The function ψ depends on the choice of the background or the parameter values of the target space of the topological array. The holomorphic anomaly equations capture the anomalous dependence of the topological partition function of the set on the anti-holomorphic Calabi-Yau modules (target space).
Calabi-Yau manifolds
The space of such choices is called the space of M Calabi-Yaua modules, with K¨ahler and complex structure modules as coordinates in the module space. That all volumes are positive restricts the modules to a K¨ahler cone, with boundaries corresponding to singular Calabi-Yau degeneracies.
Topological string theory
X =Xic can be treated as a bosonic field on the worldsheet, taking values in the target space, where i runs over the dimension of the target space. We are interested in the dependence of the amplitudes Fg on the moduli of the Calabi-Yau target space.
Chiral rings
The appropriate form of the OSV conjecture using Ψ (schematic ZBH = |Ψ|2) is exactly non-perturbative; and. As a result, we will call Ψ the nonperturbative completion of the topological string partition function for the system we consider.
The OSV conjecture
In any case, the black hole partition function should have a non-perturbative definition, so the non-perturbative implications of (2.9) are interesting. A major problem is that most existing tools for computing the topological partition function of a set use generic expansions of the form (2.8).
Realising the OSV conjecture
- Topological strings
- Black holes and two-dimensional Yang-Mills theory
- One-dimensional free fermions
- Perturbative OSV
It is worth emphasizing the distinction between the two sides of (2.9): the black hole partition function is related to counting microstates of a four-dimensional extreme black hole, while the topological string partition function is a path integral of a sigma model. with Calabi-Yau target area. For the geometry (2.8), the toric diagram is a simple line connected to itself on a periodically identified plane, the T2. As shown in Figure 2.2, small deviations from the ground state of the black hole can be seen as fermion excitations on the two Fermi surfaces, one each for positive and negative momentum states.
The disturbing OSV relation (2.32) follows from the decoupling of the two Fermi surfaces of the black hole.
Non-chiral baby universes
A non-chiral recursion relation
These excitations can be seen as a sea of holes within the sea of fermions, and thus treated as a black hole The Fermi sea of holes superimposed on a black hole Fermi sea without deep excitations. The first term on the right is the perturbative result and the second term subtracts the over-enumerated states, that is, those with one or more deep excitations.
Gravitational interpretation
The Euclidean action is proportional to the difference in entropy of the configurations and thus to the square of the charges, leading to exponential suppression at large charge. The near-horizon limit is particularly relevant in light of the interpretation [5] of ψtop (or rather its transformψP,Q) as the Hartle-Hawking wave function of the universe in the mini-superspace sector of string theory, which describes BPS (supersymmetry-protected) quantities. These wave functions are the ground states of the theory after long-time evolution, which implies the limit near the horizon.
Summing over universe number seems to imply a loss of quantum coherence for the Hartle-Hawking state for a given universe, due to the coupling of the wave functions of different universes through overall charge conservation.
Chiral completion of the topological string
Recall that the nonperturbative corrections to (2.32) are excitations deep inside the black hole Fermi sea, since such excitations can be assigned to any of the topological string partition functions. The derivation in Section 2.3.4 used exactly this approach: equation (2.30) is an exact result, and thus Ψ(t) defined by (2.31) is the candidate nonperturbative completion of the topological string partition function ψ(t). Note that the momentum of the fermions is also sign reversed, and thus Ψ−raabove is indeed chiral, as opposed to anti-chiral.
It owes its existence to the difference between ψ0 and the zero-point naive energy of the topological string splitting tower fermion.
Background independence
However, recall from Section 1.1 that K¨0 ahler structure deformations are exactly elements of H1,1(X), while the correspondence between complex structure deformations and H2,1(X) arises from the consideration of infinite-simal deformations of the holomorphic top form Ω. Deformations of the theory with marginal operators modify only the holomorphic moduli, ti → ti +ui. A related issue is the interpretation of (2.47) in terms of physical black holes, consistent with the discussion in Section 2.4.2.
The result (2.47) then represents universe-creating instantons that split only one of the AdS2 boundaries, leaving infant universes that “share” the other boundary.
Extension to other genus target spaces
Realising the OSV conjecture
This distribution function can still be interpreted as that of a sea of fermions on a circle, but the fermions now interact due to the presence of the quantum dimension. The topological series is sensitive to the choice of boundary conditions of the non-compact Calabi-Yau. The definition (2.59), after taking into account the ghost branes, is perturbatively equivalent to the perturbative topological string partition function, since the only difference is the restriction on the height of the Young diagram R.
However, the height constraint means that the OSV relation (2.58) does not suffer from overcounting, and Ψ is thus a candidate non-perturbative termination of the topological string partition function.
The chiral recursion relation
The S-type phantom branes can be physically interpreted [21] as embedding non-compact D2 branes wrapping the splitter fiber wrapped by D4 branes. However, when the baby universes are created, the non-compact charge of the D2-brane can be non-zero in each universe, as long as the sum of the charges vanishes. In the language of closed strings, the eigenvalues of S-type representations are the values of infinitely many non-normalizable K¨ahler modules for the boundary conditions at brane infinity D4.
That the classical prefactors no longer map onto simple zero-point energies is at the heart of the difficulty in finding a chiral recursion relation.
Outlook
Finally, the significance and consistency of the new anomalies with existing results, especially matrix models and grand duality, is discussed in Section 3.4. The Dirichlet boundary conditions identify the left and right moving sectors of the string theory, and hence the overhangs. By the operator-state correspondence, ψi(1) corresponds to an open string state, and thus the second term in (3.4) corresponds to the variation of the open string modulus, that is, the deformations of the D-brane.
For example, [13] argues that open-string moduli either cancel out for generic values of the bulk moduli, or do not appear as independent moduli, and thus disappear from Fg,h.
Extended holomorphic anomaly equations
- Boundaries of moduli space
- Degenerating tubes
- Degenerating thin strips
- Shrinking boundary
- Putting it together
Consider the case where, at the modulus space limit, a closed string tube becomes infinitely long and narrow. The third complex module, τ, parametrizes the handle shape, so that τ → ∞ in the limit of the module space. As in Section 3.2.2, considerations for the rest of the Riemann surface imply that the tube is replaced by a complete set of closed-string marginal basis states.
The embedding of|j*in the remainder of the Riemann surface can be written as a covariant derivative, so the amplitude of the Riemann surface is,.
Decoupling of moduli from the other model
The outer contour can now be deformed beyond boundaries on the world sheet, producing terms corresponding to all possible degeneracies of the Riemann surface, as for the ¯t¯i derivative in the previous section. Degenerating tube (Cases A and B): The tube degeneration disappears if the insertion (3.27) is not on the tube. Degenerate strip (Cases D, E and F): As in the previous section, the disappearance of these contributions follows from the assumption that open string moduli do not.
Moreover, because the insertions are not marginal operators, (3.29) cannot be written in the form of a recurrence relation for lower gender amplitudes.
Implications of the new anomalies
Conversely, the presence of the new anomalies is correlated with the breakdown of the large N duality. At the same time, these wells cancel the one-point functions of the disk, and thus the appearance of the new anomalies is correlated with invalid spacetime constructions. The limit states of the two stacks together form Ca¯ = 0, and the new anomalies do not appear.
New material in the following treatment, first reported in [15], is a proof of the open set form of Feynman's rules.
The Feynman rules
On the open string side, since D¯i∆¯j¯k ∈Sym3T ⊗ L−1 is symmetric in all three indices, it can be integrated locally to give The result can thus be interpreted as a sum of Feynman diagrams, as shown in figure 4.1, with S, Sj and Sij as propagation, ∆ and ∆j as terminators (tadpoles), and Fi(g,h)1···in as loop corrected corners. The dashed lines in Figure 4.1, corresponding to the presence of S, Sj, ∆ or ∆j, can be interpreted as the dilaton propagating [8], in contrast to the solid lines representing marginal (c, c) fields.
Thus, the propagators can be calculated in terms of the geometry of the Calabi–Yau module space and the terminators additionally depend on the boundary conditions, up to a holomorphic ambiguity in both cases.
Proof of closed string Feynman rules
By treating x and ϕ as dynamical variables, the expansion of Z can be evaluated as a perturbation expansion in λ using the usual Feynman rules. For example, F(2) is given (up to holomorphic ambiguity) by the order λ2 terms in the expansion of Z, of which the first are. This proves that ∂∂¯t¯iF(g) is the anti-holomorphic derivative of all the other terms in the Feynman rule expansion of Z in order λ2g−2 — and so that the Feynman rules give F(g) up to an arbitrary holomorphic function, the holomorphic ambiguity.
Extension to open strings
Since the shifted W satisfies the closed-string differential equation (4.17), the proof of the closed-string Feynman rules presented in Section 4.3 applies here as well. An elegant and more computationally efficient reformulation of the closed-string Feynman rules for computing topological string amplitudes was provided by [54]. The partition function of the open topological string is thus a different wave function in this phase space for each choice of world sheet boundary conditions (encoded in the form of the terminator.
In the last equation, qj−12 denotes the set of variables generated by the range of index j.
The chiral recursion relation
- Realising the OSV conjecture explicitly
- Perturbative OSV in terms of free fermions
- Decomposition of a Young diagram for a black hole microstate
- Overcounted states in the perturbative OSV relation
- Removing overcounted states recursively
- Exact OSV with modified topological string partition functions
- Decomposing the perturbative topological string partition function
- Decomposing a U ( ∞ ) representation into two finite-height representations . 42
- Open string degenerations of a Riemann surface
- The operators on a degenerating handle of a Riemann surface
- New anomalies in anti-holomorphic derivatives of amplitudes
- Near-boundary region for wrong moduli derivative
- New anomalies in wrong moduli derivatives of amplitudes
- Feynman diagram expansion of F (1,1)
As noted in [3], the eta function only contributes to genus zero and perturbative, so we need not worry that it did not appear in the perturbative Ztop. Coleman, "Black holes as red herrings: Topological fluctuations and the loss of quantum coherence," Nucl.
