Glimpses of Double Field Theory Geometry
|
|
- Camron Miles
- 5 years ago
- Views:
Transcription
1 Glimpses of Double Field Theory Geometry Strings 2012, Munich Barton Zwiebach, MIT 25 July 2012
2 1. Doubling coordinates. Viewpoints on the strong constraint. Comparison with Generalized Geometry. 2. Bosonic double field theory. 3. Generalized Riemann the geometrization of tensors. 5. Undetermined connections and α corrections. 6. Large coordinate transformations. Recent work with OLAF HOHM. Earlier work with Hull, Hohm and Kwak. 1
3 1. Introduction Closed string theory on a torus T d exhibits O(d,d,Z) symmetry. Conserved momentum and winding quantum numbers have associated coordinates, p a x a, w a x a, a = 1,2,...,d. String field theory is a double field theory: φ(x µ,{x a, x a }) Hard to uncover the geometry from the full theory, so in DFT we focus on the massless sector. S = d D x ( ge 2φ R+4( φ) 2 1 ) 12 H2 In what sense is S duality invariant? 2 Answer: Upon reduction on T d, find a global O(d,d;R) symmetry!
4 The DFT idea: Double all coordinates ) ) ( xi ( X M = x i, M = i M = 1,...,2D, is an O(D,D) index. i Raise and lower indices with the O(D,D) metric η ( ) η MN 0 1 = η 1 0 MN. Introduce doubled fields φ(x i, x i ) and write S DFT = d D xd D x L(x, x) Make manifest a global O(D,D;R), giving O(d,d;R) upon reduction. S DFT = S when fields are x independent. 3 Theory includes an O(D, D) covariant strong constraint that ensures that S DFT is (locally) physically equivalent to S.
5 Strong constraint views The constraint arises from level-matching (L 0 L 0 ) Ψ = 0 p i w i = N N = 0, for our fields. M M φ a = 0 fields φ a. This weak constraint is unavoidable. Fields still have momentum and winding excitations. A strong version seems needed to write a complete theory: M M (φ a φ b ) = 0 a,b If true, there is some dual frame (x, x ) where fields are not doubled. 4 Strongly constrained DFT displays the O(D, D) symmetry but it is not physically doubled.
6 Important caveats about the strong constraint: 1. If the strong constraint holds locally and there is a nontrivial global structure, the doubled theory may differ from the undoubled one. 2. Some relaxation of the strong constraint appear to be consistent for certain backgrounds: Massive type IIA (Hohm and Kwak). 3. Relaxation suggested by Scherk Schwarz compactifications: Aldazabal, et.al. [arxiv: ], Geissbühler, [arxiv: ]. 4. Explicit discussion of the relaxation of the strong constraint by Graña, M, and Marques, D. arxiv:
7 DFT vs Generalized Geometry Generalized geometry is a small departure from ordinary geometry: Given a manifold M it puts together vectors V i and one-forms ξ i as V +ξ TM T M. Structures on this larger space. The Courant bracket: [ V1 +ξ 1,V 2 +ξ 2 ] = [V1,V 2 ]+L V1 ξ 2 L V2 ξ d(i V 1 ξ 2 i V2 ξ 1 ) V and ξ are not treated symmetrically. Double field theory (strongly constrained) is a small departure from generalized geometry. It puts TM and T M be on similar footing by doubling the underlying manifold. ) ( ξ Gauge parameters : ξ M = i ξ i and then we have a C-bracket [ ξ1,ξ 2 ] M C ξn [1 Nξ M 2] 1 2 ξp [1 M ξ 2]P. 6 For non-doubled ξ M the C-bracket reduces to the Courant bracket.
8 2. Bosonic Double Field Theory We use a (2D 2D) generalized metric on TM T M ( ) g ij g ik b kj H MN, H MN η MP η NQ H b ik g kj g ij b ik g kl PQ H MP H PQ = δ M Q. b lj O(D,D) transformations : X M = h M N X N, h M P h N Qη MN = η PQ. H = (h 1 ) t Hh 1. Spacetime action: The action can then be written as S = d D xd D xe 2d R(H,d). R(H,d) 4H MN M N d M N H MN 4H MN M d N d+4 M H MN N d HMN M H KL N H KL 1 2 HMN M H KL K H NL. O(D, D) symmetry is manifest since indices match!
9 Gauge transformations via generalized Lie derivatives L: L ξ A M ξ P P A M +( M ξ P P ξ M )A P New term needed for trivial gauge parameters ξ M = M χ to generate no Lie derivatives: L χ A M = P χ P A M +( M P χ P M χ)a P = 0. Algebra: [ Lξ1, Lξ2 ] = L[ξ1,ξ 2 ] C, Gauge transformations δ ξ H MN = Lξ H MN, δ ξ e 2d = M (ξ M e 2d ). The action is gauge invariant because R is a generalized scalar: δ ξ R = ξ M M R. 8
10 3. Generalized Riemann and geometrization Investigation initiated by Jeon, Lee and Park in [arxiv: ] Olaf Hohm and BZ [arxiv: ]. Olaf Hohm and BZ (to appear) Introduce a Christoffel type connection through covariant derivatives: M A N = M A N Γ MN K A K, M A N = M A N +Γ MK N A K, Introduce curvatures and torsion: [ M, N ] AK = R MNK L A L T MN L L A K. It turns out that neither R nor T is a generalized tensor: A simple modification leads to a generalized curvature tensor: 9 R MNKL R MNKL +R KLMN +Γ QMN Γ Q KL. R is symmetric under exchange of first and second pair of indices.
11 Geometrization: Vector fields: X,Y,Z,W TM Connection: bilinear operator Bilinear: (X,Y) X Y TM. X (Y 1 +Y 2 ) = X Y 1 + X Y 2, X1 +X 2 Y = X1 Y + X2 Y. Moreover, given a function f on the manifold we have fx Y = f X Y X fy = X(f)Y +f X Y, X(f) X M M f Tensors are multilinear maps T(V 1,V 2,V 3,,V N ) R from vector fields to functions, with scaling 10 T(V 1,,fV i,,v N ) = f T(V 1,,V i,,v N ), i.
12 Torsion field: Torsion tensor: T(X,Y) X Y Y X [X,Y] TM. T(X,Y,Z) T(X,Y),Z = X Y Y X [X,Y],Z. Curvature operator R(X, Y): Curvature tensor: R(X,Y)Z X Y Z Y X Z [X,Y] Z TM. 11 R(X,Y,Z,W) R(X,Y)Z,W = R MNPQ X M Y N Z P W Q
13 Generalized Tensors: The Lie bracket is not quite right: If X,Y are generalized vectors, then [X,Y] is not. For this, need the Dorfman bracket [X,Y] D [X,Y] K D [X,Y] K +Y M K X M, Generalized torsion: T (X,Y,Z) X Y Y X [X,Y] D, Z + Y, Z X. 12
14 Generalized curvature: R(X,Y)Z,W ( X Y Y X [X,Y]D )Z,W +... The scaling of X is problematic. Need an extra term k Y, Zk X W, Z kz. Here Z k is a set of basis vector (fields) and Z k,z i = δ k i. Now the Z scaling has been compromised. Finally take R(X,Y,Z,W) ( X Y Y X [X,Y]D )Z,W + ( Z W W Z [Z,W]D )X,Y 13 + k Y, Zk X W, Z kz.
15 Like for ordinary Riemann, we have an algebraic Bianchi identity: 3R [MNKL] = 4 [M T NKL] +3T [MN Q T KL]Q. But find no analog of the differential Bianchi identity: cyc W,X,Y W R(X,Y,Z,V) = 0 (for zero torsion). Is there a differential Bianchi for generalized Riemann? 14
16 5. Undetermined connections, α corrections. Constraints on the connection. (1) Covariant constancy of η MN (2) Zero Generalized torsion. (3) Covariant constancy of H MN (4) Covariant divergence using dilaton density The constraints do not fully determine the connection. In fact, there exists no alternative set of covariant constraints that can determine the connection fully. The undetermined pieces can be described using the projectors: P M N = 1 2 ( δm N H M N ), P M N = 1 2 ( δm N +H M N ), 15 We introduce barred and under-barred (un-barred) indices W M P M N W N, W M P M N W N.
17 The undetermined pieces are the traceless parts (w.r.t η) of Γ MNK and Γ M N K The non-vanishing curvatures are R MNKL, R MNK L, R M N K L, R M N K L. These all contain undetermined components. The O(D,D) covariant Riemann tensor R MNPQ is a generalized tensor but is not fully determined in terms of the physical fields. 16
18 Since the undetermined connections are traceless, taking traces of Riemannian tensors can help! η NL η MK R M N K L has no undetermined connections This is the scalar curvature R(H, d) found earlier. The tracing of the (3,1) tensor is also interesting η NL R M N K L has no undetermined connections This is the Ricci-like R MK, needed for equations of motion. 17
19 The gauge invariance of the action implies a couple of differential Bianchi identities: P R 4 M R P M = 0. PR + 4 M R M P = 0. We have not found an uncontracted differential Bianchi identity for the full Riemann tensor: [M R NK]PQ 0. Perhaps there are other variants (they would be useful!) 18
20 α corrections Ashoke Sen (Phys. Lett. B271 (1991) 295) has explained that the low energy string effective action to all orders in α has an O(d,d) symmetry. One should be able to include α corrections to the DFT action while preserving duality. Curvature square terms are the leading order correction. Since generalized Riemann has undetermined connections, this is a challenge. We can attempt to add a sum of terms quadratic in curvatures and hope for the undetermined connections to drop out. It appears, however, that such a strategy needs a differential Bianchi identity that does not exist. 19 DFT should allow for the classification of all T-duality invariant terms in the effective action.
21 6. Large Coordinate Transformations Olaf Hohm and BZ (arxiv: ). The usual coordinate transformation of a gauge field is given by A M(X ) = F M N A N (X), F M N = XN X M With X M = X M ξ M (X) it gives the infinitesimal transformation: δa M A M (X) A M(X) = ξ K K A M + M ξ K A K = L ξ A M What is the full transformation that gives rise to the generalized Lie derivative? δa M = Lξ A M = ξ K K A M + ( M ξ K K ξ M )A K 20 This question, and additional consistency conditions led to a surprising answer :
22 Proposal: with F defined by F M N 1 2 A M (X ) = F M N A N (X), ( X P X P + X M X N ) X M X N X P X P. Indeed, for X = X ξ, with ξ infinitesimal it leads to δa M = A (X) A(X) = ξ K K A M + ( M ξ K K ξ M )A K 21
23 Consistency checks 1. Partial derivatives should transform consistently! M = 1 ( X P X P + X M X N ) 2 X M X N X P X P N = XN X M N. 2. The constant η MN metric is coordinate invariant: η MN = F M R F N S η RS. F M N is in fact an O(D,D) matrix. 3. Usual not-doubled coordinate transformations and b-field gauge transformations are included x i = x i (x), x i = x i, A i(x ) = xp x ia p(x) 22 x i = x i, x i = x i ζ i (x), b ij = b ij + i ζ j j ζ i.
24 Large transformations Better parameterization: X M = e ξp (X) P X M = e ξ X M. (1) Ordinary vector: One can prove that the above together with gives A (X ) = X X A(X) A (X) = exp ( L ξ ) A(X). (2) (2) is a suitable definition for large gauge transformations in the ordinary theory. It is also a good definition for the generalized theory A (X) = exp ( Lξ ) A(X). 23 Does it arise from (1) and A M(X ) = 1 ( X P X P + X M X N ) 2 X M X N X P X P A N (X)?
25 Almost. The coordinate transformation needs some readjustment: X = e Θ(ξ) X, with Θ M = ξ M (ξξl ) M ξ L +O(ξ 5 ). (construct to all orders!) Generalized Lie derivatives define a Lie algebra: ] [ Lξ1, Lξ2 = L[ξ1,ξ 2 ] c, [[ Lξ1, Lξ2 ], Lξ3 ] + [[ Lξ2, Lξ3 ], Lξ1 ] + [[ Lξ3, Lξ1 ], Lξ2 ] = 0. This happens because: J(ξ 1,ξ 2,ξ 3 ) = [ξ 1,ξ 2 ] c,ξ 3 ] c +cyc = ( ) The finite transformations form a group. where exp ( Lξ1 (X)) exp ( Lξ2 (X)) = exp ( Lξc (ξ 2,ξ 1 )), ξ c (ξ 2,ξ 1 ) = ξ 2 +ξ [ ξ2,ξ 1 ] c Group associativity works. Explicitly ξ c( ξ 3, ξ c (ξ 2,ξ 1 ) ) = ξ c( ξ c (ξ 3,ξ 2 ), ξ 1 ) 1 6 J(ξ 1,ξ 2,ξ 3 )+,
26 Consider two maps m 1 : X X ; X = e Θ(ξ 1)(X) X, m 2 : X X ; X = e Θ(ξ 2)(X ) X. The relevant map m 21 : X X is not the composition m 2 m 1. Following the fields, we get ( ) X = e Θ ξ c (ξ 2,ξ 1 )(X) X. Consider a third map That s some exotic: m 21 = m 2 m 1 m 3 : X X ; X = e Θ(ξ 3)(X ) X. We can form a map X X as m 3 (m 2 m 1 ), (m 3 m 2 ) m 1. Leading to X given by two, different options ( ) ( ) exp Θ(ξ c (ξ 3,ξ c (ξ 2,ξ 1 ))) X exp Θ(ξ c (ξ c (ξ 3,ξ 2 ),ξ 1 )) X.
27 Generalized coordinate transformations build a group when acting on fields, but The composition rule for coordinate maps does not form a group. Perhaps related to proposal about non-commutative or even non-associative coordinates: Lust (arxiv:1010:1361), Blumenhagen, et.al (arxiv: ), Mylonas, et.al (arxiv: ). - In summary, we have seen some glimpses of DFT geometry. 25 We hope this will help find a useful language to deal with non-geometric compactifications and learn more about the way Riemannian geometry is modified in string theory.
On the curious spectrum of duality-invariant higher-derivative gravitational field theories
On the curious spectrum of duality-invariant higher-derivative gravitational field theories VIII Workshop on String Field Theory and Related Aspects ICTP-SAIFR 31 May 2016 Barton Zwiebach, MIT Introduction
More informationCosmological solutions of Double field theory
Cosmological solutions of Double field theory Haitang Yang Center for Theoretical Physics Sichuan University USTC, Oct. 2013 1 / 28 Outlines 1 Quick review of double field theory 2 Duality Symmetries in
More informationRelating DFT to N=2 gauged supergravity
Relating DFT to N=2 gauged supergravity Erik Plauschinn LMU Munich Chengdu 29.07.2016 based on... This talk is based on :: Relating double field theory to the scalar potential of N=2 gauged supergravity
More informationSymmetry and Geometry in String Theory
Symmetry and Geometry in String Theory Symmetry and Extra Dimensions String/M Theory Rich theory, novel duality symmetries and exotic structures Supergravity limit - misses stringy features Infinite set
More informationDouble Field Theory and Stringy Geometry
Double Field Theory and Stringy Geometry String/M Theory Rich theory, novel duality symmetries and exotic structures Fundamental formulation? Supergravity limit - misses stringy features Infinite set of
More informationDouble Field Theory Double Fun?
Double Field Theory Double Fun? Falk Haßler based on..., 1410.6374, 1502.02428, 1509.04176,... University of North Carolina at Chapel Hill City University of New York March 3, 2016 The big picture Phenomenology
More informationM-theory and extended geometry
M-theory and extended geometry D.S.B., Chris Blair, Martin Cederwall, Axel Kleinschmidt, Hadi & Mahdi Godazgar, Kanghoon Lee, Emanuel Malek, Edvard Musaev, Malcolm Perry, Felix Rudolph, Daniel Thompson,
More informationString Geometry Beyond the Torus
String Geometry Beyond the Torus Falk Haßler based on arxiv:1410.6374 with Ralph Blumenhagen and Dieter Lüst Arnold Sommerfeld Center LMU Munich October 25, 2014 The big picture Phenomenology SUGRA target
More informationDouble Field Theory at SL(2) angles
Double Field Theory at SL(2) angles Adolfo Guarino Université Libre de Bruxelles Iberian Strings 207 January 7th, Lisbon Based on arxiv:62.05230 & arxiv:604.08602 Duality covariant approaches to strings
More informationDouble Field Theory on Group Manifolds in a Nutshell
in a Nutshell Ralph Blumenhagen Max-Planck-Institute for Physics, Munich E-mail: blumenha@mpp.mpg.de Max-Planck-Institute for Physics, Munich Arnold-Sommerfeld-Center for Theoretical Physics, Munich E-mail:
More informationThe SL(2) R + exceptional field theory and F-theory
The SL(2) R + exceptional field theory and F-theory David Berman Queen Mary University of London Based on 1512.06115 with Chris Blair, Emanuel Malek and Felix Rudolph Motivations I Exceptional Field Theory
More informationThe boundary state from open string fields. Yuji Okawa University of Tokyo, Komaba. March 9, 2009 at Nagoya
The boundary state from open string fields Yuji Okawa University of Tokyo, Komaba March 9, 2009 at Nagoya Based on arxiv:0810.1737 in collaboration with Kiermaier and Zwiebach (MIT) 1 1. Introduction Quantum
More informationNon-associative Deformations of Geometry in Double Field Theory
Non-associative Deformations of Geometry in Double Field Theory Michael Fuchs Workshop Frontiers in String Phenomenology based on JHEP 04(2014)141 or arxiv:1312.0719 by R. Blumenhagen, MF, F. Haßler, D.
More information1/2-maximal consistent truncations of EFT and the K3 / Heterotic duality
1/2-maximal consistent truncations of EFT and the K3 / Heterotic duality Emanuel Malek Arnold Sommerfeld Centre for Theoretical Physics, Ludwig-Maximilian-University Munich. Geometry and Physics, Schloss
More informationEinstein Double Field Equations
Einstein Double Field Equations Stephen Angus Ewha Woman s University based on arxiv:1804.00964 in collaboration with Kyoungho Cho and Jeong-Hyuck Park (Sogang Univ.) KIAS Workshop on Fields, Strings and
More information1 Covariant quantization of the Bosonic string
Covariant quantization of the Bosonic string The solution of the classical string equations of motion for the open string is X µ (σ) = x µ + α p µ σ 0 + i α n 0 where (α µ n) = α µ n.and the non-vanishing
More informationNon-Commutative/Non-Associative Geometry and Non-geometric String Backgrounds
Non-Commutative/Non-Associative Geometry and Non-geometric String Backgrounds DIETER LÜST (LMU, MPI) Workshop on Non-Associativity in Physics and Related Mathematical Structures, PennState, 1st. May 2014
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More informationGravity theory on Poisson manifold with R-flux
Gravity theory on Poisson manifold with R-flux Hisayoshi MURAKI (University of Tsukuba) in collaboration with Tsuguhiko ASAKAWA (Maebashi Institute of Technology) Satoshi WATAMURA (Tohoku University) References
More informationCollective T-duality transformations and non-geometric spaces
Collective T-duality transformations and non-geometric spaces Erik Plauschinn LMU Munich ESI Vienna 09.12.2015 based on... This talk is based on :: T-duality revisited On T-duality transformations for
More informationMembrane σ-models and quantization of non-geometric flux backgrounds
Membrane σ-models and quantization of non-geometric flux backgrounds Peter Schupp Jacobs University Bremen ASC workshop Geometry and Physics Munich, November 19-23, 2012 joint work with D. Mylonas and
More informationNon-Associative Flux Algebra in String and M-theory from Octonions
Non-Associative Flux Algebra in String and M-theory from Octonions DIETER LÜST (LMU, MPI) Corfu, September 15th, 2016 1 Non-Associative Flux Algebra in String and M-theory from Octonions DIETER LÜST (LMU,
More informationC-spaces, Gerbes and Patching DFT
C-spaces, Gerbes and Patching DFT George Papadopoulos King s College London Bayrischzell Workshop 2017 22-24 April Germany 23 April 2017 Material based on GP: arxiv:1402.2586; arxiv:1412.1146 PS Howe and
More informationExtended Space for. Falk Hassler. bases on. arxiv: and in collaboration with. Pascal du Bosque and Dieter Lüst
Extended Space for (half) Maximally Supersymmetric Theories Falk Hassler bases on arxiv: 1611.07978 and 1705.09304 in collaboration with Pascal du Bosque and Dieter Lüst University of North Carolina at
More informationSUPERSTRING REALIZATIONS OF SUPERGRAVITY IN TEN AND LOWER DIMENSIONS. John H. Schwarz. Dedicated to the memory of Joël Scherk
SUPERSTRING REALIZATIONS OF SUPERGRAVITY IN TEN AND LOWER DIMENSIONS John H. Schwarz Dedicated to the memory of Joël Scherk SOME FAMOUS SCHERK PAPERS Dual Models For Nonhadrons J. Scherk, J. H. Schwarz
More informationFirst structure equation
First structure equation Spin connection Let us consider the differential of the vielbvein it is not a Lorentz vector. Introduce the spin connection connection one form The quantity transforms as a vector
More informationQuantum Nambu Geometry in String Theory
in String Theory Centre for Particle Theory and Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, UK E-mail: chong-sun.chu@durham.ac.uk Proceedings of the Corfu Summer Institute
More informationD-branes as a single object. SIS Dubna, Edvard Musaev
D-branes as a single object Edvard Musaev Moscow Inst of Physics and Technology; Kazan Federal University based on works with Eric Bergshoeff (Groningen U), Chris Blair (VUB), Axel Kleinschmidt (AEI MPG),
More informationString Theory Compactifications with Background Fluxes
String Theory Compactifications with Background Fluxes Mariana Graña Service de Physique Th Journées Physique et Math ématique IHES -- Novembre 2005 Motivation One of the most important unanswered question
More informationPHYS 4390: GENERAL RELATIVITY NON-COORDINATE BASIS APPROACH
PHYS 4390: GENERAL RELATIVITY NON-COORDINATE BASIS APPROACH 1. Differential Forms To start our discussion, we will define a special class of type (0,r) tensors: Definition 1.1. A differential form of order
More informationOutline. 1 Relativistic field theory with variable space-time. 3 Extended Hamiltonians in field theory. 4 Extended canonical transformations
Outline General Relativity from Basic Principles General Relativity as an Extended Canonical Gauge Theory Jürgen Struckmeier GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany j.struckmeier@gsi.de,
More informationarxiv: v3 [hep-th] 27 Mar 2018
Point Particle Motion in DFT and a Singularity-Free Cosmological Solution Robert Brandenberger, 1, Renato Costa, 2, Guilherme Franzmann, 1, and Amanda Weltman 2, 1 Physics Department, McGill University,
More informationLECTURE 8: THE SECTIONAL AND RICCI CURVATURES
LECTURE 8: THE SECTIONAL AND RICCI CURVATURES 1. The Sectional Curvature We start with some simple linear algebra. As usual we denote by ( V ) the set of 4-tensors that is anti-symmetric with respect to
More informationLECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups
LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are
More informationIn this lecture we define tensors on a manifold, and the associated bundles, and operations on tensors.
Lecture 12. Tensors In this lecture we define tensors on a manifold, and the associated bundles, and operations on tensors. 12.1 Basic definitions We have already seen several examples of the idea we are
More informationString Theory and Generalized Geometries
String Theory and Generalized Geometries Jan Louis Universität Hamburg Special Geometries in Mathematical Physics Kühlungsborn, March 2006 2 Introduction Close and fruitful interplay between String Theory
More information7 Curvature of a connection
[under construction] 7 Curvature of a connection 7.1 Theorema Egregium Consider the derivation equations for a hypersurface in R n+1. We are mostly interested in the case n = 2, but shall start from the
More informationarxiv: v1 [hep-th] 25 May 2017
LMU-ASC 32/17 MPP-2017-105 Generalized Parallelizable Spaces from Exceptional Field Theory arxiv:1705.09304v1 [hep-th] 25 May 2017 Pascal du Bosque, a,b Falk Hassler, c Dieter Lüst, a,b a Max-Planck-Institut
More informationNon-Geometric Calabi- Yau Backgrounds
Non-Geometric Calabi- Yau Backgrounds CH, Israel and Sarti 1710.00853 A Dabolkar and CH, 2002 Duality Symmetries Supergravities: continuous classical symmetry, broken in quantum theory, and by gauging
More informationSupergeometry and unified picture of fluxes
Supergeometry and unified picture of fluxes Satoshi Watamura (Tohoku U.) based on the collaboration with T. Asakawa, N. Ikeda, Y. Kaneko, T. Kaneko, M.A. Heller, U. Carow-Watamura, H.Muraki. Ref.[JHEP02
More informationClifford Algebras and Spin Groups
Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of
More informationBranes, Wrapping Rules and Mixed-symmetry Potentials
Branes, Wrapping Rules and Mixed-symmetry Potentials Eric Bergshoeff Groningen University based on work with Fabio Riccioni Recent Advances in T/U-dualities and Generalized Geometries Zagreb, June 9 2017
More informationElements of differential geometry
Elements of differential geometry R.Beig (Univ. Vienna) ESI-EMS-IAMP School on Mathematical GR, 28.7. - 1.8. 2014 1. tensor algebra 2. manifolds, vector and covector fields 3. actions under diffeos and
More informationUNIVERSITY OF DUBLIN
UNIVERSITY OF DUBLIN TRINITY COLLEGE JS & SS Mathematics SS Theoretical Physics SS TSM Mathematics Faculty of Engineering, Mathematics and Science school of mathematics Trinity Term 2015 Module MA3429
More informationDoubled Aspects of Vaisman Algebroid and Gauge Symmetry in Double Field Theory arxiv: v1 [hep-th] 15 Jan 2019
January, 2019 Doubled Aspects of Vaisman Algebroid and Gauge Symmetry in Double Field Theory arxiv:1901.04777v1 [hep-th] 15 Jan 2019 Haruka Mori a, Shin Sasaki b and Kenta Shiozawa c Department of Physics,
More informationDifferential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18
Differential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18 Required problems (to be handed in): 2bc, 3, 5c, 5d(i). In doing any of these problems, you may assume the results
More informationHeterotic Flux Compactifications
Heterotic Flux Compactifications Mario Garcia-Fernandez Instituto de Ciencias Matemáticas, Madrid String Pheno 2017 Virginia Tech, 7 July 2017 Based on arxiv:1611.08926, and joint work with Rubio, Tipler,
More informationOne Loop Tests of Higher Spin AdS/CFT
One Loop Tests of Higher Spin AdS/CFT Simone Giombi UNC-Chapel Hill, Jan. 30 2014 Based on 1308.2337 with I. Klebanov and 1401.0825 with I. Klebanov and B. Safdi Massless higher spins Consistent interactions
More informationContents. Preface to the second edition. Preface to the first edition. Part I Introduction to gravity and supergravity 1
Table of Preface to the second edition page xxi Preface to the first edition xxv Part I Introduction to gravity and supergravity 1 1 Differential geometry 3 1.1 World tensors 3 1.2 Affinely connected spacetimes
More informationGauge Theory of Gravitation: Electro-Gravity Mixing
Gauge Theory of Gravitation: Electro-Gravity Mixing E. Sánchez-Sastre 1,2, V. Aldaya 1,3 1 Instituto de Astrofisica de Andalucía, Granada, Spain 2 Email: sastre@iaa.es, es-sastre@hotmail.com 3 Email: valdaya@iaa.es
More informationAs always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing).
An Interlude on Curvature and Hermitian Yang Mills As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing). Suppose we wanted
More informationMIFPA PiTP Lectures. Katrin Becker 1. Department of Physics, Texas A&M University, College Station, TX 77843, USA. 1
MIFPA-10-34 PiTP Lectures Katrin Becker 1 Department of Physics, Texas A&M University, College Station, TX 77843, USA 1 kbecker@physics.tamu.edu Contents 1 Introduction 2 2 String duality 3 2.1 T-duality
More informationDouble field theory of type II strings
Double field theory of type II strings The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Hohm, Olaf,
More informationHeterotic Geometry and Fluxes
Heterotic Geometry and Fluxes Li-Sheng Tseng Abstract. We begin by discussing the question, What is string geometry? We then proceed to discuss six-dimensional compactification geometry in heterotic string
More informationarxiv:hep-th/ v1 10 Apr 2006
Gravitation with Two Times arxiv:hep-th/0604076v1 10 Apr 2006 W. Chagas-Filho Departamento de Fisica, Universidade Federal de Sergipe SE, Brazil February 1, 2008 Abstract We investigate the possibility
More informationDIFFERENTIAL GEOMETRY HW 12
DIFFERENTIAL GEOMETRY HW 1 CLAY SHONKWILER 3 Find the Lie algebra so(n) of the special orthogonal group SO(n), and the explicit formula for the Lie bracket there. Proof. Since SO(n) is a subgroup of GL(n),
More information2T-physics and the Standard Model of Particles and Forces Itzhak Bars (USC)
2T-physics and the Standard Model of Particles and Forces Itzhak Bars (USC) hep-th/0606045 Success of 2T-physics for particles on worldlines. Field theory version of 2T-physics. Standard Model in 4+2 dimensions.
More informationRiemannian Manifolds
Chapter 25 Riemannian Manifolds Our ultimate goal is to study abstract surfaces that is 2-dimensional manifolds which have a notion of metric compatible with their manifold structure see Definition 2521
More informationInvariant Nonholonomic Riemannian Structures on Three-Dimensional Lie Groups
Invariant Nonholonomic Riemannian Structures on Three-Dimensional Lie Groups Dennis I. Barrett Geometry, Graphs and Control (GGC) Research Group Department of Mathematics, Rhodes University Grahamstown,
More informationNewman-Penrose formalism in higher dimensions
Newman-Penrose formalism in higher dimensions V. Pravda various parts in collaboration with: A. Coley, R. Milson, M. Ortaggio and A. Pravdová Introduction - algebraic classification in four dimensions
More informationLecture A2. conformal field theory
Lecture A conformal field theory Killing vector fields The sphere S n is invariant under the group SO(n + 1). The Minkowski space is invariant under the Poincaré group, which includes translations, rotations,
More informationA PROOF OF BRST INVARIANCE
A PROOF OF BRST INVARIANCE T. Ortín Departamento de Física Teórica C-XI Universidad Autónoma de adrid 8049 adrid, Spain ay 3, 011 Abstract Introducing a geometric normal ordering, we give a proof of BRST
More informationFabio Riccioni. 17th July 2018 New Frontiers in String Theory 2018 Yukawa Institute for Theoretical Physics, Kyoto
& & 17th July 2018 New Frontiers in String Theory 2018 Yukawa Institute for Theoretical Physics, Kyoto Based on arxiv:1803.07023 with G. Dibitetto and S. Risoli arxiv:1610.07975, 1704.08566 with D. Lombardo
More informationSymmetries Then and Now
Symmetries Then and Now Nathan Seiberg, IAS 40 th Anniversary conference Laboratoire de Physique Théorique Global symmetries are useful If unbroken Multiplets Selection rules If broken Goldstone bosons
More informationCurvature homogeneity of type (1, 3) in pseudo-riemannian manifolds
Curvature homogeneity of type (1, 3) in pseudo-riemannian manifolds Cullen McDonald August, 013 Abstract We construct two new families of pseudo-riemannian manifolds which are curvature homegeneous of
More informationTensor Hierarchies of 5- and 6-Dimensional Field Theories
IFT-UAM/CSIC-09-22 June 23 rd, 2009 Tensor Hierarchies of 5- and 6-Dimensional Field Theories arxiv:0906.4043v1 [hep-th] 22 Jun 2009 Jelle Hartong and Tomás Ortín Institute for Theoretical Physics, Sidlerstrasse
More informationBERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS
BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS SHOO SETO Abstract. These are the notes to an expository talk I plan to give at MGSC on Kähler Geometry aimed for beginning graduate students in hopes to motivate
More informationAdvanced Machine Learning & Perception
Advanced Machine Learning & Perception Instructor: Tony Jebara Topic 6 Standard Kernels Unusual Input Spaces for Kernels String Kernels Probabilistic Kernels Fisher Kernels Probability Product Kernels
More informationclass # MATH 7711, AUTUMN 2017 M-W-F 3:00 p.m., BE 128 A DAY-BY-DAY LIST OF TOPICS
class # 34477 MATH 7711, AUTUMN 2017 M-W-F 3:00 p.m., BE 128 A DAY-BY-DAY LIST OF TOPICS [DG] stands for Differential Geometry at https://people.math.osu.edu/derdzinski.1/courses/851-852-notes.pdf [DFT]
More informationLecture 8: 1-loop closed string vacuum amplitude
Lecture 8: 1-loop closed string vacuum amplitude José D. Edelstein University of Santiago de Compostela STRING THEORY Santiago de Compostela, March 5, 2013 José D. Edelstein (USC) Lecture 8: 1-loop vacuum
More information(a p (t)e i p x +a (t)e ip x p
5/29/3 Lecture outline Reading: Zwiebach chapters and. Last time: quantize KG field, φ(t, x) = (a (t)e i x +a (t)e ip x V ). 2Ep H = ( ȧ ȧ(t)+ 2E 2 E pa a) = p > E p a a. P = a a. [a p,a k ] = δ p,k, [a
More informationDerivatives in General Relativity
Derivatives in General Relativity One of the problems with curved space is in dealing with vectors how do you add a vector at one point in the surface of a sphere to a vector at a different point, and
More informationThe Conformal Algebra
The Conformal Algebra Dana Faiez June 14, 2017 Outline... Conformal Transformation/Generators 2D Conformal Algebra Global Conformal Algebra and Mobius Group Conformal Field Theory 2D Conformal Field Theory
More informationFlux Compactification of Type IIB Supergravity
Flux Compactification of Type IIB Supergravity based Klaus Behrndt, LMU Munich Based work done with: M. Cvetic and P. Gao 1) Introduction 2) Fluxes in type IIA supergravity 4) Fluxes in type IIB supergravity
More informationPAPER 52 GENERAL RELATIVITY
MATHEMATICAL TRIPOS Part III Monday, 1 June, 2015 9:00 am to 12:00 pm PAPER 52 GENERAL RELATIVITY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
More informationGeneralized complex geometry and topological sigma-models
Generalized complex geometry and topological sigma-models Anton Kapustin California Institute of Technology Generalized complex geometry and topological sigma-models p. 1/3 Outline Review of N = 2 sigma-models
More informationNon-Abelian and gravitational Chern-Simons densities
Non-Abelian and gravitational Chern-Simons densities Tigran Tchrakian School of Theoretical Physics, Dublin nstitute for Advanced Studies (DAS) and Department of Computer Science, Maynooth University,
More informationGeometry and Physics. Amer Iqbal. March 4, 2010
March 4, 2010 Many uses of Mathematics in Physics The language of the physical world is mathematics. Quantitative understanding of the world around us requires the precise language of mathematics. Symmetries
More informationWeek 6: Differential geometry I
Week 6: Differential geometry I Tensor algebra Covariant and contravariant tensors Consider two n dimensional coordinate systems x and x and assume that we can express the x i as functions of the x i,
More informationThe Algebroid Structure of Double Field Theory
The Algebroid Structure of Double Field Theory Larisa Jonke Division of Theoretical Physics Rudjer Bošković Institute, Zagreb Based on: 1802.07003 with A. Chatzistavrakidis, F. S. Khoo and R. J. Szabo
More informationChapter 7 Curved Spacetime and General Covariance
Chapter 7 Curved Spacetime and General Covariance In this chapter we generalize the discussion of preceding chapters to extend covariance to more general curved spacetimes. 145 146 CHAPTER 7. CURVED SPACETIME
More informationBrane Gravity from Bulk Vector Field
Brane Gravity from Bulk Vector Field Merab Gogberashvili Andronikashvili Institute of Physics, 6 Tamarashvili Str., Tbilisi 380077, Georgia E-mail: gogber@hotmail.com September 7, 00 Abstract It is shown
More informationOpen String Wavefunctions in Flux Compactifications. Fernando Marchesano
Open String Wavefunctions in Flux Compactifications Fernando Marchesano Open String Wavefunctions in Flux Compactifications Fernando Marchesano In collaboration with Pablo G. Cámara Motivation Two popular
More informationGRAVITATION F10. Lecture Maxwell s Equations in Curved Space-Time 1.1. Recall that Maxwell equations in Lorentz covariant form are.
GRAVITATION F0 S. G. RAJEEV Lecture. Maxwell s Equations in Curved Space-Time.. Recall that Maxwell equations in Lorentz covariant form are. µ F µν = j ν, F µν = µ A ν ν A µ... They follow from the variational
More informationRigid Holography and 6d N=(2,0) Theories on AdS 5 xs 1
Rigid Holography and 6d N=(2,0) Theories on AdS 5 xs 1 Ofer Aharony Weizmann Institute of Science 8 th Crete Regional Meeting on String Theory, Nafplion, July 9, 2015 OA, Berkooz, Rey, 1501.02904 Outline
More informationarxiv: v1 [physics.gen-ph] 17 Apr 2016
String coupling constant seems to be 1 arxiv:1604.05924v1 [physics.gen-ph] 17 Apr 2016 Youngsub Yoon Dunsan-ro 201, Seo-gu Daejeon 35242, South Korea April 21, 2016 Abstract We present a reasoning that
More informationDynamics of branes in DFT
Dynamics of branes in DFT Edvard Musaev Moscow Inst of Physics and Technology based on works with Eric Bergshoeff, Chris Blair, Axel Kleinschmidt, Fabio Riccioni Dualities Corfu, 2018 Web of (some) branes
More informationExact Quantization of a Superparticle in
21st October, 2010 Talk at SFT and Related Aspects Exact Quantization of a Superparticle in AdS 5 S 5 Tetsuo Horigane Institute of Physics, Univ. of Tokyo(Komaba) Based on arxiv : 0912.1166( Phys.Rev.
More information1 Canonical quantization conformal gauge
Contents 1 Canonical quantization conformal gauge 1.1 Free field space of states............................... 1. Constraints..................................... 3 1..1 VIRASORO ALGEBRA...........................
More information8.324 Relativistic Quantum Field Theory II MIT OpenCourseWare Lecture Notes Hong Liu, Fall 2010 Lecture 3
Lecture 3 8.324 Relativistic Quantum Field Theory II Fall 200 8.324 Relativistic Quantum Field Theory II MIT OpenCourseWare Lecture Notes Hong Liu, Fall 200 Lecture 3 We begin with some comments concerning
More informationarxiv:hep-th/ v3 21 Jul 1997
CERN-TH/96-366 hep-th/9612191 Classical Duality from Dimensional Reduction of Self Dual 4-form Maxwell Theory in 10 dimensions arxiv:hep-th/9612191v3 21 Jul 1997 David Berman Theory Division, CERN, CH
More informationA brief introduction to Semi-Riemannian geometry and general relativity. Hans Ringström
A brief introduction to Semi-Riemannian geometry and general relativity Hans Ringström May 5, 2015 2 Contents 1 Scalar product spaces 1 1.1 Scalar products...................................... 1 1.2 Orthonormal
More informationMetrisability of Painleve equations and Hamiltonian systems of hydrodynamic type
Metrisability of Painleve equations and Hamiltonian systems of hydrodynamic type Felipe Contatto Department of Applied Mathematics and Theoretical Physics University of Cambridge felipe.contatto@damtp.cam.ac.uk
More informationRiemannian Curvature Functionals: Lecture I
Riemannian Curvature Functionals: Lecture I Jeff Viaclovsky Park City athematics Institute July 16, 2013 Overview of lectures The goal of these lectures is to gain an understanding of critical points of
More informationConnections for noncommutative tori
Levi-Civita connections for noncommutative tori reference: SIGMA 9 (2013), 071 NCG Festival, TAMU, 2014 In honor of Henri, a long-time friend Connections One of the most basic notions in differential geometry
More informationOn Flux Quantization in F-Theory
On Flux Quantization in F-Theory Raffaele Savelli MPI - Munich Bad Honnef, March 2011 Based on work with A. Collinucci, arxiv: 1011.6388 Motivations Motivations The recent attempts to find UV-completions
More informationTowards new non-geometric backgrounds
Towards new non-geometric backgrounds Erik Plauschinn University of Padova Ringberg 30.07.204 this talk is based on... This talk is based on T-duality revisited [arxiv:30.494], and on some work in progress
More informationHigher dimensional Kerr-Schild spacetimes 1
Higher dimensional Kerr-Schild spacetimes 1 Marcello Ortaggio Institute of Mathematics Academy of Sciences of the Czech Republic Bremen August 2008 1 Joint work with V. Pravda and A. Pravdová, arxiv:0808.2165
More informationDouble field theory. Citation. As Published Publisher. Version
Double field theory The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Hull, Chris, and Barton Zwiebach.
More informationGeometry for Physicists
Hung Nguyen-Schafer Jan-Philip Schmidt Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers 4 i Springer Contents 1 General Basis and Bra-Ket Notation 1 1.1 Introduction to
More information