M-theory and quantum fields

Seok Kim

(Seoul National University)

Colloquium, USTC, Hefei

June 1, 2018

Quantum field theory (QFT)

• Field: physical quantities spread over space ~ wave

• Quantum fields:

- Quantum physics governs microscopic world.

- So one often needs to quantize fields.

- For instance, quantized electromagnetic fields

- Or, quantized phonons (waves of lattice vibrations)

- And so on…

• QFT is a culmination of modern physics

- “Quantum field” is a very universal notion

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QFT & string theory: 20th century

• QFT: Developed for particle physics. Almost all matters & interactions.

• String theory: includes quantum gravity

• QFT construction is flexible ↔ String theory is hard to construct.

(Lagrangian field theory)

• QFT “well understood” ↔ String theory not even well formulated.

- Can’t “solve” difficult QFTs, but it always happens in physics. (e.g. 3-body problem)

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QFT & string theory: 21st century

• Deeper studies of string theory reveal different stories.

- Novel QFTs found from string & M-theory:

novel Lagrangians, or no known Lagrangians

- duality between string theory & QFT

AdS/CFT [Maldacena]

• Needs re-interpretation of quantum fields.

- Notion of particle is emergent in QFT: in principle may not exist

- QFT often hosts emergent physics, esp. at strong coupling

• Often, ignorance on string theory ~ ignorance on QFT.

- Where we thought we understood physics but just couldn’t compute

- This feature will be more extreme in M-theory (~ strong coupling limit of strings)

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Plan of this talk

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String/M-theory Quantum field theory

New insights, often beyond current (Lagrangian) techniques

Triggers QFT developments

New advances in QFT enable concrete studies of string/M-theory & quantum gravity

E.g. physics of black holes

Naïve way to QFT from strings

• So how can string theory provide ground-breaking insights to QFT?

• Often obtains quantum fields at low E.

- Closed strings: contains gravity, Einstein’s general relativity

- Open strings: contains quarks & gluons

gauge theory ~ Yang-Mills theory (e.g. SU(N)),

All d.o.f. are 𝑁 ×𝑁 matrices

𝑁 D-brane boundaries

• Gauge theory is the natural QFT for string theory, as is general relativity.

• Quantum fields engineered this way: “part” of string theory, in general.

- Reflects the physics of string theory, so provides useful but limited tools.

- But in a delicate set-up, QFT is identical to strings. Provides a systematic way to study it.

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Strings from quantum fields

• A main example: “AdS/CFT duality” or “gauge/gravity duality”

Some QFT in d dimension = Strings in a curved spacetime in d+1 dimension

(Anti – de Sitter (AdS) spacetime)

[Maldacena] 1997

• Strong coupling QFTs can be for strings (not particles)

• A realization of “holographic” gravity [‘t Hooft] [Susskind]

• Strong coupling QFT questions studied via strings & gravity

- Holographic QCD, condensed matter systems, …

• Quantum gravity questions well posed using quantum fields.

- Emergent spacetime, microscopic thermodynamics of black holes (later), …7

M-theory

• M-theory: strong coupling limits of string theories [Hull, Townsend] [Witten ‘95]

- 10d → 11d. soliton particles (“D0-branes”) ~ momenta. Emergent space at strong coupling

- String theory: Has coupling constants. Perturbation theory.

- M-theory: no tunable coupling parameters. Strong coupling. Challenge for ~ 20 years

• Completely new way of thinking of string theory & quantum gravity.

• Also provides completely new visions on QFT (than gauge theories)

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Σ10

n D0-branes

n units of KK momenta

Quantum fields from M-theory

• M-theory has two extended objects: replace strings & other extended objects

- M2-brane ~ membrane: fundamental string (wrapped on M-theory circle)

… and so on…

- M5-brane: D4-brane (wrapped on circle), NS5-brane (unwrapped)

• Challenge to QFT:

- M-theory predicts strongly interacting d=2+1 & d=5+1 QFTs on these branes.

• The challenges are posed in two different ways.

- M2-brane: demands better understanding of QFT at strong coupling (like QCD)

- M5-brane: demands new formulations of QFT, beyond the current one

• These two examples drastically enhanced (or will enhance) our notions of strong

coupling QFT.

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M2 QFTs & challenges

• 2+1d: Yang-Mills(+matters) theories at low E ~ strong coupling (“RG flow”)

- Describes M2-brane dynamics at strong coupling.

- Classical matrix-like dynamics is wiped out at strong coupling: new emergent physics

E.g. [Bagger, Lambert], [Gustavsson] (2007); [Aharony-Bergman-Jafferis-Maldacena] (2008), ……

• E.g. large 𝑁 black membrane solution predicts shocking emergent physics

- Many physical quantities scale like ∼ 𝑁3/2, rather than ∼ 𝑁2.

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E ~ gYM2 E >> gYM

2E << gYM2

weakly interacting

IR fixed point

“M2-brane QFT”

Yang-Mills-type QFTs

• M5-brane QFTs in 5+1d is much harder.

- No microscopic formulation known.

- Profound implications to lower dimensional QFTs via compactification: “M-theory of QFTs”

- E.g. on T2 : 𝑆𝐿(2, 𝑍) and Montonen-Olive electromagnetic duality

- Other Riemann surfaces: more nontrivial 4d QFTs

[Gaiotto] et.al. (2009)

• 𝑁 black M5-branes, etc. : 𝑁3 degrees of freedom

- Much more than Yang-Mills theory

- Presumably demands new structures

M5 QFTs & challenges

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= coupling constant of Yang-Millscomplex structure 𝜏 =𝜃

2𝜋+

4𝜋𝑖

𝑔𝑌𝑀2

Real challenge: QFTs in 𝑑 > 4…?

Consistent interacting QFT in d>4 is unexpected in traditional QFT.

Classical particle: 1d path. Hard to make contact interactions in higher d.

Even after quantum fluctuations, particles rarely interact in higher dimension.

Quantum particles interact in 𝑑 ≤ 3 + 1 .

In 𝑑 > 3 + 1, interacting QFTs presumably aren’t for particles.

No known models in textbooks.

(E.g. QFTs may be for “strings” in d=6.)

These examples will completely change

the intrinsic meaning of QFT.

open membrane

5-branes

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Summary of challenges & advances

• Novel # of degrees of freedom: 𝑁3/2 or 𝑁3. Either

- suggests novel emergent phenomena of gauge theories, or

- Demands new formulations of QFT

• Some physics understood from advances in strong coupling QFT tools. E.g.

- Entanglement entropy of 3d vacuum ∝ 𝑁3/2 [Drukker, Marino, Putrov] [Klebanov et.al.] (2011)

- Vacuum energy of M5 QFT ∝ 𝑁3: [H.-C. Kim, SK] [Kallen, Minahan, Nedeline, Zabzine] (2012)

- Reconstruction of AdS4 & AdS7 graviton spectra from QFTs [SK] 2009, [H.-C.Kim, SK] 2012

- And so on …

- But in d>4, only relied on string/M-theory tools or effective field theory techniques.

- So still, one needs to find a completely new approach to such systems. (E.g. New

Lagrangians, new notion of gauge symmetry, non-Lagrangian techniques for QFT, etc.)

• Recent advances in QFT techniques managed to partly catch up with string M-

theory visions on QFT. 13

Quantum gravity from QFT?

• Now, can we now use the developed notions/techniques of QFT to provide new

& useful insights to quantum gravity & strings?

• Yes.

• I’ll focus on the use of QFT in the context of AdS/CFT duality

• So many interesting achievements on QFT → QG…

- Exact spectrum of strings using “integrability” of large N QFT

[Minahan, Zarembo, Staudacher, Beisert, ………]

- Holographic gravity using entanglement, quantum information, …

- ……

- Microscopic studies of black holes in AdS space

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Black holes & QFT

• Hawking-Page phase transition (1983)

• In QFT, confinement / deconfinement phase transition [Witten] 1998

• Qualitative aspects of this transition studied from weakly-coupled gauge theory

[Aharony, Marsano, Minwalla, Papadodimas, Raamsdonk] 2003 , … 15

thermal graviton gas in AdS: 𝑇 < 𝑇𝑐 ∼ 1/𝑅𝐴𝑑𝑆

Black hole in AdS: 𝑇 > 𝑇𝑐 ∼ 1/𝑅𝐴𝑑𝑆

transition at 𝑇 = 𝑇𝑐

confined quarks/gluons ~ gravitons plasma of quark/gluon ~ black hole

transition at 𝑇 = 𝑇𝑐

A 4d Hawking-Page transition from 3d QFT

• 2+1d M2-brane QFT: vortex solitons

- Gauge theories have such objects. [Abrikosov, Nielsen, Olesen]

- In weakly-coupled Yang-Mills: heavier than 𝑁2 elementary d.o.f.

- We study partition function of “vortex” solitons on 𝑅2+1.

• “Confining” phase: vortex mass > than other mass scales (other chemical potentials)

- Large N free energy ∼ 𝑂(𝑁0): confined phase of “gravitons”

• Novel “deconfining” phase: light vortices, emergent 𝑁3/2 d.o.f.

• Gauge theory solitons provide such a mechanism in 2+1d QFT.

[Studied from so-called “Witten index” for “BPS vortices”] [Choi, Hwang, SK] to appear.

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𝛽, 𝑇1,2,3,4 are chemical potentials

for energy & internal symmetries

Black holes from QFT: quantitative

• QFT on (sphere) x R: study their thermal free energies

• Black holes which may be understood more easily

- “Supersymmetric” black holes:

lightest black holes w/ given electric charges & spins

- Where we first microscopically understood black holes.

[Strominger, Vafa] 1996

- E.g., in AdS5 or AdS7 (related to 4d, 6d QFTs) [Gutowski, Reall, et.al], [Cvetic, et.al.]

- Their entropies are obtained from the following free energies [Zaffaroni et.al.] 2017, 2018

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black hole

𝑀 ≥1

𝑅𝐴𝑑𝑆(# σ𝐼𝑄𝐼 + # σ𝑖 𝐽𝑖)

𝑀 ≥1

𝑅𝐴𝑑𝑆(𝑄1 + 𝑄2 + 𝑄3 + 𝐽1 + 𝐽2)

𝑀 ≥1

𝑅𝐴𝑑𝑆(2𝑄1 + 2𝑄2 + 𝐽1 + 𝐽2 + 𝐽3)

string theory on 𝐴𝑑𝑆5 × 𝑆5

M-theory on 𝐴𝑑𝑆7 × 𝑆4very complicated solutions …

𝑆𝐵𝐻 =𝑁2

2

Δ1Δ2Δ3𝜔1𝜔2

+ 𝑄𝐼Δ𝐼 +𝜔𝑖𝐽𝑖

𝑆𝐵𝐻 =𝑁3

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Δ1Δ22

𝜔1𝜔2𝜔3+ 𝑄𝐼Δ𝐼 +𝜔𝑖𝐽𝑖

𝐴𝑑𝑆5 × 𝑆5

𝐴𝑑𝑆7 × 𝑆4

Legendre transformation w/ constraint σ𝐼 Δ𝐼 − σ𝑖𝜔𝑖 = 2𝜋𝑖

Black holes from anomalies

• For 𝐴𝑑𝑆2𝑛+1=𝑜𝑑𝑑, exact “chiral anomalies” of even dimensional QFTs can be

used to study various physics at strong coupling. [Adler, Bell, Jackiw] 1969, ……

• QFT effective action on 𝑆2𝑛−1 encoding these anomalies

[Minwalla et.al.] [Komargodski et.al.] [SK, Nahmgoong]

• This computes these entropy functions from QFT, thus counting supersymmetric

AdS5 and AdS7 black holes [J.Kim, SK, Nahmgoong] to appear, [SK, Nahmgoong] 2017

• For 4d QFT, also developing a QFT model of understanding their microstates,

called “Fermi liquid model” [Berkooz, Reichmann, Simon] 2006, [Berkooz, Reichmann]18

𝜕𝜇𝑗𝜇

𝐴𝜇

𝐴𝜇

𝜕𝜇𝑗𝜇 ∼ 𝜖𝜇𝜈𝜌𝜎𝐹𝜇𝜈𝐹𝜌𝜎

−log 𝑍 =𝑁2

2

Δ1Δ2Δ3𝜔1𝜔2

−log 𝑍 =𝑁3

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Δ1Δ22

𝜔1𝜔2𝜔3𝐴𝑑𝑆5 × 𝑆5 𝐴𝑑𝑆7 × 𝑆4

Perspectives: string theory & QFT

• String theory reveals that QFT is going far beyond the dynamics of particles.

- Various emergent phenomena: new d.o.f., strings, gravity, …

• M-theory branes have been posing special challenges to QFT.

- M2 & M5-branes: insights beyond traditional gauge theories.

- New emergent aspects of gauge theories. Needs new QFT formulations.

• QFT techniques provide insights back to string/M-theory & quantum gravity.

- Challenging problems in quantum gravity. Microscopic studies of black holes

- Many incomplete methods so far. Wild & sometimes crude (but working) ideas

- Excellent laboratory for developing some of the most challenging problems in physics…!

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