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 : ↔ ′ 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)(∂ñtb) + (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