nLab perturbative string theory vacuum

Idea

In perturbative quantum field theory a vacuum state is the information needed to turn a product of field observables such as $\mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y)$ into a function (or rather: generalized function/distribution) of the insertion points $x$ any $y$, namely the n-point function (here 2-point function, also called the Hadamard propagator)

$$\langle \mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y)\rangle$$

which may be regarded as the probability amplitude for a quantum in state $b$ at spacetime point $y$ to turn into a quantum in state $a$ at spacetime point $x$, in the given state that the fields are in, which is defined thereby (see at state in AQFT).

In the worldline formalism of field theories these propagators arise from a 1-dimensional field theory on the “worldline” of (virtual) particles running from $y$ to $x$.

Now by the very definition of perturbative string theory, these particles are replaced by strings whose dynamics is now encoded in a 2d field theory on the worldsheet of strings, specifically a 2d superconformal field theory (2d SCFT) of central charge 15. Hence now it is the 2d SCFT which defines the vacuum state that the perturbative string theory is in.

This is then called a perturbative string theory vacuum.

If this 2d SCFT arises from quantization of a sigma-model, then this is called a geometric background, otherwise it is a purely algebraically defined non-geometric string vacuum.

In practice full 2d SCFTs are hard to construct, and often one considers them by perturbation theory of a “sigma-model” which is defined by a spacetime manifold equipped with extra fields (e.g. the B-field etc.). It turns out that to low order these background field configurations that define sigma-model 2d SCFTs are given by solutions to equations of motion of supergravity theories (e.g. type II supergravity for type II string theory, etc.)

Therefore often such supergravity solutions equipped with some extra data that makes them consistent CFT backgrounds at higher order are referred to as vacua for string theory. But this is in general a coarse approximation. The full vacua are the full 2d SCFTs that define the worldsheet theory of the string.

The collection of all string vacua, possibly subject to some assumptions, has come to be called the landscape of string theory vacua.

Related concepts

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

References

See the references at

• string phenomenology

• landscape of string theory vacua

On (in-)stability of non-supersymmetric AdS vacua in string theory:

• Iosif Bena, Krzysztof Pilch, Nicholas Warner, Brane-Jet Instabilities, J. High Energ. Phys. 2020, 91 (2020) (arXiv:2003.02851)