Hilb

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# $Hilb$
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## Definition

A [[category]] whose [[objects]] are [[Hilbert spaces]] is typically denoted $Hilb$ or similar.

There are different choices of [[morphisms]] in use, such as all [[linear maps]] or just the [[short linear maps]] (linear maps of norm at most $1$).

One may regard $Hilb$ as a [[dagger category]] with [[morphisms]] the [[bounded operator|bounded]] linear maps between them and the dagger operation assigning [[adjoint operators]]. The [[full subcategory]] $Fin Hilb$ of [[finite-dimensional vector space|finite-dimensional]] Hilbert spaces becomes a [[dagger compact category]].

Note that either way, the [[core]] (of [[isomorphisms]] in the first case, or of [[unitary operator|unitary]] isomorphisms in the other case) is the same [[groupoid]], whose morphisms are all invertible linear maps of norm exactly $1$.

In any case, the [[forgetful functor]] from $Hilb$ to [[Vect]] is [[faithful functor|faithful]], confirming the intuition that a Hilbert space is a vector space equipped with [[extra structure]].  $Hilb$ is also a [[full subcategory]] of [[Ban]], the category of [[Banach spaces]].

### Explicit definition

Assuming [[excluded middle]], $Hilb_\mathbb{R}$ is a [[concrete category|concrete]] [[dagger category]] with

* a tensor product $\otimes$ and tensor unit $\Iota$, such that $(Hilb_\mathbb{R},\otimes,\Iota)$ is a [[symmetric monoidal dagger category]] and $\Iota$ is a [[simple object]] and a monoidal [[separator]],

* a biproduct $\oplus$ and zero object $0$ such that $(Hilb_\mathbb{R},\oplus,0)$ is a [[semiadditive dagger category]]

* a [[dagger equaliser]] $eq$ such that $(Hilb_\mathbb{R},\oplus,0,eq)$ is a [[finitely complete dagger category]]

* a [[directed colimit]] $\underrightarrow{lim} F$ for any inductive system $F$ in the wide sub-dagger category of objects and [[dagger monomorphisms]].

* a morphism $g:B \to C$ for any [[dagger monomorphism]] $f:A \to B$ such that $g = eq(f,0_B \circ 0_A^\dagger)$

However, without excluded middle, this definition fails to result in the category of Hilbert spaces over the real or complex numbers. That a dagger monomorphism $f:H \to I$ is either zero or invertible (in a symmetric monoidal dagger category where $\Iota$ is a simple monoidal separator) implies excluded middle, and [[Solèr’s theorem]] also implies excluded middle. It is currently unknown what modifications are needed to the axioms and to Solèr’s theorem to result in a definition valid without propositional excluded middle in the type theory. It is also unknown what type of real or complex numbers would be derived from the modified axioms and modified Soler's theorem in constructive mathematics.

## Related concepts

* [[quantum mechanics]]

* [[finite quantum mechanics in terms of dagger-compact categories]]

* [[tensor product of Banach spaces]]

* [[semiadditive dagger category]]

* [[compact closed dagger category]]

## References

A pedagogical description of the [[monoidal category]] structure on $Hilb$ with an emphasis on their role in [[quantum mechanics]] and their relation to [[nCob]]:

* [[John Baez]], _The monoidal category of Hilbert spaces_ ([web](http://math.ucr.edu/home/baez/quantum/node4.html))

An [[axiom|axiomatic]] characterization of the [[dagger-category]] of Hilbert spaces, with [[linear maps]] between them:

* {#HeunenKornell21} [[Chris Heunen]], [[Andre Kornell]], *Axioms for the category of Hilbert spaces* ([arXiv:2109.07418](https://arxiv.org/abs/2109.07418))

category: category

[[!redirects HilbertSpaces]]


hilb category > history

see _[[Hilb]]_


Hilbert bimodule

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#Contents#
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## Idea

The notion of _Hilbert $C^\ast$-bimodule_ adapts the notion of [[bimodules]] over [[associative algebras]] to [[operator algebra]]/[[C-star-algebra]] theory.

## Definition

+-- {: .num_defn}
###### Definition

For $A,B \in $ [[C*Alg]] two [[C-star algebras]], an $(A,B)$-*Hilbert $C^\ast$-bimodule* (or just _Hilbert bimodule_, for short) is

* a right $B$-[[Hilbert C*-module]] $(N, \langle -,-\rangle)$;

* equipped with a further left $A$-[[representation of a C-star-algebra|representation]] $A \to \mathcal{B}(N)$ by *adjointable operators*, hence such that $\langle a -,- \rangle = \langle -,a^\ast -\rangle$ for all $a \in A$.

A [[isomorphism]] between two $(A,B)$-bimodules $(N_1, \langle -,-\rangle) \to (N_2, \langle -,-\rangle_2)$ is a [[linear operator]] $N_1\to N_2$ which is [[unitary operator|unitary]] with respect to $\langle -,-\rangle_2$.

=--

+-- {: .num_defn #TensorProductOfHilbertBimodules}
###### Definition

Given an $(A,B)$-Hilbert bimodule $(N_1, \langle -,-\rangle_1)$ and a $(B,C)$-Hilbert bimodule $(N_2, \langle -,-\rangle_2)$, the **tensor product of Hilbert bimodules** $N_1 \otimes_B N_2$ is the $(A,C)$-Hilbert bimodule obtained from the ordinary (algebraic) [[tensor product of modules]] over $\mathbb{C}$ $N_1 \otimes_{\mathbb{C}} N_2$ by

1. equipping it with the $C$-valued [[inner product]] defined by

   $$
     \left\langle \xi_1 \otimes \eta_1 , \xi_2 \otimes \eta_2\right\rangle
     \coloneqq
     \left\langle \eta_1, \left\langle \xi_1, \xi_2\right\rangle \cdot \eta_2 \right\rangle
     \,.
   $$

1. forming the [[quotient]] by the submodule of elements $v$ for which $\langle v,v\rangle = 0$;

1. forming the [[completion]] of this quotient with respect to the induced [[norm]].

=--

+-- {: .num_remark}
###### Remark

Def. \ref{TensorProductOfHilbertBimodules} really does yield a kind of tensor product over $B$: elements of the form

$$
  v \cdot b \otimes w - v \otimes b \cdot w
$$

are in the submodule that it being divided out, because

$$
  \begin{aligned}
    \langle
      v \cdot b \otimes w - v \otimes b \cdot w \;,\;
      v \cdot b \otimes w - v \otimes b \cdot w
    \rangle
     & \coloneqq
    \langle w , \langle v \cdot b, v \cdot b\rangle \cdot w \rangle
     \\
     & +
     \langle b \cdot w, \langle v,v\rangle \cdot b \cdot w\rangle
     \\
     &
     -
     \langle w, \langle v\cdot b, v \rangle \cdot b \cdot w\rangle
     \\
     &
     -
     \langle b \cdot w, \langle v, v \cdot b \rangle \cdot w\rangle
     \\
     & = (1+1-1-1) \langle w , \langle v \cdot b, v \cdot b\rangle \cdot w \rangle

     \\
     & = 0
  \end{aligned}
  \,,
$$

where we use that by definition the left actions are required to have adjoints, so that for instance

$$
  \begin{aligned}
    \langle b \cdot w , \langle v,v\rangle \cdot b \cdot w\rangle
    & =
    \langle w , b^\ast \cdot \langle v,v\rangle \cdot b \cdot t\rangle
    \\
    & =
    \langle w , \langle v \cdot b,v \cdot b\rangle \cdot w\rangle
  \end{aligned}
$$

=--

+-- {: .num_prop}
###### Proposition

There is a [[(2,1)-category]] $C^\ast Alg_b$ whose

* [[objects]] are [[C*-algebras]];

* [[1-morphisms]] are Hilbert bimodules;

* [[2-morphisms]] are isomorphisms of Hilbert bimodules;

* [[vertical composition]] is the evident composition of intertwiners;

* [[horizontal composition]] is the tensor product of Hilbert bimodules, def. \ref{TensorProductOfHilbertBimodules}.

=--

([Buss-Zhu-Meyer 09](#BussZhuMeyer))

## Examples

An $(\mathbb{C},B)$-Hilbert $C^\ast$-bimodule is equialently just an $B$-[[Hilbert C*-module]].

An $(A, \mathbb{C})$-Hilbert $C^\ast$-bimodule is equivalently just as [[representation of a C-star-algebra]].

## Properties

(...)

## Related concepts

* [[amplimorphism]]

* [[generalized elliptic operator]]

* [[KK-theory]]

* [[C*-correspondence]]

## References

For instance

* [[Sergio Doplicher]], Claudia Pinzari, Rita Zuccante, _The $C^\ast$-algebra of a Hilbert Bimodule_ [[arXiv:funct-an/9707006](http://arxiv.org/abs/funct-an/9707006)]

* [[Nik Weaver]], _Hilbert bimodules with involution_, Canad. Math. Bull. **44** 3 (2001) 355–369 [[arXiv:math/9908119](http://arxiv.org/abs/math/9908119), [pdf](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/DC7055E821BEA9438350CCB325EA64CA/S0008439500014466a.pdf/hilbert-bimodules-with-involution.pdf)]

The tensor product of Hilbert bimodules and the induced [[2-category]] structure is discussed in

* {#BussZhuMeyer} Alcides Buss, [[Chenchang Zhu]], [[Ralf Meyer]], _A higher category approach to twisted actions on $C^\ast$-algebras_ ([arXiv:0908.0455](http://arxiv.org/abs/0908.0455))

[[!redirects Hilbert bimodules]]

[[!redirects Hilbert C-star-bimodule]]
[[!redirects Hilbert C-star-bimodules]]
[[!redirects Hilbert C-*-bimodule]]
[[!redirects Hilbert C-*-bimodules]]
[[!redirects Hilbert C*-bimodule]]
[[!redirects Hilbert C*-bimodules]]


Hilbert cube

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##Definition##

The Hilbert cube is the product:

$$\prod_n [0,\frac{1}{n}]\cong\prod_n [-\frac{1}{n},\frac{1}{n}] $$

It is a [[compactum|compact]] [[metrizable space]] under the sup norm, where the metric topology equals the [[product topology]] (as is easily seen).  It is variously denoted by $Q$ or $I^\omega$.

It plays a central role in [[Borsuk's shape theory]], and is the basis for the construction of Hilbert cube manifolds.  The theory of these were developed by Tom Chapman (mid 1970s) and were used in his proof of the topological invariance of [[Whitehead torsion]].

##Pseudo-interior of $Q$##
It has an important subspace known as its _pseudo-interior_.  This is the product of the corresponding open intervals,

$$s= \prod_n (-\frac{1}{n},\frac{1}{n}).$$

This plays an essential role in the [[shape theory|Chapman complement theorem]].

## Properties

Let $Q$ be the Hilbert cube.

* A space is [[second countable space|second countable]] (has a countable [[basis]]) and $T_4$ (is [[separation axioms|normal and Hausdorff]]) if and only if it is homeomorphic to a [[subspace]] of $Q$.

* A [[topological space]] is [[Polish space|Polish]] if and only if it is homeomorphic to a $G_\delta$-[[G-delta subset|subset]] of $Q$.

* The Hilbert cube has some counterintuitive properties, such as the fact that it is a [[homogeneous space]] (i.e., the group of self-homeomorphisms $Aut(Q)$ acts transitively on $Q$), even though it seems to have a "boundary". See [Halverson and Wright](#HW) for some explicit constructions.

##References##

* T.A.Chapman, _On Some Applications of Infinite Dimensional Manifolds to the Theory of Shape_, Fund. Math. &6 (1972), 181 - 193.

* T.A. Chapman,  _Lectures on Hilbert Cube Manifolds_, CBMS 28, American Mathematical Society, Providence, RI, 1975

* Denise M. Halverson, David G. Wright, _The Homogeneous Property of the Hilbert Cube_, [http://arxiv.org/pdf/1211.1363v1.pdf](http://arxiv.org/pdf/1211.1363v1.pdf)
{#HW}

[[!redirects Hilbert cube]]
[[!redirects Hilbert Cube]]


Hilbert lattice

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#Contents#
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## Idea

A __Hilbert lattice__ is the [[lattice]] of [[closed subset|closed]] [[linear subspaces]] of a [[Hilbert spaces]] (with [[preorder]] given by the inclusion) over real, complex or quaternion numbers. This is an [[orthocomplemented lattice]] and in fact an [[orthomodular lattice]].

The [[logic]] it embodies is Birkhoff-von Neumann [[quantum logic]]. See there for more.

## References

* [[Garrett Birkhoff]], [[John von Neumann]], _The logic of quantum mechanics_, Annals of Mathematics, 37: 823-843 (1936)

* A. Gleason, _Measures on the closed subspaces of a Hilbert space_, Journal of Mathematics and Mechanics __6__: 885-893 (1957)

* P. Pták and S. Pulmannová, _Orthomodular structures as quantum logics_, ser. Fundamental theories of physics. Kluwer Academic Publishers,
1991.

* Samuel S. Holland Jr., _Orthomodularity in infinite dimensions; a theorem of M. Solèr_, Bull. Amer. Math. Soc. (N.S.) __32__ (1995) 205-234, [arXiv:math.RA/9504224](http://arxiv.org/abs/math/9504224)

[[!redirects Hilbert lattices]]


Hilbert manifold

[[!redirects Hilbert manifolds]]

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#Contents#
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## Idea

A [[manifold]] locally modeled on an infinite-dimensional [[Hilbert space]].

## Related concepts

* [[Banach manifold]],

* [[Frechet manifold]]

* [[convenient manifold]]

* [[ILH manifold]]


Hilbert modular form

#Contents#
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## Idea

Hilbert modular forms are one generalization of the concept of _[[modular form]]_ to functions of more than one complex variable. (Another such generalization are [[Siegel modular forms]].)

Just as modular forms are, locally, functions on the [[moduli space of elliptic curves]] over the [[complex numbers]], soHilbert modular forms are locally functions on moduli spaces of something like complex [[abelian varieties]] with real multiplication and polarization module ([Goren, p. 49](#Goren)).

Hilbert modular forms have a slick equivalent formulation as adelic [[automorphic forms]]. (...)

## References

* Jan Hendrik Bruinier, _Hilbert modular forms and their applications_ ([arXiv:0609763](http://arxiv.org/abs/math/0609763))

* {#Goren} Eyal Z. Goren, _Lectures on Hilbert Modular Varieties and Modular Forms_

[[!redirects Hilbert modular forms]]


Hilbert module

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# Hilbert modules
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{: toc}

Hilbert module is an abbreaviation both for a Hilbert C\*-module (this entry)
and the analogous notion of a [[Hilbert Q-module]] (see there), where $Q$ is a [[quantale]] (or a [[locale]], in particular).

## Idea

The notion of _Hilbert C\*-module_ (or simply _Hilbert module_) is a generalization of the notion of _[[Hilbert space]]_ where the algebra of [[complex numbers]] is replaced by a possibly more general [[C*-algebra]] $A$. In particular a Hilbert $A$-module has an [[inner product]] which takes values not in $\mathbb{C}$, but in $A$, and such that [[complex conjugation]] is replaced by the [[star-algebra|star-operation]] in $A$.

Hilbert C\*-modules naturally appear as modules over [[groupoid convolution algebras]]. Refined to [[Hilbert C*-bimodules]] they serve as generalized [[homomorphism]] between [[C*-algebras]] in [[noncommutative topology]], and, when further equipped with a left weak [[Fredholm module]] as [[cocycles]] in [[KK-theory]].

## Definition
 {#Definition}

+-- {: .num_defn #HilbertCStarModule}
###### Definition

For $B \in $ [[C*Alg]], a [[Hilbert C*-module]] over $B$ is

1. a [[complex numbers|complex]] [[vector space]] $H$;

1. equipped with an [[action]] of $B$ from the right;

1. equipped with a [[sesquilinear map]] (linear in the second argument)

   $$
     \langle -,-\rangle \,\colon\, H \times H \to B
   $$

   (the $B$-valued [[inner product]])

such that

1. $\langle -,-\rangle$ behaves like a positive definite [[Hermitian form|Hermitian]] [[inner product]] over $B$ in that for all $x,y \in H$ and $b \in B$ we have

   1. $\langle x,y\rangle^\ast = \langle y,x\rangle$

   1. $\langle x,x\rangle \geq 0$ (in the sense of [[positive elements]] in $B$)

   1. $\langle x,x\rangle = 0$ precisely if $x = 0$;

   1. $\langle x,y \cdot b\rangle = \langle x,y \rangle \cdot b $

1. $H$ is [[complete space|complete]] with respect to the [[norm]]

   ${\Vert x \Vert_H} \coloneqq {\Vert \langle x,x\rangle\Vert_B}^{1/2}$.

=--

+-- {: .num_remark}
###### Remark

In addition to the explicit $B$-linearity in the second argument under right multiplicatojn

$$
  \langle v, w \cdot b\rangle = \langle v,w\rangle \cdot b
$$

the axioms imply conjugate $B$-linearity in the first argument and under left multiplication

$$
  \langle v \cdot b,w\rangle = b^\ast \cdot \langle v,w\rangle
  \,.
$$

Because:

$$
  \begin{aligned}
    \langle v \cdot b,w\rangle
    & =
    \langle w,  v\cdot b\rangle^\ast
    \\
    & = \left( \left\langle w,v\right\rangle \cdot b\right)^\ast
    \\
    & = b^\ast \cdot \langle w,v\rangle^\ast
    \\
    & = b^\ast \cdot \langle v,w\rangle
  \end{aligned}
  \,.
$$
=--

## Examples
 {#Examples}

First of all we have:

+-- {: .num_example }
###### Example

An ordinary complex [[Hilbert space]] is a Hilbert $\mathbb{C}$-module.

=--

The archetypical class of examples of Hilbert C\*-modules for [[commutative C*-algebras]] is the following. The general definition \ref{HilbertCStarModule} may be understood as the generalization of the structure of this example to [[noncommutative topology|non-commutative C*-algebras]]. See also remark \ref{TheTrivialHilbertBundleAndL2} below.

+-- {: .num_example #FromAHilbertSpaceBundle}
###### Example

Let $X$ be a [[locally compact topological space]] and write $C_0(X)$ for its [[C*-algebra]] of [[continuous functions]] [[vanishing at infinity]].

Let $E \to X$ be a [[fiber bundle]] of [[Hilbert spaces]] over $X$, hence a canonically [[associated bundle]] to a [[unitary group]]-[[principal bundle]]. Then the space $\Gamma_0(E)$ of continuous compactly supported [[sections]] is a Hilbert C\*-module over $C_0(X)$ with $C_0(X)$-valued [[inner product]] $\langle -,-\rangle$ the pointwise inner product in the [[Hilbert space]] [[fiber]] of $E$:

$$
  \langle \sigma_1, \sigma_2\rangle(x)
  \coloneqq
  \langle \sigma_1(y), \sigma_2(y)\rangle_{E_y}
  \;\in C_0(X)\,,
  \;\;\;\;\;\;
  \sigma_1, \sigma_2 \in \Gamma(E), \; x \in X
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

Every Hilbert $C_0(X)$-module arises, up to [[isomorphism]], as in example \ref{FromAHilbertSpaceBundle}.

=--

+-- {: .num_example #CAstAlgebraAsModuleOverItself}
###### Example

Every C\*-algebra $A$ is a Hilbert $A$-module over itself when equipped by with the $A$-valued inner product given simply by

$$
  \langle a_1,a_2\rangle \coloneqq a_1^\ast \cdot a \;\;\in A
$$

=--

+-- {: .num_remark }
###### Remark

In view of the archetypical example \ref{FromAHilbertSpaceBundle}, example \ref{CAstAlgebraAsModuleOverItself} may be interpreted as exhibiting the trivial [[complex line bundle]] over whatever space $A$ is the C\*-algebra of functions on (an actual [[topological space]] if $A$ is a [[commutative C*-algebra]] or else the [[noncommutative topology]] defined as the formal dual of $A$).

=--

+-- {: .num_example #L2}
###### Example

For $A \in $ [[C*Alg]], let $\ell^2 A$ be the space of those [[sequences]] $\{a_n \in A\}_{n \in \mathbb{N}}$ of elements in $A$ such that the [[series]] $\sum_n a_n^\ast a_n$ [[convergence|converges]]. This is a Hilbert $A$-module when equipped with the degreewise $A$-[[C*-representation]], with the $A$-valued inner product

$$
  \langle \{a_n\}, \{b_n\}\rangle
  \coloneqq
  \sum_n a_n^\ast b_n
$$

and after [[completion]] with under the induced [[norm]].

This $\ell^2 A$ is sometimes called the **standard Hilbert $A$-module**.

=--

+-- {: .num_remark #TheTrivialHilbertBundleAndL2}
###### Remark

In view of example \ref{FromAHilbertSpaceBundle} we may think of example \ref{L2} as exhibiting the trivial countably-infinite dimensional [[Hilbert space]] bundle over the space dual to $A$.

This is because the [[unitary group]] $U(\mathcal{H})$ of an infinite-dimensional [[separable Hilbert space]] $\mathcal{H}$ is [[contractible topological space|contractible]] (by [[Kuiper's theorem]]), hence so is the [[classifying space]], and so unitary $\mathcal{H}$-fiber bundles (over actual topological spaces) all trivializable. Since moreover $\mathcal{H} \simeq \ell^2(\mathbb{C})$ the Hilbert module of example \ref{FromAHilbertSpaceBundle} for the trivial $\mathcal{H}$-bundle over $C_0(X)$ is equivalent to $\ell^2(C_0(X))$. Example \ref{L2} generalizes this to arbitrary C\*-algebras $A$.

=--

## Properties

### C\*-algebras of adjointable operators on a Hilbert module

+-- {: .num_defn #AdjointableOperator}
###### Definition

For $A \in $ [[C*Alg]] and $H$ a Hilbert $A$-module, def. \ref{HilbertCStarModule}, a $\mathbb{C}$-[[linear operator]] $F \colon H \to H$ is called **adjointable** if there is an [[adjoint operator]] $F^\ast \colon H \to H$ with respect to the $A$-valued inner product in the sense that

$$
  \langle F -, -\rangle = \langle -,F^\ast -\rangle
  \,.
$$

=--

+-- {: .num_prop }
###### Proposition

The adjointable operators on a Hilbert $A$-module, def. \ref{AdjointableOperator}, form a [[Banach algebra|Banach]] [[star-algebra]].

For $A$ itself regarded as a Hilbert $A$-module as in example \ref{CAstAlgebraAsModuleOverItself}, this is the [[multiplier algebra]] of $A$.

=--

### Compact operators on a Hilbert C\*-module

+-- {: .num_defn #CompactOperator}
###### Definition

For $H_1, H_2$ two Hilbert C\*-modules, an adjointable operator $T \colon H_1 \to H_2$, def. \ref{AdjointableOperator}, is of **finite rank** if it is of the form

$$
  T \colon v \mapsto \sum_{i = 1}^n w_i \langle v_i, v\rangle
$$

for vectors $v_i \in H_1$ and $w_i \in H_2$.  $T$ is called a **generalized compact operator** if it is in the norm-closure of finite-rank operators.

=--

Typically one writes $\mathcal{K}(H_1, H_2)$ for the space of generalized complact operators.

### Fredholm operators

+-- {: .num_defn }
###### Definition

An operator $F \colon H_1 \to H_2$ is called a **generalized [[Fredholm operator]]** if there exists an operator $S \colon H_2 \to H_1$ (then called a [[parametrix]] for $F$) such that both

$ F \circ S - id_{H_2}$ and $S \circ F - id_{H_1}$

are [[compact operators]] according to def. \ref{CompactOperator}.

=--

## Applications

* Kasparov's [[KK-theory]] is formulated in terms of Hilbert (bi)modules

## Related concepts

* [[Hilbert bimodule]]

* [[Mishchenko-Fomenko index theorem]]

* [[duality between algebra and geometry]]

## References

Hilbert C\*-modules were introduced by [[Irving Kaplansky]] in

* Irving Kaplansky, _Modules over operator algebras_, Amer. J. Math. __75__ (1953) 839--853

Contemporary references are

* [[Bruce Blackadar]], _[[K-Theory for Operator Algebras]]_, section 13
* wikipedia, _[Hilbert C\*-module](http://en.wikipedia.org/wiki/Hilbert_C*-module)_
* E. Christopher Lance, _Hilbert C\*-modules, A toolkit for operator algebraists_,  London Math. Soc. Lec. Note Ser. __210__, Cambridge Univ. Press 1995

[[!redirects Hilbert module]]
[[!redirects Hilbert modules]]
[[!redirects Hilbert C*-module]]
[[!redirects Hilbert C*-modules]]
[[!redirects Hilbert C-star-module]]
[[!redirects Hilbert C-star-modules]]


Hilbert Q-module

Hilbert module over a quantale $Q$ or a __Hilbert $Q$-module__ is an analogue of a [[Hilbert module]] in the setup of quantales
and the inner product is valued in $Q$.

Hilbert $Q$-modules have been introduced by Paseka.

In the special case of [[locale]] $Q=B$, the Hilbert $B$-modules are studied geometrically in

* [[Pedro Resende]], Elias Rodrigues, _Sheaves as modules_, Appl. Categ. Struct. __18__:199--217 (2010) [arxiv/0711.4401](http://arxiv.org/abs/0711.4401) [doi]()

* [[Pedro Resende]], _Groupoid sheaves as Hilbert modules_, J. Pure Appl. Algebra __216__ (2012) 41-70 [arxiv/0807.4848](http://arxiv.org/abs/0807.4848)


Hilbert reciprocity law

A [[reciprocity law]]...

## References

Katsumi Shiratani, _Note on the Kummer-Hilbert reciprocity law_ J. Math. Soc. Japan Volume 12, Number 4 (1960), 412-421 ([Euclid](http://projecteuclid.org/euclid.jmsj/1261147731))


Hilbert scheme

#Contents#
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## Idea

Hilbert schemes are [[moduli space|moduli spaces]] of subvarieties, hence [[configuration space (mathematics)|configuration spaces]] in [[algebraic geometry]].

For instance a scheme of 0-dimensional sub-schemes is called a _[[Hilbert scheme of points]]_, etc.

Specifically for [[projective variety|quasi-projective variety]] with fixed [[Hilbert polynomial]], Hilbert schemes are well behave as moduli spaces go, in that they're actually quasi-projective varieties themselves.

The existence and construction of Hilbert schemes is due to Grothendieck (FGA).

The Hilbert scheme of $\mathbb{C}^2$ is widely studied in combinatorics and geometric representation theory for its connections to [[Macdonald polynomials]] and [[Cherednik algebra]]s.

## Properties

### Compact hyperkähler structure

The only known examples of [[compact hyperkähler manifolds]] are [[Hilbert schemes of points]] $X^{[n+1]}$ (for $n \in \mathbb{N}$) for $X$ either

1. a [[K3-surface]]

1. a [[4-torus]] (in which case the [[compact hyperkähler manifolds]] is really the [[fiber]] of $(\mathbb{T}^4)^{[n]} \to \mathbb{T}^4$)

([Beauville 83](#Beauville83))
and two exceptional examples ([O'Grady 99](compact+hyperkähler+manifold#OGrady99), [O'Grady 03](compact+hyperkähler+manifold#OGrady03)
), see [Sawon 04, Sec. 5.3](#Sawon04).

## Related concepts

* [[homotopy of rational maps]]

## References

### General

Textbook account:

* [[János Kollár]], Chapter I of: *Rational Curves on Algebraic Varieties*,  Ergebnisse der Mathematik und ihrer Grenzgebiete **32**, Springer 1996 ([doi:10.1007/978-3-662-03276-3](https://doi.org/10.1007/978-3-662-03276-3))

See also

* Wikipedia, _[Hilbert scheme](https://en.wikipedia.org/wiki/Hilbert_scheme)

### Hilbert schemes of points

* [[Hiraku Nakajima]], _Lectures on Hilbert schemes of points on surfaces_, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999 ([ams:ulect-18](https://bookstore.ams.org/ulect-18))

* [[Hiraku Nakajima]], _More lectures on Hilbert schemes of points on surfaces_, Advanced Studies in Pure Mathematics 69, 2016, Development of Moduli Theory -- Kyoto 2013, 173-205 ([arXiv:1401.6782](https://arxiv.org/abs/1401.6782))

* J. Bertin, _The punctual Hilbert scheme: An introduction_ ([pdf](https://www-fourier.ujf-grenoble.fr/sites/ifmaquette.ujf-grenoble.fr/files/bertin_rev.pdf))

* Dori Bejleri, _Hilbert schemes: Geometry, combionatorics, and representation theory_ ([pdf](http://math.harvard.edu/~bejleri/Hilbertschemes_gss.pdf))

* Barbara Bolognese, Ivan Losev, _A general introduction to the Hilbert scheme of points on the plane_ ([pdf](https://web.northeastern.edu/iloseu/Barbara_14_complete.pdf))

* Joachim Jelisiejew, _Pathologies on the Hilbert scheme of points_ ([arXiv:1812.08531](https://arxiv.org/abs/1812.08531))

Discussion in relation to the [[Fulton-MacPherson compactifications]] of [[configuration spaces of points]]:

* {#FultonMacPherson94} [[William Fulton]], [[Robert MacPherson]], p. 189 of: _A compactification of configuration spaces_, Ann. of Math. (2), 139(1):183–225, 1994 ([jstor:2946631](https://www.jstor.org/stable/2946631))

Discussion of [[Euler numbers]] of [[Hilbert schemes of points]]:

* [[Hiraku Nakajima]], _Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras_ ([arXiv:2001.03834](https://arxiv.org/abs/2001.03834))

### As moduli spaces of instantons

Discussion in their role as [[moduli spaces]] of [[instantons]]:

* Sean Pohorence, _Hilbert scheme of points and the ADHM construction_ ([pdf](http://www.ihes.fr/~celliott/Hilbert_Schemes_of_Points.pdf), [[PohorenceHilbertScheme.pdf:file]])

* [[Ian Grojnowski]], _Instantons and affine algebras I: The Hilbert scheme and vertex operators_ ([arXiv:alg-geom/9506020](https://arxiv.org/abs/alg-geom/9506020))

Specifically in relation to [[Donaldson-Thomas theory]]:

* Michele Cirafici, Annamaria Sinkovics, [[Richard Szabo]], _Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory_, Nucl. Phys. B809: 452-518, 2009 ([arXiv:0803.4188](https://arxiv.org/abs/0803.4188))

* Artan Sheshmani, _Hilbert Schemes, Donaldson-Thomas theory, Vafa-Witten and Seiberg-Witten theories_, Notices of the International Congress  of Chines Mathematics (2019) ([j.mp:2U7qd01](http://j.mp/2U7qd01), [pdf](https://scholar.harvard.edu/files/artan/files/iccm_07_02_a03.pdf))

* Artan Sheshmani, _Hilbert Schemes, Donaldson-Thomas Theory, Vafa-Witten and Seiberg Witten theories_ ([arxiv:1911.01796](https://arxiv.org/abs/1911.01796))

See also

* Jian Zhou, _K-Theory of Hilbert Schemes as a Formal Quantum Field Theory_ ([arXiv:1803.06080](https://arxiv.org/abs/1803.06080))

### Of ADE singularities

Discussion of the [[Hilbert schemes of points]] of [[ADE-singularities]]:

* [[Kenji Mohri]], _Kähler Moduli Space of a D-Brane at Orbifold Singularities_, Commun. Math. Phys. 202 (1999) 669-699 ([arXiv:hep-th/9806052](https://arxiv.org/abs/hep-th/9806052))

* [[Ron Donagi]], [[Sheldon Katz]], [[Eric Sharpe]], _Spectra of D-branes with Higgs vevs_, Adv.Theor.Math.Phys. 8 (2005) 813-859 ([arXiv:hep-th/0309270](https://arxiv.org/abs/hep-th/0309270))

* D. Maulik, A. Oblomkov, _Quantum cohomology of the Hilbert scheme of points on A_n-resolutions_, J. Amer. Math. Soc. 22 (2009), 1055-1091 ([arXiv:0802.2737](https://arxiv.org/abs/0802.2737))

See also:

* Yukari Ito, [[Hiraku Nakajima]], _McKay correspondence and Hilbert schemes in dimension three_ ([math/9803120](https://arxiv.org/abs/math/9803120))

### Of K3 surfaces

Discussion of the [[Hilbert schemes of points]] of [[K3-surfaces]]:

* {#Beauville83} [[Arnaud Beauville]], _Variétés Kähleriennes dont la premiere classe de Chern est nulle_, Jour.
Diff. Geom. 18 (1983), 755–782 ([euclid.jdg/1214438181](https://projecteuclid.org/euclid.jdg/1214438181))

* Georg Oberdieck, _Gromov-Witten invariants of the Hilbert schemes of points of a K3 surface_, Geom. Topol. 22 (2018) 323-437 ([arXiv:1406.1139](https://arxiv.org/abs/1406.1139), [[OberdieckK3HilbertSchemes.pdf:file]])

  (using [[Gromov-Witten invariants]])

Discussion of configuration spaces of _possibly coincident_ points on [[K3-surfaces]] $X$, hence of [[symmetric products]] $X^n/Sym(n)$ as [[moduli spaces]] of [[D0-D4-brane bound states]] [[wrapped brane|wrapped]] on [[K3-surfaces]]:

* [[Cumrun Vafa]], [[Edward Witten]], Section 4.1 of: _A Strong Coupling Test of S-Duality_, Nucl. Phys. B431:3-77, 1994 ([arXiv:hep-th/9408074](https://arxiv.org/abs/hep-th/9408074))

* [[Cumrun Vafa]], _Instantons on D-branes_, Nucl. Phys. B463 (1996) 435-442 ([arXiv:hep-th/9512078](https://arxiv.org/abs/hep-th/9512078))

Suggestion that this is to be resolved by the [[Hilbert scheme of points]]:

* [[Jeffrey Harvey]], [[Gregory Moore]], Section 5.1 of: _On the algebras of BPS states_, Commun. Math. Phys. 197:489-519, 1998 ([arXiv:hep-th/9609017](https://arxiv.org/abs/hep-th/9609017))

* Kazuyuki Furuuchi, Hiroshi Kunitomo, Toshio Nakatsu, p. 17 of: _Topological Field Theory and Second-Quantized Five-Branes_, Nucl.Phys. B494 (1997) 144-160 ([arXiv:hep-th/9610016](https://arxiv.org/abs/hep-th/9610016))

Hilbert schemes on K3 as [[moduli space]] of [[stable vector bundles]]:

* Laura Costa Farràs, _K3 surfaces: moduli spaces and Hilbert schemes_, Collectanea Mathematica, 1998, vol. 49, núm. 2-3, p. 273-282 ([hdl:2445/16925](	http://hdl.handle.net/2445/16925)

with an eye towards [[Rozansky-Witten theory]] ([[ground field]]-valued [[Rozansky-Witten weight systems]]):

* {#RobertsWillerton06} [[Justin Roberts]], [[Simon Willerton]], p. 17 of: _On the Rozansky-Witten weight systems_, Algebr. Geom. Topol. 10 (2010) 1455-1519 ([arXiv:math/0602653](https://arxiv.org/abs/math/0602653))

* {#Sawon04} [[Justin Sawon]], Section 5.3 of: _Rozansky-Witten invariants of hyperkähler manifold_, Cambridge 2000 ([arXiv:math/0404360](https://arxiv.org/abs/math/0404360))

### Hilbert schemes and Higgs/Coulomb branches

Identification of [[Higgs branches]]/[[Coulomb branches]] in [[D=3 N=4 super Yang-Mills theory]] with [[Hilbert schemes of points]] of [[complex curves]]:

* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], _Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes_, Nucl. Phys. B493:101-147, 1997 ([arXiv:hep-th/9611063](https://arxiv.org/abs/hep-th/9611063))

* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], Zheng Yin, _Mirror Symmetry in Three-Dimensional Gauge Theories, $SL(2,\mathbb{Z})$ and D-Brane Moduli Spaces_, Nucl. Phys. B493:148-176, 1997 ([arXiv:hep-th/9612131](https://arxiv.org/abs/hep-th/9612131))

* [[Stefano Cremonesi]], [[Amihay Hanany]], [[Alberto Zaffaroni]], sround (4.4) of: _Monopole operators and Hilbert series of Coulomb branches of 3d $\mathcal{N} = 4$ gauge theories_, JHEP 01 (2014) 005 ([arXiv:1309.2657](https://arxiv.org/abs/1309.2657))

* [[Alexander Braverman]], [[Michael Finkelberg]], [[Hiraku Nakajima]], _Line bundles over Coulomb branches_ ([arXiv:1805.11826](https://arxiv.org/abs/1805.11826))

* [[Ben Webster]], _Coherent sheaves on Hilbert schemes through the Coulomb lens_, 2018 ([[WebsterHilbertScheme18.pdf:file]])

* Mykola Dedushenko, Yale Fan, Silviu Pufu, Ran Yacoby, Section E.2 of: _Coulomb Branch Quantization and Abelianized Monopole Bubbling_, JHEP 10 (2019) 179 ([arXiv:1812.08788](https://arxiv.org/abs/1812.08788))

Discussion in the context of [[Witten indices]] and [[K-theory|K-theoretic]] [[enumerative geometry]]:

* [[Andrei Okounkov]], _Lectures on K-theoretic computations in enumerative geometry_ ([arXiv:1512.07363](https://arxiv.org/abs/1512.07363))

[[!redirects Hilbert schemes]]

[[!redirects Hilbert scheme of points]]
[[!redirects Hilbert schemes of points]]


Hilbert series

Hilbert series


Hilbert space

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#Contents#
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{:toc}

## Idea
 {#Idea}

> *Dr. von Neumann, ich möchte gerne wissen, was ist denn eigentlich ein Hilbertscher Raum ?*  [^Hilbert]

[^Hilbert]: *Dr. von Neumann, I would like to know what is a Hilbert space?* -- Question asked by [[David Hilbert]], in a 1929 talk by [[John von Neumann]] in Göttingen (cf. [von Neumann 1930, §I](#vonNeumann30)). The anecdote is narrated for instance in [MacLane 1988, §5](#MacLane88).
(We have corrected 'dann' in the original quotation to the more likely 'denn', in either case expressing a certain sense of puzzlement that is not quite captured by the direct English translation.)

A Hilbert space is (see Def. \ref{HilbertSpace} for details):

1. a ([[real vector space|real]] or,  usually, [[complex vector space|complex]]) [[vector space]], possibly of infinite [[dimension of a vector space|dimension]],

1. equipped with a positive definite [[Hermitian form|Hermitian]] [[inner product]],

2. which, as a [[topological space]], is [[complete metric space|complete]] with respect to the induced [[metric]].

Hilbert spaces are central to [[quantum physics]] and specifically to [[quantum mechanics]], where they serve as [[space of quantum states|spaces of]] [[pure quantum states]]. Here the [[inner product]] encodes the [[probability amplitudes]] for one [[pure state]] to "[[collapse of the wavefunction|collapse]]" to another one under [[quantum measurement|measurement]]. When the space of pure states is of [[finite-dimensional vector space|finite dimension]] (as is the case of interest in [[quantum information theory]]/[[quantum computation]]) then the completeness condition on a Hilbert space is automatic (see Rem. \ref{FiniteDimensionalInnerProductSpaces} below), otherwise it naturally encodes the possibility of an infinite number of [[quantum measurement|measurement]] outcomes.

Hilbert spaces with ([[bounded linear map|bounded]]) [[linear maps]] between them form a ([[dagger category|dagger]]-)[[category]], often denoted *[[Hilb]]* or similar, with the [[dagger-category|dagger]]-structure given by sending [[bounded linear maps]] to their [[adjoint operators]] with respect to the Hermitian inner product. Finite-dimensional Hilbert spaces form a [[dagger-compact category]].

See also:

* [[an elementary treatment of Hilbert spaces]].

## Definition
 {#Definition}

We first state the

* [standard formulation](#StandardDefinition)

of the definition of Hilbert spaces and then various equivalent

* [alternative formulations](#AlternativeDefinitions)

### Standard formulation
 {#StandardDefinition}

We first state the definition as such and then recall the ingredients that go into it:

\begin{definition}\label{HilbertSpace}
**(Hilbert space)**
\linebreak

A *Hilbert space* $\big(\mathscr{H}, \langle -,-\rangle\big)$ is

1. a ([[complex vector space|complex]] or [[real vector space|real]]) [[vector space]] $\mathscr{H}$ (possibly infinite-[[dimension of a vector space|dimensional]]),

1. equipped with a [[Hermitian form|Hermitian]] [[inner product]] $\langle -,- \rangle$ (Def. \ref{HermitianInnerProduct})

which is

1. complete (Def. \ref{CompleteInnerProduct});

1. positive definite (Def. \ref{DefiniteInnerProduct}).

\end{definition}

\begin{remark}\label{MorphismOfHilbertSpaces}
**(morphisms of Hilbert spaces)**
\linebreak
There are different notions of [[morphisms]] between [[Hilbert spaces]]:

At the very least, a [[morphism]] of [[Hilbert spaces]] is a [[linear map]] between the underlying [[vector spaces]], and this is the default notion usually considered. Notice that such morphisms do not need to respect the [[inner product]]-[[structure]]. Often one restricts to [[bounded linear maps]] or sometimes [[short maps]] (see at *[maps of Banach spaces](Banach+space#morphisms)*) which respect part of the inner product structure.

Instead of being fully respected by maps, the inner product on Hilbert spaces induces the [[dagger-category|dagger]] [[involution]] on their [[hom-spaces]] given by sending any [[bounded linear map]] $A \colon \mathscr{H}_1 \to \mathscr{H}_2$ to its [[Hermitian form|Hermitian]] [[adjoint operator]] $A^\dagger \colon \mathscr{H}_2 \to \mathscr{H}_1$, characterized by

\[
  \label{AdjointnessRelation}
  \langle A^\dagger(-), -\rangle_{1}
  \;=\;
  \langle -, A(-)\rangle_{2}
\]

\end{remark}

\begin{remark}\label{FiniteDimensionalInnerProductSpaces}
**([[finite-dimensional Hilbert space]])**
\linebreak
  In the case that the underlying vector space $\mathscr{H}$ happens to be [[finite dimensional vector space]], any hermitian inner product is necessarily complete. Therefore:

A *finite-dimensional Hilbert space* is equivalently a positive-definite [[hermitian inner product]]-space of [[finite set|finite]] [[dimension of a vector space|dimension]].

Finite-dimensional Hilbert spaces form a [[dagger-compact category]] and play a central role in [[quantum information theory]] and [[quantum computation]], see also at *[[quantum information theory via dagger-compact categories]]*.
\end{remark}

\linebreak

We recall now the meaning of the concepts entering Def. \ref{HilbertSpace}. In the following, let $\mathcal{H}$ be a [[vector space]] over the [[ground field]] $\mathbb{C}$ of [[complex numbers]]. For $z \in \mathbb{C}$ a [[complex number]], we write $\overline{z}$ for its [[complex conjugate]].

\begin{remark}\label{AlternativeGroundFields}
**(alternative ground fields)**
\linebreak
The following applies verbatim also for the ground field of [[real numbers]] $\mathbb{R}$, in which case the sesquilinear inner product [below](#HermitianInnerProduct) becomes [[bilinear form|bilinear]] and one speaks of *real Hilbert spaces*. Under mild assumptions, $\mathbb{C}$ and $\mathbb{R}$ are the only possible [[ground fields]] for Hilbert spaces (see [MO:a/4184099](https://math.stackexchange.com/a/4184099/58526)).
\end{remark}

\begin{definition}\label{HermitianInnerProduct}
**(Hermitian inner product)**
\linebreak
A *[[Hermitian form|Hermitian]] [[inner product]]* on $\mathscr{H}$ is a function

$$
  \langle {-},{-} \rangle
  \;\colon\;
  \mathscr{H} \times \mathscr{H} \to \mathbb{C}
$$

that is

1. **sesquilinear**:

   1. $ \langle 0, x \rangle = 0 $ and $ \langle x, 0 \rangle = 0 $;

   1.  $ \langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle $ and $ \langle x, y + z \rangle = \langle x, y \rangle + \langle x, z \rangle $;

   1.  $ \langle c x, y \rangle = \bar{c} \langle x, y \rangle $ and $ \langle x, c y \rangle = c \langle x, y \rangle $;

1. **conjugate-symmetric**:

   \[
     \label{ConjugateSymmetryOfInnerProduct}
     \langle x, y \rangle = \overline{\langle y, x \rangle}
   \]

\end{definition}

\begin{remark}\label{ConventionsForInnerProduct}
**(convention for the inner product)**
\linebreak
Def. \ref{HermitianInnerProduct} uses the _physicist\'s convention_ that the inner product is conjugate-linear in the first variable rather than in the second, instead of the _mathematician\'s convention_, which is the reverse.  The physicist\'s convention fits in a little better with $2$-[[2-Hilbert space|Hilbert spaces]].
\end{remark}

\begin{remark}
The axiom list in Def. \ref{HermitianInnerProduct} is rather redundant.  First of all, (1.1) follows from (1.3) by setting $c = 0$; besides that, (1.1--1.3) come in pairs, only one of which is needed, since each half follows from the other using (2).  It is even possible to derive (1.3) from (1.2) by supposing that $V$ is a [[topological vector space]] and that the inner product is [[continuous function|continuous]] (which, as we will see, is always true anyway for a Hilbert space).
\end{remark}

\begin{definition}\label{DefiniteInnerProduct}
**(definite inner product)**
\linebreak
Given an inner product according to Def. \ref{HermitianInnerProduct}, consider the (norm square)
function

\[
  \label{NormSquare}
  \array{
    \mathllap{
    \|{-}\|^2
    \;\colon\;
    }
    \mathscr{H} &\longrightarrow&
    \mathbb{R}
    \hookrightarrow
    \mathbb{C}
    \\
    v &\mapsto&
    \langle v, v \rangle
  }
\]

Notice that this takes only real values, by (eq:ConjugateSymmetryOfInnerProduct).

The [[Hermitian form|Hermitian]] [[inner product]] is called:

* **positive semidefinite**, or simply **positive**, if $\|v\|^2 \geq 0$ for all $v \in \mathscr{H}$;

* **non-degenerate** if $\|v\|^2 = 0$ implies that $v = 0$;

* **positive definite** if it is both positive and non-degenerate.

\end{definition}

\begin{remark}
An inner product is called _indefinite_ if some $\|v\|^2$ are positive and some are negative.
\end{remark}

\begin{definition}\label{CompleteInnerProduct}
**(complete inner product)**
\linebreak
The inner product (Def. \ref{HermitianInnerProduct}) is __complete__ if, given any infinite [[sequence]] $(v_1, v_2, \ldots)$ in $\mathscr{V}$ such that we have the [[limit of a sequence|limit]]

\[
  \label{Cauchy}
  \lim_{m,n\to\infty}
  \left\|\sum_{i=m}^{m+n} v_i\right\|^2 = 0
  \,,
\]

then there exists a (necessarily unique) *[[sum]]* $S$ such that

\[
  \label{converge}
  \lim_{n\to\infty} \left\|S
  -
  \sum_{i=1}^n v_i\right\|^2 = 0
  \,.
\]

If the inner product is definite (Def. \ref{DefiniteInnerProduct}), then this sum, if it exists, must be unique, and we write

$$
  S = \sum_{i=1}^\infty v_i
$$

(with the right-hand side undefined if no such sum exists).

\end{definition}

### Alternative formulations
 {#AlternativeDefinitions}

#### As Banach spaces and metric spaces
 {#AsBanachSpacesAndMetricSpaces}

If an inner product (Def. \ref{HermitianInnerProduct}) is positive (Def. \ref{DefiniteInnerProduct}), then there exists the principal [[square root]] of the norm square $\|v\|^2 = \langle v, v \rangle$ (eq:NormSquare) being the *[[norm]]* $\|v\|$ of $v \in \mathscr{H}$.

This norm satisfies all of the requirements of a [[Banach space]].  It additionally satisfies the _parallelogram law_

\[
  \label{ParallelogramLaw}
  \|v_1 + v_2\|^2
  +
  \|v_1 - v_2\|^2
  \;=\;
  2 \|v_1\|^2 + 2 \|v_2\|^2
  \,.
\]

which not all Banach spaces need satisfy.  (The name of this law comes from its geometric interpretation: the norms in the left-hand side are the lengths of the diagonals of a parallelogram, while the norms in the right-hand side are the lengths of the sides.)

Furthermore, any Banach space satsifying the parallelogram law (eq:ParallelogramLaw) has a unique inner product that reproduces the norm, defined by

$$
  \langle v_1, v_2 \rangle
  \;\coloneqq\;
  \frac{1}{4}
  \left(
    \|v_1 + v_2\|^2
    -
    \|v_1 - v_2\|^2
    -
    \mathrm{i}
    \|v_1 + \mathrm{i}v_2\|^2
    +
    \mathrm{i}
    \|v_1 - \mathrm{i}v_2\|^2
  \right) ,
$$

or $\frac{1}{2}(\|v_1 + v_2\|^2 - \|v_1 - v_2\|^2)$ in the real case.

Therefore: A Hilbert space is equivalently [[Banach space]] that satisfies the parallelogram law (eq:ParallelogramLaw).

This actually works a bit more generally; a positive semidefinite inner product space is a pseudonormed vector space that satisfies the parallelogram law.  (We cannot, however, recover an indefinite inner product from a norm.)

Moreover, in any positive semidefinite inner product space, let the __distance__ $d(x,y)$ be

$$
  d(x,y)
  \;\coloneqq\;
  \|y - x\|
  \,.
$$

Then $d$ is a [[pseudometric]]; and it is a complete metric if and only if we have a Hilbert space.

In fact, the axioms of a [[Banach space]] (or pseudonormed vector space) can be written entirely in terms of the metric; we can also state the parallelogram law as follows:
$$ d(x,y)^2 + d(x,-y)^2 = 2 d(x,0)^2 + 2 d(x,x+y)^2 .$$

In definitions, it is probably most common to see the metric introduced only to state the completeness requirement.  Indeed, (eq:Cauchy) says that the sequence of partial sums is a [[Cauchy sequence]], while (eq:converge) says that the sequence of partial sums converges to $S$.

## Examples

### Banach spaces

All of the $p$-parametrised examples at [[Banach space]] apply if you take $p = 2$.

In particular, the $n$-dimensional [[vector space]] $\mathbb{C}^n$ is a complex Hilbert space with
$$ \langle x, y \rangle = \sum_{u=1}^n \bar{x}_u y_u .$$
Any subfield $K$ of $\mathbb{C}$ gives a positive definite inner product space $K^n$ whose completion is either $\mathbb{R}^n$ or $\mathbb{C}^n$.  In particular, the [[cartesian space]] $\mathbb{R}^n$ is a real Hilbert space; the geometric notions of distance and angle defined above agree with ordinary [[Euclidean space|Euclidean geometry]] for this example.

### Of Lebesgue square-integrable functions over a manifold

The [[L-2-spaces|L-]] Hilbert spaces $L^2(\mathbb{R})$, $L^2([0,1])$, $L^2(\mathbb{R}^3)$, etc (real or complex) are very well known.  In general, $L^2(X)$ for $X$ a [[measure space]] consists of the almost-everywhere defined functions $f$ from $X$ to the scalar field ($\mathbb{R}$ or $\mathbb{C}$) such that $ \int |f|^2 $ converges to a finite number, with functions identified if they are equal almost everywhere; we have $\langle f, g\rangle = \int \bar{f} g$, which converges by the Cauchy--Schwarz inequality.  In the specific cases listed (and in general, when $X$ is a [[locally compact space|locally compact]] [[Hausdorff space]]), we can also get this space by completing the positive definite inner product space of compactly supported continuous functions.

### Of square-integrable half-densities

* [[canonical Hilbert space of half-densities]]

## Properties
 {#Properties}

### Bases

A basic result is that abstractly, Hilbert spaces of the same dimension are all of the same type: every Hilbert space $H$ admits an [[orthonormal basis]], meaning a [[subset]] $S \subseteq H$ whose inclusion map extends (necessarily uniquely) to an isomorphism
$$l^2(S) \to H$$
of Hilbert spaces. Here $l^2(S)$ is the vector space consisting of those [[function]]s $x$ from $S$ to the scalar field such that
$$ \|x\|^2 = \sum_{u: S} |x_u|^2 $$
converges to a finite number; this may also be obtained by completing the vector space of formal linear combinations of elements of $S$ with an inner product uniquely determined by the rule
$$\langle u, v \rangle = \delta_{u v} \qquad u, v \in S$$
in which $\delta_{u v}$ denotes [[Kronecker delta]]. We thus have, in $l^2(S)$,
$$ \langle x, y \rangle = \sum_{u: S} \bar{x}_u y_u .$$
(This sum converges by the Cauchy--Schwarz inequality.)

In general, this result uses the [[axiom of choice]] (usually in the form of [[Zorn's lemma]] and [[excluded middle]]) in its proof, and is equivalent to it.  However, the result for [[separable space|separable]] Hilbert spaces needs only [[dependent choice]] and so is [[constructive mathematics|constructive]] by most schools\' standards.  Even without dependent choice, explicit orthornormal bases for particular $L^2(X)$ can often be produced using [[approximation of the identity]] techniques, often in concert with a [[Gram-Schmidt process]].

In particular, all infinite-dimensional separable Hilbert spaces are abstractly isomorphic to $l^2(\mathbb{N})$.

### Cauchy--Schwarz inequality

The _Schwarz inequality_ (or _Cauchy--Буняковский--Schwarz inequality_, etc) is very handy:
$$ |\langle x, y \rangle| \leq \|x\| \|y\| .$$

This is really two theorems (at least): an abstract theorem that the inequality holds in any Hilbert space, and concrete theorems that it holds when the inner product and norm are defined by the formulas used in the examples $L^2(X)$ and $l^2(S)$ above.  The concrete theorems apply even to functions that don\'t belong to the Hilbert space and so prove that the inner product converges whenever the norms converge.  (A somewhat stronger result is needed to conclude this convergence constructively; it may be found in Errett Bishop\'s book.)

## Related concepts

* [[finite-dimensional Hilbert space]]

* [[rigged Hilbert space]]

* [[Hilbert C-star-module]], [[Hilbert bimodule]]

* [[measurable field of Hilbert spaces]]

* [[Kähler vector space]]

## References

Hilbert spaces were effectively introduced and used by [[David Hilbert]] and others in the context of [[integration]] theory, but the terminology and the formal definition is due to:

* {#vonNeumann30} [[John von Neumann]], §I in: *Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren*, Math. Ann. **102** (1930) 49–131 [[doi:10.1007/BF01782338](https://doi.org/10.1007/BF01782338)]

motivated from laying foundations for [[quantum mechanics]]:

* {#vonNeumann32} [[John von Neumann]]:

  *Mathematische Grundlagen der Quantenmechanik*, Springer (1932, 1971) [[doi:10.1007/978-3-642-96048-2](https://link.springer.com/book/10.1007/978-3-642-96048-2)]

  *Mathematical Foundations of Quantum Mechanics* Princeton University Press (1955) [[doi:10.1515/9781400889921](https://doi.org/10.1515/9781400889921), [Wikipedia entry](https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics)]

(where the definition occupies section II.1)

but see:

* [[Miklos Rédei]], *Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead)*, Studies in History and Philosophy of Modern Physics **27** 4 (1996) 493-510 [doi:10.1016/S1355-2198(96)00017-2]

Early history:

* {#MacLane88} [[Saunders Mac Lane]]: *Hilbert space*, §5 in: *Concepts and Categories in Perspective*, in: P. Duren, *A century of mathematics in America* Part 1, AMS (1988) 323-365. [[pdf](http://www.ams.org/samplings/math-history/hmath1-maclane25.pdf), [ISBN:0-8218-0124-4](https://www.ams.org/publicoutreach/math-history/hmath1-index)]

Textbook account:

* [[Nicolas Bourbaki]], §V in: *Topological Vector Spaces*, Chapters 1–5, Masson (1981), Springer (2003) [[doi:10.1007/978-3-642-61715-7](https://doi.org/10.1007/978-3-642-61715-7)]

Review:

* [[George Mackey]], _The Mathematical Foundations of Quamtum Mechanics_ A Lecture-note Volume, The mathematical physics monograph series. Princeton university (1963)

* E. Prugoveĉki, _Quantum mechanics in Hilbert Space_. Academic Press (1971)

An [[axiom|axiomatic]] characterization of the [[dagger-category]] [[Hilb]] of Hilbert spaces, with [[linear maps]] between them:

* {#HeunenKornell21} [[Chris Heunen]], [[Andre Kornell]], *Axioms for the category of Hilbert spaces*, PNAS **119** 9 (2022) e2117024119 [[arXiv:2109.07418](https://arxiv.org/abs/2109.07418), [doi:10.1073/pnas.2117024119](https://doi.org/10.1073/pnas.2117024119)]

* [[Chris Heunen]], [[Andre Kornell]], [[Nesta van der Schaaf]], *Axioms for the category of Hilbert spaces and linear contractions* [[arXiv:2211.02688](https://arxiv.org/abs/2211.02688)]

category: analysis

[[!redirects Hilbert spaces]]
[[!redirects category of Hilbert spaces]]
[[!redirects categories of Hilbert spaces]]


Hilbert system

# Hilbert systems

* table of contents
{: toc}

## Idea

A **Hilbert system** is a kind of [[deductive system]] characterized by the presence of many [[axioms]] and few [[rule of inference|rules]].  Usually the only [[judgment]] form is "$A \; true$" for some proposition $A$, which is usually written simply as "$A$".  Thus, a Hilbert-style deduction is a [[tree]] whose edges are formulas, whose leaves are axioms, and whose nodes are one of a very small number of rules.  The most common rule is [[modus ponens]]:

$$ \array{\arrayopts{\rowlines{solid}} A \qquad A\to B \\ B } $$

and in many Hilbert systems this is the only rule, although sometimes one finds others in use.  Then each [[logical connective]] is described by imposing [[axioms]].

## The role of implication

In many cases, the necessary axioms are obtained fairly straightforwardly from the corresponding rules by replacing derivability and entailment by the [[implication]] connective.  For instance, the rules for [[disjunction]] are
$$ \frac{\Gamma\vdash A}{\Gamma\vdash A\vee B}\qquad
\frac{\Gamma\vdash B}{\Gamma\vdash A\vee B} \qquad
\frac{\Gamma\vdash A\vee B \quad \Gamma,A\vdash C \quad \Gamma,B\vdash C}{\Gamma\vdash C}$$
and the corresponding Hilbert-style axioms are
$$ A\to (A\vee B) \qquad B\to (A\vee B) \qquad (A\vee B) \to (A\to C) \to (B\to C) \to C.$$

Of course, this gives a special role to [[implication]].  Its axioms are

$$ \array{ P \to P \\
  P \to (Q\to P) \\
  (P \to (Q\to R)) \to ((P \to Q) \to (P \to R)).
  }
$$

Note that these are precisely the types of the basic [[combinators]] in [[combinatory logic]].  In fact, under the [[propositions as types]] correspondence, Hilbert systems correspond to combinatory logic in the same way that [[natural deduction]]/[[sequent calculus]] corresponds to [[lambda-calculus]].

## On notation

In a Hilbert system one often writes rules and deductions using a turnstile $\vdash$ instead of a horizontal bar.  Thus, modus ponens would be written

$$ A, A\to B \vdash B$$

This can be somewhat confusing when comparing a Hilbert system to other systems such as [[sequent calculus]] or [[type theory]] in which $\vdash$ appears as part of *each judgment* to specify the context of that judgment.  Often the two meanings of $\vdash$ can be conflated, but not always.  For instance, in [[substructural logic]] such as [[linear logic]] or [[relevance logic]], we may have a rule (not an axiom) such as

$$ \array{\arrayopts{\rowlines{solid}} A \qquad B \\ A\& B } $$

(where $\&$ represents the [[additive conjunction]]), which in a Hilbert system would be written as $A,B\vdash A\& B$.  But the *sequent* $A,B\vdash A\& B$ is not valid in the usual sequent-calculus presentations of such logics, because the comma in the sequent context (on the left) is usually a judgment-level version of the [[multiplicative conjunction]].

## Related pages

* [[deductive system]]
* [[sequent calculus]]
* [[type theory]]
* [[natural deduction]]

[[!redirects Hilbert systems]]
[[!redirects Hilbert calculus]]
[[!redirects Hilbert calculi]]
[[!redirects Hilbert-style system]]
[[!redirects Hilbert-style systems]]
[[!redirects Hilbert-style deductive system]]
[[!redirects Hilbert-style deductive systems]]


Hilbert uniformization

* Hilbert uniformization at [string coffee](http://golem.ph.utexas.edu/string/archives/000762.html) table

* C.-F. [Bödigheimer](http://www.math.uni-bonn.de/people/cfb/publications.htm), _Configuration models for moduli spaces of Riemann surfaces with boundary_, Abh. Math. Sem. Univ. Hamburg 76 (2006), 191--233.

* C.-F. Bödigheimer, Hilbert uniformaization of Riemann surfaces: I (2004),[pdf](http://www.math.uni-bonn.de/people/cfb/PUBLICATIONS/short-hilbert.pdf)

* C.-F. Bödigheimer, _Moduli spaces of Riemann surfaces with boundary_
(2003).


Hilbert W*-module

## Definition

A __Hilbert W\*-module__ is a [[Hilbert C*-module]] $M$
over a [[von Neumann algebra]] $A$
such that:

* $M$ admits a [[predual]] $M_{\ast}$ as a [[Banach space]];

* for all $m \in M$, the map $\langle m, - \rangle_{A}: M \to A$ is normal (i.e. the dual of some map $A_{*} \to M_{*}$),

where $\langle -,- \rangle_{A}: \overline{M} \times M \rightarrow A$ is the
$A$-valued inner product on $M$, and $A_{*}$ is the predual of $A$.

The above definition is equivalent to the definitions using a notion of self-duality (See [Blecher-Merdy, Lemma 8.5.4](#BlecherMerdy2004)).

## Properties

For any [[von Neumann algebra]] $A$,
the category of [[Hilbert W*-modules]] over $A$ is equivalent to the category of [[W*-representations]] of $A$.

The equivalence is implemented by the following functors.

Given a [[Hilbert W*-module]] $M$,
we send it to the [[completion]] of $M\otimes_A L^2(A)$,
where $L^2(A)$ is the Haagerup standard form of $A$.

Given a [[W*-representation]] $R$,
we send it to the [[internal hom]] $Hom_A(L^2(A), R)$,
which is a [[Hilbert W*-module]] over $A$.

## Related concepts

* [[measurable field of Hilbert spaces]]

* [[Serre-Swan theorem]]

* [[W*-representation]]

* [[Hilbert C*-module]]

* [[von Neumann algebra]]

* [[duality between geometry and algebra]]

## References

{#Paschke1973} [[William Paschke]], _Inner Product Modules over $B^\ast$-algebras_, Trans. Amer. Math. Soc., 182, 1973 ([link](https://www.ams.org/journals/tran/1973-182-00/S0002-9947-1973-0355613-0/S0002-9947-1973-0355613-0.pdf))

{#Blecher1997} [[David Blecher]], _On Selfdual Hilbert Modules_, Fields Institute Communications, 13, 1997

{#BlecherMerdy2004} [[David Blecher]], [[Christian Le Merdy]], _Operator and Their Modules---an operator space approach_, Vol. 30, London Mathematical Society Mongraphs

[[!redirects Hilbert W*-modules]]
[[!redirects W*-modules]]
[[!redirects W*-module]]


Hilbert's basis theorem

Classical _[[affine algebraic variety|affine algebraic varieties]]_ appear as sets of zeros of a set $S = \{P_\alpha|\alpha\in A\}$ of polynomials in affine $n$-dimensional space $\mathbb{A}^n_k$ over a field $k$. The coordinate algebra of $\mathbb{A}^n_k$ is the algebra of [[polynomial]]s in $n$ variables, $k[x_1,\ldots,x_n]$, and the coordinate algebra of an affine algebraic variety is $R \coloneqq k[x_1,\ldots,x_n]/I$ where $I \coloneqq \langle S\rangle$ is the [[ideal]] generated by $S$.

The __Hilbert basis theorem__ (HBT) asserts that this ideal $I$ is finitely generated; and consequently $R$ is a [[noetherian ring]]. For a proof see standard textbooks on commutative algebra or algebraic geometry (e.g. Atiyah, MacDonald); there is also a proof on [wikipedia](http://en.wikipedia.org/wiki/Hilbert's_basis_theorem).

More generally, a finitely generated commutative algebra over a commutative [[noetherian ring]] $R$ is noetherian.  For the case of $R$ a field, this is the case in the previous paragraphs.

More at [[Noetherian ring]].

## Applications

The theorem vastly generalises and subsumes [[Paul Gordan]]'s work on [[invariant theory]], albeit in a non-constructive way. [[Emmy Noether]] wrote a short paper in 1920 that sidestepped the use of the HBT to construct a basis for, and so implying the finite generation of, a certain ring of invariants attached to any [[finite group]].

## References

* [[Emmy Noether]], _Der Endlichkeitsatz der Invarianten endlicher Gruppen_, Mathematische Annalen, vol. 77, 1920, pages 89--92, (English translation by [[Colin McLarty]], at [arXiv:1503.07849](http://arxiv.org/abs/1503.07849))

[[!redirects Hilbert basis theorem]]
[[!redirects Hilbert's basis theorem]]


Hilbert's fifth problem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

Hilbert's fifth problem, from his famous [[Hilbert's problems|list of problems]] in his address to the International Congress of Mathematicians in 1900, is conventionally understood as broadly asking

> Which [[topological group|topological groups]] admit [[Lie group]] structures?

A Lie group here is understood as a [[group object]] in the category of finite-dimensional smooth manifolds locally modeled on $\mathbb{R}^n$. Evidently not all topological groups $G$ admit Lie group structures (consider for example the compact group of $p$-adic integers under addition); a necessary condition is that the underlying space of $G$ be a topological manifold (or equivalently, using the group structure, that there is a neighborhood of the identity that is homeomorphic to a Euclidean space $\mathbb{R}^n$).

## Results

+-- {: .num_theorem}
###### Theorem 1 (Gleason; Montgomery-Zippin)
A topological group $G$ underlies a (unique) Lie group structure if and only if the underlying space of $G$ is a topological manifold.
=--

A deeper structural theorem from which theorem 1 can be deduced is

+-- {: .num_theorem}
###### Theorem 2 (Gleason-Yamabe)
Let $G$ be a locally compact group, and let $U$ be an open neighborhood of the identity in $G$. Then there exists an open subgroup $V$ of $G$, and a compact normal subgroup $H$ of $V$ contained in $U$, such that $V/H$ is isomorphic to a Lie group.
=--

An exposition of this incredible theorem was given in a series of blog posts by Terry Tao; see the reference below for a suitable entry point.

Here are some sample theorems which follow from the Gleason-Yamabe theorem.

+-- {: .num_theorem}
###### Theorem
Suppose $G$ is a locally compact group. A necessary and sufficient condition for $G$ is that it satisfies the "no small subgroups" property (NSS for short) that there exist a neighborhood the identity $U$ so small that it contains no nontrivial subgroups.
=--

+-- {: .proof}
###### Proof
First let us prove that Lie groups $G$ are NSS. Certainly $\mathbb{R}$ is NSS, and so is $\mathbb{R}^n$. This means we can find a starlike neighborhood $W$ of the origin in $Lie(G)$, small enough so that the exponential map $\exp: Lie(G) \to G$ maps diffeomorphically onto its image $U = \exp(W)$, and so that $W$ contains no line through the origin. Then $U$ contains the image of no 1-parameter subgroup $\phi \colon \mathbb{R} \to G$, otherwise we get a line $L$:

$$L = (\mathbb{R} \stackrel{\phi}{\to} U \stackrel{\exp^{-1}}{\to} Lie(G))$$

whose image is contained in $W$, and this is a contradiction. Since $U$ contains no 1-parameter subgroups, it follows easily that $U$ contains no non-trivial subgroups.

In the other direction, suppose given such a $U$. It follows from the Gleason-Yamabe theorem that there is an open subgroup $V$ of $G$ that is a Lie group. (To be completed.)
=--

## Related concepts

* [[nearby homomorphisms from compact Lie groups are conjugate]]

## References

* [[Terence Tao]], Hilbert's fifth problem and Gleason metrics. _What's new_ (weblog), June 17, 2011 ([link](http://terrytao.wordpress.com/2011/06/17/hilberts-fifth-problem-and-gleason-metrics/)).


Hilbert's operator

* PlanethMath.org: [Hilbert's epsilon operator](http://planetmath.org/hilbertsvarepsilonoperator)
* J. L. Bell, _Hilbert's ε-operator and classical logic_, J. of Philosophical Logic __22__:1 (Feb., 1993), pp. 1-18 [jstor](http://www.jstor.org/stable/30226487)
* Allen Hazen, _Natural deduction and Hilbert's ε-operator_, Journal of Philosophical Logic __16__:4 (Nov., 1987), pp. 411-421 [jstor](http://www.jstor.org/stable/30226381)

category: logic


Hilbert's problems

#Contents#
* table of contents
{:toc}

## Idea

At the _International Congress of Mathematics_ in the year 1900 in Paris, [[David Hilbert]] gave a talk in which he stated 23 open problems that he regarded central for the development of mathematics. These became famous as _Hilbert's 23 problems_ and have indeed considerably influenced mathematics
in the 20th century.

## List of problems

* [[Hilbert's third problem]]

* [[Hilbert's fifth problem]]

* [[Hilbert's sixth problem]]

* [[Hilbert's 15th problem]]

* [[Hilbert's 21st problem]]

## Related concepts

* [[Hilbert's program]]

## References

* [[David Hilbert]], _Mathematical Problems_, Bulletin of the American Mathematical Society, vol. 8, no. 10 (1902), pp. 437-479. Earlier publications (in the original German) appeared in Göttinger Nachrichten, 1900, pp. 253-297, and Archiv der Mathematik und Physik, 3dser., vol. 1 (1901), pp. 44-63, 213-237. ([AMS](http://www.ams.org/journals/bull/1902-08-10/home.html), [pdf](http://www.ams.org/journals/bull/1902-08-10/S0002-9904-1902-00923-3/S0002-9904-1902-00923-3.pdf), [html](aleph0.clarku.edu/~djoyce/hilbert/problems.html))
 {#Hilbert1901}

* Felix E. Browder. Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. XXVIII.2. American Mathematical Society.

* Wikipedia, _[Hilbert's problems](http://en.wikipedia.org/wiki/Hilbert%27s_problems)_


Hilbert's program

#Contents#
* table of contents
{:toc}

## Idea

[[David Hilbert]] had famously suggested to find a finite set of [[axioms]] to serve as a [[foundation]] for all of [[mathematics]].

The original version of Hilbert's program was overthrown by [[Gödel's incompleteness theorem]]. But suitable variants of it persist, see at _[[proof theory]]_

## Related concepts

* [[Hilbert's problems]]

## References

* Wikipedia _[Hilbert's program](https://en.wikipedia.org/wiki/Hilbert%27s_program)_

[[!redirects Hilbert problem]]


Hilbert's sixth problem

+-- {: .rightHandSide}
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

The sixth in the famous list of [[Hilbert's problems]] asks for the formalization/[[axiom|axiomatization]] of [[physics]] in [[mathematics]]. The original version in the way Hilbert stated it ([Hilbert 1901](#Hilbert1901)) is the following (translated from the original German):

> **6. Mathematical Treatment of the Axioms of Physics.** The investigations on the [[foundations]] of [[geometry]] suggest the problem: To treat in the same manner, by means of [[axioms]], those [[physics|physical sciences]] in which already today [[mathematics]] plays an important part; in the first rank are the [[probability theory|theory of probabilities]] and [[mechanics]]. ([full text](http://aleph0.clarku.edu/~djoyce/hilbert/problems.html#prob6))

Discussion of background and historical context is in the ICM06 speech ([Corry 06](#Corry06)), where it says

> From all the problems in the list, the sixth is the only one that continually engaged $[$Hilbert's$]$ efforts over a very long period, at least between 1894 and 1932.

Hence not the least due to Hilbert himself, meanwhile of course several aspects of physics have been formalized, while others are certainly still waiting to be understood in a systematic fashion.

Hilbert's original example of [[probability theory]] is considered to be fully axiomatized since the 1930s by the concepts of [[measure theory]]. For the example of _[[mechanics]]_ the status of the axiomatization crucially depends on how widely one understands the term today. The special case of [[classical mechanics]] which Hilbert was origially referring to has been fully formalized by means of [[symplectic geometry]] and [[variational calculus]]. Hilbert himself then contributed to the application of this formalization to [[gravity]] ("[[Einstein-Hilbert action]]", see [Sauer 99](#Sauer1999)) and the formalization of the refinement of classical mechanics to [[quantum mechanics]] ("[[Hilbert space]]"). Even though to this day the ontological status of quantum mechanics as a [[theory of physics]] is the topic of debates, certainly the mathematics of quantum mechanics as such is fully captured by [[functional analysis]] and [[operator algebra]] theory. But this situation changes drastically when one passes to what since around the 1950s is understood to be the more fundamental and general version of mechanics in nature, namely _[[quantum field theory]]_. Since the 1960 the [[Haag-Kastler axioms]] ("[[AQFT]]") have been proposed as an axiomatization for quantum field theory and crucially for _[[local quantum field theory]]_. While these axioms had some success in laying a foundation for structural results such as the [[PCT theorem]], their continuing lack of relevant examples ([[models]]) in dimensions greater than 2 seems to indicate that something is missing. Modern developments in [[higher algebra]] and [[higher geometry]] suggest that a refinement of the axioms to the context of [[homotopy theory]] is necessary and natural. More on this is below in _[Modern formalization of local quantum field theory](#ModernFormalizationOfLocalQuantumFieldTheory)_

## Modern formalization of local quantum field theory
 {#ModernFormalizationOfLocalQuantumFieldTheory}

(...)

## Related pages

* [[mathematical physics]]

* [[theory of everything]]

* [[Naturerkennen und Logik]]

* [[higher category theory and physics]], [[geometry of physics]]

Related expositions:

* [[geometry of physics]]

* [[fiber bundles in physics]]

* [[higher category theory and physics]]

* [[string theory FAQ]]

* [[twisted smooth cohomology in string theory]]

* [[motives in physics]]

* [[Hilbert's sixth problem]]

* [[model theory and physics]]

* [[L-infinity algebras in physics]]

* [[motivation for sheaves, cohomology and higher stacks]]

* [[applications of (higher) category theory]]

* [[motivation for higher differential geometry]]

* [[motivation for cohesion]]

## References

The original list of [[Hilbert's problems]] appears in

* [[David Hilbert]], _Mathematical Problems_, Bulletin of the American Mathematical Society, vol. 8, no. 10 (1902), pp. 437-479. Earlier publications (in the original German) appeared in Göttinger Nachrichten, 1900, pp. 253-297, and Archiv der Mathematik und Physik, 3dser., vol. 1 (1901), pp. 44-63, 213-237. ([AMS](http://www.ams.org/journals/bull/1902-08-10/home.html), [pdf](http://www.ams.org/journals/bull/1902-08-10/S0002-9904-1902-00923-3/S0002-9904-1902-00923-3.pdf), [html](aleph0.clarku.edu/~djoyce/hilbert/problems.html))
 {#Hilbert1901}

The historical context of the sixth problem is laid out in some detail in

* [[Leo Corry]], _David Hilbert and the axiomatization of physics: From Grundlagen der Geometrie to Grundlagen der Physik_, Archimedes: New Studies in the History and
Philosophy of Science and Technology 10, Kluwer Academic Publishers, Dordrecht 2004.([pdf](http://tau.ac.il/~corry/publications/articles/pdf/hilbert.pdf))

* [[Leo Corry]], _On the origins of Hilbert's sixth problem: physics and the empiricist approach to axiomatization_, Proceedings of the [International Congress of Mathematics in Madrid 2006](http://www.icm2006.org/) ([pdf](http://www.tau.ac.il/~corry/publications/articles/pdf/Hilbert%20ICM.pdf))
 {#Corry06}

Hilbert's own work on the foundations of the [[classical field theory]] of [[gravity]] ([[general relativity]]) is surveyed in

* Tilman Sauer,  _The relativity of discovery: Hilbert's first note on the foundations of physics". Arch. Hist. Exact Sci. 53 (6): 529–575. (1999). ([arXiv:physics/9811050](http://arxiv.org/abs/physics/9811050))
 {#Sauer1999}

Discussion in the context of [[higher category theory and physics]]:

* [[Yuri Ximenes Martins]], [[Rodney Josué Biezuner]], _Towards An Approach to Hilbert's Sixth Problem: A Brief Review_, 2020 ([hal:02909681](https://hal.archives-ouvertes.fr/hal-02909681), [pdf](https://hal.archives-ouvertes.fr/hal-02909681/document))

Surveys of the sixths problem include

* Wikipedia, _[Hilbert's sixth problem](http://en.wikipedia.org/wiki/Hilbert%27s_sixth_problem)_

[[!redirects Hilbert's 6th problem]]


Hilbert's Theorem 90

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A theorem in [[Galois cohomology]] due to [[David Hilbert]].

## Statement

There are actually two versions of Hilbert's theorem 90, one multiplicative and the other additive. We begin with the multiplicative version.

+-- {: .num_theorem #mult90}
###### Theorem
**(Hilbert)**
Suppose $K$ be a [[finite set|finite]] [[Galois extension]] of a [[field]] $k$, with a [[cyclic group|cyclic]] [[Galois group]] $G = \langle g \rangle$ of [[order]] $n$. Regard the [[multiplicative group]] $K^\ast$ as a $G$-[[module]]. Then the [[group cohomology]] of $G$ with [[coefficients]] in $K^\ast$ -- the [[Galois cohomology]] -- satisfies

$$
  H^1(G, K^\ast) = 0
  \,.
$$

=--

Before embarking on the proof, we recall from the article [projective resolution](http://ncatlab.org/nlab/show/projective+resolution#CohomologyOfCyclicGroups) that if $G = C_n$ is a finite cyclic group of order $n$, then there is a projective resolution of $\mathbb{Z}$ as trivial $G$-module:

$$\ldots \stackrel{N}{\to} \mathbb{Z}G \stackrel{D}{\to} \mathbb{Z}G \stackrel{N}{\to} \mathbb{Z}G \stackrel{D}{\to} \mathbb{Z}G \to \mathbb{Z} \to 0$$

where the map $\mathbb{Z}G \to \mathbb{Z}$ is induced from the trivial group homomorphism $G \to 1$ (hence is the map that forms the sum of all [[coefficients]] of all group elements), and where $D$, $N$ are multiplication by special elements in $\mathbb{Z}G$, also denoted $D$, $N$:

$$D \coloneqq g - 1,$$

$$\,$$

$$N \coloneqq 1 + g + g^2 + \ldots + g^{k-1}$$

The calculations in the proof that follows implicitly refer to this resolution as a means to defining $H^n(G, A)$ (in the case $A = K^\ast$), by taking cohomology of the induced cochain complex

$$0 \to \hom_{\mathbb{Z}G}(\mathbb{Z}, A) \to \hom_{\mathbb{Z}G}(\mathbb{Z}G, A) \to \hom_{\mathbb{Z}G}(\mathbb{Z}G, A) \to \ldots$$

+-- {: .proof}
###### Proof

Let $\sigma \in \mathbb{Z}G$ be an element of the [[group algebra]], and denote the [[action]] of $\sigma$ on an [[element]] $\beta \in K$ by exponential notation $\beta^\sigma$. The action of the element $N \in \mathbb{Z}G$ is

$$\beta^N = \beta^{1 + g + \ldots + g^{n-1}} = \beta \cdot \beta^g \cdot \ldots \beta^{g^{n-1}}$$

which is precisely the _[[norm]]_ $N(\beta)$. We are to show that if $N(\beta) = 1$, then there exists $\alpha \in K$ such that $\beta = \alpha/g(\alpha)$.

By lemma \ref{XYZ} below, the [[homomorphisms]] $1, g, \ldots, g^{n-1}: K^\ast \to K^\ast$ are, when considered as elements in a vector space of $K$-valued functions, $K$-[[linear independence|linearly independent]]. It follows in particular that

$$1 + \beta g + \beta^{1+g}g^2 + \ldots + \beta^{1 + g + \ldots + g^{n-2}}g^{n-1}$$

is not identically zero, and therefore there exists $\theta \in K^\ast$ such that the element

$$\alpha = \theta + \beta \theta^g + \beta^{1+g}\theta^{g^2} + \ldots + \beta^{1 + g + \ldots + g^{n-2}}\theta^{g^{n-1}}$$

is non-zero. Using the fact that $N(\beta) = 1$, one may easily calculate that $\beta \alpha^g = \alpha$, as was to be shown.
=--

Now we give the additive version of Hilbert's theorem 90:

+-- {: .num_theorem #add90}
###### Theorem
Under the same hypotheses given in \ref{mult90}, and regarding the additive group $K$ as a $G$-module, we have

$$
H^1(G, K) = 0
  \,.
$$

=--

+-- {: .proof}
The _trace_ of an element $\alpha \in K$ is defined by

$$Tr(\alpha) \coloneqq N \cdot \alpha = \alpha + g(\alpha) + \ldots + g^{n-1}(\alpha).$$

We want to show that if $Tr(\beta) = 0$, then there exists $\alpha \in K$ such that $\beta = \alpha - g(\alpha)$. By the theorem on linear independence of characters (following section), there exists $\theta$ such that $Tr(\theta) \neq 0$; notice $Tr(\theta)$ belongs to the ground field $k$ since $g \cdot N = N$. Put

$$\alpha \coloneqq \frac1{Tr(\theta)}(\beta g(\theta) + (\beta + g(\beta))g^2(\theta) + \ldots + (\beta + g(\beta) + \ldots + g^{n-2}(\beta))g^{n-1}(\theta).$$

One may then calculate that

$$\array{
\alpha - g(\alpha) & = & \frac1{Tr(\theta)}(\beta g(\theta) + \beta g^2(\theta) + \ldots + \beta g^{n-1}(\theta) - (g(\beta) + \ldots + g^{n-1}(\beta))\theta \\
 & = & \frac1{Tr(\theta)}(\beta g(\theta) + \beta g^2(\theta) + \ldots + \beta g^{n-1}(\theta) + \beta \theta) \\
 & = & \beta
}$$

where in the second line we used $Tr(\beta) = 0$.

=--

## Independence of characters

The next result may be thought of as establishing "independence of characters" (where "[[group character|characters]]" are valued in the [[multiplicative group]] of a field):

+-- {: .num_lemma #XYZ}
###### Lemma

Let $K$ be a [[field]], let $G$ be a [[monoid]], and let $\chi_1, \ldots, \chi_n \colon G \to K^\ast$ be distinct monoid [[homomorphisms]]. Then the [[functions]] $\chi_i$, considered as functions valued in $K$, are $K$-[[linear independence|linearly independent]].

=--

+-- {: .proof}
###### Proof
A single $\chi \colon G \to K^\ast$ obviously forms a [[linear independence|linearly independent]] set. Now suppose we have an [[equation]]

\[a_1 \chi_1 + \ldots + a_n \chi_n = 0\]

where $a_i \in K$, and assume $n$ is as small as possible. In particular, no $a_i$ is [[equality|equal]] to $0$, and $n \geq 2$. Choose $g \in G$ such that $\chi_1(g) \neq \chi_2(g)$. Then for all $h \in G$ we have

$$a_1 \chi_1(g h) + \ldots + a_n \chi_n(g h) = 0$$

so that

\[a_1 \chi_1(g) \chi_1 + \ldots + a_n \chi_n(g)\chi_n = 0.\]

Dividing equation 2 by $\chi_1(g)$ and subtracting from it equation 1, the first term cancels, and we are left with a shorter relation

$$(a_2\frac{\chi_2(g)}{\chi_1(g)} - a_2)\chi_2 + \ldots = 0$$

which is a [[contradiction]].

=--

A corollary of this result is an important result in its own right, the **normal basis heorem**.

(Will write this out later. I am puzzled that all the proofs I've so far looked at involve determinants. What happened to the battle cry, "Down with determinants!"?)

## Related concepts

* [[Kummer theory]]

## References

e.g.

* [[Günter Tamme]], section II 4.3 of _[[Introduction to Étale Cohomology]]_

[[!redirects Hilbert Theorem 90]]
[[!redirects Hilbert theorem 90]]

[[!redirects Hilbert's theorem 90]]


Hilbert-Schmidt operator

## References

* Wikipedia, _[Hilbert-Schmidt operator](https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt_operator)_

[[!redirects Hilbert-Schmidt operators]]


hilbertthm90

# Contents
* table of contents
{: toc}

# Contact {#contact}

The best way to contact me is by sending me an email at wardm4 'at' math.washington.edu

# Basic Information

I am a graduate student in math at the University of Washington studying algebraic geometry.

# Webpage

My department webpage is here

category: people

[[!redirects Matthew Ward]]
[[!redirects hilbertthm90]]


Hilbrand Groenewold

* [Wikipedia entry](http://en.wikipedia.org/wiki/Hilbrand_J._Groenewold)

category: people

[[!redirects H. Groenewold]]


Hiraku Nakajima

* [website](http://www.kurims.kyoto-u.ac.jp/~nakajima/)

## Selected writings

On [[Hilbert schemes of points]]:

* [[Hiraku Nakajima]], _Lectures on Hilbert schemes of points on surfaces_, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999 ([ams:ulect-18](https://bookstore.ams.org/ulect-18))

* [[Hiraku Nakajima]], _More lectures on Hilbert schemes of points on surfaces_, Advanced Studies in Pure Mathematics 69, 2016, Development of Moduli Theory -- Kyoto 2013, 173-205 ([arXiv:1401.6782](https://arxiv.org/abs/1401.6782))

On a mathematical definition of [[Coulomb branches]] of [[D=3 N=4 super Yang-Mills theory]]:

* [[Hiraku Nakajima]], _Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N} = 4$ gauge theories_ ([arXiv:1706.05154](http://arxiv.org/abs/1706.05154))

* [[Hiraku Nakajima]], _Towards a mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N} = 4$ gauge theories, I_ ([arXiv:1503.03676](http://arxiv.org/abs/1503.03676))

* [[Alexander Braverman]], [[Michael Finkelberg]], [[Hiraku Nakajima]], _Towards a mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N} = 4$ gauge theories, II_, Adv. Theor. Math. Phys. 22 (2018) 1071-1147 ([arXiv:1601.03586](http://arxiv.org/abs/1601.03586))

* [[Alexander Braverman]], [[Michael Finkelberg]], [[Hiraku Nakajima]], _Line bundles over Coulomb branches_ ([arXiv:1805.11826](https://arxiv.org/abs/1805.11826))

Discussion of [[Euler numbers]] of [[Hilbert schemes of points]]:

* [[Hiraku Nakajima]], _Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras_ ([arXiv:2001.03834](https://arxiv.org/abs/2001.03834))

category: people


Hiro Lee Tanaka

* [webpage](http://www.hiroleetanaka.com/)

## Selected writings

On [[A-infinity categories]]:

* Yong-Geun Oh, [[Hiro Lee Tanaka]], _$A_\infty$-categories, their $\infty$-categories, and their localizations_ ([arXiv:2003.05806](https://arxiv.org/abs/2003.05806))

On [[poset-stratified spaces]]:

* {#AFT} David Ayala, John Francis, Hiro Lee Tanaka, *Local structures on stratified spaces*, [arXiv](http://arxiv.org/abs/1409.0501)

## Related entries

* [[Lagrangian cobordism]]

* [[Fukaya category]]

category: people

[[!redirects Hiro Tanaka]]


Hiro Tannaka

* [webpage](http://math.harvard.edu/~hirolee/)

## Selected writings

On a [[stable (infinity,1)-category|stable $(\infty,1)$-category]] of [[Lagrangian cobordisms]]:

* {#NadlerTannaka11} [[David Nadler]], [[Hiro Tannaka]], _A stable infinity-category of Lagrangian cobordisms_, Adv. Math. __366__ 3 (2020) 107026 [[arXiv:1109.4835](https://arxiv.org/abs/1109.4835), [doi:10.1016/j.aim.2020.107026](https://doi.org/10.1016/j.aim.2020.107026)]

## Related entries

* [[Lagrangian cobordism]]

* [[Fukaya category]]

category: people


Hiroaki Hamanaka

* [Institute page](https://hute-rd.hyogo-u.ac.jp/profile/en.17e8533e0905287d72ffd582a40f817e.html)

* [ResearchGate page](https://www.researchgate.net/profile/Hiroaki-Hamanaka)

## Selected writings

On [[unstable topological K-theory]]:

* {#HK03} [[Hiroaki Hamanaka]], [[Akira Kono]]: *On $[X, U(n)]$ when $\text{dim}(X)$ is $2n$*, Journal of Mathematics of Kyoto University **43** 2 (2003) 333-348 [[doi:10.1215/kjm/1250283730](https://doi.org/10.1215/kjm/1250283730)[

* [[Hiroaki Hamanaka]], [[Akira Kono]]: *An application of unstable K-theory*, Journal of Mathematics of Kyoto University **44** 2 (2004) 451-456 [[doi:10.1215/kjm/1250283560](https://doi.org/10.1215/kjm/1250283560)]

* [[Hiroaki Hamanaka]]: *Adams $ e $-invariant, Toda bracket and $[X, U (n)]$*, Journal of Mathematics of Kyoto University **43** 4 (2003) 815-827 [[doi:10.1215/kjm/1250281737](https://doi.org/10.1215/kjm/1250281737)]

category: people


Hiromasa Watanabe

* [personal page](https://sites.google.com/view/hiromasa-watanabe/home)

* [inspire page](https://inspirehep.net/authors/1834554)

## Selected writings

On [[confinement]] in the [[BMN matrix model]]:

* Georg Bergner, Norbert Bodendorfer, Masanori Hanada, Stratos Pateloudis, Enrico Rinaldi, Andreas Schäfer, Pavlos Vranas, [[Hiromasa Watanabe]], *Confinement/deconfinement transition in the D0-brane matrix model -- A signature of M-theory?*, JHEP 05 (2022) 096 [[arXiv:2110.01312](https://arxiv.org/abs/2110.01312)]

On [[M5-branes]] in the [[BMN matrix model]] in a limit where aspects of [[little string theory]] on [[NS5-branes]] becomes visible:

* [[Yuhma Asano]], [[Goro Ishiki]], [[Takaki Matsumoto]], [[Shinji Shimasaki]], [[Hiromasa Watanabe]], *On the existence of the NS5-brane limit of the plane wave matrix model*, PTEP 2023 (2023) 4, 043B01 $[$[arXiv:2211.13716](https://arxiv.org/abs/2211.13716), [doi:10.1093/ptep/ptad042](https://doi.org/10.1093/ptep/ptad042)$]$

category: people


Hironari Miyazawa

* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Hironari_Miyazawa)

## Selected writings

The phenomenon of [[hadron supersymmetry]] was first noted and formalized (together with early classification of [[supersymmetry]] [[super Lie algebras]]) in:

* [[Hironari Miyazawa]], _Baryon Number Changing Currents_, Prog. Theor. Phys. 36 (1966) 6, 1266-1276 ([spire:1235194](https://inspirehep.net/literature/1235194), [doi:10.1143/PTP.36.1266](https://doi.org/10.1143/PTP.36.1266))

* [[Hironari Miyazawa]], _Spinor Currents and Symmetries of Baryons and Mesons_, Phys. Rev. 170, 1586 (1968) ([doi:10.1103/PhysRev.170.1586](https://doi.org/10.1103/PhysRev.170.1586))

category: people


Hironori Mori

* [personal page](https://skhironori.github.io/webpage/)

* [inspire page](https://inspirehep.net/authors/1591022)

## Selected writings

On [[defect branes]] in 6d understood as [[M5-brane|M5]]$\perp$[[M5-brane|M5]]-intersections (cf. [[3-brane in 6d]]):

* [[Hironori Mori]], *M-theory Perspectives on Codimension-2 Defects*, Osaka (2016) [[inspire:1519095](https://inspirehep.net/literature/1519095)]

* [[Hironori Mori]], Yuji Sugimoto, *Surface Operators from M-strings*, Phys. Rev. D **95** 026001 (2017) [[arXiv:1608.02849](https://arxiv.org/abs/1608.02849), [doi:10.1103/PhysRevD.95.026001](https://doi.org/10.1103/PhysRevD.95.026001)]

category: people


Hiroshi Itoyama

* [InSpire page](https://inspirehep.net/authors/1004761)

## Selected writings

On the early theory of [[relativistic strings]] ([[Nambu-Goto action]]) then motivated as an explanations for the "dual resonance model" of [[hadrons]] (cf. *[[Polyakov gauge-string duality]]*):

* [[Hiroshi Itoyama]], *Birth of String Theory*, Progress of Theoretical and Experimental Physics **2016** 6 (2016) 06A103 [[arXiv:1604.03701](http://arxiv.org/abs/1604.03701), [doi:10.1093/ptep/ptw063](https://doi.org/10.1093/ptep/ptw063)]

category: people


Hiroshi Kihara

* [webpage](https://www.u-aizu.ac.jp/research/faculty/detail?cd=90009&lng=en)

## Selected writings

On a [[model structure on diffeological spaces]] (beware: for non-standard diffeology on the standard [[simplices]]):

* [[Hiroshi Kihara]], _Model category of diffeological spaces_, Journal of Homotopy and Related Structures, (2018), 1-40 ([arXiv:1605.06794](https://arxiv.org/abs/1605.06794))

* [[Hiroshi Kihara]], *Quillen equivalences between the model categories of smooth spaces, simplicial sets, and arc-gengerated spaces* ([arXiv:1702.04070](https://arxiv.org/abs/1702.04070))

category: people


Hiroshi Toda

* [Wikipedia entry](https://en.wikipedia.org/wiki/Hiroshi_Toda)

## Selected writings

Computing [[homotopy groups of spheres]]:

* [[Hirosi Toda]], _Composition Methods in Homotopy Groups of Spheres_, Annals of Mathematics Studies Vol. 49, Princeton University Press (1962) ([jstor:j.ctt1bgzb5t](https://www.jstor.org/stable/j.ctt1bgzb5t))

## Related $n$Lab entries

* [[EHP spectral sequence]]

* [[Toda bracket]]

category: people

[[!redirects Hirosi Toda]]


Hiroshi Watanabe

## Selected writings

On [[distributive laws]] between [[monads]] and [[comonads]]:

* {#PowerWatanabe99} [[John Power]], [[Hiroshi Watanabe]], *Distributivity for a monad and a comonad*, Electronic Notes in Theoretical Computer Science **19** (1999) 102 [doi:10.1016/S1571-0661(05)80271-3, [[PowerWatanabe-Distributivity.pdf:file]]]

* {#PowerWatanabe02} [[John Power]], [[Hiroshi Watanabe]], *Combining a monad and a comonad*, Theoretical Computer Science **280** 1–2 (2002) 137-162 [doi:10.1016/S0304-3975(01)00024-X]

category: people


Hiroshige Kajiura

* [website](http://www2.yukawa.kyoto-u.ac.jp/~kajiura/)

category: people


Hirosi Ooguri

__Hirosi Ooguri__ is a [[string theory|string theorist]] at CalTech.

* [webpage](http://ooguri.caltech.edu/)

## Selected writings

Including [[Wilson loop observables]] in [[Chern-Simons theory as a topological string theory]]:

* [[Hirosi Ooguri]], [[Cumrun Vafa]], _Knot Invariants and Topological Strings_, Nucl. Phys. B 577:419-438, 2000 ([arXiv:hep-th/9912123](https://arxiv.org/abs/hep-th/9912123))

On [[large N limit]] of [[Chern-Simons theory]]:

* {#OoguriVafa02} [[Hirosi Ooguri]], [[Cumrun Vafa]], _Worldsheet Derivation of a Large $N$ Duality_, Nucl. Phys. B641:3-34, 2002 ([arXiv:hep-th/0205297](https://arxiv.org/abs/hep-th/0205297))

On [[mirror symmetry]] between [[Higgs branches]]/[[Coulomb branches]] of [[D=3 N=4 super Yang-Mills theory]] (with emphasis of [[Hilbert schemes of points]]):

* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], _Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes_, Nucl. Phys. B493:101-147, 1997 ([arXiv:hep-th/9611063](https://arxiv.org/abs/hep-th/9611063))

* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], Zheng Yin, _Mirror Symmetry in Three-Dimensional Gauge Theories, $SL(2,\mathbb{Z})$ and D-Brane Moduli Spaces_, Nucl. Phys. B493:148-176, 1997 ([arXiv:hep-th/9612131](https://arxiv.org/abs/hep-th/9612131))

On [[geometric engineering of QFT]] on [[D4-D6 brane intersections]] subject to the [[s-rule]]:

* {#OoguriVafa97} [[Hirosi Ooguri]], [[Cumrun Vafa]], _Geometry of $N = 1$ Dualities in Four Dimensions_, Nucl. Phys. B 500:62-74, 1997 ([arXiv:hep-th/9702180](https://arxiv.org/abs/hep-th/9702180))

* [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], Section 3 of: _Strong Coupling Dynamics of Four-Dimensional N=1 Gauge Theories from M Theory Fivebrane_, Adv. Theor. Math. Phys.1:1-52, 1998 ([arXiv:hep-th/9706082](https://arxiv.org/abs/hep-th/9706082))

On the [[AdS-CFT correspondence]]:

* {#AharonyGubserMaldacenaOoguriOz99} [[Ofer Aharony]], [[Steven Gubser]], [[Juan Maldacena]], [[Hirosi Ooguri]], [[Yaron Oz]], _Large $N$ Field Theories, String Theory and Gravity_, Phys. Rept. 323:183-386, 2000 ([arXiv:hep-th/9905111](http://arxiv.org/abs/hep-th/9905111))

On how the [[second quantization]] of the [[B-model]] yields [[Kodeira-Spencer gravity]]/[[BCOV theory]]:

* {#BCOV93} [[Michael Bershadsky]], [[Sergio Cecotti]], [[Hirosi Ooguri]], [[Cumrun Vafa]], *Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes*, Commun. Math. Phys. **165** (1994) 311-428 [[arXiv:hep-th/9309140](http://arxiv.org/abs/hep-th/9309140), [doi:10.1007/BF02099774](https://doi.org/10.1007/BF02099774)]

On [[lattice field theory]]:

* Hirosi Ooguri, _Topological lattice models in four dimensions_, Modern Phys. Lett. A 7 (1992), no. 30, 2799--2810; [MR93h:57031](http://www.ams.org/mathscinet-getitem?mr=1184565), [doi](http://dx.doi.org/10.1142/S0217732392004171)

On [[wall crossing]] in [[M-theory]]:

* [[Mina Aganagic]], [[Hirosi Ooguri]], [[Cumrun Vafa]], Masahito Yamazaki, _Wall crossing and M-theory_, [arxiv/0908.1194](http://arxiv.org/abs/0908.1194)

On the argument that after embedding into consistent [[quantum gravity]], all [[global symmetries]] must become [[local symmetries]], substantiated via [[AdS/CFT]]"

* [[Daniel Harlow]], [[Hirosi Ooguri]], _Symmetries in quantum field theory and quantum gravity_ ([arXiv:1810.05338](https://arxiv.org/abs/1810.05338))

* [[Daniel Harlow]], [[Hirosi Ooguri]], _Constraints on symmetry from holography_, Phys. Rev. Lett. 122, 191601 (2019) ([arXiv:1810.05337](https://arxiv.org/abs/1810.05337))

On the [[swampland cobordism conjecture]]:

* [[Hirosi Ooguri]], [[Tadashi Takayanagi]], _Cobordism Conjecture in AdS_ ([arXiv:2006.13953](https://arxiv.org/abs/2006.13953))

## related $n$Lab entries

* [[AdS-CFT]]

* [[black holes in string theory]]

* [[M-theory]]

* [[TCFT]]

* [[Boulatov model]]

* [[associative submanifold]]

* [[Crane-Yetter model]]

[[!redirects H. Ooguri]]


Hirota equation

Hirota equations are certain bilinear equations related to [[integrable models]]/hierarchies. The dependent variables can, in most cases, be identified with $\tau$-[[tau-function|functions]], as it was first observed by Miwa. One often speaks of Hirota's direct method in solving integrable equations. We say that an equation is put in Hirota bilinear form
if it is written as $P(D)(f\cdot f) = 0$ where $D$ is the Hirota derivative (or difference) operator. One defines it by
$$
D_x^n (f\cdot g) = (\partial_{x_1}-\partial_{x_2})^n (f(x_1)g(x_2))|_{x_1=x_2=x}
$$
The difference version of Hirota operator is obtained by exponentiating.
The Hirota equations enable easier finding of [[multisoliton]] solutions.
Recent [[Y-system|$Y$-system]]s may also be viewed as a version of generalized Hirota equations.

One can consider partition functions as made up from characters; $GL(N)$/characters are by Weyl formula given in terms of (ratios of) certain determinants. Flag minor determinants satisfy bilinear [[Pluecker relation]]s. If the tau functions are
interpreted in terms of infinite Grassmanians (fermionic approach) then the Hirota relations sometimes boil down to Pluecker relations.

## Literature

* R. Hirota, _Exact solution of the Korteweg-de Vries equation for multiple
collisions of solitons_, Phys. Rev. Lett., 27, 1192 (1971).
* R. Hirota, _Discrete analogue of a generalized Toda equation_, J. Phys. Soc. Japan 50 (1981) 3785-3791.
* T. Miwa, _On Hirota's difference equations_, Proc. Japan Acad. 58 (1982) 9-12 [euclid](http://projecteuclid.org/euclid.pja/1195516178)
* S. Saito, _String theories and Hirota's bilinear diference equation_, Pys. Rev. Lett. 59 (1987) 1798–1801.
* A. Zabrodin, _Bethe ansatz and Hirota equation in integrable models_, [arxiv/1211.4428](http://arxiv.org/abs/1211.4428)
* G. Segal, G. Wilson, _Loop groups and equations of KdV type_, Inst. Hautes Etudes Sci. Publ. Math. 61 (1985) 5–65.

#### Relation to Y-systems

* A. Kuniba, T. Nakanishi, J. Suzuki, _T-systems and Y-systems in integrable
systems_, J. Phys. A 44 (2011) 103001.
* S. Leurent, [Hirota equation and Q-functions](http://www.normalesup.org/~sleurent/talks/2011_ENS.pdf), slides

[[!redirects Hirota bilinear equation]]
[[!redirects Hirota equations]]


Hirotaka Sugawara

* [Wikipedia entry](https://en.wikipedia.org/wiki/Hirotaka_Sugawara)

## related $n$Lab entries

* [[current algebra]]

* [[M2-brane]], [[M5-brane]]

category: people


Hiroyuki Shimizu

## Selected writings

On [[D=6 N=(2,0) SCFT]] and [[D=6 N=(1,0) SCFT]] andtheir [[KK-compactification]] to [[D=5 super Yang-Mills theory]]:

* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], _$S^1/T^2$ Compactifications of 6d $\mathcal{N} = (1,0)$ Theories and Brane Webs_, J. High Energ. Phys. (2016) 2016: 24 ([arXiv:1509.03195](https://arxiv.org/abs/1509.03195))

and their [[anomaly cancellation]] via the [[Green-Schwarz mechanism]]:

* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], [[Yuji Tachikawa]], _Anomaly polynomial of E-string theories_, J. High Energ. Phys. 2014, 2 (2014) ([arXiv:1404.3887](https://arxiv.org/abs/1404.3887))

* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], [[Yuji Tachikawa]], [[Kazuya Yonekura]], _Anomaly polynomial of general 6d SCFTs_, Progress of Theoretical and Experimental Physics, Volume 2014, Issue 10, October 2014, 103B07 ([arXiv:1408.5572](https://arxiv.org/abs/1408.5572))

Discussion of [[anomaly cancellation]]:

* [[Hiroyuki Shimizu]], _Aspects of anomalies in 6d superconformal field theories_, Tokyo 2018 ([spire:1802462](https://inspirehep.net/literature/1802462))

category: people


Hirzebruch signature theorem

## Related concepts

* [[signature]]

* [[exotic smooth structure]]

## References

* Wikipedia, _[Hirzebruch signature theorem](https://en.wikipedia.org/wiki/Hirzebruch_signature_theorem)_

[[!redirects Hirzebruch's signature theorem]]

[[!redirects Hirzebruch index theorem]]
[[!redirects Hirzebruch's index theorem]]


Hirzebruch-Riemann-Roch theorem

#Contents#
* table of contents
{:toc}

## Idea

The _Hirzebruch-Riemann-Roch theorem_ identifies for a [[holomorphic vector bundle]] $E$ the alternating sum of its [[abelian sheaf cohomology]] groups (hence for trivial bundle the [[index]] of the [[Dolbeault operator]]) with the [[Chern character]] of the bundle [[cup product|cupped]] with the [[Todd class]] of the underlying [[complex manifold]].

## Related concepts

* [[Atiyah-Singer index theorem]]

* [[Grothendieck-Riemann-Roch theorem]]

* [[Todd genus]], [[geometric quantization]]

## References

* [[Friedrich Hirzebruch]], _Topological methods in algebraic geometry_, russian translation, Mir, Moscow, 1973.

* Wikipedia, _[Hirzebruch-Riemann-Roch theorem](https://en.wikipedia.org/wiki/Hirzebruch%E2%80%93Riemann%E2%80%93Roch_theorem)_


Hisao Suzuki

* [webpage](https://www2.sci.hokudai.ac.jp/faculty/en/researcher/hisao-suzuki)

## Selected writings

Introducing [[Kazama-Suzuki models]]:

* [[Yoichi Kazama]], [[Hisao Suzuki]], _$N = 2$ superconformal field theories and superstring compactification_, Nucl. Phys. B 321 (1989), pp. 232-268 ([http://inspirehep.net/record/263185)](http://inspirehep.net/record/263185), doi:10.1016/0550-3213(89)90250-2)

category: people