First order string theory and the Kodaira-Spencer equations. Integrable Systems in Quantum Theory Lorentz Center, Leiden, December 2008

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1 First order string theory and the Kodaira-Spencer equations Losev, Zeitlin, A.M.; Phys.Lett. B633 (2006) ; hep-th/ Gamayun, Losev, A.M.; 2008 Integrable Systems in Quantum Theory Lorentz Center, Leiden, December 2008

2 No integrability (multidimensional equations!), but very simple quantum theory... Original motivations: serious problems (personal unsatisfaction) with string theory in nontrivial backgrounds Non-linear two-dimensional sigma-model Σ d2 z G µν (X) X µ X ν +... Dependence of quantization upon the choice of targetspace co-ordinates;

3 To study the first-order theory instead... Sigma-model: typical expansion around the point where G µν = δ µν, far from singular backgrounds; No off-shell formulation, (k 2 = 0 for the photons and gravitons).

4 One of alternatives: to consider the first-order action S 0 = 1 2πα with the OPE Σ d2 z(p i X i + pī Xī) (1) X i (z)p j (z ) α δ i j z z +... (2) Here i, j = 1,..., D/2 and ī, j = 1,..., D/2. Just a 2d analog of the Hamiltonian formalism!

5 More strictly - an analog of the Hamiltonian formalism, which requires some (local!) choice of the target-space complex structure. Allows functions G µν (X) as perturbations, in contrast to sigma-model, where they do not exist off-shell! Free action corresponds to G µν (X) 0, i.e. to a singular background! Already some problems in the flat space - unphysical zero mode counting ( X i = 0 - (holomorphic functions H 0 (Σ)), while p i = 0 - (holomorphic one-forms H 1 (Σ))), and extra poles in the correlation functions if g Σ > 1; however the gaussian path integral can be always computed.

6 Generic perturbation of the singular first-order background by S int = Σ d2 z ( g i j p i p j + p iµ ī j X j + pī µīj Xj + b Xi X j i j (3) with g i j = g i j (X, X) etc - almost arbitrary (but transversal i g i j = 0, i µ ī = 0) functions. j ) No anomalous dimensions! Unlike ( e iqx) q 2 in secondorder theory.

7 The (complete?) set of primary operators O g = g i j (X, X)p i p j O b = b i j (X, X) X i X j (4) and (the real operator) Φ = O µ + O µ = = p i µ ī j (X, X) X j + pī µīj (X, X) X j (5) Nonvanishing components of the Zamolodchikov metric: O g O b and ΦΦ.

8 BRST cohomologies Q = Q M + Q gh = dz ( ct M + 1 ) 2 ct gh (6) where T = T M + T gh = p i X i 2 cb + c b (7) contain c co g, c cφ (transversal), c co b, with µ ī j µī j + j αi b i j b i j + j β i + i β j (8) and holomorphic currents c cp i v i (X), c cω i (X) X i, where i v i = 0 and ω i ω i + i γ.

9 Relation to the physical background fields G k l = g īj µīk µj l + g k l b k l B = g k l īj µīk µj l g k l b k l G ki = g i j µ j k g k j µ j i B ki = g k j µ j i g i j µ j k φ = log g i.e. to G µν, B µν, φ from the bosonic string spectrum. (9)

10 First four relations are classical, while the dilaton φ log g arises due to anomaly, e.g. DpD p e S g[x, X,p, p] e S[X, X]+ 1 2π Σ d2 z hr log g when coming back to the second-order theory. (10)

11 The source of anomaly: holomorphic (target-space!) reparameterization δx i = ɛv i (X), δp i = ɛp v j j Xiwith the anomalous current p i v i (X) p i v i (X) = 1 2π R iv i (X) = 1 2π R L v log Ω (11) where Ω D/2 i=1 dxi is the holomorphic top-form. Different measures in the path integrals, or #(p) #(X) = dimh 1 (Σ) dimh 0 (Σ) = (12) = g Σ 1 hr Σ

12 Algebra: the charges commute as n v = 1 2πiα dz S 1 vi (X)p i r ω = 1 2πiα dz ω i(x) X i S 1 (13) [n v1, n v2 ] = n [v2,v 1 ] + α r ω(v1,v 2 ) (14) [r ω, n v ] = r Lv ω (15) [r ω1, r ω2 ] = 0 (16) with ω n (v 1, v 2 ) = 1 2 ( kv l 2 n l v k 1 n k v l 1 lv k 2 ) α -deformation of the semi-direct product.

13 Dynamics and equations of motion OPE of two vertex operators O g (z 1 )O g (z 2 ) where the beta-function δ ɛg k l p k p l z 1 z (17) δ ɛ g k l α (g i j i j gk l i g k j j gi l ) + O(α ) (18)

14 Background equation g i j i j gk l i g k j j gi l = 0. (19) replaces the linearized gauge-fixed Einstein equation of conventional sigma-model. Integrability??? Upon i g i j = 0 it can be rewritten as the system R µν = 1 4 H µλρh λρ ν + 2 µ ν φ, µ H µνρ 2( λ φ)h λνρ = 0, 4( µ φ) 2 4 µ µ (20) φ+ +R H µνρh µνρ = 0 of the Einstein equations for the physical fields G, B and φ.

15 Source: 2d conformal invariance, e.g. in BRST-language with m 2 (V, V )? = QV = 0, m 2 (V, V ) = 0 (21) z =ɛ ( V (1,0) (z) + V (0,1) (z) ) V (0,0) (0) (22) Bilinear operation m 2 extracts the coefficients from OPE of two vertex operators, as was done above. Generally QV + m 2 (V, V ) + m 3 (V, V, V ) +... = 0 (23) with some polyvertex fields m n (V,..., V ). How to get the polyvertex contributions (higher nonlinear terms for the beta-functions)?

16 The correlation functions of first-order theory. Consider the 3-point function where O g (x)φ(y) = 2 gi j β i j x y 2 Σ Σ d2 zφ(z) = d 2 z π 1 x z 2 y z 2 g i j β i j = gk k [i µ j k] [ k µī j] (24) (25)

17 The logarithmic divergence z x,y >ɛ d 2 z π 1 x z 2 y z 2 2 x y 2 log x y 2 ɛ 2 gives contribution to the beta-function (26) δb = B k k k k = β k k + O(µ3 ) + O(µ 4 ) B (2) k k + B(3) k k + (27) O( µ 4 ) which looks like the squared Kodaira-Spencer equations.

18 The Kodaira-Spenser equations: vanishing of the Nijenhuis tensor and their squared version is F j ik [i µ j k] µ l [i l µ j k] = 0 F ī k j [ k µī j] µl [ k lµ ī j] = 0 (28) B k k? = F j ik F ī k j (29) Is this square completed by multipoint correlation functions?

19 The quantum field theory computation L = p i X i + p i µ ī j X j + c.c. = L cl + +p i X i + p i X j cl kµ ī j (X cl) X k + p i X j cl F ī k j (X cl)x k + c.c.+ where +... (30) X i = X i + µ ī k (X cl)x k (31)

20 U U The only 2-vertex 1-loop diagram which diverges logarithmically, with the vertices with M = µ(1 µ) 1. U ī j = F ī k l M k j (X cl) X l cl (32)

21 It results in δb i j log ɛ B i j (33) where B i j = F l k j F l ik M k l M k l = = B (2) i j + B (3) i j + B (3) i j = [k µ k i] [ k µk j] + + o(µ 4 ) = (34) + [k µ k i] µ l [ k lµ ī j] + [ k µk j] µ l [k l µ k i] + +O(µ 4 )

22 To extract the O(µ 3 ) contribution compute the 4-point function 1 3! Σ 2 d2 zd 2 w O g (x)φ(y)φ(z)φ(w) = 2gi j B (3) + v i j i w i x y 2 = x y 4 log ɛ 2 + c.c (35) with B (3) being the O(µ i j 3 )-contribution into the squared KS equations, and v i = µ k k kg i k k µ ī k gk k w i = [i µ k k] µ k k (36)

23 Explicit computation (free theory!) and integration by parts over the zero mode d D XF [g, µ; g, µ;...]). Important: w i = [i µ k k] µ k k = F k ik µ k k + O(µ3 ) (37) Therefore, the extra unwanted contribution v i w i = v i F k ik µ k k + O(µ4 ) (38) i.e. vanishes on KS equations modulo quartic in µ terms.

24 Properties of the first-order theory: Disappearance of the on-shell condition (as a linear equation on vertex operator); Disappearance of the higher-spin fields or Regge descendants from the theory (anomalous dimensions cannot compensate the dimensional polynomials of the derivatives of the co-ordinate fields); Also reminds a little the AdS-like gauge/string duality.

25 being similar to the (open string) phenomenon on the D- brane near the AdS throat, where metric is infinite (singular background!). y

26 Problems and applications: Global properties of the first-order systems; Relation with bosonization of the WZW theories, or the first-order systems on flag manifolds; Bosonization: p e u ξ i ve u+iv X e u η e u iv (39) Still no canonical way of constructing multiloop measure for the 10d superstring!

27 Relation with the Berkovits formulation of ten-dimensional superstring; First-order systems: the pure spinors Q m (X) = γ m αβ Xα X β = 0 i.e. set of quadrics {Q m } in P 2D/2 1 Yet to be done...

28 Conclusions First-order CFT as expansion in the singular background; Background equations of motion: immediately nonlinear, some interesting properties; Polyvertex structures and beta-functions: the Kodaira- Spencer equation from the multipoint correlation functions.