Consistent superstrings

We have found three tachyon free and non-anomalous superstring theories

Type IIA

Type IIB

Type I SO(32)

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 eiScl /h

Amplitudes are defined summing

over all world-sheets connecting

initial and final curves

Basic string interactions

The only ineractions allowed are those that arealready 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 obtainedas states of excitation of the string and all interactions arise from the single processes of the figures

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

Interactions

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.

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

2,0Im2 www

The cylinder has a conformally equivalent description in

terms of a coord.

Conformally equivalent description

1||)2exp(,)exp( ziwz

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.

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.

Perturbative expansion

Sum over compact topologies in the closed string

S2 T2

In the open string we sum over surfaces with boundaries

D2

+…+ +

C2

Vertex operators

To a given incoming or outgoing string mode with p and internal state j there corresponds a local operator Vj(p) determined by the limiting process: vertex operator

This is the state-operator mapping.

An n-particle S-matrix element is then given by

n

iiijii

iestopocompact

XWeyldiff

ppjj pVgdSVdXdg

Sinn

1

2/12

log

),...,(... ),()()exp(][

11

Euler number

Perturbative expansion in genus

SX 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.

n

iiijii

iestopocompact

XWeyldiff

ppjj pVgdSVdXdg

Sinn

1

2/12

log

),...,(... ),()()exp(][

11

Four-point tachyon interaction

Properties

Information about the spectrum

High-energy behavior

The exponential fall-off is much faster than the amplitude of anyfield theory, which fall off with power law decay and diverge.The infinite number of particles in string theory conspire to renderFinite any divergence arising from an individual particle species.

Low energy limit supergravity, SYM

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.

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.

)(1 2 ZT

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 identiticationswdwdds 2 22 www

w

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 gab= 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.

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

21

21

exp meXXedDXDe

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.

)('

e

0

)"(")(' ed

led 1

0)"(")1('

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.

le

le

0,1'

10,'

The torus

Both descriptions have analogs in the string.

Take the torus with coordinate region

with X(0,1) and gab(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?

10,10 10

),(),(),( 1010 nm

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 ua and va.

10,10 10

2012 || ddds

)(~~~ aaaa nvmu

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

= v1+i v0. The torus is now the parallelogram in the w plane with periodic boundary conditions

01 ~~ iw wdwdds 2

)( nmww

w

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.

01 w

),(),(),( 0101 nm

2012 || ddds

1

1

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) (mn, n).

And so does –1/ , replacing (m, n) (n,m). Repeated application of these two transformations

T: ’= +1, S: ’= 1/ generate

)(~ nmww

1,,,,,' bcadZdcba

dcba

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.

1||,21

Re21

:0 F

–½ ½

F0

The partition function

00

~24/

)()(LLd

qqTrqqZ

)1~(1 00

012

2

2

2

2 1 LLT qqtr

dZ

F

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

22

21

21

MxdS D

22/1det~ MeDe ESZ

tAetrtdt

A

))log(det(

2

12/2/)4(2Mt

DD et

dtVZ

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

2~'

200

2 LLM

00~LL

sLLiLLteetr

tdt

dsV

Z )~(22~'

2

1300

00

14

2/1

2/1)4(2

'21

tisi

ii eqeq 22 ,

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 F0 and this introduces an effective cutoff. After a final rescaling, the partition function is

)1~(1132

00

14

2

2

2/1

2/11)'4(2

LL qqtrd

dV

Z

)1~(1 00

012

2

2

2

2 1 LLT qqtr

dZ

F

GRACIAS!

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:

),(||),(,),()1,1( 211 TTTT ZZZZ

00~, LL

::0 n

nnL

nnn

qqqqtr nn

11

...1 2

The bosonic vacuum amplitude

Putting all these contributions together, the full spectrum gives

where is the Dedekind function.

0

4812

2

2

2

2

|)(|

11

F

dZT

1

24/1 )1()(n

nqq

It is evident that ZT(,) contains the information about the number of states of each mass level.

Divergent cosmological constant

Expanding Z(,) in powers of q one gets a power series

of the form where dij is the number of states

with m2 = i and m2 = 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.

jiij qqd

...576||

1|)(~|),( 2

48

qZT

ZT 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

),()1,(

),(),1(0101

0101

+

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:

),(),(:01

10 1001

S

),(),('),( 100101

)()(

)()(

)()(

)()(:

S

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

)()(

)()(

)()(

)()(:

T

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.

HieTr 2

HieTr 2

HieTr 2Computing

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 (+ )

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

Modular invariance

)()(

)()(

)()(

)()(:

S

)()(

)()(

)()(

)()(:

T

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.

),( 01

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

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.

HiF eTr 2)1(

HiF eTr 2)1(

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 ZT ( are phasesmodular inv.)

])[exp()(

][)(

])[exp()(

][)(

)()(

)()(

)()(

)()(

HR

HR

HNS

HNS

qFiTrZ

qTrZ

qFiTrZ

qTrZ

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 Nr = 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.

r

rr

r

r

qqTrqTrir

ir

ir

r

ir 8)1(1

rrFrr qqTrqqTr rrrr 1)1(,1

Jacobi theta functions

Therefore we can write

3 is one of the four Jacobi theta functions, defined as

)(

)|0(

)1()1()1(1

4

43

1

82/14

4

1

124/1

1

82/16/1

r

rr

r

r

r

r qqqqqq

r

ir

r

iri

riri

eqeqqe

)(2)(exp

11)()|0(

2

22/1

1

22/124

1

22

2

-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

),()1,(

),(),1(01201

01201

i

i

e

e

42

31

2/1

02/1,0)(;

0

2/10,2/1)(

0

00)(;

2/1

2/12/1)(

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.

0)|0()|0()|0( 44

43

42

0|01

The fermion partition function

The partition functions for different spin structures are

0)(

)|0()(

)()|0(

)(

)()|0(

)(

)()|0(

)(

4

41

)(

4

42

)(

4

44

)(

4

43

)(

)(

)(

)(

)(

Z

Z

Z

Z

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 e2i/3

(+) = 1. Similarly we can show that (+ ) = 1

)()|0(

)( 4

43

)(

Z

3/4

44

)()|0(

)()1()()(

ieZZ

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

)|0()|0()|0()|0()(21

)1(121

)1(121

)(

41)(

42

44

43

4

)(212

FHiFHi RNS eTreTrZ

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.

FFHiHi

FFHiHi

RR

NSNS

eTr

eTrZ

~

)(

~22

~~22

)1(121

)1(121

)(

84

83

82

244 |)|0(||)|0(||)|0(|)()(Im21

),( TZ