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Conformal Field Theory (For String Theorists)

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Conformal Field Theory (For String Theorists) YITP-SB-15-?? Conformal Field Theory (for string theorists) Christopher P. Herzog C. N. Yang Institute for Theoretical Physics, Department of Physics and Astronomy Stony Brook University, Stony Brook, NY 11794 Abstract A write up of about ten lectures on conformal field theory given as part of a first semester course on string theory. Contents 1 Opening Remarks 1 2 Conformal Transformations in One and Two Dimensions are Special 3 3 Correlation Functions are Highly Constrained by Conformal Symmetry 4 4 Noether's Theorem 6 5 Conformal Anomaly 8 6 Path Integral Approach 15 7 BRST meets CFT 19 8 From Operators to States: The Vacuum 23 8.1 Bosonization . 27 8.2 R Sector Fermions . 28 8.3 The βγ System . 30 9 From Operators to States: Virasoro and Super Virasoro 32 10 Thermal Partition Function 35 A Bosonization and Cocycles 36 1 Opening Remarks To date in this class, string theory boils down to the study four free (quadratic) quantum field theories: one for the X fields, one for the fields, one for the bc ghost system, and one for the βγ ghost system. We saw a BRST action that coupled the X and fields to world-sheet supergravity, and hence required the presence of the bc and βγ ghosts in addition to some auxiliary fields d and ∆ and also ghosts for the Weyl and super-Weyl symmetry. After some elementary path-integral manipulations, these extra fields dropped out, and we were left with simple, quadratic actions for the remaining X, , bc and βγ fields on a flat world sheet hab = ηab. The full quantum world-sheet supergravity action had a number of symmetries which are no longer evident in the gauge fixed action for X, , bc and βγ. Among other symmetries, the full quantum action had world-sheet diffeomorphism invariance, σ ! σ0(σ). Under diffeomorphisms, the metric changes in the usual way a b 0 0 @σ @σ h (σ ) = hab(σ) : (1) cd @σ0c @σ0d 1 Another symmetry was world-sheet Weyl invariance, hab ! Λ(x)hab. There were then corresponding rules for how the fields X, , bc, and βγ transform under diffeomorphisms and Weyl scaling. We gauge fixed by choosing a flat world-sheet metric hab = ηab. However, this gauge fixing is not complete. There are residual gauge transformations that are a combination of a diffeomorphism and a Weyl scaling that leave the metric ηab invariant. These residual gauge transformations are called conformal transformations: Definition. A conformal transformation is a map on coordinates σ ! σ0 that preserves the metric up to a scale factor 0 hab(σ) = Λ(σ)hab(σ) : Example. In the case when hab = ηab, two conformal transformations are • Elements of the Poincar´egroup (Lorentz group and translations) for which Λ = 1. • Dilations x ! λx, λ 2 R, for which Λ = λ2. Note in the Euclidean case, hab = δab, the Lorentz group is replaced by rotations. Both rota- tions and dilations manifestly preserve the angles between vectors, motivating the choice of word \conformal", which means preserving angles. Remark. The set of conformal transformations C forms a group when the transformation σ ! σ0 is invertible. Definition. A conformal field theory is a quantum field theory which has C as a classical symmetry of the action. Almost all the quantum field theories we study, when coupled to gravity, will be diffeomorphism invariant. The litmus test for figuring out when a quantum field theory in a fixed background space- time is a conformal field theory is then the presence of local Weyl invariance. Perhaps as a result, in the literature there is a certain carelessness and interchanging in the use of the words Weyl scaling and conformal transformation. We will try to be careful here. Having fixed hab = ηab, the field theories for X, , bc, and βγ become examples of conformal field theories. In fact, we will eventually see they are essentially all the same conformal field theory, just expressed in different variables. We can therefore use the extensive and highly developed machinery of conformal field theory to systematize our understanding of these four systems. The goal of these lectures will be four-fold: 1. To replace the cumbersome oscillator algebra manipulations with (in our view) more elegant operator product expansions. 2. To streamline calculations involving the BRST symmetry. 3. To understand how a quantum anomaly in the classical conformal symmetry restricts the types of consistent string theories. 4. To set up machinery for string scattering calculations. 2 References These lecture notes draw largely from chapters 2, 3, 6, 8, and 10 of Polchinski's classic string theory text book [1]. I have also drawn on early chapters in Di Francesco, S´en´echal, and Mathieu's classic work on conformal field theory [2] and P. van Nieuwenhuizen's unpublished string theory lecture notes [3]. Another nice publicly available reference I found are unpublished notes by M. Kreuzer [4]. 2 Conformal Transformations in One and Two Dimensions are Special 0 @x @x In one dimension, any diffeomorphism y(x) is conformal with gyy = @y @y gxx. 1 In two dimensions, for convenience, consider the Euclidean case hab = δab. We take advantage of complex numbers: z = σ1 + iσ2 ; z¯ = σ1 − iσ2 ; (2) 1 1 @ ≡ @ = (@ − i@ ) ; @¯ ≡ @ = (@ + i@ ) : (3) z 2 1 2 z¯ 2 1 2 The world sheet metric then has components 1 h = h = ; h = h = 0 : (4) zz¯ zz¯ 2 zz z¯z¯ In complex coordinates, any holomorphic transformation z ! w(z) along with its anti-holomorphic counterpartz ¯ ! w¯(¯z) is conformal: @z @z¯ @z @z¯ h0 (w; w¯) = h (z; z¯) where Λ = : (5) ww¯ @w @w¯ zz¯ @w @w¯ In more than two dimensions, the set of conformal transformations is far smaller. It is generated by • translations: xµ ! xµ + aµ. • dilations: xµ ! axµ. µ µ ν • rigid rotations: x ! M ν x . • special conformal transformations: xµ − bµx2 xµ ! : 1 − 2b · x − b2x2 In d Euclidean dimensions, these transformations generate a connected part of the Lorentz group SO+(1; d + 1). This group forms an important subgroup of the conformal transformations in d = 2, where it is isomorphic to the set of Moebius transformations on the complex plane, SO+(1; 3) = 1To return to the Lorentzian case, one can make the Wick rotation σ0 = −iσ2. 3 P SL(2; C). In particular, translations, dilations, rotations, and special conformal transformations on the plane combine to give the transformation rule az + b z ! ; (6) cz + d where a, b, c, and d 2 C. Without further conditions on a, b, c, and d, this map would be in GL(2; C). However, as multiplying a, b, c, and d by an overall scale factor does not change the transformation rule, we are free to set ad − bc = 1 and restrict to SL(2; C). Furthermore, the map is invariant under the sign flip (a; b; c; d) ! (−a; −b; −c; −d), which restricts the group to P SL(2; C). We can also consider the corresponding Lie algebra sl(2; C) for P SL(2; C). This Lie algebra has the generators conventionally labeled 2 L−1 = @z ;L0 = z@z ;L1 = z @z : (7) (There is another copy of sl(2; C) generated by complex conjugates of L0, L−1, and L1.) The operator L−1 generates a translation, L0 a combination of dilation and rotation, and L1 a special conformal transformation. These generators satisfy the standard sl(2; C) Lie algebra [L0;L−1] = −L−1 ; [L0;L1] = L1 ; [L−1;L1] = 2L0 : (8) (In quantum mechanics, we might make the replacements L0 ! Jz, L−1 ! J−, and L1 ! J+.) An infinite dimensional representation of this algebra is furnished by the monomials zn where n n n n±1 L0z = nz ;L±1z = nz : (9) At first sight there is something a bit odd about this representation; under what inner product do the eigenvectors zn have finite norm and is there a notion of Hermiticity? To obtain an inner product, we make the transformation z = e−it. Under this transformation, we find the new generators −it it L−1 = −ie @t ;L0 = −i@t ;L1 = −ie @t : (10) There is then an obvious inner product based on the orthogonality of Fourier modes on the circle, Z 2π int y imt (e ) (e )dt = 2πδn;m ; (11) 0 and under which L0 is now clearly Hermitian. Back in the z coordinate, interestingly, this inner product corresponds to a contour integral along the curve jzj = 1. This so-called plane to cylinder map z = eit along with corresponding contour integrals will play a key role as we go forward. 3 Correlation Functions are Highly Constrained by Confor- mal Symmetry The transformation properties of fields fix two and also three point functions up to some undeter- mined constants. Previously in the class, we saw examples of how X and transform infinitesimally 4 under such conformal transformations. The finite versions of those rules are as follows: @w @ X0(z; z¯) = @ X(w; w¯) ; (12) z @z w @w 1=2 0(z) = (w) : (13) @z The fields @X and are examples of primary fields. More generally we have the definition: Definition. For any meromorphic map z ! w(z), a primary field satisfies the transformation rule ¯ @w −h @w¯ −h φ0(w; w¯) = φ(z; z¯) : (14) @z @z¯ The quantity h is called the holomorphic conformal dimension, h¯ the anti-holomorphic conformal dimension. The quantities h + h¯ = ∆ are the conformal (or scaling) dimension and h − h¯ = s the spin. Applying this definition to our two examples, we find that h = 1 and h¯ = 0 for @X while h = 1=2 and h¯ = 0 for .
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