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Winding Corrected Five-Branes in Double Field Theory

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Winding Corrected Five-Branes in Double Field Theory Worldvolume Theory of Winding Corrected Five-branes in Double Field Theory Kenta Shiozawa (Kitasato University, Japan) 17 Dec. 2019 Miami2019 conference @ Fort Lauderdale JHEP07(2018)001, JHEP12(2018)095 (joint work with Tetsuji Kimura and Shin Sasaki) and work in progress (with Shin Sasaki) Menu 1 Introduction 2 Double Field Theory 3 DFT solutions 4 Instanton corrections to spacetime geometry 5 Worldvolume theory in DFT 6 Summary 2 / 40 Menu 1 Introduction 2 Double Field Theory 3 DFT solutions 4 Instanton corrections to spacetime geometry 5 Worldvolume theory in DFT 6 Summary 3 / 40 Spacetime Geometry We are interested in Spacetime Geometry considering string effects What is the difference between Einstein and string theories? Einstein gravity (Supergravity) probed by point particles described by a Riemannian manifold String Theory probed by strings described by new stringy geometry What are the characteristic quantities in stringy geometry? 4 / 40 T-duality Strings on compact spaces S1 can have momentum along compact directions ! KK mode n can wind compact directions (unlike point particles) ! winding mode w String theory is invariant under exchange of KK-mode n and winding mode w ( ) ( ) ( ) ( )2 2 2 n R n w E = + w T : $ 0 R α0 R α =R This invariance is known as T-duality. 5 / 40 T-duality orbit Background geometries are mapped into each other under T-duality. e.g. Dp-brane $ D(p − 1)-brane F1-string $ pp-wave In particular, solitonic five-brane: NS5-brane ! Taub-NUT space (KK5-brane) 6 / 40 A Puzzle for 5-branes Analysis by [Gregory-Harvey-Moore '97] Smeared NS5 on S1 ! Taub-NUT space n w tower of KK-modes tower of winding modes Smeared NS5 geometry receives KK-mode corrections There is a tower of winding modes by Taub-NUT side ! Does the Taub-NUT geometry receive string winding corrections? | yes. 7 / 40 Dual coordinate dependence Smeared NS5 becomes NS5-brane geometry by KK-mode corrections [Tong '02] NS5-brane geometry: gµν = gµν (x; y) y: Fourier dual of KK momentum Taub-NUT space is modified by string winding corrections [Harvey-Jensen '05] Modified Taub-NUT space: gµν = gµν (x; y~) y~: Fourier dual of winding momentum | has winding coordinate dependence The corrections break isometries of the geometries ! this means \Non-isometric T-duality" [Dabholkar-Hull '05] NS5-brane ! Modified Taub-NUT 8 / 40 2 Exotic 52-brane c Branes can be labelled as bn-branes c bn b: spatial dimensions of worldvolume c: number of isometry directions ∼ −n n: mass of brane (tension: Tbrane gs ) 0 1 e.g. NS5-brane = 52, KK5-brane (Taub-NUT) = 52, ... 9 / 40 2 Exotic 52-brane T-duality relation of background geometries extends as follows ! ! 2 NS5 Taub-NUT 52 [de Boer-Shigemori '10, '13] 2 52-brane geometry is modified by winding corrections [Kimura-Sasaki '13] 2 Modified 52 geometry: gµν = gµν (x; y~1) y~1: Fourier dual of winding momentum | has winding coordinate dependence \Non-isometric T-duality" relation also extends as ! ! 2 NS5-brane Modified Taub-NUT Modified 52-brane 10 / 40 Winding corrections to geometries Winding corrected geometries Modified Taub-NUT space 2 Modified 52-brane geometry are NOT solutions to type II supergravity ! what kind of solutions are they? | solutions to double field theory 11 / 40 Menu 1 Introduction 2 Double Field Theory 3 DFT solutions 4 Instanton corrections to spacetime geometry 5 Worldvolume theory in DFT 6 Summary 12 / 40 Double Field Theory (DFT) A manifestly T-duality invariant formulation of supergravity theory [Hull-Zwiebach '09] Double Field Theory DFT is a gravity theory defined on the doubled space M2D = M D × M~ D 2D M µ M is parametrized by the generalized coordinates X = (~xµ; x ) momentum p T-dual−−−−! winding momentum w Fourier conj l Fourier conj. Fourier conj l Fourier conj. ordinary coord. x winding coord. x~ M D M~ D Dynamical fields in DFT: \generalized metric" HMN (X) and \DFT dilaton" d(X) HMN is parametrized by metric g (on M D) and NSNS B-field B : ( µν ) µν − ρσ ρν HMN gµν Bµρg Bσν Bµρg = µρ µν −g Bρν g p d is rescaled the dilaton ϕ: e−2d = −ge−2ϕ 14 / 40 DFT action DFT action: represented by Einstein-Hilbert form [Hohm-Hull-Zwiebach '10] Z 2D −2d SDFT = d X e R(H; d) T-duality is a manifest symmetry of the theory DFT never requires isometries ! inherits \Non-isometric T-duality" [Dabholkar-Hull '05] The equation of motion obtained from SDFT is the generalized Einstein eq. 1 R − H R = 0 MN 2 MN DFT is formulated in the same form as Einstein gravity theory, except that the space is doubled to respect T-duality. 15 / 40 Strong Constraint DFT has extra degrees of freedom since the base space is doubled. It is necessary to reduce. The constraint that makes DFT be a physical theory is called the strong constraint MN η @M ∗ @N ∗ = 0 The strong constraint originally came from the level matching condition of closed strings By choosing one of the solution to the strong constraint @M = (0;@µ), DFT reduces to the NSNS sector of type II supergravity Z [ ] ~ p 1 S −−−!@=0 S = dDx −ge−2ϕ R + 4(@ϕ)2 − H2 DFT sugra 12 DFT ⊃ Supergravity 16 / 40 Menu 1 Introduction 2 Double Field Theory 3 DFT solutions 4 Instanton corrections to spacetime geometry 5 Worldvolume theory in DFT 6 Summary 17 / 40 Classical solutions of DFT These known supergravity solutions are classical solutions of DFT: [Berkeley-Berman-Rudolph '14] [Berman-Rudolph '15] [Bakhmatov-Kleinschmidt-Musaev '16] [and more ...] NS5-brane Taub-NUT space F-string pp-wave 2 52-brane and so on 18 / 40 Non-supergravity DFT solution We are interested in the non-supergravity solutions peculiar to DFT DFT inherits \Non-isometric T-duality" T-duality transformation without isometries is possible We find new solutions in DFT Modified Taub-NUT space 2 Modified 52-brane Locally non-geometric R-brane and so on 19 / 40 New solutions Modified Taub-NUT space 2 m n −1 9 i 2 i j ds = ηmndx dx + H (dy + bi9dy ) + Hδijdy dy ; i j B = bij dy ^ dy ; 2ϕ d e = const:; 3@[abbc] = "abcd@ H (i; j = 6; 7; 8; a; b; c; d = 6; 7; 8; 9; m; n = 0;:::; 5) Harmonic function of this solution: 2 3 ( ) X i Q 4 y~9 i n 5 H(y ; y~9) = c + 1 + exp in − jy j jyij ~ ~ n=06 R9 R9 | has winding coordinate dependence ! this exhibits exponential behavior 20 / 40 New solutions 2 Modified 52-brane [Kimura-Sasaki-K.S. '18] 2 m n α β ds = ηmn dx dx + Hδαβ dy dy H [ ] + (dy9 + b dyα)2 + (dy8 + b dyα)2 ; H2 + b2 α9 α8 89 h i α ^ β − b89 8 α ^ 9 β B = bαβ dy dy 2 2 (dy + bα8dy ) (dy + bβ9dy ) ; H + b89 2ϕ H e = 2 2 ; (α; β = 6; 7) H + b89 Harmonic function of this solution: Q µ H ' log ~ ~ jyαj 2πR8R9 2 3 s( ) ( ) ( ) Q X m 2 n 2 y~ y~ + exp 4−jyαj + + i m 8 + n 9 5 2πR~ R~ R~ R~ R~ R~ 8 9 n;m=(06 ;0) 8 9 8 9 | has winding coordinate dependence 21 / 40 New solutions Locally non-geometric R-brane [Kimura-Sasaki-K.S. '18] ds2 = η dxmdxn + H(dy6)2 mn " # ( )2 1 ^ + K−1 H2(dy + b dy6)2 + "^{|^kb (dy + b dy6) ; 2 k^ 6k^ 2 ^{|^ k^ 6k^ Hb^{|^ 6 ^ B = − (dy^{ + b6^{dy ) ^ dy|^; (^{; |;^ k = 7; 8; 9) K2 2ϕ −1 ≡ 2 2 2 2 e = HK2 ;K2 H(H + b89 + b79 + b78): Harmonic function of this solution: [ ( )] X 1 y~ y~ y~ H = const: − Cjy6j + C exp −jy6js + i l 7 + m 8 + n 9 s R~ R~ R~ l;m;n=(06 ;0;0) 7 8 9 s( ) ( ) ( ) Q l 2 m 2 n 2 C = ; s = + + 4πR~7R~8R~9 R~7 R~8 R~9 R-brane has No corresponding solution in supergravity R-brane is mysterious object 22 / 40 New solutions We obtain various five-brane solutions in DFT: [Kimura-Sasaki-K.S. '18] 1 2 3 4 52 52 52 52 52 1 2 2 3 3 4 4 codim 4 NS5 KK5 + w 52 + w 52 + w 52 + w 2 1 3 2 4 3 codim 3 sNS5 KK5 52 + w 52 + w 52 + w 2 3 1 4 2 codim 2 dsNS5 sKK5 52 52 + w 52 + w 2 3 4 codim 1 tsNS5 dsKK5 s52 52 52 2 3 4 codim 0 qsNS5 tsKK5 ds52 s52 52 black: the supergravity solutions blue: the previously known solutions red: newly obtained solutions by our calculation DFT solutions naturally have the winding coordinate dependence Winding coordinate dependence shows exponential behavior 23 / 40 The exponential behavior suggests: the winding corrections are interpreted as the instanton corrections to the spacetime − e Sinst. 24 / 40 Menu 1 Introduction 2 Double Field Theory 3 DFT solutions 4 Instanton corrections to spacetime geometry 5 Worldvolume theory in DFT 6 Summary 25 / 40 Winding dependence as worldsheet instanton effects We discuss an interpretation of the winding correction to geometries as the string worldsheet instanton effects. Worldsheet instanton is a map ' from worldsheet to 2-cycle in target space. [Wen-Witten '86] [figure from Witten] 26 / 40 Instantons in single-centered Taub-NUT Single-centered Taub-NUT space 2 2 2 −1 9 a 2 ds = dx012345 + Hdx678 + H (dx + Aadx ) ; 1 Q H = + ; r2 = (x6)2 + (x7)2 + (x8)2; (a = 6; 7; 8) g2 r Taub-NUT space is S1 fibration over R3 radius of S1 is given by H−1 at center r = 0, fibered S1 shrinks to zero asymptotic radius is H−1(1) = g2 cigar ∼ D2 n# ~v becomes an open cigar = disk D2 ~r ~ra ~ra + f(jzj)~v 27 / 40 Instantons in single-centered Taub-NUT 2 2 Worldsheet Σ = S as the set of disk DΣ and infinity: 2 [ f1 g Σ = DΣ Σ : In the Taub-NUT case, worldsheet instanton is a map from disk to cigar.
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