Born reciprocity

Born Reciprocity
and the Nature of Spacetime in String Theory
Rob Leigh
University of Illinois
Based on 1307.7080 +...
with Laurent Freidel [Perimeter] and Djordje Minic [Virginia Tech]
Rob Leigh (UIUC) Born Reciprocity Wits: March 2014 1 / 25

Quotes
Some Inspiration
Our mistake is not that we take our theories too seriously, but that we
do not take them seriously enough.
S. Weinberg (The First Three Minutes: A Modern View of the Origin of the Universe)

Quotes
Some Inspiration
Our mistake is not that we take our theories too seriously, but that we
do not take them seriously enough.
S. Weinberg (The First Three Minutes: A Modern View of the Origin of the Universe)
Pretty smart, them strings.
J. Polchinski (Combinatorics of Boundaries in String Theory, hep-th/9407031)

Born Reciprocity
Born Reciprocity
it is a familiar feature of quantum mechanics that a choice of basis
for Hilbert space is immaterial
e.g., for particle states, {|q } is just as good as {|p }
one choice may be preferred given a choice of observables
e.g., for atomic systems:
interaction with light → use energy basis
for material properties → use position basis
a change of basis is accomplished by Fourier transform
fast forward to Quantum Gravity: Max Born in the 1930’s pointed
out that this should be a feature of any QG theory too
→ there should be no invariant signiﬁcance to space-time!!
this seemingly incontravertible fact went largely ignored, and Born
himself was unable to make sense of it

Born Reciprocity
Born Reciprocity in String Theory
in string theory, we have a supposedly consistent quantum theory,
so we ask, what of Born reciprocity (BR)?
open any book on string theory, and you will find that Born
reciprocity is broken explicitly within the first few pages
Why? because we force string theory to provide us with a local
low energy theory on spacetime
Indeed, as we will see, the basic formulation of string theory as a
quantum theory is itself BR-symmetric.
It is the truncation to specific boundary conditions (which we will
relate to space-time locality) and the subsequent reduction to a low
energy sector that is non-generic.

Born Reciprocity
Born Reciprocity in String Theory
Does this picture ever break down?! String theory is consistent
quantum mechanically!
Do we know much about the nature of space-time at short
distances? Are we making unwarranted assumptions?
indeed, there are classic tests, such as
hard scattering (Gross-Mende,...)
ﬁnite temperature (Atick-Witten)
each of these rode off into the sunset with profound questions
unanswered
Clear thinking along these lines must have something profound to
say about “the short-distance nature of spacetime".

Born Reciprocity
Born Reciprocity and T-duality
of course, what often comes to mind in thinking about short
distances is T-duality
compactification gives us a way to define a short distance – i.e.,
small radii
seems to indicate that space-time looks the same at short
distances as at long distances
more precisely, short distance is governed by the long distance
properties of a dual space-time
does this apply generically? what about curvature? Are there
other questions/probes where it doesn’t apply?
if a dual space-time is involved in such a fundamental way, why do
we say that string theory is defined by maps into a space-time?
in fact, T-duality is closely related to Born reciprocity

Born Reciprocity
Closed strings
we usually define the string path integral as a summation of maps
X : Σ → M
g
Z[gαβ] =
g
[DX]e
i
λ2 Σ ηµν (∗dXµ∧dXν )
specifically, there is an assumption of single-valuedness of these
maps – the string is closed – X is periodic
is this the right prescription? Is it required for consistency, or
well-definedness?
the action would be well-defined if the integrand is single-valued,
that is, for periodic dX.
this does not mean that X(σ, τ) has to be periodic, even if M is
non-compact. Instead, it means that X must be quasi-periodic.

Born Reciprocity
Quasi-periodicity
quasi-periodicity means
Xµ
(σ + 2π, τ) = Xµ
(σ, τ) + δµ
.
If δµ is not zero, there is no a priori geometrical interpretation of a
closed string propagating in a ﬂat spacetime — periodicity goes
hand-in-hand with a space-time interpretation.
Of course, if M were compact and space-like, then δµ = 0 can
correspond to periodic X. This is interpreted as winding, and it is
not in general zero.
more generally, δµ = 0 corresponds to a tear in the embedding of
the worldsheet in the target space.
δ =
C
dX
are there operators that would induce such a tear?

Born Reciprocity
Quasi-periodicity and locality
in fact, eliminating such operators is central to the consistency of
the usual string – operators are required to be mutually local.
this is closely related to locality in space-time as well, to the
interpretation of the string path integral as giving rise to local QFT.
classically, one can motivate this through the local constraints
Constraints
H :
1
2
p2
+
1
2
δ2
= N + ˜N − 2
D : p · δ = N − ˜N
if the spectrum is level-matched, then it is consistent to take δ = 0

Phase Space Formulation
a hint that δ = 0 might be consistent: given a boundary ∂Σ
parameterized by σ, a string state |Ψ may be represented by a
Polyakov path integral
Ψ[x(σ)] =
X|∂Σ=x
[DX Dg] eiSP [X]/λ2
(1)
if X is periodic, then C dX = 0 and α p = C ∗dX
We deﬁne a Fourier transform of this state by
˜Ψ[y(σ)] ≡ [Dx(σ)] ei ∂Σ xµdyµ
Ψ[x(σ)].
In fact, this state can also be represented as a string state
associated to a dual Polyakov action: by extending y(σ) to the bulk
of the worldsheet, and integrating out X then gives
˜Ψ[y(σ)] =
Y|∂Σ=y
[DY Dg] e−iλ2SP [Y]
. (2)

The momentum may now be expressed as p = C dY, and so we
will refer to Y as coordinates in momentum space.
note also that δ = dX = λ2 ∗dY (by EOM)
so from the point of view of the Y space-time, momentum is zero,
but quasi-period is non-zero.
in the compact case, Y is the usual dual coordinate to X, and thus
the Fourier transform reproduces T-duality
this is an indication that we might generalize to arbitrary (p, δ),
even in the non-compact case (or the curved case)

Dyons
if we allow quasi-periodicity, then we can consider generic vertices
∼ eip·X+iδ·Y
there is a close analogy here with 2d electro-magneto-statics
p → elec chg, δ → mag chg
if we quantize in the usual form, then the diff constraint looks like
p · δ = N − Ñ
(usually this is trivially satisfied by δ = 0 and N = Ñ)
so here, we need to enforce the diff constraint, but not necessarily
in the usual way – in fact p · δ need only be an integer
this condition should be thought of as a Dirac quantization
condition
in E& M, one way to solve the Dirac quantization is to forbid
monopoles. But it’s not the only way.

Phase Space
it is convenient to go to a first-order formalism, in which we
integrate in worldsheet 1-forms Pµ = Pµdτ + Qµdσ
S1 =
Σ
Pµ ∧ dXµ
+
λ
2ε
ηµν
(∗Pµ ∧ Pν) .
integrate out P: back to SP[X]/λ2
integrate out X: find dP = 0, so locally P = dY, and get λ2
SP[Y]
(however, Y is quasi-periodic)
obtain a ‘phase space’ formalism if we partially integrate P
indeed, integrating out Q, and introducing P = ∂σY, we find the
Tseytlin action (c.f. Floreani-Jackiw)
1
SPS =
1
λε
∂σY · ∂τ X −
1
2ε2
∂σY · ∂σY −
1
2λ2
∂σX · ∂σX

Born Geometry
It is convenient, as suggested by the double ﬁeld formalism to
introduce a coordinate X on phase space, together with a neutral1
metric η, and a metric H.
XA
≡
Xµ/λ
Yµ/ε
, ηAB =
0 δ
δ−1 0
, HAB ≡
η 0
0 η−1
we then obtain
1
SPS =
1
2
∂τ XA
∂σXB
ηAB − ∂σXA
∂σXB
HAB .
note that the data (η, H) are not independent: J ≡ η−1H is an
involution that preserves η, i.e., JT ηJ = η or
J2
= 1 (chiral structure)
1
Here neutral means that η is of signature (d, d), while H is of signature (2, 2(d − 2)).

Dyons
if we allow quasi-periodicity, then we can consider generic vertices
∼ eiP·X where P = (p, δ)
if we quantize in the usual form, then the diff constraint looks like
p · δ = N − Ñ
(usually this is trivially satisfied by δ = 0 and N = Ñ)
so here, we need to enforce the diff constraint, but not necessarily
in the usual way – in fact p · δ need only be an integer

Hamiltonian form
Note that the momentum conjugate to XA is
ΠA =
1
2
ηAB∂σXB
and the Hamiltonian is
H = 2 dσ Π.JΠ
the canonical bracket implies
ΠA(σ), ΠB(σ ) = πηAB∂σ δ(σ − σ )
and one then ﬁnds
1
4π
H, XA
(σ) = (J∂σX)A
= ∂τ XA
− SA

Classical EOM
so the classical EOM imply
S ≡ ∂τ X − J∂σX = 0
the classical constraints from linear coordinate transformations are
W = 0, H = 1
2 ∂σX · J(∂σX),
L = 1
2 S · S, D = ∂σX · ∂σX,

Lorentz!
we note the free Tseytlin action is not worldsheet Lorentz invariant
1
SPS =
1
2
∂τ XA
∂σXB
ηAB − ∂σXA
∂σXB
HAB .
of course, the full path integral is Lorentz invariant
this shows up on-shell, because J2
= 1 and SA
= 0
(to do somewhat better, we can write the original Polyakov theory
as an integral over frames rather than metrics, and divide by Diff ×
Weyl × Lorentz)
then there are Liouville modes (θ, ρ) for both Lorentz and Weyl
dω → dω + d ∗ dρ, d ∗ ω → d ∗ ω + d ∗ dθ
(ω is worldsheet spin connection, dω its usual curvature)

The Quantum Theory and Chiral Structure
this classical structure depends crucially on J2 = 1. Defining
P± =
1
2
(1 ± J)
one finds that worldsheet chirality is paired with J-chirality
(“soldering")
one can construct the algebra of components of the stress tensor,
and one finds that T+− decouples2 and cL = cR = d iff J2 = 1
this is the quantum version of the effect of the classical constraints
and EOM which imply S = ∂τ X − J.∂σX = 0
2
Recall T+− + T−+ = 0 is the Weyl anomaly, while T+− − T−+ is the Lorentz anomaly.

Phase Space Backgrounds
conjecture: there are consistent string theories (CFTs) for which
η = η(X) and H = H(X) (etc.)
however, the phase space geometry is not arbitrary: we have
seen the importance of J2 = 1 and presumably this is to be kept in
the curved case
as well we have two notions of ‘metric’, η and H
in such a background, the equations of motion become
σSA = −
1
2
( AHBC)∂σXB
∂σXC
where has been assumed to be η-compatible
this should be supplemented by the constraints
what are solutions – what is the geometry of phase space?

Born Geometry
ingredient #1: ∃ an (almost) chiral (or para-complex) structure
(η, J)
this allows for a bi-Lagrangian structure, a choice of
decomposition TP = L ⊕ ˜L
L, ˜L are null wrt η, and J(L) = ˜L
bi-Lagrangian also characterized by K
L
= Id, K
˜L
= −Id, with
K2
= 1, JK + KJ = 0, KT
ηK = −η.
ω = ηK is a symplectic structure on P 3
I = KJ is an almost Kähler structure (I2
= −1, IT
ωI = ω)
we believe these are the primary features of the phase space
geometry
3
It is not obvious however that ω must be closed.

Born Geometry
in summary, Born geometry4 is characterized by (η, I, J, K) with
I2
= −1, J2
= +1, K2
= +1
IJ + JI = 0, IK + KI = 0, JK + KJ = 0
and possesses
η : neutral metric
ω = ηK : symplectic structure (3)
H = η−1
J = ωI : generalized metric
we’ve got something for everyone: complex, real, symplectic
geometry
4
This is a simpler name than hyper-para-Kähler, or some such.

Space-time?
there are a number of questions!
the nature of the Born geometry should be determined by
quantum consistency – we have to insist on both Weyl and
Lorentz symmetries
given such a geometry, how does a space-time emerge?
what are the observables of this theory? What is the interpretation
of the path integral in general?
does locality emerge along with the space-time?
we have a symplectic form so it makes sense to introduce
Lagrangian distributions – I’ll show you some evidence that
space-time should be thought of in those terms
another closely related idea is relative locality – each free state
carries its own notion of a spacetime (Lagrangian submﬂd)!
when strings interact, they have to ﬁrst agree on their space-times!
perhaps this structure is more or less invisible in a suitable limit
(e.g., x >> λ, or E << )

Born Geometry and Space-time
the classical constraints following from the Tseytlin action are
W = 0, H = 1
2 ∂σX · J(∂σX),
L = 1
2 S · S, D = ∂σX · ∂σX,
the diff constraint implies that ∂σX is null wrt η
thus ∂σX deﬁnes a Lagrangian L
in the ﬂat case, S = 0 and thus ∂τ X = J∂σX ∈ ˜L, where ˜L = J(L)
in the curved case, recall
σSA = −
1
2
( AHBC)∂σXB
∂σXC
so S is no longer zero, but the Lorentz constraint implies that it is
null with respect to η, as is ∂σX
one argues that S ∈ ˜L and ∂τ X ∈ ˜L, and that the induced metric on
L is g = H
L

Summary
string theory can be reformulated without putting in assumptions
of space-time properties by hand. This necessarily involves giving
up on a priori notions of locality.
claim/hope/expectation: the theory can be consistently quantized
and non-trivial backgrounds exist (these claims rely on
yet-to-be-ﬁnished anomaly calculations)
expect that supersymmetry will play an interesting role, as will the
usual non-perturbative aspects (D-branes, etc.)

Born reciprocity

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