More precisely, in the Ho dge-decomp osition of B in (5) contains a harmonic piece,

@ + B = @ h with r h = h =0:

(16)

[; ]

More generally, allowing for arbitrary couplings of the various terms in (16) to the 7

fermionic current, we are led to add a term

Z Z

2 p 2 p

g gh j +( ) gh h

L L

to the action (4). Note, that in isothermal co ordinates, for which the metric has the

form (13), the harmonics h are constant. The constant h couple to the harmonic

2 2

part of the current and are needed to recover the Thirring mo del in the limit g = g =

0 1

g . Also, we shall see that the harmonic degrees of freedom are essential to obtain

the correct thermo dynamic p otential.

Finally weintro duce a chemical potential for the conserved U (1) charge. In the

Euclidean functional approach this is equivalent to coupling the fermions to a constant

imaginary gauge p otential A [19].

As a consequence of the ab ove observations the scaling formula (12) (recall, that

F = g and G = g when the electromagnetic interaction is switched o ) is mo di ed

1 2

2 f f

D= = e e D=e ; where f = ig + g +

1 5 2

(17)

2i L

@ + i!^ D= = [g h + ] and = i :

0 0

L 2

This scaling prop erty will enable us to relate the fermionic determinants and Green's

functions of D= and D= . The spin connection! ^ in (17) vanishes for our choice of

the reference zweib ein. The dep endence of D= on the chemical p otential and the

constant harmonic eld h cannot b e gotten by the anomaly equation [20]. It must b e

computed by direct metho ds. For this we expand the fermionic eld in a orthonormal

basis of the Hilb ert space

X X

(x) = a (x)+ b (x)

n n+ n n

n n

X X

(18)

y y

(x) = a (x)+ b (x);

n n

n+ n

n n

where a ;b ;a ;b are indep endent Grassmann variables. A basis is given by

n n n n

2 1 1

i(p ;x)

e e ; where (p ) = ( + + n ); (x)=

i i i i n

(19)

L 2

and the e are the eigenvectors of . Recall that and represent the vectorial- and

5 i i 8

chiral twists (14) resp ectively. The and must ob ey the S and S b oundary

n+ n

conditions, resp ectively. These b oundary conditions x the admissible momenta p

in (19). Since the Dirac op erator maps S into S the must then ob ey the same

b oundary conditions as the .Thus (x) is obtained from (x)by exchanging

n n n

p and p . It follows then that

n n

iD= = (20)

n n

with

h i

2 1 1

= ( + a + + n ) ( + a + + n )

1 1 1 0 0 0

L 2 2

(21)

h i

2 1 1

= ( + a + n ) ( + a + n ) :

1 1 1 0 0 0

L 2 2

Here wehaveintro duced a g h . To continue we recast the in nite

pro duct for the determinant in the form

Y Y

1 1 2

= g^ ( + c + n )( + c + n );

n n

(22)

L 2 2

~n2Z

whereg ^ is the inverse of the reference metric (13) and

j j

c = a + i^ ; with (^ )= :

(23)

The logarithm of the pro duct (22) can in turn b e written as the derivative at zero

argument of a generalized zeta function. Indeed one easily veri es that for

(s) ( )

n n

(24)

wehave (formally)

+ 0

) = exp[ (s)]j : det(iD= ) (

s=0

n n

reg

(25)

However (s)isdivergent for s 1. These divergences can b e regularized as follows:

We compute (s) for s> 1 and subsequently de ne its value for s<1by analytic 9

continuation.

Assume for the moment that c is real or equivalently that there are no chiral twists

and chemical p otential . Then (s)hasawell de ned analytic continuation to

s<1 via a Poisson resummation [21]. Indeed, writing (s) as a Mellin transform

s1 t

n n

; (s)= dt t e

(26)

(s)

the generalized Poisson resummation formula

X X

exp[h (n a )(n a )] = h exp[h n n 2in a ];

(27)

Z Z

applied to the integrand in (26) yields after integration over t

(1 s) 1

2s1

(s)= )]: g (g n n ) exp[2in (c +

(28)

(s) 2

The zero mo de with n = 0 is eliminated b ecause for s>1 it do es not contribute.

After this analytic continuation (s) and (s) are now regular at s = 0. More precisely

(0) = 0 and

0 1

(0) = g (g n n ) exp[2in c ]

(29)

h i h i i h

c c 1

1 1

(0;) (0;) : = log

j ( )j c c

0 0

2 2

^ ^

Here we made use of det[c(iD= ) ] = det[(iD= ) ], which follows from (0)=0.

For complex c the Poisson resummation is not applicable and (0) cannot b e cal-

culated by direct means. To circumvent these diculties we note that the in nite

sum (24) de ning the -function for s>1 is a mereomorphic function in c.Thus we

may rst continue to s<1 for real c and then continue the result to complex values.

Using the transformation prop erties of theta functions the resulting determinant can

b e written as

h i h i

1 a + a

1 1 1 1

2( g^g^ 2i a )

1 0

det(iD= )=e (0;) (0;):

(30)

a a + j ( )j

0 0 0 0

This is the main result of this section. 10

It can b e shown that this determinantis gauge invariant, i.e. invariant under !

+ 1, but not invariant under chiral transformations, ! + 1, as exp ected.

Furthermore, it transforms covariantly under mo dular transformations ! + 1 and

!1= . In other words, det iD= is invariant under mo dular transformations if at

the same time the b oundary conditions are transformed accordingly. The exp onential

prefactor is needed for mo dular covariance and is not present in the literature [4]. It

correlates the twochiral sectors and will have imp ortant consequences. In App endix

Bwe con rm (30) with op erator metho ds.

The last step in the calculation of the fermionic generating functional is the inclu-

sion of the lo cal contributions to the auxiliary- and metric eld, i.e. the dep endence

of the determinanton , and .For this weintro duce the one-parameter family of

Dirac op erators

1=2

f f

D= = e D=e :

(31)

1=2

We take the -dep endence of the metric as g = e g^. With f as de ned in (17), this

family interp olates b etween D= and D= . The determinant of the full Dirac op erator is

then obtained byintegrating the corresp onding anomaly equation [22]:

2 q

S g

^ ^

detiD= = det(iD= ) exp + g^ 4 ;

(32)

24 2

where

^ ^

g^[R 4 ] S =

(33)

is the Liouvil le action. In deriving this result we assumed that g =0. This

constraint on the zero-mo de of (and similarly of ) will b e discussed b elow. Actually,

for our reference metric the Ricci scalar R vanishes and the Liouville action simpli es

to g^ 4. However, the ab ove formulae hold for arbitrary reference metrics

and arbitrary Riemannian surfaces. Furthermore, as exp ected for a gauge-invariant

regularization, the function and thus the longitudinal part of B do es not app ear

in the determinant.

To complete the calculation of the generating functional we need to know the

fermionic Green-functions S . Using the scaling prop erty of the Dirac op erator, eq. 11

(31), it is easy to see that in an arbitrary background eld S is related to S by

f(x) f (y )

S (x; y )= e S(x; y ) e :

Together with the relation (32) and the explicit form (29,30) for det iD= this yields

the fermionic generating functional

i i h h

c 1 c

1 1

(0;) (0;) Z [; ]=

c j()j c

0 0

(34)

R 2

1 p g

(x)S(x;y ) (y )

e exp g4] : S +

24 2

By using the scaling prop erties of the Ricci-scalar and Laplacian (see app endix A)

the exp onent can b e written in a manifest di eomorphism-invariant wayas

Z Z

1 1 g

p p

g R R + g4:

96 4 2

Here we used that on the torus R integrates to zero. On the sphere or higher genus

surfaces the last formula is mo di ed.

The Integration over the auxiliary elds then leads to the full generating functional

of the Thirring mo del. It contains all information ab out the thermo dynamic- and

conformal prop erties. This is the sub ject of the next two sections.

3.2 Thermo dynamics of the Thirring Mo del

In this chapter we derive the grand canonical p otential, equation of state and ground

state energy for the Thirring mo del. For this we need to compute the partition

function

2 S

Z = d hD D Z [==0] e ;

(35)

where Z is the fermionic generating functional (34) and S the b osonic action

F B

S =(2) g^g^ h h g 4 + 4 :

(36)

As it stands the partition function is still ill-de ned unless we constrain the zero-

mo des arti cially intro duced in the Ho dge decomp osition of B in (16). The choice

of the constraints is restricted by the symmetries of the system. In particular transla-

tion invariance (or rotation invariance on the sphere) and covariance under mo dular

transformations of the torus are symmetries whichmemaywant to preserveby the

zero-mo de constraint. The constraint measure

Z Z Z

dh dh D D ( ) () dh dh D D ; g

(37)

0 1 0 1

(and similarly for ) satis es these requirements (The normalization by the volume

in the de nition of is needed such that the constraints and hence the partition

function are b oth dimensionless). For example, one nds the dimensionless partition

function

(;4)

N D ()e =

(38)

det (4)

for free b osons, where the prime indicates the omission of the zero-eigenvalue.

Integration over the harmonics: There is no restriction on the harmonic parts

of the auxiliary elds and the Gaussian integral yields

h i

h i h i

()

c c

1 1

2 2

d h exp[(2 ) g^g^ h h ]= ;

(39)

c c

0 0

4 1+g =2

where

h i

i (n+u)(n+u)+2i(n+u)w

() = e

n2Z

is the theta function with characteristics

1 1

u = ( + i ) and w = ( + i )

1 0 0

(40)

1 0

1 1

and covariance

2 2

g 4 g 0

0 0

: = +i

(41)

2 2

4 g g 0

2 + g

0 0

0 13

Integration over and : The integral over , sub ject to the -constraint in (37),

merely contributes one inverse square-ro ot of the primed determinantof 24to the

partition function and so do es the integration over . In fact, to obtain the partition

function of the Thirring mo del we divide Z by the corresp onding partition functions

N of the free b osons, eq. (38). Using (39) and (34) we obtain

2 h i

Z 1 2 + g u

2 (1=24 +g )S

= () e ;

(42)

2 2

N j ( )j 2 + g w

where wehave also used the scaling formula for the primed determinantof 4[20 , 23]

1 det (a4)

a p]; = log a (0) = log a [ log

(43)

det (4) 4

with p b eing the numb er of zero mo des of the op erator. On the torus a = 0 and we

0 0

det ( a4)= det (4);

which pro duces the extra factor 2 + g . In the Thirring mo del limit g = g and

2 2 0

the square-ro ot in (42) disapp ears.

Zero-temp erature limit: Toinvestigate the thermo dynamics of the mo del we

assume space-time to b e at and that = i =L. Then

1 Z

= log

is the grand canonical p otential. First we analyze the low temp erature limit of . For

= 0 this yields the ground state energy.We observe that for = i =L the covariance

matrix in (41) simpli es to

h 2 2 i

g 1

g 4 g

0 0

i = Id +

(44)

2 2

4 g g

L 4 2 + g

0 0

and has eigenvalues 14

2 + g 2

= and =

1 2

(45)

L 2 L 2 + g

with corresp onding eigenvectors

v =(1;1) and v =(1;1): (46)

1 2

Also, the ^ tensor (see 23) and (see 17) in (40) simplify to

0 =L

= and = i :

L= 0

For !1 the saddle p oint approximation to the Gaussian sum (39) de ning the

theta-function b ecomes exact and therefore using that

for !1 log j ( )j !

we nd

4 L

( !1)= ( + )

6L 2+g L 2

2 + g 4 L

min fn n ( + )g +

2 1 1 (47)

n2Z

2L 2 2 + g 2

+ fn + n 2 g

1 2 1

2 + g

for the zero-temp erature grand p otential of the un-gauged mo del. Here the chemical

p otential and chiral twist enter only through the combination +L=2 . Let us now

discuss the p otential in the various limiting cases.

i) No chiral twist, =0, and vanishing chemical p otential: Then ( !

1) coincides with the ground state energy. The minimum in (47) is attained for

n = n =[ + ] and we nd

1 2 1

1 2 2

( [ E (L; ; =0) = + + ]) :

1 0 1 1 1

(48)

6L L 2 + g 2

Only for anti-p erio dic b oundary conditions, that is for = 0, do es this Casimir energy

1 15

coincide with the corresp onding result for free fermions. For g 4 the Casimir force

is always attractive whereas for g < 4 it can b e attractive or repulsive, dep ending on

the value of . The result (48) is in agreement with the literature [5]. For example, it

coincides with De Vega's and Destri's result if we make the identi cation ! =2

DD 1

and 1= =1 + g =2 in formula (42) of that pap er.

ii) Small twists and chemical p otential: For small and the minimum

is assumed for n = 0 and the p otential simpli es to

2 2

( !1)= +

(49)

6L L 2+g

and do es not dep end on the chemical p otential. For vanishing g the minimum of

(47) is attained for

1 L 1 L

n =[ + ] and n =[ + + + ];

1 1 1 2 1 1

2 2 2 2

where [x] denotes the biggest integer which is smaller or equal to x. This then leads

to the following zero temp erature p otential

2 L

= ( + )

6L L 2

L 1 L

+ [ + ]g

(50)

1 1 1 1

L 2 2 2

L 1 L

+ + + [ + + + ]g :

1 1 1 1

L 2 2 2

For = = 0 this reduces to the Casimir energy for free fermions with left-right

symmetric twists and agrees with the results in [24].

Note, however, that for 6=0 we disagree with [5]. The di erence is due to the second

term on the right in (47). Let us givetwo arguments in favour of our result: The

discrepancy arises from the prefactor app earing in the fermionic determinant (30). As

discussed earlier this prefactor implies the breakdown of holomorphic factorization,

a prop erty which has b een presupp osed in [5]. One can show that our results can

b e repro duced by starting with massive fermions and taking the limit m ! 0 (see

app endix B).

The second argument go es as follows: Supp ose that = = 0. Then (50) simpli es

1 1

to 16

2 2

2 L 2 L 1 L

( !1)= + [ + ] :

(51)

6L L 2 L 2 2 2

For massless fermions the Fermi energy is just and at T = 0 all electron states with

energies less then and all p ositron states with energies less then are lled. The

other states are empty. Since d =d is the exp ectation value of the electric charge

in the presence of we conclude that it must jump if crosses an eigenvalue of the

rst quantized Dirac Hamiltonian h. For vanishing twists the eigenvalues of h are

just E =(n )=L.From (51) one sees by insp ection that the electric charge

d 1 L

hQi = =2[ + ]= 2n for E

n n+1

d 2 2

indeed jumps at these values of .Further observe, that in the thermo dynamic limit

L !1 the density

! ;

L 2+g 2

reduces for g = 0 to the standard result for free electrons.

Equation of state: We wish to derive the equation of state for nite T in the

in nite volume limit L !1. This maybeachieved byinterchanging the roles

played by L and . More precisely, using that

h i h i

u w

2iwu 1

() = det(i ) e (i )

w u

we nd in analogy with the low temp erature limit, that for L !1 the pressure is

given by

1 Z 2 2 + g

p = lim log = +

L!1

L N 6 2

2 + g

min fn + n +2 g

1 2 0

n2Z

2 2

fn n +2 +2i g + :

2 1 0

2 +g 2

0 17

Here the minimum of the real part has to b e taken. Again the minimization arises

from the saddle p oint approximation to the theta function which b ecomes exact when

L !1.For small twists the minimum is assumed for n = 0 and then

2 2

p = ( + i )

6 2 + g 2

b ecomes indep endent on the chiral twist . As wehaveinterchanged the roles of

the temp oral and spatial twists this is consistent with the earlier result that for small

twists is indep endentof . In particular, for =0, we nd the following equation

1 0

of state

p( ; ; =0) = + :

(52)

2 2

6 2 2 + g

This result is consistent with the renormalization of the electric charge which is con-

jugate to the chemical p otential. It shows that the thermo dynamics of the Thirring

mo del is not just that of free fermions as has b een claimed in [3]. Indeed, the zero

p oint pressure is multiplied by a factor 2=(2 + g ). This mo di cation arises from

the coupling of the current to the harmonic elds. It is missed if only the lo cal

part of the auxiliary eld is considered, which is the case if one quantizes the mo del

in Minkowski space and then replaces the k -integral in the Green functions by the

Matsubara sum. This remark should also b e taken seriously in four dimensions! Fur-

thermore, we see that the 'pressure' p is real only for = 0, which is consistent with

the nite temp erature b oundary conditions .

3.3 Conformal structure

In the rst part of this section we derive the Kac-Mo o dy and Virasoro algebras of

the mo del (4) without gauge-interaction and prepare the ground for an extension,

containing in particular the minimal mo dels, in the second part.

Recall (11) that for A = 0 the theory reduces to a conformal eld theory on at

Minkowski space-time. To continue it is convenienttointro duce adapted light cone

0 1

co ordinates x = x x and the chiral comp onents of the Dirac spinor =

(1 ) . Then after substituting the classical equations of motion

This can also b e observed in the Hamiltonian formalism [25]. 18

2 2

T = ( @ @ )+2(@ ) +2(@ )

+ +

(53)

+i@ (g + g )

1 2 +

dep ends only on x and is therefore the chiral No ether current. Evaluating the

Poisson bracket of the symmetry generator T = dx f (x )T with the di erent

elds yields the classical structure

= f@ ; = f@

f f

(54)

= f@ + @ f ; =( )

f + + + f f +

j =f@ j + j @ f ; T = f@ T +2T @ f:

f f

Short Distance Expansions: Let us now determine the quantum corrections to

these classical results. These are computed within the Euclidean functional approach

from the short-distance expansions of the relevant n p oint functions. We need not

p ostulate Kac-Mo o dy and Virasoro algebras in advance as has b een done in [10 , 26 ].

These structures are derive here. When comparing the classical with the quantum

y y

results one should keep in mind that the roles of and are interchanged when

0 1

one switches from Minkowski to Euclidean space-time. In co ordinates adapted to the

holomorphic structure of the torus

0 1

x = ix + ix ; so that @ = (@ @ );

x x

the Dirac op erator and the corresp onding Greens function take the form

0 1= 0 @

+ O (1); i@= =2i and S (x ;y )=

1= 0 @ 0

where = x y , and the chiral comp onents of the energy momentum tensor and

current are given by

dg^ 1

0 0

00 01 0 1

T = (T + T )= T and j = (j j ):

xx x

2i 2i d 2i

From the conformal Ward identities 19

dz h O (x ) O(x )T i hO (x ) O(x ) O(x )i =

1 n zz 1 i n

(55)

i=1

we obtain the central charges and conformal weights directly from the correlation

functions. However, b ecause on the at torus the exp ectation value of T is constant,

we need to compute at least the 3-p oint function to read o the conformal weights.

As in the classical theory (see (9)) the symmetric energy momentum tensor measures

the change of the e ective action = log Z under arbitrary variations of the metric.

On the torus there are two indep endent contributions. One b eing due to variations of

the mo dular parameter and its conjugate which dep end implicitly on the metric.

The other is due to the variations of terms which dep end explicitly on the metric.

Since the chiral comp onent T is gotten by contracting T with dg^ =d it follows

that

i 1 @ dg^

hT i = [g; ; ] [g; ; ]: +

xx x

L @ d g (x )

g (x )

When doing metric variations it is always understo o d that we take the at space-

time limit afterwards. The variation is constant and may b e discarded in the

short distance expansion. Thus to analyze the algebraic structure we can work on

any Riemann surface. This is not true for the nite size e ects, which are global

prop erties. This asp ect will b e analyzed in section 3:4:

For example, taking three metric variations of the curvature dep endent part of log Z

with Z from (42) we nd the following short distance expansions for the three p oint

correlation function

1 1

hT T T i :

uu vv zz

3 2 2 2

(2 ) (u v ) (u z ) (v z )

Substituting this result into the Ward identity (55) we obtain the central charge and

the conformal weight of the energy momentum tensor

c =1 and h =2: (56)

Note that the the central charge as well as the conformal weight are indep endentof

the couplings g and g .

1 2

The conformal weights of the fundamental elds are obtained by computing the 20

fermionic two p oint function with stress tensor insertion

y y

Z h (x) (y )i : h (x) (y ) T i =

0 0 zz z

1 1

Since Z exp[F (R )], its metric variation vanishes after the at space-time limit

has b een taken. The variation of S can b e found in app endix A. This yields

1 2g 1 1

2 2

h = h y = + g

0 1

2 16 16 2 + g

(57)

2g 1 1

h = h y = g :

16 16 2 + g

Thus wehave repro duced the classical results supplemented by additional g and g

1 2

dep endent quantum corrections. In the Thirring mo del limit g = g = g , these terms

2 1

add up to give the known anomalous dimension app earing in the Thirring mo del [26].

Furthermore, from (57) wemay derive a condition on the couplings g ;g if we insist

1 2

on unitarity, i.e. on h 0. We nd

2 2

g :

(58)

2 + g

In particular for g 2 the conformal weights are p ositive for any real g .

1 2

Next we determine the Kac-Moody algebra of the U (1) currents. To derive the corre-

lation functions with current insertions we couple the fermions to an external vector

eld, that is consider the 'gauged' mo del without Maxwell term. For example,

1 [g; A]

= j :

A=0

A (x )A (y )

e g(x )g(y )

The e ective action with external vector eld is then obtained by shifting the auxiliary

elds in (17) as

(59)

g ! g + e' , g ! g + e ;

2 2 1 1

where A = @ ' + @ and wehave neglected the harmonic contribution to the

external vector eld, b ecause it do es not contribute to the short distance expansion.

The resulting e ective action do es not dep end on due to gauge-invariance. To relate 21

the variation w.r.t. A to that w.r.t. ' we use

T T

@ = A ; where A = A r r A

is the transverse part of A .We obtain the following short distance expansion

1 1 1

hj j i :

x y

2 2

22+g (xy)

We read o the value k of the central extension in the U (1)-Kac-Mo o dy algebra

: k =

(60)

2 + g

The precise g -dep endence of k (which can of course b e rescaled to unitybyan

appropriate rede nition of the current) is related to a nite renormalization of the

electric charge in the gauged Thirring-mo del whichwe will discuss in section 4.

Finally, from

1 1 1

hj j T i

x y zz

2 2 2 2

4 2+g (xz)(yz)

we obtain h =1.

To see how the left and right Kac Mo o dy currents act on the fermionic elds we

notice that after the integration over the auxiliary elds the A-dep endence of the

fermionic Green function factorizes as

y y

m '4' eg (x) eg (y )

h (x) (y )i = e e h (x) (y )i e ;

0 A 0 A=0

1 1

where g (x)=i (x)+ '(x), =2=(2 +g ) and m is the induced 'photon'-mass

(see(86)). Variation w.r.t. the A eld yields, after some algebraic manipulations,

the U (1) charges

2 2 1 1

(1 + (1 q = ) and q = ):

0 0 (61)

2 2

2 2 + g 2 2 + g

2 2

Wehave used the convention where the electric charge q +q is unity. In the Thirring

mo del limit we can compare (61) with the results obtained in [26]. For that we need

to rescale the currents such that the central extension (60) of the Kac-Mo o dy algebra

=2j . It is then easy to see that we agree with Furlan 1+g b ecomes unity j !

z z

et al. [26] if we make the identi cationg = g =4 1+g =2.

2 2 22

Non-Minimal Coupling: In section 3:1wehave analyzed the fermionic determi-

nant in the presence of twisted b oundary conditions. One may ask what happ ens if

weintro duce a lo cal twist instead, that is

y y (x)

(x) ! (x) ; (x) ! (x) e ; (62)

which should b e interpreted as a mo di cation of the charge neutrality condition. The

computation of the fermionic determinant in the presence of suchtwists is similar to

that for a Weyl rescaling of the background metric (31-32). Integrating the corre-

sp onding anomaly equation we nd

det(iD= )

log / R + O (( ) ):

(63)

det(iD= )

We will come back to the relation b etween the ab ove determinant and charges at

in nity at the end of this section. For the momentwe use the analogy merely as a

motivation to study the extension of the Thirring mo del obtained by coupling the

- eld non-minimally to the background geometry. That is we consider the mo del

(4) again without gauge-interaction but with an extra coupling

g R:

Then T in (53) is mo di ed,

T ! T = T +3g @ :

The corresp onding mo di cation of the classical conformal transformations (54) gen-

erated by the mo di ed generator T = dx f (x )T are

= ; = @ f

f f f f

(64)

i i

y y y

= g g @ f ; = + g g @ f:

f + f + 1 3 + f f 1 3

+ + +

2 2

Whereas and remain primary elds, do es not. This is in fact needed for

consistency. Indeed, since is not a scalar under conformal transformations generated

D= in the action is only conformally invariantiftransforms by T , the term

inhomogeneously like a spin connection.

It may b e surprising that the new symmetry transformations dep end on the cou-

pling constant g which is not present in the at space time Lagrangian . However,

3 23

the same happ ens for example in 4 dimensions, if one couples a scalar eld confor-

mally, that is non-minimally, to gravity. Although the Lagrangian for the minimally

and conformally coupled particles are the same on Minkowski space-time, their en-

ergy momentum tensors are not. The same happ ens for the conformally invariant

non ab elian To da theories which admit several energy momentum tensors and hence

several conformal structures [27].

The current still transforms as a primary with weight 1, but the energy momentum

tensor acquires a classical central charge

2 3

T = f@ T +2T @ f g @ f:

(65)

The corresp onding commutators in the quantized theory with non-minimal coupling

to gravity are calculated as explained for the minimally-coupled mo del. One nds

that the quantum corrections to (64) are identical to those of the minimally coupled

mo del and thus are g 6= 0-indep endent.

To summarize, wehave obtained the following Virasoro Kac-Mo o dy structure:

Central charge:

c =1+24g and h =2 (66)

Kac-Mo o dy level and charges:

k = ; h =1

2 + g

1 2 2 1

q = (1 + (1 ) ; q = )

0 0

2 2

2 2 + g 2 2 + g

2 2

Conformal weights:

1 1 1 2g ig g

1 3

h = + g =(h y)

0 1

2 16 16 2 + g 2

(67)

1 2g ig g 1

1 3

g =(h ) : h =

16 16 2 + g 2

Here some comment ab out unitarity is in order. It can b e shown that with resp ect

to the standard scalar pro duct [28] re ection-p ositivity holds for any real g [29].

However with resp ect to this inner pro duct the Virasoro generators are not selfadjoint.

Cho osing an alternative scalar pro duct [14 ] for which they are selfadjoint, p ositivity 24

do es not hold in general for g 6=0.We give a more detailed discussion ab out unitary

subspaces in section 3:5.

3.4 Finite size e ects

When quantizing a conformal eld theory on a space-time with nite volume one in-

tro duces a length scale. The presence of this length scale in turn breaks the conformal

invariance and gives rise to nite size e ects. It has b een conjectured [30 ] that the

nite size e ects on a Riemann surface are prop ortional to the central charge. For

example, when one stretches space time, x ! ax , then the change of the e ective

action is prop ortional to c:

= log a ;

ax x

(68)

where is the Euler numb er of the Euclidean space time. In [31 ] this conjecture has

b een proven for a wide class of conformal eld theories on spaces with b oundaries.

The only imp ortant assumption has b een that the regularization resp ects general

covariance. In this subsection we shall see that the equivalence do es hold only for a

particular zero-mo de treatment, which di ers from (37).

The only global conformal transformations on the torus are translations whichdo

not give rise to nite size e ects. Also, the Euler number vanishes and according

to (68) the nite size e ects are insensitive to the value of c. For that reason we

quantize the un-gauged mo del (4) on the sphere where the global conformal group is

the Mo ebius group.

An e ective metho d to compute nite size e ects has b een develop ed in [31]. It

is based on the following observation: Any conformal transformation z ! w (z )isa

comp osition of a di eomorphism (de ned by the same w ) and a comp ensating Weyl

transformation g ! e g with

dw (z ) dw (z)

2 0 1

e = ; z = x + ix :

dz dz

Therefore, cho osing a di eomorphism invariant regularization one has

0= = :

Dif f C onf Weyl

The change of the e ective action under Weyl rescaling is

D () det (iD= ) exp(S [g ])

g B

= log ;

Weyl

D () det (iD= ) exp(S [^g])

g^ 25

where S is the b osonic action (36). Since on the sphere there are no harmonic vector

elds the term h in S is not present. Imp osing the conditions (37) we obtain

1 V S g 8 det 4

R = log + (R ) + log :

Weyl (69)

V 24 4 4 V

det 4

Toevaluate (69) one intro duces the 1-parametric family of Laplacians

4 = e 4

interp olating b etween 4 and 4. Integrating the corresp onding anomaly equation

[20] we end up with

Z Z Z

2 q q

g 1 3 8 3

= gR g^R + g^ 4: R

(70)

4 4 V 24 24

Consider now a dilatation w (z )=az . Then, the conformal angle is constant, = log a,

and (R8=V ) = 0. Then the rst term in (70) vanishes and the nite size e ect

do es not dep end on g . It is given by

= log a g^R = log a

and do es not agree with (68) since c in (66) dep ends on g . On other Riemannian

surfaces one would nd the same result. Note that the nite size scaling comes from

the middle term log a g^R in (70) which is top ological in nature, while the short-

distance b ehaviour of the energy-momentum correlators is controlled by the remaining

two terms in (70) which are insensitive to the top ology. In that sense nite size scaling

and the central charge are complementary. There is a way to match the two results

by adding the term

g p

g R4R

to the e ective action. With this new e ective action the short distance expansion

of the energy-momentum correlators do es not dep end on g any more and the cor-

resp onding central charge equals that obtained from the nite size scaling. However

such a term would corresp ond to a non-lo cal counter term to b e added to the regu-

larized action. 26

3.5 Charge neutrality and unitary subspaces

In this subsection we showhow the equivalence b etween the central charge and nite

size scaling can b e restored, provided the partition function is replaced byanaverage

over un-normalized exp ectation values of charges at in nity. In fact it turns out that

R-term, ie. the non-minimal coupling to gravity, itself can b e given the the g

interpretation of a charge at in nity if the zero-mo de constraints (37) is replaced by

a non-translation invariant sum over charges at in nity.

The hint comes from insp ecting the fermionic weights (67), which shows that

8g (x)

(x) and (x) e (x)have the same conformal weights. We can therefore

consistently put a charge at in nity with a corresp onding mo di cation of the charge

neutrality condition. The non-vanishing two-p oint function is now h (x) (x)i. It's

coincidence limit j is again a primary eld with conformal weight h =1.

g j

On the other hand, including a charge at in nityinto the de nition of the partition

function wehave

S 8g ( )

3 0

Z = D D Z [==0] e : e :

g F

(71)

2 2

= Z exp[16 g G ( ; )] (recall that D = ()D ):

0 0 0 0

To continue we need to determine the coincidence limit of the scalar Greens function

G (x; y ), i.e. to regularize the comp osite op erator exp( ) app earing in (71). The

normal ordering prescription

(x)

: : e :=

(72)

(x)

he i

works well on the whole plane [32, 33]. On curved space wemust b e more careful

when renormalizing this op erator. The required wave function renormalization is not

unique but it is very much restricted by the following requirements: First we takeas

reference system (the denominator in (72)) one with a minimal numb er of dynamical

degrees of freedom since we do not want to lo ose information by our regularization.

Second, the renormalized op erator should havea well-de ned in nite volume limit.

Finally, the regularization should resp ect general covariance. These requirements

then force us to take as reference system the in nite plane with metric g . The at

metric is not p ermitted since it leads to a ill-de ned expression for hexp( )i.

With this choice the normal ordering in (72) is equivalent to replacing the massless

Green function in (71) by 27

reg

2 2

G (x; y ):=G (x; y )+ log [ s (x; y )]:

(73)

Here s(x; y ) denotes the geo desic distance b etween x and y . The o ccurrence of the

arbitrary mass scale comes from the ambiguities in the required ultra-violet regu-

larization. On the 2-sphere with constant Ricci scalar R wehave

1 R

reg

G (x; x)= [log [ ]+1]:

4 8

8g ( )

3 0

The exp ectation value h: e :i then transforms under a constant rescaling z !

az as

8g ( ) 8g ( ) 2

3 0 3 0

h: e :i!h:e :i exp[8g log (a)]; (74)

and therefore gives an extra contribution

24g

log (a) ; =

to the nite size scaling of the e ective action. Adding this contribution to (70) ab ove

we see that this is precisely the piece needed to restore equivalence with the central

charge for any real or imaginary g .

More generally we can de ne the functional integral as an average over all p ossible

charges at in nity: assume g imaginary. The (un-normalized) exp ectation values are

then given by

Z Z

n n

D E h i

Y Y

1 1

ik ( ) S

O (x ) D D dk :e : O (x ) e :

i i

(75)

i=1 i=1

Here denotes the U (1)-charge of the op erator O . In particular the partition

i i

function on S is

Z Z

h i

1 1

] 0 ik ( ) 8g ( ) S [

0 3 0

; D d D dk :e : : e : e Z =

where is the zero mo de and the excited mo des of (x). The middle term in the

ab oveintegrand is the zero-mo de part of S . The zero-mo de integration yields a delta

function (k + i8g ) and thus the g R-term itself acquires the interpretation of

3 3

ik ( )

acharge at in nity, due to the presence of the zero mo de. The 'extra' charges e 28

assure the charge neutrality of the partition function. For the general n-p oint function

(75) the -integration yields

(k +8ig + );

3 i

i=1

where the sum of the U (1)-charges of the op erators in (75) enters. In particular, for

neutral states, for which( +i8g = 0), k must b e zero and no extra charge at

i 3

in nity is present.

Finally, using

ik ( )

dk e = (( ));

the averaging over all p ossible charges can also b e written as

D (( )): (76)

It is easy to verify that if the action has translation invariance in the target space,

then the constraints (76) and (37) are equivalent and the correlation functions do not

dep end on the chosen base-p oint .However, in the present case (76) clearly breaks

translation invariance (or rotation invariance on S ) and the zero-mo des constraints

are inequivalent. Although wehave assumed an imaginary g , our results apply for

any g . For particular values we recover the (unitary) minimal mo dels, provided

screening charges [34 ] are included for the n-p oint function with n>2. In particular

for g =1= 48 and g = g =0 we obtain the Ising mo del with h = h y = .

3 1 2

4 Gauged Thirring-like Mo dels

In this section we extend the mo del by gauging the global U (1)-symmetry. Contrary

to what one might think, many asp ects of the gauged mo del are actually simpler

as compared to the ungauged mo del. In particular the thermo dynamical prop erties

are indep endent of external conditions likechemical p otentials and twisted b oundary

conditions. The reason is that the mo del is closely related to the Schwinger mo del, for

which the sp ectrum consist solely of a neutral, massive particle. On the other hand,

the gauge interaction complicates the analysis, b ecause the U (1)- bundle over the

torus allows for gauge eld con gurations with winding numb er, so called instantons.

These, in turn, imply fermionic zero-mo des which trigger a chiral symmetry breaking 29

and therefore a non-vanishing condensate. This is the sub ject of the second part

of this section. In the rst part we discuss the partition function to which only

top ological ly trivial con gurations contribute.

To see how the fermionic generating functional (34) is mo di ed, we decomp ose a

general gauge p otential on a torus as

A = A + t + @ @ ';

(77)

where the last 3 terms corresp ond (as for the auxiliary eld B ) to the Ho dge de-

comp osition of the single valued part of A in a given top ological sector, that is the

harmonic-, exact- and co-exact pieces. The role of the toron elds t has recently b een

emphasized within the canonical approach [35 ]. In the Hamiltonian formulation they

are quantum mechanical degrees of freedom which are needed for an understanding

of the infrared sector in gauge theories. Also, in [36 ] it has b een argued that the Z -

phases of hot pure Yang-Mills theories [37] should corresp ond to the same physical

state if the toron elds are taken into account. The rst term in (77) is an instanton

potential which gives rise to a non-vanishing quantized ux. As noted ab ove con gu-

rations with non-vanishing ux do not contribute to the partition function due to the

asso ciated fermionic zero mo des. We can therefore assume A = 0 for the moment.

The fermionic generating functional is obtained from (30) by simply shifting

g h ! et + g h = H , g ! e + g = F and g ! g + e' = G;

0 0 1 1 2 2

which leads to

h i h i

1 a + a

1 1 1 1

2( g^g^ 2i a )

1 0

Z [; ]= e (0;) (0;)

j ( )j a a +

0 0 0 0

(78)

1 1

(x)S(x;y ) (y )

e exp S + gG4G] ;

24 2

with a = H .

To compute the partition function wemust switch o the sources and in (78)

so that

2 2 S

Z = J d td hD 'D D Z [0; 0] e ;

0 F

(79)

where now 30

S = (2 ) g^g^ h h

(80)

p 1

+ g '4 ' 4 4 g R :

Note that wehavekept the non-minimal coupling of the - eld to gravityasin

section 3:3. Since S and the fermionic determinants are b oth gauge invariant and

thus indep endent of the pure gauge mo de in (77), it is natural to change variables

from A to ('; ; t ). This transformation is one to one, provided

Z Z

p p

g' = g =0 and et 2 [0; 1]:

(81)

In contrast to the auxiliary harmonic elds h in (16), the toron elds et and et + n

with integer n are to b e identi ed, due to gauge invariance [8]. The measures are

related as

D A = J dt dt D 'D ; where J =(2) det (4):

0 1

(82)

In exp ectation values of gauge invariant and thus -indep endent op erators the -

integration cancels against the normalization. This simply expresses the fact that in

QE D the ghosts decouple in the Lorentz gauge.

As we shall see shortly it is advantageous to integrate rst over the toron elds. By

using the series representation of the theta functions one computes

h i h i

1 c c

1 1

d (et) (0;) (0;)= :

(83)

c c

0 0

Since the result app ears always together with the -function factor in (34) it is con-

venienttointro duce

1 1

2 j ( )j

in the following expressions. The result (83) do es not dep end on the h- eld and hence

the h-integration in (79) b ecomes Gaussian and yields a factor 1=4 so that 31

S =24 gG4GS [h=0]

L B

Z = det (4) e D (') e ;

(84)

where we inserted the explicit expression (82) for the Jacobian. Nowwe see whywe

did well integrating over the toron elds rst. It has washed out the dep endence on

the b oundary conditions and chemical p otential in (83).

The integral over , sub ject to the condition in (37), decouples completely apart from

the non-minimal coupling to gravity which mo di es the Liouville factor and yields

one inverse square-ro ot of the determinantof 24in (84). Thus

(g +1=24 )S

Z = 2V det (4) e

(85)

gG4GS [h==0]

; D (') e

where wehave used (43). The -integration in contrast, leads to a nite renormaliza-

tion of the dynamically generated 'photon' mass

2 eV 1

2 2 2

(g +1=24 )S g'(4 m 4)'

Z = e D 'e ;

(86)

e 2

where m =

2 + g

plays the same role as the -mass in QC D [41]. The determinant obtained from the

' integration factorizes as

0 0 0

2 2 2

det (4 m 4) = det (4) det (4 + m ):

This factorization prop erty is not obvious since all determinants must b e regulated.

But it holds for commuting op erators and in the zeta-function scheme. Then the

partition function simpli es to

2 1 eV

0 0

2 2

Z = det (4)det (4 + m ) exp (g + )S :

0 L

m 24

We can go further by using the scaling formula for the determinantof 4[20 ] and the

known result for the determinantof4[21] which together yield

V 1

det (4)= Lj()j exp S :

(87)

0 L

V 32

Thus we obtain the following partition function for the general mo del (4) on curved

spaces:

e 1 1 1

Z = 2V exp ( : + g ) S

0 L

(88)

m j( )j 12

(4 + m ) det

Again wehave factored out the partition function N for free auxiliary elds. The

formula (88) shows explicitly that in the top ologicall y trivial sector the theory is

equivalent to a theory of free massless and massive b osons with mass m ,even in

curved space-time [13].

The app earance of m in (86) should b e interpreted as renormalization of the electric

charge induced by the interaction of the auxiliary elds with the fermions. After

summing over all fermion-lo ops this leads to an e ective coupling b etween the photons

and the - eld and in turn to a mo di ed e ective mass for the photons in (86). In the

limit g ! 0 this mass tends to the well-known Schwinger mo del result, m ! e=

[38].

Wehave already mentioned that the chemical p otential coupled to the electric

charge has completely disapp eared from the partition function. This do es not come

as a surprise since the only particle in the gauged Thirring mo del is a neutral b oson.

This has no charge whichmay couple to the chemical p otential. Also, if the partition

function dep ended on then the exp ectation value of the charge would not vanish, in

contradiction to the integrated Gauss law. The result (88) provides therefore another

test for our result (30) for the fermionic determinants with chemical p otential.

The nal result is also indep endent of the chiral and non-chiral twists. The normal

twists have b een wip ed out by the toron integration. In fact the chiral twists are

equivalenttoachemical p otential and therefore the ab ove remarks concerning the

chemical p otential apply here as well. Did we assume holomorphic factorization for

the fermionic determinant [5] then the partition function would dep end on the chiral

twists.

We conclude this subsection by giving the explicit formula for the partition function

on the at torus. The zeta-function regularized massive determinant is expressed by

1 1 1

2 (0)

2 2

det (4 + m ) = e ;

where 33

Vm 1

(0) = K (m L (n; n)) ;

(89)

L 4

(n; n)

n6=0

i j

and (n; n)=g ^ n n is the inner pro duct taken with the reference metric, and the

sum is over all (n ) 2 Z with the origin excluded. For g = , in which case the

partition function has the usual thermo dynamical interpretation, the result reduces

to one derived previously byAmb jorn and Wolfram [39]. In addition, if L approaches

in nitywe recover a result in [19 ]. The free energy for = 0 and on at space

simpli es then to

1 1

F = log Z = (0):

with (0) from (89) and the particular choice for the parameters.

4.1 Bosonization of the gauged Thirring mo del

We p ointed out in section 2 that for g = g = g the classical theory (4) reduces to the

1 2

gauged Thirring mo del. The same is true for the quantized theory on the torus if in

addition we set g = g . More precisely, the Hubbard-Stratonovich transform [40] of

the Thirring mo del is just the derivative coupling mo del (4) with identical couplings.

In the pro cess of showing that we shall arrive at the Bosonization formulae for the

gauged Thirring mo del on the curved torus. We shall see that only the non-harmonic

part of the fermion current can naively b e b osonized and that for this part the rules

of the un-gauged mo del on at space time [32] need just b e covariantized.

For that we calculate the partition function (79) in a di erent order. First we

integrate over the auxiliary elds. To understand the role of and weintro duce

sources for them. Thus we study the generating functional for the correlators of the

auxiliary elds

S + g [+]

Z [; ]= D(h A )e :

Here

g D= + S [g =0] S = i

B 3

is the action of the full theory. D= is the Dirac op erator in (17) with all couplings

set equal and S the b osonic action (80). Since and integrate to zero (see (37))

B 34

wemay assume the same prop erty to hold for the sources. The integration over the

auxiliary elds is Gaussian and yields

Z Z

h i

p 1 1 1 g 1 1

D ( A ) e Z = N exp g ( + )+ ( j + j ) ;

5; (90)

;

4 4 4 2 4 4

where

1 g

S = g F F i D= j j

(91)

4 4

is the action of the gauged Thirring model on curved space-time and

N =

(92)

2 det (4)

comes from the integration over the auxiliary elds.

Let us rst consider the partition function, that is set the sources to zero. Comparing

(90) with (86) and using (87) we easily nd

Z Z

1 g

S F F S

(93)

; D ( t)e = + e D ( ) e

2 4

where is the mean eld (see (37)) and we used (82) and (43). The action for the

neutral scalar eld is found to b e

Z Z

1 ie 1

p p

S = g@ @ g 4':

1+g =2

Since (93) holds for any ' (and thus for the non-harmonic part of any A , b ecause of

gauge-invariance) we read o the following bosonization rules:

i 1

j ! @

1+g =2

(94)

i 1

j @ ; !

1+g =2

where prime denotes the non-harmonic part of the currents. Thus, only the non-

harmonic parts of the currents can b e b osonized in terms of a single valued scalar 35

eld. To b osonize their harmonic parts one would have to allow for a scalar eld

with winding numb ers. On the in nite plane the harmonic part is not present and

wemay leave out the primes in (94). If we further assume space time to b e at we

recover the well-known b osonization rules in [32]. What wehave shown then, is that

for the gauged mo del on curved space time the b osonization rules are just the at

ones prop erly covariantized and with the omission of the zero-mo des.

Since (93) holds for any gauge eld the current correlators in the Thirring mo del

are correctly repro duced by the b osonization rules (94). To see that more clearly we

calculate the two-p oint functions of the auxiliary elds in the Thirring mo del (90-

92). For that we di erentiate (90) (' is treated as external eld) with resp ect to the

sources and nd

1 g

h(x)(y )i = G (x; y )+ hG (x; u)j (u)G (y; v)j (v)i

0 0 0 T

; ;

2 4

(95)

1 g

h(x)(y)i = G (x; y )+ hG (x; u)j (u)G (y; v)j (v )i ;

0 0 0 T

2 4

where G is the free massless Green-function in curved space-time and the integrations

are over the variables u and v with the invariant measure on the curved torus. Here

h:::i are vacuum exp ectation values in the Thirring mo del (91). Alternatively we can

calculate these exp ectation values from (84) and (85), where the fermionic integration

has b een p erformed and nd

h(x)(y )i = G (x; y )

2 2 2

m m m

h(x)(y )i = G (x; y )+ 1 '(x)'(y):

2 2

2e 2 e

Comparing this with the result (95) we see at once that

hG (x; u)j (u)G (y; v)j (v)i = 0 (96)

0 0 T

; ;

hG (x; u)j (u)G (y; v)j (v )i = m '(x)'(y ) G (x; y ) :

0 0 T 0

These correlators express the gauge invariance and the axial anomaly hj i = m 4'

in the gauged Thirring mo del. They can b e correctly repro duced with the b osoniza-

tion rules (94). They are not repro duced with the ones derived for the un-gauged

mo del [32]. 36

4.2 Chiral condensate

The chiral condensate is an order parameter for the chiral symmetry breaking. How-

ever, on the torus its exp ectation value, whose temp erature- and curvature dep en-

dence we will here compute would vanish if top ological ly non-trivial gauge eld con-

gurations were absent. There is a useful classi cation of the gauge con gurations

corresp onding to the numb er of fermionic zero mo des they give rise to. If we let

k = n n , where n counts the numb er of zero-mo des with p ositive/negative

chirality, then wehave

Z Z

1 1 p 1

2 2 2

k = d x a (D= ;x)= gd x F ;

(97)

5 1

2 4 2

which establishes a relation b etween the numb er of fermionic zero mo des (or, more

precisely the numb er of zero mo des with p ositivechirality minus the numb er of those

with negativechirality) and the rst Chern character of the bundle. Also from (97)

one immediately concludes that the ux must b e quantized in integer multiples of 2 .

This is really a consequence of the single valuedness of the fermionic wave function

(co cycle condition).

Recalling the decomp osition (77) of the gauge eld wenow concentrate on the

rst term A whichistheinstanton potential giving rise to a non-vanishing quantized

ux . Since 2-dimensional gauge theories are not scale or Weyl invariant, as 4-

dimensional ones are, the instantons on a conformally at space-time are not identical

to the at ones [42, 43 ]. As representative in the k -instanton sector wecho ose the, up

to gauge transformations, unique absolute minimum of the Maxwell action in a given

top ological sector. It has eld strength eE = g =V . The corresp onding p otential

can b e chosen as

1 I I I

^ ^

(x ; 0) eA = eA @ ; where eA =

(98)

is the instanton p otential on the at torus with the same ux but eld strength

g^ =V . The function is then determined (up to a constant) by

p p

g g^ = g 4 :

(99)

The solution of this equation is given by 37

1 1 1

2 2 2 (y )

(x)= ( e )(x)= d y g(y)G (x; y ) e ;

(100)

^ ^

V V

where

1 (x) (y )

G (x; y )= hxj jy i =

(101)

is the Green-function for 4. In deriving (100) wehave used that (=V ) = 0 which

follows from the sp ectral resolution (101) for the Green function in which the constant

zero mo de =1= V of 4 is missing.

Our choice for the instanton p otential (98) corresp onds to a particular trivializa-

tions of the U (1)-bundle over the torus [8]. In other words, the gauge p otentials and

0 1 0 1

fermion elds at (x ;x ) and (x ;x +L) are necessarily related bya nontrivial gauge

transformation with winding numb ers

0 1 0 1

A (x ;x + L) A (x ;x )=@ (x)

(102)

0 1 ie (x) 2i( + ) 0 1

1 1 5

(x ;x + L)=e e (x ;x ):

For the choice (98) we nd

e (x)= x :

Note that A is still p erio dic in x with p erio d L and still ob eys the rst b oundary

condition in (14). To calculate the fermionic zeromodes we use the square of the

Dirac op erator

e 1 p 1

D D 0

D R + F D= = = gg D

(103)

0 D D

g 4 2

In a pure instanton and harmonic background (' = = 0) on the at torus (103)

simpli es to

^ ^ ^

D= = g^ D D :

(104)

In other words, D= is the same in the left- and right-handed sectors, up to the constant 38

2=V .Furthermore this op erator commutes with the time translations which leads

to the following ansatz for the zero-mo des

0 1

2ic x =L 2iH x =L 1

~ = e e (x ) e ; c = + p;

p p + p

where wehave assumed k>0. The choice of c is dictated by the time-like b oundary

conditions in (14). Inserting this ansatz into the zero mo de equation D= ~ = 0 yields

2 2

d d

2 2

(j j y 2i y i ) =0;

1 p

2 4 2 2

dy L L dy L

(c H ): where y = x +

p 0

This is just the di erential equation for the ground state of a generalized harmonic

oscillator to which it reduces for = i . The solution is given by

h i

= exp fx + (c H )g :

p p 0

2iL k

These functions do not ob ey the b oundary condition (102), but the correct eigenmo des

can b e constructed as sup erp ositions of them. For that we observe that

0 1 ix = 2i H 0 1

~ (x ;x +L)= e e ~ (x ;x )

p p+k

so that the sums

1 1

(2k )

2i (n+p=k )( 2i(H ) H )

0 0 1 1

^ k

2 2

= e e e ~ e ;

p+nk +

(105)

n2Z

j jV

where p =1;:::;k, ob ey the b oundary conditions and thus are the k required zero-

mo des. Indeed, since (iD= ) in non-negative there are no zero mo des with negative

chirality b ecause of (104). With (97) we conclude then that there are exactly k zero

mo des with p ositivechirality. Mo des with di erent p in (105) are orthogonal to each

other and the overall factor normalized them to one, so that the system (105) forms

an orthonormal basis of the zero-mo de subspace. For k<0 the zero-mo des are the

same if one replaces e by e .

To compute the fermionic determinant in a given top ological sector we again intro duce

the one-parameter family of Dirac op erators 39

1=2

g^ 2i

f f I

^ ^ ^

D= = e D=e ; D= =^ @ +i!^ ieA [H + ] ;

(106)

1=2

whichinterp olates b etween D= and D= , similarly as in (31). But now

f = iF + (G + )+ ;

with F and G from (7), contains an instanton contribution. Also note that D= contains

the instanton part A .To compute it's determinantwe observe that the simple form

(104) of D= allows one to reconstruct its sp ectrum completely [20 , 8]:

0 degeneracy = k

2n=V degeneracy = 2k:

The corresp onding determinant is [20, 8]

(107)

det (iD= )= :

To relate the determinants of D= to that of D= we again integrate the anomaly

equation, whichnow reads

d log det D= a (x; D= ) d log g

2 2 y

= d x f P (x; D= )g; g f (x)+f (x)

(108)

d 2d 4

where, due to the fermionic zero-mo des, the pro jector onto the zero-mo de-subspace,

( ) () ()

2 1 () y

P (x; D= )= (x)N ( )( ) (x) , N ( )= ( ; )

0 pr

p0 p0 r0

pr r

(109)

app eared. For the deformed op erator D= the rst Seeley-deWitt co ecientis

h i

1 1

a = p g^ + g R + 4 G + (1 ) :

5 5

1 (110)

12 g V

Integrating w.r.t. [20 ] one ends up with the following formula for the determinant

in arbitrary background gravitational and gauge elds: 40

N S 1

0 0

det iD= = det det (iD= ) exp + gG^ 4G

24 2

(111)

Z Z

p 2k

gG + g ^ : exp

In deriving this result we used that g =0.

Nowwe are ready to compute the chiral condensate h P i. Observing that

the fermionic Green's function anti-commutes with one sees at once that only

con guration supp orting one fermionic zero-mo de with p ositivechirality contribute

to the chiral condensate

y S

h P i = D(:::)Z [0; 0] e ;

+ F

Z (x) (x)

0 + +

where = P . Earlier wehave seen that these are the gauge elds with ux = 2

+ +

or instanton number k =1. Thus the condensate b ecomes

J V

y S [k =1]

(112)

h P i = D (:) (x) (x) exp(:) e ;

+ 0

Z 2

where exp(:::) stands for the exp onentials in (111). First weintegrate over the toron

eld t. The t-dep endence enters only through the zero mo de and more sp eci cally

in (105) with p = 1. Using the series representation for the theta functions one nds

^ ^

: d t (x) (x)=

(113)

Note that the result do es not dep end on the chemical p otential similarly as in our

calculation of the partition function. To continue we observe that the term gG

in exp(:::)vanishes b ecause of our conditions (81) and (37) on the elds ' and .

Furthermore S [k =1]=S [k =0]+ . The remaining functional integrals are

B B

p erformed similarly as those leading to the partition function and we end up with the

following formula for the condensate

D E

2 2

y 2 2 =e V +2=V g ^ 2(g+e')(x) (x)

h P i = j ( )j e e :

(114)

V 41

The exp ectation value is evaluated with

2 2

h i

p 1 e e eg

S = g '(4 4)' 4 4' :

ef f

2 m

A formal calculation of the resulting Gaussian integrals yields

2 2

2 2 =e V +2=V g ^ (x)2 (x) y

j ( )j e e h P i =

(115)

2 4

2 m

exp [ K (x; x)] exp [ G (x; x)];

2 2

e 2 + g

where

1 1

jyi= (G (x; y ) G (x; y )) K (x; y )= hxj

0 m

(116)

2 2 2

4 m 4 m

and G ;G are the massive and massless Green-functions. Here we encounter ultra-

m 0

violet divergences since G (x; y ) is logarithmically divergent when x tends to y .To

extract a nite answer we need to renormalize the op erator exp( ) as explained in

section 3:5. This wave function renormalization is equivalent to the renormalization

of the fermion eld in the Thirring mo del and thus is very much exp ected [32, 33 ].

The at Green's function on the torus

0 1

h i

1 1 1 1 +

0 2

2 2

i ( =L) 2 L

G (x; y )= j (0;)j = j e ( ;)j ;

0 1 1

4 () 4 () L

2 L

where = x y , p ossesses the logarithmic short distance singularity

1 g^ 1

2 4

G (x; y )= log log (4 j ( )j )+O():

0 0

(117)

4 4

Furthermore

g ^ + O(): G (x; y ) G (x; y )+ 2 (x)

0 0

reg

With the prescription explained in section 3:5we nd that on the at torus G has

now the nite coincidence limit 42

2 4

1 4 j ( )j

reg

: G (x; x)= log

(118)

To determine the chiral condensate we also need to determine K (x; y ) on the diagonal.

In a rst step we shall obtain it for the at torus. Its -dep endence is then determined

in a second step. For = 0 and = i the Green function K has b een computed in

[8]. The generalization to arbitrary is found to b e

1 a 1

m K (x; x) = coth ( )+

2m L j j

0 m V

(119)

1 1

+ log j ( )j + F (L; ) H (L; ) ;

where weintro duced the dimensionless constant a = Lm j j=2 and the functions

i h

1 1

F (L; ) =

2 2

n + a

n>0

(120)

h i

1 1 1

H (L; ) = + :

2iz (n) 2iz (n)

2 2 +

e 1 e 1

n + a

n>0

We used the abbreviations

2 2

z = (n i n + a ):

1 0

(121)

j j

Substituting (119) and (118) into (115) with =0 we obtain the following exact

formula for the chiral condensate on the torus with at metricg ^ :

1 m Lj j m Lm

2 +g

h P i = exp coth

+ g^

Lj j 2 e L 2j j

(122)

h i

exp F (L; ) H (L; ) ;

where we used that on the at torus = 0 and V = V .Furthermore, we identi ed

with the natural mass scale m of the theory.

To extract the nite temp erature b ehaviour of the chiral condensate we take =

i =L where =1=T is the inverse temp erature. In the thermo dynamic limit L !1. 43

Then coth(:::) ! 1, H ! 0 and the expression for the chiral condensate simpli es

2 h i

m m 2

2+g

h P i = T exp T + F :

(123)

2 2

2T e 2 + g

Using

a 1

F ( ) ! + log + for a !1;

2 2a

where =0:57721 ::: is the Euler numb er, we obtain the zero temperature limit

m 2

2 2

y g =(2 +g )

2 2

h P i = 2 exp for T ! 0:

(124)

4 2 + g

For temp eratures large compared to the induced photon mass F vanishes. Thus we

obtain the high temperaturebehaviour

m m

2+g

2 (125)

h P i = T exp T for T !1

+ T

2T e

It is instructive to discuss the various limiting cases. For all g = 0, i.e. the Schwinger

mo del limit, the exact result (123) simpli es to

e T !0

T+F( )

h P i = Te !

+ T

(126)

T =m

Te T !1,

2 2

where now m = e = is the induced photon mass in the Schwinger mo del. This

formula for the temp erature dep endence of the chiral condensate in QE D agrees

with the earlier results in [8].

Next we wish to investigate how the self interaction of the fermions a ect the

breaking. For large coupling g and xed temp erature the exp onent in (123) vanishes

so that

h P i for T xed; g !1:

+ T 2

2 + g

Hence, for very large current-current coupling the chiral condensate vanishes. Or

in other words, the electromagnetic interaction which is resp onsible for the chiral

condensate, is shielded by the pseudo scalar-fermion interaction. 44

For intermediate temp erature and coupling g wemust retreat to numerical evalua-

tions of the sums de ning the chiral condensate in (123). The numerical results are

depicted in Fig. 1

How do es the gravitational eld a ect the chiral condensate? To answer this

question we need to know the massive Green's function, entering in (116), for non-

trivial gravitational elds (for simplicitywe assume T = 0). Let us rst consider a

space with constant negative curvature. Then G has b een computed explicitly in

[44]. Here we only need its short distance expansion, given by

1 s R 1 1

G (x; y )= f2 + log ( )+ ( + )+ ( )+O(s )g; (127)

4 8 2 2

where = + and (z ) is the Digamma function. Substituting (127) into (116)

4 R

we end up with the exact formula for the chiral condensate for constant curvature

h i

R 1 1

y y 2

h P i = h P i exp m f log ( )+ ( + )+ ( )g :

+ R + R=0

(128)

2 2

2e 2m 2 2

The asymptotic expansions for large-and smal l curvature are easily worked out in-

serting the corresp onding expansions for the Digamma function [45 ]. We nd

h i

jRj

y y

h P i = h P i exp R for 1

(129)

+ R + R=0

2 2

12e e

and

h i

jRj R

y y

2 +g

exp R for 1: h P i = h P i ( )

+ R + R=0

(130)

2 2 2 2

2m 4e 4e e

Hence the chiral condensate decays exp onentially for large curvature analogous to the

high temp erature b ehaviour. However, the pseudo-scalars do not suppress the e ect

of the curvature in contrast to (125). Comparing the exp onentials in (130) to (125)

wemay de ne the curvature induced e ective temp erature as

T = : (131)

ef f

In passing we note that if we compare the prefactors, rather than the exp onentials,

wewould write

(R)

T = : (132)

ef f

4 2 45

The latter identi cation actually coincides (up to factor of 2) with the Hawking

temp erature of free scalars in de Sitter space [6]. The correct identi cation involves

the (dynamical) mass of the gauge eld and is therefore not universal. From this

observation we learn that the temp erature asso ciated with curvature dep ends on the

matter content. Note nally that the non-minimal coupling (g ) has no e ect on

the chiral condensate. In Fig. 2 wehave plotted the chiral condensate for arbitrary

constantvalues of the curvature.

For gravitational backgrounds with non-constant curvature wehave to refer to

p erturbative metho ds for the calculation of the massive Green's function. Again

we only need the short distance expansion of G . For geo desic distances s small

compared to m the massive Green's function may b e approximated by the Seeley-

DeWitt expansion [46 ]

1 @

(2)

G (x; y ) ) H (ms); (133) a (x; y )(

m j

4i @m

j =0

(2)

where H is the Hankel function of the second kind and order zero. In particular

2 z

(2)

H (z ) ! [ log + ] for z ! 0:

i 2

Inserting (133) into (116) we end up with the following expansion for the chiral

condensate in an arbitrary background

h i

m (j 1)!

y y

h P i = h P i exp ( ) a (x) ; (134)

+ R + R=0 j

2 e m

where wehave used that a (x) = 1. The rst order contribution involves a (x)= R

0 1

and repro duces the asymptotic b ehaviour (129). Higher order contributions must b e

taken into account to uncover the e ect of variable curvature. For this one has to

substitute is the corresp onding Seeley DeWitt co ecients a into (134). These have

b een computed up to j = 5 [47].

5 Conclusions

In this pap er wehave elab orated on various features of the Thirring mo del as well as

some of its extensions. In particular we found the dep endence of the partition function

on the chemical p otential and the non-trivial b oundary conditions for the fermions 46

on the torus. For that a careful analysis of fermionic determinants has b een crucial.

Wehave found that the familiar chiral anomaly of the UV-regularized two p oint

function is also seen in the IR-sector as a breakdown of holomorphic factorization.

This fact, which has not b een prop erly taken into account previously, together with

the presence of harmonic contributions to the current, leads to a mo di cation of the

equation of state due to the current-currentinteraction. We b elieve that our results

could also b e obtained in the b osonized theory, provided the usual b osonization rules

are mo di ed to include scalar elds with winding numb ers, i.e. scalar elds with

values in a compacti ed target space.

Furthermore, wehave deformed the conformal structure by allowing for di erent cou-

plings in the transversal- and the longitudinal parts of the current-currentinteraction.

This do es not change the Virasoro- and Kac-Mo o dy algebra, but mo di es the con-

formal weights of the primaries and in particular of the fermionic elds. Not all

values of the coupling constants b elong to physical theories, since p ositivity of the

scalar pro duct imp oses certain restrictions on them. Our approach allows also for

a non-minimal coupling of the longitudinal sector to gravity. While such a coupling

may seem to b e ad-ho c wegave some arguments that it might arise naturally when

quantizing fermions in presence of a a background charge. We nd that the central

charge of the Virasoro algebra is sensitive to the non-minimal coupling. In particular

c<1 o ccurs for certain values of the coupling constant. However, wehave not b een

able to derive constraints on this extra coupling without referring to the result by

Friedan, Qiu and Shenker. We b elieve that an indep endent derivation of their result

within a fermionic mo del would b e most welcome. Wehave also established that the

central charge controls the nite size e ects only for a particular treatment of the

zero-mo des of the auxiliary elds which is equivalenttoanaverage over charges at

in nity.

Finally wehave considered the gauged Thirring model in curved space-time. We nd

that the partition function is indep endentofvectorial as well as chiral twists and the

chemical p otential. This result, which technically is due to the harmonic contributions

to the gauge- elds, is in fact exp ected as a consequence of Gauss's law. Furthermore,

using the (probably not so obvious) factorization prop erty of the zeta-function reg-

ularized determinants of commuting op erators we nd that the partition function

can b e expressed completely in terms of a single massive scalar eld. The gauged

Thirring mo del showsachiral symmetry breaking which originates in the existence of

fermionic zero-mo des and thus in con gurations with winding numb er (instantons).

Wehave obtained explicit expressions for these instantons as well as the exp ectation

value of the chiral condensate as a function of temp erature and curvature. The con- 47

densate is exp onentially suppressed for high temp eratures and/or big curvature which

is interpreted as an almost restoration of the chiral symmetry under these extreme

conditions. Although temp erature and curvature have qualitatively the same e ect

they cannot b e identi ed. In particular the identi cation with the Hawking temp er-

ature for free scalar elds in de Sitter space do es not hold in the present situation. It

follows from general arguments that the chiral symmetry can not b e restored for any

nite temp erature or curvature so an exp onential suppression is most we can exp ect.

In fact, it has b een argued earlier, that the axial U (1)-symmetry in 4 dimensional

QC D also shows an almost restoration as a function of the temp erature [49]. Our

results on the curvature dep endence could motivate a corresp onding investigation in

QC D . Finally we note that the chiral condensate is linearly suppressed for large

current-current couplings.

Acknowledgments: This work has b een partially supp orted by the Swiss National

Science Foundation and the ETH-Zuric h, where part of the research leading to the

present results has b een done. We wish to thank Arne Dettki for his collab oratio n at

the b eginning of this work and J. Frohlich, K. Gawedzky, D. O'Connor and C. Nash

for helpful discussions.

A Conventions and Variational Formulae

Our conventions for the metric and curvature agree with those of Birrell and Davies

0 1

[6]. We use the chiral representation ^ = ; ^ = i for at space with Lorentzian

1 2

M M

0 1

signature and ^ = ; ^ = in Euclidean space. Furthermore ^ = = .

1 2 5 5 3

E E

In what follows we derive some variational formulae used in the text. Here D denotes

the space-time and Lorentz covariant derivative.

Using the de nition of the Christo el symb ols it is straightforward to show that

p 1 p

a a

g = e e + e e ; g = gg g

a a

= ; g g ) e ( g = e

(135)

= g (D g + D g D g ):

For some formulae related to the variation of the tetrad let us refer to [48] 48

1 1

b a a a b

e = e g t e ; e = e g t e ;

a a b b

2 2

(136)

a a a

where t = (e e e e ):

b b

In addition wehave

! = D t ; = e e (D g D g ):

(137)

ab ab ab ab

a b

When p erforming the variation of curvature dep endent expressions wehave used the

identities

g R = ! ; where ! = g g

;

R R

(138)

p p

g! A = gfg r A r A gg : and

Dep ending on the top ology of space-time, the reference curvature R may b e di erent

from zero. In this case it is not p ossible to express the conformal angle in terms of

the curvature scalar. Nevertheless, to p erform variations of -dep endent expressions,

the identity

p p

( g R)=2( g4)

(139)

proves to b e useful.

Taking the variations of the equations

p p

g 2G(x; y )=(xy) and giD=S(x; y )=(xy)

(140)

for the scalar and fermionic Greens functions wemay derive (up to contact terms)

the following variational formulae

G = ( g g + g g )@ G(x; u) @ G(u; y ) gg

S = 2S (x; u) D S (u; y ) D [S (x; u) gg : S (u; y )]

Here all arguments and derivatives which are not made explicit in the integral refer

to the co ordinate u over whichisintegrated. Finally,we need the following formula

for the variation of the inverse Laplacian

! !

1 1 p 1 1 1

gg g f = f (4) f f;

(141)

4 4 4 2V 4

where V is the volume of space-time and f an arbitrary function. To prove this

identitywe note that for f 2 (Kern4) wehave

f =f: 4

Varying this equation yields

1 1

4( f )=f (4) f

4 4

whichmaybeinverted to give

! ! !

1 1 1 1 1

f = f (4) f + f : g

(142)

4 4 4 V 4

Varying the identity

1 1 p

g f =0

V 4

allows to replace the last term of (142) to obtain the required result (141).

B Canonical Approach to the Partition Function

In this app endix we compute the partition function for massive Dirac fermions in the

canonical formalism. In the limit m ! 0we con rm the result (30) for the fermionic

determinant with chemical p otential in chapter 3. For massive fermions one cannot

consistently imp ose chirally twisted b oundary conditions. However, from the explicit

eigenvalues (21) one sees at once that the chiral twist and the chemical p otential

are equivalent. One can easily verify that this equivalence holds also for massless

fermions in the canonical approach and that L=2 . Let us therefore compute

the partition function

:(H Q):

Z ( )=Tr[e ] (143)

for massive Dirac fermions with chemical p otential on a cylinder with (non chiral)

twisted b oundary conditions 50

(x + L; t)= e (x; t): (144)

For massive particles it is more convenient to use the Dirac representation

0 1 5 0 1

(145)

= = i ; = = :

3 2 1

The Dirac eld is expanded in terms of the eigenmo des of the rst quantized Hamil-

tonian

m i@

h =

(146)

i@ m

X X

(x; t)= b + d ;

n;+ n n;

(147)

n n

where the and are the p ositive and negative energy mo des,

n;+ n;

i! ti x i! ti x

n n n n

= e c ; = e c ;

n;+ n n; 1 n

! +m

: (148) c =(2! (! +m)L)

n n n

The momenta and frequencies ! are determined by the b oundary condition (144)

n n

to b e

2 1

2 2

= (n ) and ! = m + :

(149)

n 1 n

L 2

After normal ordering the 'p ositron' op erators with resp ect to the Fockvacuum de-

ned by H we nd

X X X

y y

(H Q)= (! )b b + (! +)d d (! +);

n n n n n

n n

(150)

n n n

where the last c-numb er term represents the in nite vacuum contribution whichmust

b e regularized. To do that we employ the zeta function regularization. That is we

de ne the zeta-function for s>1by the sum

(s)= (! +) ;

n 51

which in turn de nes an analytic function on the whole complex s-plane up to a simple

p ole at s = 1. The analytic continuation is constructed bya Poisson resummation

X X

(! + ) = F (n);

(151)

n n

where

2i( ) i y

2 2

F ( )=e dy e [ m~ + y +~]

(152)

andm ~ = Lm,~=L.Taking the Mellin transform of (152) we nd

Z Z

2 2

i y s1 t 2i( ) m~ +y t~

dy e dt t e F ( ) = e

(s)

1 d 2

2i( ) s1 t~

2 2

e dt t e K (~ + t ) (153) =

(s) dt

2 2

1 2~m K (~ + t )

2i( ) s t~

e : = dt t e

2 2

(s)

+ t

F diverges at = 0 since the Kelvin function K (z ) 1=z for small z . It follows that

the n = 0 term in (151) diverges. This divergence is regularized by subtracting the

ground state energy of the in nite volume system. Indeed, b ecause of the exp onential

decay of the Bessel function for large arguments, only the n = 0 term contributes for

in nite volume. So we nd for the regularized sum

2 2

X X

mL~ K (~m n + t )

s 2in( ) s t~

(! + ) = dt e t e :

(154)

2 2

(s)

n + t

n6=0

Nowwe p erform the limit m ! 0. Only the most singular term in the expansion of

the Bessel function contributes, hence

X X

L 1

s 2in( ) s t~

(! + ) = dt e t e

2 2

(s) (n + t )

n6=0

q s

1 1 sL

2in( ) s

2 2

1 1

e n~ S (~n); (155) =

s ;

2 2

n6=0

where S (z ) is the Lommel function [50]. In particular for s = 1 this function is

a;b

S =1=z so that nally 52

X X

1 () 2 1

reg 2 in 2

(! + ) = e ( [ + ]) : =

n 1 1

(156)

L n 6L L 2

n6=0

Inserting this into (150) then yields the regularized expression

X X

2 1

y y

: H Q := (! )b b + (! + )b d + [ + ] :

n n n n 1 1

n n

(157)

6L L 2

n n

For small the normal ordering is -indep endent so that

2 1

h0j : H Q : j0i = + [ + ] = h0j : H : j0i

(158)

1 1

6L L 2

is indep endentof and coincides with the Casimir energy [24 ].

Let us now compute the partition function. Using (158) we easily nd

:(H Q): [ ]

Z ( ) = tr [e ]=q

1 1

Y Y

1 1

(n (n ) + )

1 1

2 2

= (1 + q (1 + q e ) e )

1 1

n>[ + ] + ] n>[

1 1

2 2

1 1

Y Y

1 1

(n + ) (n )

1 1

2 2

(1 + q e ) (1 + q e )

1 1

n>[ ] n>[ ]

1 1

2 2

h i h i

1 1

= (0;) (0;); (159)

j( )j i i

2 2

where wehave used the pro duct representation of the theta functions in the last

2i 2 =L

identity and that q = e = e . A non-vanishing chiral twist can nowbe

included by shifting the chemical p otential. Thus wehave con rmed the formula (30)

in the text.

Note that for 6= 0 the zero-temp erature limit of the grand p otential is not equal

to the vacuum exp ectation value of : H Q :. For 6= 0 all states up to the -

dep endentFermi energy are lled. For example, for ! <

1 2

reduces to the exp ectation value of : H Q : in the one-electron state.

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