arXiv subjects

Albrecht Klemm

Publications and source records attributed to Albrecht Klemm.

At least 55 records · Page 3Linked to original sources

arXiv2014

Strings of Minimal 6d SCFTs

We study strings associated with minimal 6d SCFTs, which by definition have only one string charge and no Higgs branch. These theories are labelled by a number n with 1 <= n <= 8 or n = 12. Quiver theories have previously been proposed which describe strings of SCFTs for n = 1, 2. For n > 2 the strings interact with the bulk gauge symmetry. In this paper we find a quiver description for the n = 4 string using Sen's limit of F-theory and calculate its elliptic genus with localization techniques. This result is checked using the duality of F-theory with M-theory and topological string theory whose refined BPS partition function captures the elliptic genus of the SCFT strings. We use the topological string theory to gain insight into the elliptic genus for other values of n.

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arXiv2014

Knot Invariants from Topological Recursion on Augmentation Varieties

Using the duality between Wilson loop expectation values of SU(N) Chern-Simons theory on $S^3$ and topological open-string amplitudes on the local mirror of the resolved conifold, we study knots on $S^3$ and their invariants encoded in colored HOMFLY polynomials by means of topological recursion. In the context of the local mirror Calabi-Yau threefold of the resolved conifold, we generalize the topological recursion of the remodelled B-model in order to study branes beyond the class of toric Harvey-Lawson special Lagrangians -- as required for analyzing non-trivial knots on $S^3$. The basic ingredients for the proposed recursion are the spectral curve, given by the augmentation variety of the knot, and the calibrated annulus kernel, encoding the topological annulus amplitudes associated to the knot. We present an explicit construction of the calibrated annulus kernel for torus knots and demonstrate the validity of the topological recursion. We further argue that -- if an explicit form of the calibrated annulus kernel is provided for any other knot -- the proposed topological recursion should still be applicable. We study the implications of our proposal for knot theory, which exhibit interesting consequences for colored HOMFLY polynomials of mutant knots.

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arXiv2014

Landscaping with fluxes and the E8 Yukawa Point in F-theory

Integrality in the Hodge theory of Calabi-Yau fourfolds is essential to find the vacuum structure and the anomaly cancellation mechanism of four dimensional F-theory compactifications. We use the Griffiths-Frobenius geometry and homological mirror symmetry to fix the integral monodromy basis in the primitive horizontal subspace of Calabi-Yau fourfolds. The Gamma class and supersymmetric localization calculations in the 2d gauged linear sigma model on the hemisphere are used to check and extend this method. The result allows us to study the superpotential and the Weil-Petersson metric and an associated tt* structure over the full complex moduli space of compact fourfolds for the first time. We show that integral fluxes can drive the theory to N=1 supersymmetric vacua at orbifold points and argue that fluxes can be chosen that fix the complex moduli of F-theory compactifications at gauge enhancements including such with U(1) factors. Given the mechanism it is natural to start with the most generic complex structure families of elliptic Calabi-Yau 4-fold fibrations over a given base. We classify these families in toric ambient spaces and among them the ones with heterotic duals. The method also applies to the creating of matter and Yukawa structures in F-theory. We construct two SU(5) models in F-theory with a Yukawa point that have a point on the base with an $E_8$-type singularity on the fiber and explore their embeddings in the global models. The explicit resolution of the singularity introduce a higher dimensional fiber and leads to novel features.

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arXiv2014

Quantum geometry of del Pezzo surfaces in the Nekrasov-Shatashvili limit

We use mirror symmetry, quantum geometry and modularity properties of elliptic curves to calculate the refined free energies in the Nekrasov-Shatashvili limit on non-compact toric Calabi-Yau manifolds, based on del Pezzo surfaces. Quantum geometry here is to be understood as a quantum deformed version of rigid special geometry, which has its origin in the quantum mechanical behaviour of branes in the topological string B-model. We will argue that, in the Seiberg-Witten picture, only the Coulomb parameters lead to quantum corrections, while the mass parameters remain uncorrected. In certain cases we will also compute the expansion of the free energies at the orbifold point and the conifold locus. We will compute the quantum corrections order by order on $\hbar$, by deriving second order differential operators, which act on the classical periods.

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arXiv2013

Refined stable pair invariants for E-, M- and [p,q]-strings

We use mirror symmetry, the refined holomorphic anomaly equation and modularity properties of elliptic singularities to calculate the refined BPS invariants of stable pairs on non-compact Calabi-Yau manifolds, based on del Pezzo surfaces and elliptic surfaces, in particular the half K3. The BPS numbers contribute naturally to the five-dimensional N=1 supersymmetric index of M-theory, but they can be also interpreted in terms of the superconformal index in six dimensions and upon dimensional reduction the generating functions count N=2 Seiberg-Witten gauge theory instantons in four dimensions. Using the M/F-theory uplift the additional information encoded in the spin content can be used in an essential way to obtain information about BPS states in physical systems associated to small instantons, tensionless strings, gauge symmetry enhancement in F-theory by [p,q]-strings as well as M-strings.

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arXiv2013

The Omega deformed B-model for rigid N=2 theories

We give an interpretation of the Omega deformed B-model that leads naturally to the generalized holomorphic anomaly equations. Direct integration of the latter calculates topological amplitudes of four dimensional rigid N=2 theories explicitly in general Omega-backgrounds in terms of modular forms. These amplitudes encode the refined BPS spectrum as well as new gravitational couplings in the effective action of N=2 supersymmetric theories. The rigid N=2 field theories we focus on are the conformal rank one N=2 Seiberg-Witten theories. The failure of holomorphicity is milder in the conformal cases, but fixing the holomorphic ambiguity is only possible upon mass deformation. Our formalism applies irrespectively of whether a Lagrangian formulation exists. In the class of rigid N=2 theories arising from compactifications on local Calabi-Yau manifolds, we consider the theory of local P2. We calculate motivic Donaldson-Thomas invariants for this geometry and make predictions for generalized Gromov-Witten invariants at the orbifold point.

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arXiv2013

ABJM Wilson loops in the Fermi gas approach

The matrix model of ABJM theory can be formulated in terms of a Fermi gas in an external potential. We show that, in this formalism, vevs of Wilson loops correspond to averages of operators in the statistical-mechanical problem. This makes it possible to calculate these vevs at all orders in 1/N, up to exponentially small corrections, and for arbitrary Chern-Simons coupling, by using the WKB expansion. We present explicit results for the vevs of 1/6 and the 1/2 BPS Wilson loops, at any winding number, in terms of Airy functions. Our expressions are shown to reproduce the low genus results obtained previously in the 't Hooft expansion.

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arXiv2012

Torus Knots and the Topological Vertex

We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S^3. The key role is played by the SL(2,Z) transformation, which generates a general torus knot from the unknot. Applying the topological vertex to the proposed A-branes, we rederive the colored HOMFLY polynomials for torus knots, in agreement with the Rosso and Jones formula. We show that our A-model construction is mirror symmetric to the B-model analysis of Brini, Eynard and Marino. Comparing to the recent proposal by Aganagic and Vafa for knots on S^3, we demonstrate that the disk amplitude of the A-brane associated to any knot is sufficient to reconstruct the entire B-model spectral curve. Finally, the construction of toric Lagrangian A-branes is generalized to other local toric Calabi-Yau geometries, which paves the road to study knots in other three-manifolds such as lens spaces.

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arXiv2012

The refined BPS index from stable pair invariants

A refinement of the stable pair invariants of Pandharipande and Thomas for non-compact Calabi-Yau spaces is introduced based on a virtual Bialynicki-Birula decomposition with respect to a C* action on the stable pair moduli space, or alternatively the equivariant index of Nekrasov and Okounkov. This effectively calculates the refined index for M-theory reduced on these Calabi-Yau geometries. Based on physical expectations we propose a product formula for the refined invariants extending the motivic product formula of Morrison, Mozgovoy, Nagao, and Szendroi for local P^1. We explicitly compute refined invariants in low degree for local P^2 and local P^1 x P^1 and check that they agree with the predictions of the direct integration of the generalized holomorphic anomaly and with the product formula. The modularity of the expressions obtained in the direct integration approach allows us to relate the generating function of refined PT invariants on appropriate geometries to Nekrasov's partition function and a refinement of Chern-Simons theory on a lens space. We also relate our product formula to wallcrossing.

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arXiv2012

Quantum geometry of elliptic Calabi-Yau manifolds

We study the quantum geometry of the class of Calabi-Yau threefolds, which are elliptic fibrations over a two-dimensional toric base. A holomorphic anomaly equation for the topological string free energy is proposed, which is iterative in the genus expansion as well as in the curve classes in the base. T-duality on the fibre implies that the topological string free energy also captures the BPS-invariants of D4-branes wrapping the elliptic fibre and a class in the base. We verify this proposal by explicit computation of the BPS invariants of 3 D4-branes on the rational elliptic surface.

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arXiv2011

Five-Brane Superpotentials, Blow-Up Geometries and SU(3) Structure Manifolds

We investigate the dynamics of space-time filling five-branes wrapped on curves in heterotic and orientifold Calabi-Yau compactifications. We first study the leading N=1 scalar potential on the infinite deformation space of the brane-curve around a supersymmetric configuration. The higher order potential is also determined by a brane superpotential which we compute for a subset of light deformations. We argue that these deformations map to new complex structure deformations of a non-Calabi-Yau manifold which is obtained by blowing up the brane-curve into a four-cycle and by replacing the brane by background fluxes. This translates the original brane-bulk system into a unifying geometrical formulation. Using this blow-up geometry we compute the complete set of open-closed Picard-Fuchs differential equations and identify the brane superpotential at special points in the field space for five-branes in toric Calabi-Yau hypersurfaces. This has an interpretation in open mirror symmetry and enables us to list compact disk instanton invariants. As a first step towards promoting the blow-up geometry to a supersymmetric heterotic background we propose a non-Kaehler SU(3) structure and an identification of the three-form flux.

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arXiv2010

Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes

Using wall-crossing formulae and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two M5-branes wrapping a rigid divisor inside a Calabi-Yau manifold. The anomaly originates from restoring modularity of an indefinite theta-function capturing the wall-crossing of BPS invariants associated to D4-D2-D0 brane systems. We show the compatibility of this equation with anomaly equations previously observed in the context of N=4 topological Yang-Mills theory on P^2 and E-strings obtained from wrapping M5-branes on a del Pezzo surface. The non-holomorphic part is related to the contribution originating from bound-states of singly wrapped M5-branes on the divisor. We show in examples that the information provided by the anomaly is enough to compute the BPS degeneracies for certain charges. We further speculate on a natural extension of the anomaly to higher D4-brane charge.

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arXiv2010

Direct integration for general Omega backgrounds

We extend the direct integration method of the holomorphic anomaly equations to general Omega backgrounds for pure SU(2) N=2 Super-Yang-Mills theory and topological string theory on non-compact Calabi-Yau threefolds. We find that an extension of the holomorphic anomaly equation, modularity and boundary conditions provided by the perturbative terms as well as by the gap condition at the conifold are sufficient to solve the generalized theory in the above cases. In particular we use the method to solve the topological string for the general Omega backgrounds on non-compact toric Calabi-Yau spaces. The conifold boundary condition follows from that the N=2 Schwinger-Loop calculation with BPS states coupled to a self-dual and an anti-self-dual field strength. We calculate such BPS states also for the decompactification limit of Calabi-Yau spaces with regular K3 fibrations and half K3s embedded in Calabi-Yau backgrounds.

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arXiv2010

Five-Brane Superpotentials and Heterotic/F-theory Duality

Under heterotic/F-theory duality it was argued that a wide class of heterotic five-branes is mapped into the geometry of an F-theory compactification manifold. In four-dimensional compactifications this identifies a five-brane wrapped on a curve in the base of an elliptically fibered Calabi-Yau threefold with a specific F-theory Calabi-Yau fourfold containing the blow-up of the five-brane curve. We argue that this duality can be reformulated by first constructing a non-Calabi-Yau heterotic threefold by blowing up the curve of the five-brane into a divisor with five-brane flux. Employing heterotic/F-theory duality this leads us to the construction of a Calabi-Yau fourfold and four-form flux. Moreover, we obtain an explicit map between the five-brane superpotential and an F-theory flux superpotential. The map of the open-closed deformation problem of a five-brane in a compact Calabi-Yau threefold into a deformation problem of complex structures on a dual Calabi-Yau fourfold with four-form flux provides a powerful tool to explicitly compute the five-brane superpotential.

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arXiv2010

Direct Integration and Non-Perturbative Effects in Matrix Models

We show how direct integration can be used to solve the closed amplitudes of multi-cut matrix models with polynomial potentials. In the case of the cubic matrix model, we give explicit expressions for the ring of non-holomorphic modular objects that are needed to express all closed matrix model amplitudes. This allows us to integrate the holomorphic anomaly equation up to holomorphic modular terms that we fix by the gap condition up to genus four. There is an one-dimensional submanifold of the moduli space in which the spectral curve becomes the Seiberg--Witten curve and the ring reduces to the non-holomorphic modular ring of the group $Γ(2)$. On that submanifold, the gap conditions completely fix the holomorphic ambiguity and the model can be solved explicitly to very high genus. We use these results to make precision tests of the connection between the large order behavior of the 1/N expansion and non-perturbative effects due to instantons. Finally, we argue that a full understanding of the large genus asymptotics in the multi-cut case requires a new class of non-perturbative sectors in the matrix model.

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arXiv2010

Computing Brane and Flux Superpotentials in F-theory Compactifications

In four-dimensional F-theory compactifications with N=1 supersymmetry the fields describing the dynamics of space-time filling 7-branes are part of the complex structure moduli space of the internal Calabi-Yau fourfold. We explicitly compute the flux superpotential in F-theory depending on all complex structure moduli, including the 7-brane deformations and the field corresponding to the dilaton-axion. Since fluxes on the 7-branes induce 5-brane charge, a local limit allows to effectively match the F-theory results to a D5-brane in a non-compact Calabi-Yau threefold with threeform fluxes. We analyze the classical and instanton contributions to the F-theory superpotential using mirror symmetry for Calabi-Yau fourfolds. The F-theory compactifications under consideration also admit heterotic dual descriptions and we discuss the identification of the moduli in this non-perturbative duality.

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arXiv2009

Solving the Topological String on K3 Fibrations

We present solutions of the holomorphic anomaly equations for compact two-parameter Calabi-Yau manifolds which are hypersurfaces in weighted projective space. In particular we focus on K3-fibrations where due to heterotic type II duality the topological invariants in the fibre direction are encoded in certain modular forms. The formalism employed provides holomorphic expansions of topological string amplitudes everywhere in moduli space.

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arXiv2009

Seiberg-Witten theory and matrix models

We derive a family of matrix models which encode solutions to the Seiberg-Witten theory in 4 and 5 dimensions. Partition functions of these matrix models are equal to the corresponding Nekrasov partition functions, and their spectral curves are the Seiberg-Witten curves of the corresponding theories. In consequence of the geometric engineering, the 5-dimensional case provides a novel matrix model formulation of the topological string theory on a wide class of non-compact toric Calabi-Yau manifolds. This approach also unifies and generalizes other matrix models, such as the Eguchi-Yang matrix model, matrix models for bundles over $P^1$, and Chern-Simons matrix models for lens spaces, which arise as various limits of our general result.

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