work with neil lambert, to appear at some time - florence theory
TRANSCRIPT
(2,0) THEORY: DECOUPLING LIMIT OF M5-BRANE
• (2,0) Superconformal theory in 5+1 dimensions, 16 SUSYs and SO(5) R-symmetry, with ADE group G.
• Reduces to 4+1 SYM with gauge group G.
• If G abelian: 5 scalars XI, 2-form gauge field B, H=*H, H=dB.
•Non-abelian: no conventional field theory? No classical limit?
• Can construct from IIB on K3 at ADE singularity, or matrix model, or CFT dual to AdS7xS4 Other constructions?
Tuesday, 17 September 13
SPACE-TIME SIGNATURE
• Physics: Lorentzian signature, one time
• Path integral: Euclidean signature. All dimensions on same footing. Issues with e.g SUSY as some fields complexified
• Could there be regions of the universe (or multiverse) with different signature?
•Does M-theory or SUSY allow other signatures?
Tuesday, 17 September 13
COULD TIME BE PERIODIC?
• Classical physics allows e.g. flat space with compact time.
• Can a consistent quantum theory be defined?
•Quantum: can solve wave equation, do functional integral, calculate correlation functions
•Quantum interpretation? Issues with probability and measurement. Result of any “experiment” would be the same as last time!
Rn ⇥ S1
Tuesday, 17 September 13
DOES M-THEORY ALLOW COMPACT TIME?
•Does M-theory allow a time circle?
• If so, then there will be dualities associated with the time circle, just as there are for spatial circles
• Surprise: the U-dualities involving a time circle include transformations changing spacetime signature! CH
• All signatures that arise are fully consistent with SUSY
• Seek further evidence for this?
R10 ⇥ S1time
Tuesday, 17 September 13
EMERGENT TIME & SPACE
• Time and space may not be fundamental
• Cosmology: Time or Space might “emerge” from non-geometric phase e.g. matrix model
• Planck scale: physics may be discrete, with geometry emerging at larger “scales”. Discrete theory may resolve singularities
• Holography: 4D SYM gains 6 dimensions: 10D string, AdS5xS5
Tuesday, 17 September 13
A LA RECHERCHE DU TEMPS PERDU
• Can a timeless Euclidean theory gain a “hidden” time dimension in the quantum theory?
• e.g. de Sitter holography: Euclidean CFT gets holographic time
•New insights into time and dynamics?
• Issues with what is meant by Euclidean quantisation
Tuesday, 17 September 13
DIMENSIONS EMERGENT AT STRONG COUPLING
• Some strong coupling limits (e.g. IIA string) give extra space dimension.
• Can some give extra time dimension?
• Can theory in Euclidean space Rd become a theory in d+1 dimensional Minkowski space at strong coupling?
• Controlled situation to study “emergent” time, giving insight into time and emergent theory? Extra dims typically compact.
Tuesday, 17 September 13
SPACE DIMENSION FROM STRONG COUPLING
• Tower of 0-branes. BPS, so can extrapolate to strong coupling
• Infinite tower of 0-branes become massless at infinite coupling
• Interpret as decompactification of extra dimension, with 0-branes as Kaluza-Klein modes
• Further checks: e.g. compare BPS spectra, seek evidence of higher dimensional Lorentz symmetry
Tuesday, 17 September 13
M-THEORY FROM IIA STRING
• IIA string: D0 brane states
• Interpret as KK states for circle
• IIA string at strong coupling is M-theory on spatial circle
• Problem: do not have intrinsic formulation of M-theory, or of IIA string at strong coupling
M / n
g s
M / n
R
R / gs
Witten
Tuesday, 17 September 13
(2,0) THEORY FROM SYM
• Super Yang-Mills in 4+1 with 16 SUSY, SO(5) R-symmetry
• Soliton: (YM instanton in R4) x (time)
• Interpret as KK states for circle
• SYM at strong coupling is (2,0) theory on spatial S1
• Problem: do not have intrinsic formulation of (2,0) theory, or of SYM at strong coupling
M / n
R
R =g2
4⇡2
M =4⇡2|n|g2
Rozali, Witten
Tuesday, 17 September 13
5D SYM
• SYM in 4+1 non-renormalisable, 6-loop divergence
• Embed in UV complete theory, e.g. string theory.
•D4 brane world-volume theory
• At strong coupling: M5 brane wrapped on M-theory S1
• (2,0) theory (decoupling limit of) M5-brane world-volume theory
g2 = (2⇡)2↵01/2gs
Bern et al
Tuesday, 17 September 13
SYM IN 5 EUCLIDEAN DIMENSIONS
• Super Yang-Mills in 5 Euclidean dimensions with 16 SUSY, SO(5) R-symmetry
• Seek evidence that at strong coupling an extra TIME dimension opens up
• Euclidean SYM at strong coupling is (2,0) theory on timelike S1
• Simple argument: This theory and usual 4+1 SYM defined by SAME Euclidean path integral, but with different continuations back to real section. If one gets extra dimension, both do. SUSY fixes signature.
R =g2
4⇡2
Tuesday, 17 September 13
EUCLIDEAN SUSY?• Consider Lorentzian SUSY theory, e.g. SYM in 3+1
• Path integral: Wick rotate to “Euclideanised theory” with positive bosonic action. Fermions typically complex, but integrate over , not . Euclideanised theory NOT SUSY, but its complexification is.
• Euclidean theory: version in Euclidean signature with real fields, real SUSY. Can twist to get TFT. ALWAYS has some fields with kinetic term of wrong sign. Quantum: Continue to SAME Euclideanised theory.
� �?
Tuesday, 17 September 13
•e.g. N=4 SYM in 3+1. SO(3,1) Lorentz, SO(6) R-symmetry. 16 real fermions, 16 real SUSYs.
•Euclideanised SYM: SO(4) Lorentz, SO(6) R-symmetry. 16 complex fermions NOT SUSY
•Euclidean SYM: SO(4) Lorentz, SO(5,1) R-symmetry. 16 real fermions, 16 real SUSYs. Can twist to get TFT. One of scalars has wrong sign kinetic term. If , R-sym becomes SO(6)�6 ! i�6
Tuesday, 17 September 13
•3+1 N=4 SYM: Superconformal group contains SO(4,2)xSO(6). Isometry of AdS5 xS5: AdS holography
•Euclideanised N=4 SYM: conformal group SO(5,1)xSO(6). NO Superconformal group. Isometry of H5xS5: Euclideanised AdS holography
•Euclidean N=4 SYM: Superconformal group contains SO(5,1)xSO(5,1). Isometry of dS5 xH5 de Sitter holography: An extra TIME dimension at strong ‘t Hooft coupling. CH
Tuesday, 17 September 13
CONSTRUCTING EUCLIDEAN SYM
• Reduce SYM from 9+1 on d+1 dimensions gives Euclidean SYM in D=9-d with SO(d,1) R-symmetry. D=5: SO(4,1) R-symmetry
• Compactify (2,0) on time: 5+0 SYM with SO(5) R-symmetry. Want this Euclidean SYM, not one with SO(4,1) R symmetry
• Can get this theory from SYM in 5+5 dimensions by reducing on 5 time dimensions
Tuesday, 17 September 13
EUCLIDEAN SYM
Space indices i,j=1,..,5Internal indices I,J=6,7,...,10
S =1
4g2tr
Zd
5x
h14FijF
ij � 1
2DiX
ID
iX
I +1
4[XI
, X
J ]2
� i
2
T�0�iDi � 1
2
T�I [XI, ]
i
Conserved currents Tij J =1
8g2⇤ tr(F ^ F )
Tuesday, 17 September 13
EUCLIDEAN SYM
Space indices i,j=1,..,5Internal indices I,J=6,7,...,10
S =1
4g2tr
Zd
5x
h14FijF
ij � 1
2DiX
ID
iX
I +1
4[XI
, X
J ]2
� i
2
T�0�iDi � 1
2
T�I [XI, ]
i
Analytic continuation to positive bosonic action XI ! iXI
Same Euclidean action as Wick rotation of 4+1 SYM
Conserved currents Tij J =1
8g2⇤ tr(F ^ F )
Tuesday, 17 September 13
ABELIAN (2,0) THEORYS(abelian) = �
Zd6x
h 1
12Hµ⌫⇢H
µ⌫⇢ +1
2@µX
I@µXIi
S = �Z
d5xh 1
12HijkH
ijk +1
2@iX
I@iXIi
S =
Zd
5x
h14FijF
ij � 1
2@iX
I@
iX
Ii
Reduce on time
F = ⇤HDualise
Tuesday, 17 September 13
ABELIAN (2,0) THEORYS(abelian) = �
Zd6x
h 1
12Hµ⌫⇢H
µ⌫⇢ +1
2@µX
I@µXIi
S = �Z
d5xh 1
12HijkH
ijk +1
2@iX
I@iXIi
S =
Zd
5x
h14FijF
ij � 1
2@iX
I@
iX
Ii
Reduce on time
F = ⇤HDualise
5+1 EM tensor ⇥µ⌫ ⇥ij = Tij , ⇥0i = Ji
⇥00 = tr
✓1
4g4FijF
ij +1
2DiX
IDiXI � g4[XI , XJ ]2◆
SUSY, gauge inv. ⇒ true for non-abelian theory too
Tuesday, 17 September 13
4+1 SYM & NEW SPACE DIM.
(YM instanton in R4) x (time): BPS soliton
Topological current J =1
2g2⇤ tr(F ^ F )
M = |K|
K ⌘Z
d
4xJ0 =
4⇡2n
g
2Charge
Tuesday, 17 September 13
4+1 SYM & NEW SPACE DIM.
(YM instanton in R4) x (time): BPS soliton
Interpret as KK states for circle in x5 direction
Topological current J =1
2g2⇤ tr(F ^ F )
M = |K|
K ⌘Z
d
4xJ0 =
4⇡2n
g
2Charge
R =g2
4⇡2P5 =
n
RM = |P5|
P5 =
Zd
4xdx
5 ⇥05 ⇥05 / J0so ifP5 ⇠ K
Tuesday, 17 September 13
5+0 SYM & NEW TIME DIM?
(YM instanton in R4) x (R1): BPS solution extended in x5
Interpret as KK states for circle in x0 direction
Topological current J =1
2g2⇤ tr(F ^ F )
Charge
R =g2
4⇡2
K =
Zd
4xJ5 =
4⇡2n
g
2
E =n
R
If ⇥05 / J5 K ⇠Z
d
4x⇥05
Is this an energy E or a momentum P5 ? What if n -ve? Tuesday, 17 September 13
QUANTISATION IN EUCLIDEAN SIGNATURE
•One way: choose a Euclidean “time” e.g. t=x5
• Canonical formalism, Poisson brackets etc based on t
• Path integral defined by slicing wrt Euclidean “time” t
•Natural if t is Wick rotated time, but not for intrinsically Euclidean theories, as it breaks rotation invariance
• Useful for subsector extended along t direction.
Tuesday, 17 September 13
A TIMELY RESOLUTION
• Treat x5 as a Euclidean time, use canonical formalism for x5
• Interchange roles of x5, x0 in usual picture, get same story
• E is P0 in x5 formalism (has different meaning in x0 picture)
•Null state in 5+1 with x0=x5 gives world line along x0 in 4+1, or E1-brane along x5 in 5+0
• Ep-brane: extended along p spatial dimensions
Tuesday, 17 September 13
EUCLIDEAN CHARGESx
i = (⌧, xa)
Restrict to configurations that fall off sufficiently rapidly in transverse R4.
a = 1, 2, 3, 4
q =
Zd
4x j⌧
Integral over 4-surface of fixed t. Conserved: @⌧ q = 0
Pi =
Zd
4x T⌧i
Generates translations through t Poisson brackets[Pi, ] = @i
Tuesday, 17 September 13
K =1
8g2
Zd
4xJ⌧
Charge for branes from instantons in transverse R4
Q =
Zd
4x S⌧Supercharges
{Q↵, Q�} =2(�iC�1)↵�Pi � 2�↵�K + (�i�IC�1)↵�ZIi + . . .
{Q↵, Q�} =2(�µC�1)↵�Pµ + (�µ�IC�1)↵�ZIµ
+ (�µ⌫��IJC�1)↵�ZIJµ⌫�
Pµ = (P 0 = K, P i)Use K as 0-component:Whole superalgebra can be written in a way suggestive of 5+1 dimensions
Tuesday, 17 September 13
Pa = tr
Zd
4xDaX
ID⌧X
I � 1
g
4FabF⌧
b
P⌧ = tr
Zd
4x
1
4g4FabF
ab � 1
2g4F⌧aF⌧
a � g
2
4[XI
, X
J ][XI, X
J ]
+1
2D⌧X
ID⌧X
I � 1
2DaX
ID
aX
I
P0 =1
8g4tr
Z"⌧bcdeF
bcF
de
Z
I0 =
2
g
2tr
Zd
4xF⌧aD
aX
I � 4ig4D⌧XJ [XJ
, X
I ]
Z
Ia =
1
g
2tr
Zd
4x "⌧abcdF
cdD
bX
I
Z
I⌧ = 0
Z
IJ0a⌧ = �ig
2"
IJKLMtr
Zd
4x [XK
, X
L]DaXM
Z
IJ0ab = � i
2"⌧abcdtr
Zd
4x F
cd[XI, X
J ]
Z
IJab⌧ = itr
Zd
5x Fab[X
I, X
J ]
Tuesday, 17 September 13
LIFT TO 5+1x
i = (⌧, xa)
Restrict to configurations falling off rapidly in transverse R4
Expect to lift to configs in 5+1 falling off in transverse R4
a = 1, 2, 3, 4
x
µ = (x0 = t, x
a, x
5 = ⌧)
Expect quantization based on time t to lift to quantization in 5+1 based on t instead of t.t-independent charges
instead of t-independent charges
q =
Zd
4xdt j⌧
q =
Zd
4xd⌧ jt
Tuesday, 17 September 13
Pµ =
Zd
4xdt ⇥µ⌧
generates translations through t-Poisson brackets
P0 =
Zd
4xdt ⇥0⌧
reduces to K-charge in 5+0 as ⇥i0 ! Ji
As t periodic, charge is quantised P0 =
n
R
Suggests topological charge K identified with for KK modes of 5+1 (2,0) theory compactified on time
K =1
8g2
Zd
4xJ⌧ J =
1
2g2⇤ tr(F ^ F )
P0
R =g2
4⇡2
Tuesday, 17 September 13
MOMENTA
P⌧ =
Zd
4xd⌧ ⇥⌧0 =
Zd
4xdt ⇥⌧0 = P0
Pµ =
Zd
4xdt ⇥µ⌧Pµ =
Zd
4xd⌧ ⇥µt
for configurations that only depend onx
a, t+ ⌧ (a = 1, 2, 3, 4)
so quantized charge arising from instantonic E1-branes is conventional momentum in t direction but is momentum in time direction for t formalism
Tuesday, 17 September 13
TIMES AND QUANTIZATIONS• t canonical quantization breaks SO(d,1) to SO(d),
need to prove Lorentz covariance
•Well suited for states extended along time (world-lines), but perhaps issues for states (if any) localised in time?
• t canonical quantization breaks SO(d,1) to SO(d-1,1), need to establish Lorentz covariance
• Good for states extended along t, but issues for others?
• 5+0 SYM: all BPS states extended along at least one direction
Tuesday, 17 September 13
MATCHING BPS STATESCharged string of (2,0) in 5+1In abelian phase, charged self-dual strings, Charge ZiI
i=1,..,5 labels direction in spaceI=6,...,10 labels R-chargeString with charge Z56 matches with:Charged E1-brane in ESYM
Fa5 = DaX6
A5 = X6 = hX6i � QE
4⇡2r2, Aa = 0
Electrically charged, 1/2 BPS Z65 ! Z6
0 P5 =1
2|Z6
0 |
Tuesday, 17 September 13
FROM A (2,0) THEORY
• A 5+1 field theory with (2,0) SUSY, non-abelian gauge symmetry, adjoint self-dual tensors Has vector field C that takes constant expectation value
• If C spacelike, get 4+1 SYM in orthogonal R4,1
• If C timelike, get 5+0 SYM in orthogonal R5
• Gives suggestive setting of SYM in 5+1 formalism
Lambert+Papageorgakis
Tuesday, 17 September 13
TIME FOR M-THEORY
• Compactify D=11 sugra on time → IIAE sugra in 10+0 dimensions
•Does M-theory on timelike circle give sensible quantum theory? If so, gives IIAE string in 10+0, fundamental E2-brane
• At strong coupling, IIAE string gives M-theory on time circle
• RR E1-branes lift to M-theory KK modes on time circleR = ↵01/2gs
Hull
Tuesday, 17 September 13
M-THEORY, BRANES AND TIME
•M5 branes wrap time circle to give E5 branes of IIAE string
• At strong coupling, E5 branes of IIAE string become M5-branes
• E5-brane world-volume theory is ESYM with SO(5) R-symm.
• So ESYM at strong coupling should give (2,0) theory on time circle
g2 = (2⇡)2↵01/2gs R = ↵01/2gs, =) R =g2
4⇡2
Tuesday, 17 September 13
•Conversely, if Euclidean SYM gains extra time dimension, so does E5 brane and this gives evidence that the IIAE string gains extra dimension at strong coupling.
• If the IIAE string is compactified on a (spacelike) circle, T-duality changes signature -- it is T-dual to a string theory in 9+1 dimensions compactified on a timelike circle.
Tuesday, 17 September 13
CONCLUSIONS
• Evidence for conjecture that SYM in 5+0 gets an extra time dimension at strong coupling, giving (2,0) theory on time circle
• Similar to evidence for 4+1 SYM getting an extra space dim
• 5+0 and 4+1 SYM governed by same path integral, so both should grow a dimension if one does
•Detailed matching of BPS spectra in 5D and 6D
• BPS spectra SO(5,1) covariant
Tuesday, 17 September 13
• Follows from strong coupling limits of IIA,IIAE giving M-theory
•Don’t know what (2,0) theory or M-theory are, but these strong coupling limits give useful information
• Emergent time is a circle. There are issues in meaning of quantum theory with periodic time. (Probability? Measurement?)
• Intimate relation between time and quantum theory. How to quantise a theory without time (without resorting to a Euclidean time)? For SYM, use quantisation of (2,0) on time circle?
Tuesday, 17 September 13