double field theory & non-relativistic string theoryqft.web/2015/slides/meyer.pdf · double...
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Double Field Theory &Non-Relativistic String Theory
based on 1508.01121, withSung Moon Ko, Charles Melby-Thompson and Jeong-Hyuck Park
René MeyerStony Brook University
Double Field Theory?
R $ ↵0
R
T-Duality
Gµ⌫
Bµ⌫
�
Gµ⌫
Bµ⌫
�
T. H. BuscherPhys. Lett. B194B, 59 (1987)Phys. Lett. B201, 466 (1988)
Double Field Theory?
R $ ↵0
R
T-Duality
T. H. BuscherPhys. Lett. B194B, 59 (1987)Phys. Lett. B201, 466 (1988)
along X25circle
Double Field Theory?
R $ ↵0
R
T-Duality
T. H. BuscherPhys. Lett. B194B, 59 (1987)Phys. Lett. B201, 466 (1988)
Eµ⌫ = Gµ⌫ +Bµ⌫
O(D,D) T-duality transformation
Double Field Theory?
R $ ↵0
R
T-Duality
M. Duff, Tseytlin & Siegel
HAB =
✓G�1 �G�1BBG�1 G�BG�1B
◆2 O(D,D)
e�2d =p�Ge�2�
Double Field Theory!Hull & Zwiebach, Siegel, Park et. al., Hohm, …
) LDFT =p�ge�2�
�R+ 4(@�)2 � 1
12H2µ⌫⇢
�
Geometry or No Geometry
HAB =
✓G�1 �G�1BBG�1 G�BG�1B
◆2 O(D,D)
e�2d =p�Ge�2�
Globally Geometric
Everywhere Nongeometric(Non-Riemannian)
Locally Geometric &Globally Nongeometric
T-fold
Geometry or No Geometry
HAB =
✓G�1 �G�1BBG�1 G�BG�1B
◆2 O(D,D)
e�2d =p�Ge�2�
Globally Geometric
Everywhere Nongeometric(Non-Riemannian)
Locally Geometric &Globally Nongeometric
T-fold
Non-relativistic Closed StringJ. Gomis & H. Ooguri hep-th/0009181
Rx
1
B01 = B
ds
2 = c
2��dt
2 + (dx1)2�
| {z }⌘↵�dx
↵dx
�
+d~x
2
Lagrange multiplies �, �, �, � survive
DFT sigma modelK.-H. Lee & J.-H. Park hep-th/1307.8377
Section condition on fields and arbitrary products: @A@A(...) = 0
Equivalent “coordinate gauge invariance”, e.g. �(xA + �(x)@A'(x)) = �(xA)
Doubled space-time: x
A = (xµ, x⌫)
Coordinate gauge field and covariant one forms: Dx
M = dx
M �AM
AM = (Aµ, Aµ)
HMN =
0
BB@
0 0 E↵� 0
0 �ij 0 0�E↵� 0 2µ⌘↵� 00 0 0 �ij
1
CCA
� = X1 +X0 � = X1 �X0
A+ = � A� = ��
X↵ = 0
Fluctuations in DFTS.-M. Ko, C. Melby-Thompson, RM, J.-H. Park, hep-th/1508.01121
L = 18e
�2d⇥(PACPBD � PAC PBD)SABCD � 2⇤
⇤ PAB =1
2(JAB +HAB)
PAB =1
2(JAB �HAB)
SABCD =1
2(RABCD +RCDAB � �E
AB�ECD)
Fluctuations around Gomis-OoguriS.-M. Ko, C. Melby-Thompson, RM, J.-H. Park, hep-th/1508.01121
Show: Kaluza-Klein Sector of Gomis-Ooguri string is empty:
Lin. Gen. Diffeo.:
hAB = hBA HACHCB = JAB
hAB =
0
BB@
h↵� h↵i h↵� h↵
j
hi↵ hij hi↵ hi
j
h↵� h↵
j h↵� h↵j
hi↵ hi
j hi↵ hij
1
CCA =
0
BBBB@
h⌘↵� ��↵�h
�kgki �µhE↵
� h↵j
gimhmngnj �gikbkj�2µ⌘↵�h�
kgki = 0 = 0= 0 hij
1
CCCCA
Fourier expansion in plane waves:No propagating normalizable
wave packet solution if there exists no solution with
k2 6= 0 and (p� 6= 0 or p+ 6= 0)
T-dual frame and Winding ModesS.-M. Ko, C. Melby-Thompson, RM, J.-H. Park, hep-th/1508.01121
HMN =
0
BB@
0 0 E↵� 0
0 �ij 0 0�E↵� 0 2µ⌘↵� 00 0 0 �ij
1
CCA T-duality along x1:
ds
2 = �2µdt2 + 2dtdx+ (dxi)2
T-duality along x1: (lowest modes only)
E = µp✓ +k2
p✓
KK Spectrum in dual frame = Gomis-Ooguri Winding Spectrum!
Galilean Invariance in DFTS.-M. Ko, C. Melby-Thompson, RM, J.-H. Park, hep-th/1508.01121
Galilean invariant spectrum:
Generators of Bargmann algebra (central extension of Galilean):
Circle U(1):
U(1) particle number:
Galilean Boosts:
Close into the Bargmann Algebra under the C-Bracket:
Doubled Schrödinger GeometryS.-M. Ko, C. Melby-Thompson, RM, J.-H. Park, hep-th/1508.01121
Non relativistic scale invariance:
Schrödinger invariant DFT geometry: (no solution)
ConclusionsS.-M. Ko, C. Melby-Thompson, RM, J.-H. Park, hep-th/1508.01121
DFT: T-duality invariant low energy effective action for massless string modes
Geometric as well as non-geometric string backgrounds
Non-relativistic Closed String (NRCS) J. Gomis & H. Ooguri hep-th/0009181
Concrete, healthy example of a non-relativistic, Galilean invariant string vacuum
NRCS is a locally non-geometric background of DFT sigma model K.-H. Lee & J.-H. Park hep-th/1307.8377
DFT correctly captures the (empty) KK and non relativistic winding spectrum
New Schrödinger invariant background
New non-geometric string vacua for Cosmology?(Non)relativistic Holography? …