Generating Functional for Gauge Invariant Actions: Examples of Nonrelativistic Gauge Theories
GGenerating Functional for Gauge Invariant Actions: Examples ofNonrelativistic Gauge Theories
Oleg Andreev ∗ Arnold Sommerfeld Center for Theoretical Physics, LMU-M¨unchen,Theresienstrasse 37, 80333 M¨unchen, Germany
Abstract
We propose a generating functional for nonrelativistic gauge invariant actions. In partic-ular, we consider actions without the usual magnetic term. Like in the Born-Infeld theory,there is an upper bound to the electric field strength in these gauge theories.
LMU-ASC 42/09
The Wilson loops [1], with C a closed curve, A the gauge field, and the trace taken in the N -dimensional fundamental representation of SU ( N ) W ( A, C ) = 1 N tr P exp (cid:104) i (cid:73) C dx µ A µ (cid:105) , (1.1)have been studied from many points of view in gauge theories [2].The first thought about loops is the Wilson criterion for confinement. It is expected to showup in an area law for the expectation value of a loop W ( C ) = (cid:90) [ dA ] e − S YM W ( A, C ) , (1.2)with S YM the Yang-Mills action. The area law means that W ( C ) decays exponentially with anarea enclosed by the loop. There have also been many discussions of this issue in the context ofsupersymmetric gauge theories with additional fermionic and scalar fields [2].The second thought about loops is that instead of quantizing the gauge field A µ , one canquantize coordinates x µ . In terms of Euclidean path integral, that means Z ( A, C ) = (cid:90) [ dx ] e − S ( x ) W ( A, C ) , (1.3)where S ( x ) is an action for the quantized coordinates, fields x µ with µ = 0 , . . . , p . For a U (1)gauge group, such an integral appeared in the Feynman’s first-quantized formulation of scalar ∗ Also at Landau Institute for Theoretical Physics, Moscow. a r X i v : . [ h e p - t h ] O c t ED many years ago [3]. Later, it also appeared in string theory, where the non-Abelian Wilsonfactor (1.1) was inserted into the Polyakov path integral to describe an effective action for masslessstring modes [4]. The main point here is that the renormalized path integral (1.3) is to beidentified with the effective action S ( A ) = Z ( A ) . (1.4)Subsequent work [5, 7] has made it clear that this approach should be taken seriously, particularlyin the context of D -brane actions [8].In fact, one can consider the formula (1.3) as a generating functional for gauge invariantactions without any reference to string theory. In other words, given the action S ( x ), one canuse it to compute (generate) the corresponding gauge theory action via (1.3). Note that in generalsuch actions include infinitely many terms. The two classical examples are that of Schwinger [9]and non-linear electrodynamics of Born-Infeld type obtained from (1.3) in [5]. In this paper we have two main aims. First, we want to generalize the method to nonrela-tivistic gauge theories. In recent years, such theories have been of considerable interest in thecontext of strongly correlated electron systems as an effective field theory description of thelong-wavelength physics at quantum critical points with anisotropic scaling. A peculiar propertyof those is that in the Lagrangian the usual H -term [10], or alternatively the E -term [11], hasvanishing coefficient. For example, in 3 + 1 dimensions the action is of the form [10] S ( E , H ) = (cid:90) d xdt (cid:2) E − ρ ( ∇ × H ) (cid:3) , (1.5)where E = E · E and ρ is a coupling constant parameterizing a line of fixed points.Our second aim is to find some (off-critical) non-linear deformations of the effective actionsproposed in [10]. We keep in mind the Born-Infeld electrodynamics [12] that differs from that ofMaxwell by higher order terms in the field strength. Moreover, as in the case of the Born-Infeldaction where it describes the low energy dynamics of D -branes, such deformations might beuseful for studying nonrelativistic branes.Before getting to the specific examples that we will consider, let us set the basic framework.For the contour C , we take a unit circle parameterized by an angular variable ϕ . Then, following[4], we split the field x µ ( ϕ ) into a zero Fourier mode x µ and remaining non-zero modes such that x µ ( ϕ ) = x µ + ξ µ ( ϕ ) with (cid:82) π ϕ ξ µ = 0. A good pragmatic reason for doing so is that the x ’sare interpreted as classical coordinates, while the ξ ’s are quantum fluctuations which have to beintegrated out. If we let S ( x ) be quadratic in fluctuations, then, for a generic kinetic term, wecan rewrite (1.3) as S ( A ) = (cid:90) dx (cid:90) [ dξ ] exp (cid:2) − ξG − ξ (cid:3) W ( A ( x + ξ )) , (1.6)where ξG − ξ = (cid:82)(cid:82) π dϕ dϕ ξ µ ( ϕ ) G − µν ( ϕ , ϕ ) ξ ν ( ϕ ). We normalize the functional integralmeasure [ dξ ] as (cid:82) [ dξ ] exp (cid:2) − ξG − ξ (cid:3) = 1.In conclusion, we make the following remarks about formula (1.6): It may need some refinement in the presence of scalar fields. For a discussion see [6] and references therein. Some extrapolation formulas can be found in appendix of [7]. In what follows we omit the index 0 from the zero mode when it is clear from the context.
21) For S ( x ) = (cid:82) π dϕ ˙ x ˙ x with ˙ x = ∂ ϕ x that describes the free propagation of a particle, G − is a local operator. Its Green function is given by G ( ϕ ) = π (cid:80) ∞ n =1 1 n cos nϕ , where ϕ = ϕ − ϕ .(2) For the ”string” action S ( x ) = (cid:82) D d σ ∂ a x µ ∂ a x µ , with D a unit disc, G − is a non-localoperator. Note that non-locality appears after Gaussian integration over x ( σ ) in all internalpoints of the disc that provides an effective one-dimensional path integral [5]. In this case, thecorresponding Green function is given by G ( ϕ ) = π (cid:80) ∞ n =1 1 n cos nϕ .(3) Finally, there is one more issue to be mentioned here. The above construction seemssomewhat formal that may be when the path integral diverges. Following [13], we will defineone-dimensional path integral by using the Riemann ζ -function. Thus, we express all sums interms of ζ ( s ) = (cid:80) ∞ n =1 1 n s and ζ ( s, ) = (cid:80) ∞ r =1 / r s . In this section we will describe a concrete example of nonrelativistic gauge theory without theusual magnetic F ij -term in the Lagrangian. Since the Wilson factor (1.1) is Lorentz invariant,we choose a non-invariant kinetic term S ( x ) = 12 (cid:90) π dϕ (cid:104) πα (cid:48) ) ˙ x ˙ x + 12 πα (cid:48) κ ( x i − x i )( x i − x i ) (cid:105) , (2.1)or, equivalently, ξG − ξ = 1(2 πα (cid:48) ) ξ G − ξ + 12 πα (cid:48) κ ξ i G − ij ξ j . (2.2)Here G ( ϕ ) = G = 1 π ∞ (cid:88) n =1 n cos nϕ , G ij ( ϕ ) = δ ij G = 1 π δ ij ∞ (cid:88) n =1 cos nϕ , (2.3)where (cid:82) π dϕ G G − = δ + ( ϕ ). The function δ + ( ϕ ) plays a role of the δ -function on the ξ ’s.Therefore, it is given by δ + ( ϕ ) = π (cid:80) ∞ n =1 cos nϕ . The indices i , j run from 1 to p . Like in stringtheory, we use 2 πα (cid:48) for dimensional purposes. The normalization is designed to describe thegauge theory action with anisotropic scaling characterized by the dynamical critical exponent z = 2. In this case dimensions of all objects are measured in the units of spatial momenta suchthat dim x = − x i = − κ is a relative factor of dimension zero. With such a choice,we define the zero mode measure as dx = τ p dtd p x , where t = x , τ − p = g (2 πα (cid:48) ) ( p +2) / and g isa dimensionless parameter. This provides dim dx = 0.Remark (1) above makes it clear that fluctuations in time direction are described by the usualparticle term. The integral (1.6) for this case was intensively discussed in the literature. On theother hand, fluctuations in spacial directions are described by a sum of terms such that everyterm represents a white Gaussian noise. To our knowledge, there have been no studies of (1.6)with the noise term in the literature. In string theory g is proportional to the string coupling. For a review, see, e.g., [14] and references therein. Another possible way is to interpret them as a quadratic open string tachyon profile. However, a crucialdifference with [15] is that the standard ∂X i ¯ ∂X i -term is missing in our case. .1 Leading Terms To actually compute leading terms in the α (cid:48) -expansion of (1.6), one would have to expand A ( x + ξ )in powers of ξ and then compute a few Feynman diagrams shown in Figure 1. F FFF FDF DF D F Figure 1:
Feynman diagrams that contribute the leading terms in the α (cid:48) -expansion. F , F , D stand for F i , F ij , D i , respectively. With ζ ( −
2) = 0, the result, up to third order in α (cid:48) , is given by S ( A ) = τ p (cid:90) dtd p x tr (cid:104) πα (cid:48) ) (cid:16) b F i + b ( D i F ij ) + b F ij F jk F ki (cid:17) + O ( α (cid:48) ) (cid:105) ,b = − κζ (0) , b = − π κ ζ ( − , b = − iκ ζ ( − . (2.4)Here D µ = ∂ µ − iA µ is the covariant derivative and F µν = i [ D µ , D ν ] is the field strength. With ζ ( −
1) = − /
12 and ζ (0) = − /
2, the above equation becomes S ( A ) = τ p (cid:90) dtd p x tr (cid:104) κ ( πα (cid:48) ) (cid:16) F i − π κ ( D i F ij ) − iκ F ij F jk F ki (cid:17) + O ( α (cid:48) ) (cid:105) . (2.5)In the Abelian case the F term vanishes and therefore (2.5) reduces to S ( A ) = τ p (cid:90) dtd p x (cid:104) κ ( πα (cid:48) ) (cid:16) F i − π κ ( ∂ i F ij ) (cid:17) + O ( α (cid:48) ) (cid:105) . (2.6)An important remark about (2.5) and (2.6) is that the α (cid:48) -expansion doesn’t coincide with aderivative expansion.Motivated by the zero slope limit ( α (cid:48) →
0) of string theory, we would like to consider it in theproblem at hand. As known, the role of this limit is just to remove the higher order correctionsin α (cid:48) from the string effective action and, as a result, it becomes quadratic in F .So we take the limit α (cid:48) ∼ ε → g ∼ ε − ( p/ with κ held fixed. Ignoring the constant termwe get S ( A ) = ρ (cid:90) dtd p x tr (cid:104) F i − π κ ( D i F ij ) − iκ F ij F jk F ki (cid:105) , (2.7) The diagrams look like those of [16] because in both cases the vertices are the same. ρ = κ g (2 πα (cid:48) ) − ( p/ . In contrast to string theory, the resulting action now includes the DF DF and F -terms. It becomes purely quadratic in the field strength only in the Abelian casewhen the cubic term identically vanishes.Finally, it remains to be seen how the action (1.5) is recovered. For this, we take p = 3and use a description in terms of ordinary (Abelian) electric and magnetic fields: E i = F i and H k = (cid:15) kij F ij . Then a little algebra shows that (2.6) does reduce to (1.5) with ρ = − κ π . We now turn to the question of what deformations of the action (1.5) are natural in our approach.Since (1.5) describes the long wavelength physics, it is natural to consider the case of slowlyvarying, but not necessarily small, fields. For this we will ignore higher derivative terms in the ξ expansion of A ( x + ξ ).For slowly varying Abelian fields, (1.6) reduces to S ( A ) = (cid:90) dx (cid:90) [ dξ ] exp (cid:104) − ξG − ξ + i (cid:90) π dϕ ˙ ξ j (cid:16) F j ξ + 12 F ij ξ i + 13 ∂ k F ij ξ i ξ k + 18 ∂ l ∂ k F ij ξ i ξ k ξ l (cid:17)(cid:105) . (2.8)It is convenient to integrate over ξ first. Then, we get S ( A ) = (cid:90) dx (cid:90) [ dξ i ] exp (cid:104) − ξ i (cid:16) πα (cid:48) κ δ ij G − + (2 πα (cid:48) ) F i F j ¨G (cid:17) ξ j + i (cid:90) π dϕ ξ i ˙ ξ j (cid:16) F ij + 13 ∂ k F ij ξ k + 18 ∂ l ∂ k F ij ξ k ξ l (cid:17)(cid:105) , (2.9)where ¨G = ∂ ∂ G( ϕ ).As in string theory [17], it is possible to include the F ξξ and F ˙ ξξ -terms into the propagator.However, for what follows we will include only one of them. To this end, we use the identity ∂ ∂ G( ϕ ) = G − ( ϕ ) (2.10)to rewrite (2.9) as S ( A ) = (cid:90) dx (cid:90) [ dξ i ] exp (cid:104) − πα (cid:48) κ ξ i G ij G − ξ j + i (cid:90) π dϕ ξ i ˙ ξ j (cid:16) F ij + 13 ∂ k F ij ξ k + 18 ∂ l ∂ k F ij ξ k ξ l (cid:17)(cid:105) , (2.11)where a new metric is given by G ij = δ ij + (2 πα (cid:48) ) κF i F j . (2.12)Note that G depends only on the electric field.We are now ready to compute a few terms in S ( A ) by treating all the magnetic terms underthe integral in (2.11) as perturbations. With the metric G held fixed, this is equivalent to aperturbation theory in α (cid:48) . Up to third order, the result is In the process, one has to rescale the electric field E → i E by the Wick rotation. ( A ) = τ p (cid:90) dtd p x (cid:113) det( δ ij + (2 πα (cid:48) ) κF i F j ) (cid:20) − π ( πα (cid:48) κ ) ∂ n F ij ∂ m F kl G ni G mk G jl + O ( α (cid:48) ) (cid:21) . (2.13)Since we ignore derivatives of F i , we replaced the covariant derivatives (with respect to themetric G ij ) with ordinary ones.To compare (2.13) to (2.6), it is sufficient to expand √ det G and G − in α (cid:48) . Since we are onlyinterested in the leading terms, we substitute √ det G = 1 + 4( πα (cid:48) ) κF i and G ij = δ ij into (2.13)that immediately leads to the desired result.Now, specializing to four dimension and ignoring the higher α (cid:48) -corrections, we have S ( A ) = τ (cid:90) dtd x (cid:113) det( δ ij + (2 πα (cid:48) ) κ E i E j ) (cid:20) − π ( πα (cid:48) κ ) ∂ n F ij ∂ m F kl G ni G mk G jl (cid:21) , (2.14)with G ij = δ ij +(2 πα (cid:48) ) κ E i E j and F ij = (cid:15) ijk H k . It can be interpreted as a non-linear deformationof the original action (1.5) by a slowly varying electric field.To illustrate the deformation (2.14), we extend the leading order expression in the limit α (cid:48) → E → i E / √ ρ and H → H / √ ρ ) to next order, and find S = (cid:90) d xdt (cid:20) E − ρ ( ∇ × H ) + ρ α (cid:16)
12 ( E ) + ρ ( ∇ × H ) E − ρ ( E · ∇ × H ) − ρ ( ∇ × H ) · ( E × ( E · ∇ ) H ) (cid:17)(cid:21) , (2.15)where ρ α = g (2 πα (cid:48) ) .Finally, let us consider higher order terms in F ij . We have already found that there is no F ij in the action. To extend this calculation to order F n , we must compute a connected Feynmandiagram shown in Fig.2. F F FF Figure 2:
A Feynman diagram that contributes the F n -term to the action. F stands for F ij . It is easy to see that this diagram is proportional to the following integral Note that in the Abelian case there are no terms which are odd in F . n = (cid:90) π dϕ . . . (cid:90) π dϕ n ˙ G . . . ˙ G n , (2.16)where ˙ G km = ∂ k G nm . Integrating over the ϕ ’s, we get I n = 2( − ) n ζ ( − n ) = 0 . (2.17)We have used that ζ ( − n ) = 0 for any positive integer n . So, there are no F G − F . . . F G − -termsin the action and, as a consequence, there are no ( H ) n -terms for p = 3. This is the reason fornot including the F ˙ ξξ -term into the propagator in (2.11). We conclude this section with a computation that includes a more general form of the kineticterm. So we considerG = 1 π ∞ (cid:88) n =1 n a cos nϕ , G = 1 π ∞ (cid:88) n =1 n b cos nϕ , (2.18)where a and b are free parameters. Note that for a = 2 and b = 0, (2.18) reduces to (2.3).To compute the F ij -term in the action, one has to evaluate the Feynman diagram as that ofFig. 2 with n = 1 and the propagator defined by (2.18). The integral (2.17) becomes in this case I ( b ) = − ζ (2 b − . (2.19)No essentially new computation is required to get the F i -term. Replacing F ij ξ i ˙ ξ j by F j ξ ˙ ξ j leads to the following integral I ( a, b ) = (cid:90) π dϕ (cid:90) π dϕ ˙ G ˙G , (2.20)which is simply I ( a, b ) = − ζ ( a + b − . (2.21)Now we are ready to see what vanishing the F ij -term or alternatively the F i -term in theaction means. From (2.19) it follows that there is no F ij if 2 b − b to be a non-positive integer.For the non-trivial zeros, b is complex. It is given by + i t . Similarly, the coefficient in frontof the F i -term vanishes if a + b − a + b to be 2(1 − k )with k a positive integer for the trivial zeros and + it for the non-trivial zeros. According to the Riemann hypothesis, all the non-trivial zeros lie on the critical line + it . Examples with Fermions
The purpose of the present section is to add worldline fermions to the theory and do some explicitcalculations illustrating our approach.With the fermions, the Wilson factor (1.1) is typically extended toˆ W ( A, C ) = 1 N tr P exp (cid:104) i (cid:90) π dϕ (cid:16) ˙ x µ A µ − F µν ψ µ ψ ν (cid:17)(cid:105) , (3.1)such that it respects the supersymmetry transformations δx µ = ψ µ (cid:15) , δψ µ = ˙ x µ (cid:15) , (3.2)even with a non-constant parameter (cid:15) .If we formulate the theory in the superfield notation, the Wilson factor (3.1) can be writtenin the following form [18]ˆ W ( A, C ) = 1 N tr (cid:16) ∞ (cid:88) n =0 i n n (cid:89) i =1 (cid:90) d ˆ ϕ i D ˆ x µ A µ (ˆ x ) n − (cid:89) j =1 Θ( ˆ ϕ j j +1 ) (cid:17) . (3.3)Here ˆ x µ = x µ + θψ µ , D = θ∂ ϕ − ∂ θ , d ˆ ϕ = dϕdθ , Θ( ˆ ϕ ij ) = Θ( ϕ ij ) + θ i θ j δ ( ϕ ij ) , (3.4)where Θ is a step function.Given the supersymmetry transformations, one can extend the action (2.1) toˆ S ( x, ψ ) = 12 (cid:90) π dϕ (cid:104) πα (cid:48) ) (cid:0) ˙ x ˙ x + ψ ˙ ψ (cid:1) + 12 πα (cid:48) κ (cid:0) x i x i − ψ i ∂ − ψ i (cid:1)(cid:105) (3.5)by adding the fermions. Formally, it is invariant under (3.2). We consider for a moment the field x i rather than its projection on non-zero modes. The fact that ψ∂ − ψ is a non-local operatoris not in contradiction with our approach. The only restriction on S ( x, ψ ) is that it must bequadratic in fluctuations.Now we impose the antiperiodic boundary conditions on the fermions: ψ µ ( ϕ +2 π ) = − ψ µ ( ϕ ).This implies that there are no fermionic zero modes and, therefore, the ψ ’s can be interpreted asadditional quantum fluctuations. Then it is natural to combine bosonic and fermionic fluctuationsinto a single superfield ˆ ξ µ = ξ µ + θψ µ such that ˆ x µ = x µ + ˆ ξ µ .With this choice of the boundary conditions we can write (3.5), with x i subtracted from x i ,as ξG − ξ + ψK − ψ = 1(2 πα (cid:48) ) (cid:0) ξ G − ξ + ψ K − ψ (cid:1) + 12 πα (cid:48) κ (cid:0) ξ i G − ij ξ j + ψ i K − ij ψ j (cid:1) . (3.6)Here K ( ϕ ) = K = 1 π ∞ (cid:88) r =1 / r sin rϕ , K ij ( ϕ ) = δ ij K = 1 π δ ij ∞ (cid:88) r =1 / r sin rϕ . (3.7)8here (cid:82) π dϕ K K − = δ − ( ϕ ). The function δ − ( ϕ ) plays a role of the δ -function on the ψ ’s.Explicitly, it is given by δ − ( ϕ ) = π (cid:80) ∞ r =1 / cos rϕ .In terms of ˆ ξ , (3.6) readsˆ ξ ˆ G − ˆ ξ = 1(2 πα (cid:48) ) ˆ ξ ˆ G − ˆ ξ + 12 πα (cid:48) κ ˆ ξ i ˆ G − ij ˆ ξ j . (3.8)Here ˆ G ( ϕ ) = ˆG = G − θ θ K , ˆ G ij ( ϕ ) = δ ij ˆ G = δ ij (cid:0) G − θ θ K (cid:1) , (3.9)where ˆ ξ ˆ G − ˆ ξ = (cid:82)(cid:82) d ˆ ϕ d ˆ ϕ ˆ ξ µ ( ϕ ) ˆ G − µν ( ϕ , ϕ ) ˆ ξ ν ( ϕ ) and (cid:82) d ˆ ϕ ˆ G ˆ G − = θ δ − ( ϕ ) − θ δ + ( ϕ ). Once ˆ S is found, we will use it in the next subsection to study the generating functional S ( A ) = (cid:90) dx (cid:90) [ d ˆ ξ ] exp (cid:104) − ˆ ξ ˆ G − ˆ ξ (cid:105) ˆ W ( A ( x + ˆ ξ )) . (3.10)Here we normalize the functional integral measure [ d ˆ ξ ] as (cid:82) [ d ˆ ξ ] exp (cid:2) − ˆ ξ ˆ G − ˆ ξ (cid:3) = 1. We will now carry out a precisely analogous computation as that of subsection 2.1. The simplestway to do so is to use the superfield formalism. In practice, this means to replace dϕ , Θ( ϕ ), ∂ ϕ , G , etc. with d ˆ ϕ , Θ( ˆ ϕ ), D , ˆ G , etc. in the corresponding expressions for the Feynman diagramsshown in Figure 1. Using ζ ( −
2) = ζ ( − , ) = 0, the diagrams can be evaluated to give S ( A ) = τ p (cid:90) dtd p x tr (cid:104) πα (cid:48) ) (cid:16) s F i + s ( D i F ij ) + s F ij F jk F ki (cid:17) + O ( α (cid:48) ) (cid:105) ,s = − κ (cid:0) ζ (0) − ζ (0 , ) (cid:1) , s = − π κ (cid:0) ζ ( − − ζ ( − , ) (cid:1) , s = − iκ (cid:0) ζ ( − − ζ ( − , ) (cid:1) . (3.11)With ζ (0 , ) = 0 and ζ ( − , ) = 1 /
24, (3.11) takes the form S ( A ) = τ p (cid:90) dtd p x tr (cid:104) κ ( πα (cid:48) ) (cid:16) F i − π κ ( D i F ij ) − iκ F ij F jk F ki (cid:17) + O ( α (cid:48) ) (cid:105) . (3.12)A few noteworthy facts are the following. The effect of the fermions on the coefficients b i is a shift: ζ ( s ) → ζ ( s ) − ζ ( s, ). The only exception is the coefficient in front of the quadraticterm. In fact, it doesn’t get shifted because ζ ( s, ) = 0 for s = 0. In the Abelian case the F term vanishes. After taking the limit α (cid:48) → p = 3, the action reduces to (1.5) with ρ = − κ π . Note that ˆ G − = K − − θ θ G − . .2 Slowly Varying Fields Now let us explore some issues that arise in the case of slowly varying Abelian fields. We useagain the superfield formalism.For this case, (3.10) reduces to S ( A ) = (cid:90) dx (cid:90) [ d ˆ ξ ] exp (cid:104) −
12 ˆ ξ ˆ G − ˆ ξ + i (cid:90) d ˆ ϕ D ˆ ξ j (cid:16) F j ˆ ξ + 12 F ij ˆ ξ i + 13 ∂ k F ij ˆ ξ i ˆ ξ k + 18 ∂ l ∂ k F ij ˆ ξ i ˆ ξ k ˆ ξ l (cid:17)(cid:105) . (3.13)Integrating out ˆ ξ , one gets simply S ( A ) = (cid:90) dx (cid:90) [ dξ i ] exp (cid:104) −
12 ˆ ξ i (cid:16) πα (cid:48) κ δ ij ˆ G − − (2 πα (cid:48) ) F i F j D D ˆG (cid:17) ˆ ξ j + i (cid:90) d ˆ ϕ ˆ ξ i D ˆ ξ j (cid:16) F ij + 13 ∂ k F ij ˆ ξ k + 18 ∂ l ∂ k F ij ˆ ξ k ˆ ξ l (cid:17)(cid:105) , (3.14)where ˆ ξ D D ˆG ˆ ξ = (cid:82) (cid:82) d ˆ ϕ d ˆ ϕ ˆ ξ D D ˆG ˆ ξ .It is easy to see now that D D ˆG( ϕ ) = − ˆ G − ( ϕ ) (3.15)is a superspace version of (2.10). The minus sign is due to the fact that the D ’s anticommutewith each other. This identity allows us to combine the two first terms of (3.14) into a singleone. So, we have S ( A ) = (cid:90) dx (cid:90) [ dξ i ] exp (cid:104) − πα (cid:48) κ ˆ ξ i G ij ˆ G − ˆ ξ j + i (cid:90) d ˆ ϕ ˆ ξ i D ˆ ξ j (cid:16) F ij + 13 ∂ k F ij ˆ ξ k + 18 ∂ l ∂ k F ij ˆ ξ k ˆ ξ l (cid:17)(cid:105) , (3.16)with G ij defined in (2.12).Having derived (3.16), we are now in a position to compute a few terms in the action. Justas in section 2.2, keeping the metric G fixed, we find up to third order in α (cid:48) S ( A ) = τ p (cid:90) dtd p x (cid:113) det( δ ij + (2 πα (cid:48) ) κF i F j ) (cid:20) − π ( πα (cid:48) κ ) ∂ n F ij ∂ m F kl G ni G mk G jl + O ( α (cid:48) ) (cid:21) . (3.17)To get from (3.17) to (3.12), its Abelian version, we must expand G in α (cid:48) . In doing so, the F i term comes from the determinant of G , while the ( DF ) term is already in (3.17).We conclude this section by computing the higher order terms in F ij . For this, we need toevaluate the Feynman diagram of Figure 2 in the superfield formalism. This diagram is given,up to a constant multiple, by a multiple integralˆ I n = (cid:90) d ˆ ϕ . . . (cid:90) d ˆ ϕ n D ˆ G . . . D n ˆ G n . (3.18)Performing integration, we obtain 10 n = 2( − ) n +1 (cid:0) ζ ( − n ) − ζ ( − n, ) (cid:1) = 0 . (3.19)We have used that ζ ( − n ) = ζ ( − n, ) = 0 for any positive integer n . Thus, there are no F G − F . . . F G − terms in the action and, as a consequence, there are no ( H ) n -terms for p = 3. To generalize the discussion of section 2.3 to the case with the fermions, the first step is to pickup fermionic kinetic terms such thatK = 1 π ∞ (cid:88) r =1 / r a − sin rϕ , K = 1 π ∞ (cid:88) r =1 / r b − sin rϕ . (3.20)So for a = 2 and b = 0, (3.20) coincides with (3.7). Then, the corresponding superspacepropagators are constructed by combining them with the G ’s of section 2.3.Now, to compute a coefficient in front of the F ij -term in the action, one has to evaluate theFeynman diagram of Figure 2 with n = 1. In the case of interest the integral (3.18) becomesˆ I ( b ) = 2 (cid:0) ζ (2 b − − ζ (2 b − , ) (cid:1) = 4 (cid:0) − b − (cid:1) ζ (2 b − . (3.21)Here we have used the identity that ζ ( s, ) = (2 s − ζ ( s ).Similarly, one can find a coefficient in front of the F i -term. Replacing F ij ˆ ξ i D ˆ ξ j by F j ˆ ξ D ˆ ξ j leads to the following integralˆ I ( a, b ) = (cid:90) d ˆ ϕ (cid:90) d ˆ ϕ D ˆ G D ˆG . (3.22)After performing integration, one getsˆ I ( a, b ) = 2 (cid:0) ζ ( a + b − − ζ ( a + b − , ) (cid:1) = 4 (cid:0) − a + b − (cid:1) ζ ( a + b − . (3.23)Clearly, no new analysis is required to conclude what vanishing the F ij and F i -terms in theaction means. Therefore our conclusions here are similar to those of section 2.3. We will here conclude with brief observations about our approach and the non-relativistic gaugeinvariant actions derived above.(1) For F ij = 0, the Abelian actions (2.13) and (3.17) reduce to S ( A ) = τ p (cid:90) dtd p x (cid:113) det( δ ij + F i F j ) . (4.1)In order to keep the equations simple we have set (2 πα (cid:48) ) κ = 1. Clearly, the α (cid:48) -dependencecan be easily restored on dimensional grounds. It is surprising that (4.1) can be written in theBorn-Infeld form 11 ( A ) = τ p (cid:90) dtd p x (cid:113) det( δ µν + F µν ) . (4.2)To show this, we derive a useful identity. We do that in a frame where F µν has a special form F = F . . . − F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , (4.3)with F the only nontrivial component. Then, we finddet( δ ij + F i F j ) = det( δ µν + F µν ) = 1 + F i . (4.4)The relation with the Born-Infeld theory implies that, after Wick rotating to Lorentziansignature, there is an upper bound to the electric field strength. If one restores dimensions, thecritical field strength E c is given by E c = 1 (cid:112) (2 πα (cid:48) ) κ . (4.5)Note that, as in string theory, there is no upper bound in the limit α (cid:48) → ζ ( s ) combines with ζ ( s, ) to form a linearcombination ζ ( s ) − ζ ( s, ). It is very special in that it is a holomorphic function of the complexvariable s . A simple pole of ζ ( s ) at s = 1 cancels with that of ζ ( s, ). Indeed, one can easilycheck that lim s → ( ζ ( s ) − ζ ( s, )) = − ln 4. This means that the Feynman diagrams we consideredin section 3 remain finite for all values of a and b . As a result, the path integral representationfor the action (3.10) is well-defined at least up to third order in α (cid:48) .(3) Although the Gaussian noise terms describing fluctuations in spatial directions wereintroduced by hand so as to recover finally the gauge theory action (1.5), there is some indirectevidence for this. A few facts are particularly interesting:(i) This gauge theory action was proposed in [10] to describe the Rokhsar-Kivelson points ofquantum dimer models.(ii) Recently, it was claimed in [19] that the quantization scheme used for the quantum dimermodels is nothing else but a discrete analog of stochastic quantization.(iii) As known, the notion of noise plays a pivotal role in stochastic quantization.If so, then it seems natural to expect the appearance of ”noise” in our approach too. As wehave shown above this is indeed the case.(4) It is worth mentioning that a non-Abelian gauge theory action without the usual magneticterm was also proposed in [20]. It differs from ours (2.7) and (3.12) by the absence of the F -term. From our approach we don’t see any reason for dropping it. It disappears in the opensuperstring effective action ( a = b = 1), where, on the other hand, the magnetic term is presentbecause of Lorentz invariance. Note that a non-Abelian action without the usual electric term was proposed in [11].
We are grateful to A.A. Tseytlin for discussions and M. Haack for reading the manuscript.This work was supported in part by DFG within the Emmy-Noether-Program under GrantNo.HA 3448/3-1, Excellence Cluster, and the Alexander von Humboldt Foundation under GrantNo.PHYS0167. We also would like to thank D. Kharzeev and I. Zaliznyak for hospitality at theBrookhaven National Laboratory, where a portion of this work was completed.
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