Superstrings and Supergeometry

Transcription

1 Dubna, Dec 2012 Superstrings and Supergeometry by P.A.Grassi University of Eastern Piedmont DISIT, Alessandria and INFN Turin, ITALY Work in collaboration with G. Policastro, R Catenacci, D. Matessi, M. De Bernardi, and M. Marescotti.

2 Summary of the Talk Pure Spinor Superstring and Supergeometry Amplitudes and Integration Forms Integral Forms and Cartan Calculus Čech Cohomology and de Rham Cohomology Applications

3 Pure Spinor String Theory ( We start from the fields x m, θ α, p α (θ α are Majorana-Weyl spinors spinor in d = (9, 1)) and the free field action ( ) ( S = d 2 z ( x m xm + p α θ α + ˆp α ˆθ α) i) The total conformal charge is c T = (10) x + ( 32) p,θ ii) Inserting p α = p α 1 2 x mγ m αβ θβ (γm αβ θβ )(θγ m θ) in S, S p=p = S Green Schwarz iii) So, d α p α p α 0 must be identified with the fundamental constraint. Q = dzλ α d α

4 By requiring the nilpotency of Q iv) The nilpotentcy {Q, Q} = λ α (z) λ β (w)d α (z)d β (w) = λ α γ m αβλ β Π m = 0 λ α γ m αβλ β = 0 Pure Spinor Constraints The space of the zero modes for the ghost fields for the Pure Spinor String theory is Conifold Space with Base SO(10)/U(5) It is a local Calabi-Yau with conifold singularity Integration Measure for this space?

5 Viewed from String Amplitude computations: Vertex operators: dzd zv (0,0) z z dzv (1,0) z, d zv (0,1) z and U (1,1) U (1,1) 1 U (1,1) 2 U (1,1) 3 n j=1 dz j d z j V (0,0) j To compute these amplitudes we need to perform the OPE contractions until we get to the following expression M = d 16 θ 0 d 10 x 0 Dλ 0 µ(θ 0, λ 0 )M(x 0, θ 0, λ 0 ) To integrate over these variables we need to define a suitable measure for that we need a good understanding of the integration on supermanifolds

6 Multiloop Amplitudes The complete N-point g-loop amplitudes can be written in this way M g dm i 11g i=3g 3+1 [dµ λ ][dθ][dx] g [dµ w ][dd l ] l=1 Z(B mn N mn i )Z(J i ) 11 j=1 Y C j 3g 3 n=1 N p=1 µ(z n )b B (z n ) dτ p V (0) p r k=1 U (1) (z k ) where we used Z(X) = [ ] Q, Θ(X) = [ ] Q, X δ(x) Picture Raising Operators Y C = Cαθ i α δ(cαλ i α ) i {Q, } b X = Z(X)T Picture Lowering Operators

7 Viewed from String Field Theory perspective S = ( d 10 x 0 d 16 θ 0 Dλ 0 µ(λ 0, θ 0 ) Φ (1) Q (1) Φ (1) +... ) Properties: 1) BRST invariance 2) SUSY 3) Saturation of zero modes (bosonic and fermionic zero modes) The result is the following Dλ 0 = dλ α 1... dλ α 11 ɛ α1...α 16 (γ m γ n γ p γ mnp ) [α 12...α 16 ](β 1...β 3 ) µ(λ 0, θ 0 ) = 11 i=1 (C i αθ α )δ(c i αλ α ) λ β... 1 λ β 3

8 ω Ω (M) Integration on Manifolds The usual integration on standard manifolds can be viewed as follows There is a map between by identifying the forms with anticommuting variables ( M) C M dx i θ i ω = ( M) M ω C ( ) M Then, on the new space we can integrate functions Since there is a natural measure which gives ( ) ( ) M = Ω (M) ω C ( ) M µ = dx i 1... dx i n dθ i 1... dθ i n

9 ( Complexes of Superforms M M Given 1-superforms, we have commuting variables dθ i dθ j = dθ j dθ i So the complex 0 d Ω 0 is infinite. There is no top form. d Ω 1 d Ω n d ch is bounded from below, but not from abo How we can define the top form? How we can define a sensible integration theory? It is convenient to introduce a new basic object δ(dθ i )

10 The new object has the following properties If we denote by dx I the 1-forms associated to the commuting coordinates of the manifold dx I dx J = dx J dx I, dx I dθ j = dθ j dx I, dθ i dθ j = dθ j dθ i, δ(dθ) δ(dθ ) = δ(dθ ) δ(dθ), dθδ(dθ) = 0, dθδ (dθ) = δ(dθ). second and third property can be easily deduced from the above app the last three equations follow from the usual definition of the Dirac delta function and from the changing-variable formula for Dirac delta s δ(ax + by)δ(cx + dy) = Det 1 ( a b c d ) δ(x)δ(y) x y R. By setting a = 0 b = 1 c = 1 and d =

11 Then, we can introduce the new set of forms dx [K1 dx K l] dθ (i l+1 dθ i r ) δ(dθ [i r+1 ) δ(dθ i r+s] ) n r-form with picture s. All indices K are antisymmetrized and the complexes re d Ω (r q) d Ω (r+1 q) d Ω (p+1 q) p+1 q is the top form dx [K dx K + ] Form Number d d [i Picture Number Ω (p+1 q) commuti Notice that the picture number corresponds to the number of Delta functions, while the form number corresponds to the total form degree. It can be negative by considering the derivatives of Delta functions. + + ω (p+1 q) = ɛ i 1 i q θ i 1 θ iq f (x, θ) C (p+1 q) C p+1 re the last equalities is obtained by integrating on the de

12 Čech cohomology of P 1 1 f P 1 1 is the simplest Wenon-trivial describe example now of super-projective Čech coho space. It is defined as usual as an algebraic variety by quotient of the complex ( ) superspace C 2 1 with respect to a complex number different from zero. It can be covered by two patches Transition functions from one patch to another U 0 = {[z 0 ; z 1 ] P 1 : z 0 = 0}, U 1 = {[z 0 ; z 1 ] P 1 : z 1 = 0}. affine coordinates are = z 1 on Φ (γ ) = 1 γ, Φ (ψ) = ψ γ. We now consider a sheaf of differential The change of patch reflects upon the following transformation Φ δ n (d ψ) = γ n+1 δ n (dψ) γ n ψ dγ δ n+1 (dψ). w, we can proceed in calculating sheaf cohomology for

13 Results Per interi n non negativi { Ȟ 0 (P 1 1, Ω n 0 ) 0, n > 0, = C, n = 0. Ȟ 0 (P 1 1, Ω n 1 ) = C 4n+4, Ȟ 0 (P 1 1, Ω 1 1 ) = 0. Ȟ 1 (P 1 1, Ω n 0 ) = C 4n Ȟ 1 (P 1 1, Ω n 1 ) = 0, Ȟ 1 (P 1 1, Ω 1 1 ) = C. of. Notice that Ȟ 1 (P 1 1, Ω n+1 0 ) and Ȟ 0 (P 1 1, Ω n 1 ) have the same dimension. ct, in fact we can construct a pairing Ȟ 1 (P 1 1, Ω n+1 0 ) Ȟ 0 (P 1 1, Ω n 1 ) Ȟ 1 (P 1 1, Ω 1 1 ) = C of. ˇ + ( ) ( ) Defined in terms of the this pairing dγ (dψ) n, δ n (dψ) = ( 1) n n! dγ δ(dψ), (dψ) n+1, dγ δ n+1 (dψ) = ( 1) n (n + 1)! dγ δ(dψ), dγ (dψ) n, dγ δ n+1 (dψ) = (dψ) n+1, δ n (dψ) = 0. n be checked that this product descends to a pairing in cohomology. W

14 Super de Rham cohomology 1. d behaves as a differential on functions; 2. d 2 = 0; 3. d commutes with δ and its derivatives, and so d(δ (k) (dψ)) = 0. Similarly, the same operator d is defined on j, and behaves as the o { 0 n 0 { 0, n > 0, For n 0, the holomorphic de Rham cohomology groups of P 1 1 are as follows: H n 0 DR (P1 1, hol) = H n 1 DR (P 1 1, hol) = C, n = 0. { 0, n > 0, C, n = 0. In this computation H 1 1 terms of the co DR (P1 1, hol) = 0. H 0 1 of. We have given explicit descriptions DR (P1 1, hol) see formu is generated by the constant sheaf ψδ(dψ)

15 Two Theorems Given a supermanifold M n m, for i 0 we have the following isomorphism = ˇ H i j where the constant sheaf is generated by the integral forms extended globally DR (Mn m ) = Ȟi (M n m, H 0 j ). see formu ψδ(dψ) sider a ge which is the analogous of the Čech-de Rham isomorphism. tlet M M n m be abesupermanifold, a such such that that H 0 j is a constant sheaf y). Then Then the the de Rham de Rham cohomology of Mof n m Mis n m is: H j DR (Mn m ) = H DR (M) H 0 j. :

16 Now we can use the above geometrical setting in superstring amplitudes -- we define the PCO as follows µ(λ 0, θ 0 ) = 11 i=1 (C i αθ α )δ(c i αλ α ) 1) -- BRST invariant 2) -- Any change of the gauge paramenters Cα i : 3) -- It changes the picture, but what is the picture in this context? δ Cα µ = {Q, θ α C β θ β δ (C γ λ γ )} Given a vertex operator U (n) = λ α 1... λ α n A α1...α n (x, θ) (and {Q, θ α } = λ α ) Using the usual dictionary Q d and λ α dθ α we have U (n) = dθ α 1... dθ α n A α1...α n (x, θ)

17 Path integral on zero modes d n x 0 d m θ 0 Dλ 0 k U (k) Theory Integration of Superforms on Supermanifold (Baranov-Schwarz-Bernstein-Leites-Voronov) For a bosonic manifold M (n), a top form ω n such that any diff. of the manifold, namely it transforms as a measure. ω n = DetJω n For a superspace M (n m), a top form in the space of superforms that transforms as Berezinian The insertion of the cohomology sheafs θ i 0δ(dθ i 0) solves all problems.

18 Cartan Calculus d = dθ α D α + (dx m + θγ m dθ) m Even/Odd Vector fields: v = v α D α + v m m with v α v m odd/even even/odd Even ι v, ι 2 v = 0, L v = dι v + ι v d Odd ιṽ, ι 2 ṽ 0, Lṽ = dιṽ ιṽd Finally, δ(ιṽ) = dt e itι ṽ Γṽ = [ ] d, Θ(ιṽ)

19 The last equation translates into the PCO operators for multiloop string computations Z(X) = [ ] Q, Θ(X) = [ ] Q, X δ(x) where X is a generic field constructed in terms of zero modes The same analysis can be also performed for RNS superstring where the there are 2-d super-riemann surfaces It is useful for twistor string theory to explain the measure for the open sector of the theory It is important for Non-Renormalization theorems in 10d and 11d (for supermembranes and topological theories)

20 Our Results Rigorous formulation of integration theory for superforms Derivation of Cech de Rham isomorphisms for supermanifolds Thom Class and Integration on Sub-Supermanifolds (with M. Marescotti) SuperBalanced Varieties and Properties Dualities for Super-Cohomologies Embedding of a bosonic manifold in a Super-Calabi-Yau manifolds and topological strings Pure Spinor Superstring Perturbation theory and multiloop computations (with P. Vanhove). Fermionic T-dualities and Super-Cohomologies (Sigma models computations)

21 Conclusions Supergeometry is a powerful tool for several applications, here I used it to motivate the construction of multiloop amplitudes in Pure Spinor Superstrings Recent developments by E. Witten for string perturbation theory in RNS formalism Applications to Super-Calabi-Yau as the coset manifold PSU(4 4)/SU(3 4) related to AdSxS spaces.