arXiv subjects

Richard Eager

Publications and source records attributed to Richard Eager.

At least 19 recordsLinked to original sources

arXiv2024

Adinkras and Pure Spinors

The nilpotence variety for extended supersymmetric quantum mechanics is a cone over a quadric in projective space. The pure spinor correspondence, which relates the description of off-shell supermultiplets to the classification of modules over the corresponding hypersurface ring, reduces to a classical problem of linear algebra. Spinor bundles, which correspond to maximal Cohen-Macaulay modules, serve as basic building blocks. Koszul duality appears as a deformed version of the Bernstein-Gel'fand-Gel'fand correspondence that we make fully concrete. We illustrate in numerous examples the close relationship between these connections and the powerful graphical technology of Adinkras, which appear as a decategorification of special complexes on quadrics. We emphasize the role of R-symmetry for recovering higher-dimensional gauge and gravity multiplets.

math-ph↗

arXiv2022

Perspectives on the pure spinor superfield formalism

In this note, we study, formalize, and generalize the pure spinor superfield formalism from a rather nontraditional perspective. To set the stage, we review the notion of a multiplet for a general super Lie algebra, working in the context of the BV and BRST formalisms. Building on this, we explain how the pure spinor superfield formalism can be viewed as constructing a supermultiplet out of the input datum of an equivariant graded module over the ring of functions on the nilpotence variety. We use the homotopy transfer theorem and other computational techniques from homological algebra to relate these multiplets to more standard component-field formulations. Physical properties of the resulting multiplets can then be understood in terms of algebrogeometric properties of the nilpotence variety. We illustrate our discussion with many examples in various dimensions.

hep-th↗

arXiv2021

Maximally twisted eleven-dimensional supergravity

We perform the maximal twist of eleven-dimensional supergravity. This twist is partially topological and exists on manifolds of $G_2 \times SU(2)$ holonomy. Our derivation starts with an explicit description of the Batalin-Vilkovisky complex associated to the three-form multiplet in the pure spinor superfield formalism. We then determine the $L_\infty$ module structure of the supersymmetry algebra on the component fields. We twist the theory by modifying the differential of the Batalin-Vilkovisky complex to incorporate the action of a scalar supercharge. We find that the resulting free twisted theory is given by the tensor product of the de Rham and Dolbeault complexes of the respective $G_2$ and $SU(2)$ holonomy manifolds as conjectured by Costello.

hep-th↗

arXiv2019

Supersymmetric Yang--Mills Matrix Integrals Revisited

We evaluate the twisted partition function of four-dimensional $\mathcal{N} = 1$ supersymmetric Yang--Mills theory reduced to a point for all simple gauge groups. The partition function is expressed as a sum of residues. The types of residues that appear are classified by distinguished nilpotent orbits. Surprisingly, the multiplicity of residues of each type is proportional to their common value. The sum over residues has the same form as the Plancherel measure for Yang's system of particles. Intriguingly, the precise constants appearing in the Plancherel measure also appear in the formal degrees of unipotent discrete series representations of $p$-adic groups.

hep-th↗

arXiv2019

Hidden exceptional symmetry in the pure spinor superstring

The pure spinor formulation of superstring theory includes an interacting sector of central charge $c_λ=22$, which can be realized as a curved $βγ$ system on the cone over the orthogonal Grassmannian $\text{OG}^{+}(5,10)$. We find that the spectrum of the $βγ$ system organizes into representations of the $\mathfrak{g}=\mathfrak{e}_6$ affine algebra at level $-3$, whose $\mathfrak{so}(10)_{-3}\oplus {\mathfrak u}(1)_{-4}$ subalgebra encodes the rotational and ghost symmetries of the system. As a consequence, the pure spinor partition function decomposes as a sum of affine $\mathfrak{e}_6$ characters. We interpret this as an instance of a more general pattern of enhancements in curved $βγ$ systems, which also includes the cases $\mathfrak{g}=\mathfrak{so}(8)$ and $\mathfrak{e}_7$, corresponding to target spaces that are cones over the complex Grassmannian $\text{Gr}(2,4)$ and the complex Cayley plane $\mathbb{OP}^2$. We identify these curved $βγ$ systems with the chiral algebras of certain $2d$ $(0,2)$ CFTs arising from twisted compactification of 4d $\mathcal{N}=2$ SCFTs on $S^2$.

hep-th↗

arXiv2018

Local Operators from the Space of Vacua of Four Dimensional SUSY Gauge Theories

We construct local operators in short representations of supersymmetry algebras from polyvector fields on the quantum moduli space of vacua of supersymmetric gauge theories. These operators form a super Lie algebra under a natural bracket operation with structure constants determined by terms in the operator product expansion of the corresponding operators. We propose a formula for the superconformal index in terms of an index over polyvector fields on the moduli space of vacua. Along the way, we construct several models with moduli space of vacua corresponding to affine cones over smooth bases using the classical geometry of Severi varieties and the Landsberg-Manivel projective geometries corresponding to the Freudenthal magic square of exceptional Lie algebras. Curiously, we relate the Landsberg-Manivel projective geometries to the exceptional enhanced symmetry surprises of Dimofte and Gaiotto. Finally, we determine Beasley-Witten higher-derivative F-terms in new examples arising from Severi varieties and remark on their origin in classical projective duality.

hep-th↗

arXiv2018

Nilpotence varieties

We consider algebraic varieties canonically associated to any Lie superalgebra, and study them in detail for super-Poincaré algebras of physical interest. They are the locus of nilpotent elements in (the projectivized parity reversal of) the odd part of the algebra. Most of these varieties have appeared in various guises in previous literature, but we study them systematically here, from a new perspective: as the natural moduli spaces parameterizing twists of a super-Poincaré-invariant physical theory. We obtain a classification of all possible twists, as well as a systematic analysis of unbroken symmetry in twisted theories. The natural stratification of the varieties, the identification of strata with twists, and the action of Lorentz and $R$-symmetry on the varieties are emphasized. We also include a short and unconventional exposition of the pure-spinor superfield formalism, from the perspective of twisting, and demonstrate that it can be applied to construct familiar multiplets in four-dimensional minimally supersymmetric theories; in all dimensions and with any amount of supersymmetry, this technique produces BRST or BV complexes of supersymmetric theories from the Koszul complex of the cone point over the coordinate ring of the nilpotence variety, possibly tensored with a module over that coordinate ring. In addition, we remark on a natural emergence of nilpotence varieties in the Chevalley-Eilenberg cohomology of supertranslations, and give two applications related to these ideas: a calculation of Chevalley-Eilenberg cohomology for the six-dimensional $\mathcal{N}=(2,0)$ supertranslation algebra, and a BV complex matching the field content of type IIB supergravity from the coordinate ring of the corresponding nilpotence variety.

hep-th↗

arXiv2018

Holomorphic field theories and Calabi--Yau algebras

We consider the holomorphic twist of the worldvolume theory of flat D$(2k-1)$-branes transversely probing a Calabi--Yau manifold. A chain complex, constructed using the BV formalism, computes the local observables in the holomorphically twisted theory. Generalizing earlier work in the case $k=2$, we find that this complex can be identified with the Ginzburg dg algebra associated to the Calabi--Yau. However, the identification is subtle; the complex is the space of fields contributing to the holomorphic twist of the free theory, and its differential arises from interactions. For $k=1$, this holomorphically twisted theory is related to the elliptic genus. We give a general description for D1-branes probing a Calabi--Yau fourfold singularity, and for $\mathcal{N}=(0,2)$ quiver gauge theories. In addition, we propose a relation between the equivariant Hirzebruch $χ_y$ genus of large-$N$ symmetric products and cyclic homology.

hep-th↗

arXiv2017

Exponential Networks and Representations of Quivers

We study the geometric description of BPS states in supersymmetric theories with eight supercharges in terms of geodesic networks on suitable spectral curves. We lift and extend several constructions of Gaiotto-Moore-Neitzke from gauge theory to local Calabi-Yau threefolds and related models. The differential is multi-valued on the covering curve and features a new type of logarithmic singularity in order to account for D0-branes and non-compact D4-branes, respectively. We describe local rules for the three-way junctions of BPS trajectories relative to a particular framing of the curve. We reproduce BPS quivers of local geometries and illustrate the wall-crossing of finite-mass bound states in several new examples. We describe first steps toward understanding the spectrum of framed BPS states in terms of such "exponential networks."

hep-th↗

arXiv2015

Superconformal field theories and cyclic homology

One of the predictions of the AdS/CFT correspondence is the matching of protected operators between a superconformal field theory and its holographic dual. We review the spectrum of protected operators in quiver gauge theories that flow to superconformal field theories at low energies. The spectrum is determined by the cyclic homology of an algebra associated to the quiver gauge theory. Identifying the spectrum of operators with cyclic homology allows us to apply the Hochschild-Kostant-Rosenberg theorem to relate the cyclic homology groups to deRham cohomology groups. The map from cyclic homology to deRham cohomology can be viewed as a mathematical avatar of the passage from open to closed strings under the AdS/CFT correspondence.

hep-th↗

arXiv2013

Elliptic genera of 2d N=2 gauge theories

We compute the elliptic genera of general two-dimensional N=(2,2) and N=(0,2) gauge theories. We find that the elliptic genus is given by the sum of Jeffrey-Kirwan residues of a meromorphic form, representing the one-loop determinant of fields, on the moduli space of flat connections on T^2. We give several examples illustrating our formula, with both Abelian and non-Abelian gauge groups, and discuss some dualities for U(k) and SU(k) theories. This paper is a sequel to the authors' previous paper arXiv:1305.0533.

hep-th↗

arXiv2013

Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups

We compute the elliptic genera of two-dimensional N=(2,2) and N=(0,2) gauged linear sigma models via supersymmetric localization, for rank-one gauge groups. The elliptic genus is expressed as a sum over residues of a meromorphic function whose argument is the holonomy of the gauge field along both the spatial and the temporal directions of the torus. We illustrate our formulas by a few examples including the quintic Calabi-Yau, N=(2,2) SU(2) and O(2) gauge theories coupled to N fundamental chiral multiplets, and a geometric N=(0,2) model.

hep-th↗

arXiv2013

Superconformal Indices, Sasaki-Einstein Manifolds, and Cyclic Homologies

The superconformal index of the quiver gauge theory dual to type IIB string theory on the product of an arbitrary smooth Sasaki-Einstein manifold with five-dimensional AdS space is calculated both from the gauge theory and gravity viewpoints. We find complete agreement. Along the way, we find that the index on the gravity side can be expressed in terms of the Kohn-Rossi cohomology of the Sasaki-Einstein manifold and that the index of a quiver gauge theory equals the Euler characteristic of the cyclic homology of the Ginzburg dg algebra associated to the quiver.

hep-th↗

arXiv2013

Superconformal Indices and M2-Branes

We derive the superconformal index of the world-volume theory on M2-branes probing the cone over an arbitrary Sasaki-Einstein seven-manifold. The index is expressed in terms of the cohomology groups of the cone. We match our supergravity results with known results from gauge theory. Along the way we derive the spectrum of short Kaluza-Klein multiplets on generic Sasaki-Einstein seven-manifolds.

hep-th↗

arXiv2012

Dimer Models and Integrable Systems

We explore various aspects of the correspondence between dimer models and integrable systems recently introduced by Goncharov and Kenyon. Dimer models give rise to relativistic integrable systems that match those arising from 5d N=1 gauge theories studied by Nekrasov. We apply the correspondence to dimer models associated to the Y^{p,0} geometries, showing that they give rise to the relativistic generalization of the periodic Toda chain originally studied by Ruijsenaars. The correspondence reduces the calculation of all conserved charges to a straightforward combinatorial problem of enumerating non-intersecting paths in the dimer model. We show how the usual periodic Toda chain emerges in the non-relativistic limit and how the Lax operator corresponds to the Kasteleyn matrix of the dimer model. We discuss how the dimer models for general Y^{p,q} manifolds give rise to other relativistic integrable systems, generalizing the periodic Toda chain and construct the integrable systems for general Y^{p,p} explicitly. The impurities introduced in the construction of Y^{p,q} quivers are identified with impurities in twisted sl(2) XXZ spin chains. Finally we discuss how the physical concept of higgsing a dimer model provides an efficient method for producing new integrable systems starting from known ones. We illustrate this idea by constructing the integrable systems for higgsings of Y^{4,0}.

hep-th↗

arXiv2011

Colored BPS Pyramid Partition Functions, Quivers and Cluster Transformations

We investigate the connections between flavored quivers, dimer models, and BPS pyramids for generic toric Calabi-Yau threefolds from various perspectives. We introduce a purely field theoretic definition of both finite and infinite pyramids in terms of quivers with flavors. These pyramids are associated to the counting of BPS invariants for generic toric Calabi-Yau threefolds. We discuss how cluster transformations provide an efficient recursive method for computing pyramid partition functions and show that the recursion is equivalent to the multidimensional octahedron recurrence. Transitions between different pyramids are related to Seiberg dualities, and we offer complimentary characterizations of these transitions in terms of the motion of zonotopes and duality webs. Our methods apply to completely general geometries including those with vanishing 4-cycles, which are associated to chiral quivers, thus overcoming one of the main limitations in the existing literature. We illustrate our ideas with explicit results for the infinite family of L^{a,b,c} geometries, dP_2, pseudo-dP_2, and dP_3. The counting of pyramid partitions for dP_1 gives rise to the Somos-4 sequence, while dP_2 and pseudo-dP_2 generate the Somos-5 sequence. Our results for dP_3 reproduce and extend those previously obtained for this theory, which were originally obtained from dimer shuffling.

hep-th↗

arXiv2010

Brane Tilings and Non-Commutative Geometry

We derive the quiver gauge theory on the world-volume of D3-branes transverse to an L(a,b,c) singularity by computing the endomorphism algebra of a tilting object first constructed by Van den Bergh. The quiver gauge theory can be concisely specified by an embedding of a graph into a face-centered cubic lattice. In this description, planar Seiberg dualities of the gauge theory act by changing the graph embedding. We use this description of Seiberg duality to show these quiver gauge theories possess periodic Seiberg dualities whose existence was expected from the AdS/CFT correspondence.

hep-th↗

arXiv2010

Can you hear the shape of dual geometries?

We compute the sub-leading terms in the Tian-Yau-Zelditch asymptotic expansion of the partition function for dual giant gravitons on AdS5 $\times$ L5 and provide a bulk interpretation in terms of curvature invariants. We accomplish this by relating the partition function of dual giant gravitons to the Hilbert series for mesonic operators in the CFT. The coefficients of the subleading terms encode integrated curvature invariants of L5. In the same spirit of Martelli, Sparks and Yau, we are able to compute these integrated curvature invariants without explicit knowledge of the Sasaki-Einstein metric on L5. These curvature invariants contribute to the 1/N^2 corrections of the difference of the 4D anomaly coefficients a and c recently found by Liu and Minasian, which we now have a purely field theoretic method of calculating.

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