Schwarz-STRING THEORY: PROGRESS AND PROBLEMS

[yijun]

STRING THEORY: PROGRESS AND PROBLEMS

John H. Schwarz

String theory builds on the great legacy of Yukawa and Tomonaga: New degrees of freedom and control of the UV are two important themes. This talk will give an overview of some of the
progress and some of the unsolved problems that characterize string theory today. It is divided into two parts:
[list]Connecting String Theory to the Real World[/list:u]
[list]Gauge Theory/String Theory Duality[/list:u]

Two other major subjects, which I will omit, are Black Holes in String Theory and The Impact of String Theory on Mathematics.

[b][size=18]Connecting String Theory to the Real World[/size][/b]

String theory became a hot subject in the mid-1980s when it became clear that it might give a deeper understanding of the origins of the standard model and a consistent quantum theory containing gravity \cite{Green:1987sp}\cite{Polchinski:1998rq}\cite{Becker:2007}.
At that time five consistent string theories were known, each of which requires ten spacetime dimensions and supersymmetry. They are called:

[i]Type I, Type IIA, Type IIB[/i]
`SO(32)` heterotic, $E_8 \times E_8$ heterotic

Each of these theories is entirely free of adjustable dimensionless parameters. All dimensionless parameters arise in either of two possible ways:

[list] dynamically as the expectation values of scalar fields[/list:u]

[list] as integers that count something such as topological invariants, physical objects (branes), or quantized fluxes.[/list:u]


[b]Calabi--Yau Compactification[/b]


One scheme looked particularly promising in the mid-1980s. Specifically, the $E_8 \times E_8$ heterotic theory has consistent vacuum solutions in which six spatial dimensions form a compact Calabi--Yau manifold, which has SU(3) holonomy, and the other four dimensions form Minkowski spacetime \cite{Candelas:1985en}. Thus, the ten-dimensional spacetime $M_{10}$ is a direct product
$M_{10} = CY_6 \times M_{3,1}.$

The effective four-dimensional theories that characterize such solutions at low energies have the following attractive features:

[list] They have the structure of supersymmetric grand unified theories. The well-known dvantages of low-energy supersymmetry and grand unification are therefore naturally incorporated.[/list:u]

[list] Each solution has a definite number of families of quarks and leptons determined by the topology of the CY space.[/list:u]

[list]The standard model gauge symmetry is embedded in one $E_8$ factor, and there is a hidden sector, associated to the second $E_8$ factor. Supersymmetry can break dynamically (by gluino condensation) in the hidden sector. This breaking is communicated gravitationally to the visible sector. Such schemes suppress unwanted flavor-changing neutral currents, though possibly not as much as is required.[/list:u]

[list] There are several good dark-matter candidates: The lightest supersymmetric particle, called the LSP (perhaps a neutralino) is absolutely stable if there is an unbroken R symmetry, as generally supposed to be the case. A stable neutralino has the right properties for weakly interacting cold dark matter (a WIMP). Other possibilities include gravitinos, axions, and hidden-sector particles.[/list:u]

Despite the exuberance that was in the air in 1985,\footnote{I felt at the time that the subject had undergone an almost instantaneous transition from being vastly under-appreciated to being over-optimistically regarded as being on the verge of providing a [i]theory of everything[/i] } there was much that was not yet understood. The successes were qualitative, and there were many problems and puzzling questions. Some of these problems and questions were the following:

[list] Hundreds of Calabi--Yau manifolds were known (now there are many thousands). Which one of them, if any, is the [i]right one[/i]? Is there a principle (other than agreement with observations) by which the right one can be determined? Are there string-theory based
schemes other than CY compactification of the $E_8 \times E_8$ heterotic theory that can give quasi-realistic solutions?[/list:u]

[list] Why are there four other consistent superstring theories? After all, we only need one fundamental theory. Are some of them inconsistent, or else, could some of them somehow be equivalent?[/list:u]

[list] The CY compactification scenario was analyzed using perturbation theory, but there is no good reason to believe that the string coupling should be small. What new nonperturbative
features appear at strong coupling? Does the same qualitative picture continue to hold at strong coupling?[/list:u]

[list] The CY solutions typically give many massless scalars (called [i] moduli[/i]). Since the moduli have gravitational strength interactions, they are ruled out by standard tests of GR. How can we get rid of them? The effective potential does not depend on the values of the moduli, so they describe {\it flat directions}. One should somehow stabilize the moduli by generating an effective potential with isolated minima and no flat directions.[/list:u]

[list] What ensures that the vacuum energy density (or dark energy) is sufficiently small, namely of order $10^{-120}$ in Planck units? In 1985 it was generally believed to be zero, so one popular idea was to look for a symmetry principle that would enforce this. It is just as well that such a symmetry was never found, since we now know that the vacuum energy density is not exactly zero. A generic nonsupersymmetric vacuum is expected to give a vacuum energy of order one. Supersymmetry, broken at the TeV scale, cuts this down to $10^{-60}$, which is only half way to the right value on a logarithmic scale.[/list:u]

[list] What does string theory have to say about cosmology?[/list:u]


[tex][/tex]

[yijun]

[b]Lessons of the Second Superstring Revolution[/b]

In the mid-1990s there was a remarkable burst of progress in addressing some of the issues listed above. The main lessons were the following:

[list] There is just one theory! What had been viewed as five theories are actually five different (highly symmetric) corners of a space of solutions to a unique underlying theory. The five superstring theories are related by various dualities:[/list:u]

[i]T-duality[/i] ($R \to {l_{s}^2}/R$) relates the two type II superstring theories, and also the two heterotic string theories. Here, $R$ represents the radius of a circular extra dimension and $l_{s}$ is the fundamental string length scale. There are analogous dualities involving pairs of CY manifolds. The two CY spaces are said to be related by [i] mirror symmetry[/i].

[i]S-duality[/i] ($g_{s} \to 1/g_{s}$) relates the type I superstring theory and the $SO(32)$ heterotic string theory. Here, $g_{s}$ denotes the dimensionless string coupling constant,
whose value is determined by the vacuum expectation value of a certain scalar field called the [i]dilaton[/i]. The type IIB superstring theory has an $SL(2,\IZ)$ duality group, which contains
this transformation as a special case.

The type I theory can be derived from the type IIB theory by a procedure called [i] orientifold projection[/i]. This projection essentially enforces left-right symmetry on the world-sheet theory.

[list] New objects, called $p$-branes, arise nonperturbatively. Here $p$ is an integer that represents the number of spatial dimensions of the brane. [i]Brane[/i] is a made-up word derived from membrane.) Stable $p$-branes carry conserved charges and satisfy generalized Dirac quantization conditions. There are also unstable $p$-branes that do not carry conserved charges. The main categories of stable $p$-branes are D-branes, M-branes, and NS5-branes.
NS5-branes are magnetic duals of fundamental strings in the heterotic and type II theories. (Type I strings do not carry a conserved charge, which is one way of understanding why they can break.)[/list:u]

[list] D-branes are characterized by Dirichlet boundary conditions for open strings. Also, the stable D-branes in type I and type II superstring theories carry conserved charges of the Ramond--Ramond type \cite{Polchinski:1995mt}. The relation of D-branes to open strings has the crucial consequence that the world-volume theories of D$p$-branes are Yang--Mills gauge theories in $p+1$ dimensions. $p$ takes even values in the type IIA theory and odd values in the type IIB theory. A system of $N$ coincident D$p$-branes gives a $U(N)$ gauge theory. Other gauge groups can also be obtained, if the D-branes coincide with certain types of singularities.[/list:u]

[list] In units where the fundamental superstring has tension (energy density) of the form $T_{F1} \sim m_{ s}^2,$ the D$p$-branes have tension $T_{Dp} \sim m_{s}^{p+1}/g_{s}.$
This is in contrast with more conventional nonperturbative excitations, such as monopoles in gauge theory or NS5-branes in superstring theory, which have mass or tension proportional to $1/g^2$.[/list:u]

[list] The strong-coupling limit of the Type IIA superstring theory or the $E_8 \times E_8$ heterotic string theory gives 11-dimensional M-theory. The size of the 11th dimension is $R_{11} \sim g_{s} l_{ s}.$ In the $E_8 \times E_8$ heterotic case the 11th dimension is a line interval, so the 11-dimensional spacetime has two 10-dimensional boundaries. One set of $E_8$ gauge fields is localized on each boundary. This gauge group is singled out by quantum consistency (absence of anomalies).[/list:u]

After Calabi--Yau compactification of the strongly coupled $E_8
[list] $E_8$ heterotic string theory, it is possible for the volume of the CY manifold to vary along the 11th dimension, which opens up new possibilities for phenomenology. This approach is referred to as [i]heterotic M-theory[/i].[/list:u]