Seiberg-Witten Limit Research Articles

$${\mathcal {N}}$$ = $$2^*$$ Gauge Theory, Free Fermions on the Torus and Painlevé VI

In this paper we study the extension of Painlev\'e/gauge theory correspondence to circular quivers by focusing on the special case of $SU(2)$ $\mathcal{N}=2^*$ theory. We show that the Nekrasov-Okounkov partition function of this gauge theory provides an explicit combinatorial expression and a Fredholm determinant formula for the tau-function describing isomonodromic deformations of $SL_2$ flat connections on the one-punctured torus. This is achieved by reformulating the Riemann-Hilbert problem associated to the latter in terms of chiral conformal blocks of a free-fermionic algebra. This viewpoint provides the exact solution of the renormalization group flow of the $SU(2)$ $\mathcal{N}=2^*$ theory on self-dual $\Omega$-background and, in the Seiberg-Witten limit, an elegant relation between the IR and UV gauge couplings.

A bi-invariant Einstein–Hilbert action for the non-geometric string

Inspired by recent studies on string theory with non-geometric fluxes, we develop a differential geometry calculus combining usual diffeomorphisms with what we call β-diffeomorphisms. This allows us to construct a manifestly bi-invariant Einstein–Hilbert type action for the graviton, the dilaton and a dynamical (quasi-)symplectic structure. The equations of motion of this symplectic gravity theory, further generalizations and the relation to the usual form of the string effective action are discussed. The Seiberg–Witten limit, known for open strings to relate commutative with non-commutative theories, makes an interesting appearance.

Non-geometric strings, symplectic gravity and differential geometry of Lie algebroids

Based on the structure of a Lie algebroid for non-geometric fluxes in string theory, a differential-geometry calculus is developed which combines usual diffeomorphisms with so-called \beta-diffeomorphisms emanating from gauge symmetries of the Kalb-Ramond field. This allows to construct a bi-invariant action of Einstein-Hilbert type comprising a metric, a (quasi-)symplectic structure \beta and a dilaton. As a salient feature, this symplectic gravity action and the resulting equations of motion take a form which is similar to the standard action and field equations. Furthermore, the two actions turn out to be related via a field redefinition reminiscent of the Seiberg-Witten limit. Remarkably, this redefinition admits a direct generalization to higher-order \alpha'-corrections and to the additional fields and couplings appearing in the effective action of the superstring. Simple solutions to the equations of motion of the symplectic gravity action, including Calabi-Yau geometries, are discussed.

S-duality as a β-deformed Fourier transform

An attempt is made to formulate Gaiotto’s S-duality relations in an explicit quantitative form. Formally the problem is that of evaluation of the Racah coefficients for the Virasoro algebra, and we approach it with the help of the matrix model representation of the AGT-related conformal blocks and Nekrasov functions. In the Seiberg-Witten limit, this S-duality reduces to the Legendre transformation. In the simplest case, its lifting to the level of Nekrasov functions is just the Fourier transform, while corrections are related to the beta-deformation. We calculate them with the help of the matrix model approach and observe that they vanish for β = 1. Explicit evaluation of the same corrections from the U q (sl(2)) infinite-dimensional representation formulas due to B.Ponsot and J.Teshner remains an open problem.

Noncommutativity from the string perspective: modification of gravity at ammwithoutmmsized extra dimensions

We explore how the IR pathologies of noncommutative field theory are resolved when the theory is realized as open strings in background B-fields: essentially, since the IR singularities are induced by UV/IR mixing, string theory brings them under control in much the same way as it does the UV singularities. We show that at intermediate scales (where the Seiberg-Witten limit is a good approximation) the theory reproduces the noncommutative field theory with all the (un)usual features such as UV/IR mixing, but that outside this regime, in the deep infra-red, the theory flows continuously to the commutative theory and normal Wilsonian behaviour is restored. The resulting low energy physics resembles normal commutative physics, but with additional suppressed Lorentz violating operators. We also show that the phenomenon of UV/IR mixing occurs for the graviton as well, with the result that, in configurations where Planck's constant receives a significant one-loop correction (for example brane-induced gravity), the distance scale below which gravity becomes non-Newtonian can be much greater than any compact dimensions.

Comments on noncommutative gravity

We study the possibility of obtaining noncommutative gravity dynamics from string theory in the Seiberg–Witten limit. We find that the resulting low-energy theory contains more interaction terms than those proposed in noncommutative deformations of gravity. The rôle of twisted diffeomorphisms in string theory is studied and it is found that they are not standard physical symmetries. It is argued that this might be the reason why twisted diffeomorphisms are not preserved by string theory in the low energy limit. Twisted gauge transformations are also discussed.

Dissolving D0-brane into D2-brane with background B-field

D0-branes on a D2-brane with a constant background B-field are unstable due to the presence of a tachyonic mode and expected to dissolve into the D2-brane to formulate a constant D0-charge density. In this paper we study such a dissolution process in terms of a noncommutative gauge theory. Our results show that the localized D0-brane spreads out over all of space on the D2-brane as the tachyon rolls down into a stable vacuum. D0-branes on a D2-brane can be described as unstable solitons in a noncommutative gauge theory in 2+1 dimensions in the Seiberg-Witten limit. In contrast to the case of annihilation of a non-BPS D-brane, we are free from difficulty of disappearance of DOF, since there exist open strings after the tachyon condensation. We solve an equation of motion of the gauge field numerically, and our results show that the localized soliton smears over all of noncommutative space. In addition, we evaluate distributions of D-brane charge, F-string charge, and energy density via formulas derived in Matrix theory. Our results show that the initial singularities of D0-charge and energy density are resolved by turning on the tachyon, and they disperse over the whole space on the D2-brane during the tachyon condensation process.

Cardy states, factorization and idempotency in closed string field theory

We show that boundary states in the generic on-shell background satisfy a universal nonlinear equation of closed string field theory. It generalizes our previous claim for the flat background. The origin of the equation is factorization relation of boundary conformal field theory which is always true as an axiom. The equation necessarily incorporates the information of open string sector through a regularization, which implies the equivalence with Cardy condition. We also give a more direct proof by oscillator representations for some nontrivial backgrounds (torus and orbifolds). Finally we discuss some properties of the closed string star product for nonvanishing B field and find that a commutative and nonassociative product (Strachan product) appears naturally in Seiberg–Witten limit.

Modified star-products beyond the large-B limit

Derivative corrections to the Wess–Zumino couplings of open-string effective actions are computed at all orders in derivatives, taking the open-string metric into account. This leads to a set of deformed star-products beyond the Seiberg–Witten limit, and allows to reinterpret the couplings in terms of a deformed integration prescription along a Wilson line in the non-commutative set-up. Moreover, the recursive definition of the star-products induces deformations of U(1) non-commutative Yang–Mills theory.

One-loop unitarity of string theories in a constant external background and their Seiberg-Witten limit

Perturbative spectra and related factorization properties of one-loop open string amplitudes in the presence of a constant external background B are analysed in detail. While the pattern of the closed string spectrum, obtained after a careful study of the properly symmetrized amplitudes, turns out to be unaffected by the presence of B, a series of double open-string poles, which would be absent when B is turned off, can couple owing to a partial symmetry loss. These features are studied first in a bosonic setting and then generalized in the more satisfactory superstring context. When the background is of an ``electric'' type, a classical perturbative instability is produced beyond a critical value of the electric field. In the Seiberg-Witten limit this instability is the origin of the unphysical tachyonic cut occurring in the non-planar amplitudes of the corresponding noncommutative field theories.

UNITARITY OF NONCOMMUTATIVE FIELD THEORIES FROM STRING THEORY

We improve the study of the lack of perturbative unitarity of noncommutative space-time quantum field theories derived from open string theory in electric backgrounds, enforcing the universality of the mechanism by which a tachyonic branch cut appears when the Seiberg-Witten limit freezes the string in an unstable vacuum. The main example is realized in the context of the on-shell four-tachyon amplitude of the bosonic string, and the dependence of the phenomenon on the brane-worldvolume dimension is analysed. We discuss the possibility of a proof in superstring theory, and finally mention the NCOS limit in this framework.

NONCOMMUTATIVE EXTENDED WAVES AND SOLITON-LIKE CONFIGURATIONS IN N = 2 STRING THEORY

The Seiberg–Witten limit of fermionic N = 2 string theory with nonvanishing B-field is governed by noncommutative self-dual Yang–Mills theory (ncSDYM) in 2+2 dimensions. Conversely, the self-duality equations are contained in the equation of motion of N = 2 string field theory in a B-field background. Therefore finding solutions to noncommutative self-dual Yang–Mills theory on ℝ2,2 might help to improve our understanding of nonperturbative properties of string (field) theory. In this paper, we construct nonlinear soliton-like and multi-plane wave solutions of the ncSDYM equations corresponding to certain D-brane configurations by employing a solution generating technique, an extension of the so-called dressing approach. The underlying Lax pair is discussed in two different gauges, the unitary and the Hermitian gauge. Several examples and applications for both situations are considered, including Abelian solutions constructed from GMS-like projectors, noncommutative U(2) soliton-like configurations and interacting plane waves. We display a correspondence to earlier work on string field theory and argue that the solutions found here can serve as a guideline in the search for nonperturbative solutions of nonpolynomial string field theory.

General properties of non-commutative field theories

In this paper we study general properties of non-commutative field theories obtained from the Seiberg–Witten limit of string theories in the presence of an external B-field. We analyze the extension of the Wightman axioms to this context and explore their consequences, in particular, we present a proof of the CPT theorem for theories with space–space non-commutativity. We analyze as well questions associated to the spin-statistics connections, and show that non-commutative N=4 , U(1) gauge theory can be softly broken to N=0 satisfying the axioms and providing an example where the Wilsonian low energy effective action can be constructed without UV/IR problems, after a judicious choice of soft breaking parameters is made. We also assess the phenomenological prospects of such a theory, which are in fact rather negative.

Open-String Actions and Noncommutativity Beyond the Large-BLimit

In the limit of large, constant B-field (the ``Seiberg-Witten limit''), the derivative expansion for open-superstring effective actions is naturally expressed in terms of the symmetric products *n. Here, we investigate corrections around the large-B limit, for Chern-Simons couplings on the brane and to quadratic order in gauge fields. We perform a boundary-state computation in the commutative theory, and compare it with the corresponding computation on the noncommutative side. These results are then used to examine the possible role of Wilson lines beyond the Seiberg-Witten limit. To quadratic order in fields, the entire tree-level amplitude is described by a metric-dependent deformation of the *2 product, which can be interpreted in terms of a deformed (non-associative) version of the Moyal * product.

Kronecker foliation, D1-branes and Morita equivalence of Noncommutative two-tori

It is known that the physics of open strings on a D2-brane on a two-torus is realized from the viewpoint of deformation quantization in the Seiberg-Witten limit. We study its T-dual theory, i.e. D1-brane physics on two-tori. Such theory is described by Kronecker foliation. The algebra of open strings on the D1-brane is then identified with the crossed product representation of a noncommutative two-torus. The Morita equivalence of noncommutative two-tori is also realized geometrically along this line. As an application, Heisenberg modules and the tensor product between them are discussed from these geometric viewpoints. We show they are related to the homological mirror symmetry of two-tori.

Anatomy of two-loop effective action in noncommutative field theories

We compute, at two-loop order, one-particle-irreducible Green functions and effective action in noncommutative λ[ Φ 3] ★-theory for both planar ( g=0, h=3) and nonplanar ( g=1, h=1) contributions. We adopt worldline formulation of the Feynman diagrammatics so that relation to string theory diagrammatics is made transparent in the Seiberg–Witten limit. We argue that the resulting two-loop effective action is expressible via open Wilson lines: one-particle-irreducible effective action is generating functional of connected diagrams for interacting open Wilson lines.

DERIVATIVE CORRECTIONS TO DIRAC–BORN–INFELD AND CHERN–SIMON ACTIONS FROM NONCOMMUTATIVITY

We show that the higher-order derivative α′ corrections to the Dirac–Born–Infeld (DBI) and Chern–Simon actions is derived from noncommutativity in the Seiberg–Witten limit, and is shown to agree with Wyllard's (hep-th/0008125) result, as conjectured by Das et al., (hep-th/0106024). In calculating the corrections, we have expressed [Formula: see text] in terms of F, Â in terms of A up to order [Formula: see text], and made use of it.

NONCOMMUTATIVE N = 2 p - p′ SYSTEM

We analyze several open and mixed sector tree-level amplitudes in N = 2 p - p′ systems with a constant magnetic B turned on. The three-point function vanishes on-shell. The four-point function, in the Seiberg–Witten (SW) low energy limit,2 is local, indicating the possible topological nature of the theory (in the SW low energy limit) and the possible relation between noncommutativeN = 2 p - p′system in two complex dimensions and in the SW limit, and (non)commutativeN = 2 p′ - p′system in two real dimensions. We discuss three extreme noncommutativity limits (after having taken the Seiberg–Witten low energy limit) of the mixed three-point function, and get two kinds of commutative nonassociative generalized star products. We make some speculative remarks related to reproducing the above four-point tree level amplitude in the open sector, from a field theory.

Anatomy of one-loop effective action in non-commutative scalar field theories

One-loop effective action of noncommutative scalar field theory with cubic self-interaction is studied. Utilizing worldline formulation, both planar and nonplanar part of the effective action are computed explicitly. We find complete agreement of the result with Seiberg-Witten limit of string worldsheet computation and standard Feynman diagrammatics. We prove that, at low-energy and large noncommutativity limit, nonplanar part of the effective action is simplified enormously and is resummable into a quadratic action of scalar open Wilson line operators.

Dirac–Born–Infeld action, Seiberg–Witten map, and Wilson lines

We write the recently conjectured action for transformation of the ordinary Born–Infeld action under the Seiberg–Witten map with one open Wilson contour in a manifestly noncommutative gauge invariant form. This action contains the nonconstant closed string fields, higher order derivatives of the noncommutative gauge fields through the ∗ N -product, and a Wilson operator. We extend this noncommutative D 9-brane action to the action for D p -brane by transforming it under T-duality. Using this noncommutative D p -brane action we then evaluate the linear couplings of the graviton and dilaton to the brane for arbitrary noncommutative parameters. By taking the Seiberg–Witten limit we show that they reduce exactly to the known results of the energy–momentum tensor of the noncommutative super-Yang–Mills theory. We take this as an evidence that the noncommutative action in the Seiberg–Witten limit includes properly all derivative correction terms.