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Table 1: Supergravity £;_pc11—p) (R) duality symmetries, Kp maximal compact subgroups and the superstring F11_ p(11—p) (Z) discretizations is the discretized “U-duality” [12, 13] form consistent with the Dirac quantization condition, which is conjectured to survive in superstring theory.

Table 1 Supergravity £;_pc11—p) (R) duality symmetries, Kp maximal compact subgroups and the superstring F11_ p(11—p) (Z) discretizations is the discretized “U-duality” [12, 13] form consistent with the Dirac quantization condition, which is conjectured to survive in superstring theory.

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Fig. 2: 3-closed-string vertex: the splitting point is determined by the choice of time slicing, unlike the sharp identification of the interaction point in particle theory become smoothed out to string worldsheet interactions like Fig. 1: 3-point field-theoretic particle vertex with a consequent loss of sharpness in the spacetime localization of the interaction. The field-theory propagator
Fig. 2: 3-closed-string vertex: the splitting point is determined by the choice of time slicing, unlike the sharp identification of the interaction point in particle theory become smoothed out to string worldsheet interactions like Fig. 1: 3-point field-theoretic particle vertex with a consequent loss of sharpness in the spacetime localization of the interaction. The field-theory propagator
An essential feature of all relativistic systems is the freedom to choose arbitrary parametrizations for their histories. Begin with the analog of a relativistic particle, whose action is obtained geometrically from the invariant proper length of its world- line as shown in Figure 3: This yields a worldline reparametrization-invariant action 2.1 Reparametrization invariance
An essential feature of all relativistic systems is the freedom to choose arbitrary parametrizations for their histories. Begin with the analog of a relativistic particle, whose action is obtained geometrically from the invariant proper length of its world- line as shown in Figure 3: This yields a worldline reparametrization-invariant action 2.1 Reparametrization invariance
Fig. 4: Open string worldsheet Now generalize the relativistic particle action to that of a relativistic extended object with intrinsic spatial dimensionality p = 1. Instead of a worldline, one now has a 2-dimensional worldsheet as illustrated in Figure 4 for an open string; for a closed string, one needs to identify o = 0 ando =7: The string worldsheet action is then the reparametrization-invariant area of the worldsheet YW
Fig. 4: Open string worldsheet Now generalize the relativistic particle action to that of a relativistic extended object with intrinsic spatial dimensionality p = 1. Instead of a worldline, one now has a 2-dimensional worldsheet as illustrated in Figure 4 for an open string; for a closed string, one needs to identify o = 0 ando =7: The string worldsheet action is then the reparametrization-invariant area of the worldsheet YW
Fig. 5: Linear Regge trajectories relating spin and (mass)? of particle states One can now make a rough Bohr-Sommerfeld estimate of the quantum spectrum, requiring |J| =nh , n € Z and considering the excitations in their rest frames where E=M. Thenn= bl = aM? where a = 7 is the string slope parameter. The quantized states lies on linear Regge trajectories making an angle a’ in a J/h versus M? plot:
Fig. 5: Linear Regge trajectories relating spin and (mass)? of particle states One can now make a rough Bohr-Sommerfeld estimate of the quantum spectrum, requiring |J| =nh , n € Z and considering the excitations in their rest frames where E=M. Thenn= bl = aM? where a = 7 is the string slope parameter. The quantized states lies on linear Regge trajectories making an angle a’ in a J/h versus M? plot:
This symmetry is 7-duality: a string propagating on a compact direction of radius & with momentum mode n and winding mode fi is equivalent to a string propagating on a compact direction of radius a’ /@ with interchanged mode numbers: momen- tum 7 and winding n. Fig. 6: Winding modes with various fi values
This symmetry is 7-duality: a string propagating on a compact direction of radius & with momentum mode n and winding mode fi is equivalent to a string propagating on a compact direction of radius a’ /@ with interchanged mode numbers: momen- tum 7 and winding n. Fig. 6: Winding modes with various fi values
For increasing gs = (e®), the spacing between the BPS mass levels decreases, approaching a continuum as one approaches the decompactification limit of infinite S! circumference, where the full D = 11 nature of the theory becomes more and more manifest. Accordingly, the strong g, coupling limit of Type IIA string theory is hypothesized to be described by a phase whose full quantum properties remain incompletely known, but which has D = 11 supergravity as a field-theory limit. This phase of the overall picture has been called M-Theory. Fig. 7: “Not to take this web of dualities as a sign that we are on the right track would be a bit like believing that God had put fossils into the rocks in order to mislead Darwin about the evolution of life.’ Steven Hawking
For increasing gs = (e®), the spacing between the BPS mass levels decreases, approaching a continuum as one approaches the decompactification limit of infinite S! circumference, where the full D = 11 nature of the theory becomes more and more manifest. Accordingly, the strong g, coupling limit of Type IIA string theory is hypothesized to be described by a phase whose full quantum properties remain incompletely known, but which has D = 11 supergravity as a field-theory limit. This phase of the overall picture has been called M-Theory. Fig. 7: “Not to take this web of dualities as a sign that we are on the right track would be a bit like believing that God had put fossils into the rocks in order to mislead Darwin about the evolution of life.’ Steven Hawking
Fig. 9: “Brane-scan” of supergravity p-brane solutions, linked by worldvolume (di- agonal) and transverse-space (vertical) dimensional reductions
Fig. 9: “Brane-scan” of supergravity p-brane solutions, linked by worldvolume (di- agonal) and transverse-space (vertical) dimensional reductions
Fig. 10: Four out of the 50 diagrams arising in the calculation of the 4-loop diver- gences in maximal supergravity To date, these techniques have allowed the explicit calculation of maximal su- pergravity divergences to proceed up to the 4-loop level, something that would have been unthinkable using traditional Feynman diagram techniques [19].
Fig. 10: Four out of the 50 diagrams arising in the calculation of the 4-loop diver- gences in maximal supergravity To date, these techniques have allowed the explicit calculation of maximal su- pergravity divergences to proceed up to the 4-loop level, something that would have been unthinkable using traditional Feynman diagram techniques [19].
Table 2: Maximal supergravity first possible divergences & BPS degree from unitarity-based calculations. Known divergences are shown in gray. The current status of the first possible maximal supergravity ultraviolet diver- ences is summarized in Table 2, with the currently known divergences shown in sray. The BPS degree represents the degree of supersymmetry invariance of the in- egrand prior to superspace integration. The surprizing feature of these results is the ardiness of maximal supergravity in getting around to revealing its ultraviolet di- yergences. At D = 8 dimensions and L = 1, one does indeed encounter the expected °+ counterterm, but in lower spacetime dimensions this 5 BPS divergence does not secur, leaving it to significantly less constrained counterterms to be the first UV livergence candidates.
Table 2: Maximal supergravity first possible divergences & BPS degree from unitarity-based calculations. Known divergences are shown in gray. The current status of the first possible maximal supergravity ultraviolet diver- ences is summarized in Table 2, with the currently known divergences shown in sray. The BPS degree represents the degree of supersymmetry invariance of the in- egrand prior to superspace integration. The surprizing feature of these results is the ardiness of maximal supergravity in getting around to revealing its ultraviolet di- yergences. At D = 8 dimensions and L = 1, one does indeed encounter the expected °+ counterterm, but in lower spacetime dimensions this 5 BPS divergence does not secur, leaving it to significantly less constrained counterterms to be the first UV livergence candidates.
Table 3: Maximal supergravity current first divergence expectations & BPS degree. Known divergences are shown in gray and the first anticipated D = 4 and D=5 divergences are shown in black. new cancellations are now predicted at D = 4, L = 5&6. The resulting pattern of anticipated! first divergences is shown in Table 3. String Theory, Unification and Quantum Gravity Imperial/TP/12/KSS/01
Table 3: Maximal supergravity current first divergence expectations & BPS degree. Known divergences are shown in gray and the first anticipated D = 4 and D=5 divergences are shown in black. new cancellations are now predicted at D = 4, L = 5&6. The resulting pattern of anticipated! first divergences is shown in Table 3. String Theory, Unification and Quantum Gravity Imperial/TP/12/KSS/01
Fig. 12: Topology change at a boundary of moduli space
Fig. 12: Topology change at a boundary of moduli space
Fig. 13: “Fuzzball” effective geometry from an average over nonsingular geometries
Fig. 13: “Fuzzball” effective geometry from an average over nonsingular geometries
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PhysicsTheoretical PhysicsQuantum GravityString TheoryString theory (Physics)String PhenomenologyString Field Theory
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