Spinning strings and QED

Transcription

1 Spinning strings and QED James Edwards Oxford Particles and Fields Seminar January 2015 Based on arxiv: [hep-th] and arxiv: [hep-th]

2 Outline

3 Introduction Various relationships between string theory and field theory have long been active areas of research in high energy physics. 1. Flux tubes in QCD 2. String scattering amplitudes 3. Nambu [1] and Polyakov loops [2]. 4. Bern-Kosower rules [3]. 5. AdS / CFT... 1 Phys Lett B 80 2 Nucl Phys B Arχiv: v2 (Review)

4 Introduction Various relationships between string theory and field theory have long been active areas of research in high energy physics. 1. Flux tubes in QCD 2. String scattering amplitudes 3. Nambu [1] and Polyakov loops [2]. 4. Bern-Kosower rules [3]. 5. AdS / CFT... The work I shall present takes a complementary approach - a theory of interacting tensionless spinning strings provides the expectation value of a product of Wilson loops in spinor QED. ( ) W [A] := exp A dx (1) C 1 Phys Lett B 80 2 Nucl Phys B Arχiv: v2 (Review)

5 Motivation It has been shown that the classical field strength tensor of Maxwell electrodynamics can be determined from a string theory perspective [4] : F c µν (x) = 4π 2 dσ µν (X) δ 4 (x X) (2) This describes the functional average of an operator over the configurations of a string bounded by the worldline of a particle / anti-particle pair. It has some remarkable properties: Σ 4 Mansfield: Arχiv: v2

6 Motivation It has been shown that the classical field strength tensor of Maxwell electrodynamics can be determined from a string theory perspective [4] : F c µν (x) = 4π 2 dσ µν (X) δ 4 (x X) (2) This describes the functional average of an operator over the configurations of a string bounded by the worldline of a particle / anti-particle pair. It has some remarkable properties: The string theory is off-shell and not in the expected critical dimension. Vertex operators are integrated over the entire worldsheet. The key to understanding how this is possible is the geometry of functional quantisation. Σ 4 Mansfield: Arχiv: v2

7 The Weyl anomaly Polyakov s action involves the scalar fields X and intrinsic metric g: S Poly [X, g] = 1 2πα d 2 ξ gg ab a X µ b X ν G µν (3) where Σ is the worldsheet domain and X is a mapping from Σ to a space-time with metric G. It is invariant under (worldsheet) diffeomorphisms and conformal transformations. Σ Path integral quantisation requires functionally integrating over all embeddings and geometries Z = D (X, g) e S Poly[X, g]. (4) We must give a meaning to these formal integration measures and account for the overcounting due to the Diff Weyl invariance.

8 Path integral measures 5 If the space of metrics can be parameterised by local coordinates ζ i we define ( g Dg =, g ) dζ k (5) ζ i ζ j k where (, ) denotes an inner product on variations in the metric that must be chosen. An arbitrary variation in the metric can be written as δg ab = δφg ab + (a δv b) + δl A g ab l A (6) representing a change due to a Weyl scaling, reparameterisation and shift in the moduli of the surface respectively. Polyakov preserved diffeomorphism invariance with the inner product (δ 1 g, δ 2 g) = d 2 ξ ( gδ 1 g ab δ 2 g rs Ag ar g bs + Bg ab g rs) (7) at the expense of explicit Weyl invariance. 5 See Physics of Atomic Nuclei 73 5 (5), 2010 for an excellent summary

9 Path integral measures If the space of metrics can be parameterised by local coordinates ζ i we define ( g Dg =, g ) dζ k (8) ζ i ζ j k With this inner product the measure on genus zero surfaces factorises as Dg = DV Dφ det P P (9) The operator P maps vectors to traceless symmetric tensors and contains information about the geometry of the worldsheet: P ab (V ) = (a V b) g ab c V c (10)

10 Path integral measures Further subtleties to take care of: 1. P has zero modes the conformal killing vectors 2. P has zero modes these correspond to moduli 3. The determinant requires regularisation this introduces a scale Dg Dφ det P P Vol (CKV) (11) There s a similar story with DX. Integrating over the matter fields leads to a functional determinant which must be regulated. The combined effect is to generate ( D 26 Z = Z 0 Dφ exp d 2 ξ gr (ξ) G (ξ, ξ ) ) gr (ξ ) d 2 ξ 48π Σ Σ (12) where Z 0 is independent of φ.

11 The Liouville action For a metric that is conformally related to a reference metric ĝ the φ dependent term becomes S L = d 2 ξ ( ĝ φ ˆ φ + µ 2 e φ + 2φ ˆR ) (13) Σ

12 The Liouville action For a metric that is conformally related to a reference metric ĝ the φ dependent term becomes S L = d 2 ξ ( ĝ φ ˆ φ + µ 2 e φ + 2φ ˆR ) (13) Σ The problem is in the measure Dφ. It is induced by the earlier choice of inner product: (δφ, δφ) = d 2 ξ ĝ e φ (δφ) 2 (14) Nobody knows quite how to interpret this so it is unclear how to carry out the integral over φ.

13 Summary of Weyl anomaly For quantisation we had to introduce extra structure into the theory. Preserving diffeomorphism invariance cost us a conformal anomaly. If we can make the Liouville action disappear then we don t need to worry about this anomaly and we are free to calculate quantities in conformal gauge where there is no dependence on φ. This condition imposes constraints Critical dimension: D = 26 (or D = 10 for the spinning string). Seen already for the partition function

14 Summary of Weyl anomaly For quantisation we had to introduce extra structure into the theory. Preserving diffeomorphism invariance cost us a conformal anomaly. If we can make the Liouville action disappear then we don t need to worry about this anomaly and we are free to calculate quantities in conformal gauge where there is no dependence on φ. This condition imposes constraints Critical dimension: D = 26 (or D = 10 for the spinning string). Seen already for the partition function Mass shell conditions: k 2 = 0 for photon and graviton states. Required for scattering amplitudes to not contain further φ dependence.

15 Summary of Weyl anomaly For quantisation we had to introduce extra structure into the theory. Preserving diffeomorphism invariance cost us a conformal anomaly. If we can make the Liouville action disappear then we don t need to worry about this anomaly and we are free to calculate quantities in conformal gauge where there is no dependence on φ. This condition imposes constraints Critical dimension: D = 26 (or D = 10 for the spinning string). Seen already for the partition function Mass shell conditions: k 2 = 0 for photon and graviton states. Required for scattering amplitudes to not contain further φ dependence. Transversality conditions: ɛ µ k µ = 0 for photon and ζ µν k µ = for gravitons. Also required to avoid unwanted φ dependence.

16 From worldlines to strings Reminder of the stringy expression for the classical field strength tensor for the worldlines of oppositely charged particles F c µν (x) = 4π 2 dσ µν (X) δ 4 (x X) (15) Σ Remarkably the dependence on the conformal scale of the worldsheet metric decouples from the physical information in the expectation value despite the appearance of states with arbitrary momenta throughout the calculation.

17 The main results Take a set of curves {w i } and introduce bosonic strings whose endpoints are fixed to these curves. The strings interact via the action S = i Goal: S Poly [X i, g i ] + ij q 2 4 dσ µν i (X i ) δ 4 (X i X j ) dσ µν j (X j ) We proposed that the partition function of the string theory coincides with the expectation value of a product of Wilson loops (16)

18 The main results Take a set of curves {w i } and introduce bosonic strings whose endpoints are fixed to these curves. The strings interact via the action S = i Goal: S Poly [X i, g i ] + ij q 2 4 dσ µν i (X i ) δ 4 (X i X j ) dσ µν j (X j ) We proposed that the partition function of the string theory coincides with the expectation value of a product of Wilson loops (16) N i=1 D(X i, g i ) DA e S = Z 0 N e S gf i e i dw i A (17)

19 Spinor matter For spinor QED we deal with the super-wilson loop W [A] = dw A + 1 dξ ψ µ F µν ψ ν (18) 2 We generalise to the spinning string with gauge fixed action S = 1 ( ) 4πα d 2 zd 2 θ DX µ DX µ dx Ψ Ψ y=0 (19) where D = θ + θ z, D = z + θ z and X is the superfield X µ = X µ + θψ µ + θ Ψ µ ( +θ θb µ ) (20)

20 From strings to fields To reformulate the field theory we generalise the interaction and impose boundary conditions: The supersymmetric generalisation of the interaction term is q 2 ( ) d 2 θ i d 2 z i Di X [µ i D ix ν] i dx i θ i θi Ψ[µ i Ψν] i δ 4 (X i X j ) y ( i=0 ) d 2 θ j d 2 z j Dj X [µ j D jx ν] j dx j θ j θj Ψ[µ j Ψν] j (21) y j=0 We fix the worldsheet to the boundary by generalising the previous Dirichlet boundary conditions X µ y=0 = w µ, ( Ψ µ + Ψ µ) y=0 = ψ µ. (22)

21 Vertex operators To determine the partition function we proceed by pertubatively expanding the interaction term. We find insertions of vertex operators inside the path integral: DX [µ DX ν] δ 4 (X X ) DX [µ DX ν] = d 4 k 1 (2π) 4 4 V µν (k) V µν ( k) V µν (k) = DX [µ DX ν] e ik X. (23) This seems to be inconsistent with the mass-shell condition required to avoid the Weyl anomaly. Divergences appear when these insertions approach one another regulate with a cut-off, ɛ, which breaks conformal invariance.

22 Results for the spinning string There are three important configurations of these insertions: When the insertions are close together in the bulk we find possible divergences: 1 ɛ ( ɛ d 2 z n+1 : e ik X(zn+1) : y 2 n+1 ) α K 2 /4 (24) and K µν ɛ ( d 2 z n+1 : Ψ ɛ µ Ψ ν e ik X(zn+1) : y 2 n+1 ) α K 2 /4 (25)

23 Results for the spinning string There are three important configurations of these insertions: When the insertions are close together in the bulk we find possible divergences: 1 ɛ ( ɛ d 2 z n+1 : e ik X(zn+1) : y 2 n+1 ) α K 2 /4 (24) and K µν ɛ ( d 2 z n+1 : Ψ ɛ µ Ψ ν e ik X(zn+1) : y 2 n+1 ) α K 2 /4 (25) Supersymmetry helps with the first one and direct calculation determined the second could not be present

24 Results for the spinning string There are three important configurations of these insertions: When the insertions are also close to the boundary the supersymmetry allows us to constrain the form of the result. There are two possible divergences 1 dx e ik X K ρ and dx ( Ψ + Ψ ) ρ e ik X (26) ɛ ɛ 1 4 There s also a possible finite piece invariant under the supersymmetry which does not generate the Wilson loop content we are looking for dx e ik X ( dx µ /dx + ik (Ψ + Ψ)(Ψ + Ψ) µ) (27)

25 Results for the spinning string There are three important configurations of these insertions: When the insertions are also close to the boundary the supersymmetry allows us to constrain the form of the result. There are two possible divergences 1 dx e ik X K ρ and dx ( Ψ + Ψ ) ρ e ik X (26) ɛ ɛ 1 4 There s also a possible finite piece invariant under the supersymmetry which does not generate the Wilson loop content we are looking for dx e ik X ( dx µ /dx + ik (Ψ + Ψ)(Ψ + Ψ) µ) (27) A generalised Gauss law comes to the rescue.

26 Results for the spinning string There are three important configurations of these insertions: When the insertions are close to the boundary and separated from one another we find the super-wilson loop we sought q 2N N j=1 B dx j dx j ( ) e ikj (wj w j) dwj k 2 + ik j ψ j ψ j dx j ( ) dw j ik j ψ jψ j dx j This is independent of the cut-off, ɛ, and the string tension, α. (28)

27 The classical action and the conformal scale The conformal scale of the worldsheet metric has decoupled from the calculation. String data appear only in a prefactor ( 1 exp 2πα S [X c] D 10 ) 48π S L [φ, χ]] (29) We deal with these in turn: The tensionless limit α removes the dependence on the classical action There are a number of ways to handle the Liouville theory Appeal to it cancelling out when we normalise against the free theory partition function Assume the existence of further internal degrees of freedom to take us into a critical theory

28 The final result The structure provided by worldsheet supersymmetry ensured that no divergences or finite corrections were encountered so in this case the partition function coincides with the expectation value of a product of super Wilson-loops: n D(g, X, w, ψ) j e S S B = j Z 0 n D(w, ψ) j DA N e S A S B W [A]. (30) This is the result of our work. j j

29 Non-Abelian gauge theory We ve taken spinor QED as the field theory we wish to reformulate. We dealt with the Liouville mode in a slightly unsatisfactory way. What could these internal degrees of freedom be? We can use them to provide the extra details required for a field theory with a non-abelian symmetry. The super Wilson loop now takes the form { ( W [A] = P exp dw A A τ A + 1 )} dξ ψ µ F A 2 µντ A ψ ν (31) There is an interesting method for dealing with group representations and chirality on worldlines [6] by introducing further degrees of freedom ϕ and ϕ. 6 Arχiv:

30 Non-Abelian gauge theory We therefore introduce new superfields Y and Ỹ and modify the interaction to q 2 d 2 θ i d 2 z i e Φ A DY i τabdỹ k i B D i X [µ i D ix ν] i δd (X i X j ) d 2 θ j d 2 z j Dj X [µ j D jx ν] j DY R j τ k RSDỸ S j e Φ (32) The equations of motion for Y and Ỹ are first order and get matched to boundary fields which impose the path ordering. The boundary contribution comes from the classical piece and provides factors of the form e φ/2 ϕ A τ k AB ϕb.

31 Non-Abelian gauge theory We stand to pick up other terms from the quantum fluctuations of Y and Ỹ perhaps they may even break the conformal invariance we ve worked so hard to preserve: DY A 1 τ k ABDỸ B 1 DY R 2 τ l RSDỸ S 2 = D 1 D 2 G 12 D 1 D2 G 12 τ m AA, (33) but this vanishes if the generators of the symmetry group are traceless. The take-home message is that it is possible to introduce further degrees of freedom on the worldsheet to deal with non-abelian gauge symmetries. Avoiding an anomaly and generating the correct expression for the Wilson lines imposes a constraint on the allowed gauge groups.

32 Conclusion We have presented a reformulation of spinor QED where the fundamental degrees of freedom generating the gauge interactions are tensionless spinning strings interacting on contact. This string theory is unusual in a number of ways 1. The string world-sheets correspond to the trajectories of lines of electric flux joined to charged particles. 2. It is off-shell and we have open string vertex operators integrated throughout the worldsheet. 3. The conformal scale decouples so there is no Weyl anomaly. We couldn t do this for the bosonic string so in this way our model prefers spinor matter. 4. The tensionless limit means that the string length-scale is large compared to the size of the Wilson loops. 5. The non-abelian generalisation requires new string degrees of freedom and provides constraints on the underlying gauge group.

33 Conclusion We have presented a reformulation of spinor QED where the fundamental degrees of freedom generating the gauge interactions are tensionless spinning strings interacting on contact. This string theory is unusual in a number of ways 1. The string world-sheets correspond to the trajectories of lines of electric flux joined to charged particles. 2. It is off-shell and we have open string vertex operators integrated throughout the worldsheet. 3. The conformal scale decouples so there is no Weyl anomaly. We couldn t do this for the bosonic string so in this way our model prefers spinor matter. 4. The tensionless limit means that the string length-scale is large compared to the size of the Wilson loops. 5. The non-abelian generalisation requires new string degrees of freedom and provides constraints on the underlying gauge group. Thank you for your attention.