Locality, Quantum Gravity and Metastrings

Rob Leigh University of Illinois at Urbana-Champaign

based on a series of papers, past and future, with Laurent Freidel (Perimeter) and Djordje Minic (VaTech) (see 1502.08005 for details)

August 12, 2015

CERN-CKC TH Institute on Duality Symmetries in String and M-theories Metastrings

- in this talk, I want to introduce you to the concept of metastrings

- as you will see, it is topical to this workshop, as some of its features are reminiscent of double ﬁeld theory, and concepts such as T-duality are central • however, this is meant to be a full string theory, not a truncation to a few modes or restricted to certain compactiﬁcations

- my motivations are more primitive • in fact our development of this theory arose from thinking about the issue of locality in theories of quantum gravity more generally, and in particular what sorts of prime principles we should use in building such a theory • now many would say that string theory, as we usually conceive of it, is already a theory of quantum gravity, so why should we think about replacing it by something else? • is string theory a good enough theory of quantum gravity?

Metastrings and Locality 2 Generalized Geometry

- string theory, at the very least, is rather successful in generating interesting things to think about

- at this workshop, there has been a focus on building up (non-) geometric structures that might underly the robust dualities of the theory

- a paradigm that we have seen is obtained by considering toroidal compactiﬁcations — attempts to think about stringy states (such as winding modes) in a geometric context, in such a way that dualities are realized linearly. • in particular one focuses on some small sub-sector of the theory, at weak coupling, and attempts to describe it in ﬁeld theoretic terms

Metastrings and Locality 3 - although it is interesting to explore the possibilities, forcing string theory to behave like field theory (or quantum to behave as classical) will likely miss the most remarkable features • string theory was developed (in its perturbative definition) as an S- matrix theory for particle states in a flat space-time background

CFT consistency Space-time consistency (symmetry, mutual locality, modular inv) (causality, unitarity, locality)

• we have a well-deﬁned quantum theory, with an overlay of target space interpretation

- I’m going to ask: is there a mistake there? — is that interpretation built in too deeply? • the guiding principle will be locality • we’ll obtain a new string theory in which the target is no longer space- time T duality Born reciprocity $ Metastrings and Locality 4 Locality

- locality plays a central guiding role in modern physics • local effective ﬁeld theories, locality of interactions, asymptotic states • structure of the renormalization group — separation of scales

- QM possesses a variety of non-local features • Heisenberg xp 1 ~ 2 • entanglement, Bell inequalities, measurement • Aharonov-Bohm, interference

- we expect that any theory of quantum gravity will involve non-locality • in GR, no local diff-invariant observables • gravity is holographic • strings (or D-branes, …) are non-local probes

Metastrings and Locality 5 Locality

- QM possesses a variety of non-local features • Heisenberg xp 1 ~ 2 • entanglement, Bell inequalities, measurement respects local causality • Aharonov-Bohm, interference } - we expect that any theory of quantum gravity will involve non-locality • in GR, no local diff-invariant observables • gravity is holographic • strings (or D-branes, …) are non-local probes

Metastrings and Locality 5 Locality

- QM possesses a variety of non-local features • Heisenberg xp 1 ~ 2 • entanglement, Bell inequalities, measurement • Aharonov-Bohm, interference

- we expect that any theory of quantum gravity will involve non-locality • in GR, no local diff-invariant observables • gravity is holographic • strings (or D-branes, …) are non-local probes

Metastrings and Locality 5 Locality

- QM possesses a variety of non-local features • Heisenberg xp 1 ~ 2 • entanglement, Bell inequalities, measurement • Aharonov-Bohm, interference

- we expect that any theory of quantum gravity will involve non-locality • in GR, no local diff-invariant observables • gravity is holographic • strings (or D-branes, …) are non-local probes messing with locality must be done very carefully!

Metastrings and Locality 5 - there is ample evidence in string theory that the local space-time picture must break down • “string bits” (Klebanov-Susskind): X2 2 h µi⇠ • regulate string on lattice: ﬁnd that rms distance diverges logarithmically • strings become long and space-time wants to discretize • at odds with picture of small strings in smooth space-time • hard scattering (Gross-Mende): • ﬁnite temperature (Atick-Witten): • string dualities

Metastrings and Locality 6 Locality: absolute vs. relative

Absolute locality: space-time is independent of its probes • this is the version of locality that we use implicitly • appropriate to classical, point-like probes Relative locality: space-time depends on the (quantum) nature of probe • e.g., energy or other quantum numbers

Metastrings and Locality 7 Locality: absolute vs. relative

high energy probe low energy probe

STM visualization of quantum crystals at different (S. Davis) probe energies

disordered ordered translation inv’t

Metastrings and Locality 7 - same system, but physics looks different when probed at different energies! • the classical question is it ordered or disordered? is ill-deﬁned in quantum theory!

- if this analogy (crystal space-time, electrons probes) makes any sense, suggests that! if locality is relative, the proper! setting for quantum geometry is a phase space P • need both space-time and energy-momentum for complete description • we should expect energy-momentum to be curved • this is unusual — usually we think of this as a linear space

- suggests that there ought to be two scales a • will induce in string theory, and interpret A X / X = P Ya/" 2 " = ~, = ↵0 ✓ ◆ " Metastrings and Locality 8 Born Reciprocity

- in perturbative string theory, the elementary probes are strings, and the degrees of freedom are maps X :⌃ M ! - on the other hand, 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 | i | i • 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

- applying this to string theory, we interpret the identiﬁcation of M as space-time as breaking Born reciprocity [M. Born, 1930’s] • by taking the target to be space-time, we’ve chosen a basis

Metastrings and Locality 9 Born Reciprocity

- this choice of basis is directly associated with asking the string theory to give local effective ﬁeld theories

- but the underlying structure of the string is BR-symmetric • diff constraints: 1 1 H : p2 + 2 = N + N˜ 2 2 2 D : p = N N˜ · - what is non-generic is solving the constraints in a particular way • this is a choice of boundary condition — periodicity of the string • this is where locality is built in • of course, in toroidal compactiﬁcations, we see the full structure and the resulting duality

Metastrings and Locality 10 Born Reciprocity and T-duality

- usually, in toroidal compactiﬁcations, we interpret short distance (radius) as long distance in a dual space-time

- however we can directly relate this to a Fourier transform

- consider a string state ↵0p = dX 2 C ⇤ iSP [X ]/ Z x(σ) [x()] = [DX Dg ] e X =x = dX Z |@⌃ C - deﬁne a Fourier transform of this state by Z i/ xµdy ˜ [y()] [Dx()]e ~ @⌃ µ [x()] ⌘ Z R - extending y ( ) to the world sheet, we integrate out X to obtain a dual Polyakov path integral 2 iS [Y ]/" = ↵0 dY ˜ [y()] = [DY Dg ] e P ⇤ ZC Y @⌃=y Z | p = dY ZC Metastrings and Locality 11 Born Reciprocity and T-duality

- usually, in toroidal compactiﬁcations, we interpret short distance (radius) as long distance in a dual space-time

- however we can directly relate this to a Fourier transform

- consider a string state ↵0p = dX 2 C ⇤ iSP [X ]/ Z x(σ) [x()] = [DX Dg ] e X =x = dX Z |@⌃ C - deﬁne a Fourier transform of this state by Z i/ xµdy ˜ [y()] [Dx()]e ~ @⌃ µ [x()] ⌘ Z R - extending y ( ) to the world sheet, we integrate out X to obtain a dual Polyakov path integral 2 iS [Y ]/" = ↵0 dY ˜ [y()] = [DY Dg ] e P ⇤ ZC Y @⌃=y Z | p = dY and p exchange their roles ZC Metastrings and Locality 11 Born Reciprocity and T-duality - the Fourier transform induces T-duality; the relation dX = dY here is an equation of motion " ⇤

- if X were compact, then Y is a coordinate on the dual space

- if X is non-compact, then (p, )=(p, 0) (˜p, ˜) = (0, p) ! - this interchange suggests that we should at least entertain the possibility of generic (p, ) X µ( +2⇡, ⌧)=X µ(, ⌧)+µ • we refer to as a quasi-period, or monodromy

- note that for the action to be well-deﬁned, it is dX that should be periodic (periodicity of X is not required) — allows monodromies

Metastrings and Locality 12 Monodromies and Locality

- if is not zero, there is no a priori geometrical interpretation of a closed string propagating in a ﬂat spacetime

- there is no space-time interpretation — • in the usual interpretation, the worldsheet embeddings appear to be ‘torn’ • instead we will regard the target space as a phase space (or doubled space) and allow all monodromies

- one can show that (classically) there is a proper definition of closed strings even if monodromies are allowed • for a given Riemann surface constructed by gluing flat strips together, we define a closed string by gluing two edges together to form a loop and requiring that the symplectic flux be continuous across the cut • this definition does not require monodromies to vanish

Metastrings and Locality 13 Monodromies and Locality

- if is not zero, there is no a priori geometrical interpretation of a closed string propagating in a ﬂat spacetime

- one can show that (classically) there is a proper deﬁnition of closed strings even if monodromies are allowed • for a given Riemann surface constructed by gluing ﬂat strips together, we

define a closed string by gluing two edges together to form a loop and Technical Lorentzianpoint: requiring that the symplectic flux be continuous across the cut one needs a decomposition of • this definition does not require monodromies to vanish arbitrary worldsheets

Metastrings and Locality 13 Metastring Path Integral

- convenient to go to a ﬁrst-order formalism

• integrate in worldsheet 1-form Pµ = Pµd⌧ + Qµd S = P dX µ + ⌘µ⌫ ( P P ) 1 µ ^ 2" ⇤ µ ^ ⌫ Z⌃ ✓ ◆ ➡ integrate out P: back to original Polyakov theory ➡ integrate out X: implies dP=0, solve locally P=dY; get dual Polyakov theory (with quasi-periodic Y)

➡ integrate out Q and writing P = @ Y , we ﬁnd Floreanini-Jackiw-Tseytlin

1 1 1 1 S = @ Y @ X @ Y @ Y @ X @ X PS " · ⌧ 2"2 · 22 · ~ Z  - described here classically, but there is a path integral version • the details of arbitrary genus path integral are being worked out

Metastrings and Locality 14 Metastring Path Integral

- the notation used by double ﬁeld theory is useful here

µ A X / 0 h 0 0 X , ⌘AB = T , HAB 1 , !AB T ⌘ Yµ/" 0 ⌘ 0 h ⌘ 0 ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆

1 A B A B S = @⌧ X (⌘AB + !AB )@X @X HAB @X 4⇡ Z ⇣ ⌘ - the ! term is a total derivative, but it’s kept because of monodromies

- worldsheet Lorentz invariance is broken (by construction) or at least not manifest • in fact Lorentz and Weyl constraints go hand in hand • generally, Liouville sector would be doubled

Metastrings and Locality 15 Born Geometry

µ A X / 0 h 0 0 X , ⌘AB = T , HAB 1 , !AB T ⌘ Yµ/" 0 ⌘ 0 h ⌘ 0 ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ d=26 signature (d, d) signature (2, 2(d 1)) symplectic ‘polarization metric’ ‘quantum metric’

- in this case, points ( X µ , 0) give a null Lagrangian subspace L P ⇢ - (null with respect to ⌘ ) equivalently, L = ker (⌘ + !)

- similarly (0, Y µ ) give a transversal Lagrangian L˜ P ⇢ - H provides a metric on L, g = H L | - ⌘ deﬁnes a bi-Lagrangian structure P = L L˜

Metastrings and Locality 16 Born Geometry

- we can regard this as a sigma-model for a certain geometry

1 2 T - deﬁne the involution J = ⌘ H J =1 J ⌘J = ⌘ - we say that ( ⌘ , J ) deﬁnes a chiral structure - ! provides a symplectic structure - under mild assumptions, in fact the geometry is ‘hyper-para-Kahler’ I,J,K

- we expect that both ⌘ and H must be dynamical • QM: H kinematical, ⌘ (polarization) can evolve unitarily • GR: ⌘ kinematical (picks out space-time), H dynamical (metric) this is in fact borne out by direct CFT calculations • conformal perturbation theory yields linearized equations for ⌘ and H

Metastrings and Locality 17 Born Geometry

- we can regard this as a sigma-model for a certain geometry

1 2 T - deﬁne the involution J = ⌘ H J =1 J ⌘J = ⌘ - we say that ( ⌘ , J ) deﬁnes a chiral structure

- ! provides a symplectic⇤H H structureAB =0=⇤H ⌘AB ++ H = 4(H H) - under mild assumptions,⇤⌘ inAB fact the geometryAB isAB ‘hyper-para-Kahler’ ++ ⇤⌘⌘AB = 4(HAB + HAB) I,J,K HAC @ H =0=HAC @ ⌘ - we expect that both ⌘ andC H ABmust be dynamicalC AB AB AB • QM: H kinematical,H ⌘ (polarization)⌘AB =0= canH evolveH ABunitarily — B. Shoshany • GR: ⌘ kinematical (picks out space-time), H dynamical (metric) this is in fact borne out by direct CFT calculations • conformal perturbation theory yields linearized equations for ⌘ and H

Metastrings and Locality 17 Metastring: classical aspects

- we’ve worked out the details of the classical theory, including the gauge invariant observables and the Poisson algebra of symmetries

- a central element is that the momentum conjugate to X is

P = @X

- thus the string algebra is non-commutative! A B AB X (), X (0) = ⌘ ✓( 0). - this basic resultn has many manifestationso

- a basic observable, supported on a single loop, is

A ⇠ (X)= d⇠A(X)@X “stringy gauge ﬁeld” h i - this can be thought as generatingI the entire space of observables

Metastrings and Locality 18 - the Poisson algebra of the constraints generates Diff(S1)xDiff(S1)

1 L± = (H D ) N 2 N ± N 1 L±, L± = L± DN = N@X @X N N0 N@ N0 N0@ N 2 · 1 ¯ HN¯ = 2 N@X J@X · A - gauge-invariance of the observable ⇠ ( X )= d ⇠ A ( X ) @ X h i requires (in presence of monodromy ) I ⇠(X +)=⇠(X) ⇤ 2 - natural then to Fourier transform, and consider the vertices A iP X A V (X) d e · @X P 2⇡⇤⇤ P ⌘ 2 I

Metastrings and Locality 19 Poisson Algebra of Observables

- remarkably, the Poisson bracket of observables yields A A

⇠ , ⇠0 = [⇠, ⇠0] } ⇠ @ ⇠0 + @ ⇠ @ ⇠0 {h i h i} h ih A i hh A ii

Siegel C-bracket Hull & Zwiebach

- the last term is stringy; it is supported on a pair of loops

A AB @ ⇠ @ ⇠0 = dd0 ⌘ [@ ⇠()✓( 0)@ ⇠0(0)] hh A ii A B II - can get rid of this term by forcing the functions to depend only on a restricted set of coordinates, essentially TP TL = TL T ⇤L ! - this projectibility imposes absolute locality — singles out a preferred subspace L P ⇢

Metastrings and Locality 20 A Classical Anomaly

- problem: the Poisson bracket of two observables is not observable

- (recall gauge invariance required periodicity)

- this problem is removed by the projectibility condition

Metastrings and Locality 21 A Classical Anomaly

- problem: the Poisson bracket of two observables is not observable

- (recall gauge invariance required periodicity)

- this problem is removed by the projectibility condition

- remarkably, this ‘classical anomaly’ disappears in the quantum theory!

Metastrings and Locality 21 A Classical Anomaly

- problem: the Poisson bracket of two observables is not observable

- (recall gauge invariance required periodicity)

- this problem is removed by the projectibility condition

- remarkably, this ‘classical anomaly’ disappears in the quantum theory!

this will lead to a quantum notion of space-time

Metastrings and Locality 21 - in the quantum theory, we construct normal-ordered operators

- iP Q ✓( 0) one ﬁnds VP()VQ(0)=e · VQ(0)VP() - the phase is removed (and thus mutual locality ensured) if P, Q ⇤ 2 - mutual locality, modular invariance, etc. require an integral Lorentzian even self-dual lattice

• this is essentially unique P ⇤=⇧1,25 ⇧1,25 2 ⇥ • the ﬂat metastring symmetry algebra is closely related to products of Borcherds algebras. Duality symmetries contained within.

Metastrings and Locality 22 Modular Space-time

- but isn’t this a disaster? • the traditional interpretation of this result would be that the string propagates on a ﬂat Lorentzian compact space-time • surely at odds with a causal interpretation

- we will offer a novel interpretation • in the language of non-commutative geometry, space-time can be thought of as a maximal commutative sub-algebra • we know what this means in classical geometry — a Lagrangian subspace

i i • e.g., in a phase space coordinatized by (pi,q ), the q Poisson-commute amongst themselves and coordinatize a Lagrangian subspace

Metastrings and Locality 23 Modular Space-time

- when the commutator of quantum observables is carefully defined, one finds that the ⇤ -periodicity condition is sufficient • no projectibility required! quantum generalization of ⇤ -periodic field = field on space-time

- recall that the target space is non-commutative • another way to express the notion of a Lagrangian submanifold, algebraically, is to identify a commutative subalgebra of the full operator algebra • classically, projectibility results in a classical Lagrangian sub-manifold L P ⇢ • but the ⇤ -periodicity suggests another commutative sub-algebra, given by modular variables

Metastrings and Locality 24 Modular quantum variables

- the exponential operator is periodic

• deﬁne modular momentum [ˆp]=ˆp mod h/L introduced in QM by Aharonov to describe purely-quantum phenomena • similarly, modular position [ˆx]=ˆx mod L (interference, etc.)

- for any f(x) possessing a Fourier transform eiLpˆ/~e2⇡ixˆ/L0 f (x)=e2⇡iL0/Le2⇡ixˆ/L0 eiLpˆ/~f (x)

- thus if L’=L, [ˆ x ]and[ˆ p ] commute! • could specify both [ˆ x ]and[ˆ p ] simultaneously p • usual uncertainty principle: q • can’t specify x and p within a cell • modular uncertainty principle: • no knowledge of which cell!

Metastrings and Locality 25 Modular quantum variables

- x’s and p’s are ﬁne for analyzing single wave packets • need modular variables whenever we analyze more

- NB: quantization replaces Poisson bracket by commutator

• however f ( x ), g ( p ) = f 0 ( x ) g 0 ( p ) vanishes whenever one of the functions {is constant } • but in the quantum theory [ f (ˆ x ), g (ˆ p )] = 0 whenever f,g are periodic

ei↵pêixˆ = ei~↵eixêi↵pˆ ↵ =2⇡/~ - these two statements are (simplified versions of) what happens in the operator algebra of the metastring • replace q by the pair ([q],[p])

Metastrings and Locality 26 Modular Space-time and Extensiﬁcation

- the quantum Lagrangian has ‘area’ given by ~ = " - a classical limit is a truncation to a subset of observables

quantum Lagrangian

classical Lagrangian - classical limit = extensiﬁcation (preserves area in phase space) • many such extensiﬁcations exist — re-emergence of moduli

Metastrings and Locality 27 Remarks, open questions

- in QM, time t can be replaced by a modular pair ([t],[E]). These are periodic, but causality is not violated. • presumably then the modular interpretation of space-time does not conﬂict with causality

- presumably, all known string compactifications can be obtained by suitable extensification limits • is there some dynamics that drives extensification? • just as hard a question as ever

- target geometry — expect conformal invariance to imply equations for target space geometry • but here, it should be a “double renormalization group” (Lorentz+Weyl) a full quantum formulation of the metastring path integral • generalizes the Polyakov path integral; seems that direct Lorentzian worldsheet methods are required

Metastrings and Locality 28