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 : ↔ ′ R α′ 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 ↕ Fourier conj. Fourier conj ↕ 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 √ 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] ∫ 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 ∫ [ ] ˜ √ 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] = εabcd∂ H (i, j = 6, 7, 8; a, b, c, d = 6, 7, 8, 9; m, n = 0,..., 5)
Harmonic function of this solution: ( ) ∑ i Q y˜9 i n H(y , y˜9) = c + 1 + exp in − |y | |yi| ˜ ˜ n=0̸ 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 [ ] 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 ˜ ˜ |yα| 2πR8R9 √( ) ( ) ( ) Q ∑ m 2 n 2 y˜ y˜ + exp −|yα| + + i m 8 + n 9 2πR˜ R˜ R˜ R˜ R˜ R˜ 8 9 n,m=(0̸ ,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 ˆ + K−1 H2(dy + b dy6)2 + εˆıȷˆkb (dy + b dy6) , 2 kˆ 6kˆ 2 ˆıȷˆ kˆ 6kˆ
Hbˆıȷˆ 6 ˆ B = − (dyˆı + b6ˆıdy ) ∧ dyȷˆ, (ˆı, ȷ,ˆ k = 7, 8, 9) K2 2ϕ −1 ≡ 2 2 2 2 e = HK2 ,K2 H(H + b89 + b79 + b78). Harmonic function of this solution: [ ( )] ∑ 1 y˜ y˜ y˜ H = const. − C|y6| + C exp −|y6|s + i l 7 + m 8 + n 9 s R˜ R˜ R˜ l,m,n=(0̸ ,0,0) 7 8 9 √( ) ( ) ( ) 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 φ 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(∞) = g2
cigar ∼ D2
nϑ ~v becomes an open cigar = disk D2
~r ~ra ~ra + f(|z|)~v
27 / 40 Instantons in single-centered Taub-NUT
2 2 Worldsheet Σ = S as the set of disk DΣ and infinity: 2 ∪ {∞ } Σ = DΣ Σ .
In the Taub-NUT case, worldsheet instanton is a map from disk to cigar.
2 2 DΣ cigar ∼ D nϑ 1 mapping =⇒ ~v
|z| = A ~r ~ra ~ra + f(|z|)~v
2 { ∈ C | | } → { | | 9 } DΣ = x : z < 1 nCa = ⃗r(z) = ⃗ra + f( z )⃗v, x (z) = n arg(z) , f(0) = 0, f(1) = ∞.
This is called the “Disk instanton”. 28 / 40 Instantons in single-centered Taub-NUT
2 2 Worldsheet Σ = S as the set of disk DΣ and infinity: 2 ∪ {∞ } Σ = DΣ Σ . Taub-NUT cigar In the limit g → 0, the cigar is closed at infinity. The cigar becomes topologically S2.
We have the following relation:
[Disk instanton] −−−→ [Worldsheet instanton] g→0
It reproduces the modified Taub-NUT solution in DFT [Kimura-Sasaki-K.S. ’18] ( ) ∑ ′ Q x˜9 ′ n H(r , x˜9) = c + 1 + exp in − r r′ ˜ ˜ n=0̸ R9 R9
28 / 40 2 Instantons in 52-brane geometry
2 52-brane geometry [de Boer-Shigemori ’13] 2 2 2 2 2 −1 8 2 9 2 ds = dx012345 + H(dρ + ρ dθ ) + HK ((dx ) + (dx ) ), µ H = h + σ log , ρ2 = (x6)2 + (x7)2,K = H2 + (σθ)2 0 ρ This situation is similar to the Taub-NUT case. this geometry is T 2 = S1 × S1 fibration over R2. the physical radius of each S1 is given by HK−1. at the origin ρ → 0, we have HK−1 → 0 −1 h0 at the cutoff scale ρ = µ, HK = 2 2 < ∞ is finite h0+A8
29 / 40 2 Instantons in 52-brane geometry
2 2 Analogous to Taub-NUT case, in 52 geometry, a general 1-cycle in T (n1, n2) is fibered over the segment ρ ∈ [0, µ] and defines an open cigar.
We give a generalization of the disk instanton [Kimura-Sasaki-K.S. ’18]
D2 = {z ∈ C : |z| < 1} Σ { } 8 9 −→ (n1, n2)C = ⃗ρ(z) = ⃗ρ0 + f(|z|)⃗v, x (z) = n1 arg(z), x (z) = n2 arg(z) , f(0) = 0, f(1) = µ.
T 2 fiber 2 DΣ
(n1, n2)-cycle mapping 1 =⇒ ~v
~ρ |z| = A ~ρa ~ρa + f(|z|)~v
In the limit h0 → ∞, the cigars are closed at ρ ∼ µ 29 / 40 n Instanton corrections to the 52 -brane geometry
The discussion of worldsheet instanton corrections to R-brane geometry is completely parallel
3 3 R R-brane geometry (52-brane): T -fibration over
3 The homology H1(T ) = Z ⊕ Z ⊕ Z: Disk instantons are labelled by three integers (n1, n2, n3)
4 This picture fails when we move to space-filling 52-branes
30 / 40 Gauged Linear Sigma Model
The worldsheet instanton effects are well understand in terms of 2D GLSMs
Some GLSMs are UV completions of string worldsheet NLSMs in specific target space GLSM ====IR limit⇒ String NLSM
The gauge instantons in GLSM is reduced to the worldsheet instantons in the IR limit
2D gauge theory instanton ====IR limit⇒ Worldsheet instanton
[Witten ’93, Tong ’02]
31 / 40 GLSM for 5-branes of codimension two
We construct full GLSM for 5-branes of codimension two [Kimura-Sasaki-K.S. ’18]
2d N = (4, 4) SUSY theory
2 describes defect NS5-branes, KK-vortices, and exotic 52-branes simultaneously (“semi-doubled” GLSM)
reproduces NLSM for these brane backgrounds in IR regime
instanton calculus is possible
32 / 40 Menu
1 Introduction
2 Double Field Theory
3 DFT solutions
4 Instanton corrections to spacetime geometry
5 Worldvolume theory in DFT
6 Summary
33 / 40 Five-brane worldvolume effective theory in DFT
In the above discussion, we obtained how the winding dependence is given
However, we did not discuss physical properties of locally non-geometric solutions (locally non-geometric means that a solution depends on the winding coordinate)
In particular, we would like to understand about the energy gained by the winding coordinate dependence
We expect that winding depended five-branes have fluctuation along the winding space directions
We focus on the brane fluctuation effective action in DFT
34 / 40 DFT five-brane fluctuation effective action
We decompose the generalized metric as follows 2 2 2 dsDFT = dsdoubled worldvolume + dsdoubled transverse space H Mˆ Mˆ Nˆ H Mˇ A Mˆ Mˇ Nˇ = Mˆ Nˆ (X )dX dX + Mˇ Nˇ (Y , t (X ))dY dY XMˆ (Mˆ = 1,..., 12): doubled worldvolume coordinates Y Mˇ (Mˇ = 1,..., 8): doubled transverse coordinates H Mˆ A Mˇ Nˇ in transverse dir. depends on X through parameters t
A a → parameters t = (t , t˜a) represent scalar fields on the worldvolume
Under this decomposition of HMN , DFT action becomes ∫ ∫ [ ( ) ( ) 1 ∂ ∂ S = d12X d8Y e−2d HMˆ Nˆ (∂ tA) HKˇ Lˇ (∂ tB) H 8 Mˆ ∂tA Nˆ ∂tB Kˇ Lˇ ] + Rˆ(X) + Rˇ(t(X),Y )
Rˆ(X), Rˇ(t(X),Y ) are potential term → now, we focus on the kinetic term of tA 35 / 40 DFT five-brane fluctuation effective action
We consider the DFT five-brane solution (it contains NS5-, KK5-, 52-, R- and 54-brane simultaneously) ( 2 2 ) ( ) − −2 c −1 b mn H H(δab H bacb b) H ba HMˆ Nˆ η 0 Mˇ Nˇ = −1 a −1 ab , = −H b b H δ 0 ηmn Q H(y, t) = c + , (a = {6, 7, 8, 9}) (ya − ta)2
−2d −2ϕ0 e = He , ϕ0 = const. We obtain the doubled action of scalar fields on 5-branes in DFT ∫ [ ] − 1 1 S = d6xd6x˜ e 2ϕ0 ηmn(∂ ta)(∂ tb) + η (∂˜mta)(∂˜ntb) + (potential) 2 m n 2 mn This action contains [work in progress] NS5-brane: (t6, t7, t8, t9) = (X6, X7, X8, X9) 6 7 8 9 6 7 8 KK5-brane: (t , t , t , t ) = (X , X , X , X˜9) 2 6 7 8 9 6 7 ˜ ˜ 52-brane: (t , t , t , t ) = (X , X , X8, X9) 6 7 8 9 6 R-brane: (t , t , t , t ) = (X , X˜7, X˜8, X˜9) 4 6 7 8 9 ˜ ˜ ˜ ˜ 52-brane: (t , t , t , t ) = (X6, X7, X8, X9) 35 / 40 DFT five-brane worldvolume effective action
The above discussion agrees with the discussion of five-brane DBI actions in DFT [Blair-Musaev ’17]
Winding depended five-branes are locally non-geometric and mysterious objects → it may be able to explore these physical properties by using w.v. action
However, the above discussion is only for the kinetic term of scalar fields → It is necessary to discuss the potential and the gauge field term
Therefore, the worldvolume theory of winding depended five-brane in DFT is under construction and is incomplete
In supergravity, winding direction is isometry, but it breaks in this case → we expect that scalar fields corresponding winding coordinate are interpreted as Goldstone modes
36 / 40 Menu
1 Introduction
2 Double Field Theory
3 DFT solutions
4 Instanton corrections to spacetime geometry
5 Worldvolume theory in DFT
6 Summary
37 / 40 Summary
There is many mysterious branes in string theory. The winding coordinate dependence is necessary to understand these branes.
DFT is useful tool to examine the winding coordinate dependence.
We obtained the DFT solutions that naturally has the winding coordinate dependence.
We interpreted the winding coordinate dependence of DFT solutions as the string worldsheet instanton corrections to supergravity solutions.
We discussed the action of scalar fields on the five-branes in DFT.
38 / 40 Future direction
Geometry behind DFT (doubled geometry) [Mori-Sasaki-K.S. ’19]
U-duality — Exceptional Field Theories (EFT)
∼ −3 −4 Heavy branes are mysterious objects gs , gs ,... Winding dependence — deformed supergravities, integrable deformations, ...
Poisson-Lie T-duality — DFT on Group manifolds
Acknowledgment: This talk is supported by Sasakawa Scientific Research Grant from Japan Science Society.
39 / 40 Thank you for your attention
40 / 40 Extra Material
41 / 40 Buscher Rule
The Buscher rule for T-duality along X9 direction: − ′ − G9iG9j B9iB9j ′ B9i ′ 1 Gij = Gij ,G9i = ,G99 = , G99 G99 G99 − ′ − G9iB9j B9iG9j ′ G9i Bij = Bij ,B9i = . G99 G99
In DFT, the Buscher rule is encoded by O(D,D) transf. Factorized T-duality along k-th dir.: ( ) 1 − ek ek hk = ∈ O(D,D), ek = diag(0,..., 0, 1, 0,..., 0) ek 1 − ek
The Buscher rule:
′ ′ K L HMN (G ,B ) = (hk)M (hk)N HKL(G, B)
42 / 40