SISSA ISAS SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI - INTERNATIONAL SCHOOL FOR ADVANCED STUDIES I-34014 Trieste ITALY - Via Beirut 4 - Tel. [+]39-40-37871 - Telex:460269 SISSA I - Fax: [+]39-40-3787528 INTERNATIONAL SCHOOL FOR ADVANCED STUDIES TRIESTE Thesis submitted for the degree of Doctor Philosophiæ Aspects of D-brane dynamics in superstring theory Candidate Supervisor Claudio A. Scrucca Prof. Roberto Iengo Accademic year 1997/98 Contents Introduction 3 1 Superstrings and D-branes 12 1.1 Strings....................................... 12 1.1.1 Openstrings ............................... 13 1.1.2 Closedstrings............................... 17 1.1.3 T-duality ................................. 20 1.2 D-branes...................................... 23 1.2.1 Supersymmetry.............................. 23 1.2.2 Effectiveaction.............................. 25 1.2.3 Chargequantization ........................... 27 1.2.4 Low energy interpretation . 28 2 D-brane dynamics 30 2.1 StaticD-branes.................................. 30 2.1.1 Dp-Dp interaction . 33 2.1.2 Dp-D(p+4) interaction . 36 2.2 RotatedandboostedD-branes . 38 2.2.1 Dp-Dpdynamics ............................. 41 2.2.2 Dp-D(p+4)dynamics .......................... 43 2.3 D-branes with electromagnetic fluxes . ...... 45 3 Boundary state formalism 48 3.1 World-sheetduality. .. .. .. .. .. .. .. .. .. 48 3.2 Boundarystates ................................. 50 3.2.1 StaticDp-brane.............................. 51 3.2.2 RotatedandboostedDp-brane . 53 3.2.3 Dp-branewithfluxes........................... 54 1 3.3 Interactions .................................... 55 4 Compactification and point-like D-branes 59 4.1 Toroidal and orbifold compactifications . ........ 59 4.2 Interaction of point-like D-branes . ....... 62 4.2.1 Dimensionally reduced D0-branes . 64 4.2.2 WrappedD3-branes ........................... 66 4.2.3 Non-relativistic behavior versus supersymmetry . ......... 67 4.2.4 Field theory interpretation . 68 4.3 Closedstringradiation. 71 4.3.1 Axion ................................... 79 4.3.2 Dilaton .................................. 81 4.3.3 Graviton ................................. 82 4.3.4 Non-relativistic behavior versus supersymmetry . ......... 84 4.3.5 Field theory interpretation . 85 5 Point-like D-branes as black holes 87 5.1 RR interaction for dual Dp-D(6 p)-branes .................. 87 − 5.1.1 Interactions of charges, monopoles and dyons . ...... 88 5.1.2 Compactification ............................. 90 5.1.3 The interactions in string theory . 93 5.2 WrappedD3-branesasdyons . 95 5.2.1 D3-branesintendimensions . 95 6 6 5.2.2 D3-branes on T and T /ZZ3 ....................... 97 5.3 R-N black hole as D3-branes wrapped on CY threefolds . ....... 102 5.3.1 Black hole and CY compactification . 102 6 5.3.2 The 3-brane wrapped on T /ZZ3 as a SUGRA solution . 103 6 5.3.3 The D3-brane wrapped on T /ZZ3 in string theory . 106 6 Spin effects in D-brane dynamics 111 6.1 Boundary states in the G-S formalism . 111 6.2 One-pointfunctions .............................. 117 6.3 Leading interactions and spin effects . 119 6.3.1 Dp-Dpsystem .............................. 119 6.3.2 Dp-D(p+4)system............................ 123 6.4 Field theory interpretation . 126 6.5 Scale-invariance and the SYM-SUGRA correspondence . ......... 130 2 Conclusion 132 Acknowledgments 133 A ϑ-functions 135 a A.1 ϑ b -functions. .................................. 135 A.1.1 Definition ................................. 135 A.1.2 Transformationproperties . 135 A.1.3 Riemannidentity............................. 136 A.2 ϑα-functions. ................................... 136 A.2.1 Definition ................................. 136 A.2.2 Transformationproperties . 137 A.2.3 Identities ................................. 137 B Partition functions 139 B.1 Untwisted partition functions . 139 B.2 Twistedpartitionfunctions . 140 C Field-theory computations 141 C.1 AsymptoticfieldsofaDp-brane. 141 C.2 Interaction between static D-branes . ....... 142 C.2.1 Dp-Dp static interaction . 142 C.2.2 Dp-D(p+4) static interaction . 143 C.3 Interaction between moving D-branes . 143 C.3.1 Dp-Dpinteraction ............................ 143 C.3.2 Dp-D(p+4) interaction . 144 C.4 U(1)Effectiveactions ............................. 144 D Boundary states and propagators 147 D.1 Boundary state and partition functions . ....... 147 D.1.1 Non-compactpart ............................ 147 D.1.2 Compactpart............................... 150 D.2 Two-pointfunctions .............................. 154 Bibliography 159 3 Introduction One of the most fascinating and intriguing issues ever addressed in theoretical physics is the search for a consistent and unified quantum theory of fundamental interactions. The first major difficulty in this attempt is that there seems to be no consistent quantum field theory associated to Einstein’s classical theory of general relativity [1], whereas all other fundamental interactions have instead been successfully formulated in this language. This suggests that perhaps quantum field theory is not the correct framework for the formula- tion of a so-called theory of everything. Moreover, the astonishing and appealing possibility of unifying gravity and gauge theories through the Kaluza-Klein mechanism of compact- ification [2, 3] has given strong support to the idea that actually our world might have more than four spacetime dimensions. All these arguments, together with supersymmetry [4, 5], have led to the formulation of supergravity [6, 7] and especially superstring theory [8, 9, 10, 11, 12], which is undoubtly the most promising candidate to this date for a unified theory of fundamental interactions. Superstring theory describes one-dimensional extended strings, rather than point-like particles as does quantum field theory. The infinitely many vibrational modes of the string can be regarded as particle excitations with growing mass and spin, belonging to a so far unknown (and probably very complicated) field theory with an infinite tower of elementary fields. The tension T = 1/(2πα′) of the string (energy per unit length) introduces a length scale ls = √α′ in the theory, so that the typical mass of the modes is ms = 1/√α′. For energies much below ms, only the lowest lying massless modes are relevant, and their dynamics is encoded in a low-energy effective action (LEEA) obtained by integrating out all the massive modes. The effective field theories obtained in this way are various versions of super Yang-Mills (SYM) and supergravity (SUGRA) theories for open and closed strings respectively, α′-corrections appearing through higher dimensional effective operators. The first quantized version of string theory is defined by assigning a conformally invariant world-sheet action, weighting the free propagation. More precisely, this action is in general a non-linear σ-model defining an embedding form the world-sheet Σ to a generic spacetime (or super spacetime for the Green-Schwarz formulation of the superstring) . The scalar M fields appearing in the action are the spacetime coordinates of the string, whereas its spin is encoded in additional fermionic degrees of freedom. Free propagation of open and closed strings corresponds to world-sheets with the topology of a strip and a cylinder respectively. Interactions can instead be associated in a natural and geometric way to topologically more complex world-sheets representing the splitting and joining of strings. Thanks to the coupling of the dilaton background field φ to the world-sheet scalar curvature, whose integral gives (in two dimensions) the Euler characteristic χΣ of the world-sheet Σ, the 4 amplitude corresponding to Σ is automatically weighted by its topology through a factor φ χ e−h i Σ involving the vacuum expectation value (VEV) of φ. It is therefore natural to φ χΣ identify the string coupling as gs = e−h i, so that Σ is weighted by gs− . The second quantized theory can then be built perturbatively, `ala Feynman, assum- ing that the coupling constant gs is small. Despite the powerful underlying mathematical structure inherited from conformal invariance [13], the lack of a truly second-quantized formulation of the theory is a severe limitation which is responsible for the difficulty of studying non-perturbative effects. Consistency at the quantum level requires D=10 space- time dimensions, giving therefore a interesting prediction for the spacetime dimensionality. However, the theory is not unique, as one might have hoped. In fact, five apparently different consistent string theories are known: Type I • N=1 SUSY, open strings with gauge group SO(32) and closed strings, unoriented. The LEEA is N=1 SUGRA coupled to SO(32) SYM. Type IIA,B • N=2 non-chiral (A) or chiral (B) SUSY, closed strings only. The LEEA is N=2A,B SUGRA. Heterotic SO(32) and E E • 8 × 8 N=1 SUSY, closed strings with gauge groups SO(32) or E E . 8 × 8 The LEEA is N=1 SUGRA coupled to SO(32) or E E SYM. 8 × 8 Phenomenologically interesting models can be obtained upon compactification. More pre- cisely, one makes a Kaluza-Klein ansatz of the form = IR3,1 for the ten dimen- M10 ⊗ M6 sional spacetime background, where IR3,1 is four-dimensional flat Minkowski’s space and a compact manifold. The condition for this background to be an acceptable vacuum M6 solution of the theory translates into the requirement that the non-linear σ-model describing string propagation be at a conformal fixed-point. For this to be true it is enough to chose to be Ricci-flat. In order to preserve some of the original ten-dimensional