String theory formula
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String Theory Formula: Insights and Developments
Higher-Loop Amplitude Monodromy Relations in String Theory
Monodromy relations in string theory have been pivotal in understanding tree-level field theory amplitudes, particularly through the color-kinematics duality. This duality has significantly advanced the computation of loop amplitudes in quantum field theory. Recent research has extended these monodromy relations to higher loops in open string theory. By employing a contour deformation argument of the open string diagram integrands, new identities have been derived that relate planar and nonplanar topologies in string theory. Explicit one and two-loop monodromy formulas have been written at any multiplicity, and these formulas have been validated by unitarity checks in the case of the four-point N=4 super-Yang-Mills amplitude1.
Determinants, Torsion, and Anomaly Cancellation in String Theory
The application of determinants of Dirac operators to string theory has led to significant findings, such as the holomorphic factorization of Belavin and Knizhik for the bosonic string. Additionally, Witten's global anomaly formula has provided sufficient conditions for anomaly cancellation in the heterotic string, even in arbitrary background spacetimes. This has been achieved by developing torsion invariants related to characteristic classes of vector bundles and index theory2.
Effective Theory of Long Strings
The low-energy effective theory on long strings in quantum field theory has been explored, revealing that the energy levels of long (closed) strings, when expanded in powers of 1/L (where L is the string length), are universal up to order 1/L^5. For excited strings in dimensions greater than three, a universal deviation at order 1/L^5 from the naive formula used to fit lattice results has been identified. These findings have been corroborated by explicit computations of the effective action on long strings in confining theories with a gravitational dual5.
New Ambitwistor String Theories
New ambitwistor string theories have been described, which give rise to recent amplitude formulas for various theories, including Einstein-Yang-Mills and (Dirac)-Born-Infeld. A novel worldsheet conformal field theory has been introduced, providing the appropriate color factors without the spurious multitrace terms of earlier models. This advancement is crucial for obtaining the correct multitrace terms when Yang-Mills is coupled to gravity6.
k-String Tensions in SU(N) Gauge Theories
In the context of four-dimensional SU(N) gauge theories, the spectrum of confining strings has been studied. For the SU(6) gauge theory formulated on a lattice, the three independent string tensions σ_k related to sources with Z_N charge k=1,2,3 have been computed using Monte Carlo simulations. The results are consistent with the sine formula for the ratio between σ_k and the standard string tension σ, showing deviations from Casimir scaling. This sine formula is known to emerge in supersymmetric SU(N) gauge theories and M-theory7.
Semiclassical Quantization and Regge Trajectories
An effective string theory for long-distance QCD has been evaluated through semiclassical expansion about a classical rotating string solution. After renormalization, the zero-point energy of the string fluctuations remains finite as the quark masses approach zero, maintaining conformal invariance in any spacetime dimension D. For D=26, the energy spectrum of the rotating string coincides with that of the open string in classical bosonic string theory, though its physical origin differs. For D=4, the first semiclassical correction adds a constant to the classical Regge formula8.
Topological Strings and Nekrasov's Formulas
The method of geometric transition has been applied to compute all genus topological closed string amplitudes compactified on local F0, using Chern-Simons gauge theory. The results align with the formula proposed by Nekrasov for N=2 SU(2) gauge theory, encoding all information of topological string amplitudes on local F0, including the number of holomorphic curves at arbitrary genus. By taking suitable limits, one can recover the four-dimensional Seiberg-Witten theory and its coupling to external graviphoton fields9.
Conclusion
The advancements in string theory formulas, from higher-loop amplitude monodromy relations to effective theories of long strings and new ambitwistor string theories, have significantly deepened our understanding of the fundamental properties and behaviors of strings in various theoretical frameworks. These developments not only enhance our theoretical knowledge but also provide new tools and methods for further exploration in the field of string theory.
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Most relevant research papers on this topic
Higher-Loop Amplitude Monodromy Relations in String and Gauge Theory.
This paper extends monodromy relations to higher loops in open string theory, leading to new identities relating planar and nonplanar topologies, and demonstrates their utility in understanding tree-level field theory amplitudes in quantum field theory.
Determinants, torsion, and strings
This paper applies the results of [BF1, BF2] on determinants of Dirac operators to String Theory, recovering the "holomorphic factorization" of Belavin and Knizhik and proving anomaly cancellation in heterotic strings.
Highly CitedString theory
For each n > 0, two alternative string theory axiomatizations are presented, both of which are synonymous with each other and Peano arithmetic.
Highly CitedComments on the Nekrasov-type formula for E-string theory
The Nekrasov-type formula for E-string theory is valid for cases with three and four Wilson lines, and the Seiberg-Witten curves are reproduced from the formula, with their dependence on the Wilson lines.
The effective theory of long strings
Long strings in quantum field theory have universal energy levels up to order 1/L5, but non-universal terms appear at order 1/L7.
Literature ReviewHighly CitedNew ambitwistor string theories
New ambitwistor string theories provide accurate amplitude formulae for Einstein-Yang-Mills, (Dirac)-Born-Infeld, Galileons, and others, with a novel worldsheet conformal field theory.
Highly Citedk-string tensions in SU(N) gauge theories
The sine formula sigma_k/sigma in four-dimensional SU(N) gauge theories reveals deviations from the Casimir scaling and is consistent with the spectrum of confining string tensions in SU(N) gauge theories.
Semiclassical quantization of effective string theory and Regge trajectories
Effective string theory for long distance QCD remains conformally invariant in any spacetime dimension, with a semiclassical spectrum valid for large values of angular momentum.
Topological strings and Nekrasov's formulas
Nekrasov's formula accurately encodes topological string amplitudes on local F0, allowing for the recovery of Seiberg-Witten theory and its coupling to external graviphoton fields.
Highly Cited