Introduction to Superstring Theory

Transcription

1 Introduction to Superstring Theory Lecture Notes Spring 005 Esko Keski-Vakkuri S. Fawad Hassan 1

2 Contents 1 Introduction 5 The bosonic string 5.1 The Nambu-Goto action Strings The Polyakov action Classical symmetries of the Polyakov action Gauge fixing Conformal invariance: Equations of motion and boundary conditions Mode expansion and quantization Constraints Low-lying string states The light-cone gauge Lowest lying states Open strings Path integral quantization of the bosonic string Conformal field theory (CFT) Commutators in CFT and Radial Ordering Operator Product Expansions Correlation Functions Wick s Theorem Operator-state Correspondence Tree-level Bosonic String Interactions Scattering of Open String Tachyons Tree-level Scattering of Closed String Tachyons Strings in Background Fields Weyl Invariance and the Weyl Anomaly The Bosonic String Beta Functions and the Effective Action An Example of a One-loop Amplitude: the Vacuum-to-vacuum Amplitude, i.e., the Partition Function The Vacuum Energy Superstrings ( Where It Begins Again ) The Superstring Action Equations of Motion and Boundary Conditions Mode Expansions and Quantization Constraints on Physical States Emergence of Spacetime Spinors

3 3.5.1 Chirality The Spin Field Lowest Lying Excitations of Closed Superstrings NS-NS Sector R-NS Sector NS-R Sector R-R Sector Problems with the Spectrum The GSO projection (GSO=Gliozzi-Scherk-Olive) NS Sector R Sector Type IIA and Type IIB Superstrings Gamma Matrix Conventions R-R Ground States Type IIA and IIB Supergravity Toroidal Compactification and T-duality General Idea of Compactification Scalar Field Theory Compactified on S Main Features of Field Theory Compactifications on S String Theory Compactified on S Features of String Theory on S T-duality The T-duality Map T-duality in Superstring Theory T-duality Action on Ramond States Gauge Symmetry Enhancement in Circle Compactification Momentum and Winding as Abelian Charges Lattices and Torii Rectangular Torus Construction of a General Torus Metric Integer and Even Lattices Dual Lattice Heterotic String Theory Equations of Motion Boundary Conditions Quantization Mass Conditions (Spectrum) Extra Massless States Massless Sector of the Heterotic String Spectrum Type I Superstrings

4 Chan-Paton Factors (Works for Bosonic and Superstrings) D-branes Multiple D-branes String Dualities Type I - Heterotic SO(3) Duality Type IIA - IIB Duality Heterotic SO(3) - Heterotic E 8 E 8 Duality Type IIA - M-theory Duality Heterotic E 8 E 8 - M-theory Duality

5 1 Introduction At the moment, string theory is the most promising candidate for a unified theory of all fundamental particles and forces, including gravity. Furthermore, it unifies all the known forces with the laws of quantum mechanics. Traditional quantum gauge field theories have been a successful framework for describing elementary particles up to currently testable energy scales. However, in the big picture, they are to be considered as effective theories, approximations to the final theory that lies somewhere underneath. For example, the Standard Model contains 19 parameters, whose values are set by hand to agree with experiments. String theory is based on the idea of replacing particles as fundamental constituents by one-dimensional extended objects, strings. There is only one parameter in the theory, the length of the string, l s. The string length is thought to be of the order of Planck length cm. The hope is that at larger scales, string theory determines how the symmetries of Nature are selected and broken, and what is the ladder of effective field theories that emerge, containing the Standard Model. As a theory of quantum gravity, string theory is also hoped to teach us what is superseded by space and time at truly small scales where quantum effects render these concepts fuzzy. It is also hoped to answer some fundamental problems related to quantum behavior of black holes. In this course we will see some examples of how our usual concepts of spacetime are modified. During the past years, string theory itself has developed rapidly. Our current understanding of it is now much more rich. The term string theory no longer refers to only strings and their interactions - it now refers to many kinds of different extended objects and to a large web different but related techniques (including traditional gauge field theories) and so on. The hope is that this course will serve as an introduction to the basic concepts, and enable and encourage to further study this fascinating field. Editorial note. These lectures have been compiled from various sources (I will provide a more detailed list of references when I remember them). Therefore I have had to struggle with different conventions and notations some confusion is bound to remain even at this stage. Please let me know of any typos and mistakes that you find. A portion of these notes was developed by Fawad Hassan, currently at Stockholm University. The typesetting is mostly thanks to the efforts of Moundheur Zarroug and Niko Jokela and I highly appreciate their work. The bosonic string For someone used to field theory, our starting point may perhaps look a bit peculiar at first. In field theory, the particles appear as quanta of the field so one begins 5

6 with a multiparticle description. Here, in contrast, we will begin by investigating the dynamics of a single string. Different oscillations of the string will turn out to correspond to different particles. However, it will take a while before we will quantize the string. We begin by studying its classical behavior..1 The Nambu-Goto action A string is simply a one-dimensional extended object moving and vibrating across spacetime. As an introduction, it is useful first to consider the motion of relativistic massive point particle. As a particle moves through the spacetime, it sweeps out a curve, the worldline: X 0 particle worldline X i Figure 1: Worldline of a point particle. If we denote by τ the parameter along the worldline (the proper time that the particle would measure if it carried a watch), we can parameterize the curve of the worldline by X µ = X µ (τ), µ = 0, 1,..., D 1 (.1) in a D-dimensional spacetime. If the spacetime is a flat Minkowski space, it has a metric ds = η µν (X)dX µ dx ν = (dx 0 ) + (dx 1 ) + (dx ) + + (dx D 1 ). (.) (I m using the signature) 6

7 The action that controls the dynamics of the point particle is simply proportional to the proper length of the worldline: S = m ds. (.3) We can rewrite this using the embedding X µ (τ) and the proper time τ: ds = ( (Ẋ0 ) + (Ẋ1 ) + + (ẊD 1 ) )dτ = η µν Ẋ µ Ẋ ν dτ (. d dτ ) (.4) S = m η µν Ẋ µẋν dτ. (.5) The classical equations of motion are minima of the action, so they are found by the variational principle Since the path integral contains δs = 0 (.6) { c = 1 = 1 (.7) we get thus [length] = [mass] 1 = [energy] 1. Since [dτ] = [length], we must have [mdτ] = [length] 0, so m has the units of a mass (of the particle). If the particle is moving in a curved spacetime, we replace the flat metric η µν by the curved metric g µν (X). Then the action is S = m g µν (X)ẊµẊν dτ. (.8) You can check that the variational principle gives the geodesic equation as the (relativistic) equation of motion of the point particle. Recall that massive point particles are supposed to follow timelike geodesics in curved spacetime..1.1 Strings Now consider a string moving in spacetime. It traces out a two-dimensional surface, called the worldsheet. We have two natural choices for a string: an open string and a closed string. Their worldsheets are depicted in Figure. The D-dimensional spacetime where the string moves is often called the target space. Just as the action for a relativistic point particle was proportional to the length of the worldline, the logical guess for the action of a relativistic string is the area of its worldsheet: S = T d s. (.9) 7

8 0 X 0 X i X i X open string closed string Figure : Open and closed string worldsheets. Now we just have to be a bit careful about what is the integration measure here, i.e. what is meant by the infinitesimal area element d s. First of all, we need two parameters to parameterize the worldsheet. In addition to a proper time τ we need a spacelike coordinate σ which parameterizes the string. For an open string we choose σ [0, π] (σ = 0, π are the end points), for a closed string we choose σ [0, π] with σ = 0 identified with σ = π (Fig. 3): π σ π 0 Figure 3: Parameterization of a closed string. Note that the strings are oriented (imagine an arrow pointing to the direction of increasing σ). The worldsheet is characterized by its embedding to the spacetime: X µ = X µ (τ, σ). (.10) 8

9 For a closed string we need a periodicity condition X µ (τ, σ) = X µ (τ, σ + π). If the worldsheet was a flat strip or cylinder, the area would be simply Area = dτ π or π 0 dσ. (.11) However, the embedding to the spacetime allows the string to wiggle and bend (see Fig..), this induces a curved metric h αβ (X) to the worldsheet to characterize distances. To find it, we substitute the embedding (.10) to the metric of the spacetime. Let us take the spacetime to be a flat Minkowski space with metric η µν. Then, by substituting (.10): ds = ( η µν dx µ dx ν = η µν d(x ) ( µ (τ, σ))d(x ν (τ, σ)) ) X µ Xµ X ν = η µν dτ + τ σ dσ Xν dτ + τ σ dσ = η µν ( α X µ )( β X ν )dσ α dσ β h αβ (X)dσ α dσ β (.1) where I used the notation σ α = { σ 0 τ (α = 0) σ 1 σ (α = 1) (.13) for the worldsheet coordinates τ, σ, and α -dimensional surface with curved metric. So the string worldsheet is a σ α h αβ (X) = η µν α X µ β X ν. What is its infinitesimal area element? In curved space, at every point we can introduce ( local orthonormal ) coordinates ξ α where the metric looks flat: ds W S = h αβ dσ α dσ β = (dξ 0 ) + (dξ 1 ). (.14) In these coordinates, the infinitesimal area element is simply dξ 0 dξ 1. So we only need to evaluate the Jacobian in transforming back to the original coordinates σ α : dξ 0 dξ 1 = J(ξ, σ)dσ 0 dσ 1. (.15) You can convince yourself that the Jacobian is J(ξ, σ) = det(h αβ ) h. (.16) So now we are ready to write down the action of a relativistic string: S NG = T dσ 0 dσ 1 det(h αβ ) = T dτdσ det(η µν α X µ β X ν ). (.17) 9

10 The action (.17) is the Nambu-Goto action. For S NG to be a number ( = 1), T must scale like [mass] or [length]. One can check that T can be identified as the tension of the string. The string is elastic. We can trade the tension to an another parameter, the string length l s : l s = 1 πt. (.18) Another equivalent parameterization uses the Regge slope α (pronounced alpha prime ): T = 1 πα. (.19) Note that the Nambu-Goto action (.17) contains a square root. So the action is nonpolynomial, and it is no surprise that it would be very hard to proceed to quantize it. We will therefore use a trick and move to an alternative description, the Polyakov action, which is quadratic (in X µ ).. The Polyakov action The action (.17) can be thought as the action for a 1+1 dimensional field theory of D scalar fields X µ (τ, σ). Now we make a trick. We introduce a set of (a matrix of) auxiliary fields h αβ (τ, σ) (to be identified with the worldsheet metric, but not yet) and consider the action S p = T dτdσ h h αβ α X µ β X ν η µν. (.0) This is the Polyakov action. Since h αβ are nonpropagating degrees of freedom, their equation of motion δs δh αβ = 0 (.1) plays the role of a constraint. The equation of motion can be written as the equation T αβ 1 T 1 δs h δh = 1 αβ ( αx µ β X ν 1 h αβh γδ γ X µ δ X ν )η µν = 0. (.) We can use (.) to eliminate 1 the auxiliary fields h αβ. Let us denote h αβ α X µ β X ν η µν and h = det( h αβ ). From (.): h αβ = 1 h αβh γδ = h γδ {}}{ γ X µ δ X ν η µν (.3) 1 Note that the equation (.) really introduces only non-trivial constraints. T αβ is symmetric so it has only 3 independent components. On the other hand it turns out to be automatically traceless (see next sections), so there is one linear relation between to components meaning that only are independent. 10

11 h = 1 4 h(hγδ hγδ ). (.4) Substituting the square root of (.4) to (.0) yields S p = T dτdσ h h αβ hαβ (.5) = T dτdσ h = S NG. (.6) Thus, by eliminating the auxiliary fields h αβ recover the Nambu-Goto action (.17). from the Polyakov action (.0), we ***** END OF LECTURE 1 *****.3 Classical symmetries of the Polyakov action The Polyakov action is invariant under the following three different groups of symmetry transformations: (i) Global symmetries: The D-dimensional spacetime is invariant under Poincaré transformations. From the point of view of the -dimensional string action, these transformations are global symmetry transformations acting on the D scalar fields X µ : { X µ X µ + a µ (translations) X µ X µ + ω µ νx ν (ω µν = ω νµ ) (Lorentz) We will ignore these for the moment. (.7) (ii) Worldsheet diffeomorphisms: on the worldsheet, These are general coordinate transformations σ α σ α (σ β ) (.8) or, infinitesimally: σ α = σ α + ɛ α (σ β ). (.9) These are also called reparametrizations of the world sheet. Under (.8), the infinitesimal area element hdσ 0 dσ 1 remains invariant 3 d σ h = d σ h. (.30) Note that (.9) is characterized by local parameters ɛ α. 3 They are both related to the simple area element in the local orthonormal frame. 11

12 To see how the worldsheet metric transforms, use h αβ (σ)dσ α dσ β = h αβ ( σ)d σ α d σ β = h γδ (σ) dσγ d σ α dσ δ d σ β d σα d σ β h αβ ( σ) = h γδ (σ) dσγ d σ α dσ δ d σ β. (.31) This transformation is compensated by the transformation of α X µ β X ν η µν, so that h αβ α X µ β X ν η µν is invariant. Thus S P is invariant under (.8). Weyl transformation means rescaling of the (world- (iii) Weyl transformations: sheet) metric h αβ (σ) Λ(σ)h αβ (σ) (.3) by a local scale factor Λ(σ). (.3) implies ( ) Λh00 Λh h det 01 = Λ Λh 10 Λh h 11 h αβ Λ 1 h αβ. (.33) So the combination hh αβ is invariant 4. Note that a Weyl transformation only acts on h αβ, not on X µ or α. This is in contrast with the reparametrizations which also transform α. The Weyl transformation (.3) is characterized by 1 local parameter. Thus, the different groups of transformations (i)-(iii) involve altogether +1 =3 local parameters. Recall that the Polyakov action involves 3 auxiliary local variables h αβ (τ, σ). We can use the symmetry transformations (ii)-(iii) to remove the 3 auxiliary variables and gauge fix h αβ..3.1 Gauge fixing 1) First, any -dimensional metric can be related to a flat (Minkowski) metric by a suitable coordinate transformation (reparametrization, symmetry (ii)), up to an overall local scale factor. In other words, using (ii) we can write ( ) 1 0 (h αβ ) = Λ(τ, σ). (.34) 0 1 This is called the conformal gauge. For a proof, see e.g. Nakahara. ) Second, we can perform a Weyl transformation (iii) to remove the overall scale factor Λ(τ, σ). Then, ( ) 1 0 (h αβ ) = (η αβ ). (.35) Note that in general in n dimensions this is not true: hh αβ Λ n/ Λ 1 hh αβ. 1

13 This is called the covariant gauge. Thus, we have fixed the gauge by specifying to a worldsheet coordinate system where h αβ takes the form of a flat Minkowski metric η αβ..4 Conformal invariance: Actually, the above procedure does not fix the gauge completely. In other words, we did not completely specify the worldsheet coordinates by demanding (.35). There is a class of coordinates where (.35) continues to hold, and these are related by conformal transformations followed by Weyl transformations. Recall that a conformal transformation is a special case of a reparametrization which satisfies h αβ = η αβ h αβ = h γδ (σ) dσγ d σ α dσ δ d σ β = Λ(σα )h αβ. (.36) Then, we can perform a Weyl transformation which cancels the scale factor: h αβ hαβ = Λ 1 (σ α ) h αβ = η αβ (.37) to still stay in the covariant gauge h αβ = η αβ. The combinations of (.36) and (.37) { σ α σ α (σ β ) η αβ η αβ (.38) are the residual gauge transformations. It is useful to introduce light-cone coordinates on the worldsheet: { σ + = σ 0 + σ 1 = τ + σ σ = σ 0 σ 1 = τ σ. (.39) Then and the flat metric becomes ± σ ± = 1 ( τ ± σ ), (.40) ds = η αβ dσ α dσ β = dσ + dσ (.41) so ( ) η++ η + = η + η ( ) 0 1/ 1/ 0. (.4) Then the conformal transformations are reparametrizations { σ + σ + (σ + ) σ σ (σ ) (.43) 13

14 They satisfy d σ + d σ = σ + σ dσ + dσ Λ(σ α )dσ + dσ. (.44) [There is some abuse of language in the literature, sometimes the term conformal transformation means the residual gauge transformation (.38) which includes a Weyl transformation.] When we plug the covariant gauge worldsheet metric h αβ = η αβ into the Polyakov action (.0), it takes a very simple form: S p = T d σ( τ X µ τ X ν σ X µ σ X ν )η µν. (.45) This looks like an action for D free massless scalar fields X µ (τ, σ). However, X 0 comes with a wrong sign because of the Minkowski signature η µν = ( ). Since h αβ has been gauged away, we have to remember the constraint T αβ = 0 (.46) which followed from the equation of motion. In light-cone coordinates (.17), the gauge fixed action is S p = T d σ + X µ X ν η µν (.47) and the constraint (.46) is replaced by T ±± = 0 (.48) T + = 0. (.49) Actually, (.49) is not a constraint. Recall that T αβ is traceless, η αβ T αβ = 0. In an exercise you will show that the tracelessness follows from the Weyl invariance (under (.3)) of the action. In light-cone coordinates the tracelessness reads η + T + = 0 T + = 0. (.50) So (.49) reflects the Weyl invariance. The two equations (.48) are real constraint equations. They are called the Virasoro constraints..5 Equations of motion and boundary conditions Consider now the field variation X µ X µ + δx µ. The variation of the action (.45) is (integrating by parts) 14

15 δs P = T + dτ 0 T π or π +T dσ[( σ τ )X µ ]δx µ dτ[( σ X µ σ=π or π )δx µ ] σ=0 dσ[( τ X µ )δx µ ] τ=+ τ= (.51) where X µ δx µ X µ δx ν η µν. As usual, we take δx µ = 0 at τ = ± so the last term = 0. The second term in (.51) is a surface term. We need to impose boundary conditions for X µ. 1) For a closed string, we needed a periodicity condition X µ (τ, σ) = X µ (τ, σ + π) (see after eqn (.10)), so τ X µ (τ, 0) = τ X µ (τ, π). The variations δx µ must also be periodic in σ σ + π. Thus the second term in (.51) vanishes (δx µ (τ, π) δx µ (τ, 0) = 0). ) For an open string we need a different boundary condition. Now the second term in (.51) is T { σ X µ (τ, σ = π)δx µ (τ, σ = π) σ X µ (τ, σ = 0)δX µ (τ, σ = 0)} This vanishes, if we impose the boundary conditions σ X µ (τ, σ = 0) = σ X µ (τ, σ = π) = 0. (.5) These are called Neumann boundary conditions. The end points of the open string are free to vibrate: Figure 4: Neumann boundary conditions for open string. (There are other possible alternative open string boundary conditions, namely keeping one or both endpoints fixed: δx µ = 0. These Dirichlet boundary conditions will be discussed in the end of the course: they are associated with the existence of other extended objects called D-branes in string theory.) Then, what remains in (.51) is the first term. Setting δs P equations = 0 gives the field [ τ σ]x µ (τ, σ) = 0, (.53) i.e., X µ = 0 where = η αβ α β is the d Alembertian. The eqn (.53) is the good old wave equation for a massless scalar field X µ in 1+1 dimensions. Of course we 15

16 must remember the constraints T αβ = 0. They are now written as T 00 = T 11 = 1 4 ( τx µ τ X µ + σ X µ σ X µ ) = 0 (.54) T 01 = T 10 = 1 τx µ σ X µ = 0 (.55) The tracelessness is h αβ T αβ = T 00 + T 11 = 0 (.56).6 Mode expansion and quantization Let us first consider the closed string. A general solution to the wave equation has the schematic structure X µ (τ, σ) = x µ + a µ τ + b µ σ+[superpositions of plane waves]. The periodic b.c. kills the term linear in σ. The general solutions consistent with the periodic boundary condition can be written as X µ (τ, σ) = x µ + l s pµ τ + i l s n 0 { 1 n αµ ne in(τ σ) + 1 n αµ ne in(τ+σ) }. (.57) Since X µ has the dimension of length, we have included the dimensional parameter l s, the string length (.0). The factors 1 have been introduced by convention and n convenience into Fourier coefficients 1 n αµ n, 1 n αµ n. An alternative way to write the general solution is to use the light-cone coordinates σ ±. The wave equation reads now and the solution decomposes into + X µ = 0 (.58) X µ = X µ L (σ ) + X µ R (σ+ ), (.59) a superposition of a leftmoving wave X µ L which depends only on σ = τ σ and a rightmoving wave X µ R which depends only on σ+ = τ + σ. Schematically: The left and right movers can be expanded as and the periodic b.c. requires X µ L (σ ) = 1 xµ + l s pµ L σ + il s n αµ ne inσ n 0 X µ R (σ+ ) = 1 xµ + l s pµ R σ+ + il s n αµ ne inσ+, (.60) n 0 p µ L = pµ R 1 pµ. (.61) 16

17 X L X R Figure 5: Left- and rightmoving excitations on closed string. Quantization. So far we discussed classical features. We now proceed to quantize the fields X µ using the standard canonical quantization procedure. We promote the fields X µ to operators, and impose canonical commutation relations. First, we need to find the canonical momentum P µ (τ, σ) conjugate to X µ. Following the standard definition of P µ : P µ (τ, σ) = δl δx = T X µ (τ, σ). (.6) µ Then, we interpret P µ (τ, σ) and X µ (τ, σ) as Heisenberg operators, and impose the equal time τ canonical commutation relations and [P µ (τ, σ), P ν (τ, σ )] = [X µ (τ, σ), X ν (τ, σ )] = 0 (.63) [P µ (τ, σ), X ν (τ, σ )] = T [Ẋµ (τ, σ), X ν (τ, σ )] = iη µν δ(σ σ ). (.64) In an exercise you will show that after substituting the mode expansion (.57) into (.63) and (.64), you recover the commutation relations [p µ, x ν ] = iη µν (.65) [α µ m, α ν n] = mη µν δ m+n,0 (.66) [ α µ m, α ν n] = mη µν δ m+n,0 (.67) 17

18 for the mode coefficients. (Note: Bailin and Love have opposite signs for (.65)-(.67) since they use η µν = (+,,,,,..., ).) We can make the following interpretations: x µ = (target space) center-of-mass coordinate of the string. p µ = (target space) center-of-mass momentum of the string. This can be justified as follows. To obtain the total momentum, integrate the momentum density along the string: P µ dσp µ (τ, σ) (.6) = T π Ẋ µ dσ = πt 1 0 l sp µ = p µ. α µ m, α µ m create and annihilate oscillation degrees of freedom of the string. Since X µ is a Hermitean operator, (X µ ) = X µ, the center-of-mass coordinate x µ and momentum p µ are also Hermitean. Furthermore, the oscillator coefficients α µ m, α µ m must satisfy the following relations: (α µ m) = α µ m (.68) Therefore, we can rescale them and denote ( α µ m) = α µ m. (.69) a µ m 1 m α µ m ; ã µ m 1 m α µ m. (.70) Now the commutation relations (.66), (.67) become [a µ m, (a ν n) ] = η µν δ m,n (.71) [ã µ m, (ã ν n) ] = η µν δ m,n (.7) These are the standard commutation relations for harmonic oscillator creation and annihilation operators! Note however that because of η µν, the µ = ν = 0 components have a wrong sign. Let us define a vacuum state 0 : a µ n 0 = 0 n > 0. (.73) We can then create oscillation modes for the string by acting with a creation operator (a ν n) = a ν n. However, before moving forward, we make the following two observations: 18

19 1) Even if the string is not oscillating, it is moving in target space with some center-of-mass momentum k µ. We need to take this into account in the definition of the vacuum. So a more accurate notation for it is 0, k µ, with p µ 0, k µ = k µ 0, k µ (.74) The vacuum 0, k µ is an eigenstate of the c.o.m. momentum operator p µ which appears in the expansion of the field operators X µ : X µ (τ, σ) = x µ + 1 l sp µ τ +... ) Because of [a 0 n, (a 0 m) ] = η 00 δ n,m = δ n,m, the spectrum contains states with a negative squared norm: consider e.g. (a 0 m) 0, k µ : (a 0 m) 0, k µ = 0, k µ a 0 m(a 0 ) m 0, k µ = 1. Such states, called ghosts, are unphysical and we need to find a way to exclude them from the spectrum. For this we need the Virasoro constraints. 5.7 Constraints ***** END OF LECTURE ***** We have seen that if we describe string dynamics with the Polyakov action, we had to include (the Virasoro) constraints. When quantizing constrained systems, one has two natural alternatives to proceed. Either one can first quantize all degrees of freedom, and then apply the constraints to extract out the real physical states. Or, one can first solve the constraints and find the real classical degrees of freedom, and then quantize only those. For the string, we have been following the first road, called the covariant quantization. So now we must take into account the Virasoro constraints. In light-cone coordinates, they were (see (.48)) As with the X µ s, we will use Fourier expansions: T ±± (σ ± ) = 1 ±X µ ± X µ = 0. (.75) T (σ ) = l s 4 T ++ (σ + ) = l s 4 n= n= L n e inσ (.76) L n e inσ+ (.77) 5 An analogous situation exists in Quantum Electrodynamics: the timelike polarization of a photon gives rise to ghost states, but they can be excluded from the spectrum by applying the Gauss Law constraint. 19

20 where L n = π dσe in(σ ) T πls (σ ) τ=0 (.78) 0 etc. If you substitute (.60) and (.75) into (.78) (exercise) 6, you can derive the expressions for L n s in terms of the oscillator coefficients α µ n. The results are L n = 1 m= α µ n mα ν mη µν (.79) where we have used a notation L n = 1 m= α µ n m α ν mη µν (.80) α µ 0 1 l sp µ α µ 0 (.81) to express the results in a neat form. The Hamiltonian density H of the string is obtained by a Legendre transformation: H = P µ Ẋ µ L (.45),(.6) = (where. τ, σ ). Comparing with (.54), you see that T {Ẋµ Ẋ µ + X µ X µ} (.8) H = T T 00 = T {T ++ + T }. (.83) If we integrate H along the string, we obtain the Hamiltonian H: π H = dσh = { l S 0 πls 4 L n dσe in(τ σ) + (.84) n ls 4 L n dσe in(τ+σ) } = L 0 + L 0. (.85) Substituting (.79) and (.80): 6 H = 1 n m= (α µ mα µm + α µ m α µm ). (.86) T ++ = 1 (T 00 + T 11 ) T = 1 (T 00 T 11 ) 0

21 Recall that the α µ ms with m < 0 were creation operators. It is standard to express the Hamiltonian in the form where all the creation operators have been commuted to the left of all the annihilation operators. Using the commutation relations (.66), (.67) will then result to a constant term which is an infinite series. We write H = 1 α α 0 + (α nα µ µn + α µ n α µn ) a (.87) n=1 where a denotes the series. Formally, it is an infinite sum. In quantum field theory, this is known as the zero-point or vacuum energy. Without gravity, this would not be a problem, because experiments measure differences in energy (with respect to vacuum). So there one could set a = 0. However, here a cannot be chosen freely. In string theory, it is fixed by the requirement that unphysical states are removed. We will return to this issue. The Virasoro constraints at classical level were T ++ = T = 0. At quantum level, these are replaced by conditions on expectation values on the physical states: This implies (using (.76)): phys T ++ phys = phys T phys = 0. (.88) L m phys = L m phys = 0 n > 0. (.89) For L 0, L 0, we have to be a bit more careful, because of the zero-point energy. First, we redefine L 0, L 0 to denote only the normal ordered parts of the oscillator expansion: 7 L 0 = 1 αµ 0α µ0 + 1 : n 0 α µ nα µn : = 1 αµ 0α µ0 + L 0 = 1 αµ 0 α µ0 + n=1 α µ nα µn α µ n α µn n=1 (.90) Then, 7 For L n, n 0, L n =: L n : trivially. H = L 0 + L 0 a. (.91) 1

22 The constraint T ++ = T yields the level matching condition L 0 phys = L 0 phys (.9) and the vanishing of H T ++ + T = 0 means that in total we get (L 0 a) phys = ( L 0 a) phys = 0 (.93) We can now check if the vacuum, as it was defined in (.73), is a physical state: L m 0, k µ = 0 for m > 0, but (L 0 a) 0, k µ = ( 1 αµ 0α µ0 + 0 a) 0, k µ (.94) = ( l s 8 p a) 0, k µ >= 0 (.95) This is satisfied if k = 8a/l s. We will interpret this later. The operators L n have a special importance, and they have a name: they are called Virasoro operators. Using the commutation relations (.65)-(.67), and the oscillator expansions (.79),(.80), (.90) we can derive commutation relations for the L n s. For n + m 0, it is straightforward to derive [L n, L m ] = (n m)l n+m (n + m 0) (.96) For the case n + m = 0, it is most convenient to first write an ansatz [L n, L m ] = (n m)l n+m + b(n)δ n+m,0 (.97) where b(n) denotes the expected additional contribution, arising from the infinite constant contributions in the normal ordering. The b(n) can then be evaluated by considering the expectation value b(n) = 0 [L n, L n ] 0 = (exercise...) = D 1 n(n 1). (.98) (See Bailin and Love.) So, all told, the commutation relations for L n s are [L n, L m ] = (n m)l n+m + D 1 n(n 1)δ n+m,0 (.99)

23 This is called the Virasoro algebra. The L m s have similar commutation relations. Moreover, L n, L m commute with each other: [L n, L m ] = 0 n, m. (.100) So the closed string contains two Virasoro algebras, one for the left and one for the right movers. Now let us return to the conditions (.93). They give a mass formula for the string excitations. The string moves through the target space with c.o.m momentum k µ. Then, as seen in the target space, the string has a rest mass M, with M = k µ k µ. (.101) From the point of view of the worldsheet, the energy of a string was a constraint H phys = 0. That means (L 0 + L 0 a) phys = 0. (.10) This gives a relation between the target space rest mass, c.o.m momentum, and oscillations of the string. Recall: L 0 = 1 α 0 + n=1 α n α n l s 8 p + N L (.103) and similarly L 0 = l S 8 p + N R. Then, for a physical state with momentum k µ, (.10) becomes { } l S 4 k + N L + N R a phys = 0. (.104) So we get the mass formula M = 4 (N ls L + N R ) 8a ls (.105) where N L = n=1 αµ nα µn, N R = n=1 αµ n α µn are called the level numbers. For a physical state, level matching condition (.9) requires N L = N R. The interpretation is that different excitations of string will correspond to different sorts of particles in the target space, with the rest mass given by (.105). So far we have carried along two parameters D, a, and we have not checked that the constraints (.89) really 3

24 remove the unphysical states with negative squared norm, called the ghosts, from the spectrum. The proof of this no-ghost theorem is technical, and I will skip it. The idea is to check that ghosts have a vanishing overlap with any physical state: phys ghosts = 0. Then they form a subspace of the Fock space which is orthogonal to that of physical states: H = {(α µ 1 1) i1 (α µ n n) i n ( α ν 1 1) j1 ( α ν m m ) j m 0, k µ } = { phys } { ghost } (.106) and hence can be ignored. However, this is possible only when D = 6 and a = 1 or D 5 and a < 1. In the latter case the ghosts will show up again at one-loop level as unphysical poles in string scattering amplitudes. Therefore we will focus on the first case, critical string theory. The latter case is called the non-critical (bosonic) string theory. In our case, the string then propagates in 6 spacetime dimensions!.8 Low-lying string states. With a = 1, the mass formula (.105) becomes M = 4 (N ls L + N R ) 8. Let us check ls the lowest mass states in the spectrum. Recall that the level matching condition requires N L = N R. 1)N L = N R = 0 : This is just the vacuum 0, k µ with M = k = 8/lS. This was required for the vacuum to be a physical state. Thus, from the target space point of view, a string which is not oscillating corresponds to a particle with a negative rest mass. It is called the tachyon. ) N L = N R = 1 : This is the first excited level, α µ 1 α ν 1 0, k µ (.107) with M = 4 (1 + 1) 8 = 0. Any D D matrix A µν = α µ ls ls 1 α 1 ν can be decomposed into three parts as follows: A µν = 1 (Aµν + A νµ ) tra D ηνµ }{{} symmetric traceless G µν + 1 (Aµν A νµ ) }{{} antisymmetric traceless B µν + tra D ηµν trace (.108) These massless particles correspond to the graviton, the antisymmetric tensor, and the dilaton. 4

25 3) N L = N R = : α µ 1 1α µ 1 α ν 1 1 α ν 1 0, k µ α µ α ν 0, k µ (.109) These have M = 8. They correspond to very massive particles (M m ls P l ). Polarizations: Let us consider the effect of the constraints. Consider again the massless level N L = N R = 1. We introduce a polarization tensor ɛ µν (k) and write the states as ɛ µν (k)α µ 1 1α µ 1 0, k µ. (.110) [Compare with QED: quantize the U(1) gauge field A µ and introduce a polarization vector ɛ µ (k) for the photon state ɛ µ (k)a µ k 0.] The momentum vector satisfies The constraint implies: k = M = 0. (.111) L 1 ɛ µν α µ 1 α ν 1 0, k µ = 0 (.11) 1 {α 1α + α 0 α 1 + α 1 α 0 + α α 1 + }ɛα 1 α 1 0 = ɛ µν η γδ α γ 1α δ 0α µ 1 α ν 1 0, k µ = l s ɛ µνη γδ k δ α ν 1α γ 1α µ 1 0, k µ = l s ɛ µνk µ α ν 1 0, k µ = 0, (.113) where α γ 1α µ 1 = [α γ 1, α µ 1] = η γµ. So the polarization tensor and the momentum satisfy [I suppressed the ν index on the RHS.] Similarly, L 1 ɛ µν α µ 1 α ν 1 0, k µ = 0 yields k ɛ k µ ɛ µν = 0. (.114) ɛ k = ɛ µν k ν = 0. (.115) Let us choose a frame where (k µ ) = (k 0, k 1, 0,..., 0). Then (.111) requires k 0 = k 1 k. Suppose that (.111) corresponds to a graviton, then ɛ µν must be symmetric and traceless. Then, (.114) and (.115) mean that the polarization tensor can be reduced into the form (after decoupling the longitudinal polarizations ɛ µν k µ k ν ) (ɛ µν ) = 0 0 ε ii ε ij (.116) 0 0 ε ji ε jj 5

26 where ε is a symmetric, traceless (D ) (D ) matrix. This is a higher dimensional analogue of the transverse and traceless gauge for the graviton in four spacetime dimensions. The polarization can be chosen to be transverse to the direction of propagation. The number of true physical degrees of freedom is manifestly (D )(D 1) 1. [The QED analogue is, a photon has momentum k µ = (k 0, k 1, 0,..., 0). The constraint µ A µ = 0 gives the condition e µ k µ = 0 for the polarization vector. So the photon has only two physical degrees of freedom, corresponding to the two transverse polarizations.] So, we have seen that the constraints do reduce the number of degrees of freedom to the real physical one. Now we will discuss an alternative to the covariant quantization. We first solve the constraints and find the real physical degrees of freedom, and quantize only them. ***** END OF LECTURE 3 *****.9 The light-cone gauge Recall that the covariant gauge h αβ = η αβ still allowed the freedom of residual gauge transformations (.38), which were combinations of conformal transformations (.36) and Weyl transformations (.37). Such transformation σ ± σ ± (σ ± ) (.117) means in a Cartesian frame τ = ( σ + + σ )/, σ = ( σ + σ )/ that the new time is a superposition of an arbitrary function of σ + and an arbitrary function of σ : Hence it satisfies the dimensional wave equation τ = f(σ + ) + g(σ ). (.118) ( τ σ) τ = 0. (.119) Since all X µ satisfy the same wave equation, we could pick one of them and set it to be equal to τ by a suitable transformation (.117). It would be natural to pick X 0 : X 0 = τ. (.10) This choice is sometimes used and known as the static gauge. However, for our purposes there is a more useful pick. Define the light cone coordinates X ± for the string, X ± 1 (X 0 ± X D 1 ). (.11) 6

27 Since X ± also satisfy the wave equation, we can pick e.g. X + to be proportional to τ (I drop the tilde now) X + (τ, σ) = x πt p+ τ, (.1) where we introduced the constants x +, p + for convenience. This choice is called the light-cone gauge. Recall the constraints (.54): T 00 = T 11 = 1 ( τx µ τ X µ + σ X µ σ X µ ) = 0 (.13) T 01 = T 10 = τ X µ σ X µ = 0. (.14) In light-cone coordinates X µ Y µ = X + Y X Y + + X i Y i. The equation (.1) becomes 1 πt p+ τ X = τ X + τ X + σ X }{{ + } σ X = 1 {( τ X i ) + ( σ X i ) } (.15) and (.13) becomes =0 i 1 πt p+ σ X = τ X + σ X + τ X σ X }{{ + = } =0 i τ X i σ X i. (.16) Thus, if we substitute the mode solution for X i : X i (τ, σ) = x i + l s pi τ + i l s { 1 n αi ne in(σ ) + 1 n αi ne in(σ+) }, n 0 we can use (.15) and (.16) to solve for X in terms of α i ns, p i, p + and an integration constant x. So, in the light-cone gauge (.1) we can solve the constraints, and find that the real degrees of freedom are the X i (or the α i n, α i n, x i, p i )! Now recall that previously we would have written X as X = x + l s p τ +, (.17) but we can use (.15) again to solve for p. Again, commuting α ns i to the left of αns i introduces a zero point contribution a. We find p + p = [ ] (p i ) + 4 (α l nα i n i + α n i α n) i a. (.18) i s n=1 Then, the mass shell condition M = p = p + p p i p i becomes i 7

28 [ M = 4 ls i ] (α nα i n i + α n i α n) i a. n=1 (.19) Since we have solved the constraints, the Fock space now contains only physical states: H LC = {(α i 1 1 ) P1 (α i n n ) P n ( α i 1 1 ) q1 ( α jn n) q n 0, k µ >}. (.130) Since we have already solved the constraints, now we need a different condition to decide what are the allowed values of a and D. In solving the constraints we paid the following price. In the light cone gauge we have lost the manifest D-dimensional Lorentz invariance of the target space. If we construct the generators for D-dimensional target space Lorentz transformations and demand that they do satisfy the correct commutation relations of SO(1, D) (the D-dim. Lorentz algebra), we find that this only works for D = 6, a = 1. (.131) In this case one can also check that the previous space of physical states is the same as the state space in the light-cone gauge, H phys = { phys covariant gauge } = H LC..10 Lowest lying states In the light-cone gauge, the lowest lying string excitations are again 1) N L = N R = 0: 0, k µ, tachyon ) N L = N R = 1: { 1 (αi 1 α j 1 + α j 1 α 1) i δ klα 1 k α 1 l } 0, k µ graviton (.13) D 1 (αi 1 α j 1 α j 1 α 1) 0, i k µ antisymm.tensor (.133) δ kl α 1 k α 1 l δ ij 0, k µ dilaton. (.134) D Now the graviton has explicitly (D )(D 1) 1 degrees of freedom. 8

29 .11 Open strings So far we have focused on the closed string. For open strings we can proceed in a similar fashion. The difference is that the boundary conditions glue the left- and rightmoving waves together to standing waves. Thus, in the covariant gauge X µ = x µ + l sp µ τ + il s n 0 1 n αµ ne inτ cos(nτ). The quantization again yields [p µ, x ν ] = iη µν [α µ m, α ν n] = mη µν δ m+n,0. (.135) There is only one set of oscillator coefficients α n. µ The Virasoro generators L n are defined by L n = T For n 0 they are π 0 dσ{e in(τ+σ) T ++ + e in(τ σ) T }. (.136) L n = 1 α µ n mα µm (.137) m= where we have now defined α µ 0 = l s p µ. (.138) Note that (.138) differs by a factor of from the definition in the closed string case. The L 0 generator is again defined to be normal ordered: L 0 = 1 : α mα µ µm := 1 αµ 0α µ0 + α mα µ µm. m= m= 9

30 The Hamiltonian is H = L 0 a. (.139) and the physical state conditions are L m phys = 0 m > 0 (.140a) (L 0 a) phys = 0. (.140b) Now there is no level matching condition. The no-ghost theorem again requires D = 6, a = 1. The mass shell condition from (.140a) becomes M = l S n=1 α nα µ µn a ls (.141) The Virasoro generators (.137), (.139) satisfy the same Virasoro algebra commutation relations as in (.99). The lowest lying states are 1) N = 0: the vacuum 0, k µ with M =. So it is again a tachyon. ls ) N = 1: the first excited states e µ α 1 0, µ k µ are massless: M = 0. This corresponds to a massless gauge particle. The physical state condition L 1 = 0 gives the polarization condition e k = 0. So the physical degrees of freedom are (D ) transverse polarizations, just like in QED. In the light cone gauge M = l S ( ) α nα ı n i a n=1 and the real degrees of freedom are explicit. i (.14).1 Path integral quantization of the bosonic string So far we have been discussing the old fashioned canonical quantization approach. A more modern approach to quantize is via a path integral. In QFT, the idea is to account for quantum fluctuations by considering all possible field configurations 30

31 (not only solutions of classical equations of motion) and weighting them by their contribution to the action. The sum is the path integral, Z = Dφe i S(φ). (.143) In string theory we sum over all possible worldsheets and their embeddings, so integral runs over all worldsheet metrics h αβ and all embeddings X µ : Z = DhDXe is P (h,x), (.144) where S P is the Polyakov action. The path integral gives vacuum-to-vacuum amplitude, so in closed string theory that means that we are summing over all possible two dimensional Riemann surfaces Figure 6: Two-dimensional surfaces. Figure 6 depicts that surfaces are not only deformed in shape, but one can also add holes. The latter correspond to loop corrections. Note that in contrast to field theory, the action is still a free theory. So the string interactions are introduced by different surfaces, instead of adding nonlinear terms to the action! It turns out that there are some important terms we can add into the Polyakov action, if we are considering strings moving in more generic backgrounds than just an empty flat Minkowski space. One important addition is with (in the Euclidean signature 8 ) S = S P + λχ (.145) χ = 1 4π d σ hr, (.146) 8 As usual, the path integral is best defined in the Euclidean signature and one then has to continue back to Minkowski signature. 31

32 where R is the Ricci scalar curvature of the worldsheet metric. Note that χ is a two-dimensional version of the Einstein-Hilbert action. However, in two spacetime dimensions χ does not really depend on the metric it only depends on the topology of the surface. The quantity χ is a topological invariant, the Euler number, equal to (1 g) (for manifolds without boundaries) where g is the number of holes or handles (0 for sphere, 1 for torus, etc.). The factor λ looks like an arbitrary parameter. Yet I said that in string theory theres only one, the string tension or length. In fact λ depends on the background and is thought to be set dynamically. I will get back to this later. The importance of the term λχ is that (in Euclidean continuation), the path integral has the factor e S P λχ = e S P e λ e gλ. (.147) Thus, increasing the genus g by one, by adding a handle, the path integral picks up an additional factor of e λ. Now think of the addition of a handle as a sequence of closed string interactions a closed string is first emitted, then propagates along the handle, and then is reabsorbed. The emission and absorption should be characterized by the strength of the closed string interactions. In other words, they should be associated with one power of a closed string coupling constant g c. Thus, adding a handle corresponds to adding two powers of string coupling constant, and we are thus lead to identify g c e λ. (.148) For open strings the path integral is defined in a similar manner, only now the surfaces must have a different topology. Recall that the (tree level) open string world sheet looked like a strip of width π, so it looks like there are two boundaries associated with the two open strin endpoints. However, taking into account the point at infinity, the infinite strip has only one boundary and it can be conformally mapped to the unit disk on a complex plane. For an open string, adding loops in the path integral then corresponds to adding holes into surfaces that are topologically like the unit disk. In other words, adding loops corresponds to adding boundaries. For example, at one loop level the surfaces are topologically equivalent to the annulus, which has two boundaries. A more general formula for the Euler number, which also counts boundaries, is χ = g b c. (.149) Here b counts the number of boundaries, eg. b = for the annulus, and c counts something called crosscaps, you can forget that for now and consider c = 0. Now you can see that adding a boundary to the open worldsheet introduces a factor e λ. Thinking of this as an emission and reabsorption of an open string, we are lead to identify g o = e λ, (.150) 3

33 where g o is the open string coupling constant. Alltogether then g o = g c = e λ. (.151) Gauge fixing. Recall that the action S P was invariant under worldsheet reparameterizations (diffeomorphisms) and Weyl transformations. Denote the group of such transformations by Diff Weyl. Then the integral of surfaces induces a huge overcounting: all surfaces (or h αβ and X µ s) related by Diff Weyl are counted redundantly, since they all contribute equally (to S P ). This is just like in gauge field theory: if the action is invariant under a gauge group G (say, SU() of a non-abelian field A a µ), all gauge field configurations A a µ which are equivalent by gauge transformations contribute equally. So we have to remove the overcounting from the path integral. The standard way is to use the Faddeev-Popov method. Let ξ symbolize a Diff Weyl transformation: h h ξ : h ξ σγ σ δ αβ ( σ) = eρ(σ) σ α σ h γδ(σ). (.15) β In particular, h αβ could be the worldsheet metric in the covariant gauge: (η αβ ) = diag( 1, 1) and h ξ αβ anything else. The F-P trick is to insert a special 1 into the path integral: ( ) δf (h 1 = dξδ[f (h ξ ξ ) )] det ξ=0 (.153) δξ where F (h ξ ) h ξ αβ h αβ. (.154) Let me denote det( δf (hξ ) ) δξ ξ=0 (h). Insert (.153) into the path integral (.143): Z = dξ DhDXδ(h ξ h) (h)e is(x,h) R Dh = dξ DX (h ξ )e is(x,hξ ) (.155) Since the action is invariant under Diff Weyl, S(X, h ξ ) = S(X, η). One can show that also (h ξ ) is invariant, so Z = (h ξ ) = (h ξ ) = (η) dξ DX (η)e is(x,η) = V ol(diff Weyl) }{{} R dξ DX (η)e is(x,η). (.156a) (.156b) 33

34 The integral dξ over Diff Weyl transformations was trivial and just gave the (infinite) volume of the group. It can be dropped from the path integral by a normalization convention. Now we only need to evaluate the Jacobian (η). I will simplify this by taking a step back. Let us not fix the Weyl transformations: we only fix to the covariant gauge h (c) αβ = eρ η αβ. That means, we use S(X, h ξ ) = S(X, h (c) ) and (h ξ ) = (h (c) ). then instead of (.156b), Z = V ol(diff) }{{} Drop Dρ DX (h (c) )e is(x,h(c)). (.157) It is simpler to evaluate (h (c) ). Under an infinitesimal Diff transformation: the metric transforms: σ α σ α + ξ α, (.158) h αβ h ξ αβ = h αβ + δh αβ, (.159a) where δh αβ = α ξ β β ξ α (.159b) (See e.g. Nakahara for the mathematics or some gravity textbook.) In particular, in the worldsheet light-cone coordinates, δh ξ ++ = + ξ + (.160a) δh ξ = ξ. (.160b) So where det( δf δξ ) = det(δf ++ ) det( δf ), (.161) δξ + δξ det( δf ±±(τ, σ ) δξ ± (τ, σ) ) = det( ± δ(τ τ)(σ σ)). (.16) The determinant can be exponentiated into the action by introducing anticommuting fields b, c callled Faddeev-Popov ghosts: { } i det(b) = DcDb exp dτ dσ dτdσ c(τ, σ )B(τ, σ, τ, σ)b(τ, σ). (.163) π So the path integral (??) can be written in the form Z = DρDXDc + Db ++ Dc Db e i[s P (h c,x)+s gh (b,c)], (.164) 34

35 where S gh is ghost action S gh = 1 π d σ(c + b + c + b ++ ). (.165) The conventional labeling (c, b etc.) for the ghost fields may look a bit strange, it can be motivated by looking at their properties under conformal transformations. Note: Since S P is invariant under Weyl transformations, the covariant derivatives in S gh reduce to ordinary derivatives, so it appears to be independent of ρ: S gh = 1 d σ(c + b + c + b ++ ). (.166) π So the whole path integral looks independent of ρ. That would mean that Weyl invariance is a symmetry at quantum as well as classical level 9. However, a more careful investigation of the measure Dρ shows that to be case only in D = 6. For D 6 Weyl invariance is broken at quantum level ( Weyl anomaly ) and ρ reappears in the action. It is then called the Liouville field. D 6 is the noncritical bosonic string..13 Conformal field theory (CFT) In the discussion of the path integral, we mentioned the string interactions for the first time. Before proceeding to discuss string interactions in more detail, it is useful to go through some other issues which may seem slightly abstract at first. We need to discuss some generic features of conformally invariant -dimensional field theories. Conformal symmetry is a rather powerful feature. It does not appear only in string theory, but it is also encountered in statistical mechanics and in some condensed matter systems. For example, in statistical mechanics it arises in systems which have a second order phase transition, at the critical point. The reason why conformal invariance is so important in string theory is that it is a gauge symmetry. Often in gauge theories, gauge transformations are used to eliminate unphysical degrees of freedom (like the timelike photon in QED). Suppose that the gauge invariance is then broken at quantum level. Then the unphysical degrees of freedom may return and spoil the theory. So in general we like to preserve gauge symmetries at quantum level too. Recall that string theory had reparametrization and Weyl invariance as local (gauge) symmetries. Going to covariant gauge did not completely fix the gauge symmetry. We still had the freedom to make conformal transformations combined with Weyl transformations. We used this residual gauge freedom to go to the lightcone gauge, where we eliminated all unphysical degrees of freedom and completely fixed the gauge. Thus, conformal symmetry played a central role in eliminating the unphysical degrees of freedom. Now suppose that the conformal symmetry is broken 9 And it looks like we could do the remaining Dρ functional integral in Z to pick up V ol(weyl). 35