Theta Series, Wall-Crossing and Quantum Dilogarithm Identities (original) (raw)
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Journal of High Energy Physics, 2019
We propose quantum states for Little String Theories (LSTs) arising from M5 branes probing A- and D-type singularities. This extends Witten’s picture of M5 brane partition functions as theta functions to this more general setup. Compactifying the world-volume of the five-branes on a two-torus, we find that the corresponding theta functions are sections of line bundles over complex 4-tori. This formalism allows us to derive Seiberg-Witten curves for the resulting four-dimensional theories. Along the way, we prove a duality for LSTs observed by Iqbal, Hohenegger and Rey.
The present paper is a review, a thesis of some very important contributes of E. Witten, C. Beasley, R. Ricci, B. Basso et al. regarding various applications concerning the Jones polynomials, the Wilson loops and the cusp anomaly and integrability from string theory. In this work, in the Section 1, we have described some equations concerning the knot polynomials, the Chern-Simons from four dimensions, the D3-NS5 system with a theta-angle, the Wick rotation, the comparison to topological field theory, the Wilson loops, the localization and the boundary formula. We have described also some equations concerning electric-magnetic duality to N = 4 super Yang-Mills theory, the gravitational coupling and the framing anomaly for knots. Furthermore, we have described some equations concerning the gauge theory description, relation to Morse theory and the action. In the Section 2, we have described some equations concerning the applications of non-abelian localization to analyze the Chern-Simons path integral including Wilson loop insertions. In the Section 3, we have described some equations concerning the cusp anomaly and integrability from String theory and some equations concerning the cusp anomalous dimension in the transition regime from strong to weak coupling. In the Section 4, we have described also some equations concerning the "fractal" behaviour of the partition function. Also here, we have described some mathematical connections between various equation described in the paper and (i) the Ramanujan"s modular equations regarding the physical vibrations of the bosonic strings and the superstrings, thence the relationship with the Palumbo-Nardelli model, (ii) the mathematical connections with the Ramanujan"s equations concerning π and, in conclusion, (iii) the mathematical connections with the aurea ratio v1 26.09.2011 - v2 21.03.2020 - REVISITED AND UPDATED VERSION 25.10.2020
D1-Strings in Large . . . Quantum Nambu Geometry and M5-Branes in C-Field
2011
We consider D1-branes in a RR flux background and show that there is a low energy-large flux double scaling limit where the D1-branes action is dominated by a Chern-Simons-Myers coupling term. As a classical solution to the matrix model, we find a novel quantized geometry characterized by a quantum Nambu 3-bracket. Infinite dimensional representations of the quantum Nambu geometry are constructed which demonstrate that the quantum Nambu geometry is intrinsically different from the ordinary Lie algebra type noncommutative geometry. Matrix models for the IIB string, IIA string and M-theory in the corresponding backgrounds are constructed. A classical solution of a quantum Nambu geometry in the IIA Matrix string theory gives rise to an expansion of the fundamental strings into a system of multiple D4-branes and the fluctuation is found to describe an action for a non-abelian 3-form field strength which is a natural non-abelian generalization of the PST action for a single D4-brane. In view of the recent proposals [1, 2] of the M5-branes theory in terms of the D4-branes, we suggest a natural way to include all the KK modes and propose an action for the the multiple M5-branes in a constant C-field. The worldvolume of the M5-branes in a C-field is found to be described by a quantum Nambu geometry with self-dual parameters. It is intriguing that our action is naturally formulated in terms of a 1-form gauge field living on a six dimensional quantum Nambu geometry.
Instanton-monopole correspondence from M-branes on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">mml:msup<mml:mi mathvariant="double-struck">Smml:mn1and little string theory
Physical review, 2016
We study BPS excitations in M5-M2-brane configurations with a compact transverse direction, which are also relevant for type IIa and IIb little string theories. These configurations are dual to a class of toric elliptically fibered Calabi-Yau manifolds X N with manifest SL(2, Z)×SL(2, Z) modular symmetry. They admit two dual gauge theory descriptions. For both, the non-perturbative partition function can be written as an expansion of the topological string partition function of X N with respect to either of the two modular parameters. We analyze the resulting BPS counting functions in detail and find that they can be fully constructed as linear combinations of the BPS counting functions of M5-M2-brane configurations with non-compact transverse directions. For certain M2-brane configurations, we also find that the free energies in the two dual theories agree with each other, which points to a new correspondence between instanton and monopole configurations. These results are also a manifestation of T-duality between type IIa and IIb little string theories.
Fermion wavefunctions in magnetized branes: Theta identities and Yukawa couplings
Nuclear Physics B, 2009
Computation of Yukawa couplings, determining superpotentials as well as the Kähler metric, with oblique (non-commuting) fluxes in magnetized brane constructions is an interesting unresolved issue, in view of the importance of such fluxes for obtaining phenomenologically viable models. In order to perform this task, fermion (scalar) wavefunctions on toroidally compactified spaces are presented for general fluxes, parameterized by Hermitian matrices with eigenvalues of arbitrary signatures. We also give explicit mappings among fermion wavefunctions, of different internal chiralities on the tori, which interchange the role of the flux components with the complex structure of the torus. By evaluating the overlap integral of the wavefunctions, we give the expressions for Yukawa couplings among chiral multiplets arising from an arbitrary set of branes (or their orientifold images). The method is based on constructing certain mathematical identities for general Riemann theta functions with matrix valued modular parameter. We briefly discuss an application of the result, for the mass generation of non-chiral fermions, in the SU (5) GUT model presented by us in arXiv: 0709.2799.
BRANES, STRINGS, AND ODD QUANTUM NAMBU BRACKETS
The dynamics of topological open branes is controlled by Nambu Brackets. Thus, they might be quantized through the consistent quantization of the underlying Nambu brackets, including odd ones: these are reachable systematically from even brackets, whose more tractable properties have been detailed before.
Instanton-monopole correspondence from M-branes onS1and little string theory
Physical Review D, 2016
We study BPS excitations in M5-M2-brane configurations with a compact transverse direction, which are also relevant for type IIa and IIb little string theories. These configurations are dual to a class of toric elliptically fibered Calabi-Yau manifolds X N with manifest SL(2, Z)×SL(2, Z) modular symmetry. They admit two dual gauge theory descriptions. For both, the non-perturbative partition function can be written as an expansion of the topological string partition function of X N with respect to either of the two modular parameters. We analyze the resulting BPS counting functions in detail and find that they can be fully constructed as linear combinations of the BPS counting functions of M5-M2-brane configurations with non-compact transverse directions. For certain M2-brane configurations, we also find that the free energies in the two dual theories agree with each other, which points to a new correspondence between instanton and monopole configurations. These results are also a manifestation of T-duality between type IIa and IIb little string theories.
D1-strings in large RR 3-form flux, quantum Nambu geometry and M5-branes in the C -field
Journal of Physics A: Mathematical and Theoretical, 2012
We consider D1-branes in a RR flux background and show that there is a low energy-large flux double scaling limit where the D1-branes action is dominated by a Chern-Simons-Myers coupling term. As a classical solution to the matrix model, we find a novel quantized geometry characterized by a quantum Nambu 3-bracket. Infinite dimensional representations of the quantum Nambu geometry are constructed which demonstrate that the quantum Nambu geometry is intrinsically different from the ordinary Lie algebra type noncommutative geometry. Matrix models for the IIB string, IIA string and M-theory in the corresponding backgrounds are constructed. A classical solution of a quantum Nambu geometry in the IIA Matrix string theory gives rise to an expansion of the fundamental strings into a system of multiple D4-branes and the fluctuation is found to describe an action for a non-abelian 3-form field strength which is a natural non-abelian generalization of the PST action for a single D4-brane. In view of the recent proposals [1, 2] of the M5-branes theory in terms of the D4-branes, we suggest a natural way to include all the KK modes and propose an action for the the multiple M5-branes in a constant C-field. The worldvolume of the M5-branes in a C-field is found to be described by a quantum Nambu geometry with self-dual parameters. It is intriguing that our action is naturally formulated in terms of a 1-form gauge field living on a six dimensional quantum Nambu geometry.
On certain aspects of string theory/gauge theory correspondence
N = 2 supersymmetric Yang-Mills theories for all classical gauge groups, that is, for SU (N), SO(N), and Sp(N) is considered. The formal expression for almost all models accepted by the asymptotic freedom are obtained. The equations which define the Seiberg-Witten curve are proposed. In some cases they are solved. It is shown that for all considered the 1-instanton corrections which follows from these equations agree with the direct computations. Also they agree with the computations based on Seiberg-Witten curves which come from the M-theory consideration. It is shown that for a large class of models the M-theory predictions matches with the direct compuatations. It is done for all considered models at the 1-instanton level. For some models it is shown at the level of the Seiberg-Witten curves. In section 2.4 we introduce the instanton counting parameter q which is related to τ , g and Θ as follows: q = e 2πiτ = e − 8π 2 g 2 e iΘ. some other sources by the factor 2πi. The complete Wilsonian effective action does contain other terms, for example the next one, which contains four derivatives and eight fermions can be expressed with the help of a real function H(a, a) as the N = 2 D-term [44, 20, 74, 59, 92, 93, 26]: S 4−deriv = d 4 xd 4 θd 4θ H(Ψ,Ψ).
Spinor-vector duality in heterotic string vacua
Nuclear Physics B, 2008
Classification of the N = 1 space-time supersymmetric fermionic Z 2 × Z 2 heterotic-string vacua with symmetric internal shifts, revealed a novel spinorvector duality symmetry over the entire space of vacua, where the S t ↔ V duality interchanges the spinor plus anti-spinor representations with vector representations. In this paper we demonstrate that the spinor-vector duality exists also in fermionic Z 2 heterotic string models, which preserve N = 2 space-time supersymmetry. In this case the interchange is between spinorial and vectorial representations of the unbroken SO(12) GUT symmetry. We provide a general algebraic proof for the existence of the S t ↔ V duality map. We present a novel basis to generate the free fermionic models in which the ten dimensional gauge degrees of freedom are grouped into four groups of four, each generating an SO(8) modular block. In the new basis the GUT symmetries are produced by generators arising from the trivial and non-trivial sectors, and due to the triality property of the SO(8) representations. Thus, while in the new basis the appearance of GUT symmetries is more cumbersome, it may be more instrumental in revealing the duality symmetries that underly the string vacua.