1. Consistent superstrings We have found three tachyon free and non-anomalous superstring theories Type IIA Type IIB Type I SO(32)

2. Interactions  The natural way to introduce interactions in string theory is the Feynman path integral.  Amplitudes are given by summing over all possible histories interpolating between initial and final states.  Each history is weighted by e iS cl /h  Amplitudes are defined summing over all world-sheets connecting initial and final curves

3. Basic string interactions The only ineractions allowed are those that are already implicit in the sum over world-sheets: one string decaying into two or two strings merging into one. These are the basic interactions. All particle interactions are obtained as states of excitation of the string and all interactions arise from the single processes of the figures

4. Always a free theory  There is no distinguished point where the interaction occurs. The interaction arises only from the global topology of the world-sheet and the local properties of the world-sheet are as in the free theory.  It is this smearing of the interaction that cuts off the short distance divergencies of gravity.  Sum over topologies

5. Interactions

6. Scattering amplitudes  The idea to sum over all world-sheets bounded by initial and final curves seems natural. But it is difficult to define it consistently with the local world- sheet symmetries and the resulting amplitudes are rather complicated.  There is one special case where the amplitudes simplify: the limit when the sources are taken to infinity  Scattering amplitude: an S-matrix element with incoming and outgoing strings specified.  Most string calculations confine to S-matrix computations and are on-shell.

7. Conformal invariance  So we consider processes where the sources are pulled to infinity, like  Each of the incoming and outgoing legs is a long cylinder which can be described with a complex coordinate w=  + i   The limit corresponding to the scattering process is   

8.  The cylinder has a conformally equivalent description in terms of a coord. Conformally equivalent description  In this picture the long cylinder is mapped into the unit disk In the limit    the tiny circles shrink to points and the world-sheet reduces to a sphere with a point-like insertion for each external state.

9. Four ways to define the sum over world-sheets  There are different string theories, depending on which topologies we include in the sum over world-sheets:  Closed oriented: all oriented world-sheets without boundary  Closed unoriented: All world-sheets without boundary  Closed + open oriented: All oriented world-sheets with any number of boundaries  Closed + open unoriented: All world-sheets with any number of boundaries.

10. Perturbative expansion  Sum over compact topologies in the closed string S 2 T 2  In the open string we sum over surfaces with boundaries D 2 +… ++ C2C2

11. Vertex operators  To a given incoming or outgoing string mode with p  and internal state j there corresponds a local operator V j (p) determined by the limiting process: vertex operator  This is the state-operator mapping.  An n-particle S-matrix element is then given by

12. Euler number 

13. Perturbative expansion in genus  S X refers to the action without gauge fixing  One has to divide by the volume of the symmetry group this is the origin of the ghosts in the theory.  In the bosonic string these are the anticommuting bc ghosts of reparametrizations; here there will be commuting ghosts of susy ,   Finally the vertex operators create physical states  Thus the scattering amplitudes are expectation values of a product of vertex operators.

14. Four-point tachyon interaction

15. Properties

16. Information about the spectrum

17. High-energy behavior The exponential fall-off is much faster than the amplitude of any field theory, which fall off with power law decay and diverge. The infinite number of particles in string theory conspire to render Finite any divergence arising from an individual particle species. Low energy limit  supergravity, SYM

18. One loop surfaces  At one loop there are four Riemann surfaces with Euler number zero.  The torus is the only closed oriented surface with Euler number zero.  If we include unoriented surfaces we have also the Klein bottle for the closed string (two crosscaps).  In the open string we have surfaces with boundaries: the cylinder and the Mobius strip.

19. The torus amplitude  We now compute the simplest one-loop amplitude in the closed oriented string theory: the partition function or vacuum amplitude.  There is a great deal of physics in the amplitude with no physical operators. Essentially it determines the full perturbative spectrum:  The possibility to assign to the world-sheet fermions periodic or antiperiodic boundary conditions leads to the concept of spin structures.  The GSO projection is then shown to be the geometric constraint of modular invariance.

20. The torus  To describe a torus we need to identify two periods.  Alternatively we can cut the torus along the two cycles and map it to the plane.  Thus we describe it as the complex plane with metric and identitications w

21. Gauge fixing  We wrote the S-matrix as a path integral. We now want to reduce the path integral to gauge-fixed form.  We would like to choose one configuration from each (diff  Weyl)-equivalent set.  Locally we did this by fixing g ab =  ab (and  a ), but globally there is a mismatch between the space of metrics and the world-sheet gauge group.  Let us look at the equivalent situation for the point particle.

22. The circle  Consider de path integral: Take a path forming a closed loop in spacetime, so the topology is a circle. The parameter  can be taken to run from 0 to 1 with the end points identified  that is X  (  ) and e(  ) are periodic on 0    1. The tetrad e(  ) has one component and there is one local symmetry, the choice of parameter  enough symmetry to fix the tetrad

23. The periodicity is not preserved by the gauge choice  The tetrad transforms as e’d  ’ = e d   The gauge choice e’=1 then gives a differential equation for  ’ (  ):  Integrating this with the boundary condition  ’(0) = 0 determines  The complication is that in general  ’ (1)  1 so the periodicity is not preserved. In fact is the invariant length of the circle.

24. Two possibilities  So we cannot simultaneously set e ’=1 and keep the coordinate region fixed.  We can hold the coordinate region fixed and set e’ to the constant value e ’= l or set e ’=1 and let the coordinate region vary:  In either case, after fixing the gauge invariance we are left with an ordinary integral over l.  Not all tetrads on the circle are diff-equivalent. There is a one parameter family of inequivalent tetrads parametrized by l.

25. The torus  Both descriptions have analogs in the string.  Take the torus with coordinate region with X  (  0,  1 ) and g ab (  0,  1 ) periodic in both directions.  Equivalently we can think of this as the plane with the identification of points for integer m and n.  To what extent is the field space diff  Weyl redundant?

26. Two possibilities on the torus  Theorem: it is not possible to bring a general metric to unit form by a diff  Weyl transformation that leaves invariant the periodicity, But it is possible to bring it to the form where  is a complex const. For  = i this would be  ab.  Alternatively one can take the flat metric. By coordinate and Weyl transformations we can keep the metric flat but it is not guaranteed that this will leave the periodicity unchanged. Rather we may have with general translation vectors u a and v a.

27. The parallelogram  By rotating and rescaling the coordinate system accompanied by a shift in the Weyl factor we can always set u =(1,0).  This leaves two parameters, the components of v.  Thus defining the metric is and the periodicity is where  = v 1 +i v 0. The torus is now the parallelogram in the w plane with periodic boundary conditions w

28. The square  Alternatively we can define  The original periodicity is preserved but the metric takes the more general form  The integration over metrics reduces to two ordinary integrals over the real and imaginary parts of .  The metric is invariant under complex conjugation of , so we can restrict attention to Im  > 0. 1 1

29. The modular group  As in the case of the circle we can put these parameters  either in the metric or the periodicity.  The parameter  is known as a modulus or Teichmuller parameter.  There is some additional redundance that does not have an analogue in the point particle case. The value  +1 generates the same set of identifications as  replacing ( m, n )  ( m  n, n ).  And so does –1/ , replacing ( m, n )  ( n,  m ). Repeated application of these two transformations T:  ’=  +1, S:  ’=  1/  generate

30. The fundamental region  Using the modular transformations it can be shown that every  is equivalent to exactly one point in the region  This is called the fundamental region and it is one representation of the moduli space of (diff  Weyl)-inequivalent metrics. –½ ½ F0F0

31. The partition function 

32. The partition function of a scalar field  In a field theory in D dimensions, for which  The path integral defines the vacuum energy Z as  Using the identity where  is an ultraviolet cutoff and t a Schwinger parameter, we find

33. The partition function of the bosonic string  Apply this formula to the closed bosonic string in D=26, whose spectrum is encoded in subject to the constraint where we have introduced the  -function constraint.  Define the complex Schwinger parameter and let

34. The partition function of the bosonic string  Then we can write  As we have seen, at one-loop a closed string sweeps a torus, whose Teichmuller parameter  is naturally identified with the complex Schwinger parameter.  But not all values of  correspond to distinct torii. We have to restrict the integration to F 0 and this introduces an effective cutoff. After a final rescaling, the partition function is

35. GRACIAS!

36. Modular invariance  This expression is modular covariant so that its transformations compensate those of the measure.  Recall the explicit expression for the vacuum amplitude in the bosonic string. Recall that are number operators for two infinite sets of harmonic oscillators.  For each spacetime dimension  And for each n:

37. The bosonic vacuum amplitude  Putting all these contributions together, the full spectrum gives where is the Dedekind function.  It is evident that Z T ( ,  ) contains the information about the number of states of each mass level.

38. Divergent cosmological constant  Expanding Z( ,  ) in powers of q one gets a power series of the form where d ij is the number of states with m 2 = i and m 2 = j.  The first few terms of the expansion are The first term corresponds to the negative (mass) 2 tachyon and the constant term to the massless string states (graviton, dilaton and antisymmetric tensor).  Due to the tachyon pole, the one-loop cosmological constant for the closed bosonic string is infinite.

39. Z T for the superstring  The four possible boundary conditions for fermions lead to four spin structures  Recall that periodic boundary conditions in  1 correspond to the R sector and antiperiodic boundary conditions to the NS sector.  The boundary conditions may be separately periodic or antiperiodic in the  1 and  0 directions.  We denote the spin structure with periodic b.c. in  0 and antiperiodic b.c. in  1 by + 

40. S transformation on spin structures  Under an S transformation with modular matrix that is, basically  1 and  0 are exchanged. This means that the fermions transform as  From this we easily derive the action of S on the spin structures:

41. T transformation on spin structures  Similarly, under  + 1 the torus transforms as  Which leads to the following action of T on the spin structures:   +1

42.  Loop amplitudes contain the factor for propagation through imaginary time  It is essential to remember now that in the path integral formulation of quantum statistical mechanics, the partition function of the fermions is computed using antiperiodic b.c. in time  is represented by path integral with antiperiodic (  ) b.c. in  0. Computing  Therefore in the absence of GSO projection, the contribution of the NS sector to a loop amplitude corresponds to (   ) while the contribution of the R sector corresponds to ( +  )

43.  The combination of partition functions (   ) and ( +  ) is not modular invariant. To get a modular invariant theory, they must be supplemented by (  + ) Modular invariance

44. Correlator for fermions  But (  + ) is a partition function for NS states (  b.c. in  1 ) with an insertion of (  1) F ( + b.c. in  0 )  The periodicity conditions in time are rather unusual in the context of field theory, but they may be expressed as conditions on correlation functions on the torus.  Consider a generic correlation function for fermions where  stands for the product of an odd number of fermion fields at various positions, so that the correlator is nonzero (correlator of odd number of fermions is zero since they are Grassmann numbers). Take this fermion from its position  0 to  0 + 1.

45. Periodic boundary conditions  Within the operator formalism this means that  will go through all possible instants of time and will have to be passed over all the other fermions in  in succession, because of the time ordering.  Since a minus sign is generated each time, there will be an overall factor of  1 generated by this translation, and therefore the usual correspondence between the path integral and the Hamiltonian approach leads naturally to the antiperiodic condition when the theory is defined on a torus.  To implement the periodic condition we need to modify the usual correspondence by inserting an operator that anticommutes with  (  1) F

46. Computing  If we want to compute where (  1) F is the operator used in the GSO projection that counts the number of world-sheet fermions modulo 2, then we must use the + b.c. in the  0 direction.

47. Partition function in all sectors  To make sure that this feature is built into the partition function we simply insert (  1 ) F in the definition of the partition function within the trace in the time-periodic case  This prescription implies the following expessions for the holomorphic part of Z T (  are phases  modular inv.)

48. Computing the trace  The calcultation is completely analogous to the bosonic case only that the occupation numbers are now restricted by the Pauli principle to N r = 0 and 1.  For the fermionic oscillators we have since the Pauli exclusion principle allows at most one fermion in each of these states. For a given fermionic mode there are only two states   This expression actually applies to both NS and R sectors, provided r is turned into an integer for R.

49. Jacobi theta functions  Therefore we can write   3 is one of the four Jacobi theta functions, defined as

50.  -functions  boundary conditions  Through the one-loop partition function the  functions for arbitrary  and  are in correspondence to the b.c. for fermions  The different spin structures then correspond to

51. Riemann theta identities  The Jacobi theta functions and their generalization to higher genus Riemann surfaces play an important role in string theory. They satisfy many amazing identities.  At one-loop one of the most important ones is  It is easy to see that  In the same way that we have derived the partition function for the (   ) spin structure we can easily derive the rest.

52. The fermion partition function  The partition functions for different spin structures are

53. Determining the phases  First require that the spin structure sum is modular invariant separately both in the right and left moving sectors.  Since only the relative phases are relevant we arbitrarily set  (  ) =  1, i.e.  Using the transformation rules of the theta and eta functions we find  The eight transverse bosonic degrees of freedom contribute  (  )  which gives an extra factor of e  2  i/3   (  +) =  1. Similarly we can show that  (+  ) =  1

54. Spacetime supersymmetry   (+ +) cannot be determined from modular invariance  So finally  The relative sign between the two sectors reflects the fact that states in NS are bosons and in R are fermions  The factor in the NS sector is just the GSO projection (  1 ) F = +1.  Due to the Riemann identity and the vanishing of  1 (0), the partition function vanishes. This reflects a supersymmetric spectrum: the contributions from spacetime bosons and fermions cancel

55. Modular invariance  There is another modular invariant combination of boundary conditions: summing over left and right movers with the same b.c.  This leads to  Including the contribution from the bosons we get  This theory has only spacetime bosons and contains a tachyon. The GSO projection in NS allows the tachyon.